Subgroups — #27163742

Order 12736.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²⋊(C₁₉₉⋊C₈)	C₂	1	2	yes	maximal, complemented
2	C₂×C₂²⋊C₄×C₁₉₉	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₉₉⋊C₈	—	2	2	no	—
4	C₂⁴×C₁₉₉	C₄	1	1	yes	characteristic
5	C₂²×C₄×C₁₉₉	C₄	1	1	yes	characteristic
6	C₂²×C₄×C₁₉₉	C₂²	1	1	yes	characteristic
7	C₂²⋊C₄×C₁₉₉	—	2	2	no	—
8	C₁₉₉⋊C₈	—	4	2	no	—
9	C₂³×C₁₉₉	—	2	1	no	—
10	C₂³×C₁₉₉	—	2	1	no	—
11	C₂³×C₁₉₉	—	4	1	no	—
12	C₂³×C₁₉₉	—	4	1	no	—
13	C₂³×C₁₉₉	C₈	1	2	yes	complemented
14	C₂³×C₁₉₉	C₂×C₄	1	1	yes	characteristic
15	C₂×C₄×C₁₉₉	—	2	1	no	—
16	C₂×C₄×C₁₉₉	—	4	1	no	—
17	C₂×C₄×C₁₉₉	—	4	1	no	—
18	C₂×C₄×C₁₉₉	D₄	1	2	yes	—
19	C₂²×C₁₉₉	—	2	1	no	—
20	C₂²×C₁₉₉	—	2	1	no	—
21	C₂²×C₁₉₉	—	2	2	no	—
22	C₂²×C₁₉₉	—	4	2	no	—
23	C₂²×C₁₉₉	—	4	2	no	—
24	C₂²×C₁₉₉	—	4	1	no	—
25	C₂²×C₁₉₉	—	4	1	no	—
26	C₂²×C₁₉₉	C₂²⋊C₄	1	1	yes	characteristic
27	C₂²×C₁₉₉	C₂×C₈	1	1	yes	characteristic
28	C₂²×C₁₉₉	M₁₆	1	1	yes	characteristic
29	C₄×C₁₉₉	—	2	2	no	—
30	C₄×C₁₉₉	—	4	1	no	—
31	C₂×C₁₉₉	—	2	1	no	—
32	C₂×C₁₉₉	—	2	1	no	—
33	C₂×C₁₉₉	—	4	2	no	—
34	C₂×C₁₉₉	C₂²⋊C₈	1	1	yes	characteristic
35	C₂×C₁₉₉	#99	1	1	yes	characteristic
36	C₂×C₁₉₉	#100	1	1	yes	characteristic
37	C₁₉₉	#323	1	1	yes	characteristic, complemented, Sylow
38	#323	—	199	1	no	maximal, Sylow
39	C₂×C₂²⋊C₄	D₁₉₉	1	1	yes	characteristic
40	C₂²⋊C₈	—	199	2	no	—
41	C₂⁴	Dic₁₉₉	1	1	yes	characteristic
42	C₂²×C₄	Dic₁₉₉	1	1	yes	characteristic
43	C₂²×C₄	C₂×D₁₉₉	1	1	yes	characteristic
44	C₂×C₈	—	398	2	no	—
45	C₂²⋊C₄	—	2	2	no	—
46	C₂³	—	2	1	no	—
47	C₂³	—	2	1	no	—
48	C₂³	—	4	1	no	—
49	C₂³	—	4	1	no	—
50	C₂³	C₁₉₉⋊C₈	1	2	yes	complemented
51	C₂³	C₂×Dic₁₉₉	1	1	yes	characteristic
52	C₂×C₄	—	2	1	no	—
53	C₂×C₄	—	4	1	no	—
54	C₂×C₄	—	4	1	no	—
55	C₂×C₄	C₂²⋊D₁₉₉	1	2	yes	—
56	C₈	—	796	2	no	—
57	C₂²	—	2	1	no	—
58	C₂²	—	2	1	no	—
59	C₂²	—	2	2	no	—
60	C₂²	—	4	2	no	—
61	C₂²	—	4	2	no	—
62	C₂²	—	4	1	no	—
63	C₂²	—	4	1	no	—
64	C₂²	C₂²⋊Dic₁₉₉	1	1	yes	characteristic
65	C₂²	C₂×C₁₉₉⋊C₈	1	1	yes	characteristic
66	C₂²	#15691870	1	1	yes	characteristic
67	C₄	—	2	2	no	—
68	C₄	—	4	1	no	—
69	C₂	—	2	1	no	—
70	C₂	—	2	1	no	—
71	C₂	—	4	2	no	—
72	C₂	C₂²⋊(C₁₉₉⋊C₈)	1	1	yes	characteristic
73	C₂	#20263870	1	1	yes	characteristic
74	C₂	#20263872	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²⋊Dic₁₉₉
commutator	C₂²×C₁₉₉	C₂×C₈
Frattini	C₂²×C₄	C₂×D₁₉₉
Fitting	C₂×C₂²⋊C₄×C₁₉₉	C₂
socle	C₂²×C₁₉₉	C₂²⋊C₄
prime radical	C₂⁴×C₁₉₉	C₄
cosocle	C₂²×C₄×C₁₉₉	C₂²
hypercenter	C₂×C₂²⋊C₄	D₁₉₉
hypocenter	C₁₉₉	#323
LCS	G ▷ C₂²×C₁₉₉ ▷ C₂×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
1. C₂²⋊(C₁₉₉⋊C₈)	C₂	semidirect
13. C₂³×C₁₉₉	C₈	semidirect
37. C₁₉₉	#323	semidirect
50. C₂³	C₁₉₉⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₂²⋊(C₁₉₉⋊C₈)	2
C₂×C₂²⋊C₄×C₁₉₉	2
#323	199