Subgroups — #11727505

Order 896.

240 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₇×C₈	C₂	1	1	yes	maximal, characteristic
2	#89704	C₂	1	1	yes	maximal, characteristic
3	C₂×#6377	C₂	1	1	yes	maximal, characteristic
4	#89345	C₂	1	4	yes	maximal
5	C₂²×C₇×C₈	C₂²	1	2	yes	—
6	C₂×C₄²×C₇	C₂²	1	1	yes	characteristic
7	C₂×C₄×Dic₇	C₂²	1	2	yes	—
8	C₂²×C₇⋊C₈	C₂²	1	2	yes	—
9	C₄×C₇×C₈	C₄	1	4	yes	complemented
10	C₇⋊#95	C₄	1	4	yes	—
11	#6377	C₄	1	4	yes	complemented
12	#1746	—	7	1	no	maximal, Sylow
13	C₂²×C₄×C₇	C₂×C₄	1	2	yes	—
14	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
15	C₂²×Dic₇	C₂×C₄	1	4	yes	—
16	C₂×C₇×C₈	—	2	4	no	—
17	C₂×C₇×C₈	C₂×C₄	1	4	yes	—
18	C₄²×C₇	C₂×C₄	1	2	yes	—
19	C₄²×C₇	C₂×C₄	1	2	yes	—
20	C₄×Dic₇	—	2	4	no	—
21	C₂×C₇⋊C₈	—	2	4	no	—
22	C₂×C₇⋊C₈	C₂×C₄	1	4	yes	—
23	C₂×#97	—	7	1	no	—
24	C₂×C₄×C₈	D₇	1	1	yes	characteristic
25	#376	—	7	1	no	—
26	#336	—	7	4	no	—
27	C₂³×C₇	C₂²×C₄	1	1	yes	characteristic
28	C₂×C₄×C₇	—	2	4	no	—
29	C₂×C₄×C₇	C₄²	1	4	yes	—
30	C₂×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
31	C₂×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
32	C₂×C₄×C₇	#41	1	4	yes	—
33	C₂×Dic₇	—	2	4	no	—
34	C₂×Dic₇	—	2	8	no	—
35	C₇×C₈	—	2	4	no	—
36	C₇⋊C₈	—	2	4	no	—
37	C₂²×C₈	—	7	2	no	—
38	C₂²×C₈	C₂×D₇	1	2	yes	—
39	C₂×C₄²	—	7	2	no	—
40	C₂×C₄²	C₂×D₇	1	1	yes	characteristic
41	#97	—	7	4	no	—
42	C₄×C₈	Dic₇	1	4	yes	complemented
43	#95	—	7	4	no	—
44	C₂²×C₇	C₂×C₄²	1	1	yes	characteristic
45	C₂²×C₇	#117	1	2	yes	—
46	C₂²×C₇	#131	1	4	yes	—
47	C₄×C₇	—	2	4	no	—
48	C₄×C₇	#117	1	2	yes	—
49	C₄×C₇	#117	1	2	yes	—
50	Dic₇	—	4	4	no	—
51	C₂²×C₄	—	7	4	no	—
52	C₂²×C₄	C₄×D₇	1	2	yes	—
53	C₂²×C₄	C₂²×D₇	1	1	yes	characteristic
54	C₂×C₈	—	2	4	no	—
55	C₂×C₈	—	7	4	no	—
56	C₂×C₈	—	14	4	no	—
57	C₂×C₈	C₂×Dic₇	1	4	yes	—
58	C₄²	—	14	4	no	—
59	C₄²	C₄×D₇	1	2	yes	—
60	C₄²	C₂×Dic₇	1	2	yes	—
61	C₂×C₇	#376	1	1	yes	characteristic
62	C₂×C₇	#405	1	2	yes	—
63	C₂×C₇	#433	1	4	yes	—
64	C₂³	C₂×C₄×D₇	1	1	yes	characteristic
65	C₂×C₄	—	2	4	no	—
66	C₂×C₄	—	14	4	no	—
67	C₂×C₄	—	14	8	no	—
68	C₂×C₄	C₄×Dic₇	1	4	yes	—
69	C₂×C₄	C₂×C₄×D₇	1	1	yes	characteristic
70	C₂×C₄	#1157	1	4	yes	—
71	C₂×C₄	C₂²×Dic₇	1	1	yes	characteristic
72	C₈	—	2	4	no	—
73	C₈	—	14	4	no	—
74	C₇	#1746	1	1	yes	characteristic, complemented, Sylow
75	C₂²	#6435	1	2	yes	—
76	C₂²	#6464	1	4	yes	—
77	C₂²	C₂×C₄×Dic₇	1	1	yes	characteristic
78	C₄	—	2	4	no	—
79	C₄	—	28	4	no	—
80	C₄	#6435	1	2	yes	—
81	C₄	#6489	1	2	yes	—
82	C₂	#89462	1	4	yes	—
83	C₂	#89704	1	1	yes	characteristic
84	C₂	#89873	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₇
commutator	C₂²×C₇	C₂×C₄²
Fitting	C₂×C₄×C₇×C₈	C₂
socle, prime radical	C₂³×C₇	C₂²×C₄
cosocle	C₂²×C₄×C₇	C₂³
hypercenter	C₂×C₄×C₈	D₇
hypocenter	C₇	#1746
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
9. C₄×C₇×C₈	C₄	semidirect
11. #6377	C₄	semidirect
42. C₄×C₈	Dic₇	semidirect
74. C₇	#1746	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₇×C₈	2
#89704	2
C₂×#6377	2
#89345	2
#1746	7