Subgroups — #27164149

Order 12736.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊(C₁₉₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
2	C₄⋊(C₁₉₉⋊C₈)	C₂	1	2	yes	maximal
3	#20263866	C₂	1	2	yes	maximal
4	C₄×C₁₉₉⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₄×Q₈×C₁₉₉	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₉₉⋊C₈	C₂²	1	1	yes	characteristic
7	C₂×C₁₉₉⋊C₈	C₂²	1	2	yes	—
8	C₂×C₁₉₉⋊C₈	C₂²	1	1	yes	characteristic
9	C₂×Q₈×C₁₉₉	C₄	1	1	yes	characteristic
10	C₄⋊C₄×C₁₉₉	C₄	1	2	yes	—
11	C₄⋊C₄×C₁₉₉	C₄	1	1	yes	characteristic
12	C₄²×C₁₉₉	C₂²	1	1	yes	characteristic
13	C₄²×C₁₉₉	C₂²	1	2	yes	—
14	C₁₉₉⋊C₈	—	2	1	no	—
15	C₁₉₉⋊C₈	—	2	2	no	—
16	C₁₉₉⋊C₈	Q₈	1	2	yes	complemented
17	Q₈×C₁₉₉	—	2	2	no	—
18	C₂×C₄×C₁₉₉	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×C₁₉₉	C₂×C₄	1	2	yes	—
20	C₂×C₄×C₁₉₉	C₂×C₄	1	2	yes	—
21	C₂×C₄×C₁₉₉	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×C₁₉₉	C₂³	1	1	yes	characteristic
23	C₂²×C₁₉₉	C₂²×C₄	1	1	yes	characteristic
24	C₄×C₁₉₉	—	2	2	no	—
25	C₄×C₁₉₉	—	2	1	no	—
26	C₄×C₁₉₉	—	2	2	no	—
27	C₄×C₁₉₉	M₁₆	1	2	yes	—
28	C₄×C₁₉₉	C₂×Q₈	1	1	yes	characteristic
29	C₄×C₁₉₉	#41	1	1	yes	characteristic
30	C₂×C₁₉₉	C₄×Q₈	1	1	yes	characteristic
31	C₂×C₁₉₉	C₂×M₁₆	1	1	yes	characteristic
32	C₂×C₁₉₉	#131	1	1	yes	characteristic
33	C₁₉₉	#446	1	1	yes	characteristic, complemented, Sylow
34	#446	—	199	1	no	maximal, Sylow
35	C₄×Q₈	D₁₉₉	1	1	yes	characteristic
36	C₄⋊C₈	—	199	1	no	—
37	C₄⋊C₈	—	199	2	no	—
38	#97	—	199	2	no	—
39	C₄×C₈	—	199	1	no	—
40	C₂×Q₈	Dic₁₉₉	1	1	yes	characteristic
41	C₂×C₈	—	199	1	no	—
42	C₂×C₈	—	199	2	no	—
43	C₂×C₈	—	199	1	no	—
44	C₄⋊C₄	Dic₁₉₉	1	2	yes	—
45	C₄⋊C₄	Dic₁₉₉	1	1	yes	characteristic
46	C₄²	C₂×D₁₉₉	1	1	yes	characteristic
47	C₄²	C₂×D₁₉₉	1	2	yes	—
48	Q₈	—	2	2	no	—
49	C₂×C₄	C₂×Dic₁₉₉	1	1	yes	characteristic
50	C₂×C₄	C₂×Dic₁₉₉	1	2	yes	—
51	C₂×C₄	C₂×Dic₁₉₉	1	2	yes	—
52	C₂×C₄	C₂×Dic₁₉₉	1	1	yes	characteristic
53	C₂×C₄	C₂²×D₁₉₉	1	1	yes	characteristic
54	C₈	—	398	1	no	—
55	C₈	—	398	2	no	—
56	C₈	—	199	2	no	—
57	C₂²	C₂²×Dic₁₉₉	1	1	yes	characteristic
58	C₄	—	2	2	no	—
59	C₄	—	2	1	no	—
60	C₄	—	2	2	no	—
61	C₄	#15691870	1	2	yes	—
62	C₄	Q₈×D₁₉₉	1	1	yes	characteristic
63	C₄	#15691887	1	1	yes	characteristic
64	C₂	Q₈×Dic₁₉₉	1	1	yes	characteristic
65	C₂	C₂×#15691870	1	1	yes	characteristic
66	C₂	#20263962	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₁₉₉
commutator, socle, prime radical	C₂²×C₁₉₉	C₂²×C₄
Fitting	C₄×Q₈×C₁₉₉	C₂
cosocle	C₂×C₄×C₁₉₉	C₂³
hypercenter	C₄×Q₈	D₁₉₉
hypocenter	C₁₉₉	#446
LCS	G ▷ C₂²×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
16. C₁₉₉⋊C₈	Q₈	semidirect
33. C₁₉₉	#446	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊(C₁₉₉⋊C₈)	2
#20263866	2
C₄×C₁₉₉⋊C₈	2
C₄×Q₈×C₁₉₉	2
#446	199