Group #37582868 — subgroups

Subgroups — #37582868

Order 24448.

392 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 133 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	#27133213	C₂	1	1	yes	maximal, characteristic
2	#27133033	C₂	1	2	yes	maximal, complemented
3	#27133029	C₂	1	2	yes	maximal, complemented
4	#27133025	C₂	1	4	yes	maximal, complemented
5	#27132657	C₂	1	4	yes	maximal, complemented
6	C₂×#20259233	C₂	1	1	yes	maximal, characteristic, complemented
7	#533×C₁₉₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#15690287	C₂²	1	2	yes	—
9	C₂²×C₁₉₁⋊C₈	C₂²	1	2	yes	—
10	#20259322	C₂²	1	1	yes	characteristic
11	#20259319	C₂²	1	1	yes	characteristic
12	#20259310	C₂²	1	2	yes	—
13	#20259296	—	2	2	no	—
14	C₄×D₇₆₄	—	2	2	no	—
15	#20259245	C₂²	1	8	yes	—
16	#20259242	C₂²	1	8	yes	—
17	#20259233	C₂²	1	2	yes	—
18	#20259233	C₂²	1	2	yes	—
19	C₄⋊Q₈×C₁₉₁	C₂²	1	2	yes	—
20	#125×C₁₉₁	C₂²	1	2	yes	—
21	C₂²⋊Q₈×C₁₉₁	—	2	2	no	—
22	C₄×Q₈×C₁₉₁	—	2	2	no	—
23	#117×C₁₉₁	C₂²	1	2	yes	—
24	C₂×C₄²×C₁₉₁	C₂²	1	1	yes	characteristic
25	#15690287	—	2	8	no	—
26	C₂×Dic₃₈₂	C₂³	1	2	yes	—
27	C₂×D₇₆₄	—	2	2	no	—
28	C₂×D₇₆₄	C₂³	1	2	yes	—
29	C₂×C₂²⋊D₁₉₁	—	2	2	no	—
30	C₂×C₄×D₁₉₁	—	2	2	no	—
31	C₂×C₁₉₁⋊C₈	—	2	4	no	—
32	C₂×C₁₉₁⋊C₈	C₂³	1	4	yes	—
33	C₄⋊Dic₁₉₁	—	2	2	no	—
34	(C₂×D₁₉₁)⋊C₄	—	2	4	no	—
35	C₂×Q₈×C₁₉₁	—	2	2	no	—
36	C₂²×C₄×C₁₉₁	D₄	1	2	yes	—
37	C₂²×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
38	C₄⋊C₄×C₁₉₁	—	2	4	no	—
39	C₄⋊C₄×C₁₉₁	—	2	2	no	—
40	C₄⋊C₄×C₁₉₁	C₂³	1	4	yes	—
41	C₂²⋊C₄×C₁₉₁	—	2	2	no	—
42	C₄²×C₁₉₁	—	2	2	no	—
43	C₄²×C₁₉₁	D₄	1	2	yes	—
44	C₄²×C₁₉₁	C₂³	1	1	yes	characteristic
45	C₄²×C₁₉₁	C₂³	1	1	yes	characteristic
46	C₂²×D₁₉₁	—	2	2	no	—
47	Dic₃₈₂	—	2	4	no	—
48	D₇₆₄	—	2	4	no	—
49	D₇₆₄	—	4	4	no	—
50	C₂²⋊D₁₉₁	—	4	4	no	—
51	C₂×Dic₁₉₁	—	2	2	no	—
52	C₄×D₁₉₁	—	4	4	no	—
53	C₁₉₁⋊C₈	—	2	4	no	—
54	C₂³×C₁₉₁	C₂×D₄	1	1	yes	characteristic
55	Q₈×C₁₉₁	—	4	4	no	—
56	C₂×C₄×C₁₉₁	—	2	4	no	—
57	C₂×C₄×C₁₉₁	—	4	2	no	—
58	C₂×C₄×C₁₉₁	C₂×D₄	1	2	yes	—
59	C₂×C₄×C₁₉₁	C₂×D₄	1	2	yes	—
60	C₂×C₄×C₁₉₁	C₂×D₄	1	1	yes	characteristic
61	C₂×C₄×C₁₉₁	#41	1	4	yes	—
62	C₂×C₄×C₁₉₁	C₂⁴	1	1	yes	characteristic
63	C₂²×C₁₉₁	—	2	2	no	—
64	C₂²×C₁₉₁	#124	1	2	yes	—
65	C₂²×C₁₉₁	C₂²×D₄	1	1	yes	characteristic
66	C₂×D₁₉₁	—	4	2	no	—
67	C₂×D₁₉₁	—	4	4	no	—
68	C₄×C₁₉₁	—	2	2	no	—
69	C₄×C₁₉₁	—	2	2	no	—
70	C₄×C₁₉₁	—	4	4	no	—
71	C₄×C₁₉₁	#124	1	2	yes	—
72	C₄×C₁₉₁	C₂×#41	1	2	yes	—
73	Dic₁₉₁	—	4	2	no	—
74	C₂×C₁₉₁	—	2	2	no	—
75	C₂×C₁₉₁	C₂×#124	1	1	yes	characteristic
76	C₂×C₁₉₁	#575	1	2	yes	—
77	D₁₉₁	—	8	2	no	—
78	C₁₉₁	#3135	1	1	yes	characteristic, complemented, Sylow
79	#3135	—	191	1	no	maximal, Sylow
80	#533	D₁₉₁	1	1	yes	characteristic, complemented
81	#532	—	191	1	no	—
82	#491	—	191	2	no	—
83	#490	—	191	2	no	—
84	#489	—	191	4	no	—
85	#416	—	191	4	no	—
86	C₂×#97	—	191	1	no	—
87	C₂×#41	—	191	2	no	—
88	C₂²×C₈	—	191	2	no	—
89	C₄⋊Q₈	—	191	1	no	—
90	C₄⋊Q₈	C₂×D₁₉₁	1	2	yes	—
91	#127	—	191	1	no	—
92	#125	C₂×D₁₉₁	1	2	yes	—
93	#124	—	191	2	no	—
94	C₂²⋊Q₈	—	2	2	no	—
95	#121	—	382	2	no	—
96	C₄×Q₈	—	2	2	no	—
97	C₄×D₄	—	382	2	no	—
98	#117	C₂×D₁₉₁	1	2	yes	—
99	C₂×C₄²	C₂×D₁₉₁	1	1	yes	characteristic
100	#103	—	191	8	no	—
101	#102	—	191	8	no	—
102	#97	—	191	2	no	—
103	#97	—	191	2	no	—
104	#41	—	382	8	no	—
105	C₂×Q₈	—	2	2	no	—
106	C₂×Q₈	—	191	2	no	—
107	C₂×D₄	—	382	2	no	—
108	C₂×D₄	—	382	2	no	—
109	C₂×D₄	—	191	2	no	—
110	C₂²×C₄	—	382	2	no	—
111	C₂²×C₄	C₂²⋊D₁₉₁	1	2	yes	—
112	C₂²×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
113	C₂×C₈	—	382	4	no	—
114	C₂×C₈	—	191	4	no	—
115	C₄⋊C₄	—	2	4	no	—
116	C₄⋊C₄	—	2	2	no	—
117	C₄⋊C₄	—	382	2	no	—
118	C₄⋊C₄	C₂²×D₁₉₁	1	4	yes	—
119	C₂²⋊C₄	—	2	2	no	—
120	C₂²⋊C₄	—	382	4	no	—
121	C₄²	—	2	2	no	—
122	C₄²	C₂²⋊D₁₉₁	1	2	yes	—
123	C₄²	C₂²×D₁₉₁	1	1	yes	characteristic
124	C₄²	C₂²×D₁₉₁	1	1	yes	characteristic
125	C₂³	—	382	2	no	—
126	C₂³	C₂×C₂²⋊D₁₉₁	1	1	yes	characteristic
127	Q₈	—	4	4	no	—
128	Q₈	—	382	4	no	—
129	D₄	—	382	4	no	—
130	D₄	—	764	4	no	—
131	D₄	—	764	4	no	—
132	C₂×C₄	—	2	4	no	—
133	C₂×C₄	—	4	2	no	—
134	C₂×C₄	—	382	2	no	—
135	C₂×C₄	—	764	4	no	—
136	C₂×C₄	C₂×C₂²⋊D₁₉₁	1	2	yes	—
137	C₂×C₄	C₂×C₂²⋊D₁₉₁	1	2	yes	—
138	C₂×C₄	C₂×C₂²⋊D₁₉₁	1	1	yes	characteristic
139	C₂×C₄	#15690288	1	4	yes	—
140	C₂×C₄	C₂³×D₁₉₁	1	1	yes	characteristic
141	C₈	—	382	4	no	—
142	C₂²	—	2	2	no	—
143	C₂²	—	764	4	no	—
144	C₂²	—	764	2	no	—
145	C₂²	#20259311	1	2	yes	—
146	C₂²	C₂²×C₂²⋊D₁₉₁	1	1	yes	characteristic
147	C₄	—	2	2	no	—
148	C₄	—	2	2	no	—
149	C₄	—	4	4	no	—
150	C₄	—	764	2	no	—
151	C₄	#20259311	1	2	yes	—
152	C₄	C₂×#15690288	1	2	yes	—
153	C₂	—	2	2	no	—
154	C₂	—	1528	2	no	—
155	C₂	C₂×#20259311	1	1	yes	characteristic
156	C₂	#27133461	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#27133213

C₂

yes

maximal, characteristic

#27133033

C₂

yes

maximal, complemented

#27133029

C₂

yes

maximal, complemented

#27133025

C₂

yes

maximal, complemented

#27132657

C₂

yes

maximal, complemented

C₂×#20259233

C₂

yes

maximal, characteristic, complemented

#533×C₁₉₁

C₂

yes

maximal, characteristic, complemented

C₂×#15690287

C₂²

yes

—

C₂²×C₁₉₁⋊C₈

C₂²

yes

—

#20259322

C₂²

yes

characteristic

#20259319

C₂²

yes

characteristic

#20259310

C₂²

yes

—

#20259296

—

C₄×D₇₆₄

—

#20259245

C₂²

yes

—

#20259242

C₂²

yes

—

#20259233

C₂²

yes

—

#20259233

C₂²

yes

—

C₄⋊Q₈×C₁₉₁

C₂²

yes

—

#125×C₁₉₁

C₂²

yes

—

C₂²⋊Q₈×C₁₉₁

—

C₄×Q₈×C₁₉₁

—

#117×C₁₉₁

C₂²

yes

—

C₂×C₄²×C₁₉₁

C₂²

yes

characteristic

#15690287

—

C₂×Dic₃₈₂

C₂³

yes

—

C₂×D₇₆₄

—

C₂×D₇₆₄

C₂³

yes

—

C₂×C₂²⋊D₁₉₁

—

C₂×C₄×D₁₉₁

—

C₂×C₁₉₁⋊C₈

—

C₂×C₁₉₁⋊C₈

C₂³

yes

—

C₄⋊Dic₁₉₁

—

(C₂×D₁₉₁)⋊C₄

—

C₂×Q₈×C₁₉₁

—

C₂²×C₄×C₁₉₁

D₄

yes

—

C₂²×C₄×C₁₉₁

C₂³

yes

characteristic

C₄⋊C₄×C₁₉₁

—

C₄⋊C₄×C₁₉₁

—

C₄⋊C₄×C₁₉₁

C₂³

yes

—

C₂²⋊C₄×C₁₉₁

—

C₄²×C₁₉₁

—

C₄²×C₁₉₁

D₄

yes

—

C₄²×C₁₉₁

C₂³

yes

characteristic

C₄²×C₁₉₁

C₂³

yes

characteristic

C₂²×D₁₉₁

—

Dic₃₈₂

—

D₇₆₄

—

D₇₆₄

—

C₂²⋊D₁₉₁

—

C₂×Dic₁₉₁

—

C₄×D₁₉₁

—

C₁₉₁⋊C₈

—

C₂³×C₁₉₁

C₂×D₄

yes

characteristic

Q₈×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

C₂×D₄

yes

—

C₂×C₄×C₁₉₁

C₂×D₄

yes

—

C₂×C₄×C₁₉₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₁

#41

yes

—

C₂×C₄×C₁₉₁

C₂⁴

yes

characteristic

C₂²×C₁₉₁

—

C₂²×C₁₉₁

#124

yes

—

C₂²×C₁₉₁

C₂²×D₄

yes

characteristic

C₂×D₁₉₁

—

C₂×D₁₉₁

—

C₄×C₁₉₁

—

C₄×C₁₉₁

—

C₄×C₁₉₁

—

C₄×C₁₉₁

#124

yes

—

C₄×C₁₉₁

C₂×#41

yes

—

Dic₁₉₁

—

C₂×C₁₉₁

—

C₂×C₁₉₁

C₂×#124

yes

characteristic

C₂×C₁₉₁

#575

yes

—

D₁₉₁

—

C₁₉₁

#3135

yes

characteristic, complemented, Sylow

#3135

—

191

maximal, Sylow

#533

D₁₉₁

yes

characteristic, complemented

#532

—

191

—

#491

—

191

—

#490

—

191

—

#489

—

191

—

#416

—

191

—

C₂×#97

—

191

—

C₂×#41

—

191

—

C₂²×C₈

—

191

—

C₄⋊Q₈

—

191

—

C₄⋊Q₈

C₂×D₁₉₁

yes

—

#127

—

191

—

#125

C₂×D₁₉₁

yes

—

#124

—

191

—

C₂²⋊Q₈

—

#121

—

382

—

C₄×Q₈

—

C₄×D₄

—

382

—

#117

C₂×D₁₉₁

yes

—

C₂×C₄²

C₂×D₁₉₁

yes

characteristic

100

#103

—

191

—

101

#102

—

191

—

102

#97

—

191

—

103

#97

—

191

—

104

#41

—

382

—

105

C₂×Q₈

—

106

C₂×Q₈

—

191

—

107

C₂×D₄

—

382

—

108

C₂×D₄

—

382

—

109

C₂×D₄

—

191

—

110

C₂²×C₄

—

382

—

111

C₂²×C₄

C₂²⋊D₁₉₁

yes

—

112

C₂²×C₄

C₂²×D₁₉₁

yes

characteristic

113

C₂×C₈

—

382

—

114

C₂×C₈

—

191

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

382

—

118

C₄⋊C₄

C₂²×D₁₉₁

yes

—

119

C₂²⋊C₄

—

120

C₂²⋊C₄

—

382

—

121

C₄²

—

122

C₄²

C₂²⋊D₁₉₁

yes

—

123

C₄²

C₂²×D₁₉₁

yes

characteristic

124

C₄²

C₂²×D₁₉₁

yes

characteristic

125

C₂³

—

382

—

126

C₂³

C₂×C₂²⋊D₁₉₁

yes

characteristic

127

Q₈

—

128

Q₈

—

382

—

129

D₄

—

382

—

130

D₄

—

764

—

131

D₄

—

764

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

382

—

135

C₂×C₄

—

764

—

136

C₂×C₄

C₂×C₂²⋊D₁₉₁

yes

—

137

C₂×C₄

C₂×C₂²⋊D₁₉₁

yes

—

138

C₂×C₄

C₂×C₂²⋊D₁₉₁

yes

characteristic

139

C₂×C₄

#15690288

yes

—

140

C₂×C₄

C₂³×D₁₉₁

yes

characteristic

141

C₈

—

382

—

142

C₂²

—

143

C₂²

—

764

—

144

C₂²

—

764

—

145

C₂²

#20259311

yes

—

146

C₂²

C₂²×C₂²⋊D₁₉₁

yes

characteristic

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

—

764

—

151

C₄

#20259311

yes

—

152

C₄

C₂×#15690288

yes

—

153

C₂

—

154

C₂

—

1528

—

155

C₂

C₂×#20259311

yes

characteristic

156

C₂

#27133461

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₁₉₁
commutator, cosocle	C₂×C₄×C₁₉₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₉₁
Fitting	#533×C₁₉₁	C₂
socle	C₂²×C₁₉₁	C₂²×D₄
prime radical	#27133213	C₂
hypercenter	#533	D₁₉₁
hypocenter	C₁₉₁	#3135
LCS	G ▷ C₂×C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #533

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₁₉₁

commutator, cosocle

C₂×C₄×C₁₉₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₉₁

Fitting

#533×C₁₉₁

C₂

socle

C₂²×C₁₉₁

C₂²×D₄

C₂

D₁₉₁

C₁₉₁

G ▷ C₂×C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #533

N	H ≅ G/N	type
2. #27133033	C₂	semidirect
3. #27133029	C₂	semidirect
4. #27133025	C₂	semidirect
5. #27132657	C₂	semidirect
6. C₂×#20259233	C₂	semidirect
7. #533×C₁₉₁	C₂	semidirect
78. C₁₉₁	#3135	semidirect
80. #533	D₁₉₁	semidirect

H ≅ G/N

type

2. #27133033

C₂

semidirect

3. #27133029

C₂

semidirect

4. #27133025

C₂

semidirect

5. #27132657

C₂

semidirect

6. C₂×#20259233

C₂

semidirect

7. #533×C₁₉₁

C₂

semidirect

78. C₁₉₁

#3135

semidirect

80. #533

D₁₉₁

semidirect

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

subgroup	index
#27133213	2
#27133033	2
#27133029	2
#27133025	2
#27132657	2
C₂×#20259233	2
#533×C₁₉₁	2
#3135	191

subgroup

index

#533×C₁₉₁

191

Subgroups — #37582868

Conjugacy classes of subgroups

Complements

Maximal subgroups