Subgroups — #27164451

Order 12736.

84 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20263978	C₂	1	2	yes	maximal, complemented
2	#20263898	C₂	1	2	yes	maximal, complemented
3	D₃₁₈₄	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₆×D₁₉₉	C₂	1	1	yes	maximal, characteristic, complemented
5	Q₃₂×C₁₉₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#15691887	—	2	2	no	—
7	#15691873	—	2	2	no	—
8	D₁₅₉₂	C₂²	1	2	yes	—
9	C₈×D₁₉₉	C₂²	1	1	yes	characteristic
10	C₁₉₉⋊C₁₆	C₂²	1	1	yes	characteristic
11	Q₁₆×C₁₉₉	C₂²	1	2	yes	—
12	C₁₆×C₁₉₉	C₂²	1	1	yes	characteristic
13	D₇₉₆	—	2	2	no	—
14	D₇₉₆	—	4	2	no	—
15	C₄×D₁₉₉	—	4	2	no	—
16	C₄×D₁₉₉	D₄	1	1	yes	characteristic
17	C₁₉₉⋊C₈	D₄	1	1	yes	characteristic
18	Q₈×C₁₉₉	—	2	2	no	—
19	C₈×C₁₉₉	C₂³	1	1	yes	characteristic
20	C₂×D₁₉₉	—	4	2	no	—
21	C₂×D₁₉₉	D₈	1	1	yes	characteristic
22	C₄×C₁₉₉	—	4	2	no	—
23	C₄×C₁₉₉	C₂×D₄	1	1	yes	characteristic
24	Dic₁₉₉	D₈	1	1	yes	characteristic
25	C₂×C₁₉₉	C₂×D₈	1	1	yes	characteristic
26	D₁₉₉	—	2	1	no	—
27	D₁₉₉	—	8	2	no	—
28	C₁₉₉	#508	1	1	yes	characteristic, complemented, Sylow
29	#508	—	199	1	no	maximal, Sylow
30	#135	—	199	2	no	—
31	Q₃₂	D₁₉₉	1	1	yes	characteristic, complemented
32	SD₃₂	—	199	2	no	—
33	D₁₆	—	199	1	no	—
34	C₂×C₁₆	—	199	1	no	—
35	#41	—	398	2	no	—
36	Q₁₆	C₂×D₁₉₉	1	2	yes	—
37	SD₁₆	—	398	2	no	—
38	D₈	—	199	2	no	—
39	C₂×C₈	—	199	1	no	—
40	C₁₆	—	199	1	no	—
41	C₁₆	C₂×D₁₉₉	1	1	yes	characteristic
42	Q₈	—	2	2	no	—
43	D₄	—	796	2	no	—
44	D₄	—	398	2	no	—
45	C₂×C₄	—	796	2	no	—
46	C₂×C₄	—	199	1	no	—
47	C₈	—	199	1	no	—
48	C₈	C₂²×D₁₉₉	1	1	yes	characteristic
49	C₂²	—	796	2	no	—
50	C₂²	—	199	1	no	—
51	C₄	—	4	2	no	—
52	C₄	—	199	1	no	—
53	C₄	D₄×D₁₉₉	1	1	yes	characteristic
54	C₂	—	1592	2	no	—
55	C₂	—	398	1	no	—
56	C₂	D₈×D₁₉₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₈×D₁₉₉
commutator, cosocle	C₈×C₁₉₉	C₂³
Frattini	C₈	C₂²×D₁₉₉
Fitting	Q₃₂×C₁₉₉	C₂
socle	C₂×C₁₉₉	C₂×D₈
hypercenter	Q₃₂	D₁₉₉
hypocenter	C₁₉₉	#508
LCS	G ▷ C₈×C₁₉₉ ▷ C₄×C₁₉₉ ▷ C₂×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ Q₃₂

Complements

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

N	H ≅ G/N	type
1. #20263978	C₂	semidirect
2. #20263898	C₂	semidirect
3. D₃₁₈₄	C₂	semidirect
4. C₁₆×D₁₉₉	C₂	semidirect
5. Q₃₂×C₁₉₉	C₂	semidirect
28. C₁₉₉	#508	semidirect
31. Q₃₂	D₁₉₉	semidirect

Maximal subgroups

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

subgroup	index
#20263978	2
#20263898	2
D₃₁₈₄	2
C₁₆×D₁₉₉	2
Q₃₂×C₁₉₉	2
#508	199