Group #18005025 — subgroups

Subgroups — #18005025

Order 4736.

572 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 145 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×#11924559	C₂	1	4	yes	maximal, complemented
2	#14558084	C₂	1	2	yes	maximal, complemented
3	#14557918	C₂	1	4	yes	maximal, complemented
4	#14557910	C₂	1	1	yes	maximal, characteristic, complemented
5	#14557907	C₂	1	2	yes	maximal, complemented
6	C₄×C₈×D₃₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₇×#494	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#10596966	C₂²	1	4	yes	—
9	#11924559	—	2	16	no	—
10	C₂×#10596941	C₂²	1	8	yes	—
11	C₂×D₂₉₆	C₂²	1	4	yes	—
12	C₂×C₈×D₃₇	C₂²	1	2	yes	—
13	#11924533	C₂²	1	2	yes	—
14	#11924525	C₂²	1	4	yes	—
15	#11924511	—	2	4	no	—
16	#11924498	—	2	4	no	—
17	C₄²×D₃₇	C₂²	1	1	yes	characteristic
18	C₄×#64436	C₂²	1	1	yes	characteristic
19	C₈×Dic₃₇	C₂²	1	2	yes	—
20	C₂×Q₁₆×C₃₇	C₂²	1	4	yes	—
21	C₄⋊Q₈×C₃₇	C₂²	1	2	yes	—
22	C₄×C₈×C₃₇	C₂²	1	1	yes	characteristic
23	C₂×Q₈×C₃₇	C₂³	1	4	yes	—
24	#10596966	—	2	16	no	—
25	C₂×D₁₄₈	—	2	4	no	—
26	C₂×D₁₄₈	—	2	4	no	—
27	C₂×D₁₄₈	C₂³	1	4	yes	—
28	C₂×C₄×D₃₇	—	2	4	no	—
29	C₂×C₄×D₃₇	D₄	1	2	yes	—
30	C₂×C₄×D₃₇	C₂³	1	1	yes	characteristic
31	Q₁₆×C₃₇	—	2	8	no	—
32	C₂×C₈×C₃₇	C₂³	1	2	yes	—
33	C₄⋊C₄×C₃₇	—	2	4	no	—
34	C₄²×C₃₇	C₂³	1	1	yes	characteristic
35	#10596941	—	2	16	no	—
36	#10596938	—	2	8	no	—
37	C₄×Dic₃₇	D₄	1	2	yes	—
38	C₄×Dic₃₇	C₂³	1	1	yes	characteristic
39	C₂×#64436	C₂³	1	2	yes	—
40	D₂₉₆	—	2	8	no	—
41	C₈×D₃₇	D₄	1	8	yes	complemented
42	C₂²×D₃₇	—	2	4	no	—
43	C₂²×D₃₇	C₂×D₄	1	1	yes	characteristic
44	Q₈×C₃₇	—	2	8	no	—
45	C₂×C₄×C₃₇	—	2	4	no	—
46	C₂×C₄×C₃₇	C₂×D₄	1	2	yes	—
47	C₂×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
48	C₂×Dic₃₇	C₂×D₄	1	2	yes	—
49	C₂×Dic₃₇	C₂×D₄	1	1	yes	characteristic
50	D₁₄₈	—	2	8	no	—
51	D₁₄₈	—	4	8	no	—
52	D₁₄₈	—	4	8	no	—
53	C₄×D₃₇	—	2	4	no	—
54	C₄×D₃₇	—	4	8	no	—
55	C₄×D₃₇	C₂×D₄	1	4	yes	—
56	C₈×C₃₇	C₂×D₄	1	4	yes	—
57	#64436	C₂×D₄	1	4	yes	—
58	C₂²×C₃₇	C₂²×D₄	1	1	yes	characteristic
59	C₂×D₃₇	—	2	2	no	—
60	C₂×D₃₇	—	4	4	no	—
61	C₂×D₃₇	—	4	8	no	—
62	C₂×D₃₇	#127	1	2	yes	—
63	C₄×C₃₇	—	4	4	no	—
64	C₄×C₃₇	#135	1	4	yes	complemented
65	C₄×C₃₇	C₂²×D₄	1	2	yes	—
66	Dic₃₇	—	2	2	no	—
67	Dic₃₇	#127	1	2	yes	—
68	#3147	—	37	1	no	maximal, Sylow
69	C₂×C₃₇	C₂×#127	1	1	yes	characteristic
70	C₂×C₃₇	C₂×#135	1	2	yes	—
71	D₃₇	—	2	2	no	—
72	D₃₇	—	8	4	no	—
73	C₂×#135	—	37	4	no	—
74	#532	—	37	2	no	—
75	#495	—	37	4	no	—
76	#494	D₃₇	1	1	yes	characteristic, complemented
77	#493	—	37	1	no	—
78	#492	—	37	2	no	—
79	C₂×C₄×C₈	—	37	1	no	—
80	C₃₇	#3147	1	1	yes	characteristic, complemented, Sylow
81	C₂×#41	—	37	4	no	—
82	#135	—	74	16	no	—
83	C₂×Q₁₆	C₂×D₃₇	1	4	yes	—
84	C₂×SD₁₆	—	37	8	no	—
85	C₂×D₈	—	37	4	no	—
86	C₂²×C₈	—	37	2	no	—
87	C₄⋊Q₈	C₂×D₃₇	1	2	yes	—
88	#127	—	37	2	no	—
89	#124	—	37	4	no	—
90	#121	—	74	4	no	—
91	C₄×D₄	—	74	4	no	—
92	C₂×C₄²	—	37	1	no	—
93	C₄×C₈	—	37	2	no	—
94	C₄×C₈	—	37	1	no	—
95	C₄×C₈	C₂×D₃₇	1	1	yes	characteristic
96	#41	—	74	16	no	—
97	C₂×Q₈	C₂²×D₃₇	1	4	yes	—
98	C₂×D₄	—	37	4	no	—
99	C₂×D₄	—	74	4	no	—
100	C₂×D₄	—	74	4	no	—
101	C₂²×C₄	—	37	1	no	—
102	C₂²×C₄	—	37	2	no	—
103	C₂²×C₄	—	74	4	no	—
104	Q₁₆	—	2	8	no	—
105	SD₁₆	—	74	16	no	—
106	D₈	—	74	8	no	—
107	C₂×C₈	—	37	2	no	—
108	C₂×C₈	—	37	8	no	—
109	C₂×C₈	C₂²×D₃₇	1	2	yes	—
110	C₄⋊C₄	—	2	4	no	—
111	C₂²⋊C₄	—	74	8	no	—
112	C₄²	—	37	2	no	—
113	C₄²	—	37	1	no	—
114	C₄²	C₂²×D₃₇	1	1	yes	characteristic
115	C₂³	—	37	1	no	—
116	C₂³	—	74	4	no	—
117	Q₈	—	2	8	no	—
118	D₄	—	74	8	no	—
119	D₄	—	148	8	no	—
120	D₄	—	148	8	no	—
121	C₂×C₄	—	2	4	no	—
122	C₂×C₄	—	37	1	no	—
123	C₂×C₄	—	37	4	no	—
124	C₂×C₄	—	37	2	no	—
125	C₂×C₄	—	74	4	no	—
126	C₂×C₄	—	148	8	no	—
127	C₂×C₄	D₄×D₃₇	1	2	yes	—
128	C₂×C₄	C₂³×D₃₇	1	1	yes	characteristic
129	C₈	—	37	4	no	—
130	C₈	D₄×D₃₇	1	4	yes	—
131	C₂²	—	37	2	no	—
132	C₂²	—	74	2	no	—
133	C₂²	—	148	8	no	—
134	C₂²	—	148	4	no	—
135	C₂²	C₂×D₄×D₃₇	1	1	yes	characteristic
136	C₄	—	4	4	no	—
137	C₄	—	37	2	no	—
138	C₄	—	74	2	no	—
139	C₄	#11924559	1	4	yes	complemented
140	C₄	C₂×D₄×D₃₇	1	2	yes	—
141	C₂	—	296	4	no	—
142	C₂	—	74	2	no	—
143	C₂	#127×D₃₇	1	1	yes	characteristic
144	C₂	C₂×#11924559	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11924559

C₂

yes

maximal, complemented

#14558084

C₂

yes

maximal, complemented

#14557918

C₂

yes

maximal, complemented

#14557910

C₂

yes

maximal, characteristic, complemented

#14557907

C₂

yes

maximal, complemented

C₄×C₈×D₃₇

C₂

yes

maximal, characteristic, complemented

C₃₇×#494

C₂

yes

maximal, characteristic, complemented

C₂×#10596966

C₂²

yes

—

#11924559

—

C₂×#10596941

C₂²

yes

—

C₂×D₂₉₆

C₂²

yes

—

C₂×C₈×D₃₇

C₂²

yes

—

#11924533

C₂²

yes

—

#11924525

C₂²

yes

—

#11924511

—

#11924498

—

C₄²×D₃₇

C₂²

yes

characteristic

C₄×#64436

C₂²

yes

characteristic

C₈×Dic₃₇

C₂²

yes

—

C₂×Q₁₆×C₃₇

C₂²

yes

—

C₄⋊Q₈×C₃₇

C₂²

yes

—

C₄×C₈×C₃₇

C₂²

yes

characteristic

C₂×Q₈×C₃₇

C₂³

yes

—

#10596966

—

C₂×D₁₄₈

—

C₂×D₁₄₈

—

C₂×D₁₄₈

C₂³

yes

—

C₂×C₄×D₃₇

—

C₂×C₄×D₃₇

D₄

yes

—

C₂×C₄×D₃₇

C₂³

yes

characteristic

Q₁₆×C₃₇

—

C₂×C₈×C₃₇

C₂³

yes

—

C₄⋊C₄×C₃₇

—

C₄²×C₃₇

C₂³

yes

characteristic

#10596941

—

#10596938

—

C₄×Dic₃₇

D₄

yes

—

C₄×Dic₃₇

C₂³

yes

characteristic

C₂×#64436

C₂³

yes

—

D₂₉₆

—

C₈×D₃₇

D₄

yes

complemented

C₂²×D₃₇

—

C₂²×D₃₇

C₂×D₄

yes

characteristic

Q₈×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₂×D₄

yes

—

C₂×C₄×C₃₇

C₂⁴

yes

characteristic

C₂×Dic₃₇

C₂×D₄

yes

—

C₂×Dic₃₇

C₂×D₄

yes

characteristic

D₁₄₈

—

D₁₄₈

—

D₁₄₈

—

C₄×D₃₇

—

C₄×D₃₇

—

C₄×D₃₇

C₂×D₄

yes

—

C₈×C₃₇

C₂×D₄

yes

—

#64436

C₂×D₄

yes

—

C₂²×C₃₇

C₂²×D₄

yes

characteristic

C₂×D₃₇

—

C₂×D₃₇

—

C₂×D₃₇

—

C₂×D₃₇

#127

yes

—

C₄×C₃₇

—

C₄×C₃₇

#135

yes

complemented

C₄×C₃₇

C₂²×D₄

yes

—

Dic₃₇

—

Dic₃₇

#127

yes

—

—

C₂×C₃₇

yes

C₂×C₃₇

yes

—

D₃₇

—

D₃₇

—

—

—

—

D₃₇

yes

characteristic, complemented

#493

—

#492

—

C₂×C₄×C₈

—

C₃₇

#3147

yes

characteristic, complemented, Sylow

C₂×#41

—

#135

—

C₂×Q₁₆

C₂×D₃₇

yes

—

C₂×SD₁₆

—

C₂×D₈

—

C₂²×C₈

—

C₄⋊Q₈

C₂×D₃₇

yes

—

#127

—

#124

—

#121

—

C₄×D₄

—

C₂×C₄²

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₃₇

yes

characteristic

#41

—

C₂×Q₈

C₂²×D₃₇

yes

—

C₂×D₄

—

C₂×D₄

—

100

C₂×D₄

—

101

C₂²×C₄

—

102

C₂²×C₄

—

103

C₂²×C₄

—

104

Q₁₆

—

105

SD₁₆

—

106

D₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

C₂²×D₃₇

yes

—

110

C₄⋊C₄

—

111

C₂²⋊C₄

—

112

C₄²

—

113

C₄²

—

114

C₄²

C₂²×D₃₇

yes

characteristic

115

C₂³

—

116

C₂³

—

117

Q₈

—

118

D₄

—

119

D₄

—

148

—

120

D₄

—

148

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

148

—

127

C₂×C₄

D₄×D₃₇

yes

—

128

C₂×C₄

C₂³×D₃₇

yes

characteristic

129

C₈

—

130

C₈

D₄×D₃₇

yes

—

131

C₂²

—

132

C₂²

—

133

C₂²

—

148

—

134

C₂²

—

148

—

135

C₂²

C₂×D₄×D₃₇

yes

characteristic

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

#11924559

yes

complemented

140

C₄

C₂×D₄×D₃₇

yes

—

141

C₂

—

296

—

142

C₂

—

143

C₂

#127×D₃₇

yes

characteristic

144

C₂

C₂×#11924559

yes

—

	subgroup	G/H
center	C₂²	C₂×D₄×D₃₇
commutator, cosocle	C₂×C₄×C₃₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₃₇
Fitting	C₃₇×#494	C₂
socle	C₂²×C₃₇	C₂²×D₄
hypercenter	#494	D₃₇
hypocenter	C₃₇	#3147
LCS	G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #494

subgroup

G/H

center

C₂²

C₂×D₄×D₃₇

commutator, cosocle

C₂×C₄×C₃₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₃₇

Fitting

C₃₇×#494

C₂

socle

C₂²×C₃₇

C₂²×D₄

D₃₇

C₃₇

G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #494

N	H ≅ G/N	type
1. C₂×#11924559	C₂	semidirect
2. #14558084	C₂	semidirect
3. #14557918	C₂	semidirect
4. #14557910	C₂	semidirect
5. #14557907	C₂	semidirect
6. C₄×C₈×D₃₇	C₂	semidirect
7. C₃₇×#494	C₂	semidirect
41. C₈×D₃₇	D₄	semidirect
64. C₄×C₃₇	#135	semidirect
76. #494	D₃₇	semidirect
80. C₃₇	#3147	semidirect
139. C₄	#11924559	semidirect

H ≅ G/N

type

1. C₂×#11924559

C₂

semidirect

2. #14558084

C₂

semidirect

3. #14557918

C₂

semidirect

4. #14557910

C₂

semidirect

5. #14557907

C₂

semidirect

6. C₄×C₈×D₃₇

C₂

semidirect

7. C₃₇×#494

C₂

semidirect

41. C₈×D₃₇

D₄

semidirect

64. C₄×C₃₇

#135

semidirect

76. #494

D₃₇

semidirect

80. C₃₇

#3147

semidirect

139. C₄

#11924559

semidirect

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

subgroup	index
C₂×#11924559	2
#14558084	2
#14557918	2
#14557910	2
#14557907	2
C₄×C₈×D₃₇	2
C₃₇×#494	2
#3147	37

subgroup

index

C₄×C₈×D₃₇

C₃₇×#494

#3147

Subgroups — #18005025

Conjugacy classes of subgroups

Complements

Maximal subgroups