Group #18011562 — subgroups

Subgroups — C₂×#14558592 #18011562

Order 4736.

368 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 202 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	#14558592	C₂	1	8	yes	maximal, direct product factor
2	C₂×#11924611	C₂	1	2	yes	maximal, complemented
3	C₂×#11924610	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₃₇⋊C₁₆	C₂	1	2	yes	maximal, complemented
5	C₂²×#10596927	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11924544	C₂	1	1	yes	maximal, characteristic
7	#11924611	C₂²	1	8	yes	complemented
8	#11924610	C₂²	1	4	yes	complemented
9	D₃₇⋊C₁₆	C₂²	1	8	yes	complemented
10	C₂×#10596927	C₂²	1	1	yes	characteristic
11	C₂×#10596927	C₂²	1	1	yes	characteristic
12	C₂×#10596927	C₂²	1	2	yes	—
13	C₂×#10596927	C₂²	1	4	yes	complemented
14	C₂×#10596962	C₄	1	1	yes	characteristic
15	#11924544	C₄	1	4	yes	—
16	#11924544	C₂²	1	4	yes	—
17	C₂×#10596929	C₄	1	2	yes	—
18	C₂²×#64436	C₂²	1	1	yes	characteristic
19	C₂×C₈×D₃₇	C₂²	1	2	yes	—
20	C₂×M₁₆×C₃₇	C₄	1	1	yes	characteristic
21	C₂²×C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
22	C₂×C₂²⋊D₃₇	C₈	1	2	yes	—
23	#10596962	C₈	1	4	yes	—
24	#10596962	C₂×C₄	1	4	yes	—
25	C₂×D₁₄₈	C₈	1	1	yes	characteristic
26	C₂×C₄×D₃₇	C₂×C₄	1	2	yes	—
27	C₂×Dic₇₄	C₈	1	1	yes	characteristic
28	M₁₆×C₃₇	C₂×C₄	1	4	yes	—
29	C₂×C₈×C₃₇	C₂×C₄	1	2	yes	—
30	C₂×#64436	C₂×C₄	1	1	yes	characteristic
31	C₂×#64436	C₂×C₄	1	2	yes	—
32	C₂×#64436	C₂³	1	1	yes	characteristic
33	C₂×#64436	C₂³	1	2	yes	—
34	#10596929	C₂×C₄	1	8	yes	—
35	C₈×D₃₇	C₂×C₄	1	4	yes	—
36	C₈×D₃₇	C₂³	1	4	yes	—
37	#10596927	C₂³	1	2	yes	—
38	#10596927	C₂³	1	2	yes	—
39	#10596927	C₂³	1	4	yes	—
40	C₂³×C₃₇	C₂×C₈	1	1	yes	characteristic
41	C₂²×D₃₇	C₂×C₈	1	2	yes	—
42	C₂×C₄×C₃₇	C₂×C₈	1	1	yes	characteristic
43	C₂×C₄×C₃₇	C₂×C₈	1	2	yes	—
44	C₂×C₄×C₃₇	C₂²×C₄	1	2	yes	—
45	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
46	C₂²⋊D₃₇	C₂×C₈	1	8	yes	—
47	C₂×Dic₃₇	C₂×C₈	1	2	yes	—
48	D₁₄₈	C₂×C₈	1	4	yes	—
49	C₄×D₃₇	C₂×C₈	1	4	yes	—
50	C₄×D₃₇	C₂²×C₄	1	4	yes	—
51	Dic₇₄	C₂×C₈	1	4	yes	—
52	C₈×C₃₇	C₂²×C₄	1	4	yes	—
53	#64436	C₂²×C₄	1	1	yes	characteristic
54	#64436	C₂²×C₄	1	2	yes	—
55	#64436	C₂⁴	1	1	yes	characteristic
56	C₂²×C₃₇	—	2	2	no	—
57	C₂²×C₃₇	C₂²×C₈	1	2	yes	—
58	C₂²×C₃₇	C₂²×C₈	1	1	yes	characteristic
59	C₂×D₃₇	—	2	4	no	—
60	C₂×D₃₇	C₂²×C₈	1	4	yes	—
61	C₄×C₃₇	C₂²×C₈	1	2	yes	—
62	C₄×C₃₇	C₂²×C₈	1	1	yes	characteristic
63	C₄×C₃₇	C₂³×C₄	1	1	yes	characteristic
64	Dic₃₇	C₂²×C₈	1	4	yes	—
65	C₂×#504	—	37	1	no	maximal, Sylow
66	C₂×C₃₇	—	2	2	no	—
67	C₂×C₃₇	#504	1	2	yes	complemented
68	C₂×C₃₇	C₂³×C₈	1	1	yes	characteristic
69	D₃₇	—	2	4	no	—
70	C₂×#131	—	37	1	no	—
71	#504	—	37	8	no	—
72	C₂×M₃₂	—	37	1	no	—
73	C₂×M₃₂	—	37	2	no	—
74	C₂²×C₁₆	—	37	1	no	—
75	C₂²×C₁₆	—	37	2	no	—
76	C₃₇	C₂×#504	1	1	yes	characteristic, complemented, Sylow
77	C₂×#41	—	37	1	no	—
78	#131	—	37	4	no	—
79	#131	—	37	4	no	—
80	C₂×M₁₆	—	37	2	no	—
81	C₂×M₁₆	#3866	1	1	yes	characteristic
82	C₂²×C₈	—	37	1	no	—
83	C₂²×C₈	—	37	2	no	—
84	M₃₂	—	37	4	no	—
85	M₃₂	—	37	8	no	—
86	C₂×C₁₆	—	37	1	no	—
87	C₂×C₁₆	—	37	1	no	—
88	C₂×C₁₆	—	37	2	no	—
89	C₂×C₁₆	—	37	4	no	—
90	C₂×C₁₆	—	37	8	no	—
91	#41	—	37	4	no	—
92	#41	—	37	4	no	—
93	C₂×Q₈	—	37	1	no	—
94	C₂×D₄	—	37	2	no	—
95	C₂×D₄	—	37	1	no	—
96	C₂²×C₄	—	37	2	no	—
97	C₂²×C₄	C₂×#3866	1	1	yes	characteristic
98	M₁₆	—	37	8	no	—
99	M₁₆	C₂×#3866	1	4	yes	—
100	C₂×C₈	—	37	2	no	—
101	C₂×C₈	—	37	2	no	—
102	C₂×C₈	—	37	1	no	—
103	C₂×C₈	—	37	1	no	—
104	C₂×C₈	—	37	4	no	—
105	C₂×C₈	—	37	4	no	—
106	C₂×C₈	C₂×#3866	1	2	yes	—
107	C₁₆	—	37	2	no	—
108	C₁₆	—	37	2	no	—
109	C₁₆	—	37	4	no	—
110	C₂³	—	37	2	no	—
111	C₂³	D₃₇⋊C₈	1	1	yes	characteristic
112	Q₈	—	37	4	no	—
113	D₄	—	37	8	no	—
114	D₄	—	37	4	no	—
115	C₂×C₄	—	37	2	no	—
116	C₂×C₄	—	37	4	no	—
117	C₂×C₄	—	37	4	no	—
118	C₂×C₄	D₃₇⋊C₈	1	1	yes	characteristic
119	C₂×C₄	D₃₇⋊C₈	1	2	yes	—
120	C₂×C₄	C₂²×#3866	1	2	yes	—
121	C₂×C₄	C₂²×#3866	1	1	yes	characteristic
122	C₈	—	37	1	no	—
123	C₈	—	37	1	no	—
124	C₈	—	37	2	no	—
125	C₈	C₂²×#3866	1	4	yes	—
126	C₂²	—	2	2	no	—
127	C₂²	—	37	4	no	—
128	C₂²	—	74	4	no	—
129	C₂²	C₂×D₃₇⋊C₈	1	2	yes	—
130	C₂²	C₂×D₃₇⋊C₈	1	1	yes	characteristic
131	C₄	—	37	4	no	—
132	C₄	C₂×D₃₇⋊C₈	1	1	yes	characteristic
133	C₄	C₂×D₃₇⋊C₈	1	2	yes	—
134	C₄	C₂³×#3866	1	1	yes	characteristic
135	C₂	—	2	2	no	—
136	C₂	—	74	4	no	—
137	C₂	#14558592	1	2	yes	direct product factor
138	C₂	C₂²×D₃₇⋊C₈	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14558592

C₂

yes

maximal, direct product factor

C₂×#11924611

C₂

yes

maximal, complemented

C₂×#11924610

C₂

yes

maximal, characteristic, complemented

C₂×D₃₇⋊C₁₆

C₂

yes

maximal, complemented

C₂²×#10596927

C₂

yes

maximal, characteristic, complemented

C₂×#11924544

C₂

yes

maximal, characteristic

#11924611

C₂²

yes

complemented

#11924610

C₂²

yes

complemented

D₃₇⋊C₁₆

C₂²

yes

complemented

C₂×#10596927

C₂²

yes

characteristic

C₂×#10596927

C₂²

yes

characteristic

C₂×#10596927

C₂²

yes

—

C₂×#10596927

C₂²

yes

complemented

C₂×#10596962

C₄

yes

characteristic

#11924544

C₄

yes

—

#11924544

C₂²

yes

—

C₂×#10596929

C₄

yes

—

C₂²×#64436

C₂²

yes

characteristic

C₂×C₈×D₃₇

C₂²

yes

—

C₂×M₁₆×C₃₇

C₄

yes

characteristic

C₂²×C₄×C₃₇

C₂×C₄

yes

characteristic

C₂×C₂²⋊D₃₇

C₈

yes

—

#10596962

C₈

yes

—

#10596962

C₂×C₄

yes

—

C₂×D₁₄₈

C₈

yes

characteristic

C₂×C₄×D₃₇

C₂×C₄

yes

—

C₂×Dic₇₄

C₈

yes

characteristic

M₁₆×C₃₇

C₂×C₄

yes

—

C₂×C₈×C₃₇

C₂×C₄

yes

—

C₂×#64436

C₂×C₄

yes

characteristic

C₂×#64436

C₂×C₄

yes

—

C₂×#64436

C₂³

yes

characteristic

C₂×#64436

C₂³

yes

—

#10596929

C₂×C₄

yes

—

C₈×D₃₇

C₂×C₄

yes

—

C₈×D₃₇

C₂³

yes

—

#10596927

C₂³

yes

—

#10596927

C₂³

yes

—

#10596927

C₂³

yes

—

C₂³×C₃₇

C₂×C₈

yes

characteristic

C₂²×D₃₇

C₂×C₈

yes

—

C₂×C₄×C₃₇

C₂×C₈

yes

characteristic

C₂×C₄×C₃₇

C₂×C₈

yes

—

C₂×C₄×C₃₇

C₂²×C₄

yes

—

C₂×C₄×C₃₇

C₂²×C₄

yes

characteristic

C₂²⋊D₃₇

C₂×C₈

yes

—

C₂×Dic₃₇

C₂×C₈

yes

—

D₁₄₈

C₂×C₈

yes

—

C₄×D₃₇

C₂×C₈

yes

—

C₄×D₃₇

C₂²×C₄

yes

—

Dic₇₄

C₂×C₈

yes

—

C₈×C₃₇

C₂²×C₄

yes

—

#64436

C₂²×C₄

yes

characteristic

#64436

C₂²×C₄

yes

—

#64436

C₂⁴

yes

characteristic

C₂²×C₃₇

—

C₂²×C₃₇

C₂²×C₈

yes

—

C₂²×C₃₇

C₂²×C₈

yes

characteristic

C₂×D₃₇

—

C₂×D₃₇

C₂²×C₈

yes

—

C₄×C₃₇

C₂²×C₈

yes

—

C₄×C₃₇

C₂²×C₈

yes

characteristic

C₄×C₃₇

C₂³×C₄

yes

characteristic

Dic₃₇

C₂²×C₈

yes

—

C₂×#504

—

maximal, Sylow

C₂×C₃₇

—

C₂×C₃₇

#504

yes

complemented

C₂×C₃₇

C₂³×C₈

yes

characteristic

D₃₇

—

C₂×#131

—

#504

—

C₂×M₃₂

—

C₂×M₃₂

—

C₂²×C₁₆

—

C₂²×C₁₆

—

C₃₇

C₂×#504

yes

characteristic, complemented, Sylow

C₂×#41

—

#131

—

#131

—

C₂×M₁₆

—

C₂×M₁₆

#3866

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

#41

—

#41

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

C₂×#3866

yes

characteristic

M₁₆

—

M₁₆

C₂×#3866

yes

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

C₂×#3866

yes

—

107

C₁₆

—

108

C₁₆

—

109

C₁₆

—

110

C₂³

—

111

C₂³

D₃₇⋊C₈

yes

characteristic

112

Q₈

—

113

D₄

—

114

D₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

D₃₇⋊C₈

yes

characteristic

119

C₂×C₄

D₃₇⋊C₈

yes

—

120

C₂×C₄

C₂²×#3866

yes

—

121

C₂×C₄

C₂²×#3866

yes

characteristic

122

C₈

—

123

C₈

—

124

C₈

—

125

C₈

C₂²×#3866

yes

—

126

C₂²

—

127

C₂²

—

128

C₂²

—

129

C₂²

C₂×D₃₇⋊C₈

yes

—

130

C₂²

C₂×D₃₇⋊C₈

yes

characteristic

131

C₄

—

132

C₄

C₂×D₃₇⋊C₈

yes

characteristic

133

C₄

C₂×D₃₇⋊C₈

yes

—

134

C₄

C₂³×#3866

yes

characteristic

135

C₂

—

136

C₂

—

137

C₂

#14558592

yes

direct product factor

138

C₂

C₂²×D₃₇⋊C₈

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×#3866
commutator	C₂×C₃₇	C₂³×C₈
Frattini	C₄	C₂³×#3866
Fitting	C₂×M₁₆×C₃₇	C₄
socle	C₂²×C₃₇	C₂²×C₈
prime radical	C₂×#10596962	C₄
cosocle	#64436	C₂⁴
hypercenter	C₂×M₁₆	#3866
hypocenter	C₃₇	C₂×#504
LCS	G ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×M₁₆

subgroup

G/H

center

C₂×C₄

C₂²×#3866

commutator

C₂×C₃₇

C₂³×C₈

Frattini

C₄

C₂³×#3866

Fitting

C₂×M₁₆×C₃₇

C₄

socle

C₂²×C₃₇

C₂²×C₈

C₄

C₂⁴

C₂×M₁₆

C₃₇

G ▷ C₂×C₃₇ ▷ C₃₇

C₁ ◁ C₂×C₄ ◁ C₂×M₁₆

N	H ≅ G/N	type
1. #14558592	C₂	direct (N × H)
2. C₂×#11924611	C₂	semidirect
3. C₂×#11924610	C₂	semidirect
4. C₂×D₃₇⋊C₁₆	C₂	semidirect
5. C₂²×#10596927	C₂	semidirect
7. #11924611	C₂²	semidirect
8. #11924610	C₂²	semidirect
9. D₃₇⋊C₁₆	C₂²	semidirect
13. C₂×#10596927	C₂²	semidirect
67. C₂×C₃₇	#504	semidirect
76. C₃₇	C₂×#504	semidirect
137. C₂	#14558592	direct (N × H)

H ≅ G/N

type

1. #14558592

C₂

direct (N × H)

2. C₂×#11924611

C₂

semidirect

3. C₂×#11924610

C₂

semidirect

4. C₂×D₃₇⋊C₁₆

C₂

semidirect

5. C₂²×#10596927

C₂

semidirect

7. #11924611

C₂²

semidirect

8. #11924610

C₂²

semidirect

9. D₃₇⋊C₁₆

C₂²

semidirect

13. C₂×#10596927

C₂²

semidirect

67. C₂×C₃₇

#504

semidirect

76. C₃₇

C₂×#504

semidirect

137. C₂

#14558592

direct (N × H)

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

subgroup	index
#14558592	2
C₂×#11924611	2
C₂×#11924610	2
C₂×D₃₇⋊C₁₆	2
C₂²×#10596927	2
C₂×#11924544	2
C₂×#504	37

subgroup

index

C₂×D₃₇⋊C₁₆

Subgroups — C₂×#14558592 #18011562

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#14558592 #18011562

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#14558592 #18011562