Subgroups — C₇×#61284 #13238500

Order 1792.

1882 conjugacy classes of subgroups.

The subgroup lattice has 550 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	2	yes	maximal, complemented
2	C₇×#3355	C₂	1	4	yes	maximal, complemented
3	C₇×#3354	C₂	1	4	yes	maximal, complemented
4	C₇×#3341	C₂	1	2	yes	maximal, complemented
5	C₇×#3339	C₂	1	4	yes	maximal, complemented
6	C₇×#3334	C₂	1	2	yes	maximal, complemented
7	C₇×#3160	C₂	1	1	yes	maximal, characteristic, complemented
8	C₇×#3096	C₂	1	4	yes	maximal, complemented
9	C₇×#3077	C₂	1	2	yes	maximal, complemented
10	C₇×#3058	C₂	1	2	yes	maximal, complemented
11	C₂×C₄×C₇×D₈	C₂	1	1	yes	maximal, characteristic, complemented
12	C₇×#2931	C₂	1	1	yes	maximal, characteristic, complemented
13	C₇×#2913	C₂	1	2	yes	maximal, complemented
14	C₂²×C₇×D₈	C₂²	1	1	yes	characteristic
15	C₂×C₇×#131	C₂²	1	2	yes	—
16	C₇×#559	—	2	4	no	—
17	C₇×#556	—	2	2	no	—
18	C₇×#555	C₂²	1	4	yes	complemented
19	C₇×#553	C₂²	1	8	yes	complemented
20	C₇×#550	—	2	2	no	—
21	C₇×#547	C₂²	1	4	yes	complemented
22	C₇×#538	—	2	4	no	—
23	C₇×#535	C₂²	1	4	yes	—
24	C₇×#533	C₂²	1	2	yes	—
25	C₇×#532	C₂²	1	1	yes	characteristic
26	C₇×#525	—	2	4	no	—
27	C₇×#520	C₂²	1	4	yes	—
28	C₂×C₄×C₇×D₄	C₂²	1	2	yes	—
29	C₇×#500	C₂²	1	1	yes	characteristic, complemented
30	C₇×#500	C₂²	1	1	yes	characteristic, complemented
31	C₇×#499	C₂²	1	2	yes	complemented
32	C₇×#499	C₂²	1	2	yes	complemented
33	C₇×#498	C₂²	1	2	yes	complemented
34	C₇×#482	C₂²	1	8	yes	complemented
35	C₇×#482	C₂²	1	8	yes	complemented
36	C₇×#481	C₂²	1	8	yes	complemented
37	C₇×#480	C₂²	1	8	yes	complemented
38	C₇×#478	C₂²	1	8	yes	complemented
39	C₇×#476	C₂²	1	4	yes	complemented
40	C₇×#474	C₂²	1	4	yes	complemented
41	C₇×#469	C₂²	1	8	yes	—
42	C₇×#466	C₂²	1	4	yes	—
43	C₇×#466	C₂²	1	4	yes	—
44	C₄×C₇×D₈	C₂²	1	2	yes	—
45	C₄×C₇×D₈	C₂²	1	2	yes	—
46	C₄×C₇×D₈	C₂²	1	2	yes	—
47	C₄×C₇×D₈	C₂²	1	2	yes	—
48	C₇×#436	C₂²	1	2	yes	complemented
49	C₇×#436	C₂²	1	2	yes	complemented
50	C₇×C₈×D₄	C₂²	1	2	yes	complemented
51	C₇×C₈×D₄	C₂²	1	2	yes	complemented
52	C₇×#428	C₂²	1	4	yes	—
53	C₇×#428	C₂²	1	8	yes	complemented
54	C₇×#427	C₂²	1	2	yes	—
55	C₇×#427	C₂²	1	2	yes	—
56	C₇×#427	C₂²	1	4	yes	complemented
57	C₂×C₇×#107	C₂²	1	1	yes	characteristic
58	C₂×C₇×#107	C₂²	1	4	yes	complemented
59	C₇×#423	C₂²	1	1	yes	characteristic
60	C₇×#418	C₂²	1	4	yes	—
61	C₂×C₇×#102	C₂²	1	2	yes	—
62	C₇×#408	C₂²	1	2	yes	—
63	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
64	#61284	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
65	C₂×C₇×#41	—	2	4	no	—
66	C₂×C₇×#41	C₂³	1	2	yes	—
67	C₂²×C₇×D₄	C₂³	1	2	yes	—
68	C₂³×C₄×C₇	—	2	2	no	—
69	C₂×C₇×D₈	—	2	4	no	—
70	C₂×C₇×D₈	C₂³	1	2	yes	—
71	C₂×C₇×D₈	C₂³	1	2	yes	—
72	C₇×#131	—	2	8	no	—
73	C₂×C₇×M₁₆	C₂³	1	4	yes	—
74	C₂×C₇×M₁₆	C₂³	1	2	yes	—
75	C₂²×C₇×C₈	C₂³	1	4	yes	—
76	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
77	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
78	C₇×C₄⋊Q₈	D₄	1	1	yes	characteristic
79	C₇×C₄⋊Q₈	C₂³	1	2	yes	—
80	C₇×C₄⋊Q₈	C₂³	1	2	yes	—
81	C₇×#127	—	2	4	no	—
82	C₇×#127	D₄	1	1	yes	characteristic
83	C₇×#126	—	2	8	no	—
84	C₇×#125	C₂³	1	4	yes	—
85	C₇×#124	—	2	4	no	—
86	C₇×#124	—	2	8	no	—
87	C₇×#124	D₄	1	2	yes	—
88	C₇×#123	—	2	8	no	—
89	C₇×#123	—	2	8	no	—
90	C₇×C₂²⋊Q₈	—	2	2	no	—
91	C₇×C₂²⋊Q₈	—	2	2	no	—
92	C₇×C₂²⋊Q₈	—	2	4	no	—
93	C₇×#121	—	2	4	no	—
94	C₇×#121	—	2	8	no	—
95	C₇×#121	D₄	1	2	yes	—
96	C₇×#121	D₄	1	2	yes	—
97	C₇×#121	C₂³	1	8	yes	—
98	C₇×#121	C₂³	1	8	yes	—
99	C₇×#120	—	2	8	no	—
100	C₄×C₇×Q₈	—	2	4	no	—
101	C₄×C₇×Q₈	—	2	2	no	—
102	C₄×C₇×Q₈	—	2	2	no	—
103	C₄×C₇×D₄	—	2	8	no	—
104	C₄×C₇×D₄	—	2	4	no	—
105	C₄×C₇×D₄	—	2	8	no	—
106	C₄×C₇×D₄	—	2	2	no	—
107	C₄×C₇×D₄	—	2	2	no	—
108	C₄×C₇×D₄	C₂³	1	4	yes	—
109	C₄×C₇×D₄	C₂³	1	4	yes	—
110	C₄×C₇×D₄	C₂³	1	2	yes	—
111	C₄×C₇×D₄	C₂³	1	2	yes	—
112	C₇×#117	—	2	4	no	—
113	C₇×#117	C₂³	1	8	yes	—
114	C₇×#117	C₂³	1	4	yes	—
115	C₂×C₇×C₄⋊C₄	C₂³	1	8	yes	—
116	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
117	C₂×C₇×C₂²⋊C₄	—	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×#102	C₂³	1	8	yes	—
132	C₇×#102	C₂³	1	8	yes	—
133	C₇×#102	C₂³	1	4	yes	—
134	C₇×#102	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	#3506	C₂×C₇	1	2	yes	complemented
142	#3355	C₂×C₇	1	4	yes	complemented
143	#3354	C₂×C₇	1	4	yes	complemented
144	#3341	C₂×C₇	1	2	yes	complemented
145	#3339	C₂×C₇	1	4	yes	complemented
146	#3334	C₂×C₇	1	2	yes	complemented
147	#3160	C₂×C₇	1	1	yes	characteristic, complemented
148	#3096	C₂×C₇	1	4	yes	complemented
149	#3077	C₂×C₇	1	2	yes	complemented
150	#3058	C₂×C₇	1	2	yes	complemented
151	C₂×C₄×D₈	C₂×C₇	1	1	yes	characteristic, complemented
152	#2931	C₂×C₇	1	1	yes	characteristic, complemented
153	#2913	C₂×C₇	1	2	yes	complemented
154	C₂⁴×C₇	—	2	2	no	—
155	C₇×#41	—	2	4	no	—
156	C₇×#41	—	2	4	no	—
157	C₇×#41	—	8	8	no	—
158	C₂×C₇×Q₈	—	2	2	no	—
159	C₂×C₇×Q₈	—	2	2	no	—
160	C₂×C₇×Q₈	C₂×D₄	1	2	yes	—
161	C₂×C₇×D₄	—	2	8	no	—
162	C₂×C₇×D₄	—	2	8	no	—
163	C₂×C₇×D₄	—	2	2	no	—
164	C₂×C₇×D₄	—	2	4	no	—
165	C₂×C₇×D₄	—	2	4	no	—
166	C₂×C₇×D₄	C₂×D₄	1	4	yes	—
167	C₂×C₇×D₄	C₂×D₄	1	2	yes	—
168	C₂×C₇×D₄	C₂×D₄	1	2	yes	—
169	C₂×C₇×D₄	C₂×D₄	1	2	yes	—
170	C₂×C₇×D₄	C₂⁴	1	4	yes	—
171	C₂²×C₄×C₇	—	2	2	no	—
172	C₂²×C₄×C₇	—	2	8	no	—
173	C₂²×C₄×C₇	—	2	4	no	—
174	C₂²×C₄×C₇	—	2	4	no	—
175	C₂²×C₄×C₇	—	8	4	no	—
176	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
177	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
178	C₂²×C₄×C₇	C₂⁴	1	4	yes	—
179	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
180	C₇×D₈	—	2	4	no	—
181	C₇×D₈	—	2	4	no	—
182	C₇×M₁₆	—	2	4	no	—
183	C₇×M₁₆	—	2	8	no	—
184	C₂×C₇×C₈	—	2	2	no	—
185	C₂×C₇×C₈	—	2	8	no	—
186	C₂×C₇×C₈	#41	1	4	yes	complemented
187	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
188	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
189	C₂×C₇×C₈	C₂⁴	1	4	yes	—
190	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
191	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
192	C₇×C₄⋊C₄	—	2	2	no	—
193	C₇×C₄⋊C₄	—	2	8	no	—
194	C₇×C₄⋊C₄	—	2	2	no	—
195	C₇×C₄⋊C₄	—	2	4	no	—
196	C₇×C₄⋊C₄	—	2	4	no	—
197	C₇×C₄⋊C₄	—	2	2	no	—
198	C₇×C₄⋊C₄	C₂×D₄	1	2	yes	—
199	C₇×C₄⋊C₄	C₂×D₄	1	2	yes	—
200	C₇×C₄⋊C₄	C₂⁴	1	4	yes	—
201	C₇×C₄⋊C₄	C₂⁴	1	4	yes	—
202	C₇×C₄⋊C₄	C₂⁴	1	4	yes	—
203	C₇×C₂²⋊C₄	—	2	4	no	—
204	C₇×C₂²⋊C₄	—	2	8	no	—
205	C₇×C₂²⋊C₄	—	2	4	no	—
206	C₇×C₂²⋊C₄	—	2	4	no	—
207	C₇×C₂²⋊C₄	—	2	8	no	—
208	C₇×C₂²⋊C₄	—	2	8	no	—
209	C₇×C₂²⋊C₄	C₂×D₄	1	4	yes	—
210	C₇×C₂²⋊C₄	C₂×D₄	1	4	yes	—
211	C₄²×C₇	—	2	8	no	—
212	C₄²×C₇	—	2	4	no	—
213	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
214	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
215	C₄²×C₇	C₂⁴	1	1	yes	characteristic
216	C₄²×C₇	C₂⁴	1	1	yes	characteristic
217	C₂²×D₈	C₂²×C₇	1	1	yes	characteristic
218	C₂×#131	C₂²×C₇	1	2	yes	—
219	#559	—	2	4	no	—
220	#556	—	2	2	no	—
221	#555	C₂²×C₇	1	4	yes	complemented
222	#553	C₂²×C₇	1	8	yes	complemented
223	#550	—	2	2	no	—
224	#547	C₂²×C₇	1	4	yes	complemented
225	#538	—	2	4	no	—
226	#535	C₂²×C₇	1	4	yes	—
227	#533	C₂²×C₇	1	2	yes	—
228	#532	C₂²×C₇	1	1	yes	characteristic
229	#525	—	2	4	no	—
230	#520	C₂²×C₇	1	4	yes	—
231	C₂×C₄×D₄	C₂²×C₇	1	2	yes	—
232	#500	C₂²×C₇	1	1	yes	characteristic, complemented
233	#500	C₂²×C₇	1	1	yes	characteristic, complemented
234	#499	C₂²×C₇	1	2	yes	complemented
235	#499	C₂²×C₇	1	2	yes	complemented
236	#498	C₂²×C₇	1	2	yes	complemented
237	#482	C₂²×C₇	1	8	yes	complemented
238	#482	C₂²×C₇	1	8	yes	complemented
239	#481	C₂²×C₇	1	8	yes	complemented
240	#480	C₂²×C₇	1	8	yes	complemented
241	#478	C₂²×C₇	1	8	yes	complemented
242	#476	C₂²×C₇	1	4	yes	complemented
243	#474	C₂²×C₇	1	4	yes	complemented
244	#469	C₂²×C₇	1	8	yes	—
245	#466	C₂²×C₇	1	4	yes	—
246	#466	C₂²×C₇	1	4	yes	—
247	C₄×D₈	C₂²×C₇	1	2	yes	—
248	C₄×D₈	C₂²×C₇	1	2	yes	—
249	C₄×D₈	C₂²×C₇	1	2	yes	—
250	C₄×D₈	C₂²×C₇	1	2	yes	—
251	#436	C₂²×C₇	1	2	yes	complemented
252	#436	C₂²×C₇	1	2	yes	complemented
253	C₈×D₄	C₂²×C₇	1	2	yes	complemented
254	C₈×D₄	C₂²×C₇	1	2	yes	complemented
255	#428	C₂²×C₇	1	4	yes	—
256	#428	C₂²×C₇	1	8	yes	complemented
257	#427	C₂²×C₇	1	2	yes	—
258	#427	C₂²×C₇	1	2	yes	—
259	#427	C₂²×C₇	1	4	yes	complemented
260	C₂×#107	C₂²×C₇	1	1	yes	characteristic
261	C₂×#107	C₂²×C₇	1	4	yes	complemented
262	#423	C₂²×C₇	1	1	yes	characteristic
263	#418	C₂²×C₇	1	4	yes	—
264	C₂×#102	C₂²×C₇	1	2	yes	—
265	#408	C₂²×C₇	1	2	yes	—
266	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
267	C₂³×C₇	—	2	4	no	—
268	C₂³×C₇	—	4	4	no	—
269	C₂³×C₇	—	8	2	no	—
270	C₂³×C₇	—	8	2	no	—
271	C₂³×C₇	C₂²×D₄	1	4	yes	—
272	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
273	C₇×Q₈	—	2	4	no	—
274	C₇×Q₈	—	4	4	no	—
275	C₇×Q₈	—	8	2	no	—
276	C₇×D₄	—	2	8	no	—
277	C₇×D₄	—	2	4	no	—
278	C₇×D₄	—	2	4	no	—
279	C₇×D₄	—	2	4	no	—
280	C₇×D₄	—	2	4	no	—
281	C₇×D₄	—	4	2	no	—
282	C₇×D₄	—	4	4	no	—
283	C₇×D₄	—	8	8	no	—
284	C₇×D₄	—	8	8	no	—
285	C₂×C₄×C₇	—	2	2	no	—
286	C₂×C₄×C₇	—	2	4	no	—
287	C₂×C₄×C₇	—	2	4	no	—
288	C₂×C₄×C₇	—	2	4	no	—
289	C₂×C₄×C₇	—	2	4	no	—
290	C₂×C₄×C₇	—	2	4	no	—
291	C₂×C₄×C₇	—	4	1	no	—
292	C₂×C₄×C₇	—	4	8	no	—
293	C₂×C₄×C₇	—	8	4	no	—
294	C₂×C₄×C₇	—	8	2	no	—
295	C₂×C₄×C₇	—	8	8	no	—
296	C₂×C₄×C₇	C₂²×D₄	1	4	yes	—
297	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
298	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
299	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
300	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
301	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
302	C₂×C₄×C₇	C₂×#41	1	2	yes	—
303	C₂×C₄×C₇	2₋¹⁺⁴	1	2	yes	—
304	C₂×C₄×C₇	C₂⁵	1	1	yes	characteristic
305	C₇×C₈	—	2	4	no	—
306	C₇×C₈	—	2	1	no	—
307	C₇×C₈	—	2	1	no	—
308	C₇×C₈	C₂×#41	1	2	yes	—
309	C₇×C₈	C₂×#41	1	2	yes	—
310	C₂×#41	—	2	4	no	—
311	C₂×#41	C₂³×C₇	1	2	yes	—
312	C₂²×D₄	C₂³×C₇	1	2	yes	—
313	C₂³×C₄	—	2	2	no	—
314	C₂×D₈	—	2	4	no	—
315	C₂×D₈	C₂³×C₇	1	2	yes	—
316	C₂×D₈	C₂³×C₇	1	2	yes	—
317	#131	—	2	8	no	—
318	C₂×M₁₆	C₂³×C₇	1	2	yes	—
319	C₂×M₁₆	C₂³×C₇	1	4	yes	—
320	C₂²×C₈	C₂³×C₇	1	4	yes	—
321	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
322	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
323	C₄⋊Q₈	C₇×D₄	1	1	yes	characteristic
324	C₄⋊Q₈	C₂³×C₇	1	2	yes	—
325	C₄⋊Q₈	C₂³×C₇	1	2	yes	—
326	#127	—	2	4	no	—
327	#127	C₇×D₄	1	1	yes	characteristic
328	#126	—	2	8	no	—
329	#125	C₂³×C₇	1	4	yes	—
330	#124	—	2	4	no	—
331	#124	—	2	8	no	—
332	#124	C₇×D₄	1	2	yes	—
333	#123	—	2	8	no	—
334	#123	—	2	8	no	—
335	C₂²⋊Q₈	—	2	2	no	—
336	C₂²⋊Q₈	—	2	2	no	—
337	C₂²⋊Q₈	—	2	4	no	—
338	#121	—	2	4	no	—
339	#121	—	2	8	no	—
340	#121	C₇×D₄	1	2	yes	—
341	#121	C₇×D₄	1	2	yes	—
342	#121	C₂³×C₇	1	8	yes	—
343	#121	C₂³×C₇	1	8	yes	—
344	#120	—	2	8	no	—
345	C₄×Q₈	—	2	4	no	—
346	C₄×Q₈	—	2	2	no	—
347	C₄×Q₈	—	2	2	no	—
348	C₄×D₄	—	2	8	no	—
349	C₄×D₄	—	2	4	no	—
350	C₄×D₄	—	2	8	no	—
351	C₄×D₄	—	2	2	no	—
352	C₄×D₄	—	2	2	no	—
353	C₄×D₄	C₂³×C₇	1	4	yes	—
354	C₄×D₄	C₂³×C₇	1	4	yes	—
355	C₄×D₄	C₂³×C₇	1	2	yes	—
356	C₄×D₄	C₂³×C₇	1	2	yes	—
357	#117	—	2	4	no	—
358	#117	C₂³×C₇	1	4	yes	—
359	#117	C₂³×C₇	1	8	yes	—
360	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
361	C₂×C₄⋊C₄	C₂³×C₇	1	8	yes	—
362	C₂×C₂²⋊C₄	—	2	2	no	—
363	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
364	#107	C₂³×C₇	1	2	yes	—
365	#107	C₂³×C₇	1	2	yes	—
366	#107	C₂³×C₇	1	2	yes	—
367	#107	C₂³×C₇	1	8	yes	—
368	#107	C₂³×C₇	1	2	yes	—
369	#107	C₂³×C₇	1	2	yes	—
370	#107	C₂³×C₇	1	2	yes	—
371	#106	C₂³×C₇	1	4	yes	—
372	#106	C₂³×C₇	1	4	yes	—
373	#106	C₂³×C₇	1	8	yes	—
374	C₄⋊C₈	C₂³×C₇	1	2	yes	—
375	C₄⋊C₈	C₂³×C₇	1	2	yes	—
376	#102	C₂³×C₇	1	4	yes	—
377	#102	C₂³×C₇	1	4	yes	—
378	#102	C₂³×C₇	1	8	yes	—
379	#102	C₂³×C₇	1	8	yes	—
380	C₂²⋊C₈	C₂³×C₇	1	4	yes	—
381	C₂²⋊C₈	C₂³×C₇	1	4	yes	—
382	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
383	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
384	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
385	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
386	C₂²×C₇	—	2	1	no	—
387	C₂²×C₇	—	2	1	no	—
388	C₂²×C₇	—	2	4	no	—
389	C₂²×C₇	—	2	4	no	—
390	C₂²×C₇	—	2	4	no	—
391	C₂²×C₇	—	4	4	no	—
392	C₂²×C₇	—	4	4	no	—
393	C₂²×C₇	—	4	4	no	—
394	C₂²×C₇	—	8	8	no	—
395	C₂²×C₇	#547	1	2	yes	—
396	C₂²×C₇	C₂³×D₄	1	1	yes	characteristic
397	C₄×C₇	—	2	1	no	—
398	C₄×C₇	—	2	1	no	—
399	C₄×C₇	—	2	4	no	—
400	C₄×C₇	—	2	1	no	—
401	C₄×C₇	—	4	4	no	—
402	C₄×C₇	—	4	4	no	—
403	C₄×C₇	—	4	4	no	—
404	C₄×C₇	#547	1	2	yes	—
405	C₄×C₇	#576	1	2	yes	—
406	C₄×C₇	C₂²×#41	1	1	yes	characteristic
407	C₄×C₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
408	C₂⁴	—	2	2	no	—
409	#41	—	2	4	no	—
410	#41	—	2	4	no	—
411	#41	—	8	8	no	—
412	C₂×Q₈	—	2	2	no	—
413	C₂×Q₈	—	2	2	no	—
414	C₂×Q₈	C₂×C₇×D₄	1	2	yes	—
415	C₂×D₄	—	2	8	no	—
416	C₂×D₄	—	2	8	no	—
417	C₂×D₄	—	2	2	no	—
418	C₂×D₄	—	2	4	no	—
419	C₂×D₄	—	2	4	no	—
420	C₂×D₄	C₂×C₇×D₄	1	4	yes	—
421	C₂×D₄	C₂×C₇×D₄	1	2	yes	—
422	C₂×D₄	C₂×C₇×D₄	1	2	yes	—
423	C₂×D₄	C₂×C₇×D₄	1	2	yes	—
424	C₂×D₄	C₂⁴×C₇	1	4	yes	—
425	C₂²×C₄	—	2	2	no	—
426	C₂²×C₄	—	2	8	no	—
427	C₂²×C₄	—	2	4	no	—
428	C₂²×C₄	—	2	4	no	—
429	C₂²×C₄	—	8	4	no	—
430	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
431	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
432	C₂²×C₄	C₂⁴×C₇	1	4	yes	—
433	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
434	D₈	—	2	4	no	—
435	D₈	—	2	4	no	—
436	M₁₆	—	2	4	no	—
437	M₁₆	—	2	8	no	—
438	C₂×C₈	—	2	2	no	—
439	C₂×C₈	—	2	8	no	—
440	C₂×C₈	C₇×#41	1	4	yes	complemented
441	C₂×C₈	C₂⁴×C₇	1	1	yes	characteristic
442	C₂×C₈	C₂⁴×C₇	1	1	yes	characteristic
443	C₂×C₈	C₂⁴×C₇	1	4	yes	—
444	C₂×C₈	C₂⁴×C₇	1	1	yes	characteristic
445	C₂×C₈	C₂⁴×C₇	1	1	yes	characteristic
446	C₄⋊C₄	—	2	2	no	—
447	C₄⋊C₄	—	2	8	no	—
448	C₄⋊C₄	—	2	2	no	—
449	C₄⋊C₄	—	2	4	no	—
450	C₄⋊C₄	—	2	4	no	—
451	C₄⋊C₄	—	2	2	no	—
452	C₄⋊C₄	C₂×C₇×D₄	1	2	yes	—
453	C₄⋊C₄	C₂×C₇×D₄	1	2	yes	—
454	C₄⋊C₄	C₂⁴×C₇	1	4	yes	—
455	C₄⋊C₄	C₂⁴×C₇	1	4	yes	—
456	C₄⋊C₄	C₂⁴×C₇	1	4	yes	—
457	C₂²⋊C₄	—	2	4	no	—
458	C₂²⋊C₄	—	2	8	no	—
459	C₂²⋊C₄	—	2	4	no	—
460	C₂²⋊C₄	—	2	4	no	—
461	C₂²⋊C₄	—	2	8	no	—
462	C₂²⋊C₄	—	2	8	no	—
463	C₂²⋊C₄	C₂×C₇×D₄	1	4	yes	—
464	C₂²⋊C₄	C₂×C₇×D₄	1	4	yes	—
465	C₄²	—	2	8	no	—
466	C₄²	—	2	4	no	—
467	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
468	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
469	C₄²	C₂⁴×C₇	1	1	yes	characteristic
470	C₄²	C₂⁴×C₇	1	1	yes	characteristic
471	C₂×C₇	—	2	2	no	—
472	C₂×C₇	—	4	4	no	—
473	C₂×C₇	—	8	4	no	—
474	C₂×C₇	C₂×#547	1	1	yes	characteristic
475	C₂×C₇	C₂×#576	1	1	yes	characteristic
476	C₂×C₇	#3584	1	1	yes	characteristic
477	C₂³	—	2	4	no	—
478	C₂³	—	4	4	no	—
479	C₂³	—	8	2	no	—
480	C₂³	—	8	2	no	—
481	C₂³	C₂²×C₇×D₄	1	4	yes	—
482	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
483	Q₈	—	2	4	no	—
484	Q₈	—	4	4	no	—
485	Q₈	—	8	2	no	—
486	D₄	—	2	8	no	—
487	D₄	—	2	4	no	—
488	D₄	—	2	4	no	—
489	D₄	—	2	4	no	—
490	D₄	—	2	4	no	—
491	D₄	—	4	2	no	—
492	D₄	—	4	4	no	—
493	D₄	—	8	8	no	—
494	D₄	—	8	8	no	—
495	C₂×C₄	—	2	4	no	—
496	C₂×C₄	—	2	4	no	—
497	C₂×C₄	—	2	2	no	—
498	C₂×C₄	—	2	4	no	—
499	C₂×C₄	—	2	4	no	—
500	C₂×C₄	—	2	4	no	—
501	C₂×C₄	—	4	1	no	—
502	C₂×C₄	—	4	8	no	—
503	C₂×C₄	—	8	4	no	—
504	C₂×C₄	—	8	2	no	—
505	C₂×C₄	—	8	8	no	—
506	C₂×C₄	C₂²×C₇×D₄	1	4	yes	—
507	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
508	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
509	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
510	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
511	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
512	C₂×C₄	C₂×C₇×#41	1	2	yes	—
513	C₂×C₄	C₇×2₋¹⁺⁴	1	2	yes	—
514	C₂×C₄	C₂⁵×C₇	1	1	yes	characteristic
515	C₈	—	2	4	no	—
516	C₈	—	2	1	no	—
517	C₈	—	2	1	no	—
518	C₈	C₂×C₇×#41	1	2	yes	—
519	C₈	C₂×C₇×#41	1	2	yes	—
520	C₇	#61284	1	1	yes	characteristic, direct product factor, Sylow
521	C₂²	—	2	1	no	—
522	C₂²	—	2	1	no	—
523	C₂²	—	2	4	no	—
524	C₂²	—	2	4	no	—
525	C₂²	—	2	4	no	—
526	C₂²	—	4	4	no	—
527	C₂²	—	4	4	no	—
528	C₂²	—	4	4	no	—
529	C₂²	—	8	8	no	—
530	C₂²	C₇×#547	1	2	yes	—
531	C₂²	C₂³×C₇×D₄	1	1	yes	characteristic
532	C₄	—	2	1	no	—
533	C₄	—	2	1	no	—
534	C₄	—	2	1	no	—
535	C₄	—	2	4	no	—
536	C₄	—	4	4	no	—
537	C₄	—	4	4	no	—
538	C₄	—	4	4	no	—
539	C₄	C₇×#547	1	2	yes	—
540	C₄	C₇×#576	1	2	yes	—
541	C₄	C₂²×C₇×#41	1	1	yes	characteristic
542	C₄	C₂×C₇×2₋¹⁺⁴	1	1	yes	characteristic
543	C₂	—	2	2	no	—
544	C₂	—	4	4	no	—
545	C₂	—	8	4	no	—
546	C₂	C₂×C₇×#547	1	1	yes	characteristic
547	C₂	C₂×C₇×#576	1	1	yes	characteristic
548	C₂	C₇×#3584	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×#3506	C₂	semidirect
2. C₇×#3355	C₂	semidirect
3. C₇×#3354	C₂	semidirect
4. C₇×#3341	C₂	semidirect
5. C₇×#3339	C₂	semidirect
6. C₇×#3334	C₂	semidirect
7. C₇×#3160	C₂	semidirect
8. C₇×#3096	C₂	semidirect
9. C₇×#3077	C₂	semidirect
10. C₇×#3058	C₂	semidirect
11. C₂×C₄×C₇×D₈	C₂	semidirect
12. C₇×#2931	C₂	semidirect
13. C₇×#2913	C₂	semidirect
18. C₇×#555	C₂²	semidirect
19. C₇×#553	C₂²	semidirect
21. C₇×#547	C₂²	semidirect
29. C₇×#500	C₂²	semidirect
30. C₇×#500	C₂²	semidirect
31. C₇×#499	C₂²	semidirect
32. C₇×#499	C₂²	semidirect
33. C₇×#498	C₂²	semidirect
34. C₇×#482	C₂²	semidirect
35. C₇×#482	C₂²	semidirect
36. C₇×#481	C₂²	semidirect
37. C₇×#480	C₂²	semidirect
38. C₇×#478	C₂²	semidirect
39. C₇×#476	C₂²	semidirect
40. C₇×#474	C₂²	semidirect
48. C₇×#436	C₂²	semidirect
49. C₇×#436	C₂²	semidirect
50. C₇×C₈×D₄	C₂²	semidirect
51. C₇×C₈×D₄	C₂²	semidirect
53. C₇×#428	C₂²	semidirect
56. C₇×#427	C₂²	semidirect
58. C₂×C₇×#107	C₂²	semidirect
64. #61284	C₇	direct (N × H)
141. #3506	C₂×C₇	semidirect
142. #3355	C₂×C₇	semidirect
143. #3354	C₂×C₇	semidirect
144. #3341	C₂×C₇	semidirect
145. #3339	C₂×C₇	semidirect
146. #3334	C₂×C₇	semidirect
147. #3160	C₂×C₇	semidirect
148. #3096	C₂×C₇	semidirect
149. #3077	C₂×C₇	semidirect
150. #3058	C₂×C₇	semidirect
151. C₂×C₄×D₈	C₂×C₇	semidirect
152. #2931	C₂×C₇	semidirect
153. #2913	C₂×C₇	semidirect
186. C₂×C₇×C₈	#41	semidirect
221. #555	C₂²×C₇	semidirect
222. #553	C₂²×C₇	semidirect
224. #547	C₂²×C₇	semidirect
232. #500	C₂²×C₇	semidirect
233. #500	C₂²×C₇	semidirect
234. #499	C₂²×C₇	semidirect
235. #499	C₂²×C₇	semidirect
236. #498	C₂²×C₇	semidirect
237. #482	C₂²×C₇	semidirect
238. #482	C₂²×C₇	semidirect
239. #481	C₂²×C₇	semidirect
240. #480	C₂²×C₇	semidirect
241. #478	C₂²×C₇	semidirect
242. #476	C₂²×C₇	semidirect
243. #474	C₂²×C₇	semidirect
251. #436	C₂²×C₇	semidirect
252. #436	C₂²×C₇	semidirect
253. C₈×D₄	C₂²×C₇	semidirect
254. C₈×D₄	C₂²×C₇	semidirect
256. #428	C₂²×C₇	semidirect
259. #427	C₂²×C₇	semidirect
261. C₂×#107	C₂²×C₇	semidirect
440. C₂×C₈	C₇×#41	semidirect
520. C₇	#61284	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×#3355	2
C₇×#3354	2
C₇×#3341	2
C₇×#3339	2
C₇×#3334	2
C₇×#3160	2
C₇×#3096	2
C₇×#3077	2
C₇×#3058	2
C₂×C₄×C₇×D₈	2
C₇×#2931	2
C₇×#2913	2
#61284	7

Subgroups — C7×#61284 #13238500

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#61284 #13238500