Subgroups — #27164351

Order 12736.

104 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊Dic₁₉₉	C₂	1	1	yes	maximal, characteristic
2	#20263888	C₂	1	2	yes	maximal, complemented
3	#20263879	C₂	1	2	yes	maximal, complemented
4	C₂²⋊(C₁₉₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
5	C₂²⋊Q₈×C₁₉₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₉₉	—	2	1	no	—
7	C₂×C₁₉₉⋊C₈	C₂²	1	2	yes	—
8	C₄⋊Dic₁₉₉	—	2	1	no	—
9	C₄⋊Dic₁₉₉	C₂²	1	2	yes	—
10	C₂×Q₈×C₁₉₉	C₂²	1	1	yes	characteristic
11	C₂²×C₄×C₁₉₉	C₂²	1	1	yes	characteristic
12	C₄⋊C₄×C₁₉₉	—	2	1	no	—
13	C₄⋊C₄×C₁₉₉	C₂²	1	1	yes	characteristic
14	C₂²⋊C₄×C₁₉₉	—	2	1	no	—
15	C₂×Dic₁₉₉	—	2	2	no	—
16	C₂×Dic₁₉₉	—	4	2	no	—
17	C₁₉₉⋊C₈	—	2	2	no	—
18	C₂³×C₁₉₉	D₄	1	1	yes	characteristic
19	Q₈×C₁₉₉	—	2	2	no	—
20	C₂×C₄×C₁₉₉	—	2	2	no	—
21	C₂×C₄×C₁₉₉	—	2	1	no	—
22	C₂×C₄×C₁₉₉	—	2	1	no	—
23	C₂×C₄×C₁₉₉	D₄	1	1	yes	characteristic
24	C₂×C₄×C₁₉₉	C₂³	1	1	yes	characteristic
25	C₂²×C₁₉₉	—	2	1	no	—
26	C₂²×C₁₉₉	—	2	1	no	—
27	C₂²×C₁₉₉	Q₁₆	1	2	yes	—
28	C₂²×C₁₉₉	C₂×D₄	1	1	yes	characteristic
29	C₄×C₁₉₉	—	2	1	no	—
30	C₄×C₁₉₉	—	4	1	no	—
31	C₄×C₁₉₉	—	4	1	no	—
32	C₄×C₁₉₉	#41	1	2	yes	—
33	Dic₁₉₉	—	4	2	no	—
34	C₂×C₁₉₉	—	2	2	no	—
35	C₂×C₁₉₉	#123	1	1	yes	characteristic
36	C₂×C₁₉₉	C₂×Q₁₆	1	1	yes	characteristic
37	C₂×C₁₉₉	AGL(1,ℤ/8)	1	1	yes	characteristic
38	C₁₉₉	#484	1	1	yes	characteristic, complemented, Sylow
39	#484	—	199	1	no	maximal, Sylow
40	C₂²⋊Q₈	D₁₉₉	1	1	yes	characteristic
41	C₂×C₄⋊C₄	—	199	1	no	—
42	#107	—	199	2	no	—
43	#103	—	199	2	no	—
44	C₂²⋊C₈	—	199	1	no	—
45	C₂×Q₈	C₂×D₁₉₉	1	1	yes	characteristic
46	C₂²×C₄	—	398	1	no	—
47	C₂²×C₄	C₂×D₁₉₉	1	1	yes	characteristic
48	C₂×C₈	—	199	2	no	—
49	C₄⋊C₄	—	2	1	no	—
50	C₄⋊C₄	—	398	1	no	—
51	C₄⋊C₄	—	199	2	no	—
52	C₄⋊C₄	C₂×D₁₉₉	1	1	yes	characteristic
53	C₂²⋊C₄	—	2	1	no	—
54	C₂³	C₂²⋊D₁₉₉	1	1	yes	characteristic
55	Q₈	—	2	2	no	—
56	C₂×C₄	—	2	2	no	—
57	C₂×C₄	—	2	1	no	—
58	C₂×C₄	—	2	1	no	—
59	C₂×C₄	—	796	2	no	—
60	C₂×C₄	—	398	2	no	—
61	C₂×C₄	C₂²⋊D₁₉₉	1	1	yes	characteristic
62	C₂×C₄	C₂²×D₁₉₉	1	1	yes	characteristic
63	C₈	—	398	2	no	—
64	C₂²	—	2	1	no	—
65	C₂²	—	2	1	no	—
66	C₂²	#15691876	1	2	yes	—
67	C₂²	C₂×C₂²⋊D₁₉₉	1	1	yes	characteristic
68	C₄	—	2	1	no	—
69	C₄	—	4	1	no	—
70	C₄	—	4	1	no	—
71	C₄	—	796	2	no	—
72	C₄	#15691885	1	2	yes	—
73	C₂	—	2	2	no	—
74	C₂	#20263936	1	1	yes	characteristic
75	C₂	C₂×#15691876	1	1	yes	characteristic
76	C₂	#20263980	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₁₉₉
commutator, cosocle	C₂×C₄×C₁₉₉	C₂³
Frattini	C₂×C₄	C₂²×D₁₉₉
Fitting	C₂²⋊Q₈×C₁₉₉	C₂
socle	C₂²×C₁₉₉	C₂×D₄
prime radical	C₂³×C₁₉₉	D₄
hypercenter	C₂²⋊Q₈	D₁₉₉
hypocenter	C₁₉₉	#484
LCS	G ▷ C₂×C₄×C₁₉₉ ▷ C₂×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²⋊Q₈

Complements

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

N	H ≅ G/N	type
2. #20263888	C₂	semidirect
3. #20263879	C₂	semidirect
38. C₁₉₉	#484	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊Dic₁₉₉	2
#20263888	2
#20263879	2
C₂²⋊(C₁₉₉⋊C₈)	2
C₂²⋊Q₈×C₁₉₉	2
#484	199