Subgroups — #27135465

Order 12352.

120 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15690544	C₂	1	2	yes	maximal
2	#20259899	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×D₇₇₂	C₂	1	1	yes	maximal, characteristic
4	#20259824	C₂	1	2	yes	maximal, complemented
5	C₄⋊C₈×C₁₉₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Dic₃₈₆	C₂²	1	2	yes	—
7	C₂×D₇₇₂	C₂²	1	1	yes	characteristic
8	C₂×C₄×D₁₉₃	—	2	1	no	—
9	#15690544	—	2	4	no	—
10	C₄⋊Dic₁₉₃	—	2	1	no	—
11	C₄⋊Dic₁₉₃	—	2	1	no	—
12	C₄⋊Dic₁₉₃	C₂²	1	1	yes	characteristic
13	#15690532	—	2	1	no	—
14	C₂×C₈×C₁₉₃	C₂²	1	2	yes	—
15	C₄²×C₁₉₃	C₂²	1	1	yes	characteristic
16	C₂²×D₁₉₃	—	2	1	no	—
17	Dic₃₈₆	—	2	2	no	—
18	Dic₃₈₆	—	2	2	no	—
19	D₇₇₂	—	2	1	no	—
20	D₇₇₂	D₄	1	2	yes	—
21	C₂×Dic₁₉₃	—	2	1	no	—
22	C₂×Dic₁₉₃	—	2	2	no	—
23	C₄×D₁₉₃	—	4	2	no	—
24	C₂×C₄×C₁₉₃	D₄	1	1	yes	characteristic
25	C₂×C₄×C₁₉₃	D₄	1	1	yes	characteristic
26	C₂×C₄×C₁₉₃	C₂³	1	1	yes	characteristic
27	C₈×C₁₉₃	—	2	2	no	—
28	C₂²×C₁₉₃	C₂×D₄	1	1	yes	characteristic
29	C₂×D₁₉₃	—	2	2	no	—
30	C₂×D₁₉₃	—	4	1	no	—
31	C₂×D₁₉₃	—	4	1	no	—
32	C₄×C₁₉₃	—	2	1	no	—
33	C₄×C₁₉₃	SD₁₆	1	2	yes	complemented
34	C₄×C₁₉₃	C₂×D₄	1	1	yes	characteristic
35	C₄×C₁₉₃	#41	1	1	yes	characteristic
36	Dic₁₉₃	—	4	2	no	—
37	Dic₁₉₃	—	4	1	no	—
38	C₂×C₁₉₃	#121	1	1	yes	characteristic
39	C₂×C₁₉₃	C₂×SD₁₆	1	1	yes	characteristic
40	C₂×C₁₉₃	#137	1	1	yes	characteristic
41	D₁₉₃	—	4	2	no	—
42	C₁₉₃	#461	1	1	yes	characteristic, complemented, Sylow
43	#461	—	193	1	no	maximal, Sylow
44	C₂×SD₁₆	—	193	2	no	—
45	C₄⋊Q₈	—	193	1	no	—
46	C₄×D₄	—	193	1	no	—
47	C₄⋊C₈	D₁₉₃	1	1	yes	characteristic, complemented
48	#103	—	193	2	no	—
49	C₂×Q₈	—	193	2	no	—
50	C₂×D₄	—	193	1	no	—
51	C₂²×C₄	—	386	1	no	—
52	SD₁₆	—	386	4	no	—
53	C₂×C₈	C₂×D₁₉₃	1	2	yes	—
54	C₄⋊C₄	—	386	1	no	—
55	C₄⋊C₄	—	386	1	no	—
56	C₄⋊C₄	—	193	1	no	—
57	C₂²⋊C₄	—	386	1	no	—
58	C₄²	C₂×D₁₉₃	1	1	yes	characteristic
59	C₂³	—	386	1	no	—
60	Q₈	—	386	2	no	—
61	Q₈	—	386	2	no	—
62	D₄	—	386	1	no	—
63	D₄	—	193	2	no	—
64	C₂×C₄	—	772	2	no	—
65	C₂×C₄	—	386	1	no	—
66	C₂×C₄	—	386	2	no	—
67	C₂×C₄	D₇₇₂	1	1	yes	characteristic
68	C₂×C₄	D₇₇₂	1	1	yes	characteristic
69	C₂×C₄	C₂²×D₁₉₃	1	1	yes	characteristic
70	C₈	—	2	2	no	—
71	C₂²	—	772	1	no	—
72	C₂²	—	772	1	no	—
73	C₂²	—	386	2	no	—
74	C₂²	C₂×D₇₇₂	1	1	yes	characteristic
75	C₄	—	2	1	no	—
76	C₄	—	772	2	no	—
77	C₄	—	772	1	no	—
78	C₄	#15690544	1	2	yes	complemented
79	C₄	D₄×D₁₉₃	1	1	yes	characteristic
80	C₄	#15690556	1	1	yes	characteristic
81	C₂	—	772	2	no	—
82	C₂	#20259874	1	1	yes	characteristic
83	C₂	C₂×#15690544	1	1	yes	characteristic
84	C₂	#20259932	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₇₇₂
commutator, cosocle	C₂×C₄×C₁₉₃	C₂³
Frattini	C₂×C₄	C₂²×D₁₉₃
Fitting	C₄⋊C₈×C₁₉₃	C₂
socle	C₂²×C₁₉₃	C₂×D₄
prime radical	C₂×D₇₇₂	C₂²
hypercenter	C₄⋊C₈	D₁₉₃
hypocenter	C₁₉₃	#461
LCS	G ▷ C₂×C₄×C₁₉₃ ▷ C₂×C₁₉₃ ▷ C₁₉₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
2. #20259899	C₂	semidirect
4. #20259824	C₂	semidirect
5. C₄⋊C₈×C₁₉₃	C₂	semidirect
33. C₄×C₁₉₃	SD₁₆	semidirect
42. C₁₉₃	#461	semidirect
47. C₄⋊C₈	D₁₉₃	semidirect
78. C₄	#15690544	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15690544	2
#20259899	2
C₄×D₇₇₂	2
#20259824	2
C₄⋊C₈×C₁₉₃	2
#461	193