Subgroups — #27136007

Order 12352.

220 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20259887	C₂	1	2	yes	maximal, complemented
2	#20259884	C₂	1	4	yes	maximal, complemented
3	#20259877	C₂	1	4	yes	maximal, complemented
4	#20259874	C₂	1	4	yes	maximal, complemented
5	#125×C₁₉₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Dic₃₈₆	C₂²	1	2	yes	—
7	C₂×D₇₇₂	C₂²	1	4	yes	—
8	C₂×D₇₇₂	C₂²	1	2	yes	—
9	C₂×C₄×D₁₉₃	C₂²	1	4	yes	—
10	#15690534	C₂²	1	4	yes	—
11	#15690532	C₂²	1	4	yes	—
12	#15690532	C₂²	1	4	yes	—
13	#15690532	C₂²	1	4	yes	—
14	C₄⋊C₄×C₁₉₃	C₂²	1	4	yes	—
15	C₄⋊C₄×C₁₉₃	C₂²	1	2	yes	—
16	C₄²×C₁₉₃	C₂²	1	1	yes	characteristic
17	C₂²×D₁₉₃	C₂³	1	4	yes	—
18	Dic₃₈₆	—	4	2	no	—
19	D₇₇₂	—	4	4	no	—
20	D₇₇₂	—	4	2	no	—
21	C₂×Dic₁₉₃	C₂³	1	4	yes	—
22	C₄×D₁₉₃	—	4	4	no	—
23	C₂×C₄×C₁₉₃	C₂³	1	1	yes	characteristic
24	C₂×C₄×C₁₉₃	C₂³	1	2	yes	—
25	C₂×C₄×C₁₉₃	C₂³	1	4	yes	—
26	C₂²×C₁₉₃	C₂⁴	1	1	yes	characteristic
27	C₂×D₁₉₃	—	2	4	no	—
28	C₂×D₁₉₃	—	2	4	no	—
29	C₂×D₁₉₃	—	2	4	no	—
30	C₄×C₁₉₃	—	2	4	no	—
31	C₄×C₁₉₃	—	2	2	no	—
32	C₄×C₁₉₃	—	2	1	no	—
33	Dic₁₉₃	—	2	4	no	—
34	C₂×C₁₉₃	H(5,𝔽₂)	1	2	yes	—
35	C₂×C₁₉₃	2₋¹⁺⁴	1	1	yes	characteristic
36	D₁₉₃	—	4	4	no	—
37	C₁₉₃	#562	1	1	yes	characteristic, complemented, Sylow
38	#562	—	193	1	no	maximal, Sylow
39	#125	D₁₉₃	1	1	yes	characteristic, complemented
40	#124	—	193	2	no	—
41	#123	—	193	4	no	—
42	C₂²⋊Q₈	—	193	4	no	—
43	#121	—	193	4	no	—
44	C₂×Q₈	—	193	2	no	—
45	C₂×D₄	—	193	4	no	—
46	C₂×D₄	—	193	2	no	—
47	C₂²×C₄	—	193	4	no	—
48	C₄⋊C₄	—	193	4	no	—
49	C₄⋊C₄	C₂×D₁₉₃	1	4	yes	—
50	C₄⋊C₄	C₂×D₁₉₃	1	2	yes	—
51	C₂²⋊C₄	—	193	4	no	—
52	C₂²⋊C₄	—	193	4	no	—
53	C₂²⋊C₄	—	193	4	no	—
54	C₄²	C₂×D₁₉₃	1	1	yes	characteristic
55	C₂³	—	193	4	no	—
56	Q₈	—	772	2	no	—
57	D₄	—	772	4	no	—
58	D₄	—	772	2	no	—
59	C₂×C₄	—	772	4	no	—
60	C₂×C₄	—	193	4	no	—
61	C₂×C₄	C₂²×D₁₉₃	1	1	yes	characteristic
62	C₂×C₄	C₂²×D₁₉₃	1	2	yes	—
63	C₂×C₄	C₂²×D₁₉₃	1	4	yes	—
64	C₂²	—	386	4	no	—
65	C₂²	—	386	4	no	—
66	C₂²	—	386	4	no	—
67	C₂²	C₂³×D₁₉₃	1	1	yes	characteristic
68	C₄	—	2	4	no	—
69	C₄	—	2	2	no	—
70	C₄	—	2	1	no	—
71	C₄	—	386	4	no	—
72	C₂	—	772	4	no	—
73	C₂	#20259945	1	2	yes	—
74	C₂	#20259947	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₉₃
commutator, socle, cosocle	C₂²×C₁₉₃	C₂⁴
Fitting	#125×C₁₉₃	C₂
hypercenter	#125	D₁₉₃
hypocenter	C₁₉₃	#562
LCS	G ▷ C₂²×C₁₉₃ ▷ C₁₉₃
UCS	C₁ ◁ C₂² ◁ #125

Complements

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

N	H ≅ G/N	type
1. #20259887	C₂	semidirect
2. #20259884	C₂	semidirect
3. #20259877	C₂	semidirect
4. #20259874	C₂	semidirect
5. #125×C₁₉₃	C₂	semidirect
37. C₁₉₃	#562	semidirect
39. #125	D₁₉₃	semidirect

Maximal subgroups

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

subgroup	index
#20259887	2
#20259884	2
#20259877	2
#20259874	2
#125×C₁₉₃	2
#562	193