Subgroups — #27164453

Order 12736.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₁₅₉₂	C₂	1	1	yes	maximal, characteristic, complemented
2	#20263898	C₂	1	2	yes	maximal, complemented
3	#20263896	C₂	1	2	yes	maximal, complemented
4	#20263895	C₂	1	1	yes	maximal, characteristic, complemented
5	#135×C₁₉₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₇₉₆	—	2	1	no	—
7	D₁₅₉₂	—	2	1	no	—
8	D₁₅₉₂	C₂²	1	2	yes	—
9	C₁₉₉⋊C₁₆	C₂²	1	2	yes	complemented
10	#41×C₁₉₉	—	2	1	no	—
11	Q₁₆×C₁₉₉	C₂²	1	1	yes	characteristic, complemented
12	SD₁₆×C₁₉₉	—	2	1	no	—
13	D₈×C₁₉₉	C₂²	1	1	yes	characteristic, complemented
14	C₂×C₈×C₁₉₉	C₂²	1	1	yes	characteristic
15	C₂²×D₁₉₉	—	4	1	no	—
16	D₇₉₆	—	2	2	no	—
17	D₇₉₆	—	4	1	no	—
18	Q₈×C₁₉₉	—	2	1	no	—
19	D₄×C₁₉₉	—	2	1	no	—
20	D₄×C₁₉₉	—	4	1	no	—
21	C₂×C₄×C₁₉₉	—	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	—	4	1	no	—
26	C₂²×C₁₉₉	D₈	1	1	yes	characteristic
27	C₂×D₁₉₉	—	4	2	no	—
28	C₂×D₁₉₉	—	8	2	no	—
29	C₄×C₁₉₉	—	4	1	no	—
30	C₄×C₁₉₉	D₈	1	1	yes	characteristic
31	C₄×C₁₉₉	C₂×D₄	1	1	yes	characteristic
32	C₂×C₁₉₉	—	2	1	no	—
33	C₂×C₁₉₉	—	8	1	no	—
34	C₂×C₁₉₉	C₂×D₈	1	1	yes	characteristic
35	D₁₉₉	—	8	2	no	—
36	C₁₉₉	#509	1	1	yes	characteristic, complemented, Sylow
37	#509	—	199	1	no	maximal, Sylow
38	#135	D₁₉₉	1	1	yes	characteristic, complemented
39	C₂×D₈	—	199	1	no	—
40	SD₃₂	—	199	2	no	—
41	D₁₆	—	199	2	no	—
42	M₃₂	—	199	1	no	—
43	#41	—	2	1	no	—
44	C₂×D₄	—	398	1	no	—
45	Q₁₆	C₂×D₁₉₉	1	1	yes	characteristic, complemented
46	SD₁₆	—	2	1	no	—
47	D₈	—	398	1	no	—
48	D₈	—	199	2	no	—
49	D₈	C₂×D₁₉₉	1	1	yes	characteristic, complemented
50	C₂×C₈	C₂×D₁₉₉	1	1	yes	characteristic
51	C₁₆	—	199	2	no	—
52	C₂³	—	796	1	no	—
53	Q₈	—	2	1	no	—
54	D₄	—	2	1	no	—
55	D₄	—	4	1	no	—
56	D₄	—	796	1	no	—
57	D₄	—	398	2	no	—
58	C₂×C₄	—	4	1	no	—
59	C₂×C₄	C₂²⋊D₁₉₉	1	1	yes	characteristic
60	C₈	C₂²⋊D₁₉₉	1	1	yes	characteristic
61	C₈	C₂²×D₁₉₉	1	1	yes	characteristic
62	C₂²	—	4	1	no	—
63	C₂²	—	796	2	no	—
64	C₂²	—	1592	2	no	—
65	C₂²	#15691871	1	1	yes	characteristic
66	C₄	—	4	1	no	—
67	C₄	#15691871	1	1	yes	characteristic
68	C₄	C₂×C₂²⋊D₁₉₉	1	1	yes	characteristic
69	C₂	—	2	1	no	—
70	C₂	—	8	1	no	—
71	C₂	—	1592	2	no	—
72	C₂	C₂×#15691871	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×#15691871
commutator, cosocle	C₈×C₁₉₉	C₂³
Frattini	C₈	C₂²×D₁₉₉
Fitting	#135×C₁₉₉	C₂
socle	C₂×C₁₉₉	C₂×D₈
hypercenter	#135	D₁₉₉
hypocenter	C₁₉₉	#509
LCS	G ▷ C₈×C₁₉₉ ▷ C₄×C₁₉₉ ▷ C₂×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #135

Complements

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

N	H ≅ G/N	type
1. C₂×D₁₅₉₂	C₂	semidirect
2. #20263898	C₂	semidirect
3. #20263896	C₂	semidirect
4. #20263895	C₂	semidirect
5. #135×C₁₉₉	C₂	semidirect
9. C₁₉₉⋊C₁₆	C₂²	semidirect
11. Q₁₆×C₁₉₉	C₂²	semidirect
13. D₈×C₁₉₉	C₂²	semidirect
36. C₁₉₉	#509	semidirect
38. #135	D₁₉₉	semidirect
45. Q₁₆	C₂×D₁₉₉	semidirect
49. D₈	C₂×D₁₉₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₁₅₉₂	2
#20263898	2
#20263896	2
#20263895	2
#135×C₁₉₉	2
#509	199