Group #37574571 — subgroups

Subgroups — #37574571

Order 24448.

154 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 154 conjugacy classes of subgroups — too many to draw. Only its 55 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×Dic₁₅₂₈	C₂	1	2	yes	maximal
2	#27133043	C₂	1	1	yes	maximal, characteristic
3	C₄×Dic₇₆₄	C₂	1	1	yes	maximal, characteristic
4	#27132468	C₂	1	2	yes	maximal
5	C₄⋊C₁₆×C₁₉₁	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₇₆₄	—	2	1	no	—
7	C₂×Dic₇₆₄	C₂²	1	2	yes	—
8	C₂×Dic₇₆₄	C₂²	1	1	yes	characteristic
9	#20259322	—	2	1	no	—
10	C₄×Dic₃₈₂	—	2	1	no	—
11	Dic₁₅₂₈	—	2	4	no	—
12	#20259256	C₂²	1	1	yes	characteristic
13	#20259247	—	2	1	no	—
14	C₂×C₁₆×C₁₉₁	C₂²	1	2	yes	—
15	C₄×C₈×C₁₉₁	C₂²	1	1	yes	characteristic
16	C₂×Dic₃₈₂	—	2	1	no	—
17	C₂×Dic₃₈₂	—	2	2	no	—
18	Dic₇₆₄	—	2	2	no	—
19	Dic₇₆₄	—	2	1	no	—
20	Dic₇₆₄	—	2	2	no	—
21	Dic₇₆₄	—	4	2	no	—
22	Dic₇₆₄	D₄	1	2	yes	—
23	C₄⋊Dic₁₉₁	—	2	1	no	—
24	C₄⋊Dic₁₉₁	—	4	1	no	—
25	C₄⋊Dic₁₉₁	—	4	1	no	—
26	Dic₁₉₁⋊C₄	—	4	1	no	—
27	C₄×Dic₁₉₁	—	4	1	no	—
28	C₂×C₈×C₁₉₁	D₄	1	1	yes	characteristic
29	C₂×C₈×C₁₉₁	C₂³	1	1	yes	characteristic
30	C₄²×C₁₉₁	D₄	1	1	yes	characteristic
31	C₁₆×C₁₉₁	—	2	2	no	—
32	Dic₃₈₂	—	2	2	no	—
33	Dic₃₈₂	—	4	1	no	—
34	Dic₃₈₂	—	4	2	no	—
35	Dic₃₈₂	—	4	2	no	—
36	C₂×Dic₁₉₁	—	4	1	no	—
37	C₂×Dic₁₉₁	—	4	1	no	—
38	C₂×Dic₁₉₁	—	4	2	no	—
39	C₂×C₄×C₁₉₁	D₈	1	1	yes	characteristic
40	C₂×C₄×C₁₉₁	D₈	1	1	yes	characteristic
41	C₂×C₄×C₁₉₁	C₂×D₄	1	1	yes	characteristic
42	C₈×C₁₉₁	—	2	1	no	—
43	C₈×C₁₉₁	C₂×D₄	1	1	yes	characteristic
44	C₈×C₁₉₁	#41	1	1	yes	characteristic
45	C₂²×C₁₉₁	C₂×D₈	1	1	yes	characteristic
46	C₄×C₁₉₁	—	2	1	no	—
47	C₄×C₁₉₁	Q₃₂	1	2	yes	complemented
48	C₄×C₁₉₁	#121	1	1	yes	characteristic
49	C₄×C₁₉₁	AGL(1,ℤ/8)	1	1	yes	characteristic
50	Dic₁₉₁	—	4	2	no	—
51	Dic₁₉₁	—	8	2	no	—
52	Dic₁₉₁	—	8	1	no	—
53	C₂×C₁₉₁	#459	1	1	yes	characteristic
54	C₂×C₁₉₁	C₂×Q₃₂	1	1	yes	characteristic
55	C₂×C₁₉₁	#510	1	1	yes	characteristic
56	C₁₉₁	#2209	1	1	yes	characteristic, complemented, Sylow
57	#2209	—	191	1	no	maximal, Sylow
58	C₂×Q₃₂	—	191	2	no	—
59	#494	—	191	1	no	—
60	C₄×Q₁₆	—	191	1	no	—
61	C₄⋊C₁₆	D₁₉₁	1	1	yes	characteristic
62	#358	—	191	2	no	—
63	C₂×Q₁₆	—	382	1	no	—
64	C₂×Q₁₆	—	191	2	no	—
65	C₂×Q₁₆	—	191	1	no	—
66	C₄⋊Q₈	—	382	1	no	—
67	C₄×Q₈	—	382	1	no	—
68	Q₃₂	—	382	4	no	—
69	C₂×C₁₆	C₂×D₁₉₁	1	2	yes	—
70	#107	—	191	1	no	—
71	#103	—	382	1	no	—
72	C₄×C₈	C₂×D₁₉₁	1	1	yes	characteristic
73	C₂×Q₈	—	382	1	no	—
74	C₂×Q₈	—	382	2	no	—
75	Q₁₆	—	382	2	no	—
76	Q₁₆	—	382	1	no	—
77	Q₁₆	—	382	2	no	—
78	Q₁₆	—	191	2	no	—
79	Q₁₆	—	764	2	no	—
80	C₂×C₈	D₇₆₄	1	1	yes	characteristic
81	C₂×C₈	C₂²×D₁₉₁	1	1	yes	characteristic
82	C₄⋊C₄	—	382	1	no	—
83	C₄⋊C₄	—	764	1	no	—
84	C₄⋊C₄	—	764	1	no	—
85	C₄⋊C₄	—	764	1	no	—
86	C₄²	—	764	1	no	—
87	C₄²	D₇₆₄	1	1	yes	characteristic
88	C₁₆	—	2	2	no	—
89	Q₈	—	382	2	no	—
90	Q₈	—	764	2	no	—
91	Q₈	—	764	2	no	—
92	Q₈	—	764	1	no	—
93	C₂×C₄	—	764	1	no	—
94	C₂×C₄	—	764	2	no	—
95	C₂×C₄	—	764	1	no	—
96	C₂×C₄	D₁₅₂₈	1	1	yes	characteristic
97	C₂×C₄	D₁₅₂₈	1	1	yes	characteristic
98	C₂×C₄	C₂×D₇₆₄	1	1	yes	characteristic
99	C₈	—	2	1	no	—
100	C₈	D₄×D₁₉₁	1	1	yes	characteristic
101	C₈	#15690288	1	1	yes	characteristic
102	C₂²	C₂×D₁₅₂₈	1	1	yes	characteristic
103	C₄	—	2	1	no	—
104	C₄	—	1528	1	no	—
105	C₄	—	1528	2	no	—
106	C₄	—	764	2	no	—
107	C₄	Dic₁₅₂₈	1	2	yes	complemented
108	C₄	#20259297	1	1	yes	characteristic
109	C₄	#20259350	1	1	yes	characteristic
110	C₂	#27132859	1	1	yes	characteristic
111	C₂	C₂×Dic₁₅₂₈	1	1	yes	characteristic
112	C₂	#27133106	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Dic₁₅₂₈

C₂

yes

maximal

#27133043

C₂

yes

maximal, characteristic

C₄×Dic₇₆₄

C₂

yes

maximal, characteristic

#27132468

C₂

yes

maximal

C₄⋊C₁₆×C₁₉₁

C₂

yes

maximal, characteristic

C₂×Dic₇₆₄

—

C₂×Dic₇₆₄

C₂²

yes

—

C₂×Dic₇₆₄

C₂²

yes

characteristic

#20259322

—

C₄×Dic₃₈₂

—

Dic₁₅₂₈

—

#20259256

C₂²

yes

characteristic

#20259247

—

C₂×C₁₆×C₁₉₁

C₂²

yes

—

C₄×C₈×C₁₉₁

C₂²

yes

characteristic

C₂×Dic₃₈₂

—

C₂×Dic₃₈₂

—

Dic₇₆₄

—

Dic₇₆₄

—

Dic₇₆₄

—

Dic₇₆₄

—

Dic₇₆₄

D₄

yes

—

C₄⋊Dic₁₉₁

—

C₄⋊Dic₁₉₁

—

C₄⋊Dic₁₉₁

—

Dic₁₉₁⋊C₄

—

C₄×Dic₁₉₁

—

C₂×C₈×C₁₉₁

D₄

yes

characteristic

C₂×C₈×C₁₉₁

C₂³

yes

characteristic

C₄²×C₁₉₁

D₄

yes

characteristic

C₁₆×C₁₉₁

—

Dic₃₈₂

—

Dic₃₈₂

—

Dic₃₈₂

—

Dic₃₈₂

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

—

C₂×C₄×C₁₉₁

D₈

yes

characteristic

C₂×C₄×C₁₉₁

D₈

yes

characteristic

C₂×C₄×C₁₉₁

C₂×D₄

yes

characteristic

C₈×C₁₉₁

—

C₈×C₁₉₁

C₂×D₄

yes

characteristic

C₈×C₁₉₁

#41

yes

characteristic

C₂²×C₁₉₁

C₂×D₈

yes

characteristic

C₄×C₁₉₁

—

C₄×C₁₉₁

Q₃₂

yes

C₄×C₁₉₁

yes

C₄×C₁₉₁

yes

Dic₁₉₁

—

Dic₁₉₁

—

Dic₁₉₁

—

C₂×C₁₉₁

#459

yes

characteristic

C₂×C₁₉₁

C₂×Q₃₂

yes

C₂×C₁₉₁

yes

C₁₉₁

yes

characteristic, complemented, Sylow

#2209

—

191

maximal, Sylow

C₂×Q₃₂

—

191

—

#494

—

191

—

C₄×Q₁₆

—

191

—

C₄⋊C₁₆

D₁₉₁

yes

characteristic

#358

—

191

—

C₂×Q₁₆

—

382

—

C₂×Q₁₆

—

191

—

C₂×Q₁₆

—

191

—

C₄⋊Q₈

—

382

—

C₄×Q₈

—

382

—

Q₃₂

—

382

—

C₂×C₁₆

C₂×D₁₉₁

yes

—

#107

—

191

—

#103

—

382

—

C₄×C₈

C₂×D₁₉₁

yes

characteristic

C₂×Q₈

—

382

—

C₂×Q₈

—

382

—

Q₁₆

—

382

—

Q₁₆

—

382

—

Q₁₆

—

382

—

Q₁₆

—

191

—

Q₁₆

—

764

—

C₂×C₈

D₇₆₄

yes

characteristic

C₂×C₈

C₂²×D₁₉₁

yes

characteristic

C₄⋊C₄

—

382

—

C₄⋊C₄

—

764

—

C₄⋊C₄

—

764

—

C₄⋊C₄

—

764

—

C₄²

—

764

—

C₄²

D₇₆₄

yes

characteristic

C₁₆

—

Q₈

—

382

—

Q₈

—

764

—

Q₈

—

764

—

Q₈

—

764

—

C₂×C₄

—

764

—

C₂×C₄

—

764

—

C₂×C₄

—

764

—

C₂×C₄

D₁₅₂₈

yes

characteristic

C₂×C₄

D₁₅₂₈

yes

characteristic

C₂×C₄

C₂×D₇₆₄

yes

characteristic

C₈

—

100

C₈

D₄×D₁₉₁

yes

characteristic

101

C₈

#15690288

yes

characteristic

102

C₂²

C₂×D₁₅₂₈

yes

characteristic

103

C₄

—

104

C₄

—

1528

—

105

C₄

—

1528

—

106

C₄

—

764

—

107

C₄

Dic₁₅₂₈

yes

complemented

108

C₄

#20259297

yes

characteristic

109

C₄

#20259350

yes

characteristic

110

C₂

#27132859

yes

characteristic

111

C₂

C₂×Dic₁₅₂₈

yes

characteristic

112

C₂

#27133106

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₁₅₂₈
commutator, cosocle	C₂×C₈×C₁₉₁	C₂³
Frattini	C₂×C₈	C₂²×D₁₉₁
Fitting	C₄⋊C₁₆×C₁₉₁	C₂
socle, prime radical	C₂²×C₁₉₁	C₂×D₈
hypercenter	C₄⋊C₁₆	D₁₉₁
hypocenter	C₁₉₁	#2209
LCS	G ▷ C₂×C₈×C₁₉₁ ▷ C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄⋊C₁₆

subgroup

G/H

center

C₂²

C₂×D₁₅₂₈

commutator, cosocle

C₂×C₈×C₁₉₁

C₂³

Frattini

C₂×C₈

C₂²×D₁₉₁

Fitting

C₄⋊C₁₆×C₁₉₁

C₂

socle, prime radical

C₂²×C₁₉₁

C₂×D₈

hypercenter

C₄⋊C₁₆

D₁₉₁

hypocenter

C₁₉₁

#2209

LCS

G ▷ C₂×C₈×C₁₉₁ ▷ C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄⋊C₁₆

N	H ≅ G/N	type
47. C₄×C₁₉₁	Q₃₂	semidirect
56. C₁₉₁	#2209	semidirect
107. C₄	Dic₁₅₂₈	semidirect

H ≅ G/N

type

47. C₄×C₁₉₁

Q₃₂

semidirect

56. C₁₉₁

#2209

semidirect

107. C₄

Dic₁₅₂₈

semidirect

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

subgroup	index
C₂×Dic₁₅₂₈	2
#27133043	2
C₄×Dic₇₆₄	2
#27132468	2
C₄⋊C₁₆×C₁₉₁	2
#2209	191

subgroup

index

C₂×Dic₁₅₂₈

#27133043

C₄×Dic₇₆₄

#27132468

C₄⋊C₁₆×C₁₉₁

#2209

191

Subgroups — #37574571

Conjugacy classes of subgroups

Complements

Maximal subgroups