Group #18010112 — subgroups

Subgroups — #18010112

Order 4736.

240 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 70 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	#14558551	C₂	1	1	yes	maximal, characteristic
2	#14558538	C₂	1	1	yes	maximal, characteristic
3	#14558392	C₂	1	4	yes	maximal, complemented
4	C₄⋊Q₈×D₃₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#10596953	—	2	1	no	—
6	#11924603	—	2	1	no	—
7	#11924603	C₂²	1	2	yes	—
8	#11924603	C₂²	1	2	yes	—
9	#11924593	—	2	1	no	—
10	C₄×#64438	—	2	1	no	—
11	C₂×Q₈×D₃₇	—	2	1	no	—
12	#11924536	C₄	1	1	yes	characteristic
13	#11924535	—	2	1	no	—
14	#11924535	C₂²	1	2	yes	—
15	#11924534	—	2	1	no	—
16	C₄⋊C₄×D₃₇	—	2	1	no	—
17	C₄⋊C₄×D₃₇	C₄	1	2	yes	—
18	C₄²×D₃₇	C₂²	1	1	yes	characteristic
19	C₄⋊Q₈×C₃₇	C₄	1	1	yes	characteristic
20	C₂×Q₈×C₃₇	—	2	1	no	—
21	Q₈×D₃₇	—	4	4	no	—
22	C₂×C₄×D₃₇	—	2	2	no	—
23	C₂×C₄×D₃₇	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×D₃₇	D₄	1	1	yes	characteristic
25	C₂×C₄×D₃₇	D₄	1	1	yes	characteristic
26	C₂×Dic₇₄	—	2	1	no	—
27	C₂×Dic₇₄	—	2	2	no	—
28	C₂×#64438	—	2	2	no	—
29	C₂×#64438	—	2	2	no	—
30	#10596953	—	4	2	no	—
31	C₄⋊C₄×C₃₇	—	2	1	no	—
32	C₄⋊C₄×C₃₇	C₂×C₄	1	2	yes	—
33	C₄²×C₃₇	C₂×C₄	1	1	yes	characteristic
34	C₄⋊Dic₃₇	—	2	1	no	—
35	C₄⋊Dic₃₇	C₂×C₄	1	2	yes	—
36	#10596936	—	2	2	no	—
37	#10596936	—	2	2	no	—
38	C₄×Dic₃₇	D₄	1	1	yes	characteristic
39	C₄×Dic₃₇	D₄	1	1	yes	characteristic
40	C₄×Dic₃₇	C₂³	1	1	yes	characteristic
41	C₂²×D₃₇	C₂²⋊C₄	1	1	yes	characteristic
42	Q₈×C₃₇	—	4	2	no	—
43	C₂×C₄×C₃₇	—	2	2	no	—
44	C₂×C₄×C₃₇	C₂²⋊C₄	1	1	yes	characteristic
45	C₂×C₄×C₃₇	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
47	C₂×Dic₃₇	—	2	2	no	—
48	C₂×Dic₃₇	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×Dic₃₇	C₂×D₄	1	1	yes	characteristic
50	C₂×Dic₃₇	C₂×D₄	1	1	yes	characteristic
51	C₄×D₃₇	—	2	2	no	—
52	C₄×D₃₇	—	2	2	no	—
53	C₄×D₃₇	—	4	4	no	—
54	C₄×D₃₇	SD₁₆	1	2	yes	—
55	C₄×D₃₇	Q₁₆	1	2	yes	—
56	Dic₇₄	—	4	2	no	—
57	Dic₇₄	—	4	4	no	—
58	#64438	—	4	2	no	—
59	#64438	—	4	2	no	—
60	C₂²×C₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
61	C₂×D₃₇	—	2	1	no	—
62	C₂×D₃₇	—	2	1	no	—
63	C₂×D₃₇	#103	1	2	yes	—
64	C₄×C₃₇	—	2	1	no	—
65	C₄×C₃₇	—	2	1	no	—
66	C₄×C₃₇	—	4	2	no	—
67	C₄×C₃₇	#103	1	2	yes	—
68	Dic₃₇	—	4	2	no	—
69	Dic₃₇	#101	1	2	yes	—
70	Dic₃₇	C₂×SD₁₆	1	1	yes	characteristic
71	Dic₃₇	C₂×Q₁₆	1	1	yes	characteristic
72	Dic₃₇	AGL(1,ℤ/8)	1	2	yes	—
73	#1549	—	37	1	no	maximal, Sylow
74	C₂×C₃₇	C₂×#101	1	1	yes	characteristic
75	C₂×C₃₇	C₂×#103	1	1	yes	characteristic
76	C₂×C₃₇	#418	1	1	yes	characteristic
77	D₃₇	—	2	2	no	—
78	C₂×C₄⋊Q₈	—	37	1	no	—
79	#431	—	37	1	no	—
80	#423	—	37	1	no	—
81	#332	—	37	4	no	—
82	C₃₇	#1549	1	1	yes	characteristic, complemented, Sylow
83	C₂²×Q₈	—	74	1	no	—
84	C₂×M₁₆	—	74	1	no	—
85	C₄⋊Q₈	—	37	2	no	—
86	C₄⋊Q₈	—	37	1	no	—
87	C₄⋊Q₈	—	74	1	no	—
88	C₄⋊Q₈	—	74	1	no	—
89	C₄⋊Q₈	#3866	1	1	yes	characteristic
90	C₂×C₄⋊C₄	—	37	2	no	—
91	C₂×C₄⋊C₄	—	74	1	no	—
92	C₂×C₄²	—	37	1	no	—
93	C₄⋊C₈	—	37	2	no	—
94	C₄⋊C₈	—	37	2	no	—
95	C₄⋊C₈	—	74	1	no	—
96	C₂²⋊C₈	—	74	1	no	—
97	C₄×C₈	—	74	1	no	—
98	C₂×Q₈	—	2	1	no	—
99	C₂×Q₈	—	74	2	no	—
100	C₂×Q₈	—	74	1	no	—
101	C₂×Q₈	—	148	4	no	—
102	C₂²×C₄	—	37	1	no	—
103	C₂²×C₄	—	37	1	no	—
104	C₂²×C₄	—	37	1	no	—
105	C₂²×C₄	—	74	2	no	—
106	M₁₆	—	148	2	no	—
107	C₂×C₈	—	74	2	no	—
108	C₂×C₈	—	74	2	no	—
109	C₄⋊C₄	—	2	1	no	—
110	C₄⋊C₄	—	37	2	no	—
111	C₄⋊C₄	—	74	2	no	—
112	C₄⋊C₄	—	74	2	no	—
113	C₄⋊C₄	—	74	1	no	—
114	C₄⋊C₄	C₂×#3866	1	2	yes	—
115	C₄²	—	37	1	no	—
116	C₄²	—	37	1	no	—
117	C₄²	—	37	1	no	—
118	C₄²	C₂×#3866	1	1	yes	characteristic
119	C₂³	—	37	1	no	—
120	Q₈	—	4	2	no	—
121	Q₈	—	148	2	no	—
122	Q₈	—	148	4	no	—
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	37	1	no	—
125	C₂×C₄	—	37	1	no	—
126	C₂×C₄	—	37	1	no	—
127	C₂×C₄	—	37	2	no	—
128	C₂×C₄	—	37	2	no	—
129	C₂×C₄	—	74	2	no	—
130	C₂×C₄	—	74	2	no	—
131	C₂×C₄	—	74	2	no	—
132	C₂×C₄	—	148	4	no	—
133	C₂×C₄	C₂²⋊#3866	1	1	yes	characteristic
134	C₂×C₄	C₂²⋊#3866	1	1	yes	characteristic
135	C₂×C₄	C₂²×#3866	1	1	yes	characteristic
136	C₈	—	148	2	no	—
137	C₈	—	148	2	no	—
138	C₂²	—	37	2	no	—
139	C₂²	—	74	1	no	—
140	C₂²	—	74	1	no	—
141	C₂²	C₂×C₂²⋊#3866	1	1	yes	characteristic
142	C₄	—	2	1	no	—
143	C₄	—	2	1	no	—
144	C₄	—	4	2	no	—
145	C₄	—	37	2	no	—
146	C₄	—	37	1	no	—
147	C₄	—	37	1	no	—
148	C₄	—	37	2	no	—
149	C₄	—	148	2	no	—
150	C₄	#11924599	1	2	yes	—
151	C₂	—	74	2	no	—
152	C₂	D₃₇⋊#101	1	1	yes	characteristic
153	C₂	C₂×#11924599	1	1	yes	characteristic
154	C₂	#14558525	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14558551

C₂

yes

maximal, characteristic

#14558538

C₂

yes

maximal, characteristic

#14558392

C₂

yes

maximal, complemented

C₄⋊Q₈×D₃₇

C₂

yes

maximal, characteristic

C₂×#10596953

—

#11924603

—

#11924603

C₂²

yes

—

#11924603

C₂²

yes

—

#11924593

—

C₄×#64438

—

C₂×Q₈×D₃₇

—

#11924536

C₄

yes

characteristic

#11924535

—

#11924535

C₂²

yes

—

#11924534

—

C₄⋊C₄×D₃₇

—

C₄⋊C₄×D₃₇

C₄

yes

—

C₄²×D₃₇

C₂²

yes

characteristic

C₄⋊Q₈×C₃₇

C₄

yes

characteristic

C₂×Q₈×C₃₇

—

Q₈×D₃₇

—

C₂×C₄×D₃₇

—

C₂×C₄×D₃₇

C₂×C₄

yes

characteristic

C₂×C₄×D₃₇

D₄

yes

characteristic

C₂×C₄×D₃₇

D₄

yes

characteristic

C₂×Dic₇₄

—

C₂×Dic₇₄

—

C₂×#64438

—

C₂×#64438

—

#10596953

—

C₄⋊C₄×C₃₇

—

C₄⋊C₄×C₃₇

C₂×C₄

yes

—

C₄²×C₃₇

C₂×C₄

yes

characteristic

C₄⋊Dic₃₇

—

C₄⋊Dic₃₇

C₂×C₄

yes

—

#10596936

—

#10596936

—

C₄×Dic₃₇

D₄

yes

characteristic

C₄×Dic₃₇

D₄

yes

characteristic

C₄×Dic₃₇

C₂³

yes

characteristic

C₂²×D₃₇

C₂²⋊C₄

yes

characteristic

Q₈×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₃₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₃₇

C₂²×C₄

yes

characteristic

C₂×Dic₃₇

—

C₂×Dic₃₇

C₂²⋊C₄

yes

characteristic

C₂×Dic₃₇

C₂×D₄

yes

characteristic

C₂×Dic₃₇

C₂×D₄

yes

characteristic

C₄×D₃₇

—

C₄×D₃₇

—

C₄×D₃₇

—

C₄×D₃₇

SD₁₆

yes

—

C₄×D₃₇

Q₁₆

yes

—

Dic₇₄

—

Dic₇₄

—

#64438

—

#64438

—

C₂²×C₃₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₃₇

—

C₂×D₃₇

—

C₂×D₃₇

#103

yes

—

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

#103

yes

—

Dic₃₇

—

Dic₃₇

#101

yes

—

Dic₃₇

C₂×SD₁₆

yes

characteristic

Dic₃₇

C₂×Q₁₆

yes

characteristic

Dic₃₇

AGL(1,ℤ/8)

yes

—

—

C₂×C₃₇

yes

C₂×C₃₇

yes

C₂×C₃₇

yes

D₃₇

—

C₂×C₄⋊Q₈

—

#431

—

#423

—

#332

—

C₃₇

#1549

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂×M₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

#3866

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

100

C₂×Q₈

—

101

C₂×Q₈

—

148

—

102

C₂²×C₄

—

103

C₂²×C₄

—

104

C₂²×C₄

—

105

C₂²×C₄

—

106

M₁₆

—

148

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

C₂×#3866

yes

—

115

C₄²

—

116

C₄²

—

117

C₄²

—

118

C₄²

C₂×#3866

yes

characteristic

119

C₂³

—

120

Q₈

—

121

Q₈

—

148

—

122

Q₈

—

148

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

148

—

133

C₂×C₄

C₂²⋊#3866

yes

characteristic

134

C₂×C₄

C₂²⋊#3866

yes

characteristic

135

C₂×C₄

C₂²×#3866

yes

characteristic

136

C₈

—

148

—

137

C₈

—

148

—

138

C₂²

—

139

C₂²

—

140

C₂²

—

141

C₂²

C₂×C₂²⋊#3866

yes

characteristic

142

C₄

—

143

C₄

—

144

C₄

—

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

—

148

—

150

C₄

#11924599

yes

—

151

C₂

—

152

C₂

D₃₇⋊#101

yes

characteristic

153

C₂

C₂×#11924599

yes

characteristic

154

C₂

#14558525

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊#3866
commutator	C₂×C₄×C₃₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×#3866
Fitting	C₄⋊Q₈×C₃₇	C₄
socle	C₂²×C₃₇	C₂×C₂²⋊C₄
prime radical	C₂²×D₃₇	C₂²⋊C₄
cosocle	C₄×Dic₃₇	C₂³
hypercenter	C₄⋊Q₈	#3866
hypocenter	C₃₇	#1549
LCS	G ▷ C₂×C₄×C₃₇ ▷ C₂²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

subgroup

G/H

center

C₂²

C₂×C₂²⋊#3866

commutator

C₂×C₄×C₃₇

C₂²×C₄

Frattini

C₂×C₄

C₂²×#3866

Fitting

C₄⋊Q₈×C₃₇

C₄

socle

C₂²×C₃₇

C₂×C₂²⋊C₄

prime radical

C₂²×D₃₇

C₂²⋊C₄

cosocle

C₄×Dic₃₇

C₂³

C₄⋊Q₈

C₃₇

G ▷ C₂×C₄×C₃₇ ▷ C₂²×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

N	H ≅ G/N	type
3. #14558392	C₂	semidirect
82. C₃₇	#1549	semidirect

H ≅ G/N

type

3. #14558392

C₂

semidirect

82. C₃₇

#1549

semidirect

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

subgroup	index
#14558551	2
#14558538	2
#14558392	2
C₄⋊Q₈×D₃₇	2
#1549	37

subgroup

index

C₄⋊Q₈×D₃₇

Subgroups — #18010112

Conjugacy classes of subgroups

Complements

Maximal subgroups