Group #18010806 — subgroups

Subgroups — #18010806

Order 4736.

184 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 102 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×#11924611	C₂	1	1	yes	maximal, characteristic
2	C₂×D₃₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	#14558422	C₂	1	1	yes	maximal, characteristic, complemented
4	#14558421	C₂	1	1	yes	maximal, characteristic, complemented
5	#14558419	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×#10596927	C₂	1	1	yes	maximal, characteristic, complemented
7	#14557490	C₂	1	1	yes	maximal, characteristic
8	#11924611	C₄	1	4	yes	complemented
9	D₃₇⋊C₁₆	C₄	1	4	yes	complemented
10	C₂×#10596927	C₂²	1	1	yes	characteristic
11	C₂×#10596927	C₂²	1	1	yes	characteristic
12	C₂×#10596927	C₂²	1	1	yes	characteristic
13	C₂×#10596927	C₂²	1	1	yes	characteristic
14	C₂×#10596929	C₄	1	1	yes	characteristic
15	C₂×C₈×D₃₇	C₂²	1	1	yes	characteristic
16	#11924493	C₄	1	1	yes	characteristic
17	#11924446	C₄	1	1	yes	characteristic
18	C₄×#64436	C₂²	1	1	yes	characteristic
19	C₈×Dic₃₇	C₂²	1	1	yes	characteristic
20	#97×C₃₇	C₄	1	1	yes	characteristic
21	C₂×C₄×D₃₇	C₂×C₄	1	1	yes	characteristic
22	C₂×C₈×C₃₇	C₂×C₄	1	1	yes	characteristic
23	C₂×C₈×C₃₇	C₂×C₄	1	1	yes	characteristic
24	C₄²×C₃₇	C₂×C₄	1	1	yes	characteristic
25	#10596938	C₈	1	2	yes	—
26	#10596936	C₈	1	2	yes	—
27	C₄×Dic₃₇	C₂×C₄	1	1	yes	characteristic
28	C₂×#64436	C₂×C₄	1	1	yes	characteristic
29	C₂×#64436	C₂³	1	1	yes	characteristic
30	#10596929	C₈	1	4	yes	—
31	C₈×D₃₇	C₂×C₄	1	2	yes	—
32	C₈×D₃₇	C₂×C₄	1	2	yes	—
33	#10596927	C₂×C₄	1	2	yes	—
34	#10596927	C₂×C₄	1	2	yes	—
35	#10596927	C₂×C₄	1	2	yes	—
36	#10596927	C₂×C₄	1	2	yes	—
37	C₂²×D₃₇	C₂×C₈	1	1	yes	characteristic
38	C₂×C₄×C₃₇	C₂×C₈	1	2	yes	—
39	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
40	C₂×Dic₃₇	C₂×C₈	1	2	yes	—
41	C₂×Dic₃₇	C₂×C₈	1	1	yes	characteristic
42	C₄×D₃₇	C₄²	1	2	yes	—
43	C₄×D₃₇	C₂×C₈	1	2	yes	—
44	C₈×C₃₇	C₄²	1	2	yes	—
45	C₈×C₃₇	C₂×C₈	1	2	yes	—
46	#64436	C₂×C₈	1	2	yes	—
47	#64436	C₂²×C₄	1	1	yes	characteristic
48	#64436	C₂²×C₄	1	1	yes	characteristic
49	C₂²×C₃₇	C₂²×C₈	1	1	yes	characteristic
50	C₂×D₃₇	—	2	2	no	—
51	C₂×D₃₇	C₄×C₈	1	2	yes	—
52	C₄×C₃₇	—	2	2	no	—
53	C₄×C₃₇	C₂×C₄²	1	1	yes	characteristic
54	C₄×C₃₇	C₂²×C₈	1	1	yes	characteristic
55	Dic₃₇	—	2	2	no	—
56	Dic₃₇	C₄×C₈	1	2	yes	—
57	#2108	—	37	1	no	maximal, Sylow
58	C₂×C₃₇	C₂×C₄×C₈	1	1	yes	characteristic
59	C₂×C₃₇	#504	1	2	yes	—
60	D₃₇	—	2	2	no	—
61	C₂×M₃₂	—	37	1	no	—
62	C₂²×C₁₆	—	37	1	no	—
63	#405	—	37	1	no	—
64	#346	—	37	1	no	—
65	#346	—	37	1	no	—
66	C₄×C₁₆	—	37	1	no	—
67	C₄×C₁₆	—	37	1	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	4	no	—
73	C₂×C₁₆	—	37	1	no	—
74	C₂×C₁₆	—	37	1	no	—
75	C₂×C₁₆	—	37	1	no	—
76	C₂×C₁₆	—	37	1	no	—
77	C₂×C₁₆	—	37	4	no	—
78	#97	—	37	1	no	—
79	#97	#3866	1	1	yes	characteristic
80	C₄×C₈	—	37	1	no	—
81	C₄×C₈	—	37	1	no	—
82	C₂²×C₄	—	37	1	no	—
83	M₁₆	—	37	4	no	—
84	C₂×C₈	—	37	1	no	—
85	C₂×C₈	—	37	1	no	—
86	C₂×C₈	—	37	2	no	—
87	C₂×C₈	—	37	2	no	—
88	C₂×C₈	C₂×#3866	1	1	yes	characteristic
89	C₂×C₈	C₂×#3866	1	1	yes	characteristic
90	C₄⋊C₄	—	37	2	no	—
91	C₂²⋊C₄	—	37	2	no	—
92	C₄²	—	37	1	no	—
93	C₄²	C₂×#3866	1	1	yes	characteristic
94	C₁₆	—	37	2	no	—
95	C₁₆	—	37	2	no	—
96	C₁₆	—	37	2	no	—
97	C₁₆	—	37	2	no	—
98	C₂³	—	37	1	no	—
99	C₂×C₄	—	37	1	no	—
100	C₂×C₄	—	37	2	no	—
101	C₂×C₄	—	37	2	no	—
102	C₂×C₄	—	37	2	no	—
103	C₂×C₄	D₃₇⋊C₈	1	2	yes	—
104	C₂×C₄	C₂²×#3866	1	1	yes	characteristic
105	C₈	—	37	1	no	—
106	C₈	—	37	1	no	—
107	C₈	—	37	2	no	—
108	C₈	D₃₇⋊C₈	1	2	yes	—
109	C₈	C₄×#3866	1	2	yes	—
110	C₂²	—	37	2	no	—
111	C₂²	—	74	2	no	—
112	C₂²	C₂×D₃₇⋊C₈	1	1	yes	characteristic
113	C₄	—	2	2	no	—
114	C₄	—	37	2	no	—
115	C₄	—	74	2	no	—
116	C₄	C₂×C₄×#3866	1	1	yes	characteristic
117	C₄	C₂×D₃₇⋊C₈	1	1	yes	characteristic
118	C₂	—	74	2	no	—
119	C₂	C₄×D₃₇⋊C₈	1	1	yes	characteristic
120	C₂	#14558592	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11924611

C₂

yes

maximal, characteristic

C₂×D₃₇⋊C₁₆

C₂

yes

maximal, characteristic

#14558422

C₂

yes

maximal, characteristic, complemented

#14558421

C₂

yes

maximal, characteristic, complemented

#14558419

C₂

yes

maximal, characteristic, complemented

C₄×#10596927

C₂

yes

maximal, characteristic, complemented

#14557490

C₂

yes

maximal, characteristic

#11924611

C₄

yes

complemented

D₃₇⋊C₁₆

C₄

yes

complemented

C₂×#10596927

C₂²

yes

characteristic

C₂×#10596927

C₂²

yes

characteristic

C₂×#10596927

C₂²

yes

characteristic

C₂×#10596927

C₂²

yes

characteristic

C₂×#10596929

C₄

yes

characteristic

C₂×C₈×D₃₇

C₂²

yes

characteristic

#11924493

C₄

yes

characteristic

#11924446

C₄

yes

characteristic

C₄×#64436

C₂²

yes

characteristic

C₈×Dic₃₇

C₂²

yes

characteristic

#97×C₃₇

C₄

yes

characteristic

C₂×C₄×D₃₇

C₂×C₄

yes

characteristic

C₂×C₈×C₃₇

C₂×C₄

yes

characteristic

C₂×C₈×C₃₇

C₂×C₄

yes

characteristic

C₄²×C₃₇

C₂×C₄

yes

characteristic

#10596938

C₈

yes

—

#10596936

C₈

yes

—

C₄×Dic₃₇

C₂×C₄

yes

characteristic

C₂×#64436

C₂×C₄

yes

characteristic

C₂×#64436

C₂³

yes

characteristic

#10596929

C₈

yes

—

C₈×D₃₇

C₂×C₄

yes

—

C₈×D₃₇

C₂×C₄

yes

—

#10596927

C₂×C₄

yes

—

#10596927

C₂×C₄

yes

—

#10596927

C₂×C₄

yes

—

#10596927

C₂×C₄

yes

—

C₂²×D₃₇

C₂×C₈

yes

characteristic

C₂×C₄×C₃₇

C₂×C₈

yes

—

C₂×C₄×C₃₇

C₂²×C₄

yes

characteristic

C₂×Dic₃₇

C₂×C₈

yes

—

C₂×Dic₃₇

C₂×C₈

yes

characteristic

C₄×D₃₇

C₄²

yes

—

C₄×D₃₇

C₂×C₈

yes

—

C₈×C₃₇

C₄²

yes

—

C₈×C₃₇

C₂×C₈

yes

—

#64436

C₂×C₈

yes

—

#64436

C₂²×C₄

yes

characteristic

#64436

C₂²×C₄

yes

characteristic

C₂²×C₃₇

C₂²×C₈

yes

characteristic

C₂×D₃₇

—

C₂×D₃₇

C₄×C₈

yes

—

C₄×C₃₇

—

C₄×C₃₇

C₂×C₄²

yes

characteristic

C₄×C₃₇

C₂²×C₈

yes

characteristic

Dic₃₇

—

Dic₃₇

C₄×C₈

yes

—

#2108

—

maximal, Sylow

C₂×C₃₇

C₂×C₄×C₈

yes

characteristic

C₂×C₃₇

#504

yes

—

D₃₇

—

C₂×M₃₂

—

C₂²×C₁₆

—

#405

—

#346

—

#346

—

C₄×C₁₆

—

C₄×C₁₆

—

C₃₇

#2108

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂²×C₈

—

#117

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

—

yes

C₄×C₈

—

C₄×C₈

—

C₂²×C₄

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

yes

C₂×C₈

yes

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₂×#3866

yes

characteristic

C₁₆

—

C₁₆

—

C₁₆

—

C₁₆

—

C₂³

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

D₃₇⋊C₈

yes

—

104

C₂×C₄

C₂²×#3866

yes

characteristic

105

C₈

—

106

C₈

—

107

C₈

—

108

C₈

D₃₇⋊C₈

yes

—

109

C₈

C₄×#3866

yes

—

110

C₂²

—

111

C₂²

—

112

C₂²

C₂×D₃₇⋊C₈

yes

characteristic

113

C₄

—

114

C₄

—

115

C₄

—

116

C₄

C₂×C₄×#3866

yes

characteristic

117

C₄

C₂×D₃₇⋊C₈

yes

characteristic

118

C₂

—

119

C₂

C₄×D₃₇⋊C₈

yes

characteristic

120

C₂

#14558592

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×#3866
commutator	C₂×C₃₇	C₂×C₄×C₈
Fitting	#97×C₃₇	C₄
socle	C₂²×C₃₇	C₂²×C₈
prime radical	C₂²×D₃₇	C₂×C₈
cosocle	C₂×#64436	C₂³
hypercenter	#97	#3866
hypocenter	C₃₇	#2108
LCS	G ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂×C₄ ◁ #97

subgroup

G/H

center, Frattini

C₂×C₄

C₂²×#3866

commutator

C₂×C₃₇

C₂×C₄×C₈

Fitting

#97×C₃₇

C₄

socle

C₂²×C₃₇

C₂²×C₈

prime radical

C₂²×D₃₇

C₂×C₈

C₂³

C₃₇

G ▷ C₂×C₃₇ ▷ C₃₇

C₁ ◁ C₂×C₄ ◁ #97

N	H ≅ G/N	type
3. #14558422	C₂	semidirect
4. #14558421	C₂	semidirect
5. #14558419	C₂	semidirect
6. C₄×#10596927	C₂	semidirect
8. #11924611	C₄	semidirect
9. D₃₇⋊C₁₆	C₄	semidirect
68. C₃₇	#2108	semidirect

H ≅ G/N

type

3. #14558422

C₂

semidirect

4. #14558421

C₂

semidirect

5. #14558419

C₂

semidirect

6. C₄×#10596927

C₂

semidirect

8. #11924611

C₄

semidirect

9. D₃₇⋊C₁₆

C₄

semidirect

68. C₃₇

#2108

semidirect

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

subgroup	index
C₂×#11924611	2
C₂×D₃₇⋊C₁₆	2
#14558422	2
#14558421	2
#14558419	2
C₄×#10596927	2
#14557490	2
#2108	37

subgroup

index

C₂×D₃₇⋊C₁₆

Subgroups — #18010806

Conjugacy classes of subgroups

Complements

Maximal subgroups