Subgroups — #27164456

Order 12736.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20263977	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₁₅₉₂	C₂	1	1	yes	maximal, characteristic, complemented
3	#20263900	C₂	1	2	yes	maximal, complemented
4	D₃₁₈₄	C₂	1	2	yes	maximal, complemented
5	M₃₂×C₁₉₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#15691886	—	2	1	no	—
7	C₂×D₇₉₆	—	2	1	no	—
8	Dic₇₉₆	C₂²	1	1	yes	characteristic, complemented
9	#15691875	—	2	1	no	—
10	D₁₅₉₂	—	2	1	no	—
11	D₁₅₉₂	C₂²	1	2	yes	—
12	D₁₅₉₂	C₂²	1	1	yes	characteristic, complemented
13	C₂×C₈×C₁₉₉	C₂²	1	1	yes	characteristic
14	C₁₆×C₁₉₉	C₂²	1	2	yes	complemented
15	C₂²×D₁₉₉	—	4	1	no	—
16	Dic₃₉₈	—	2	1	no	—
17	D₇₉₆	—	2	2	no	—
18	D₇₉₆	—	2	1	no	—
19	D₇₉₆	—	4	1	no	—
20	C₂²⋊D₁₉₉	—	4	1	no	—
21	C₄×D₁₉₉	—	4	1	no	—
22	C₂×C₄×C₁₉₉	D₄	1	1	yes	characteristic
23	C₈×C₁₉₉	D₄	1	1	yes	characteristic
24	C₈×C₁₉₉	C₂³	1	1	yes	characteristic
25	C₂²×C₁₉₉	D₈	1	1	yes	characteristic
26	C₂×D₁₉₉	—	4	1	no	—
27	C₂×D₁₉₉	—	4	2	no	—
28	C₂×D₁₉₉	—	8	2	no	—
29	C₄×C₁₉₉	D₈	1	1	yes	characteristic
30	C₄×C₁₉₉	C₂×D₄	1	1	yes	characteristic
31	Dic₁₉₉	—	4	1	no	—
32	C₂×C₁₉₉	—	2	1	no	—
33	C₂×C₁₉₉	C₂×D₈	1	1	yes	characteristic
34	D₁₉₉	—	8	1	no	—
35	D₁₉₉	—	8	2	no	—
36	C₁₉₉	#509	1	1	yes	characteristic, complemented, Sylow
37	#509	—	199	1	no	maximal, Sylow
38	#135	—	199	1	no	—
39	C₂×D₈	—	199	1	no	—
40	SD₃₂	—	199	2	no	—
41	D₁₆	—	199	2	no	—
42	M₃₂	D₁₉₉	1	1	yes	characteristic, complemented
43	#41	—	398	1	no	—
44	C₂×D₄	—	398	1	no	—
45	Q₁₆	—	199	1	no	—
46	SD₁₆	—	398	1	no	—
47	D₈	—	398	1	no	—
48	D₈	—	199	2	no	—
49	D₈	—	199	1	no	—
50	C₂×C₈	C₂×D₁₉₉	1	1	yes	characteristic
51	C₁₆	C₂×D₁₉₉	1	2	yes	complemented
52	C₂³	—	796	1	no	—
53	Q₈	—	398	1	no	—
54	D₄	—	796	1	no	—
55	D₄	—	796	1	no	—
56	D₄	—	398	2	no	—
57	D₄	—	398	1	no	—
58	C₂×C₄	—	796	1	no	—
59	C₂×C₄	D₇₉₆	1	1	yes	characteristic
60	C₈	D₇₉₆	1	1	yes	characteristic
61	C₈	C₂²×D₁₉₉	1	1	yes	characteristic
62	C₂²	—	796	1	no	—
63	C₂²	—	796	2	no	—
64	C₂²	—	1592	2	no	—
65	C₂²	D₁₅₉₂	1	1	yes	characteristic
66	C₄	—	796	1	no	—
67	C₄	D₁₅₉₂	1	1	yes	characteristic
68	C₄	C₂×D₇₉₆	1	1	yes	characteristic
69	C₂	—	2	1	no	—
70	C₂	—	1592	2	no	—
71	C₂	—	1592	1	no	—
72	C₂	C₂×D₁₅₉₂	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₁₅₉₂
commutator, cosocle	C₈×C₁₉₉	C₂³
Frattini	C₈	C₂²×D₁₉₉
Fitting	M₃₂×C₁₉₉	C₂
socle	C₂×C₁₉₉	C₂×D₈
hypercenter	M₃₂	D₁₉₉
hypocenter	C₁₉₉	#509
LCS	G ▷ C₈×C₁₉₉ ▷ C₄×C₁₉₉ ▷ C₂×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ M₃₂

Complements

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

N	H ≅ G/N	type
1. #20263977	C₂	semidirect
2. C₂×D₁₅₉₂	C₂	semidirect
3. #20263900	C₂	semidirect
4. D₃₁₈₄	C₂	semidirect
5. M₃₂×C₁₉₉	C₂	semidirect
8. Dic₇₉₆	C₂²	semidirect
12. D₁₅₉₂	C₂²	semidirect
14. C₁₆×C₁₉₉	C₂²	semidirect
36. C₁₉₉	#509	semidirect
42. M₃₂	D₁₉₉	semidirect
51. C₁₆	C₂×D₁₉₉	semidirect

Maximal subgroups

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

subgroup	index
#20263977	2
C₂×D₁₅₉₂	2
#20263900	2
D₃₁₈₄	2
M₃₂×C₁₉₉	2
#509	199