Subgroups — #27136140

Order 12352.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15690564	C₂	1	1	yes	maximal, characteristic
2	C₂×#15690564	C₂	1	1	yes	maximal, characteristic
3	#20259855	C₂	1	1	yes	maximal, characteristic
4	#15690564	—	2	1	no	—
5	#15690564	—	2	1	no	—
6	#15690564	—	2	1	no	—
7	#15690564	C₄	1	2	yes	complemented
8	C₂×#13095400	—	2	1	no	—
9	C₂×#13095400	C₄	1	2	yes	complemented
10	C₂²×Dic₁₉₃	C₂²	1	1	yes	characteristic
11	#15690534	—	2	1	no	—
12	C₂²⋊Dic₁₉₃	C₄	1	1	yes	characteristic
13	C₄×Dic₁₉₃	—	2	1	no	—
14	C₂²⋊C₄×C₁₉₃	C₄	1	1	yes	characteristic
15	#13095400	—	2	2	no	—
16	#13095400	—	2	2	no	—
17	C₂×Dic₁₉₃	—	2	1	no	—
18	C₂×Dic₁₉₃	C₂×C₄	1	1	yes	characteristic
19	C₂×Dic₁₉₃	C₂×C₄	1	1	yes	characteristic
20	C₂×Dic₁₉₃	D₄	1	1	yes	characteristic
21	C₂×Dic₁₉₃	D₄	1	1	yes	characteristic
22	C₂×Dic₁₉₃	D₄	1	1	yes	characteristic
23	C₂×Dic₁₉₃	Q₈	1	1	yes	characteristic
24	C₂³×C₁₉₃	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×C₁₉₃	—	2	1	no	—
26	C₂²×C₁₉₃	—	4	1	no	—
27	C₂²×C₁₉₃	C₄²	1	1	yes	characteristic
28	C₂²×C₁₉₃	C₂²⋊C₄	1	1	yes	characteristic
29	C₂²×C₁₉₃	C₄⋊C₄	1	1	yes	characteristic
30	C₄×C₁₉₃	—	4	1	no	—
31	Dic₁₉₃	—	4	1	no	—
32	Dic₁₉₃	C₂²⋊C₄	1	1	yes	characteristic
33	Dic₁₉₃	C₂²⋊C₄	1	1	yes	characteristic
34	Dic₁₉₃	C₄⋊C₄	1	1	yes	characteristic
35	Dic₁₉₃	C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₁₉₃	—	2	1	no	—
37	C₂×C₁₉₃	—	2	1	no	—
38	C₂×C₁₉₃	—	2	1	no	—
39	C₂×C₁₉₃	#95	1	1	yes	characteristic
40	C₁₉₃	#344	1	1	yes	characteristic, complemented, Sylow
41	#344	—	193	1	no	maximal, Sylow
42	C₂×M₁₆	—	193	1	no	—
43	C₂×M₁₆	—	193	1	no	—
44	#117	—	193	1	no	—
45	C₂²×C₄	—	193	1	no	—
46	M₁₆	—	386	1	no	—
47	M₁₆	—	386	1	no	—
48	M₁₆	—	386	1	no	—
49	M₁₆	—	193	2	no	—
50	C₂×C₈	—	386	1	no	—
51	C₂×C₈	—	193	2	no	—
52	C₄⋊C₄	—	386	1	no	—
53	C₂²⋊C₄	—	193	1	no	—
54	C₂²⋊C₄	#11715871	1	1	yes	characteristic
55	C₄²	—	386	1	no	—
56	C₂³	C₂×#11715871	1	1	yes	characteristic
57	C₂×C₄	—	2	1	no	—
58	C₂×C₄	—	386	1	no	—
59	C₂×C₄	—	193	1	no	—
60	C₂×C₄	—	193	1	no	—
61	C₂×C₄	—	193	1	no	—
62	C₂×C₄	—	193	1	no	—
63	C₂×C₄	—	193	1	no	—
64	C₂×C₄	—	193	1	no	—
65	C₈	—	386	2	no	—
66	C₈	—	386	2	no	—
67	C₂²	—	4	1	no	—
68	C₂²	C₄×#11715871	1	1	yes	characteristic
69	C₂²	C₂²⋊#11715871	1	1	yes	characteristic
70	C₂²	C₄⋊#11715871	1	1	yes	characteristic
71	C₄	—	4	1	no	—
72	C₄	—	772	1	no	—
73	C₄	—	193	1	no	—
74	C₄	—	193	1	no	—
75	C₄	—	193	1	no	—
76	C₄	—	193	1	no	—
77	C₂	—	2	1	no	—
78	C₂	—	2	1	no	—
79	C₂	—	2	1	no	—
80	C₂	(C₂×#11715871)⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂	(C₂×#11715871)⋊C₄
commutator	C₂²×C₁₉₃	C₄²
Frattini	C₂³	C₂×#11715871
Fitting	C₂²⋊C₄×C₁₉₃	C₄
socle	C₂×C₁₉₃	#95
prime radical	C₂³×C₁₉₃	C₂×C₄
cosocle	C₂²×Dic₁₉₃	C₂²
hypercenter	C₂²⋊C₄	#11715871
hypocenter	C₁₉₃	#344
LCS	G ▷ C₂²×C₁₉₃ ▷ C₂×C₁₉₃ ▷ C₁₉₃
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
7. #15690564	C₄	semidirect
9. C₂×#13095400	C₄	semidirect
40. C₁₉₃	#344	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15690564	2
#20259855	2
#344	193