Subgroups — C₇×#60937 #13238153

Order 1792.

1878 conjugacy classes of subgroups.

The subgroup lattice has 726 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3494	C₂	1	2	yes	maximal, complemented
2	C₇×#3263	C₂	1	2	yes	maximal, complemented
3	C₇×#3259	C₂	1	2	yes	maximal, complemented
4	C₇×#3257	C₂	1	4	yes	maximal, complemented
5	C₇×#3256	C₂	1	2	yes	maximal, complemented
6	C₇×#3253	C₂	1	2	yes	maximal, complemented
7	C₇×#3251	C₂	1	4	yes	maximal, complemented
8	C₇×#3084	C₂	1	2	yes	maximal, complemented
9	C₇×#3081	C₂	1	2	yes	maximal, complemented
10	C₇×#3055	C₂	1	2	yes	maximal, complemented
11	C₇×#3054	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₇×#469	C₂	1	1	yes	maximal, characteristic, complemented
13	C₇×#3045	C₂	1	2	yes	maximal, complemented
14	C₇×#3041	C₂	1	2	yes	maximal, complemented
15	C₇×#2905	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×C₇×AGL(1,ℤ/8)	C₂²	1	2	yes	—
17	C₂×C₇×#135	C₂²	1	1	yes	characteristic
18	C₂²×C₇×D₈	C₂²	1	1	yes	characteristic
19	C₂²×C₇×M₁₆	C₂²	1	1	yes	characteristic
20	C₂²×C₇×M₁₆	C₂²	1	1	yes	characteristic
21	C₇×#555	—	2	2	no	—
22	C₇×#553	—	2	2	no	—
23	C₇×#553	—	2	2	no	—
24	C₇×#551	—	2	4	no	—
25	C₇×#547	—	2	2	no	—
26	C₇×#546	—	2	4	no	—
27	C₇×#544	C₂²	1	2	yes	—
28	C₇×#542	C₂²	1	4	yes	complemented
29	C₇×#540	C₂²	1	4	yes	complemented
30	C₇×#535	C₂²	1	2	yes	—
31	C₇×#532	C₂²	1	4	yes	complemented
32	C₇×#529	C₂²	1	4	yes	complemented
33	C₇×#525	C₂²	1	2	yes	—
34	C₂×C₇×#121	C₂²	1	2	yes	—
35	C₇×#520	C₂²	1	2	yes	—
36	C₇×#518	C₂²	1	2	yes	—
37	C₂×C₇×#117	C₂²	1	1	yes	characteristic
38	C₇×#479	C₂²	1	4	yes	complemented
39	C₇×#478	C₂²	1	4	yes	complemented
40	C₇×#478	C₂²	1	4	yes	complemented
41	C₇×#478	C₂²	1	4	yes	complemented
42	C₇×#476	C₂²	1	4	yes	complemented
43	C₇×#476	C₂²	1	4	yes	complemented
44	C₇×#475	C₂²	1	4	yes	complemented
45	C₇×#474	C₂²	1	4	yes	complemented
46	C₇×#470	C₂²	1	1	yes	characteristic, complemented
47	C₇×#470	C₂²	1	1	yes	characteristic, complemented
48	C₇×#469	C₂²	1	2	yes	—
49	C₇×#469	C₂²	1	2	yes	—
50	C₇×#469	C₂²	1	2	yes	—
51	C₇×#469	C₂²	1	2	yes	—
52	C₇×#469	C₂²	1	4	yes	complemented
53	C₇×#469	C₂²	1	1	yes	characteristic, complemented
54	C₇×#469	C₂²	1	1	yes	characteristic, complemented
55	C₇×#468	C₂²	1	2	yes	complemented
56	C₇×#468	C₂²	1	4	yes	complemented
57	C₇×#468	C₂²	1	2	yes	complemented
58	C₇×#466	C₂²	1	4	yes	complemented
59	C₇×#465	C₂²	1	4	yes	complemented
60	C₇×#464	C₂²	1	4	yes	complemented
61	C₇×#463	C₂²	1	4	yes	complemented
62	C₇×#463	C₂²	1	4	yes	complemented
63	C₇×#463	C₂²	1	4	yes	complemented
64	C₇×#460	C₂²	1	4	yes	complemented
65	C₇×#459	C₂²	1	4	yes	complemented
66	C₇×#459	C₂²	1	4	yes	complemented
67	C₇×#459	C₂²	1	4	yes	complemented
68	C₇×#428	C₂²	1	2	yes	—
69	C₇×#427	C₂²	1	1	yes	characteristic
70	C₂×C₇×#106	C₂²	1	1	yes	characteristic
71	C₇×#424	C₂²	1	2	yes	—
72	C₇×#424	C₂²	1	2	yes	—
73	C₇×#424	C₂²	1	2	yes	complemented
74	C₇×#424	C₂²	1	2	yes	complemented
75	C₇×#423	C₂²	1	2	yes	complemented
76	C₇×#423	C₂²	1	2	yes	complemented
77	C₇×#418	C₂²	1	2	yes	—
78	C₇×#417	C₂²	1	2	yes	—
79	C₇×#416	C₂²	1	2	yes	—
80	C₂×C₇×#102	C₂²	1	2	yes	—
81	#60937	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
82	C₂×C₇×#41	—	2	4	no	—
83	C₂×C₇×#41	C₂³	1	2	yes	—
84	C₂²×C₇×D₄	—	2	2	no	—
85	C₂²×C₇×D₄	C₂³	1	2	yes	—
86	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
87	C₇×AGL(1,ℤ/8)	—	4	4	no	—
88	C₇×#135	—	4	2	no	—
89	C₂×C₇×Q₁₆	C₂³	1	1	yes	characteristic
90	C₂×C₇×SD₁₆	C₂³	1	2	yes	—
91	C₂×C₇×SD₁₆	C₂³	1	4	yes	—
92	C₂×C₇×D₈	—	4	2	no	—
93	C₂×C₇×D₈	C₂³	1	4	yes	—
94	C₂×C₇×D₈	C₂³	1	2	yes	—
95	C₂×C₇×D₈	C₂³	1	2	yes	—
96	C₂×C₇×D₈	C₂³	1	1	yes	characteristic
97	C₂×C₇×M₁₆	—	4	2	no	—
98	C₂×C₇×M₁₆	—	4	2	no	—
99	C₂×C₇×M₁₆	C₂³	1	2	yes	—
100	C₂×C₇×M₁₆	C₂³	1	1	yes	characteristic
101	C₂×C₇×M₁₆	C₂³	1	1	yes	characteristic
102	C₂×C₇×M₁₆	C₂³	1	2	yes	—
103	C₂×C₇×M₁₆	C₂³	1	1	yes	characteristic
104	C₂×C₇×M₁₆	C₂³	1	1	yes	characteristic
105	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
106	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
107	C₂²×C₇×C₈	C₂³	1	2	yes	—
108	C₇×C₄⋊Q₈	—	2	2	no	—
109	C₇×C₄⋊Q₈	C₂³	1	4	yes	—
110	C₇×#127	C₂³	1	4	yes	—
111	C₇×#126	—	2	4	no	—
112	C₇×#126	—	2	4	no	—
113	C₇×#125	—	2	2	no	—
114	C₇×#125	C₂³	1	4	yes	—
115	C₇×#124	—	2	4	no	—
116	C₇×#124	—	2	4	no	—
117	C₇×#124	C₂³	1	4	yes	—
118	C₇×#123	—	2	2	no	—
119	C₇×#123	—	2	4	no	—
120	C₇×#123	—	2	4	no	—
121	C₇×#123	—	2	4	no	—
122	C₇×#123	—	2	4	no	—
123	C₇×C₂²⋊Q₈	—	2	4	no	—
124	C₇×C₂²⋊Q₈	—	2	4	no	—
125	C₇×C₂²⋊Q₈	C₂³	1	2	yes	—
126	C₇×C₂²⋊Q₈	C₂³	1	2	yes	—
127	C₇×#121	—	2	2	no	—
128	C₇×#121	—	2	2	no	—
129	C₇×#121	—	2	4	no	—
130	C₇×#121	—	2	2	no	—
131	C₇×#121	—	2	2	no	—
132	C₇×#121	—	2	4	no	—
133	C₇×#121	—	2	4	no	—
134	C₇×#121	C₂³	1	2	yes	—
135	C₇×#121	C₂³	1	4	yes	—
136	C₇×#121	C₂³	1	4	yes	—
137	C₇×#121	C₂³	1	2	yes	—
138	C₇×#120	—	2	2	no	—
139	C₇×#120	—	2	4	no	—
140	C₄×C₇×Q₈	C₂³	1	4	yes	—
141	C₄×C₇×D₄	—	2	2	no	—
142	C₄×C₇×D₄	—	2	2	no	—
143	C₄×C₇×D₄	—	2	4	no	—
144	C₄×C₇×D₄	—	2	2	no	—
145	C₄×C₇×D₄	—	2	2	no	—
146	C₄×C₇×D₄	—	2	4	no	—
147	C₄×C₇×D₄	—	2	4	no	—
148	C₄×C₇×D₄	C₂³	1	4	yes	—
149	C₄×C₇×D₄	C₂³	1	4	yes	—
150	C₄×C₇×D₄	C₂³	1	4	yes	—
151	C₇×#117	—	2	2	no	—
152	C₇×#117	—	2	4	no	—
153	C₇×#117	D₄	1	2	yes	—
154	C₇×#117	D₄	1	1	yes	characteristic
155	C₇×#117	D₄	1	1	yes	characteristic
156	C₇×#117	C₂³	1	2	yes	—
157	C₇×#117	C₂³	1	2	yes	—
158	C₇×#117	C₂³	1	2	yes	—
159	C₂×C₇×C₄⋊C₄	—	2	2	no	—
160	C₂×C₇×C₄⋊C₄	D₄	1	2	yes	—
161	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
162	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
163	C₂×C₇×C₂²⋊C₄	—	2	4	no	—
164	C₂×C₇×C₂²⋊C₄	D₄	1	2	yes	—
165	C₂×C₄²×C₇	C₂³	1	2	yes	—
166	C₇×#107	C₂³	1	4	yes	—
167	C₇×#107	C₂³	1	2	yes	—
168	C₇×#106	C₂³	1	2	yes	—
169	C₇×#106	C₂³	1	2	yes	—
170	C₇×#106	C₂³	1	1	yes	characteristic
171	C₇×#106	C₂³	1	1	yes	characteristic
172	C₇×#106	C₂³	1	4	yes	—
173	C₇×C₄⋊C₈	C₂³	1	4	yes	—
174	C₇×C₄⋊C₈	C₂³	1	4	yes	—
175	C₇×C₄⋊C₈	C₂³	1	4	yes	—
176	C₇×C₄⋊C₈	C₂³	1	4	yes	—
177	C₇×#103	C₂³	1	2	yes	—
178	C₇×#103	C₂³	1	2	yes	—
179	C₇×#103	C₂³	1	4	yes	—
180	C₇×#102	C₂³	1	4	yes	—
181	C₇×#102	C₂³	1	4	yes	—
182	C₇×#102	C₂³	1	2	yes	—
183	C₇×#102	C₂³	1	2	yes	—
184	C₇×#102	C₂³	1	4	yes	—
185	C₇×#102	C₂³	1	4	yes	—
186	C₇×#102	C₂³	1	4	yes	—
187	#3494	C₂×C₇	1	2	yes	complemented
188	#3263	C₂×C₇	1	2	yes	complemented
189	#3259	C₂×C₇	1	2	yes	complemented
190	#3257	C₂×C₇	1	4	yes	complemented
191	#3256	C₂×C₇	1	2	yes	complemented
192	#3253	C₂×C₇	1	2	yes	complemented
193	#3251	C₂×C₇	1	4	yes	complemented
194	#3084	C₂×C₇	1	2	yes	complemented
195	#3081	C₂×C₇	1	2	yes	complemented
196	#3055	C₂×C₇	1	2	yes	complemented
197	#3054	C₂×C₇	1	1	yes	characteristic, complemented
198	C₂×#469	C₂×C₇	1	1	yes	characteristic, complemented
199	#3045	C₂×C₇	1	2	yes	complemented
200	#3041	C₂×C₇	1	2	yes	complemented
201	#2905	C₂×C₇	1	1	yes	characteristic, complemented
202	C₂⁴×C₇	—	2	2	no	—
203	C₂⁴×C₇	C₂×D₄	1	1	yes	characteristic
204	C₇×#41	—	2	4	no	—
205	C₇×#41	—	2	4	no	—
206	C₇×#41	—	8	8	no	—
207	C₂×C₇×Q₈	—	2	4	no	—
208	C₂×C₇×Q₈	C₂⁴	1	2	yes	—
209	C₂×C₇×D₄	—	2	2	no	—
210	C₂×C₇×D₄	—	2	2	no	—
211	C₂×C₇×D₄	—	2	4	no	—
212	C₂×C₇×D₄	—	2	2	no	—
213	C₂×C₇×D₄	—	2	4	no	—
214	C₂×C₇×D₄	—	2	2	no	—
215	C₂×C₇×D₄	—	2	4	no	—
216	C₂×C₇×D₄	—	2	4	no	—
217	C₂×C₇×D₄	—	2	4	no	—
218	C₂×C₇×D₄	—	2	4	no	—
219	C₂×C₇×D₄	—	8	4	no	—
220	C₂×C₇×D₄	C₂⁴	1	4	yes	—
221	C₂×C₇×D₄	C₂⁴	1	2	yes	—
222	C₂²×C₄×C₇	—	2	2	no	—
223	C₂²×C₄×C₇	—	2	2	no	—
224	C₂²×C₄×C₇	—	2	2	no	—
225	C₂²×C₄×C₇	—	2	2	no	—
226	C₂²×C₄×C₇	—	2	4	no	—
227	C₂²×C₄×C₇	—	2	4	no	—
228	C₂²×C₄×C₇	—	2	4	no	—
229	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
230	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
231	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
232	C₂²×C₄×C₇	C₂×D₄	1	4	yes	—
233	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
234	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
235	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
236	C₇×Q₁₆	—	4	1	no	—
237	C₇×SD₁₆	—	4	2	no	—
238	C₇×SD₁₆	—	4	4	no	—
239	C₇×D₈	—	4	2	no	—
240	C₇×D₈	—	4	2	no	—
241	C₇×D₈	—	4	1	no	—
242	C₇×D₈	—	4	4	no	—
243	C₇×M₁₆	—	2	4	no	—
244	C₇×M₁₆	—	4	1	no	—
245	C₇×M₁₆	—	4	1	no	—
246	C₇×M₁₆	—	4	1	no	—
247	C₇×M₁₆	—	4	1	no	—
248	C₇×M₁₆	—	4	2	no	—
249	C₂×C₇×C₈	—	2	2	no	—
250	C₂×C₇×C₈	—	2	2	no	—
251	C₂×C₇×C₈	—	4	2	no	—
252	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
253	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
254	C₂×C₇×C₈	C₂⁴	1	4	yes	—
255	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
256	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
257	C₇×C₄⋊C₄	—	2	2	no	—
258	C₇×C₄⋊C₄	—	2	4	no	—
259	C₇×C₄⋊C₄	—	2	2	no	—
260	C₇×C₄⋊C₄	—	2	4	no	—
261	C₇×C₄⋊C₄	—	2	4	no	—
262	C₇×C₄⋊C₄	—	2	4	no	—
263	C₇×C₄⋊C₄	—	2	2	no	—
264	C₇×C₄⋊C₄	C₂×D₄	1	2	yes	—
265	C₇×C₄⋊C₄	C₂×D₄	1	2	yes	—
266	C₇×C₄⋊C₄	C₂×D₄	1	4	yes	—
267	C₇×C₄⋊C₄	C₂⁴	1	4	yes	—
268	C₇×C₄⋊C₄	C₂⁴	1	2	yes	—
269	C₇×C₄⋊C₄	C₂⁴	1	2	yes	—
270	C₇×C₂²⋊C₄	—	2	2	no	—
271	C₇×C₂²⋊C₄	—	2	2	no	—
272	C₇×C₂²⋊C₄	—	2	2	no	—
273	C₇×C₂²⋊C₄	—	2	4	no	—
274	C₇×C₂²⋊C₄	—	2	4	no	—
275	C₇×C₂²⋊C₄	—	2	4	no	—
276	C₇×C₂²⋊C₄	—	2	4	no	—
277	C₇×C₂²⋊C₄	—	2	4	no	—
278	C₇×C₂²⋊C₄	—	2	4	no	—
279	C₇×C₂²⋊C₄	—	2	2	no	—
280	C₇×C₂²⋊C₄	—	2	4	no	—
281	C₇×C₂²⋊C₄	—	2	2	no	—
282	C₇×C₂²⋊C₄	C₂×D₄	1	4	yes	—
283	C₇×C₂²⋊C₄	C₂×D₄	1	2	yes	—
284	C₇×C₂²⋊C₄	C₂×D₄	1	2	yes	—
285	C₄²×C₇	—	2	4	no	—
286	C₄²×C₇	—	2	2	no	—
287	C₄²×C₇	C₂×D₄	1	2	yes	—
288	C₄²×C₇	C₂×D₄	1	2	yes	—
289	C₄²×C₇	C₂⁴	1	4	yes	—
290	C₂×AGL(1,ℤ/8)	C₂²×C₇	1	2	yes	—
291	C₂×#135	C₂²×C₇	1	1	yes	characteristic
292	C₂²×D₈	C₂²×C₇	1	1	yes	characteristic
293	C₂²×M₁₆	C₂²×C₇	1	1	yes	characteristic
294	C₂²×M₁₆	C₂²×C₇	1	1	yes	characteristic
295	#555	—	2	2	no	—
296	#553	—	2	2	no	—
297	#553	—	2	2	no	—
298	#551	—	2	4	no	—
299	#547	—	2	2	no	—
300	#546	—	2	4	no	—
301	#544	C₂²×C₇	1	2	yes	—
302	#542	C₂²×C₇	1	4	yes	complemented
303	#540	C₂²×C₇	1	4	yes	complemented
304	#535	C₂²×C₇	1	2	yes	—
305	#532	C₂²×C₇	1	4	yes	complemented
306	#529	C₂²×C₇	1	4	yes	complemented
307	#525	C₂²×C₇	1	2	yes	—
308	C₂×#121	C₂²×C₇	1	2	yes	—
309	#520	C₂²×C₇	1	2	yes	—
310	#518	C₂²×C₇	1	2	yes	—
311	C₂×#117	C₂²×C₇	1	1	yes	characteristic
312	#479	C₂²×C₇	1	4	yes	complemented
313	#478	C₂²×C₇	1	4	yes	complemented
314	#478	C₂²×C₇	1	4	yes	complemented
315	#478	C₂²×C₇	1	4	yes	complemented
316	#476	C₂²×C₇	1	4	yes	complemented
317	#476	C₂²×C₇	1	4	yes	complemented
318	#475	C₂²×C₇	1	4	yes	complemented
319	#474	C₂²×C₇	1	4	yes	complemented
320	#470	C₂²×C₇	1	1	yes	characteristic, complemented
321	#470	C₂²×C₇	1	1	yes	characteristic, complemented
322	#469	C₂²×C₇	1	2	yes	—
323	#469	C₂²×C₇	1	2	yes	—
324	#469	C₂²×C₇	1	2	yes	—
325	#469	C₂²×C₇	1	2	yes	—
326	#469	C₂²×C₇	1	4	yes	complemented
327	#469	C₂²×C₇	1	1	yes	characteristic, complemented
328	#469	C₂²×C₇	1	1	yes	characteristic, complemented
329	#468	C₂²×C₇	1	2	yes	complemented
330	#468	C₂²×C₇	1	4	yes	complemented
331	#468	C₂²×C₇	1	2	yes	complemented
332	#466	C₂²×C₇	1	4	yes	complemented
333	#465	C₂²×C₇	1	4	yes	complemented
334	#464	C₂²×C₇	1	4	yes	complemented
335	#463	C₂²×C₇	1	4	yes	complemented
336	#463	C₂²×C₇	1	4	yes	complemented
337	#463	C₂²×C₇	1	4	yes	complemented
338	#460	C₂²×C₇	1	4	yes	complemented
339	#459	C₂²×C₇	1	4	yes	complemented
340	#459	C₂²×C₇	1	4	yes	complemented
341	#459	C₂²×C₇	1	4	yes	complemented
342	#428	C₂²×C₇	1	2	yes	—
343	#427	C₂²×C₇	1	1	yes	characteristic
344	C₂×#106	C₂²×C₇	1	1	yes	characteristic
345	#424	C₂²×C₇	1	2	yes	—
346	#424	C₂²×C₇	1	2	yes	—
347	#424	C₂²×C₇	1	2	yes	complemented
348	#424	C₂²×C₇	1	2	yes	complemented
349	#423	C₂²×C₇	1	2	yes	complemented
350	#423	C₂²×C₇	1	2	yes	complemented
351	#418	C₂²×C₇	1	2	yes	—
352	#417	C₂²×C₇	1	2	yes	—
353	#416	C₂²×C₇	1	2	yes	—
354	C₂×#102	C₂²×C₇	1	2	yes	—
355	C₂³×C₇	—	2	2	no	—
356	C₂³×C₇	—	2	4	no	—
357	C₂³×C₇	—	2	2	no	—
358	C₂³×C₇	—	4	2	no	—
359	C₂³×C₇	—	4	4	no	—
360	C₂³×C₇	—	8	2	no	—
361	C₂³×C₇	—	8	2	no	—
362	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
363	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
364	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
365	C₇×Q₈	—	2	2	no	—
366	C₇×Q₈	—	2	2	no	—
367	C₇×Q₈	—	8	4	no	—
368	C₇×D₄	—	2	4	no	—
369	C₇×D₄	—	2	2	no	—
370	C₇×D₄	—	2	4	no	—
371	C₇×D₄	—	2	2	no	—
372	C₇×D₄	—	4	8	no	—
373	C₇×D₄	—	4	4	no	—
374	C₇×D₄	—	4	4	no	—
375	C₇×D₄	—	4	4	no	—
376	C₇×D₄	—	8	4	no	—
377	C₇×D₄	—	8	4	no	—
378	C₇×D₄	—	8	2	no	—
379	C₇×D₄	—	8	4	no	—
380	C₂×C₄×C₇	—	2	2	no	—
381	C₂×C₄×C₇	—	2	2	no	—
382	C₂×C₄×C₇	—	2	2	no	—
383	C₂×C₄×C₇	—	2	1	no	—
384	C₂×C₄×C₇	—	2	1	no	—
385	C₂×C₄×C₇	—	2	1	no	—
386	C₂×C₄×C₇	—	2	1	no	—
387	C₂×C₄×C₇	—	2	2	no	—
388	C₂×C₄×C₇	—	2	2	no	—
389	C₂×C₄×C₇	—	2	2	no	—
390	C₂×C₄×C₇	—	2	4	no	—
391	C₂×C₄×C₇	—	4	4	no	—
392	C₂×C₄×C₇	—	4	4	no	—
393	C₂×C₄×C₇	—	8	4	no	—
394	C₂×C₄×C₇	—	8	4	no	—
395	C₂×C₄×C₇	—	8	4	no	—
396	C₂×C₄×C₇	—	8	2	no	—
397	C₂×C₄×C₇	C₂²×D₄	1	4	yes	—
398	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
399	C₂×C₄×C₇	C₂²×D₄	1	4	yes	—
400	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
401	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
402	C₂×C₄×C₇	H(5,𝔽₂)	1	2	yes	—
403	C₂×C₄×C₇	2₋¹⁺⁴	1	2	yes	—
404	C₂×C₄×C₇	C₂⁵	1	1	yes	characteristic
405	C₇×C₈	—	2	1	no	—
406	C₇×C₈	—	2	1	no	—
407	C₇×C₈	—	2	1	no	—
408	C₇×C₈	—	2	1	no	—
409	C₇×C₈	—	2	4	no	—
410	C₂×#41	—	2	4	no	—
411	C₂×#41	C₂³×C₇	1	2	yes	—
412	C₂²×D₄	—	2	2	no	—
413	C₂²×D₄	C₂³×C₇	1	2	yes	—
414	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
415	AGL(1,ℤ/8)	—	4	4	no	—
416	#135	—	4	2	no	—
417	C₂×Q₁₆	C₂³×C₇	1	1	yes	characteristic
418	C₂×SD₁₆	C₂³×C₇	1	2	yes	—
419	C₂×SD₁₆	C₂³×C₇	1	4	yes	—
420	C₂×D₈	—	4	2	no	—
421	C₂×D₈	C₂³×C₇	1	2	yes	—
422	C₂×D₈	C₂³×C₇	1	2	yes	—
423	C₂×D₈	C₂³×C₇	1	1	yes	characteristic
424	C₂×D₈	C₂³×C₇	1	4	yes	—
425	C₂×M₁₆	—	4	2	no	—
426	C₂×M₁₆	—	4	2	no	—
427	C₂×M₁₆	C₂³×C₇	1	2	yes	—
428	C₂×M₁₆	C₂³×C₇	1	1	yes	characteristic
429	C₂×M₁₆	C₂³×C₇	1	2	yes	—
430	C₂×M₁₆	C₂³×C₇	1	1	yes	characteristic
431	C₂×M₁₆	C₂³×C₇	1	1	yes	characteristic
432	C₂×M₁₆	C₂³×C₇	1	1	yes	characteristic
433	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
434	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
435	C₂²×C₈	C₂³×C₇	1	2	yes	—
436	C₄⋊Q₈	—	2	2	no	—
437	C₄⋊Q₈	C₂³×C₇	1	4	yes	—
438	#127	C₂³×C₇	1	4	yes	—
439	#126	—	2	4	no	—
440	#126	—	2	4	no	—
441	#125	—	2	2	no	—
442	#125	C₂³×C₇	1	4	yes	—
443	#124	—	2	4	no	—
444	#124	—	2	4	no	—
445	#124	C₂³×C₇	1	4	yes	—
446	#123	—	2	2	no	—
447	#123	—	2	4	no	—
448	#123	—	2	4	no	—
449	#123	—	2	4	no	—
450	#123	—	2	4	no	—
451	C₂²⋊Q₈	—	2	4	no	—
452	C₂²⋊Q₈	—	2	4	no	—
453	C₂²⋊Q₈	C₂³×C₇	1	2	yes	—
454	C₂²⋊Q₈	C₂³×C₇	1	2	yes	—
455	#121	—	2	2	no	—
456	#121	—	2	2	no	—
457	#121	—	2	4	no	—
458	#121	—	2	2	no	—
459	#121	—	2	2	no	—
460	#121	—	2	4	no	—
461	#121	—	2	4	no	—
462	#121	C₂³×C₇	1	4	yes	—
463	#121	C₂³×C₇	1	4	yes	—
464	#121	C₂³×C₇	1	2	yes	—
465	#121	C₂³×C₇	1	2	yes	—
466	#120	—	2	2	no	—
467	#120	—	2	4	no	—
468	C₄×Q₈	C₂³×C₇	1	4	yes	—
469	C₄×D₄	—	2	2	no	—
470	C₄×D₄	—	2	2	no	—
471	C₄×D₄	—	2	4	no	—
472	C₄×D₄	—	2	2	no	—
473	C₄×D₄	—	2	2	no	—
474	C₄×D₄	—	2	4	no	—
475	C₄×D₄	—	2	4	no	—
476	C₄×D₄	C₂³×C₇	1	4	yes	—
477	C₄×D₄	C₂³×C₇	1	4	yes	—
478	C₄×D₄	C₂³×C₇	1	4	yes	—
479	#117	—	2	2	no	—
480	#117	—	2	4	no	—
481	#117	C₇×D₄	1	2	yes	—
482	#117	C₇×D₄	1	1	yes	characteristic
483	#117	C₇×D₄	1	1	yes	characteristic
484	#117	C₂³×C₇	1	2	yes	—
485	#117	C₂³×C₇	1	2	yes	—
486	#117	C₂³×C₇	1	2	yes	—
487	C₂×C₄⋊C₄	—	2	2	no	—
488	C₂×C₄⋊C₄	C₇×D₄	1	2	yes	—
489	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
490	C₂×C₂²⋊C₄	—	2	2	no	—
491	C₂×C₂²⋊C₄	—	2	4	no	—
492	C₂×C₂²⋊C₄	C₇×D₄	1	2	yes	—
493	C₂×C₄²	C₂³×C₇	1	2	yes	—
494	#107	C₂³×C₇	1	2	yes	—
495	#107	C₂³×C₇	1	4	yes	—
496	#106	C₂³×C₇	1	2	yes	—
497	#106	C₂³×C₇	1	2	yes	—
498	#106	C₂³×C₇	1	1	yes	characteristic
499	#106	C₂³×C₇	1	1	yes	characteristic
500	#106	C₂³×C₇	1	4	yes	—
501	C₄⋊C₈	C₂³×C₇	1	4	yes	—
502	C₄⋊C₈	C₂³×C₇	1	4	yes	—
503	C₄⋊C₈	C₂³×C₇	1	4	yes	—
504	C₄⋊C₈	C₂³×C₇	1	4	yes	—
505	#103	C₂³×C₇	1	2	yes	—
506	#103	C₂³×C₇	1	2	yes	—
507	#103	C₂³×C₇	1	4	yes	—
508	#102	C₂³×C₇	1	4	yes	—
509	#102	C₂³×C₇	1	4	yes	—
510	#102	C₂³×C₇	1	2	yes	—
511	#102	C₂³×C₇	1	2	yes	—
512	#102	C₂³×C₇	1	4	yes	—
513	#102	C₂³×C₇	1	4	yes	—
514	#102	C₂³×C₇	1	4	yes	—
515	C₂²×C₇	—	2	2	no	—
516	C₂²×C₇	—	2	2	no	—
517	C₂²×C₇	—	2	2	no	—
518	C₂²×C₇	—	2	1	no	—
519	C₂²×C₇	—	2	1	no	—
520	C₂²×C₇	—	4	2	no	—
521	C₂²×C₇	—	4	4	no	—
522	C₂²×C₇	—	4	4	no	—
523	C₂²×C₇	—	4	2	no	—
524	C₂²×C₇	—	4	2	no	—
525	C₂²×C₇	—	4	4	no	—
526	C₂²×C₇	—	8	2	no	—
527	C₂²×C₇	—	16	4	no	—
528	C₂²×C₇	#537	1	2	yes	—
529	C₂²×C₇	C₂³×D₄	1	1	yes	characteristic
530	C₄×C₇	—	2	4	no	—
531	C₄×C₇	—	2	4	no	—
532	C₄×C₇	—	2	1	no	—
533	C₄×C₇	—	2	1	no	—
534	C₄×C₇	—	4	4	no	—
535	C₄×C₇	—	4	2	no	—
536	C₄×C₇	—	4	2	no	—
537	C₄×C₇	—	4	2	no	—
538	C₄×C₇	#537	1	2	yes	—
539	C₄×C₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
540	C₄×C₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
541	C₂⁴	—	2	2	no	—
542	C₂⁴	C₂×C₇×D₄	1	1	yes	characteristic
543	#41	—	2	4	no	—
544	#41	—	2	4	no	—
545	#41	—	8	8	no	—
546	C₂×Q₈	—	2	4	no	—
547	C₂×Q₈	C₂⁴×C₇	1	2	yes	—
548	C₂×D₄	—	2	2	no	—
549	C₂×D₄	—	2	2	no	—
550	C₂×D₄	—	2	4	no	—
551	C₂×D₄	—	2	2	no	—
552	C₂×D₄	—	2	4	no	—
553	C₂×D₄	—	2	2	no	—
554	C₂×D₄	—	2	4	no	—
555	C₂×D₄	—	2	4	no	—
556	C₂×D₄	—	2	4	no	—
557	C₂×D₄	—	2	4	no	—
558	C₂×D₄	—	8	4	no	—
559	C₂×D₄	C₂⁴×C₇	1	4	yes	—
560	C₂×D₄	C₂⁴×C₇	1	2	yes	—
561	C₂²×C₄	—	2	2	no	—
562	C₂²×C₄	—	2	2	no	—
563	C₂²×C₄	—	2	2	no	—
564	C₂²×C₄	—	2	2	no	—
565	C₂²×C₄	—	2	4	no	—
566	C₂²×C₄	—	2	4	no	—
567	C₂²×C₄	—	2	4	no	—
568	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
569	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
570	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
571	C₂²×C₄	C₂×C₇×D₄	1	4	yes	—
572	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
573	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
574	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
575	Q₁₆	—	4	1	no	—
576	SD₁₆	—	4	2	no	—
577	SD₁₆	—	4	4	no	—
578	D₈	—	4	2	no	—
579	D₈	—	4	2	no	—
580	D₈	—	4	1	no	—
581	D₈	—	4	4	no	—
582	M₁₆	—	2	4	no	—
583	M₁₆	—	4	1	no	—
584	M₁₆	—	4	1	no	—
585	M₁₆	—	4	1	no	—
586	M₁₆	—	4	1	no	—
587	M₁₆	—	4	2	no	—
588	C₂×C₈	—	2	2	no	—
589	C₂×C₈	—	2	2	no	—
590	C₂×C₈	—	4	2	no	—
591	C₂×C₈	C₂⁴×C₇	1	1	yes	characteristic
592	C₂×C₈	C₂⁴×C₇	1	1	yes	characteristic
593	C₂×C₈	C₂⁴×C₇	1	4	yes	—
594	C₂×C₈	C₂⁴×C₇	1	1	yes	characteristic
595	C₂×C₈	C₂⁴×C₇	1	1	yes	characteristic
596	C₄⋊C₄	—	2	2	no	—
597	C₄⋊C₄	—	2	4	no	—
598	C₄⋊C₄	—	2	2	no	—
599	C₄⋊C₄	—	2	4	no	—
600	C₄⋊C₄	—	2	4	no	—
601	C₄⋊C₄	—	2	4	no	—
602	C₄⋊C₄	—	2	2	no	—
603	C₄⋊C₄	C₂×C₇×D₄	1	2	yes	—
604	C₄⋊C₄	C₂×C₇×D₄	1	2	yes	—
605	C₄⋊C₄	C₂×C₇×D₄	1	4	yes	—
606	C₄⋊C₄	C₂⁴×C₇	1	4	yes	—
607	C₄⋊C₄	C₂⁴×C₇	1	2	yes	—
608	C₄⋊C₄	C₂⁴×C₇	1	2	yes	—
609	C₂²⋊C₄	—	2	2	no	—
610	C₂²⋊C₄	—	2	2	no	—
611	C₂²⋊C₄	—	2	2	no	—
612	C₂²⋊C₄	—	2	4	no	—
613	C₂²⋊C₄	—	2	4	no	—
614	C₂²⋊C₄	—	2	4	no	—
615	C₂²⋊C₄	—	2	4	no	—
616	C₂²⋊C₄	—	2	4	no	—
617	C₂²⋊C₄	—	2	4	no	—
618	C₂²⋊C₄	—	2	2	no	—
619	C₂²⋊C₄	—	2	4	no	—
620	C₂²⋊C₄	—	2	2	no	—
621	C₂²⋊C₄	C₂×C₇×D₄	1	4	yes	—
622	C₂²⋊C₄	C₂×C₇×D₄	1	2	yes	—
623	C₂²⋊C₄	C₂×C₇×D₄	1	2	yes	—
624	C₄²	—	2	4	no	—
625	C₄²	—	2	2	no	—
626	C₄²	C₂×C₇×D₄	1	2	yes	—
627	C₄²	C₂×C₇×D₄	1	2	yes	—
628	C₄²	C₂⁴×C₇	1	4	yes	—
629	C₂×C₇	—	2	2	no	—
630	C₂×C₇	—	4	1	no	—
631	C₂×C₇	—	4	1	no	—
632	C₂×C₇	—	8	2	no	—
633	C₂×C₇	—	8	4	no	—
634	C₂×C₇	C₂×#537	1	1	yes	characteristic
635	C₂×C₇	#3585	1	2	yes	—
636	C₂³	—	2	2	no	—
637	C₂³	—	2	4	no	—
638	C₂³	—	2	2	no	—
639	C₂³	—	4	2	no	—
640	C₂³	—	4	4	no	—
641	C₂³	—	8	2	no	—
642	C₂³	—	8	2	no	—
643	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
644	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
645	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
646	Q₈	—	2	2	no	—
647	Q₈	—	2	2	no	—
648	Q₈	—	8	4	no	—
649	D₄	—	2	4	no	—
650	D₄	—	2	2	no	—
651	D₄	—	2	4	no	—
652	D₄	—	2	2	no	—
653	D₄	—	4	8	no	—
654	D₄	—	4	4	no	—
655	D₄	—	4	4	no	—
656	D₄	—	4	4	no	—
657	D₄	—	8	4	no	—
658	D₄	—	8	4	no	—
659	D₄	—	8	2	no	—
660	D₄	—	8	4	no	—
661	C₂×C₄	—	2	2	no	—
662	C₂×C₄	—	2	1	no	—
663	C₂×C₄	—	2	1	no	—
664	C₂×C₄	—	2	1	no	—
665	C₂×C₄	—	2	1	no	—
666	C₂×C₄	—	2	2	no	—
667	C₂×C₄	—	2	2	no	—
668	C₂×C₄	—	2	2	no	—
669	C₂×C₄	—	2	2	no	—
670	C₂×C₄	—	2	2	no	—
671	C₂×C₄	—	2	4	no	—
672	C₂×C₄	—	4	4	no	—
673	C₂×C₄	—	4	4	no	—
674	C₂×C₄	—	8	4	no	—
675	C₂×C₄	—	8	4	no	—
676	C₂×C₄	—	8	4	no	—
677	C₂×C₄	—	8	2	no	—
678	C₂×C₄	C₂²×C₇×D₄	1	4	yes	—
679	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
680	C₂×C₄	C₂²×C₇×D₄	1	4	yes	—
681	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
682	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
683	C₂×C₄	C₇×H(5,𝔽₂)	1	2	yes	—
684	C₂×C₄	C₇×2₋¹⁺⁴	1	2	yes	—
685	C₂×C₄	C₂⁵×C₇	1	1	yes	characteristic
686	C₈	—	2	1	no	—
687	C₈	—	2	1	no	—
688	C₈	—	2	1	no	—
689	C₈	—	2	1	no	—
690	C₈	—	2	4	no	—
691	C₇	#60937	1	1	yes	characteristic, direct product factor, Sylow
692	C₂²	—	2	2	no	—
693	C₂²	—	2	2	no	—
694	C₂²	—	2	2	no	—
695	C₂²	—	2	1	no	—
696	C₂²	—	2	1	no	—
697	C₂²	—	4	2	no	—
698	C₂²	—	4	4	no	—
699	C₂²	—	4	4	no	—
700	C₂²	—	4	2	no	—
701	C₂²	—	4	2	no	—
702	C₂²	—	4	4	no	—
703	C₂²	—	8	2	no	—
704	C₂²	—	16	4	no	—
705	C₂²	C₇×#537	1	2	yes	—
706	C₂²	C₂³×C₇×D₄	1	1	yes	characteristic
707	C₄	—	2	4	no	—
708	C₄	—	2	4	no	—
709	C₄	—	2	1	no	—
710	C₄	—	2	1	no	—
711	C₄	—	4	4	no	—
712	C₄	—	4	2	no	—
713	C₄	—	4	2	no	—
714	C₄	—	4	2	no	—
715	C₄	C₇×#537	1	2	yes	—
716	C₄	C₂×C₇×H(5,𝔽₂)	1	1	yes	characteristic
717	C₄	C₂×C₇×2₋¹⁺⁴	1	1	yes	characteristic
718	C₂	—	2	2	no	—
719	C₂	—	4	1	no	—
720	C₂	—	4	1	no	—
721	C₂	—	8	4	no	—
722	C₂	—	8	2	no	—
723	C₂	C₂×C₇×#537	1	1	yes	characteristic
724	C₂	C₇×#3585	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₇	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₇
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₇×#117 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₇×#3494	C₂	semidirect
2. C₇×#3263	C₂	semidirect
3. C₇×#3259	C₂	semidirect
4. C₇×#3257	C₂	semidirect
5. C₇×#3256	C₂	semidirect
6. C₇×#3253	C₂	semidirect
7. C₇×#3251	C₂	semidirect
8. C₇×#3084	C₂	semidirect
9. C₇×#3081	C₂	semidirect
10. C₇×#3055	C₂	semidirect
11. C₇×#3054	C₂	semidirect
12. C₂×C₇×#469	C₂	semidirect
13. C₇×#3045	C₂	semidirect
14. C₇×#3041	C₂	semidirect
15. C₇×#2905	C₂	semidirect
28. C₇×#542	C₂²	semidirect
29. C₇×#540	C₂²	semidirect
31. C₇×#532	C₂²	semidirect
32. C₇×#529	C₂²	semidirect
38. C₇×#479	C₂²	semidirect
39. C₇×#478	C₂²	semidirect
40. C₇×#478	C₂²	semidirect
41. C₇×#478	C₂²	semidirect
42. C₇×#476	C₂²	semidirect
43. C₇×#476	C₂²	semidirect
44. C₇×#475	C₂²	semidirect
45. C₇×#474	C₂²	semidirect
46. C₇×#470	C₂²	semidirect
47. C₇×#470	C₂²	semidirect
52. C₇×#469	C₂²	semidirect
53. C₇×#469	C₂²	semidirect
54. C₇×#469	C₂²	semidirect
55. C₇×#468	C₂²	semidirect
56. C₇×#468	C₂²	semidirect
57. C₇×#468	C₂²	semidirect
58. C₇×#466	C₂²	semidirect
59. C₇×#465	C₂²	semidirect
60. C₇×#464	C₂²	semidirect
61. C₇×#463	C₂²	semidirect
62. C₇×#463	C₂²	semidirect
63. C₇×#463	C₂²	semidirect
64. C₇×#460	C₂²	semidirect
65. C₇×#459	C₂²	semidirect
66. C₇×#459	C₂²	semidirect
67. C₇×#459	C₂²	semidirect
73. C₇×#424	C₂²	semidirect
74. C₇×#424	C₂²	semidirect
75. C₇×#423	C₂²	semidirect
76. C₇×#423	C₂²	semidirect
81. #60937	C₇	direct (N × H)
187. #3494	C₂×C₇	semidirect
188. #3263	C₂×C₇	semidirect
189. #3259	C₂×C₇	semidirect
190. #3257	C₂×C₇	semidirect
191. #3256	C₂×C₇	semidirect
192. #3253	C₂×C₇	semidirect
193. #3251	C₂×C₇	semidirect
194. #3084	C₂×C₇	semidirect
195. #3081	C₂×C₇	semidirect
196. #3055	C₂×C₇	semidirect
197. #3054	C₂×C₇	semidirect
198. C₂×#469	C₂×C₇	semidirect
199. #3045	C₂×C₇	semidirect
200. #3041	C₂×C₇	semidirect
201. #2905	C₂×C₇	semidirect
302. #542	C₂²×C₇	semidirect
303. #540	C₂²×C₇	semidirect
305. #532	C₂²×C₇	semidirect
306. #529	C₂²×C₇	semidirect
312. #479	C₂²×C₇	semidirect
313. #478	C₂²×C₇	semidirect
314. #478	C₂²×C₇	semidirect
315. #478	C₂²×C₇	semidirect
316. #476	C₂²×C₇	semidirect
317. #476	C₂²×C₇	semidirect
318. #475	C₂²×C₇	semidirect
319. #474	C₂²×C₇	semidirect
320. #470	C₂²×C₇	semidirect
321. #470	C₂²×C₇	semidirect
326. #469	C₂²×C₇	semidirect
327. #469	C₂²×C₇	semidirect
328. #469	C₂²×C₇	semidirect
329. #468	C₂²×C₇	semidirect
330. #468	C₂²×C₇	semidirect
331. #468	C₂²×C₇	semidirect
332. #466	C₂²×C₇	semidirect
333. #465	C₂²×C₇	semidirect
334. #464	C₂²×C₇	semidirect
335. #463	C₂²×C₇	semidirect
336. #463	C₂²×C₇	semidirect
337. #463	C₂²×C₇	semidirect
338. #460	C₂²×C₇	semidirect
339. #459	C₂²×C₇	semidirect
340. #459	C₂²×C₇	semidirect
341. #459	C₂²×C₇	semidirect
347. #424	C₂²×C₇	semidirect
348. #424	C₂²×C₇	semidirect
349. #423	C₂²×C₇	semidirect
350. #423	C₂²×C₇	semidirect
691. C₇	#60937	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₇×#3494	2
C₇×#3263	2
C₇×#3259	2
C₇×#3257	2
C₇×#3256	2
C₇×#3253	2
C₇×#3251	2
C₇×#3084	2
C₇×#3081	2
C₇×#3055	2
C₇×#3054	2
C₂×C₇×#469	2
C₇×#3045	2
C₇×#3041	2
C₇×#2905	2
#60937	7

Subgroups — C7×#60937 #13238153

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#60937 #13238153