Group #37573622 — subgroups

Subgroups — #37573622

Order 24448.

370 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 99 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	2	yes	maximal
2	C₂×#20259322	C₂	1	1	yes	maximal, characteristic
3	C₂×#20259247	C₂	1	2	yes	maximal
4	#27132569	C₂	1	1	yes	maximal, characteristic
5	#336×C₁₉₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₃₈₂	C₂²	1	1	yes	characteristic
7	C₂²×Dic₃₈₂	C₂²	1	2	yes	—
8	C₂×Dic₇₆₄	—	2	4	no	—
9	C₂×Dic₇₆₄	—	2	4	no	—
10	C₂×Dic₇₆₄	—	2	4	no	—
11	#20259322	—	2	4	no	—
12	C₂×C₄⋊Dic₁₉₁	—	2	2	no	—
13	C₂×C₄⋊Dic₁₉₁	C₂²	1	1	yes	characteristic
14	#20259247	—	2	4	no	—
15	C₁₉₁⋊#95	—	2	2	no	—
16	C₂²×C₈×C₁₉₁	C₂²	1	2	yes	—
17	C₂×C₄²×C₁₉₁	C₂²	1	1	yes	characteristic
18	C₂×Dic₃₈₂	—	2	2	no	—
19	C₂×Dic₃₈₂	—	2	2	no	—
20	C₂×Dic₃₈₂	—	2	2	no	—
21	C₂×Dic₃₈₂	—	2	2	no	—
22	C₂×Dic₃₈₂	—	2	2	no	—
23	C₂×Dic₃₈₂	—	2	2	no	—
24	C₂×Dic₃₈₂	—	2	1	no	—
25	C₂×Dic₃₈₂	—	2	2	no	—
26	C₂×Dic₃₈₂	D₄	1	2	yes	—
27	C₂×Dic₃₈₂	D₄	1	4	yes	—
28	C₂²×Dic₁₉₁	—	2	1	no	—
29	C₂²×Dic₁₉₁	—	2	2	no	—
30	C₂²×Dic₁₉₁	—	2	1	no	—
31	Dic₇₆₄	—	2	16	no	—
32	C₄⋊Dic₁₉₁	—	2	8	no	—
33	C₄⋊Dic₁₉₁	—	2	2	no	—
34	C₂²×C₄×C₁₉₁	D₄	1	2	yes	—
35	C₂²×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
36	C₂×C₈×C₁₉₁	—	2	2	no	—
37	C₂×C₈×C₁₉₁	—	2	2	no	—
38	C₂×C₈×C₁₉₁	D₄	1	4	yes	—
39	C₄²×C₁₉₁	—	2	2	no	—
40	Dic₃₈₂	—	2	4	no	—
41	Dic₃₈₂	—	2	4	no	—
42	Dic₃₈₂	—	2	4	no	—
43	Dic₃₈₂	—	4	2	no	—
44	Dic₃₈₂	—	4	2	no	—
45	Dic₃₈₂	—	4	2	no	—
46	C₂×Dic₁₉₁	—	2	4	no	—
47	C₂×Dic₁₉₁	—	2	2	no	—
48	C₂×Dic₁₉₁	—	4	1	no	—
49	C₂×Dic₁₉₁	—	4	2	no	—
50	C₂×Dic₁₉₁	—	4	1	no	—
51	C₂×Dic₁₉₁	—	4	2	no	—
52	C₂×Dic₁₉₁	—	4	1	no	—
53	C₂×Dic₁₉₁	—	4	2	no	—
54	C₂³×C₁₉₁	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₁₉₁	—	2	2	no	—
56	C₂×C₄×C₁₉₁	—	2	2	no	—
57	C₂×C₄×C₁₉₁	Q₁₆	1	4	yes	complemented
58	C₂×C₄×C₁₉₁	C₂×D₄	1	1	yes	characteristic
59	C₂×C₄×C₁₉₁	C₂×D₄	1	2	yes	—
60	C₂×C₄×C₁₉₁	C₂×D₄	1	2	yes	—
61	C₂×C₄×C₁₉₁	#41	1	1	yes	characteristic
62	C₈×C₁₉₁	—	2	4	no	—
63	C₂²×C₁₉₁	#120	1	1	yes	characteristic
64	C₂²×C₁₉₁	#121	1	1	yes	characteristic
65	C₂²×C₁₉₁	#127	1	1	yes	characteristic
66	C₂²×C₁₉₁	C₂×Q₁₆	1	2	yes	—
67	C₂²×C₁₉₁	#137	1	2	yes	—
68	C₄×C₁₉₁	—	2	4	no	—
69	C₄×C₁₉₁	#120	1	2	yes	—
70	C₄×C₁₉₁	#121	1	2	yes	—
71	Dic₁₉₁	—	4	4	no	—
72	Dic₁₉₁	—	4	2	no	—
73	Dic₁₉₁	—	8	1	no	—
74	C₂×C₁₉₁	#392	1	1	yes	characteristic
75	C₂×C₁₉₁	#451	1	2	yes	—
76	C₂×C₁₉₁	#462	1	2	yes	—
77	C₂×C₁₉₁	#494	1	1	yes	characteristic
78	C₂×C₁₉₁	#497	1	1	yes	characteristic
79	C₁₉₁	#2016	1	1	yes	characteristic, complemented, Sylow
80	#2016	—	191	1	no	maximal, Sylow
81	C₂²×Q₁₆	—	191	2	no	—
82	C₂×C₄⋊Q₈	—	191	1	no	—
83	C₂×#103	—	191	2	no	—
84	#391	—	191	1	no	—
85	#336	D₁₉₁	1	1	yes	characteristic
86	C₂²×Q₈	—	191	2	no	—
87	C₂²×Q₈	—	191	1	no	—
88	C₂×Q₁₆	—	382	4	no	—
89	C₂×Q₁₆	—	382	4	no	—
90	C₂×Q₁₆	—	382	4	no	—
91	C₂²×C₈	C₂×D₁₉₁	1	2	yes	—
92	C₄⋊Q₈	—	382	4	no	—
93	C₂×C₄⋊C₄	—	382	2	no	—
94	C₂×C₄⋊C₄	—	191	1	no	—
95	C₂×C₄²	C₂×D₁₉₁	1	1	yes	characteristic
96	#103	—	382	4	no	—
97	#95	—	382	2	no	—
98	C₂×Q₈	—	382	2	no	—
99	C₂×Q₈	—	382	1	no	—
100	C₂×Q₈	—	382	2	no	—
101	C₂×Q₈	—	382	2	no	—
102	C₂×Q₈	—	382	2	no	—
103	C₂×Q₈	—	382	2	no	—
104	C₂×Q₈	—	382	2	no	—
105	C₂×Q₈	—	382	2	no	—
106	C₂×Q₈	—	191	4	no	—
107	C₂×Q₈	—	191	2	no	—
108	C₂²×C₄	—	382	1	no	—
109	C₂²×C₄	—	382	2	no	—
110	C₂²×C₄	—	382	1	no	—
111	C₂²×C₄	D₇₆₄	1	2	yes	—
112	C₂²×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
113	Q₁₆	—	382	16	no	—
114	C₂×C₈	—	2	2	no	—
115	C₂×C₈	—	2	2	no	—
116	C₂×C₈	D₇₆₄	1	4	yes	—
117	C₄⋊C₄	—	382	8	no	—
118	C₄⋊C₄	—	382	2	no	—
119	C₄²	—	2	2	no	—
120	C₂³	C₂×D₇₆₄	1	1	yes	characteristic
121	Q₈	—	382	4	no	—
122	Q₈	—	382	4	no	—
123	Q₈	—	382	4	no	—
124	Q₈	—	764	2	no	—
125	Q₈	—	764	2	no	—
126	Q₈	—	764	2	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	—	382	4	no	—
130	C₂×C₄	—	382	2	no	—
131	C₂×C₄	—	764	2	no	—
132	C₂×C₄	—	764	1	no	—
133	C₂×C₄	—	764	1	no	—
134	C₂×C₄	—	764	2	no	—
135	C₂×C₄	—	764	1	no	—
136	C₂×C₄	—	764	2	no	—
137	C₂×C₄	Dic₇₆₄	1	4	yes	complemented
138	C₂×C₄	D₄×D₁₉₁	1	1	yes	characteristic
139	C₂×C₄	D₄×D₁₉₁	1	2	yes	—
140	C₂×C₄	C₂×D₇₆₄	1	2	yes	—
141	C₂×C₄	#15690288	1	1	yes	characteristic
142	C₈	—	2	4	no	—
143	C₂²	#20259292	1	1	yes	characteristic
144	C₂²	#20259297	1	1	yes	characteristic
145	C₂²	#20259319	1	1	yes	characteristic
146	C₂²	C₂×Dic₇₆₄	1	2	yes	—
147	C₂²	#20259355	1	2	yes	—
148	C₄	—	2	4	no	—
149	C₄	—	1528	1	no	—
150	C₄	—	764	4	no	—
151	C₄	—	764	2	no	—
152	C₄	#20259292	1	2	yes	—
153	C₄	#20259297	1	2	yes	—
154	C₂	#27132573	1	1	yes	characteristic
155	C₂	#27132821	1	2	yes	—
156	C₂	#27132874	1	2	yes	—
157	C₂	#27133043	1	1	yes	characteristic
158	C₂	#27133057	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₇₆₄

C₂

yes

maximal

C₂×#20259322

C₂

yes

maximal, characteristic

C₂×#20259247

C₂

yes

maximal

#27132569

C₂

yes

maximal, characteristic

#336×C₁₉₁

C₂

yes

maximal, characteristic

C₂²×Dic₃₈₂

C₂²

yes

characteristic

C₂²×Dic₃₈₂

C₂²

yes

—

C₂×Dic₇₆₄

—

C₂×Dic₇₆₄

—

C₂×Dic₇₆₄

—

#20259322

—

C₂×C₄⋊Dic₁₉₁

—

C₂×C₄⋊Dic₁₉₁

C₂²

yes

characteristic

#20259247

—

C₁₉₁⋊#95

—

C₂²×C₈×C₁₉₁

C₂²

yes

—

C₂×C₄²×C₁₉₁

C₂²

yes

characteristic

C₂×Dic₃₈₂

—

C₂×Dic₃₈₂

—

C₂×Dic₃₈₂

—

C₂×Dic₃₈₂

—

C₂×Dic₃₈₂

—

C₂×Dic₃₈₂

—

C₂×Dic₃₈₂

—

C₂×Dic₃₈₂

—

C₂×Dic₃₈₂

D₄

yes

—

C₂×Dic₃₈₂

D₄

yes

—

C₂²×Dic₁₉₁

—

C₂²×Dic₁₉₁

—

C₂²×Dic₁₉₁

—

Dic₇₆₄

—

C₄⋊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

—

C₄²×C₁₉₁

—

Dic₃₈₂

—

Dic₃₈₂

—

Dic₃₈₂

—

Dic₃₈₂

—

Dic₃₈₂

—

Dic₃₈₂

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

—

C₂³×C₁₉₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

Q₁₆

yes

complemented

C₂×C₄×C₁₉₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₁

C₂×D₄

yes

—

C₂×C₄×C₁₉₁

C₂×D₄

yes

—

C₂×C₄×C₁₉₁

#41

yes

characteristic

C₈×C₁₉₁

—

C₂²×C₁₉₁

yes

C₂²×C₁₉₁

yes

C₂²×C₁₉₁

yes

C₂²×C₁₉₁

C₂×Q₁₆

yes

—

C₂²×C₁₉₁

#137

yes

—

C₄×C₁₉₁

—

C₄×C₁₉₁

#120

yes

—

C₄×C₁₉₁

#121

yes

—

Dic₁₉₁

—

Dic₁₉₁

—

Dic₁₉₁

—

C₂×C₁₉₁

#392

yes

characteristic

C₂×C₁₉₁

#451

yes

—

C₂×C₁₉₁

#462

yes

—

C₂×C₁₉₁

yes

C₂×C₁₉₁

yes

C₁₉₁

yes

characteristic, complemented, Sylow

#2016

—

191

maximal, Sylow

C₂²×Q₁₆

—

191

—

C₂×C₄⋊Q₈

—

191

—

C₂×#103

—

191

—

#391

—

191

—

#336

D₁₉₁

yes

characteristic

C₂²×Q₈

—

191

—

C₂²×Q₈

—

191

—

C₂×Q₁₆

—

382

—

C₂×Q₁₆

—

382

—

C₂×Q₁₆

—

382

—

C₂²×C₈

C₂×D₁₉₁

yes

—

C₄⋊Q₈

—

382

—

C₂×C₄⋊C₄

—

382

—

C₂×C₄⋊C₄

—

191

—

C₂×C₄²

C₂×D₁₉₁

yes

characteristic

#103

—

382

—

#95

—

382

—

C₂×Q₈

—

382

—

C₂×Q₈

—

382

—

100

C₂×Q₈

—

382

—

101

C₂×Q₈

—

382

—

102

C₂×Q₈

—

382

—

103

C₂×Q₈

—

382

—

104

C₂×Q₈

—

382

—

105

C₂×Q₈

—

382

—

106

C₂×Q₈

—

191

—

107

C₂×Q₈

—

191

—

108

C₂²×C₄

—

382

—

109

C₂²×C₄

—

382

—

110

C₂²×C₄

—

382

—

111

C₂²×C₄

D₇₆₄

yes

—

112

C₂²×C₄

C₂²×D₁₉₁

yes

characteristic

113

Q₁₆

—

382

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

D₇₆₄

yes

—

117

C₄⋊C₄

—

382

—

118

C₄⋊C₄

—

382

—

119

C₄²

—

120

C₂³

C₂×D₇₆₄

yes

characteristic

121

Q₈

—

382

—

122

Q₈

—

382

—

123

Q₈

—

382

—

124

Q₈

—

764

—

125

Q₈

—

764

—

126

Q₈

—

764

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

382

—

130

C₂×C₄

—

382

—

131

C₂×C₄

—

764

—

132

C₂×C₄

—

764

—

133

C₂×C₄

—

764

—

134

C₂×C₄

—

764

—

135

C₂×C₄

—

764

—

136

C₂×C₄

—

764

—

137

C₂×C₄

Dic₇₆₄

yes

complemented

138

C₂×C₄

D₄×D₁₉₁

yes

characteristic

139

C₂×C₄

D₄×D₁₉₁

yes

—

140

C₂×C₄

C₂×D₇₆₄

yes

—

141

C₂×C₄

#15690288

yes

characteristic

142

C₈

—

143

C₂²

#20259292

yes

characteristic

144

C₂²

#20259297

yes

characteristic

145

C₂²

#20259319

yes

characteristic

146

C₂²

C₂×Dic₇₆₄

yes

—

147

C₂²

#20259355

yes

—

148

C₄

—

149

C₄

—

1528

—

150

C₄

—

764

—

151

C₄

—

764

—

152

C₄

#20259292

yes

—

153

C₄

#20259297

yes

—

154

C₂

#27132573

yes

characteristic

155

C₂

#27132821

yes

—

156

C₂

#27132874

yes

—

157

C₂

#27133043

yes

characteristic

158

C₂

#27133057

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₇₆₄
commutator, cosocle	C₂²×C₄×C₁₉₁	C₂³
Frattini	C₂²×C₄	C₂²×D₁₉₁
Fitting	#336×C₁₉₁	C₂
socle, prime radical	C₂³×C₁₉₁	C₂×D₄
hypercenter	#336	D₁₉₁
hypocenter	C₁₉₁	#2016
LCS	G ▷ C₂²×C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

subgroup

G/H

center

C₂³

C₂×D₇₆₄

commutator, cosocle

C₂²×C₄×C₁₉₁

C₂³

Frattini

C₂²×C₄

C₂²×D₁₉₁

Fitting

#336×C₁₉₁

C₂

socle, prime radical

C₂³×C₁₉₁

C₂×D₄

D₁₉₁

C₁₉₁

G ▷ C₂²×C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

N	H ≅ G/N	type
57. C₂×C₄×C₁₉₁	Q₁₆	semidirect
79. C₁₉₁	#2016	semidirect
137. C₂×C₄	Dic₇₆₄	semidirect

H ≅ G/N

type

57. C₂×C₄×C₁₉₁

Q₁₆

semidirect

79. C₁₉₁

#2016

semidirect

137. C₂×C₄

Dic₇₆₄

semidirect

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

subgroup	index
C₂²×Dic₇₆₄	2
C₂×#20259322	2
C₂×#20259247	2
#27132569	2
#336×C₁₉₁	2
#2016	191

subgroup

index

C₂²×Dic₇₆₄

#336×C₁₉₁

191

Subgroups — #37573622

Conjugacy classes of subgroups

Complements

Maximal subgroups