Group #18010808 — subgroups

Subgroups — #18010808

Order 4736.

184 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 96 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×D₃₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	#14558496	C₂	1	1	yes	maximal, characteristic
3	#14558420	C₂	1	2	yes	maximal, complemented
4	C₁₆×#3866	C₂	1	2	yes	maximal, complemented
5	M₃₂×D₃₇	C₂	1	1	yes	maximal, characteristic
6	D₃₇⋊M₁₆	C₄	1	1	yes	characteristic
7	#11924615	C₄	1	1	yes	characteristic
8	D₃₇⋊C₁₆	C₄	1	2	yes	complemented
9	D₃₇⋊C₁₆	C₂²	1	1	yes	characteristic
10	D₃₇⋊C₁₆	C₂²	1	1	yes	characteristic
11	C₂×#10596927	C₄	1	2	yes	complemented
12	#11924590	C₄	1	2	yes	—
13	C₈×#3866	C₂²	1	2	yes	—
14	C₂×C₈×D₃₇	C₂²	1	1	yes	characteristic
15	#11924476	C₄	1	1	yes	characteristic, complemented
16	#11924475	C₄	1	2	yes	complemented
17	C₁₆×D₃₇	C₂²	1	2	yes	—
18	M₃₂×C₃₇	C₄	1	1	yes	characteristic, complemented
19	C₂×C₄×D₃₇	C₂×C₄	1	1	yes	characteristic
20	C₂²⋊#3866	C₈	1	2	yes	—
21	#10596957	C₈	1	2	yes	—
22	C₄⋊#3866	C₈	1	2	yes	—
23	C₄×#3866	C₂×C₄	1	2	yes	—
24	#10596953	C₈	1	2	yes	—
25	D₃₇⋊C₈	C₂×C₄	1	2	yes	—
26	C₂×C₈×C₃₇	C₂×C₄	1	1	yes	characteristic
27	C₂×#64436	C₂×C₄	1	1	yes	characteristic
28	C₈×D₃₇	C₂×C₄	1	1	yes	characteristic
29	C₈×D₃₇	C₂×C₄	1	1	yes	characteristic
30	C₈×D₃₇	C₂×C₄	1	1	yes	characteristic
31	C₈×D₃₇	C₂³	1	1	yes	characteristic
32	#10596927	C₂×C₄	1	2	yes	—
33	#10596927	C₂×C₄	1	2	yes	—
34	C₁₆×C₃₇	C₂×C₄	1	2	yes	—
35	#10596925	C₂×C₄	1	2	yes	—
36	C₂²×D₃₇	C₂×C₈	1	1	yes	characteristic
37	C₂×#3866	C₂×C₈	1	4	yes	—
38	C₂×C₄×C₃₇	C₄²	1	1	yes	characteristic
39	C₂×Dic₃₇	C₂×C₈	1	1	yes	characteristic
40	C₄×D₃₇	C₄²	1	1	yes	characteristic
41	C₄×D₃₇	C₂×C₈	1	1	yes	characteristic
42	C₄×D₃₇	C₂×C₈	1	1	yes	characteristic
43	C₄×D₃₇	C₂²×C₄	1	1	yes	characteristic
44	#64438	C₂×C₈	1	4	yes	—
45	C₈×C₃₇	C₄²	1	1	yes	characteristic
46	C₈×C₃₇	C₂²×C₄	1	1	yes	characteristic
47	#64436	C₄²	1	1	yes	characteristic
48	#64436	C₂²×C₄	1	1	yes	characteristic
49	C₂²×C₃₇	C₄×C₈	1	1	yes	characteristic
50	C₂×D₃₇	—	2	2	no	—
51	C₂×D₃₇	C₄×C₈	1	1	yes	characteristic
52	C₂×D₃₇	C₂²×C₈	1	1	yes	characteristic
53	#3866	—	2	4	no	—
54	C₄×C₃₇	C₄×C₈	1	1	yes	characteristic
55	C₄×C₃₇	C₂×C₄²	1	1	yes	characteristic
56	Dic₃₇	C₄×C₈	1	1	yes	characteristic
57	Dic₃₇	C₂²×C₈	1	1	yes	characteristic
58	#2108	—	37	1	no	maximal, Sylow
59	C₂×C₃₇	—	2	1	no	—
60	C₂×C₃₇	C₂×C₄×C₈	1	1	yes	characteristic
61	D₃₇	—	2	1	no	—
62	D₃₇	#504	1	2	yes	—
63	C₂×M₃₂	—	37	1	no	—
64	C₂²×C₁₆	—	37	1	no	—
65	#405	—	37	1	no	—
66	#346	—	37	2	no	—
67	C₄×C₁₆	—	37	2	no	—
68	C₃₇	#2108	1	1	yes	characteristic, complemented, Sylow
69	C₂×M₁₆	—	37	1	no	—
70	C₂²×C₈	—	37	1	no	—
71	#117	—	37	1	no	—
72	M₃₂	—	37	1	no	—
73	M₃₂	—	37	2	no	—
74	M₃₂	#3866	1	1	yes	characteristic, complemented
75	C₂×C₁₆	—	37	1	no	—
76	C₂×C₁₆	—	37	1	no	—
77	C₂×C₁₆	—	37	2	no	—
78	C₂×C₁₆	—	37	2	no	—
79	C₂×C₁₆	—	37	2	no	—
80	#97	—	37	2	no	—
81	C₄×C₈	—	37	2	no	—
82	C₂²×C₄	—	37	1	no	—
83	M₁₆	—	37	2	no	—
84	M₁₆	—	37	2	no	—
85	C₂×C₈	—	37	2	no	—
86	C₂×C₈	—	37	1	no	—
87	C₂×C₈	—	37	1	no	—
88	C₂×C₈	—	37	1	no	—
89	C₂×C₈	—	37	1	no	—
90	C₂×C₈	—	37	1	no	—
91	C₂×C₈	C₂×#3866	1	1	yes	characteristic
92	C₄⋊C₄	—	37	2	no	—
93	C₂²⋊C₄	—	37	2	no	—
94	C₄²	—	37	2	no	—
95	C₁₆	—	37	2	no	—
96	C₁₆	—	37	2	no	—
97	C₁₆	—	37	2	no	—
98	C₁₆	C₂×#3866	1	2	yes	—
99	C₂³	—	37	1	no	—
100	C₂×C₄	—	37	1	no	—
101	C₂×C₄	—	37	1	no	—
102	C₂×C₄	—	37	1	no	—
103	C₂×C₄	—	37	1	no	—
104	C₂×C₄	—	37	1	no	—
105	C₂×C₄	—	37	4	no	—
106	C₂×C₄	C₄×#3866	1	1	yes	characteristic
107	C₈	—	37	4	no	—
108	C₈	—	37	1	no	—
109	C₈	—	37	1	no	—
110	C₈	C₄×#3866	1	1	yes	characteristic
111	C₈	C₂²×#3866	1	1	yes	characteristic
112	C₂²	—	37	1	no	—
113	C₂²	—	37	1	no	—
114	C₂²	—	74	2	no	—
115	C₂²	C₈×#3866	1	1	yes	characteristic
116	C₄	—	37	1	no	—
117	C₄	—	37	1	no	—
118	C₄	—	74	4	no	—
119	C₄	C₈×#3866	1	1	yes	characteristic
120	C₄	C₂×C₄×#3866	1	1	yes	characteristic
121	C₂	—	2	1	no	—
122	C₂	—	37	2	no	—
123	C₂	—	74	1	no	—
124	C₂	C₂×C₈×#3866	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×D₃₇⋊C₁₆

C₂

yes

maximal, characteristic

#14558496

C₂

yes

maximal, characteristic

#14558420

C₂

yes

maximal, complemented

C₁₆×#3866

C₂

yes

maximal, complemented

M₃₂×D₃₇

C₂

yes

maximal, characteristic

D₃₇⋊M₁₆

C₄

yes

characteristic

#11924615

C₄

yes

characteristic

D₃₇⋊C₁₆

C₄

yes

complemented

D₃₇⋊C₁₆

C₂²

yes

characteristic

D₃₇⋊C₁₆

C₂²

yes

characteristic

C₂×#10596927

C₄

yes

complemented

#11924590

C₄

yes

—

C₈×#3866

C₂²

yes

—

C₂×C₈×D₃₇

C₂²

yes

characteristic

#11924476

C₄

yes

characteristic, complemented

#11924475

C₄

yes

complemented

C₁₆×D₃₇

C₂²

yes

—

M₃₂×C₃₇

C₄

yes

characteristic, complemented

C₂×C₄×D₃₇

C₂×C₄

yes

characteristic

C₂²⋊#3866

C₈

yes

—

#10596957

C₈

yes

—

C₄⋊#3866

C₈

yes

—

C₄×#3866

C₂×C₄

yes

—

#10596953

C₈

yes

—

D₃₇⋊C₈

C₂×C₄

yes

—

C₂×C₈×C₃₇

C₂×C₄

yes

characteristic

C₂×#64436

C₂×C₄

yes

characteristic

C₈×D₃₇

C₂×C₄

yes

characteristic

C₈×D₃₇

C₂×C₄

yes

characteristic

C₈×D₃₇

C₂×C₄

yes

characteristic

C₈×D₃₇

C₂³

yes

characteristic

#10596927

C₂×C₄

yes

—

#10596927

C₂×C₄

yes

—

C₁₆×C₃₇

C₂×C₄

yes

—

#10596925

C₂×C₄

yes

—

C₂²×D₃₇

C₂×C₈

yes

characteristic

C₂×#3866

C₂×C₈

yes

—

C₂×C₄×C₃₇

C₄²

yes

characteristic

C₂×Dic₃₇

C₂×C₈

yes

characteristic

C₄×D₃₇

C₄²

yes

characteristic

C₄×D₃₇

C₂×C₈

yes

characteristic

C₄×D₃₇

C₂×C₈

yes

characteristic

C₄×D₃₇

C₂²×C₄

yes

characteristic

#64438

C₂×C₈

yes

—

C₈×C₃₇

C₄²

yes

characteristic

C₈×C₃₇

C₂²×C₄

yes

characteristic

#64436

C₄²

yes

characteristic

#64436

C₂²×C₄

yes

characteristic

C₂²×C₃₇

C₄×C₈

yes

characteristic

C₂×D₃₇

—

C₂×D₃₇

C₄×C₈

yes

characteristic

C₂×D₃₇

C₂²×C₈

yes

characteristic

#3866

—

C₄×C₃₇

C₄×C₈

yes

characteristic

C₄×C₃₇

C₂×C₄²

yes

characteristic

Dic₃₇

C₄×C₈

yes

characteristic

Dic₃₇

C₂²×C₈

yes

characteristic

#2108

—

maximal, Sylow

C₂×C₃₇

—

C₂×C₃₇

C₂×C₄×C₈

yes

characteristic

D₃₇

—

D₃₇

#504

yes

—

C₂×M₃₂

—

C₂²×C₁₆

—

#405

—

#346

—

C₄×C₁₆

—

C₃₇

#2108

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂²×C₈

—

#117

—

M₃₂

—

M₃₂

—

M₃₂

#3866

yes

characteristic, complemented

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

#97

—

C₄×C₈

—

C₂²×C₄

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×#3866

yes

characteristic

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₁₆

—

C₁₆

—

C₁₆

—

C₁₆

C₂×#3866

yes

—

C₂³

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

C₄×#3866

yes

characteristic

107

C₈

—

108

C₈

—

109

C₈

—

110

C₈

C₄×#3866

yes

characteristic

111

C₈

C₂²×#3866

yes

characteristic

112

C₂²

—

113

C₂²

—

114

C₂²

—

115

C₂²

C₈×#3866

yes

characteristic

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

C₈×#3866

yes

characteristic

120

C₄

C₂×C₄×#3866

yes

characteristic

121

C₂

—

122

C₂

—

123

C₂

—

124

C₂

C₂×C₈×#3866

yes

characteristic

	subgroup	G/H
center, Frattini	C₈	C₂²×#3866
commutator, socle	C₂×C₃₇	C₂×C₄×C₈
Fitting	M₃₂×C₃₇	C₄
prime radical	C₂²×D₃₇	C₂×C₈
cosocle	C₈×D₃₇	C₂³
hypercenter	M₃₂	#3866
hypocenter	C₃₇	#2108
LCS	G ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₈ ◁ M₃₂

subgroup

G/H

center, Frattini

C₈

C₂²×#3866

commutator, socle

C₂×C₃₇

C₂×C₄×C₈

Fitting

M₃₂×C₃₇

C₄

prime radical

C₂²×D₃₇

C₂×C₈

cosocle

C₈×D₃₇

C₂³

M₃₂

C₃₇

G ▷ C₂×C₃₇ ▷ C₃₇

C₁ ◁ C₈ ◁ M₃₂

N	H ≅ G/N	type
3. #14558420	C₂	semidirect
4. C₁₆×#3866	C₂	semidirect
8. D₃₇⋊C₁₆	C₄	semidirect
11. C₂×#10596927	C₄	semidirect
15. #11924476	C₄	semidirect
16. #11924475	C₄	semidirect
18. M₃₂×C₃₇	C₄	semidirect
68. C₃₇	#2108	semidirect
74. M₃₂	#3866	semidirect

H ≅ G/N

type

3. #14558420

C₂

semidirect

4. C₁₆×#3866

C₂

semidirect

8. D₃₇⋊C₁₆

C₄

semidirect

11. C₂×#10596927

C₄

semidirect

15. #11924476

C₄

semidirect

16. #11924475

C₄

semidirect

18. M₃₂×C₃₇

C₄

semidirect

68. C₃₇

#2108

semidirect

74. M₃₂

#3866

semidirect

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

subgroup	index
C₂×D₃₇⋊C₁₆	2
#14558496	2
#14558420	2
C₁₆×#3866	2
M₃₂×D₃₇	2
#2108	37

subgroup

index

C₂×D₃₇⋊C₁₆

#14558496

#14558420

C₁₆×#3866

M₃₂×D₃₇

#2108

Subgroups — #18010808

Conjugacy classes of subgroups

Complements

Maximal subgroups