Subgroups — #27136102

Order 12352.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²⋊#13095400	C₂	1	2	yes	maximal, complemented
2	C₂×C₂²⋊Dic₁₉₃	C₂	1	1	yes	maximal, characteristic
3	C₂×#13095400	—	2	2	no	—
4	C₂²×Dic₁₉₃	C₄	1	1	yes	characteristic
5	C₂²×Dic₁₉₃	C₂²	1	1	yes	characteristic
6	C₂²⋊Dic₁₉₃	—	2	2	no	—
7	C₂⁴×C₁₉₃	C₄	1	1	yes	characteristic
8	#13095400	—	4	2	no	—
9	C₂×Dic₁₉₃	—	2	1	no	—
10	C₂×Dic₁₉₃	—	4	1	no	—
11	C₂×Dic₁₉₃	—	4	1	no	—
12	C₂×Dic₁₉₃	D₄	1	2	yes	—
13	C₂³×C₁₉₃	—	2	1	no	—
14	C₂³×C₁₉₃	—	2	1	no	—
15	C₂³×C₁₉₃	—	4	1	no	—
16	C₂³×C₁₉₃	—	4	1	no	—
17	C₂³×C₁₉₃	C₈	1	2	yes	complemented
18	C₂³×C₁₉₃	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₁₉₃	—	2	1	no	—
20	C₂²×C₁₉₃	—	2	1	no	—
21	C₂²×C₁₉₃	—	2	2	no	—
22	C₂²×C₁₉₃	—	4	2	no	—
23	C₂²×C₁₉₃	—	4	2	no	—
24	C₂²×C₁₉₃	—	4	1	no	—
25	C₂²×C₁₉₃	—	4	1	no	—
26	C₂²×C₁₉₃	C₂²⋊C₄	1	1	yes	characteristic
27	C₂²×C₁₉₃	C₂×C₈	1	1	yes	characteristic
28	C₂²×C₁₉₃	M₁₆	1	1	yes	characteristic
29	Dic₁₉₃	—	2	2	no	—
30	Dic₁₉₃	—	4	1	no	—
31	C₂×C₁₉₃	—	2	1	no	—
32	C₂×C₁₉₃	—	2	1	no	—
33	C₂×C₁₉₃	—	4	2	no	—
34	C₂×C₁₉₃	C₂²⋊C₈	1	1	yes	characteristic
35	C₂×C₁₉₃	#99	1	1	yes	characteristic
36	C₂×C₁₉₃	#100	1	1	yes	characteristic
37	C₁₉₃	#323	1	1	yes	characteristic, complemented, Sylow
38	#323	—	193	1	no	maximal, Sylow
39	C₂×C₂²⋊C₄	—	193	1	no	—
40	C₂²⋊C₈	—	193	2	no	—
41	C₂⁴	#11715871	1	1	yes	characteristic
42	C₂²×C₄	—	193	1	no	—
43	C₂²×C₄	—	193	1	no	—
44	C₂×C₈	—	386	2	no	—
45	C₂²⋊C₄	—	386	2	no	—
46	C₂³	—	2	1	no	—
47	C₂³	—	2	1	no	—
48	C₂³	—	4	1	no	—
49	C₂³	—	4	1	no	—
50	C₂³	#13095400	1	2	yes	complemented
51	C₂³	C₂×#11715871	1	1	yes	characteristic
52	C₂×C₄	—	772	1	no	—
53	C₂×C₄	—	772	1	no	—
54	C₂×C₄	—	386	1	no	—
55	C₂×C₄	—	193	2	no	—
56	C₈	—	772	2	no	—
57	C₂²	—	2	1	no	—
58	C₂²	—	2	1	no	—
59	C₂²	—	2	2	no	—
60	C₂²	—	4	2	no	—
61	C₂²	—	4	2	no	—
62	C₂²	—	4	1	no	—
63	C₂²	—	4	1	no	—
64	C₂²	C₂²⋊#11715871	1	1	yes	characteristic
65	C₂²	C₂×#13095400	1	1	yes	characteristic
66	C₂²	#15690564	1	1	yes	characteristic
67	C₄	—	772	1	no	—
68	C₄	—	386	2	no	—
69	C₂	—	2	1	no	—
70	C₂	—	2	1	no	—
71	C₂	—	4	2	no	—
72	C₂	C₂²⋊#13095400	1	1	yes	characteristic
73	C₂	#20259957	1	1	yes	characteristic
74	C₂	#20259959	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²⋊#11715871
commutator	C₂²×C₁₉₃	C₂×C₈
Frattini	C₂³	C₂×#11715871
Fitting, prime radical	C₂⁴×C₁₉₃	C₄
socle	C₂²×C₁₉₃	C₂²⋊C₄
cosocle	C₂²×Dic₁₉₃	C₂²
hypercenter	C₂⁴	#11715871
hypocenter	C₁₉₃	#323
LCS	G ▷ C₂²×C₁₉₃ ▷ C₂×C₁₉₃ ▷ C₁₉₃
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂⁴

Complements

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

N	H ≅ G/N	type
1. C₂²⋊#13095400	C₂	semidirect
17. C₂³×C₁₉₃	C₈	semidirect
37. C₁₉₃	#323	semidirect
50. C₂³	#13095400	semidirect

Maximal subgroups

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

subgroup	index
C₂²⋊#13095400	2
C₂×C₂²⋊Dic₁₉₃	2
#323	193