Subgroups — #27163747

Order 12736.

82 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×D₇₉₆	C₂	1	1	yes	maximal, characteristic
2	C₄⋊(C₁₉₉⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×C₈×C₁₉₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₇₉₆	C₄	1	1	yes	characteristic
5	C₂×C₄×D₁₉₉	—	2	1	no	—
6	C₂×C₁₉₉⋊C₈	—	2	1	no	—
7	C₄⋊Dic₁₉₉	C₄	1	1	yes	characteristic
8	(C₂×D₁₉₉)⋊C₄	—	2	1	no	—
9	C₂×C₈×C₁₉₉	—	2	1	no	—
10	C₄²×C₁₉₉	C₂²	1	1	yes	characteristic
11	C₂²×D₁₉₉	—	2	1	no	—
12	D₇₉₆	—	2	1	no	—
13	D₇₉₆	C₈	1	2	yes	complemented
14	C₂×Dic₁₉₉	—	2	1	no	—
15	C₄×D₁₉₉	—	4	1	no	—
16	C₄×D₁₉₉	—	4	1	no	—
17	C₁₉₉⋊C₈	—	4	1	no	—
18	C₂×C₄×C₁₉₉	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×C₁₉₉	D₄	1	1	yes	characteristic
20	C₂×C₄×C₁₉₉	D₄	1	1	yes	characteristic
21	C₈×C₁₉₉	—	2	2	no	—
22	C₂²×C₁₉₉	C₂²⋊C₄	1	1	yes	characteristic
23	C₂×D₁₉₉	—	2	2	no	—
24	C₂×D₁₉₉	—	4	1	no	—
25	C₂×D₁₉₉	—	4	1	no	—
26	C₄×C₁₉₉	—	2	1	no	—
27	C₄×C₁₉₉	C₂×C₈	1	1	yes	characteristic
28	C₄×C₁₉₉	M₁₆	1	1	yes	characteristic
29	C₄×C₁₉₉	D₈	1	1	yes	characteristic
30	C₄×C₁₉₉	SD₁₆	1	1	yes	characteristic
31	Dic₁₉₉	—	4	1	no	—
32	C₂×C₁₉₉	C₂²⋊C₈	1	1	yes	characteristic
33	C₂×C₁₉₉	#102	1	1	yes	characteristic
34	C₂×C₁₉₉	#104	1	1	yes	characteristic
35	D₁₉₉	—	4	2	no	—
36	C₁₉₉	D₄⋊C₈	1	1	yes	characteristic, complemented, Sylow
37	D₄⋊C₈	—	199	1	no	maximal, Sylow
38	C₄×D₄	—	199	1	no	—
39	C₄⋊C₈	—	199	1	no	—
40	C₄×C₈	D₁₉₉	1	1	yes	characteristic, complemented
41	C₂×D₄	—	199	1	no	—
42	C₂²×C₄	—	398	1	no	—
43	C₂×C₈	—	2	1	no	—
44	C₂×C₈	—	398	1	no	—
45	C₄⋊C₄	—	199	1	no	—
46	C₂²⋊C₄	—	398	1	no	—
47	C₄²	C₂×D₁₉₉	1	1	yes	characteristic
48	C₂³	—	398	1	no	—
49	D₄	—	398	1	no	—
50	D₄	—	199	2	no	—
51	C₂×C₄	—	796	1	no	—
52	C₂×C₄	—	796	1	no	—
53	C₂×C₄	—	398	1	no	—
54	C₂×C₄	C₄×D₁₉₉	1	1	yes	characteristic
55	C₂×C₄	C₂²⋊D₁₉₉	1	1	yes	characteristic
56	C₂×C₄	D₇₉₆	1	1	yes	characteristic
57	C₈	—	2	2	no	—
58	C₈	—	796	1	no	—
59	C₂²	—	796	1	no	—
60	C₂²	—	796	1	no	—
61	C₂²	—	398	2	no	—
62	C₂²	(C₂×D₁₉₉)⋊C₄	1	1	yes	characteristic
63	C₄	—	2	1	no	—
64	C₄	—	796	1	no	—
65	C₄	C₈×D₁₉₉	1	1	yes	characteristic
66	C₄	#15691869	1	1	yes	characteristic
67	C₄	D₁₅₉₂	1	1	yes	characteristic
68	C₄	#15691875	1	1	yes	characteristic
69	C₂	—	796	2	no	—
70	C₂	(C₂×D₁₉₉)⋊C₈	1	1	yes	characteristic
71	C₂	#20263877	1	1	yes	characteristic
72	C₂	#20263883	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₇₉₆
commutator	C₄×C₁₉₉	C₂×C₈
Frattini	C₄²	C₂×D₁₉₉
Fitting	C₄×C₈×C₁₉₉	C₂
socle	C₂²×C₁₉₉	C₂²⋊C₄
prime radical	C₂×D₇₉₆	C₄
cosocle	C₄²×C₁₉₉	C₂²
hypercenter	C₄×C₈	D₁₉₉
hypocenter	C₁₉₉	D₄⋊C₈
LCS	G ▷ C₄×C₁₉₉ ▷ C₂×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
2. C₄⋊(C₁₉₉⋊C₈)	C₂	semidirect
3. C₄×C₈×C₁₉₉	C₂	semidirect
13. D₇₉₆	C₈	semidirect
36. C₁₉₉	D₄⋊C₈	semidirect
40. 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
C₄×D₇₉₆	2
C₄⋊(C₁₉₉⋊C₈)	2
C₄×C₈×C₁₉₉	2
D₄⋊C₈	199