Subgroups — C₇×#61286 #13238502

Order 1792.

1754 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3507	C₂	1	2	yes	maximal
2	C₇×#3356	C₂	1	4	yes	maximal, complemented
3	C₇×#3352	C₂	1	4	yes	maximal, complemented
4	C₇×#3342	C₂	1	2	yes	maximal, complemented
5	C₇×#3338	C₂	1	2	yes	maximal, complemented
6	C₇×#3336	C₂	1	4	yes	maximal, complemented
7	C₇×#3160	C₂	1	1	yes	maximal, characteristic, complemented
8	C₇×#3097	C₂	1	4	yes	maximal, complemented
9	C₇×#3078	C₂	1	2	yes	maximal, complemented
10	C₇×#3059	C₂	1	2	yes	maximal, complemented
11	C₂×C₄×C₇×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
12	C₇×#2931	C₂	1	1	yes	maximal, characteristic
13	C₇×#2913	C₂	1	2	yes	maximal, complemented
14	C₂²×C₇×Q₁₆	C₂²	1	1	yes	characteristic
15	C₂×C₇×#131	C₂²	1	2	yes	—
16	C₇×#559	—	2	2	no	—
17	C₇×#557	—	2	2	no	—
18	C₇×#556	—	2	4	no	—
19	C₇×#556	C₂²	1	4	yes	—
20	C₇×#554	C₂²	1	4	yes	—
21	C₇×#552	C₂²	1	8	yes	—
22	C₇×#539	—	2	4	no	—
23	C₇×#536	C₂²	1	4	yes	—
24	C₇×#533	C₂²	1	2	yes	—
25	C₇×#532	C₂²	1	1	yes	characteristic
26	C₇×#529	—	2	4	no	—
27	C₇×#520	C₂²	1	4	yes	—
28	C₂×C₄×C₇×Q₈	C₂²	1	2	yes	—
29	C₇×#500	C₂²	1	1	yes	characteristic
30	C₇×#500	C₂²	1	1	yes	characteristic
31	C₇×#499	C₂²	1	2	yes	—
32	C₇×#499	C₂²	1	2	yes	—
33	C₇×#498	C₂²	1	2	yes	—
34	C₇×#485	C₂²	1	8	yes	—
35	C₇×#485	C₂²	1	8	yes	—
36	C₇×#484	C₂²	1	8	yes	—
37	C₇×#483	C₂²	1	8	yes	—
38	C₇×#479	C₂²	1	8	yes	—
39	C₇×#477	C₂²	1	4	yes	—
40	C₇×#475	C₂²	1	4	yes	—
41	C₇×#470	C₂²	1	8	yes	—
42	C₇×#467	C₂²	1	4	yes	—
43	C₇×#467	C₂²	1	4	yes	—
44	C₄×C₇×Q₁₆	C₂²	1	2	yes	—
45	C₄×C₇×Q₁₆	C₂²	1	2	yes	—
46	C₄×C₇×Q₁₆	C₂²	1	2	yes	—
47	C₄×C₇×Q₁₆	C₂²	1	2	yes	—
48	C₇×#436	C₂²	1	2	yes	—
49	C₇×#436	C₂²	1	2	yes	—
50	C₇×C₈×D₄	C₂²	1	2	yes	—
51	C₇×C₈×D₄	C₂²	1	2	yes	—
52	C₇×#428	C₂²	1	4	yes	—
53	C₇×#428	C₂²	1	8	yes	—
54	C₇×#427	C₂²	1	2	yes	—
55	C₇×#427	C₂²	1	2	yes	—
56	C₇×#427	C₂²	1	4	yes	—
57	C₂×C₇×#107	C₂²	1	1	yes	characteristic
58	C₂×C₇×#107	C₂²	1	4	yes	—
59	C₇×#423	C₂²	1	1	yes	characteristic
60	C₇×#419	C₂²	1	4	yes	—
61	C₂×C₇×#103	C₂²	1	2	yes	—
62	C₇×#408	C₂²	1	2	yes	—
63	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
64	#61286	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
65	C₂×C₇×#41	C₂³	1	2	yes	—
66	C₂²×C₇×Q₈	C₂³	1	2	yes	—
67	C₂×C₇×Q₁₆	—	2	4	no	—
68	C₂×C₇×Q₁₆	C₂³	1	2	yes	—
69	C₂×C₇×Q₁₆	C₂³	1	2	yes	—
70	C₇×#131	—	2	8	no	—
71	C₂×C₇×M₁₆	C₂³	1	4	yes	—
72	C₂×C₇×M₁₆	C₂³	1	2	yes	—
73	C₂²×C₇×C₈	C₂³	1	4	yes	—
74	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
75	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
76	C₇×C₄⋊Q₈	—	2	4	no	—
77	C₇×C₄⋊Q₈	D₄	1	1	yes	characteristic
78	C₇×C₄⋊Q₈	C₂³	1	2	yes	—
79	C₇×C₄⋊Q₈	C₂³	1	2	yes	—
80	C₇×#127	D₄	1	1	yes	characteristic
81	C₇×#126	—	2	8	no	—
82	C₇×#126	—	2	8	no	—
83	C₇×#125	—	2	4	no	—
84	C₇×#125	—	2	8	no	—
85	C₇×#125	C₂³	1	4	yes	—
86	C₇×#124	—	2	4	no	—
87	C₇×#124	—	2	4	no	—
88	C₇×#124	D₄	1	2	yes	—
89	C₇×#123	—	2	8	no	—
90	C₇×#123	—	2	8	no	—
91	C₇×C₂²⋊Q₈	—	2	2	no	—
92	C₇×C₂²⋊Q₈	—	2	2	no	—
93	C₇×C₂²⋊Q₈	—	2	8	no	—
94	C₇×C₂²⋊Q₈	C₂³	1	8	yes	—
95	C₇×C₂²⋊Q₈	C₂³	1	8	yes	—
96	C₇×#121	D₄	1	2	yes	—
97	C₇×#121	D₄	1	2	yes	—
98	C₄×C₇×Q₈	—	2	4	no	—
99	C₄×C₇×Q₈	—	2	4	no	—
100	C₄×C₇×Q₈	—	2	2	no	—
101	C₄×C₇×Q₈	—	2	2	no	—
102	C₄×C₇×Q₈	—	2	2	no	—
103	C₄×C₇×Q₈	—	2	2	no	—
104	C₄×C₇×Q₈	C₂³	1	4	yes	—
105	C₄×C₇×Q₈	C₂³	1	4	yes	—
106	C₄×C₇×D₄	—	2	8	no	—
107	C₄×C₇×D₄	—	2	4	no	—
108	C₄×C₇×D₄	—	2	4	no	—
109	C₄×C₇×D₄	C₂³	1	2	yes	—
110	C₄×C₇×D₄	C₂³	1	2	yes	—
111	C₇×#117	—	2	8	no	—
112	C₇×#117	C₂³	1	8	yes	—
113	C₇×#117	C₂³	1	4	yes	—
114	C₂×C₇×C₄⋊C₄	—	2	2	no	—
115	C₂×C₇×C₄⋊C₄	C₂³	1	8	yes	—
116	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
117	C₂×C₄²×C₇	—	2	2	no	—
118	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
119	C₇×#107	C₂³	1	2	yes	—
120	C₇×#107	C₂³	1	2	yes	—
121	C₇×#107	C₂³	1	8	yes	—
122	C₇×#107	C₂³	1	2	yes	—
123	C₇×#107	C₂³	1	2	yes	—
124	C₇×#107	C₂³	1	2	yes	—
125	C₇×#107	C₂³	1	2	yes	—
126	C₇×#106	C₂³	1	4	yes	—
127	C₇×#106	C₂³	1	8	yes	—
128	C₇×#106	C₂³	1	4	yes	—
129	C₇×C₄⋊C₈	C₂³	1	2	yes	—
130	C₇×C₄⋊C₈	C₂³	1	2	yes	—
131	C₇×#103	C₂³	1	8	yes	—
132	C₇×#103	C₂³	1	8	yes	—
133	C₇×#103	C₂³	1	4	yes	—
134	C₇×#103	C₂³	1	4	yes	—
135	C₇×C₂²⋊C₈	C₂³	1	4	yes	—
136	C₇×C₂²⋊C₈	C₂³	1	4	yes	—
137	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
138	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
139	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
140	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
141	#3507	C₂×C₇	1	2	yes	—
142	#3356	C₂×C₇	1	4	yes	complemented
143	#3352	C₂×C₇	1	4	yes	complemented
144	#3342	C₂×C₇	1	2	yes	complemented
145	#3338	C₂×C₇	1	2	yes	complemented
146	#3336	C₂×C₇	1	4	yes	complemented
147	#3160	C₂×C₇	1	1	yes	characteristic, complemented
148	#3097	C₂×C₇	1	4	yes	complemented
149	#3078	C₂×C₇	1	2	yes	complemented
150	#3059	C₂×C₇	1	2	yes	complemented
151	C₂×C₄×Q₁₆	C₂×C₇	1	1	yes	characteristic, complemented
152	#2931	C₂×C₇	1	1	yes	characteristic
153	#2913	C₂×C₇	1	2	yes	complemented
154	C₇×#41	—	2	4	no	—
155	C₇×#41	—	2	4	no	—
156	C₂×C₇×Q₈	—	2	2	no	—
157	C₂×C₇×Q₈	—	2	2	no	—
158	C₂×C₇×Q₈	—	2	2	no	—
159	C₂×C₇×Q₈	—	2	4	no	—
160	C₂×C₇×Q₈	—	2	4	no	—
161	C₂×C₇×Q₈	C₂×D₄	1	2	yes	—
162	C₂×C₇×Q₈	C₂⁴	1	4	yes	—
163	C₂×C₇×D₄	C₂×D₄	1	2	yes	—
164	C₂×C₇×D₄	C₂×D₄	1	4	yes	—
165	C₂×C₇×D₄	C₂×D₄	1	2	yes	—
166	C₂×C₇×D₄	C₂×D₄	1	2	yes	—
167	C₂²×C₄×C₇	—	2	2	no	—
168	C₂²×C₄×C₇	—	2	2	no	—
169	C₂²×C₄×C₇	—	2	8	no	—
170	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
171	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
172	C₂²×C₄×C₇	C₂⁴	1	4	yes	—
173	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
174	C₇×Q₁₆	—	2	4	no	—
175	C₇×Q₁₆	—	2	4	no	—
176	C₇×M₁₆	—	2	4	no	—
177	C₇×M₁₆	—	2	8	no	—
178	C₂×C₇×C₈	—	2	2	no	—
179	C₂×C₇×C₈	—	2	8	no	—
180	C₂×C₇×C₈	#41	1	4	yes	—
181	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
182	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
183	C₂×C₇×C₈	C₂⁴	1	4	yes	—
184	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
185	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
186	C₇×C₄⋊C₄	—	2	4	no	—
187	C₇×C₄⋊C₄	—	2	4	no	—
188	C₇×C₄⋊C₄	—	2	2	no	—
189	C₇×C₄⋊C₄	—	2	8	no	—
190	C₇×C₄⋊C₄	—	2	4	no	—
191	C₇×C₄⋊C₄	—	2	8	no	—
192	C₇×C₄⋊C₄	—	2	2	no	—
193	C₇×C₄⋊C₄	—	2	4	no	—
194	C₇×C₄⋊C₄	—	2	2	no	—
195	C₇×C₄⋊C₄	—	2	8	no	—
196	C₇×C₄⋊C₄	C₂×D₄	1	2	yes	—
197	C₇×C₄⋊C₄	C₂×D₄	1	2	yes	—
198	C₇×C₄⋊C₄	C₂⁴	1	4	yes	—
199	C₇×C₄⋊C₄	C₂⁴	1	4	yes	—
200	C₇×C₄⋊C₄	C₂⁴	1	4	yes	—
201	C₇×C₂²⋊C₄	—	2	8	no	—
202	C₇×C₂²⋊C₄	—	2	4	no	—
203	C₇×C₂²⋊C₄	—	2	8	no	—
204	C₇×C₂²⋊C₄	—	2	8	no	—
205	C₇×C₂²⋊C₄	C₂×D₄	1	4	yes	—
206	C₇×C₂²⋊C₄	C₂×D₄	1	4	yes	—
207	C₄²×C₇	—	2	4	no	—
208	C₄²×C₇	—	2	4	no	—
209	C₄²×C₇	—	2	8	no	—
210	C₄²×C₇	—	2	4	no	—
211	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
212	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
213	C₄²×C₇	C₂⁴	1	1	yes	characteristic
214	C₄²×C₇	C₂⁴	1	1	yes	characteristic
215	C₂²×Q₁₆	C₂²×C₇	1	1	yes	characteristic
216	C₂×#131	C₂²×C₇	1	2	yes	—
217	#559	—	2	2	no	—
218	#557	—	2	2	no	—
219	#556	—	2	4	no	—
220	#556	C₂²×C₇	1	4	yes	—
221	#554	C₂²×C₇	1	4	yes	—
222	#552	C₂²×C₇	1	8	yes	—
223	#539	—	2	4	no	—
224	#536	C₂²×C₇	1	4	yes	—
225	#533	C₂²×C₇	1	2	yes	—
226	#532	C₂²×C₇	1	1	yes	characteristic
227	#529	—	2	4	no	—
228	#520	C₂²×C₇	1	4	yes	—
229	C₂×C₄×Q₈	C₂²×C₇	1	2	yes	—
230	#500	C₂²×C₇	1	1	yes	characteristic
231	#500	C₂²×C₇	1	1	yes	characteristic
232	#499	C₂²×C₇	1	2	yes	—
233	#499	C₂²×C₇	1	2	yes	—
234	#498	C₂²×C₇	1	2	yes	—
235	#485	C₂²×C₇	1	8	yes	—
236	#485	C₂²×C₇	1	8	yes	—
237	#484	C₂²×C₇	1	8	yes	—
238	#483	C₂²×C₇	1	8	yes	—
239	#479	C₂²×C₇	1	8	yes	—
240	#477	C₂²×C₇	1	4	yes	—
241	#475	C₂²×C₇	1	4	yes	—
242	#470	C₂²×C₇	1	8	yes	—
243	#467	C₂²×C₇	1	4	yes	—
244	#467	C₂²×C₇	1	4	yes	—
245	C₄×Q₁₆	C₂²×C₇	1	2	yes	—
246	C₄×Q₁₆	C₂²×C₇	1	2	yes	—
247	C₄×Q₁₆	C₂²×C₇	1	2	yes	—
248	C₄×Q₁₆	C₂²×C₇	1	2	yes	—
249	#436	C₂²×C₇	1	2	yes	—
250	#436	C₂²×C₇	1	2	yes	—
251	C₈×D₄	C₂²×C₇	1	2	yes	—
252	C₈×D₄	C₂²×C₇	1	2	yes	—
253	#428	C₂²×C₇	1	4	yes	—
254	#428	C₂²×C₇	1	8	yes	—
255	#427	C₂²×C₇	1	4	yes	—
256	#427	C₂²×C₇	1	2	yes	—
257	#427	C₂²×C₇	1	2	yes	—
258	C₂×#107	C₂²×C₇	1	4	yes	—
259	C₂×#107	C₂²×C₇	1	1	yes	characteristic
260	#423	C₂²×C₇	1	1	yes	characteristic
261	#419	C₂²×C₇	1	4	yes	—
262	C₂×#103	C₂²×C₇	1	2	yes	—
263	#408	C₂²×C₇	1	2	yes	—
264	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
265	C₂³×C₇	C₂²×D₄	1	4	yes	—
266	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
267	C₇×Q₈	—	2	4	no	—
268	C₇×Q₈	—	2	4	no	—
269	C₇×Q₈	—	2	4	no	—
270	C₇×Q₈	—	4	4	no	—
271	C₇×Q₈	—	4	4	no	—
272	C₇×Q₈	—	8	2	no	—
273	C₇×D₄	—	2	8	no	—
274	C₇×D₄	—	2	4	no	—
275	C₇×D₄	—	2	4	no	—
276	C₇×D₄	—	4	2	no	—
277	C₂×C₄×C₇	—	2	2	no	—
278	C₂×C₄×C₇	—	2	4	no	—
279	C₂×C₄×C₇	—	2	4	no	—
280	C₂×C₄×C₇	—	2	4	no	—
281	C₂×C₄×C₇	—	2	4	no	—
282	C₂×C₄×C₇	—	2	4	no	—
283	C₂×C₄×C₇	—	2	4	no	—
284	C₂×C₄×C₇	—	4	1	no	—
285	C₂×C₄×C₇	—	4	4	no	—
286	C₂×C₄×C₇	—	8	2	no	—
287	C₂×C₄×C₇	—	8	8	no	—
288	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
289	C₂×C₄×C₇	C₂²×D₄	1	4	yes	—
290	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
291	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
292	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
293	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
294	C₂×C₄×C₇	C₂×#41	1	2	yes	—
295	C₂×C₄×C₇	2₋¹⁺⁴	1	2	yes	—
296	C₂×C₄×C₇	C₂⁵	1	1	yes	characteristic
297	C₇×C₈	—	2	4	no	—
298	C₇×C₈	—	2	1	no	—
299	C₇×C₈	—	2	1	no	—
300	C₇×C₈	C₂×#41	1	2	yes	—
301	C₇×C₈	C₂×#41	1	2	yes	—
302	C₂×#41	C₂³×C₇	1	2	yes	—
303	C₂²×Q₈	C₂³×C₇	1	2	yes	—
304	C₂×Q₁₆	—	2	4	no	—
305	C₂×Q₁₆	C₂³×C₇	1	2	yes	—
306	C₂×Q₁₆	C₂³×C₇	1	2	yes	—
307	#131	—	2	8	no	—
308	C₂×M₁₆	C₂³×C₇	1	2	yes	—
309	C₂×M₁₆	C₂³×C₇	1	4	yes	—
310	C₂²×C₈	C₂³×C₇	1	4	yes	—
311	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
312	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
313	C₄⋊Q₈	—	2	4	no	—
314	C₄⋊Q₈	C₇×D₄	1	1	yes	characteristic
315	C₄⋊Q₈	C₂³×C₇	1	2	yes	—
316	C₄⋊Q₈	C₂³×C₇	1	2	yes	—
317	#127	C₇×D₄	1	1	yes	characteristic
318	#126	—	2	8	no	—
319	#126	—	2	8	no	—
320	#125	—	2	4	no	—
321	#125	—	2	8	no	—
322	#125	C₂³×C₇	1	4	yes	—
323	#124	—	2	4	no	—
324	#124	—	2	4	no	—
325	#124	C₇×D₄	1	2	yes	—
326	#123	—	2	8	no	—
327	#123	—	2	8	no	—
328	C₂²⋊Q₈	—	2	2	no	—
329	C₂²⋊Q₈	—	2	2	no	—
330	C₂²⋊Q₈	—	2	8	no	—
331	C₂²⋊Q₈	C₂³×C₇	1	8	yes	—
332	C₂²⋊Q₈	C₂³×C₇	1	8	yes	—
333	#121	C₇×D₄	1	2	yes	—
334	#121	C₇×D₄	1	2	yes	—
335	C₄×Q₈	—	2	4	no	—
336	C₄×Q₈	—	2	4	no	—
337	C₄×Q₈	—	2	2	no	—
338	C₄×Q₈	—	2	2	no	—
339	C₄×Q₈	—	2	2	no	—
340	C₄×Q₈	—	2	2	no	—
341	C₄×Q₈	C₂³×C₇	1	4	yes	—
342	C₄×Q₈	C₂³×C₇	1	4	yes	—
343	C₄×D₄	—	2	8	no	—
344	C₄×D₄	—	2	4	no	—
345	C₄×D₄	—	2	4	no	—
346	C₄×D₄	C₂³×C₇	1	2	yes	—
347	C₄×D₄	C₂³×C₇	1	2	yes	—
348	#117	—	2	8	no	—
349	#117	C₂³×C₇	1	4	yes	—
350	#117	C₂³×C₇	1	8	yes	—
351	C₂×C₄⋊C₄	—	2	2	no	—
352	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
353	C₂×C₄⋊C₄	C₂³×C₇	1	8	yes	—
354	C₂×C₄²	—	2	2	no	—
355	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
356	#107	C₂³×C₇	1	2	yes	—
357	#107	C₂³×C₇	1	2	yes	—
358	#107	C₂³×C₇	1	2	yes	—
359	#107	C₂³×C₇	1	8	yes	—
360	#107	C₂³×C₇	1	2	yes	—
361	#107	C₂³×C₇	1	2	yes	—
362	#107	C₂³×C₇	1	2	yes	—
363	#106	C₂³×C₇	1	4	yes	—
364	#106	C₂³×C₇	1	8	yes	—
365	#106	C₂³×C₇	1	4	yes	—
366	C₄⋊C₈	C₂³×C₇	1	2	yes	—
367	C₄⋊C₈	C₂³×C₇	1	2	yes	—
368	#103	C₂³×C₇	1	4	yes	—
369	#103	C₂³×C₇	1	4	yes	—
370	#103	C₂³×C₇	1	8	yes	—
371	#103	C₂³×C₇	1	8	yes	—
372	C₂²⋊C₈	C₂³×C₇	1	4	yes	—
373	C₂²⋊C₈	C₂³×C₇	1	4	yes	—
374	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
375	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
376	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
377	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
378	C₂²×C₇	—	2	1	no	—
379	C₂²×C₇	—	2	1	no	—
380	C₂²×C₇	—	2	4	no	—
381	C₂²×C₇	—	2	4	no	—
382	C₂²×C₇	—	2	4	no	—
383	C₂²×C₇	#547	1	2	yes	—
384	C₂²×C₇	C₂³×D₄	1	1	yes	characteristic
385	C₄×C₇	—	2	1	no	—
386	C₄×C₇	—	2	1	no	—
387	C₄×C₇	—	2	4	no	—
388	C₄×C₇	—	2	1	no	—
389	C₄×C₇	—	4	4	no	—
390	C₄×C₇	—	4	4	no	—
391	C₄×C₇	—	4	4	no	—
392	C₄×C₇	—	4	4	no	—
393	C₄×C₇	#547	1	2	yes	—
394	C₄×C₇	#578	1	2	yes	—
395	C₄×C₇	C₂²×#41	1	1	yes	characteristic
396	C₄×C₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
397	#41	—	2	4	no	—
398	#41	—	2	4	no	—
399	C₂×Q₈	—	2	2	no	—
400	C₂×Q₈	—	2	2	no	—
401	C₂×Q₈	—	2	2	no	—
402	C₂×Q₈	—	2	4	no	—
403	C₂×Q₈	—	2	4	no	—
404	C₂×Q₈	C₂×C₇×D₄	1	2	yes	—
405	C₂×Q₈	C₂⁴×C₇	1	4	yes	—
406	C₂×D₄	C₂×C₇×D₄	1	2	yes	—
407	C₂×D₄	C₂×C₇×D₄	1	4	yes	—
408	C₂×D₄	C₂×C₇×D₄	1	2	yes	—
409	C₂×D₄	C₂×C₇×D₄	1	2	yes	—
410	C₂²×C₄	—	2	2	no	—
411	C₂²×C₄	—	2	2	no	—
412	C₂²×C₄	—	2	8	no	—
413	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
414	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
415	C₂²×C₄	C₂⁴×C₇	1	4	yes	—
416	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
417	Q₁₆	—	2	4	no	—
418	Q₁₆	—	2	4	no	—
419	M₁₆	—	2	4	no	—
420	M₁₆	—	2	8	no	—
421	C₂×C₈	—	2	2	no	—
422	C₂×C₈	—	2	8	no	—
423	C₂×C₈	C₇×#41	1	4	yes	—
424	C₂×C₈	C₂⁴×C₇	1	1	yes	characteristic
425	C₂×C₈	C₂⁴×C₇	1	1	yes	characteristic
426	C₂×C₈	C₂⁴×C₇	1	4	yes	—
427	C₂×C₈	C₂⁴×C₇	1	1	yes	characteristic
428	C₂×C₈	C₂⁴×C₇	1	1	yes	characteristic
429	C₄⋊C₄	—	2	4	no	—
430	C₄⋊C₄	—	2	4	no	—
431	C₄⋊C₄	—	2	2	no	—
432	C₄⋊C₄	—	2	8	no	—
433	C₄⋊C₄	—	2	4	no	—
434	C₄⋊C₄	—	2	8	no	—
435	C₄⋊C₄	—	2	2	no	—
436	C₄⋊C₄	—	2	4	no	—
437	C₄⋊C₄	—	2	2	no	—
438	C₄⋊C₄	—	2	8	no	—
439	C₄⋊C₄	C₂×C₇×D₄	1	2	yes	—
440	C₄⋊C₄	C₂×C₇×D₄	1	2	yes	—
441	C₄⋊C₄	C₂⁴×C₇	1	4	yes	—
442	C₄⋊C₄	C₂⁴×C₇	1	4	yes	—
443	C₄⋊C₄	C₂⁴×C₇	1	4	yes	—
444	C₂²⋊C₄	—	2	8	no	—
445	C₂²⋊C₄	—	2	4	no	—
446	C₂²⋊C₄	—	2	8	no	—
447	C₂²⋊C₄	—	2	8	no	—
448	C₂²⋊C₄	C₂×C₇×D₄	1	4	yes	—
449	C₂²⋊C₄	C₂×C₇×D₄	1	4	yes	—
450	C₄²	—	2	4	no	—
451	C₄²	—	2	4	no	—
452	C₄²	—	2	8	no	—
453	C₄²	—	2	4	no	—
454	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
455	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
456	C₄²	C₂⁴×C₇	1	1	yes	characteristic
457	C₄²	C₂⁴×C₇	1	1	yes	characteristic
458	C₂×C₇	—	2	2	no	—
459	C₂×C₇	—	4	4	no	—
460	C₂×C₇	C₂×#547	1	1	yes	characteristic
461	C₂×C₇	C₂×#578	1	1	yes	characteristic
462	C₂×C₇	#3584	1	1	yes	characteristic
463	C₂³	C₂²×C₇×D₄	1	4	yes	—
464	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
465	Q₈	—	2	4	no	—
466	Q₈	—	2	4	no	—
467	Q₈	—	2	4	no	—
468	Q₈	—	4	4	no	—
469	Q₈	—	4	4	no	—
470	Q₈	—	8	2	no	—
471	D₄	—	2	8	no	—
472	D₄	—	2	4	no	—
473	D₄	—	2	4	no	—
474	D₄	—	4	2	no	—
475	C₂×C₄	—	2	4	no	—
476	C₂×C₄	—	2	4	no	—
477	C₂×C₄	—	2	2	no	—
478	C₂×C₄	—	2	4	no	—
479	C₂×C₄	—	2	4	no	—
480	C₂×C₄	—	2	4	no	—
481	C₂×C₄	—	2	4	no	—
482	C₂×C₄	—	4	1	no	—
483	C₂×C₄	—	4	4	no	—
484	C₂×C₄	—	8	2	no	—
485	C₂×C₄	—	8	8	no	—
486	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
487	C₂×C₄	C₂²×C₇×D₄	1	4	yes	—
488	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
489	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
490	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
491	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
492	C₂×C₄	C₂×C₇×#41	1	2	yes	—
493	C₂×C₄	C₇×2₋¹⁺⁴	1	2	yes	—
494	C₂×C₄	C₂⁵×C₇	1	1	yes	characteristic
495	C₈	—	2	4	no	—
496	C₈	—	2	1	no	—
497	C₈	—	2	1	no	—
498	C₈	C₂×C₇×#41	1	2	yes	—
499	C₈	C₂×C₇×#41	1	2	yes	—
500	C₇	#61286	1	1	yes	characteristic, direct product factor, Sylow
501	C₂²	—	2	1	no	—
502	C₂²	—	2	1	no	—
503	C₂²	—	2	4	no	—
504	C₂²	—	2	4	no	—
505	C₂²	—	2	4	no	—
506	C₂²	C₇×#547	1	2	yes	—
507	C₂²	C₂³×C₇×D₄	1	1	yes	characteristic
508	C₄	—	2	1	no	—
509	C₄	—	2	1	no	—
510	C₄	—	2	1	no	—
511	C₄	—	2	4	no	—
512	C₄	—	4	4	no	—
513	C₄	—	4	4	no	—
514	C₄	—	4	4	no	—
515	C₄	—	4	4	no	—
516	C₄	C₇×#547	1	2	yes	—
517	C₄	C₇×#578	1	2	yes	—
518	C₄	C₂²×C₇×#41	1	1	yes	characteristic
519	C₄	C₂×C₇×2₋¹⁺⁴	1	1	yes	characteristic
520	C₂	—	2	2	no	—
521	C₂	—	4	4	no	—
522	C₂	C₂×C₇×#547	1	1	yes	characteristic
523	C₂	C₂×C₇×#578	1	1	yes	characteristic
524	C₂	C₇×#3584	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₇	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₇
prime radical	C₇×#532	C₂²
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
2. C₇×#3356	C₂	semidirect
3. C₇×#3352	C₂	semidirect
4. C₇×#3342	C₂	semidirect
5. C₇×#3338	C₂	semidirect
6. C₇×#3336	C₂	semidirect
7. C₇×#3160	C₂	semidirect
8. C₇×#3097	C₂	semidirect
9. C₇×#3078	C₂	semidirect
10. C₇×#3059	C₂	semidirect
11. C₂×C₄×C₇×Q₁₆	C₂	semidirect
13. C₇×#2913	C₂	semidirect
64. #61286	C₇	direct (N × H)
142. #3356	C₂×C₇	semidirect
143. #3352	C₂×C₇	semidirect
144. #3342	C₂×C₇	semidirect
145. #3338	C₂×C₇	semidirect
146. #3336	C₂×C₇	semidirect
147. #3160	C₂×C₇	semidirect
148. #3097	C₂×C₇	semidirect
149. #3078	C₂×C₇	semidirect
150. #3059	C₂×C₇	semidirect
151. C₂×C₄×Q₁₆	C₂×C₇	semidirect
153. #2913	C₂×C₇	semidirect
500. C₇	#61286	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×#3507	2
C₇×#3356	2
C₇×#3352	2
C₇×#3342	2
C₇×#3338	2
C₇×#3336	2
C₇×#3160	2
C₇×#3097	2
C₇×#3078	2
C₇×#3059	2
C₂×C₄×C₇×Q₁₆	2
C₇×#2931	2
C₇×#2913	2
#61286	7

Subgroups — C7×#61286 #13238502

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#61286 #13238502