Subgroups — #27163816

Order 12736.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20263953	C₂	1	1	yes	maximal, characteristic
2	#20263874	C₂	1	1	yes	maximal, characteristic
3	#101×C₁₉₉	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₃₉₈	C₄	1	1	yes	characteristic
5	#15691870	—	2	1	no	—
6	Dic₁₉₉⋊C₄	—	2	1	no	—
7	Dic₁₉₉⋊C₄	—	2	1	no	—
8	C₄×Dic₁₉₉	C₄	1	1	yes	characteristic
9	C₂×Q₈×C₁₉₉	C₂²	1	1	yes	characteristic
10	M₁₆×C₁₉₉	—	2	1	no	—
11	Dic₃₉₈	—	4	1	no	—
12	C₂×Dic₁₉₉	—	2	1	no	—
13	C₂×Dic₁₉₉	—	2	1	no	—
14	C₁₉₉⋊C₈	—	4	1	no	—
15	Q₈×C₁₉₉	—	4	1	no	—
16	C₂×C₄×C₁₉₉	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄×C₁₉₉	D₄	1	1	yes	characteristic
18	C₂×C₄×C₁₉₉	D₄	1	1	yes	characteristic
19	C₈×C₁₉₉	—	4	1	no	—
20	C₂²×C₁₉₉	C₂²⋊C₄	1	1	yes	characteristic
21	C₄×C₁₉₉	—	2	1	no	—
22	C₄×C₁₉₉	—	2	1	no	—
23	C₄×C₁₉₉	—	2	1	no	—
24	Dic₁₉₉	—	2	2	no	—
25	Dic₁₉₉	—	4	1	no	—
26	C₂×C₁₉₉	—	2	1	no	—
27	C₂×C₁₉₉	#99	1	1	yes	characteristic
28	C₁₉₉	#356	1	1	yes	characteristic, complemented, Sylow
29	#356	—	199	1	no	maximal, Sylow
30	C₄⋊Q₈	—	199	1	no	—
31	#101	—	199	1	no	—
32	#101	D₁₉₉	1	1	yes	characteristic
33	C₂×Q₈	—	199	1	no	—
34	C₂×Q₈	C₂×D₁₉₉	1	1	yes	characteristic
35	M₁₆	—	2	1	no	—
36	M₁₆	—	398	1	no	—
37	C₄⋊C₄	—	398	1	no	—
38	C₄⋊C₄	—	398	1	no	—
39	C₄²	—	199	1	no	—
40	Q₈	—	4	1	no	—
41	Q₈	—	796	1	no	—
42	C₂×C₄	—	398	1	no	—
43	C₂×C₄	—	398	1	no	—
44	C₂×C₄	C₄×D₁₉₉	1	1	yes	characteristic
45	C₂×C₄	C₂²⋊D₁₉₉	1	1	yes	characteristic
46	C₂×C₄	D₇₉₆	1	1	yes	characteristic
47	C₈	—	4	1	no	—
48	C₈	—	796	1	no	—
49	C₂²	(C₂×D₁₉₉)⋊C₄	1	1	yes	characteristic
50	C₄	—	2	1	no	—
51	C₄	—	2	1	no	—
52	C₄	—	2	1	no	—
53	C₄	—	796	1	no	—
54	C₄	—	398	2	no	—
55	C₂	—	2	1	no	—
56	C₂	#20263869	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20263869
commutator	C₂×C₄×C₁₉₉	C₂×C₄
Frattini	C₂×Q₈	C₂×D₁₉₉
Fitting	#101×C₁₉₉	C₂
socle	C₂×C₁₉₉	#99
prime radical	C₂²×C₁₉₉	C₂²⋊C₄
cosocle	C₂×Q₈×C₁₉₉	C₂²
hypercenter	#101	D₁₉₉
hypocenter	C₁₉₉	#356
LCS	G ▷ C₂×C₄×C₁₉₉ ▷ C₂²×C₁₉₉ ▷ C₂×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×Q₈ ◁ #101

Complements

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

N	H ≅ G/N	type
28. C₁₉₉	#356	semidirect

Maximal subgroups

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

subgroup	index
#20263953	2
#20263874	2
#101×C₁₉₉	2
#356	199