Subgroups — C₇×#61287 #13238503

Order 1792.

1846 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3511	C₂	1	2	yes	maximal, complemented
2	C₇×#3357	C₂	1	4	yes	maximal, complemented
3	C₇×#3353	C₂	1	4	yes	maximal, complemented
4	C₇×#3342	C₂	1	2	yes	maximal, complemented
5	C₇×#3341	C₂	1	2	yes	maximal, complemented
6	C₇×#3338	C₂	1	2	yes	maximal, complemented
7	C₇×#3334	C₂	1	2	yes	maximal, complemented
8	C₂×C₇×#500	C₂	1	1	yes	maximal, characteristic, complemented
9	C₇×#3098	C₂	1	4	yes	maximal, complemented
10	C₇×#3079	C₂	1	2	yes	maximal, complemented
11	C₇×#3061	C₂	1	2	yes	maximal, complemented
12	C₄×C₇×#135	C₂	1	1	yes	maximal, characteristic, complemented
13	C₇×#2931	C₂	1	1	yes	maximal, characteristic, complemented
14	C₇×#2913	C₂	1	2	yes	maximal, complemented
15	C₂×C₇×#135	C₂²	1	1	yes	characteristic
16	C₂×C₇×#131	C₂²	1	2	yes	—
17	C₇×#556	C₂²	1	4	yes	—
18	C₇×#555	C₂²	1	4	yes	—
19	C₇×#554	C₂²	1	4	yes	—
20	C₇×#553	—	2	4	no	—
21	C₇×#548	—	2	4	no	—
22	C₇×#547	C₂²	1	4	yes	—
23	C₇×#542	—	2	4	no	—
24	C₇×#537	C₂²	1	4	yes	—
25	C₇×#533	—	2	2	no	—
26	C₇×#532	—	2	2	no	—
27	C₇×#532	C₂²	1	1	yes	characteristic
28	C₂×C₇×C₄⋊Q₈	C₂²	1	2	yes	—
29	C₇×#520	C₂²	1	4	yes	—
30	C₄×C₇×#41	C₂²	1	2	yes	—
31	C₇×#500	C₂²	1	2	yes	—
32	C₇×#500	C₂²	1	2	yes	—
33	C₇×#500	C₂²	1	2	yes	—
34	C₇×#500	C₂²	1	2	yes	—
35	C₇×#485	C₂²	1	8	yes	—
36	C₇×#484	C₂²	1	8	yes	—
37	C₇×#482	C₂²	1	8	yes	—
38	C₇×#480	C₂²	1	8	yes	—
39	C₇×#477	C₂²	1	4	yes	—
40	C₇×#476	C₂²	1	4	yes	—
41	C₇×#475	C₂²	1	4	yes	—
42	C₇×#474	C₂²	1	4	yes	—
43	C₇×#468	C₂²	1	8	yes	—
44	C₇×#467	C₂²	1	4	yes	—
45	C₇×#466	C₂²	1	4	yes	—
46	C₄×C₇×Q₁₆	C₂²	1	1	yes	characteristic
47	C₄×C₇×Q₁₆	C₂²	1	1	yes	characteristic
48	C₄×C₇×SD₁₆	C₂²	1	2	yes	—
49	C₄×C₇×SD₁₆	C₂²	1	2	yes	—
50	C₄×C₇×D₈	C₂²	1	1	yes	characteristic
51	C₄×C₇×D₈	C₂²	1	1	yes	characteristic
52	C₇×#436	C₂²	1	2	yes	—
53	C₇×#436	C₂²	1	2	yes	—
54	C₇×C₈×D₄	C₂²	1	2	yes	—
55	C₇×C₈×D₄	C₂²	1	2	yes	—
56	C₇×#428	C₂²	1	4	yes	—
57	C₇×#428	C₂²	1	8	yes	—
58	C₇×#427	C₂²	1	1	yes	characteristic
59	C₇×#427	C₂²	1	4	yes	—
60	C₂×C₇×#107	C₂²	1	2	yes	—
61	C₂×C₇×#107	C₂²	1	2	yes	—
62	C₂×C₇×#107	C₂²	1	4	yes	—
63	C₇×#423	C₂²	1	1	yes	characteristic
64	C₇×#417	C₂²	1	4	yes	—
65	C₇×#416	C₂²	1	2	yes	—
66	C₇×#408	C₂²	1	2	yes	—
67	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
68	#61287	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
69	C₂×C₇×#41	—	2	4	no	—
70	C₂×C₇×#41	C₂³	1	2	yes	—
71	C₂×C₇×#41	C₂³	1	2	yes	—
72	C₂²×C₇×Q₈	—	2	2	no	—
73	C₇×#135	—	2	4	no	—
74	C₂×C₇×Q₁₆	C₂³	1	1	yes	characteristic
75	C₂×C₇×SD₁₆	C₂³	1	2	yes	—
76	C₂×C₇×D₈	C₂³	1	1	yes	characteristic
77	C₇×#131	—	2	8	no	—
78	C₂×C₇×M₁₆	C₂³	1	4	yes	—
79	C₂×C₇×M₁₆	C₂³	1	2	yes	—
80	C₂²×C₇×C₈	C₂³	1	4	yes	—
81	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
82	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
83	C₇×C₄⋊Q₈	—	2	2	no	—
84	C₇×C₄⋊Q₈	—	2	2	no	—
85	C₇×C₄⋊Q₈	D₄	1	1	yes	characteristic
86	C₇×C₄⋊Q₈	C₂³	1	4	yes	—
87	C₇×C₄⋊Q₈	C₂³	1	4	yes	—
88	C₇×#127	D₄	1	1	yes	characteristic
89	C₇×#126	—	2	8	no	—
90	C₇×#125	—	2	4	no	—
91	C₇×#124	—	2	4	no	—
92	C₇×#124	—	2	8	no	—
93	C₇×#124	D₄	1	2	yes	—
94	C₇×#123	—	2	8	no	—
95	C₇×C₂²⋊Q₈	—	2	4	no	—
96	C₇×C₂²⋊Q₈	—	2	4	no	—
97	C₇×C₂²⋊Q₈	—	2	8	no	—
98	C₇×C₂²⋊Q₈	C₂³	1	8	yes	—
99	C₇×#121	—	2	4	no	—
100	C₇×#121	—	2	4	no	—
101	C₇×#121	—	2	8	no	—
102	C₇×#121	D₄	1	2	yes	—
103	C₇×#121	D₄	1	2	yes	—
104	C₇×#121	C₂³	1	8	yes	—
105	C₄×C₇×Q₈	—	2	4	no	—
106	C₄×C₇×Q₈	—	2	4	no	—
107	C₄×C₇×Q₈	C₂³	1	2	yes	—
108	C₄×C₇×Q₈	C₂³	1	2	yes	—
109	C₄×C₇×D₄	—	2	4	no	—
110	C₄×C₇×D₄	—	2	8	no	—
111	C₄×C₇×D₄	—	2	4	no	—
112	C₄×C₇×D₄	—	2	2	no	—
113	C₄×C₇×D₄	—	2	2	no	—
114	C₄×C₇×D₄	C₂³	1	2	yes	—
115	C₄×C₇×D₄	C₂³	1	2	yes	—
116	C₄×C₇×D₄	C₂³	1	2	yes	—
117	C₄×C₇×D₄	C₂³	1	2	yes	—
118	C₇×#117	—	2	4	no	—
119	C₇×#117	—	2	2	no	—
120	C₇×#117	C₂³	1	8	yes	—
121	C₇×#117	C₂³	1	2	yes	—
122	C₂×C₇×C₄⋊C₄	—	2	2	no	—
123	C₂×C₇×C₄⋊C₄	C₂³	1	8	yes	—
124	C₂×C₇×C₄⋊C₄	C₂³	1	4	yes	—
125	C₂×C₄²×C₇	—	2	2	no	—
126	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
127	C₇×#107	C₂³	1	4	yes	—
128	C₇×#107	C₂³	1	4	yes	—
129	C₇×#107	C₂³	1	8	yes	—
130	C₇×#107	C₂³	1	4	yes	—
131	C₇×#107	C₂³	1	4	yes	—
132	C₇×#107	C₂³	1	1	yes	characteristic
133	C₇×#107	C₂³	1	1	yes	characteristic
134	C₇×#106	C₂³	1	8	yes	—
135	C₇×#106	C₂³	1	2	yes	—
136	C₇×C₄⋊C₈	C₂³	1	2	yes	—
137	C₇×C₄⋊C₈	C₂³	1	2	yes	—
138	C₇×#103	C₂³	1	8	yes	—
139	C₇×#103	C₂³	1	2	yes	—
140	C₇×#103	C₂³	1	2	yes	—
141	C₇×#102	C₂³	1	8	yes	—
142	C₇×#102	C₂³	1	2	yes	—
143	C₇×#102	C₂³	1	2	yes	—
144	C₇×C₂²⋊C₈	C₂³	1	4	yes	—
145	C₇×C₂²⋊C₈	C₂³	1	4	yes	—
146	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
147	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
148	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
149	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
150	#3511	C₂×C₇	1	2	yes	complemented
151	#3357	C₂×C₇	1	4	yes	complemented
152	#3353	C₂×C₇	1	4	yes	complemented
153	#3342	C₂×C₇	1	2	yes	complemented
154	#3341	C₂×C₇	1	2	yes	complemented
155	#3338	C₂×C₇	1	2	yes	complemented
156	#3334	C₂×C₇	1	2	yes	complemented
157	C₂×#500	C₂×C₇	1	1	yes	characteristic, complemented
158	#3098	C₂×C₇	1	4	yes	complemented
159	#3079	C₂×C₇	1	2	yes	complemented
160	#3061	C₂×C₇	1	2	yes	complemented
161	C₄×#135	C₂×C₇	1	1	yes	characteristic, complemented
162	#2931	C₂×C₇	1	1	yes	characteristic, complemented
163	#2913	C₂×C₇	1	2	yes	complemented
164	C₇×#41	—	2	4	no	—
165	C₇×#41	—	2	4	no	—
166	C₇×#41	—	2	4	no	—
167	C₇×#41	—	2	4	no	—
168	C₇×#41	—	8	8	no	—
169	C₂×C₇×Q₈	—	2	4	no	—
170	C₂×C₇×Q₈	—	2	4	no	—
171	C₂×C₇×Q₈	—	8	4	no	—
172	C₂×C₇×Q₈	C₂×D₄	1	2	yes	—
173	C₂×C₇×Q₈	C₂⁴	1	2	yes	—
174	C₂×C₇×D₄	—	2	2	no	—
175	C₂×C₇×D₄	—	2	8	no	—
176	C₂×C₇×D₄	C₂×D₄	1	2	yes	—
177	C₂×C₇×D₄	C₂×D₄	1	4	yes	—
178	C₂×C₇×D₄	C₂×D₄	1	2	yes	—
179	C₂×C₇×D₄	C₂×D₄	1	2	yes	—
180	C₂×C₇×D₄	C₂⁴	1	2	yes	—
181	C₂²×C₄×C₇	—	2	4	no	—
182	C₂²×C₄×C₇	—	2	8	no	—
183	C₂²×C₄×C₇	—	2	2	no	—
184	C₂²×C₄×C₇	—	2	2	no	—
185	C₂²×C₄×C₇	—	2	2	no	—
186	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
187	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
188	C₂²×C₄×C₇	C₂⁴	1	4	yes	—
189	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
190	C₇×Q₁₆	—	2	2	no	—
191	C₇×SD₁₆	—	2	4	no	—
192	C₇×D₈	—	2	2	no	—
193	C₇×M₁₆	—	2	4	no	—
194	C₇×M₁₆	—	2	8	no	—
195	C₂×C₇×C₈	—	2	2	no	—
196	C₂×C₇×C₈	—	2	8	no	—
197	C₂×C₇×C₈	#41	1	4	yes	complemented
198	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
199	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
200	C₂×C₇×C₈	C₂⁴	1	4	yes	—
201	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
202	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
203	C₇×C₄⋊C₄	—	2	4	no	—
204	C₇×C₄⋊C₄	—	2	4	no	—
205	C₇×C₄⋊C₄	—	2	4	no	—
206	C₇×C₄⋊C₄	—	2	8	no	—
207	C₇×C₄⋊C₄	—	2	2	no	—
208	C₇×C₄⋊C₄	—	2	2	no	—
209	C₇×C₄⋊C₄	—	2	8	no	—
210	C₇×C₄⋊C₄	C₂×D₄	1	2	yes	—
211	C₇×C₄⋊C₄	C₂×D₄	1	2	yes	—
212	C₇×C₄⋊C₄	C₂⁴	1	2	yes	—
213	C₇×C₄⋊C₄	C₂⁴	1	2	yes	—
214	C₇×C₄⋊C₄	C₂⁴	1	8	yes	—
215	C₇×C₂²⋊C₄	—	2	2	no	—
216	C₇×C₂²⋊C₄	—	2	8	no	—
217	C₇×C₂²⋊C₄	—	2	8	no	—
218	C₇×C₂²⋊C₄	—	2	2	no	—
219	C₇×C₂²⋊C₄	—	2	2	no	—
220	C₇×C₂²⋊C₄	—	2	8	no	—
221	C₇×C₂²⋊C₄	C₂×D₄	1	4	yes	—
222	C₇×C₂²⋊C₄	C₂×D₄	1	4	yes	—
223	C₄²×C₇	—	2	2	no	—
224	C₄²×C₇	—	2	2	no	—
225	C₄²×C₇	—	2	2	no	—
226	C₄²×C₇	—	2	8	no	—
227	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
228	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
229	C₄²×C₇	C₂⁴	1	1	yes	characteristic
230	C₄²×C₇	C₂⁴	1	1	yes	characteristic
231	C₂×#135	C₂²×C₇	1	1	yes	characteristic
232	C₂×#131	C₂²×C₇	1	2	yes	—
233	#556	C₂²×C₇	1	4	yes	—
234	#555	C₂²×C₇	1	4	yes	—
235	#554	C₂²×C₇	1	4	yes	—
236	#553	—	2	4	no	—
237	#548	—	2	4	no	—
238	#547	C₂²×C₇	1	4	yes	—
239	#542	—	2	4	no	—
240	#537	C₂²×C₇	1	4	yes	—
241	#533	—	2	2	no	—
242	#532	—	2	2	no	—
243	#532	C₂²×C₇	1	1	yes	characteristic
244	C₂×C₄⋊Q₈	C₂²×C₇	1	2	yes	—
245	#520	C₂²×C₇	1	4	yes	—
246	C₄×#41	C₂²×C₇	1	2	yes	—
247	#500	C₂²×C₇	1	2	yes	—
248	#500	C₂²×C₇	1	2	yes	—
249	#500	C₂²×C₇	1	2	yes	—
250	#500	C₂²×C₇	1	2	yes	—
251	#485	C₂²×C₇	1	8	yes	—
252	#484	C₂²×C₇	1	8	yes	—
253	#482	C₂²×C₇	1	8	yes	—
254	#480	C₂²×C₇	1	8	yes	—
255	#477	C₂²×C₇	1	4	yes	—
256	#476	C₂²×C₇	1	4	yes	—
257	#475	C₂²×C₇	1	4	yes	—
258	#474	C₂²×C₇	1	4	yes	—
259	#468	C₂²×C₇	1	8	yes	—
260	#467	C₂²×C₇	1	4	yes	—
261	#466	C₂²×C₇	1	4	yes	—
262	C₄×Q₁₆	C₂²×C₇	1	1	yes	characteristic
263	C₄×Q₁₆	C₂²×C₇	1	1	yes	characteristic
264	C₄×SD₁₆	C₂²×C₇	1	2	yes	—
265	C₄×SD₁₆	C₂²×C₇	1	2	yes	—
266	C₄×D₈	C₂²×C₇	1	1	yes	characteristic
267	C₄×D₈	C₂²×C₇	1	1	yes	characteristic
268	#436	C₂²×C₇	1	2	yes	—
269	#436	C₂²×C₇	1	2	yes	—
270	C₈×D₄	C₂²×C₇	1	2	yes	—
271	C₈×D₄	C₂²×C₇	1	2	yes	—
272	#428	C₂²×C₇	1	4	yes	—
273	#428	C₂²×C₇	1	8	yes	—
274	#427	C₂²×C₇	1	4	yes	—
275	#427	C₂²×C₇	1	1	yes	characteristic
276	C₂×#107	C₂²×C₇	1	4	yes	—
277	C₂×#107	C₂²×C₇	1	2	yes	—
278	C₂×#107	C₂²×C₇	1	2	yes	—
279	#423	C₂²×C₇	1	1	yes	characteristic
280	#417	C₂²×C₇	1	4	yes	—
281	#416	C₂²×C₇	1	2	yes	—
282	#408	C₂²×C₇	1	2	yes	—
283	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
284	C₂³×C₇	—	2	2	no	—
285	C₂³×C₇	C₂²×D₄	1	4	yes	—
286	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
287	C₇×Q₈	—	2	4	no	—
288	C₇×Q₈	—	2	2	no	—
289	C₇×Q₈	—	2	2	no	—
290	C₇×Q₈	—	4	8	no	—
291	C₇×Q₈	—	8	4	no	—
292	C₇×D₄	—	2	8	no	—
293	C₇×D₄	—	2	4	no	—
294	C₇×D₄	—	2	4	no	—
295	C₇×D₄	—	2	4	no	—
296	C₇×D₄	—	2	2	no	—
297	C₇×D₄	—	2	2	no	—
298	C₇×D₄	—	4	4	no	—
299	C₇×D₄	—	8	8	no	—
300	C₂×C₄×C₇	—	2	2	no	—
301	C₂×C₄×C₇	—	2	2	no	—
302	C₂×C₄×C₇	—	2	4	no	—
303	C₂×C₄×C₇	—	2	4	no	—
304	C₂×C₄×C₇	—	2	2	no	—
305	C₂×C₄×C₇	—	2	2	no	—
306	C₂×C₄×C₇	—	2	2	no	—
307	C₂×C₄×C₇	—	2	8	no	—
308	C₂×C₄×C₇	—	4	4	no	—
309	C₂×C₄×C₇	—	4	4	no	—
310	C₂×C₄×C₇	—	4	4	no	—
311	C₂×C₄×C₇	—	8	4	no	—
312	C₂×C₄×C₇	—	8	8	no	—
313	C₂×C₄×C₇	C₂²×D₄	1	4	yes	—
314	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
315	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
316	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
317	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
318	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
319	C₂×C₄×C₇	C₂×#41	1	2	yes	—
320	C₂×C₄×C₇	2₋¹⁺⁴	1	2	yes	—
321	C₂×C₄×C₇	C₂⁵	1	1	yes	characteristic
322	C₇×C₈	—	2	4	no	—
323	C₇×C₈	—	2	1	no	—
324	C₇×C₈	—	2	1	no	—
325	C₇×C₈	C₂×#41	1	2	yes	—
326	C₇×C₈	C₂×#41	1	2	yes	—
327	C₂×#41	—	2	4	no	—
328	C₂×#41	C₂³×C₇	1	2	yes	—
329	C₂×#41	C₂³×C₇	1	2	yes	—
330	C₂²×Q₈	—	2	2	no	—
331	#135	—	2	4	no	—
332	C₂×Q₁₆	C₂³×C₇	1	1	yes	characteristic
333	C₂×SD₁₆	C₂³×C₇	1	2	yes	—
334	C₂×D₈	C₂³×C₇	1	1	yes	characteristic
335	#131	—	2	8	no	—
336	C₂×M₁₆	C₂³×C₇	1	2	yes	—
337	C₂×M₁₆	C₂³×C₇	1	4	yes	—
338	C₂²×C₈	C₂³×C₇	1	4	yes	—
339	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
340	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
341	C₄⋊Q₈	—	2	2	no	—
342	C₄⋊Q₈	—	2	2	no	—
343	C₄⋊Q₈	C₇×D₄	1	1	yes	characteristic
344	C₄⋊Q₈	C₂³×C₇	1	4	yes	—
345	C₄⋊Q₈	C₂³×C₇	1	4	yes	—
346	#127	C₇×D₄	1	1	yes	characteristic
347	#126	—	2	8	no	—
348	#125	—	2	4	no	—
349	#124	—	2	4	no	—
350	#124	—	2	8	no	—
351	#124	C₇×D₄	1	2	yes	—
352	#123	—	2	8	no	—
353	C₂²⋊Q₈	—	2	4	no	—
354	C₂²⋊Q₈	—	2	4	no	—
355	C₂²⋊Q₈	—	2	8	no	—
356	C₂²⋊Q₈	C₂³×C₇	1	8	yes	—
357	#121	—	2	4	no	—
358	#121	—	2	4	no	—
359	#121	—	2	8	no	—
360	#121	C₇×D₄	1	2	yes	—
361	#121	C₇×D₄	1	2	yes	—
362	#121	C₂³×C₇	1	8	yes	—
363	C₄×Q₈	—	2	4	no	—
364	C₄×Q₈	—	2	4	no	—
365	C₄×Q₈	C₂³×C₇	1	2	yes	—
366	C₄×Q₈	C₂³×C₇	1	2	yes	—
367	C₄×D₄	—	2	4	no	—
368	C₄×D₄	—	2	8	no	—
369	C₄×D₄	—	2	4	no	—
370	C₄×D₄	—	2	2	no	—
371	C₄×D₄	—	2	2	no	—
372	C₄×D₄	C₂³×C₇	1	2	yes	—
373	C₄×D₄	C₂³×C₇	1	2	yes	—
374	C₄×D₄	C₂³×C₇	1	2	yes	—
375	C₄×D₄	C₂³×C₇	1	2	yes	—
376	#117	—	2	4	no	—
377	#117	—	2	2	no	—
378	#117	C₂³×C₇	1	2	yes	—
379	#117	C₂³×C₇	1	8	yes	—
380	C₂×C₄⋊C₄	—	2	2	no	—
381	C₂×C₄⋊C₄	C₂³×C₇	1	4	yes	—
382	C₂×C₄⋊C₄	C₂³×C₇	1	8	yes	—
383	C₂×C₄²	—	2	2	no	—
384	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
385	#107	C₂³×C₇	1	1	yes	characteristic
386	#107	C₂³×C₇	1	4	yes	—
387	#107	C₂³×C₇	1	4	yes	—
388	#107	C₂³×C₇	1	8	yes	—
389	#107	C₂³×C₇	1	4	yes	—
390	#107	C₂³×C₇	1	4	yes	—
391	#107	C₂³×C₇	1	1	yes	characteristic
392	#106	C₂³×C₇	1	2	yes	—
393	#106	C₂³×C₇	1	8	yes	—
394	C₄⋊C₈	C₂³×C₇	1	2	yes	—
395	C₄⋊C₈	C₂³×C₇	1	2	yes	—
396	#103	C₂³×C₇	1	2	yes	—
397	#103	C₂³×C₇	1	2	yes	—
398	#103	C₂³×C₇	1	8	yes	—
399	#102	C₂³×C₇	1	2	yes	—
400	#102	C₂³×C₇	1	2	yes	—
401	#102	C₂³×C₇	1	8	yes	—
402	C₂²⋊C₈	C₂³×C₇	1	4	yes	—
403	C₂²⋊C₈	C₂³×C₇	1	4	yes	—
404	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
405	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
406	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
407	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
408	C₂²×C₇	—	2	1	no	—
409	C₂²×C₇	—	2	1	no	—
410	C₂²×C₇	—	2	4	no	—
411	C₂²×C₇	—	2	4	no	—
412	C₂²×C₇	—	2	4	no	—
413	C₂²×C₇	—	4	2	no	—
414	C₂²×C₇	—	4	2	no	—
415	C₂²×C₇	—	4	2	no	—
416	C₂²×C₇	#547	1	2	yes	—
417	C₂²×C₇	C₂³×D₄	1	1	yes	characteristic
418	C₄×C₇	—	2	4	no	—
419	C₄×C₇	—	2	1	no	—
420	C₄×C₇	—	2	1	no	—
421	C₄×C₇	—	4	2	no	—
422	C₄×C₇	—	4	2	no	—
423	C₄×C₇	—	4	8	no	—
424	C₄×C₇	—	4	2	no	—
425	C₄×C₇	#547	1	2	yes	—
426	C₄×C₇	#576	1	2	yes	—
427	C₄×C₇	#578	1	2	yes	—
428	C₄×C₇	C₂²×#41	1	1	yes	characteristic
429	C₄×C₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
430	#41	—	2	4	no	—
431	#41	—	2	4	no	—
432	#41	—	2	4	no	—
433	#41	—	2	4	no	—
434	#41	—	8	8	no	—
435	C₂×Q₈	—	2	4	no	—
436	C₂×Q₈	—	2	4	no	—
437	C₂×Q₈	—	8	4	no	—
438	C₂×Q₈	C₂×C₇×D₄	1	2	yes	—
439	C₂×Q₈	C₂⁴×C₇	1	2	yes	—
440	C₂×D₄	—	2	2	no	—
441	C₂×D₄	—	2	8	no	—
442	C₂×D₄	C₂×C₇×D₄	1	2	yes	—
443	C₂×D₄	C₂×C₇×D₄	1	4	yes	—
444	C₂×D₄	C₂×C₇×D₄	1	2	yes	—
445	C₂×D₄	C₂×C₇×D₄	1	2	yes	—
446	C₂×D₄	C₂⁴×C₇	1	2	yes	—
447	C₂²×C₄	—	2	4	no	—
448	C₂²×C₄	—	2	8	no	—
449	C₂²×C₄	—	2	2	no	—
450	C₂²×C₄	—	2	2	no	—
451	C₂²×C₄	—	2	2	no	—
452	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
453	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
454	C₂²×C₄	C₂⁴×C₇	1	4	yes	—
455	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
456	Q₁₆	—	2	2	no	—
457	SD₁₆	—	2	4	no	—
458	D₈	—	2	2	no	—
459	M₁₆	—	2	4	no	—
460	M₁₆	—	2	8	no	—
461	C₂×C₈	—	2	2	no	—
462	C₂×C₈	—	2	8	no	—
463	C₂×C₈	C₇×#41	1	4	yes	complemented
464	C₂×C₈	C₂⁴×C₇	1	1	yes	characteristic
465	C₂×C₈	C₂⁴×C₇	1	1	yes	characteristic
466	C₂×C₈	C₂⁴×C₇	1	4	yes	—
467	C₂×C₈	C₂⁴×C₇	1	1	yes	characteristic
468	C₂×C₈	C₂⁴×C₇	1	1	yes	characteristic
469	C₄⋊C₄	—	2	4	no	—
470	C₄⋊C₄	—	2	4	no	—
471	C₄⋊C₄	—	2	4	no	—
472	C₄⋊C₄	—	2	8	no	—
473	C₄⋊C₄	—	2	2	no	—
474	C₄⋊C₄	—	2	2	no	—
475	C₄⋊C₄	—	2	8	no	—
476	C₄⋊C₄	C₂×C₇×D₄	1	2	yes	—
477	C₄⋊C₄	C₂×C₇×D₄	1	2	yes	—
478	C₄⋊C₄	C₂⁴×C₇	1	2	yes	—
479	C₄⋊C₄	C₂⁴×C₇	1	2	yes	—
480	C₄⋊C₄	C₂⁴×C₇	1	8	yes	—
481	C₂²⋊C₄	—	2	2	no	—
482	C₂²⋊C₄	—	2	8	no	—
483	C₂²⋊C₄	—	2	8	no	—
484	C₂²⋊C₄	—	2	2	no	—
485	C₂²⋊C₄	—	2	2	no	—
486	C₂²⋊C₄	—	2	8	no	—
487	C₂²⋊C₄	C₂×C₇×D₄	1	4	yes	—
488	C₂²⋊C₄	C₂×C₇×D₄	1	4	yes	—
489	C₄²	—	2	2	no	—
490	C₄²	—	2	2	no	—
491	C₄²	—	2	2	no	—
492	C₄²	—	2	8	no	—
493	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
494	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
495	C₄²	C₂⁴×C₇	1	1	yes	characteristic
496	C₄²	C₂⁴×C₇	1	1	yes	characteristic
497	C₂×C₇	—	2	2	no	—
498	C₂×C₇	—	4	4	no	—
499	C₂×C₇	—	8	2	no	—
500	C₂×C₇	C₂×#547	1	1	yes	characteristic
501	C₂×C₇	C₂×#576	1	1	yes	characteristic
502	C₂×C₇	C₂×#578	1	1	yes	characteristic
503	C₂³	—	2	2	no	—
504	C₂³	C₂²×C₇×D₄	1	4	yes	—
505	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
506	Q₈	—	2	4	no	—
507	Q₈	—	2	2	no	—
508	Q₈	—	2	2	no	—
509	Q₈	—	4	8	no	—
510	Q₈	—	8	4	no	—
511	D₄	—	2	8	no	—
512	D₄	—	2	2	no	—
513	D₄	—	2	2	no	—
514	D₄	—	2	4	no	—
515	D₄	—	2	4	no	—
516	D₄	—	2	4	no	—
517	D₄	—	4	4	no	—
518	D₄	—	8	8	no	—
519	C₂×C₄	—	2	4	no	—
520	C₂×C₄	—	2	4	no	—
521	C₂×C₄	—	2	2	no	—
522	C₂×C₄	—	2	2	no	—
523	C₂×C₄	—	2	2	no	—
524	C₂×C₄	—	2	2	no	—
525	C₂×C₄	—	2	2	no	—
526	C₂×C₄	—	2	8	no	—
527	C₂×C₄	—	4	4	no	—
528	C₂×C₄	—	4	4	no	—
529	C₂×C₄	—	4	4	no	—
530	C₂×C₄	—	8	4	no	—
531	C₂×C₄	—	8	8	no	—
532	C₂×C₄	C₂²×C₇×D₄	1	4	yes	—
533	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
534	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
535	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
536	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
537	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
538	C₂×C₄	C₂×C₇×#41	1	2	yes	—
539	C₂×C₄	C₇×2₋¹⁺⁴	1	2	yes	—
540	C₂×C₄	C₂⁵×C₇	1	1	yes	characteristic
541	C₈	—	2	4	no	—
542	C₈	—	2	1	no	—
543	C₈	—	2	1	no	—
544	C₈	C₂×C₇×#41	1	2	yes	—
545	C₈	C₂×C₇×#41	1	2	yes	—
546	C₇	#61287	1	1	yes	characteristic, direct product factor, Sylow
547	C₂²	—	2	1	no	—
548	C₂²	—	2	1	no	—
549	C₂²	—	2	4	no	—
550	C₂²	—	2	4	no	—
551	C₂²	—	2	4	no	—
552	C₂²	—	4	2	no	—
553	C₂²	—	4	2	no	—
554	C₂²	—	4	2	no	—
555	C₂²	C₇×#547	1	2	yes	—
556	C₂²	C₂³×C₇×D₄	1	1	yes	characteristic
557	C₄	—	2	1	no	—
558	C₄	—	2	1	no	—
559	C₄	—	2	4	no	—
560	C₄	—	4	2	no	—
561	C₄	—	4	8	no	—
562	C₄	—	4	2	no	—
563	C₄	—	4	2	no	—
564	C₄	C₇×#547	1	2	yes	—
565	C₄	C₇×#576	1	2	yes	—
566	C₄	C₇×#578	1	2	yes	—
567	C₄	C₂²×C₇×#41	1	1	yes	characteristic
568	C₄	C₂×C₇×2₋¹⁺⁴	1	1	yes	characteristic
569	C₂	—	2	2	no	—
570	C₂	—	4	4	no	—
571	C₂	—	8	2	no	—
572	C₂	C₂×C₇×#547	1	1	yes	characteristic
573	C₂	C₂×C₇×#576	1	1	yes	characteristic
574	C₂	C₂×C₇×#578	1	1	yes	characteristic

	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×#532 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₇×#3511	C₂	semidirect
2. C₇×#3357	C₂	semidirect
3. C₇×#3353	C₂	semidirect
4. C₇×#3342	C₂	semidirect
5. C₇×#3341	C₂	semidirect
6. C₇×#3338	C₂	semidirect
7. C₇×#3334	C₂	semidirect
8. C₂×C₇×#500	C₂	semidirect
9. C₇×#3098	C₂	semidirect
10. C₇×#3079	C₂	semidirect
11. C₇×#3061	C₂	semidirect
12. C₄×C₇×#135	C₂	semidirect
13. C₇×#2931	C₂	semidirect
14. C₇×#2913	C₂	semidirect
68. #61287	C₇	direct (N × H)
150. #3511	C₂×C₇	semidirect
151. #3357	C₂×C₇	semidirect
152. #3353	C₂×C₇	semidirect
153. #3342	C₂×C₇	semidirect
154. #3341	C₂×C₇	semidirect
155. #3338	C₂×C₇	semidirect
156. #3334	C₂×C₇	semidirect
157. C₂×#500	C₂×C₇	semidirect
158. #3098	C₂×C₇	semidirect
159. #3079	C₂×C₇	semidirect
160. #3061	C₂×C₇	semidirect
161. C₄×#135	C₂×C₇	semidirect
162. #2931	C₂×C₇	semidirect
163. #2913	C₂×C₇	semidirect
197. C₂×C₇×C₈	#41	semidirect
463. C₂×C₈	C₇×#41	semidirect
546. C₇	#61287	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×#3511	2
C₇×#3357	2
C₇×#3353	2
C₇×#3342	2
C₇×#3341	2
C₇×#3338	2
C₇×#3334	2
C₂×C₇×#500	2
C₇×#3098	2
C₇×#3079	2
C₇×#3061	2
C₄×C₇×#135	2
C₇×#2931	2
C₇×#2913	2
#61287	7

Subgroups — C7×#61287 #13238503

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#61287 #13238503