Group #18005098 — subgroups

Subgroups — C₂×#14557933 #18005098

Order 4736.

360 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 165 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×#11924527	C₂	1	2	yes	maximal
2	#14557933	C₂	1	8	yes	maximal, direct product factor
3	C₂×#11924469	C₂	1	2	yes	maximal
4	C₂×#11924468	C₂	1	1	yes	maximal, characteristic
5	C₂×C₈×Dic₃₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#106×C₃₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×#64436	C₂²	1	1	yes	characteristic
8	#11924527	—	2	4	no	—
9	#11924527	C₂²	1	8	yes	—
10	C₂×C₄⋊Dic₃₇	—	2	2	no	—
11	C₂×C₄⋊Dic₃₇	C₂²	1	2	yes	—
12	C₂×#10596936	—	2	2	no	—
13	C₂×C₄×Dic₃₇	C₂²	1	1	yes	characteristic
14	#11924469	C₂²	1	8	yes	—
15	#11924468	C₂²	1	4	yes	—
16	C₈×Dic₃₇	C₂²	1	4	yes	—
17	C₂²×C₈×C₃₇	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₄×C₃₇	C₂²	1	2	yes	—
19	#106×C₃₇	C₂²	1	4	yes	—
20	C₂²×C₄×C₃₇	—	2	2	no	—
21	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
22	C₂²×Dic₃₇	—	2	2	no	—
23	C₂²×Dic₃₇	D₄	1	2	yes	—
24	C₂×C₈×C₃₇	Q₈	1	4	yes	—
25	C₂×C₈×C₃₇	C₂³	1	2	yes	—
26	C₄⋊C₄×C₃₇	—	2	2	no	—
27	C₄⋊C₄×C₃₇	C₂³	1	4	yes	—
28	C₄⋊Dic₃₇	—	2	2	no	—
29	C₄⋊Dic₃₇	—	2	8	no	—
30	C₄⋊Dic₃₇	C₂³	1	4	yes	—
31	#10596936	—	2	8	no	—
32	C₄×Dic₃₇	D₄	1	2	yes	—
33	C₄×Dic₃₇	C₂³	1	2	yes	—
34	C₂×#64436	Q₈	1	4	yes	—
35	C₂×#64436	C₂³	1	2	yes	—
36	C₂³×C₃₇	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₃₇	—	2	4	no	—
38	C₂×C₄×C₃₇	—	4	4	no	—
39	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₃₇	C₂×Q₈	1	2	yes	—
41	C₂×C₄×C₃₇	C₂×Q₈	1	2	yes	—
42	C₂×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
43	C₂×Dic₃₇	—	2	4	no	—
44	C₂×Dic₃₇	—	2	4	no	—
45	C₂×Dic₃₇	—	4	4	no	—
46	C₂×Dic₃₇	C₂×D₄	1	4	yes	—
47	C₈×C₃₇	C₂×Q₈	1	4	yes	—
48	#64436	C₂×Q₈	1	4	yes	—
49	C₂²×C₃₇	C₄⋊Q₈	1	2	yes	—
50	C₂²×C₃₇	#135	1	4	yes	—
51	C₂²×C₃₇	C₂²×D₄	1	1	yes	characteristic
52	C₄×C₃₇	—	4	4	no	—
53	C₄×C₃₇	C₄⋊Q₈	1	2	yes	—
54	C₄×C₃₇	C₂²×Q₈	1	1	yes	characteristic
55	C₄×C₃₇	C₂²×Q₈	1	1	yes	characteristic
56	Dic₃₇	—	2	4	no	—
57	Dic₃₇	—	4	4	no	—
58	C₂×#499	—	37	1	no	maximal, Sylow
59	C₂×C₃₇	#499	1	4	yes	complemented
60	C₂×C₃₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
61	C₂×C₃₇	C₂×#135	1	2	yes	—
62	C₂×#125	—	37	2	no	—
63	#499	—	37	8	no	—
64	C₂×#107	—	37	2	no	—
65	C₂×#106	—	37	1	no	—
66	C₂×#106	D₃₇	1	1	yes	characteristic
67	C₂×C₄×C₈	—	37	1	no	—
68	C₃₇	C₂×#499	1	1	yes	characteristic, complemented, Sylow
69	C₂²×C₈	—	37	1	no	—
70	C₂²×C₈	C₂×D₃₇	1	1	yes	characteristic
71	#125	—	37	8	no	—
72	#125	—	74	4	no	—
73	C₂×C₄⋊C₄	—	37	2	no	—
74	C₂×C₄⋊C₄	—	74	2	no	—
75	C₂×C₄⋊C₄	—	74	2	no	—
76	C₂×C₄⋊C₄	C₂×D₃₇	1	2	yes	—
77	C₂×C₄²	—	37	1	no	—
78	#107	—	37	8	no	—
79	#106	—	37	4	no	—
80	#106	C₂×D₃₇	1	4	yes	—
81	C₄×C₈	—	37	4	no	—
82	C₂²×C₄	—	2	2	no	—
83	C₂²×C₄	—	37	2	no	—
84	C₂²×C₄	—	74	2	no	—
85	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
86	C₂×C₈	—	37	4	no	—
87	C₂×C₈	—	37	2	no	—
88	C₂×C₈	Dic₇₄	1	4	yes	—
89	C₂×C₈	C₂²×D₃₇	1	2	yes	—
90	C₄⋊C₄	—	2	2	no	—
91	C₄⋊C₄	—	37	4	no	—
92	C₄⋊C₄	—	74	8	no	—
93	C₄⋊C₄	—	74	8	no	—
94	C₄⋊C₄	—	74	2	no	—
95	C₄⋊C₄	C₂²×D₃₇	1	4	yes	—
96	C₄²	—	37	2	no	—
97	C₄²	—	37	2	no	—
98	C₂³	D₄×D₃₇	1	1	yes	characteristic
99	C₂×C₄	—	2	4	no	—
100	C₂×C₄	—	4	4	no	—
101	C₂×C₄	—	37	4	no	—
102	C₂×C₄	—	74	4	no	—
103	C₂×C₄	—	74	4	no	—
104	C₂×C₄	—	148	4	no	—
105	C₂×C₄	C₂×Dic₇₄	1	2	yes	—
106	C₂×C₄	D₄×D₃₇	1	1	yes	characteristic
107	C₂×C₄	Q₈×D₃₇	1	2	yes	—
108	C₂×C₄	C₂³×D₃₇	1	1	yes	characteristic
109	C₈	—	37	4	no	—
110	C₈	C₂×Dic₇₄	1	4	yes	—
111	C₂²	#11924535	1	2	yes	—
112	C₂²	#11924557	1	4	yes	—
113	C₂²	C₂×D₄×D₃₇	1	1	yes	characteristic
114	C₄	—	4	4	no	—
115	C₄	—	74	4	no	—
116	C₄	—	148	4	no	—
117	C₄	#11924535	1	2	yes	—
118	C₄	C₂×Q₈×D₃₇	1	1	yes	characteristic
119	C₄	C₂²×Dic₇₄	1	1	yes	characteristic
120	C₂	#14557933	1	4	yes	direct product factor
121	C₂	C₂×#11924535	1	1	yes	characteristic
122	C₂	C₂×#11924557	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11924527

C₂

yes

maximal

#14557933

C₂

yes

maximal, direct product factor

C₂×#11924469

C₂

yes

maximal

C₂×#11924468

C₂

yes

maximal, characteristic

C₂×C₈×Dic₃₇

C₂

yes

maximal, characteristic

C₂×#106×C₃₇

C₂

yes

maximal, characteristic

C₂²×#64436

C₂²

yes

characteristic

#11924527

—

#11924527

C₂²

yes

—

C₂×C₄⋊Dic₃₇

—

C₂×C₄⋊Dic₃₇

C₂²

yes

—

C₂×#10596936

—

C₂×C₄×Dic₃₇

C₂²

yes

characteristic

#11924469

C₂²

yes

—

#11924468

C₂²

yes

—

C₈×Dic₃₇

C₂²

yes

—

C₂²×C₈×C₃₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₃₇

C₂²

yes

—

#106×C₃₇

C₂²

yes

—

C₂²×C₄×C₃₇

—

C₂²×C₄×C₃₇

C₂³

yes

characteristic

C₂²×Dic₃₇

—

C₂²×Dic₃₇

D₄

yes

—

C₂×C₈×C₃₇

Q₈

yes

—

C₂×C₈×C₃₇

C₂³

yes

—

C₄⋊C₄×C₃₇

—

C₄⋊C₄×C₃₇

C₂³

yes

—

C₄⋊Dic₃₇

—

C₄⋊Dic₃₇

—

C₄⋊Dic₃₇

C₂³

yes

—

#10596936

—

C₄×Dic₃₇

D₄

yes

—

C₄×Dic₃₇

C₂³

yes

—

C₂×#64436

Q₈

yes

—

C₂×#64436

C₂³

yes

—

C₂³×C₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₃₇

C₂×Q₈

yes

—

C₂×C₄×C₃₇

C₂×Q₈

yes

—

C₂×C₄×C₃₇

C₂⁴

yes

characteristic

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

C₂×D₄

yes

—

C₈×C₃₇

C₂×Q₈

yes

—

#64436

C₂×Q₈

yes

—

C₂²×C₃₇

C₄⋊Q₈

yes

—

C₂²×C₃₇

#135

yes

—

C₂²×C₃₇

C₂²×D₄

yes

characteristic

C₄×C₃₇

—

C₄×C₃₇

C₄⋊Q₈

yes

—

C₄×C₃₇

C₂²×Q₈

yes

characteristic

C₄×C₃₇

C₂²×Q₈

yes

characteristic

Dic₃₇

—

Dic₃₇

—

—

C₂×C₃₇

yes

C₂×C₃₇

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₃₇

C₂×#135

yes

—

—

—

—

—

D₃₇

yes

characteristic

C₂×C₄×C₈

—

C₃₇

C₂×#499

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

C₂×D₃₇

yes

characteristic

#125

—

#125

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₃₇

yes

—

C₂×C₄²

—

#107

—

#106

—

#106

C₂×D₃₇

yes

—

C₄×C₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₃₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

Dic₇₄

yes

—

C₂×C₈

C₂²×D₃₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂²×D₃₇

yes

—

C₄²

—

C₄²

—

C₂³

D₄×D₃₇

yes

characteristic

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

148

—

105

C₂×C₄

C₂×Dic₇₄

yes

—

106

C₂×C₄

D₄×D₃₇

yes

characteristic

107

C₂×C₄

Q₈×D₃₇

yes

—

108

C₂×C₄

C₂³×D₃₇

yes

characteristic

109

C₈

—

110

C₈

C₂×Dic₇₄

yes

—

111

C₂²

#11924535

yes

—

112

C₂²

#11924557

yes

—

113

C₂²

C₂×D₄×D₃₇

yes

characteristic

114

C₄

—

115

C₄

—

116

C₄

—

148

—

117

C₄

#11924535

yes

—

118

C₄

C₂×Q₈×D₃₇

yes

characteristic

119

C₄

C₂²×Dic₇₄

yes

characteristic

120

C₂

#14557933

yes

direct product factor

121

C₂

C₂×#11924535

yes

characteristic

122

C₂

C₂×#11924557

yes

—

	subgroup	G/H
center	C₂³	D₄×D₃₇
commutator, cosocle	C₂×C₄×C₃₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₃₇
Fitting	C₂×#106×C₃₇	C₂
socle, prime radical	C₂³×C₃₇	C₂×D₄
hypercenter	C₂×#106	D₃₇
hypocenter	C₃₇	C₂×#499
LCS	G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#106

subgroup

G/H

center

C₂³

D₄×D₃₇

commutator, cosocle

C₂×C₄×C₃₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₃₇

Fitting

C₂×#106×C₃₇

C₂

socle, prime radical

C₂³×C₃₇

C₂×D₄

D₃₇

C₃₇

G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#106

N	H ≅ G/N	type
2. #14557933	C₂	direct (N × H)
59. C₂×C₃₇	#499	semidirect
68. C₃₇	C₂×#499	semidirect
120. C₂	#14557933	direct (N × H)

H ≅ G/N

type

2. #14557933

C₂

direct (N × H)

59. C₂×C₃₇

#499

semidirect

68. C₃₇

C₂×#499

semidirect

120. C₂

#14557933

direct (N × H)

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

subgroup	index
C₂×#11924527	2
#14557933	2
C₂×#11924469	2
C₂×#11924468	2
C₂×C₈×Dic₃₇	2
C₂×#106×C₃₇	2
C₂×#499	37

subgroup

index

C₂×C₈×Dic₃₇

C₂×#106×C₃₇

C₂×#499

Subgroups — C₂×#14557933 #18005098

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#14557933 #18005098

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#14557933 #18005098