Subgroups — #27164823

Order 12736.

268 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20264001	C₂	1	2	yes	maximal, complemented
2	#20263983	C₂	1	6	yes	maximal, complemented
3	#20263977	C₂	1	3	yes	maximal, complemented
4	C₂×D₁₅₉₂	C₂	1	3	yes	maximal, complemented
5	#131×C₁₉₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#15691887	—	2	2	no	—
7	#15691886	C₂²	1	6	yes	complemented
8	C₂×D₇₉₆	C₂²	1	6	yes	complemented
9	D₄×D₁₉₉	—	2	6	no	—
10	Dic₇₉₆	C₂²	1	1	yes	characteristic, complemented
11	#15691875	C₂²	1	6	yes	complemented
12	D₁₅₉₂	C₂²	1	3	yes	complemented
13	D₁₅₉₂	C₂²	1	6	yes	complemented
14	#41×C₁₉₉	C₂²	1	1	yes	characteristic
15	M₁₆×C₁₉₉	C₂²	1	3	yes	complemented
16	C₂×C₈×C₁₉₉	C₂²	1	3	yes	complemented
17	C₂²×D₁₉₉	—	2	6	no	—
18	Dic₃₉₈	C₂³	1	2	yes	—
19	D₇₉₆	—	2	6	no	—
20	D₇₉₆	C₂³	1	6	yes	—
21	C₂²⋊D₁₉₉	—	2	6	no	—
22	C₄×D₁₉₉	—	2	6	no	—
23	Q₈×C₁₉₉	D₄	1	1	yes	characteristic
24	D₄×C₁₉₉	D₄	1	3	yes	—
25	C₂×C₄×C₁₉₉	C₂³	1	3	yes	—
26	C₈×C₁₉₉	C₂³	1	1	yes	characteristic
27	C₈×C₁₉₉	C₂³	1	3	yes	—
28	C₂²×C₁₉₉	C₂×D₄	1	3	yes	—
29	C₂×D₁₉₉	—	2	6	no	—
30	C₂×D₁₉₉	—	8	6	no	—
31	C₄×C₁₉₉	C₂×D₄	1	3	yes	—
32	C₄×C₁₉₉	C₂⁴	1	1	yes	characteristic
33	Dic₁₉₉	—	2	2	no	—
34	C₂×C₁₉₉	—	2	3	no	—
35	C₂×C₁₉₉	C₂²×D₄	1	1	yes	characteristic
36	D₁₉₉	—	4	6	no	—
37	C₁₉₉	#576	1	1	yes	characteristic, complemented, Sylow
38	#576	—	199	1	no	maximal, Sylow
39	H(5,𝔽₂)	—	199	2	no	—
40	AGL(1,ℤ/8)	—	199	6	no	—
41	#135	—	199	3	no	—
42	C₂×D₈	—	199	3	no	—
43	#131	D₁₉₉	1	1	yes	characteristic, complemented
44	#41	—	398	2	no	—
45	#41	—	199	6	no	—
46	#41	C₂×D₁₉₉	1	1	yes	characteristic
47	C₂×D₄	—	398	6	no	—
48	C₂×D₄	—	199	6	no	—
49	Q₁₆	—	199	1	no	—
50	SD₁₆	—	199	6	no	—
51	D₈	—	199	3	no	—
52	D₈	—	199	6	no	—
53	M₁₆	C₂×D₁₉₉	1	3	yes	complemented
54	C₂×C₈	C₂×D₁₉₉	1	3	yes	complemented
55	C₂³	—	398	6	no	—
56	Q₈	—	199	2	no	—
57	Q₈	D₇₉₆	1	1	yes	characteristic
58	D₄	—	398	6	no	—
59	D₄	—	398	6	no	—
60	D₄	—	199	6	no	—
61	D₄	D₇₉₆	1	3	yes	—
62	C₂×C₄	—	398	6	no	—
63	C₂×C₄	C₂²×D₁₉₉	1	3	yes	—
64	C₈	C₂²×D₁₉₉	1	1	yes	characteristic
65	C₈	C₂²×D₁₉₉	1	3	yes	—
66	C₂²	—	1592	6	no	—
67	C₂²	—	398	6	no	—
68	C₂²	C₂×D₇₉₆	1	3	yes	—
69	C₄	—	398	2	no	—
70	C₄	C₂×D₇₉₆	1	3	yes	—
71	C₄	C₂³×D₁₉₉	1	1	yes	characteristic
72	C₂	—	2	3	no	—
73	C₂	—	796	6	no	—
74	C₂	C₂²×D₇₉₆	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂²×D₇₉₆
commutator, cosocle	C₄×C₁₉₉	C₂⁴
Frattini	C₄	C₂³×D₁₉₉
Fitting	#131×C₁₉₉	C₂
socle	C₂×C₁₉₉	C₂²×D₄
hypercenter	#131	D₁₉₉
hypocenter	C₁₉₉	#576
LCS	G ▷ C₄×C₁₉₉ ▷ C₂×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂ ◁ #41 ◁ #131

Complements

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

N	H ≅ G/N	type
1. #20264001	C₂	semidirect
2. #20263983	C₂	semidirect
3. #20263977	C₂	semidirect
4. C₂×D₁₅₉₂	C₂	semidirect
5. #131×C₁₉₉	C₂	semidirect
7. #15691886	C₂²	semidirect
8. C₂×D₇₉₆	C₂²	semidirect
10. Dic₇₉₆	C₂²	semidirect
11. #15691875	C₂²	semidirect
12. D₁₅₉₂	C₂²	semidirect
13. D₁₅₉₂	C₂²	semidirect
15. M₁₆×C₁₉₉	C₂²	semidirect
16. C₂×C₈×C₁₉₉	C₂²	semidirect
37. C₁₉₉	#576	semidirect
43. #131	D₁₉₉	semidirect
53. M₁₆	C₂×D₁₉₉	semidirect
54. C₂×C₈	C₂×D₁₉₉	semidirect

Maximal subgroups

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

subgroup	index
#20264001	2
#20263983	2
#20263977	2
C₂×D₁₅₉₂	2
#131×C₁₉₉	2
#576	199