Group #18004002 — subgroups

Subgroups — #18004002

Order 4736.

716 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 716 conjugacy classes of subgroups — too many to draw. Only its 161 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	#14558104	C₂	1	3	yes	maximal, complemented
2	#14557723	C₂	1	8	yes	maximal, complemented
3	#14557507	C₂	1	3	yes	maximal, complemented
4	C₃₇×#585	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₇₄	C₂²	1	1	yes	characteristic, complemented
6	#11924534	—	2	3	no	—
7	#11924523	—	2	3	no	—
8	#11924517	C₂²	1	12	yes	complemented
9	#11924515	—	2	6	no	—
10	#11924495	C₂²	1	3	yes	complemented
11	#11924451	C₂²	1	12	yes	complemented
12	2₋¹⁺⁴×C₃₇	—	2	3	no	—
13	H(5,𝔽₂)×C₃₇	—	2	3	no	—
14	H(5,𝔽₂)×C₃₇	C₂²	1	4	yes	—
15	C₂×#41×C₃₇	—	2	6	no	—
16	C₂×#41×C₃₇	C₂²	1	3	yes	—
17	#41×C₃₇	—	2	4	no	—
18	#41×C₃₇	—	2	12	no	—
19	#41×C₃₇	—	2	6	no	—
20	#41×C₃₇	—	2	12	no	—
21	#41×C₃₇	—	2	6	no	—
22	C₂×Q₈×C₃₇	—	2	6	no	—
23	C₂×Q₈×C₃₇	D₄	1	3	yes	—
24	C₂×D₄×C₃₇	—	2	6	no	—
25	C₂×D₄×C₃₇	—	2	12	no	—
26	C₂×D₄×C₃₇	D₄	1	3	yes	—
27	C₂×D₄×C₃₇	C₂³	1	6	yes	—
28	C₂²×C₄×C₃₇	—	2	4	no	—
29	C₂²×C₄×C₃₇	D₄	1	6	yes	—
30	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
31	C₂²×Dic₃₇	C₂³	1	2	yes	—
32	C₂×Dic₇₄	—	2	3	no	—
33	C₂×Dic₇₄	—	2	3	no	—
34	C₂²⋊Dic₃₇	—	2	12	no	—
35	C₂²⋊Dic₃₇	C₂³	1	6	yes	—
36	C₄⋊Dic₃₇	—	2	3	no	—
37	#10596936	—	2	12	no	—
38	C₄×Dic₃₇	—	2	3	no	—
39	C₂³×C₃₇	—	2	4	no	—
40	C₂³×C₃₇	C₂×D₄	1	6	yes	—
41	C₂³×C₃₇	C₂⁴	1	1	yes	characteristic
42	Q₈×C₃₇	—	2	4	no	—
43	Q₈×C₃₇	—	2	6	no	—
44	D₄×C₃₇	—	2	12	no	—
45	D₄×C₃₇	—	2	12	no	—
46	D₄×C₃₇	—	2	6	no	—
47	C₂×C₄×C₃₇	—	2	12	no	—
48	C₂×C₄×C₃₇	—	2	6	no	—
49	C₂×C₄×C₃₇	—	2	6	no	—
50	C₂×C₄×C₃₇	C₂×D₄	1	6	yes	—
51	C₂×C₄×C₃₇	C₂×D₄	1	3	yes	—
52	C₂×C₄×C₃₇	C₂×D₄	1	3	yes	—
53	C₂×Dic₃₇	—	2	6	no	—
54	C₂×Dic₃₇	—	2	6	no	—
55	Dic₇₄	—	4	1	no	—
56	Dic₇₄	—	4	3	no	—
57	C₂²×C₃₇	—	2	6	no	—
58	C₂²×C₃₇	—	4	1	no	—
59	C₂²×C₃₇	—	4	6	no	—
60	C₂²×C₃₇	—	8	4	no	—
61	C₂²×C₃₇	C₂²×D₄	1	3	yes	—
62	C₄×C₃₇	—	2	6	no	—
63	C₄×C₃₇	#120	1	1	yes	characteristic
64	C₄×C₃₇	#120	1	3	yes	—
65	Dic₃₇	—	4	6	no	—
66	Dic₃₇	—	4	2	no	—
67	#3028	—	37	1	no	maximal, Sylow
68	C₂×C₃₇	—	2	3	no	—
69	C₂×C₃₇	—	4	6	no	—
70	C₂×C₃₇	C₂×#120	1	1	yes	characteristic
71	#585	D₃₇	1	1	yes	characteristic
72	#536	—	37	3	no	—
73	#458	—	37	8	no	—
74	#410	—	37	3	no	—
75	C₃₇	#3028	1	1	yes	characteristic, complemented, Sylow
76	2₋¹⁺⁴	—	2	3	no	—
77	H(5,𝔽₂)	—	2	3	no	—
78	H(5,𝔽₂)	C₂×D₃₇	1	4	yes	—
79	C₂×#41	—	2	6	no	—
80	C₂×#41	C₂×D₃₇	1	3	yes	—
81	C₂²×Q₈	—	37	1	no	—
82	C₄⋊Q₈	—	74	3	no	—
83	#124	—	74	3	no	—
84	#123	—	37	12	no	—
85	C₂²⋊Q₈	—	74	6	no	—
86	#117	—	37	3	no	—
87	#99	—	37	12	no	—
88	#41	—	2	4	no	—
89	#41	—	2	12	no	—
90	#41	—	2	6	no	—
91	#41	—	2	12	no	—
92	#41	—	2	6	no	—
93	C₂×Q₈	—	2	6	no	—
94	C₂×Q₈	—	74	3	no	—
95	C₂×Q₈	—	74	3	no	—
96	C₂×Q₈	C₂²⋊D₃₇	1	3	yes	—
97	C₂×D₄	—	2	6	no	—
98	C₂×D₄	—	2	12	no	—
99	C₂×D₄	C₂²⋊D₃₇	1	3	yes	—
100	C₂×D₄	C₂²×D₃₇	1	6	yes	—
101	C₂²×C₄	—	2	4	no	—
102	C₂²×C₄	—	37	2	no	—
103	C₂²×C₄	C₂²⋊D₃₇	1	6	yes	—
104	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
105	C₄⋊C₄	—	74	12	no	—
106	C₄⋊C₄	—	74	3	no	—
107	C₂²⋊C₄	—	37	6	no	—
108	C₂²⋊C₄	—	74	12	no	—
109	C₄²	—	74	3	no	—
110	C₂³	—	2	4	no	—
111	C₂³	C₂×C₂²⋊D₃₇	1	6	yes	—
112	C₂³	C₂³×D₃₇	1	1	yes	characteristic
113	Q₈	—	2	4	no	—
114	Q₈	—	2	6	no	—
115	Q₈	—	148	1	no	—
116	Q₈	—	148	3	no	—
117	D₄	—	2	12	no	—
118	D₄	—	2	12	no	—
119	D₄	—	2	6	no	—
120	C₂×C₄	—	2	12	no	—
121	C₂×C₄	—	2	6	no	—
122	C₂×C₄	—	2	6	no	—
123	C₂×C₄	—	74	6	no	—
124	C₂×C₄	—	74	6	no	—
125	C₂×C₄	C₂×C₂²⋊D₃₇	1	6	yes	—
126	C₂×C₄	C₂×C₂²⋊D₃₇	1	3	yes	—
127	C₂×C₄	C₂×C₂²⋊D₃₇	1	3	yes	—
128	C₂²	—	2	6	no	—
129	C₂²	—	4	1	no	—
130	C₂²	—	4	6	no	—
131	C₂²	—	8	4	no	—
132	C₂²	C₂²×C₂²⋊D₃₇	1	3	yes	—
133	C₄	—	2	6	no	—
134	C₄	—	148	6	no	—
135	C₄	—	148	2	no	—
136	C₄	#11924505	1	1	yes	characteristic
137	C₄	#11924505	1	3	yes	—
138	C₂	—	2	3	no	—
139	C₂	—	4	6	no	—
140	C₂	C₂×#11924505	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14558104

C₂

yes

maximal, complemented

#14557723

C₂

yes

maximal, complemented

#14557507

C₂

yes

maximal, complemented

C₃₇×#585

C₂

yes

maximal, characteristic

C₂²×Dic₇₄

C₂²

yes

characteristic, complemented

#11924534

—

#11924523

—

#11924517

C₂²

yes

complemented

#11924515

—

#11924495

C₂²

yes

complemented

#11924451

C₂²

yes

complemented

2₋¹⁺⁴×C₃₇

—

H(5,𝔽₂)×C₃₇

—

H(5,𝔽₂)×C₃₇

C₂²

yes

—

C₂×#41×C₃₇

—

C₂×#41×C₃₇

C₂²

yes

—

#41×C₃₇

—

#41×C₃₇

—

#41×C₃₇

—

#41×C₃₇

—

#41×C₃₇

—

C₂×Q₈×C₃₇

—

C₂×Q₈×C₃₇

D₄

yes

—

C₂×D₄×C₃₇

—

C₂×D₄×C₃₇

—

C₂×D₄×C₃₇

D₄

yes

—

C₂×D₄×C₃₇

C₂³

yes

—

C₂²×C₄×C₃₇

—

C₂²×C₄×C₃₇

D₄

yes

—

C₂²×C₄×C₃₇

C₂³

yes

characteristic

C₂²×Dic₃₇

C₂³

yes

—

C₂×Dic₇₄

—

C₂×Dic₇₄

—

C₂²⋊Dic₃₇

—

C₂²⋊Dic₃₇

C₂³

yes

—

C₄⋊Dic₃₇

—

#10596936

—

C₄×Dic₃₇

—

C₂³×C₃₇

—

C₂³×C₃₇

C₂×D₄

yes

—

C₂³×C₃₇

C₂⁴

yes

characteristic

Q₈×C₃₇

—

Q₈×C₃₇

—

D₄×C₃₇

—

D₄×C₃₇

—

D₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₂×D₄

yes

—

C₂×C₄×C₃₇

C₂×D₄

yes

—

C₂×C₄×C₃₇

C₂×D₄

yes

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

Dic₇₄

—

Dic₇₄

—

C₂²×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

C₂²×D₄

yes

—

C₄×C₃₇

—

C₄×C₃₇

#120

yes

characteristic

C₄×C₃₇

#120

yes

—

Dic₃₇

—

Dic₃₇

—

#3028

—

maximal, Sylow

C₂×C₃₇

—

C₂×C₃₇

—

C₂×C₃₇

C₂×#120

yes

characteristic

#585

D₃₇

yes

—

—

—

C₃₇

#3028

yes

characteristic, complemented, Sylow

2₋¹⁺⁴

—

H(5,𝔽₂)

—

H(5,𝔽₂)

C₂×D₃₇

yes

—

C₂×#41

—

C₂×#41

C₂×D₃₇

yes

—

C₂²×Q₈

—

C₄⋊Q₈

—

#124

—

#123

—

C₂²⋊Q₈

—

—

—

—

—

—

—

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂²⋊D₃₇

yes

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂²⋊D₃₇

yes

—

100

C₂×D₄

C₂²×D₃₇

yes

—

101

C₂²×C₄

—

102

C₂²×C₄

—

103

C₂²×C₄

C₂²⋊D₃₇

yes

—

104

C₂²×C₄

C₂²×D₃₇

yes

characteristic

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₂²⋊C₄

—

108

C₂²⋊C₄

—

109

C₄²

—

110

C₂³

—

111

C₂³

C₂×C₂²⋊D₃₇

yes

—

112

C₂³

C₂³×D₃₇

yes

characteristic

113

Q₈

—

114

Q₈

—

115

Q₈

—

148

—

116

Q₈

—

148

—

117

D₄

—

118

D₄

—

119

D₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

C₂×C₂²⋊D₃₇

yes

—

126

C₂×C₄

C₂×C₂²⋊D₃₇

yes

—

127

C₂×C₄

C₂×C₂²⋊D₃₇

yes

—

128

C₂²

—

129

C₂²

—

130

C₂²

—

131

C₂²

—

132

C₂²

C₂²×C₂²⋊D₃₇

yes

—

133

C₄

—

134

C₄

—

148

—

135

C₄

—

148

—

136

C₄

#11924505

yes

characteristic

137

C₄

#11924505

yes

—

138

C₂

—

139

C₂

—

140

C₂

C₂×#11924505

yes

characteristic

	subgroup	G/H
center	C₂	C₂×#11924505
commutator, cosocle	C₂³×C₃₇	C₂⁴
Frattini	C₂³	C₂³×D₃₇
Fitting, prime radical	C₃₇×#585	C₂
socle	C₂×C₃₇	C₂×#120
hypercenter	#585	D₃₇
hypocenter	C₃₇	#3028
LCS	G ▷ C₂³×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #585

subgroup

G/H

center

C₂

C₂×#11924505

commutator, cosocle

C₂³×C₃₇

C₂⁴

Frattini

C₂³

C₂³×D₃₇

Fitting, prime radical

C₃₇×#585

C₂

C₂×C₃₇

D₃₇

C₃₇

G ▷ C₂³×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #585

N	H ≅ G/N	type
1. #14558104	C₂	semidirect
2. #14557723	C₂	semidirect
3. #14557507	C₂	semidirect
5. C₂²×Dic₇₄	C₂²	semidirect
8. #11924517	C₂²	semidirect
10. #11924495	C₂²	semidirect
11. #11924451	C₂²	semidirect
75. C₃₇	#3028	semidirect

H ≅ G/N

type

1. #14558104

C₂

semidirect

2. #14557723

C₂

semidirect

3. #14557507

C₂

semidirect

5. C₂²×Dic₇₄

C₂²

semidirect

8. #11924517

C₂²

semidirect

10. #11924495

C₂²

semidirect

11. #11924451

C₂²

semidirect

75. C₃₇

#3028

semidirect

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

subgroup	index
#14558104	2
#14557723	2
#14557507	2
C₃₇×#585	2
#3028	37

subgroup

index

C₃₇×#585

Subgroups — #18004002

Conjugacy classes of subgroups

Complements

Maximal subgroups