Group #37573344 — subgroups

Subgroups — #37573344

Order 24448.

412 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 109 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×#20259319	C₂	1	1	yes	maximal, characteristic
2	C₂×#20259244	C₂	1	4	yes	maximal, complemented
3	#27132529	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#97×C₁₉₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×D₇₆₄	—	2	2	no	—
6	C₂²×D₇₆₄	C₂²	1	2	yes	—
7	#20259319	—	2	2	no	—
8	#20259319	C₄	1	4	yes	complemented
9	C₂×C₄⋊Dic₁₉₁	C₂²	1	2	yes	—
10	#20259244	—	2	8	no	—
11	C₁₉₁⋊#95	—	2	2	no	—
12	C₂²×C₈×C₁₉₁	C₂²	1	2	yes	—
13	C₂×C₄²×C₁₉₁	C₂²	1	1	yes	characteristic
14	#97×C₁₉₁	—	2	2	no	—
15	C₂³×D₁₉₁	—	2	2	no	—
16	C₂×D₇₆₄	—	2	8	no	—
17	C₂×D₇₆₄	—	2	4	no	—
18	C₂×D₇₆₄	—	2	2	no	—
19	C₂×D₇₆₄	—	2	4	no	—
20	C₂×D₇₆₄	—	4	8	no	—
21	C₂×D₇₆₄	C₂×C₄	1	4	yes	—
22	C₂²×Dic₁₉₁	—	2	2	no	—
23	C₄⋊Dic₁₉₁	—	2	4	no	—
24	C₂²×C₄×C₁₉₁	D₄	1	2	yes	—
25	C₂²×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
26	C₂×C₈×C₁₉₁	—	2	2	no	—
27	C₂×C₈×C₁₉₁	—	2	2	no	—
28	C₂×C₈×C₁₉₁	D₄	1	4	yes	complemented
29	C₄²×C₁₉₁	C₂×C₄	1	2	yes	—
30	C₄²×C₁₉₁	D₄	1	2	yes	—
31	C₂²×D₁₉₁	—	2	4	no	—
32	C₂²×D₁₉₁	—	4	2	no	—
33	C₂²×D₁₉₁	—	4	2	no	—
34	C₂²×D₁₉₁	—	4	8	no	—
35	D₇₆₄	—	2	8	no	—
36	D₇₆₄	—	4	16	no	—
37	D₇₆₄	—	4	4	no	—
38	C₂×Dic₁₉₁	—	4	2	no	—
39	C₂×Dic₁₉₁	—	4	4	no	—
40	C₂³×C₁₉₁	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₁₉₁	—	2	4	no	—
42	C₂×C₄×C₁₉₁	C₂²⋊C₄	1	4	yes	—
43	C₂×C₄×C₁₉₁	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₄×C₁₉₁	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₁₉₁	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₁₉₁	#41	1	2	yes	—
47	C₈×C₁₉₁	—	2	4	no	—
48	C₂²×C₁₉₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
49	C₂²×C₁₉₁	#124	1	1	yes	characteristic
50	C₂²×C₁₉₁	#127	1	1	yes	characteristic
51	C₂²×C₁₉₁	AGL(1,ℤ/8)	1	4	yes	—
52	C₂×D₁₉₁	—	4	4	no	—
53	C₂×D₁₉₁	—	4	8	no	—
54	C₂×D₁₉₁	—	8	4	no	—
55	C₂×D₁₉₁	—	8	4	no	—
56	C₄×C₁₉₁	—	2	4	no	—
57	C₄×C₁₉₁	C₄×D₄	1	2	yes	—
58	C₄×C₁₉₁	#121	1	2	yes	—
59	Dic₁₉₁	—	8	2	no	—
60	C₂×C₁₉₁	#390	1	1	yes	characteristic
61	C₂×C₁₉₁	#418	1	2	yes	—
62	C₂×C₁₉₁	#490	1	2	yes	—
63	C₂×C₁₉₁	#496	1	2	yes	—
64	D₁₉₁	—	8	4	no	—
65	C₁₉₁	#1961	1	1	yes	characteristic, complemented, Sylow
66	#1961	—	191	1	no	maximal, Sylow
67	C₂×#127	—	191	1	no	—
68	C₂×#102	—	191	4	no	—
69	C₂×#97	D₁₉₁	1	1	yes	characteristic, complemented
70	#382	—	191	1	no	—
71	C₂²×D₄	—	382	2	no	—
72	C₂²×D₄	—	191	2	no	—
73	C₂²×C₈	C₂×D₁₉₁	1	2	yes	—
74	#127	—	382	2	no	—
75	#127	—	191	4	no	—
76	C₂×C₄⋊C₄	—	191	2	no	—
77	C₂×C₄²	C₂×D₁₉₁	1	1	yes	characteristic
78	#102	—	382	8	no	—
79	#97	—	2	2	no	—
80	#95	—	382	2	no	—
81	C₂⁴	—	382	2	no	—
82	C₂×D₄	—	382	8	no	—
83	C₂×D₄	—	382	2	no	—
84	C₂×D₄	—	382	4	no	—
85	C₂×D₄	—	382	4	no	—
86	C₂×D₄	—	191	4	no	—
87	C₂×D₄	—	764	8	no	—
88	C₂²×C₄	—	382	2	no	—
89	C₂²×C₄	D₇₆₄	1	2	yes	—
90	C₂²×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
91	C₂×C₈	—	2	2	no	—
92	C₂×C₈	—	2	2	no	—
93	C₂×C₈	D₇₆₄	1	4	yes	complemented
94	C₄⋊C₄	—	382	4	no	—
95	C₄²	C₄×D₁₉₁	1	2	yes	—
96	C₄²	C₂²⋊D₁₉₁	1	2	yes	—
97	C₂³	—	382	4	no	—
98	C₂³	—	764	8	no	—
99	C₂³	—	764	2	no	—
100	C₂³	—	764	2	no	—
101	C₂³	C₂×D₇₆₄	1	1	yes	characteristic
102	D₄	—	382	8	no	—
103	D₄	—	764	16	no	—
104	D₄	—	764	4	no	—
105	C₂×C₄	—	2	4	no	—
106	C₂×C₄	—	764	4	no	—
107	C₂×C₄	—	764	2	no	—
108	C₂×C₄	(C₂×D₁₉₁)⋊C₄	1	4	yes	—
109	C₂×C₄	C₂×C₄×D₁₉₁	1	1	yes	characteristic
110	C₂×C₄	C₂×C₂²⋊D₁₉₁	1	1	yes	characteristic
111	C₂×C₄	C₂×D₇₆₄	1	2	yes	—
112	C₂×C₄	#15690287	1	2	yes	—
113	C₈	—	2	4	no	—
114	C₂²	—	1528	4	no	—
115	C₂²	—	1528	4	no	—
116	C₂²	—	764	4	no	—
117	C₂²	—	764	8	no	—
118	C₂²	C₂×(C₂×D₁₉₁)⋊C₄	1	1	yes	characteristic
119	C₂²	#20259310	1	1	yes	characteristic
120	C₂²	#20259319	1	1	yes	characteristic
121	C₂²	#20259350	1	4	yes	—
122	C₄	—	2	4	no	—
123	C₄	—	1528	2	no	—
124	C₄	C₄×D₇₆₄	1	2	yes	—
125	C₄	#20259296	1	2	yes	—
126	C₂	—	1528	4	no	—
127	C₂	#27132565	1	1	yes	characteristic
128	C₂	#27132669	1	2	yes	—
129	C₂	#27133028	1	2	yes	—
130	C₂	#27133052	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#20259319

C₂

yes

maximal, characteristic

C₂×#20259244

C₂

yes

maximal, complemented

#27132529

C₂

yes

maximal, characteristic, complemented

C₂×#97×C₁₉₁

C₂

yes

maximal, characteristic, complemented

C₂²×D₇₆₄

—

C₂²×D₇₆₄

C₂²

yes

—

#20259319

—

#20259319

C₄

yes

complemented

C₂×C₄⋊Dic₁₉₁

C₂²

yes

—

#20259244

—

C₁₉₁⋊#95

—

C₂²×C₈×C₁₉₁

C₂²

yes

—

C₂×C₄²×C₁₉₁

C₂²

yes

characteristic

#97×C₁₉₁

—

C₂³×D₁₉₁

—

C₂×D₇₆₄

—

C₂×D₇₆₄

—

C₂×D₇₆₄

—

C₂×D₇₆₄

—

C₂×D₇₆₄

—

C₂×D₇₆₄

C₂×C₄

yes

—

C₂²×Dic₁₉₁

—

C₄⋊Dic₁₉₁

—

C₂²×C₄×C₁₉₁

D₄

yes

—

C₂²×C₄×C₁₉₁

C₂³

yes

characteristic

C₂×C₈×C₁₉₁

—

C₂×C₈×C₁₉₁

—

C₂×C₈×C₁₉₁

D₄

yes

complemented

C₄²×C₁₉₁

C₂×C₄

yes

—

C₄²×C₁₉₁

D₄

yes

—

C₂²×D₁₉₁

—

C₂²×D₁₉₁

—

C₂²×D₁₉₁

—

C₂²×D₁₉₁

—

D₇₆₄

—

D₇₆₄

—

D₇₆₄

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

—

C₂³×C₁₉₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

C₂²⋊C₄

yes

—

C₂×C₄×C₁₉₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₉₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₁

C₂×D₄

yes

—

C₂×C₄×C₁₉₁

#41

yes

—

C₈×C₁₉₁

—

C₂²×C₁₉₁

C₂×C₂²⋊C₄

yes

C₂²×C₁₉₁

yes

C₂²×C₁₉₁

yes

C₂²×C₁₉₁

yes

—

C₂×D₁₉₁

—

C₂×D₁₉₁

—

C₂×D₁₉₁

—

C₂×D₁₉₁

—

C₄×C₁₉₁

—

C₄×C₁₉₁

C₄×D₄

yes

—

C₄×C₁₉₁

#121

yes

—

Dic₁₉₁

—

C₂×C₁₉₁

#390

yes

characteristic

C₂×C₁₉₁

#418

yes

—

C₂×C₁₉₁

#490

yes

—

C₂×C₁₉₁

#496

yes

—

D₁₉₁

—

C₁₉₁

#1961

yes

characteristic, complemented, Sylow

#1961

—

191

maximal, Sylow

C₂×#127

—

191

—

C₂×#102

—

191

—

C₂×#97

D₁₉₁

yes

characteristic, complemented

#382

—

191

—

C₂²×D₄

—

382

—

C₂²×D₄

—

191

—

C₂²×C₈

C₂×D₁₉₁

yes

—

#127

—

382

—

#127

—

191

—

C₂×C₄⋊C₄

—

191

—

C₂×C₄²

C₂×D₁₉₁

yes

characteristic

#102

—

382

—

#97

—

#95

—

382

—

C₂⁴

—

382

—

C₂×D₄

—

382

—

C₂×D₄

—

382

—

C₂×D₄

—

382

—

C₂×D₄

—

382

—

C₂×D₄

—

191

—

C₂×D₄

—

764

—

C₂²×C₄

—

382

—

C₂²×C₄

D₇₆₄

yes

—

C₂²×C₄

C₂²×D₁₉₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

D₇₆₄

yes

complemented

C₄⋊C₄

—

382

—

C₄²

C₄×D₁₉₁

yes

—

C₄²

C₂²⋊D₁₉₁

yes

—

C₂³

—

382

—

C₂³

—

764

—

C₂³

—

764

—

100

C₂³

—

764

—

101

C₂³

C₂×D₇₆₄

yes

characteristic

102

D₄

—

382

—

103

D₄

—

764

—

104

D₄

—

764

—

105

C₂×C₄

—

106

C₂×C₄

—

764

—

107

C₂×C₄

—

764

—

108

C₂×C₄

(C₂×D₁₉₁)⋊C₄

yes

—

109

C₂×C₄

C₂×C₄×D₁₉₁

yes

characteristic

110

C₂×C₄

C₂×C₂²⋊D₁₉₁

yes

characteristic

111

C₂×C₄

C₂×D₇₆₄

yes

—

112

C₂×C₄

#15690287

yes

—

113

C₈

—

114

C₂²

—

1528

—

115

C₂²

—

1528

—

116

C₂²

—

764

—

117

C₂²

—

764

—

118

C₂²

C₂×(C₂×D₁₉₁)⋊C₄

yes

characteristic

119

C₂²

#20259310

yes

characteristic

120

C₂²

#20259319

yes

characteristic

121

C₂²

#20259350

yes

—

122

C₄

—

123

C₄

—

1528

—

124

C₄

C₄×D₇₆₄

yes

—

125

C₄

#20259296

yes

—

126

C₂

—

1528

—

127

C₂

#27132565

yes

characteristic

128

C₂

#27132669

yes

—

129

C₂

#27133028

yes

—

130

C₂

#27133052

yes

—

	subgroup	G/H
center	C₂³	C₂×D₇₆₄
commutator	C₂×C₄×C₁₉₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₉₁
Fitting	C₂×#97×C₁₉₁	C₂
socle	C₂³×C₁₉₁	C₂×D₄
prime radical	C₂×#20259319	C₂
cosocle	C₂²×C₄×C₁₉₁	C₂³
hypercenter	C₂×#97	D₁₉₁
hypocenter	C₁₉₁	#1961
LCS	G ▷ C₂×C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

subgroup

G/H

center

C₂³

C₂×D₇₆₄

commutator

C₂×C₄×C₁₉₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₉₁

Fitting

C₂×#97×C₁₉₁

C₂

socle

C₂³×C₁₉₁

C₂×D₄

prime radical

C₂×#20259319

C₂

cosocle

C₂²×C₄×C₁₉₁

C₂³

hypercenter

C₂×#97

D₁₉₁

hypocenter

C₁₉₁

#1961

LCS

G ▷ C₂×C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

N	H ≅ G/N	type
2. C₂×#20259244	C₂	semidirect
3. #27132529	C₂	semidirect
4. C₂×#97×C₁₉₁	C₂	semidirect
8. #20259319	C₄	semidirect
28. C₂×C₈×C₁₉₁	D₄	semidirect
65. C₁₉₁	#1961	semidirect
69. C₂×#97	D₁₉₁	semidirect
93. C₂×C₈	D₇₆₄	semidirect

H ≅ G/N

type

2. C₂×#20259244

C₂

semidirect

3. #27132529

C₂

semidirect

4. C₂×#97×C₁₉₁

C₂

semidirect

8. #20259319

C₄

semidirect

28. C₂×C₈×C₁₉₁

D₄

semidirect

65. C₁₉₁

#1961

semidirect

69. C₂×#97

D₁₉₁

semidirect

93. C₂×C₈

D₇₆₄

semidirect

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

subgroup	index
C₂×#20259319	2
C₂×#20259244	2
#27132529	2
C₂×#97×C₁₉₁	2
#1961	191

subgroup

index

C₂×#97×C₁₉₁

191

Subgroups — #37573344

Conjugacy classes of subgroups

Complements

Maximal subgroups