Subgroups — C₇×#2491 #11741800

Order 896.

632 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₇×#124	C₂	1	2	yes	maximal, complemented
3	C₂×C₇×#121	C₂	1	1	yes	maximal, characteristic
4	C₇×#394	C₂	1	2	yes	maximal, complemented
5	C₇×#393	C₂	1	2	yes	maximal, complemented
6	C₇×#390	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇×#390	C₂	1	1	yes	maximal, characteristic, complemented
8	C₇×#388	C₂	1	2	yes	maximal, complemented
9	C₇×#380	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×C₇×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×C₇×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×C₇×Q₈	C₂²	1	2	yes	—
13	C₂²×C₇×D₄	C₂²	1	1	yes	characteristic
14	C₂²×C₇×D₄	C₂²	1	2	yes	—
15	C₂³×C₄×C₇	C₂²	1	1	yes	characteristic
16	C₇×C₄⋊Q₈	—	2	4	no	—
17	C₇×#124	—	2	8	no	—
18	C₇×#121	—	2	4	no	—
19	C₂×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
20	C₂×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
21	C₂×C₇×C₄⋊C₄	C₂²	1	2	yes	—
22	C₂×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
23	C₂×C₇×C₂²⋊C₄	C₂²	1	1	yes	characteristic
24	C₂×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
25	C₂×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
26	C₂×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
27	C₂×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
28	C₂×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
29	C₂×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
30	C₂×C₇×C₂²⋊C₄	C₂²	1	1	yes	characteristic
31	C₂×C₄²×C₇	C₂²	1	1	yes	characteristic
32	C₂×C₄²×C₇	C₂²	1	1	yes	characteristic
33	C₂×C₄²×C₇	C₂²	1	2	yes	—
34	C₇×#95	C₂²	1	1	yes	characteristic
35	C₇×#95	C₂²	1	1	yes	characteristic
36	C₇×#95	C₂²	1	2	yes	—
37	C₇×#95	C₂²	1	2	yes	—
38	#2491	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
39	C₂⁴×C₇	C₂³	1	2	yes	—
40	C₂⁴×C₇	C₂³	1	1	yes	characteristic
41	C₂×C₇×Q₈	—	2	4	no	—
42	C₂×C₇×Q₈	—	2	4	no	—
43	C₂×C₇×Q₈	—	4	2	no	—
44	C₂×C₇×D₄	—	2	4	no	—
45	C₂×C₇×D₄	—	2	4	no	—
46	C₂×C₇×D₄	—	2	2	no	—
47	C₂×C₇×D₄	—	2	2	no	—
48	C₂×C₇×D₄	—	2	2	no	—
49	C₂×C₇×D₄	—	4	2	no	—
50	C₂²×C₄×C₇	—	2	2	no	—
51	C₂²×C₄×C₇	—	2	2	no	—
52	C₂²×C₄×C₇	—	2	2	no	—
53	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
54	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
55	C₂²×C₄×C₇	C₂³	1	2	yes	—
56	C₂²×C₄×C₇	C₂³	1	2	yes	—
57	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
58	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
59	C₂²×C₄×C₇	C₂³	1	2	yes	—
60	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
61	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
62	C₇×C₄⋊C₄	—	2	4	no	—
63	C₇×C₄⋊C₄	—	2	2	no	—
64	C₇×C₄⋊C₄	—	2	2	no	—
65	C₇×C₄⋊C₄	D₄	1	4	yes	—
66	C₇×C₂²⋊C₄	—	2	4	no	—
67	C₇×C₂²⋊C₄	—	2	4	no	—
68	C₇×C₂²⋊C₄	—	2	4	no	—
69	C₇×C₂²⋊C₄	—	2	4	no	—
70	C₇×C₂²⋊C₄	—	2	2	no	—
71	C₇×C₂²⋊C₄	—	2	4	no	—
72	C₇×C₂²⋊C₄	—	4	2	no	—
73	C₇×C₂²⋊C₄	D₄	1	4	yes	complemented
74	C₄²×C₇	—	2	4	no	—
75	C₄²×C₇	—	2	2	no	—
76	C₄²×C₇	—	2	2	no	—
77	C₂×C₄⋊Q₈	C₂×C₇	1	1	yes	characteristic, complemented
78	C₂×#124	C₂×C₇	1	2	yes	complemented
79	C₂×#121	C₂×C₇	1	1	yes	characteristic
80	#394	C₂×C₇	1	2	yes	complemented
81	#393	C₂×C₇	1	2	yes	complemented
82	#390	C₂×C₇	1	1	yes	characteristic, complemented
83	#390	C₂×C₇	1	1	yes	characteristic, complemented
84	#388	C₂×C₇	1	2	yes	complemented
85	#380	C₂×C₇	1	1	yes	characteristic, complemented
86	C₄×C₄⋊C₄	C₂×C₇	1	1	yes	characteristic, complemented
87	C₄×C₂²⋊C₄	C₂×C₇	1	1	yes	characteristic, complemented
88	C₂³×C₇	—	2	1	no	—
89	C₂³×C₇	—	2	1	no	—
90	C₂³×C₇	—	2	1	no	—
91	C₂³×C₇	—	2	1	no	—
92	C₂³×C₇	—	2	1	no	—
93	C₂³×C₇	—	2	1	no	—
94	C₂³×C₇	—	2	2	no	—
95	C₂³×C₇	—	2	2	no	—
96	C₂³×C₇	—	2	2	no	—
97	C₂³×C₇	—	2	2	no	—
98	C₂³×C₇	—	2	2	no	—
99	C₂³×C₇	—	2	2	no	—
100	C₂³×C₇	—	2	2	no	—
101	C₂³×C₇	C₂×D₄	1	2	yes	—
102	C₂³×C₇	C₂⁴	1	1	yes	characteristic
103	C₇×Q₈	—	4	8	no	—
104	C₇×D₄	—	4	8	no	—
105	C₇×D₄	—	4	4	no	—
106	C₂×C₄×C₇	—	2	1	no	—
107	C₂×C₄×C₇	—	2	2	no	—
108	C₂×C₄×C₇	—	2	1	no	—
109	C₂×C₄×C₇	—	2	1	no	—
110	C₂×C₄×C₇	—	2	2	no	—
111	C₂×C₄×C₇	—	2	1	no	—
112	C₂×C₄×C₇	—	2	1	no	—
113	C₂×C₄×C₇	—	2	1	no	—
114	C₂×C₄×C₇	—	2	2	no	—
115	C₂×C₄×C₇	—	2	1	no	—
116	C₂×C₄×C₇	—	2	2	no	—
117	C₂×C₄×C₇	—	2	2	no	—
118	C₂×C₄×C₇	—	2	1	no	—
119	C₂×C₄×C₇	—	2	2	no	—
120	C₂×C₄×C₇	—	2	1	no	—
121	C₂×C₄×C₇	—	2	2	no	—
122	C₂×C₄×C₇	—	2	1	no	—
123	C₂×C₄×C₇	—	2	1	no	—
124	C₂×C₄×C₇	—	2	1	no	—
125	C₂×C₄×C₇	—	2	2	no	—
126	C₂×C₄×C₇	—	4	4	no	—
127	C₂×C₄×C₇	C₂×D₄	1	2	yes	—
128	C₂×C₄×C₇	C₂×D₄	1	2	yes	—
129	C₂×C₄×C₇	C₂×D₄	1	4	yes	—
130	C₂×C₄×C₇	C₂×D₄	1	2	yes	—
131	C₂×C₄×C₇	#41	1	2	yes	—
132	C₂×C₄×C₇	#41	1	2	yes	—
133	C₂×C₄×C₇	#41	1	2	yes	—
134	C₂²×Q₈	C₂²×C₇	1	2	yes	—
135	C₂²×D₄	C₂²×C₇	1	1	yes	characteristic
136	C₂²×D₄	C₂²×C₇	1	2	yes	—
137	C₂³×C₄	C₂²×C₇	1	1	yes	characteristic
138	C₄⋊Q₈	—	2	4	no	—
139	#124	—	2	8	no	—
140	#121	—	2	4	no	—
141	C₂×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
142	C₂×C₄⋊C₄	C₂²×C₇	1	2	yes	—
143	C₂×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
144	C₂×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
145	C₂×C₂²⋊C₄	C₂²×C₇	1	2	yes	—
146	C₂×C₂²⋊C₄	C₂²×C₇	1	2	yes	—
147	C₂×C₂²⋊C₄	C₂²×C₇	1	2	yes	—
148	C₂×C₂²⋊C₄	C₂²×C₇	1	2	yes	—
149	C₂×C₂²⋊C₄	C₂²×C₇	1	1	yes	characteristic
150	C₂×C₂²⋊C₄	C₂²×C₇	1	2	yes	—
151	C₂×C₂²⋊C₄	C₂²×C₇	1	2	yes	—
152	C₂×C₂²⋊C₄	C₂²×C₇	1	1	yes	characteristic
153	C₂×C₄²	C₂²×C₇	1	2	yes	—
154	C₂×C₄²	C₂²×C₇	1	1	yes	characteristic
155	C₂×C₄²	C₂²×C₇	1	1	yes	characteristic
156	#95	C₂²×C₇	1	1	yes	characteristic
157	#95	C₂²×C₇	1	1	yes	characteristic
158	#95	C₂²×C₇	1	2	yes	—
159	#95	C₂²×C₇	1	2	yes	—
160	C₂²×C₇	—	2	2	no	—
161	C₂²×C₇	—	2	2	no	—
162	C₂²×C₇	—	2	2	no	—
163	C₂²×C₇	—	4	1	no	—
164	C₂²×C₇	—	4	1	no	—
165	C₂²×C₇	—	4	1	no	—
166	C₂²×C₇	—	4	1	no	—
167	C₂²×C₇	—	4	2	no	—
168	C₂²×C₇	—	4	2	no	—
169	C₂²×C₇	—	4	2	no	—
170	C₂²×C₇	—	4	2	no	—
171	C₂²×C₇	—	4	2	no	—
172	C₂²×C₇	—	4	2	no	—
173	C₂²×C₇	—	4	2	no	—
174	C₂²×C₇	C₂²×D₄	1	1	yes	characteristic
175	C₂²×C₇	C₂²×D₄	1	1	yes	characteristic
176	C₂²×C₇	C₂×#41	1	1	yes	characteristic
177	C₂²×C₇	C₂×#41	1	1	yes	characteristic
178	C₂²×C₇	C₂×#41	1	1	yes	characteristic
179	C₂²×C₇	H(5,𝔽₂)	1	1	yes	characteristic
180	C₂²×C₇	2₋¹⁺⁴	1	1	yes	characteristic
181	C₄×C₇	—	2	4	no	—
182	C₄×C₇	—	2	2	no	—
183	C₄×C₇	—	2	2	no	—
184	C₄×C₇	—	2	2	no	—
185	C₄×C₇	—	4	1	no	—
186	C₄×C₇	—	4	2	no	—
187	C₄×C₇	—	4	2	no	—
188	C₄×C₇	—	4	1	no	—
189	C₄×C₇	#124	1	4	yes	complemented
190	C₂⁴	C₂³×C₇	1	2	yes	—
191	C₂⁴	C₂³×C₇	1	1	yes	characteristic
192	C₂×Q₈	—	2	4	no	—
193	C₂×Q₈	—	2	4	no	—
194	C₂×Q₈	—	4	2	no	—
195	C₂×D₄	—	2	4	no	—
196	C₂×D₄	—	2	4	no	—
197	C₂×D₄	—	2	2	no	—
198	C₂×D₄	—	2	2	no	—
199	C₂×D₄	—	2	2	no	—
200	C₂×D₄	—	4	2	no	—
201	C₂²×C₄	—	2	2	no	—
202	C₂²×C₄	—	2	2	no	—
203	C₂²×C₄	—	2	2	no	—
204	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
205	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
206	C₂²×C₄	C₂³×C₇	1	2	yes	—
207	C₂²×C₄	C₂³×C₇	1	2	yes	—
208	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
209	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
210	C₂²×C₄	C₂³×C₇	1	2	yes	—
211	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
212	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
213	C₄⋊C₄	—	2	4	no	—
214	C₄⋊C₄	—	2	2	no	—
215	C₄⋊C₄	—	2	2	no	—
216	C₄⋊C₄	C₇×D₄	1	4	yes	—
217	C₂²⋊C₄	—	2	4	no	—
218	C₂²⋊C₄	—	2	4	no	—
219	C₂²⋊C₄	—	2	4	no	—
220	C₂²⋊C₄	—	2	4	no	—
221	C₂²⋊C₄	—	2	2	no	—
222	C₂²⋊C₄	—	2	4	no	—
223	C₂²⋊C₄	—	4	2	no	—
224	C₂²⋊C₄	C₇×D₄	1	4	yes	complemented
225	C₄²	—	2	4	no	—
226	C₄²	—	2	2	no	—
227	C₄²	—	2	2	no	—
228	C₂×C₇	—	4	2	no	—
229	C₂×C₇	—	8	2	no	—
230	C₂×C₇	C₂×#124	1	1	yes	characteristic
231	C₂×C₇	#532	1	1	yes	characteristic
232	C₂×C₇	#542	1	1	yes	characteristic
233	C₂×C₇	D₄²	1	1	yes	characteristic
234	C₂×C₇	#547	1	1	yes	characteristic
235	C₂×C₇	#548	1	1	yes	characteristic
236	C₂×C₇	#555	1	1	yes	characteristic
237	C₂³	—	2	1	no	—
238	C₂³	—	2	1	no	—
239	C₂³	—	2	1	no	—
240	C₂³	—	2	1	no	—
241	C₂³	—	2	1	no	—
242	C₂³	—	2	1	no	—
243	C₂³	—	2	2	no	—
244	C₂³	—	2	2	no	—
245	C₂³	—	2	2	no	—
246	C₂³	—	2	2	no	—
247	C₂³	—	2	2	no	—
248	C₂³	—	2	2	no	—
249	C₂³	—	2	2	no	—
250	C₂³	C₂×C₇×D₄	1	2	yes	—
251	C₂³	C₂⁴×C₇	1	1	yes	characteristic
252	Q₈	—	4	8	no	—
253	D₄	—	4	8	no	—
254	D₄	—	4	4	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	4	4	no	—
276	C₂×C₄	C₂×C₇×D₄	1	2	yes	—
277	C₂×C₄	C₂×C₇×D₄	1	2	yes	—
278	C₂×C₄	C₂×C₇×D₄	1	4	yes	—
279	C₂×C₄	C₂×C₇×D₄	1	2	yes	—
280	C₂×C₄	C₇×#41	1	2	yes	—
281	C₂×C₄	C₇×#41	1	2	yes	—
282	C₂×C₄	C₇×#41	1	2	yes	—
283	C₇	#2491	1	1	yes	characteristic, direct product factor, Sylow
284	C₂²	—	2	2	no	—
285	C₂²	—	2	2	no	—
286	C₂²	—	2	2	no	—
287	C₂²	—	4	1	no	—
288	C₂²	—	4	1	no	—
289	C₂²	—	4	1	no	—
290	C₂²	—	4	1	no	—
291	C₂²	—	4	2	no	—
292	C₂²	—	4	2	no	—
293	C₂²	—	4	2	no	—
294	C₂²	—	4	2	no	—
295	C₂²	—	4	2	no	—
296	C₂²	—	4	2	no	—
297	C₂²	—	4	2	no	—
298	C₂²	C₂²×C₇×D₄	1	1	yes	characteristic
299	C₂²	C₂²×C₇×D₄	1	1	yes	characteristic
300	C₂²	C₂×C₇×#41	1	1	yes	characteristic
301	C₂²	C₂×C₇×#41	1	1	yes	characteristic
302	C₂²	C₂×C₇×#41	1	1	yes	characteristic
303	C₂²	C₇×H(5,𝔽₂)	1	1	yes	characteristic
304	C₂²	C₇×2₋¹⁺⁴	1	1	yes	characteristic
305	C₄	—	2	4	no	—
306	C₄	—	2	2	no	—
307	C₄	—	2	2	no	—
308	C₄	—	2	2	no	—
309	C₄	—	4	1	no	—
310	C₄	—	4	2	no	—
311	C₄	—	4	2	no	—
312	C₄	—	4	1	no	—
313	C₄	C₇×#124	1	4	yes	complemented
314	C₂	—	4	2	no	—
315	C₂	—	8	2	no	—
316	C₂	C₂×C₇×#124	1	1	yes	characteristic
317	C₂	C₇×#532	1	1	yes	characteristic
318	C₂	C₇×#542	1	1	yes	characteristic
319	C₂	C₇×D₄²	1	1	yes	characteristic
320	C₂	C₇×#547	1	1	yes	characteristic
321	C₂	C₇×#548	1	1	yes	characteristic
322	C₂	C₇×#555	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₇	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₇
prime radical	C₂×C₇×#121	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₇×C₄⋊Q₈	C₂	semidirect
2. C₂×C₇×#124	C₂	semidirect
4. C₇×#394	C₂	semidirect
5. C₇×#393	C₂	semidirect
6. C₇×#390	C₂	semidirect
7. C₇×#390	C₂	semidirect
8. C₇×#388	C₂	semidirect
9. C₇×#380	C₂	semidirect
10. C₄×C₇×C₄⋊C₄	C₂	semidirect
11. C₄×C₇×C₂²⋊C₄	C₂	semidirect
38. #2491	C₇	direct (N × H)
73. C₇×C₂²⋊C₄	D₄	semidirect
77. C₂×C₄⋊Q₈	C₂×C₇	semidirect
78. C₂×#124	C₂×C₇	semidirect
80. #394	C₂×C₇	semidirect
81. #393	C₂×C₇	semidirect
82. #390	C₂×C₇	semidirect
83. #390	C₂×C₇	semidirect
84. #388	C₂×C₇	semidirect
85. #380	C₂×C₇	semidirect
86. C₄×C₄⋊C₄	C₂×C₇	semidirect
87. C₄×C₂²⋊C₄	C₂×C₇	semidirect
189. C₄×C₇	#124	semidirect
224. C₂²⋊C₄	C₇×D₄	semidirect
283. C₇	#2491	direct (N × H)
313. C₄	C₇×#124	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₇×C₄⋊Q₈	2
C₂×C₇×#124	2
C₂×C₇×#121	2
C₇×#394	2
C₇×#393	2
C₇×#390	2
C₇×#388	2
C₇×#380	2
C₄×C₇×C₄⋊C₄	2
C₄×C₇×C₂²⋊C₄	2
#2491	7

Subgroups — C7×#2491 #11741800

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#2491 #11741800