Group #18010252 — subgroups

Subgroups — #18010252

Order 4736.

328 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 328 conjugacy classes of subgroups — too many to draw. Only its 158 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	#14558492	C₂	1	1	yes	maximal, characteristic
2	C₂×#11924590	C₂	1	2	yes	maximal
3	C₂×C₈×#3866	C₂	1	2	yes	maximal
4	#14558469	C₂	1	1	yes	maximal, characteristic
5	C₄×C₈×D₃₇	C₂	1	1	yes	maximal, characteristic
6	C₂×D₃₇⋊C₈	C₂²	1	2	yes	—
7	C₂×C₄×#3866	C₂²	1	2	yes	—
8	#11924591	C₄	1	4	yes	complemented
9	#11924590	C₄	1	8	yes	complemented
10	C₈×#3866	C₄	1	8	yes	complemented
11	(C₂×#3866)⋊C₄	C₄	1	4	yes	—
12	C₂×C₈×D₃₇	C₂²	1	1	yes	characteristic
13	C₂×C₈×D₃₇	C₂²	1	1	yes	characteristic
14	C₄²×D₃₇	C₂²	1	1	yes	characteristic
15	C₄×#64436	C₄	1	1	yes	characteristic, complemented
16	C₈×Dic₃₇	C₄	1	1	yes	characteristic, complemented
17	C₈×Dic₃₇	C₄	1	1	yes	characteristic, complemented
18	C₄×C₈×C₃₇	C₄	1	1	yes	characteristic, complemented
19	C₂²×#3866	C₂×C₄	1	4	yes	—
20	C₂×C₄×D₃₇	C₂×C₄	1	2	yes	—
21	C₂×C₄×D₃₇	C₂³	1	1	yes	characteristic
22	C₂×#64438	C₂×C₄	1	4	yes	—
23	C₄×#3866	C₂×C₄	1	4	yes	—
24	C₄×#3866	C₂×C₄	1	4	yes	—
25	D₃₇⋊C₈	C₂×C₄	1	4	yes	—
26	D₃₇⋊C₈	C₂×C₄	1	4	yes	—
27	C₂×C₈×C₃₇	C₂×C₄	1	1	yes	characteristic
28	C₂×C₈×C₃₇	C₂×C₄	1	1	yes	characteristic
29	C₄²×C₃₇	C₂×C₄	1	1	yes	characteristic
30	C₄×Dic₃₇	C₂×C₄	1	1	yes	characteristic
31	C₄×Dic₃₇	C₂×C₄	1	2	yes	—
32	C₂×#64436	C₂×C₄	1	1	yes	characteristic
33	C₂×#64436	C₂×C₄	1	1	yes	characteristic
34	C₈×D₃₇	C₂×C₄	1	4	yes	—
35	C₈×D₃₇	C₂×C₄	1	4	yes	—
36	C₂²×D₃₇	C₂²×C₄	1	1	yes	characteristic
37	C₂×#3866	—	2	8	no	—
38	C₂×#3866	C₄²	1	8	yes	—
39	C₂×C₄×C₃₇	C₄²	1	2	yes	—
40	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
41	C₂×Dic₃₇	C₄²	1	2	yes	—
42	C₂×Dic₃₇	C₂²×C₄	1	1	yes	characteristic
43	C₄×D₃₇	—	2	4	no	—
44	C₄×D₃₇	C₂²×C₄	1	2	yes	—
45	C₄×D₃₇	C₂²×C₄	1	2	yes	—
46	#64438	C₄²	1	8	yes	—
47	C₈×C₃₇	C₄²	1	2	yes	—
48	C₈×C₃₇	C₄²	1	2	yes	—
49	#64436	C₄²	1	2	yes	—
50	#64436	C₄²	1	2	yes	—
51	C₂²×C₃₇	C₂×C₄²	1	1	yes	characteristic
52	C₂×D₃₇	C₂×C₄²	1	2	yes	—
53	C₂×D₃₇	#131	1	4	yes	—
54	#3866	—	2	8	no	—
55	C₄×C₃₇	—	2	2	no	—
56	C₄×C₃₇	C₂×C₄²	1	1	yes	characteristic
57	C₄×C₃₇	C₂×C₄²	1	1	yes	characteristic
58	Dic₃₇	—	2	2	no	—
59	Dic₃₇	C₂×C₄²	1	2	yes	—
60	#1726	—	37	1	no	maximal, Sylow
61	C₂×C₃₇	C₄³	1	1	yes	characteristic
62	C₂×C₃₇	#405	1	2	yes	—
63	D₃₇	#405	1	4	yes	—
64	C₂×#97	—	37	2	no	—
65	C₂×#97	—	37	1	no	—
66	C₂×C₄×C₈	—	37	2	no	—
67	C₂×C₄×C₈	—	37	1	no	—
68	#376	—	37	1	no	—
69	C₃₇	#1726	1	1	yes	characteristic, complemented, Sylow
70	C₂²×C₈	—	37	2	no	—
71	C₂²×C₈	—	37	1	no	—
72	C₂²×C₈	—	37	1	no	—
73	C₂×C₄²	—	37	2	no	—
74	C₂×C₄²	—	37	1	no	—
75	#97	—	37	8	no	—
76	#97	—	37	4	no	—
77	C₄×C₈	—	37	8	no	—
78	C₄×C₈	—	37	1	no	—
79	C₄×C₈	—	37	1	no	—
80	C₄×C₈	—	37	1	no	—
81	C₄×C₈	#3866	1	1	yes	characteristic, complemented
82	#95	—	37	4	no	—
83	C₂²×C₄	—	37	1	no	—
84	C₂²×C₄	—	37	4	no	—
85	C₂²×C₄	—	37	2	no	—
86	C₂×C₈	—	37	4	no	—
87	C₂×C₈	—	37	4	no	—
88	C₂×C₈	—	37	4	no	—
89	C₂×C₈	—	37	1	no	—
90	C₂×C₈	—	37	1	no	—
91	C₂×C₈	—	37	4	no	—
92	C₂×C₈	—	37	4	no	—
93	C₂×C₈	C₂×#3866	1	1	yes	characteristic
94	C₂×C₈	C₂×#3866	1	1	yes	characteristic
95	C₄²	—	37	4	no	—
96	C₄²	—	37	4	no	—
97	C₄²	—	37	2	no	—
98	C₄²	—	37	1	no	—
99	C₄²	C₂×#3866	1	1	yes	characteristic
100	C₂³	—	37	1	no	—
101	C₂×C₄	—	37	1	no	—
102	C₂×C₄	—	37	2	no	—
103	C₂×C₄	—	37	2	no	—
104	C₂×C₄	—	37	8	no	—
105	C₂×C₄	—	37	2	no	—
106	C₂×C₄	—	74	4	no	—
107	C₂×C₄	—	74	8	no	—
108	C₂×C₄	C₄×#3866	1	2	yes	—
109	C₂×C₄	C₂²×#3866	1	1	yes	characteristic
110	C₈	—	37	8	no	—
111	C₈	—	37	2	no	—
112	C₈	—	37	2	no	—
113	C₈	C₄×#3866	1	2	yes	—
114	C₈	C₄×#3866	1	2	yes	—
115	C₂²	—	37	2	no	—
116	C₂²	—	37	4	no	—
117	C₂²	C₂×C₄×#3866	1	1	yes	characteristic
118	C₄	—	2	2	no	—
119	C₄	—	37	2	no	—
120	C₄	—	74	8	no	—
121	C₄	—	74	2	no	—
122	C₄	C₂×C₄×#3866	1	1	yes	characteristic
123	C₄	C₂×C₄×#3866	1	1	yes	characteristic
124	C₂	—	37	4	no	—
125	C₂	C₄²×#3866	1	1	yes	characteristic
126	C₂	#14558496	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14558492

C₂

yes

maximal, characteristic

C₂×#11924590

C₂

yes

maximal

C₂×C₈×#3866

C₂

yes

maximal

#14558469

C₂

yes

maximal, characteristic

C₄×C₈×D₃₇

C₂

yes

maximal, characteristic

C₂×D₃₇⋊C₈

C₂²

yes

—

C₂×C₄×#3866

C₂²

yes

—

#11924591

C₄

yes

complemented

#11924590

C₄

yes

complemented

C₈×#3866

C₄

yes

complemented

(C₂×#3866)⋊C₄

C₄

yes

—

C₂×C₈×D₃₇

C₂²

yes

characteristic

C₂×C₈×D₃₇

C₂²

yes

characteristic

C₄²×D₃₇

C₂²

yes

characteristic

C₄×#64436

C₄

yes

characteristic, complemented

C₈×Dic₃₇

C₄

yes

characteristic, complemented

C₈×Dic₃₇

C₄

yes

characteristic, complemented

C₄×C₈×C₃₇

C₄

yes

characteristic, complemented

C₂²×#3866

C₂×C₄

yes

—

C₂×C₄×D₃₇

C₂×C₄

yes

—

C₂×C₄×D₃₇

C₂³

yes

characteristic

C₂×#64438

C₂×C₄

yes

—

C₄×#3866

C₂×C₄

yes

—

C₄×#3866

C₂×C₄

yes

—

D₃₇⋊C₈

C₂×C₄

yes

—

D₃₇⋊C₈

C₂×C₄

yes

—

C₂×C₈×C₃₇

C₂×C₄

yes

characteristic

C₂×C₈×C₃₇

C₂×C₄

yes

characteristic

C₄²×C₃₇

C₂×C₄

yes

characteristic

C₄×Dic₃₇

C₂×C₄

yes

characteristic

C₄×Dic₃₇

C₂×C₄

yes

—

C₂×#64436

C₂×C₄

yes

characteristic

C₂×#64436

C₂×C₄

yes

characteristic

C₈×D₃₇

C₂×C₄

yes

—

C₈×D₃₇

C₂×C₄

yes

—

C₂²×D₃₇

C₂²×C₄

yes

characteristic

C₂×#3866

—

C₂×#3866

C₄²

yes

—

C₂×C₄×C₃₇

C₄²

yes

—

C₂×C₄×C₃₇

C₂²×C₄

yes

characteristic

C₂×Dic₃₇

C₄²

yes

—

C₂×Dic₃₇

C₂²×C₄

yes

characteristic

C₄×D₃₇

—

C₄×D₃₇

C₂²×C₄

yes

—

C₄×D₃₇

C₂²×C₄

yes

—

#64438

C₄²

yes

—

C₈×C₃₇

C₄²

yes

—

C₈×C₃₇

C₄²

yes

—

#64436

C₄²

yes

—

#64436

C₄²

yes

—

C₂²×C₃₇

C₂×C₄²

yes

characteristic

C₂×D₃₇

C₂×C₄²

yes

—

C₂×D₃₇

#131

yes

—

#3866

—

C₄×C₃₇

—

C₄×C₃₇

C₂×C₄²

yes

characteristic

C₄×C₃₇

C₂×C₄²

yes

characteristic

Dic₃₇

—

Dic₃₇

C₂×C₄²

yes

—

#1726

—

maximal, Sylow

C₂×C₃₇

C₄³

yes

characteristic

C₂×C₃₇

#405

yes

—

D₃₇

#405

yes

—

C₂×#97

—

C₂×#97

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

#376

—

C₃₇

#1726

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

C₂×C₄²

—

#97

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

#3866

yes

characteristic, complemented

#95

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

yes

C₂×C₈

yes

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₂×#3866

yes

characteristic

100

C₂³

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

C₄×#3866

yes

—

109

C₂×C₄

C₂²×#3866

yes

characteristic

110

C₈

—

111

C₈

—

112

C₈

—

113

C₈

C₄×#3866

yes

—

114

C₈

C₄×#3866

yes

—

115

C₂²

—

116

C₂²

—

117

C₂²

C₂×C₄×#3866

yes

characteristic

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

—

122

C₄

C₂×C₄×#3866

yes

characteristic

123

C₄

C₂×C₄×#3866

yes

characteristic

124

C₂

—

125

C₂

C₄²×#3866

yes

characteristic

126

C₂

#14558496

yes

—

	subgroup	G/H
center	C₂×C₈	C₂×#3866
commutator	C₂×C₃₇	C₄³
Frattini	C₂×C₄	C₂²×#3866
Fitting	C₄×C₈×C₃₇	C₄
socle	C₂²×C₃₇	C₂×C₄²
prime radical	C₂²×D₃₇	C₂²×C₄
cosocle	C₂×C₄×D₃₇	C₂³
hypercenter	C₄×C₈	#3866
hypocenter	C₃₇	#1726
LCS	G ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈

subgroup

G/H

center

C₂×C₈

C₂×#3866

commutator

C₂×C₃₇

C₄³

Frattini

C₂×C₄

C₂²×#3866

Fitting

C₄×C₈×C₃₇

C₄

socle

C₂²×C₃₇

C₂×C₄²

prime radical

C₂²×D₃₇

C₂²×C₄

cosocle

C₂×C₄×D₃₇

C₂³

C₄×C₈

C₃₇

G ▷ C₂×C₃₇ ▷ C₃₇

C₁ ◁ C₂×C₈ ◁ C₄×C₈

N	H ≅ G/N	type
8. #11924591	C₄	semidirect
9. #11924590	C₄	semidirect
10. C₈×#3866	C₄	semidirect
15. C₄×#64436	C₄	semidirect
16. C₈×Dic₃₇	C₄	semidirect
17. C₈×Dic₃₇	C₄	semidirect
18. C₄×C₈×C₃₇	C₄	semidirect
69. C₃₇	#1726	semidirect
81. C₄×C₈	#3866	semidirect

H ≅ G/N

type

8. #11924591

C₄

semidirect

9. #11924590

C₄

semidirect

10. C₈×#3866

C₄

semidirect

15. C₄×#64436

C₄

semidirect

16. C₈×Dic₃₇

C₄

semidirect

17. C₈×Dic₃₇

C₄

semidirect

18. C₄×C₈×C₃₇

C₄

semidirect

69. C₃₇

#1726

semidirect

81. C₄×C₈

#3866

semidirect

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

subgroup	index
#14558492	2
C₂×#11924590	2
C₂×C₈×#3866	2
#14558469	2
C₄×C₈×D₃₇	2
#1726	37

subgroup

index

C₂×C₈×#3866

C₄×C₈×D₃₇

Subgroups — #18010252

Conjugacy classes of subgroups

Complements

Maximal subgroups