Subgroups — #27135351

Order 12352.

110 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15690545	C₂	1	1	yes	maximal, characteristic
2	Q₈×Dic₁₉₃	C₂	1	1	yes	maximal, characteristic
3	#20259864	C₂	1	1	yes	maximal, characteristic
4	#20259832	C₂	1	1	yes	maximal, characteristic
5	#20259824	C₂	1	1	yes	maximal, characteristic
6	C₈×Dic₁₉₃	C₂	1	1	yes	maximal, characteristic
7	#103×C₁₉₃	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₃₈₆	C₂²	1	1	yes	characteristic
9	#15690545	C₄	1	4	yes	complemented
10	C₂×#13095393	C₂²	1	1	yes	characteristic
11	C₄⋊Dic₁₉₃	—	2	1	no	—
12	C₄⋊Dic₁₉₃	C₂²	1	1	yes	characteristic
13	#15690534	—	2	1	no	—
14	C₄×Dic₁₉₃	—	2	1	no	—
15	C₄×Dic₁₉₃	—	2	1	no	—
16	C₄×Dic₁₉₃	C₂²	1	1	yes	characteristic
17	C₂×Q₈×C₁₉₃	C₂²	1	1	yes	characteristic
18	C₂×C₈×C₁₉₃	C₂²	1	1	yes	characteristic
19	C₄⋊C₄×C₁₉₃	C₂²	1	1	yes	characteristic
20	Dic₃₈₆	—	2	1	no	—
21	Dic₃₈₆	C₂×C₄	1	2	yes	—
22	C₂×Dic₁₉₃	—	2	1	no	—
23	C₂×Dic₁₉₃	—	2	1	no	—
24	C₂×Dic₁₉₃	D₄	1	1	yes	characteristic
25	C₂×Dic₁₉₃	D₄	1	1	yes	characteristic
26	#13095393	C₂×C₄	1	2	yes	—
27	Q₈×C₁₉₃	—	2	1	no	—
28	Q₈×C₁₉₃	C₂×C₄	1	2	yes	—
29	C₂×C₄×C₁₉₃	—	2	1	no	—
30	C₂×C₄×C₁₉₃	—	2	1	no	—
31	C₂×C₄×C₁₉₃	C₂³	1	1	yes	characteristic
32	C₈×C₁₉₃	—	2	1	no	—
33	C₂²×C₁₉₃	C₂×D₄	1	1	yes	characteristic
34	C₄×C₁₉₃	—	2	2	no	—
35	C₄×C₁₉₃	—	4	1	no	—
36	C₄×C₁₉₃	C₂²×C₄	1	1	yes	characteristic
37	C₄×C₁₉₃	#41	1	1	yes	characteristic
38	Dic₁₉₃	—	2	2	no	—
39	Dic₁₉₃	—	2	1	no	—
40	Dic₁₉₃	—	4	1	no	—
41	Dic₁₉₃	Q₁₆	1	2	yes	complemented
42	C₂×C₁₉₃	C₄×D₄	1	1	yes	characteristic
43	C₂×C₁₉₃	C₂×Q₁₆	1	1	yes	characteristic
44	C₂×C₁₉₃	#135	1	1	yes	characteristic
45	C₁₉₃	C₄×Q₁₆	1	1	yes	characteristic, complemented, Sylow
46	C₄×Q₁₆	—	193	1	no	maximal, Sylow
47	C₂×Q₁₆	—	193	1	no	—
48	C₄×Q₈	—	193	1	no	—
49	C₄×Q₈	—	193	1	no	—
50	#107	—	193	1	no	—
51	#103	—	193	1	no	—
52	#103	D₁₉₃	1	1	yes	characteristic
53	C₄×C₈	—	193	1	no	—
54	C₂×Q₈	—	193	1	no	—
55	C₂×Q₈	C₂×D₁₉₃	1	1	yes	characteristic
56	Q₁₆	—	193	4	no	—
57	C₂×C₈	—	193	1	no	—
58	C₂×C₈	C₂×D₁₉₃	1	1	yes	characteristic
59	C₄⋊C₄	—	386	1	no	—
60	C₄⋊C₄	—	386	1	no	—
61	C₄⋊C₄	—	193	1	no	—
62	C₄⋊C₄	C₂×D₁₉₃	1	1	yes	characteristic
63	C₄²	—	386	1	no	—
64	C₄²	—	386	1	no	—
65	C₄²	—	193	1	no	—
66	Q₈	—	2	1	no	—
67	Q₈	—	386	1	no	—
68	Q₈	—	193	2	no	—
69	Q₈	C₄×D₁₉₃	1	2	yes	—
70	C₂×C₄	—	2	1	no	—
71	C₂×C₄	—	2	1	no	—
72	C₂×C₄	—	386	1	no	—
73	C₂×C₄	—	386	1	no	—
74	C₂×C₄	—	193	1	no	—
75	C₂×C₄	—	193	1	no	—
76	C₂×C₄	C₂²×D₁₉₃	1	1	yes	characteristic
77	C₈	—	2	1	no	—
78	C₈	—	193	2	no	—
79	C₂²	D₄×D₁₉₃	1	1	yes	characteristic
80	C₄	—	2	2	no	—
81	C₄	—	4	1	no	—
82	C₄	—	772	1	no	—
83	C₄	—	386	2	no	—
84	C₄	—	386	1	no	—
85	C₄	—	193	2	no	—
86	C₄	C₂×C₄×D₁₉₃	1	1	yes	characteristic
87	C₄	#15690554	1	1	yes	characteristic
88	C₂	#20259859	1	1	yes	characteristic
89	C₂	Q₁₆×D₁₉₃	1	1	yes	characteristic
90	C₂	#20259920	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₁₉₃
commutator	C₄×C₁₉₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₁₉₃
Fitting	#103×C₁₉₃	C₂
socle, prime radical	C₂²×C₁₉₃	C₂×D₄
cosocle	C₂×C₄×C₁₉₃	C₂³
hypercenter	#103	D₁₉₃
hypocenter	C₁₉₃	C₄×Q₁₆
LCS	G ▷ C₄×C₁₉₃ ▷ C₂×C₁₉₃ ▷ C₁₉₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #103

Complements

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

N	H ≅ G/N	type
9. #15690545	C₄	semidirect
41. Dic₁₉₃	Q₁₆	semidirect
45. C₁₉₃	C₄×Q₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15690545	2
Q₈×Dic₁₉₃	2
#20259864	2
#20259832	2
#20259824	2
C₈×Dic₁₉₃	2
#103×C₁₉₃	2
C₄×Q₁₆	193