Subgroups — #13268508

Order 1792.

398 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11726814	C₂	1	4	yes	maximal, complemented
2	#11725816	C₂	1	2	yes	maximal, complemented
3	C₇×C₈×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₁₁₂	—	2	8	no	—
5	#89467	—	2	4	no	—
6	#89467	C₂²	1	4	yes	—
7	C₄×C₇×C₁₆	C₂²	1	2	yes	—
8	C₇×C₈²	C₂²	1	1	yes	characteristic
9	#12515	—	7	1	no	maximal, Sylow
10	C₂×D₅₆	—	2	8	no	—
11	C₂×D₅₆	—	4	8	no	—
12	#6437	—	2	4	no	—
13	C₂×C₇×C₁₆	D₄	1	4	yes	—
14	C₄×C₇×C₈	D₄	1	2	yes	—
15	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
16	D₁₁₂	—	4	16	no	—
17	#2246	—	7	4	no	—
18	#1712	—	7	2	no	—
19	C₈×C₁₆	D₇	1	1	yes	characteristic, complemented
20	C₂×D₂₈	—	4	4	no	—
21	C₂×D₂₈	—	4	8	no	—
22	C₂×C₇×C₈	D₈	1	4	yes	—
23	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
24	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
25	D₅₆	—	4	16	no	—
26	D₅₆	—	8	16	no	—
27	C₇×C₁₆	D₈	1	8	yes	complemented
28	C₂×D₁₆	—	14	8	no	—
29	#493	—	7	4	no	—
30	#493	—	14	4	no	—
31	C₄×C₁₆	C₂×D₇	1	2	yes	—
32	C₈²	C₂×D₇	1	1	yes	characteristic
33	C₂²×D₇	—	8	4	no	—
34	C₂×C₄×C₇	#127	1	1	yes	characteristic
35	C₂×C₄×C₇	C₂×D₈	1	2	yes	—
36	D₂₈	—	8	8	no	—
37	D₂₈	—	8	16	no	—
38	C₇×C₈	D₁₆	1	8	yes	complemented
39	C₇×C₈	C₂×D₈	1	4	yes	—
40	C₂×D₈	—	14	8	no	—
41	C₂×D₈	—	28	8	no	—
42	#127	—	14	4	no	—
43	D₁₆	—	28	16	no	—
44	C₂×C₁₆	D₂₈	1	4	yes	—
45	C₄×C₈	D₂₈	1	2	yes	—
46	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
47	C₂²×C₇	#493	1	1	yes	characteristic
48	C₂×D₇	—	16	4	no	—
49	C₂×D₇	—	16	8	no	—
50	C₄×C₇	#493	1	2	yes	—
51	C₄×C₇	C₂×D₁₆	1	4	yes	—
52	C₂×D₄	—	28	4	no	—
53	C₂×D₄	—	28	8	no	—
54	D₈	—	28	16	no	—
55	D₈	—	56	16	no	—
56	C₂×C₈	D₅₆	1	4	yes	—
57	C₂×C₈	C₂×D₂₈	1	2	yes	—
58	C₄²	C₂×D₂₈	1	1	yes	characteristic
59	C₁₆	D₅₆	1	8	yes	complemented
60	C₂×C₇	#1712	1	1	yes	characteristic
61	C₂×C₇	#2246	1	2	yes	—
62	D₇	—	32	4	no	—
63	C₂³	—	56	4	no	—
64	D₄	—	56	16	no	—
65	D₄	—	56	8	no	—
66	C₂×C₄	#6437	1	1	yes	characteristic
67	C₂×C₄	C₂×D₅₆	1	2	yes	—
68	C₈	D₁₁₂	1	8	yes	complemented
69	C₈	C₂×D₅₆	1	4	yes	—
70	C₇	#12515	1	1	yes	characteristic, complemented, Sylow
71	C₂²	—	112	8	no	—
72	C₂²	—	112	4	no	—
73	C₂²	#89467	1	1	yes	characteristic
74	C₄	#89467	1	2	yes	—
75	C₄	C₂×D₁₁₂	1	4	yes	—
76	C₂	—	224	4	no	—
77	C₂	#11725816	1	1	yes	characteristic
78	C₂	#11726814	1	2	yes	—

	subgroup	G/H
center	C₂²	#89467
commutator, cosocle	C₄×C₇×C₈	C₂³
Frattini	C₄×C₈	C₂²×D₇
Fitting	C₇×C₈×C₁₆	C₂
socle	C₂²×C₇	#493
hypercenter	C₈×C₁₆	D₇
hypocenter	C₇	#12515
LCS	G ▷ C₄×C₇×C₈ ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ C₈×C₁₆

Complements

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

N	H ≅ G/N	type
1. #11726814	C₂	semidirect
2. #11725816	C₂	semidirect
3. C₇×C₈×C₁₆	C₂	semidirect
19. C₈×C₁₆	D₇	semidirect
27. C₇×C₁₆	D₈	semidirect
38. C₇×C₈	D₁₆	semidirect
59. C₁₆	D₅₆	semidirect
68. C₈	D₁₁₂	semidirect
70. C₇	#12515	semidirect

Maximal subgroups

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

subgroup	index
#11726814	2
#11725816	2
C₇×C₈×C₁₆	2
#12515	7