Group #18005009 — subgroups

Subgroups — #494×D₃₇ #18005009

Order 4736.

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

normal subgroups only

This group has 552 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×Q₁₆×D₃₇	C₂	1	4	yes	maximal
2	C₄⋊Q₈×D₃₇	C₂	1	2	yes	maximal
3	#14557913	C₂	1	1	yes	maximal, characteristic, complemented
4	#14557912	C₂	1	2	yes	maximal, complemented
5	#14557911	C₂	1	4	yes	maximal, complemented
6	C₄×C₈×D₃₇	C₂	1	1	yes	maximal, characteristic
7	C₃₇×#494	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Q₈×D₃₇	C₂²	1	4	yes	—
9	C₂×Dic₁₄₈	C₂²	1	4	yes	—
10	C₂×#10596942	C₂²	1	8	yes	—
11	Q₁₆×D₃₇	—	2	16	no	—
12	C₂×C₈×D₃₇	C₂²	1	2	yes	—
13	#11924536	C₂²	1	2	yes	—
14	#11924535	—	2	4	no	—
15	#11924534	C₂²	1	4	yes	—
16	C₄⋊C₄×D₃₇	—	2	4	no	—
17	C₄²×D₃₇	C₂²	1	1	yes	characteristic
18	C₄×#64436	C₂²	1	1	yes	characteristic
19	C₈×Dic₃₇	C₂²	1	2	yes	—
20	C₂×Q₁₆×C₃₇	C₂²	1	4	yes	—
21	C₄⋊Q₈×C₃₇	C₂²	1	2	yes	—
22	C₄×C₈×C₃₇	C₂²	1	1	yes	characteristic
23	C₂×Q₈×C₃₇	C₂³	1	4	yes	—
24	Q₈×D₃₇	—	2	16	no	—
25	C₂×C₄×D₃₇	—	2	4	no	—
26	C₂×C₄×D₃₇	D₄	1	2	yes	—
27	C₂×C₄×D₃₇	C₂³	1	1	yes	characteristic
28	C₂×Dic₇₄	—	2	4	no	—
29	C₂×Dic₇₄	C₂³	1	4	yes	—
30	Q₁₆×C₃₇	—	2	8	no	—
31	C₂×C₈×C₃₇	C₂³	1	2	yes	—
32	C₄⋊C₄×C₃₇	—	2	4	no	—
33	C₄²×C₃₇	C₂³	1	1	yes	characteristic
34	#10596942	—	2	16	no	—
35	C₄⋊Dic₃₇	—	2	4	no	—
36	#10596936	—	2	8	no	—
37	C₄×Dic₃₇	D₄	1	2	yes	—
38	C₄×Dic₃₇	C₂³	1	1	yes	characteristic
39	C₂×#64436	C₂³	1	2	yes	—
40	Dic₁₄₈	—	2	8	no	—
41	C₈×D₃₇	D₄	1	8	yes	—
42	C₂²×D₃₇	C₂×D₄	1	1	yes	characteristic
43	Q₈×C₃₇	—	2	8	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	1	yes	characteristic
49	C₂×Dic₃₇	C₂×D₄	1	2	yes	—
50	C₄×D₃₇	—	4	8	no	—
51	C₄×D₃₇	Q₁₆	1	8	yes	complemented
52	C₄×D₃₇	C₂×D₄	1	4	yes	—
53	Dic₇₄	—	2	8	no	—
54	Dic₇₄	—	4	8	no	—
55	C₈×C₃₇	C₂×D₄	1	4	yes	—
56	#64436	C₂×D₄	1	4	yes	—
57	C₂²×C₃₇	C₂²×D₄	1	1	yes	characteristic
58	C₂×D₃₇	#127	1	2	yes	—
59	C₂×D₃₇	C₂×Q₁₆	1	4	yes	—
60	C₄×C₃₇	—	4	4	no	—
61	C₄×C₃₇	C₂×Q₁₆	1	4	yes	complemented
62	C₄×C₃₇	C₂²×D₄	1	2	yes	—
63	Dic₃₇	—	4	4	no	—
64	Dic₃₇	#127	1	2	yes	—
65	Dic₃₇	C₂×Q₁₆	1	4	yes	complemented
66	C₂×#494	—	37	1	no	maximal, Sylow
67	C₂×C₃₇	C₂×#127	1	1	yes	characteristic
68	C₂×C₃₇	C₂²×Q₁₆	1	2	yes	—
69	D₃₇	#494	1	4	yes	direct product factor
70	C₂²×Q₁₆	—	37	4	no	—
71	C₂×C₄⋊Q₈	—	37	2	no	—
72	#494	—	37	4	no	—
73	#494	—	37	2	no	—
74	#494	—	37	1	no	—
75	#494	D₃₇	1	1	yes	characteristic, direct product factor
76	C₂×C₄×C₈	—	37	1	no	—
77	C₃₇	C₂×#494	1	1	yes	characteristic, complemented, Sylow
78	C₂²×Q₈	—	37	4	no	—
79	C₂×Q₁₆	—	37	8	no	—
80	C₂×Q₁₆	—	37	4	no	—
81	C₂×Q₁₆	—	74	16	no	—
82	C₂×Q₁₆	C₂×D₃₇	1	4	yes	—
83	C₂²×C₈	—	37	2	no	—
84	C₄⋊Q₈	—	37	4	no	—
85	C₄⋊Q₈	—	37	2	no	—
86	C₄⋊Q₈	—	74	4	no	—
87	C₄⋊Q₈	C₂×D₃₇	1	2	yes	—
88	C₂×C₄⋊C₄	—	74	4	no	—
89	C₂×C₄²	—	37	1	no	—
90	C₄×C₈	—	37	2	no	—
91	C₄×C₈	—	37	1	no	—
92	C₄×C₈	C₂×D₃₇	1	1	yes	characteristic
93	C₂×Q₈	—	37	4	no	—
94	C₂×Q₈	—	74	4	no	—
95	C₂×Q₈	—	74	16	no	—
96	C₂×Q₈	C₂²×D₃₇	1	4	yes	—
97	C₂²×C₄	—	37	2	no	—
98	C₂²×C₄	—	37	1	no	—
99	C₂²×C₄	—	74	4	no	—
100	Q₁₆	—	2	8	no	—
101	Q₁₆	—	74	16	no	—
102	Q₁₆	—	74	8	no	—
103	C₂×C₈	—	37	2	no	—
104	C₂×C₈	—	37	8	no	—
105	C₂×C₈	C₂²×D₃₇	1	2	yes	—
106	C₄⋊C₄	—	2	4	no	—
107	C₄⋊C₄	—	74	8	no	—
108	C₄⋊C₄	—	74	4	no	—
109	C₄²	—	37	2	no	—
110	C₄²	—	37	1	no	—
111	C₄²	C₂²×D₃₇	1	1	yes	characteristic
112	C₂³	—	37	1	no	—
113	Q₈	—	2	8	no	—
114	Q₈	—	74	8	no	—
115	Q₈	—	148	8	no	—
116	C₂×C₄	—	2	4	no	—
117	C₂×C₄	—	37	2	no	—
118	C₂×C₄	—	37	1	no	—
119	C₂×C₄	—	37	8	no	—
120	C₂×C₄	—	37	4	no	—
121	C₂×C₄	—	74	4	no	—
122	C₂×C₄	—	148	8	no	—
123	C₂×C₄	D₄×D₃₇	1	2	yes	—
124	C₂×C₄	C₂³×D₃₇	1	1	yes	characteristic
125	C₈	—	37	4	no	—
126	C₈	D₄×D₃₇	1	4	yes	—
127	C₂²	—	37	4	no	—
128	C₂²	—	37	2	no	—
129	C₂²	C₂×D₄×D₃₇	1	1	yes	characteristic
130	C₄	—	4	4	no	—
131	C₄	—	37	4	no	—
132	C₄	—	37	2	no	—
133	C₄	—	148	4	no	—
134	C₄	Q₁₆×D₃₇	1	4	yes	complemented
135	C₄	C₂×D₄×D₃₇	1	2	yes	—
136	C₂	—	37	4	no	—
137	C₂	#127×D₃₇	1	1	yes	characteristic
138	C₂	C₂×Q₁₆×D₃₇	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Q₁₆×D₃₇

C₂

yes

maximal

C₄⋊Q₈×D₃₇

C₂

yes

maximal

#14557913

C₂

yes

maximal, characteristic, complemented

#14557912

C₂

yes

maximal, complemented

#14557911

C₂

yes

maximal, complemented

C₄×C₈×D₃₇

C₂

yes

maximal, characteristic

C₃₇×#494

C₂

yes

maximal, characteristic, complemented

C₂×Q₈×D₃₇

C₂²

yes

—

C₂×Dic₁₄₈

C₂²

yes

—

C₂×#10596942

C₂²

yes

—

Q₁₆×D₃₇

—

C₂×C₈×D₃₇

C₂²

yes

—

#11924536

C₂²

yes

—

#11924535

—

#11924534

C₂²

yes

—

C₄⋊C₄×D₃₇

—

C₄²×D₃₇

C₂²

yes

characteristic

C₄×#64436

C₂²

yes

characteristic

C₈×Dic₃₇

C₂²

yes

—

C₂×Q₁₆×C₃₇

C₂²

yes

—

C₄⋊Q₈×C₃₇

C₂²

yes

—

C₄×C₈×C₃₇

C₂²

yes

characteristic

C₂×Q₈×C₃₇

C₂³

yes

—

Q₈×D₃₇

—

C₂×C₄×D₃₇

—

C₂×C₄×D₃₇

D₄

yes

—

C₂×C₄×D₃₇

C₂³

yes

characteristic

C₂×Dic₇₄

—

C₂×Dic₇₄

C₂³

yes

—

Q₁₆×C₃₇

—

C₂×C₈×C₃₇

C₂³

yes

—

C₄⋊C₄×C₃₇

—

C₄²×C₃₇

C₂³

yes

characteristic

#10596942

—

C₄⋊Dic₃₇

—

#10596936

—

C₄×Dic₃₇

D₄

yes

—

C₄×Dic₃₇

C₂³

yes

characteristic

C₂×#64436

C₂³

yes

—

Dic₁₄₈

—

C₈×D₃₇

D₄

yes

—

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

characteristic

C₂×Dic₃₇

C₂×D₄

yes

—

C₄×D₃₇

—

C₄×D₃₇

Q₁₆

yes

complemented

C₄×D₃₇

C₂×D₄

yes

—

Dic₇₄

—

Dic₇₄

—

C₈×C₃₇

C₂×D₄

yes

—

#64436

C₂×D₄

yes

—

C₂²×C₃₇

C₂²×D₄

yes

characteristic

C₂×D₃₇

#127

yes

—

C₂×D₃₇

C₂×Q₁₆

yes

—

C₄×C₃₇

—

C₄×C₃₇

C₂×Q₁₆

yes

complemented

C₄×C₃₇

C₂²×D₄

yes

—

Dic₃₇

—

Dic₃₇

#127

yes

—

Dic₃₇

C₂×Q₁₆

yes

—

C₂×C₃₇

yes

C₂×C₃₇

C₂²×Q₁₆

yes

—

D₃₇

#494

yes

direct product factor

C₂²×Q₁₆

—

C₂×C₄⋊Q₈

—

—

—

—

D₃₇

yes

characteristic, direct product factor

C₂×C₄×C₈

—

C₃₇

C₂×#494

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

C₂×D₃₇

yes

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₃₇

yes

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₃₇

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂²×D₃₇

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

100

Q₁₆

—

101

Q₁₆

—

102

Q₁₆

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

C₂²×D₃₇

yes

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄²

—

110

C₄²

—

111

C₄²

C₂²×D₃₇

yes

characteristic

112

C₂³

—

113

Q₈

—

114

Q₈

—

115

Q₈

—

148

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

148

—

123

C₂×C₄

D₄×D₃₇

yes

—

124

C₂×C₄

C₂³×D₃₇

yes

characteristic

125

C₈

—

126

C₈

D₄×D₃₇

yes

—

127

C₂²

—

128

C₂²

—

129

C₂²

C₂×D₄×D₃₇

yes

characteristic

130

C₄

—

131

C₄

—

132

C₄

—

133

C₄

—

148

—

134

C₄

Q₁₆×D₃₇

yes

complemented

135

C₄

C₂×D₄×D₃₇

yes

—

136

C₂

—

137

C₂

#127×D₃₇

yes

characteristic

138

C₂

C₂×Q₁₆×D₃₇

yes

—

	subgroup	G/H
center	C₂²	C₂×D₄×D₃₇
commutator, cosocle	C₂×C₄×C₃₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₃₇
Fitting	C₃₇×#494	C₂
socle	C₂²×C₃₇	C₂²×D₄
prime radical	C₂²×D₃₇	C₂×D₄
hypercenter	#494	D₃₇
hypocenter	C₃₇	C₂×#494
LCS	G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #494

subgroup

G/H

center

C₂²

C₂×D₄×D₃₇

commutator, cosocle

C₂×C₄×C₃₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₃₇

Fitting

C₃₇×#494

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 ◁ #494

N	H ≅ G/N	type
3. #14557913	C₂	semidirect
4. #14557912	C₂	semidirect
5. #14557911	C₂	semidirect
7. C₃₇×#494	C₂	semidirect
51. C₄×D₃₇	Q₁₆	semidirect
61. C₄×C₃₇	C₂×Q₁₆	semidirect
65. Dic₃₇	C₂×Q₁₆	semidirect
69. D₃₇	#494	direct (N × H)
75. #494	D₃₇	direct (N × H)
77. C₃₇	C₂×#494	semidirect
134. C₄	Q₁₆×D₃₇	semidirect

H ≅ G/N

type

3. #14557913

C₂

semidirect

4. #14557912

C₂

semidirect

5. #14557911

C₂

semidirect

7. C₃₇×#494

C₂

semidirect

51. C₄×D₃₇

Q₁₆

semidirect

61. C₄×C₃₇

C₂×Q₁₆

semidirect

65. Dic₃₇

C₂×Q₁₆

semidirect

69. D₃₇

#494

direct (N × H)

75. #494

D₃₇

direct (N × H)

77. C₃₇

C₂×#494

semidirect

134. C₄

Q₁₆×D₃₇

semidirect

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

subgroup	index
C₂×Q₁₆×D₃₇	2
C₄⋊Q₈×D₃₇	2
#14557913	2
#14557912	2
#14557911	2
C₄×C₈×D₃₇	2
C₃₇×#494	2
C₂×#494	37

subgroup

index

C₂×Q₁₆×D₃₇

C₄⋊Q₈×D₃₇

#14557913

#14557912

#14557911

C₄×C₈×D₃₇

C₃₇×#494

C₂×#494

Subgroups — #494×D₃₇ #18005009

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #494×D37 #18005009

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #494×D₃₇ #18005009