Group #37572647 — subgroups

Subgroups — #37572647

Order 24448.

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

normal subgroups only

This group has 248 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	C₂×C₄×C₁₉₁⋊C₈	C₂	1	1	yes	maximal, characteristic
2	#27132533	C₂	1	1	yes	maximal, characteristic
3	#27132425	C₂	1	4	yes	maximal
4	#384×C₁₉₁	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₁₉₁⋊C₈	C₂²	1	2	yes	—
6	C₂×C₄⋊Dic₁₉₁	—	2	1	no	—
7	C₂×C₄⋊Dic₁₉₁	—	2	1	no	—
8	C₂×C₄⋊Dic₁₉₁	C₂²	1	2	yes	—
9	C₄×C₁₉₁⋊C₈	C₄	1	4	yes	complemented
10	C₂×C₄⋊C₄×C₁₉₁	—	2	1	no	—
11	C₂×C₄⋊C₄×C₁₉₁	—	2	1	no	—
12	C₂×C₄⋊C₄×C₁₉₁	C₂²	1	2	yes	—
13	C₂×C₄²×C₁₉₁	C₂²	1	1	yes	characteristic
14	C₂²×Dic₁₉₁	—	2	2	no	—
15	C₂×C₁₉₁⋊C₈	—	2	2	no	—
16	C₂×C₁₉₁⋊C₈	—	2	2	no	—
17	C₂×C₁₉₁⋊C₈	C₂×C₄	1	4	yes	—
18	C₄⋊Dic₁₉₁	—	2	4	no	—
19	C₄⋊Dic₁₉₁	—	4	2	no	—
20	C₄⋊Dic₁₉₁	—	4	2	no	—
21	C₂²×C₄×C₁₉₁	—	2	2	no	—
22	C₂²×C₄×C₁₉₁	D₄	1	1	yes	characteristic
23	C₂²×C₄×C₁₉₁	D₄	1	1	yes	characteristic
24	C₂²×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
25	C₄⋊C₄×C₁₉₁	—	2	4	no	—
26	C₄⋊C₄×C₁₉₁	—	4	2	no	—
27	C₄⋊C₄×C₁₉₁	—	4	2	no	—
28	C₄²×C₁₉₁	C₂×C₄	1	2	yes	—
29	C₄²×C₁₉₁	Q₈	1	2	yes	—
30	C₂×Dic₁₉₁	—	4	2	no	—
31	C₂×Dic₁₉₁	—	4	2	no	—
32	C₂×Dic₁₉₁	—	4	2	no	—
33	C₁₉₁⋊C₈	—	2	4	no	—
34	C₂³×C₁₉₁	C₂×D₄	1	1	yes	characteristic
35	C₂×C₄×C₁₉₁	—	4	2	no	—
36	C₂×C₄×C₁₉₁	—	4	2	no	—
37	C₂×C₄×C₁₉₁	—	4	2	no	—
38	C₂×C₄×C₁₉₁	C₄⋊C₄	1	2	yes	—
39	C₂×C₄×C₁₉₁	C₄⋊C₄	1	2	yes	—
40	C₂×C₄×C₁₉₁	D₈	1	2	yes	—
41	C₂×C₄×C₁₉₁	SD₁₆	1	2	yes	—
42	C₂×C₄×C₁₉₁	SD₁₆	1	2	yes	—
43	C₂×C₄×C₁₉₁	Q₁₆	1	2	yes	—
44	C₂×C₄×C₁₉₁	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₁₉₁	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₄×C₁₉₁	#41	1	2	yes	—
47	C₂×C₄×C₁₉₁	#41	1	2	yes	—
48	C₂²×C₁₉₁	C₂×C₄⋊C₄	1	1	yes	characteristic
49	C₂²×C₁₉₁	#124	1	1	yes	characteristic
50	C₂²×C₁₉₁	#124	1	1	yes	characteristic
51	C₂²×C₁₉₁	C₂×D₈	1	1	yes	characteristic
52	C₂²×C₁₉₁	C₂×SD₁₆	1	1	yes	characteristic
53	C₂²×C₁₉₁	C₂×SD₁₆	1	1	yes	characteristic
54	C₂²×C₁₉₁	C₂×Q₁₆	1	1	yes	characteristic
55	C₄×C₁₉₁	—	8	2	no	—
56	C₄×C₁₉₁	#106	1	4	yes	—
57	C₄×C₁₉₁	#107	1	4	yes	—
58	C₄×C₁₉₁	#117	1	2	yes	—
59	C₄×C₁₉₁	#125	1	2	yes	—
60	Dic₁₉₁	—	8	2	no	—
61	C₂×C₁₉₁	#382	1	1	yes	characteristic
62	C₂×C₁₉₁	C₂×#106	1	1	yes	characteristic
63	C₂×C₁₉₁	C₂×#107	1	1	yes	characteristic
64	C₂×C₁₉₁	#486	1	1	yes	characteristic
65	C₂×C₁₉₁	#486	1	1	yes	characteristic
66	C₂×C₁₉₁	#487	1	1	yes	characteristic
67	C₂×C₁₉₁	#487	1	1	yes	characteristic
68	C₁₉₁	#1834	1	1	yes	characteristic, complemented, Sylow
69	#1834	—	191	1	no	maximal, Sylow
70	C₂×C₄×C₈	—	191	1	no	—
71	#384	—	191	1	no	—
72	#384	D₁₉₁	1	1	yes	characteristic
73	#340	—	191	4	no	—
74	C₂²×C₈	—	191	2	no	—
75	C₂×C₄⋊C₄	—	2	1	no	—
76	C₂×C₄⋊C₄	—	2	1	no	—
77	C₂×C₄⋊C₄	—	382	1	no	—
78	C₂×C₄⋊C₄	—	382	1	no	—
79	C₂×C₄⋊C₄	—	191	2	no	—
80	C₂×C₄⋊C₄	C₂×D₁₉₁	1	2	yes	—
81	C₂×C₄²	C₂×D₁₉₁	1	1	yes	characteristic
82	C₄×C₈	—	191	4	no	—
83	C₂²×C₄	—	2	2	no	—
84	C₂²×C₄	—	382	2	no	—
85	C₂²×C₄	C₂²⋊D₁₉₁	1	1	yes	characteristic
86	C₂²×C₄	C₂²⋊D₁₉₁	1	1	yes	characteristic
87	C₂²×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
88	C₂×C₈	—	382	2	no	—
89	C₂×C₈	—	382	2	no	—
90	C₂×C₈	—	191	4	no	—
91	C₄⋊C₄	—	2	4	no	—
92	C₄⋊C₄	—	4	2	no	—
93	C₄⋊C₄	—	4	2	no	—
94	C₄⋊C₄	—	382	4	no	—
95	C₄⋊C₄	—	764	2	no	—
96	C₄⋊C₄	—	764	2	no	—
97	C₄²	C₄×D₁₉₁	1	2	yes	—
98	C₄²	Dic₃₈₂	1	2	yes	—
99	C₂³	C₂×C₂²⋊D₁₉₁	1	1	yes	characteristic
100	C₂×C₄	—	4	2	no	—
101	C₂×C₄	—	4	2	no	—
102	C₂×C₄	—	4	2	no	—
103	C₂×C₄	—	764	2	no	—
104	C₂×C₄	—	764	2	no	—
105	C₂×C₄	—	764	2	no	—
106	C₂×C₄	Dic₁₉₁⋊C₄	1	2	yes	—
107	C₂×C₄	Dic₁₉₁⋊C₄	1	2	yes	—
108	C₂×C₄	#15690272	1	2	yes	—
109	C₂×C₄	#15690274	1	2	yes	—
110	C₂×C₄	#15690275	1	2	yes	—
111	C₂×C₄	#15690277	1	2	yes	—
112	C₂×C₄	C₂×C₄×D₁₉₁	1	1	yes	characteristic
113	C₂×C₄	C₂×Dic₃₈₂	1	1	yes	characteristic
114	C₂×C₄	#15690286	1	2	yes	—
115	C₂×C₄	#15690288	1	2	yes	—
116	C₈	—	382	4	no	—
117	C₂²	C₂×Dic₁₉₁⋊C₄	1	1	yes	characteristic
118	C₂²	#20259309	1	1	yes	characteristic
119	C₂²	#20259311	1	1	yes	characteristic
120	C₂²	C₂×#15690272	1	1	yes	characteristic
121	C₂²	C₂×#15690274	1	1	yes	characteristic
122	C₂²	C₂×#15690275	1	1	yes	characteristic
123	C₂²	C₂×#15690277	1	1	yes	characteristic
124	C₄	—	8	2	no	—
125	C₄	—	1528	2	no	—
126	C₄	#20259253	1	4	yes	—
127	C₄	#20259255	1	4	yes	—
128	C₄	#20259280	1	2	yes	—
129	C₄	#20259313	1	2	yes	—
130	C₂	#27132528	1	1	yes	characteristic
131	C₂	C₂×#20259253	1	1	yes	characteristic
132	C₂	C₂×#20259255	1	1	yes	characteristic
133	C₂	#27133009	1	1	yes	characteristic
134	C₂	#27133011	1	1	yes	characteristic
135	C₂	#27133013	1	1	yes	characteristic
136	C₂	#27133015	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×C₁₉₁⋊C₈

C₂

yes

maximal, characteristic

#27132533

C₂

yes

maximal, characteristic

#27132425

C₂

yes

maximal

#384×C₁₉₁

C₂

yes

maximal, characteristic

C₂²×C₁₉₁⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₁₉₁

—

C₂×C₄⋊Dic₁₉₁

—

C₂×C₄⋊Dic₁₉₁

C₂²

yes

—

C₄×C₁₉₁⋊C₈

C₄

yes

complemented

C₂×C₄⋊C₄×C₁₉₁

—

C₂×C₄⋊C₄×C₁₉₁

—

C₂×C₄⋊C₄×C₁₉₁

C₂²

yes

—

C₂×C₄²×C₁₉₁

C₂²

yes

characteristic

C₂²×Dic₁₉₁

—

C₂×C₁₉₁⋊C₈

—

C₂×C₁₉₁⋊C₈

—

C₂×C₁₉₁⋊C₈

C₂×C₄

yes

—

C₄⋊Dic₁₉₁

—

C₄⋊Dic₁₉₁

—

C₄⋊Dic₁₉₁

—

C₂²×C₄×C₁₉₁

—

C₂²×C₄×C₁₉₁

D₄

yes

characteristic

C₂²×C₄×C₁₉₁

D₄

yes

characteristic

C₂²×C₄×C₁₉₁

C₂³

yes

characteristic

C₄⋊C₄×C₁₉₁

—

C₄⋊C₄×C₁₉₁

—

C₄⋊C₄×C₁₉₁

—

C₄²×C₁₉₁

C₂×C₄

yes

—

C₄²×C₁₉₁

Q₈

yes

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

—

C₁₉₁⋊C₈

—

C₂³×C₁₉₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

C₄⋊C₄

yes

—

C₂×C₄×C₁₉₁

C₄⋊C₄

yes

—

C₂×C₄×C₁₉₁

D₈

yes

—

C₂×C₄×C₁₉₁

SD₁₆

yes

—

C₂×C₄×C₁₉₁

SD₁₆

yes

—

C₂×C₄×C₁₉₁

Q₁₆

yes

—

C₂×C₄×C₁₉₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₉₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₉₁

#41

yes

—

C₂×C₄×C₁₉₁

#41

yes

—

C₂²×C₁₉₁

C₂×C₄⋊C₄

yes

C₂²×C₁₉₁

yes

C₂²×C₁₉₁

yes

C₂²×C₁₉₁

C₂×D₈

yes

characteristic

C₂²×C₁₉₁

C₂×SD₁₆

yes

characteristic

C₂²×C₁₉₁

C₂×SD₁₆

yes

characteristic

C₂²×C₁₉₁

C₂×Q₁₆

yes

characteristic

C₄×C₁₉₁

—

C₄×C₁₉₁

#106

yes

—

C₄×C₁₉₁

#107

yes

—

C₄×C₁₉₁

#117

yes

—

C₄×C₁₉₁

#125

yes

—

Dic₁₉₁

—

C₂×C₁₉₁

yes

C₂×C₁₉₁

yes

C₂×C₁₉₁

yes

C₂×C₁₉₁

yes

C₂×C₁₉₁

yes

C₂×C₁₉₁

yes

C₂×C₁₉₁

yes

C₁₉₁

yes

characteristic, complemented, Sylow

#1834

—

191

maximal, Sylow

C₂×C₄×C₈

—

191

—

#384

—

191

—

#384

D₁₉₁

yes

characteristic

#340

—

191

—

C₂²×C₈

—

191

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

382

—

C₂×C₄⋊C₄

—

382

—

C₂×C₄⋊C₄

—

191

—

C₂×C₄⋊C₄

C₂×D₁₉₁

yes

—

C₂×C₄²

C₂×D₁₉₁

yes

characteristic

C₄×C₈

—

191

—

C₂²×C₄

—

C₂²×C₄

—

382

—

C₂²×C₄

C₂²⋊D₁₉₁

yes

characteristic

C₂²×C₄

C₂²⋊D₁₉₁

yes

characteristic

C₂²×C₄

C₂²×D₁₉₁

yes

characteristic

C₂×C₈

—

382

—

C₂×C₈

—

382

—

C₂×C₈

—

191

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

382

—

C₄⋊C₄

—

764

—

C₄⋊C₄

—

764

—

C₄²

C₄×D₁₉₁

yes

—

C₄²

Dic₃₈₂

yes

—

C₂³

C₂×C₂²⋊D₁₉₁

yes

characteristic

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

764

—

104

C₂×C₄

—

764

—

105

C₂×C₄

—

764

—

106

C₂×C₄

Dic₁₉₁⋊C₄

yes

—

107

C₂×C₄

Dic₁₉₁⋊C₄

yes

—

108

C₂×C₄

#15690272

yes

—

109

C₂×C₄

#15690274

yes

—

110

C₂×C₄

#15690275

yes

—

111

C₂×C₄

#15690277

yes

—

112

C₂×C₄

C₂×C₄×D₁₉₁

yes

characteristic

113

C₂×C₄

C₂×Dic₃₈₂

yes

characteristic

114

C₂×C₄

#15690286

yes

—

115

C₂×C₄

#15690288

yes

—

116

C₈

—

382

—

117

C₂²

C₂×Dic₁₉₁⋊C₄

yes

characteristic

118

C₂²

#20259309

yes

characteristic

119

C₂²

#20259311

yes

characteristic

120

C₂²

C₂×#15690272

yes

characteristic

121

C₂²

C₂×#15690274

yes

characteristic

122

C₂²

C₂×#15690275

yes

characteristic

123

C₂²

C₂×#15690277

yes

characteristic

124

C₄

—

125

C₄

—

1528

—

126

C₄

#20259253

yes

—

127

C₄

#20259255

yes

—

128

C₄

#20259280

yes

—

129

C₄

#20259313

yes

—

130

C₂

#27132528

yes

characteristic

131

C₂

C₂×#20259253

yes

characteristic

132

C₂

C₂×#20259255

yes

characteristic

133

C₂

#27133009

yes

characteristic

134

C₂

#27133011

yes

characteristic

135

C₂

#27133013

yes

characteristic

136

C₂

#27133015

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₉₁
commutator	C₂×C₄×C₁₉₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₉₁
Fitting	#384×C₁₉₁	C₂
socle, prime radical	C₂³×C₁₉₁	C₂×D₄
cosocle	C₂²×C₄×C₁₉₁	C₂³
hypercenter	#384	D₁₉₁
hypocenter	C₁₉₁	#1834
LCS	G ▷ C₂×C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₉₁

commutator

C₂×C₄×C₁₉₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₉₁

Fitting

#384×C₁₉₁

C₂

socle, prime radical

C₂³×C₁₉₁

C₂×D₄

cosocle

C₂²×C₄×C₁₉₁

C₂³

D₁₉₁

C₁₉₁

G ▷ C₂×C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

N	H ≅ G/N	type
9. C₄×C₁₉₁⋊C₈	C₄	semidirect
68. C₁₉₁	#1834	semidirect

H ≅ G/N

type

9. C₄×C₁₉₁⋊C₈

C₄

semidirect

68. C₁₉₁

#1834

semidirect

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

subgroup	index
C₂×C₄×C₁₉₁⋊C₈	2
#27132533	2
#27132425	2
#384×C₁₉₁	2
#1834	191

subgroup

index

C₂×C₄×C₁₉₁⋊C₈

#27132533

#27132425

#384×C₁₉₁

#1834

191

Subgroups — #37572647

Conjugacy classes of subgroups

Complements

Maximal subgroups