Subgroups — #27136009

Order 12352.

196 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20259897	C₂	1	2	yes	maximal, complemented
2	#20259892	C₂	1	4	yes	maximal, complemented
3	#20259891	C₂	1	2	yes	maximal, complemented
4	#20259880	C₂	1	2	yes	maximal
5	#20259876	C₂	1	4	yes	maximal, complemented
6	#124×C₁₉₃	C₂	1	1	yes	maximal, characteristic
7	C₂×Dic₃₈₆	C₂²	1	2	yes	—
8	C₂²×Dic₁₉₃	C₂²	1	2	yes	—
9	C₄⋊Dic₁₉₃	C₂²	1	2	yes	—
10	C₄⋊Dic₁₉₃	C₂²	1	2	yes	—
11	#15690534	C₂²	1	4	yes	—
12	#15690534	C₂²	1	4	yes	—
13	#15690534	C₂²	1	4	yes	—
14	C₂²⋊Dic₁₉₃	C₂²	1	4	yes	—
15	C₂²⋊Dic₁₉₃	C₂²	1	2	yes	—
16	C₄×Dic₁₉₃	C₂²	1	2	yes	—
17	C₂×Q₈×C₁₉₃	C₂²	1	1	yes	characteristic
18	C₂×D₄×C₁₉₃	C₂²	1	1	yes	characteristic
19	C₂²⋊C₄×C₁₉₃	C₂²	1	4	yes	—
20	C₄²×C₁₉₃	C₂²	1	1	yes	characteristic
21	Dic₃₈₆	—	4	2	no	—
22	C₂×Dic₁₉₃	—	4	2	no	—
23	C₂×Dic₁₉₃	C₂³	1	4	yes	—
24	C₂×Dic₁₉₃	C₂³	1	4	yes	—
25	C₂³×C₁₉₃	C₂³	1	2	yes	—
26	Q₈×C₁₉₃	—	4	1	no	—
27	D₄×C₁₉₃	—	4	1	no	—
28	C₂×C₄×C₁₉₃	C₂³	1	2	yes	—
29	C₂×C₄×C₁₉₃	C₂³	1	1	yes	characteristic
30	C₂×C₄×C₁₉₃	C₂³	1	2	yes	—
31	C₂²×C₁₉₃	—	2	2	no	—
32	C₂²×C₁₉₃	—	2	4	no	—
33	C₂²×C₁₉₃	C₂⁴	1	1	yes	characteristic
34	C₄×C₁₉₃	—	2	1	no	—
35	C₄×C₁₉₃	—	2	2	no	—
36	C₄×C₁₉₃	—	2	2	no	—
37	Dic₁₉₃	—	2	4	no	—
38	Dic₁₉₃	—	2	4	no	—
39	C₂×C₁₉₃	—	4	2	no	—
40	C₂×C₁₉₃	H(5,𝔽₂)	1	1	yes	characteristic
41	C₂×C₁₉₃	2₋¹⁺⁴	1	2	yes	—
42	C₁₉₃	#563	1	1	yes	characteristic, complemented, Sylow
43	#563	—	193	1	no	maximal, Sylow
44	C₄⋊Q₈	—	193	2	no	—
45	#126	—	193	4	no	—
46	#125	—	193	2	no	—
47	#124	D₁₉₃	1	1	yes	characteristic
48	#123	—	193	2	no	—
49	C₂²⋊Q₈	—	193	4	no	—
50	C₂×Q₈	—	193	2	no	—
51	C₂×Q₈	C₂×D₁₉₃	1	1	yes	characteristic
52	C₂×D₄	C₂×D₁₉₃	1	1	yes	characteristic
53	C₂²×C₄	—	193	2	no	—
54	C₄⋊C₄	—	193	4	no	—
55	C₄⋊C₄	—	193	4	no	—
56	C₄⋊C₄	—	193	4	no	—
57	C₄⋊C₄	—	193	2	no	—
58	C₄⋊C₄	—	193	2	no	—
59	C₂²⋊C₄	—	193	4	no	—
60	C₂²⋊C₄	—	193	2	no	—
61	C₂²⋊C₄	C₂×D₁₉₃	1	4	yes	—
62	C₄²	—	193	2	no	—
63	C₄²	C₂×D₁₉₃	1	1	yes	characteristic
64	C₂³	C₂²×D₁₉₃	1	2	yes	—
65	Q₈	—	4	1	no	—
66	Q₈	—	772	2	no	—
67	D₄	—	4	1	no	—
68	C₂×C₄	—	772	2	no	—
69	C₂×C₄	—	193	4	no	—
70	C₂×C₄	—	193	4	no	—
71	C₂×C₄	C₂²×D₁₉₃	1	2	yes	—
72	C₂×C₄	C₂²×D₁₉₃	1	2	yes	—
73	C₂×C₄	C₂²×D₁₉₃	1	1	yes	characteristic
74	C₂²	—	2	2	no	—
75	C₂²	—	2	4	no	—
76	C₂²	C₂³×D₁₉₃	1	1	yes	characteristic
77	C₄	—	2	1	no	—
78	C₄	—	2	2	no	—
79	C₄	—	2	2	no	—
80	C₄	—	386	4	no	—
81	C₄	—	386	4	no	—
82	C₂	—	4	2	no	—
83	C₂	#20259944	1	1	yes	characteristic
84	C₂	#20259946	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₉₃
commutator, socle, cosocle	C₂²×C₁₉₃	C₂⁴
Fitting	#124×C₁₉₃	C₂
prime radical	C₂×D₄×C₁₉₃	C₂²
hypercenter	#124	D₁₉₃
hypocenter	C₁₉₃	#563
LCS	G ▷ C₂²×C₁₉₃ ▷ C₁₉₃
UCS	C₁ ◁ C₂² ◁ #124

Complements

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

N	H ≅ G/N	type
1. #20259897	C₂	semidirect
2. #20259892	C₂	semidirect
3. #20259891	C₂	semidirect
5. #20259876	C₂	semidirect
42. C₁₉₃	#563	semidirect

Maximal subgroups

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

subgroup	index
#20259897	2
#20259892	2
#20259891	2
#20259880	2
#20259876	2
#124×C₁₉₃	2
#563	193