Subgroups — #27135429

Order 12352.

168 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20259944	C₂	1	1	yes	maximal, characteristic, complemented
2	#20259927	C₂	1	2	yes	maximal, complemented
3	#20259895	C₂	1	1	yes	maximal, characteristic, complemented
4	#20259825	C₂	1	2	yes	maximal, complemented
5	#100×C₁₉₃	C₂	1	1	yes	maximal, characteristic, complemented
6	#15690555	C₂²	1	2	yes	—
7	#15690554	—	2	2	no	—
8	C₂×D₇₇₂	C₂²	1	1	yes	characteristic, complemented
9	C₂×C₂²⋊D₁₉₃	—	2	2	no	—
10	C₂×C₂²⋊D₁₉₃	—	2	2	no	—
11	D₄×D₁₉₃	—	2	2	no	—
12	#15690544	—	2	2	no	—
13	D₁₅₄₄	—	2	2	no	—
14	C₄×Dic₁₉₃	C₂²	1	1	yes	characteristic, complemented
15	C₂×D₄×C₁₉₃	C₂²	1	1	yes	characteristic
16	M₁₆×C₁₉₃	C₂²	1	2	yes	complemented
17	C₂²×D₁₉₃	—	2	2	no	—
18	C₂²×D₁₉₃	—	2	1	no	—
19	Dic₃₈₆	D₄	1	2	yes	complemented
20	D₇₇₂	—	2	2	no	—
21	D₇₇₂	D₄	1	2	yes	complemented
22	C₂²⋊D₁₉₃	—	2	4	no	—
23	C₂²⋊D₁₉₃	—	2	2	no	—
24	C₂²⋊D₁₉₃	—	4	4	no	—
25	C₂×Dic₁₉₃	—	2	2	no	—
26	C₂×Dic₁₉₃	—	2	1	no	—
27	C₄×D₁₉₃	—	2	2	no	—
28	C₂³×C₁₉₃	D₄	1	2	yes	—
29	D₄×C₁₉₃	—	2	2	no	—
30	C₂×C₄×C₁₉₃	C₂³	1	1	yes	characteristic
31	C₈×C₁₉₃	—	2	2	no	—
32	C₂²×C₁₉₃	—	2	2	no	—
33	C₂²×C₁₉₃	—	4	2	no	—
34	C₂²×C₁₉₃	C₂×D₄	1	1	yes	characteristic
35	C₂×D₁₉₃	—	2	2	no	—
36	C₂×D₁₉₃	—	4	2	no	—
37	C₂×D₁₉₃	—	4	1	no	—
38	C₂×D₁₉₃	—	8	2	no	—
39	C₄×C₁₉₃	C₂×D₄	1	2	yes	—
40	Dic₁₉₃	—	2	2	no	—
41	Dic₁₉₃	—	2	2	no	—
42	C₂×C₁₉₃	—	2	1	no	—
43	C₂×C₁₉₃	—	4	2	no	—
44	C₂×C₁₉₃	#120	1	1	yes	characteristic
45	D₁₉₃	—	4	2	no	—
46	D₁₉₃	—	8	1	no	—
47	C₁₉₃	#453	1	1	yes	characteristic, complemented, Sylow
48	#453	—	193	1	no	maximal, Sylow
49	H(5,𝔽₂)	—	193	1	no	—
50	AGL(1,ℤ/8)	—	193	2	no	—
51	#127	—	193	1	no	—
52	#104	—	193	2	no	—
53	#100	D₁₉₃	1	1	yes	characteristic, complemented
54	#41	—	386	2	no	—
55	#41	—	193	2	no	—
56	C₂×D₄	—	386	2	no	—
57	C₂×D₄	—	386	2	no	—
58	C₂×D₄	—	386	2	no	—
59	C₂×D₄	—	193	1	no	—
60	C₂×D₄	C₂×D₁₉₃	1	1	yes	characteristic
61	SD₁₆	—	386	2	no	—
62	D₈	—	386	2	no	—
63	M₁₆	C₂×D₁₉₃	1	2	yes	complemented
64	C₄²	—	193	1	no	—
65	C₂³	—	386	2	no	—
66	C₂³	—	386	1	no	—
67	C₂³	D₇₇₂	1	2	yes	—
68	Q₈	—	193	2	no	—
69	D₄	—	2	2	no	—
70	D₄	—	772	4	no	—
71	D₄	—	386	4	no	—
72	D₄	—	386	2	no	—
73	D₄	—	386	2	no	—
74	D₄	—	193	2	no	—
75	C₂×C₄	—	386	2	no	—
76	C₂×C₄	—	386	1	no	—
77	C₂×C₄	—	386	2	no	—
78	C₂×C₄	C₂²×D₁₉₃	1	1	yes	characteristic
79	C₈	—	2	2	no	—
80	C₂²	—	2	2	no	—
81	C₂²	—	4	2	no	—
82	C₂²	—	772	2	no	—
83	C₂²	—	772	1	no	—
84	C₂²	—	1544	2	no	—
85	C₂²	—	386	2	no	—
86	C₂²	C₂×D₇₇₂	1	1	yes	characteristic
87	C₄	—	386	2	no	—
88	C₄	—	386	2	no	—
89	C₄	D₄×D₁₉₃	1	2	yes	—
90	C₂	—	2	1	no	—
91	C₂	—	4	2	no	—
92	C₂	—	772	2	no	—
93	C₂	—	1544	1	no	—
94	C₂	#20259869	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20259869
commutator, cosocle	C₂×C₄×C₁₉₃	C₂³
Frattini	C₂×C₄	C₂²×D₁₉₃
Fitting	#100×C₁₉₃	C₂
socle	C₂×C₁₉₃	#120
hypercenter	#100	D₁₉₃
hypocenter	C₁₉₃	#453
LCS	G ▷ C₂×C₄×C₁₉₃ ▷ C₂×C₁₉₃ ▷ C₁₉₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #100

Complements

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

N	H ≅ G/N	type
1. #20259944	C₂	semidirect
2. #20259927	C₂	semidirect
3. #20259895	C₂	semidirect
4. #20259825	C₂	semidirect
5. #100×C₁₉₃	C₂	semidirect
8. C₂×D₇₇₂	C₂²	semidirect
14. C₄×Dic₁₉₃	C₂²	semidirect
16. M₁₆×C₁₉₃	C₂²	semidirect
19. Dic₃₈₆	D₄	semidirect
21. D₇₇₂	D₄	semidirect
47. C₁₉₃	#453	semidirect
53. #100	D₁₉₃	semidirect
63. M₁₆	C₂×D₁₉₃	semidirect

Maximal subgroups

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

subgroup	index
#20259944	2
#20259927	2
#20259895	2
#20259825	2
#100×C₁₉₃	2
#453	193