Group #18005037 — subgroups

Subgroups — C₂×#14557928 #18005037

Order 4736.

564 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 165 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	C₂²×#10596942	C₂	1	2	yes	maximal, complemented
2	C₂²×#10596940	C₂	1	2	yes	maximal
3	C₂×#11924536	C₂	1	1	yes	maximal, characteristic, complemented
4	#14557928	C₂	1	8	yes	maximal, direct product factor
5	C₂×#11924447	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#124×C₃₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₇₄	C₂²	1	2	yes	—
8	C₂×#10596942	—	2	8	no	—
9	C₂×#10596942	C₂²	1	8	yes	complemented
10	C₂×#10596940	—	2	8	no	—
11	C₂×#10596940	C₂²	1	8	yes	—
12	C₂²×#64436	C₂²	1	2	yes	—
13	#11924536	—	2	2	no	—
14	#11924536	C₂²	1	4	yes	complemented
15	C₂×C₄⋊Dic₃₇	—	2	2	no	—
16	#11924447	C₂²	1	4	yes	complemented
17	C₂²×Q₈×C₃₇	C₂²	1	1	yes	characteristic
18	C₂²×D₄×C₃₇	C₂²	1	1	yes	characteristic
19	#124×C₃₇	—	2	2	no	—
20	#124×C₃₇	C₂²	1	4	yes	—
21	C₂×C₂²⋊C₄×C₃₇	—	2	2	no	—
22	C₂×C₄²×C₃₇	C₂²	1	1	yes	characteristic
23	C₂⁴×C₃₇	—	2	1	no	—
24	C₂×Q₈×C₃₇	—	2	1	no	—
25	C₂×Q₈×C₃₇	—	2	4	no	—
26	C₂×Q₈×C₃₇	C₂³	1	2	yes	—
27	C₂×D₄×C₃₇	—	2	4	no	—
28	C₂×D₄×C₃₇	—	2	1	no	—
29	C₂×D₄×C₃₇	C₂³	1	2	yes	—
30	C₂²×C₄×C₃₇	—	2	1	no	—
31	C₂²×C₄×C₃₇	D₄	1	2	yes	—
32	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
33	C₂²×Dic₃₇	—	2	2	no	—
34	C₂×Dic₇₄	—	2	4	no	—
35	C₂×Dic₇₄	—	2	4	no	—
36	C₂×Dic₇₄	—	2	2	no	—
37	C₂×Dic₇₄	C₂³	1	4	yes	—
38	C₂²⋊C₄×C₃₇	—	2	8	no	—
39	C₄²×C₃₇	D₄	1	2	yes	—
40	C₄²×C₃₇	C₂³	1	2	yes	—
41	#10596942	—	2	16	no	—
42	#10596940	—	2	16	no	—
43	C₄⋊Dic₃₇	—	2	8	no	—
44	C₂×#64436	D₄	1	8	yes	—
45	C₂×#64436	C₂³	1	4	yes	—
46	C₂³×C₃₇	—	2	2	no	—
47	C₂³×C₃₇	—	4	2	no	—
48	C₂³×C₃₇	—	4	4	no	—
49	C₂³×C₃₇	C₂×D₄	1	1	yes	characteristic
50	Q₈×C₃₇	—	2	4	no	—
51	Q₈×C₃₇	—	4	2	no	—
52	D₄×C₃₇	—	2	4	no	—
53	D₄×C₃₇	—	4	2	no	—
54	C₂×C₄×C₃₇	—	2	4	no	—
55	C₂×C₄×C₃₇	—	2	2	no	—
56	C₂×C₄×C₃₇	—	4	2	no	—
57	C₂×C₄×C₃₇	C₂×D₄	1	4	yes	—
58	C₂×C₄×C₃₇	C₂×D₄	1	4	yes	—
59	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
60	C₂×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
61	C₂×Dic₃₇	—	2	4	no	—
62	C₂×Dic₃₇	—	4	4	no	—
63	Dic₇₄	—	2	4	no	—
64	Dic₇₄	—	2	4	no	—
65	Dic₇₄	—	4	4	no	—
66	#64436	C₂×D₄	1	8	yes	—
67	C₂²×C₃₇	—	4	2	no	—
68	C₂²×C₃₇	—	4	4	no	—
69	C₂²×C₃₇	—	8	4	no	—
70	C₂²×C₃₇	#127	1	2	yes	—
71	C₂²×C₃₇	#137	1	4	yes	—
72	C₂²×C₃₇	C₂²×D₄	1	1	yes	characteristic
73	C₄×C₃₇	—	2	4	no	—
74	C₄×C₃₇	—	4	2	no	—
75	C₄×C₃₇	#127	1	2	yes	—
76	C₄×C₃₇	C₂²×D₄	1	2	yes	—
77	Dic₃₇	—	4	4	no	—
78	C₂×#497	—	37	1	no	maximal, Sylow
79	C₂×C₃₇	—	8	2	no	—
80	C₂×C₃₇	#497	1	4	yes	complemented
81	C₂×C₃₇	C₂×#127	1	1	yes	characteristic
82	C₂×C₃₇	C₂×#137	1	2	yes	—
83	C₂²×Q₁₆	—	37	2	no	—
84	C₂²×SD₁₆	—	37	2	no	—
85	C₂×C₄⋊Q₈	—	37	1	no	—
86	C₂×#124	D₃₇	1	1	yes	characteristic
87	#497	—	37	8	no	—
88	C₂×#97	—	37	1	no	—
89	C₃₇	C₂×#497	1	1	yes	characteristic, complemented, Sylow
90	C₂²×Q₈	—	37	2	no	—
91	C₂²×Q₈	C₂×D₃₇	1	1	yes	characteristic
92	C₂²×D₄	C₂×D₃₇	1	1	yes	characteristic
93	C₂×Q₁₆	—	37	8	no	—
94	C₂×Q₁₆	—	74	8	no	—
95	C₂×SD₁₆	—	37	8	no	—
96	C₂×SD₁₆	—	74	8	no	—
97	C₂²×C₈	—	37	2	no	—
98	C₄⋊Q₈	—	37	4	no	—
99	C₄⋊Q₈	—	74	2	no	—
100	#124	—	2	2	no	—
101	#124	C₂×D₃₇	1	4	yes	—
102	C₂×C₄⋊C₄	—	74	2	no	—
103	C₂×C₂²⋊C₄	—	2	2	no	—
104	C₂×C₄²	C₂×D₃₇	1	1	yes	characteristic
105	#97	—	37	4	no	—
106	C₂⁴	—	2	1	no	—
107	C₂×Q₈	—	2	1	no	—
108	C₂×Q₈	—	2	4	no	—
109	C₂×Q₈	—	37	4	no	—
110	C₂×Q₈	—	74	2	no	—
111	C₂×Q₈	—	74	4	no	—
112	C₂×Q₈	—	74	4	no	—
113	C₂×Q₈	C₂²×D₃₇	1	2	yes	—
114	C₂×D₄	—	2	4	no	—
115	C₂×D₄	—	2	1	no	—
116	C₂×D₄	C₂²×D₃₇	1	2	yes	—
117	C₂²×C₄	—	2	1	no	—
118	C₂²×C₄	—	74	2	no	—
119	C₂²×C₄	C₂²⋊D₃₇	1	2	yes	—
120	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
121	Q₁₆	—	74	16	no	—
122	SD₁₆	—	74	16	no	—
123	C₂×C₈	—	37	8	no	—
124	C₂×C₈	—	37	4	no	—
125	C₄⋊C₄	—	74	8	no	—
126	C₂²⋊C₄	—	2	8	no	—
127	C₄²	C₂²⋊D₃₇	1	2	yes	—
128	C₄²	C₂²×D₃₇	1	2	yes	—
129	C₂³	—	2	2	no	—
130	C₂³	—	4	2	no	—
131	C₂³	—	4	4	no	—
132	C₂³	C₂×C₂²⋊D₃₇	1	1	yes	characteristic
133	Q₈	—	2	4	no	—
134	Q₈	—	4	2	no	—
135	Q₈	—	74	4	no	—
136	Q₈	—	74	4	no	—
137	Q₈	—	148	4	no	—
138	D₄	—	2	4	no	—
139	D₄	—	4	2	no	—
140	C₂×C₄	—	2	4	no	—
141	C₂×C₄	—	2	2	no	—
142	C₂×C₄	—	4	2	no	—
143	C₂×C₄	—	74	4	no	—
144	C₂×C₄	—	148	4	no	—
145	C₂×C₄	D₄×D₃₇	1	4	yes	—
146	C₂×C₄	C₂×C₂²⋊D₃₇	1	4	yes	—
147	C₂×C₄	C₂×C₂²⋊D₃₇	1	1	yes	characteristic
148	C₂×C₄	C₂³×D₃₇	1	1	yes	characteristic
149	C₈	—	37	8	no	—
150	C₂²	—	4	4	no	—
151	C₂²	—	4	2	no	—
152	C₂²	—	8	4	no	—
153	C₂²	#11924532	1	2	yes	—
154	C₂²	#11924566	1	4	yes	—
155	C₂²	C₂²×C₂²⋊D₃₇	1	1	yes	characteristic
156	C₄	—	2	4	no	—
157	C₄	—	4	2	no	—
158	C₄	—	148	4	no	—
159	C₄	#11924532	1	2	yes	—
160	C₄	C₂×D₄×D₃₇	1	2	yes	—
161	C₂	—	8	2	no	—
162	C₂	#14557928	1	4	yes	direct product factor
163	C₂	C₂×#11924532	1	1	yes	characteristic
164	C₂	C₂×#11924566	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#10596942

C₂

yes

maximal, complemented

C₂²×#10596940

C₂

yes

maximal

C₂×#11924536

C₂

yes

maximal, characteristic, complemented

#14557928

C₂

yes

maximal, direct product factor

C₂×#11924447

C₂

yes

maximal, characteristic, complemented

C₂×#124×C₃₇

C₂

yes

maximal, characteristic

C₂²×Dic₇₄

C₂²

yes

—

C₂×#10596942

—

C₂×#10596942

C₂²

yes

complemented

C₂×#10596940

—

C₂×#10596940

C₂²

yes

—

C₂²×#64436

C₂²

yes

—

#11924536

—

#11924536

C₂²

yes

complemented

C₂×C₄⋊Dic₃₇

—

#11924447

C₂²

yes

complemented

C₂²×Q₈×C₃₇

C₂²

yes

characteristic

C₂²×D₄×C₃₇

C₂²

yes

characteristic

#124×C₃₇

—

#124×C₃₇

C₂²

yes

—

C₂×C₂²⋊C₄×C₃₇

—

C₂×C₄²×C₃₇

C₂²

yes

characteristic

C₂⁴×C₃₇

—

C₂×Q₈×C₃₇

—

C₂×Q₈×C₃₇

—

C₂×Q₈×C₃₇

C₂³

yes

—

C₂×D₄×C₃₇

—

C₂×D₄×C₃₇

—

C₂×D₄×C₃₇

C₂³

yes

—

C₂²×C₄×C₃₇

—

C₂²×C₄×C₃₇

D₄

yes

—

C₂²×C₄×C₃₇

C₂³

yes

characteristic

C₂²×Dic₃₇

—

C₂×Dic₇₄

—

C₂×Dic₇₄

—

C₂×Dic₇₄

—

C₂×Dic₇₄

C₂³

yes

—

C₂²⋊C₄×C₃₇

—

C₄²×C₃₇

D₄

yes

—

C₄²×C₃₇

C₂³

yes

—

#10596942

—

#10596940

—

C₄⋊Dic₃₇

—

C₂×#64436

D₄

yes

—

C₂×#64436

C₂³

yes

—

C₂³×C₃₇

—

C₂³×C₃₇

—

C₂³×C₃₇

—

C₂³×C₃₇

C₂×D₄

yes

characteristic

Q₈×C₃₇

—

Q₈×C₃₇

—

D₄×C₃₇

—

D₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₂×D₄

yes

—

C₂×C₄×C₃₇

C₂×D₄

yes

—

C₂×C₄×C₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₃₇

C₂⁴

yes

characteristic

C₂×Dic₃₇

—

C₂×Dic₃₇

—

Dic₇₄

—

Dic₇₄

—

Dic₇₄

—

#64436

C₂×D₄

yes

—

C₂²×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

#127

yes

—

C₂²×C₃₇

#137

yes

—

C₂²×C₃₇

C₂²×D₄

yes

characteristic

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

#127

yes

—

C₄×C₃₇

C₂²×D₄

yes

—

Dic₃₇

—

C₂×#497

—

maximal, Sylow

C₂×C₃₇

—

C₂×C₃₇

yes

C₂×C₃₇

yes

C₂×C₃₇

yes

—

C₂²×Q₁₆

—

C₂²×SD₁₆

—

C₂×C₄⋊Q₈

—

C₂×#124

D₃₇

yes

characteristic

#497

—

C₂×#97

—

C₃₇

C₂×#497

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×Q₈

C₂×D₃₇

yes

characteristic

C₂²×D₄

C₂×D₃₇

yes

characteristic

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×SD₁₆

—

C₂×SD₁₆

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

100

#124

—

101

#124

C₂×D₃₇

yes

—

102

C₂×C₄⋊C₄

—

103

C₂×C₂²⋊C₄

—

104

C₂×C₄²

C₂×D₃₇

yes

characteristic

105

#97

—

106

C₂⁴

—

107

C₂×Q₈

—

108

C₂×Q₈

—

109

C₂×Q₈

—

110

C₂×Q₈

—

111

C₂×Q₈

—

112

C₂×Q₈

—

113

C₂×Q₈

C₂²×D₃₇

yes

—

114

C₂×D₄

—

115

C₂×D₄

—

116

C₂×D₄

C₂²×D₃₇

yes

—

117

C₂²×C₄

—

118

C₂²×C₄

—

119

C₂²×C₄

C₂²⋊D₃₇

yes

—

120

C₂²×C₄

C₂²×D₃₇

yes

characteristic

121

Q₁₆

—

122

SD₁₆

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₄⋊C₄

—

126

C₂²⋊C₄

—

127

C₄²

C₂²⋊D₃₇

yes

—

128

C₄²

C₂²×D₃₇

yes

—

129

C₂³

—

130

C₂³

—

131

C₂³

—

132

C₂³

C₂×C₂²⋊D₃₇

yes

characteristic

133

Q₈

—

134

Q₈

—

135

Q₈

—

136

Q₈

—

137

Q₈

—

148

—

138

D₄

—

139

D₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

148

—

145

C₂×C₄

D₄×D₃₇

yes

—

146

C₂×C₄

C₂×C₂²⋊D₃₇

yes

—

147

C₂×C₄

C₂×C₂²⋊D₃₇

yes

characteristic

148

C₂×C₄

C₂³×D₃₇

yes

characteristic

149

C₈

—

150

C₂²

—

151

C₂²

—

152

C₂²

—

153

C₂²

#11924532

yes

—

154

C₂²

#11924566

yes

—

155

C₂²

C₂²×C₂²⋊D₃₇

yes

characteristic

156

C₄

—

157

C₄

—

158

C₄

—

148

—

159

C₄

#11924532

yes

—

160

C₄

C₂×D₄×D₃₇

yes

—

161

C₂

—

162

C₂

#14557928

yes

direct product factor

163

C₂

C₂×#11924532

yes

characteristic

164

C₂

C₂×#11924566

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₃₇
commutator, cosocle	C₂×C₄×C₃₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₃₇
Fitting	C₂×#124×C₃₇	C₂
socle	C₂³×C₃₇	C₂×D₄
prime radical	C₂²×D₄×C₃₇	C₂²
hypercenter	C₂×#124	D₃₇
hypocenter	C₃₇	C₂×#497
LCS	G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#124

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₃₇

commutator, cosocle

C₂×C₄×C₃₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₃₇

Fitting

C₂×#124×C₃₇

C₂

socle

C₂³×C₃₇

C₂×D₄

prime radical

C₂²×D₄×C₃₇

C₂²

D₃₇

C₃₇

G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#124

N	H ≅ G/N	type
1. C₂²×#10596942	C₂	semidirect
3. C₂×#11924536	C₂	semidirect
4. #14557928	C₂	direct (N × H)
5. C₂×#11924447	C₂	semidirect
9. C₂×#10596942	C₂²	semidirect
14. #11924536	C₂²	semidirect
16. #11924447	C₂²	semidirect
80. C₂×C₃₇	#497	semidirect
89. C₃₇	C₂×#497	semidirect
162. C₂	#14557928	direct (N × H)

H ≅ G/N

type

1. C₂²×#10596942

C₂

semidirect

3. C₂×#11924536

C₂

semidirect

4. #14557928

C₂

direct (N × H)

5. C₂×#11924447

C₂

semidirect

9. C₂×#10596942

C₂²

semidirect

14. #11924536

C₂²

semidirect

16. #11924447

C₂²

semidirect

80. C₂×C₃₇

#497

semidirect

89. C₃₇

C₂×#497

semidirect

162. C₂

#14557928

direct (N × H)

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

subgroup	index
C₂²×#10596942	2
C₂²×#10596940	2
C₂×#11924536	2
#14557928	2
C₂×#11924447	2
C₂×#124×C₃₇	2
C₂×#497	37

subgroup

index

C₂×#124×C₃₇

Subgroups — C₂×#14557928 #18005037

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#14557928 #18005037

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#14557928 #18005037