Group #37583259 — subgroups

Subgroups — #37583259

Order 24448.

446 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 125 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	#27133214	C₂	1	2	yes	maximal, complemented
2	#27133040	C₂	1	1	yes	maximal, characteristic, complemented
3	#27133011	C₂	1	2	yes	maximal, complemented
4	#27132917	C₂	1	4	yes	maximal, complemented
5	#27132774	C₂	1	4	yes	maximal, complemented
6	C₄×#15690270	C₂	1	1	yes	maximal, characteristic, complemented
7	#498×C₁₉₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#15690288	—	2	2	no	—
9	C₂×D₁₅₂₈	C₂²	1	4	yes	—
10	C₂×#15690270	C₂²	1	2	yes	—
11	#20259319	C₂²	1	2	yes	—
12	#20259311	—	2	2	no	—
13	#20259297	—	2	2	no	—
14	#20259297	C₂²	1	4	yes	—
15	#20259284	—	2	2	no	—
16	#20259284	C₂²	1	4	yes	—
17	C₄²×D₁₉₁	C₂²	1	1	yes	characteristic
18	#20259242	C₂²	1	8	yes	—
19	#20259232	C₂²	1	2	yes	—
20	C₄×C₁₉₁⋊C₈	C₂²	1	1	yes	characteristic
21	C₄⋊Q₈×C₁₉₁	C₂²	1	2	yes	—
22	#106×C₁₉₁	C₂²	1	4	yes	—
23	C₄×C₈×C₁₉₁	C₂²	1	1	yes	characteristic
24	#15690288	—	4	8	no	—
25	C₂×D₇₆₄	—	2	4	no	—
26	C₂×D₇₆₄	—	2	2	no	—
27	C₂×D₇₆₄	—	2	2	no	—
28	C₂×D₇₆₄	C₂³	1	4	yes	—
29	C₂×C₄×D₁₉₁	—	2	4	no	—
30	C₂×C₄×D₁₉₁	D₄	1	2	yes	—
31	C₂×C₄×D₁₉₁	C₂³	1	1	yes	characteristic
32	D₁₅₂₈	—	2	8	no	—
33	#15690270	—	2	4	no	—
34	C₂×C₁₉₁⋊C₈	C₂³	1	2	yes	—
35	(C₂×D₁₉₁)⋊C₄	—	2	4	no	—
36	(C₂×D₁₉₁)⋊C₄	—	2	4	no	—
37	C₄×Dic₁₉₁	D₄	1	2	yes	—
38	C₄×Dic₁₉₁	C₂³	1	1	yes	characteristic
39	C₂×Q₈×C₁₉₁	—	2	2	no	—
40	C₂×C₈×C₁₉₁	C₂³	1	2	yes	—
41	C₄⋊C₄×C₁₉₁	—	2	2	no	—
42	C₄⋊C₄×C₁₉₁	C₂³	1	4	yes	—
43	C₄²×C₁₉₁	C₂³	1	1	yes	characteristic
44	C₂²×D₁₉₁	—	2	4	no	—
45	C₂²×D₁₉₁	C₂×D₄	1	1	yes	characteristic
46	D₇₆₄	—	2	8	no	—
47	D₇₆₄	—	4	8	no	—
48	D₇₆₄	—	4	4	no	—
49	D₇₆₄	—	4	4	no	—
50	C₂×Dic₁₉₁	C₂×D₄	1	1	yes	characteristic
51	C₂×Dic₁₉₁	C₂×D₄	1	2	yes	—
52	C₄×D₁₉₁	—	2	4	no	—
53	C₄×D₁₉₁	—	2	2	no	—
54	C₄×D₁₉₁	—	4	8	no	—
55	C₁₉₁⋊C₈	—	2	2	no	—
56	Q₈×C₁₉₁	—	4	4	no	—
57	C₂×C₄×C₁₉₁	—	2	4	no	—
58	C₂×C₄×C₁₉₁	C₂×D₄	1	2	yes	—
59	C₂×C₄×C₁₉₁	C₂⁴	1	1	yes	characteristic
60	C₈×C₁₉₁	#41	1	4	yes	complemented
61	C₂²×C₁₉₁	C₂²×D₄	1	1	yes	characteristic
62	C₂×D₁₉₁	—	2	1	no	—
63	C₂×D₁₉₁	—	2	2	no	—
64	C₂×D₁₉₁	—	4	4	no	—
65	C₂×D₁₉₁	—	4	4	no	—
66	C₂×D₁₉₁	—	4	4	no	—
67	C₄×C₁₉₁	—	4	4	no	—
68	C₄×C₁₉₁	AGL(1,ℤ/8)	1	4	yes	—
69	C₄×C₁₉₁	C₂×#41	1	2	yes	—
70	Dic₁₉₁	—	2	1	no	—
71	Dic₁₉₁	—	2	2	no	—
72	C₂×C₁₉₁	#532	1	1	yes	characteristic
73	C₂×C₁₉₁	C₂×AGL(1,ℤ/8)	1	2	yes	—
74	D₁₉₁	—	4	1	no	—
75	D₁₉₁	—	8	4	no	—
76	C₁₉₁	#3184	1	1	yes	characteristic, complemented, Sylow
77	#3184	—	191	1	no	maximal, Sylow
78	#532	—	191	2	no	—
79	#498	D₁₉₁	1	1	yes	characteristic, complemented
80	#493	—	191	1	no	—
81	#486	—	191	2	no	—
82	#469	—	191	4	no	—
83	#442	—	191	4	no	—
84	C₄×M₁₆	—	191	1	no	—
85	C₂×#41	—	382	2	no	—
86	C₂×D₈	—	191	4	no	—
87	C₂×M₁₆	—	191	2	no	—
88	C₄⋊Q₈	C₂×D₁₉₁	1	2	yes	—
89	#127	—	191	2	no	—
90	#124	—	382	2	no	—
91	#121	—	382	2	no	—
92	#121	—	191	4	no	—
93	C₄×D₄	—	382	2	no	—
94	C₄×D₄	—	191	4	no	—
95	C₂×C₄²	—	191	1	no	—
96	#106	C₂×D₁₉₁	1	4	yes	—
97	#102	—	191	8	no	—
98	#97	—	191	2	no	—
99	C₄×C₈	—	191	1	no	—
100	C₄×C₈	C₂×D₁₉₁	1	1	yes	characteristic
101	#41	—	764	8	no	—
102	C₂×Q₈	—	2	2	no	—
103	C₂×D₄	—	382	4	no	—
104	C₂×D₄	—	382	2	no	—
105	C₂×D₄	—	382	2	no	—
106	C₂×D₄	—	191	4	no	—
107	C₂²×C₄	—	382	4	no	—
108	C₂²×C₄	—	191	2	no	—
109	C₂²×C₄	—	191	1	no	—
110	D₈	—	382	8	no	—
111	M₁₆	—	382	4	no	—
112	C₂×C₈	—	191	2	no	—
113	C₂×C₈	C₂²×D₁₉₁	1	2	yes	—
114	C₄⋊C₄	—	2	2	no	—
115	C₄⋊C₄	C₂²×D₁₉₁	1	4	yes	—
116	C₂²⋊C₄	—	382	4	no	—
117	C₂²⋊C₄	—	382	4	no	—
118	C₄²	—	191	2	no	—
119	C₄²	—	191	1	no	—
120	C₄²	C₂²×D₁₉₁	1	1	yes	characteristic
121	C₂³	—	382	4	no	—
122	C₂³	—	191	1	no	—
123	Q₈	—	4	4	no	—
124	D₄	—	382	8	no	—
125	D₄	—	764	8	no	—
126	D₄	—	764	4	no	—
127	D₄	—	764	4	no	—
128	C₂×C₄	—	2	4	no	—
129	C₂×C₄	—	382	4	no	—
130	C₂×C₄	—	382	2	no	—
131	C₂×C₄	—	191	1	no	—
132	C₂×C₄	—	191	2	no	—
133	C₂×C₄	—	764	8	no	—
134	C₂×C₄	D₄×D₁₉₁	1	2	yes	—
135	C₂×C₄	C₂³×D₁₉₁	1	1	yes	characteristic
136	C₈	—	382	2	no	—
137	C₈	#15690288	1	4	yes	complemented
138	C₂²	—	382	2	no	—
139	C₂²	—	382	1	no	—
140	C₂²	—	764	4	no	—
141	C₂²	—	764	4	no	—
142	C₂²	—	764	4	no	—
143	C₂²	C₂×D₄×D₁₉₁	1	1	yes	characteristic
144	C₄	—	4	4	no	—
145	C₄	—	382	2	no	—
146	C₄	—	382	1	no	—
147	C₄	#20259349	1	4	yes	—
148	C₄	C₂×#15690288	1	2	yes	—
149	C₂	—	1528	4	no	—
150	C₂	—	764	1	no	—
151	C₂	#27133214	1	1	yes	characteristic
152	C₂	C₂×#20259349	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#27133214

C₂

yes

maximal, complemented

#27133040

C₂

yes

maximal, characteristic, complemented

#27133011

C₂

yes

maximal, complemented

#27132917

C₂

yes

maximal, complemented

#27132774

C₂

yes

maximal, complemented

C₄×#15690270

C₂

yes

maximal, characteristic, complemented

#498×C₁₉₁

C₂

yes

maximal, characteristic, complemented

C₂×#15690288

—

C₂×D₁₅₂₈

C₂²

yes

—

C₂×#15690270

C₂²

yes

—

#20259319

C₂²

yes

—

#20259311

—

#20259297

—

#20259297

C₂²

yes

—

#20259284

—

#20259284

C₂²

yes

—

C₄²×D₁₉₁

C₂²

yes

characteristic

#20259242

C₂²

yes

—

#20259232

C₂²

yes

—

C₄×C₁₉₁⋊C₈

C₂²

yes

characteristic

C₄⋊Q₈×C₁₉₁

C₂²

yes

—

#106×C₁₉₁

C₂²

yes

—

C₄×C₈×C₁₉₁

C₂²

yes

characteristic

#15690288

—

C₂×D₇₆₄

—

C₂×D₇₆₄

—

C₂×D₇₆₄

—

C₂×D₇₆₄

C₂³

yes

—

C₂×C₄×D₁₉₁

—

C₂×C₄×D₁₉₁

D₄

yes

—

C₂×C₄×D₁₉₁

C₂³

yes

characteristic

D₁₅₂₈

—

#15690270

—

C₂×C₁₉₁⋊C₈

C₂³

yes

—

(C₂×D₁₉₁)⋊C₄

—

(C₂×D₁₉₁)⋊C₄

—

C₄×Dic₁₉₁

D₄

yes

—

C₄×Dic₁₉₁

C₂³

yes

characteristic

C₂×Q₈×C₁₉₁

—

C₂×C₈×C₁₉₁

C₂³

yes

—

C₄⋊C₄×C₁₉₁

—

C₄⋊C₄×C₁₉₁

C₂³

yes

—

C₄²×C₁₉₁

C₂³

yes

characteristic

C₂²×D₁₉₁

—

C₂²×D₁₉₁

C₂×D₄

yes

characteristic

D₇₆₄

—

D₇₆₄

—

D₇₆₄

—

D₇₆₄

—

C₂×Dic₁₉₁

C₂×D₄

yes

characteristic

C₂×Dic₁₉₁

C₂×D₄

yes

—

C₄×D₁₉₁

—

C₄×D₁₉₁

—

C₄×D₁₉₁

—

C₁₉₁⋊C₈

—

Q₈×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

C₂×D₄

yes

—

C₂×C₄×C₁₉₁

C₂⁴

yes

characteristic

C₈×C₁₉₁

#41

yes

complemented

C₂²×C₁₉₁

C₂²×D₄

yes

characteristic

C₂×D₁₉₁

—

C₂×D₁₉₁

—

C₂×D₁₉₁

—

C₂×D₁₉₁

—

C₂×D₁₉₁

—

C₄×C₁₉₁

—

C₄×C₁₉₁

AGL(1,ℤ/8)

yes

—

C₄×C₁₉₁

C₂×#41

yes

—

Dic₁₉₁

—

Dic₁₉₁

—

C₂×C₁₉₁

#532

yes

characteristic

C₂×C₁₉₁

C₂×AGL(1,ℤ/8)

yes

—

D₁₉₁

—

D₁₉₁

—

C₁₉₁

#3184

yes

characteristic, complemented, Sylow

#3184

—

191

maximal, Sylow

#532

—

191

—

#498

D₁₉₁

yes

characteristic, complemented

#493

—

191

—

#486

—

191

—

#469

—

191

—

#442

—

191

—

C₄×M₁₆

—

191

—

C₂×#41

—

382

—

C₂×D₈

—

191

—

C₂×M₁₆

—

191

—

C₄⋊Q₈

C₂×D₁₉₁

yes

—

#127

—

191

—

#124

—

382

—

#121

—

382

—

#121

—

191

—

C₄×D₄

—

382

—

C₄×D₄

—

191

—

C₂×C₄²

—

191

—

#106

C₂×D₁₉₁

yes

—

#102

—

191

—

#97

—

191

—

C₄×C₈

—

191

—

100

C₄×C₈

C₂×D₁₉₁

yes

characteristic

101

#41

—

764

—

102

C₂×Q₈

—

103

C₂×D₄

—

382

—

104

C₂×D₄

—

382

—

105

C₂×D₄

—

382

—

106

C₂×D₄

—

191

—

107

C₂²×C₄

—

382

—

108

C₂²×C₄

—

191

—

109

C₂²×C₄

—

191

—

110

D₈

—

382

—

111

M₁₆

—

382

—

112

C₂×C₈

—

191

—

113

C₂×C₈

C₂²×D₁₉₁

yes

—

114

C₄⋊C₄

—

115

C₄⋊C₄

C₂²×D₁₉₁

yes

—

116

C₂²⋊C₄

—

382

—

117

C₂²⋊C₄

—

382

—

118

C₄²

—

191

—

119

C₄²

—

191

—

120

C₄²

C₂²×D₁₉₁

yes

characteristic

121

C₂³

—

382

—

122

C₂³

—

191

—

123

Q₈

—

124

D₄

—

382

—

125

D₄

—

764

—

126

D₄

—

764

—

127

D₄

—

764

—

128

C₂×C₄

—

129

C₂×C₄

—

382

—

130

C₂×C₄

—

382

—

131

C₂×C₄

—

191

—

132

C₂×C₄

—

191

—

133

C₂×C₄

—

764

—

134

C₂×C₄

D₄×D₁₉₁

yes

—

135

C₂×C₄

C₂³×D₁₉₁

yes

characteristic

136

C₈

—

382

—

137

C₈

#15690288

yes

complemented

138

C₂²

—

382

—

139

C₂²

—

382

—

140

C₂²

—

764

—

141

C₂²

—

764

—

142

C₂²

—

764

—

143

C₂²

C₂×D₄×D₁₉₁

yes

characteristic

144

C₄

—

145

C₄

—

382

—

146

C₄

—

382

—

147

C₄

#20259349

yes

—

148

C₄

C₂×#15690288

yes

—

149

C₂

—

1528

—

150

C₂

—

764

—

151

C₂

#27133214

yes

characteristic

152

C₂

C₂×#20259349

yes

—

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₉₁
commutator, cosocle	C₂×C₄×C₁₉₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₉₁
Fitting	#498×C₁₉₁	C₂
socle	C₂²×C₁₉₁	C₂²×D₄
hypercenter	#498	D₁₉₁
hypocenter	C₁₉₁	#3184
LCS	G ▷ C₂×C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #498

subgroup

G/H

center

C₂²

C₂×D₄×D₁₉₁

commutator, cosocle

C₂×C₄×C₁₉₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₉₁

Fitting

#498×C₁₉₁

C₂

socle

C₂²×C₁₉₁

C₂²×D₄

D₁₉₁

C₁₉₁

G ▷ C₂×C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #498

N	H ≅ G/N	type
1. #27133214	C₂	semidirect
2. #27133040	C₂	semidirect
3. #27133011	C₂	semidirect
4. #27132917	C₂	semidirect
5. #27132774	C₂	semidirect
6. C₄×#15690270	C₂	semidirect
7. #498×C₁₉₁	C₂	semidirect
60. C₈×C₁₉₁	#41	semidirect
76. C₁₉₁	#3184	semidirect
79. #498	D₁₉₁	semidirect
137. C₈	#15690288	semidirect

H ≅ G/N

type

1. #27133214

C₂

semidirect

2. #27133040

C₂

semidirect

3. #27133011

C₂

semidirect

4. #27132917

C₂

semidirect

5. #27132774

C₂

semidirect

6. C₄×#15690270

C₂

semidirect

7. #498×C₁₉₁

C₂

semidirect

60. C₈×C₁₉₁

#41

semidirect

76. C₁₉₁

#3184

semidirect

79. #498

D₁₉₁

semidirect

137. C₈

#15690288

semidirect

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

subgroup	index
#27133214	2
#27133040	2
#27133011	2
#27132917	2
#27132774	2
C₄×#15690270	2
#498×C₁₉₁	2
#3184	191

subgroup

index

#498×C₁₉₁

191

Subgroups — #37583259

Conjugacy classes of subgroups

Complements

Maximal subgroups