Group #37583344 — subgroups

Subgroups — #37583344

Order 24448.

410 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 125 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×#20259301	C₂	1	2	yes	maximal
2	#27132994	C₂	1	4	yes	maximal, complemented
3	#27132921	C₂	1	4	yes	maximal, complemented
4	#27132821	C₂	1	4	yes	maximal, complemented
5	#407×C₁₉₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₃₈₂	C₂²	1	2	yes	—
7	C₂×Dic₇₆₄	C₂²	1	4	yes	complemented
8	#20259301	—	2	2	no	—
9	#20259301	—	2	2	no	—
10	#20259301	C₂²	1	4	yes	—
11	#20259301	C₂²	1	4	yes	—
12	C₂×C₄⋊Dic₁₉₁	C₂²	1	2	yes	—
13	C₂×Dic₁₉₁⋊C₄	—	2	2	no	—
14	C₂×C₂²⋊Dic₁₉₁	—	2	2	no	—
15	#20259254	C₂²	1	4	yes	complemented
16	#20259247	C₂²	1	8	yes	complemented
17	C₂³×C₄×C₁₉₁	C₂²	1	1	yes	characteristic
18	C₂×M₁₆×C₁₉₁	C₂²	1	2	yes	—
19	C₂²⋊C₈×C₁₉₁	C₂²	1	4	yes	—
20	C₂×Dic₃₈₂	—	2	2	no	—
21	C₂×Dic₃₈₂	—	4	4	no	—
22	C₂×Dic₃₈₂	C₂³	1	4	yes	—
23	C₂²×Dic₁₉₁	—	2	2	no	—
24	C₂²×Dic₁₉₁	—	2	2	no	—
25	Dic₇₆₄	—	4	4	no	—
26	C₄⋊Dic₁₉₁	—	2	2	no	—
27	C₄⋊Dic₁₉₁	C₂³	1	4	yes	—
28	Dic₁₉₁⋊C₄	—	2	4	no	—
29	Dic₁₉₁⋊C₄	—	2	4	no	—
30	C₂²⋊Dic₁₉₁	—	2	4	no	—
31	C₂²⋊Dic₁₉₁	—	2	4	no	—
32	C₂⁴×C₁₉₁	D₄	1	1	yes	characteristic
33	C₂²×C₄×C₁₉₁	—	4	2	no	—
34	C₂²×C₄×C₁₉₁	D₄	1	2	yes	—
35	C₂²×C₄×C₁₉₁	D₄	1	1	yes	characteristic
36	C₂²×C₄×C₁₉₁	C₂³	1	2	yes	—
37	C₂²×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
38	M₁₆×C₁₉₁	—	4	2	no	—
39	C₂×C₈×C₁₉₁	C₂³	1	4	yes	—
40	Dic₃₈₂	—	2	8	no	—
41	Dic₃₈₂	—	4	4	no	—
42	C₂×Dic₁₉₁	—	2	4	no	—
43	C₂×Dic₁₉₁	—	2	4	no	—
44	C₂×Dic₁₉₁	—	4	4	no	—
45	C₂×Dic₁₉₁	—	4	4	no	—
46	C₂³×C₁₉₁	—	2	4	no	—
47	C₂³×C₁₉₁	—	4	1	no	—
48	C₂³×C₁₉₁	C₂×D₄	1	1	yes	characteristic
49	C₂³×C₁₉₁	C₂×D₄	1	2	yes	—
50	C₂×C₄×C₁₉₁	—	2	2	no	—
51	C₂×C₄×C₁₉₁	—	2	4	no	—
52	C₂×C₄×C₁₉₁	—	4	2	no	—
53	C₂×C₄×C₁₉₁	—	4	1	no	—
54	C₂×C₄×C₁₉₁	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₁₉₁	C₂×D₄	1	2	yes	—
56	C₂×C₄×C₁₉₁	C₂⁴	1	1	yes	characteristic
57	C₈×C₁₉₁	—	2	4	no	—
58	C₂²×C₁₉₁	—	2	2	no	—
59	C₂²×C₁₉₁	—	2	2	no	—
60	C₂²×C₁₉₁	—	2	1	no	—
61	C₂²×C₁₉₁	—	2	2	no	—
62	C₂²×C₁₉₁	—	4	4	no	—
63	C₂²×C₁₉₁	#137	1	4	yes	—
64	C₂²×C₁₉₁	C₂²×D₄	1	1	yes	characteristic
65	C₄×C₁₉₁	—	2	1	no	—
66	C₄×C₁₉₁	—	2	2	no	—
67	C₄×C₁₉₁	H(5,𝔽₂)	1	2	yes	—
68	Dic₁₉₁	—	4	4	no	—
69	Dic₁₉₁	—	4	4	no	—
70	C₂×C₁₉₁	—	2	4	no	—
71	C₂×C₁₉₁	—	4	1	no	—
72	C₂×C₁₉₁	#534	1	1	yes	characteristic
73	C₂×C₁₉₁	C₂×#137	1	2	yes	—
74	C₁₉₁	#3195	1	1	yes	characteristic, complemented, Sylow
75	#3195	—	191	1	no	maximal, Sylow
76	C₂×C₂²⋊Q₈	—	191	2	no	—
77	#483	—	191	4	no	—
78	#470	—	191	4	no	—
79	#451	—	191	4	no	—
80	#407	D₁₉₁	1	1	yes	characteristic
81	C₂²×Q₈	—	191	2	no	—
82	C₂³×C₄	C₂×D₁₉₁	1	1	yes	characteristic
83	C₂×Q₁₆	—	191	4	no	—
84	C₂×M₁₆	C₂×D₁₉₁	1	2	yes	—
85	C₂²⋊Q₈	—	382	2	no	—
86	C₂²⋊Q₈	—	382	2	no	—
87	C₂²⋊Q₈	—	191	4	no	—
88	C₂²⋊Q₈	—	191	4	no	—
89	C₂×C₄⋊C₄	—	382	2	no	—
90	C₂×C₄⋊C₄	—	191	2	no	—
91	C₂×C₂²⋊C₄	—	382	2	no	—
92	#106	—	191	4	no	—
93	#103	—	191	8	no	—
94	C₂²⋊C₈	C₂×D₁₉₁	1	4	yes	—
95	C₂⁴	D₇₆₄	1	1	yes	characteristic
96	C₂×Q₈	—	382	2	no	—
97	C₂×Q₈	—	191	4	no	—
98	C₂×Q₈	—	764	4	no	—
99	C₂²×C₄	—	4	2	no	—
100	C₂²×C₄	—	382	2	no	—
101	C₂²×C₄	—	382	2	no	—
102	C₂²×C₄	D₇₆₄	1	2	yes	—
103	C₂²×C₄	D₇₆₄	1	1	yes	characteristic
104	C₂²×C₄	C₂²×D₁₉₁	1	2	yes	—
105	C₂²×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
106	Q₁₆	—	764	4	no	—
107	M₁₆	—	4	2	no	—
108	C₂×C₈	C₂²×D₁₉₁	1	4	yes	—
109	C₄⋊C₄	—	382	4	no	—
110	C₄⋊C₄	—	382	4	no	—
111	C₄⋊C₄	—	382	2	no	—
112	C₄⋊C₄	—	191	4	no	—
113	C₂²⋊C₄	—	382	4	no	—
114	C₂²⋊C₄	—	382	4	no	—
115	C₂³	—	2	4	no	—
116	C₂³	—	4	1	no	—
117	C₂³	C₂×D₇₆₄	1	1	yes	characteristic
118	C₂³	C₂×D₇₆₄	1	2	yes	—
119	Q₈	—	382	8	no	—
120	Q₈	—	764	4	no	—
121	C₂×C₄	—	2	2	no	—
122	C₂×C₄	—	2	4	no	—
123	C₂×C₄	—	4	1	no	—
124	C₂×C₄	—	4	2	no	—
125	C₂×C₄	—	382	4	no	—
126	C₂×C₄	—	382	4	no	—
127	C₂×C₄	—	764	4	no	—
128	C₂×C₄	—	764	4	no	—
129	C₂×C₄	C₂×D₇₆₄	1	1	yes	characteristic
130	C₂×C₄	C₂×D₇₆₄	1	2	yes	—
131	C₂×C₄	C₂³×D₁₉₁	1	1	yes	characteristic
132	C₈	—	2	4	no	—
133	C₂²	—	2	2	no	—
134	C₂²	—	2	2	no	—
135	C₂²	—	2	2	no	—
136	C₂²	—	2	1	no	—
137	C₂²	—	4	4	no	—
138	C₂²	#20259355	1	4	yes	—
139	C₂²	C₂²×D₇₆₄	1	1	yes	characteristic
140	C₄	—	2	2	no	—
141	C₄	—	2	1	no	—
142	C₄	—	764	4	no	—
143	C₄	—	764	4	no	—
144	C₄	#20259367	1	2	yes	—
145	C₂	—	2	4	no	—
146	C₂	—	4	1	no	—
147	C₂	#27133225	1	1	yes	characteristic
148	C₂	C₂×#20259355	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#20259301

C₂

yes

maximal

#27132994

C₂

yes

maximal, complemented

#27132921

C₂

yes

maximal, complemented

#27132821

C₂

yes

maximal, complemented

#407×C₁₉₁

C₂

yes

maximal, characteristic

C₂²×Dic₃₈₂

C₂²

yes

—

C₂×Dic₇₆₄

C₂²

yes

—

—

C₂²

yes

—

#20259301

C₂²

yes

—

C₂×C₄⋊Dic₁₉₁

C₂²

yes

—

C₂×Dic₁₉₁⋊C₄

—

C₂×C₂²⋊Dic₁₉₁

—

#20259254

C₂²

yes

complemented

#20259247

C₂²

yes

complemented

C₂³×C₄×C₁₉₁

C₂²

yes

characteristic

C₂×M₁₆×C₁₉₁

C₂²

yes

—

C₂²⋊C₈×C₁₉₁

C₂²

yes

—

C₂×Dic₃₈₂

—

C₂×Dic₃₈₂

—

C₂×Dic₃₈₂

C₂³

yes

—

C₂²×Dic₁₉₁

—

C₂²×Dic₁₉₁

—

Dic₇₆₄

—

C₄⋊Dic₁₉₁

—

C₄⋊Dic₁₉₁

C₂³

yes

—

Dic₁₉₁⋊C₄

—

Dic₁₉₁⋊C₄

—

C₂²⋊Dic₁₉₁

—

C₂²⋊Dic₁₉₁

—

C₂⁴×C₁₉₁

D₄

yes

characteristic

C₂²×C₄×C₁₉₁

—

C₂²×C₄×C₁₉₁

D₄

yes

—

C₂²×C₄×C₁₉₁

D₄

yes

characteristic

C₂²×C₄×C₁₉₁

C₂³

yes

—

C₂²×C₄×C₁₉₁

C₂³

yes

characteristic

M₁₆×C₁₉₁

—

C₂×C₈×C₁₉₁

C₂³

yes

—

Dic₃₈₂

—

Dic₃₈₂

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

—

C₂³×C₁₉₁

—

C₂³×C₁₉₁

—

C₂³×C₁₉₁

C₂×D₄

yes

characteristic

C₂³×C₁₉₁

C₂×D₄

yes

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₁

C₂×D₄

yes

—

C₂×C₄×C₁₉₁

C₂⁴

yes

characteristic

C₈×C₁₉₁

—

C₂²×C₁₉₁

—

C₂²×C₁₉₁

—

C₂²×C₁₉₁

—

C₂²×C₁₉₁

—

C₂²×C₁₉₁

—

C₂²×C₁₉₁

#137

yes

—

C₂²×C₁₉₁

C₂²×D₄

yes

characteristic

C₄×C₁₉₁

—

C₄×C₁₉₁

—

C₄×C₁₉₁

H(5,𝔽₂)

yes

—

Dic₁₉₁

—

Dic₁₉₁

—

C₂×C₁₉₁

—

C₂×C₁₉₁

—

C₂×C₁₉₁

#534

yes

characteristic

C₂×C₁₉₁

C₂×#137

yes

—

C₁₉₁

#3195

yes

characteristic, complemented, Sylow

#3195

—

191

maximal, Sylow

C₂×C₂²⋊Q₈

—

191

—

#483

—

191

—

#470

—

191

—

#451

—

191

—

#407

D₁₉₁

yes

characteristic

C₂²×Q₈

—

191

—

C₂³×C₄

C₂×D₁₉₁

yes

characteristic

C₂×Q₁₆

—

191

—

C₂×M₁₆

C₂×D₁₉₁

yes

—

C₂²⋊Q₈

—

382

—

C₂²⋊Q₈

—

382

—

C₂²⋊Q₈

—

191

—

C₂²⋊Q₈

—

191

—

C₂×C₄⋊C₄

—

382

—

C₂×C₄⋊C₄

—

191

—

C₂×C₂²⋊C₄

—

382

—

#106

—

191

—

#103

—

191

—

C₂²⋊C₈

C₂×D₁₉₁

yes

—

C₂⁴

D₇₆₄

yes

characteristic

C₂×Q₈

—

382

—

C₂×Q₈

—

191

—

C₂×Q₈

—

764

—

C₂²×C₄

—

100

C₂²×C₄

—

382

—

101

C₂²×C₄

—

382

—

102

C₂²×C₄

D₇₆₄

yes

—

103

C₂²×C₄

D₇₆₄

yes

characteristic

104

C₂²×C₄

C₂²×D₁₉₁

yes

—

105

C₂²×C₄

C₂²×D₁₉₁

yes

characteristic

106

Q₁₆

—

764

—

107

M₁₆

—

108

C₂×C₈

C₂²×D₁₉₁

yes

—

109

C₄⋊C₄

—

382

—

110

C₄⋊C₄

—

382

—

111

C₄⋊C₄

—

382

—

112

C₄⋊C₄

—

191

—

113

C₂²⋊C₄

—

382

—

114

C₂²⋊C₄

—

382

—

115

C₂³

—

116

C₂³

—

117

C₂³

C₂×D₇₆₄

yes

characteristic

118

C₂³

C₂×D₇₆₄

yes

—

119

Q₈

—

382

—

120

Q₈

—

764

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

382

—

126

C₂×C₄

—

382

—

127

C₂×C₄

—

764

—

128

C₂×C₄

—

764

—

129

C₂×C₄

C₂×D₇₆₄

yes

characteristic

130

C₂×C₄

C₂×D₇₆₄

yes

—

131

C₂×C₄

C₂³×D₁₉₁

yes

characteristic

132

C₈

—

133

C₂²

—

134

C₂²

—

135

C₂²

—

136

C₂²

—

137

C₂²

—

138

C₂²

#20259355

yes

—

139

C₂²

C₂²×D₇₆₄

yes

characteristic

140

C₄

—

141

C₄

—

142

C₄

—

764

—

143

C₄

—

764

—

144

C₄

#20259367

yes

—

145

C₂

—

146

C₂

—

147

C₂

#27133225

yes

characteristic

148

C₂

C₂×#20259355

yes

—

	subgroup	G/H
center	C₂²	C₂²×D₇₆₄
commutator, cosocle	C₂×C₄×C₁₉₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₉₁
Fitting	#407×C₁₉₁	C₂
socle	C₂²×C₁₉₁	C₂²×D₄
prime radical	C₂⁴×C₁₉₁	D₄
hypercenter	#407	D₁₉₁
hypocenter	C₁₉₁	#3195
LCS	G ▷ C₂×C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #407

subgroup

G/H

center

C₂²

C₂²×D₇₆₄

commutator, cosocle

C₂×C₄×C₁₉₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₉₁

Fitting

#407×C₁₉₁

C₂

socle

C₂²×C₁₉₁

C₂²×D₄

prime radical

C₂⁴×C₁₉₁

D₄

D₁₉₁

C₁₉₁

G ▷ C₂×C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁

UCS

C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #407

N	H ≅ G/N	type
2. #27132994	C₂	semidirect
3. #27132921	C₂	semidirect
4. #27132821	C₂	semidirect
7. C₂×Dic₇₆₄	C₂²	semidirect
15. #20259254	C₂²	semidirect
16. #20259247	C₂²	semidirect
74. C₁₉₁	#3195	semidirect

H ≅ G/N

type

2. #27132994

C₂

semidirect

3. #27132921

C₂

semidirect

4. #27132821

C₂

semidirect

7. C₂×Dic₇₆₄

C₂²

semidirect

15. #20259254

C₂²

semidirect

16. #20259247

C₂²

semidirect

74. C₁₉₁

#3195

semidirect

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

subgroup	index
C₂×#20259301	2
#27132994	2
#27132921	2
#27132821	2
#407×C₁₉₁	2
#3195	191

subgroup

index

#407×C₁₉₁

191

Subgroups — #37583344

Conjugacy classes of subgroups

Complements

Maximal subgroups