Subgroups — #27135427

Order 12352.

168 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20259924	C₂	1	2	yes	maximal, complemented
2	#20259895	C₂	1	1	yes	maximal, characteristic, complemented
3	#20259826	C₂	1	2	yes	maximal, complemented
4	#20259816	C₂	1	1	yes	maximal, characteristic, complemented
5	H(5,𝔽₂)×C₁₉₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₇₇₂	C₂²	1	1	yes	characteristic, complemented
7	C₂×C₂²⋊D₁₉₃	—	2	2	no	—
8	#15690542	—	2	2	no	—
9	#15690540	—	2	2	no	—
10	#15690539	C₂²	1	2	yes	complemented
11	C₄×Dic₁₉₃	C₂²	1	1	yes	characteristic, complemented
12	#41×C₁₉₃	—	2	2	no	—
13	#41×C₁₉₃	C₂²	1	2	yes	—
14	C₂×D₄×C₁₉₃	—	2	2	no	—
15	C₂×D₄×C₁₉₃	—	2	2	no	—
16	C₂×D₄×C₁₉₃	C₂²	1	1	yes	characteristic
17	C₂²×D₁₉₃	—	2	1	no	—
18	D₇₇₂	—	2	2	no	—
19	C₂²⋊D₁₉₃	—	4	4	no	—
20	C₂×Dic₁₉₃	—	2	1	no	—
21	#13095393	—	2	2	no	—
22	C₂³×C₁₉₃	—	2	2	no	—
23	C₂³×C₁₉₃	D₄	1	2	yes	—
24	Q₈×C₁₉₃	D₄	1	2	yes	complemented
25	D₄×C₁₉₃	—	2	4	no	—
26	D₄×C₁₉₃	—	2	2	no	—
27	D₄×C₁₉₃	—	2	2	no	—
28	D₄×C₁₉₃	D₄	1	2	yes	complemented
29	C₂×C₄×C₁₉₃	—	2	2	no	—
30	C₂×C₄×C₁₉₃	—	2	2	no	—
31	C₂×C₄×C₁₉₃	C₂³	1	1	yes	characteristic
32	C₂²×C₁₉₃	—	2	2	no	—
33	C₂²×C₁₉₃	—	2	2	no	—
34	C₂²×C₁₉₃	—	4	2	no	—
35	C₂²×C₁₉₃	—	8	2	no	—
36	C₂²×C₁₉₃	C₂×D₄	1	1	yes	characteristic
37	C₂×D₁₉₃	—	4	2	no	—
38	C₂×D₁₉₃	—	4	1	no	—
39	C₄×C₁₉₃	—	2	2	no	—
40	C₄×C₁₉₃	C₂×D₄	1	2	yes	—
41	Dic₁₉₃	—	2	2	no	—
42	C₂×C₁₉₃	—	2	1	no	—
43	C₂×C₁₉₃	—	4	2	no	—
44	C₂×C₁₉₃	—	4	2	no	—
45	C₂×C₁₉₃	#120	1	1	yes	characteristic
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)	D₁₉₃	1	1	yes	characteristic, complemented
50	AGL(1,ℤ/8)	—	193	2	no	—
51	#127	—	193	1	no	—
52	#104	—	193	2	no	—
53	#100	—	193	1	no	—
54	#41	—	2	2	no	—
55	#41	C₂×D₁₉₃	1	2	yes	—
56	C₂×D₄	—	2	2	no	—
57	C₂×D₄	—	2	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	—	193	2	no	—
64	C₄²	—	193	1	no	—
65	C₂³	—	2	2	no	—
66	C₂³	—	386	1	no	—
67	C₂³	C₂²⋊D₁₉₃	1	2	yes	—
68	Q₈	C₂²⋊D₁₉₃	1	2	yes	complemented
69	D₄	—	2	4	no	—
70	D₄	—	2	2	no	—
71	D₄	—	2	2	no	—
72	D₄	—	772	4	no	—
73	D₄	—	386	2	no	—
74	D₄	C₂²⋊D₁₉₃	1	2	yes	complemented
75	C₂×C₄	—	2	2	no	—
76	C₂×C₄	—	2	2	no	—
77	C₂×C₄	—	386	1	no	—
78	C₂×C₄	C₂²×D₁₉₃	1	1	yes	characteristic
79	C₈	—	386	2	no	—
80	C₂²	—	2	2	no	—
81	C₂²	—	2	2	no	—
82	C₂²	—	4	2	no	—
83	C₂²	—	772	2	no	—
84	C₂²	—	772	1	no	—
85	C₂²	—	8	2	no	—
86	C₂²	C₂×C₂²⋊D₁₉₃	1	1	yes	characteristic
87	C₄	—	2	2	no	—
88	C₄	—	386	2	no	—
89	C₄	C₂×C₂²⋊D₁₉₃	1	2	yes	—
90	C₂	—	2	1	no	—
91	C₂	—	4	2	no	—
92	C₂	—	4	2	no	—
93	C₂	—	1544	1	no	—
94	C₂	#20259868	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20259868
commutator, cosocle	C₂×C₄×C₁₉₃	C₂³
Frattini	C₂×C₄	C₂²×D₁₉₃
Fitting	H(5,𝔽₂)×C₁₉₃	C₂
socle	C₂×C₁₉₃	#120
hypercenter	H(5,𝔽₂)	D₁₉₃
hypocenter	C₁₉₃	#453
LCS	G ▷ C₂×C₄×C₁₉₃ ▷ C₂×C₁₉₃ ▷ C₁₉₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ H(5,𝔽₂)

Complements

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

N	H ≅ G/N	type
1. #20259924	C₂	semidirect
2. #20259895	C₂	semidirect
3. #20259826	C₂	semidirect
4. #20259816	C₂	semidirect
5. H(5,𝔽₂)×C₁₉₃	C₂	semidirect
6. C₂×D₇₇₂	C₂²	semidirect
10. #15690539	C₂²	semidirect
11. C₄×Dic₁₉₃	C₂²	semidirect
24. Q₈×C₁₉₃	D₄	semidirect
28. D₄×C₁₉₃	D₄	semidirect
47. C₁₉₃	#453	semidirect
49. H(5,𝔽₂)	D₁₉₃	semidirect
68. Q₈	C₂²⋊D₁₉₃	semidirect
74. D₄	C₂²⋊D₁₉₃	semidirect

Maximal subgroups

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

subgroup	index
#20259924	2
#20259895	2
#20259826	2
#20259816	2
H(5,𝔽₂)×C₁₉₃	2
#453	193