Subgroups — #27163835

Order 12736.

54 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₇₉₆	C₂	1	1	yes	maximal, characteristic
2	#20263891	C₂	1	1	yes	maximal, characteristic
3	M₃₂×C₁₉₉	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₃₉₈	—	2	1	no	—
5	Dic₇₉₆	—	2	1	no	—
6	Dic₇₉₆	C₄	1	2	yes	—
7	#15691870	—	2	1	no	—
8	C₂×C₈×C₁₉₉	C₂²	1	1	yes	characteristic
9	C₁₆×C₁₉₉	—	2	1	no	—
10	Dic₃₉₈	—	2	2	no	—
11	Dic₃₉₈	—	4	1	no	—
12	C₂×Dic₁₉₉	—	4	1	no	—
13	C₁₉₉⋊C₈	—	4	1	no	—
14	C₂×C₄×C₁₉₉	D₄	1	1	yes	characteristic
15	C₈×C₁₉₉	C₂×C₄	1	1	yes	characteristic
16	C₈×C₁₉₉	D₄	1	1	yes	characteristic
17	C₂²×C₁₉₉	SD₁₆	1	1	yes	characteristic
18	C₄×C₁₉₉	C₂²⋊C₄	1	1	yes	characteristic
19	C₄×C₁₉₉	D₈	1	1	yes	characteristic
20	Dic₁₉₉	—	4	2	no	—
21	C₂×C₁₉₉	—	2	1	no	—
22	C₂×C₁₉₉	#102	1	1	yes	characteristic
23	C₁₉₉	#362	1	1	yes	characteristic, complemented, Sylow
24	#362	—	199	1	no	maximal, Sylow
25	C₂×Q₁₆	—	199	1	no	—
26	M₃₂	D₁₉₉	1	1	yes	characteristic
27	#108	—	199	1	no	—
28	C₂×Q₈	—	398	1	no	—
29	Q₁₆	—	398	1	no	—
30	Q₁₆	—	199	2	no	—
31	M₁₆	—	398	1	no	—
32	C₂×C₈	C₂×D₁₉₉	1	1	yes	characteristic
33	C₁₆	—	2	1	no	—
34	Q₈	—	796	1	no	—
35	Q₈	—	398	2	no	—
36	C₂×C₄	—	796	1	no	—
37	C₂×C₄	D₇₉₆	1	1	yes	characteristic
38	C₈	—	796	1	no	—
39	C₈	C₄×D₁₉₉	1	1	yes	characteristic
40	C₈	C₂²⋊D₁₉₉	1	1	yes	characteristic
41	C₂²	#15691875	1	1	yes	characteristic
42	C₄	—	796	2	no	—
43	C₄	(C₂×D₁₉₉)⋊C₄	1	1	yes	characteristic
44	C₄	D₁₅₉₂	1	1	yes	characteristic
45	C₂	—	2	1	no	—
46	C₂	#20263877	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20263877
commutator	C₈×C₁₉₉	C₂×C₄
Frattini	C₂×C₈	C₂×D₁₉₉
Fitting	M₃₂×C₁₉₉	C₂
socle	C₂×C₁₉₉	#102
prime radical	C₂²×C₁₉₉	SD₁₆
cosocle	C₂×C₈×C₁₉₉	C₂²
hypercenter	M₃₂	D₁₉₉
hypocenter	C₁₉₉	#362
LCS	G ▷ C₈×C₁₉₉ ▷ C₄×C₁₉₉ ▷ C₂×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ M₃₂

Complements

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

N	H ≅ G/N	type
23. C₁₉₉	#362	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₇₉₆	2
#20263891	2
M₃₂×C₁₉₉	2
#362	199