Subgroups — #27163808

Order 12736.

68 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20263872	C₂	1	1	yes	maximal, characteristic
2	#20263870	C₂	1	1	yes	maximal, characteristic
3	#123×C₁₉₉	C₂	1	1	yes	maximal, characteristic
4	#15691870	—	2	1	no	—
5	C₂²⋊Dic₁₉₉	—	2	1	no	—
6	C₂×D₄×C₁₉₉	C₂²	1	1	yes	characteristic
7	C₂²×C₄×C₁₉₉	C₄	1	1	yes	characteristic, complemented
8	C₄⋊C₄×C₁₉₉	—	2	1	no	—
9	C₂²⋊C₄×C₁₉₉	—	2	1	no	—
10	C₂²⋊C₄×C₁₉₉	C₄	1	1	yes	characteristic, complemented
11	C₂×Dic₁₉₉	—	4	1	no	—
12	C₁₉₉⋊C₈	—	4	1	no	—
13	C₂³×C₁₉₉	C₂×C₄	1	1	yes	characteristic
14	C₂³×C₁₉₉	D₄	1	1	yes	characteristic
15	D₄×C₁₉₉	—	4	1	no	—
16	C₂×C₄×C₁₉₉	—	2	1	no	—
17	C₂×C₄×C₁₉₉	—	2	1	no	—
18	C₂×C₄×C₁₉₉	—	4	1	no	—
19	C₂×C₄×C₁₉₉	D₄	1	1	yes	characteristic
20	C₂²×C₁₉₉	—	2	1	no	—
21	C₂²×C₁₉₉	—	2	1	no	—
22	C₂²×C₁₉₉	—	4	1	no	—
23	C₂²×C₁₉₉	—	4	1	no	—
24	C₂²×C₁₉₉	C₂²⋊C₄	1	1	yes	characteristic
25	C₄×C₁₉₉	—	2	1	no	—
26	C₄×C₁₉₉	—	4	1	no	—
27	C₄×C₁₉₉	—	4	1	no	—
28	Dic₁₉₉	—	8	1	no	—
29	C₂×C₁₉₉	—	2	1	no	—
30	C₂×C₁₉₉	—	4	1	no	—
31	C₂×C₁₉₉	—	4	1	no	—
32	C₂×C₁₉₉	#99	1	1	yes	characteristic
33	C₁₉₉	#352	1	1	yes	characteristic, complemented, Sylow
34	#352	—	199	1	no	maximal, Sylow
35	#123	D₁₉₉	1	1	yes	characteristic
36	#100	—	199	1	no	—
37	#99	—	199	1	no	—
38	C₂×D₄	C₂×D₁₉₉	1	1	yes	characteristic
39	C₂²×C₄	Dic₁₉₉	1	1	yes	characteristic, complemented
40	M₁₆	—	398	1	no	—
41	C₄⋊C₄	—	2	1	no	—
42	C₂²⋊C₄	—	2	1	no	—
43	C₂²⋊C₄	—	398	1	no	—
44	C₂²⋊C₄	Dic₁₉₉	1	1	yes	characteristic, complemented
45	C₂³	C₂×Dic₁₉₉	1	1	yes	characteristic
46	C₂³	C₂²⋊D₁₉₉	1	1	yes	characteristic
47	D₄	—	4	1	no	—
48	C₂×C₄	—	2	1	no	—
49	C₂×C₄	—	2	1	no	—
50	C₂×C₄	—	4	1	no	—
51	C₂×C₄	—	796	1	no	—
52	C₂×C₄	C₂²⋊D₁₉₉	1	1	yes	characteristic
53	C₈	—	796	1	no	—
54	C₂²	—	2	1	no	—
55	C₂²	—	2	1	no	—
56	C₂²	—	4	1	no	—
57	C₂²	—	4	1	no	—
58	C₂²	C₂²⋊Dic₁₉₉	1	1	yes	characteristic
59	C₄	—	2	1	no	—
60	C₄	—	4	1	no	—
61	C₄	—	4	1	no	—
62	C₄	—	1592	1	no	—
63	C₂	—	2	1	no	—
64	C₂	—	4	1	no	—
65	C₂	—	4	1	no	—
66	C₂	#20263870	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20263870
commutator	C₂³×C₁₉₉	C₂×C₄
Frattini	C₂×D₄	C₂×D₁₉₉
Fitting	#123×C₁₉₉	C₂
socle	C₂×C₁₉₉	#99
prime radical, cosocle	C₂×D₄×C₁₉₉	C₂²
hypercenter	#123	D₁₉₉
hypocenter	C₁₉₉	#352
LCS	G ▷ C₂³×C₁₉₉ ▷ C₂²×C₁₉₉ ▷ C₂×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×D₄ ◁ #123

Complements

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

N	H ≅ G/N	type
7. C₂²×C₄×C₁₉₉	C₄	semidirect
10. C₂²⋊C₄×C₁₉₉	C₄	semidirect
33. C₁₉₉	#352	semidirect
39. C₂²×C₄	Dic₁₉₉	semidirect
44. C₂²⋊C₄	Dic₁₉₉	semidirect

Maximal subgroups

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

subgroup	index
#20263872	2
#20263870	2
#123×C₁₉₉	2
#352	199