Group #18010723 — subgroups

Subgroups — #18010723

Order 4736.

332 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 98 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×#11924599	C₂	1	4	yes	maximal
2	C₄×D₃₇⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#14558479	C₂	1	1	yes	maximal, characteristic
4	C₄⋊Q₈×D₃₇	C₂	1	1	yes	maximal, characteristic
5	C₂×D₃₇⋊C₈	C₂²	1	2	yes	—
6	C₂×C₄⋊#3866	—	2	2	no	—
7	C₂×C₄⋊#3866	C₂²	1	2	yes	—
8	#11924599	—	2	8	no	—
9	C₄×#64438	—	2	2	no	—
10	C₂×Q₈×D₃₇	C₂²	1	2	yes	—
11	#11924536	C₄	1	1	yes	characteristic, complemented
12	#11924535	—	2	2	no	—
13	#11924534	C₄	1	2	yes	complemented
14	C₄⋊C₄×D₃₇	—	2	2	no	—
15	C₄²×D₃₇	C₂²	1	1	yes	characteristic
16	C₄⋊Q₈×C₃₇	C₄	1	1	yes	characteristic, complemented
17	C₂²×#3866	—	2	2	no	—
18	C₂×Q₈×C₃₇	C₂×C₄	1	2	yes	—
19	Q₈×D₃₇	—	2	4	no	—
20	Q₈×D₃₇	—	2	4	no	—
21	C₂×C₄×D₃₇	—	2	2	no	—
22	C₂×C₄×D₃₇	D₄	1	2	yes	—
23	C₂×C₄×D₃₇	C₂³	1	1	yes	characteristic
24	C₂×Dic₇₄	—	2	2	no	—
25	C₂×Dic₇₄	C₂×C₄	1	2	yes	—
26	C₂×#64438	—	2	2	no	—
27	C₄⋊#3866	—	2	4	no	—
28	C₄⋊#3866	—	4	4	no	—
29	D₃₇⋊C₈	—	2	2	no	—
30	D₃₇⋊C₈	D₄	1	4	yes	—
31	C₄⋊C₄×C₃₇	—	2	2	no	—
32	C₄²×C₃₇	C₂×C₄	1	1	yes	characteristic
33	C₄⋊Dic₃₇	—	2	2	no	—
34	#10596936	—	2	4	no	—
35	C₄×Dic₃₇	C₂×C₄	1	1	yes	characteristic
36	C₄×Dic₃₇	D₄	1	2	yes	—
37	C₂²×D₃₇	C₂×D₄	1	1	yes	characteristic
38	C₂×#3866	—	4	2	no	—
39	C₂×#3866	—	4	4	no	—
40	Q₈×C₃₇	—	2	4	no	—
41	C₂×C₄×C₃₇	—	2	2	no	—
42	C₂×C₄×C₃₇	C₂²⋊C₄	1	2	yes	—
43	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
44	C₂×Dic₃₇	—	2	2	no	—
45	C₂×Dic₃₇	C₂²⋊C₄	1	2	yes	—
46	C₂×Dic₃₇	C₂×D₄	1	1	yes	characteristic
47	C₄×D₃₇	—	4	2	no	—
48	C₄×D₃₇	—	4	2	no	—
49	C₄×D₃₇	SD₁₆	1	4	yes	—
50	C₄×D₃₇	Q₁₆	1	4	yes	—
51	C₄×D₃₇	C₂×D₄	1	2	yes	—
52	C₄×D₃₇	#41	1	2	yes	—
53	Dic₇₄	—	2	4	no	—
54	Dic₇₄	—	4	4	no	—
55	#64438	—	2	4	no	—
56	C₂²×C₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
57	C₂×D₃₇	#124	1	1	yes	characteristic
58	C₂×D₃₇	#127	1	1	yes	characteristic
59	C₂×D₃₇	C₂×SD₁₆	1	2	yes	—
60	C₂×D₃₇	C₂×Q₁₆	1	2	yes	—
61	#3866	—	8	2	no	—
62	C₄×C₃₇	—	4	2	no	—
63	C₄×C₃₇	#103	1	4	yes	complemented
64	C₄×C₃₇	C₄×D₄	1	2	yes	—
65	Dic₃₇	—	4	2	no	—
66	Dic₃₇	#103	1	4	yes	complemented
67	Dic₃₇	#121	1	2	yes	—
68	#1956	—	37	1	no	maximal, Sylow
69	C₂×C₃₇	#390	1	1	yes	characteristic
70	C₂×C₃₇	C₂×#103	1	2	yes	—
71	D₃₇	#487	1	2	yes	—
72	D₃₇	#492	1	1	yes	characteristic
73	D₃₇	#494	1	1	yes	characteristic
74	C₂×C₄⋊Q₈	—	37	1	no	—
75	C₂×#103	—	37	4	no	—
76	C₂×C₄×C₈	—	37	1	no	—
77	#384	—	37	1	no	—
78	C₃₇	#1956	1	1	yes	characteristic, complemented, Sylow
79	C₂²×Q₈	—	37	2	no	—
80	C₂²×C₈	—	37	2	no	—
81	C₄⋊Q₈	—	37	2	no	—
82	C₄⋊Q₈	—	37	1	no	—
83	C₄⋊Q₈	—	74	2	no	—
84	C₄⋊Q₈	#3866	1	1	yes	characteristic, complemented
85	C₂×C₄⋊C₄	—	37	2	no	—
86	C₂×C₄⋊C₄	—	74	2	no	—
87	C₂×C₄⋊C₄	—	74	2	no	—
88	C₂×C₄²	—	37	1	no	—
89	#103	—	74	8	no	—
90	C₄×C₈	—	74	2	no	—
91	C₂×Q₈	—	37	2	no	—
92	C₂×Q₈	—	74	2	no	—
93	C₂×Q₈	—	74	4	no	—
94	C₂×Q₈	—	74	4	no	—
95	C₂×Q₈	C₂×#3866	1	2	yes	—
96	C₂²×C₄	—	37	2	no	—
97	C₂²×C₄	—	37	1	no	—
98	C₂²×C₄	—	74	2	no	—
99	C₂²×C₄	—	74	2	no	—
100	C₂×C₈	—	37	4	no	—
101	C₂×C₈	—	74	2	no	—
102	C₂×C₈	—	74	2	no	—
103	C₄⋊C₄	—	2	2	no	—
104	C₄⋊C₄	—	74	4	no	—
105	C₄⋊C₄	—	74	4	no	—
106	C₄⋊C₄	—	74	2	no	—
107	C₄⋊C₄	—	148	4	no	—
108	C₄²	—	37	2	no	—
109	C₄²	—	37	1	no	—
110	C₄²	C₂×#3866	1	1	yes	characteristic
111	C₂³	—	37	1	no	—
112	Q₈	—	2	4	no	—
113	Q₈	—	74	4	no	—
114	Q₈	—	148	4	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	37	2	no	—
117	C₂×C₄	—	37	1	no	—
118	C₂×C₄	—	37	4	no	—
119	C₂×C₄	—	37	4	no	—
120	C₂×C₄	—	37	2	no	—
121	C₂×C₄	—	37	2	no	—
122	C₂×C₄	—	74	2	no	—
123	C₂×C₄	—	148	2	no	—
124	C₂×C₄	—	148	2	no	—
125	C₂×C₄	—	148	2	no	—
126	C₂×C₄	—	148	4	no	—
127	C₂×C₄	C₂²⋊#3866	1	2	yes	—
128	C₂×C₄	C₂²×#3866	1	1	yes	characteristic
129	C₈	—	74	4	no	—
130	C₂²	—	37	1	no	—
131	C₂²	—	37	1	no	—
132	C₂²	—	37	2	no	—
133	C₂²	—	37	2	no	—
134	C₂²	C₂×C₂²⋊#3866	1	1	yes	characteristic
135	C₄	—	4	2	no	—
136	C₄	—	37	2	no	—
137	C₄	—	37	4	no	—
138	C₄	—	296	2	no	—
139	C₄	—	148	2	no	—
140	C₄	#11924599	1	4	yes	complemented
141	C₄	D₄×#3866	1	2	yes	—
142	C₂	—	37	1	no	—
143	C₂	—	37	2	no	—
144	C₂	—	37	1	no	—
145	C₂	#14558485	1	1	yes	characteristic
146	C₂	C₂×#11924599	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11924599

C₂

yes

maximal

C₄×D₃₇⋊C₈

C₂

yes

maximal, characteristic

#14558479

C₂

yes

maximal, characteristic

C₄⋊Q₈×D₃₇

C₂

yes

maximal, characteristic

C₂×D₃₇⋊C₈

C₂²

yes

—

C₂×C₄⋊#3866

—

C₂×C₄⋊#3866

C₂²

yes

—

#11924599

—

C₄×#64438

—

C₂×Q₈×D₃₇

C₂²

yes

—

#11924536

C₄

yes

characteristic, complemented

#11924535

—

#11924534

C₄

yes

complemented

C₄⋊C₄×D₃₇

—

C₄²×D₃₇

C₂²

yes

characteristic

C₄⋊Q₈×C₃₇

C₄

yes

characteristic, complemented

C₂²×#3866

—

C₂×Q₈×C₃₇

C₂×C₄

yes

—

Q₈×D₃₇

—

Q₈×D₃₇

—

C₂×C₄×D₃₇

—

C₂×C₄×D₃₇

D₄

yes

—

C₂×C₄×D₃₇

C₂³

yes

characteristic

C₂×Dic₇₄

—

C₂×Dic₇₄

C₂×C₄

yes

—

C₂×#64438

—

C₄⋊#3866

—

C₄⋊#3866

—

D₃₇⋊C₈

—

D₃₇⋊C₈

D₄

yes

—

C₄⋊C₄×C₃₇

—

C₄²×C₃₇

C₂×C₄

yes

characteristic

C₄⋊Dic₃₇

—

#10596936

—

C₄×Dic₃₇

C₂×C₄

yes

characteristic

C₄×Dic₃₇

D₄

yes

—

C₂²×D₃₇

C₂×D₄

yes

characteristic

C₂×#3866

—

C₂×#3866

—

Q₈×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₂²⋊C₄

yes

—

C₂×C₄×C₃₇

C₂²×C₄

yes

characteristic

C₂×Dic₃₇

—

C₂×Dic₃₇

C₂²⋊C₄

yes

—

C₂×Dic₃₇

C₂×D₄

yes

characteristic

C₄×D₃₇

—

C₄×D₃₇

—

C₄×D₃₇

SD₁₆

yes

—

C₄×D₃₇

Q₁₆

yes

—

C₄×D₃₇

C₂×D₄

yes

—

C₄×D₃₇

#41

yes

—

Dic₇₄

—

Dic₇₄

—

#64438

—

C₂²×C₃₇

C₂×C₂²⋊C₄

yes

C₂×D₃₇

yes

C₂×D₃₇

yes

C₂×D₃₇

C₂×SD₁₆

yes

—

C₂×D₃₇

C₂×Q₁₆

yes

—

#3866

—

C₄×C₃₇

—

C₄×C₃₇

#103

yes

complemented

C₄×C₃₇

C₄×D₄

yes

—

Dic₃₇

—

Dic₃₇

#103

yes

complemented

Dic₃₇

#121

yes

—

—

C₂×C₃₇

yes

C₂×C₃₇

yes

—

D₃₇

#487

yes

—

D₃₇

yes

D₃₇

yes

C₂×C₄⋊Q₈

—

C₂×#103

—

C₂×C₄×C₈

—

#384

—

C₃₇

#1956

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

#3866

yes

characteristic, complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

#103

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×#3866

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

148

—

108

C₄²

—

109

C₄²

—

110

C₄²

C₂×#3866

yes

characteristic

111

C₂³

—

112

Q₈

—

113

Q₈

—

114

Q₈

—

148

—

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

—

148

—

124

C₂×C₄

—

148

—

125

C₂×C₄

—

148

—

126

C₂×C₄

—

148

—

127

C₂×C₄

C₂²⋊#3866

yes

—

128

C₂×C₄

C₂²×#3866

yes

characteristic

129

C₈

—

130

C₂²

—

131

C₂²

—

132

C₂²

—

133

C₂²

—

134

C₂²

C₂×C₂²⋊#3866

yes

characteristic

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

—

296

—

139

C₄

—

148

—

140

C₄

#11924599

yes

complemented

141

C₄

D₄×#3866

yes

—

142

C₂

—

143

C₂

—

144

C₂

—

145

C₂

#14558485

yes

characteristic

146

C₂

C₂×#11924599

yes

—

	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	C₂×D₄
cosocle	C₂×C₄×D₃₇	C₂³
hypercenter	C₄⋊Q₈	#3866
hypocenter	C₃₇	#1956
LCS	G ▷ C₂×C₄×C₃₇ ▷ C₂×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

C₂×D₄

cosocle

C₂×C₄×D₃₇

C₂³

C₄⋊Q₈

C₃₇

G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

N	H ≅ G/N	type
11. #11924536	C₄	semidirect
13. #11924534	C₄	semidirect
16. C₄⋊Q₈×C₃₇	C₄	semidirect
63. C₄×C₃₇	#103	semidirect
66. Dic₃₇	#103	semidirect
78. C₃₇	#1956	semidirect
84. C₄⋊Q₈	#3866	semidirect
140. C₄	#11924599	semidirect

H ≅ G/N

type

11. #11924536

C₄

semidirect

13. #11924534

C₄

semidirect

16. C₄⋊Q₈×C₃₇

C₄

semidirect

63. C₄×C₃₇

#103

semidirect

66. Dic₃₇

#103

semidirect

78. C₃₇

#1956

semidirect

84. C₄⋊Q₈

#3866

semidirect

140. C₄

#11924599

semidirect

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

subgroup	index
C₂×#11924599	2
C₄×D₃₇⋊C₈	2
#14558479	2
C₄⋊Q₈×D₃₇	2
#1956	37

subgroup

index

C₂×#11924599

C₄×D₃₇⋊C₈

#14558479

C₄⋊Q₈×D₃₇

#1956

Subgroups — #18010723

Conjugacy classes of subgroups

Complements

Maximal subgroups