Group #37573244 — subgroups

Subgroups — #37573244

Order 24448.

248 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 121 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×C₄⋊(C₁₉₁⋊C₈)	C₂	1	2	yes	maximal
2	C₂×C₄⋊(C₁₉₁⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	C₂×#20259233	C₂	1	1	yes	maximal, characteristic
4	#27132418	C₂	1	2	yes	maximal
5	C₄×C₄⋊C₄×C₁₉₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₉₁⋊C₈	C₂²	1	2	yes	—
7	C₂²×C₁₉₁⋊C₈	C₂²	1	1	yes	characteristic
8	C₂²×C₁₉₁⋊C₈	C₂²	1	1	yes	characteristic
9	C₄⋊(C₁₉₁⋊C₈)	—	2	4	no	—
10	C₄⋊(C₁₉₁⋊C₈)	—	2	2	no	—
11	#20259233	—	2	2	no	—
12	C₂×C₄⋊C₄×C₁₉₁	C₄	1	2	yes	—
13	C₂×C₄²×C₁₉₁	C₂²	1	2	yes	—
14	C₂×C₄²×C₁₉₁	C₂²	1	1	yes	characteristic
15	#95×C₁₉₁	C₄	1	1	yes	characteristic
16	#95×C₁₉₁	C₄	1	1	yes	characteristic
17	C₂×C₁₉₁⋊C₈	—	2	1	no	—
18	C₂×C₁₉₁⋊C₈	—	2	1	no	—
19	C₂×C₁₉₁⋊C₈	—	2	2	no	—
20	C₂×C₁₉₁⋊C₈	—	2	2	no	—
21	C₂×C₁₉₁⋊C₈	—	2	1	no	—
22	C₂×C₁₉₁⋊C₈	—	2	1	no	—
23	C₂×C₁₉₁⋊C₈	—	2	2	no	—
24	C₂×C₁₉₁⋊C₈	D₄	1	2	yes	—
25	C₂×C₁₉₁⋊C₈	Q₈	1	2	yes	—
26	C₂²×C₄×C₁₉₁	C₂×C₄	1	2	yes	—
27	C₂²×C₄×C₁₉₁	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄×C₁₉₁	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₄×C₁₉₁	C₂×C₄	1	2	yes	—
30	C₂²×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
31	C₄⋊C₄×C₁₉₁	—	2	4	no	—
32	C₄²×C₁₉₁	—	2	2	no	—
33	C₄²×C₁₉₁	—	2	2	no	—
34	C₄²×C₁₉₁	D₄	1	2	yes	—
35	C₄²×C₁₉₁	Q₈	1	2	yes	—
36	C₁₉₁⋊C₈	—	2	2	no	—
37	C₁₉₁⋊C₈	—	2	2	no	—
38	C₁₉₁⋊C₈	—	4	2	no	—
39	C₂³×C₁₉₁	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₄×C₁₉₁	—	2	2	no	—
41	C₂×C₄×C₁₉₁	—	2	2	no	—
42	C₂×C₄×C₁₉₁	—	2	2	no	—
43	C₂×C₄×C₁₉₁	—	2	2	no	—
44	C₂×C₄×C₁₉₁	—	2	2	no	—
45	C₂×C₄×C₁₉₁	—	2	2	no	—
46	C₂×C₄×C₁₉₁	—	2	2	no	—
47	C₂×C₄×C₁₉₁	C₄⋊C₄	1	2	yes	—
48	C₂×C₄×C₁₉₁	C₄⋊C₄	1	2	yes	—
49	C₂×C₄×C₁₉₁	M₁₆	1	4	yes	—
50	C₂×C₄×C₁₉₁	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₁₉₁	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₁₉₁	C₂×Q₈	1	1	yes	characteristic
53	C₂×C₄×C₁₉₁	C₂×Q₈	1	1	yes	characteristic
54	C₂×C₄×C₁₉₁	#41	1	1	yes	characteristic
55	C₂×C₄×C₁₉₁	#41	1	1	yes	characteristic
56	C₂²×C₁₉₁	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₂²×C₁₉₁	C₄×D₄	1	1	yes	characteristic
58	C₂²×C₁₉₁	C₄×Q₈	1	1	yes	characteristic
59	C₂²×C₁₉₁	C₂×M₁₆	1	2	yes	—
60	C₂²×C₁₉₁	#131	1	2	yes	—
61	C₄×C₁₉₁	—	2	2	no	—
62	C₄×C₁₉₁	—	2	2	no	—
63	C₄×C₁₉₁	—	2	4	no	—
64	C₄×C₁₉₁	—	4	2	no	—
65	C₄×C₁₉₁	#121	1	1	yes	characteristic
66	C₄×C₁₉₁	C₂²⋊Q₈	1	1	yes	characteristic
67	C₄×C₁₉₁	#125	1	1	yes	characteristic
68	C₄×C₁₉₁	C₄⋊Q₈	1	1	yes	characteristic
69	C₂×C₁₉₁	#389	1	1	yes	characteristic
70	C₂×C₁₉₁	#423	1	1	yes	characteristic
71	C₂×C₁₉₁	#424	1	1	yes	characteristic
72	C₂×C₁₉₁	#435	1	2	yes	—
73	C₂×C₁₉₁	#446	1	2	yes	—
74	C₁₉₁	#1940	1	1	yes	characteristic, complemented, Sylow
75	#1940	—	191	1	no	maximal, Sylow
76	C₂×C₄⋊C₈	—	191	2	no	—
77	C₂×C₄⋊C₈	—	191	1	no	—
78	C₂×#97	—	191	1	no	—
79	C₄×C₄⋊C₄	D₁₉₁	1	1	yes	characteristic
80	#336	—	191	2	no	—
81	C₂²×C₈	—	191	2	no	—
82	C₂²×C₈	—	191	1	no	—
83	C₂²×C₈	—	191	1	no	—
84	C₂×C₄⋊C₄	Dic₁₉₁	1	2	yes	—
85	C₂×C₄²	C₂×D₁₉₁	1	2	yes	—
86	C₂×C₄²	C₂×D₁₉₁	1	1	yes	characteristic
87	C₄⋊C₈	—	382	4	no	—
88	C₄⋊C₈	—	382	2	no	—
89	#97	—	382	2	no	—
90	#95	Dic₁₉₁	1	1	yes	characteristic
91	#95	Dic₁₉₁	1	1	yes	characteristic
92	C₂²×C₄	C₂×Dic₁₉₁	1	2	yes	—
93	C₂²×C₄	C₂×Dic₁₉₁	1	1	yes	characteristic
94	C₂²×C₄	C₂×Dic₁₉₁	1	1	yes	characteristic
95	C₂²×C₄	C₂×Dic₁₉₁	1	2	yes	—
96	C₂²×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
97	C₂×C₈	—	382	1	no	—
98	C₂×C₈	—	382	1	no	—
99	C₂×C₈	—	382	1	no	—
100	C₂×C₈	—	382	1	no	—
101	C₂×C₈	—	382	2	no	—
102	C₂×C₈	—	382	2	no	—
103	C₂×C₈	—	382	2	no	—
104	C₂×C₈	—	191	2	no	—
105	C₂×C₈	—	191	2	no	—
106	C₄⋊C₄	—	2	4	no	—
107	C₄²	—	2	2	no	—
108	C₄²	—	2	2	no	—
109	C₄²	D₇₆₄	1	2	yes	—
110	C₄²	Dic₃₈₂	1	2	yes	—
111	C₂³	C₂²×Dic₁₉₁	1	1	yes	characteristic
112	C₂×C₄	—	2	2	no	—
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	2	2	no	—
119	C₂×C₄	C₄⋊Dic₁₉₁	1	2	yes	—
120	C₂×C₄	C₄⋊Dic₁₉₁	1	2	yes	—
121	C₂×C₄	#15690271	1	4	yes	—
122	C₂×C₄	D₄×D₁₉₁	1	1	yes	characteristic
123	C₂×C₄	C₂×D₇₆₄	1	1	yes	characteristic
124	C₂×C₄	Q₈×D₁₉₁	1	1	yes	characteristic
125	C₂×C₄	C₂×Dic₃₈₂	1	1	yes	characteristic
126	C₂×C₄	#15690286	1	1	yes	characteristic
127	C₂×C₄	#15690288	1	1	yes	characteristic
128	C₈	—	382	2	no	—
129	C₈	—	382	2	no	—
130	C₈	—	764	2	no	—
131	C₂²	C₂×C₄⋊Dic₁₉₁	1	1	yes	characteristic
132	C₂²	D₄×Dic₁₉₁	1	1	yes	characteristic
133	C₂²	Q₈×Dic₁₉₁	1	1	yes	characteristic
134	C₂²	C₂×#15690271	1	2	yes	—
135	C₂²	#20259329	1	2	yes	—
136	C₄	—	2	2	no	—
137	C₄	—	2	2	no	—
138	C₄	—	2	4	no	—
139	C₄	—	4	2	no	—
140	C₄	#20259297	1	1	yes	characteristic
141	C₄	#20259302	1	1	yes	characteristic
142	C₄	#20259313	1	1	yes	characteristic
143	C₄	#20259321	1	1	yes	characteristic
144	C₂	#27132562	1	1	yes	characteristic
145	C₂	#27132687	1	1	yes	characteristic
146	C₂	#27132690	1	1	yes	characteristic
147	C₂	#27132735	1	2	yes	—
148	C₂	#27132794	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊(C₁₉₁⋊C₈)

C₂

yes

maximal

C₂×C₄⋊(C₁₉₁⋊C₈)

C₂

yes

maximal, characteristic

C₂×#20259233

C₂

yes

maximal, characteristic

#27132418

C₂

yes

maximal

C₄×C₄⋊C₄×C₁₉₁

C₂

yes

maximal, characteristic

C₂²×C₁₉₁⋊C₈

C₂²

yes

—

C₂²×C₁₉₁⋊C₈

C₂²

yes

characteristic

C₂²×C₁₉₁⋊C₈

C₂²

yes

characteristic

C₄⋊(C₁₉₁⋊C₈)

—

C₄⋊(C₁₉₁⋊C₈)

—

#20259233

—

C₂×C₄⋊C₄×C₁₉₁

C₄

yes

—

C₂×C₄²×C₁₉₁

C₂²

yes

—

C₂×C₄²×C₁₉₁

C₂²

yes

characteristic

#95×C₁₉₁

C₄

yes

characteristic

#95×C₁₉₁

C₄

yes

characteristic

C₂×C₁₉₁⋊C₈

—

C₂×C₁₉₁⋊C₈

—

C₂×C₁₉₁⋊C₈

—

C₂×C₁₉₁⋊C₈

—

C₂×C₁₉₁⋊C₈

—

C₂×C₁₉₁⋊C₈

—

C₂×C₁₉₁⋊C₈

—

C₂×C₁₉₁⋊C₈

D₄

yes

—

C₂×C₁₉₁⋊C₈

Q₈

yes

—

C₂²×C₄×C₁₉₁

C₂×C₄

yes

—

C₂²×C₄×C₁₉₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₉₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₉₁

C₂×C₄

yes

—

C₂²×C₄×C₁₉₁

C₂³

yes

characteristic

C₄⋊C₄×C₁₉₁

—

C₄²×C₁₉₁

—

C₄²×C₁₉₁

—

C₄²×C₁₉₁

D₄

yes

—

C₄²×C₁₉₁

Q₈

yes

—

C₁₉₁⋊C₈

—

C₁₉₁⋊C₈

—

C₁₉₁⋊C₈

—

C₂³×C₁₉₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

C₄⋊C₄

yes

—

C₂×C₄×C₁₉₁

C₄⋊C₄

yes

—

C₂×C₄×C₁₉₁

M₁₆

yes

—

C₂×C₄×C₁₉₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₉₁

C₂×Q₈

yes

C₂×C₄×C₁₉₁

yes

C₂×C₄×C₁₉₁

yes

C₂²×C₁₉₁

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₉₁

C₄×D₄

yes

characteristic

C₂²×C₁₉₁

C₄×Q₈

yes

characteristic

C₂²×C₁₉₁

C₂×M₁₆

yes

—

C₂²×C₁₉₁

#131

yes

—

C₄×C₁₉₁

—

C₄×C₁₉₁

—

C₄×C₁₉₁

—

C₄×C₁₉₁

—

C₄×C₁₉₁

#121

yes

characteristic

C₄×C₁₉₁

C₂²⋊Q₈

yes

characteristic

C₄×C₁₉₁

#125

yes

characteristic

C₄×C₁₉₁

C₄⋊Q₈

yes

C₂×C₁₉₁

yes

C₂×C₁₉₁

yes

C₂×C₁₉₁

yes

C₂×C₁₉₁

yes

—

C₂×C₁₉₁

#446

yes

—

C₁₉₁

#1940

yes

characteristic, complemented, Sylow

#1940

—

191

maximal, Sylow

C₂×C₄⋊C₈

—

191

—

C₂×C₄⋊C₈

—

191

—

C₂×#97

—

191

—

C₄×C₄⋊C₄

D₁₉₁

yes

characteristic

#336

—

191

—

C₂²×C₈

—

191

—

C₂²×C₈

—

191

—

C₂²×C₈

—

191

—

C₂×C₄⋊C₄

Dic₁₉₁

yes

—

C₂×C₄²

C₂×D₁₉₁

yes

—

C₂×C₄²

C₂×D₁₉₁

yes

characteristic

C₄⋊C₈

—

382

—

C₄⋊C₈

—

382

—

#97

—

382

—

#95

Dic₁₉₁

yes

characteristic

#95

Dic₁₉₁

yes

characteristic

C₂²×C₄

C₂×Dic₁₉₁

yes

—

C₂²×C₄

C₂×Dic₁₉₁

yes

characteristic

C₂²×C₄

C₂×Dic₁₉₁

yes

characteristic

C₂²×C₄

C₂×Dic₁₉₁

yes

—

C₂²×C₄

C₂²×D₁₉₁

yes

characteristic

C₂×C₈

—

382

—

C₂×C₈

—

382

—

C₂×C₈

—

382

—

100

C₂×C₈

—

382

—

101

C₂×C₈

—

382

—

102

C₂×C₈

—

382

—

103

C₂×C₈

—

382

—

104

C₂×C₈

—

191

—

105

C₂×C₈

—

191

—

106

C₄⋊C₄

—

107

C₄²

—

108

C₄²

—

109

C₄²

D₇₆₄

yes

—

110

C₄²

Dic₃₈₂

yes

—

111

C₂³

C₂²×Dic₁₉₁

yes

characteristic

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

C₄⋊Dic₁₉₁

yes

—

120

C₂×C₄

C₄⋊Dic₁₉₁

yes

—

121

C₂×C₄

#15690271

yes

—

122

C₂×C₄

D₄×D₁₉₁

yes

characteristic

123

C₂×C₄

C₂×D₇₆₄

yes

characteristic

124

C₂×C₄

Q₈×D₁₉₁

yes

characteristic

125

C₂×C₄

C₂×Dic₃₈₂

yes

characteristic

126

C₂×C₄

#15690286

yes

characteristic

127

C₂×C₄

#15690288

yes

characteristic

128

C₈

—

382

—

129

C₈

—

382

—

130

C₈

—

764

—

131

C₂²

C₂×C₄⋊Dic₁₉₁

yes

characteristic

132

C₂²

D₄×Dic₁₉₁

yes

characteristic

133

C₂²

Q₈×Dic₁₉₁

yes

characteristic

134

C₂²

C₂×#15690271

yes

—

135

C₂²

#20259329

yes

—

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

#20259297

yes

characteristic

141

C₄

#20259302

yes

characteristic

142

C₄

#20259313

yes

characteristic

143

C₄

#20259321

yes

characteristic

144

C₂

#27132562

yes

characteristic

145

C₂

#27132687

yes

characteristic

146

C₂

#27132690

yes

characteristic

147

C₂

#27132735

yes

—

148

C₂

#27132794

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₁₉₁
commutator, socle, prime radical	C₂³×C₁₉₁	C₂²×C₄
Fitting	C₄×C₄⋊C₄×C₁₉₁	C₂
cosocle	C₂²×C₄×C₁₉₁	C₂³
hypercenter	C₄×C₄⋊C₄	D₁₉₁
hypocenter	C₁₉₁	#1940
LCS	G ▷ C₂³×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₄⋊C₄

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₁₉₁

commutator, socle, prime radical

C₂³×C₁₉₁

C₂²×C₄

Fitting

C₄×C₄⋊C₄×C₁₉₁

C₂

cosocle

C₂²×C₄×C₁₉₁

C₂³

hypercenter

C₄×C₄⋊C₄

D₁₉₁

C₁₉₁

G ▷ C₂³×C₁₉₁ ▷ C₁₉₁

C₁ ◁ C₂²×C₄ ◁ C₄×C₄⋊C₄

N	H ≅ G/N	type
74. C₁₉₁	#1940	semidirect

H ≅ G/N

type

74. C₁₉₁

#1940

semidirect

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

subgroup	index
C₂×C₄⋊(C₁₉₁⋊C₈)	2
C₂×#20259233	2
#27132418	2
C₄×C₄⋊C₄×C₁₉₁	2
#1940	191

subgroup

index

C₂×C₄⋊(C₁₉₁⋊C₈)

C₂×#20259233

#27132418

C₄×C₄⋊C₄×C₁₉₁

#1940

191

Subgroups — #37573244

Conjugacy classes of subgroups

Complements

Maximal subgroups