Subgroups — #27135051

Order 12352.

86 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20259895	C₂	1	1	yes	maximal, characteristic
2	#20259814	C₂	1	1	yes	maximal, characteristic, complemented
3	#99×C₁₉₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₇₇₂	C₄	1	1	yes	characteristic, complemented
5	C₂×C₂²⋊D₁₉₃	—	2	1	no	—
6	C₂×C₂²⋊D₁₉₃	—	2	1	no	—
7	C₂²⋊Dic₁₉₃	—	2	1	no	—
8	C₄×Dic₁₉₃	C₄	1	1	yes	characteristic, complemented
9	C₂×D₄×C₁₉₃	C₂²	1	1	yes	characteristic
10	C₂²⋊C₄×C₁₉₃	—	2	1	no	—
11	C₂²×D₁₉₃	—	2	1	no	—
12	D₇₇₂	—	4	1	no	—
13	C₂²⋊D₁₉₃	—	4	2	no	—
14	C₂²⋊D₁₉₃	—	4	2	no	—
15	C₂×Dic₁₉₃	—	2	1	no	—
16	C₂×Dic₁₉₃	—	4	1	no	—
17	C₂³×C₁₉₃	D₄	1	1	yes	characteristic
18	C₂³×C₁₉₃	D₄	1	1	yes	characteristic
19	D₄×C₁₉₃	—	4	1	no	—
20	C₂×C₄×C₁₉₃	—	4	1	no	—
21	C₂×C₄×C₁₉₃	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₁₉₃	—	2	1	no	—
23	C₂²×C₁₉₃	—	2	1	no	—
24	C₂²×C₁₉₃	—	4	1	no	—
25	C₂²×C₁₉₃	—	4	1	no	—
26	C₂²×C₁₉₃	C₂²⋊C₄	1	1	yes	characteristic
27	C₂×D₁₉₃	—	4	1	no	—
28	C₂×D₁₉₃	—	4	1	no	—
29	C₂×D₁₉₃	—	4	1	no	—
30	C₄×C₁₉₃	—	2	1	no	—
31	C₄×C₁₉₃	—	8	1	no	—
32	Dic₁₉₃	—	2	2	no	—
33	Dic₁₉₃	—	8	1	no	—
34	C₂×C₁₉₃	—	2	1	no	—
35	C₂×C₁₉₃	—	4	1	no	—
36	C₂×C₁₉₃	—	4	1	no	—
37	C₂×C₁₉₃	#99	1	1	yes	characteristic
38	D₁₉₃	—	8	1	no	—
39	C₁₉₃	#353	1	1	yes	characteristic, complemented, Sylow
40	#353	—	193	1	no	maximal, Sylow
41	#127	—	193	1	no	—
42	#99	—	193	1	no	—
43	#99	D₁₉₃	1	1	yes	characteristic, complemented
44	C₂×D₄	—	386	1	no	—
45	C₂×D₄	—	386	1	no	—
46	C₂×D₄	—	193	1	no	—
47	C₂×D₄	C₂×D₁₉₃	1	1	yes	characteristic
48	C₂²⋊C₄	—	2	1	no	—
49	C₂²⋊C₄	—	386	1	no	—
50	C₄²	—	193	1	no	—
51	C₂³	—	386	1	no	—
52	C₂³	C₂²⋊D₁₉₃	1	1	yes	characteristic
53	C₂³	D₇₇₂	1	1	yes	characteristic
54	D₄	—	4	1	no	—
55	D₄	—	772	1	no	—
56	D₄	—	772	2	no	—
57	D₄	—	772	2	no	—
58	C₂×C₄	—	4	1	no	—
59	C₂×C₄	—	772	1	no	—
60	C₂×C₄	—	386	1	no	—
61	C₂×C₄	C₄×D₁₉₃	1	1	yes	characteristic
62	C₂²	—	2	1	no	—
63	C₂²	—	2	1	no	—
64	C₂²	—	4	1	no	—
65	C₂²	—	4	1	no	—
66	C₂²	—	772	1	no	—
67	C₂²	—	772	1	no	—
68	C₂²	—	772	1	no	—
69	C₂²	#15690532	1	1	yes	characteristic
70	C₄	—	2	1	no	—
71	C₄	—	8	1	no	—
72	C₄	—	1544	1	no	—
73	C₄	—	386	2	no	—
74	C₂	—	2	1	no	—
75	C₂	—	4	1	no	—
76	C₂	—	4	1	no	—
77	C₂	—	1544	1	no	—
78	C₂	#20259813	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20259813
commutator	C₂×C₄×C₁₉₃	C₂×C₄
Frattini	C₂×D₄	C₂×D₁₉₃
Fitting	#99×C₁₉₃	C₂
socle	C₂×C₁₉₃	#99
prime radical	#20259895	C₂
cosocle	C₂×D₄×C₁₉₃	C₂²
hypercenter	#99	D₁₉₃
hypocenter	C₁₉₃	#353
LCS	G ▷ C₂×C₄×C₁₉₃ ▷ C₂²×C₁₉₃ ▷ C₂×C₁₉₃ ▷ C₁₉₃
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×D₄ ◁ #99

Complements

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

N	H ≅ G/N	type
2. #20259814	C₂	semidirect
3. #99×C₁₉₃	C₂	semidirect
4. C₂×D₇₇₂	C₄	semidirect
8. C₄×Dic₁₉₃	C₄	semidirect
39. C₁₉₃	#353	semidirect
43. #99	D₁₉₃	semidirect

Maximal subgroups

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

subgroup	index
#20259895	2
#20259814	2
#99×C₁₉₃	2
#353	193