Subgroups — #27135446

Order 12352.

198 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20259867	C₂	1	3	yes	maximal, complemented
2	#20259814	C₂	1	3	yes	maximal, complemented
3	H(5,𝔽₂)×C₁₉₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂³×D₁₉₃	C₂²	1	1	yes	characteristic, complemented
5	C₂×C₂²⋊D₁₉₃	—	2	3	no	—
6	C₂²⋊Dic₁₉₃	C₂²	1	3	yes	complemented
7	#15690532	—	2	3	no	—
8	#41×C₁₉₃	—	2	3	no	—
9	C₂×D₄×C₁₉₃	—	2	3	no	—
10	C₂×D₄×C₁₉₃	C₂²	1	3	yes	—
11	C₂²×D₁₉₃	—	2	3	no	—
12	C₂²×D₁₉₃	—	4	2	no	—
13	C₂²⋊D₁₉₃	—	4	6	no	—
14	C₂×Dic₁₉₃	—	2	3	no	—
15	C₂³×C₁₉₃	—	2	1	no	—
16	C₂³×C₁₉₃	D₄	1	3	yes	complemented
17	C₂³×C₁₉₃	C₂³	1	1	yes	characteristic
18	Q₈×C₁₉₃	—	2	1	no	—
19	D₄×C₁₉₃	—	2	3	no	—
20	D₄×C₁₉₃	—	2	6	no	—
21	C₂×C₄×C₁₉₃	—	2	3	no	—
22	C₂×C₄×C₁₉₃	D₄	1	3	yes	complemented
23	C₂²×C₁₉₃	—	2	3	no	—
24	C₂²×C₁₉₃	—	4	1	no	—
25	C₂²×C₁₉₃	—	4	3	no	—
26	C₂²×C₁₉₃	—	8	1	no	—
27	C₂²×C₁₉₃	C₂×D₄	1	3	yes	—
28	C₂×D₁₉₃	—	2	6	no	—
29	C₂×D₁₉₃	—	4	3	no	—
30	C₂×D₁₉₃	—	4	1	no	—
31	C₄×C₁₉₃	—	2	3	no	—
32	Dic₁₉₃	—	4	3	no	—
33	C₂×C₁₉₃	—	2	3	no	—
34	C₂×C₁₉₃	—	4	3	no	—
35	C₂×C₁₉₃	#120	1	1	yes	characteristic
36	D₁₉₃	—	4	2	no	—
37	C₁₉₃	B(4,𝔽₂)	1	1	yes	characteristic, complemented, Sylow
38	B(4,𝔽₂)	—	193	1	no	maximal, Sylow
39	H(5,𝔽₂)	D₁₉₃	1	1	yes	characteristic, complemented
40	#120	—	193	3	no	—
41	#99	—	193	3	no	—
42	C₂⁴	—	193	1	no	—
43	#41	—	2	3	no	—
44	C₂×D₄	—	2	3	no	—
45	C₂×D₄	—	386	3	no	—
46	C₂×D₄	C₂×D₁₉₃	1	3	yes	—
47	C₂²⋊C₄	—	386	3	no	—
48	C₂²⋊C₄	—	193	3	no	—
49	C₂³	—	2	1	no	—
50	C₂³	—	772	2	no	—
51	C₂³	—	386	3	no	—
52	C₂³	C₂²⋊D₁₉₃	1	3	yes	complemented
53	C₂³	C₂²×D₁₉₃	1	1	yes	characteristic
54	Q₈	—	2	1	no	—
55	D₄	—	2	3	no	—
56	D₄	—	2	6	no	—
57	D₄	—	772	6	no	—
58	C₂×C₄	—	2	3	no	—
59	C₂×C₄	—	386	3	no	—
60	C₂×C₄	C₂²⋊D₁₉₃	1	3	yes	complemented
61	C₂²	—	2	3	no	—
62	C₂²	—	4	1	no	—
63	C₂²	—	4	3	no	—
64	C₂²	—	772	3	no	—
65	C₂²	—	772	1	no	—
66	C₂²	—	8	1	no	—
67	C₂²	—	386	6	no	—
68	C₂²	C₂×C₂²⋊D₁₉₃	1	3	yes	—
69	C₄	—	2	3	no	—
70	C₄	—	772	3	no	—
71	C₂	—	2	3	no	—
72	C₂	—	4	3	no	—
73	C₂	—	772	2	no	—
74	C₂	#20259868	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20259868
commutator, cosocle	C₂³×C₁₉₃	C₂³
Frattini	C₂³	C₂²×D₁₉₃
Fitting	H(5,𝔽₂)×C₁₉₃	C₂
socle	C₂×C₁₉₃	#120
hypercenter	H(5,𝔽₂)	D₁₉₃
hypocenter	C₁₉₃	B(4,𝔽₂)
LCS	G ▷ C₂³×C₁₉₃ ▷ C₂×C₁₉₃ ▷ C₁₉₃
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ H(5,𝔽₂)

Complements

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

N	H ≅ G/N	type
1. #20259867	C₂	semidirect
2. #20259814	C₂	semidirect
3. H(5,𝔽₂)×C₁₉₃	C₂	semidirect
4. C₂³×D₁₉₃	C₂²	semidirect
6. C₂²⋊Dic₁₉₃	C₂²	semidirect
16. C₂³×C₁₉₃	D₄	semidirect
22. C₂×C₄×C₁₉₃	D₄	semidirect
37. C₁₉₃	B(4,𝔽₂)	semidirect
39. H(5,𝔽₂)	D₁₉₃	semidirect
52. C₂³	C₂²⋊D₁₉₃	semidirect
60. C₂×C₄	C₂²⋊D₁₉₃	semidirect

Maximal subgroups

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

subgroup	index
#20259867	2
#20259814	2
H(5,𝔽₂)×C₁₉₃	2
B(4,𝔽₂)	193