Group #37574926 — subgroups

Subgroups — #37574926

Order 24448.

656 conjugacy classes of subgroups; only the 245 normal subgroups are shown.

normal subgroups only

This group has 656 conjugacy classes of subgroups — too many to draw. Only its 245 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×#20259280	C₂	1	2	yes	maximal
2	C₂×C₄²×D₁₉₁	C₂	1	1	yes	maximal, characteristic
3	#27132532	C₂	1	2	yes	maximal, complemented
4	#27132529	C₂	1	1	yes	maximal, characteristic, complemented
5	#27132522	C₂	1	4	yes	maximal
6	#27132514	C₂	1	4	yes	maximal, complemented
7	#382×C₁₉₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₄×D₁₉₁	C₂²	1	2	yes	—
9	C₂²×C₄×D₁₉₁	C₂²	1	1	yes	characteristic
10	#20259280	—	2	8	no	—
11	C₂×C₄⋊Dic₁₉₁	C₂²	1	2	yes	—
12	C₂×Dic₁₉₁⋊C₄	C₂²	1	8	yes	—
13	C₂×(C₂×D₁₉₁)⋊C₄	C₂²	1	4	yes	—
14	C₂×C₄×Dic₁₉₁	C₂²	1	4	yes	—
15	C₂×C₄×Dic₁₉₁	C₂²	1	2	yes	—
16	C₂×C₄×Dic₁₉₁	C₂²	1	1	yes	characteristic
17	C₄²×D₁₉₁	C₄	1	8	yes	complemented
18	C₁₉₁⋊#95	C₂²	1	4	yes	—
19	C₂×C₄⋊C₄×C₁₉₁	C₂²	1	2	yes	—
20	C₂×C₄²×C₁₉₁	C₂²	1	1	yes	characteristic
21	#95×C₁₉₁	C₂²	1	4	yes	—
22	C₂³×D₁₉₁	C₂³	1	1	yes	characteristic
23	C₂²×Dic₁₉₁	C₂³	1	4	yes	—
24	C₂²×Dic₁₉₁	C₂³	1	1	yes	characteristic
25	C₂²×Dic₁₉₁	C₂³	1	2	yes	—
26	C₂×C₄×D₁₉₁	—	2	8	no	—
27	C₂×C₄×D₁₉₁	C₂×C₄	1	8	yes	—
28	C₂×C₄×D₁₉₁	C₂×C₄	1	4	yes	—
29	C₂×C₄×D₁₉₁	D₄	1	4	yes	—
30	C₂×C₄×D₁₉₁	Q₈	1	4	yes	—
31	C₄⋊Dic₁₉₁	—	2	4	no	—
32	Dic₁₉₁⋊C₄	—	2	16	no	—
33	(C₂×D₁₉₁)⋊C₄	—	2	8	no	—
34	C₄×Dic₁₉₁	—	2	8	no	—
35	C₄×Dic₁₉₁	C₂×C₄	1	8	yes	—
36	C₄×Dic₁₉₁	C₂×C₄	1	4	yes	—
37	C₂²×C₄×C₁₉₁	C₂³	1	4	yes	—
38	C₂²×C₄×C₁₉₁	C₂³	1	2	yes	—
39	C₂²×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
40	C₄⋊C₄×C₁₉₁	—	2	4	no	—
41	C₄²×C₁₉₁	C₂×C₄	1	4	yes	—
42	C₂²×D₁₉₁	—	2	4	no	—
43	C₂²×D₁₉₁	C₂²×C₄	1	2	yes	—
44	C₂²×D₁₉₁	C₂×D₄	1	2	yes	—
45	C₂²×D₁₉₁	C₂×Q₈	1	2	yes	—
46	C₂×Dic₁₉₁	—	2	4	no	—
47	C₂×Dic₁₉₁	—	2	8	no	—
48	C₂×Dic₁₉₁	C₂²×C₄	1	2	yes	—
49	C₂×Dic₁₉₁	C₂²×C₄	1	4	yes	—
50	C₂×Dic₁₉₁	C₂×D₄	1	2	yes	—
51	C₂×Dic₁₉₁	C₂×Q₈	1	2	yes	—
52	C₂×Dic₁₉₁	#41	1	8	yes	—
53	C₄×D₁₉₁	—	2	16	no	—
54	C₄×D₁₉₁	C₄⋊C₄	1	16	yes	complemented
55	C₂³×C₁₉₁	C₂⁴	1	1	yes	characteristic
56	C₂×C₄×C₁₉₁	—	2	4	no	—
57	C₂×C₄×C₁₉₁	—	2	4	no	—
58	C₂×C₄×C₁₉₁	—	2	8	no	—
59	C₂×C₄×C₁₉₁	C₂²×C₄	1	4	yes	—
60	C₂×C₄×C₁₉₁	C₂²×C₄	1	2	yes	—
61	C₂×C₄×C₁₉₁	C₂×D₄	1	2	yes	—
62	C₂×C₄×C₁₉₁	C₂×Q₈	1	2	yes	—
63	C₂²×C₁₉₁	C₂³×C₄	1	1	yes	characteristic
64	C₂²×C₁₉₁	C₂²×D₄	1	1	yes	characteristic
65	C₂²×C₁₉₁	C₂²×Q₈	1	1	yes	characteristic
66	C₂²×C₁₉₁	C₂×#41	1	4	yes	—
67	C₂×D₁₉₁	—	2	4	no	—
68	C₂×D₁₉₁	—	2	4	no	—
69	C₂×D₁₉₁	—	2	4	no	—
70	C₂×D₁₉₁	C₂×C₄⋊C₄	1	4	yes	—
71	C₄×C₁₉₁	—	2	4	no	—
72	C₄×C₁₉₁	—	4	4	no	—
73	C₄×C₁₉₁	C₂×C₄⋊C₄	1	4	yes	—
74	Dic₁₉₁	—	4	4	no	—
75	Dic₁₉₁	C₂×C₄⋊C₄	1	4	yes	—
76	Dic₁₉₁	#117	1	8	yes	complemented
77	C₂×C₁₉₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
78	C₂×C₁₉₁	C₂×#117	1	2	yes	—
79	C₂×C₁₉₁	#532	1	2	yes	—
80	C₂×C₁₉₁	#533	1	2	yes	—
81	D₁₉₁	—	2	4	no	—
82	C₁₉₁	#2285	1	1	yes	characteristic, complemented, Sylow
83	#2285	—	191	1	no	maximal, Sylow
84	C₂×#117	—	191	2	no	—
85	C₂²×C₄²	—	191	1	no	—
86	#384	—	191	2	no	—
87	#382	—	191	1	no	—
88	#382	D₁₉₁	1	1	yes	characteristic, complemented
89	#380	—	191	4	no	—
90	C₄×C₄⋊C₄	—	191	4	no	—
91	C₂³×C₄	—	191	2	no	—
92	C₂³×C₄	—	191	1	no	—
93	#117	—	382	8	no	—
94	C₂×C₄⋊C₄	—	191	2	no	—
95	C₂×C₄⋊C₄	—	191	8	no	—
96	C₂×C₄⋊C₄	C₂×D₁₉₁	1	2	yes	—
97	C₂×C₂²⋊C₄	—	191	4	no	—
98	C₂×C₄²	—	191	4	no	—
99	C₂×C₄²	—	191	2	no	—
100	C₂×C₄²	—	191	1	no	—
101	C₂×C₄²	—	191	8	no	—
102	C₂×C₄²	C₂×D₁₉₁	1	1	yes	characteristic
103	#95	—	191	4	no	—
104	#95	C₂×D₁₉₁	1	4	yes	—
105	C₂⁴	—	191	1	no	—
106	C₂²×C₄	—	382	8	no	—
107	C₂²×C₄	—	191	4	no	—
108	C₂²×C₄	—	191	8	no	—
109	C₂²×C₄	—	191	1	no	—
110	C₂²×C₄	—	191	2	no	—
111	C₂²×C₄	—	191	4	no	—
112	C₂²×C₄	—	191	4	no	—
113	C₂²×C₄	—	191	4	no	—
114	C₂²×C₄	C₂²×D₁₉₁	1	4	yes	—
115	C₂²×C₄	C₂²×D₁₉₁	1	2	yes	—
116	C₂²×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
117	C₄⋊C₄	—	2	4	no	—
118	C₄⋊C₄	—	382	16	no	—
119	C₄⋊C₄	—	382	4	no	—
120	C₂²⋊C₄	—	382	8	no	—
121	C₄²	—	382	8	no	—
122	C₄²	—	191	8	no	—
123	C₄²	—	191	4	no	—
124	C₄²	C₄×D₁₉₁	1	4	yes	—
125	C₂³	—	382	4	no	—
126	C₂³	—	191	2	no	—
127	C₂³	—	191	2	no	—
128	C₂³	—	191	2	no	—
129	C₂³	C₂³×D₁₉₁	1	1	yes	characteristic
130	C₂×C₄	—	2	4	no	—
131	C₂×C₄	—	2	4	no	—
132	C₂×C₄	—	2	8	no	—
133	C₂×C₄	—	382	16	no	—
134	C₂×C₄	—	382	4	no	—
135	C₂×C₄	—	382	8	no	—
136	C₂×C₄	—	191	2	no	—
137	C₂×C₄	—	191	4	no	—
138	C₂×C₄	—	191	2	no	—
139	C₂×C₄	—	191	2	no	—
140	C₂×C₄	—	191	8	no	—
141	C₂×C₄	—	191	16	no	—
142	C₂×C₄	C₂×C₄×D₁₉₁	1	4	yes	—
143	C₂×C₄	C₂×C₄×D₁₉₁	1	2	yes	—
144	C₂×C₄	D₄×D₁₉₁	1	2	yes	—
145	C₂×C₄	Q₈×D₁₉₁	1	2	yes	—
146	C₂²	—	382	4	no	—
147	C₂²	—	382	4	no	—
148	C₂²	—	382	4	no	—
149	C₂²	—	191	4	no	—
150	C₂²	C₂²×C₄×D₁₉₁	1	1	yes	characteristic
151	C₂²	C₂×D₄×D₁₉₁	1	1	yes	characteristic
152	C₂²	C₂×Q₈×D₁₉₁	1	1	yes	characteristic
153	C₂²	#41×D₁₉₁	1	4	yes	—
154	C₄	—	2	4	no	—
155	C₄	—	4	4	no	—
156	C₄	—	191	4	no	—
157	C₄	—	191	8	no	—
158	C₄	—	764	4	no	—
159	C₄	C₄⋊C₄×D₁₉₁	1	4	yes	—
160	C₂	—	382	4	no	—
161	C₂	C₂×C₄⋊C₄×D₁₉₁	1	1	yes	characteristic
162	C₂	#117×D₁₉₁	1	2	yes	—
163	C₂	#27133212	1	2	yes	—
164	C₂	#27133219	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#20259280

C₂

yes

maximal

C₂×C₄²×D₁₉₁

C₂

yes

maximal, characteristic

#27132532

C₂

yes

maximal, complemented

#27132529

C₂

yes

maximal, characteristic, complemented

#27132522

C₂

yes

maximal

#27132514

C₂

yes

maximal, complemented

#382×C₁₉₁

C₂

yes

maximal, characteristic, complemented

C₂²×C₄×D₁₉₁

C₂²

yes

—

C₂²×C₄×D₁₉₁

C₂²

yes

characteristic

#20259280

—

C₂×C₄⋊Dic₁₉₁

C₂²

yes

—

C₂×Dic₁₉₁⋊C₄

C₂²

yes

—

C₂×(C₂×D₁₉₁)⋊C₄

C₂²

yes

—

C₂×C₄×Dic₁₉₁

C₂²

yes

—

C₂×C₄×Dic₁₉₁

C₂²

yes

—

C₂×C₄×Dic₁₉₁

C₂²

yes

characteristic

C₄²×D₁₉₁

C₄

yes

complemented

C₁₉₁⋊#95

C₂²

yes

—

C₂×C₄⋊C₄×C₁₉₁

C₂²

yes

—

C₂×C₄²×C₁₉₁

C₂²

yes

characteristic

#95×C₁₉₁

C₂²

yes

—

C₂³×D₁₉₁

C₂³

yes

characteristic

C₂²×Dic₁₉₁

C₂³

yes

—

C₂²×Dic₁₉₁

C₂³

yes

characteristic

C₂²×Dic₁₉₁

C₂³

yes

—

C₂×C₄×D₁₉₁

—

C₂×C₄×D₁₉₁

C₂×C₄

yes

—

C₂×C₄×D₁₉₁

C₂×C₄

yes

—

C₂×C₄×D₁₉₁

D₄

yes

—

C₂×C₄×D₁₉₁

Q₈

yes

—

C₄⋊Dic₁₉₁

—

Dic₁₉₁⋊C₄

—

(C₂×D₁₉₁)⋊C₄

—

C₄×Dic₁₉₁

—

C₄×Dic₁₉₁

C₂×C₄

yes

—

C₄×Dic₁₉₁

C₂×C₄

yes

—

C₂²×C₄×C₁₉₁

C₂³

yes

—

C₂²×C₄×C₁₉₁

C₂³

yes

—

C₂²×C₄×C₁₉₁

C₂³

yes

characteristic

C₄⋊C₄×C₁₉₁

—

C₄²×C₁₉₁

C₂×C₄

yes

—

C₂²×D₁₉₁

—

C₂²×D₁₉₁

C₂²×C₄

yes

—

C₂²×D₁₉₁

C₂×D₄

yes

—

C₂²×D₁₉₁

C₂×Q₈

yes

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

C₂²×C₄

yes

—

C₂×Dic₁₉₁

C₂²×C₄

yes

—

C₂×Dic₁₉₁

C₂×D₄

yes

—

C₂×Dic₁₉₁

C₂×Q₈

yes

—

C₂×Dic₁₉₁

#41

yes

—

C₄×D₁₉₁

—

C₄×D₁₉₁

C₄⋊C₄

yes

complemented

C₂³×C₁₉₁

C₂⁴

yes

characteristic

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

C₂²×C₄

yes

—

C₂×C₄×C₁₉₁

C₂²×C₄

yes

—

C₂×C₄×C₁₉₁

C₂×D₄

yes

—

C₂×C₄×C₁₉₁

C₂×Q₈

yes

—

C₂²×C₁₉₁

C₂³×C₄

yes

characteristic

C₂²×C₁₉₁

C₂²×D₄

yes

characteristic

C₂²×C₁₉₁

C₂²×Q₈

yes

characteristic

C₂²×C₁₉₁

C₂×#41

yes

—

C₂×D₁₉₁

—

C₂×D₁₉₁

—

C₂×D₁₉₁

—

C₂×D₁₉₁

C₂×C₄⋊C₄

yes

—

C₄×C₁₉₁

—

C₄×C₁₉₁

—

C₄×C₁₉₁

C₂×C₄⋊C₄

yes

—

Dic₁₉₁

—

Dic₁₉₁

C₂×C₄⋊C₄

yes

—

Dic₁₉₁

#117

yes

complemented

C₂×C₁₉₁

C₂²×C₄⋊C₄

yes

characteristic

C₂×C₁₉₁

C₂×#117

yes

—

C₂×C₁₉₁

#532

yes

—

C₂×C₁₉₁

#533

yes

—

D₁₉₁

—

C₁₉₁

#2285

yes

characteristic, complemented, Sylow

#2285

—

191

maximal, Sylow

C₂×#117

—

191

—

C₂²×C₄²

—

191

—

#384

—

191

—

#382

—

191

—

#382

D₁₉₁

yes

characteristic, complemented

#380

—

191

—

C₄×C₄⋊C₄

—

191

—

C₂³×C₄

—

191

—

C₂³×C₄

—

191

—

#117

—

382

—

C₂×C₄⋊C₄

—

191

—

C₂×C₄⋊C₄

—

191

—

C₂×C₄⋊C₄

C₂×D₁₉₁

yes

—

C₂×C₂²⋊C₄

—

191

—

C₂×C₄²

—

191

—

C₂×C₄²

—

191

—

100

C₂×C₄²

—

191

—

101

C₂×C₄²

—

191

—

102

C₂×C₄²

C₂×D₁₉₁

yes

characteristic

103

#95

—

191

—

104

#95

C₂×D₁₉₁

yes

—

105

C₂⁴

—

191

—

106

C₂²×C₄

—

382

—

107

C₂²×C₄

—

191

—

108

C₂²×C₄

—

191

—

109

C₂²×C₄

—

191

—

110

C₂²×C₄

—

191

—

111

C₂²×C₄

—

191

—

112

C₂²×C₄

—

191

—

113

C₂²×C₄

—

191

—

114

C₂²×C₄

C₂²×D₁₉₁

yes

—

115

C₂²×C₄

C₂²×D₁₉₁

yes

—

116

C₂²×C₄

C₂²×D₁₉₁

yes

characteristic

117

C₄⋊C₄

—

118

C₄⋊C₄

—

382

—

119

C₄⋊C₄

—

382

—

120

C₂²⋊C₄

—

382

—

121

C₄²

—

382

—

122

C₄²

—

191

—

123

C₄²

—

191

—

124

C₄²

C₄×D₁₉₁

yes

—

125

C₂³

—

382

—

126

C₂³

—

191

—

127

C₂³

—

191

—

128

C₂³

—

191

—

129

C₂³

C₂³×D₁₉₁

yes

characteristic

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

382

—

134

C₂×C₄

—

382

—

135

C₂×C₄

—

382

—

136

C₂×C₄

—

191

—

137

C₂×C₄

—

191

—

138

C₂×C₄

—

191

—

139

C₂×C₄

—

191

—

140

C₂×C₄

—

191

—

141

C₂×C₄

—

191

—

142

C₂×C₄

C₂×C₄×D₁₉₁

yes

—

143

C₂×C₄

C₂×C₄×D₁₉₁

yes

—

144

C₂×C₄

D₄×D₁₉₁

yes

—

145

C₂×C₄

Q₈×D₁₉₁

yes

—

146

C₂²

—

382

—

147

C₂²

—

382

—

148

C₂²

—

382

—

149

C₂²

—

191

—

150

C₂²

C₂²×C₄×D₁₉₁

yes

characteristic

151

C₂²

C₂×D₄×D₁₉₁

yes

characteristic

152

C₂²

C₂×Q₈×D₁₉₁

yes

characteristic

153

C₂²

#41×D₁₉₁

yes

—

154

C₄

—

155

C₄

—

156

C₄

—

191

—

157

C₄

—

191

—

158

C₄

—

764

—

159

C₄

C₄⋊C₄×D₁₉₁

yes

—

160

C₂

—

382

—

161

C₂

C₂×C₄⋊C₄×D₁₉₁

yes

characteristic

162

C₂

#117×D₁₉₁

yes

—

163

C₂

#27133212

yes

—

164

C₂

#27133219

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₉₁
commutator	C₂²×C₁₉₁	C₂³×C₄
Fitting	#382×C₁₉₁	C₂
socle, cosocle	C₂³×C₁₉₁	C₂⁴
prime radical	C₂³×D₁₉₁	C₂³
hypercenter	#382	D₁₉₁
hypocenter	C₁₉₁	#2285
LCS	G ▷ C₂²×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂³ ◁ #382

subgroup

G/H

center, Frattini

C₂³

C₂³×D₁₉₁

commutator

C₂²×C₁₉₁

C₂³×C₄

Fitting

#382×C₁₉₁

C₂

socle, cosocle

C₂³×C₁₉₁

C₂⁴

prime radical

C₂³×D₁₉₁

C₂³

D₁₉₁

C₁₉₁

G ▷ C₂²×C₁₉₁ ▷ C₁₉₁

C₁ ◁ C₂³ ◁ #382

N	H ≅ G/N	type
3. #27132532	C₂	semidirect
4. #27132529	C₂	semidirect
6. #27132514	C₂	semidirect
7. #382×C₁₉₁	C₂	semidirect
17. C₄²×D₁₉₁	C₄	semidirect
54. C₄×D₁₉₁	C₄⋊C₄	semidirect
76. Dic₁₉₁	#117	semidirect
82. C₁₉₁	#2285	semidirect
88. #382	D₁₉₁	semidirect

H ≅ G/N

type

3. #27132532

C₂

semidirect

4. #27132529

C₂

semidirect

6. #27132514

C₂

semidirect

7. #382×C₁₉₁

C₂

semidirect

17. C₄²×D₁₉₁

C₄

semidirect

54. C₄×D₁₉₁

C₄⋊C₄

semidirect

76. Dic₁₉₁

#117

semidirect

82. C₁₉₁

#2285

semidirect

88. #382

D₁₉₁

semidirect

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

subgroup	index
C₂×#20259280	2
C₂×C₄²×D₁₉₁	2
#27132532	2
#27132529	2
#27132522	2
#27132514	2
#382×C₁₉₁	2
#2285	191

subgroup

index

C₂×C₄²×D₁₉₁

#382×C₁₉₁

191

Subgroups — #37574926

Conjugacy classes of subgroups

Complements

Maximal subgroups