Subgroups — #27163829

Order 12736.

58 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20263977	C₂	1	1	yes	maximal, characteristic
2	#20263887	C₂	1	1	yes	maximal, characteristic, complemented
3	M₃₂×C₁₉₉	C₂	1	1	yes	maximal, characteristic, complemented
4	#15691886	—	2	1	no	—
5	Dic₇₉₆	C₄	1	1	yes	characteristic, complemented
6	#15691875	—	2	1	no	—
7	D₁₅₉₂	C₄	1	1	yes	characteristic, complemented
8	C₄⋊Dic₁₉₉	—	2	1	no	—
9	C₂×C₈×C₁₉₉	C₂²	1	1	yes	characteristic
10	C₁₆×C₁₉₉	—	2	1	no	—
11	Dic₃₉₈	—	2	1	no	—
12	D₇₉₆	—	2	1	no	—
13	C₂²⋊D₁₉₉	—	4	1	no	—
14	C₂×Dic₁₉₉	—	4	1	no	—
15	C₄×D₁₉₉	—	4	1	no	—
16	C₂×C₄×C₁₉₉	D₄	1	1	yes	characteristic
17	C₈×C₁₉₉	C₂×C₄	1	1	yes	characteristic
18	C₈×C₁₉₉	D₄	1	1	yes	characteristic
19	C₂²×C₁₉₉	D₈	1	1	yes	characteristic
20	C₂×D₁₉₉	—	4	1	no	—
21	C₄×C₁₉₉	C₂²⋊C₄	1	1	yes	characteristic
22	C₄×C₁₉₉	SD₁₆	1	1	yes	characteristic
23	Dic₁₉₉	—	4	1	no	—
24	Dic₁₉₉	—	8	1	no	—
25	C₂×C₁₉₉	—	2	1	no	—
26	C₂×C₁₉₉	#102	1	1	yes	characteristic
27	D₁₉₉	—	8	1	no	—
28	C₁₉₉	#360	1	1	yes	characteristic, complemented, Sylow
29	#360	—	199	1	no	maximal, Sylow
30	#135	—	199	1	no	—
31	M₃₂	D₁₉₉	1	1	yes	characteristic, complemented
32	#106	—	199	1	no	—
33	#41	—	398	1	no	—
34	Q₁₆	—	199	1	no	—
35	SD₁₆	—	398	1	no	—
36	D₈	—	199	1	no	—
37	C₂×C₈	C₂×D₁₉₉	1	1	yes	characteristic
38	C₄⋊C₄	—	398	1	no	—
39	C₁₆	—	2	1	no	—
40	Q₈	—	398	1	no	—
41	D₄	—	796	1	no	—
42	D₄	—	398	1	no	—
43	C₂×C₄	—	796	1	no	—
44	C₂×C₄	—	796	1	no	—
45	C₂×C₄	D₇₉₆	1	1	yes	characteristic
46	C₈	C₄×D₁₉₉	1	1	yes	characteristic
47	C₈	C₂²⋊D₁₉₉	1	1	yes	characteristic
48	C₂²	—	796	1	no	—
49	C₂²	D₁₅₉₂	1	1	yes	characteristic
50	C₄	—	796	1	no	—
51	C₄	—	1592	1	no	—
52	C₄	(C₂×D₁₉₉)⋊C₄	1	1	yes	characteristic
53	C₄	#15691875	1	1	yes	characteristic
54	C₂	—	2	1	no	—
55	C₂	—	1592	1	no	—
56	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	#20263977	C₂
cosocle	C₂×C₈×C₁₉₉	C₂²
hypercenter	M₃₂	D₁₉₉
hypocenter	C₁₉₉	#360
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
2. #20263887	C₂	semidirect
3. M₃₂×C₁₉₉	C₂	semidirect
5. Dic₇₉₆	C₄	semidirect
7. D₁₅₉₂	C₄	semidirect
28. C₁₉₉	#360	semidirect
31. M₃₂	D₁₉₉	semidirect

Maximal subgroups

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

subgroup	index
#20263977	2
#20263887	2
M₃₂×C₁₉₉	2
#360	199