Group #37573352 — subgroups

Subgroups — #37573352

Order 24448.

320 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 93 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	#27132687	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#20259250	C₂	1	4	yes	maximal, complemented
4	#376×C₁₉₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#15690287	C₂²	1	2	yes	—
6	C₂×#15690271	C₂²	1	2	yes	—
7	#20259322	C₄	1	1	yes	characteristic, complemented
8	#20259319	C₄	1	1	yes	characteristic, complemented
9	#20259310	C₄	1	2	yes	complemented
10	#20259296	—	2	2	no	—
11	C₄×D₇₆₄	—	2	2	no	—
12	C₄⋊(C₁₉₁⋊C₈)	—	2	2	no	—
13	#20259250	—	2	8	no	—
14	C₂×C₄²×C₁₉₁	C₂²	1	2	yes	—
15	C₂×C₄²×C₁₉₁	C₂²	1	1	yes	characteristic
16	#95×C₁₉₁	—	2	2	no	—
17	#15690287	—	2	4	no	—
18	#15690287	—	2	4	no	—
19	C₂×Dic₃₈₂	C₂×C₄	1	2	yes	—
20	C₂×D₇₆₄	—	2	2	no	—
21	C₂×D₇₆₄	C₂×C₄	1	2	yes	—
22	C₂×C₂²⋊D₁₉₁	—	2	2	no	—
23	C₂×C₄×D₁₉₁	—	2	2	no	—
24	#15690271	—	2	4	no	—
25	C₂×C₁₉₁⋊C₈	—	2	2	no	—
26	C₄⋊Dic₁₉₁	—	2	2	no	—
27	(C₂×D₁₉₁)⋊C₄	—	2	4	no	—
28	C₂²×C₄×C₁₉₁	—	2	2	no	—
29	C₂²×C₄×C₁₉₁	D₄	1	2	yes	—
30	C₂²×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
31	C₄²×C₁₉₁	—	2	2	no	—
32	C₄²×C₁₉₁	C₂×C₄	1	1	yes	characteristic
33	C₄²×C₁₉₁	C₂×C₄	1	1	yes	characteristic
34	C₄²×C₁₉₁	D₄	1	2	yes	—
35	C₄²×C₁₉₁	D₄	1	4	yes	complemented
36	C₂²×D₁₉₁	—	2	2	no	—
37	Dic₃₈₂	—	2	4	no	—
38	D₇₆₄	—	2	4	no	—
39	D₇₆₄	—	4	4	no	—
40	C₂²⋊D₁₉₁	—	4	4	no	—
41	C₂×Dic₁₉₁	—	2	2	no	—
42	C₄×D₁₉₁	—	4	2	no	—
43	C₄×D₁₉₁	—	4	2	no	—
44	C₁₉₁⋊C₈	—	4	2	no	—
45	C₂³×C₁₉₁	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₁₉₁	—	2	4	no	—
47	C₂×C₄×C₁₉₁	—	4	2	no	—
48	C₂×C₄×C₁₉₁	—	4	4	no	—
49	C₂×C₄×C₁₉₁	C₂²⋊C₄	1	2	yes	—
50	C₂×C₄×C₁₉₁	C₂²⋊C₄	1	2	yes	—
51	C₂×C₄×C₁₉₁	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₄×C₁₉₁	C₂×D₄	1	2	yes	—
53	C₂×C₄×C₁₉₁	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₁₉₁	#41	1	2	yes	—
55	C₂²×C₁₉₁	—	2	2	no	—
56	C₂²×C₁₉₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
57	C₂²×C₁₉₁	#121	1	2	yes	—
58	C₂×D₁₉₁	—	4	2	no	—
59	C₂×D₁₉₁	—	4	4	no	—
60	C₄×C₁₉₁	—	2	2	no	—
61	C₄×C₁₉₁	—	2	2	no	—
62	C₄×C₁₉₁	—	4	4	no	—
63	C₄×C₁₉₁	C₄×D₄	1	2	yes	—
64	C₄×C₁₉₁	#124	1	1	yes	characteristic
65	C₄×C₁₉₁	#127	1	1	yes	characteristic
66	Dic₁₉₁	—	4	2	no	—
67	C₂×C₁₉₁	—	2	2	no	—
68	C₂×C₁₉₁	#390	1	1	yes	characteristic
69	C₂×C₁₉₁	#421	1	2	yes	—
70	D₁₉₁	—	8	2	no	—
71	C₁₉₁	#1963	1	1	yes	characteristic, complemented, Sylow
72	#1963	—	191	1	no	maximal, Sylow
73	#532	—	191	1	no	—
74	#423	—	191	1	no	—
75	C₂×#104	—	191	4	no	—
76	#376	D₁₉₁	1	1	yes	characteristic, complemented
77	C₂×#41	—	191	2	no	—
78	C₂×M₁₆	—	191	2	no	—
79	C₄⋊Q₈	—	191	1	no	—
80	#127	—	191	1	no	—
81	#124	—	191	2	no	—
82	#121	—	382	2	no	—
83	C₄×D₄	—	382	2	no	—
84	C₂×C₄²	C₂×D₁₉₁	1	2	yes	—
85	C₂×C₄²	C₂×D₁₉₁	1	1	yes	characteristic
86	C₄⋊C₈	—	382	2	no	—
87	#104	—	382	8	no	—
88	#95	—	2	2	no	—
89	#41	—	382	4	no	—
90	#41	—	382	4	no	—
91	C₂×Q₈	—	191	2	no	—
92	C₂×D₄	—	382	2	no	—
93	C₂×D₄	—	382	2	no	—
94	C₂×D₄	—	191	2	no	—
95	C₂²×C₄	—	2	2	no	—
96	C₂²×C₄	—	382	2	no	—
97	C₂²×C₄	C₂²⋊D₁₉₁	1	2	yes	—
98	C₂²×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
99	M₁₆	—	382	4	no	—
100	C₂×C₈	—	382	2	no	—
101	C₄⋊C₄	—	382	2	no	—
102	C₂²⋊C₄	—	382	4	no	—
103	C₄²	—	2	2	no	—
104	C₄²	C₄×D₁₉₁	1	1	yes	characteristic
105	C₄²	C₄×D₁₉₁	1	1	yes	characteristic
106	C₄²	D₇₆₄	1	2	yes	—
107	C₄²	D₇₆₄	1	4	yes	complemented
108	C₂³	—	382	2	no	—
109	C₂³	C₂×C₂²⋊D₁₉₁	1	1	yes	characteristic
110	Q₈	—	382	4	no	—
111	D₄	—	382	4	no	—
112	D₄	—	764	4	no	—
113	D₄	—	764	4	no	—
114	C₂×C₄	—	2	4	no	—
115	C₂×C₄	—	4	2	no	—
116	C₂×C₄	—	4	4	no	—
117	C₂×C₄	—	382	2	no	—
118	C₂×C₄	—	764	2	no	—
119	C₂×C₄	—	764	2	no	—
120	C₂×C₄	(C₂×D₁₉₁)⋊C₄	1	2	yes	—
121	C₂×C₄	(C₂×D₁₉₁)⋊C₄	1	2	yes	—
122	C₂×C₄	C₂×C₄×D₁₉₁	1	1	yes	characteristic
123	C₂×C₄	C₂×D₇₆₄	1	2	yes	—
124	C₂×C₄	C₂×D₇₆₄	1	1	yes	characteristic
125	C₂×C₄	#15690287	1	2	yes	—
126	C₈	—	764	2	no	—
127	C₂²	—	2	2	no	—
128	C₂²	—	764	4	no	—
129	C₂²	—	764	2	no	—
130	C₂²	C₂×(C₂×D₁₉₁)⋊C₄	1	1	yes	characteristic
131	C₂²	#20259296	1	2	yes	—
132	C₄	—	2	2	no	—
133	C₄	—	2	2	no	—
134	C₄	—	4	4	no	—
135	C₄	—	764	2	no	—
136	C₄	C₄×D₇₆₄	1	2	yes	—
137	C₄	#20259310	1	1	yes	characteristic
138	C₄	#20259319	1	1	yes	characteristic
139	C₂	—	2	2	no	—
140	C₂	—	1528	2	no	—
141	C₂	#27132565	1	1	yes	characteristic
142	C₂	#27132681	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#27133213

C₂

yes

maximal, characteristic

#27132687

C₂

yes

maximal, characteristic, complemented

C₂×#20259250

C₂

yes

maximal, complemented

#376×C₁₉₁

C₂

yes

maximal, characteristic, complemented

C₂×#15690287

C₂²

yes

—

C₂×#15690271

C₂²

yes

—

#20259322

C₄

yes

characteristic, complemented

#20259319

C₄

yes

characteristic, complemented

#20259310

C₄

yes

complemented

#20259296

—

C₄×D₇₆₄

—

C₄⋊(C₁₉₁⋊C₈)

—

#20259250

—

C₂×C₄²×C₁₉₁

C₂²

yes

—

C₂×C₄²×C₁₉₁

C₂²

yes

characteristic

#95×C₁₉₁

—

#15690287

—

#15690287

—

C₂×Dic₃₈₂

C₂×C₄

yes

—

C₂×D₇₆₄

—

C₂×D₇₆₄

C₂×C₄

yes

—

C₂×C₂²⋊D₁₉₁

—

C₂×C₄×D₁₉₁

—

#15690271

—

C₂×C₁₉₁⋊C₈

—

C₄⋊Dic₁₉₁

—

(C₂×D₁₉₁)⋊C₄

—

C₂²×C₄×C₁₉₁

—

C₂²×C₄×C₁₉₁

D₄

yes

—

C₂²×C₄×C₁₉₁

C₂³

yes

characteristic

C₄²×C₁₉₁

—

C₄²×C₁₉₁

C₂×C₄

yes

characteristic

C₄²×C₁₉₁

C₂×C₄

yes

characteristic

C₄²×C₁₉₁

D₄

yes

—

C₄²×C₁₉₁

D₄

yes

complemented

C₂²×D₁₉₁

—

Dic₃₈₂

—

D₇₆₄

—

D₇₆₄

—

C₂²⋊D₁₉₁

—

C₂×Dic₁₉₁

—

C₄×D₁₉₁

—

C₄×D₁₉₁

—

C₁₉₁⋊C₈

—

C₂³×C₁₉₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

C₂²⋊C₄

yes

—

C₂×C₄×C₁₉₁

C₂²⋊C₄

yes

—

C₂×C₄×C₁₉₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₉₁

C₂×D₄

yes

—

C₂×C₄×C₁₉₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₁

#41

yes

—

C₂²×C₁₉₁

—

C₂²×C₁₉₁

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₉₁

#121

yes

—

C₂×D₁₉₁

—

C₂×D₁₉₁

—

C₄×C₁₉₁

—

C₄×C₁₉₁

—

C₄×C₁₉₁

—

C₄×C₁₉₁

C₄×D₄

yes

—

C₄×C₁₉₁

yes

C₄×C₁₉₁

yes

Dic₁₉₁

—

C₂×C₁₉₁

—

C₂×C₁₉₁

#390

yes

characteristic

C₂×C₁₉₁

#421

yes

—

D₁₉₁

—

C₁₉₁

#1963

yes

characteristic, complemented, Sylow

#1963

—

191

maximal, Sylow

#532

—

191

—

#423

—

191

—

C₂×#104

—

191

—

#376

D₁₉₁

yes

characteristic, complemented

C₂×#41

—

191

—

C₂×M₁₆

—

191

—

C₄⋊Q₈

—

191

—

#127

—

191

—

#124

—

191

—

#121

—

382

—

C₄×D₄

—

382

—

C₂×C₄²

C₂×D₁₉₁

yes

—

C₂×C₄²

C₂×D₁₉₁

yes

characteristic

C₄⋊C₈

—

382

—

#104

—

382

—

#95

—

#41

—

382

—

#41

—

382

—

C₂×Q₈

—

191

—

C₂×D₄

—

382

—

C₂×D₄

—

382

—

C₂×D₄

—

191

—

C₂²×C₄

—

C₂²×C₄

—

382

—

C₂²×C₄

C₂²⋊D₁₉₁

yes

—

C₂²×C₄

C₂²×D₁₉₁

yes

characteristic

M₁₆

—

382

—

100

C₂×C₈

—

382

—

101

C₄⋊C₄

—

382

—

102

C₂²⋊C₄

—

382

—

103

C₄²

—

104

C₄²

C₄×D₁₉₁

yes

characteristic

105

C₄²

C₄×D₁₉₁

yes

characteristic

106

C₄²

D₇₆₄

yes

—

107

C₄²

D₇₆₄

yes

complemented

108

C₂³

—

382

—

109

C₂³

C₂×C₂²⋊D₁₉₁

yes

characteristic

110

Q₈

—

382

—

111

D₄

—

382

—

112

D₄

—

764

—

113

D₄

—

764

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

382

—

118

C₂×C₄

—

764

—

119

C₂×C₄

—

764

—

120

C₂×C₄

(C₂×D₁₉₁)⋊C₄

yes

—

121

C₂×C₄

(C₂×D₁₉₁)⋊C₄

yes

—

122

C₂×C₄

C₂×C₄×D₁₉₁

yes

characteristic

123

C₂×C₄

C₂×D₇₆₄

yes

—

124

C₂×C₄

C₂×D₇₆₄

yes

characteristic

125

C₂×C₄

#15690287

yes

—

126

C₈

—

764

—

127

C₂²

—

128

C₂²

—

764

—

129

C₂²

—

764

—

130

C₂²

C₂×(C₂×D₁₉₁)⋊C₄

yes

characteristic

131

C₂²

#20259296

yes

—

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

—

764

—

136

C₄

C₄×D₇₆₄

yes

—

137

C₄

#20259310

yes

characteristic

138

C₄

#20259319

yes

characteristic

139

C₂

—

140

C₂

—

1528

—

141

C₂

#27132565

yes

characteristic

142

C₂

#27132681

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×D₇₆₄
commutator	C₂×C₄×C₁₉₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₉₁
Fitting	#376×C₁₉₁	C₂
socle	C₂²×C₁₉₁	C₂×C₂²⋊C₄
prime radical	#27133213	C₂
cosocle	C₂²×C₄×C₁₉₁	C₂³
hypercenter	#376	D₁₉₁
hypocenter	C₁₉₁	#1963
LCS	G ▷ C₂×C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #376

subgroup

G/H

center

C₂×C₄

C₂×D₇₆₄

commutator

C₂×C₄×C₁₉₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₉₁

Fitting

#376×C₁₉₁

C₂

socle

C₂²×C₁₉₁

C₂×C₂²⋊C₄

prime radical

#27133213

C₂

cosocle

C₂²×C₄×C₁₉₁

C₂³

D₁₉₁

C₁₉₁

G ▷ C₂×C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #376

N	H ≅ G/N	type
2. #27132687	C₂	semidirect
3. C₂×#20259250	C₂	semidirect
4. #376×C₁₉₁	C₂	semidirect
7. #20259322	C₄	semidirect
8. #20259319	C₄	semidirect
9. #20259310	C₄	semidirect
35. C₄²×C₁₉₁	D₄	semidirect
71. C₁₉₁	#1963	semidirect
76. #376	D₁₉₁	semidirect
107. C₄²	D₇₆₄	semidirect

H ≅ G/N

type

2. #27132687

C₂

semidirect

3. C₂×#20259250

C₂

semidirect

4. #376×C₁₉₁

C₂

semidirect

7. #20259322

C₄

semidirect

8. #20259319

C₄

semidirect

9. #20259310

C₄

semidirect

35. C₄²×C₁₉₁

D₄

semidirect

71. C₁₉₁

#1963

semidirect

76. #376

D₁₉₁

semidirect

107. C₄²

D₇₆₄

semidirect

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

subgroup	index
#27133213	2
#27132687	2
C₂×#20259250	2
#376×C₁₉₁	2
#1963	191

subgroup

index

#376×C₁₉₁

191

Subgroups — #37573352

Conjugacy classes of subgroups

Complements

Maximal subgroups