Group #17995427 — subgroups

Subgroups — #17995427

Order 4736.

320 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 89 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	#14558083	C₂	1	1	yes	maximal, characteristic
2	C₂×#11924462	C₂	1	4	yes	maximal, complemented
3	#14557379	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₇×#423	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#10596962	C₂²	1	2	yes	—
6	#11924536	C₄	1	1	yes	characteristic, complemented
7	#11924533	C₄	1	1	yes	characteristic, complemented
8	#11924524	C₄	1	2	yes	complemented
9	#11924510	—	2	2	no	—
10	C₄×D₁₄₈	—	2	2	no	—
11	C₂×C₄×Dic₃₇	C₂²	1	2	yes	—
12	#11924462	—	2	8	no	—
13	#11924443	—	2	2	no	—
14	C₂×M₁₆×C₃₇	C₂²	1	2	yes	—
15	C₂×C₄²×C₃₇	C₂²	1	1	yes	characteristic
16	C₄⋊C₈×C₃₇	—	2	2	no	—
17	C₂²×C₄×C₃₇	D₄	1	2	yes	—
18	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
19	C₂×C₂²⋊D₃₇	—	2	2	no	—
20	C₂²×Dic₃₇	—	2	2	no	—
21	#10596962	—	2	4	no	—
22	#10596962	—	2	4	no	—
23	C₂×D₁₄₈	—	2	2	no	—
24	C₂×D₁₄₈	C₂×C₄	1	2	yes	—
25	C₂×C₄×D₃₇	—	2	2	no	—
26	C₂×Dic₇₄	C₂×C₄	1	2	yes	—
27	M₁₆×C₃₇	—	2	4	no	—
28	C₂×C₈×C₃₇	—	2	2	no	—
29	C₄²×C₃₇	C₂×C₄	1	1	yes	characteristic
30	C₄²×C₃₇	C₂×C₄	1	1	yes	characteristic
31	C₄²×C₃₇	D₄	1	2	yes	—
32	#10596938	—	2	4	no	—
33	C₄⋊Dic₃₇	—	2	2	no	—
34	C₄×Dic₃₇	—	2	2	no	—
35	C₄×Dic₃₇	D₄	1	4	yes	complemented
36	C₂³×C₃₇	C₂×D₄	1	1	yes	characteristic
37	C₂²×D₃₇	—	2	2	no	—
38	C₂×C₄×C₃₇	—	4	2	no	—
39	C₂×C₄×C₃₇	C₂²⋊C₄	1	2	yes	—
40	C₂×C₄×C₃₇	C₂²⋊C₄	1	2	yes	—
41	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₄×C₃₇	C₂×D₄	1	2	yes	—
43	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₃₇	#41	1	2	yes	—
45	C₂²⋊D₃₇	—	4	4	no	—
46	C₂×Dic₃₇	—	2	2	no	—
47	C₂×Dic₃₇	—	2	4	no	—
48	C₂×Dic₃₇	—	4	4	no	—
49	D₁₄₈	—	2	4	no	—
50	D₁₄₈	—	4	4	no	—
51	C₄×D₃₇	—	4	2	no	—
52	C₄×D₃₇	—	4	2	no	—
53	Dic₇₄	—	2	4	no	—
54	C₈×C₃₇	—	4	2	no	—
55	C₂²×C₃₇	—	2	2	no	—
56	C₂²×C₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
57	C₂²×C₃₇	#121	1	2	yes	—
58	C₂×D₃₇	—	4	2	no	—
59	C₂×D₃₇	—	4	4	no	—
60	C₄×C₃₇	—	2	2	no	—
61	C₄×C₃₇	—	2	2	no	—
62	C₄×C₃₇	C₄×D₄	1	2	yes	—
63	C₄×C₃₇	#124	1	1	yes	characteristic
64	C₄×C₃₇	#127	1	1	yes	characteristic
65	Dic₃₇	—	4	4	no	—
66	Dic₃₇	—	4	2	no	—
67	#1963	—	37	1	no	maximal, Sylow
68	C₂×C₃₇	—	2	2	no	—
69	C₂×C₃₇	#390	1	1	yes	characteristic
70	C₂×C₃₇	#421	1	2	yes	—
71	D₃₇	—	8	2	no	—
72	#532	—	37	1	no	—
73	#423	D₃₇	1	1	yes	characteristic, complemented
74	C₂×#104	—	37	4	no	—
75	#376	—	37	1	no	—
76	C₃₇	#1963	1	1	yes	characteristic, complemented, Sylow
77	C₂×#41	—	37	2	no	—
78	C₂×M₁₆	C₂×D₃₇	1	2	yes	—
79	C₄⋊Q₈	—	37	1	no	—
80	#127	—	37	1	no	—
81	#124	—	37	2	no	—
82	#121	—	74	2	no	—
83	C₄×D₄	—	74	2	no	—
84	C₂×C₄²	—	37	2	no	—
85	C₂×C₄²	C₂×D₃₇	1	1	yes	characteristic
86	C₄⋊C₈	—	2	2	no	—
87	#104	—	74	8	no	—
88	#95	—	74	2	no	—
89	#41	—	74	4	no	—
90	#41	—	74	4	no	—
91	C₂×Q₈	—	37	2	no	—
92	C₂×D₄	—	37	2	no	—
93	C₂×D₄	—	74	2	no	—
94	C₂×D₄	—	74	2	no	—
95	C₂²×C₄	—	74	2	no	—
96	C₂²×C₄	—	74	2	no	—
97	C₂²×C₄	D₁₄₈	1	2	yes	—
98	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
99	M₁₆	—	2	4	no	—
100	C₂×C₈	—	2	2	no	—
101	C₄⋊C₄	—	74	2	no	—
102	C₂²⋊C₄	—	74	4	no	—
103	C₄²	—	37	4	no	—
104	C₄²	—	74	2	no	—
105	C₄²	C₄×D₃₇	1	1	yes	characteristic
106	C₄²	C₄×D₃₇	1	1	yes	characteristic
107	C₄²	C₂²⋊D₃₇	1	2	yes	—
108	C₂³	—	74	2	no	—
109	C₂³	C₂×D₁₄₈	1	1	yes	characteristic
110	Q₈	—	74	4	no	—
111	D₄	—	74	4	no	—
112	D₄	—	148	4	no	—
113	D₄	—	148	4	no	—
114	C₂×C₄	—	4	2	no	—
115	C₂×C₄	—	74	2	no	—
116	C₂×C₄	—	74	4	no	—
117	C₂×C₄	—	148	4	no	—
118	C₂×C₄	—	148	2	no	—
119	C₂×C₄	—	148	2	no	—
120	C₂×C₄	#10596938	1	2	yes	—
121	C₂×C₄	#10596938	1	2	yes	—
122	C₂×C₄	C₂×C₄×D₃₇	1	1	yes	characteristic
123	C₂×C₄	D₄×D₃₇	1	2	yes	—
124	C₂×C₄	#10596966	1	2	yes	—
125	C₂×C₄	C₂×C₂²⋊D₃₇	1	1	yes	characteristic
126	C₈	—	4	2	no	—
127	C₂²	—	2	2	no	—
128	C₂²	—	148	4	no	—
129	C₂²	—	148	2	no	—
130	C₂²	C₂×#10596938	1	1	yes	characteristic
131	C₂²	#11924511	1	2	yes	—
132	C₄	—	2	2	no	—
133	C₄	—	2	2	no	—
134	C₄	—	148	4	no	—
135	C₄	—	148	2	no	—
136	C₄	#11924498	1	2	yes	—
137	C₄	#11924525	1	1	yes	characteristic
138	C₄	#11924532	1	1	yes	characteristic
139	C₂	—	2	2	no	—
140	C₂	—	296	2	no	—
141	C₂	#14557434	1	1	yes	characteristic
142	C₂	#14557549	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14558083

C₂

yes

maximal, characteristic

C₂×#11924462

C₂

yes

maximal, complemented

#14557379

C₂

yes

maximal, characteristic, complemented

C₃₇×#423

C₂

yes

maximal, characteristic, complemented

C₂×#10596962

C₂²

yes

—

#11924536

C₄

yes

characteristic, complemented

#11924533

C₄

yes

characteristic, complemented

#11924524

C₄

yes

complemented

#11924510

—

C₄×D₁₄₈

—

C₂×C₄×Dic₃₇

C₂²

yes

—

#11924462

—

#11924443

—

C₂×M₁₆×C₃₇

C₂²

yes

—

C₂×C₄²×C₃₇

C₂²

yes

characteristic

C₄⋊C₈×C₃₇

—

C₂²×C₄×C₃₇

D₄

yes

—

C₂²×C₄×C₃₇

C₂³

yes

characteristic

C₂×C₂²⋊D₃₇

—

C₂²×Dic₃₇

—

#10596962

—

#10596962

—

C₂×D₁₄₈

—

C₂×D₁₄₈

C₂×C₄

yes

—

C₂×C₄×D₃₇

—

C₂×Dic₇₄

C₂×C₄

yes

—

M₁₆×C₃₇

—

C₂×C₈×C₃₇

—

C₄²×C₃₇

C₂×C₄

yes

characteristic

C₄²×C₃₇

C₂×C₄

yes

characteristic

C₄²×C₃₇

D₄

yes

—

#10596938

—

C₄⋊Dic₃₇

—

C₄×Dic₃₇

—

C₄×Dic₃₇

D₄

yes

complemented

C₂³×C₃₇

C₂×D₄

yes

characteristic

C₂²×D₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₂²⋊C₄

yes

—

C₂×C₄×C₃₇

C₂²⋊C₄

yes

—

C₂×C₄×C₃₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₇

C₂×D₄

yes

—

C₂×C₄×C₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₃₇

#41

yes

—

C₂²⋊D₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

D₁₄₈

—

D₁₄₈

—

C₄×D₃₇

—

C₄×D₃₇

—

Dic₇₄

—

C₈×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₃₇

#121

yes

—

C₂×D₃₇

—

C₂×D₃₇

—

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

C₄×D₄

yes

—

C₄×C₃₇

yes

C₄×C₃₇

yes

Dic₃₇

—

Dic₃₇

—

#1963

—

maximal, Sylow

C₂×C₃₇

—

C₂×C₃₇

#390

yes

characteristic

C₂×C₃₇

#421

yes

—

D₃₇

—

#532

—

#423

D₃₇

yes

characteristic, complemented

C₂×#104

—

#376

—

C₃₇

#1963

yes

characteristic, complemented, Sylow

C₂×#41

—

C₂×M₁₆

C₂×D₃₇

yes

—

C₄⋊Q₈

—

#127

—

#124

—

#121

—

C₄×D₄

—

C₂×C₄²

—

C₂×C₄²

C₂×D₃₇

yes

characteristic

C₄⋊C₈

—

—

—

—

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

D₁₄₈

yes

—

C₂²×C₄

C₂²×D₃₇

yes

characteristic

M₁₆

—

100

C₂×C₈

—

101

C₄⋊C₄

—

102

C₂²⋊C₄

—

103

C₄²

—

104

C₄²

—

105

C₄²

C₄×D₃₇

yes

characteristic

106

C₄²

C₄×D₃₇

yes

characteristic

107

C₄²

C₂²⋊D₃₇

yes

—

108

C₂³

—

109

C₂³

C₂×D₁₄₈

yes

characteristic

110

Q₈

—

111

D₄

—

112

D₄

—

148

—

113

D₄

—

148

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

148

—

118

C₂×C₄

—

148

—

119

C₂×C₄

—

148

—

120

C₂×C₄

#10596938

yes

—

121

C₂×C₄

#10596938

yes

—

122

C₂×C₄

C₂×C₄×D₃₇

yes

characteristic

123

C₂×C₄

D₄×D₃₇

yes

—

124

C₂×C₄

#10596966

yes

—

125

C₂×C₄

C₂×C₂²⋊D₃₇

yes

characteristic

126

C₈

—

127

C₂²

—

128

C₂²

—

148

—

129

C₂²

—

148

—

130

C₂²

C₂×#10596938

yes

characteristic

131

C₂²

#11924511

yes

—

132

C₄

—

133

C₄

—

134

C₄

—

148

—

135

C₄

—

148

—

136

C₄

#11924498

yes

—

137

C₄

#11924525

yes

characteristic

138

C₄

#11924532

yes

characteristic

139

C₂

—

140

C₂

—

296

—

141

C₂

#14557434

yes

characteristic

142

C₂

#14557549

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₃₇
commutator	C₂×C₄×C₃₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₃₇
Fitting	C₃₇×#423	C₂
socle	C₂²×C₃₇	C₂×C₂²⋊C₄
prime radical	#14558083	C₂
cosocle	C₂²×C₄×C₃₇	C₂³
hypercenter	#423	D₃₇
hypocenter	C₃₇	#1963
LCS	G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₃₇

commutator

C₂×C₄×C₃₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₃₇

Fitting

C₃₇×#423

C₂

socle

C₂²×C₃₇

C₂×C₂²⋊C₄

prime radical

#14558083

C₂

cosocle

C₂²×C₄×C₃₇

C₂³

D₃₇

C₃₇

G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

N	H ≅ G/N	type
2. C₂×#11924462	C₂	semidirect
3. #14557379	C₂	semidirect
4. C₃₇×#423	C₂	semidirect
6. #11924536	C₄	semidirect
7. #11924533	C₄	semidirect
8. #11924524	C₄	semidirect
35. C₄×Dic₃₇	D₄	semidirect
73. #423	D₃₇	semidirect
76. C₃₇	#1963	semidirect

H ≅ G/N

type

2. C₂×#11924462

C₂

semidirect

3. #14557379

C₂

semidirect

4. C₃₇×#423

C₂

semidirect

6. #11924536

C₄

semidirect

7. #11924533

C₄

semidirect

8. #11924524

C₄

semidirect

35. C₄×Dic₃₇

D₄

semidirect

73. #423

D₃₇

semidirect

76. C₃₇

#1963

semidirect

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

subgroup	index
#14558083	2
C₂×#11924462	2
#14557379	2
C₃₇×#423	2
#1963	37

subgroup

index

C₃₇×#423

Subgroups — #17995427

Conjugacy classes of subgroups

Complements

Maximal subgroups