Group #17994990 — subgroups

Subgroups — #17994990

Order 4736.

324 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 97 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	#14558011	C₂	1	1	yes	maximal, characteristic
2	#14557542	C₂	1	2	yes	maximal
3	C₂×#11924455	C₂	1	1	yes	maximal, characteristic
4	#14557293	C₂	1	2	yes	maximal
5	C₂×C₄⋊Q₈×C₃₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#10596934	C₂²	1	2	yes	—
7	Q₈×Dic₃₇	—	2	4	no	—
8	#11924495	—	2	2	no	—
9	#11924495	C₂²	1	2	yes	—
10	#11924460	—	2	4	no	—
11	#11924455	—	2	2	no	—
12	C₂²×Q₈×C₃₇	C₂²	1	1	yes	characteristic
13	C₂²×Q₈×C₃₇	C₂²	1	1	yes	characteristic
14	C₄⋊Q₈×C₃₇	—	2	2	no	—
15	C₄⋊Q₈×C₃₇	C₄	1	4	yes	complemented
16	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
17	C₂×C₄²×C₃₇	C₂²	1	1	yes	characteristic
18	C₂×Q₈×C₃₇	—	2	2	no	—
19	C₂×Q₈×C₃₇	—	2	1	no	—
20	C₂×Q₈×C₃₇	—	2	2	no	—
21	C₂×Q₈×C₃₇	—	2	2	no	—
22	C₂×Q₈×C₃₇	—	2	2	no	—
23	C₂×Q₈×C₃₇	C₂×C₄	1	2	yes	—
24	C₂×Q₈×C₃₇	C₂×C₄	1	2	yes	—
25	C₂×Q₈×C₃₇	D₄	1	2	yes	—
26	C₂²×C₄×C₃₇	—	2	1	no	—
27	C₂²×C₄×C₃₇	—	2	1	no	—
28	C₂²×C₄×C₃₇	D₄	1	2	yes	—
29	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
30	C₄⋊C₄×C₃₇	—	2	8	no	—
31	C₄²×C₃₇	—	2	1	no	—
32	C₄²×C₃₇	C₂×C₄	1	2	yes	—
33	C₂²⋊Dic₃₇	—	2	2	no	—
34	C₄⋊Dic₃₇	—	2	4	no	—
35	C₄⋊Dic₃₇	D₄	1	4	yes	—
36	C₄×Dic₃₇	—	2	4	no	—
37	C₄×Dic₃₇	—	2	2	no	—
38	#10596934	—	2	2	no	—
39	#10596934	—	2	2	no	—
40	C₂×#64436	—	2	2	no	—
41	C₂³×C₃₇	C₂×D₄	1	1	yes	characteristic
42	Q₈×C₃₇	—	2	4	no	—
43	Q₈×C₃₇	—	2	4	no	—
44	Q₈×C₃₇	—	2	4	no	—
45	Q₈×C₃₇	—	4	2	no	—
46	C₂×C₄×C₃₇	—	2	4	no	—
47	C₂×C₄×C₃₇	—	2	2	no	—
48	C₂×C₄×C₃₇	—	2	2	no	—
49	C₂×C₄×C₃₇	—	2	2	no	—
50	C₂×C₄×C₃₇	—	2	2	no	—
51	C₂×C₄×C₃₇	—	4	1	no	—
52	C₂×C₄×C₃₇	—	4	1	no	—
53	C₂×C₄×C₃₇	C₂²⋊C₄	1	4	yes	—
54	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₄×C₃₇	C₂×D₄	1	2	yes	—
56	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×C₃₇	#41	1	2	yes	—
58	C₂×Dic₃₇	—	2	4	no	—
59	#64436	—	4	2	no	—
60	C₂²×C₃₇	—	2	2	no	—
61	C₂²×C₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₃₇	#120	1	1	yes	characteristic
63	C₂²×C₃₇	#123	1	1	yes	characteristic
64	C₄×C₃₇	—	2	4	no	—
65	C₄×C₃₇	—	2	4	no	—
66	C₄×C₃₇	—	4	2	no	—
67	C₄×C₃₇	C₄×D₄	1	2	yes	—
68	C₄×C₃₇	#121	1	2	yes	—
69	Dic₃₇	—	4	4	no	—
70	#1886	—	37	1	no	maximal, Sylow
71	C₂×C₃₇	—	2	1	no	—
72	C₂×C₃₇	—	2	1	no	—
73	C₂×C₃₇	#386	1	1	yes	characteristic
74	C₂×C₃₇	#456	1	2	yes	—
75	C₂×C₄⋊Q₈	D₃₇	1	1	yes	characteristic
76	#519	—	37	1	no	—
77	#419	—	37	2	no	—
78	C₂×#101	—	37	1	no	—
79	#339	—	37	2	no	—
80	C₃₇	#1886	1	1	yes	characteristic, complemented, Sylow
81	C₂²×Q₈	C₂×D₃₇	1	1	yes	characteristic
82	C₂²×Q₈	C₂×D₃₇	1	1	yes	characteristic
83	C₂×M₁₆	—	37	2	no	—
84	C₄⋊Q₈	—	2	2	no	—
85	C₄⋊Q₈	Dic₃₇	1	4	yes	complemented
86	C₄×Q₈	—	74	4	no	—
87	#117	—	37	2	no	—
88	#117	—	74	2	no	—
89	C₂×C₄⋊C₄	—	2	2	no	—
90	C₂×C₄²	C₂×D₃₇	1	1	yes	characteristic
91	#103	—	74	4	no	—
92	#101	—	74	2	no	—
93	C₂×Q₈	—	2	2	no	—
94	C₂×Q₈	—	2	2	no	—
95	C₂×Q₈	—	2	1	no	—
96	C₂×Q₈	—	2	2	no	—
97	C₂×Q₈	—	2	2	no	—
98	C₂×Q₈	C₂×Dic₃₇	1	2	yes	—
99	C₂×Q₈	C₂×Dic₃₇	1	2	yes	—
100	C₂×Q₈	C₂²⋊D₃₇	1	2	yes	—
101	C₂²×C₄	—	2	1	no	—
102	C₂²×C₄	—	2	1	no	—
103	C₂²×C₄	C₂²⋊D₃₇	1	2	yes	—
104	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
105	M₁₆	—	74	2	no	—
106	M₁₆	—	74	2	no	—
107	C₂×C₈	—	74	2	no	—
108	C₄⋊C₄	—	2	8	no	—
109	C₄⋊C₄	—	37	4	no	—
110	C₄⋊C₄	—	74	4	no	—
111	C₂²⋊C₄	—	74	2	no	—
112	C₄²	—	2	1	no	—
113	C₄²	—	74	4	no	—
114	C₄²	—	74	2	no	—
115	C₄²	C₂×Dic₃₇	1	2	yes	—
116	C₂³	C₂×C₂²⋊D₃₇	1	1	yes	characteristic
117	Q₈	—	2	4	no	—
118	Q₈	—	2	4	no	—
119	Q₈	—	2	4	no	—
120	Q₈	—	4	2	no	—
121	C₂×C₄	—	2	4	no	—
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	—	4	1	no	—
127	C₂×C₄	—	4	1	no	—
128	C₂×C₄	—	74	4	no	—
129	C₂×C₄	C₂²⋊Dic₃₇	1	4	yes	—
130	C₂×C₄	D₄×D₃₇	1	2	yes	—
131	C₂×C₄	#10596964	1	2	yes	—
132	C₂×C₄	C₂²×Dic₃₇	1	1	yes	characteristic
133	C₂×C₄	C₂×C₂²⋊D₃₇	1	1	yes	characteristic
134	C₈	—	148	2	no	—
135	C₂²	—	2	2	no	—
136	C₂²	C₂×C₂²⋊Dic₃₇	1	1	yes	characteristic
137	C₂²	#11924504	1	1	yes	characteristic
138	C₂²	#11924517	1	1	yes	characteristic
139	C₄	—	2	4	no	—
140	C₄	—	2	4	no	—
141	C₄	—	4	2	no	—
142	C₄	—	148	4	no	—
143	C₄	D₄×Dic₃₇	1	2	yes	—
144	C₄	#11924508	1	2	yes	—
145	C₂	—	2	1	no	—
146	C₂	—	2	1	no	—
147	C₂	#14557413	1	1	yes	characteristic
148	C₂	#14557715	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14558011

C₂

yes

maximal, characteristic

#14557542

C₂

yes

maximal

C₂×#11924455

C₂

yes

maximal, characteristic

#14557293

C₂

yes

maximal

C₂×C₄⋊Q₈×C₃₇

C₂

yes

maximal, characteristic

C₂×#10596934

C₂²

yes

—

Q₈×Dic₃₇

—

#11924495

—

#11924495

C₂²

yes

—

#11924460

—

#11924455

—

C₂²×Q₈×C₃₇

C₂²

yes

characteristic

C₂²×Q₈×C₃₇

C₂²

yes

characteristic

C₄⋊Q₈×C₃₇

—

C₄⋊Q₈×C₃₇

C₄

yes

complemented

C₂×C₄⋊C₄×C₃₇

—

C₂×C₄²×C₃₇

C₂²

yes

characteristic

C₂×Q₈×C₃₇

—

C₂×Q₈×C₃₇

—

C₂×Q₈×C₃₇

—

C₂×Q₈×C₃₇

—

C₂×Q₈×C₃₇

—

C₂×Q₈×C₃₇

C₂×C₄

yes

—

C₂×Q₈×C₃₇

C₂×C₄

yes

—

C₂×Q₈×C₃₇

D₄

yes

—

C₂²×C₄×C₃₇

—

C₂²×C₄×C₃₇

—

C₂²×C₄×C₃₇

D₄

yes

—

C₂²×C₄×C₃₇

C₂³

yes

characteristic

C₄⋊C₄×C₃₇

—

C₄²×C₃₇

—

C₄²×C₃₇

C₂×C₄

yes

—

C₂²⋊Dic₃₇

—

C₄⋊Dic₃₇

—

C₄⋊Dic₃₇

D₄

yes

—

C₄×Dic₃₇

—

C₄×Dic₃₇

—

#10596934

—

#10596934

—

C₂×#64436

—

C₂³×C₃₇

C₂×D₄

yes

characteristic

Q₈×C₃₇

—

Q₈×C₃₇

—

Q₈×C₃₇

—

Q₈×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

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

—

C₂×Dic₃₇

—

#64436

—

C₂²×C₃₇

—

C₂²×C₃₇

C₂×C₂²⋊C₄

yes

C₂²×C₃₇

yes

C₂²×C₃₇

yes

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

C₄×D₄

yes

—

C₄×C₃₇

#121

yes

—

Dic₃₇

—

#1886

—

maximal, Sylow

C₂×C₃₇

—

C₂×C₃₇

—

C₂×C₃₇

#386

yes

characteristic

C₂×C₃₇

#456

yes

—

C₂×C₄⋊Q₈

D₃₇

yes

—

—

—

—

C₃₇

#1886

yes

characteristic, complemented, Sylow

C₂²×Q₈

C₂×D₃₇

yes

characteristic

C₂²×Q₈

C₂×D₃₇

yes

characteristic

C₂×M₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

Dic₃₇

yes

complemented

C₄×Q₈

—

#117

—

#117

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₃₇

yes

characteristic

#103

—

#101

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×Dic₃₇

yes

—

C₂×Q₈

C₂×Dic₃₇

yes

—

100

C₂×Q₈

C₂²⋊D₃₇

yes

—

101

C₂²×C₄

—

102

C₂²×C₄

—

103

C₂²×C₄

C₂²⋊D₃₇

yes

—

104

C₂²×C₄

C₂²×D₃₇

yes

characteristic

105

M₁₆

—

106

M₁₆

—

107

C₂×C₈

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₂²⋊C₄

—

112

C₄²

—

113

C₄²

—

114

C₄²

—

115

C₄²

C₂×Dic₃₇

yes

—

116

C₂³

C₂×C₂²⋊D₃₇

yes

characteristic

117

Q₈

—

118

Q₈

—

119

Q₈

—

120

Q₈

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

C₂²⋊Dic₃₇

yes

—

130

C₂×C₄

D₄×D₃₇

yes

—

131

C₂×C₄

#10596964

yes

—

132

C₂×C₄

C₂²×Dic₃₇

yes

characteristic

133

C₂×C₄

C₂×C₂²⋊D₃₇

yes

characteristic

134

C₈

—

148

—

135

C₂²

—

136

C₂²

C₂×C₂²⋊Dic₃₇

yes

characteristic

137

C₂²

#11924504

yes

characteristic

138

C₂²

#11924517

yes

characteristic

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

—

148

—

143

C₄

D₄×Dic₃₇

yes

—

144

C₄

#11924508

yes

—

145

C₂

—

146

C₂

—

147

C₂

#14557413

yes

characteristic

148

C₂

#14557715

yes

—

	subgroup	G/H
center	C₂²	#11924504
commutator	C₂×C₄×C₃₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₃₇
Fitting	C₂×C₄⋊Q₈×C₃₇	C₂
socle	C₂²×C₃₇	#120
prime radical	C₂³×C₃₇	C₂×D₄
cosocle	C₂²×C₄×C₃₇	C₂³
hypercenter	C₂×C₄⋊Q₈	D₃₇
hypocenter	C₃₇	#1886
LCS	G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂²

#11924504

commutator

C₂×C₄×C₃₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₃₇

Fitting

C₂×C₄⋊Q₈×C₃₇

C₂

socle

C₂²×C₃₇

#120

prime radical

C₂³×C₃₇

C₂×D₄

cosocle

C₂²×C₄×C₃₇

C₂³

hypercenter

C₂×C₄⋊Q₈

D₃₇

hypocenter

C₃₇

#1886

LCS

G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
15. C₄⋊Q₈×C₃₇	C₄	semidirect
80. C₃₇	#1886	semidirect
85. C₄⋊Q₈	Dic₃₇	semidirect

H ≅ G/N

type

15. C₄⋊Q₈×C₃₇

C₄

semidirect

80. C₃₇

#1886

semidirect

85. C₄⋊Q₈

Dic₃₇

semidirect

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

subgroup	index
#14558011	2
#14557542	2
C₂×#11924455	2
#14557293	2
C₂×C₄⋊Q₈×C₃₇	2
#1886	37

subgroup

index

C₂×C₄⋊Q₈×C₃₇

Subgroups — #17994990

Conjugacy classes of subgroups

Complements

Maximal subgroups