Subgroups — #27164228

Order 12736.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15691876	C₂	1	2	yes	maximal
2	#20263955	C₂	1	1	yes	maximal, characteristic
3	C₄⋊(C₁₉₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
4	#20263878	C₂	1	2	yes	maximal
5	C₄×Q₈×C₁₉₉	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₃₉₈	C₂²	1	2	yes	—
7	#15691876	—	2	4	no	—
8	C₂×C₁₉₉⋊C₈	C₂²	1	2	yes	—
9	C₄⋊Dic₁₉₉	—	2	1	no	—
10	C₄⋊Dic₁₉₉	—	2	1	no	—
11	C₂×Q₈×C₁₉₉	C₂²	1	1	yes	characteristic
12	C₄⋊C₄×C₁₉₉	—	2	1	no	—
13	C₄⋊C₄×C₁₉₉	C₂²	1	1	yes	characteristic
14	C₄²×C₁₉₉	—	2	1	no	—
15	C₄²×C₁₉₉	C₂²	1	1	yes	characteristic
16	Dic₃₉₈	—	2	2	no	—
17	Dic₃₉₈	—	2	2	no	—
18	C₂×Dic₁₉₉	—	2	2	no	—
19	C₁₉₉⋊C₈	—	2	2	no	—
20	Q₈×C₁₉₉	—	2	1	no	—
21	Q₈×C₁₉₉	D₄	1	2	yes	—
22	C₂×C₄×C₁₉₉	—	2	1	no	—
23	C₂×C₄×C₁₉₉	—	2	1	no	—
24	C₂×C₄×C₁₉₉	D₄	1	1	yes	characteristic
25	C₂×C₄×C₁₉₉	D₄	1	1	yes	characteristic
26	C₂×C₄×C₁₉₉	C₂³	1	1	yes	characteristic
27	C₂²×C₁₉₉	C₂×D₄	1	1	yes	characteristic
28	C₄×C₁₉₉	—	2	2	no	—
29	C₄×C₁₉₉	—	2	1	no	—
30	C₄×C₁₉₉	—	4	1	no	—
31	C₄×C₁₉₉	Q₁₆	1	2	yes	complemented
32	C₄×C₁₉₉	C₂×D₄	1	1	yes	characteristic
33	C₄×C₁₉₉	#41	1	1	yes	characteristic
34	Dic₁₉₉	—	4	2	no	—
35	C₂×C₁₉₉	#121	1	1	yes	characteristic
36	C₂×C₁₉₉	C₂×Q₁₆	1	1	yes	characteristic
37	C₂×C₁₉₉	#137	1	1	yes	characteristic
38	C₁₉₉	#462	1	1	yes	characteristic, complemented, Sylow
39	#462	—	199	1	no	maximal, Sylow
40	C₂×Q₁₆	—	199	2	no	—
41	C₄⋊Q₈	—	199	1	no	—
42	C₄×Q₈	D₁₉₉	1	1	yes	characteristic
43	C₄⋊C₈	—	199	1	no	—
44	#103	—	199	2	no	—
45	C₂×Q₈	—	199	2	no	—
46	C₂×Q₈	C₂×D₁₉₉	1	1	yes	characteristic
47	Q₁₆	—	398	4	no	—
48	C₂×C₈	—	199	2	no	—
49	C₄⋊C₄	—	2	1	no	—
50	C₄⋊C₄	—	398	1	no	—
51	C₄⋊C₄	—	398	1	no	—
52	C₄⋊C₄	C₂×D₁₉₉	1	1	yes	characteristic
53	C₄²	—	2	1	no	—
54	C₄²	C₂×D₁₉₉	1	1	yes	characteristic
55	Q₈	—	2	1	no	—
56	Q₈	—	398	2	no	—
57	Q₈	—	398	2	no	—
58	Q₈	D₇₉₆	1	2	yes	—
59	C₂×C₄	—	2	1	no	—
60	C₂×C₄	—	2	1	no	—
61	C₂×C₄	—	398	2	no	—
62	C₂×C₄	C₂²⋊D₁₉₉	1	1	yes	characteristic
63	C₂×C₄	C₂²⋊D₁₉₉	1	1	yes	characteristic
64	C₂×C₄	C₂²×D₁₉₉	1	1	yes	characteristic
65	C₈	—	398	2	no	—
66	C₂²	C₂×C₂²⋊D₁₉₉	1	1	yes	characteristic
67	C₄	—	2	1	no	—
68	C₄	—	2	2	no	—
69	C₄	—	4	1	no	—
70	C₄	—	796	2	no	—
71	C₄	#15691876	1	2	yes	complemented
72	C₄	C₂×D₇₉₆	1	1	yes	characteristic
73	C₄	#15691886	1	1	yes	characteristic
74	C₂	#20263929	1	1	yes	characteristic
75	C₂	C₂×#15691876	1	1	yes	characteristic
76	C₂	#20263985	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₁₉₉
commutator, cosocle	C₂×C₄×C₁₉₉	C₂³
Frattini	C₂×C₄	C₂²×D₁₉₉
Fitting	C₄×Q₈×C₁₉₉	C₂
socle, prime radical	C₂²×C₁₉₉	C₂×D₄
hypercenter	C₄×Q₈	D₁₉₉
hypocenter	C₁₉₉	#462
LCS	G ▷ C₂×C₄×C₁₉₉ ▷ C₂×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
31. C₄×C₁₉₉	Q₁₆	semidirect
38. C₁₉₉	#462	semidirect
71. C₄	#15691876	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15691876	2
#20263955	2
C₄⋊(C₁₉₉⋊C₈)	2
#20263878	2
C₄×Q₈×C₁₉₉	2
#462	199