Subgroups — #27163824

Order 12736.

56 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	C₂×C₁₉₉⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#108×C₁₉₉	C₂	1	1	yes	maximal, characteristic, complemented
4	#15691886	—	2	1	no	—
5	Dic₇₉₆	C₄	1	1	yes	characteristic
6	#15691875	—	2	1	no	—
7	D₁₅₉₂	C₄	1	1	yes	characteristic
8	C₁₉₉⋊C₁₆	—	2	1	no	—
9	M₁₆×C₁₉₉	—	2	1	no	—
10	C₂×C₈×C₁₉₉	C₂²	1	1	yes	characteristic
11	Dic₃₉₈	—	2	1	no	—
12	D₇₉₆	—	2	1	no	—
13	C₂²⋊D₁₉₉	—	4	1	no	—
14	C₄×D₁₉₉	—	4	1	no	—
15	C₂×C₄×C₁₉₉	D₄	1	1	yes	characteristic
16	C₈×C₁₉₉	—	4	1	no	—
17	C₈×C₁₉₉	C₂×C₄	1	1	yes	characteristic
18	C₈×C₁₉₉	D₄	1	1	yes	characteristic
19	C₂²×C₁₉₉	SD₁₆	1	1	yes	characteristic
20	C₂×D₁₉₉	—	4	1	no	—
21	C₄×C₁₉₉	C₂²⋊C₄	1	1	yes	characteristic
22	C₄×C₁₉₉	D₈	1	1	yes	characteristic
23	Dic₁₉₉	—	4	1	no	—
24	C₂×C₁₉₉	—	2	1	no	—
25	C₂×C₁₉₉	#102	1	1	yes	characteristic
26	D₁₉₉	—	8	1	no	—
27	C₁₉₉	#359	1	1	yes	characteristic, complemented, Sylow
28	#359	—	199	1	no	maximal, Sylow
29	#135	—	199	1	no	—
30	C₂×C₁₆	—	199	1	no	—
31	#108	D₁₉₉	1	1	yes	characteristic, complemented
32	#41	—	398	1	no	—
33	Q₁₆	—	199	1	no	—
34	SD₁₆	—	398	1	no	—
35	D₈	—	199	1	no	—
36	M₁₆	—	2	1	no	—
37	C₂×C₈	C₂×D₁₉₉	1	1	yes	characteristic
38	C₁₆	—	398	1	no	—
39	Q₈	—	398	1	no	—
40	D₄	—	796	1	no	—
41	D₄	—	398	1	no	—
42	C₂×C₄	—	796	1	no	—
43	C₂×C₄	C₂²⋊D₁₉₉	1	1	yes	characteristic
44	C₈	—	4	1	no	—
45	C₈	C₄×D₁₉₉	1	1	yes	characteristic
46	C₈	D₇₉₆	1	1	yes	characteristic
47	C₂²	—	796	1	no	—
48	C₂²	#15691873	1	1	yes	characteristic
49	C₄	—	796	1	no	—
50	C₄	(C₂×D₁₉₉)⋊C₄	1	1	yes	characteristic
51	C₄	#15691871	1	1	yes	characteristic
52	C₂	—	2	1	no	—
53	C₂	—	1592	1	no	—
54	C₂	#20263875	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#15691871
commutator	C₈×C₁₉₉	C₂×C₄
Frattini	C₂×C₈	C₂×D₁₉₉
Fitting	#108×C₁₉₉	C₂
socle	C₂×C₁₉₉	#102
prime radical	#20263977	C₂
cosocle	C₂×C₈×C₁₉₉	C₂²
hypercenter	#108	D₁₉₉
hypocenter	C₁₉₉	#359
LCS	G ▷ C₈×C₁₉₉ ▷ C₄×C₁₉₉ ▷ C₂×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #108

Complements

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

N	H ≅ G/N	type
2. C₂×C₁₉₉⋊C₁₆	C₂	semidirect
3. #108×C₁₉₉	C₂	semidirect
27. C₁₉₉	#359	semidirect
31. #108	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
C₂×C₁₉₉⋊C₁₆	2
#108×C₁₉₉	2
#359	199