Group #18010309 — subgroups

Subgroups — #18010309

Order 4736.

296 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 126 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	C₄×D₃₇⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₄²×#3866	C₂	1	1	yes	maximal, characteristic
3	(C₂×#3866)⋊C₈	C₂	1	4	yes	maximal
4	C₄×C₈×D₃₇	C₂	1	1	yes	maximal, characteristic
5	C₂×D₃₇⋊C₈	C₂²	1	2	yes	—
6	C₂×C₄×#3866	C₄	1	2	yes	—
7	C₂×C₄×#3866	C₄	1	2	yes	—
8	C₂×C₄×#3866	C₂²	1	2	yes	—
9	C₄×#64438	C₄	1	2	yes	complemented
10	C₄×#64438	C₄	1	2	yes	complemented
11	C₂×C₈×D₃₇	C₂²	1	2	yes	—
12	C₄²×D₃₇	C₂²	1	1	yes	characteristic
13	C₄×#64436	C₄	1	1	yes	characteristic, complemented
14	C₈×Dic₃₇	C₄	1	2	yes	complemented
15	C₄×C₈×C₃₇	C₄	1	1	yes	characteristic, complemented
16	C₂²×#3866	C₂×C₄	1	4	yes	—
17	C₂×C₄×D₃₇	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×D₃₇	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×D₃₇	C₂³	1	1	yes	characteristic
20	C₂×#64438	C₂×C₄	1	4	yes	—
21	C₄×#3866	—	2	4	no	—
22	C₄×#3866	—	2	4	no	—
23	C₄×#3866	C₈	1	8	yes	complemented
24	D₃₇⋊C₈	—	2	4	no	—
25	C₂×C₈×C₃₇	C₂×C₄	1	2	yes	—
26	C₄²×C₃₇	C₂×C₄	1	1	yes	characteristic
27	C₄×Dic₃₇	C₂×C₄	1	1	yes	characteristic
28	C₄×Dic₃₇	C₂×C₄	1	1	yes	characteristic
29	C₄×Dic₃₇	C₂×C₄	1	1	yes	characteristic
30	C₂×#64436	C₂×C₄	1	2	yes	—
31	C₈×D₃₇	—	2	4	no	—
32	C₂²×D₃₇	C₂²×C₄	1	1	yes	characteristic
33	C₂×#3866	—	2	4	no	—
34	C₂×#3866	—	2	4	no	—
35	C₂×#3866	C₂×C₈	1	8	yes	—
36	C₂×C₄×C₃₇	C₄²	1	1	yes	characteristic
37	C₂×C₄×C₃₇	C₄²	1	1	yes	characteristic
38	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
39	C₂×Dic₃₇	C₄²	1	1	yes	characteristic
40	C₂×Dic₃₇	C₄²	1	1	yes	characteristic
41	C₂×Dic₃₇	C₂²×C₄	1	1	yes	characteristic
42	C₄×D₃₇	C₂×C₈	1	4	yes	—
43	C₄×D₃₇	M₁₆	1	4	yes	—
44	C₄×D₃₇	#41	1	4	yes	—
45	#64438	—	2	4	no	—
46	C₈×C₃₇	—	2	2	no	—
47	#64436	—	2	2	no	—
48	C₂²×C₃₇	C₂×C₄²	1	1	yes	characteristic
49	C₂×D₃₇	#117	1	2	yes	—
50	C₂×D₃₇	C₂²×C₈	1	2	yes	—
51	C₂×D₃₇	C₂×M₁₆	1	2	yes	—
52	#3866	—	2	8	no	—
53	C₄×C₃₇	C₄×C₈	1	2	yes	—
54	C₄×C₃₇	#97	1	2	yes	—
55	C₄×C₃₇	#117	1	2	yes	—
56	Dic₃₇	C₄×C₈	1	2	yes	—
57	Dic₃₇	#97	1	2	yes	—
58	Dic₃₇	#117	1	2	yes	—
59	#1744	—	37	1	no	maximal, Sylow
60	C₂×C₃₇	#376	1	1	yes	characteristic
61	C₂×C₃₇	C₂×C₄×C₈	1	1	yes	characteristic
62	C₂×C₃₇	C₂×#97	1	1	yes	characteristic
63	D₃₇	#431	1	4	yes	—
64	C₂×C₄×C₈	—	37	1	no	—
65	C₂×C₄×C₈	—	37	1	no	—
66	C₄³	—	37	1	no	—
67	#336	—	37	4	no	—
68	C₃₇	#1744	1	1	yes	characteristic, complemented, Sylow
69	C₂²×C₈	—	37	2	no	—
70	C₂²×C₈	—	37	2	no	—
71	C₂×C₄²	—	37	2	no	—
72	C₂×C₄²	—	37	1	no	—
73	C₂×C₄²	—	37	2	no	—
74	C₂×C₄²	—	37	2	no	—
75	C₄×C₈	—	37	2	no	—
76	C₄×C₈	—	37	2	no	—
77	C₄×C₈	—	37	2	no	—
78	C₄×C₈	—	37	1	no	—
79	C₄×C₈	#3866	1	1	yes	characteristic, complemented
80	C₂²×C₄	—	37	1	no	—
81	C₂²×C₄	—	37	1	no	—
82	C₂²×C₄	—	37	1	no	—
83	C₂²×C₄	—	37	4	no	—
84	C₂×C₈	—	37	4	no	—
85	C₂×C₈	—	37	2	no	—
86	C₂×C₈	—	74	4	no	—
87	C₂×C₈	—	74	4	no	—
88	C₂×C₈	C₂×#3866	1	2	yes	—
89	C₄²	—	37	8	no	—
90	C₄²	—	37	1	no	—
91	C₄²	—	37	1	no	—
92	C₄²	—	37	1	no	—
93	C₄²	—	74	4	no	—
94	C₄²	—	74	4	no	—
95	C₄²	C₂×#3866	1	1	yes	characteristic
96	C₂³	—	37	1	no	—
97	C₂×C₄	—	37	1	no	—
98	C₂×C₄	—	37	1	no	—
99	C₂×C₄	—	37	1	no	—
100	C₂×C₄	—	37	4	no	—
101	C₂×C₄	—	37	4	no	—
102	C₂×C₄	—	37	4	no	—
103	C₂×C₄	—	37	8	no	—
104	C₂×C₄	—	74	4	no	—
105	C₂×C₄	—	74	4	no	—
106	C₂×C₄	C₄×#3866	1	1	yes	characteristic
107	C₂×C₄	C₄×#3866	1	1	yes	characteristic
108	C₂×C₄	C₂²×#3866	1	1	yes	characteristic
109	C₈	—	2	2	no	—
110	C₈	—	74	4	no	—
111	C₈	—	74	2	no	—
112	C₂²	—	37	2	no	—
113	C₂²	—	37	2	no	—
114	C₂²	—	37	2	no	—
115	C₂²	C₂×C₄×#3866	1	1	yes	characteristic
116	C₄	—	37	2	no	—
117	C₄	—	37	2	no	—
118	C₄	—	37	2	no	—
119	C₄	—	74	8	no	—
120	C₄	C₈×#3866	1	2	yes	—
121	C₄	#11924590	1	2	yes	—
122	C₄	#11924615	1	2	yes	—
123	C₂	—	37	4	no	—
124	C₂	#14558469	1	1	yes	characteristic
125	C₂	C₂×C₈×#3866	1	1	yes	characteristic
126	C₂	C₂×#11924590	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×D₃₇⋊C₈

C₂

yes

maximal, characteristic

C₄²×#3866

C₂

yes

maximal, characteristic

(C₂×#3866)⋊C₈

C₂

yes

maximal

C₄×C₈×D₃₇

C₂

yes

maximal, characteristic

C₂×D₃₇⋊C₈

C₂²

yes

—

C₂×C₄×#3866

C₄

yes

—

C₂×C₄×#3866

C₄

yes

—

C₂×C₄×#3866

C₂²

yes

—

C₄×#64438

C₄

yes

complemented

C₄×#64438

C₄

yes

complemented

C₂×C₈×D₃₇

C₂²

yes

—

C₄²×D₃₇

C₂²

yes

characteristic

C₄×#64436

C₄

yes

characteristic, complemented

C₈×Dic₃₇

C₄

yes

complemented

C₄×C₈×C₃₇

C₄

yes

characteristic, complemented

C₂²×#3866

C₂×C₄

yes

—

C₂×C₄×D₃₇

C₂×C₄

yes

characteristic

C₂×C₄×D₃₇

C₂×C₄

yes

characteristic

C₂×C₄×D₃₇

C₂³

yes

characteristic

C₂×#64438

C₂×C₄

yes

—

C₄×#3866

—

C₄×#3866

—

C₄×#3866

C₈

yes

complemented

D₃₇⋊C₈

—

C₂×C₈×C₃₇

C₂×C₄

yes

—

C₄²×C₃₇

C₂×C₄

yes

characteristic

C₄×Dic₃₇

C₂×C₄

yes

characteristic

C₄×Dic₃₇

C₂×C₄

yes

characteristic

C₄×Dic₃₇

C₂×C₄

yes

characteristic

C₂×#64436

C₂×C₄

yes

—

C₈×D₃₇

—

C₂²×D₃₇

C₂²×C₄

yes

—

—

C₂×C₈

yes

—

C₂×C₄×C₃₇

C₄²

yes

characteristic

C₂×C₄×C₃₇

C₄²

yes

characteristic

C₂×C₄×C₃₇

C₂²×C₄

yes

characteristic

C₂×Dic₃₇

C₄²

yes

characteristic

C₂×Dic₃₇

C₄²

yes

characteristic

C₂×Dic₃₇

C₂²×C₄

yes

characteristic

C₄×D₃₇

C₂×C₈

yes

—

C₄×D₃₇

M₁₆

yes

—

C₄×D₃₇

#41

yes

—

#64438

—

C₈×C₃₇

—

#64436

—

C₂²×C₃₇

C₂×C₄²

yes

characteristic

C₂×D₃₇

#117

yes

—

C₂×D₃₇

C₂²×C₈

yes

—

C₂×D₃₇

C₂×M₁₆

yes

—

#3866

—

C₄×C₃₇

C₄×C₈

yes

—

C₄×C₃₇

#97

yes

—

C₄×C₃₇

#117

yes

—

Dic₃₇

C₄×C₈

yes

—

Dic₃₇

#97

yes

—

Dic₃₇

#117

yes

—

—

C₂×C₃₇

yes

C₂×C₃₇

C₂×C₄×C₈

yes

characteristic

C₂×C₃₇

C₂×#97

yes

characteristic

D₃₇

#431

yes

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₄³

—

#336

—

C₃₇

#1744

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

#3866

yes

characteristic, complemented

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×#3866

yes

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₂×#3866

yes

characteristic

C₂³

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

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

C₂×C₄

C₄×#3866

yes

characteristic

108

C₂×C₄

C₂²×#3866

yes

characteristic

109

C₈

—

110

C₈

—

111

C₈

—

112

C₂²

—

113

C₂²

—

114

C₂²

—

115

C₂²

C₂×C₄×#3866

yes

characteristic

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

C₈×#3866

yes

—

121

C₄

#11924590

yes

—

122

C₄

#11924615

yes

—

123

C₂

—

124

C₂

#14558469

yes

characteristic

125

C₂

C₂×C₈×#3866

yes

characteristic

126

C₂

C₂×#11924590

yes

characteristic

	subgroup	G/H
center	C₄²	C₂×#3866
commutator	C₂×C₃₇	C₂×C₄×C₈
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	#1744
LCS	G ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₄² ◁ C₄×C₈

subgroup

G/H

center

C₄²

C₂×#3866

commutator

C₂×C₃₇

C₂×C₄×C₈

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 ◁ C₄² ◁ C₄×C₈

N	H ≅ G/N	type
9. C₄×#64438	C₄	semidirect
10. C₄×#64438	C₄	semidirect
13. C₄×#64436	C₄	semidirect
14. C₈×Dic₃₇	C₄	semidirect
15. C₄×C₈×C₃₇	C₄	semidirect
23. C₄×#3866	C₈	semidirect
68. C₃₇	#1744	semidirect
79. C₄×C₈	#3866	semidirect

H ≅ G/N

type

9. C₄×#64438

C₄

semidirect

10. C₄×#64438

C₄

semidirect

13. C₄×#64436

C₄

semidirect

14. C₈×Dic₃₇

C₄

semidirect

15. C₄×C₈×C₃₇

C₄

semidirect

23. C₄×#3866

C₈

semidirect

68. C₃₇

#1744

semidirect

79. C₄×C₈

#3866

semidirect

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

subgroup	index
C₄×D₃₇⋊C₈	2
C₄²×#3866	2
(C₂×#3866)⋊C₈	2
C₄×C₈×D₃₇	2
#1744	37

subgroup

index

C₄×D₃₇⋊C₈

C₄²×#3866

(C₂×#3866)⋊C₈

C₄×C₈×D₃₇

#1744

Subgroups — #18010309

Conjugacy classes of subgroups

Complements

Maximal subgroups