Subgroups — #11727099

Order 896.

152 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₇×C₁₆	C₂	1	1	yes	maximal, characteristic
2	#89652	C₂	1	2	yes	maximal
3	C₇×#367	C₂	1	1	yes	maximal, characteristic, complemented
4	#89300	C₂	1	1	yes	maximal, characteristic, complemented
5	#89284	C₂	1	2	yes	maximal, complemented
6	C₂×C₈×D₇	C₂²	1	1	yes	characteristic
7	#6455	—	2	2	no	—
8	C₂×C₇×C₁₆	C₂²	1	1	yes	characteristic
9	C₇×#107	C₂²	1	2	yes	—
10	#6392	C₂²	1	2	yes	—
11	C₂×C₇⋊C₁₆	C₂²	1	1	yes	characteristic
12	#6382	—	2	2	no	—
13	D₇×C₁₆	C₄	1	4	yes	complemented
14	#2158	—	7	1	no	maximal, Sylow
15	C₂×C₄×D₇	D₄	1	1	yes	characteristic
16	C₂×C₇×C₈	C₂³	1	1	yes	characteristic
17	C₇×C₄⋊C₄	—	2	2	no	—
18	(C₂×D₇)⋊C₄	—	4	2	no	—
19	C₄⋊Dic₇	—	2	2	no	—
20	C₄×Dic₇	—	4	2	no	—
21	C₂×C₇⋊C₈	D₄	1	1	yes	characteristic
22	C₈×D₇	C₂×C₄	1	2	yes	—
23	C₈×D₇	Q₈	1	2	yes	—
24	C₇×C₁₆	C₂×C₄	1	2	yes	—
25	C₇⋊C₁₆	C₂×C₄	1	2	yes	—
26	C₂²×C₁₆	—	7	1	no	—
27	#427	—	7	2	no	—
28	#367	—	7	1	no	—
29	#367	D₇	1	1	yes	characteristic, complemented
30	#366	—	7	2	no	—
31	C₂²×D₇	D₈	1	1	yes	characteristic
32	C₂×C₄×C₇	—	4	2	no	—
33	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
34	C₂×Dic₇	—	4	2	no	—
35	C₂×Dic₇	D₈	1	1	yes	characteristic
36	C₄×D₇	C₄⋊C₄	1	2	yes	—
37	C₄×D₇	Q₁₆	1	2	yes	—
38	C₇×C₈	C₂²×C₄	1	1	yes	characteristic
39	C₇×C₈	C₂×Q₈	1	1	yes	characteristic
40	C₇⋊C₈	C₄⋊C₄	1	2	yes	—
41	C₂²×C₈	—	7	1	no	—
42	#117	—	14	2	no	—
43	C₂×C₁₆	—	7	1	no	—
44	C₂×C₁₆	—	7	4	no	—
45	C₂×C₁₆	C₂×D₇	1	1	yes	characteristic
46	#107	—	7	2	no	—
47	#107	C₂×D₇	1	2	yes	—
48	#106	—	14	2	no	—
49	C₂²×C₇	C₂×D₈	1	1	yes	characteristic
50	C₂×D₇	—	2	2	no	—
51	C₂×D₇	#107	1	2	yes	—
52	C₄×C₇	—	8	2	no	—
53	C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₄×C₇	C₂×Q₁₆	1	1	yes	characteristic
55	Dic₇	—	8	2	no	—
56	Dic₇	#107	1	2	yes	—
57	C₂²×C₄	—	7	1	no	—
58	C₂×C₈	—	7	1	no	—
59	C₂×C₈	—	7	2	no	—
60	C₂×C₈	—	7	2	no	—
61	C₂×C₈	C₂²×D₇	1	1	yes	characteristic
62	C₄⋊C₄	—	2	2	no	—
63	C₄⋊C₄	—	14	2	no	—
64	C₂²⋊C₄	—	28	2	no	—
65	C₄²	—	28	2	no	—
66	C₁₆	—	7	2	no	—
67	C₁₆	C₄×D₇	1	2	yes	—
68	C₂×C₇	C₂×#107	1	1	yes	characteristic
69	C₂×C₇	#508	1	2	yes	—
70	D₇	—	2	2	no	—
71	C₂³	—	7	1	no	—
72	C₂×C₄	—	4	2	no	—
73	C₂×C₄	—	7	1	no	—
74	C₂×C₄	—	7	2	no	—
75	C₂×C₄	—	7	2	no	—
76	C₂×C₄	—	28	2	no	—
77	C₂×C₄	D₄×D₇	1	1	yes	characteristic
78	C₈	—	7	2	no	—
79	C₈	C₂×C₄×D₇	1	1	yes	characteristic
80	C₈	Q₈×D₇	1	1	yes	characteristic
81	C₇	#2158	1	1	yes	characteristic, complemented, Sylow
82	C₂²	—	7	2	no	—
83	C₂²	—	14	2	no	—
84	C₂²	D₇×D₈	1	1	yes	characteristic
85	C₄	—	7	2	no	—
86	C₄	—	8	2	no	—
87	C₄	—	56	2	no	—
88	C₄	D₇×C₄⋊C₄	1	1	yes	characteristic
89	C₄	D₇×Q₁₆	1	1	yes	characteristic
90	C₂	—	14	2	no	—
91	C₂	D₇×#107	1	1	yes	characteristic
92	C₂	#89688	1	2	yes	—

	subgroup	G/H
center	C₂²	D₇×D₈
commutator	C₇×C₈	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₇
Fitting	C₇×#367	C₂
socle	C₂²×C₇	C₂×D₈
prime radical	C₂²×D₇	D₈
cosocle	C₂×C₇×C₈	C₂³
hypercenter	#367	D₇
hypocenter	C₇	#2158
LCS	G ▷ C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #367

Complements

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

N	H ≅ G/N	type
3. C₇×#367	C₂	semidirect
4. #89300	C₂	semidirect
5. #89284	C₂	semidirect
13. D₇×C₁₆	C₄	semidirect
29. #367	D₇	semidirect
81. C₇	#2158	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₇×C₁₆	2
#89652	2
C₇×#367	2
#89300	2
#89284	2
#2158	7