Group #37573681 — subgroups

Subgroups — #37573681

Order 24448.

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

normal subgroups only

This group has 246 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
2	#27132672	C₂	1	2	yes	maximal
3	#27132432	C₂	1	1	yes	maximal, characteristic
4	#27132423	C₂	1	2	yes	maximal
5	#391×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	#20259246	—	2	2	no	—
12	#20259246	—	2	2	no	—
13	C₂²×Q₈×C₁₉₁	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₄×C₁₉₁	C₂²	1	1	yes	characteristic
15	C₂×C₄²×C₁₉₁	C₂²	1	1	yes	characteristic
16	#95×C₁₉₁	—	2	2	no	—
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	C₂×Q₈×C₁₉₁	—	2	1	no	—
28	C₂×Q₈×C₁₉₁	—	2	2	no	—
29	C₂×Q₈×C₁₉₁	—	2	2	no	—
30	C₂×Q₈×C₁₉₁	Q₈	1	2	yes	complemented
31	C₂²×C₄×C₁₉₁	—	2	1	no	—
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	C₄²×C₁₉₁	—	2	2	no	—
37	Dic₃₈₂	—	8	2	no	—
38	C₂×Dic₁₉₁	—	2	2	no	—
39	C₂×Dic₁₉₁	—	2	2	no	—
40	C₁₉₁⋊C₈	—	4	2	no	—
41	C₂³×C₁₉₁	C₂×D₄	1	1	yes	characteristic
42	Q₈×C₁₉₁	—	2	4	no	—
43	Q₈×C₁₉₁	—	4	2	no	—
44	C₂×C₄×C₁₉₁	—	2	2	no	—
45	C₂×C₄×C₁₉₁	—	2	2	no	—
46	C₂×C₄×C₁₉₁	—	2	2	no	—
47	C₂×C₄×C₁₉₁	—	4	1	no	—
48	C₂×C₄×C₁₉₁	—	4	1	no	—
49	C₂×C₄×C₁₉₁	—	4	1	no	—
50	C₂×C₄×C₁₉₁	—	4	1	no	—
51	C₂×C₄×C₁₉₁	—	4	2	no	—
52	C₂×C₄×C₁₉₁	C₂×D₄	1	2	yes	—
53	C₂×C₄×C₁₉₁	C₂×Q₈	1	1	yes	characteristic
54	C₂×C₄×C₁₉₁	#41	1	2	yes	—
55	C₂×C₄×C₁₉₁	#41	1	1	yes	characteristic
56	C₂²×C₁₉₁	—	2	1	no	—
57	C₂²×C₁₉₁	—	2	1	no	—
58	C₂²×C₁₉₁	#120	1	1	yes	characteristic
59	C₂²×C₁₉₁	C₂²⋊Q₈	1	1	yes	characteristic
60	C₂²×C₁₉₁	#124	1	1	yes	characteristic
61	C₄×C₁₉₁	—	4	2	no	—
62	C₄×C₁₉₁	—	4	2	no	—
63	C₄×C₁₉₁	—	4	2	no	—
64	C₄×C₁₉₁	C₂²⋊Q₈	1	2	yes	—
65	C₄×C₁₉₁	#124	1	2	yes	—
66	Dic₁₉₁	—	4	2	no	—
67	Dic₁₉₁	—	4	2	no	—
68	C₂×C₁₉₁	—	2	1	no	—
69	C₂×C₁₉₁	—	2	1	no	—
70	C₂×C₁₉₁	#393	1	1	yes	characteristic
71	C₂×C₁₉₁	#455	1	1	yes	characteristic
72	C₂×C₁₉₁	#456	1	1	yes	characteristic
73	C₁₉₁	#2028	1	1	yes	characteristic, complemented, Sylow
74	#2028	—	191	1	no	maximal, Sylow
75	#544	—	191	1	no	—
76	#419	—	191	2	no	—
77	#391	D₁₉₁	1	1	yes	characteristic
78	#343	—	191	1	no	—
79	#339	—	191	2	no	—
80	C₂²×Q₈	C₂×D₁₉₁	1	1	yes	characteristic
81	C₂×M₁₆	—	191	2	no	—
82	C₄⋊Q₈	—	382	2	no	—
83	#125	—	382	2	no	—
84	C₂²⋊Q₈	—	382	2	no	—
85	#117	—	191	2	no	—
86	C₂×C₄⋊C₄	C₂×D₁₉₁	1	1	yes	characteristic
87	C₂×C₄²	C₂×D₁₉₁	1	1	yes	characteristic
88	#103	—	382	2	no	—
89	#103	—	382	2	no	—
90	#95	—	2	2	no	—
91	C₂×Q₈	—	2	1	no	—
92	C₂×Q₈	—	2	2	no	—
93	C₂×Q₈	—	2	2	no	—
94	C₂×Q₈	—	382	2	no	—
95	C₂×Q₈	Dic₃₈₂	1	2	yes	complemented
96	C₂²×C₄	—	2	1	no	—
97	C₂²×C₄	—	2	1	no	—
98	C₂²×C₄	C₂²⋊D₁₉₁	1	2	yes	—
99	C₂²×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
100	M₁₆	—	382	2	no	—
101	M₁₆	—	382	2	no	—
102	C₂×C₈	—	382	2	no	—
103	C₄⋊C₄	—	2	2	no	—
104	C₄⋊C₄	—	382	4	no	—
105	C₄⋊C₄	—	382	2	no	—
106	C₄⋊C₄	—	191	2	no	—
107	C₄⋊C₄	—	191	2	no	—
108	C₂²⋊C₄	—	382	2	no	—
109	C₄²	—	2	2	no	—
110	C₄²	—	382	2	no	—
111	C₂³	C₂×C₂²⋊D₁₉₁	1	1	yes	characteristic
112	Q₈	—	2	4	no	—
113	Q₈	—	4	2	no	—
114	Q₈	—	1528	2	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	4	1	no	—
119	C₂×C₄	—	4	1	no	—
120	C₂×C₄	—	4	1	no	—
121	C₂×C₄	—	4	1	no	—
122	C₂×C₄	—	4	2	no	—
123	C₂×C₄	—	382	2	no	—
124	C₂×C₄	—	382	2	no	—
125	C₂×C₄	D₄×D₁₉₁	1	2	yes	—
126	C₂×C₄	C₂×Dic₃₈₂	1	1	yes	characteristic
127	C₂×C₄	#15690286	1	2	yes	—
128	C₂×C₄	#15690287	1	1	yes	characteristic
129	C₈	—	764	2	no	—
130	C₂²	—	2	1	no	—
131	C₂²	—	2	1	no	—
132	C₂²	#20259290	1	1	yes	characteristic
133	C₂²	#20259301	1	1	yes	characteristic
134	C₂²	#20259309	1	1	yes	characteristic
135	C₄	—	4	2	no	—
136	C₄	—	4	2	no	—
137	C₄	—	4	2	no	—
138	C₄	—	764	2	no	—
139	C₄	—	764	2	no	—
140	C₄	#20259299	1	2	yes	—
141	C₄	#20259308	1	2	yes	—
142	C₂	—	2	1	no	—
143	C₂	—	2	1	no	—
144	C₂	#27132575	1	1	yes	characteristic
145	C₂	#27132840	1	1	yes	characteristic
146	C₂	#27132845	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#27133294

C₂

yes

maximal, characteristic

#27132672

C₂

yes

maximal

#27132432

C₂

yes

maximal, characteristic

#27132423

C₂

yes

maximal

#391×C₁₉₁

C₂

yes

maximal, characteristic

C₂×#15690271

C₂²

yes

—

—

—

—

C₂²

yes

—

#20259246

—

#20259246

—

C₂²×Q₈×C₁₉₁

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₁₉₁

C₂²

yes

characteristic

C₂×C₄²×C₁₉₁

C₂²

yes

characteristic

#95×C₁₉₁

—

C₂×Dic₃₈₂

—

#15690271

—

#15690271

—

C₂×C₁₉₁⋊C₈

—

C₄⋊Dic₁₉₁

—

C₄⋊Dic₁₉₁

D₄

yes

—

C₄⋊Dic₁₉₁

D₄

yes

—

Dic₁₉₁⋊C₄

—

C₂²⋊Dic₁₉₁

—

C₄×Dic₁₉₁

—

C₂×Q₈×C₁₉₁

—

C₂×Q₈×C₁₉₁

—

C₂×Q₈×C₁₉₁

—

C₂×Q₈×C₁₉₁

Q₈

yes

complemented

C₂²×C₄×C₁₉₁

—

C₂²×C₄×C₁₉₁

—

C₂²×C₄×C₁₉₁

D₄

yes

—

C₂²×C₄×C₁₉₁

C₂³

yes

characteristic

C₄⋊C₄×C₁₉₁

—

C₄²×C₁₉₁

—

Dic₃₈₂

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

—

C₁₉₁⋊C₈

—

C₂³×C₁₉₁

C₂×D₄

yes

characteristic

Q₈×C₁₉₁

—

Q₈×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×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

#120

yes

characteristic

C₂²×C₁₉₁

C₂²⋊Q₈

yes

characteristic

C₂²×C₁₉₁

#124

yes

characteristic

C₄×C₁₉₁

—

C₄×C₁₉₁

—

C₄×C₁₉₁

—

C₄×C₁₉₁

C₂²⋊Q₈

yes

—

C₄×C₁₉₁

#124

yes

—

Dic₁₉₁

—

Dic₁₉₁

—

C₂×C₁₉₁

—

C₂×C₁₉₁

—

C₂×C₁₉₁

yes

C₂×C₁₉₁

yes

C₂×C₁₉₁

yes

C₁₉₁

yes

characteristic, complemented, Sylow

#2028

—

191

maximal, Sylow

#544

—

191

—

#419

—

191

—

#391

D₁₉₁

yes

characteristic

#343

—

191

—

#339

—

191

—

C₂²×Q₈

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×C₄²

C₂×D₁₉₁

yes

characteristic

#103

—

382

—

#103

—

382

—

#95

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

382

—

C₂×Q₈

Dic₃₈₂

yes

complemented

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₁₉₁

yes

—

C₂²×C₄

C₂²×D₁₉₁

yes

characteristic

100

M₁₆

—

382

—

101

M₁₆

—

382

—

102

C₂×C₈

—

382

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

382

—

105

C₄⋊C₄

—

382

—

106

C₄⋊C₄

—

191

—

107

C₄⋊C₄

—

191

—

108

C₂²⋊C₄

—

382

—

109

C₄²

—

110

C₄²

—

382

—

111

C₂³

C₂×C₂²⋊D₁₉₁

yes

characteristic

112

Q₈

—

113

Q₈

—

114

Q₈

—

1528

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

382

—

124

C₂×C₄

—

382

—

125

C₂×C₄

D₄×D₁₉₁

yes

—

126

C₂×C₄

C₂×Dic₃₈₂

yes

characteristic

127

C₂×C₄

#15690286

yes

—

128

C₂×C₄

#15690287

yes

characteristic

129

C₈

—

764

—

130

C₂²

—

131

C₂²

—

132

C₂²

#20259290

yes

characteristic

133

C₂²

#20259301

yes

characteristic

134

C₂²

#20259309

yes

characteristic

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

—

764

—

139

C₄

—

764

—

140

C₄

#20259299

yes

—

141

C₄

#20259308

yes

—

142

C₂

—

143

C₂

—

144

C₂

#27132575

yes

characteristic

145

C₂

#27132840

yes

characteristic

146

C₂

#27132845

yes

characteristic

	subgroup	G/H
center	C₂²	#20259290
commutator, cosocle	C₂²×C₄×C₁₉₁	C₂³
Frattini	C₂²×C₄	C₂²×D₁₉₁
Fitting	#391×C₁₉₁	C₂
socle	C₂²×C₁₉₁	#120
prime radical	C₂³×C₁₉₁	C₂×D₄
hypercenter	#391	D₁₉₁
hypocenter	C₁₉₁	#2028
LCS	G ▷ C₂²×C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #391

subgroup

G/H

center

C₂²

#20259290

commutator, cosocle

C₂²×C₄×C₁₉₁

C₂³

Frattini

C₂²×C₄

C₂²×D₁₉₁

Fitting

#391×C₁₉₁

C₂

socle

C₂²×C₁₉₁

#120

prime radical

C₂³×C₁₉₁

C₂×D₄

D₁₉₁

C₁₉₁

G ▷ C₂²×C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #391

N	H ≅ G/N	type
30. C₂×Q₈×C₁₉₁	Q₈	semidirect
73. C₁₉₁	#2028	semidirect
95. C₂×Q₈	Dic₃₈₂	semidirect

H ≅ G/N

type

30. C₂×Q₈×C₁₉₁

Q₈

semidirect

73. C₁₉₁

#2028

semidirect

95. C₂×Q₈

Dic₃₈₂

semidirect

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

subgroup	index
#27133294	2
#27132672	2
#27132432	2
#27132423	2
#391×C₁₉₁	2
#2028	191

subgroup

index

#391×C₁₉₁

191

Subgroups — #37573681

Conjugacy classes of subgroups

Complements

Maximal subgroups