Group #18011565 — subgroups

Subgroups — #18011565

Order 4736.

360 conjugacy classes of subgroups; only the 198 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 198 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, complemented
2	D₃₇⋊M₃₂	C₂	1	4	yes	maximal, complemented
3	C₂×#11924610	C₂	1	2	yes	maximal, complemented
4	C₂×#11924544	C₂	1	1	yes	maximal, characteristic
5	#11924611	C₂²	1	8	yes	complemented
6	#11924610	C₂²	1	4	yes	—
7	#11924610	C₂²	1	4	yes	complemented
8	D₃₇⋊C₁₆	C₂²	1	8	yes	complemented
9	C₂×#10596927	C₂²	1	4	yes	complemented
10	C₂×#10596962	C₄	1	1	yes	characteristic
11	#11924544	C₄	1	4	yes	—
12	#11924544	C₂²	1	4	yes	—
13	C₂×#10596929	C₄	1	2	yes	—
14	C₂²×#64436	C₂²	1	1	yes	characteristic
15	C₂×C₈×D₃₇	C₂²	1	2	yes	—
16	C₂×M₁₆×C₃₇	C₄	1	1	yes	characteristic
17	C₂²×C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
18	C₂×C₂²⋊D₃₇	C₈	1	2	yes	—
19	#10596962	C₈	1	4	yes	—
20	#10596962	C₂×C₄	1	4	yes	—
21	C₂×D₁₄₈	C₈	1	1	yes	characteristic
22	C₂×C₄×D₃₇	C₂×C₄	1	2	yes	—
23	C₂×Dic₇₄	C₈	1	1	yes	characteristic
24	M₁₆×C₃₇	C₂×C₄	1	4	yes	—
25	C₂×C₈×C₃₇	C₂×C₄	1	2	yes	—
26	C₂×#64436	C₂×C₄	1	1	yes	characteristic
27	C₂×#64436	C₂×C₄	1	2	yes	—
28	C₂×#64436	C₂³	1	1	yes	characteristic
29	C₂×#64436	C₂³	1	2	yes	—
30	#10596929	C₂×C₄	1	8	yes	—
31	C₈×D₃₇	C₂×C₄	1	4	yes	—
32	C₈×D₃₇	C₂³	1	4	yes	—
33	#10596927	C₂³	1	8	yes	—
34	C₂³×C₃₇	C₂×C₈	1	1	yes	characteristic
35	C₂²×D₃₇	C₂×C₈	1	2	yes	—
36	C₂×C₄×C₃₇	C₂×C₈	1	1	yes	characteristic
37	C₂×C₄×C₃₇	C₂×C₈	1	2	yes	—
38	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₄×C₃₇	C₂²×C₄	1	2	yes	—
40	C₂²⋊D₃₇	C₂×C₈	1	8	yes	—
41	C₂×Dic₃₇	C₂×C₈	1	2	yes	—
42	D₁₄₈	C₂×C₈	1	4	yes	—
43	C₄×D₃₇	C₂×C₈	1	4	yes	—
44	C₄×D₃₇	C₂²×C₄	1	4	yes	—
45	Dic₇₄	C₂×C₈	1	4	yes	—
46	C₈×C₃₇	C₂²×C₄	1	4	yes	—
47	#64436	C₂²×C₄	1	1	yes	characteristic
48	#64436	C₂²×C₄	1	2	yes	—
49	#64436	C₂⁴	1	1	yes	characteristic
50	C₂²×C₃₇	—	4	1	no	—
51	C₂²×C₃₇	C₂²×C₈	1	1	yes	characteristic
52	C₂²×C₃₇	C₂²×C₈	1	2	yes	—
53	C₂×D₃₇	—	4	2	no	—
54	C₂×D₃₇	C₂²×C₈	1	4	yes	—
55	C₄×C₃₇	C₂²×C₈	1	2	yes	—
56	C₄×C₃₇	C₂²×C₈	1	1	yes	characteristic
57	C₄×C₃₇	C₂³×C₄	1	1	yes	characteristic
58	Dic₃₇	C₂²×C₈	1	4	yes	—
59	#3407	—	37	1	no	maximal, Sylow
60	C₂×C₃₇	—	2	1	no	—
61	C₂×C₃₇	—	2	2	no	—
62	C₂×C₃₇	C₂³×C₈	1	1	yes	characteristic
63	D₃₇	—	2	4	no	—
64	C₂×#131	—	37	1	no	—
65	#504	—	37	8	no	—
66	C₂×M₃₂	—	37	2	no	—
67	C₂×M₃₂	—	37	4	no	—
68	C₃₇	#3407	1	1	yes	characteristic, complemented, Sylow
69	C₂×#41	—	37	1	no	—
70	#131	—	37	4	no	—
71	#131	—	37	4	no	—
72	C₂×M₁₆	—	37	2	no	—
73	C₂×M₁₆	#3866	1	1	yes	characteristic
74	C₂²×C₈	—	37	1	no	—
75	C₂²×C₈	—	37	2	no	—
76	M₃₂	—	37	4	no	—
77	M₃₂	—	37	4	no	—
78	M₃₂	—	37	8	no	—
79	C₂×C₁₆	—	37	4	no	—
80	C₂×C₁₆	—	37	8	no	—
81	#41	—	37	4	no	—
82	#41	—	37	4	no	—
83	C₂×Q₈	—	37	1	no	—
84	C₂×D₄	—	37	2	no	—
85	C₂×D₄	—	37	1	no	—
86	C₂²×C₄	—	37	2	no	—
87	C₂²×C₄	C₂×#3866	1	1	yes	characteristic
88	M₁₆	—	37	8	no	—
89	M₁₆	C₂×#3866	1	4	yes	—
90	C₂×C₈	—	37	1	no	—
91	C₂×C₈	—	37	2	no	—
92	C₂×C₈	—	37	2	no	—
93	C₂×C₈	—	37	1	no	—
94	C₂×C₈	—	37	4	no	—
95	C₂×C₈	—	37	4	no	—
96	C₂×C₈	C₂×#3866	1	2	yes	—
97	C₁₆	—	37	8	no	—
98	C₂³	—	37	2	no	—
99	C₂³	D₃₇⋊C₈	1	1	yes	characteristic
100	Q₈	—	37	4	no	—
101	D₄	—	37	8	no	—
102	D₄	—	37	4	no	—
103	C₂×C₄	—	37	2	no	—
104	C₂×C₄	—	37	4	no	—
105	C₂×C₄	—	37	4	no	—
106	C₂×C₄	D₃₇⋊C₈	1	1	yes	characteristic
107	C₂×C₄	D₃₇⋊C₈	1	2	yes	—
108	C₂×C₄	C₂²×#3866	1	1	yes	characteristic
109	C₂×C₄	C₂²×#3866	1	2	yes	—
110	C₈	—	37	1	no	—
111	C₈	—	37	1	no	—
112	C₈	—	37	2	no	—
113	C₈	C₂²×#3866	1	4	yes	—
114	C₂²	—	4	1	no	—
115	C₂²	—	37	4	no	—
116	C₂²	—	148	2	no	—
117	C₂²	C₂×D₃₇⋊C₈	1	1	yes	characteristic
118	C₂²	C₂×D₃₇⋊C₈	1	2	yes	—
119	C₄	—	37	4	no	—
120	C₄	C₂×D₃₇⋊C₈	1	1	yes	characteristic
121	C₄	C₂×D₃₇⋊C₈	1	2	yes	—
122	C₄	C₂³×#3866	1	1	yes	characteristic
123	C₂	—	2	1	no	—
124	C₂	—	2	2	no	—
125	C₂	—	74	4	no	—
126	C₂	C₂²×D₃₇⋊C₈	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14558592

C₂

yes

maximal, complemented

D₃₇⋊M₃₂

C₂

yes

maximal, complemented

C₂×#11924610

C₂

yes

maximal, complemented

C₂×#11924544

C₂

yes

maximal, characteristic

#11924611

C₂²

yes

complemented

#11924610

C₂²

yes

—

#11924610

C₂²

yes

complemented

D₃₇⋊C₁₆

C₂²

yes

complemented

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

—

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

characteristic

C₂×C₄×C₃₇

C₂²×C₄

yes

—

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

characteristic

C₂²×C₃₇

C₂²×C₈

yes

—

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

—

#3407

—

maximal, Sylow

C₂×C₃₇

—

C₂×C₃₇

—

C₂×C₃₇

C₂³×C₈

yes

characteristic

D₃₇

—

C₂×#131

—

#504

—

C₂×M₃₂

—

C₂×M₃₂

—

C₃₇

#3407

yes

characteristic, complemented, Sylow

C₂×#41

—

#131

—

#131

—

C₂×M₁₆

—

C₂×M₁₆

#3866

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

M₃₂

—

M₃₂

—

M₃₂

—

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

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×#3866

yes

—

C₁₆

—

C₂³

—

C₂³

D₃₇⋊C₈

yes

characteristic

100

Q₈

—

101

D₄

—

102

D₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

D₃₇⋊C₈

yes

characteristic

107

C₂×C₄

D₃₇⋊C₈

yes

—

108

C₂×C₄

C₂²×#3866

yes

characteristic

109

C₂×C₄

C₂²×#3866

yes

—

110

C₈

—

111

C₈

—

112

C₈

—

113

C₈

C₂²×#3866

yes

—

114

C₂²

—

115

C₂²

—

116

C₂²

—

148

—

117

C₂²

C₂×D₃₇⋊C₈

yes

characteristic

118

C₂²

C₂×D₃₇⋊C₈

yes

—

119

C₄

—

120

C₄

C₂×D₃₇⋊C₈

yes

characteristic

121

C₄

C₂×D₃₇⋊C₈

yes

—

122

C₄

C₂³×#3866

yes

characteristic

123

C₂

—

124

C₂

—

125

C₂

—

126

C₂

C₂²×D₃₇⋊C₈

yes

characteristic

	subgroup	G/H
center, Frattini	C₄	C₂³×#3866
commutator, socle	C₂×C₃₇	C₂³×C₈
Fitting	C₂×M₁₆×C₃₇	C₄
prime radical	C₂×#10596962	C₄
cosocle	#64436	C₂⁴
hypercenter	C₂×M₁₆	#3866
hypocenter	C₃₇	#3407
LCS	G ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆

subgroup

G/H

center, Frattini

C₄

C₂³×#3866

commutator, socle

C₂×C₃₇

C₂³×C₈

Fitting

C₂×M₁₆×C₃₇

C₄

C₄

C₂⁴

C₂×M₁₆

C₃₇

G ▷ C₂×C₃₇ ▷ C₃₇

C₁ ◁ C₄ ◁ C₂×M₁₆

N	H ≅ G/N	type
1. #14558592	C₂	semidirect
2. D₃₇⋊M₃₂	C₂	semidirect
3. C₂×#11924610	C₂	semidirect
5. #11924611	C₂²	semidirect
7. #11924610	C₂²	semidirect
8. D₃₇⋊C₁₆	C₂²	semidirect
9. C₂×#10596927	C₂²	semidirect
68. C₃₇	#3407	semidirect

H ≅ G/N

type

1. #14558592

C₂

semidirect

2. D₃₇⋊M₃₂

C₂

semidirect

3. C₂×#11924610

C₂

semidirect

5. #11924611

C₂²

semidirect

7. #11924610

C₂²

semidirect

8. D₃₇⋊C₁₆

C₂²

semidirect

9. C₂×#10596927

C₂²

semidirect

68. C₃₇

#3407

semidirect

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

subgroup	index
#14558592	2
D₃₇⋊M₃₂	2
C₂×#11924610	2
C₂×#11924544	2
#3407	37

subgroup

index

D₃₇⋊M₃₂

Subgroups — #18011565

Conjugacy classes of subgroups

Complements

Maximal subgroups