Subgroups — #27163821

Order 12736.

58 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₇₉₆	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₉₉⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	#107×C₁₉₉	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₃₉₈	—	2	1	no	—
5	Dic₇₉₆	—	2	1	no	—
6	Dic₇₉₆	C₄	1	2	yes	complemented
7	C₁₉₉⋊C₁₆	—	2	1	no	—
8	C₂×C₈×C₁₉₉	C₂²	1	1	yes	characteristic
9	C₄⋊C₄×C₁₉₉	—	2	1	no	—
10	Dic₃₉₈	—	2	2	no	—
11	Dic₃₉₈	—	4	1	no	—
12	C₂×Dic₁₉₉	—	4	1	no	—
13	C₂×C₄×C₁₉₉	—	4	1	no	—
14	C₂×C₄×C₁₉₉	D₄	1	1	yes	characteristic
15	C₈×C₁₉₉	C₂×C₄	1	1	yes	characteristic
16	C₈×C₁₉₉	D₄	1	1	yes	characteristic
17	C₂²×C₁₉₉	D₈	1	1	yes	characteristic
18	C₄×C₁₉₉	—	8	1	no	—
19	C₄×C₁₉₉	C₂²⋊C₄	1	1	yes	characteristic
20	C₄×C₁₉₉	SD₁₆	1	1	yes	characteristic
21	Dic₁₉₉	—	4	2	no	—
22	C₂×C₁₉₉	#102	1	1	yes	characteristic
23	C₂×C₁₉₉	SD₃₂	1	1	yes	characteristic
24	C₂×C₁₉₉	Q₃₂	1	1	yes	characteristic
25	C₁₉₉	#358	1	1	yes	characteristic, complemented, Sylow
26	#358	—	199	1	no	maximal, Sylow
27	C₂×Q₁₆	—	199	1	no	—
28	C₂×C₁₆	—	199	1	no	—
29	#107	D₁₉₉	1	1	yes	characteristic
30	C₂×Q₈	—	398	1	no	—
31	Q₁₆	—	398	1	no	—
32	Q₁₆	—	199	2	no	—
33	C₂×C₈	C₂×D₁₉₉	1	1	yes	characteristic
34	C₄⋊C₄	—	2	1	no	—
35	C₁₆	—	398	1	no	—
36	Q₈	—	796	1	no	—
37	Q₈	—	398	2	no	—
38	C₂×C₄	—	4	1	no	—
39	C₂×C₄	—	796	1	no	—
40	C₂×C₄	C₂²⋊D₁₉₉	1	1	yes	characteristic
41	C₈	C₄×D₁₉₉	1	1	yes	characteristic
42	C₈	D₇₉₆	1	1	yes	characteristic
43	C₂²	#15691871	1	1	yes	characteristic
44	C₄	—	8	1	no	—
45	C₄	—	796	2	no	—
46	C₄	(C₂×D₁₉₉)⋊C₄	1	1	yes	characteristic
47	C₄	#15691873	1	1	yes	characteristic
48	C₂	#20263875	1	1	yes	characteristic
49	C₂	#20263899	1	1	yes	characteristic
50	C₂	#20263901	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#15691871
commutator	C₈×C₁₉₉	C₂×C₄
Frattini	C₂×C₈	C₂×D₁₉₉
Fitting	#107×C₁₉₉	C₂
socle, prime radical	C₂²×C₁₉₉	D₈
cosocle	C₂×C₈×C₁₉₉	C₂²
hypercenter	#107	D₁₉₉
hypocenter	C₁₉₉	#358
LCS	G ▷ C₈×C₁₉₉ ▷ C₄×C₁₉₉ ▷ C₂×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #107

Complements

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

N	H ≅ G/N	type
6. Dic₇₉₆	C₄	semidirect
25. C₁₉₉	#358	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₇₉₆	2
C₂×C₁₉₉⋊C₁₆	2
#107×C₁₉₉	2
#358	199