Group #17996644 — subgroups

Subgroups — #17996644

Order 4736.

154 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 154 conjugacy classes of subgroups — too many to draw. Only its 51 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×#11924482	C₂	1	1	yes	maximal, characteristic
2	#14557911	C₂	1	1	yes	maximal, characteristic
3	Q₁₆×Dic₃₇	C₂	1	1	yes	maximal, characteristic
4	#14557351	C₂	1	1	yes	maximal, characteristic
5	#14557338	C₂	1	1	yes	maximal, characteristic
6	#14557337	C₂	1	1	yes	maximal, characteristic
7	C₂×Q₃₂×C₃₇	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₁₄₈	C₂²	1	1	yes	characteristic
9	C₂×#10596942	—	2	1	no	—
10	#11924534	—	2	1	no	—
11	Q₈×Dic₃₇	—	2	1	no	—
12	#11924482	—	2	2	no	—
13	C₂×#10596925	C₂²	1	1	yes	characteristic
14	#11924470	C₂²	1	1	yes	characteristic
15	#11924460	—	2	1	no	—
16	C₈×Dic₃₇	C₂²	1	1	yes	characteristic
17	C₂×Q₁₆×C₃₇	C₂²	1	1	yes	characteristic
18	C₂×Q₁₆×C₃₇	C₂²	1	1	yes	characteristic
19	Q₃₂×C₃₇	—	2	2	no	—
20	C₂×C₁₆×C₃₇	C₂²	1	1	yes	characteristic
21	C₂×Q₈×C₃₇	—	2	1	no	—
22	C₂×Q₈×C₃₇	—	2	1	no	—
23	C₂×Dic₇₄	—	2	1	no	—
24	Q₁₆×C₃₇	—	2	1	no	—
25	Q₁₆×C₃₇	—	2	1	no	—
26	Q₁₆×C₃₇	—	2	1	no	—
27	Q₁₆×C₃₇	D₄	1	2	yes	—
28	C₂×C₈×C₃₇	C₂³	1	1	yes	characteristic
29	#10596942	—	4	2	no	—
30	C₄⋊Dic₃₇	—	2	1	no	—
31	C₄⋊Dic₃₇	—	4	1	no	—
32	#10596936	—	4	1	no	—
33	#10596936	—	4	1	no	—
34	C₄×Dic₃₇	—	4	1	no	—
35	C₄×Dic₃₇	D₄	1	1	yes	characteristic
36	C₂×#64436	D₄	1	1	yes	characteristic
37	Dic₁₄₈	—	2	1	no	—
38	Dic₁₄₈	—	2	1	no	—
39	C₁₆×C₃₇	—	2	1	no	—
40	#10596925	—	2	1	no	—
41	Q₈×C₃₇	—	2	2	no	—
42	Q₈×C₃₇	—	4	1	no	—
43	Q₈×C₃₇	—	4	1	no	—
44	Q₈×C₃₇	—	4	1	no	—
45	C₂×C₄×C₃₇	—	4	1	no	—
46	C₂×C₄×C₃₇	—	4	1	no	—
47	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
48	C₂×Dic₃₇	—	4	1	no	—
49	C₂×Dic₃₇	—	4	1	no	—
50	C₂×Dic₃₇	D₈	1	1	yes	characteristic
51	C₂×Dic₃₇	D₈	1	1	yes	characteristic
52	Dic₇₄	—	4	1	no	—
53	Dic₇₄	—	4	1	no	—
54	C₈×C₃₇	C₂×D₄	1	1	yes	characteristic
55	C₈×C₃₇	#41	1	1	yes	characteristic
56	#64436	—	2	1	no	—
57	C₂²×C₃₇	C₂×D₈	1	1	yes	characteristic
58	C₄×C₃₇	—	4	2	no	—
59	C₄×C₃₇	—	8	1	no	—
60	C₄×C₃₇	#121	1	1	yes	characteristic
61	C₄×C₃₇	AGL(1,ℤ/8)	1	1	yes	characteristic
62	Dic₃₇	—	2	1	no	—
63	Dic₃₇	—	8	1	no	—
64	Dic₃₇	—	8	1	no	—
65	Dic₃₇	Q₃₂	1	2	yes	complemented
66	#2209	—	37	1	no	maximal, Sylow
67	C₂×C₃₇	#459	1	1	yes	characteristic
68	C₂×C₃₇	C₂×Q₃₂	1	1	yes	characteristic
69	C₂×C₃₇	#510	1	1	yes	characteristic
70	C₂×Q₃₂	—	37	1	no	—
71	C₂×Q₃₂	D₃₇	1	1	yes	characteristic
72	#494	—	37	1	no	—
73	C₄×Q₁₆	—	37	1	no	—
74	C₄⋊C₁₆	—	37	1	no	—
75	#358	—	37	1	no	—
76	#358	—	37	1	no	—
77	C₃₇	#2209	1	1	yes	characteristic, complemented, Sylow
78	C₂×Q₁₆	—	37	1	no	—
79	C₂×Q₁₆	—	74	1	no	—
80	C₂×Q₁₆	C₂×D₃₇	1	1	yes	characteristic
81	C₂×Q₁₆	C₂×D₃₇	1	1	yes	characteristic
82	C₄⋊Q₈	—	74	1	no	—
83	C₄×Q₈	—	74	1	no	—
84	Q₃₂	—	2	2	no	—
85	Q₃₂	—	74	2	no	—
86	C₂×C₁₆	—	37	1	no	—
87	C₂×C₁₆	C₂×D₃₇	1	1	yes	characteristic
88	#107	—	37	1	no	—
89	#103	—	74	1	no	—
90	C₄×C₈	—	37	1	no	—
91	C₂×Q₈	—	2	1	no	—
92	C₂×Q₈	—	2	1	no	—
93	C₂×Q₈	—	74	1	no	—
94	Q₁₆	—	2	1	no	—
95	Q₁₆	—	2	1	no	—
96	Q₁₆	—	2	1	no	—
97	Q₁₆	—	74	1	no	—
98	Q₁₆	—	74	1	no	—
99	Q₁₆	—	148	2	no	—
100	Q₁₆	C₂²⋊D₃₇	1	2	yes	—
101	C₂×C₈	—	37	1	no	—
102	C₂×C₈	C₂²×D₃₇	1	1	yes	characteristic
103	C₄⋊C₄	—	74	1	no	—
104	C₄⋊C₄	—	148	1	no	—
105	C₄⋊C₄	—	148	1	no	—
106	C₄⋊C₄	—	148	1	no	—
107	C₄²	—	37	1	no	—
108	C₄²	—	148	1	no	—
109	C₁₆	—	2	1	no	—
110	C₁₆	—	74	1	no	—
111	Q₈	—	2	2	no	—
112	Q₈	—	4	1	no	—
113	Q₈	—	4	1	no	—
114	Q₈	—	4	1	no	—
115	Q₈	—	148	1	no	—
116	Q₈	—	148	1	no	—
117	C₂×C₄	—	4	1	no	—
118	C₂×C₄	—	4	1	no	—
119	C₂×C₄	—	37	1	no	—
120	C₂×C₄	—	37	1	no	—
121	C₂×C₄	—	148	1	no	—
122	C₂×C₄	—	148	1	no	—
123	C₂×C₄	D₄×D₃₇	1	1	yes	characteristic
124	C₈	—	74	1	no	—
125	C₈	#10596964	1	1	yes	characteristic
126	C₈	C₂×C₂²⋊D₃₇	1	1	yes	characteristic
127	C₂²	D₈×D₃₇	1	1	yes	characteristic
128	C₄	—	4	2	no	—
129	C₄	—	8	1	no	—
130	C₄	—	37	2	no	—
131	C₄	—	296	1	no	—
132	C₄	—	296	1	no	—
133	C₄	—	74	1	no	—
134	C₄	#11924508	1	1	yes	characteristic
135	C₄	#11924560	1	1	yes	characteristic
136	C₂	#14557726	1	1	yes	characteristic
137	C₂	Q₃₂×D₃₇	1	1	yes	characteristic
138	C₂	#14557975	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11924482

C₂

yes

maximal, characteristic

#14557911

C₂

yes

maximal, characteristic

Q₁₆×Dic₃₇

C₂

yes

maximal, characteristic

#14557351

C₂

yes

maximal, characteristic

#14557338

C₂

yes

maximal, characteristic

#14557337

C₂

yes

maximal, characteristic

C₂×Q₃₂×C₃₇

C₂

yes

maximal, characteristic

C₂×Dic₁₄₈

C₂²

yes

characteristic

C₂×#10596942

—

#11924534

—

Q₈×Dic₃₇

—

#11924482

—

C₂×#10596925

C₂²

yes

characteristic

#11924470

C₂²

yes

characteristic

#11924460

—

C₈×Dic₃₇

C₂²

yes

characteristic

C₂×Q₁₆×C₃₇

C₂²

yes

characteristic

C₂×Q₁₆×C₃₇

C₂²

yes

characteristic

Q₃₂×C₃₇

—

C₂×C₁₆×C₃₇

C₂²

yes

characteristic

C₂×Q₈×C₃₇

—

C₂×Q₈×C₃₇

—

C₂×Dic₇₄

—

Q₁₆×C₃₇

—

Q₁₆×C₃₇

—

Q₁₆×C₃₇

—

Q₁₆×C₃₇

D₄

yes

—

C₂×C₈×C₃₇

C₂³

yes

characteristic

#10596942

—

C₄⋊Dic₃₇

—

C₄⋊Dic₃₇

—

#10596936

—

#10596936

—

C₄×Dic₃₇

—

C₄×Dic₃₇

D₄

yes

characteristic

C₂×#64436

D₄

yes

characteristic

Dic₁₄₈

—

Dic₁₄₈

—

C₁₆×C₃₇

—

#10596925

—

Q₈×C₃₇

—

Q₈×C₃₇

—

Q₈×C₃₇

—

Q₈×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₂×D₄

yes

characteristic

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

D₈

yes

characteristic

C₂×Dic₃₇

D₈

yes

characteristic

Dic₇₄

—

Dic₇₄

—

C₈×C₃₇

C₂×D₄

yes

C₈×C₃₇

yes

—

C₂²×C₃₇

C₂×D₈

yes

characteristic

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

yes

C₄×C₃₇

yes

Dic₃₇

—

Dic₃₇

—

Dic₃₇

—

Dic₃₇

Q₃₂

yes

—

C₂×C₃₇

yes

C₂×C₃₇

C₂×Q₃₂

yes

characteristic

C₂×C₃₇

#510

yes

characteristic

C₂×Q₃₂

—

C₂×Q₃₂

D₃₇

yes

characteristic

#494

—

C₄×Q₁₆

—

C₄⋊C₁₆

—

#358

—

#358

—

C₃₇

#2209

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

C₂×D₃₇

yes

characteristic

C₂×Q₁₆

C₂×D₃₇

yes

characteristic

C₄⋊Q₈

—

C₄×Q₈

—

Q₃₂

—

Q₃₂

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₃₇

yes

characteristic

#107

—

#103

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

148

—

100

Q₁₆

C₂²⋊D₃₇

yes

—

101

C₂×C₈

—

102

C₂×C₈

C₂²×D₃₇

yes

characteristic

103

C₄⋊C₄

—

104

C₄⋊C₄

—

148

—

105

C₄⋊C₄

—

148

—

106

C₄⋊C₄

—

148

—

107

C₄²

—

108

C₄²

—

148

—

109

C₁₆

—

110

C₁₆

—

111

Q₈

—

112

Q₈

—

113

Q₈

—

114

Q₈

—

115

Q₈

—

148

—

116

Q₈

—

148

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

148

—

122

C₂×C₄

—

148

—

123

C₂×C₄

D₄×D₃₇

yes

characteristic

124

C₈

—

125

C₈

#10596964

yes

characteristic

126

C₈

C₂×C₂²⋊D₃₇

yes

characteristic

127

C₂²

D₈×D₃₇

yes

characteristic

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

—

296

—

132

C₄

—

296

—

133

C₄

—

134

C₄

#11924508

yes

characteristic

135

C₄

#11924560

yes

characteristic

136

C₂

#14557726

yes

characteristic

137

C₂

Q₃₂×D₃₇

yes

characteristic

138

C₂

#14557975

yes

characteristic

	subgroup	G/H
center	C₂²	D₈×D₃₇
commutator, cosocle	C₂×C₈×C₃₇	C₂³
Frattini	C₂×C₈	C₂²×D₃₇
Fitting	C₂×Q₃₂×C₃₇	C₂
socle, prime radical	C₂²×C₃₇	C₂×D₈
hypercenter	C₂×Q₃₂	D₃₇
hypocenter	C₃₇	#2209
LCS	G ▷ C₂×C₈×C₃₇ ▷ C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×Q₃₂

subgroup

G/H

center

C₂²

D₈×D₃₇

commutator, cosocle

C₂×C₈×C₃₇

C₂³

Frattini

C₂×C₈

C₂²×D₃₇

Fitting

C₂×Q₃₂×C₃₇

C₂

socle, prime radical

C₂²×C₃₇

C₂×D₈

hypercenter

C₂×Q₃₂

D₃₇

hypocenter

C₃₇

#2209

LCS

G ▷ C₂×C₈×C₃₇ ▷ C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×Q₃₂

N	H ≅ G/N	type
65. Dic₃₇	Q₃₂	semidirect
77. C₃₇	#2209	semidirect

H ≅ G/N

type

65. Dic₃₇

Q₃₂

semidirect

77. C₃₇

#2209

semidirect

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

subgroup	index
C₂×#11924482	2
#14557911	2
Q₁₆×Dic₃₇	2
#14557351	2
#14557338	2
#14557337	2
C₂×Q₃₂×C₃₇	2
#2209	37

subgroup

index

Q₁₆×Dic₃₇

C₂×Q₃₂×C₃₇

Subgroups — #17996644

Conjugacy classes of subgroups

Complements

Maximal subgroups