Group #37574082 — subgroups

Subgroups — #37574082

Order 24448.

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

normal subgroups only

This group has 214 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	#27132561	C₂	1	1	yes	maximal, characteristic
2	#27132425	C₂	1	2	yes	maximal
3	#27132425	C₂	1	2	yes	maximal
4	#27132418	C₂	1	1	yes	maximal, characteristic
5	#384×C₁₉₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₉₁⋊C₈	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₁₉₁	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊Dic₁₉₁	C₂²	1	1	yes	characteristic
9	C₂×Dic₁₉₁⋊C₄	—	2	1	no	—
10	C₁₉₁⋊#95	—	2	1	no	—
11	C₂×C₄⋊C₄×C₁₉₁	—	2	1	no	—
12	C₂×C₄⋊C₄×C₁₉₁	—	2	1	no	—
13	C₂×C₄⋊C₄×C₁₉₁	C₂²	1	2	yes	—
14	C₂×C₄²×C₁₉₁	C₂²	1	1	yes	characteristic
15	C₂²×Dic₁₉₁	—	2	1	no	—
16	C₂²×Dic₁₉₁	—	2	1	no	—
17	C₂×C₁₉₁⋊C₈	—	2	2	no	—
18	C₂×C₁₉₁⋊C₈	—	2	2	no	—
19	C₂×C₁₉₁⋊C₈	—	2	2	no	—
20	C₄⋊Dic₁₉₁	—	2	1	no	—
21	C₄⋊Dic₁₉₁	—	2	2	no	—
22	C₄⋊Dic₁₉₁	Q₈	1	2	yes	—
23	Dic₁₉₁⋊C₄	—	4	2	no	—
24	C₂²×C₄×C₁₉₁	—	2	2	no	—
25	C₂²×C₄×C₁₉₁	D₄	1	1	yes	characteristic
26	C₂²×C₄×C₁₉₁	D₄	1	1	yes	characteristic
27	C₂²×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
28	C₄⋊C₄×C₁₉₁	—	2	4	no	—
29	C₄⋊C₄×C₁₉₁	—	4	2	no	—
30	C₄⋊C₄×C₁₉₁	—	4	2	no	—
31	C₄²×C₁₉₁	—	2	2	no	—
32	C₂×Dic₁₉₁	—	2	2	no	—
33	C₂×Dic₁₉₁	—	4	1	no	—
34	C₂×Dic₁₉₁	—	4	1	no	—
35	C₂×Dic₁₉₁	—	4	1	no	—
36	C₂×Dic₁₉₁	—	4	1	no	—
37	C₂×Dic₁₉₁	—	4	1	no	—
38	C₁₉₁⋊C₈	—	4	2	no	—
39	C₂³×C₁₉₁	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₁₉₁	—	2	1	no	—
41	C₂×C₄×C₁₉₁	—	2	1	no	—
42	C₂×C₄×C₁₉₁	—	2	1	no	—
43	C₂×C₄×C₁₉₁	—	2	1	no	—
44	C₂×C₄×C₁₉₁	—	4	2	no	—
45	C₂×C₄×C₁₉₁	—	4	2	no	—
46	C₂×C₄×C₁₉₁	—	4	2	no	—
47	C₂×C₄×C₁₉₁	SD₁₆	1	2	yes	—
48	C₂×C₄×C₁₉₁	Q₁₆	1	2	yes	—
49	C₂×C₄×C₁₉₁	C₂×Q₈	1	1	yes	characteristic
50	C₂×C₄×C₁₉₁	#41	1	2	yes	—
51	C₂×C₄×C₁₉₁	#41	1	2	yes	—
52	C₂×C₄×C₁₉₁	#41	1	1	yes	characteristic
53	C₂²×C₁₉₁	C₂²⋊Q₈	1	1	yes	characteristic
54	C₂²×C₁₉₁	#123	1	1	yes	characteristic
55	C₂²×C₁₉₁	#124	1	1	yes	characteristic
56	C₂²×C₁₉₁	C₂×SD₁₆	1	1	yes	characteristic
57	C₂²×C₁₉₁	C₂×Q₁₆	1	1	yes	characteristic
58	C₂²×C₁₉₁	AGL(1,ℤ/8)	1	1	yes	characteristic
59	C₂²×C₁₉₁	AGL(1,ℤ/8)	1	1	yes	characteristic
60	C₄×C₁₉₁	—	2	2	no	—
61	C₄×C₁₉₁	—	2	2	no	—
62	C₄×C₁₉₁	—	8	2	no	—
63	C₄×C₁₉₁	#125	1	2	yes	—
64	C₄×C₁₉₁	#126	1	2	yes	—
65	Dic₁₉₁	—	4	2	no	—
66	Dic₁₉₁	—	8	1	no	—
67	C₂×C₁₉₁	#400	1	1	yes	characteristic
68	C₂×C₁₉₁	#476	1	1	yes	characteristic
69	C₂×C₁₉₁	#477	1	1	yes	characteristic
70	C₂×C₁₉₁	#481	1	1	yes	characteristic
71	C₂×C₁₉₁	#484	1	1	yes	characteristic
72	C₂×C₁₉₁	#487	1	1	yes	characteristic
73	C₂×C₁₉₁	#490	1	1	yes	characteristic
74	C₁₉₁	#2100	1	1	yes	characteristic, complemented, Sylow
75	#2100	—	191	1	no	maximal, Sylow
76	#389	—	191	1	no	—
77	#384	D₁₉₁	1	1	yes	characteristic
78	#340	—	191	2	no	—
79	#340	—	191	2	no	—
80	#336	—	191	1	no	—
81	C₂²×C₈	—	191	2	no	—
82	C₂×C₄⋊C₄	—	2	1	no	—
83	C₂×C₄⋊C₄	—	2	1	no	—
84	C₂×C₄⋊C₄	—	382	1	no	—
85	C₂×C₄⋊C₄	—	191	1	no	—
86	C₂×C₄⋊C₄	—	191	1	no	—
87	C₂×C₄⋊C₄	C₂×D₁₉₁	1	2	yes	—
88	C₂×C₄²	C₂×D₁₉₁	1	1	yes	characteristic
89	#95	—	382	1	no	—
90	C₂²×C₄	—	2	2	no	—
91	C₂²×C₄	—	382	1	no	—
92	C₂²×C₄	—	382	1	no	—
93	C₂²×C₄	C₂²⋊D₁₉₁	1	1	yes	characteristic
94	C₂²×C₄	C₂²⋊D₁₉₁	1	1	yes	characteristic
95	C₂²×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
96	C₂×C₈	—	382	2	no	—
97	C₂×C₈	—	382	2	no	—
98	C₂×C₈	—	382	2	no	—
99	C₄⋊C₄	—	2	4	no	—
100	C₄⋊C₄	—	4	2	no	—
101	C₄⋊C₄	—	4	2	no	—
102	C₄⋊C₄	—	382	1	no	—
103	C₄⋊C₄	—	382	2	no	—
104	C₄⋊C₄	—	191	2	no	—
105	C₄⋊C₄	—	764	2	no	—
106	C₄²	—	2	2	no	—
107	C₂³	C₂×C₂²⋊D₁₉₁	1	1	yes	characteristic
108	C₂×C₄	—	2	1	no	—
109	C₂×C₄	—	2	1	no	—
110	C₂×C₄	—	2	1	no	—
111	C₂×C₄	—	2	1	no	—
112	C₂×C₄	—	4	2	no	—
113	C₂×C₄	—	4	2	no	—
114	C₂×C₄	—	4	2	no	—
115	C₂×C₄	—	382	2	no	—
116	C₂×C₄	—	764	1	no	—
117	C₂×C₄	—	764	1	no	—
118	C₂×C₄	—	764	1	no	—
119	C₂×C₄	—	764	1	no	—
120	C₂×C₄	—	764	1	no	—
121	C₂×C₄	#15690274	1	2	yes	—
122	C₂×C₄	#15690277	1	2	yes	—
123	C₂×C₄	Q₈×D₁₉₁	1	1	yes	characteristic
124	C₂×C₄	#15690286	1	2	yes	—
125	C₂×C₄	#15690287	1	2	yes	—
126	C₂×C₄	#15690288	1	1	yes	characteristic
127	C₈	—	764	2	no	—
128	C₂²	#20259298	1	1	yes	characteristic
129	C₂²	#20259305	1	1	yes	characteristic
130	C₂²	#20259309	1	1	yes	characteristic
131	C₂²	C₂×#15690274	1	1	yes	characteristic
132	C₂²	C₂×#15690277	1	1	yes	characteristic
133	C₂²	#20259348	1	1	yes	characteristic
134	C₂²	#20259348	1	1	yes	characteristic
135	C₄	—	2	2	no	—
136	C₄	—	2	2	no	—
137	C₄	—	8	2	no	—
138	C₄	—	1528	1	no	—
139	C₄	—	764	2	no	—
140	C₄	#20259312	1	2	yes	—
141	C₄	#20259317	1	2	yes	—
142	C₂	#27132601	1	1	yes	characteristic
143	C₂	#27132950	1	1	yes	characteristic
144	C₂	#27132955	1	1	yes	characteristic
145	C₂	#27132981	1	1	yes	characteristic
146	C₂	#27132998	1	1	yes	characteristic
147	C₂	#27133013	1	1	yes	characteristic
148	C₂	#27133027	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#27132561

C₂

yes

maximal, characteristic

#27132425

C₂

yes

maximal

#27132425

C₂

yes

maximal

#27132418

C₂

yes

maximal, characteristic

#384×C₁₉₁

C₂

yes

maximal, characteristic

C₂²×C₁₉₁⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₁₉₁

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₉₁

C₂²

yes

characteristic

C₂×Dic₁₉₁⋊C₄

—

C₁₉₁⋊#95

—

C₂×C₄⋊C₄×C₁₉₁

—

C₂×C₄⋊C₄×C₁₉₁

—

C₂×C₄⋊C₄×C₁₉₁

C₂²

yes

—

C₂×C₄²×C₁₉₁

C₂²

yes

characteristic

C₂²×Dic₁₉₁

—

C₂²×Dic₁₉₁

—

C₂×C₁₉₁⋊C₈

—

C₂×C₁₉₁⋊C₈

—

C₂×C₁₉₁⋊C₈

—

C₄⋊Dic₁₉₁

—

C₄⋊Dic₁₉₁

—

C₄⋊Dic₁₉₁

Q₈

yes

—

Dic₁₉₁⋊C₄

—

C₂²×C₄×C₁₉₁

—

C₂²×C₄×C₁₉₁

D₄

yes

characteristic

C₂²×C₄×C₁₉₁

D₄

yes

characteristic

C₂²×C₄×C₁₉₁

C₂³

yes

characteristic

C₄⋊C₄×C₁₉₁

—

C₄⋊C₄×C₁₉₁

—

C₄⋊C₄×C₁₉₁

—

C₄²×C₁₉₁

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

—

C₁₉₁⋊C₈

—

C₂³×C₁₉₁

C₂×D₄

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

SD₁₆

yes

—

C₂×C₄×C₁₉₁

Q₁₆

yes

—

C₂×C₄×C₁₉₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₉₁

#41

yes

—

C₂×C₄×C₁₉₁

#41

yes

—

C₂×C₄×C₁₉₁

#41

yes

characteristic

C₂²×C₁₉₁

C₂²⋊Q₈

yes

C₂²×C₁₉₁

yes

C₂²×C₁₉₁

yes

C₂²×C₁₉₁

C₂×SD₁₆

yes

characteristic

C₂²×C₁₉₁

C₂×Q₁₆

yes

C₂²×C₁₉₁

yes

C₂²×C₁₉₁

yes

C₄×C₁₉₁

—

C₄×C₁₉₁

—

C₄×C₁₉₁

—

C₄×C₁₉₁

#125

yes

—

C₄×C₁₉₁

#126

yes

—

Dic₁₉₁

—

Dic₁₉₁

—

C₂×C₁₉₁

yes

C₂×C₁₉₁

yes

C₂×C₁₉₁

yes

C₂×C₁₉₁

yes

C₂×C₁₉₁

yes

C₂×C₁₉₁

yes

C₂×C₁₉₁

yes

C₁₉₁

yes

characteristic, complemented, Sylow

#2100

—

191

maximal, Sylow

#389

—

191

—

#384

D₁₉₁

yes

characteristic

#340

—

191

—

#340

—

191

—

#336

—

191

—

C₂²×C₈

—

191

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

382

—

C₂×C₄⋊C₄

—

191

—

C₂×C₄⋊C₄

—

191

—

C₂×C₄⋊C₄

C₂×D₁₉₁

yes

—

C₂×C₄²

C₂×D₁₉₁

yes

characteristic

#95

—

382

—

C₂²×C₄

—

C₂²×C₄

—

382

—

C₂²×C₄

—

382

—

C₂²×C₄

C₂²⋊D₁₉₁

yes

characteristic

C₂²×C₄

C₂²⋊D₁₉₁

yes

characteristic

C₂²×C₄

C₂²×D₁₉₁

yes

characteristic

C₂×C₈

—

382

—

C₂×C₈

—

382

—

C₂×C₈

—

382

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

382

—

103

C₄⋊C₄

—

382

—

104

C₄⋊C₄

—

191

—

105

C₄⋊C₄

—

764

—

106

C₄²

—

107

C₂³

C₂×C₂²⋊D₁₉₁

yes

characteristic

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

382

—

116

C₂×C₄

—

764

—

117

C₂×C₄

—

764

—

118

C₂×C₄

—

764

—

119

C₂×C₄

—

764

—

120

C₂×C₄

—

764

—

121

C₂×C₄

#15690274

yes

—

122

C₂×C₄

#15690277

yes

—

123

C₂×C₄

Q₈×D₁₉₁

yes

characteristic

124

C₂×C₄

#15690286

yes

—

125

C₂×C₄

#15690287

yes

—

126

C₂×C₄

#15690288

yes

characteristic

127

C₈

—

764

—

128

C₂²

#20259298

yes

characteristic

129

C₂²

#20259305

yes

characteristic

130

C₂²

#20259309

yes

characteristic

131

C₂²

C₂×#15690274

yes

characteristic

132

C₂²

C₂×#15690277

yes

characteristic

133

C₂²

#20259348

yes

characteristic

134

C₂²

#20259348

yes

characteristic

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

—

1528

—

139

C₄

—

764

—

140

C₄

#20259312

yes

—

141

C₄

#20259317

yes

—

142

C₂

#27132601

yes

characteristic

143

C₂

#27132950

yes

characteristic

144

C₂

#27132955

yes

characteristic

145

C₂

#27132981

yes

characteristic

146

C₂

#27132998

yes

characteristic

147

C₂

#27133013

yes

characteristic

148

C₂

#27133027

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₉₁
commutator, cosocle	C₂²×C₄×C₁₉₁	C₂³
Frattini	C₂²×C₄	C₂²×D₁₉₁
Fitting	#384×C₁₉₁	C₂
socle, prime radical	C₂³×C₁₉₁	C₂×D₄
hypercenter	#384	D₁₉₁
hypocenter	C₁₉₁	#2100
LCS	G ▷ C₂²×C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₉₁

commutator, cosocle

C₂²×C₄×C₁₉₁

C₂³

Frattini

C₂²×C₄

C₂²×D₁₉₁

Fitting

#384×C₁₉₁

C₂

socle, prime radical

C₂³×C₁₉₁

C₂×D₄

D₁₉₁

C₁₉₁

G ▷ C₂²×C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

N	H ≅ G/N	type
74. C₁₉₁	#2100	semidirect

H ≅ G/N

type

74. C₁₉₁

#2100

semidirect

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

subgroup	index
#27132561	2
#27132425	2
#27132418	2
#384×C₁₉₁	2
#2100	191

subgroup

index

#384×C₁₉₁

191

Subgroups — #37574082

Conjugacy classes of subgroups

Complements

Maximal subgroups