Group #18010388 — subgroups

Subgroups — #18010388

Order 4736.

296 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 134 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×#11924603	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#64438	C₂	1	2	yes	maximal
3	C₂×C₄×#64438	C₂	1	1	yes	maximal, characteristic
4	#14558399	C₂	1	2	yes	maximal
5	#14557384	C₂	1	1	yes	maximal, characteristic
6	C₂²×#64438	C₂²	1	2	yes	—
7	C₂²×#64438	C₂²	1	1	yes	characteristic
8	C₂²×#64438	C₂²	1	1	yes	characteristic
9	#11924603	C₄	1	4	yes	complemented
10	C₄×#64438	—	2	4	no	—
11	C₄×#64438	C₄	1	4	yes	complemented
12	C₂×#10596936	C₄	1	2	yes	—
13	C₂×C₄×Dic₃₇	C₂²	1	2	yes	—
14	C₂×C₄×Dic₃₇	C₂²	1	1	yes	characteristic
15	#11924443	C₄	1	1	yes	characteristic
16	#95×C₃₇	C₄	1	1	yes	characteristic
17	C₂²×C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₄×C₃₇	C₂×C₄	1	2	yes	—
19	C₂²×Dic₃₇	C₂×C₄	1	2	yes	—
20	C₂²×Dic₃₇	C₂×C₄	1	1	yes	characteristic
21	C₂²×Dic₃₇	C₂³	1	1	yes	characteristic
22	C₂×#64438	—	2	2	no	—
23	C₂×#64438	—	2	1	no	—
24	C₂×#64438	—	2	1	no	—
25	C₂×#64438	—	2	2	no	—
26	C₂×#64438	C₂×C₄	1	2	yes	—
27	C₂×#64438	C₂×C₄	1	4	yes	—
28	C₂×#64438	C₂×C₄	1	2	yes	—
29	C₂×#64438	D₄	1	2	yes	—
30	C₂×#64438	Q₈	1	2	yes	—
31	#10596936	C₈	1	8	yes	complemented
32	C₄×Dic₃₇	—	2	2	no	—
33	C₄×Dic₃₇	—	2	2	no	—
34	C₄×Dic₃₇	C₂×C₄	1	2	yes	—
35	C₄×Dic₃₇	C₂×C₄	1	2	yes	—
36	C₂³×C₃₇	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₄×C₃₇	—	2	2	no	—
38	C₂×C₄×C₃₇	—	2	2	no	—
39	C₂×C₄×C₃₇	—	2	2	no	—
40	C₂×C₄×C₃₇	C₄²	1	2	yes	—
41	C₂×C₄×C₃₇	C₂×C₈	1	4	yes	—
42	C₂×Dic₃₇	—	2	2	no	—
43	C₂×Dic₃₇	—	2	2	no	—
44	C₂×Dic₃₇	C₄²	1	2	yes	—
45	C₂×Dic₃₇	C₂×C₈	1	4	yes	—
46	C₂×Dic₃₇	C₂×C₈	1	4	yes	—
47	C₂×Dic₃₇	C₂²×C₄	1	1	yes	characteristic
48	C₂×Dic₃₇	C₂²×C₄	1	1	yes	characteristic
49	C₂×Dic₃₇	C₂×D₄	1	1	yes	characteristic
50	C₂×Dic₃₇	C₂×Q₈	1	1	yes	characteristic
51	C₂×Dic₃₇	#41	1	1	yes	characteristic
52	C₂×Dic₃₇	#41	1	1	yes	characteristic
53	#64438	—	2	2	no	—
54	#64438	—	2	4	no	—
55	#64438	C₄⋊C₄	1	4	yes	complemented
56	C₂²×C₃₇	C₂×C₄²	1	1	yes	characteristic
57	C₂²×C₃₇	C₄×D₄	1	1	yes	characteristic
58	C₂²×C₃₇	C₄×Q₈	1	1	yes	characteristic
59	C₂²×C₃₇	C₂²×C₈	1	2	yes	—
60	C₂²×C₃₇	#131	1	2	yes	—
61	C₄×C₃₇	—	2	2	no	—
62	C₄×C₃₇	—	2	4	no	—
63	Dic₃₇	—	2	4	no	—
64	Dic₃₇	C₄×C₈	1	4	yes	complemented
65	Dic₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
66	Dic₃₇	#117	1	1	yes	characteristic
67	Dic₃₇	C₄×D₄	1	1	yes	characteristic
68	Dic₃₇	C₄×Q₈	1	1	yes	characteristic
69	C₈×C₄⋊C₄	—	37	1	no	maximal, Sylow
70	C₂×C₃₇	C₄×C₄⋊C₄	1	1	yes	characteristic
71	C₂×C₃₇	C₂×C₄×C₈	1	1	yes	characteristic
72	C₂×C₃₇	#405	1	1	yes	characteristic
73	C₂×C₃₇	C₈×D₄	1	2	yes	—
74	C₂×C₃₇	C₈×Q₈	1	2	yes	—
75	C₂×C₄⋊C₈	—	37	1	no	—
76	C₂×C₄×C₈	—	37	1	no	—
77	C₂×C₄×C₈	—	37	2	no	—
78	C₄×C₄⋊C₄	—	37	1	no	—
79	#336	—	37	2	no	—
80	C₃₇	C₈×C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
81	C₂²×C₈	—	37	1	no	—
82	C₂²×C₈	—	37	1	no	—
83	C₂²×C₈	—	37	2	no	—
84	C₂×C₄⋊C₄	—	37	2	no	—
85	C₂×C₄²	—	37	1	no	—
86	C₂×C₄²	—	37	2	no	—
87	C₄⋊C₈	—	37	4	no	—
88	C₄×C₈	—	37	4	no	—
89	C₄×C₈	—	74	4	no	—
90	#95	—	37	1	no	—
91	#95	#3866	1	1	yes	characteristic
92	C₂²×C₄	—	37	1	no	—
93	C₂²×C₄	—	37	1	no	—
94	C₂²×C₄	—	37	2	no	—
95	C₂²×C₄	C₂×#3866	1	2	yes	—
96	C₂²×C₄	C₂×#3866	1	1	yes	characteristic
97	C₂×C₈	—	37	2	no	—
98	C₂×C₈	—	37	2	no	—
99	C₂×C₈	—	37	2	no	—
100	C₂×C₈	—	37	2	no	—
101	C₂×C₈	—	37	4	no	—
102	C₂×C₈	—	74	2	no	—
103	C₂×C₈	—	74	1	no	—
104	C₂×C₈	—	74	1	no	—
105	C₂×C₈	—	74	2	no	—
106	C₄⋊C₄	—	37	8	no	—
107	C₄²	—	37	2	no	—
108	C₄²	—	37	2	no	—
109	C₄²	—	74	2	no	—
110	C₄²	—	74	2	no	—
111	C₂³	C₂²×#3866	1	1	yes	characteristic
112	C₂×C₄	—	2	2	no	—
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	37	1	no	—
116	C₂×C₄	—	37	1	no	—
117	C₂×C₄	—	37	2	no	—
118	C₂×C₄	—	37	1	no	—
119	C₂×C₄	—	37	1	no	—
120	C₂×C₄	—	37	1	no	—
121	C₂×C₄	—	37	1	no	—
122	C₂×C₄	—	37	4	no	—
123	C₂×C₄	—	37	4	no	—
124	C₂×C₄	—	74	2	no	—
125	C₂×C₄	—	74	2	no	—
126	C₂×C₄	D₃₇⋊C₈	1	4	yes	—
127	C₂×C₄	C₄×#3866	1	2	yes	—
128	C₈	—	37	4	no	—
129	C₈	—	74	4	no	—
130	C₈	—	74	2	no	—
131	C₂²	C₂×C₄×#3866	1	1	yes	characteristic
132	C₂²	D₄×#3866	1	1	yes	characteristic
133	C₂²	Q₈×#3866	1	1	yes	characteristic
134	C₂²	C₂×D₃₇⋊C₈	1	2	yes	—
135	C₂²	#11924623	1	2	yes	—
136	C₄	—	2	2	no	—
137	C₄	—	2	4	no	—
138	C₄	—	37	1	no	—
139	C₄	—	37	1	no	—
140	C₄	—	37	1	no	—
141	C₄	—	37	1	no	—
142	C₄	—	37	4	no	—
143	C₄	—	74	4	no	—
144	C₂	C₄⋊C₄×#3866	1	1	yes	characteristic
145	C₂	C₄×D₃₇⋊C₈	1	1	yes	characteristic
146	C₂	#14558497	1	1	yes	characteristic
147	C₂	#14558560	1	2	yes	—
148	C₂	#14558582	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11924603

C₂

yes

maximal, characteristic

C₂×C₄×#64438

C₂

yes

maximal

C₂×C₄×#64438

C₂

yes

maximal, characteristic

#14558399

C₂

yes

maximal

#14557384

C₂

yes

maximal, characteristic

C₂²×#64438

C₂²

yes

—

C₂²×#64438

C₂²

yes

characteristic

C₂²×#64438

C₂²

yes

characteristic

#11924603

C₄

yes

complemented

C₄×#64438

—

C₄×#64438

C₄

yes

complemented

C₂×#10596936

C₄

yes

—

C₂×C₄×Dic₃₇

C₂²

yes

—

C₂×C₄×Dic₃₇

C₂²

yes

characteristic

#11924443

C₄

yes

characteristic

#95×C₃₇

C₄

yes

characteristic

C₂²×C₄×C₃₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₃₇

C₂×C₄

yes

—

C₂²×Dic₃₇

C₂×C₄

yes

—

C₂²×Dic₃₇

C₂×C₄

yes

characteristic

C₂²×Dic₃₇

C₂³

yes

—

—

—

—

C₂×C₄

yes

—

C₂×#64438

C₂×C₄

yes

—

C₂×#64438

C₂×C₄

yes

—

C₂×#64438

D₄

yes

—

C₂×#64438

Q₈

yes

—

#10596936

C₈

yes

complemented

C₄×Dic₃₇

—

C₄×Dic₃₇

—

C₄×Dic₃₇

C₂×C₄

yes

—

C₄×Dic₃₇

C₂×C₄

yes

—

C₂³×C₃₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₄²

yes

—

C₂×C₄×C₃₇

C₂×C₈

yes

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

C₄²

yes

—

C₂×Dic₃₇

C₂×C₈

yes

—

C₂×Dic₃₇

C₂×C₈

yes

—

C₂×Dic₃₇

C₂²×C₄

yes

characteristic

C₂×Dic₃₇

C₂²×C₄

yes

characteristic

C₂×Dic₃₇

C₂×D₄

yes

characteristic

C₂×Dic₃₇

C₂×Q₈

yes

C₂×Dic₃₇

yes

C₂×Dic₃₇

yes

—

—

C₄⋊C₄

yes

complemented

C₂²×C₃₇

C₂×C₄²

yes

characteristic

C₂²×C₃₇

C₄×D₄

yes

characteristic

C₂²×C₃₇

C₄×Q₈

yes

characteristic

C₂²×C₃₇

C₂²×C₈

yes

—

C₂²×C₃₇

#131

yes

—

C₄×C₃₇

—

C₄×C₃₇

—

Dic₃₇

—

Dic₃₇

C₄×C₈

yes

complemented

Dic₃₇

C₂×C₄⋊C₄

yes

characteristic

Dic₃₇

#117

yes

characteristic

Dic₃₇

C₄×D₄

yes

characteristic

Dic₃₇

C₄×Q₈

yes

characteristic

C₈×C₄⋊C₄

—

maximal, Sylow

C₂×C₃₇

C₄×C₄⋊C₄

yes

characteristic

C₂×C₃₇

C₂×C₄×C₈

yes

characteristic

C₂×C₃₇

#405

yes

characteristic

C₂×C₃₇

C₈×D₄

yes

—

C₂×C₃₇

C₈×Q₈

yes

—

C₂×C₄⋊C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₄×C₄⋊C₄

—

#336

—

C₃₇

C₈×C₄⋊C₄

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

—

C₄⋊C₈

—

C₄×C₈

—

C₄×C₈

—

—

yes

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×#3866

yes

—

C₂²×C₄

C₂×#3866

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₄⋊C₄

—

107

C₄²

—

108

C₄²

—

109

C₄²

—

110

C₄²

—

111

C₂³

C₂²×#3866

yes

characteristic

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

D₃₇⋊C₈

yes

—

127

C₂×C₄

C₄×#3866

yes

—

128

C₈

—

129

C₈

—

130

C₈

—

131

C₂²

C₂×C₄×#3866

yes

characteristic

132

C₂²

D₄×#3866

yes

characteristic

133

C₂²

Q₈×#3866

yes

characteristic

134

C₂²

C₂×D₃₇⋊C₈

yes

—

135

C₂²

#11924623

yes

—

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

—

144

C₂

C₄⋊C₄×#3866

yes

characteristic

145

C₂

C₄×D₃₇⋊C₈

yes

characteristic

146

C₂

#14558497

yes

characteristic

147

C₂

#14558560

yes

—

148

C₂

#14558582

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂²×#3866
commutator	C₂×C₃₇	C₂×C₄×C₈
Fitting	#95×C₃₇	C₄
socle, prime radical	C₂³×C₃₇	C₂²×C₄
cosocle	C₂²×Dic₃₇	C₂³
hypercenter	#95	#3866
hypocenter	C₃₇	C₈×C₄⋊C₄
LCS	G ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂³ ◁ #95

subgroup

G/H

center, Frattini

C₂³

C₂²×#3866

commutator

C₂×C₃₇

C₂×C₄×C₈

Fitting

#95×C₃₇

C₄

socle, prime radical

C₂³×C₃₇

C₂²×C₄

cosocle

C₂²×Dic₃₇

C₂³

C₃₇

C₈×C₄⋊C₄

LCS

G ▷ C₂×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂³ ◁ #95

N	H ≅ G/N	type
9. #11924603	C₄	semidirect
11. C₄×#64438	C₄	semidirect
31. #10596936	C₈	semidirect
55. #64438	C₄⋊C₄	semidirect
64. Dic₃₇	C₄×C₈	semidirect
80. C₃₇	C₈×C₄⋊C₄	semidirect

H ≅ G/N

type

9. #11924603

C₄

semidirect

11. C₄×#64438

C₄

semidirect

31. #10596936

C₈

semidirect

55. #64438

C₄⋊C₄

semidirect

64. Dic₃₇

C₄×C₈

semidirect

80. C₃₇

C₈×C₄⋊C₄

semidirect

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

subgroup	index
C₂×#11924603	2
C₂×C₄×#64438	2
#14558399	2
#14557384	2
C₈×C₄⋊C₄	37

subgroup

index

C₂×#11924603

C₂×C₄×#64438

#14558399

#14557384

C₈×C₄⋊C₄

Subgroups — #18010388

Conjugacy classes of subgroups

Complements

Maximal subgroups