Subgroups — #27136129

Order 12352.

110 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15690565	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#11715871	C₂	1	1	yes	maximal, characteristic
3	#20259857	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11715871	—	2	1	no	—
5	#15690565	—	2	1	no	—
6	#15690565	C₄	1	2	yes	complemented
7	C₂×#13095400	—	2	1	no	—
8	C₄×#11715871	—	2	1	no	—
9	C₄×#11715871	C₄	1	2	yes	complemented
10	C₂×C₄×D₁₉₃	C₂²	1	1	yes	characteristic
11	C₄⋊Dic₁₉₃	C₄	1	1	yes	characteristic, complemented
12	#15690532	—	2	1	no	—
13	C₄×Dic₁₉₃	—	2	1	no	—
14	C₄⋊C₄×C₁₉₃	C₄	1	1	yes	characteristic, complemented
15	C₂×#11715871	—	2	2	no	—
16	C₂×#11715871	—	2	2	no	—
17	C₂×#11715871	—	2	2	no	—
18	#13095400	—	2	2	no	—
19	C₂²×D₁₉₃	D₄	1	1	yes	characteristic
20	C₂×Dic₁₉₃	—	2	1	no	—
21	C₂×Dic₁₉₃	D₄	1	1	yes	characteristic
22	C₄×D₁₉₃	C₂×C₄	1	1	yes	characteristic
23	C₄×D₁₉₃	C₂×C₄	1	1	yes	characteristic
24	C₄×D₁₉₃	D₄	1	1	yes	characteristic
25	C₄×D₁₉₃	Q₈	1	1	yes	characteristic
26	C₂×C₄×C₁₉₃	—	2	1	no	—
27	C₂×C₄×C₁₉₃	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₁₉₃	C₂²⋊C₄	1	1	yes	characteristic
29	C₂×D₁₉₃	—	2	1	no	—
30	C₂×D₁₉₃	—	2	1	no	—
31	C₂×D₁₉₃	C₂²⋊C₄	1	1	yes	characteristic
32	C₂×D₁₉₃	C₄⋊C₄	1	1	yes	characteristic
33	#11715871	—	2	4	no	—
34	C₄×C₁₉₃	—	4	1	no	—
35	C₄×C₁₉₃	C₄²	1	1	yes	characteristic
36	C₄×C₁₉₃	C₄⋊C₄	1	1	yes	characteristic
37	Dic₁₉₃	—	4	1	no	—
38	Dic₁₉₃	C₂²⋊C₄	1	1	yes	characteristic
39	Dic₁₉₃	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₁₉₃	#95	1	1	yes	characteristic
41	C₂×C₁₉₃	#104	1	1	yes	characteristic
42	C₂×C₁₉₃	#104	1	1	yes	characteristic
43	D₁₉₃	—	2	1	no	—
44	D₁₉₃	—	2	1	no	—
45	C₁₉₃	#339	1	1	yes	characteristic, complemented, Sylow
46	#339	—	193	1	no	maximal, Sylow
47	C₂×M₁₆	—	193	1	no	—
48	#117	—	193	1	no	—
49	C₂×C₄²	—	193	1	no	—
50	C₂²×C₄	—	386	1	no	—
51	C₂²×C₄	—	193	1	no	—
52	M₁₆	—	386	1	no	—
53	M₁₆	—	193	2	no	—
54	C₂×C₈	—	386	1	no	—
55	C₄⋊C₄	—	193	1	no	—
56	C₄⋊C₄	#11715871	1	1	yes	characteristic, complemented
57	C₂²⋊C₄	—	386	1	no	—
58	C₄²	—	386	1	no	—
59	C₄²	—	386	1	no	—
60	C₄²	—	193	2	no	—
61	C₂³	—	193	1	no	—
62	C₂×C₄	—	2	1	no	—
63	C₂×C₄	—	386	1	no	—
64	C₂×C₄	—	386	2	no	—
65	C₂×C₄	—	386	2	no	—
66	C₂×C₄	—	386	2	no	—
67	C₂×C₄	—	193	1	no	—
68	C₂×C₄	—	193	1	no	—
69	C₂×C₄	—	193	1	no	—
70	C₂×C₄	—	193	1	no	—
71	C₂×C₄	—	193	1	no	—
72	C₂×C₄	C₂×#11715871	1	1	yes	characteristic
73	C₈	—	386	2	no	—
74	C₂²	—	386	1	no	—
75	C₂²	—	386	1	no	—
76	C₂²	—	193	1	no	—
77	C₂²	—	193	1	no	—
78	C₂²	C₂²⋊#11715871	1	1	yes	characteristic
79	C₄	—	4	1	no	—
80	C₄	—	772	1	no	—
81	C₄	—	386	4	no	—
82	C₄	—	193	1	no	—
83	C₄	—	193	1	no	—
84	C₄	C₄×#11715871	1	1	yes	characteristic
85	C₄	C₄⋊#11715871	1	1	yes	characteristic
86	C₂	—	386	1	no	—
87	C₂	—	386	1	no	—
88	C₂	(C₂×#11715871)⋊C₄	1	1	yes	characteristic
89	C₂	#20259963	1	1	yes	characteristic
90	C₂	#20259964	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²⋊#11715871
commutator	C₄×C₁₉₃	C₄²
Frattini	C₂×C₄	C₂×#11715871
Fitting	C₄⋊C₄×C₁₉₃	C₄
socle	C₂²×C₁₉₃	C₂²⋊C₄
prime radical	C₂²×D₁₉₃	D₄
cosocle	C₂×C₄×D₁₉₃	C₂²
hypercenter	C₄⋊C₄	#11715871
hypocenter	C₁₉₃	#339
LCS	G ▷ C₄×C₁₉₃ ▷ C₂×C₁₉₃ ▷ C₁₉₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₄⋊C₄

Complements

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

N	H ≅ G/N	type
6. #15690565	C₄	semidirect
9. C₄×#11715871	C₄	semidirect
11. C₄⋊Dic₁₉₃	C₄	semidirect
14. C₄⋊C₄×C₁₉₃	C₄	semidirect
45. C₁₉₃	#339	semidirect
56. C₄⋊C₄	#11715871	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15690565	2
C₂×C₄×#11715871	2
#20259857	2
#339	193