Subgroups — C₈×D₇₉₆ #27164085

Order 12736.

134 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×D₁₉₉	C₂	1	2	yes	maximal, complemented
2	C₄×D₇₉₆	C₂	1	1	yes	maximal, characteristic
3	C₄⋊(C₁₉₉⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
4	(C₂×D₁₉₉)⋊C₈	C₂	1	2	yes	maximal, complemented
5	C₄×C₈×C₁₉₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₇₉₆	C₄	1	1	yes	characteristic
7	C₂×C₄×D₁₉₉	C₂²	1	2	yes	—
8	C₂×C₁₉₉⋊C₈	C₂²	1	2	yes	—
9	C₈×D₁₉₉	—	2	4	no	—
10	C₄⋊Dic₁₉₉	C₄	1	1	yes	characteristic
11	(C₂×D₁₉₉)⋊C₄	C₄	1	2	yes	—
12	C₂×C₈×C₁₉₉	C₂²	1	1	yes	characteristic
13	C₂×C₈×C₁₉₉	C₂²	1	1	yes	characteristic
14	C₄²×C₁₉₉	C₂²	1	1	yes	characteristic
15	C₂²×D₁₉₉	C₂×C₄	1	2	yes	—
16	D₇₉₆	C₈	1	4	yes	direct product factor
17	C₂×Dic₁₉₉	C₂×C₄	1	2	yes	—
18	C₄×D₁₉₉	—	2	2	no	—
19	C₄×D₁₉₉	—	2	2	no	—
20	C₁₉₉⋊C₈	—	2	2	no	—
21	C₂×C₄×C₁₉₉	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×C₁₉₉	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×C₁₉₉	C₂³	1	1	yes	characteristic
24	C₈×C₁₉₉	—	2	1	no	—
25	C₈×C₁₉₉	D₄	1	2	yes	complemented
26	C₂²×C₁₉₉	C₂²×C₄	1	1	yes	characteristic
27	C₂×D₁₉₉	—	2	2	no	—
28	C₂×D₁₉₉	—	2	2	no	—
29	C₂×D₁₉₉	C₂×C₈	1	4	yes	complemented
30	C₄×C₁₉₉	—	2	1	no	—
31	C₄×C₁₉₉	C₂×C₈	1	2	yes	complemented
32	C₄×C₁₉₉	C₂×D₄	1	1	yes	characteristic
33	C₄×C₁₉₉	#41	1	1	yes	characteristic
34	Dic₁₉₉	—	2	2	no	—
35	C₂×C₁₉₉	C₄×D₄	1	1	yes	characteristic
36	C₂×C₁₉₉	C₂²×C₈	1	1	yes	characteristic
37	C₂×C₁₉₉	#131	1	1	yes	characteristic
38	D₁₉₉	—	2	4	no	—
39	C₁₉₉	C₈×D₄	1	1	yes	characteristic, complemented, Sylow
40	C₈×D₄	—	199	1	no	maximal, Sylow
41	C₂²×C₈	—	199	2	no	—
42	C₄×D₄	—	199	1	no	—
43	C₄⋊C₈	—	199	1	no	—
44	C₂²⋊C₈	—	199	2	no	—
45	C₄×C₈	D₁₉₉	1	1	yes	characteristic, complemented
46	C₂×D₄	—	199	1	no	—
47	C₂²×C₄	—	199	2	no	—
48	C₂×C₈	—	398	4	no	—
49	C₂×C₈	—	199	2	no	—
50	C₂×C₈	C₂×D₁₉₉	1	1	yes	characteristic
51	C₂×C₈	C₂×D₁₉₉	1	1	yes	characteristic
52	C₄⋊C₄	—	199	1	no	—
53	C₂²⋊C₄	—	199	2	no	—
54	C₄²	C₂×D₁₉₉	1	1	yes	characteristic
55	C₂³	—	199	2	no	—
56	D₄	—	199	4	no	—
57	C₂×C₄	—	398	2	no	—
58	C₂×C₄	—	398	2	no	—
59	C₂×C₄	—	199	2	no	—
60	C₂×C₄	C₄×D₁₉₉	1	1	yes	characteristic
61	C₂×C₄	C₄×D₁₉₉	1	1	yes	characteristic
62	C₂×C₄	C₂²×D₁₉₉	1	1	yes	characteristic
63	C₈	—	2	1	no	—
64	C₈	—	398	2	no	—
65	C₈	D₇₉₆	1	2	yes	direct product factor
66	C₂²	—	398	2	no	—
67	C₂²	—	398	2	no	—
68	C₂²	—	199	4	no	—
69	C₂²	C₂×C₄×D₁₉₉	1	1	yes	characteristic
70	C₄	—	2	1	no	—
71	C₄	—	398	2	no	—
72	C₄	C₈×D₁₉₉	1	2	yes	complemented
73	C₄	C₂×D₇₉₆	1	1	yes	characteristic
74	C₄	#15691886	1	1	yes	characteristic
75	C₂	—	398	4	no	—
76	C₂	C₄×D₇₉₆	1	1	yes	characteristic
77	C₂	C₂×C₈×D₁₉₉	1	1	yes	characteristic
78	C₂	#20263961	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×D₁₉₉
commutator	C₂×C₁₉₉	C₂²×C₈
Frattini	C₂×C₄	C₂²×D₁₉₉
Fitting	C₄×C₈×C₁₉₉	C₂
socle	C₂²×C₁₉₉	C₂²×C₄
prime radical	C₂×D₇₉₆	C₄
cosocle	C₂×C₄×C₁₉₉	C₂³
hypercenter	C₄×C₈	D₁₉₉
hypocenter	C₁₉₉	C₈×D₄
LCS	G ▷ C₂×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
1. C₂×C₈×D₁₉₉	C₂	semidirect
3. C₄⋊(C₁₉₉⋊C₈)	C₂	semidirect
4. (C₂×D₁₉₉)⋊C₈	C₂	semidirect
5. C₄×C₈×C₁₉₉	C₂	semidirect
16. D₇₉₆	C₈	direct (N × H)
25. C₈×C₁₉₉	D₄	semidirect
29. C₂×D₁₉₉	C₂×C₈	semidirect
31. C₄×C₁₉₉	C₂×C₈	semidirect
39. C₁₉₉	C₈×D₄	semidirect
45. C₄×C₈	D₁₉₉	semidirect
65. C₈	D₇₉₆	direct (N × H)
72. C₄	C₈×D₁₉₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×D₁₉₉	2
C₄×D₇₉₆	2
C₄⋊(C₁₉₉⋊C₈)	2
(C₂×D₁₉₉)⋊C₈	2
C₄×C₈×C₁₉₉	2
C₈×D₄	199

Subgroups — C8×D796 #27164085

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈×D₇₉₆ #27164085