Group #37572515 — subgroups

Subgroups — #37572515

Order 24448.

272 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 87 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×#20259287	C₂	1	1	yes	maximal, characteristic
2	#27132686	C₂	1	2	yes	maximal
3	C₂×#20259240	C₂	1	2	yes	maximal
4	#27132432	C₂	1	1	yes	maximal, characteristic
5	#343×C₁₉₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₃₈₂	C₂²	1	1	yes	characteristic
7	C₂×#15690271	C₂²	1	2	yes	—
8	#20259287	—	2	2	no	—
9	#20259287	—	2	2	no	—
10	C₂×Dic₁₉₁⋊C₄	C₄	1	2	yes	—
11	C₂×C₄×Dic₁₉₁	C₄	1	2	yes	—
12	C₂×C₄×Dic₁₉₁	C₂²	1	1	yes	characteristic
13	Dic₁₉₁⋊C₈	—	2	2	no	—
14	Dic₁₉₁⋊C₈	—	2	2	no	—
15	#20259240	—	2	4	no	—
16	C₂×M₁₆×C₁₉₁	C₂²	1	2	yes	—
17	C₂×C₄⋊C₄×C₁₉₁	C₂²	1	1	yes	characteristic
18	C₂×Dic₃₈₂	—	2	2	no	—
19	C₂×Dic₃₈₂	—	2	2	no	—
20	C₂×Dic₃₈₂	D₄	1	2	yes	—
21	C₂×Dic₃₈₂	Q₈	1	2	yes	—
22	C₂²×Dic₁₉₁	C₂×C₄	1	1	yes	characteristic
23	C₂²×Dic₁₉₁	C₂×C₄	1	2	yes	—
24	C₂²×Dic₁₉₁	C₂×C₄	1	1	yes	characteristic
25	#15690271	—	2	2	no	—
26	#15690271	—	2	2	no	—
27	C₂×C₁₉₁⋊C₈	—	2	2	no	—
28	Dic₁₉₁⋊C₄	—	2	4	no	—
29	C₄×Dic₁₉₁	—	2	2	no	—
30	C₄×Dic₁₉₁	—	2	2	no	—
31	C₄×Dic₁₉₁	D₄	1	2	yes	—
32	C₄×Dic₁₉₁	D₄	1	2	yes	—
33	C₂²×C₄×C₁₉₁	C₂×C₄	1	2	yes	—
34	C₂²×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
35	M₁₆×C₁₉₁	—	2	2	no	—
36	M₁₆×C₁₉₁	—	2	2	no	—
37	C₂×C₈×C₁₉₁	—	2	2	no	—
38	C₄⋊C₄×C₁₉₁	—	2	2	no	—
39	Dic₃₈₂	—	2	4	no	—
40	Dic₃₈₂	—	2	4	no	—
41	C₂×Dic₁₉₁	—	2	2	no	—
42	C₂×Dic₁₉₁	—	2	2	no	—
43	C₂×Dic₁₉₁	—	2	2	no	—
44	C₂×Dic₁₉₁	—	4	1	no	—
45	C₂×Dic₁₉₁	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×Dic₁₉₁	C₂²⋊C₄	1	2	yes	—
47	C₂×Dic₁₉₁	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×Dic₁₉₁	C₄⋊C₄	1	4	yes	—
49	C₁₉₁⋊C₈	—	4	2	no	—
50	C₂³×C₁₉₁	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₄×C₁₉₁	—	2	2	no	—
52	C₂×C₄×C₁₉₁	—	4	2	no	—
53	C₂×C₄×C₁₉₁	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₁₉₁	C₂×D₄	1	2	yes	—
55	C₂×C₄×C₁₉₁	C₂×Q₈	1	1	yes	characteristic
56	C₂×C₄×C₁₉₁	#41	1	2	yes	—
57	C₈×C₁₉₁	—	4	2	no	—
58	C₂²×C₁₉₁	—	2	1	no	—
59	C₂²×C₁₉₁	—	2	1	no	—
60	C₂²×C₁₉₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₁₉₁	C₂×C₄⋊C₄	1	1	yes	characteristic
62	C₂²×C₁₉₁	#117	1	1	yes	characteristic
63	C₄×C₁₉₁	—	4	2	no	—
64	C₄×C₁₉₁	#121	1	2	yes	—
65	C₄×C₁₉₁	C₂²⋊Q₈	1	2	yes	—
66	Dic₁₉₁	—	2	4	no	—
67	Dic₁₉₁	—	2	1	no	—
68	Dic₁₉₁	—	2	2	no	—
69	Dic₁₉₁	#101	1	2	yes	complemented
70	C₂×C₁₉₁	—	2	1	no	—
71	C₂×C₁₉₁	—	2	1	no	—
72	C₂×C₁₉₁	#380	1	1	yes	characteristic
73	C₂×C₁₉₁	C₂×#101	1	1	yes	characteristic
74	C₂×C₁₉₁	#413	1	1	yes	characteristic
75	C₁₉₁	#1801	1	1	yes	characteristic, complemented, Sylow
76	#1801	—	191	1	no	maximal, Sylow
77	C₂×C₄×Q₈	—	191	1	no	—
78	#423	—	191	2	no	—
79	C₂×#101	—	191	2	no	—
80	#343	—	191	1	no	—
81	#343	D₁₉₁	1	1	yes	characteristic
82	C₂²×Q₈	—	191	1	no	—
83	C₂×M₁₆	—	191	2	no	—
84	C₂×M₁₆	C₂×D₁₉₁	1	2	yes	—
85	C₄×Q₈	—	382	2	no	—
86	C₄×Q₈	—	382	2	no	—
87	C₂×C₄⋊C₄	—	191	2	no	—
88	C₂×C₄⋊C₄	C₂×D₁₉₁	1	1	yes	characteristic
89	C₂×C₄²	—	191	2	no	—
90	C₂×C₄²	—	191	1	no	—
91	C₄⋊C₈	—	382	2	no	—
92	C₄⋊C₈	—	382	2	no	—
93	#101	—	382	4	no	—
94	C₂×Q₈	—	382	2	no	—
95	C₂×Q₈	—	382	2	no	—
96	C₂×Q₈	—	191	2	no	—
97	C₂×Q₈	—	191	2	no	—
98	C₂²×C₄	—	191	1	no	—
99	C₂²×C₄	—	191	2	no	—
100	C₂²×C₄	—	191	1	no	—
101	C₂²×C₄	C₄×D₁₉₁	1	2	yes	—
102	C₂²×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
103	M₁₆	—	2	2	no	—
104	M₁₆	—	2	2	no	—
105	M₁₆	—	382	2	no	—
106	M₁₆	—	382	2	no	—
107	C₂×C₈	—	2	2	no	—
108	C₂×C₈	—	382	2	no	—
109	C₄⋊C₄	—	2	2	no	—
110	C₄⋊C₄	—	382	4	no	—
111	C₄²	—	382	2	no	—
112	C₄²	—	382	2	no	—
113	C₄²	—	191	2	no	—
114	C₄²	—	191	2	no	—
115	C₂³	C₂×C₄×D₁₉₁	1	1	yes	characteristic
116	Q₈	—	382	4	no	—
117	Q₈	—	382	4	no	—
118	C₂×C₄	—	2	2	no	—
119	C₂×C₄	—	4	2	no	—
120	C₂×C₄	—	382	2	no	—
121	C₂×C₄	—	382	2	no	—
122	C₂×C₄	—	382	2	no	—
123	C₂×C₄	—	191	1	no	—
124	C₂×C₄	—	191	2	no	—
125	C₂×C₄	—	191	4	no	—
126	C₂×C₄	—	191	1	no	—
127	C₂×C₄	—	764	1	no	—
128	C₂×C₄	D₄×D₁₉₁	1	1	yes	characteristic
129	C₂×C₄	D₄×D₁₉₁	1	2	yes	—
130	C₂×C₄	Q₈×D₁₉₁	1	1	yes	characteristic
131	C₂×C₄	#15690287	1	2	yes	—
132	C₈	—	4	2	no	—
133	C₈	—	764	2	no	—
134	C₂²	—	2	1	no	—
135	C₂²	—	2	1	no	—
136	C₂²	C₂²⋊C₄×D₁₉₁	1	1	yes	characteristic
137	C₂²	C₄⋊C₄×D₁₉₁	1	1	yes	characteristic
138	C₂²	#20259279	1	1	yes	characteristic
139	C₄	—	4	2	no	—
140	C₄	—	382	4	no	—
141	C₄	—	382	1	no	—
142	C₄	—	382	2	no	—
143	C₄	—	191	2	no	—
144	C₄	#20259295	1	2	yes	—
145	C₄	#20259300	1	2	yes	—
146	C₂	—	2	1	no	—
147	C₂	—	2	1	no	—
148	C₂	#27132522	1	1	yes	characteristic
149	C₂	#101×D₁₉₁	1	1	yes	characteristic
150	C₂	#27132644	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#20259287

C₂

yes

maximal, characteristic

#27132686

C₂

yes

maximal

C₂×#20259240

C₂

yes

maximal

#27132432

C₂

yes

maximal, characteristic

#343×C₁₉₁

C₂

yes

maximal, characteristic

C₂²×Dic₃₈₂

C₂²

yes

characteristic

C₂×#15690271

C₂²

yes

—

#20259287

—

#20259287

—

C₂×Dic₁₉₁⋊C₄

C₄

yes

—

C₂×C₄×Dic₁₉₁

C₄

yes

—

C₂×C₄×Dic₁₉₁

C₂²

yes

characteristic

Dic₁₉₁⋊C₈

—

Dic₁₉₁⋊C₈

—

#20259240

—

C₂×M₁₆×C₁₉₁

C₂²

yes

—

C₂×C₄⋊C₄×C₁₉₁

C₂²

yes

characteristic

C₂×Dic₃₈₂

—

C₂×Dic₃₈₂

—

C₂×Dic₃₈₂

D₄

yes

—

C₂×Dic₃₈₂

Q₈

yes

—

C₂²×Dic₁₉₁

C₂×C₄

yes

characteristic

C₂²×Dic₁₉₁

C₂×C₄

yes

—

C₂²×Dic₁₉₁

C₂×C₄

yes

characteristic

#15690271

—

#15690271

—

C₂×C₁₉₁⋊C₈

—

Dic₁₉₁⋊C₄

—

C₄×Dic₁₉₁

—

C₄×Dic₁₉₁

—

C₄×Dic₁₉₁

D₄

yes

—

C₄×Dic₁₉₁

D₄

yes

—

C₂²×C₄×C₁₉₁

C₂×C₄

yes

—

C₂²×C₄×C₁₉₁

C₂³

yes

characteristic

M₁₆×C₁₉₁

—

M₁₆×C₁₉₁

—

C₂×C₈×C₁₉₁

—

C₄⋊C₄×C₁₉₁

—

Dic₃₈₂

—

Dic₃₈₂

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

C₂²⋊C₄

yes

characteristic

C₂×Dic₁₉₁

C₂²⋊C₄

yes

—

C₂×Dic₁₉₁

C₂²⋊C₄

yes

characteristic

C₂×Dic₁₉₁

C₄⋊C₄

yes

—

C₁₉₁⋊C₈

—

C₂³×C₁₉₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₁

C₂×D₄

yes

—

C₂×C₄×C₁₉₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₉₁

#41

yes

—

C₈×C₁₉₁

—

C₂²×C₁₉₁

—

C₂²×C₁₉₁

—

C₂²×C₁₉₁

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₉₁

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₉₁

#117

yes

characteristic

C₄×C₁₉₁

—

C₄×C₁₉₁

#121

yes

—

C₄×C₁₉₁

C₂²⋊Q₈

yes

—

Dic₁₉₁

—

Dic₁₉₁

—

Dic₁₉₁

—

Dic₁₉₁

#101

yes

complemented

C₂×C₁₉₁

—

C₂×C₁₉₁

—

C₂×C₁₉₁

yes

C₂×C₁₉₁

yes

C₂×C₁₉₁

yes

C₁₉₁

yes

characteristic, complemented, Sylow

#1801

—

191

maximal, Sylow

C₂×C₄×Q₈

—

191

—

#423

—

191

—

C₂×#101

—

191

—

#343

—

191

—

#343

D₁₉₁

yes

characteristic

C₂²×Q₈

—

191

—

C₂×M₁₆

—

191

—

C₂×M₁₆

C₂×D₁₉₁

yes

—

C₄×Q₈

—

382

—

C₄×Q₈

—

382

—

C₂×C₄⋊C₄

—

191

—

C₂×C₄⋊C₄

C₂×D₁₉₁

yes

characteristic

C₂×C₄²

—

191

—

C₂×C₄²

—

191

—

C₄⋊C₈

—

382

—

C₄⋊C₈

—

382

—

#101

—

382

—

C₂×Q₈

—

382

—

C₂×Q₈

—

382

—

C₂×Q₈

—

191

—

C₂×Q₈

—

191

—

C₂²×C₄

—

191

—

C₂²×C₄

—

191

—

100

C₂²×C₄

—

191

—

101

C₂²×C₄

C₄×D₁₉₁

yes

—

102

C₂²×C₄

C₂²×D₁₉₁

yes

characteristic

103

M₁₆

—

104

M₁₆

—

105

M₁₆

—

382

—

106

M₁₆

—

382

—

107

C₂×C₈

—

108

C₂×C₈

—

382

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

382

—

111

C₄²

—

382

—

112

C₄²

—

382

—

113

C₄²

—

191

—

114

C₄²

—

191

—

115

C₂³

C₂×C₄×D₁₉₁

yes

characteristic

116

Q₈

—

382

—

117

Q₈

—

382

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

382

—

121

C₂×C₄

—

382

—

122

C₂×C₄

—

382

—

123

C₂×C₄

—

191

—

124

C₂×C₄

—

191

—

125

C₂×C₄

—

191

—

126

C₂×C₄

—

191

—

127

C₂×C₄

—

764

—

128

C₂×C₄

D₄×D₁₉₁

yes

characteristic

129

C₂×C₄

D₄×D₁₉₁

yes

—

130

C₂×C₄

Q₈×D₁₉₁

yes

characteristic

131

C₂×C₄

#15690287

yes

—

132

C₈

—

133

C₈

—

764

—

134

C₂²

—

135

C₂²

—

136

C₂²

C₂²⋊C₄×D₁₉₁

yes

characteristic

137

C₂²

C₄⋊C₄×D₁₉₁

yes

characteristic

138

C₂²

#20259279

yes

characteristic

139

C₄

—

140

C₄

—

382

—

141

C₄

—

382

—

142

C₄

—

382

—

143

C₄

—

191

—

144

C₄

#20259295

yes

—

145

C₄

#20259300

yes

—

146

C₂

—

147

C₂

—

148

C₂

#27132522

yes

characteristic

149

C₂

#101×D₁₉₁

yes

characteristic

150

C₂

#27132644

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²⋊C₄×D₁₉₁
commutator, prime radical	C₂³×C₁₉₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₉₁
Fitting	#343×C₁₉₁	C₂
socle	C₂²×C₁₉₁	C₂×C₂²⋊C₄
cosocle	C₂²×C₄×C₁₉₁	C₂³
hypercenter	#343	D₁₉₁
hypocenter	C₁₉₁	#1801
LCS	G ▷ C₂³×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #343

subgroup

G/H

center

C₂²

C₂²⋊C₄×D₁₉₁

commutator, prime radical

C₂³×C₁₉₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₉₁

Fitting

#343×C₁₉₁

C₂

socle

C₂²×C₁₉₁

C₂×C₂²⋊C₄

cosocle

C₂²×C₄×C₁₉₁

C₂³

D₁₉₁

C₁₉₁

G ▷ C₂³×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #343

N	H ≅ G/N	type
69. Dic₁₉₁	#101	semidirect
75. C₁₉₁	#1801	semidirect

H ≅ G/N

type

69. Dic₁₉₁

#101

semidirect

75. C₁₉₁

#1801

semidirect

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

subgroup	index
C₂×#20259287	2
#27132686	2
C₂×#20259240	2
#27132432	2
#343×C₁₉₁	2
#1801	191

subgroup

index

#343×C₁₉₁

191

Subgroups — #37572515

Conjugacy classes of subgroups

Complements

Maximal subgroups