Group #18003223 — subgroups

Subgroups — C₂×#14557595 #18003223

Order 4736.

412 conjugacy classes of subgroups; only the 189 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 189 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×#11924493	C₂	1	1	yes	maximal, characteristic
2	#14557595	C₂	1	8	yes	maximal, direct product factor
3	C₂×#11924465	C₂	1	2	yes	maximal, complemented
4	C₂×#11924448	C₂	1	2	yes	maximal
5	C₂×C₄×#64436	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#97×C₃₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₄×D₃₇	C₂²	1	1	yes	characteristic
8	C₂²×#64436	C₂²	1	2	yes	—
9	#11924493	C₄	1	4	yes	—
10	#11924493	C₂²	1	4	yes	—
11	C₂×#10596936	C₄	1	2	yes	—
12	C₂×#10596938	C₄	1	2	yes	—
13	C₂×C₄×Dic₃₇	C₂²	1	1	yes	characteristic
14	#11924465	C₂²	1	8	yes	complemented
15	#11924448	C₂²	1	8	yes	—
16	C₄×#64436	C₂²	1	4	yes	complemented
17	C₂²×C₈×C₃₇	C₂²	1	2	yes	—
18	C₂×C₄²×C₃₇	C₂²	1	1	yes	characteristic
19	#97×C₃₇	C₂²	1	4	yes	complemented
20	C₂³×D₃₇	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₃₇	C₂×C₄	1	2	yes	—
22	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
23	C₂²×Dic₃₇	C₂×C₄	1	2	yes	—
24	C₂²×Dic₃₇	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×D₃₇	—	2	4	no	—
26	C₂×C₄×D₃₇	C₂×C₄	1	2	yes	—
27	C₂×C₄×D₃₇	C₂³	1	2	yes	—
28	C₂×C₈×C₃₇	—	2	4	no	—
29	C₂×C₈×C₃₇	C₂³	1	4	yes	—
30	C₄²×C₃₇	C₂×C₄	1	2	yes	—
31	C₄²×C₃₇	C₂³	1	2	yes	—
32	#10596938	C₂×C₄	1	8	yes	—
33	#10596936	C₂×C₄	1	8	yes	—
34	C₄×Dic₃₇	C₂×C₄	1	2	yes	—
35	C₄×Dic₃₇	C₂³	1	2	yes	—
36	C₂×#64436	—	2	4	no	—
37	C₂×#64436	C₂³	1	4	yes	—
38	C₂³×C₃₇	C₂²×C₄	1	1	yes	characteristic
39	C₂²×D₃₇	—	2	2	no	—
40	C₂²×D₃₇	—	2	4	no	—
41	C₂²×D₃₇	C₂²×C₄	1	2	yes	—
42	C₂×C₄×C₃₇	—	2	4	no	—
43	C₂×C₄×C₃₇	C₂²×C₄	1	4	yes	—
44	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₃₇	#41	1	4	yes	—
46	C₂×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
47	C₂×Dic₃₇	—	2	2	no	—
48	C₂×Dic₃₇	—	2	4	no	—
49	C₂×Dic₃₇	C₂²×C₄	1	4	yes	—
50	C₂×Dic₃₇	C₂²×C₄	1	2	yes	—
51	C₄×D₃₇	—	2	4	no	—
52	C₄×D₃₇	—	2	4	no	—
53	C₈×C₃₇	—	2	4	no	—
54	#64436	—	2	4	no	—
55	C₂²×C₃₇	#117	1	2	yes	—
56	C₂²×C₃₇	#131	1	4	yes	—
57	C₂²×C₃₇	C₂³×C₄	1	1	yes	characteristic
58	C₂×D₃₇	—	2	2	no	—
59	C₂×D₃₇	—	2	4	no	—
60	C₂×D₃₇	—	4	4	no	—
61	C₄×C₃₇	—	2	4	no	—
62	C₄×C₃₇	#117	1	2	yes	—
63	C₄×C₃₇	C₂×#41	1	2	yes	—
64	Dic₃₇	—	2	2	no	—
65	Dic₃₇	—	2	4	no	—
66	C₂×#433	—	37	1	no	maximal, Sylow
67	C₂×C₃₇	#433	1	4	yes	complemented
68	C₂×C₃₇	C₂×#117	1	1	yes	characteristic
69	C₂×C₃₇	C₂×#131	1	2	yes	—
70	D₃₇	—	4	2	no	—
71	C₂×#117	—	37	1	no	—
72	#433	—	37	8	no	—
73	C₂×C₄⋊C₈	—	37	2	no	—
74	C₂×C₂²⋊C₈	—	37	2	no	—
75	C₂×#97	D₃₇	1	1	yes	characteristic, complemented
76	C₂×C₄×C₈	—	37	1	no	—
77	C₃₇	C₂×#433	1	1	yes	characteristic, complemented, Sylow
78	C₂³×C₄	—	37	1	no	—
79	C₂²×C₈	—	37	2	no	—
80	C₂²×C₈	C₂×D₃₇	1	2	yes	—
81	#117	—	37	4	no	—
82	#117	—	37	4	no	—
83	C₂×C₄⋊C₄	—	37	2	no	—
84	C₂×C₂²⋊C₄	—	37	2	no	—
85	C₂×C₄²	—	37	1	no	—
86	C₂×C₄²	C₂×D₃₇	1	1	yes	characteristic
87	C₄⋊C₈	—	37	8	no	—
88	C₂²⋊C₈	—	37	8	no	—
89	#97	C₂×D₃₇	1	4	yes	complemented
90	C₄×C₈	—	37	4	no	—
91	C₂⁴	—	37	1	no	—
92	C₂²×C₄	—	37	2	no	—
93	C₂²×C₄	—	37	1	no	—
94	C₂²×C₄	—	37	2	no	—
95	C₂²×C₄	—	37	2	no	—
96	C₂²×C₄	—	74	4	no	—
97	C₂²×C₄	C₄×D₃₇	1	2	yes	—
98	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
99	C₂×C₈	—	2	4	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	2	no	—
106	C₄²	—	37	2	no	—
107	C₄²	C₄×D₃₇	1	2	yes	—
108	C₄²	C₂²×D₃₇	1	2	yes	—
109	C₂³	—	37	2	no	—
110	C₂³	—	74	2	no	—
111	C₂³	—	74	4	no	—
112	C₂³	C₂×C₄×D₃₇	1	1	yes	characteristic
113	C₂×C₄	—	2	4	no	—
114	C₂×C₄	—	37	4	no	—
115	C₂×C₄	—	37	2	no	—
116	C₂×C₄	—	74	2	no	—
117	C₂×C₄	—	74	4	no	—
118	C₂×C₄	—	74	4	no	—
119	C₂×C₄	—	74	4	no	—
120	C₂×C₄	C₂×C₄×D₃₇	1	4	yes	—
121	C₂×C₄	C₂×C₄×D₃₇	1	1	yes	characteristic
122	C₂×C₄	#10596962	1	4	yes	—
123	C₂×C₄	C₂³×D₃₇	1	1	yes	characteristic
124	C₈	—	2	4	no	—
125	C₈	—	74	4	no	—
126	C₂²	—	74	2	no	—
127	C₂²	—	74	4	no	—
128	C₂²	—	148	4	no	—
129	C₂²	#11924493	1	2	yes	—
130	C₂²	#11924544	1	4	yes	—
131	C₂²	C₂²×C₄×D₃₇	1	1	yes	characteristic
132	C₄	—	2	4	no	—
133	C₄	—	74	2	no	—
134	C₄	—	74	4	no	—
135	C₄	#11924493	1	2	yes	—
136	C₄	C₂×#10596962	1	2	yes	—
137	C₂	—	148	2	no	—
138	C₂	#14557595	1	4	yes	direct product factor
139	C₂	C₂×#11924493	1	1	yes	characteristic
140	C₂	C₂×#11924544	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11924493

C₂

yes

maximal, characteristic

#14557595

C₂

yes

maximal, direct product factor

C₂×#11924465

C₂

yes

maximal, complemented

C₂×#11924448

C₂

yes

maximal

C₂×C₄×#64436

C₂

yes

maximal, characteristic, complemented

C₂×#97×C₃₇

C₂

yes

maximal, characteristic, complemented

C₂²×C₄×D₃₇

C₂²

yes

characteristic

C₂²×#64436

C₂²

yes

—

#11924493

C₄

yes

—

#11924493

C₂²

yes

—

C₂×#10596936

C₄

yes

—

C₂×#10596938

C₄

yes

—

C₂×C₄×Dic₃₇

C₂²

yes

characteristic

#11924465

C₂²

yes

complemented

#11924448

C₂²

yes

—

C₄×#64436

C₂²

yes

complemented

C₂²×C₈×C₃₇

C₂²

yes

—

C₂×C₄²×C₃₇

C₂²

yes

characteristic

#97×C₃₇

C₂²

yes

complemented

C₂³×D₃₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₃₇

C₂×C₄

yes

—

C₂²×C₄×C₃₇

C₂³

yes

characteristic

C₂²×Dic₃₇

C₂×C₄

yes

—

C₂²×Dic₃₇

C₂×C₄

yes

characteristic

C₂×C₄×D₃₇

—

C₂×C₄×D₃₇

C₂×C₄

yes

—

C₂×C₄×D₃₇

C₂³

yes

—

C₂×C₈×C₃₇

—

C₂×C₈×C₃₇

C₂³

yes

—

C₄²×C₃₇

C₂×C₄

yes

—

C₄²×C₃₇

C₂³

yes

—

#10596938

C₂×C₄

yes

—

#10596936

C₂×C₄

yes

—

C₄×Dic₃₇

C₂×C₄

yes

—

C₄×Dic₃₇

C₂³

yes

—

C₂×#64436

—

C₂×#64436

C₂³

yes

—

C₂³×C₃₇

C₂²×C₄

yes

characteristic

C₂²×D₃₇

—

C₂²×D₃₇

—

C₂²×D₃₇

C₂²×C₄

yes

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₂²×C₄

yes

—

C₂×C₄×C₃₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₇

#41

yes

—

C₂×C₄×C₃₇

C₂⁴

yes

characteristic

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

C₂²×C₄

yes

—

C₂×Dic₃₇

C₂²×C₄

yes

—

C₄×D₃₇

—

C₄×D₃₇

—

C₈×C₃₇

—

#64436

—

C₂²×C₃₇

#117

yes

—

C₂²×C₃₇

#131

yes

—

C₂²×C₃₇

C₂³×C₄

yes

characteristic

C₂×D₃₇

—

C₂×D₃₇

—

C₂×D₃₇

—

C₄×C₃₇

—

C₄×C₃₇

#117

yes

—

C₄×C₃₇

C₂×#41

yes

—

Dic₃₇

—

Dic₃₇

—

—

C₂×C₃₇

yes

C₂×C₃₇

yes

C₂×C₃₇

yes

—

D₃₇

—

C₂×#117

—

#433

—

C₂×C₄⋊C₈

—

C₂×C₂²⋊C₈

—

C₂×#97

D₃₇

yes

characteristic, complemented

C₂×C₄×C₈

—

C₃₇

C₂×#433

yes

characteristic, complemented, Sylow

C₂³×C₄

—

C₂²×C₈

—

C₂²×C₈

C₂×D₃₇

yes

—

#117

—

#117

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₄²

—

C₂×C₄²

C₂×D₃₇

yes

characteristic

C₄⋊C₈

—

C₂²⋊C₈

—

#97

C₂×D₃₇

yes

complemented

C₄×C₈

—

C₂⁴

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₄×D₃₇

yes

—

C₂²×C₄

C₂²×D₃₇

yes

characteristic

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

C₄²

C₄×D₃₇

yes

—

108

C₄²

C₂²×D₃₇

yes

—

109

C₂³

—

110

C₂³

—

111

C₂³

—

112

C₂³

C₂×C₄×D₃₇

yes

characteristic

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

C₂×C₄×D₃₇

yes

—

121

C₂×C₄

C₂×C₄×D₃₇

yes

characteristic

122

C₂×C₄

#10596962

yes

—

123

C₂×C₄

C₂³×D₃₇

yes

characteristic

124

C₈

—

125

C₈

—

126

C₂²

—

127

C₂²

—

128

C₂²

—

148

—

129

C₂²

#11924493

yes

—

130

C₂²

#11924544

yes

—

131

C₂²

C₂²×C₄×D₃₇

yes

characteristic

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

#11924493

yes

—

136

C₄

C₂×#10596962

yes

—

137

C₂

—

148

—

138

C₂

#14557595

yes

direct product factor

139

C₂

C₂×#11924493

yes

characteristic

140

C₂

C₂×#11924544

yes

—

	subgroup	G/H
center	C₂²×C₄	C₂²×D₃₇
commutator	C₂²×C₃₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₃₇
Fitting	C₂×#97×C₃₇	C₂
socle	C₂³×C₃₇	C₂²×C₄
prime radical	C₂³×D₃₇	C₂×C₄
cosocle	C₂×C₄×C₃₇	C₂⁴
hypercenter	C₂×#97	D₃₇
hypocenter	C₃₇	C₂×#433
LCS	G ▷ C₂²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#97

subgroup

G/H

center

C₂²×C₄

C₂²×D₃₇

commutator

C₂²×C₃₇

C₂³×C₄

Frattini

C₂×C₄

C₂³×D₃₇

Fitting

C₂×#97×C₃₇

C₂

socle

C₂³×C₃₇

C₂²×C₄

prime radical

C₂³×D₃₇

C₂×C₄

cosocle

C₂×C₄×C₃₇

C₂⁴

hypercenter

C₂×#97

D₃₇

C₃₇

G ▷ C₂²×C₃₇ ▷ C₃₇

C₁ ◁ C₂²×C₄ ◁ C₂×#97

N	H ≅ G/N	type
2. #14557595	C₂	direct (N × H)
3. C₂×#11924465	C₂	semidirect
5. C₂×C₄×#64436	C₂	semidirect
6. C₂×#97×C₃₇	C₂	semidirect
14. #11924465	C₂²	semidirect
16. C₄×#64436	C₂²	semidirect
19. #97×C₃₇	C₂²	semidirect
67. C₂×C₃₇	#433	semidirect
75. C₂×#97	D₃₇	semidirect
77. C₃₇	C₂×#433	semidirect
89. #97	C₂×D₃₇	semidirect
138. C₂	#14557595	direct (N × H)

H ≅ G/N

type

2. #14557595

C₂

direct (N × H)

3. C₂×#11924465

C₂

semidirect

5. C₂×C₄×#64436

C₂

semidirect

6. C₂×#97×C₃₇

C₂

semidirect

14. #11924465

C₂²

semidirect

16. C₄×#64436

C₂²

semidirect

19. #97×C₃₇

C₂²

semidirect

67. C₂×C₃₇

#433

semidirect

75. C₂×#97

D₃₇

semidirect

77. C₃₇

C₂×#433

semidirect

89. #97

C₂×D₃₇

semidirect

138. C₂

#14557595

direct (N × H)

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

subgroup	index
C₂×#11924493	2
#14557595	2
C₂×#11924465	2
C₂×#11924448	2
C₂×C₄×#64436	2
C₂×#97×C₃₇	2
C₂×#433	37

subgroup

index

C₂×C₄×#64436

C₂×#97×C₃₇

C₂×#433

Subgroups — C₂×#14557595 #18003223

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#14557595 #18003223

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#14557595 #18003223