Group #37572673 — subgroups

Subgroups — #37572673

Order 24448.

256 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 117 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊(C₁₉₁⋊C₈)	C₂	1	1	yes	maximal, characteristic
2	C₂×Dic₁₉₁⋊C₈	C₂	1	4	yes	maximal
3	#27132509	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊C₈×C₁₉₁	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₁₉₁⋊C₈	C₂²	1	2	yes	—
6	C₂×C₄×Dic₁₉₁	C₂²	1	2	yes	—
7	C₄⋊(C₁₉₁⋊C₈)	—	2	2	no	—
8	Dic₁₉₁⋊C₈	—	2	8	no	—
9	C₁₉₁⋊#95	C₄	1	2	yes	—
10	C₁₉₁⋊#95	C₄	1	2	yes	—
11	C₂²×C₈×C₁₉₁	C₂²	1	2	yes	—
12	C₂×C₄²×C₁₉₁	C₂²	1	1	yes	characteristic
13	C₄⋊C₈×C₁₉₁	—	2	2	no	—
14	C₂²×Dic₁₉₁	C₂×C₄	1	4	yes	—
15	C₂×C₁₉₁⋊C₈	—	2	4	no	—
16	C₂×C₁₉₁⋊C₈	—	2	2	no	—
17	C₄×Dic₁₉₁	D₄	1	4	yes	—
18	C₄×Dic₁₉₁	Q₈	1	4	yes	—
19	C₂²×C₄×C₁₉₁	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₄×C₁₉₁	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
22	C₂×C₈×C₁₉₁	—	2	2	no	—
23	C₂×C₈×C₁₉₁	—	2	4	no	—
24	C₄²×C₁₉₁	D₄	1	2	yes	—
25	C₄²×C₁₉₁	Q₈	1	2	yes	—
26	C₂×Dic₁₉₁	—	2	4	no	—
27	C₂×Dic₁₉₁	—	2	4	no	—
28	C₂×Dic₁₉₁	C₄⋊C₄	1	8	yes	—
29	C₁₉₁⋊C₈	—	4	2	no	—
30	C₂³×C₁₉₁	C₂²×C₄	1	1	yes	characteristic
31	C₂×C₄×C₁₉₁	—	2	2	no	—
32	C₂×C₄×C₁₉₁	—	2	2	no	—
33	C₂×C₄×C₁₉₁	C₄⋊C₄	1	2	yes	—
34	C₂×C₄×C₁₉₁	C₄⋊C₄	1	2	yes	—
35	C₂×C₄×C₁₉₁	C₂×D₄	1	2	yes	—
36	C₂×C₄×C₁₉₁	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₁₉₁	C₂×Q₈	1	2	yes	—
38	C₂×C₄×C₁₉₁	C₂×Q₈	1	1	yes	characteristic
39	C₈×C₁₉₁	—	4	2	no	—
40	C₂²×C₁₉₁	C₂×C₄⋊C₄	1	2	yes	—
41	C₂²×C₁₉₁	C₂×C₄⋊C₄	1	1	yes	characteristic
42	C₂²×C₁₉₁	#131	1	2	yes	—
43	C₂²×C₁₉₁	#131	1	2	yes	—
44	C₄×C₁₉₁	—	2	2	no	—
45	C₄×C₁₉₁	—	2	2	no	—
46	C₄×C₁₉₁	#127	1	1	yes	characteristic
47	C₄×C₁₉₁	C₄⋊Q₈	1	1	yes	characteristic
48	C₄×C₁₉₁	C₄⋊Q₈	1	2	yes	—
49	Dic₁₉₁	—	2	8	no	—
50	C₂×C₁₉₁	#384	1	1	yes	characteristic
51	C₂×C₁₉₁	#424	1	4	yes	—
52	C₂×C₁₉₁	#424	1	1	yes	characteristic
53	C₂×C₁₉₁	#424	1	1	yes	characteristic
54	C₁₉₁	#1843	1	1	yes	characteristic, complemented, Sylow
55	#1843	—	191	1	no	maximal, Sylow
56	C₂×C₄⋊C₈	—	191	4	no	—
57	C₂×C₄⋊C₈	—	191	1	no	—
58	C₂×C₄⋊C₈	D₁₉₁	1	1	yes	characteristic
59	#376	—	191	1	no	—
60	C₂²×C₈	—	191	2	no	—
61	C₂²×C₈	C₂×D₁₉₁	1	2	yes	—
62	C₂×C₄²	—	191	2	no	—
63	C₂×C₄²	C₂×D₁₉₁	1	1	yes	characteristic
64	C₄⋊C₈	—	2	2	no	—
65	C₄⋊C₈	—	382	8	no	—
66	C₄⋊C₈	—	382	2	no	—
67	#95	—	191	2	no	—
68	#95	—	191	2	no	—
69	C₂²×C₄	—	191	4	no	—
70	C₂²×C₄	C₄×D₁₉₁	1	1	yes	characteristic
71	C₂²×C₄	C₄×D₁₉₁	1	1	yes	characteristic
72	C₂²×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
73	C₂×C₈	—	2	4	no	—
74	C₂×C₈	—	2	2	no	—
75	C₂×C₈	—	382	4	no	—
76	C₂×C₈	—	382	2	no	—
77	C₄²	—	191	4	no	—
78	C₄²	—	191	4	no	—
79	C₄²	C₂²⋊D₁₉₁	1	2	yes	—
80	C₄²	Dic₃₈₂	1	2	yes	—
81	C₂³	C₂×C₄×D₁₉₁	1	1	yes	characteristic
82	C₂×C₄	—	2	2	no	—
83	C₂×C₄	—	2	2	no	—
84	C₂×C₄	—	382	4	no	—
85	C₂×C₄	—	382	4	no	—
86	C₂×C₄	—	191	8	no	—
87	C₂×C₄	Dic₁₉₁⋊C₄	1	2	yes	—
88	C₂×C₄	Dic₁₉₁⋊C₄	1	2	yes	—
89	C₂×C₄	D₄×D₁₉₁	1	2	yes	—
90	C₂×C₄	C₂×C₂²⋊D₁₉₁	1	1	yes	characteristic
91	C₂×C₄	Q₈×D₁₉₁	1	2	yes	—
92	C₂×C₄	C₂×Dic₃₈₂	1	1	yes	characteristic
93	C₈	—	4	2	no	—
94	C₈	—	764	2	no	—
95	C₂²	C₄⋊C₄×D₁₉₁	1	2	yes	—
96	C₂²	C₂×Dic₁₉₁⋊C₄	1	1	yes	characteristic
97	C₂²	#20259328	1	2	yes	—
98	C₂²	#20259330	1	2	yes	—
99	C₄	—	2	2	no	—
100	C₄	—	2	2	no	—
101	C₄	—	382	8	no	—
102	C₄	#20259318	1	1	yes	characteristic
103	C₄	#20259320	1	1	yes	characteristic
104	C₄	#20259321	1	2	yes	—
105	C₂	#27132532	1	1	yes	characteristic
106	C₂	#27132688	1	4	yes	—
107	C₂	#27132689	1	1	yes	characteristic
108	C₂	#27132691	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊(C₁₉₁⋊C₈)

C₂

yes

maximal, characteristic

C₂×Dic₁₉₁⋊C₈

C₂

yes

maximal

#27132509

C₂

yes

maximal, characteristic

C₂×C₄⋊C₈×C₁₉₁

C₂

yes

maximal, characteristic

C₂²×C₁₉₁⋊C₈

C₂²

yes

—

C₂×C₄×Dic₁₉₁

C₂²

yes

—

C₄⋊(C₁₉₁⋊C₈)

—

Dic₁₉₁⋊C₈

—

C₁₉₁⋊#95

C₄

yes

—

C₁₉₁⋊#95

C₄

yes

—

C₂²×C₈×C₁₉₁

C₂²

yes

—

C₂×C₄²×C₁₉₁

C₂²

yes

characteristic

C₄⋊C₈×C₁₉₁

—

C₂²×Dic₁₉₁

C₂×C₄

yes

—

C₂×C₁₉₁⋊C₈

—

C₂×C₁₉₁⋊C₈

—

C₄×Dic₁₉₁

D₄

yes

—

C₄×Dic₁₉₁

Q₈

yes

—

C₂²×C₄×C₁₉₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₉₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₉₁

C₂³

yes

characteristic

C₂×C₈×C₁₉₁

—

C₂×C₈×C₁₉₁

—

C₄²×C₁₉₁

D₄

yes

—

C₄²×C₁₉₁

Q₈

yes

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

C₄⋊C₄

yes

—

C₁₉₁⋊C₈

—

C₂³×C₁₉₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

C₄⋊C₄

yes

—

C₂×C₄×C₁₉₁

C₄⋊C₄

yes

—

C₂×C₄×C₁₉₁

C₂×D₄

yes

—

C₂×C₄×C₁₉₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₁

C₂×Q₈

yes

—

C₂×C₄×C₁₉₁

C₂×Q₈

yes

characteristic

C₈×C₁₉₁

—

C₂²×C₁₉₁

C₂×C₄⋊C₄

yes

—

C₂²×C₁₉₁

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₉₁

#131

yes

—

C₂²×C₁₉₁

#131

yes

—

C₄×C₁₉₁

—

C₄×C₁₉₁

—

C₄×C₁₉₁

#127

yes

characteristic

C₄×C₁₉₁

C₄⋊Q₈

yes

characteristic

C₄×C₁₉₁

C₄⋊Q₈

yes

—

Dic₁₉₁

—

C₂×C₁₉₁

#384

yes

characteristic

C₂×C₁₉₁

#424

yes

—

C₂×C₁₉₁

yes

C₂×C₁₉₁

yes

C₁₉₁

yes

characteristic, complemented, Sylow

#1843

—

191

maximal, Sylow

C₂×C₄⋊C₈

—

191

—

C₂×C₄⋊C₈

—

191

—

C₂×C₄⋊C₈

D₁₉₁

yes

characteristic

#376

—

191

—

C₂²×C₈

—

191

—

C₂²×C₈

C₂×D₁₉₁

yes

—

C₂×C₄²

—

191

—

C₂×C₄²

C₂×D₁₉₁

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

382

—

C₄⋊C₈

—

382

—

#95

—

191

—

#95

—

191

—

C₂²×C₄

—

191

—

C₂²×C₄

C₄×D₁₉₁

yes

characteristic

C₂²×C₄

C₄×D₁₉₁

yes

characteristic

C₂²×C₄

C₂²×D₁₉₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

382

—

C₂×C₈

—

382

—

C₄²

—

191

—

C₄²

—

191

—

C₄²

C₂²⋊D₁₉₁

yes

—

C₄²

Dic₃₈₂

yes

—

C₂³

C₂×C₄×D₁₉₁

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

382

—

C₂×C₄

—

382

—

C₂×C₄

—

191

—

C₂×C₄

Dic₁₉₁⋊C₄

yes

—

C₂×C₄

Dic₁₉₁⋊C₄

yes

—

C₂×C₄

D₄×D₁₉₁

yes

—

C₂×C₄

C₂×C₂²⋊D₁₉₁

yes

characteristic

C₂×C₄

Q₈×D₁₉₁

yes

—

C₂×C₄

C₂×Dic₃₈₂

yes

characteristic

C₈

—

C₈

—

764

—

C₂²

C₄⋊C₄×D₁₉₁

yes

—

C₂²

C₂×Dic₁₉₁⋊C₄

yes

characteristic

C₂²

#20259328

yes

—

C₂²

#20259330

yes

—

C₄

—

100

C₄

—

101

C₄

—

382

—

102

C₄

#20259318

yes

characteristic

103

C₄

#20259320

yes

characteristic

104

C₄

#20259321

yes

—

105

C₂

#27132532

yes

characteristic

106

C₂

#27132688

yes

—

107

C₂

#27132689

yes

characteristic

108

C₂

#27132691

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₁₉₁
commutator, socle, prime radical	C₂³×C₁₉₁	C₂²×C₄
Fitting	C₂×C₄⋊C₈×C₁₉₁	C₂
cosocle	C₂²×C₄×C₁₉₁	C₂³
hypercenter	C₂×C₄⋊C₈	D₁₉₁
hypocenter	C₁₉₁	#1843
LCS	G ▷ C₂³×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₁₉₁

commutator, socle, prime radical

C₂³×C₁₉₁

C₂²×C₄

Fitting

C₂×C₄⋊C₈×C₁₉₁

C₂

cosocle

C₂²×C₄×C₁₉₁

C₂³

hypercenter

C₂×C₄⋊C₈

D₁₉₁

C₁₉₁

G ▷ C₂³×C₁₉₁ ▷ C₁₉₁

C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
54. C₁₉₁	#1843	semidirect

H ≅ G/N

type

54. C₁₉₁

#1843

semidirect

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

subgroup	index
C₂×C₄⋊(C₁₉₁⋊C₈)	2
C₂×Dic₁₉₁⋊C₈	2
#27132509	2
C₂×C₄⋊C₈×C₁₉₁	2
#1843	191

subgroup

index

C₂×C₄⋊(C₁₉₁⋊C₈)

C₂×Dic₁₉₁⋊C₈

#27132509

C₂×C₄⋊C₈×C₁₉₁

#1843

191

Subgroups — #37572673

Conjugacy classes of subgroups

Complements

Maximal subgroups