Subgroups — C₇×#61334 #13238550

Order 1792.

1794 conjugacy classes of subgroups.

The subgroup lattice has 752 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3506	C₂	1	1	yes	maximal, characteristic
2	C₇×#3502	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#3377	C₂	1	4	yes	maximal, complemented
4	C₇×#3375	C₂	1	4	yes	maximal, complemented
5	C₇×#3371	C₂	1	4	yes	maximal, complemented
6	C₇×#3368	C₂	1	4	yes	maximal, complemented
7	C₇×#3136	C₂	1	2	yes	maximal, complemented
8	C₇×#3135	C₂	1	1	yes	maximal, characteristic, complemented
9	C₇×#3083	C₂	1	2	yes	maximal, complemented
10	C₇×#3077	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇×#3043	C₂	1	2	yes	maximal, complemented
12	C₇×#3037	C₂	1	1	yes	maximal, characteristic, complemented
13	C₇×#2961	C₂	1	1	yes	maximal, characteristic, complemented
14	C₇×#2947	C₂	1	2	yes	maximal, complemented
15	C₂×C₇×#440	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×C₇×#137	C₂²	1	2	yes	—
17	C₂²×C₇×SD₁₆	C₂²	1	1	yes	characteristic
18	C₇×#559	C₂²	1	4	yes	—
19	C₇×Q₈²	—	2	1	no	—
20	C₇×Q₈²	C₂²	1	2	yes	complemented
21	C₇×#557	—	2	1	no	—
22	C₇×#557	—	2	2	no	—
23	C₇×#557	C₂²	1	4	yes	complemented
24	C₇×#557	C₂²	1	2	yes	complemented
25	C₇×#556	C₂²	1	2	yes	—
26	C₇×#555	—	2	1	no	—
27	C₇×#553	—	2	2	no	—
28	C₇×#550	C₂²	1	2	yes	—
29	C₇×#547	—	2	1	no	—
30	C₇×#544	—	2	2	no	—
31	C₇×#544	C₂²	1	2	yes	—
32	C₇×#538	—	2	2	no	—
33	C₇×#535	C₂²	1	2	yes	—
34	C₇×#533	—	2	2	no	—
35	C₇×#533	C₂²	1	1	yes	characteristic
36	C₇×#533	C₂²	1	1	yes	characteristic
37	C₇×#532	C₂²	1	1	yes	characteristic
38	C₇×#525	—	2	2	no	—
39	C₇×#520	C₂²	1	2	yes	—
40	C₇×#519	C₂²	1	2	yes	—
41	C₂×C₄×C₇×Q₈	C₂²	1	1	yes	characteristic
42	C₂×C₄×C₇×D₄	C₂²	1	1	yes	characteristic
43	C₇×#491	C₂²	1	2	yes	complemented
44	C₇×#490	C₂²	1	4	yes	—
45	C₇×#490	C₂²	1	2	yes	complemented
46	C₇×#489	C₂²	1	4	yes	—
47	C₇×#489	C₂²	1	2	yes	complemented
48	C₇×#489	C₂²	1	2	yes	complemented
49	C₇×#488	C₂²	1	4	yes	—
50	C₇×#486	C₂²	1	4	yes	—
51	C₇×#479	C₂²	1	4	yes	complemented
52	C₇×#478	C₂²	1	4	yes	—
53	C₇×#478	C₂²	1	4	yes	complemented
54	C₇×#477	C₂²	1	4	yes	complemented
55	C₇×#476	C₂²	1	2	yes	—
56	C₇×#474	C₂²	1	2	yes	—
57	C₇×#474	C₂²	1	4	yes	complemented
58	C₇×#464	C₂²	1	4	yes	—
59	C₇×#464	C₂²	1	4	yes	—
60	C₇×#464	C₂²	1	4	yes	complemented
61	C₇×#462	C₂²	1	2	yes	complemented
62	C₇×#460	C₂²	1	4	yes	—
63	C₇×#460	C₂²	1	4	yes	—
64	C₇×#460	C₂²	1	2	yes	complemented
65	C₇×#446	C₂²	1	4	yes	complemented
66	C₇×#446	C₂²	1	4	yes	complemented
67	C₇×#441	C₂²	1	4	yes	complemented
68	C₇×#440	C₂²	1	4	yes	—
69	C₇×#440	C₂²	1	4	yes	—
70	C₇×#440	C₂²	1	4	yes	complemented
71	C₄×C₇×Q₁₆	C₂²	1	4	yes	complemented
72	C₄×C₇×SD₁₆	C₂²	1	4	yes	complemented
73	C₇×#428	C₂²	1	2	yes	—
74	C₂×C₇×#107	C₂²	1	1	yes	characteristic
75	C₇×#424	C₂²	1	2	yes	—
76	C₇×#423	C₂²	1	1	yes	characteristic
77	C₇×#418	C₂²	1	2	yes	—
78	C₇×#417	C₂²	1	2	yes	—
79	C₇×#416	C₂²	1	2	yes	—
80	C₇×#416	C₂²	1	2	yes	—
81	C₂×C₇×#103	C₂²	1	1	yes	characteristic
82	C₂×C₇×#102	C₂²	1	1	yes	characteristic
83	C₇×#405	C₂²	1	2	yes	—
84	C₂×C₇×#97	C₂²	1	1	yes	characteristic
85	#61334	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
86	C₂×C₇×#41	—	2	2	no	—
87	C₂×C₇×#41	C₂³	1	2	yes	—
88	C₂²×C₇×Q₈	C₂³	1	1	yes	characteristic
89	C₂²×C₇×D₄	C₂³	1	1	yes	characteristic
90	C₂³×C₄×C₇	—	2	1	no	—
91	C₇×#137	—	2	8	no	—
92	C₂×C₇×Q₁₆	C₂³	1	4	yes	—
93	C₂×C₇×SD₁₆	—	2	4	no	—
94	C₂×C₇×SD₁₆	C₂³	1	4	yes	—
95	C₂×C₇×SD₁₆	C₂³	1	4	yes	—
96	C₂×C₇×M₁₆	C₂³	1	2	yes	—
97	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
98	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
99	C₇×C₄⋊Q₈	—	2	2	no	—
100	C₇×C₄⋊Q₈	—	2	4	no	—
101	C₇×C₄⋊Q₈	—	2	2	no	—
102	C₇×C₄⋊Q₈	—	2	4	no	—
103	C₇×C₄⋊Q₈	—	2	2	no	—
104	C₇×C₄⋊Q₈	D₄	1	1	yes	characteristic
105	C₇×C₄⋊Q₈	D₄	1	1	yes	characteristic
106	C₇×C₄⋊Q₈	C₂³	1	1	yes	characteristic
107	C₇×C₄⋊Q₈	C₂³	1	1	yes	characteristic
108	C₇×C₄⋊Q₈	C₂³	1	4	yes	—
109	C₇×C₄⋊Q₈	C₂³	1	1	yes	characteristic
110	C₇×#127	C₂³	1	4	yes	—
111	C₇×#127	C₂³	1	1	yes	characteristic
112	C₇×#126	—	2	4	no	—
113	C₇×#125	—	2	4	no	—
114	C₇×#125	—	2	4	no	—
115	C₇×#125	—	2	4	no	—
116	C₇×#125	—	2	2	no	—
117	C₇×#125	D₄	1	2	yes	—
118	C₇×#125	C₂³	1	2	yes	—
119	C₇×#125	C₂³	1	4	yes	—
120	C₇×#124	—	2	4	no	—
121	C₇×#124	C₂³	1	4	yes	—
122	C₇×#124	C₂³	1	2	yes	—
123	C₇×#123	—	2	4	no	—
124	C₇×#123	—	2	4	no	—
125	C₇×C₂²⋊Q₈	—	2	2	no	—
126	C₇×C₂²⋊Q₈	—	2	1	no	—
127	C₇×C₂²⋊Q₈	—	2	1	no	—
128	C₇×C₂²⋊Q₈	—	2	4	no	—
129	C₇×C₂²⋊Q₈	D₄	1	2	yes	—
130	C₇×C₂²⋊Q₈	D₄	1	2	yes	—
131	C₇×#121	—	2	1	no	—
132	C₇×#121	—	2	1	no	—
133	C₇×#121	—	2	2	no	—
134	C₇×#121	—	2	2	no	—
135	C₇×#121	—	2	2	no	—
136	C₇×#121	—	2	4	no	—
137	C₇×#120	—	2	4	no	—
138	C₄×C₇×Q₈	—	2	2	no	—
139	C₄×C₇×Q₈	—	2	2	no	—
140	C₄×C₇×Q₈	—	2	2	no	—
141	C₄×C₇×Q₈	—	2	4	no	—
142	C₄×C₇×Q₈	—	2	2	no	—
143	C₄×C₇×Q₈	—	2	1	no	—
144	C₄×C₇×Q₈	—	2	1	no	—
145	C₄×C₇×Q₈	—	2	1	no	—
146	C₄×C₇×Q₈	—	2	1	no	—
147	C₄×C₇×Q₈	C₂³	1	4	yes	—
148	C₄×C₇×Q₈	C₂³	1	2	yes	—
149	C₄×C₇×Q₈	C₂³	1	2	yes	—
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	4	no	—
155	C₄×C₇×D₄	—	2	4	no	—
156	C₄×C₇×D₄	—	2	1	no	—
157	C₄×C₇×D₄	—	2	1	no	—
158	C₄×C₇×D₄	—	2	1	no	—
159	C₄×C₇×D₄	—	2	1	no	—
160	C₄×C₇×D₄	C₂³	1	2	yes	—
161	C₄×C₇×D₄	C₂³	1	4	yes	—
162	C₄×C₇×D₄	C₂³	1	2	yes	—
163	C₇×#117	—	2	2	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	2	yes	—
168	C₂×C₇×C₄⋊C₄	—	2	1	no	—
169	C₂×C₇×C₄⋊C₄	—	2	2	no	—
170	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
171	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
172	C₂×C₇×C₂²⋊C₄	—	2	1	no	—
173	C₂×C₄²×C₇	—	2	1	no	—
174	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
175	C₇×#107	C₂³	1	4	yes	—
176	C₇×#107	C₂³	1	4	yes	—
177	C₇×#106	C₂³	1	4	yes	—
178	C₇×C₄⋊C₈	C₂³	1	2	yes	—
179	C₇×C₄⋊C₈	C₂³	1	4	yes	—
180	C₇×C₄⋊C₈	C₂³	1	4	yes	—
181	C₇×C₄⋊C₈	C₂³	1	2	yes	—
182	C₇×#103	C₂³	1	2	yes	—
183	C₇×#103	C₂³	1	4	yes	—
184	C₇×#103	C₂³	1	2	yes	—
185	C₇×#103	C₂³	1	2	yes	—
186	C₇×#103	C₂³	1	2	yes	—
187	C₇×#103	C₂³	1	4	yes	—
188	C₇×#102	C₂³	1	4	yes	—
189	C₇×#102	C₂³	1	2	yes	—
190	C₇×#102	C₂³	1	4	yes	—
191	C₇×#102	C₂³	1	4	yes	—
192	C₇×#102	C₂³	1	2	yes	—
193	C₇×#102	C₂³	1	2	yes	—
194	C₇×#102	C₂³	1	4	yes	—
195	C₇×#102	C₂³	1	2	yes	—
196	C₇×#97	C₂³	1	4	yes	—
197	C₇×#97	C₂³	1	2	yes	—
198	C₇×#97	C₂³	1	2	yes	—
199	C₄×C₇×C₈	C₂³	1	4	yes	—
200	#3506	C₂×C₇	1	1	yes	characteristic
201	#3502	C₂×C₇	1	1	yes	characteristic, complemented
202	#3377	C₂×C₇	1	4	yes	complemented
203	#3375	C₂×C₇	1	4	yes	complemented
204	#3371	C₂×C₇	1	4	yes	complemented
205	#3368	C₂×C₇	1	4	yes	complemented
206	#3136	C₂×C₇	1	2	yes	complemented
207	#3135	C₂×C₇	1	1	yes	characteristic, complemented
208	#3083	C₂×C₇	1	2	yes	complemented
209	#3077	C₂×C₇	1	1	yes	characteristic, complemented
210	#3043	C₂×C₇	1	2	yes	complemented
211	#3037	C₂×C₇	1	1	yes	characteristic, complemented
212	#2961	C₂×C₇	1	1	yes	characteristic, complemented
213	#2947	C₂×C₇	1	2	yes	complemented
214	C₂×#440	C₂×C₇	1	1	yes	characteristic, complemented
215	C₂⁴×C₇	—	2	1	no	—
216	C₇×#41	—	2	4	no	—
217	C₇×#41	—	2	4	no	—
218	C₇×#41	—	8	4	no	—
219	C₂×C₇×Q₈	—	2	1	no	—
220	C₂×C₇×Q₈	—	2	1	no	—
221	C₂×C₇×Q₈	—	2	1	no	—
222	C₂×C₇×Q₈	—	2	4	no	—
223	C₂×C₇×Q₈	—	2	2	no	—
224	C₂×C₇×Q₈	C₂×D₄	1	2	yes	—
225	C₂×C₇×Q₈	C₂×D₄	1	2	yes	—
226	C₂×C₇×Q₈	#41	1	4	yes	complemented
227	C₂×C₇×Q₈	C₂⁴	1	2	yes	—
228	C₂×C₇×Q₈	C₂⁴	1	2	yes	—
229	C₂×C₇×D₄	—	2	2	no	—
230	C₂×C₇×D₄	—	2	1	no	—
231	C₂×C₇×D₄	—	2	1	no	—
232	C₂×C₇×D₄	—	2	4	no	—
233	C₂×C₇×D₄	—	2	1	no	—
234	C₂×C₇×D₄	—	2	4	no	—
235	C₂×C₇×D₄	—	2	2	no	—
236	C₂×C₇×D₄	—	2	2	no	—
237	C₂×C₇×D₄	C₂⁴	1	2	yes	—
238	C₂×C₇×D₄	C₂⁴	1	2	yes	—
239	C₂²×C₄×C₇	—	2	1	no	—
240	C₂²×C₄×C₇	—	2	1	no	—
241	C₂²×C₄×C₇	—	2	1	no	—
242	C₂²×C₄×C₇	—	2	2	no	—
243	C₂²×C₄×C₇	—	2	2	no	—
244	C₂²×C₄×C₇	—	2	4	no	—
245	C₂²×C₄×C₇	—	2	2	no	—
246	C₂²×C₄×C₇	—	8	2	no	—
247	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
248	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
249	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
250	C₇×Q₁₆	—	2	8	no	—
251	C₇×SD₁₆	—	2	8	no	—
252	C₇×SD₁₆	—	2	8	no	—
253	C₇×M₁₆	—	2	4	no	—
254	C₂×C₇×C₈	—	2	2	no	—
255	C₂×C₇×C₈	—	2	2	no	—
256	C₂×C₇×C₈	C₂⁴	1	2	yes	—
257	C₂×C₇×C₈	C₂⁴	1	4	yes	—
258	C₂×C₇×C₈	C₂⁴	1	2	yes	—
259	C₇×C₄⋊C₄	—	2	1	no	—
260	C₇×C₄⋊C₄	—	2	4	no	—
261	C₇×C₄⋊C₄	—	2	2	no	—
262	C₇×C₄⋊C₄	—	2	2	no	—
263	C₇×C₄⋊C₄	—	2	1	no	—
264	C₇×C₄⋊C₄	—	2	1	no	—
265	C₇×C₄⋊C₄	—	2	4	no	—
266	C₇×C₄⋊C₄	—	2	4	no	—
267	C₇×C₄⋊C₄	—	2	1	no	—
268	C₇×C₄⋊C₄	—	2	2	no	—
269	C₇×C₄⋊C₄	—	2	1	no	—
270	C₇×C₄⋊C₄	—	2	1	no	—
271	C₇×C₄⋊C₄	—	2	4	no	—
272	C₇×C₄⋊C₄	—	2	4	no	—
273	C₇×C₄⋊C₄	—	2	2	no	—
274	C₇×C₄⋊C₄	—	2	4	no	—
275	C₇×C₄⋊C₄	C₂×D₄	1	2	yes	—
276	C₇×C₄⋊C₄	C₂×D₄	1	2	yes	—
277	C₇×C₄⋊C₄	C₂×D₄	1	2	yes	—
278	C₇×C₄⋊C₄	C₂×D₄	1	4	yes	—
279	C₇×C₄⋊C₄	C₂×D₄	1	4	yes	—
280	C₇×C₄⋊C₄	C₂×D₄	1	2	yes	—
281	C₇×C₄⋊C₄	C₂⁴	1	2	yes	—
282	C₇×C₄⋊C₄	C₂⁴	1	2	yes	—
283	C₇×C₄⋊C₄	C₂⁴	1	2	yes	—
284	C₇×C₄⋊C₄	C₂⁴	1	2	yes	—
285	C₇×C₂²⋊C₄	—	2	2	no	—
286	C₇×C₂²⋊C₄	—	2	2	no	—
287	C₇×C₂²⋊C₄	—	2	2	no	—
288	C₇×C₂²⋊C₄	—	2	4	no	—
289	C₇×C₂²⋊C₄	—	2	4	no	—
290	C₇×C₂²⋊C₄	—	2	2	no	—
291	C₇×C₂²⋊C₄	—	2	4	no	—
292	C₇×C₂²⋊C₄	—	2	2	no	—
293	C₇×C₂²⋊C₄	C₂×D₄	1	4	yes	—
294	C₄²×C₇	—	2	2	no	—
295	C₄²×C₇	—	2	2	no	—
296	C₄²×C₇	—	2	4	no	—
297	C₄²×C₇	—	2	4	no	—
298	C₄²×C₇	—	2	4	no	—
299	C₄²×C₇	—	2	2	no	—
300	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
301	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
302	C₄²×C₇	C₂⁴	1	4	yes	—
303	C₄²×C₇	C₂⁴	1	1	yes	characteristic
304	C₄²×C₇	C₂⁴	1	1	yes	characteristic
305	C₂×#137	C₂²×C₇	1	2	yes	—
306	C₂²×SD₁₆	C₂²×C₇	1	1	yes	characteristic
307	#559	C₂²×C₇	1	4	yes	—
308	Q₈²	—	2	1	no	—
309	Q₈²	C₂²×C₇	1	2	yes	complemented
310	#557	—	2	1	no	—
311	#557	—	2	2	no	—
312	#557	C₂²×C₇	1	4	yes	complemented
313	#557	C₂²×C₇	1	2	yes	complemented
314	#556	C₂²×C₇	1	2	yes	—
315	#555	—	2	1	no	—
316	#553	—	2	2	no	—
317	#550	C₂²×C₇	1	2	yes	—
318	#547	—	2	1	no	—
319	#544	—	2	2	no	—
320	#544	C₂²×C₇	1	2	yes	—
321	#538	—	2	2	no	—
322	#535	C₂²×C₇	1	2	yes	—
323	#533	—	2	2	no	—
324	#533	C₂²×C₇	1	1	yes	characteristic
325	#533	C₂²×C₇	1	1	yes	characteristic
326	#532	C₂²×C₇	1	1	yes	characteristic
327	#525	—	2	2	no	—
328	#520	C₂²×C₇	1	2	yes	—
329	#519	C₂²×C₇	1	2	yes	—
330	C₂×C₄×Q₈	C₂²×C₇	1	1	yes	characteristic
331	C₂×C₄×D₄	C₂²×C₇	1	1	yes	characteristic
332	#491	C₂²×C₇	1	2	yes	complemented
333	#490	C₂²×C₇	1	4	yes	—
334	#490	C₂²×C₇	1	2	yes	complemented
335	#489	C₂²×C₇	1	4	yes	—
336	#489	C₂²×C₇	1	2	yes	complemented
337	#489	C₂²×C₇	1	2	yes	complemented
338	#488	C₂²×C₇	1	4	yes	—
339	#486	C₂²×C₇	1	4	yes	—
340	#479	C₂²×C₇	1	4	yes	complemented
341	#478	C₂²×C₇	1	4	yes	—
342	#478	C₂²×C₇	1	4	yes	complemented
343	#477	C₂²×C₇	1	4	yes	complemented
344	#476	C₂²×C₇	1	2	yes	—
345	#474	C₂²×C₇	1	2	yes	—
346	#474	C₂²×C₇	1	4	yes	complemented
347	#464	C₂²×C₇	1	4	yes	—
348	#464	C₂²×C₇	1	4	yes	—
349	#464	C₂²×C₇	1	4	yes	complemented
350	#462	C₂²×C₇	1	2	yes	complemented
351	#460	C₂²×C₇	1	4	yes	—
352	#460	C₂²×C₇	1	4	yes	—
353	#460	C₂²×C₇	1	2	yes	complemented
354	#446	C₂²×C₇	1	4	yes	complemented
355	#446	C₂²×C₇	1	4	yes	complemented
356	#441	C₂²×C₇	1	4	yes	complemented
357	#440	C₂²×C₇	1	4	yes	—
358	#440	C₂²×C₇	1	4	yes	—
359	#440	C₂²×C₇	1	4	yes	complemented
360	C₄×Q₁₆	C₂²×C₇	1	4	yes	complemented
361	C₄×SD₁₆	C₂²×C₇	1	4	yes	complemented
362	#428	C₂²×C₇	1	2	yes	—
363	C₂×#107	C₂²×C₇	1	1	yes	characteristic
364	#424	C₂²×C₇	1	2	yes	—
365	#423	C₂²×C₇	1	1	yes	characteristic
366	#418	C₂²×C₇	1	2	yes	—
367	#417	C₂²×C₇	1	2	yes	—
368	#416	C₂²×C₇	1	2	yes	—
369	#416	C₂²×C₇	1	2	yes	—
370	C₂×#103	C₂²×C₇	1	1	yes	characteristic
371	C₂×#102	C₂²×C₇	1	1	yes	characteristic
372	#405	C₂²×C₇	1	2	yes	—
373	C₂×#97	C₂²×C₇	1	1	yes	characteristic
374	C₂³×C₇	—	2	2	no	—
375	C₂³×C₇	—	2	2	no	—
376	C₂³×C₇	—	4	2	no	—
377	C₂³×C₇	—	8	1	no	—
378	C₂³×C₇	—	8	1	no	—
379	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
380	C₇×Q₈	—	2	4	no	—
381	C₇×Q₈	—	2	2	no	—
382	C₇×Q₈	—	2	2	no	—
383	C₇×Q₈	—	2	2	no	—
384	C₇×Q₈	—	4	2	no	—
385	C₇×Q₈	—	4	2	no	—
386	C₇×Q₈	—	4	2	no	—
387	C₇×Q₈	—	8	1	no	—
388	C₇×Q₈	—	8	4	no	—
389	C₇×Q₈	C₂×#41	1	4	yes	—
390	C₇×D₄	—	2	2	no	—
391	C₇×D₄	—	2	2	no	—
392	C₇×D₄	—	2	2	no	—
393	C₇×D₄	—	2	2	no	—
394	C₇×D₄	—	4	2	no	—
395	C₇×D₄	—	4	4	no	—
396	C₇×D₄	—	4	2	no	—
397	C₇×D₄	—	8	1	no	—
398	C₇×D₄	—	8	4	no	—
399	C₇×D₄	—	8	4	no	—
400	C₂×C₄×C₇	—	2	4	no	—
401	C₂×C₄×C₇	—	2	2	no	—
402	C₂×C₄×C₇	—	2	2	no	—
403	C₂×C₄×C₇	—	2	2	no	—
404	C₂×C₄×C₇	—	2	2	no	—
405	C₂×C₄×C₇	—	2	2	no	—
406	C₂×C₄×C₇	—	2	2	no	—
407	C₂×C₄×C₇	—	4	1	no	—
408	C₂×C₄×C₇	—	4	1	no	—
409	C₂×C₄×C₇	—	4	4	no	—
410	C₂×C₄×C₇	—	4	4	no	—
411	C₂×C₄×C₇	—	8	2	no	—
412	C₂×C₄×C₇	—	8	4	no	—
413	C₂×C₄×C₇	—	8	1	no	—
414	C₂×C₄×C₇	—	8	1	no	—
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	4	yes	—
418	C₂×C₄×C₇	C₂²×D₄	1	4	yes	—
419	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
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	2₋¹⁺⁴	1	2	yes	—
424	C₂×C₄×C₇	C₂⁵	1	1	yes	characteristic
425	C₇×C₈	—	2	2	no	—
426	C₇×C₈	—	2	4	no	—
427	C₇×C₈	—	2	2	no	—
428	C₂×#41	—	2	2	no	—
429	C₂×#41	C₂³×C₇	1	2	yes	—
430	C₂²×Q₈	C₂³×C₇	1	1	yes	characteristic
431	C₂²×D₄	C₂³×C₇	1	1	yes	characteristic
432	C₂³×C₄	—	2	1	no	—
433	#137	—	2	8	no	—
434	C₂×Q₁₆	C₂³×C₇	1	4	yes	—
435	C₂×SD₁₆	—	2	4	no	—
436	C₂×SD₁₆	C₂³×C₇	1	4	yes	—
437	C₂×SD₁₆	C₂³×C₇	1	4	yes	—
438	C₂×M₁₆	C₂³×C₇	1	2	yes	—
439	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
440	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
441	C₄⋊Q₈	—	2	2	no	—
442	C₄⋊Q₈	—	2	4	no	—
443	C₄⋊Q₈	—	2	2	no	—
444	C₄⋊Q₈	—	2	4	no	—
445	C₄⋊Q₈	—	2	2	no	—
446	C₄⋊Q₈	C₇×D₄	1	1	yes	characteristic
447	C₄⋊Q₈	C₇×D₄	1	1	yes	characteristic
448	C₄⋊Q₈	C₂³×C₇	1	1	yes	characteristic
449	C₄⋊Q₈	C₂³×C₇	1	1	yes	characteristic
450	C₄⋊Q₈	C₂³×C₇	1	1	yes	characteristic
451	C₄⋊Q₈	C₂³×C₇	1	4	yes	—
452	#127	C₂³×C₇	1	1	yes	characteristic
453	#127	C₂³×C₇	1	4	yes	—
454	#126	—	2	4	no	—
455	#125	—	2	4	no	—
456	#125	—	2	4	no	—
457	#125	—	2	4	no	—
458	#125	—	2	2	no	—
459	#125	C₇×D₄	1	2	yes	—
460	#125	C₂³×C₇	1	2	yes	—
461	#125	C₂³×C₇	1	4	yes	—
462	#124	—	2	4	no	—
463	#124	C₂³×C₇	1	2	yes	—
464	#124	C₂³×C₇	1	4	yes	—
465	#123	—	2	4	no	—
466	#123	—	2	4	no	—
467	C₂²⋊Q₈	—	2	2	no	—
468	C₂²⋊Q₈	—	2	1	no	—
469	C₂²⋊Q₈	—	2	1	no	—
470	C₂²⋊Q₈	—	2	4	no	—
471	C₂²⋊Q₈	C₇×D₄	1	2	yes	—
472	C₂²⋊Q₈	C₇×D₄	1	2	yes	—
473	#121	—	2	1	no	—
474	#121	—	2	1	no	—
475	#121	—	2	2	no	—
476	#121	—	2	2	no	—
477	#121	—	2	2	no	—
478	#121	—	2	4	no	—
479	#120	—	2	4	no	—
480	C₄×Q₈	—	2	2	no	—
481	C₄×Q₈	—	2	2	no	—
482	C₄×Q₈	—	2	2	no	—
483	C₄×Q₈	—	2	4	no	—
484	C₄×Q₈	—	2	2	no	—
485	C₄×Q₈	—	2	1	no	—
486	C₄×Q₈	—	2	1	no	—
487	C₄×Q₈	—	2	1	no	—
488	C₄×Q₈	—	2	1	no	—
489	C₄×Q₈	C₂³×C₇	1	2	yes	—
490	C₄×Q₈	C₂³×C₇	1	2	yes	—
491	C₄×Q₈	C₂³×C₇	1	4	yes	—
492	C₄×Q₈	C₂³×C₇	1	4	yes	—
493	C₄×Q₈	C₂³×C₇	1	4	yes	—
494	C₄×Q₈	C₂³×C₇	1	2	yes	—
495	C₄×Q₈	C₂³×C₇	1	2	yes	—
496	C₄×D₄	—	2	4	no	—
497	C₄×D₄	—	2	4	no	—
498	C₄×D₄	—	2	1	no	—
499	C₄×D₄	—	2	1	no	—
500	C₄×D₄	—	2	1	no	—
501	C₄×D₄	—	2	1	no	—
502	C₄×D₄	C₂³×C₇	1	2	yes	—
503	C₄×D₄	C₂³×C₇	1	2	yes	—
504	C₄×D₄	C₂³×C₇	1	4	yes	—
505	#117	—	2	2	no	—
506	#117	—	2	2	no	—
507	#117	—	2	4	no	—
508	#117	C₂³×C₇	1	2	yes	—
509	#117	C₂³×C₇	1	2	yes	—
510	C₂×C₄⋊C₄	—	2	1	no	—
511	C₂×C₄⋊C₄	—	2	2	no	—
512	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
513	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
514	C₂×C₂²⋊C₄	—	2	1	no	—
515	C₂×C₄²	—	2	1	no	—
516	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
517	#107	C₂³×C₇	1	4	yes	—
518	#107	C₂³×C₇	1	4	yes	—
519	#106	C₂³×C₇	1	4	yes	—
520	C₄⋊C₈	C₂³×C₇	1	2	yes	—
521	C₄⋊C₈	C₂³×C₇	1	2	yes	—
522	C₄⋊C₈	C₂³×C₇	1	4	yes	—
523	C₄⋊C₈	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	2	yes	—
528	#103	C₂³×C₇	1	2	yes	—
529	#103	C₂³×C₇	1	4	yes	—
530	#102	C₂³×C₇	1	4	yes	—
531	#102	C₂³×C₇	1	2	yes	—
532	#102	C₂³×C₇	1	2	yes	—
533	#102	C₂³×C₇	1	2	yes	—
534	#102	C₂³×C₇	1	2	yes	—
535	#102	C₂³×C₇	1	4	yes	—
536	#102	C₂³×C₇	1	4	yes	—
537	#102	C₂³×C₇	1	4	yes	—
538	#97	C₂³×C₇	1	2	yes	—
539	#97	C₂³×C₇	1	2	yes	—
540	#97	C₂³×C₇	1	4	yes	—
541	C₄×C₈	C₂³×C₇	1	4	yes	—
542	C₂²×C₇	—	2	1	no	—
543	C₂²×C₇	—	2	1	no	—
544	C₂²×C₇	—	4	2	no	—
545	C₂²×C₇	—	4	2	no	—
546	C₂²×C₇	—	4	2	no	—
547	C₂²×C₇	—	4	2	no	—
548	C₂²×C₇	—	4	2	no	—
549	C₂²×C₇	—	4	2	no	—
550	C₂²×C₇	—	8	4	no	—
551	C₂²×C₇	#548	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	4	no	—
556	C₄×C₇	—	2	1	no	—
557	C₄×C₇	—	2	1	no	—
558	C₄×C₇	—	2	4	no	—
559	C₄×C₇	—	2	4	no	—
560	C₄×C₇	—	4	2	no	—
561	C₄×C₇	—	4	2	no	—
562	C₄×C₇	—	4	2	no	—
563	C₄×C₇	—	4	2	no	—
564	C₄×C₇	—	4	2	no	—
565	C₄×C₇	#548	1	2	yes	—
566	C₄×C₇	C₂²×#41	1	1	yes	characteristic
567	C₄×C₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
568	C₂⁴	—	2	1	no	—
569	#41	—	2	4	no	—
570	#41	—	2	4	no	—
571	#41	—	8	4	no	—
572	C₂×Q₈	—	2	1	no	—
573	C₂×Q₈	—	2	1	no	—
574	C₂×Q₈	—	2	1	no	—
575	C₂×Q₈	—	2	4	no	—
576	C₂×Q₈	—	2	2	no	—
577	C₂×Q₈	C₂×C₇×D₄	1	2	yes	—
578	C₂×Q₈	C₂×C₇×D₄	1	2	yes	—
579	C₂×Q₈	C₇×#41	1	4	yes	complemented
580	C₂×Q₈	C₂⁴×C₇	1	2	yes	—
581	C₂×Q₈	C₂⁴×C₇	1	2	yes	—
582	C₂×D₄	—	2	2	no	—
583	C₂×D₄	—	2	1	no	—
584	C₂×D₄	—	2	1	no	—
585	C₂×D₄	—	2	4	no	—
586	C₂×D₄	—	2	1	no	—
587	C₂×D₄	—	2	4	no	—
588	C₂×D₄	—	2	2	no	—
589	C₂×D₄	—	2	2	no	—
590	C₂×D₄	C₂⁴×C₇	1	2	yes	—
591	C₂×D₄	C₂⁴×C₇	1	2	yes	—
592	C₂²×C₄	—	2	1	no	—
593	C₂²×C₄	—	2	1	no	—
594	C₂²×C₄	—	2	1	no	—
595	C₂²×C₄	—	2	2	no	—
596	C₂²×C₄	—	2	2	no	—
597	C₂²×C₄	—	2	4	no	—
598	C₂²×C₄	—	2	2	no	—
599	C₂²×C₄	—	8	2	no	—
600	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
601	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
602	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
603	Q₁₆	—	2	8	no	—
604	SD₁₆	—	2	8	no	—
605	SD₁₆	—	2	8	no	—
606	M₁₆	—	2	4	no	—
607	C₂×C₈	—	2	2	no	—
608	C₂×C₈	—	2	2	no	—
609	C₂×C₈	C₂⁴×C₇	1	2	yes	—
610	C₂×C₈	C₂⁴×C₇	1	4	yes	—
611	C₂×C₈	C₂⁴×C₇	1	2	yes	—
612	C₄⋊C₄	—	2	1	no	—
613	C₄⋊C₄	—	2	4	no	—
614	C₄⋊C₄	—	2	2	no	—
615	C₄⋊C₄	—	2	2	no	—
616	C₄⋊C₄	—	2	1	no	—
617	C₄⋊C₄	—	2	1	no	—
618	C₄⋊C₄	—	2	4	no	—
619	C₄⋊C₄	—	2	4	no	—
620	C₄⋊C₄	—	2	1	no	—
621	C₄⋊C₄	—	2	2	no	—
622	C₄⋊C₄	—	2	1	no	—
623	C₄⋊C₄	—	2	1	no	—
624	C₄⋊C₄	—	2	4	no	—
625	C₄⋊C₄	—	2	4	no	—
626	C₄⋊C₄	—	2	2	no	—
627	C₄⋊C₄	—	2	4	no	—
628	C₄⋊C₄	C₂×C₇×D₄	1	2	yes	—
629	C₄⋊C₄	C₂×C₇×D₄	1	2	yes	—
630	C₄⋊C₄	C₂×C₇×D₄	1	2	yes	—
631	C₄⋊C₄	C₂×C₇×D₄	1	4	yes	—
632	C₄⋊C₄	C₂×C₇×D₄	1	4	yes	—
633	C₄⋊C₄	C₂×C₇×D₄	1	2	yes	—
634	C₄⋊C₄	C₂⁴×C₇	1	2	yes	—
635	C₄⋊C₄	C₂⁴×C₇	1	2	yes	—
636	C₄⋊C₄	C₂⁴×C₇	1	2	yes	—
637	C₄⋊C₄	C₂⁴×C₇	1	2	yes	—
638	C₂²⋊C₄	—	2	2	no	—
639	C₂²⋊C₄	—	2	2	no	—
640	C₂²⋊C₄	—	2	2	no	—
641	C₂²⋊C₄	—	2	4	no	—
642	C₂²⋊C₄	—	2	4	no	—
643	C₂²⋊C₄	—	2	2	no	—
644	C₂²⋊C₄	—	2	4	no	—
645	C₂²⋊C₄	—	2	2	no	—
646	C₂²⋊C₄	C₂×C₇×D₄	1	4	yes	—
647	C₄²	—	2	2	no	—
648	C₄²	—	2	2	no	—
649	C₄²	—	2	4	no	—
650	C₄²	—	2	4	no	—
651	C₄²	—	2	4	no	—
652	C₄²	—	2	2	no	—
653	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
654	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
655	C₄²	C₂⁴×C₇	1	4	yes	—
656	C₄²	C₂⁴×C₇	1	1	yes	characteristic
657	C₄²	C₂⁴×C₇	1	1	yes	characteristic
658	C₂×C₇	—	2	2	no	—
659	C₂×C₇	—	8	2	no	—
660	C₂×C₇	—	8	2	no	—
661	C₂×C₇	C₂×#548	1	1	yes	characteristic
662	C₂×C₇	#3585	1	1	yes	characteristic
663	C₂×C₇	#3586	1	1	yes	characteristic
664	C₂³	—	2	2	no	—
665	C₂³	—	2	2	no	—
666	C₂³	—	4	2	no	—
667	C₂³	—	8	1	no	—
668	C₂³	—	8	1	no	—
669	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
670	Q₈	—	2	2	no	—
671	Q₈	—	2	4	no	—
672	Q₈	—	2	2	no	—
673	Q₈	—	2	2	no	—
674	Q₈	—	4	2	no	—
675	Q₈	—	4	2	no	—
676	Q₈	—	4	2	no	—
677	Q₈	—	8	1	no	—
678	Q₈	—	8	4	no	—
679	Q₈	C₂×C₇×#41	1	4	yes	—
680	D₄	—	2	2	no	—
681	D₄	—	2	2	no	—
682	D₄	—	2	2	no	—
683	D₄	—	2	2	no	—
684	D₄	—	4	2	no	—
685	D₄	—	4	4	no	—
686	D₄	—	4	2	no	—
687	D₄	—	8	1	no	—
688	D₄	—	8	4	no	—
689	D₄	—	8	4	no	—
690	C₂×C₄	—	2	2	no	—
691	C₂×C₄	—	2	4	no	—
692	C₂×C₄	—	2	2	no	—
693	C₂×C₄	—	2	2	no	—
694	C₂×C₄	—	2	2	no	—
695	C₂×C₄	—	2	2	no	—
696	C₂×C₄	—	2	2	no	—
697	C₂×C₄	—	4	1	no	—
698	C₂×C₄	—	4	1	no	—
699	C₂×C₄	—	4	4	no	—
700	C₂×C₄	—	4	4	no	—
701	C₂×C₄	—	8	2	no	—
702	C₂×C₄	—	8	4	no	—
703	C₂×C₄	—	8	1	no	—
704	C₂×C₄	—	8	1	no	—
705	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
706	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
707	C₂×C₄	C₂²×C₇×D₄	1	4	yes	—
708	C₂×C₄	C₂²×C₇×D₄	1	4	yes	—
709	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
710	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
711	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
712	C₂×C₄	C₂×C₇×#41	1	2	yes	—
713	C₂×C₄	C₇×2₋¹⁺⁴	1	2	yes	—
714	C₂×C₄	C₂⁵×C₇	1	1	yes	characteristic
715	C₈	—	2	2	no	—
716	C₈	—	2	4	no	—
717	C₈	—	2	2	no	—
718	C₇	#61334	1	1	yes	characteristic, direct product factor, Sylow
719	C₂²	—	2	1	no	—
720	C₂²	—	2	1	no	—
721	C₂²	—	4	2	no	—
722	C₂²	—	4	2	no	—
723	C₂²	—	4	2	no	—
724	C₂²	—	4	2	no	—
725	C₂²	—	4	2	no	—
726	C₂²	—	4	2	no	—
727	C₂²	—	8	4	no	—
728	C₂²	C₇×#548	1	2	yes	—
729	C₂²	C₂³×C₇×D₄	1	1	yes	characteristic
730	C₄	—	2	1	no	—
731	C₄	—	2	1	no	—
732	C₄	—	2	4	no	—
733	C₄	—	2	1	no	—
734	C₄	—	2	1	no	—
735	C₄	—	2	4	no	—
736	C₄	—	2	4	no	—
737	C₄	—	4	2	no	—
738	C₄	—	4	2	no	—
739	C₄	—	4	2	no	—
740	C₄	—	4	2	no	—
741	C₄	—	4	2	no	—
742	C₄	C₇×#548	1	2	yes	—
743	C₄	C₂²×C₇×#41	1	1	yes	characteristic
744	C₄	C₂×C₇×2₋¹⁺⁴	1	1	yes	characteristic
745	C₂	—	2	2	no	—
746	C₂	—	8	2	no	—
747	C₂	—	8	2	no	—
748	C₂	C₂×C₇×#548	1	1	yes	characteristic
749	C₂	C₇×#3585	1	1	yes	characteristic
750	C₂	C₇×#3586	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₇	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₇
prime radical	C₇×#3506	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₇×#533 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₇×#3502	C₂	semidirect
3. C₇×#3377	C₂	semidirect
4. C₇×#3375	C₂	semidirect
5. C₇×#3371	C₂	semidirect
6. C₇×#3368	C₂	semidirect
7. C₇×#3136	C₂	semidirect
8. C₇×#3135	C₂	semidirect
9. C₇×#3083	C₂	semidirect
10. C₇×#3077	C₂	semidirect
11. C₇×#3043	C₂	semidirect
12. C₇×#3037	C₂	semidirect
13. C₇×#2961	C₂	semidirect
14. C₇×#2947	C₂	semidirect
15. C₂×C₇×#440	C₂	semidirect
20. C₇×Q₈²	C₂²	semidirect
23. C₇×#557	C₂²	semidirect
24. C₇×#557	C₂²	semidirect
43. C₇×#491	C₂²	semidirect
45. C₇×#490	C₂²	semidirect
47. C₇×#489	C₂²	semidirect
48. C₇×#489	C₂²	semidirect
51. C₇×#479	C₂²	semidirect
53. C₇×#478	C₂²	semidirect
54. C₇×#477	C₂²	semidirect
57. C₇×#474	C₂²	semidirect
60. C₇×#464	C₂²	semidirect
61. C₇×#462	C₂²	semidirect
64. C₇×#460	C₂²	semidirect
65. C₇×#446	C₂²	semidirect
66. C₇×#446	C₂²	semidirect
67. C₇×#441	C₂²	semidirect
70. C₇×#440	C₂²	semidirect
71. C₄×C₇×Q₁₆	C₂²	semidirect
72. C₄×C₇×SD₁₆	C₂²	semidirect
85. #61334	C₇	direct (N × H)
201. #3502	C₂×C₇	semidirect
202. #3377	C₂×C₇	semidirect
203. #3375	C₂×C₇	semidirect
204. #3371	C₂×C₇	semidirect
205. #3368	C₂×C₇	semidirect
206. #3136	C₂×C₇	semidirect
207. #3135	C₂×C₇	semidirect
208. #3083	C₂×C₇	semidirect
209. #3077	C₂×C₇	semidirect
210. #3043	C₂×C₇	semidirect
211. #3037	C₂×C₇	semidirect
212. #2961	C₂×C₇	semidirect
213. #2947	C₂×C₇	semidirect
214. C₂×#440	C₂×C₇	semidirect
226. C₂×C₇×Q₈	#41	semidirect
309. Q₈²	C₂²×C₇	semidirect
312. #557	C₂²×C₇	semidirect
313. #557	C₂²×C₇	semidirect
332. #491	C₂²×C₇	semidirect
334. #490	C₂²×C₇	semidirect
336. #489	C₂²×C₇	semidirect
337. #489	C₂²×C₇	semidirect
340. #479	C₂²×C₇	semidirect
342. #478	C₂²×C₇	semidirect
343. #477	C₂²×C₇	semidirect
346. #474	C₂²×C₇	semidirect
349. #464	C₂²×C₇	semidirect
350. #462	C₂²×C₇	semidirect
353. #460	C₂²×C₇	semidirect
354. #446	C₂²×C₇	semidirect
355. #446	C₂²×C₇	semidirect
356. #441	C₂²×C₇	semidirect
359. #440	C₂²×C₇	semidirect
360. C₄×Q₁₆	C₂²×C₇	semidirect
361. C₄×SD₁₆	C₂²×C₇	semidirect
579. C₂×Q₈	C₇×#41	semidirect
718. C₇	#61334	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×#3506	2
C₇×#3502	2
C₇×#3377	2
C₇×#3375	2
C₇×#3371	2
C₇×#3368	2
C₇×#3136	2
C₇×#3135	2
C₇×#3083	2
C₇×#3077	2
C₇×#3043	2
C₇×#3037	2
C₇×#2961	2
C₇×#2947	2
C₂×C₇×#440	2
#61334	7

Subgroups — C7×#61334 #13238550

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#61334 #13238550