Subgroups — #27136064

Order 12352.

268 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20259945	C₂	1	2	yes	maximal, complemented
2	#20259927	C₂	1	6	yes	maximal, complemented
3	#20259921	C₂	1	3	yes	maximal, complemented
4	C₂×D₁₅₄₄	C₂	1	3	yes	maximal, complemented
5	#131×C₁₉₃	C₂	1	1	yes	maximal, characteristic, complemented
6	#15690556	—	2	2	no	—
7	#15690555	C₂²	1	6	yes	complemented
8	C₂×D₇₇₂	C₂²	1	6	yes	complemented
9	D₄×D₁₉₃	—	2	6	no	—
10	Dic₇₇₂	C₂²	1	1	yes	characteristic, complemented
11	#15690544	C₂²	1	6	yes	complemented
12	D₁₅₄₄	C₂²	1	3	yes	complemented
13	D₁₅₄₄	C₂²	1	6	yes	complemented
14	#41×C₁₉₃	C₂²	1	1	yes	characteristic
15	M₁₆×C₁₉₃	C₂²	1	3	yes	complemented
16	C₂×C₈×C₁₉₃	C₂²	1	3	yes	complemented
17	C₂²×D₁₉₃	—	2	6	no	—
18	Dic₃₈₆	C₂³	1	2	yes	—
19	D₇₇₂	—	2	6	no	—
20	D₇₇₂	C₂³	1	6	yes	—
21	C₂²⋊D₁₉₃	—	2	6	no	—
22	C₄×D₁₉₃	—	2	6	no	—
23	Q₈×C₁₉₃	D₄	1	1	yes	characteristic
24	D₄×C₁₉₃	D₄	1	3	yes	—
25	C₂×C₄×C₁₉₃	C₂³	1	3	yes	—
26	C₈×C₁₉₃	C₂³	1	1	yes	characteristic
27	C₈×C₁₉₃	C₂³	1	3	yes	—
28	C₂²×C₁₉₃	C₂×D₄	1	3	yes	—
29	C₂×D₁₉₃	—	2	6	no	—
30	C₂×D₁₉₃	—	8	6	no	—
31	C₄×C₁₉₃	C₂×D₄	1	3	yes	—
32	C₄×C₁₉₃	C₂⁴	1	1	yes	characteristic
33	Dic₁₉₃	—	2	2	no	—
34	C₂×C₁₉₃	—	2	3	no	—
35	C₂×C₁₉₃	C₂²×D₄	1	1	yes	characteristic
36	D₁₉₃	—	4	6	no	—
37	C₁₉₃	#576	1	1	yes	characteristic, complemented, Sylow
38	#576	—	193	1	no	maximal, Sylow
39	H(5,𝔽₂)	—	193	2	no	—
40	AGL(1,ℤ/8)	—	193	6	no	—
41	#135	—	193	3	no	—
42	C₂×D₈	—	193	3	no	—
43	#131	D₁₉₃	1	1	yes	characteristic, complemented
44	#41	—	386	2	no	—
45	#41	—	193	6	no	—
46	#41	C₂×D₁₉₃	1	1	yes	characteristic
47	C₂×D₄	—	386	6	no	—
48	C₂×D₄	—	193	6	no	—
49	Q₁₆	—	193	1	no	—
50	SD₁₆	—	193	6	no	—
51	D₈	—	193	3	no	—
52	D₈	—	193	6	no	—
53	M₁₆	C₂×D₁₉₃	1	3	yes	complemented
54	C₂×C₈	C₂×D₁₉₃	1	3	yes	complemented
55	C₂³	—	386	6	no	—
56	Q₈	—	193	2	no	—
57	Q₈	D₇₇₂	1	1	yes	characteristic
58	D₄	—	386	6	no	—
59	D₄	—	386	6	no	—
60	D₄	—	193	6	no	—
61	D₄	D₇₇₂	1	3	yes	—
62	C₂×C₄	—	386	6	no	—
63	C₂×C₄	C₂²×D₁₉₃	1	3	yes	—
64	C₈	C₂²×D₁₉₃	1	1	yes	characteristic
65	C₈	C₂²×D₁₉₃	1	3	yes	—
66	C₂²	—	1544	6	no	—
67	C₂²	—	386	6	no	—
68	C₂²	C₂×D₇₇₂	1	3	yes	—
69	C₄	—	386	2	no	—
70	C₄	C₂×D₇₇₂	1	3	yes	—
71	C₄	C₂³×D₁₉₃	1	1	yes	characteristic
72	C₂	—	2	3	no	—
73	C₂	—	772	6	no	—
74	C₂	C₂²×D₇₇₂	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂²×D₇₇₂
commutator, cosocle	C₄×C₁₉₃	C₂⁴
Frattini	C₄	C₂³×D₁₉₃
Fitting	#131×C₁₉₃	C₂
socle	C₂×C₁₉₃	C₂²×D₄
hypercenter	#131	D₁₉₃
hypocenter	C₁₉₃	#576
LCS	G ▷ C₄×C₁₉₃ ▷ C₂×C₁₉₃ ▷ C₁₉₃
UCS	C₁ ◁ C₂ ◁ #41 ◁ #131

Complements

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

N	H ≅ G/N	type
1. #20259945	C₂	semidirect
2. #20259927	C₂	semidirect
3. #20259921	C₂	semidirect
4. C₂×D₁₅₄₄	C₂	semidirect
5. #131×C₁₉₃	C₂	semidirect
7. #15690555	C₂²	semidirect
8. C₂×D₇₇₂	C₂²	semidirect
10. Dic₇₇₂	C₂²	semidirect
11. #15690544	C₂²	semidirect
12. D₁₅₄₄	C₂²	semidirect
13. D₁₅₄₄	C₂²	semidirect
15. M₁₆×C₁₉₃	C₂²	semidirect
16. C₂×C₈×C₁₉₃	C₂²	semidirect
37. C₁₉₃	#576	semidirect
43. #131	D₁₉₃	semidirect
53. M₁₆	C₂×D₁₉₃	semidirect
54. C₂×C₈	C₂×D₁₉₃	semidirect

Maximal subgroups

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

subgroup	index
#20259945	2
#20259927	2
#20259921	2
C₂×D₁₅₄₄	2
#131×C₁₉₃	2
#576	193