Group #37574239 — subgroups

Subgroups — #37574239

Order 24448.

220 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 85 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	C₂²×Dic₇₆₄	C₂	1	1	yes	maximal, characteristic
2	C₂×#20259262	C₂	1	1	yes	maximal, characteristic
3	#27132466	C₂	1	4	yes	maximal, complemented
4	#427×C₁₉₁	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₃₈₂	—	2	1	no	—
6	C₂×Dic₇₆₄	—	2	1	no	—
7	C₂×Dic₇₆₄	—	2	2	no	—
8	C₂×Dic₇₆₄	C₄	1	4	yes	complemented
9	C₂×Dic₇₆₄	C₂²	1	2	yes	—
10	#20259262	—	2	2	no	—
11	C₂×C₁₉₁⋊C₁₆	C₂²	1	2	yes	—
12	C₂²×C₈×C₁₉₁	C₂²	1	1	yes	characteristic
13	#117×C₁₉₁	—	2	1	no	—
14	#107×C₁₉₁	C₂²	1	2	yes	—
15	#106×C₁₉₁	—	2	1	no	—
16	C₂×Dic₃₈₂	—	2	4	no	—
17	C₂×Dic₃₈₂	—	2	2	no	—
18	C₂×Dic₃₈₂	—	4	1	no	—
19	C₂×Dic₃₈₂	—	4	2	no	—
20	C₂²×Dic₁₉₁	—	4	1	no	—
21	Dic₇₆₄	—	2	4	no	—
22	Dic₇₆₄	—	2	2	no	—
23	Dic₇₆₄	C₂×C₄	1	4	yes	—
24	C₁₉₁⋊C₁₆	—	2	2	no	—
25	C₂²×C₄×C₁₉₁	D₄	1	1	yes	characteristic
26	C₂×C₈×C₁₉₁	C₂×C₄	1	2	yes	—
27	C₂×C₈×C₁₉₁	D₄	1	1	yes	characteristic
28	C₂×C₈×C₁₉₁	D₄	1	2	yes	—
29	C₂×C₈×C₁₉₁	C₂³	1	1	yes	characteristic
30	C₄⋊C₄×C₁₉₁	—	2	2	no	—
31	C₂²⋊C₄×C₁₉₁	—	4	1	no	—
32	C₄²×C₁₉₁	—	4	1	no	—
33	Dic₃₈₂	—	2	4	no	—
34	Dic₃₈₂	—	4	4	no	—
35	Dic₃₈₂	—	4	2	no	—
36	C₂×Dic₁₉₁	—	4	4	no	—
37	C₂×Dic₁₉₁	—	4	2	no	—
38	C₂³×C₁₉₁	D₈	1	1	yes	characteristic
39	C₂×C₄×C₁₉₁	—	4	2	no	—
40	C₂×C₄×C₁₉₁	C₂²⋊C₄	1	2	yes	—
41	C₂×C₄×C₁₉₁	D₈	1	1	yes	characteristic
42	C₂×C₄×C₁₉₁	SD₁₆	1	2	yes	—
43	C₂×C₄×C₁₉₁	C₂×D₄	1	1	yes	characteristic
44	C₈×C₁₉₁	C₂²⋊C₄	1	2	yes	—
45	C₈×C₁₉₁	C₂²×C₄	1	1	yes	characteristic
46	C₈×C₁₉₁	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₁₉₁	—	2	2	no	—
48	C₂²×C₁₉₁	#102	1	2	yes	—
49	C₂²×C₁₉₁	C₂×D₈	1	1	yes	characteristic
50	C₄×C₁₉₁	—	8	2	no	—
51	C₄×C₁₉₁	#102	1	2	yes	—
52	C₄×C₁₉₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
53	C₄×C₁₉₁	C₂×SD₁₆	1	1	yes	characteristic
54	Dic₁₉₁	—	4	4	no	—
55	C₂×C₁₉₁	—	2	2	no	—
56	C₂×C₁₉₁	C₂×#102	1	1	yes	characteristic
57	C₂×C₁₉₁	#510	1	2	yes	—
58	C₁₉₁	#2141	1	1	yes	characteristic, complemented, Sylow
59	#2141	—	191	1	no	maximal, Sylow
60	C₂²×Q₁₆	—	191	1	no	—
61	C₂×M₃₂	—	191	1	no	—
62	#427	D₁₉₁	1	1	yes	characteristic
63	#358	—	191	4	no	—
64	C₂²×Q₈	—	382	1	no	—
65	C₂×Q₁₆	—	382	1	no	—
66	C₂×Q₁₆	—	382	2	no	—
67	C₂×Q₁₆	—	191	4	no	—
68	C₂×Q₁₆	—	191	2	no	—
69	C₂²×C₈	C₂×D₁₉₁	1	1	yes	characteristic
70	#117	—	2	1	no	—
71	M₃₂	—	382	2	no	—
72	C₂×C₁₆	—	191	2	no	—
73	#107	C₂×D₁₉₁	1	2	yes	—
74	#106	—	2	1	no	—
75	C₂×Q₈	—	382	4	no	—
76	C₂×Q₈	—	382	2	no	—
77	C₂×Q₈	—	764	1	no	—
78	C₂×Q₈	—	764	2	no	—
79	C₂²×C₄	—	764	1	no	—
80	C₂²×C₄	C₂²⋊D₁₉₁	1	1	yes	characteristic
81	Q₁₆	—	382	4	no	—
82	Q₁₆	—	382	2	no	—
83	Q₁₆	—	191	4	no	—
84	C₂×C₈	C₄×D₁₉₁	1	2	yes	—
85	C₂×C₈	C₂²⋊D₁₉₁	1	1	yes	characteristic
86	C₂×C₈	D₇₆₄	1	2	yes	—
87	C₂×C₈	C₂²×D₁₉₁	1	1	yes	characteristic
88	C₄⋊C₄	—	2	2	no	—
89	C₂²⋊C₄	—	4	1	no	—
90	C₄²	—	4	1	no	—
91	C₁₆	—	382	2	no	—
92	C₂³	#15690272	1	1	yes	characteristic
93	Q₈	—	382	4	no	—
94	Q₈	—	764	2	no	—
95	Q₈	—	764	4	no	—
96	C₂×C₄	—	4	2	no	—
97	C₂×C₄	—	764	2	no	—
98	C₂×C₄	—	764	4	no	—
99	C₂×C₄	(C₂×D₁₉₁)⋊C₄	1	2	yes	—
100	C₂×C₄	#15690272	1	1	yes	characteristic
101	C₂×C₄	#15690274	1	2	yes	—
102	C₂×C₄	C₂×C₂²⋊D₁₉₁	1	1	yes	characteristic
103	C₈	(C₂×D₁₉₁)⋊C₄	1	2	yes	—
104	C₈	C₂×C₄×D₁₉₁	1	1	yes	characteristic
105	C₈	C₂×D₇₆₄	1	1	yes	characteristic
106	C₂²	—	2	2	no	—
107	C₂²	#20259242	1	2	yes	—
108	C₂²	C₂×#15690272	1	1	yes	characteristic
109	C₄	—	8	2	no	—
110	C₄	—	764	4	no	—
111	C₄	#20259242	1	2	yes	—
112	C₄	C₂×(C₂×D₁₉₁)⋊C₄	1	1	yes	characteristic
113	C₄	C₂×#15690274	1	1	yes	characteristic
114	C₂	—	2	2	no	—
115	C₂	C₂×#20259242	1	1	yes	characteristic
116	C₂	#27133103	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₇₆₄

C₂

yes

maximal, characteristic

C₂×#20259262

C₂

yes

maximal, characteristic

#27132466

C₂

yes

maximal, complemented

#427×C₁₉₁

C₂

yes

maximal, characteristic

C₂²×Dic₃₈₂

—

C₂×Dic₇₆₄

—

C₂×Dic₇₆₄

—

C₂×Dic₇₆₄

C₄

yes

complemented

C₂×Dic₇₆₄

C₂²

yes

—

#20259262

—

C₂×C₁₉₁⋊C₁₆

C₂²

yes

—

C₂²×C₈×C₁₉₁

C₂²

yes

characteristic

#117×C₁₉₁

—

#107×C₁₉₁

C₂²

yes

—

#106×C₁₉₁

—

C₂×Dic₃₈₂

—

C₂×Dic₃₈₂

—

C₂×Dic₃₈₂

—

C₂×Dic₃₈₂

—

C₂²×Dic₁₉₁

—

Dic₇₆₄

—

Dic₇₆₄

—

Dic₇₆₄

C₂×C₄

yes

—

C₁₉₁⋊C₁₆

—

C₂²×C₄×C₁₉₁

D₄

yes

characteristic

C₂×C₈×C₁₉₁

C₂×C₄

yes

—

C₂×C₈×C₁₉₁

D₄

yes

characteristic

C₂×C₈×C₁₉₁

D₄

yes

—

C₂×C₈×C₁₉₁

C₂³

yes

characteristic

C₄⋊C₄×C₁₉₁

—

C₂²⋊C₄×C₁₉₁

—

C₄²×C₁₉₁

—

Dic₃₈₂

—

Dic₃₈₂

—

Dic₃₈₂

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

—

C₂³×C₁₉₁

D₈

yes

characteristic

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

C₂²⋊C₄

yes

—

C₂×C₄×C₁₉₁

D₈

yes

characteristic

C₂×C₄×C₁₉₁

SD₁₆

yes

—

C₂×C₄×C₁₉₁

C₂×D₄

yes

characteristic

C₈×C₁₉₁

C₂²⋊C₄

yes

—

C₈×C₁₉₁

C₂²×C₄

yes

characteristic

C₈×C₁₉₁

C₂×D₄

yes

characteristic

C₂²×C₁₉₁

—

C₂²×C₁₉₁

#102

yes

—

C₂²×C₁₉₁

C₂×D₈

yes

characteristic

C₄×C₁₉₁

—

C₄×C₁₉₁

#102

yes

—

C₄×C₁₉₁

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₁₉₁

C₂×SD₁₆

yes

characteristic

Dic₁₉₁

—

C₂×C₁₉₁

—

C₂×C₁₉₁

C₂×#102

yes

characteristic

C₂×C₁₉₁

#510

yes

—

C₁₉₁

#2141

yes

characteristic, complemented, Sylow

#2141

—

191

maximal, Sylow

C₂²×Q₁₆

—

191

—

C₂×M₃₂

—

191

—

#427

D₁₉₁

yes

characteristic

#358

—

191

—

C₂²×Q₈

—

382

—

C₂×Q₁₆

—

382

—

C₂×Q₁₆

—

382

—

C₂×Q₁₆

—

191

—

C₂×Q₁₆

—

191

—

C₂²×C₈

C₂×D₁₉₁

yes

characteristic

#117

—

M₃₂

—

382

—

C₂×C₁₆

—

191

—

#107

C₂×D₁₉₁

yes

—

#106

—

C₂×Q₈

—

382

—

C₂×Q₈

—

382

—

C₂×Q₈

—

764

—

C₂×Q₈

—

764

—

C₂²×C₄

—

764

—

C₂²×C₄

C₂²⋊D₁₉₁

yes

characteristic

Q₁₆

—

382

—

Q₁₆

—

382

—

Q₁₆

—

191

—

C₂×C₈

C₄×D₁₉₁

yes

—

C₂×C₈

C₂²⋊D₁₉₁

yes

characteristic

C₂×C₈

D₇₆₄

yes

—

C₂×C₈

C₂²×D₁₉₁

yes

characteristic

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₁₆

—

382

—

C₂³

#15690272

yes

characteristic

Q₈

—

382

—

Q₈

—

764

—

Q₈

—

764

—

C₂×C₄

—

C₂×C₄

—

764

—

C₂×C₄

—

764

—

C₂×C₄

(C₂×D₁₉₁)⋊C₄

yes

—

100

C₂×C₄

#15690272

yes

characteristic

101

C₂×C₄

#15690274

yes

—

102

C₂×C₄

C₂×C₂²⋊D₁₉₁

yes

characteristic

103

C₈

(C₂×D₁₉₁)⋊C₄

yes

—

104

C₈

C₂×C₄×D₁₉₁

yes

characteristic

105

C₈

C₂×D₇₆₄

yes

characteristic

106

C₂²

—

107

C₂²

#20259242

yes

—

108

C₂²

C₂×#15690272

yes

characteristic

109

C₄

—

110

C₄

—

764

—

111

C₄

#20259242

yes

—

112

C₄

C₂×(C₂×D₁₉₁)⋊C₄

yes

characteristic

113

C₄

C₂×#15690274

yes

characteristic

114

C₂

—

115

C₂

C₂×#20259242

yes

characteristic

116

C₂

#27133103

yes

—

	subgroup	G/H
center	C₂²	C₂×#15690272
commutator	C₈×C₁₉₁	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₁₉₁
Fitting	#427×C₁₉₁	C₂
socle	C₂²×C₁₉₁	C₂×D₈
prime radical	C₂³×C₁₉₁	D₈
cosocle	C₂×C₈×C₁₉₁	C₂³
hypercenter	#427	D₁₉₁
hypocenter	C₁₉₁	#2141
LCS	G ▷ C₈×C₁₉₁ ▷ C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #427

subgroup

G/H

center

C₂²

C₂×#15690272

commutator

C₈×C₁₉₁

C₂²×C₄

Frattini

C₂×C₈

C₂²×D₁₉₁

Fitting

#427×C₁₉₁

C₂

socle

C₂²×C₁₉₁

C₂×D₈

prime radical

C₂³×C₁₉₁

D₈

cosocle

C₂×C₈×C₁₉₁

C₂³

D₁₉₁

C₁₉₁

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

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #427

N	H ≅ G/N	type
3. #27132466	C₂	semidirect
8. C₂×Dic₇₆₄	C₄	semidirect
58. C₁₉₁	#2141	semidirect

H ≅ G/N

type

3. #27132466

C₂

semidirect

8. C₂×Dic₇₆₄

C₄

semidirect

58. C₁₉₁

#2141

semidirect

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

subgroup	index
C₂²×Dic₇₆₄	2
C₂×#20259262	2
#27132466	2
#427×C₁₉₁	2
#2141	191

subgroup

index

C₂²×Dic₇₆₄

C₂×#20259262

#27132466

#427×C₁₉₁

#2141

191

Subgroups — #37574239

Conjugacy classes of subgroups

Complements

Maximal subgroups