Subgroups — C₇×#60696 #13237912

Order 1792.

1850 conjugacy classes of subgroups.

The subgroup lattice has 748 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3461	C₂	1	2	yes	maximal, complemented
2	C₇×#3178	C₂	1	4	yes	maximal, complemented
3	C₇×#3177	C₂	1	2	yes	maximal, complemented
4	C₇×#3175	C₂	1	2	yes	maximal, complemented
5	C₇×#3172	C₂	1	4	yes	maximal, complemented
6	C₇×#3170	C₂	1	2	yes	maximal, complemented
7	C₇×#3168	C₂	1	2	yes	maximal, complemented
8	C₇×#3164	C₂	1	1	yes	maximal, characteristic, complemented
9	C₇×#3156	C₂	1	1	yes	maximal, characteristic, complemented
10	C₇×#3137	C₂	1	2	yes	maximal, complemented
11	C₇×#3063	C₂	1	2	yes	maximal, complemented
12	C₇×#3061	C₂	1	1	yes	maximal, characteristic
13	C₇×#3059	C₂	1	1	yes	maximal, characteristic, complemented
14	C₄×C₇×#137	C₂	1	2	yes	maximal, complemented
15	C₄×C₇×#135	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×C₄×C₇×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
17	C₄×C₇×#131	C₂	1	1	yes	maximal, characteristic, complemented
18	C₂×C₇×#137	C₂²	1	2	yes	—
19	C₂×C₇×#135	C₂²	1	1	yes	characteristic
20	C₂²×C₇×Q₁₆	C₂²	1	1	yes	characteristic
21	C₂×C₇×#131	C₂²	1	1	yes	characteristic
22	C₂×C₇×#131	C₂²	1	1	yes	characteristic
23	C₇×#544	C₂²	1	2	yes	—
24	C₇×#542	—	2	2	no	—
25	C₇×#542	C₂²	1	4	yes	—
26	C₇×#541	—	2	2	no	—
27	C₇×#541	C₂²	1	4	yes	—
28	C₇×#539	—	2	4	no	—
29	C₇×#537	C₂²	1	2	yes	—
30	C₇×#536	C₂²	1	2	yes	—
31	C₇×#536	C₂²	1	2	yes	—
32	C₇×#533	—	2	2	no	—
33	C₇×#533	C₂²	1	4	yes	—
34	C₇×#529	—	2	4	no	—
35	C₇×#529	—	2	2	no	—
36	C₇×#529	C₂²	1	4	yes	—
37	C₄×C₇×#41	C₂²	1	2	yes	—
38	C₄×C₇×#41	C₂²	1	1	yes	characteristic
39	C₂×C₄×C₇×Q₈	C₂²	1	2	yes	—
40	C₇×#501	C₂²	1	4	yes	—
41	C₇×#500	C₂²	1	2	yes	—
42	C₇×#499	C₂²	1	2	yes	—
43	C₇×#497	C₂²	1	4	yes	—
44	C₇×#495	C₂²	1	2	yes	—
45	C₇×#494	C₂²	1	2	yes	—
46	C₇×#491	C₂²	1	4	yes	—
47	C₇×#489	C₂²	1	4	yes	—
48	C₇×#488	C₂²	1	4	yes	—
49	C₇×#487	C₂²	1	4	yes	—
50	C₇×#470	C₂²	1	4	yes	—
51	C₇×#470	C₂²	1	4	yes	—
52	C₇×#468	C₂²	1	4	yes	—
53	C₇×#468	C₂²	1	4	yes	—
54	C₇×#467	C₂²	1	2	yes	—
55	C₇×#467	C₂²	1	2	yes	—
56	C₇×#467	C₂²	1	2	yes	—
57	C₇×#467	C₂²	1	4	yes	—
58	C₇×#466	C₂²	1	2	yes	—
59	C₇×#465	C₂²	1	4	yes	—
60	C₇×#441	C₂²	1	4	yes	—
61	C₇×#440	C₂²	1	4	yes	—
62	C₄×C₇×Q₁₆	C₂²	1	2	yes	—
63	C₄×C₇×Q₁₆	C₂²	1	1	yes	characteristic
64	C₄×C₇×Q₁₆	C₂²	1	4	yes	—
65	C₄×C₇×Q₁₆	C₂²	1	2	yes	—
66	C₄×C₇×Q₁₆	C₂²	1	2	yes	—
67	C₄×C₇×Q₁₆	C₂²	1	1	yes	characteristic
68	C₄×C₇×Q₁₆	C₂²	1	2	yes	—
69	C₄×C₇×SD₁₆	C₂²	1	4	yes	—
70	C₄×C₇×SD₁₆	C₂²	1	2	yes	—
71	C₄×C₇×SD₁₆	C₂²	1	2	yes	—
72	C₄×C₇×D₈	C₂²	1	1	yes	characteristic
73	C₄×C₇×D₈	C₂²	1	1	yes	characteristic
74	C₇×#428	C₂²	1	2	yes	—
75	C₇×#427	C₂²	1	1	yes	characteristic
76	C₂×C₇×#107	C₂²	1	1	yes	characteristic
77	C₇×#419	C₂²	1	2	yes	—
78	C₇×#417	C₂²	1	2	yes	—
79	C₇×#416	C₂²	1	2	yes	—
80	C₂×C₇×#103	C₂²	1	2	yes	—
81	C₇×#405	C₂²	1	1	yes	characteristic
82	C₇×#405	C₂²	1	2	yes	—
83	C₇×#405	C₂²	1	2	yes	—
84	C₇×#405	C₂²	1	1	yes	characteristic
85	C₄×C₇×M₁₆	C₂²	1	2	yes	—
86	C₄×C₇×M₁₆	C₂²	1	1	yes	characteristic
87	C₂×C₄×C₇×C₈	C₂²	1	2	yes	—
88	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
89	#60696	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
90	C₂×C₇×#41	C₂³	1	2	yes	—
91	C₂×C₇×#41	C₂³	1	1	yes	characteristic
92	C₂²×C₇×Q₈	C₂³	1	2	yes	—
93	C₇×#137	—	2	8	no	—
94	C₇×#135	—	2	4	no	—
95	C₂×C₇×Q₁₆	—	2	4	no	—
96	C₂×C₇×Q₁₆	C₂³	1	1	yes	characteristic
97	C₂×C₇×Q₁₆	C₂³	1	2	yes	—
98	C₂×C₇×Q₁₆	C₂³	1	4	yes	—
99	C₂×C₇×Q₁₆	C₂³	1	2	yes	—
100	C₂×C₇×SD₁₆	C₂³	1	2	yes	—
101	C₂×C₇×SD₁₆	C₂³	1	4	yes	—
102	C₂×C₇×D₈	C₂³	1	1	yes	characteristic
103	C₇×#131	—	2	4	no	—
104	C₇×#131	—	2	4	no	—
105	C₂×C₇×M₁₆	C₂³	1	2	yes	—
106	C₂×C₇×M₁₆	C₂³	1	2	yes	—
107	C₂×C₇×M₁₆	C₂³	1	1	yes	characteristic
108	C₂×C₇×M₁₆	C₂³	1	1	yes	characteristic
109	C₂²×C₇×C₈	C₂³	1	2	yes	—
110	C₂²×C₇×C₈	C₂³	1	2	yes	—
111	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
112	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
113	C₇×C₄⋊Q₈	—	2	2	no	—
114	C₇×C₄⋊Q₈	—	2	2	no	—
115	C₇×C₄⋊Q₈	C₂³	1	4	yes	—
116	C₇×C₄⋊Q₈	C₂³	1	4	yes	—
117	C₇×#126	—	2	4	no	—
118	C₇×#126	—	2	4	no	—
119	C₇×#126	—	2	4	no	—
120	C₇×#126	—	2	4	no	—
121	C₇×#125	—	2	4	no	—
122	C₇×#125	—	2	2	no	—
123	C₇×#125	—	2	4	no	—
124	C₇×#125	—	2	4	no	—
125	C₇×#125	C₂³	1	4	yes	—
126	C₇×#124	—	2	2	no	—
127	C₇×#124	—	2	4	no	—
128	C₇×#124	C₂³	1	4	yes	—
129	C₇×#123	—	2	4	no	—
130	C₇×#123	—	2	4	no	—
131	C₇×#123	—	2	4	no	—
132	C₇×#123	—	2	4	no	—
133	C₇×C₂²⋊Q₈	—	2	2	no	—
134	C₇×C₂²⋊Q₈	—	2	4	no	—
135	C₇×C₂²⋊Q₈	—	2	2	no	—
136	C₇×C₂²⋊Q₈	—	2	2	no	—
137	C₇×C₂²⋊Q₈	—	2	4	no	—
138	C₇×C₂²⋊Q₈	—	2	4	no	—
139	C₇×C₂²⋊Q₈	C₂³	1	4	yes	—
140	C₇×C₂²⋊Q₈	C₂³	1	4	yes	—
141	C₇×C₂²⋊Q₈	C₂³	1	4	yes	—
142	C₇×#121	—	2	4	no	—
143	C₇×#121	—	2	2	no	—
144	C₇×#121	C₂³	1	4	yes	—
145	C₄×C₇×Q₈	—	2	2	no	—
146	C₄×C₇×Q₈	—	2	2	no	—
147	C₄×C₇×Q₈	—	2	4	no	—
148	C₄×C₇×Q₈	D₄	1	1	yes	characteristic
149	C₄×C₇×Q₈	D₄	1	1	yes	characteristic
150	C₄×C₇×Q₈	C₂³	1	4	yes	—
151	C₄×C₇×Q₈	C₂³	1	4	yes	—
152	C₄×C₇×Q₈	C₂³	1	2	yes	—
153	C₄×C₇×Q₈	C₂³	1	2	yes	—
154	C₄×C₇×D₄	—	2	2	no	—
155	C₄×C₇×D₄	—	2	2	no	—
156	C₄×C₇×D₄	—	2	4	no	—
157	C₄×C₇×D₄	D₄	1	1	yes	characteristic
158	C₄×C₇×D₄	D₄	1	2	yes	—
159	C₄×C₇×D₄	D₄	1	2	yes	—
160	C₄×C₇×D₄	D₄	1	1	yes	characteristic
161	C₄×C₇×D₄	C₂³	1	2	yes	—
162	C₄×C₇×D₄	C₂³	1	2	yes	—
163	C₇×#117	—	2	4	no	—
164	C₇×#117	—	2	2	no	—
165	C₇×#117	—	2	4	no	—
166	C₇×#117	C₂³	1	2	yes	—
167	C₇×#117	C₂³	1	1	yes	characteristic
168	C₇×#117	C₂³	1	2	yes	—
169	C₂×C₇×C₄⋊C₄	—	2	2	no	—
170	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
171	C₂×C₄²×C₇	—	2	2	no	—
172	C₂×C₄²×C₇	—	2	2	no	—
173	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
174	C₂×C₄²×C₇	C₂³	1	2	yes	—
175	C₇×#107	C₂³	1	2	yes	—
176	C₇×#107	C₂³	1	1	yes	characteristic
177	C₇×#107	C₂³	1	1	yes	characteristic
178	C₇×#107	C₂³	1	4	yes	—
179	C₇×#107	C₂³	1	2	yes	—
180	C₇×#106	C₂³	1	2	yes	—
181	C₇×#106	C₂³	1	4	yes	—
182	C₇×#103	C₂³	1	2	yes	—
183	C₇×#103	C₂³	1	4	yes	—
184	C₇×#103	C₂³	1	4	yes	—
185	C₇×#103	C₂³	1	4	yes	—
186	C₇×#103	C₂³	1	4	yes	—
187	C₇×#103	C₂³	1	4	yes	—
188	C₇×#103	C₂³	1	2	yes	—
189	C₇×#102	C₂³	1	4	yes	—
190	C₇×#102	C₂³	1	2	yes	—
191	C₇×#102	C₂³	1	2	yes	—
192	C₇×#97	C₂³	1	2	yes	—
193	C₇×#97	C₂³	1	2	yes	—
194	C₇×#97	C₂³	1	2	yes	—
195	C₄×C₇×C₈	C₂³	1	2	yes	—
196	C₄×C₇×C₈	C₂³	1	2	yes	—
197	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
198	C₄×C₇×C₈	C₂³	1	2	yes	—
199	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
200	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
201	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
202	#3461	C₂×C₇	1	2	yes	complemented
203	#3178	C₂×C₇	1	4	yes	complemented
204	#3177	C₂×C₇	1	2	yes	complemented
205	#3175	C₂×C₇	1	2	yes	complemented
206	#3172	C₂×C₇	1	4	yes	complemented
207	#3170	C₂×C₇	1	2	yes	complemented
208	#3168	C₂×C₇	1	2	yes	complemented
209	#3164	C₂×C₇	1	1	yes	characteristic, complemented
210	#3156	C₂×C₇	1	1	yes	characteristic, complemented
211	#3137	C₂×C₇	1	2	yes	complemented
212	#3063	C₂×C₇	1	2	yes	complemented
213	#3061	C₂×C₇	1	1	yes	characteristic
214	#3059	C₂×C₇	1	1	yes	characteristic, complemented
215	C₄×#137	C₂×C₇	1	2	yes	complemented
216	C₄×#135	C₂×C₇	1	1	yes	characteristic, complemented
217	C₂×C₄×Q₁₆	C₂×C₇	1	1	yes	characteristic, complemented
218	C₄×#131	C₂×C₇	1	1	yes	characteristic, complemented
219	C₇×#41	—	2	2	no	—
220	C₇×#41	—	2	2	no	—
221	C₇×#41	—	2	4	no	—
222	C₇×#41	—	2	4	no	—
223	C₂×C₇×Q₈	—	2	2	no	—
224	C₂×C₇×Q₈	—	2	4	no	—
225	C₂×C₇×Q₈	—	2	4	no	—
226	C₂×C₇×Q₈	—	2	4	no	—
227	C₂×C₇×Q₈	C₂×D₄	1	1	yes	characteristic
228	C₂×C₇×Q₈	C₂⁴	1	4	yes	—
229	C₂×C₇×Q₈	C₂⁴	1	2	yes	—
230	C₂×C₇×D₄	—	2	2	no	—
231	C₂×C₇×D₄	—	2	4	no	—
232	C₂×C₇×D₄	C₂×D₄	1	2	yes	—
233	C₂×C₇×D₄	C₂×D₄	1	1	yes	characteristic
234	C₂×C₇×D₄	C₂⁴	1	2	yes	—
235	C₂²×C₄×C₇	—	2	2	no	—
236	C₂²×C₄×C₇	—	2	2	no	—
237	C₂²×C₄×C₇	—	2	2	no	—
238	C₂²×C₄×C₇	—	2	2	no	—
239	C₂²×C₄×C₇	—	2	2	no	—
240	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
241	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
242	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
243	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
244	C₂²×C₄×C₇	C₂⁴	1	2	yes	—
245	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
246	C₇×Q₁₆	—	2	4	no	—
247	C₇×Q₁₆	—	2	4	no	—
248	C₇×Q₁₆	—	2	2	no	—
249	C₇×Q₁₆	—	2	8	no	—
250	C₇×SD₁₆	—	2	4	no	—
251	C₇×SD₁₆	—	2	8	no	—
252	C₇×D₈	—	2	2	no	—
253	C₇×M₁₆	—	2	2	no	—
254	C₇×M₁₆	—	2	4	no	—
255	C₇×M₁₆	—	2	4	no	—
256	C₇×M₁₆	#41	1	4	yes	—
257	C₂×C₇×C₈	—	2	2	no	—
258	C₂×C₇×C₈	—	2	4	no	—
259	C₂×C₇×C₈	—	2	4	no	—
260	C₂×C₇×C₈	#41	1	4	yes	—
261	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
262	C₂×C₇×C₈	C₂⁴	1	2	yes	—
263	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
264	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
265	C₂×C₇×C₈	C₂⁴	1	2	yes	—
266	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
267	C₇×C₄⋊C₄	—	2	4	no	—
268	C₇×C₄⋊C₄	—	2	4	no	—
269	C₇×C₄⋊C₄	—	2	2	no	—
270	C₇×C₄⋊C₄	—	2	2	no	—
271	C₇×C₄⋊C₄	—	2	2	no	—
272	C₇×C₄⋊C₄	—	2	4	no	—
273	C₇×C₄⋊C₄	—	2	4	no	—
274	C₇×C₄⋊C₄	—	2	4	no	—
275	C₇×C₄⋊C₄	—	2	4	no	—
276	C₇×C₄⋊C₄	—	2	4	no	—
277	C₇×C₄⋊C₄	—	2	4	no	—
278	C₇×C₄⋊C₄	—	2	4	no	—
279	C₇×C₄⋊C₄	—	2	4	no	—
280	C₇×C₄⋊C₄	C₂×D₄	1	2	yes	—
281	C₇×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
282	C₇×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
283	C₇×C₄⋊C₄	C₂×D₄	1	2	yes	—
284	C₇×C₄⋊C₄	C₂⁴	1	2	yes	—
285	C₇×C₄⋊C₄	C₂⁴	1	4	yes	—
286	C₇×C₄⋊C₄	C₂⁴	1	2	yes	—
287	C₇×C₂²⋊C₄	—	2	4	no	—
288	C₇×C₂²⋊C₄	—	2	4	no	—
289	C₇×C₂²⋊C₄	—	2	2	no	—
290	C₇×C₂²⋊C₄	—	2	4	no	—
291	C₇×C₂²⋊C₄	—	2	2	no	—
292	C₇×C₂²⋊C₄	—	2	2	no	—
293	C₇×C₂²⋊C₄	—	2	4	no	—
294	C₇×C₂²⋊C₄	—	2	4	no	—
295	C₇×C₂²⋊C₄	—	2	4	no	—
296	C₇×C₂²⋊C₄	C₂×D₄	1	2	yes	—
297	C₇×C₂²⋊C₄	C₂×D₄	1	2	yes	—
298	C₇×C₂²⋊C₄	C₂×D₄	1	2	yes	—
299	C₄²×C₇	—	2	2	no	—
300	C₄²×C₇	—	2	2	no	—
301	C₄²×C₇	—	2	4	no	—
302	C₄²×C₇	—	2	4	no	—
303	C₄²×C₇	—	2	2	no	—
304	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
305	C₄²×C₇	C₂×D₄	1	2	yes	—
306	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
307	C₄²×C₇	C₂×D₄	1	2	yes	—
308	C₄²×C₇	C₂⁴	1	1	yes	characteristic
309	C₄²×C₇	C₂⁴	1	2	yes	—
310	C₄²×C₇	C₂⁴	1	1	yes	characteristic
311	C₂×#137	C₂²×C₇	1	2	yes	—
312	C₂×#135	C₂²×C₇	1	1	yes	characteristic
313	C₂²×Q₁₆	C₂²×C₇	1	1	yes	characteristic
314	C₂×#131	C₂²×C₇	1	1	yes	characteristic
315	C₂×#131	C₂²×C₇	1	1	yes	characteristic
316	#544	C₂²×C₇	1	2	yes	—
317	#542	—	2	2	no	—
318	#542	C₂²×C₇	1	4	yes	—
319	#541	—	2	2	no	—
320	#541	C₂²×C₇	1	4	yes	—
321	#539	—	2	4	no	—
322	#537	C₂²×C₇	1	2	yes	—
323	#536	C₂²×C₇	1	2	yes	—
324	#536	C₂²×C₇	1	2	yes	—
325	#533	—	2	2	no	—
326	#533	C₂²×C₇	1	4	yes	—
327	#529	—	2	4	no	—
328	#529	—	2	2	no	—
329	#529	C₂²×C₇	1	4	yes	—
330	C₄×#41	C₂²×C₇	1	2	yes	—
331	C₄×#41	C₂²×C₇	1	1	yes	characteristic
332	C₂×C₄×Q₈	C₂²×C₇	1	2	yes	—
333	#501	C₂²×C₇	1	4	yes	—
334	#500	C₂²×C₇	1	2	yes	—
335	#499	C₂²×C₇	1	2	yes	—
336	#497	C₂²×C₇	1	4	yes	—
337	#495	C₂²×C₇	1	2	yes	—
338	#494	C₂²×C₇	1	2	yes	—
339	#491	C₂²×C₇	1	4	yes	—
340	#489	C₂²×C₇	1	4	yes	—
341	#488	C₂²×C₇	1	4	yes	—
342	#487	C₂²×C₇	1	4	yes	—
343	#470	C₂²×C₇	1	4	yes	—
344	#470	C₂²×C₇	1	4	yes	—
345	#468	C₂²×C₇	1	4	yes	—
346	#468	C₂²×C₇	1	4	yes	—
347	#467	C₂²×C₇	1	2	yes	—
348	#467	C₂²×C₇	1	2	yes	—
349	#467	C₂²×C₇	1	2	yes	—
350	#467	C₂²×C₇	1	4	yes	—
351	#466	C₂²×C₇	1	2	yes	—
352	#465	C₂²×C₇	1	4	yes	—
353	#441	C₂²×C₇	1	4	yes	—
354	#440	C₂²×C₇	1	4	yes	—
355	C₄×Q₁₆	C₂²×C₇	1	2	yes	—
356	C₄×Q₁₆	C₂²×C₇	1	1	yes	characteristic
357	C₄×Q₁₆	C₂²×C₇	1	4	yes	—
358	C₄×Q₁₆	C₂²×C₇	1	2	yes	—
359	C₄×Q₁₆	C₂²×C₇	1	2	yes	—
360	C₄×Q₁₆	C₂²×C₇	1	1	yes	characteristic
361	C₄×Q₁₆	C₂²×C₇	1	2	yes	—
362	C₄×SD₁₆	C₂²×C₇	1	4	yes	—
363	C₄×SD₁₆	C₂²×C₇	1	2	yes	—
364	C₄×SD₁₆	C₂²×C₇	1	2	yes	—
365	C₄×D₈	C₂²×C₇	1	1	yes	characteristic
366	C₄×D₈	C₂²×C₇	1	1	yes	characteristic
367	#428	C₂²×C₇	1	2	yes	—
368	#427	C₂²×C₇	1	1	yes	characteristic
369	C₂×#107	C₂²×C₇	1	1	yes	characteristic
370	#419	C₂²×C₇	1	2	yes	—
371	#417	C₂²×C₇	1	2	yes	—
372	#416	C₂²×C₇	1	2	yes	—
373	C₂×#103	C₂²×C₇	1	2	yes	—
374	#405	C₂²×C₇	1	1	yes	characteristic
375	#405	C₂²×C₇	1	1	yes	characteristic
376	#405	C₂²×C₇	1	2	yes	—
377	#405	C₂²×C₇	1	2	yes	—
378	C₄×M₁₆	C₂²×C₇	1	2	yes	—
379	C₄×M₁₆	C₂²×C₇	1	1	yes	characteristic
380	C₂×C₄×C₈	C₂²×C₇	1	2	yes	—
381	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
382	C₂³×C₇	—	2	2	no	—
383	C₂³×C₇	C₂²×D₄	1	2	yes	—
384	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
385	C₇×Q₈	—	2	2	no	—
386	C₇×Q₈	—	2	4	no	—
387	C₇×Q₈	—	2	2	no	—
388	C₇×Q₈	—	2	4	no	—
389	C₇×Q₈	—	2	2	no	—
390	C₇×Q₈	—	4	4	no	—
391	C₇×Q₈	—	8	4	no	—
392	C₇×D₄	—	2	4	no	—
393	C₇×D₄	—	2	2	no	—
394	C₇×D₄	—	2	2	no	—
395	C₇×D₄	—	2	2	no	—
396	C₇×D₄	—	4	4	no	—
397	C₇×D₄	—	8	4	no	—
398	C₂×C₄×C₇	—	2	2	no	—
399	C₂×C₄×C₇	—	2	4	no	—
400	C₂×C₄×C₇	—	2	2	no	—
401	C₂×C₄×C₇	—	2	2	no	—
402	C₂×C₄×C₇	—	2	2	no	—
403	C₂×C₄×C₇	—	2	4	no	—
404	C₂×C₄×C₇	—	2	2	no	—
405	C₂×C₄×C₇	—	2	2	no	—
406	C₂×C₄×C₇	—	2	4	no	—
407	C₂×C₄×C₇	—	4	1	no	—
408	C₂×C₄×C₇	—	4	2	no	—
409	C₂×C₄×C₇	—	4	4	no	—
410	C₂×C₄×C₇	—	4	4	no	—
411	C₂×C₄×C₇	—	4	4	no	—
412	C₂×C₄×C₇	—	8	2	no	—
413	C₂×C₄×C₇	—	8	2	no	—
414	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
415	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
416	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
417	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
418	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
419	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
420	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
421	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
422	C₂×C₄×C₇	C₂×#41	1	2	yes	—
423	C₂×C₄×C₇	C₂×#41	1	2	yes	—
424	C₂×C₄×C₇	C₂⁵	1	1	yes	characteristic
425	C₇×C₈	—	2	1	no	—
426	C₇×C₈	—	2	1	no	—
427	C₇×C₈	—	2	2	no	—
428	C₇×C₈	C₂×#41	1	2	yes	—
429	C₇×C₈	C₂×#41	1	4	yes	—
430	C₇×C₈	C₂×#41	1	2	yes	—
431	C₂×#41	C₂³×C₇	1	2	yes	—
432	C₂×#41	C₂³×C₇	1	1	yes	characteristic
433	C₂²×Q₈	C₂³×C₇	1	2	yes	—
434	#137	—	2	8	no	—
435	#135	—	2	4	no	—
436	C₂×Q₁₆	—	2	4	no	—
437	C₂×Q₁₆	C₂³×C₇	1	2	yes	—
438	C₂×Q₁₆	C₂³×C₇	1	2	yes	—
439	C₂×Q₁₆	C₂³×C₇	1	1	yes	characteristic
440	C₂×Q₁₆	C₂³×C₇	1	4	yes	—
441	C₂×SD₁₆	C₂³×C₇	1	2	yes	—
442	C₂×SD₁₆	C₂³×C₇	1	4	yes	—
443	C₂×D₈	C₂³×C₇	1	1	yes	characteristic
444	#131	—	2	4	no	—
445	#131	—	2	4	no	—
446	C₂×M₁₆	C₂³×C₇	1	1	yes	characteristic
447	C₂×M₁₆	C₂³×C₇	1	2	yes	—
448	C₂×M₁₆	C₂³×C₇	1	2	yes	—
449	C₂×M₁₆	C₂³×C₇	1	1	yes	characteristic
450	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
451	C₂²×C₈	C₂³×C₇	1	2	yes	—
452	C₂²×C₈	C₂³×C₇	1	2	yes	—
453	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
454	C₄⋊Q₈	—	2	2	no	—
455	C₄⋊Q₈	—	2	2	no	—
456	C₄⋊Q₈	C₂³×C₇	1	4	yes	—
457	C₄⋊Q₈	C₂³×C₇	1	4	yes	—
458	#126	—	2	4	no	—
459	#126	—	2	4	no	—
460	#126	—	2	4	no	—
461	#126	—	2	4	no	—
462	#125	—	2	4	no	—
463	#125	—	2	2	no	—
464	#125	—	2	4	no	—
465	#125	—	2	4	no	—
466	#125	C₂³×C₇	1	4	yes	—
467	#124	—	2	2	no	—
468	#124	—	2	4	no	—
469	#124	C₂³×C₇	1	4	yes	—
470	#123	—	2	4	no	—
471	#123	—	2	4	no	—
472	#123	—	2	4	no	—
473	#123	—	2	4	no	—
474	C₂²⋊Q₈	—	2	2	no	—
475	C₂²⋊Q₈	—	2	4	no	—
476	C₂²⋊Q₈	—	2	2	no	—
477	C₂²⋊Q₈	—	2	2	no	—
478	C₂²⋊Q₈	—	2	4	no	—
479	C₂²⋊Q₈	—	2	4	no	—
480	C₂²⋊Q₈	C₂³×C₇	1	4	yes	—
481	C₂²⋊Q₈	C₂³×C₇	1	4	yes	—
482	C₂²⋊Q₈	C₂³×C₇	1	4	yes	—
483	#121	—	2	4	no	—
484	#121	—	2	2	no	—
485	#121	C₂³×C₇	1	4	yes	—
486	C₄×Q₈	—	2	2	no	—
487	C₄×Q₈	—	2	2	no	—
488	C₄×Q₈	—	2	4	no	—
489	C₄×Q₈	C₇×D₄	1	1	yes	characteristic
490	C₄×Q₈	C₇×D₄	1	1	yes	characteristic
491	C₄×Q₈	C₂³×C₇	1	4	yes	—
492	C₄×Q₈	C₂³×C₇	1	4	yes	—
493	C₄×Q₈	C₂³×C₇	1	2	yes	—
494	C₄×Q₈	C₂³×C₇	1	2	yes	—
495	C₄×D₄	—	2	2	no	—
496	C₄×D₄	—	2	2	no	—
497	C₄×D₄	—	2	4	no	—
498	C₄×D₄	C₇×D₄	1	1	yes	characteristic
499	C₄×D₄	C₇×D₄	1	2	yes	—
500	C₄×D₄	C₇×D₄	1	2	yes	—
501	C₄×D₄	C₇×D₄	1	1	yes	characteristic
502	C₄×D₄	C₂³×C₇	1	2	yes	—
503	C₄×D₄	C₂³×C₇	1	2	yes	—
504	#117	—	2	4	no	—
505	#117	—	2	2	no	—
506	#117	—	2	4	no	—
507	#117	C₂³×C₇	1	2	yes	—
508	#117	C₂³×C₇	1	1	yes	characteristic
509	#117	C₂³×C₇	1	2	yes	—
510	C₂×C₄⋊C₄	—	2	2	no	—
511	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
512	C₂×C₄²	—	2	2	no	—
513	C₂×C₄²	—	2	2	no	—
514	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
515	C₂×C₄²	C₂³×C₇	1	2	yes	—
516	#107	C₂³×C₇	1	2	yes	—
517	#107	C₂³×C₇	1	2	yes	—
518	#107	C₂³×C₇	1	1	yes	characteristic
519	#107	C₂³×C₇	1	1	yes	characteristic
520	#107	C₂³×C₇	1	4	yes	—
521	#106	C₂³×C₇	1	2	yes	—
522	#106	C₂³×C₇	1	4	yes	—
523	#103	C₂³×C₇	1	4	yes	—
524	#103	C₂³×C₇	1	4	yes	—
525	#103	C₂³×C₇	1	2	yes	—
526	#103	C₂³×C₇	1	2	yes	—
527	#103	C₂³×C₇	1	4	yes	—
528	#103	C₂³×C₇	1	4	yes	—
529	#103	C₂³×C₇	1	4	yes	—
530	#102	C₂³×C₇	1	2	yes	—
531	#102	C₂³×C₇	1	2	yes	—
532	#102	C₂³×C₇	1	4	yes	—
533	#97	C₂³×C₇	1	2	yes	—
534	#97	C₂³×C₇	1	2	yes	—
535	#97	C₂³×C₇	1	2	yes	—
536	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
537	C₄×C₈	C₂³×C₇	1	2	yes	—
538	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
539	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
540	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
541	C₄×C₈	C₂³×C₇	1	2	yes	—
542	C₄×C₈	C₂³×C₇	1	2	yes	—
543	C₂²×C₇	—	2	1	no	—
544	C₂²×C₇	—	2	1	no	—
545	C₂²×C₇	—	2	2	no	—
546	C₂²×C₇	—	2	2	no	—
547	C₂²×C₇	—	2	2	no	—
548	C₂²×C₇	—	4	2	no	—
549	C₂²×C₇	—	4	2	no	—
550	C₂²×C₇	—	4	2	no	—
551	C₂²×C₇	#532	1	2	yes	—
552	C₂²×C₇	C₂³×D₄	1	1	yes	characteristic
553	C₄×C₇	—	2	1	no	—
554	C₄×C₇	—	2	1	no	—
555	C₄×C₇	—	2	2	no	—
556	C₄×C₇	—	2	2	no	—
557	C₄×C₇	—	2	1	no	—
558	C₄×C₇	—	2	2	no	—
559	C₄×C₇	—	4	4	no	—
560	C₄×C₇	—	4	2	no	—
561	C₄×C₇	—	4	2	no	—
562	C₄×C₇	—	4	4	no	—
563	C₄×C₇	—	4	2	no	—
564	C₄×C₇	#532	1	2	yes	—
565	C₄×C₇	#578	1	2	yes	—
566	C₄×C₇	C₂²×#41	1	1	yes	characteristic
567	C₄×C₇	C₂²×#41	1	1	yes	characteristic
568	#41	—	2	2	no	—
569	#41	—	2	2	no	—
570	#41	—	2	4	no	—
571	#41	—	2	4	no	—
572	C₂×Q₈	—	2	2	no	—
573	C₂×Q₈	—	2	4	no	—
574	C₂×Q₈	—	2	4	no	—
575	C₂×Q₈	—	2	4	no	—
576	C₂×Q₈	C₂×C₇×D₄	1	1	yes	characteristic
577	C₂×Q₈	C₂⁴×C₇	1	4	yes	—
578	C₂×Q₈	C₂⁴×C₇	1	2	yes	—
579	C₂×D₄	—	2	2	no	—
580	C₂×D₄	—	2	4	no	—
581	C₂×D₄	C₂×C₇×D₄	1	2	yes	—
582	C₂×D₄	C₂×C₇×D₄	1	1	yes	characteristic
583	C₂×D₄	C₂⁴×C₇	1	2	yes	—
584	C₂²×C₄	—	2	2	no	—
585	C₂²×C₄	—	2	2	no	—
586	C₂²×C₄	—	2	2	no	—
587	C₂²×C₄	—	2	2	no	—
588	C₂²×C₄	—	2	2	no	—
589	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
590	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
591	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
592	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
593	C₂²×C₄	C₂⁴×C₇	1	2	yes	—
594	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
595	Q₁₆	—	2	4	no	—
596	Q₁₆	—	2	4	no	—
597	Q₁₆	—	2	2	no	—
598	Q₁₆	—	2	8	no	—
599	SD₁₆	—	2	4	no	—
600	SD₁₆	—	2	8	no	—
601	D₈	—	2	2	no	—
602	M₁₆	—	2	2	no	—
603	M₁₆	—	2	4	no	—
604	M₁₆	—	2	4	no	—
605	M₁₆	C₇×#41	1	4	yes	—
606	C₂×C₈	—	2	2	no	—
607	C₂×C₈	—	2	4	no	—
608	C₂×C₈	—	2	4	no	—
609	C₂×C₈	C₇×#41	1	4	yes	—
610	C₂×C₈	C₂⁴×C₇	1	1	yes	characteristic
611	C₂×C₈	C₂⁴×C₇	1	1	yes	characteristic
612	C₂×C₈	C₂⁴×C₇	1	2	yes	—
613	C₂×C₈	C₂⁴×C₇	1	1	yes	characteristic
614	C₂×C₈	C₂⁴×C₇	1	2	yes	—
615	C₂×C₈	C₂⁴×C₇	1	1	yes	characteristic
616	C₄⋊C₄	—	2	4	no	—
617	C₄⋊C₄	—	2	4	no	—
618	C₄⋊C₄	—	2	2	no	—
619	C₄⋊C₄	—	2	2	no	—
620	C₄⋊C₄	—	2	2	no	—
621	C₄⋊C₄	—	2	4	no	—
622	C₄⋊C₄	—	2	4	no	—
623	C₄⋊C₄	—	2	4	no	—
624	C₄⋊C₄	—	2	4	no	—
625	C₄⋊C₄	—	2	4	no	—
626	C₄⋊C₄	—	2	4	no	—
627	C₄⋊C₄	—	2	4	no	—
628	C₄⋊C₄	—	2	4	no	—
629	C₄⋊C₄	C₂×C₇×D₄	1	2	yes	—
630	C₄⋊C₄	C₂×C₇×D₄	1	1	yes	characteristic
631	C₄⋊C₄	C₂×C₇×D₄	1	1	yes	characteristic
632	C₄⋊C₄	C₂×C₇×D₄	1	2	yes	—
633	C₄⋊C₄	C₂⁴×C₇	1	2	yes	—
634	C₄⋊C₄	C₂⁴×C₇	1	4	yes	—
635	C₄⋊C₄	C₂⁴×C₇	1	2	yes	—
636	C₂²⋊C₄	—	2	4	no	—
637	C₂²⋊C₄	—	2	4	no	—
638	C₂²⋊C₄	—	2	2	no	—
639	C₂²⋊C₄	—	2	4	no	—
640	C₂²⋊C₄	—	2	2	no	—
641	C₂²⋊C₄	—	2	2	no	—
642	C₂²⋊C₄	—	2	4	no	—
643	C₂²⋊C₄	—	2	4	no	—
644	C₂²⋊C₄	—	2	4	no	—
645	C₂²⋊C₄	C₂×C₇×D₄	1	2	yes	—
646	C₂²⋊C₄	C₂×C₇×D₄	1	2	yes	—
647	C₂²⋊C₄	C₂×C₇×D₄	1	2	yes	—
648	C₄²	—	2	4	no	—
649	C₄²	—	2	2	no	—
650	C₄²	—	2	2	no	—
651	C₄²	—	2	2	no	—
652	C₄²	—	2	4	no	—
653	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
654	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
655	C₄²	C₂×C₇×D₄	1	2	yes	—
656	C₄²	C₂×C₇×D₄	1	2	yes	—
657	C₄²	C₂⁴×C₇	1	1	yes	characteristic
658	C₄²	C₂⁴×C₇	1	1	yes	characteristic
659	C₄²	C₂⁴×C₇	1	2	yes	—
660	C₂×C₇	—	2	2	no	—
661	C₂×C₇	—	4	2	no	—
662	C₂×C₇	—	8	2	no	—
663	C₂×C₇	C₂×#532	1	1	yes	characteristic
664	C₂×C₇	C₂×#578	1	1	yes	characteristic
665	C₂×C₇	#3584	1	1	yes	characteristic
666	C₂³	—	2	2	no	—
667	C₂³	C₂²×C₇×D₄	1	2	yes	—
668	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
669	Q₈	—	2	2	no	—
670	Q₈	—	2	4	no	—
671	Q₈	—	2	2	no	—
672	Q₈	—	2	4	no	—
673	Q₈	—	2	2	no	—
674	Q₈	—	4	4	no	—
675	Q₈	—	8	4	no	—
676	D₄	—	2	4	no	—
677	D₄	—	2	2	no	—
678	D₄	—	2	2	no	—
679	D₄	—	2	2	no	—
680	D₄	—	4	4	no	—
681	D₄	—	8	4	no	—
682	C₂×C₄	—	2	2	no	—
683	C₂×C₄	—	2	2	no	—
684	C₂×C₄	—	2	2	no	—
685	C₂×C₄	—	2	4	no	—
686	C₂×C₄	—	2	2	no	—
687	C₂×C₄	—	2	4	no	—
688	C₂×C₄	—	2	2	no	—
689	C₂×C₄	—	2	2	no	—
690	C₂×C₄	—	2	4	no	—
691	C₂×C₄	—	4	1	no	—
692	C₂×C₄	—	4	2	no	—
693	C₂×C₄	—	4	4	no	—
694	C₂×C₄	—	4	4	no	—
695	C₂×C₄	—	4	4	no	—
696	C₂×C₄	—	8	2	no	—
697	C₂×C₄	—	8	2	no	—
698	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
699	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
700	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
701	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
702	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
703	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
704	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
705	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
706	C₂×C₄	C₂×C₇×#41	1	2	yes	—
707	C₂×C₄	C₂×C₇×#41	1	2	yes	—
708	C₂×C₄	C₂⁵×C₇	1	1	yes	characteristic
709	C₈	—	2	1	no	—
710	C₈	—	2	1	no	—
711	C₈	—	2	2	no	—
712	C₈	C₂×C₇×#41	1	2	yes	—
713	C₈	C₂×C₇×#41	1	4	yes	—
714	C₈	C₂×C₇×#41	1	2	yes	—
715	C₇	#60696	1	1	yes	characteristic, direct product factor, Sylow
716	C₂²	—	2	1	no	—
717	C₂²	—	2	1	no	—
718	C₂²	—	2	2	no	—
719	C₂²	—	2	2	no	—
720	C₂²	—	2	2	no	—
721	C₂²	—	4	2	no	—
722	C₂²	—	4	2	no	—
723	C₂²	—	4	2	no	—
724	C₂²	C₇×#532	1	2	yes	—
725	C₂²	C₂³×C₇×D₄	1	1	yes	characteristic
726	C₄	—	2	1	no	—
727	C₄	—	2	2	no	—
728	C₄	—	2	1	no	—
729	C₄	—	2	1	no	—
730	C₄	—	2	2	no	—
731	C₄	—	2	2	no	—
732	C₄	—	4	2	no	—
733	C₄	—	4	4	no	—
734	C₄	—	4	2	no	—
735	C₄	—	4	2	no	—
736	C₄	—	4	4	no	—
737	C₄	C₇×#532	1	2	yes	—
738	C₄	C₇×#578	1	2	yes	—
739	C₄	C₂²×C₇×#41	1	1	yes	characteristic
740	C₄	C₂²×C₇×#41	1	1	yes	characteristic
741	C₂	—	2	2	no	—
742	C₂	—	4	2	no	—
743	C₂	—	8	2	no	—
744	C₂	C₂×C₇×#532	1	1	yes	characteristic
745	C₂	C₂×C₇×#578	1	1	yes	characteristic
746	C₂	C₇×#3584	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₇	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₇
socle	C₂²×C₇	C₂³×D₄
prime radical	C₇×#3061	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇ ◁ C₄×C₇×#41 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₇×#3461	C₂	semidirect
2. C₇×#3178	C₂	semidirect
3. C₇×#3177	C₂	semidirect
4. C₇×#3175	C₂	semidirect
5. C₇×#3172	C₂	semidirect
6. C₇×#3170	C₂	semidirect
7. C₇×#3168	C₂	semidirect
8. C₇×#3164	C₂	semidirect
9. C₇×#3156	C₂	semidirect
10. C₇×#3137	C₂	semidirect
11. C₇×#3063	C₂	semidirect
13. C₇×#3059	C₂	semidirect
14. C₄×C₇×#137	C₂	semidirect
15. C₄×C₇×#135	C₂	semidirect
16. C₂×C₄×C₇×Q₁₆	C₂	semidirect
17. C₄×C₇×#131	C₂	semidirect
89. #60696	C₇	direct (N × H)
202. #3461	C₂×C₇	semidirect
203. #3178	C₂×C₇	semidirect
204. #3177	C₂×C₇	semidirect
205. #3175	C₂×C₇	semidirect
206. #3172	C₂×C₇	semidirect
207. #3170	C₂×C₇	semidirect
208. #3168	C₂×C₇	semidirect
209. #3164	C₂×C₇	semidirect
210. #3156	C₂×C₇	semidirect
211. #3137	C₂×C₇	semidirect
212. #3063	C₂×C₇	semidirect
214. #3059	C₂×C₇	semidirect
215. C₄×#137	C₂×C₇	semidirect
216. C₄×#135	C₂×C₇	semidirect
217. C₂×C₄×Q₁₆	C₂×C₇	semidirect
218. C₄×#131	C₂×C₇	semidirect
715. C₇	#60696	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×#3461	2
C₇×#3178	2
C₇×#3177	2
C₇×#3175	2
C₇×#3172	2
C₇×#3170	2
C₇×#3168	2
C₇×#3164	2
C₇×#3156	2
C₇×#3137	2
C₇×#3063	2
C₇×#3061	2
C₇×#3059	2
C₄×C₇×#137	2
C₄×C₇×#135	2
C₂×C₄×C₇×Q₁₆	2
C₄×C₇×#131	2
#60696	7

Subgroups — C7×#60696 #13237912

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#60696 #13237912