Group #37573676 — subgroups

Subgroups — #37573676

Order 24448.

282 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 282 conjugacy classes of subgroups — too many to draw. Only its 69 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	#27133294	C₂	1	1	yes	maximal, characteristic, complemented
2	#27132668	C₂	1	2	yes	maximal
3	#27132432	C₂	1	1	yes	maximal, characteristic, complemented
4	#27132423	C₂	1	2	yes	maximal, complemented
5	#390×C₁₉₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#15690271	C₂²	1	2	yes	—
7	#20259321	—	2	2	no	—
8	#20259313	—	2	2	no	—
9	#20259301	—	2	2	no	—
10	#20259281	C₂²	1	2	yes	—
11	#20259243	—	2	2	no	—
12	#20259243	—	2	2	no	—
13	C₂²×D₄×C₁₉₁	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₄×C₁₉₁	C₂²	1	1	yes	characteristic
15	C₂×C₂²⋊C₄×C₁₉₁	—	2	2	no	—
16	C₂×C₄²×C₁₉₁	C₂²	1	1	yes	characteristic
17	C₂×Dic₃₈₂	—	2	2	no	—
18	#15690271	—	2	2	no	—
19	#15690271	—	2	2	no	—
20	C₂×C₁₉₁⋊C₈	—	2	2	no	—
21	C₄⋊Dic₁₉₁	—	2	2	no	—
22	C₄⋊Dic₁₉₁	D₄	1	2	yes	—
23	C₄⋊Dic₁₉₁	D₄	1	2	yes	—
24	Dic₁₉₁⋊C₄	—	2	4	no	—
25	C₂²⋊Dic₁₉₁	—	2	2	no	—
26	C₄×Dic₁₉₁	—	2	2	no	—
27	C₂⁴×C₁₉₁	—	2	1	no	—
28	C₂×D₄×C₁₉₁	—	2	2	no	—
29	C₂×D₄×C₁₉₁	—	2	2	no	—
30	C₂×D₄×C₁₉₁	—	2	1	no	—
31	C₂×D₄×C₁₉₁	Q₈	1	2	yes	complemented
32	C₂²×C₄×C₁₉₁	—	2	1	no	—
33	C₂²×C₄×C₁₉₁	D₄	1	2	yes	—
34	C₂²×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
35	C₄⋊C₄×C₁₉₁	—	2	2	no	—
36	C₂²⋊C₄×C₁₉₁	—	4	4	no	—
37	C₄²×C₁₉₁	—	2	2	no	—
38	Dic₃₈₂	—	8	2	no	—
39	C₂×Dic₁₉₁	—	2	2	no	—
40	C₂×Dic₁₉₁	—	2	2	no	—
41	C₁₉₁⋊C₈	—	4	2	no	—
42	C₂³×C₁₉₁	—	2	2	no	—
43	C₂³×C₁₉₁	—	4	2	no	—
44	C₂³×C₁₉₁	—	4	2	no	—
45	C₂³×C₁₉₁	—	4	1	no	—
46	C₂³×C₁₉₁	—	4	1	no	—
47	C₂³×C₁₉₁	C₂×D₄	1	1	yes	characteristic
48	D₄×C₁₉₁	—	2	4	no	—
49	D₄×C₁₉₁	—	4	2	no	—
50	C₂×C₄×C₁₉₁	—	2	2	no	—
51	C₂×C₄×C₁₉₁	—	2	2	no	—
52	C₂×C₄×C₁₉₁	—	4	1	no	—
53	C₂×C₄×C₁₉₁	—	4	1	no	—
54	C₂×C₄×C₁₉₁	—	4	2	no	—
55	C₂×C₄×C₁₉₁	C₂×D₄	1	2	yes	—
56	C₂×C₄×C₁₉₁	C₂×Q₈	1	1	yes	characteristic
57	C₂×C₄×C₁₉₁	#41	1	2	yes	—
58	C₂×C₄×C₁₉₁	#41	1	1	yes	characteristic
59	C₂²×C₁₉₁	—	2	1	no	—
60	C₂²×C₁₉₁	—	2	1	no	—
61	C₂²×C₁₉₁	—	4	4	no	—
62	C₂²×C₁₉₁	—	4	2	no	—
63	C₂²×C₁₉₁	—	8	2	no	—
64	C₂²×C₁₉₁	—	8	1	no	—
65	C₂²×C₁₉₁	—	8	1	no	—
66	C₂²×C₁₉₁	#120	1	1	yes	characteristic
67	C₂²×C₁₉₁	C₂²⋊Q₈	1	1	yes	characteristic
68	C₂²×C₁₉₁	#124	1	1	yes	characteristic
69	C₄×C₁₉₁	—	4	2	no	—
70	C₄×C₁₉₁	—	4	2	no	—
71	C₄×C₁₉₁	C₂²⋊Q₈	1	2	yes	—
72	C₄×C₁₉₁	#124	1	2	yes	—
73	Dic₁₉₁	—	4	2	no	—
74	Dic₁₉₁	—	4	2	no	—
75	C₂×C₁₉₁	—	2	1	no	—
76	C₂×C₁₉₁	—	2	1	no	—
77	C₂×C₁₉₁	—	8	2	no	—
78	C₂×C₁₉₁	#393	1	1	yes	characteristic
79	C₂×C₁₉₁	#453	1	1	yes	characteristic
80	C₂×C₁₉₁	#454	1	1	yes	characteristic
81	C₁₉₁	#2027	1	1	yes	characteristic, complemented, Sylow
82	#2027	—	191	1	no	maximal, Sylow
83	#544	—	191	1	no	—
84	#418	—	191	2	no	—
85	#390	D₁₉₁	1	1	yes	characteristic
86	#343	—	191	1	no	—
87	#339	—	191	2	no	—
88	C₂²×D₄	C₂×D₁₉₁	1	1	yes	characteristic
89	C₂×M₁₆	—	191	2	no	—
90	C₄⋊Q₈	—	382	2	no	—
91	#125	—	382	2	no	—
92	C₂²⋊Q₈	—	382	2	no	—
93	#117	—	191	2	no	—
94	C₂×C₄⋊C₄	C₂×D₁₉₁	1	1	yes	characteristic
95	C₂×C₂²⋊C₄	—	2	2	no	—
96	C₂×C₄²	C₂×D₁₉₁	1	1	yes	characteristic
97	#102	—	382	2	no	—
98	#102	—	382	2	no	—
99	C₂⁴	—	2	1	no	—
100	C₂×Q₈	—	382	2	no	—
101	C₂×D₄	—	2	2	no	—
102	C₂×D₄	—	2	2	no	—
103	C₂×D₄	—	2	1	no	—
104	C₂×D₄	Dic₃₈₂	1	2	yes	complemented
105	C₂²×C₄	—	2	1	no	—
106	C₂²×C₄	C₂²⋊D₁₉₁	1	2	yes	—
107	C₂²×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
108	M₁₆	—	382	2	no	—
109	M₁₆	—	382	2	no	—
110	C₂×C₈	—	382	2	no	—
111	C₄⋊C₄	—	2	2	no	—
112	C₄⋊C₄	—	382	4	no	—
113	C₄⋊C₄	—	382	2	no	—
114	C₄⋊C₄	—	191	2	no	—
115	C₄⋊C₄	—	191	2	no	—
116	C₂²⋊C₄	—	4	4	no	—
117	C₂²⋊C₄	—	382	2	no	—
118	C₄²	—	2	2	no	—
119	C₄²	—	382	2	no	—
120	C₂³	—	2	2	no	—
121	C₂³	—	4	2	no	—
122	C₂³	—	4	2	no	—
123	C₂³	—	4	1	no	—
124	C₂³	—	4	1	no	—
125	C₂³	C₂×C₂²⋊D₁₉₁	1	1	yes	characteristic
126	Q₈	—	1528	2	no	—
127	D₄	—	2	4	no	—
128	D₄	—	4	2	no	—
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	4	1	no	—
132	C₂×C₄	—	4	1	no	—
133	C₂×C₄	—	4	2	no	—
134	C₂×C₄	—	382	2	no	—
135	C₂×C₄	—	382	2	no	—
136	C₂×C₄	D₄×D₁₉₁	1	2	yes	—
137	C₂×C₄	C₂×Dic₃₈₂	1	1	yes	characteristic
138	C₂×C₄	#15690286	1	2	yes	—
139	C₂×C₄	#15690287	1	1	yes	characteristic
140	C₈	—	764	2	no	—
141	C₂²	—	2	1	no	—
142	C₂²	—	2	1	no	—
143	C₂²	—	4	4	no	—
144	C₂²	—	4	2	no	—
145	C₂²	—	8	2	no	—
146	C₂²	—	8	1	no	—
147	C₂²	—	8	1	no	—
148	C₂²	#20259290	1	1	yes	characteristic
149	C₂²	#20259301	1	1	yes	characteristic
150	C₂²	#20259309	1	1	yes	characteristic
151	C₄	—	4	2	no	—
152	C₄	—	4	2	no	—
153	C₄	—	764	2	no	—
154	C₄	—	764	2	no	—
155	C₄	#20259299	1	2	yes	—
156	C₄	#20259308	1	2	yes	—
157	C₂	—	2	1	no	—
158	C₂	—	2	1	no	—
159	C₂	—	8	2	no	—
160	C₂	#27132575	1	1	yes	characteristic
161	C₂	#27132830	1	1	yes	characteristic
162	C₂	#27132835	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#27133294

C₂

yes

maximal, characteristic, complemented

#27132668

C₂

yes

maximal

#27132432

C₂

yes

maximal, characteristic, complemented

#27132423

C₂

yes

maximal, complemented

#390×C₁₉₁

C₂

yes

maximal, characteristic

C₂×#15690271

C₂²

yes

—

—

—

—

C₂²

yes

—

#20259243

—

#20259243

—

C₂²×D₄×C₁₉₁

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₁₉₁

C₂²

yes

characteristic

C₂×C₂²⋊C₄×C₁₉₁

—

C₂×C₄²×C₁₉₁

C₂²

yes

characteristic

C₂×Dic₃₈₂

—

#15690271

—

#15690271

—

C₂×C₁₉₁⋊C₈

—

C₄⋊Dic₁₉₁

—

C₄⋊Dic₁₉₁

D₄

yes

—

C₄⋊Dic₁₉₁

D₄

yes

—

Dic₁₉₁⋊C₄

—

C₂²⋊Dic₁₉₁

—

C₄×Dic₁₉₁

—

C₂⁴×C₁₉₁

—

C₂×D₄×C₁₉₁

—

C₂×D₄×C₁₉₁

—

C₂×D₄×C₁₉₁

—

C₂×D₄×C₁₉₁

Q₈

yes

complemented

C₂²×C₄×C₁₉₁

—

C₂²×C₄×C₁₉₁

D₄

yes

—

C₂²×C₄×C₁₉₁

C₂³

yes

characteristic

C₄⋊C₄×C₁₉₁

—

C₂²⋊C₄×C₁₉₁

—

C₄²×C₁₉₁

—

Dic₃₈₂

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

—

C₁₉₁⋊C₈

—

C₂³×C₁₉₁

—

C₂³×C₁₉₁

—

C₂³×C₁₉₁

—

C₂³×C₁₉₁

—

C₂³×C₁₉₁

—

C₂³×C₁₉₁

C₂×D₄

yes

characteristic

D₄×C₁₉₁

—

D₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

C₂×D₄

yes

—

C₂×C₄×C₁₉₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₉₁

#41

yes

—

C₂×C₄×C₁₉₁

#41

yes

characteristic

C₂²×C₁₉₁

—

C₂²×C₁₉₁

—

C₂²×C₁₉₁

—

C₂²×C₁₉₁

—

C₂²×C₁₉₁

—

C₂²×C₁₉₁

—

C₂²×C₁₉₁

—

C₂²×C₁₉₁

#120

yes

characteristic

C₂²×C₁₉₁

C₂²⋊Q₈

yes

characteristic

C₂²×C₁₉₁

#124

yes

characteristic

C₄×C₁₉₁

—

C₄×C₁₉₁

—

C₄×C₁₉₁

C₂²⋊Q₈

yes

—

C₄×C₁₉₁

#124

yes

—

Dic₁₉₁

—

Dic₁₉₁

—

C₂×C₁₉₁

—

C₂×C₁₉₁

—

C₂×C₁₉₁

—

C₂×C₁₉₁

yes

C₂×C₁₉₁

yes

C₂×C₁₉₁

yes

C₁₉₁

yes

characteristic, complemented, Sylow

#2027

—

191

maximal, Sylow

#544

—

191

—

#418

—

191

—

#390

D₁₉₁

yes

characteristic

#343

—

191

—

#339

—

191

—

C₂²×D₄

C₂×D₁₉₁

yes

characteristic

C₂×M₁₆

—

191

—

C₄⋊Q₈

—

382

—

#125

—

382

—

C₂²⋊Q₈

—

382

—

#117

—

191

—

C₂×C₄⋊C₄

C₂×D₁₉₁

yes

characteristic

C₂×C₂²⋊C₄

—

C₂×C₄²

C₂×D₁₉₁

yes

characteristic

#102

—

382

—

#102

—

382

—

C₂⁴

—

100

C₂×Q₈

—

382

—

101

C₂×D₄

—

102

C₂×D₄

—

103

C₂×D₄

—

104

C₂×D₄

Dic₃₈₂

yes

complemented

105

C₂²×C₄

—

106

C₂²×C₄

C₂²⋊D₁₉₁

yes

—

107

C₂²×C₄

C₂²×D₁₉₁

yes

characteristic

108

M₁₆

—

382

—

109

M₁₆

—

382

—

110

C₂×C₈

—

382

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

382

—

113

C₄⋊C₄

—

382

—

114

C₄⋊C₄

—

191

—

115

C₄⋊C₄

—

191

—

116

C₂²⋊C₄

—

117

C₂²⋊C₄

—

382

—

118

C₄²

—

119

C₄²

—

382

—

120

C₂³

—

121

C₂³

—

122

C₂³

—

123

C₂³

—

124

C₂³

—

125

C₂³

C₂×C₂²⋊D₁₉₁

yes

characteristic

126

Q₈

—

1528

—

127

D₄

—

128

D₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

382

—

135

C₂×C₄

—

382

—

136

C₂×C₄

D₄×D₁₉₁

yes

—

137

C₂×C₄

C₂×Dic₃₈₂

yes

characteristic

138

C₂×C₄

#15690286

yes

—

139

C₂×C₄

#15690287

yes

characteristic

140

C₈

—

764

—

141

C₂²

—

142

C₂²

—

143

C₂²

—

144

C₂²

—

145

C₂²

—

146

C₂²

—

147

C₂²

—

148

C₂²

#20259290

yes

characteristic

149

C₂²

#20259301

yes

characteristic

150

C₂²

#20259309

yes

characteristic

151

C₄

—

152

C₄

—

153

C₄

—

764

—

154

C₄

—

764

—

155

C₄

#20259299

yes

—

156

C₄

#20259308

yes

—

157

C₂

—

158

C₂

—

159

C₂

—

160

C₂

#27132575

yes

characteristic

161

C₂

#27132830

yes

characteristic

162

C₂

#27132835

yes

characteristic

	subgroup	G/H
center	C₂²	#20259290
commutator, cosocle	C₂²×C₄×C₁₉₁	C₂³
Frattini	C₂²×C₄	C₂²×D₁₉₁
Fitting	#390×C₁₉₁	C₂
socle	C₂²×C₁₉₁	#120
prime radical	C₂²×D₄×C₁₉₁	C₂²
hypercenter	#390	D₁₉₁
hypocenter	C₁₉₁	#2027
LCS	G ▷ C₂²×C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #390

subgroup

G/H

center

C₂²

#20259290

commutator, cosocle

C₂²×C₄×C₁₉₁

C₂³

Frattini

C₂²×C₄

C₂²×D₁₉₁

Fitting

#390×C₁₉₁

C₂

socle

C₂²×C₁₉₁

#120

prime radical

C₂²×D₄×C₁₉₁

C₂²

D₁₉₁

C₁₉₁

G ▷ C₂²×C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #390

N	H ≅ G/N	type
1. #27133294	C₂	semidirect
3. #27132432	C₂	semidirect
4. #27132423	C₂	semidirect
31. C₂×D₄×C₁₉₁	Q₈	semidirect
81. C₁₉₁	#2027	semidirect
104. C₂×D₄	Dic₃₈₂	semidirect

H ≅ G/N

type

1. #27133294

C₂

semidirect

3. #27132432

C₂

semidirect

4. #27132423

C₂

semidirect

31. C₂×D₄×C₁₉₁

Q₈

semidirect

81. C₁₉₁

#2027

semidirect

104. C₂×D₄

Dic₃₈₂

semidirect

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

subgroup	index
#27133294	2
#27132668	2
#27132432	2
#27132423	2
#390×C₁₉₁	2
#2027	191

subgroup

index

#390×C₁₉₁

191

Subgroups — #37573676

Conjugacy classes of subgroups

Complements

Maximal subgroups