Subgroups — #27135263

Order 12352.

190 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₇₇₂	C₂	1	1	yes	maximal, characteristic
2	C₂×#15690539	C₂	1	1	yes	maximal, characteristic, complemented
3	#20259819	C₂	1	4	yes	maximal, complemented
4	#117×C₁₉₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂³×D₁₉₃	—	2	1	no	—
6	C₂×D₇₇₂	—	2	1	no	—
7	C₂×D₇₇₂	—	2	2	no	—
8	C₂×D₇₇₂	C₄	1	4	yes	complemented
9	C₂×D₇₇₂	C₂²	1	2	yes	—
10	#15690539	—	2	2	no	—
11	C₂×#13095393	C₂²	1	2	yes	complemented
12	C₂²×C₄×C₁₉₃	C₂²	1	1	yes	characteristic
13	C₄⋊C₄×C₁₉₃	C₂²	1	2	yes	complemented
14	C₂²⋊C₄×C₁₉₃	—	2	1	no	—
15	C₄²×C₁₉₃	—	2	1	no	—
16	C₂²×D₁₉₃	—	2	4	no	—
17	C₂²×D₁₉₃	—	2	2	no	—
18	C₂²×D₁₉₃	—	4	4	no	—
19	D₇₇₂	—	2	4	no	—
20	D₇₇₂	—	2	2	no	—
21	D₇₇₂	C₂×C₄	1	4	yes	—
22	#13095393	—	2	2	no	—
23	C₂³×C₁₉₃	D₄	1	1	yes	characteristic
24	C₂×C₄×C₁₉₃	—	2	2	no	—
25	C₂×C₄×C₁₉₃	C₂×C₄	1	2	yes	—
26	C₂×C₄×C₁₉₃	D₄	1	1	yes	characteristic
27	C₂×C₄×C₁₉₃	D₄	1	2	yes	—
28	C₂×C₄×C₁₉₃	C₂³	1	1	yes	characteristic
29	C₂²×C₁₉₃	—	2	2	no	—
30	C₂²×C₁₉₃	C₂²⋊C₄	1	2	yes	—
31	C₂²×C₁₉₃	C₂×D₄	1	1	yes	characteristic
32	C₂×D₁₉₃	—	2	4	no	—
33	C₂×D₁₉₃	—	4	8	no	—
34	C₂×D₁₉₃	—	4	4	no	—
35	C₄×C₁₉₃	—	4	2	no	—
36	C₄×C₁₉₃	C₂²⋊C₄	1	2	yes	—
37	C₄×C₁₉₃	C₂²×C₄	1	1	yes	characteristic
38	C₄×C₁₉₃	C₂×D₄	1	1	yes	characteristic
39	C₂×C₁₉₃	—	2	2	no	—
40	C₂×C₁₉₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
41	C₂×C₁₉₃	AGL(1,ℤ/8)	1	2	yes	—
42	D₁₉₃	—	4	4	no	—
43	C₁₉₃	#418	1	1	yes	characteristic, complemented, Sylow
44	#418	—	193	1	no	maximal, Sylow
45	C₂²×D₄	—	193	1	no	—
46	C₂×M₁₆	—	193	1	no	—
47	#117	D₁₉₃	1	1	yes	characteristic, complemented
48	#102	—	193	4	no	—
49	C₂⁴	—	386	1	no	—
50	C₂×D₄	—	386	1	no	—
51	C₂×D₄	—	386	2	no	—
52	C₂×D₄	—	193	4	no	—
53	C₂×D₄	—	193	2	no	—
54	C₂²×C₄	C₂×D₁₉₃	1	1	yes	characteristic
55	M₁₆	—	386	2	no	—
56	C₂×C₈	—	193	2	no	—
57	C₄⋊C₄	C₂×D₁₉₃	1	2	yes	complemented
58	C₂²⋊C₄	—	2	1	no	—
59	C₄²	—	2	1	no	—
60	C₂³	—	772	4	no	—
61	C₂³	—	386	4	no	—
62	C₂³	—	386	2	no	—
63	C₂³	C₂²⋊D₁₉₃	1	1	yes	characteristic
64	D₄	—	386	4	no	—
65	D₄	—	386	2	no	—
66	D₄	—	193	4	no	—
67	C₂×C₄	—	2	2	no	—
68	C₂×C₄	C₄×D₁₉₃	1	2	yes	—
69	C₂×C₄	C₂²⋊D₁₉₃	1	1	yes	characteristic
70	C₂×C₄	D₇₇₂	1	2	yes	—
71	C₂×C₄	C₂²×D₁₉₃	1	1	yes	characteristic
72	C₈	—	386	2	no	—
73	C₂²	—	2	2	no	—
74	C₂²	—	772	8	no	—
75	C₂²	—	772	4	no	—
76	C₂²	—	386	4	no	—
77	C₂²	#15690532	1	2	yes	—
78	C₂²	C₂×C₂²⋊D₁₉₃	1	1	yes	characteristic
79	C₄	—	4	2	no	—
80	C₄	#15690532	1	2	yes	—
81	C₄	C₂×C₄×D₁₉₃	1	1	yes	characteristic
82	C₄	C₂×D₇₇₂	1	1	yes	characteristic
83	C₂	—	2	2	no	—
84	C₂	—	772	4	no	—
85	C₂	C₂×#15690532	1	1	yes	characteristic
86	C₂	#20259924	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₁₉₃
commutator	C₄×C₁₉₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₁₉₃
Fitting	#117×C₁₉₃	C₂
socle	C₂²×C₁₉₃	C₂×D₄
prime radical	C₂²×D₇₇₂	C₂
cosocle	C₂×C₄×C₁₉₃	C₂³
hypercenter	#117	D₁₉₃
hypocenter	C₁₉₃	#418
LCS	G ▷ C₄×C₁₉₃ ▷ C₂×C₁₉₃ ▷ C₁₉₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #117

Complements

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

N	H ≅ G/N	type
2. C₂×#15690539	C₂	semidirect
3. #20259819	C₂	semidirect
4. #117×C₁₉₃	C₂	semidirect
8. C₂×D₇₇₂	C₄	semidirect
11. C₂×#13095393	C₂²	semidirect
13. C₄⋊C₄×C₁₉₃	C₂²	semidirect
43. C₁₉₃	#418	semidirect
47. #117	D₁₉₃	semidirect
57. C₄⋊C₄	C₂×D₁₉₃	semidirect

Maximal subgroups

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

subgroup	index
C₂²×D₇₇₂	2
C₂×#15690539	2
#20259819	2
#117×C₁₉₃	2
#418	193