Group #17995551 — subgroups

Subgroups — #17995551

Order 4736.

260 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 105 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×#11924528	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊Dic₃₇	C₂	1	1	yes	maximal, characteristic
3	#14557296	C₂	1	2	yes	maximal
4	#14557296	C₂	1	2	yes	maximal
5	C₂×C₄⋊C₈×C₃₇	C₂	1	1	yes	maximal, characteristic
6	#11924528	—	2	2	no	—
7	#11924528	C₄	1	4	yes	complemented
8	C₂×C₄⋊Dic₃₇	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Dic₃₇	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₃₇	C₂²	1	2	yes	—
11	C₂×#10596936	—	2	1	no	—
12	C₂×#10596936	—	2	1	no	—
13	C₂×C₄×Dic₃₇	—	2	1	no	—
14	#11924443	—	2	1	no	—
15	C₂²×C₈×C₃₇	C₂²	1	2	yes	—
16	C₂×C₄²×C₃₇	C₂²	1	1	yes	characteristic
17	C₄⋊C₈×C₃₇	—	2	2	no	—
18	C₂²×C₄×C₃₇	D₄	1	1	yes	characteristic
19	C₂²×C₄×C₃₇	D₄	1	1	yes	characteristic
20	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
21	C₂²×Dic₃₇	—	2	2	no	—
22	C₂²×Dic₃₇	—	2	1	no	—
23	C₂²×Dic₃₇	—	2	1	no	—
24	C₂×C₈×C₃₇	—	2	2	no	—
25	C₂×C₈×C₃₇	—	2	4	no	—
26	C₄²×C₃₇	C₂×C₄	1	2	yes	—
27	C₄²×C₃₇	D₄	1	2	yes	—
28	C₄⋊Dic₃₇	—	2	1	no	—
29	C₄⋊Dic₃₇	—	2	1	no	—
30	C₄⋊Dic₃₇	—	2	2	no	—
31	C₄⋊Dic₃₇	C₂×C₄	1	2	yes	—
32	C₄⋊Dic₃₇	C₂×C₄	1	2	yes	—
33	C₄⋊Dic₃₇	Q₈	1	4	yes	—
34	#10596936	—	2	4	no	—
35	#10596936	—	2	4	no	—
36	C₄×Dic₃₇	—	4	2	no	—
37	C₂³×C₃₇	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₃₇	—	2	2	no	—
39	C₂×C₄×C₃₇	—	2	1	no	—
40	C₂×C₄×C₃₇	—	2	1	no	—
41	C₂×C₄×C₃₇	C₂²⋊C₄	1	2	yes	—
42	C₂×C₄×C₃₇	C₂²⋊C₄	1	2	yes	—
43	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₃₇	C₂×Q₈	1	2	yes	—
46	C₂×C₄×C₃₇	#41	1	2	yes	—
47	C₂×Dic₃₇	—	2	2	no	—
48	C₂×Dic₃₇	—	2	2	no	—
49	C₂×Dic₃₇	—	2	4	no	—
50	C₂×Dic₃₇	—	4	2	no	—
51	C₂×Dic₃₇	—	4	2	no	—
52	C₂×Dic₃₇	—	4	2	no	—
53	C₂×Dic₃₇	—	4	1	no	—
54	C₂×Dic₃₇	—	4	1	no	—
55	C₈×C₃₇	—	4	2	no	—
56	C₂²×C₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
57	C₂²×C₃₇	C₂²⋊Q₈	1	2	yes	—
58	C₂²×C₃₇	#135	1	2	yes	—
59	C₂²×C₃₇	AGL(1,ℤ/8)	1	1	yes	characteristic
60	C₂²×C₃₇	#137	1	1	yes	characteristic
61	C₄×C₃₇	—	2	2	no	—
62	C₄×C₃₇	—	2	2	no	—
63	C₄×C₃₇	C₄×Q₈	1	2	yes	—
64	C₄×C₃₇	#124	1	1	yes	characteristic
65	C₄×C₃₇	C₄⋊Q₈	1	1	yes	characteristic
66	Dic₃₇	—	4	2	no	—
67	Dic₃₇	—	4	2	no	—
68	Dic₃₇	—	4	4	no	—
69	#1989	—	37	1	no	maximal, Sylow
70	C₂×C₃₇	#391	1	1	yes	characteristic
71	C₂×C₃₇	#416	1	1	yes	characteristic
72	C₂×C₃₇	#417	1	1	yes	characteristic
73	C₂×C₃₇	#478	1	2	yes	—
74	C₂×C₃₇	#479	1	2	yes	—
75	C₂×#125	—	37	1	no	—
76	C₂×C₄⋊C₈	D₃₇	1	1	yes	characteristic
77	C₄×C₄⋊C₄	—	37	1	no	—
78	#340	—	37	2	no	—
79	#340	—	37	2	no	—
80	C₃₇	#1989	1	1	yes	characteristic, complemented, Sylow
81	C₂²×C₈	C₂×D₃₇	1	2	yes	—
82	#125	—	37	4	no	—
83	#125	—	74	2	no	—
84	C₂×C₄⋊C₄	—	37	1	no	—
85	C₂×C₄⋊C₄	—	37	1	no	—
86	C₂×C₄⋊C₄	—	37	2	no	—
87	C₂×C₄⋊C₄	—	74	1	no	—
88	C₂×C₄⋊C₄	—	74	1	no	—
89	C₂×C₄²	—	74	1	no	—
90	C₂×C₄²	C₂×D₃₇	1	1	yes	characteristic
91	C₄⋊C₈	—	2	2	no	—
92	#95	—	74	1	no	—
93	C₂²×C₄	—	74	2	no	—
94	C₂²×C₄	—	74	1	no	—
95	C₂²×C₄	—	74	1	no	—
96	C₂²×C₄	D₁₄₈	1	1	yes	characteristic
97	C₂²×C₄	D₁₄₈	1	1	yes	characteristic
98	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
99	C₂×C₈	—	2	4	no	—
100	C₂×C₈	—	2	2	no	—
101	C₄⋊C₄	—	37	2	no	—
102	C₄⋊C₄	—	37	2	no	—
103	C₄⋊C₄	—	37	4	no	—
104	C₄⋊C₄	—	74	4	no	—
105	C₄⋊C₄	—	74	4	no	—
106	C₄⋊C₄	—	74	2	no	—
107	C₄⋊C₄	—	74	1	no	—
108	C₄⋊C₄	—	74	1	no	—
109	C₄²	—	148	2	no	—
110	C₄²	C₄×D₃₇	1	2	yes	—
111	C₄²	C₂²⋊D₃₇	1	2	yes	—
112	C₂³	C₂×D₁₄₈	1	1	yes	characteristic
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	—	2	1	no	—
115	C₂×C₄	—	2	1	no	—
116	C₂×C₄	—	74	2	no	—
117	C₂×C₄	—	74	2	no	—
118	C₂×C₄	—	74	4	no	—
119	C₂×C₄	—	148	2	no	—
120	C₂×C₄	—	148	2	no	—
121	C₂×C₄	—	148	2	no	—
122	C₂×C₄	—	148	1	no	—
123	C₂×C₄	—	148	1	no	—
124	C₂×C₄	#10596938	1	2	yes	—
125	C₂×C₄	#10596938	1	2	yes	—
126	C₂×C₄	C₂×C₄×D₃₇	1	1	yes	characteristic
127	C₂×C₄	#10596964	1	2	yes	—
128	C₂×C₄	Q₈×D₃₇	1	2	yes	—
129	C₂×C₄	C₂×C₂²⋊D₃₇	1	1	yes	characteristic
130	C₈	—	4	2	no	—
131	C₂²	C₂×#10596938	1	1	yes	characteristic
132	C₂²	#11924516	1	2	yes	—
133	C₂²	#11924558	1	2	yes	—
134	C₂²	#11924564	1	1	yes	characteristic
135	C₂²	#11924569	1	1	yes	characteristic
136	C₄	—	2	2	no	—
137	C₄	—	2	2	no	—
138	C₄	—	148	2	no	—
139	C₄	—	148	2	no	—
140	C₄	—	148	4	no	—
141	C₄	#11924501	1	2	yes	—
142	C₄	#11924523	1	1	yes	characteristic
143	C₄	#11924534	1	1	yes	characteristic
144	C₂	#14557438	1	1	yes	characteristic
145	C₂	#14557530	1	1	yes	characteristic
146	C₂	#14557535	1	1	yes	characteristic
147	C₂	#14557834	1	2	yes	—
148	C₂	#14557841	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11924528

C₂

yes

maximal, characteristic

C₄×C₄⋊Dic₃₇

C₂

yes

maximal, characteristic

#14557296

C₂

yes

maximal

#14557296

C₂

yes

maximal

C₂×C₄⋊C₈×C₃₇

C₂

yes

maximal, characteristic

#11924528

—

#11924528

C₄

yes

complemented

C₂×C₄⋊Dic₃₇

C₂²

yes

characteristic

C₂×C₄⋊Dic₃₇

C₂²

yes

characteristic

C₂×C₄⋊Dic₃₇

C₂²

yes

—

C₂×#10596936

—

C₂×#10596936

—

C₂×C₄×Dic₃₇

—

#11924443

—

C₂²×C₈×C₃₇

C₂²

yes

—

C₂×C₄²×C₃₇

C₂²

yes

characteristic

C₄⋊C₈×C₃₇

—

C₂²×C₄×C₃₇

D₄

yes

characteristic

C₂²×C₄×C₃₇

D₄

yes

characteristic

C₂²×C₄×C₃₇

C₂³

yes

characteristic

C₂²×Dic₃₇

—

C₂²×Dic₃₇

—

C₂²×Dic₃₇

—

C₂×C₈×C₃₇

—

C₂×C₈×C₃₇

—

C₄²×C₃₇

C₂×C₄

yes

—

C₄²×C₃₇

D₄

yes

—

C₄⋊Dic₃₇

—

C₄⋊Dic₃₇

—

C₄⋊Dic₃₇

—

C₄⋊Dic₃₇

C₂×C₄

yes

—

C₄⋊Dic₃₇

C₂×C₄

yes

—

C₄⋊Dic₃₇

Q₈

yes

—

#10596936

—

#10596936

—

C₄×Dic₃₇

—

C₂³×C₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

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

characteristic

C₂×C₄×C₃₇

C₂×Q₈

yes

—

C₂×C₄×C₃₇

#41

yes

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₈×C₃₇

—

C₂²×C₃₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₃₇

C₂²⋊Q₈

yes

—

C₂²×C₃₇

#135

yes

—

C₂²×C₃₇

yes

C₂²×C₃₇

yes

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

C₄×Q₈

yes

—

C₄×C₃₇

#124

yes

characteristic

C₄×C₃₇

C₄⋊Q₈

yes

characteristic

Dic₃₇

—

Dic₃₇

—

Dic₃₇

—

—

C₂×C₃₇

yes

C₂×C₃₇

yes

C₂×C₃₇

yes

C₂×C₃₇

yes

—

C₂×C₃₇

#479

yes

—

C₂×#125

—

C₂×C₄⋊C₈

D₃₇

yes

characteristic

C₄×C₄⋊C₄

—

#340

—

#340

—

C₃₇

#1989

yes

characteristic, complemented, Sylow

C₂²×C₈

C₂×D₃₇

yes

—

#125

—

#125

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

C₂×D₃₇

yes

characteristic

C₄⋊C₈

—

#95

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

D₁₄₈

yes

characteristic

C₂²×C₄

D₁₄₈

yes

characteristic

C₂²×C₄

C₂²×D₃₇

yes

characteristic

C₂×C₈

—

100

C₂×C₈

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄²

—

148

—

110

C₄²

C₄×D₃₇

yes

—

111

C₄²

C₂²⋊D₃₇

yes

—

112

C₂³

C₂×D₁₄₈

yes

characteristic

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

148

—

120

C₂×C₄

—

148

—

121

C₂×C₄

—

148

—

122

C₂×C₄

—

148

—

123

C₂×C₄

—

148

—

124

C₂×C₄

#10596938

yes

—

125

C₂×C₄

#10596938

yes

—

126

C₂×C₄

C₂×C₄×D₃₇

yes

characteristic

127

C₂×C₄

#10596964

yes

—

128

C₂×C₄

Q₈×D₃₇

yes

—

129

C₂×C₄

C₂×C₂²⋊D₃₇

yes

characteristic

130

C₈

—

131

C₂²

C₂×#10596938

yes

characteristic

132

C₂²

#11924516

yes

—

133

C₂²

#11924558

yes

—

134

C₂²

#11924564

yes

characteristic

135

C₂²

#11924569

yes

characteristic

136

C₄

—

137

C₄

—

138

C₄

—

148

—

139

C₄

—

148

—

140

C₄

—

148

—

141

C₄

#11924501

yes

—

142

C₄

#11924523

yes

characteristic

143

C₄

#11924534

yes

characteristic

144

C₂

#14557438

yes

characteristic

145

C₂

#14557530

yes

characteristic

146

C₂

#14557535

yes

characteristic

147

C₂

#14557834

yes

—

148

C₂

#14557841

yes

—

	subgroup	G/H
center	C₂³	C₂×D₁₄₈
commutator	C₂×C₄×C₃₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₃₇
Fitting	C₂×C₄⋊C₈×C₃₇	C₂
socle, prime radical	C₂³×C₃₇	C₂×D₄
cosocle	C₂²×C₄×C₃₇	C₂³
hypercenter	C₂×C₄⋊C₈	D₃₇
hypocenter	C₃₇	#1989
LCS	G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

subgroup

G/H

center

C₂³

C₂×D₁₄₈

commutator

C₂×C₄×C₃₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₃₇

Fitting

C₂×C₄⋊C₈×C₃₇

C₂

socle, prime radical

C₂³×C₃₇

C₂×D₄

cosocle

C₂²×C₄×C₃₇

C₂³

hypercenter

C₂×C₄⋊C₈

D₃₇

hypocenter

C₃₇

#1989

LCS

G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
7. #11924528	C₄	semidirect
80. C₃₇	#1989	semidirect

H ≅ G/N

type

7. #11924528

C₄

semidirect

80. C₃₇

#1989

semidirect

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

subgroup	index
C₂×#11924528	2
C₄×C₄⋊Dic₃₇	2
#14557296	2
C₂×C₄⋊C₈×C₃₇	2
#1989	37

subgroup

index

C₂×#11924528

C₄×C₄⋊Dic₃₇

#14557296

C₂×C₄⋊C₈×C₃₇

#1989

Subgroups — #17995551

Conjugacy classes of subgroups

Complements

Maximal subgroups