Subgroups — #11740358

Order 896.

420 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#525	C₂	1	1	yes	maximal, characteristic
2	C₂×#6489	C₂	1	1	yes	maximal, characteristic, complemented
3	#90005	C₂	1	2	yes	maximal, complemented
4	C₂×C₂²⋊(C₇⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
5	#89817	C₂	1	2	yes	maximal, complemented
6	#89816	C₂	1	1	yes	maximal, characteristic, complemented
7	#89815	C₂	1	1	yes	maximal, characteristic, complemented
8	#89807	C₂	1	2	yes	maximal, complemented
9	#89806	C₂	1	1	yes	maximal, characteristic, complemented
10	#89805	C₂	1	1	yes	maximal, characteristic, complemented
11	#89770	C₂	1	2	yes	maximal, complemented
12	C₂×C₇×#41	C₂²	1	1	yes	characteristic
13	C₂³×C₄×C₇	C₂²	1	1	yes	characteristic
14	C₇×#123	—	2	1	no	—
15	C₇×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
16	C₇×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
17	C₇×#121	C₂²	1	1	yes	characteristic
18	C₇×#121	C₂²	1	1	yes	characteristic
19	C₇×#120	—	2	1	no	—
20	C₄×C₇×D₄	—	2	1	no	—
21	C₄×C₇×D₄	—	2	1	no	—
22	C₇×#117	C₂²	1	1	yes	characteristic
23	C₂×C₂²⋊Dic₇	—	2	1	no	—
24	#6489	—	2	1	no	—
25	#6489	—	2	1	no	—
26	#6489	C₂²	1	2	yes	—
27	#6489	C₂²	1	2	yes	—
28	C₂×C₄⋊Dic₇	C₂²	1	1	yes	characteristic
29	C₂×C₄⋊Dic₇	C₂²	1	1	yes	characteristic
30	C₂×C₄×Dic₇	—	2	1	no	—
31	C₂²×C₇⋊C₈	C₂²	1	2	yes	—
32	#6409	C₂²	1	2	yes	complemented
33	#6409	C₂²	1	2	yes	complemented
34	#6406	C₂²	1	2	yes	complemented
35	#6406	C₂²	1	2	yes	complemented
36	C₂²⋊(C₇⋊C₈)	C₂²	1	1	yes	characteristic
37	C₂²⋊(C₇⋊C₈)	C₂²	1	1	yes	characteristic
38	C₂²⋊(C₇⋊C₈)	C₂²	1	2	yes	—
39	#6382	C₂²	1	2	yes	complemented
40	#6382	C₂²	1	2	yes	complemented
41	#6381	C₂²	1	2	yes	complemented
42	#6381	C₂²	1	2	yes	complemented
43	#3091	—	7	1	no	maximal, Sylow
44	C₂⁴×C₇	D₄	1	1	yes	characteristic
45	C₇×#41	—	2	4	no	—
46	C₂×C₇×Q₈	C₂³	1	1	yes	characteristic
47	C₂×C₇×D₄	—	2	1	no	—
48	C₂×C₇×D₄	—	2	1	no	—
49	C₂×C₇×D₄	—	2	1	no	—
50	C₂×C₇×D₄	C₂³	1	1	yes	characteristic
51	C₂²×C₄×C₇	—	2	4	no	—
52	C₂²×C₄×C₇	—	2	1	no	—
53	C₂²×C₄×C₇	D₄	1	1	yes	characteristic
54	C₂²×C₄×C₇	D₄	1	1	yes	characteristic
55	C₂²×C₄×C₇	D₄	1	1	yes	characteristic
56	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
57	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
58	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
59	C₂²×Dic₇	—	2	1	no	—
60	C₂²×Dic₇	—	2	1	no	—
61	C₇×C₄⋊C₄	—	2	1	no	—
62	C₇×C₄⋊C₄	—	2	1	no	—
63	C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
64	C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
65	C₇×C₂²⋊C₄	—	2	1	no	—
66	C₇×C₂²⋊C₄	—	2	1	no	—
67	C₇×C₂²⋊C₄	—	2	1	no	—
68	C₇×C₂²⋊C₄	—	2	1	no	—
69	C₇×C₂²⋊C₄	—	2	1	no	—
70	C₄²×C₇	—	2	1	no	—
71	C₂²⋊Dic₇	—	2	2	no	—
72	C₂²⋊Dic₇	—	2	2	no	—
73	C₄⋊Dic₇	—	2	1	no	—
74	C₄⋊Dic₇	—	2	1	no	—
75	C₄⋊Dic₇	C₂³	1	2	yes	—
76	C₄⋊Dic₇	C₂³	1	2	yes	—
77	C₄×Dic₇	—	2	2	no	—
78	C₄×Dic₇	—	2	2	no	—
79	C₂×C₇⋊C₈	—	2	4	no	—
80	C₂×C₇⋊C₈	C₂³	1	2	yes	—
81	C₂×C₇⋊C₈	C₂³	1	2	yes	—
82	#525	D₇	1	1	yes	characteristic
83	C₂×#117	—	7	1	no	—
84	#485	—	7	2	no	—
85	#484	—	7	1	no	—
86	#483	—	7	1	no	—
87	#482	—	7	2	no	—
88	#481	—	7	1	no	—
89	#480	—	7	1	no	—
90	#427	—	7	2	no	—
91	#416	—	7	2	no	—
92	C₂×C₂²⋊C₈	—	7	1	no	—
93	C₂³×C₇	—	2	2	no	—
94	C₂³×C₇	—	2	2	no	—
95	C₂³×C₇	—	2	1	no	—
96	C₂³×C₇	—	4	1	no	—
97	C₂³×C₇	C₂×D₄	1	1	yes	characteristic
98	C₂³×C₇	C₂×D₄	1	1	yes	characteristic
99	C₂³×C₇	C₂×D₄	1	1	yes	characteristic
100	C₇×Q₈	—	2	2	no	—
101	C₇×D₄	—	2	2	no	—
102	C₇×D₄	—	4	2	no	—
103	C₇×D₄	—	4	2	no	—
104	C₇×D₄	—	4	2	no	—
105	C₂×C₄×C₇	—	2	2	no	—
106	C₂×C₄×C₇	—	2	2	no	—
107	C₂×C₄×C₇	—	2	2	no	—
108	C₂×C₄×C₇	—	2	2	no	—
109	C₂×C₄×C₇	—	2	2	no	—
110	C₂×C₄×C₇	—	2	1	no	—
111	C₂×C₄×C₇	—	2	1	no	—
112	C₂×C₄×C₇	—	2	1	no	—
113	C₂×C₄×C₇	—	4	2	no	—
114	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
115	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
116	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
117	C₂×C₄×C₇	#41	1	4	yes	—
118	C₂×C₄×C₇	C₂⁴	1	1	yes	characteristic
119	C₂×Dic₇	—	2	2	no	—
120	C₂×Dic₇	—	2	2	no	—
121	C₂×Dic₇	—	4	2	no	—
122	C₂×Dic₇	—	4	2	no	—
123	C₇⋊C₈	—	2	2	no	—
124	C₇⋊C₈	—	2	2	no	—
125	C₂×#41	C₂×D₇	1	1	yes	characteristic
126	C₂³×C₄	C₂×D₇	1	1	yes	characteristic
127	C₂²×C₈	—	7	2	no	—
128	#123	—	2	1	no	—
129	C₂²⋊Q₈	C₂×D₇	1	1	yes	characteristic
130	C₂²⋊Q₈	C₂×D₇	1	1	yes	characteristic
131	#121	C₂×D₇	1	1	yes	characteristic
132	#121	C₂×D₇	1	1	yes	characteristic
133	#120	—	2	1	no	—
134	C₄×D₄	—	2	1	no	—
135	C₄×D₄	—	2	1	no	—
136	#117	—	7	2	no	—
137	#117	—	7	2	no	—
138	#117	—	14	1	no	—
139	#117	—	14	1	no	—
140	#117	C₂×D₇	1	1	yes	characteristic
141	C₂×C₄⋊C₄	—	7	1	no	—
142	C₂×C₄⋊C₄	—	7	1	no	—
143	C₂×C₂²⋊C₄	—	14	1	no	—
144	C₂×C₄²	—	14	1	no	—
145	#107	—	7	2	no	—
146	#107	—	7	2	no	—
147	#106	—	7	2	no	—
148	#106	—	7	2	no	—
149	#103	—	7	2	no	—
150	#103	—	7	2	no	—
151	#102	—	7	2	no	—
152	#102	—	7	2	no	—
153	C₂²⋊C₈	—	7	1	no	—
154	C₂²⋊C₈	—	7	1	no	—
155	C₂²⋊C₈	—	7	2	no	—
156	C₂²×C₇	—	2	1	no	—
157	C₂²×C₇	—	2	1	no	—
158	C₂²×C₇	—	2	1	no	—
159	C₂²×C₇	—	2	1	no	—
160	C₂²×C₇	—	2	1	no	—
161	C₂²×C₇	—	2	1	no	—
162	C₂²×C₇	—	2	1	no	—
163	C₂²×C₇	—	4	4	no	—
164	C₂²×C₇	—	4	1	no	—
165	C₂²×C₇	—	4	1	no	—
166	C₂²×C₇	—	4	1	no	—
167	C₂²×C₇	#123	1	2	yes	—
168	C₂²×C₇	#135	1	2	yes	—
169	C₂²×C₇	C₂²×D₄	1	1	yes	characteristic
170	C₄×C₇	—	2	1	no	—
171	C₄×C₇	—	2	1	no	—
172	C₄×C₇	—	4	1	no	—
173	C₄×C₇	—	4	1	no	—
174	C₄×C₇	—	4	1	no	—
175	C₄×C₇	#123	1	2	yes	—
176	C₄×C₇	C₂×#41	1	2	yes	—
177	Dic₇	—	4	2	no	—
178	Dic₇	—	4	2	no	—
179	C₂⁴	C₂²⋊D₇	1	1	yes	characteristic
180	#41	—	2	4	no	—
181	C₂×Q₈	C₂²×D₇	1	1	yes	characteristic
182	C₂×D₄	—	2	1	no	—
183	C₂×D₄	—	2	1	no	—
184	C₂×D₄	—	2	1	no	—
185	C₂×D₄	C₂²×D₇	1	1	yes	characteristic
186	C₂²×C₄	—	2	4	no	—
187	C₂²×C₄	—	2	1	no	—
188	C₂²×C₄	—	14	1	no	—
189	C₂²×C₄	—	14	1	no	—
190	C₂²×C₄	C₂²⋊D₇	1	1	yes	characteristic
191	C₂²×C₄	C₂²⋊D₇	1	1	yes	characteristic
192	C₂²×C₄	C₂²⋊D₇	1	1	yes	characteristic
193	C₂²×C₄	C₂²×D₇	1	1	yes	characteristic
194	C₂²×C₄	C₂²×D₇	1	1	yes	characteristic
195	C₂²×C₄	C₂²×D₇	1	1	yes	characteristic
196	C₂×C₈	—	7	2	no	—
197	C₂×C₈	—	7	2	no	—
198	C₂×C₈	—	14	4	no	—
199	C₄⋊C₄	—	2	1	no	—
200	C₄⋊C₄	—	2	1	no	—
201	C₄⋊C₄	—	7	2	no	—
202	C₄⋊C₄	—	7	2	no	—
203	C₄⋊C₄	—	14	1	no	—
204	C₄⋊C₄	—	14	1	no	—
205	C₄⋊C₄	C₂²×D₇	1	1	yes	characteristic
206	C₄⋊C₄	C₂²×D₇	1	1	yes	characteristic
207	C₂²⋊C₄	—	2	1	no	—
208	C₂²⋊C₄	—	2	1	no	—
209	C₂²⋊C₄	—	2	1	no	—
210	C₂²⋊C₄	—	2	1	no	—
211	C₂²⋊C₄	—	2	1	no	—
212	C₂²⋊C₄	—	14	2	no	—
213	C₂²⋊C₄	—	14	2	no	—
214	C₄²	—	2	1	no	—
215	C₄²	—	14	2	no	—
216	C₄²	—	14	2	no	—
217	C₂×C₇	—	2	2	no	—
218	C₂×C₇	—	2	2	no	—
219	C₂×C₇	—	4	1	no	—
220	C₂×C₇	—	8	1	no	—
221	C₂×C₇	C₂×#123	1	1	yes	characteristic
222	C₂×C₇	C₂×#135	1	1	yes	characteristic
223	C₂×C₇	#575	1	1	yes	characteristic
224	C₂³	—	2	2	no	—
225	C₂³	—	2	2	no	—
226	C₂³	—	2	1	no	—
227	C₂³	—	4	1	no	—
228	C₂³	C₂×C₂²⋊D₇	1	1	yes	characteristic
229	C₂³	C₂×C₂²⋊D₇	1	1	yes	characteristic
230	C₂³	C₂×C₂²⋊D₇	1	1	yes	characteristic
231	Q₈	—	2	2	no	—
232	D₄	—	2	2	no	—
233	D₄	—	4	2	no	—
234	D₄	—	4	2	no	—
235	D₄	—	4	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	4	2	no	—
245	C₂×C₄	—	14	2	no	—
246	C₂×C₄	—	14	2	no	—
247	C₂×C₄	—	28	2	no	—
248	C₂×C₄	—	28	2	no	—
249	C₂×C₄	#1159	1	4	yes	—
250	C₂×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
251	C₂×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
252	C₂×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
253	C₂×C₄	C₂³×D₇	1	1	yes	characteristic
254	C₈	—	14	2	no	—
255	C₈	—	14	2	no	—
256	C₇	#3091	1	1	yes	characteristic, complemented, Sylow
257	C₂²	—	2	1	no	—
258	C₂²	—	2	1	no	—
259	C₂²	—	2	1	no	—
260	C₂²	—	2	1	no	—
261	C₂²	—	2	1	no	—
262	C₂²	—	2	1	no	—
263	C₂²	—	2	1	no	—
264	C₂²	—	4	4	no	—
265	C₂²	—	4	1	no	—
266	C₂²	—	4	1	no	—
267	C₂²	—	4	1	no	—
268	C₂²	#6498	1	2	yes	—
269	C₂²	#6513	1	2	yes	—
270	C₂²	C₂²×C₂²⋊D₇	1	1	yes	characteristic
271	C₄	—	2	1	no	—
272	C₄	—	2	1	no	—
273	C₄	—	4	1	no	—
274	C₄	—	4	1	no	—
275	C₄	—	4	1	no	—
276	C₄	—	28	2	no	—
277	C₄	—	28	2	no	—
278	C₄	#6498	1	2	yes	—
279	C₄	C₂×#1159	1	2	yes	—
280	C₂	—	2	2	no	—
281	C₂	—	2	2	no	—
282	C₂	—	4	1	no	—
283	C₂	—	8	1	no	—
284	C₂	C₂×#6498	1	1	yes	characteristic
285	C₂	C₂×#6513	1	1	yes	characteristic
286	C₂	#90513	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₇
commutator, cosocle	C₂×C₄×C₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₇
Fitting, prime radical	C₇×#525	C₂
socle	C₂²×C₇	C₂²×D₄
hypercenter	#525	D₇
hypocenter	C₇	#3091
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #525

Complements

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

N	H ≅ G/N	type
2. C₂×#6489	C₂	semidirect
3. #90005	C₂	semidirect
4. C₂×C₂²⋊(C₇⋊C₈)	C₂	semidirect
5. #89817	C₂	semidirect
6. #89816	C₂	semidirect
7. #89815	C₂	semidirect
8. #89807	C₂	semidirect
9. #89806	C₂	semidirect
10. #89805	C₂	semidirect
11. #89770	C₂	semidirect
32. #6409	C₂²	semidirect
33. #6409	C₂²	semidirect
34. #6406	C₂²	semidirect
35. #6406	C₂²	semidirect
39. #6382	C₂²	semidirect
40. #6382	C₂²	semidirect
41. #6381	C₂²	semidirect
42. #6381	C₂²	semidirect
256. C₇	#3091	semidirect

Maximal subgroups

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

subgroup	index
C₇×#525	2
C₂×#6489	2
#90005	2
C₂×C₂²⋊(C₇⋊C₈)	2
#89817	2
#89816	2
#89815	2
#89807	2
#89806	2
#89805	2
#89770	2
#3091	7