Group #18003154 — subgroups

Subgroups — C₂×#14557559 #18003154

Order 4736.

512 conjugacy classes of subgroups; only the 269 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 269 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×#10596934	C₂	1	1	yes	maximal, characteristic
2	C₂×#11924495	C₂	1	1	yes	maximal, characteristic
3	#14557559	C₂	1	8	yes	maximal, direct product factor
4	C₂×#11924465	C₂	1	4	yes	maximal, complemented
5	C₂³×C₈×C₃₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#10596934	—	2	4	no	—
7	C₂×#10596934	C₂²	1	4	yes	—
8	C₂²×#64436	C₂²	1	2	yes	—
9	#11924495	C₄	1	4	yes	—
10	#11924495	C₂²	1	4	yes	—
11	C₂×C₄⋊Dic₃₇	C₄	1	2	yes	—
12	C₂×C₂²⋊Dic₃₇	C₄	1	2	yes	—
13	C₂×C₄×Dic₃₇	C₂²	1	2	yes	—
14	#11924465	C₂²	1	16	yes	complemented
15	C₂³×C₄×C₃₇	C₂²	1	1	yes	characteristic
16	C₂²×C₈×C₃₇	—	2	4	no	—
17	C₂²×C₈×C₃₇	C₂²	1	2	yes	—
18	C₂²×C₈×C₃₇	C₂²	1	4	yes	—
19	C₂⁴×C₃₇	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₃₇	C₂×C₄	1	2	yes	—
22	C₂²×C₄×C₃₇	D₄	1	4	yes	—
23	C₂²×C₄×C₃₇	Q₈	1	4	yes	—
24	C₂²×C₄×C₃₇	C₂³	1	2	yes	—
25	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
26	C₂²×Dic₃₇	C₂×C₄	1	4	yes	—
27	C₂×C₈×C₃₇	—	2	4	no	—
28	C₂×C₈×C₃₇	—	2	8	no	—
29	C₂×C₈×C₃₇	C₂³	1	4	yes	—
30	C₂²⋊Dic₃₇	C₂×C₄	1	8	yes	—
31	C₄⋊Dic₃₇	C₂×C₄	1	8	yes	—
32	C₄×Dic₃₇	C₂×C₄	1	4	yes	—
33	C₄×Dic₃₇	C₂³	1	4	yes	—
34	#10596934	—	2	8	no	—
35	C₂×#64436	—	2	4	no	—
36	C₂×#64436	C₂³	1	4	yes	—
37	C₂³×C₃₇	—	2	4	no	—
38	C₂³×C₃₇	C₄⋊C₄	1	4	yes	—
39	C₂³×C₃₇	C₂²×C₄	1	2	yes	—
40	C₂³×C₃₇	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₃₇	C₄⋊C₄	1	8	yes	—
42	C₂×C₄×C₃₇	C₄⋊C₄	1	4	yes	—
43	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₄×C₃₇	C₂²×C₄	1	2	yes	—
45	C₂×C₄×C₃₇	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₃₇	C₂×D₄	1	4	yes	—
47	C₂×C₄×C₃₇	C₂×Q₈	1	4	yes	—
48	C₂×C₄×C₃₇	C₂×Q₈	1	2	yes	—
49	C₂×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
50	C₂×Dic₃₇	—	2	8	no	—
51	C₂×Dic₃₇	C₂²×C₄	1	8	yes	—
52	C₈×C₃₇	—	2	4	no	—
53	#64436	—	2	4	no	—
54	C₂²×C₃₇	—	2	4	no	—
55	C₂²×C₃₇	—	2	8	no	—
56	C₂²×C₃₇	C₂×C₄⋊C₄	1	4	yes	—
57	C₂²×C₃₇	C₂×C₄⋊C₄	1	2	yes	—
58	C₂²×C₃₇	#131	1	4	yes	—
59	C₂²×C₃₇	C₂³×C₄	1	1	yes	characteristic
60	C₄×C₃₇	C₂×C₄⋊C₄	1	2	yes	—
61	C₄×C₃₇	C₂×C₄⋊C₄	1	4	yes	—
62	C₄×C₃₇	C₂²×D₄	1	1	yes	characteristic
63	C₄×C₃₇	C₂²×Q₈	1	1	yes	characteristic
64	Dic₃₇	—	2	8	no	—
65	C₂×#424	—	37	1	no	maximal, Sylow
66	C₂×C₃₇	—	2	4	no	—
67	C₂×C₃₇	#424	1	4	yes	complemented
68	C₂×C₃₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
69	C₂×C₃₇	C₂×#131	1	2	yes	—
70	C₂²×M₁₆	—	37	1	no	—
71	C₂³×C₈	D₃₇	1	1	yes	characteristic
72	C₂×#117	—	37	1	no	—
73	#424	—	37	8	no	—
74	C₂×C₄⋊C₈	—	37	4	no	—
75	C₃₇	C₂×#424	1	1	yes	characteristic, complemented, Sylow
76	C₂³×C₄	C₂×D₃₇	1	1	yes	characteristic
77	C₂×M₁₆	—	37	4	no	—
78	C₂×M₁₆	—	74	4	no	—
79	C₂²×C₈	—	2	4	no	—
80	C₂²×C₈	—	37	2	no	—
81	C₂²×C₈	C₂×D₃₇	1	4	yes	—
82	C₂²×C₈	C₂×D₃₇	1	2	yes	—
83	#117	—	37	4	no	—
84	#117	—	37	4	no	—
85	C₂×C₄⋊C₄	—	37	2	no	—
86	C₂×C₂²⋊C₄	—	37	2	no	—
87	C₂×C₄²	—	37	2	no	—
88	C₄⋊C₈	—	37	16	no	—
89	C₂⁴	C₄×D₃₇	1	1	yes	characteristic
90	C₂²×C₄	—	37	4	no	—
91	C₂²×C₄	Dic₇₄	1	4	yes	—
92	C₂²×C₄	C₄×D₃₇	1	1	yes	characteristic
93	C₂²×C₄	C₄×D₃₇	1	2	yes	—
94	C₂²×C₄	C₂²⋊D₃₇	1	4	yes	—
95	C₂²×C₄	C₂²×D₃₇	1	2	yes	—
96	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
97	M₁₆	—	74	8	no	—
98	C₂×C₈	—	2	4	no	—
99	C₂×C₈	—	2	8	no	—
100	C₂×C₈	—	37	4	no	—
101	C₂×C₈	—	74	4	no	—
102	C₂×C₈	C₂²×D₃₇	1	4	yes	—
103	C₄⋊C₄	—	37	8	no	—
104	C₂²⋊C₄	—	37	8	no	—
105	C₄²	—	37	4	no	—
106	C₄²	—	37	4	no	—
107	C₂³	—	2	4	no	—
108	C₂³	#10596936	1	4	yes	—
109	C₂³	C₂×C₄×D₃₇	1	2	yes	—
110	C₂³	C₂×C₄×D₃₇	1	1	yes	characteristic
111	C₂×C₄	—	37	8	no	—
112	C₂×C₄	—	74	8	no	—
113	C₂×C₄	#10596936	1	8	yes	—
114	C₂×C₄	#10596936	1	4	yes	—
115	C₂×C₄	C₂×Dic₇₄	1	4	yes	—
116	C₂×C₄	C₂×Dic₇₄	1	2	yes	—
117	C₂×C₄	C₂×C₄×D₃₇	1	1	yes	characteristic
118	C₂×C₄	C₂×C₄×D₃₇	1	2	yes	—
119	C₂×C₄	C₂×C₂²⋊D₃₇	1	2	yes	—
120	C₂×C₄	C₂×C₂²⋊D₃₇	1	4	yes	—
121	C₂×C₄	C₂³×D₃₇	1	1	yes	characteristic
122	C₈	—	2	4	no	—
123	C₈	—	74	4	no	—
124	C₂²	—	2	4	no	—
125	C₂²	—	2	8	no	—
126	C₂²	C₂×#10596936	1	4	yes	—
127	C₂²	C₂×#10596936	1	2	yes	—
128	C₂²	#11924542	1	4	yes	—
129	C₂²	C₂²×C₄×D₃₇	1	1	yes	characteristic
130	C₄	—	74	8	no	—
131	C₄	C₂×#10596936	1	4	yes	—
132	C₄	C₂×#10596936	1	2	yes	—
133	C₄	C₂²×C₂²⋊D₃₇	1	1	yes	characteristic
134	C₄	C₂²×Dic₇₄	1	1	yes	characteristic
135	C₂	—	2	4	no	—
136	C₂	#14557559	1	4	yes	direct product factor
137	C₂	C₂²×#10596936	1	1	yes	characteristic
138	C₂	C₂×#11924542	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#10596934

C₂

yes

maximal, characteristic

C₂×#11924495

C₂

yes

maximal, characteristic

#14557559

C₂

yes

maximal, direct product factor

C₂×#11924465

C₂

yes

maximal, complemented

C₂³×C₈×C₃₇

C₂

yes

maximal, characteristic

C₂×#10596934

—

C₂×#10596934

C₂²

yes

—

C₂²×#64436

C₂²

yes

—

#11924495

C₄

yes

—

#11924495

C₂²

yes

—

C₂×C₄⋊Dic₃₇

C₄

yes

—

C₂×C₂²⋊Dic₃₇

C₄

yes

—

C₂×C₄×Dic₃₇

C₂²

yes

—

#11924465

C₂²

yes

complemented

C₂³×C₄×C₃₇

C₂²

yes

characteristic

C₂²×C₈×C₃₇

—

C₂²×C₈×C₃₇

C₂²

yes

—

C₂²×C₈×C₃₇

C₂²

yes

—

C₂⁴×C₃₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₃₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₃₇

C₂×C₄

yes

—

C₂²×C₄×C₃₇

D₄

yes

—

C₂²×C₄×C₃₇

Q₈

yes

—

C₂²×C₄×C₃₇

C₂³

yes

—

C₂²×C₄×C₃₇

C₂³

yes

characteristic

C₂²×Dic₃₇

C₂×C₄

yes

—

C₂×C₈×C₃₇

—

C₂×C₈×C₃₇

—

C₂×C₈×C₃₇

C₂³

yes

—

C₂²⋊Dic₃₇

C₂×C₄

yes

—

C₄⋊Dic₃₇

C₂×C₄

yes

—

C₄×Dic₃₇

C₂×C₄

yes

—

C₄×Dic₃₇

C₂³

yes

—

#10596934

—

C₂×#64436

—

C₂×#64436

C₂³

yes

—

C₂³×C₃₇

—

C₂³×C₃₇

C₄⋊C₄

yes

—

C₂³×C₃₇

C₂²×C₄

yes

—

C₂³×C₃₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₇

C₄⋊C₄

yes

—

C₂×C₄×C₃₇

C₄⋊C₄

yes

—

C₂×C₄×C₃₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₇

C₂²×C₄

yes

—

C₂×C₄×C₃₇

C₂×D₄

yes

—

C₂×C₄×C₃₇

C₂×D₄

yes

—

C₂×C₄×C₃₇

C₂×Q₈

yes

—

C₂×C₄×C₃₇

C₂×Q₈

yes

—

C₂×C₄×C₃₇

C₂⁴

yes

characteristic

C₂×Dic₃₇

—

C₂×Dic₃₇

C₂²×C₄

yes

—

C₈×C₃₇

—

#64436

—

C₂²×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

C₂×C₄⋊C₄

yes

—

C₂²×C₃₇

C₂×C₄⋊C₄

yes

—

C₂²×C₃₇

#131

yes

—

C₂²×C₃₇

C₂³×C₄

yes

characteristic

C₄×C₃₇

C₂×C₄⋊C₄

yes

—

C₄×C₃₇

C₂×C₄⋊C₄

yes

—

C₄×C₃₇

C₂²×D₄

yes

characteristic

C₄×C₃₇

C₂²×Q₈

yes

characteristic

Dic₃₇

—

C₂×#424

—

maximal, Sylow

C₂×C₃₇

—

C₂×C₃₇

#424

yes

complemented

C₂×C₃₇

C₂²×C₄⋊C₄

yes

characteristic

C₂×C₃₇

C₂×#131

yes

—

C₂²×M₁₆

—

C₂³×C₈

D₃₇

yes

characteristic

C₂×#117

—

#424

—

C₂×C₄⋊C₈

—

C₃₇

C₂×#424

yes

characteristic, complemented, Sylow

C₂³×C₄

C₂×D₃₇

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂×D₃₇

yes

—

C₂²×C₈

C₂×D₃₇

yes

—

#117

—

#117

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₄²

—

C₄⋊C₈

—

C₂⁴

C₄×D₃₇

yes

characteristic

C₂²×C₄

—

C₂²×C₄

Dic₇₄

yes

—

C₂²×C₄

C₄×D₃₇

yes

characteristic

C₂²×C₄

C₄×D₃₇

yes

—

C₂²×C₄

C₂²⋊D₃₇

yes

—

C₂²×C₄

C₂²×D₃₇

yes

—

C₂²×C₄

C₂²×D₃₇

yes

characteristic

M₁₆

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

C₂²×D₃₇

yes

—

103

C₄⋊C₄

—

104

C₂²⋊C₄

—

105

C₄²

—

106

C₄²

—

107

C₂³

—

108

C₂³

#10596936

yes

—

109

C₂³

C₂×C₄×D₃₇

yes

—

110

C₂³

C₂×C₄×D₃₇

yes

characteristic

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

#10596936

yes

—

114

C₂×C₄

#10596936

yes

—

115

C₂×C₄

C₂×Dic₇₄

yes

—

116

C₂×C₄

C₂×Dic₇₄

yes

—

117

C₂×C₄

C₂×C₄×D₃₇

yes

characteristic

118

C₂×C₄

C₂×C₄×D₃₇

yes

—

119

C₂×C₄

C₂×C₂²⋊D₃₇

yes

—

120

C₂×C₄

C₂×C₂²⋊D₃₇

yes

—

121

C₂×C₄

C₂³×D₃₇

yes

characteristic

122

C₈

—

123

C₈

—

124

C₂²

—

125

C₂²

—

126

C₂²

C₂×#10596936

yes

—

127

C₂²

C₂×#10596936

yes

—

128

C₂²

#11924542

yes

—

129

C₂²

C₂²×C₄×D₃₇

yes

characteristic

130

C₄

—

131

C₄

C₂×#10596936

yes

—

132

C₄

C₂×#10596936

yes

—

133

C₄

C₂²×C₂²⋊D₃₇

yes

characteristic

134

C₄

C₂²×Dic₇₄

yes

characteristic

135

C₂

—

136

C₂

#14557559

yes

direct product factor

137

C₂

C₂²×#10596936

yes

characteristic

138

C₂

C₂×#11924542

yes

—

	subgroup	G/H
center	C₂²×C₄	C₂²×D₃₇
commutator	C₂²×C₃₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₃₇
Fitting	C₂³×C₈×C₃₇	C₂
socle	C₂³×C₃₇	C₂²×C₄
prime radical	C₂⁴×C₃₇	C₂×C₄
cosocle	C₂×C₄×C₃₇	C₂⁴
hypercenter	C₂³×C₈	D₃₇
hypocenter	C₃₇	C₂×#424
LCS	G ▷ C₂²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₈

subgroup

G/H

center

C₂²×C₄

C₂²×D₃₇

commutator

C₂²×C₃₇

C₂³×C₄

Frattini

C₂×C₄

C₂³×D₃₇

Fitting

C₂³×C₈×C₃₇

C₂

socle

C₂³×C₃₇

C₂²×C₄

prime radical

C₂⁴×C₃₇

C₂×C₄

cosocle

C₂×C₄×C₃₇

C₂⁴

hypercenter

C₂³×C₈

D₃₇

C₃₇

G ▷ C₂²×C₃₇ ▷ C₃₇

C₁ ◁ C₂²×C₄ ◁ C₂³×C₈

N	H ≅ G/N	type
3. #14557559	C₂	direct (N × H)
4. C₂×#11924465	C₂	semidirect
14. #11924465	C₂²	semidirect
67. C₂×C₃₇	#424	semidirect
75. C₃₇	C₂×#424	semidirect
136. C₂	#14557559	direct (N × H)

H ≅ G/N

type

3. #14557559

C₂

direct (N × H)

4. C₂×#11924465

C₂

semidirect

14. #11924465

C₂²

semidirect

67. C₂×C₃₇

#424

semidirect

75. C₃₇

C₂×#424

semidirect

136. C₂

#14557559

direct (N × H)

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

subgroup	index
C₂²×#10596934	2
C₂×#11924495	2
#14557559	2
C₂×#11924465	2
C₂³×C₈×C₃₇	2
C₂×#424	37

subgroup

index

C₂³×C₈×C₃₇

Subgroups — C₂×#14557559 #18003154

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#14557559 #18003154

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#14557559 #18003154