Group #18005105 — subgroups

Subgroups — #18005105

Order 4736.

356 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 145 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	#14558088	C₂	1	2	yes	maximal
2	#14557936	C₂	1	2	yes	maximal, complemented
3	#14557933	C₂	1	4	yes	maximal, complemented
4	#14557931	C₂	1	1	yes	maximal, characteristic, complemented
5	#14557568	C₂	1	4	yes	maximal
6	C₄×C₈×D₃₇	C₂	1	1	yes	maximal, characteristic
7	C₃₇×#498	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₈×D₃₇	C₂²	1	2	yes	—
9	#11924536	C₂²	1	2	yes	—
10	#11924527	C₂²	1	4	yes	—
11	#11924516	—	2	2	no	—
12	#11924512	—	2	2	no	—
13	Q₈×Dic₃₇	—	2	2	no	—
14	#11924501	—	2	2	no	—
15	#11924494	C₂²	1	4	yes	—
16	C₄²×D₃₇	C₂²	1	1	yes	characteristic
17	#11924469	C₂²	1	8	yes	—
18	#11924468	C₂²	1	4	yes	—
19	C₄×#64436	C₂²	1	1	yes	characteristic
20	C₈×Dic₃₇	C₂²	1	2	yes	—
21	C₄⋊Q₈×C₃₇	C₂²	1	2	yes	—
22	#106×C₃₇	C₂²	1	4	yes	—
23	C₄×C₈×C₃₇	C₂²	1	1	yes	characteristic
24	C₂×Q₈×C₃₇	—	2	2	no	—
25	C₂×C₄×D₃₇	D₄	1	2	yes	—
26	C₂×C₄×D₃₇	C₂³	1	1	yes	characteristic
27	C₂×Dic₇₄	—	2	2	no	—
28	C₂×C₈×C₃₇	C₂³	1	2	yes	—
29	C₄⋊C₄×C₃₇	—	2	2	no	—
30	C₄⋊C₄×C₃₇	C₂³	1	4	yes	—
31	C₄²×C₃₇	C₂³	1	1	yes	characteristic
32	#10596938	—	2	4	no	—
33	C₄⋊Dic₃₇	—	2	2	no	—
34	C₄⋊Dic₃₇	—	2	4	no	—
35	C₄⋊Dic₃₇	C₂³	1	4	yes	—
36	#10596936	—	2	4	no	—
37	C₄×Dic₃₇	—	2	4	no	—
38	C₄×Dic₃₇	D₄	1	2	yes	—
39	C₄×Dic₃₇	C₂³	1	1	yes	characteristic
40	C₂×#64436	C₂³	1	2	yes	—
41	C₈×D₃₇	Q₈	1	8	yes	complemented
42	C₂²×D₃₇	C₂×D₄	1	1	yes	characteristic
43	Q₈×C₃₇	—	4	4	no	—
44	C₂×C₄×C₃₇	—	2	4	no	—
45	C₂×C₄×C₃₇	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
47	C₂×Dic₃₇	—	2	4	no	—
48	C₂×Dic₃₇	C₂×D₄	1	2	yes	—
49	C₂×Dic₃₇	C₂×D₄	1	1	yes	characteristic
50	C₄×D₃₇	—	2	4	no	—
51	C₄×D₃₇	C₂×Q₈	1	4	yes	—
52	Dic₇₄	—	4	4	no	—
53	C₈×C₃₇	C₂×Q₈	1	4	yes	—
54	#64436	C₂×Q₈	1	4	yes	—
55	C₂²×C₃₇	C₂²×D₄	1	1	yes	characteristic
56	C₂×D₃₇	—	2	2	no	—
57	C₂×D₃₇	C₄⋊Q₈	1	2	yes	—
58	C₄×C₃₇	—	4	4	no	—
59	C₄×C₃₇	#135	1	4	yes	—
60	C₄×C₃₇	C₂²×Q₈	1	2	yes	—
61	Dic₃₇	—	2	2	no	—
62	Dic₃₇	—	4	4	no	—
63	Dic₃₇	C₄⋊Q₈	1	2	yes	—
64	#3160	—	37	1	no	maximal, Sylow
65	C₂×C₃₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
66	C₂×C₃₇	C₂×#135	1	2	yes	—
67	D₃₇	—	2	2	no	—
68	#533	—	37	2	no	—
69	#500	—	37	2	no	—
70	#499	—	37	4	no	—
71	#498	—	37	1	no	—
72	#498	D₃₇	1	1	yes	characteristic, complemented
73	#427	—	37	4	no	—
74	C₂×C₄×C₈	—	37	1	no	—
75	C₃₇	#3160	1	1	yes	characteristic, complemented, Sylow
76	C₂²×C₈	—	37	2	no	—
77	C₄⋊Q₈	—	37	2	no	—
78	C₄⋊Q₈	C₂×D₃₇	1	2	yes	—
79	#125	—	37	4	no	—
80	C₂²⋊Q₈	—	74	2	no	—
81	C₂²⋊Q₈	—	74	2	no	—
82	C₄×Q₈	—	74	2	no	—
83	C₄×Q₈	—	74	2	no	—
84	#117	—	37	4	no	—
85	C₂×C₄²	—	37	1	no	—
86	#107	—	37	8	no	—
87	#106	—	37	4	no	—
88	#106	C₂×D₃₇	1	4	yes	—
89	C₄×C₈	—	37	2	no	—
90	C₄×C₈	—	37	1	no	—
91	C₄×C₈	C₂×D₃₇	1	1	yes	characteristic
92	C₂×Q₈	—	2	2	no	—
93	C₂×Q₈	—	74	2	no	—
94	C₂²×C₄	—	37	1	no	—
95	C₂²×C₄	—	37	2	no	—
96	C₂×C₈	—	37	2	no	—
97	C₂×C₈	—	37	8	no	—
98	C₂×C₈	C₂²×D₃₇	1	2	yes	—
99	C₄⋊C₄	—	2	2	no	—
100	C₄⋊C₄	—	37	4	no	—
101	C₄⋊C₄	—	74	4	no	—
102	C₄⋊C₄	—	74	4	no	—
103	C₄⋊C₄	—	74	2	no	—
104	C₄⋊C₄	C₂²×D₃₇	1	4	yes	—
105	C₂²⋊C₄	—	74	4	no	—
106	C₄²	—	37	2	no	—
107	C₄²	—	37	1	no	—
108	C₄²	—	74	4	no	—
109	C₄²	C₂²×D₃₇	1	1	yes	characteristic
110	C₂³	—	37	1	no	—
111	Q₈	—	4	4	no	—
112	Q₈	—	148	4	no	—
113	C₂×C₄	—	2	4	no	—
114	C₂×C₄	—	37	1	no	—
115	C₂×C₄	—	37	4	no	—
116	C₂×C₄	—	37	2	no	—
117	C₂×C₄	—	74	4	no	—
118	C₂×C₄	—	74	4	no	—
119	C₂×C₄	D₄×D₃₇	1	2	yes	—
120	C₂×C₄	C₂³×D₃₇	1	1	yes	characteristic
121	C₈	—	37	4	no	—
122	C₈	Q₈×D₃₇	1	4	yes	—
123	C₂²	—	37	2	no	—
124	C₂²	—	74	2	no	—
125	C₂²	C₂×D₄×D₃₇	1	1	yes	characteristic
126	C₄	—	4	4	no	—
127	C₄	—	37	2	no	—
128	C₄	—	74	2	no	—
129	C₄	—	148	4	no	—
130	C₄	#11924557	1	4	yes	—
131	C₄	C₂×Q₈×D₃₇	1	2	yes	—
132	C₂	—	74	2	no	—
133	C₂	C₄⋊Q₈×D₃₇	1	1	yes	characteristic
134	C₂	C₂×#11924557	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14558088

C₂

yes

maximal

#14557936

C₂

yes

maximal, complemented

#14557933

C₂

yes

maximal, complemented

#14557931

C₂

yes

maximal, characteristic, complemented

#14557568

C₂

yes

maximal

C₄×C₈×D₃₇

C₂

yes

maximal, characteristic

C₃₇×#498

C₂

yes

maximal, characteristic, complemented

C₂×C₈×D₃₇

C₂²

yes

—

#11924536

C₂²

yes

—

#11924527

C₂²

yes

—

#11924516

—

#11924512

—

Q₈×Dic₃₇

—

#11924501

—

#11924494

C₂²

yes

—

C₄²×D₃₇

C₂²

yes

characteristic

#11924469

C₂²

yes

—

#11924468

C₂²

yes

—

C₄×#64436

C₂²

yes

characteristic

C₈×Dic₃₇

C₂²

yes

—

C₄⋊Q₈×C₃₇

C₂²

yes

—

#106×C₃₇

C₂²

yes

—

C₄×C₈×C₃₇

C₂²

yes

characteristic

C₂×Q₈×C₃₇

—

C₂×C₄×D₃₇

D₄

yes

—

C₂×C₄×D₃₇

C₂³

yes

characteristic

C₂×Dic₇₄

—

C₂×C₈×C₃₇

C₂³

yes

—

C₄⋊C₄×C₃₇

—

C₄⋊C₄×C₃₇

C₂³

yes

—

C₄²×C₃₇

C₂³

yes

characteristic

#10596938

—

C₄⋊Dic₃₇

—

C₄⋊Dic₃₇

—

C₄⋊Dic₃₇

C₂³

yes

—

#10596936

—

C₄×Dic₃₇

—

C₄×Dic₃₇

D₄

yes

—

C₄×Dic₃₇

C₂³

yes

characteristic

C₂×#64436

C₂³

yes

—

C₈×D₃₇

Q₈

yes

complemented

C₂²×D₃₇

C₂×D₄

yes

characteristic

Q₈×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₂×D₄

yes

—

C₂×C₄×C₃₇

C₂⁴

yes

characteristic

C₂×Dic₃₇

—

C₂×Dic₃₇

C₂×D₄

yes

—

C₂×Dic₃₇

C₂×D₄

yes

characteristic

C₄×D₃₇

—

C₄×D₃₇

C₂×Q₈

yes

—

Dic₇₄

—

C₈×C₃₇

C₂×Q₈

yes

—

#64436

C₂×Q₈

yes

—

C₂²×C₃₇

C₂²×D₄

yes

characteristic

C₂×D₃₇

—

C₂×D₃₇

C₄⋊Q₈

yes

—

C₄×C₃₇

—

C₄×C₃₇

#135

yes

—

C₄×C₃₇

C₂²×Q₈

yes

—

Dic₃₇

—

Dic₃₇

—

Dic₃₇

C₄⋊Q₈

yes

—

#3160

—

maximal, Sylow

C₂×C₃₇

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₃₇

C₂×#135

yes

—

D₃₇

—

—

—

—

—

D₃₇

yes

characteristic, complemented

#427

—

C₂×C₄×C₈

—

C₃₇

#3160

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₃₇

yes

—

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

—

#107

—

#106

—

#106

C₂×D₃₇

yes

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₃₇

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²×D₃₇

yes

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

C₂²×D₃₇

yes

—

105

C₂²⋊C₄

—

106

C₄²

—

107

C₄²

—

108

C₄²

—

109

C₄²

C₂²×D₃₇

yes

characteristic

110

C₂³

—

111

Q₈

—

112

Q₈

—

148

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

D₄×D₃₇

yes

—

120

C₂×C₄

C₂³×D₃₇

yes

characteristic

121

C₈

—

122

C₈

Q₈×D₃₇

yes

—

123

C₂²

—

124

C₂²

—

125

C₂²

C₂×D₄×D₃₇

yes

characteristic

126

C₄

—

127

C₄

—

128

C₄

—

129

C₄

—

148

—

130

C₄

#11924557

yes

—

131

C₄

C₂×Q₈×D₃₇

yes

—

132

C₂

—

133

C₂

C₄⋊Q₈×D₃₇

yes

characteristic

134

C₂

C₂×#11924557

yes

—

	subgroup	G/H
center	C₂²	C₂×D₄×D₃₇
commutator, cosocle	C₂×C₄×C₃₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₃₇
Fitting	C₃₇×#498	C₂
socle	C₂²×C₃₇	C₂²×D₄
prime radical	C₂²×D₃₇	C₂×D₄
hypercenter	#498	D₃₇
hypocenter	C₃₇	#3160
LCS	G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #498

subgroup

G/H

center

C₂²

C₂×D₄×D₃₇

commutator, cosocle

C₂×C₄×C₃₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₃₇

Fitting

C₃₇×#498

C₂

socle

C₂²×C₃₇

C₂²×D₄

prime radical

C₂²×D₃₇

C₂×D₄

D₃₇

C₃₇

G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #498

N	H ≅ G/N	type
2. #14557936	C₂	semidirect
3. #14557933	C₂	semidirect
4. #14557931	C₂	semidirect
7. C₃₇×#498	C₂	semidirect
41. C₈×D₃₇	Q₈	semidirect
72. #498	D₃₇	semidirect
75. C₃₇	#3160	semidirect

H ≅ G/N

type

2. #14557936

C₂

semidirect

3. #14557933

C₂

semidirect

4. #14557931

C₂

semidirect

7. C₃₇×#498

C₂

semidirect

41. C₈×D₃₇

Q₈

semidirect

72. #498

D₃₇

semidirect

75. C₃₇

#3160

semidirect

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

subgroup	index
#14558088	2
#14557936	2
#14557933	2
#14557931	2
#14557568	2
C₄×C₈×D₃₇	2
C₃₇×#498	2
#3160	37

subgroup

index

C₄×C₈×D₃₇

C₃₇×#498

#3160

Subgroups — #18005105

Conjugacy classes of subgroups

Complements

Maximal subgroups