Subgroups — C₇×#61260 #13238476

Order 1792.

1782 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3493	C₂	1	2	yes	maximal, complemented
2	C₇×#3349	C₂	1	4	yes	maximal, complemented
3	C₇×#3347	C₂	1	4	yes	maximal, complemented
4	C₇×#3340	C₂	1	4	yes	maximal, complemented
5	C₇×#3337	C₂	1	4	yes	maximal, complemented
6	C₇×#3163	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇×#3098	C₂	1	2	yes	maximal, complemented
8	C₇×#3093	C₂	1	2	yes	maximal, complemented
9	C₇×#3083	C₂	1	2	yes	maximal, complemented
10	C₇×#3060	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇×#3050	C₂	1	1	yes	maximal, characteristic, complemented
12	C₇×#2952	C₂	1	1	yes	maximal, characteristic, complemented
13	C₇×#2929	C₂	1	1	yes	maximal, characteristic, complemented
14	C₇×#2912	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂×C₇×#427	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×C₇×#135	C₂²	1	1	yes	characteristic
17	C₂×C₇×#131	C₂²	1	1	yes	characteristic
18	C₂³×C₇×C₈	C₂²	1	1	yes	characteristic
19	C₇×#556	—	2	2	no	—
20	C₇×#556	C₂²	1	4	yes	—
21	C₇×#555	—	2	2	no	—
22	C₇×#555	C₂²	1	4	yes	—
23	C₇×#553	—	2	2	no	—
24	C₇×#553	C₂²	1	4	yes	—
25	C₇×#552	—	2	2	no	—
26	C₇×#552	C₂²	1	4	yes	—
27	C₇×#544	C₂²	1	2	yes	—
28	C₇×#538	—	2	4	no	—
29	C₇×#537	C₂²	1	2	yes	—
30	C₇×#529	—	2	4	no	—
31	C₇×#525	C₂²	1	2	yes	—
32	C₇×#525	C₂²	1	1	yes	characteristic
33	C₇×#520	C₂²	1	2	yes	—
34	C₇×#520	C₂²	1	2	yes	—
35	C₂×C₇×#117	C₂²	1	2	yes	—
36	C₇×#501	C₂²	1	4	yes	complemented
37	C₇×#499	C₂²	1	2	yes	complemented
38	C₇×#498	C₂²	1	2	yes	complemented
39	C₇×#485	C₂²	1	4	yes	—
40	C₇×#485	C₂²	1	4	yes	—
41	C₇×#485	C₂²	1	4	yes	complemented
42	C₇×#484	C₂²	1	4	yes	complemented
43	C₇×#482	C₂²	1	4	yes	—
44	C₇×#482	C₂²	1	4	yes	—
45	C₇×#482	C₂²	1	4	yes	complemented
46	C₇×#480	C₂²	1	4	yes	complemented
47	C₇×#479	C₂²	1	4	yes	complemented
48	C₇×#478	C₂²	1	4	yes	complemented
49	C₇×#477	C₂²	1	4	yes	complemented
50	C₇×#474	C₂²	1	4	yes	complemented
51	C₇×#470	C₂²	1	2	yes	complemented
52	C₇×#469	C₂²	1	2	yes	complemented
53	C₇×#467	C₂²	1	2	yes	—
54	C₇×#466	C₂²	1	2	yes	—
55	C₇×#465	C₂²	1	2	yes	—
56	C₇×#465	C₂²	1	2	yes	—
57	C₇×#465	C₂²	1	4	yes	complemented
58	C₇×#442	C₂²	1	2	yes	complemented
59	C₇×#441	C₂²	1	2	yes	complemented
60	C₄×C₇×SD₁₆	C₂²	1	4	yes	complemented
61	C₇×#435	C₂²	1	4	yes	—
62	C₇×C₈×D₄	C₂²	1	4	yes	—
63	C₇×#428	C₂²	1	2	yes	—
64	C₇×#428	C₂²	1	2	yes	—
65	C₇×#428	C₂²	1	4	yes	complemented
66	C₇×#427	C₂²	1	1	yes	characteristic
67	C₇×#427	C₂²	1	4	yes	—
68	C₇×#427	C₂²	1	4	yes	—
69	C₇×#427	C₂²	1	2	yes	complemented
70	C₂×C₇×#107	C₂²	1	2	yes	—
71	C₂×C₇×#106	C₂²	1	1	yes	characteristic
72	C₂×C₇×#106	C₂²	1	1	yes	characteristic
73	C₂×C₇×#106	C₂²	1	2	yes	complemented
74	C₇×#424	C₂²	1	1	yes	characteristic
75	C₇×#417	C₂²	1	2	yes	—
76	C₇×#417	C₂²	1	2	yes	—
77	C₇×#416	C₂²	1	2	yes	—
78	C₇×#408	C₂²	1	1	yes	characteristic
79	C₇×#407	C₂²	1	1	yes	characteristic
80	C₇×#405	C₂²	1	1	yes	characteristic
81	#61260	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
82	C₂×C₇×#41	C₂³	1	2	yes	—
83	C₂×C₇×#41	C₂³	1	1	yes	characteristic
84	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
85	C₇×#135	—	2	4	no	—
86	C₂×C₇×Q₁₆	C₂³	1	1	yes	characteristic
87	C₂×C₇×SD₁₆	C₂³	1	2	yes	—
88	C₂×C₇×D₈	C₂³	1	1	yes	characteristic
89	C₇×#131	—	2	4	no	—
90	C₂×C₇×M₁₆	C₂³	1	2	yes	—
91	C₂×C₇×M₁₆	C₂³	1	1	yes	characteristic
92	C₂×C₇×M₁₆	C₂³	1	1	yes	characteristic
93	C₂²×C₇×C₈	—	2	4	no	—
94	C₂²×C₇×C₈	C₂³	1	2	yes	—
95	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
96	C₂²×C₇×C₈	C₂³	1	2	yes	—
97	C₂²×C₇×C₈	C₂³	1	2	yes	—
98	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
99	C₇×C₄⋊Q₈	—	2	2	no	—
100	C₇×C₄⋊Q₈	C₂³	1	4	yes	—
101	C₇×#126	—	2	4	no	—
102	C₇×#126	—	2	4	no	—
103	C₇×#126	—	2	4	no	—
104	C₇×#126	—	2	4	no	—
105	C₇×#125	—	2	4	no	—
106	C₇×#125	—	2	2	no	—
107	C₇×#125	C₂³	1	4	yes	—
108	C₇×#124	—	2	4	no	—
109	C₇×#124	—	2	4	no	—
110	C₇×#124	—	2	4	no	—
111	C₇×#123	—	2	4	no	—
112	C₇×#123	—	2	2	no	—
113	C₇×#123	—	2	4	no	—
114	C₇×#123	D₄	1	2	yes	—
115	C₇×C₂²⋊Q₈	—	2	2	no	—
116	C₇×C₂²⋊Q₈	—	2	4	no	—
117	C₇×C₂²⋊Q₈	D₄	1	2	yes	—
118	C₇×C₂²⋊Q₈	C₂³	1	2	yes	—
119	C₇×C₂²⋊Q₈	C₂³	1	2	yes	—
120	C₇×C₂²⋊Q₈	C₂³	1	4	yes	—
121	C₇×#121	—	2	4	no	—
122	C₇×#121	—	2	2	no	—
123	C₇×#121	—	2	4	no	—
124	C₇×#121	—	2	4	no	—
125	C₇×#121	D₄	1	2	yes	—
126	C₇×#121	C₂³	1	2	yes	—
127	C₇×#121	C₂³	1	2	yes	—
128	C₇×#121	C₂³	1	4	yes	—
129	C₇×#120	—	2	2	no	—
130	C₇×#120	D₄	1	2	yes	—
131	C₄×C₇×Q₈	—	2	2	no	—
132	C₄×C₇×Q₈	—	2	4	no	—
133	C₄×C₇×Q₈	C₂³	1	4	yes	—
134	C₄×C₇×D₄	—	2	4	no	—
135	C₄×C₇×D₄	—	2	2	no	—
136	C₄×C₇×D₄	—	2	2	no	—
137	C₄×C₇×D₄	—	2	2	no	—
138	C₄×C₇×D₄	—	2	4	no	—
139	C₄×C₇×D₄	—	2	4	no	—
140	C₄×C₇×D₄	C₂³	1	4	yes	—
141	C₄×C₇×D₄	C₂³	1	4	yes	—
142	C₇×#117	—	2	2	no	—
143	C₇×#117	—	2	2	no	—
144	C₇×#117	—	2	2	no	—
145	C₇×#117	—	2	4	no	—
146	C₇×#117	—	2	4	no	—
147	C₇×#117	C₂³	1	4	yes	—
148	C₇×#117	C₂³	1	4	yes	—
149	C₇×#117	C₂³	1	4	yes	—
150	C₇×#117	C₂³	1	2	yes	—
151	C₇×#117	C₂³	1	1	yes	characteristic
152	C₂×C₇×C₄⋊C₄	—	2	2	no	—
153	C₂×C₇×C₄⋊C₄	C₂³	1	4	yes	—
154	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
155	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
156	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
157	C₂×C₄²×C₇	—	2	2	no	—
158	C₇×#107	C₂³	1	2	yes	—
159	C₇×#107	C₂³	1	4	yes	—
160	C₇×#107	C₂³	1	4	yes	—
161	C₇×#107	C₂³	1	4	yes	—
162	C₇×#107	C₂³	1	4	yes	—
163	C₇×#106	C₂³	1	2	yes	—
164	C₇×#106	C₂³	1	4	yes	—
165	C₇×#106	C₂³	1	4	yes	—
166	C₇×#106	C₂³	1	1	yes	characteristic
167	C₇×#106	C₂³	1	2	yes	—
168	C₇×#106	C₂³	1	1	yes	characteristic
169	C₇×#106	C₂³	1	2	yes	—
170	C₇×#106	C₂³	1	2	yes	—
171	C₇×C₄⋊C₈	C₂³	1	4	yes	—
172	C₇×#103	C₂³	1	2	yes	—
173	C₇×#103	C₂³	1	2	yes	—
174	C₇×#103	C₂³	1	4	yes	—
175	C₇×#103	C₂³	1	4	yes	—
176	C₇×#102	C₂³	1	2	yes	—
177	C₇×#102	C₂³	1	2	yes	—
178	C₇×#102	C₂³	1	4	yes	—
179	C₇×#102	C₂³	1	4	yes	—
180	C₇×C₂²⋊C₈	C₂³	1	4	yes	—
181	C₇×C₂²⋊C₈	C₂³	1	4	yes	—
182	C₇×#97	C₂³	1	2	yes	—
183	C₄×C₇×C₈	C₂³	1	2	yes	—
184	#3493	C₂×C₇	1	2	yes	complemented
185	#3349	C₂×C₇	1	4	yes	complemented
186	#3347	C₂×C₇	1	4	yes	complemented
187	#3340	C₂×C₇	1	4	yes	complemented
188	#3337	C₂×C₇	1	4	yes	complemented
189	#3163	C₂×C₇	1	1	yes	characteristic, complemented
190	#3098	C₂×C₇	1	2	yes	complemented
191	#3093	C₂×C₇	1	2	yes	complemented
192	#3083	C₂×C₇	1	2	yes	complemented
193	#3060	C₂×C₇	1	1	yes	characteristic, complemented
194	#3050	C₂×C₇	1	1	yes	characteristic, complemented
195	#2952	C₂×C₇	1	1	yes	characteristic, complemented
196	#2929	C₂×C₇	1	1	yes	characteristic, complemented
197	#2912	C₂×C₇	1	1	yes	characteristic, complemented
198	C₂×#427	C₂×C₇	1	1	yes	characteristic, complemented
199	C₂⁴×C₇	C₂×D₄	1	1	yes	characteristic
200	C₇×#41	—	2	2	no	—
201	C₇×#41	—	2	2	no	—
202	C₇×#41	—	2	4	no	—
203	C₇×#41	—	2	4	no	—
204	C₂×C₇×Q₈	—	2	4	no	—
205	C₂×C₇×Q₈	C₂×D₄	1	1	yes	characteristic
206	C₂×C₇×Q₈	C₂⁴	1	2	yes	—
207	C₂×C₇×D₄	—	2	2	no	—
208	C₂×C₇×D₄	—	2	2	no	—
209	C₂×C₇×D₄	—	2	2	no	—
210	C₂×C₇×D₄	—	2	4	no	—
211	C₂×C₇×D₄	C₂×D₄	1	2	yes	—
212	C₂×C₇×D₄	C₂×D₄	1	1	yes	characteristic
213	C₂×C₇×D₄	C₂×D₄	1	4	yes	—
214	C₂×C₇×D₄	C₂⁴	1	2	yes	—
215	C₂²×C₄×C₇	—	2	2	no	—
216	C₂²×C₄×C₇	—	2	2	no	—
217	C₂²×C₄×C₇	—	2	2	no	—
218	C₂²×C₄×C₇	—	2	2	no	—
219	C₂²×C₄×C₇	—	2	4	no	—
220	C₂²×C₄×C₇	—	2	2	no	—
221	C₂²×C₄×C₇	—	2	4	no	—
222	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
223	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
224	C₂²×C₄×C₇	C₂⁴	1	2	yes	—
225	C₂²×C₄×C₇	C₂⁴	1	2	yes	—
226	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
227	C₇×Q₁₆	—	2	2	no	—
228	C₇×SD₁₆	—	2	4	no	—
229	C₇×D₈	—	2	2	no	—
230	C₇×M₁₆	—	2	2	no	—
231	C₇×M₁₆	—	2	2	no	—
232	C₇×M₁₆	—	2	4	no	—
233	C₂×C₇×C₈	—	2	2	no	—
234	C₂×C₇×C₈	—	2	4	no	—
235	C₂×C₇×C₈	—	2	4	no	—
236	C₂×C₇×C₈	—	2	4	no	—
237	C₂×C₇×C₈	#41	1	4	yes	—
238	C₂×C₇×C₈	C₂⁴	1	2	yes	—
239	C₂×C₇×C₈	C₂⁴	1	2	yes	—
240	C₂×C₇×C₈	C₂⁴	1	2	yes	—
241	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
242	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
243	C₇×C₄⋊C₄	—	2	2	no	—
244	C₇×C₄⋊C₄	—	2	4	no	—
245	C₇×C₄⋊C₄	—	2	2	no	—
246	C₇×C₄⋊C₄	—	2	4	no	—
247	C₇×C₄⋊C₄	—	2	2	no	—
248	C₇×C₄⋊C₄	—	2	4	no	—
249	C₇×C₄⋊C₄	—	2	4	no	—
250	C₇×C₄⋊C₄	—	2	2	no	—
251	C₇×C₄⋊C₄	—	2	4	no	—
252	C₇×C₄⋊C₄	—	2	4	no	—
253	C₇×C₄⋊C₄	C₂×D₄	1	2	yes	—
254	C₇×C₄⋊C₄	C₂×D₄	1	4	yes	—
255	C₇×C₄⋊C₄	C₂⁴	1	4	yes	—
256	C₇×C₄⋊C₄	C₂⁴	1	2	yes	—
257	C₇×C₄⋊C₄	C₂⁴	1	4	yes	—
258	C₇×C₄⋊C₄	C₂⁴	1	2	yes	—
259	C₇×C₂²⋊C₄	—	2	4	no	—
260	C₇×C₂²⋊C₄	—	2	4	no	—
261	C₇×C₂²⋊C₄	—	2	4	no	—
262	C₇×C₂²⋊C₄	—	2	2	no	—
263	C₇×C₂²⋊C₄	—	2	4	no	—
264	C₇×C₂²⋊C₄	—	2	2	no	—
265	C₇×C₂²⋊C₄	—	2	2	no	—
266	C₇×C₂²⋊C₄	—	2	2	no	—
267	C₇×C₂²⋊C₄	—	2	4	no	—
268	C₇×C₂²⋊C₄	—	2	4	no	—
269	C₇×C₂²⋊C₄	—	2	2	no	—
270	C₇×C₂²⋊C₄	C₂×D₄	1	4	yes	—
271	C₇×C₂²⋊C₄	C₂×D₄	1	2	yes	—
272	C₇×C₂²⋊C₄	C₂×D₄	1	4	yes	—
273	C₄²×C₇	—	2	4	no	—
274	C₄²×C₇	—	2	4	no	—
275	C₄²×C₇	—	2	4	no	—
276	C₄²×C₇	—	2	4	no	—
277	C₄²×C₇	—	2	2	no	—
278	C₄²×C₇	C₂⁴	1	2	yes	—
279	C₂×#135	C₂²×C₇	1	1	yes	characteristic
280	C₂×#131	C₂²×C₇	1	1	yes	characteristic
281	C₂³×C₈	C₂²×C₇	1	1	yes	characteristic
282	#556	—	2	2	no	—
283	#556	C₂²×C₇	1	4	yes	—
284	#555	—	2	2	no	—
285	#555	C₂²×C₇	1	4	yes	—
286	#553	—	2	2	no	—
287	#553	C₂²×C₇	1	4	yes	—
288	#552	—	2	2	no	—
289	#552	C₂²×C₇	1	4	yes	—
290	#544	C₂²×C₇	1	2	yes	—
291	#538	—	2	4	no	—
292	#537	C₂²×C₇	1	2	yes	—
293	#529	—	2	4	no	—
294	#525	C₂²×C₇	1	2	yes	—
295	#525	C₂²×C₇	1	1	yes	characteristic
296	#520	C₂²×C₇	1	2	yes	—
297	#520	C₂²×C₇	1	2	yes	—
298	C₂×#117	C₂²×C₇	1	2	yes	—
299	#501	C₂²×C₇	1	4	yes	complemented
300	#499	C₂²×C₇	1	2	yes	complemented
301	#498	C₂²×C₇	1	2	yes	complemented
302	#485	C₂²×C₇	1	4	yes	—
303	#485	C₂²×C₇	1	4	yes	—
304	#485	C₂²×C₇	1	4	yes	complemented
305	#484	C₂²×C₇	1	4	yes	complemented
306	#482	C₂²×C₇	1	4	yes	—
307	#482	C₂²×C₇	1	4	yes	—
308	#482	C₂²×C₇	1	4	yes	complemented
309	#480	C₂²×C₇	1	4	yes	complemented
310	#479	C₂²×C₇	1	4	yes	complemented
311	#478	C₂²×C₇	1	4	yes	complemented
312	#477	C₂²×C₇	1	4	yes	complemented
313	#474	C₂²×C₇	1	4	yes	complemented
314	#470	C₂²×C₇	1	2	yes	complemented
315	#469	C₂²×C₇	1	2	yes	complemented
316	#467	C₂²×C₇	1	2	yes	—
317	#466	C₂²×C₇	1	2	yes	—
318	#465	C₂²×C₇	1	2	yes	—
319	#465	C₂²×C₇	1	2	yes	—
320	#465	C₂²×C₇	1	4	yes	complemented
321	#442	C₂²×C₇	1	2	yes	complemented
322	#441	C₂²×C₇	1	2	yes	complemented
323	C₄×SD₁₆	C₂²×C₇	1	4	yes	complemented
324	#435	C₂²×C₇	1	4	yes	—
325	C₈×D₄	C₂²×C₇	1	4	yes	—
326	#428	C₂²×C₇	1	2	yes	—
327	#428	C₂²×C₇	1	2	yes	—
328	#428	C₂²×C₇	1	4	yes	complemented
329	#427	C₂²×C₇	1	4	yes	—
330	#427	C₂²×C₇	1	4	yes	—
331	#427	C₂²×C₇	1	1	yes	characteristic
332	#427	C₂²×C₇	1	2	yes	complemented
333	C₂×#107	C₂²×C₇	1	2	yes	—
334	C₂×#106	C₂²×C₇	1	1	yes	characteristic
335	C₂×#106	C₂²×C₇	1	1	yes	characteristic
336	C₂×#106	C₂²×C₇	1	2	yes	complemented
337	#424	C₂²×C₇	1	1	yes	characteristic
338	#417	C₂²×C₇	1	2	yes	—
339	#417	C₂²×C₇	1	2	yes	—
340	#416	C₂²×C₇	1	2	yes	—
341	#408	C₂²×C₇	1	1	yes	characteristic
342	#407	C₂²×C₇	1	1	yes	characteristic
343	#405	C₂²×C₇	1	1	yes	characteristic
344	C₂³×C₇	—	2	2	no	—
345	C₂³×C₇	—	2	2	no	—
346	C₂³×C₇	—	4	2	no	—
347	C₂³×C₇	C₂²×D₄	1	2	yes	—
348	C₂³×C₇	C₂²×D₄	1	2	yes	—
349	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
350	C₇×Q₈	—	2	2	no	—
351	C₇×Q₈	—	2	2	no	—
352	C₇×Q₈	—	2	2	no	—
353	C₇×Q₈	—	8	4	no	—
354	C₇×D₄	—	2	4	no	—
355	C₇×D₄	—	2	2	no	—
356	C₇×D₄	—	2	2	no	—
357	C₇×D₄	—	2	2	no	—
358	C₇×D₄	—	4	4	no	—
359	C₇×D₄	—	4	4	no	—
360	C₇×D₄	—	8	2	no	—
361	C₇×D₄	—	8	4	no	—
362	C₇×D₄	—	8	2	no	—
363	C₂×C₄×C₇	—	2	2	no	—
364	C₂×C₄×C₇	—	2	4	no	—
365	C₂×C₄×C₇	—	2	2	no	—
366	C₂×C₄×C₇	—	2	2	no	—
367	C₂×C₄×C₇	—	2	2	no	—
368	C₂×C₄×C₇	—	2	2	no	—
369	C₂×C₄×C₇	—	2	4	no	—
370	C₂×C₄×C₇	—	2	2	no	—
371	C₂×C₄×C₇	—	2	2	no	—
372	C₂×C₄×C₇	—	2	4	no	—
373	C₂×C₄×C₇	—	2	2	no	—
374	C₂×C₄×C₇	—	4	4	no	—
375	C₂×C₄×C₇	—	8	4	no	—
376	C₂×C₄×C₇	—	8	2	no	—
377	C₂×C₄×C₇	—	8	4	no	—
378	C₂×C₄×C₇	—	8	2	no	—
379	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
380	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
381	C₂×C₄×C₇	C₂²×D₄	1	4	yes	—
382	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
383	C₂×C₄×C₇	C₂×#41	1	2	yes	—
384	C₂×C₄×C₇	2₋¹⁺⁴	1	2	yes	—
385	C₂×C₄×C₇	C₂⁵	1	1	yes	characteristic
386	C₇×C₈	—	2	2	no	—
387	C₇×C₈	—	2	2	no	—
388	C₇×C₈	—	2	2	no	—
389	C₇×C₈	C₂×#41	1	2	yes	—
390	C₇×C₈	C₂×#41	1	2	yes	—
391	C₂×#41	C₂³×C₇	1	2	yes	—
392	C₂×#41	C₂³×C₇	1	1	yes	characteristic
393	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
394	#135	—	2	4	no	—
395	C₂×Q₁₆	C₂³×C₇	1	1	yes	characteristic
396	C₂×SD₁₆	C₂³×C₇	1	2	yes	—
397	C₂×D₈	C₂³×C₇	1	1	yes	characteristic
398	#131	—	2	4	no	—
399	C₂×M₁₆	C₂³×C₇	1	1	yes	characteristic
400	C₂×M₁₆	C₂³×C₇	1	2	yes	—
401	C₂×M₁₆	C₂³×C₇	1	1	yes	characteristic
402	C₂²×C₈	—	2	4	no	—
403	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
404	C₂²×C₈	C₂³×C₇	1	2	yes	—
405	C₂²×C₈	C₂³×C₇	1	2	yes	—
406	C₂²×C₈	C₂³×C₇	1	2	yes	—
407	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
408	C₄⋊Q₈	—	2	2	no	—
409	C₄⋊Q₈	C₂³×C₇	1	4	yes	—
410	#126	—	2	4	no	—
411	#126	—	2	4	no	—
412	#126	—	2	4	no	—
413	#126	—	2	4	no	—
414	#125	—	2	4	no	—
415	#125	—	2	2	no	—
416	#125	C₂³×C₇	1	4	yes	—
417	#124	—	2	4	no	—
418	#124	—	2	4	no	—
419	#124	—	2	4	no	—
420	#123	—	2	4	no	—
421	#123	—	2	2	no	—
422	#123	—	2	4	no	—
423	#123	C₇×D₄	1	2	yes	—
424	C₂²⋊Q₈	—	2	2	no	—
425	C₂²⋊Q₈	—	2	4	no	—
426	C₂²⋊Q₈	C₇×D₄	1	2	yes	—
427	C₂²⋊Q₈	C₂³×C₇	1	2	yes	—
428	C₂²⋊Q₈	C₂³×C₇	1	2	yes	—
429	C₂²⋊Q₈	C₂³×C₇	1	4	yes	—
430	#121	—	2	4	no	—
431	#121	—	2	2	no	—
432	#121	—	2	4	no	—
433	#121	—	2	4	no	—
434	#121	C₇×D₄	1	2	yes	—
435	#121	C₂³×C₇	1	2	yes	—
436	#121	C₂³×C₇	1	2	yes	—
437	#121	C₂³×C₇	1	4	yes	—
438	#120	—	2	2	no	—
439	#120	C₇×D₄	1	2	yes	—
440	C₄×Q₈	—	2	2	no	—
441	C₄×Q₈	—	2	4	no	—
442	C₄×Q₈	C₂³×C₇	1	4	yes	—
443	C₄×D₄	—	2	4	no	—
444	C₄×D₄	—	2	2	no	—
445	C₄×D₄	—	2	2	no	—
446	C₄×D₄	—	2	2	no	—
447	C₄×D₄	—	2	4	no	—
448	C₄×D₄	—	2	4	no	—
449	C₄×D₄	C₂³×C₇	1	4	yes	—
450	C₄×D₄	C₂³×C₇	1	4	yes	—
451	#117	—	2	2	no	—
452	#117	—	2	2	no	—
453	#117	—	2	2	no	—
454	#117	—	2	4	no	—
455	#117	—	2	4	no	—
456	#117	C₂³×C₇	1	4	yes	—
457	#117	C₂³×C₇	1	4	yes	—
458	#117	C₂³×C₇	1	2	yes	—
459	#117	C₂³×C₇	1	4	yes	—
460	#117	C₂³×C₇	1	1	yes	characteristic
461	C₂×C₄⋊C₄	—	2	2	no	—
462	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
463	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
464	C₂×C₄⋊C₄	C₂³×C₇	1	4	yes	—
465	C₂×C₂²⋊C₄	—	2	2	no	—
466	C₂×C₄²	—	2	2	no	—
467	#107	C₂³×C₇	1	4	yes	—
468	#107	C₂³×C₇	1	4	yes	—
469	#107	C₂³×C₇	1	2	yes	—
470	#107	C₂³×C₇	1	4	yes	—
471	#107	C₂³×C₇	1	4	yes	—
472	#106	C₂³×C₇	1	2	yes	—
473	#106	C₂³×C₇	1	2	yes	—
474	#106	C₂³×C₇	1	2	yes	—
475	#106	C₂³×C₇	1	2	yes	—
476	#106	C₂³×C₇	1	1	yes	characteristic
477	#106	C₂³×C₇	1	4	yes	—
478	#106	C₂³×C₇	1	1	yes	characteristic
479	#106	C₂³×C₇	1	4	yes	—
480	C₄⋊C₈	C₂³×C₇	1	4	yes	—
481	#103	C₂³×C₇	1	2	yes	—
482	#103	C₂³×C₇	1	2	yes	—
483	#103	C₂³×C₇	1	4	yes	—
484	#103	C₂³×C₇	1	4	yes	—
485	#102	C₂³×C₇	1	2	yes	—
486	#102	C₂³×C₇	1	2	yes	—
487	#102	C₂³×C₇	1	4	yes	—
488	#102	C₂³×C₇	1	4	yes	—
489	C₂²⋊C₈	C₂³×C₇	1	4	yes	—
490	C₂²⋊C₈	C₂³×C₇	1	4	yes	—
491	#97	C₂³×C₇	1	2	yes	—
492	C₄×C₈	C₂³×C₇	1	2	yes	—
493	C₂²×C₇	—	2	2	no	—
494	C₂²×C₇	—	2	4	no	—
495	C₂²×C₇	—	2	2	no	—
496	C₂²×C₇	—	2	2	no	—
497	C₂²×C₇	—	2	2	no	—
498	C₂²×C₇	—	2	2	no	—
499	C₂²×C₇	—	4	4	no	—
500	C₂²×C₇	—	4	2	no	—
501	C₂²×C₇	—	4	2	no	—
502	C₂²×C₇	—	4	2	no	—
503	C₂²×C₇	#547	1	2	yes	—
504	C₂²×C₇	C₂³×D₄	1	1	yes	characteristic
505	C₄×C₇	—	2	4	no	—
506	C₄×C₇	—	2	2	no	—
507	C₄×C₇	—	2	2	no	—
508	C₄×C₇	—	4	4	no	—
509	C₄×C₇	—	4	2	no	—
510	C₄×C₇	—	4	4	no	—
511	C₄×C₇	—	4	2	no	—
512	C₄×C₇	—	4	2	no	—
513	C₄×C₇	#547	1	2	yes	—
514	C₄×C₇	C₂²×#41	1	1	yes	characteristic
515	C₄×C₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
516	C₂⁴	C₂×C₇×D₄	1	1	yes	characteristic
517	#41	—	2	2	no	—
518	#41	—	2	2	no	—
519	#41	—	2	4	no	—
520	#41	—	2	4	no	—
521	C₂×Q₈	—	2	4	no	—
522	C₂×Q₈	C₂×C₇×D₄	1	1	yes	characteristic
523	C₂×Q₈	C₂⁴×C₇	1	2	yes	—
524	C₂×D₄	—	2	2	no	—
525	C₂×D₄	—	2	2	no	—
526	C₂×D₄	—	2	2	no	—
527	C₂×D₄	—	2	4	no	—
528	C₂×D₄	C₂×C₇×D₄	1	2	yes	—
529	C₂×D₄	C₂×C₇×D₄	1	1	yes	characteristic
530	C₂×D₄	C₂×C₇×D₄	1	4	yes	—
531	C₂×D₄	C₂⁴×C₇	1	2	yes	—
532	C₂²×C₄	—	2	2	no	—
533	C₂²×C₄	—	2	2	no	—
534	C₂²×C₄	—	2	2	no	—
535	C₂²×C₄	—	2	2	no	—
536	C₂²×C₄	—	2	4	no	—
537	C₂²×C₄	—	2	2	no	—
538	C₂²×C₄	—	2	4	no	—
539	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
540	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
541	C₂²×C₄	C₂⁴×C₇	1	2	yes	—
542	C₂²×C₄	C₂⁴×C₇	1	2	yes	—
543	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
544	Q₁₆	—	2	2	no	—
545	SD₁₆	—	2	4	no	—
546	D₈	—	2	2	no	—
547	M₁₆	—	2	2	no	—
548	M₁₆	—	2	2	no	—
549	M₁₆	—	2	4	no	—
550	C₂×C₈	—	2	2	no	—
551	C₂×C₈	—	2	4	no	—
552	C₂×C₈	—	2	4	no	—
553	C₂×C₈	—	2	4	no	—
554	C₂×C₈	C₇×#41	1	4	yes	—
555	C₂×C₈	C₂⁴×C₇	1	2	yes	—
556	C₂×C₈	C₂⁴×C₇	1	2	yes	—
557	C₂×C₈	C₂⁴×C₇	1	2	yes	—
558	C₂×C₈	C₂⁴×C₇	1	1	yes	characteristic
559	C₂×C₈	C₂⁴×C₇	1	1	yes	characteristic
560	C₄⋊C₄	—	2	2	no	—
561	C₄⋊C₄	—	2	4	no	—
562	C₄⋊C₄	—	2	2	no	—
563	C₄⋊C₄	—	2	4	no	—
564	C₄⋊C₄	—	2	2	no	—
565	C₄⋊C₄	—	2	4	no	—
566	C₄⋊C₄	—	2	4	no	—
567	C₄⋊C₄	—	2	2	no	—
568	C₄⋊C₄	—	2	4	no	—
569	C₄⋊C₄	—	2	4	no	—
570	C₄⋊C₄	C₂×C₇×D₄	1	2	yes	—
571	C₄⋊C₄	C₂×C₇×D₄	1	4	yes	—
572	C₄⋊C₄	C₂⁴×C₇	1	4	yes	—
573	C₄⋊C₄	C₂⁴×C₇	1	2	yes	—
574	C₄⋊C₄	C₂⁴×C₇	1	4	yes	—
575	C₄⋊C₄	C₂⁴×C₇	1	2	yes	—
576	C₂²⋊C₄	—	2	4	no	—
577	C₂²⋊C₄	—	2	4	no	—
578	C₂²⋊C₄	—	2	4	no	—
579	C₂²⋊C₄	—	2	2	no	—
580	C₂²⋊C₄	—	2	4	no	—
581	C₂²⋊C₄	—	2	2	no	—
582	C₂²⋊C₄	—	2	2	no	—
583	C₂²⋊C₄	—	2	2	no	—
584	C₂²⋊C₄	—	2	4	no	—
585	C₂²⋊C₄	—	2	4	no	—
586	C₂²⋊C₄	—	2	2	no	—
587	C₂²⋊C₄	C₂×C₇×D₄	1	4	yes	—
588	C₂²⋊C₄	C₂×C₇×D₄	1	2	yes	—
589	C₂²⋊C₄	C₂×C₇×D₄	1	4	yes	—
590	C₄²	—	2	4	no	—
591	C₄²	—	2	4	no	—
592	C₄²	—	2	4	no	—
593	C₄²	—	2	4	no	—
594	C₄²	—	2	2	no	—
595	C₄²	C₂⁴×C₇	1	2	yes	—
596	C₂×C₇	—	2	2	no	—
597	C₂×C₇	—	4	2	no	—
598	C₂×C₇	—	4	2	no	—
599	C₂×C₇	—	8	2	no	—
600	C₂×C₇	C₂×#547	1	1	yes	characteristic
601	C₂×C₇	#3584	1	2	yes	—
602	C₂³	—	2	2	no	—
603	C₂³	—	2	2	no	—
604	C₂³	—	4	2	no	—
605	C₂³	C₂²×C₇×D₄	1	2	yes	—
606	C₂³	C₂²×C₇×D₄	1	2	yes	—
607	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
608	Q₈	—	2	2	no	—
609	Q₈	—	2	2	no	—
610	Q₈	—	2	2	no	—
611	Q₈	—	8	4	no	—
612	D₄	—	2	4	no	—
613	D₄	—	2	2	no	—
614	D₄	—	2	2	no	—
615	D₄	—	2	2	no	—
616	D₄	—	4	4	no	—
617	D₄	—	4	4	no	—
618	D₄	—	8	2	no	—
619	D₄	—	8	4	no	—
620	D₄	—	8	2	no	—
621	C₂×C₄	—	2	2	no	—
622	C₂×C₄	—	2	2	no	—
623	C₂×C₄	—	2	2	no	—
624	C₂×C₄	—	2	2	no	—
625	C₂×C₄	—	2	2	no	—
626	C₂×C₄	—	2	4	no	—
627	C₂×C₄	—	2	4	no	—
628	C₂×C₄	—	2	2	no	—
629	C₂×C₄	—	2	2	no	—
630	C₂×C₄	—	2	4	no	—
631	C₂×C₄	—	2	2	no	—
632	C₂×C₄	—	4	4	no	—
633	C₂×C₄	—	8	4	no	—
634	C₂×C₄	—	8	2	no	—
635	C₂×C₄	—	8	4	no	—
636	C₂×C₄	—	8	2	no	—
637	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
638	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
639	C₂×C₄	C₂²×C₇×D₄	1	4	yes	—
640	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
641	C₂×C₄	C₂×C₇×#41	1	2	yes	—
642	C₂×C₄	C₇×2₋¹⁺⁴	1	2	yes	—
643	C₂×C₄	C₂⁵×C₇	1	1	yes	characteristic
644	C₈	—	2	2	no	—
645	C₈	—	2	2	no	—
646	C₈	—	2	2	no	—
647	C₈	C₂×C₇×#41	1	2	yes	—
648	C₈	C₂×C₇×#41	1	2	yes	—
649	C₇	#61260	1	1	yes	characteristic, direct product factor, Sylow
650	C₂²	—	2	2	no	—
651	C₂²	—	2	4	no	—
652	C₂²	—	2	2	no	—
653	C₂²	—	2	2	no	—
654	C₂²	—	2	2	no	—
655	C₂²	—	2	2	no	—
656	C₂²	—	4	4	no	—
657	C₂²	—	4	2	no	—
658	C₂²	—	4	2	no	—
659	C₂²	—	4	2	no	—
660	C₂²	C₇×#547	1	2	yes	—
661	C₂²	C₂³×C₇×D₄	1	1	yes	characteristic
662	C₄	—	2	4	no	—
663	C₄	—	2	2	no	—
664	C₄	—	2	2	no	—
665	C₄	—	4	4	no	—
666	C₄	—	4	2	no	—
667	C₄	—	4	4	no	—
668	C₄	—	4	2	no	—
669	C₄	—	4	2	no	—
670	C₄	C₇×#547	1	2	yes	—
671	C₄	C₂²×C₇×#41	1	1	yes	characteristic
672	C₄	C₂×C₇×2₋¹⁺⁴	1	1	yes	characteristic
673	C₂	—	2	2	no	—
674	C₂	—	4	2	no	—
675	C₂	—	4	2	no	—
676	C₂	—	8	2	no	—
677	C₂	C₂×C₇×#547	1	1	yes	characteristic
678	C₂	C₇×#3584	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 ◁ C₇×#525 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₇×#3493	C₂	semidirect
2. C₇×#3349	C₂	semidirect
3. C₇×#3347	C₂	semidirect
4. C₇×#3340	C₂	semidirect
5. C₇×#3337	C₂	semidirect
6. C₇×#3163	C₂	semidirect
7. C₇×#3098	C₂	semidirect
8. C₇×#3093	C₂	semidirect
9. C₇×#3083	C₂	semidirect
10. C₇×#3060	C₂	semidirect
11. C₇×#3050	C₂	semidirect
12. C₇×#2952	C₂	semidirect
13. C₇×#2929	C₂	semidirect
14. C₇×#2912	C₂	semidirect
15. C₂×C₇×#427	C₂	semidirect
36. C₇×#501	C₂²	semidirect
37. C₇×#499	C₂²	semidirect
38. C₇×#498	C₂²	semidirect
41. C₇×#485	C₂²	semidirect
42. C₇×#484	C₂²	semidirect
45. C₇×#482	C₂²	semidirect
46. C₇×#480	C₂²	semidirect
47. C₇×#479	C₂²	semidirect
48. C₇×#478	C₂²	semidirect
49. C₇×#477	C₂²	semidirect
50. C₇×#474	C₂²	semidirect
51. C₇×#470	C₂²	semidirect
52. C₇×#469	C₂²	semidirect
57. C₇×#465	C₂²	semidirect
58. C₇×#442	C₂²	semidirect
59. C₇×#441	C₂²	semidirect
60. C₄×C₇×SD₁₆	C₂²	semidirect
65. C₇×#428	C₂²	semidirect
69. C₇×#427	C₂²	semidirect
73. C₂×C₇×#106	C₂²	semidirect
81. #61260	C₇	direct (N × H)
184. #3493	C₂×C₇	semidirect
185. #3349	C₂×C₇	semidirect
186. #3347	C₂×C₇	semidirect
187. #3340	C₂×C₇	semidirect
188. #3337	C₂×C₇	semidirect
189. #3163	C₂×C₇	semidirect
190. #3098	C₂×C₇	semidirect
191. #3093	C₂×C₇	semidirect
192. #3083	C₂×C₇	semidirect
193. #3060	C₂×C₇	semidirect
194. #3050	C₂×C₇	semidirect
195. #2952	C₂×C₇	semidirect
196. #2929	C₂×C₇	semidirect
197. #2912	C₂×C₇	semidirect
198. C₂×#427	C₂×C₇	semidirect
299. #501	C₂²×C₇	semidirect
300. #499	C₂²×C₇	semidirect
301. #498	C₂²×C₇	semidirect
304. #485	C₂²×C₇	semidirect
305. #484	C₂²×C₇	semidirect
308. #482	C₂²×C₇	semidirect
309. #480	C₂²×C₇	semidirect
310. #479	C₂²×C₇	semidirect
311. #478	C₂²×C₇	semidirect
312. #477	C₂²×C₇	semidirect
313. #474	C₂²×C₇	semidirect
314. #470	C₂²×C₇	semidirect
315. #469	C₂²×C₇	semidirect
320. #465	C₂²×C₇	semidirect
321. #442	C₂²×C₇	semidirect
322. #441	C₂²×C₇	semidirect
323. C₄×SD₁₆	C₂²×C₇	semidirect
328. #428	C₂²×C₇	semidirect
332. #427	C₂²×C₇	semidirect
336. C₂×#106	C₂²×C₇	semidirect
649. C₇	#61260	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×#3493	2
C₇×#3349	2
C₇×#3347	2
C₇×#3340	2
C₇×#3337	2
C₇×#3163	2
C₇×#3098	2
C₇×#3093	2
C₇×#3083	2
C₇×#3060	2
C₇×#3050	2
C₇×#2952	2
C₇×#2929	2
C₇×#2912	2
C₂×C₇×#427	2
#61260	7

Subgroups — C7×#61260 #13238476

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#61260 #13238476