Group #17994705 — subgroups

Subgroups — #17994705

Order 4736.

272 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 81 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	#14558008	C₂	1	1	yes	maximal, characteristic
2	C₂×#11924453	C₂	1	1	yes	maximal, characteristic
3	#14557305	C₂	1	2	yes	maximal, complemented
4	#14557304	C₂	1	2	yes	maximal, complemented
5	C₂×#100×C₃₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#10596934	C₂²	1	2	yes	—
7	D₄×Dic₃₇	—	2	2	no	—
8	D₄×Dic₃₇	—	2	2	no	—
9	#11924495	C₂²	1	1	yes	characteristic
10	#11924495	C₂²	1	1	yes	characteristic
11	C₂×C₂²⋊Dic₃₇	C₄	1	4	yes	—
12	#11924453	—	2	2	no	—
13	C₂²×D₄×C₃₇	C₂²	1	1	yes	characteristic
14	C₂×M₁₆×C₃₇	C₂²	1	2	yes	—
15	#100×C₃₇	—	2	2	no	—
16	C₂⁴×C₃₇	C₂×C₄	1	2	yes	—
17	C₂×D₄×C₃₇	—	2	2	no	—
18	C₂×D₄×C₃₇	—	2	2	no	—
19	C₂×D₄×C₃₇	D₄	1	2	yes	—
20	C₂×D₄×C₃₇	D₄	1	2	yes	—
21	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
22	C₂²×Dic₃₇	—	2	4	no	—
23	M₁₆×C₃₇	—	2	2	no	—
24	M₁₆×C₃₇	—	2	2	no	—
25	C₂×C₈×C₃₇	—	2	2	no	—
26	C₂²⋊Dic₃₇	—	2	4	no	—
27	C₂²⋊Dic₃₇	C₂×C₄	1	2	yes	—
28	C₂²⋊Dic₃₇	C₂×C₄	1	2	yes	—
29	C₄⋊Dic₃₇	—	2	1	no	—
30	C₄⋊Dic₃₇	—	2	1	no	—
31	C₄×Dic₃₇	—	2	1	no	—
32	C₄×Dic₃₇	—	2	1	no	—
33	#10596934	—	2	2	no	—
34	#10596934	—	2	2	no	—
35	C₂×#64436	—	2	2	no	—
36	C₂³×C₃₇	—	2	2	no	—
37	C₂³×C₃₇	—	2	2	no	—
38	C₂³×C₃₇	—	8	2	no	—
39	C₂³×C₃₇	C₂²⋊C₄	1	4	yes	—
40	C₂³×C₃₇	C₂²×C₄	1	1	yes	characteristic
41	D₄×C₃₇	—	2	4	no	—
42	D₄×C₃₇	—	2	4	no	—
43	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₃₇	#41	1	2	yes	—
46	C₂×C₄×C₃₇	#41	1	2	yes	—
47	C₂×Dic₃₇	—	2	2	no	—
48	C₂×Dic₃₇	—	2	2	no	—
49	C₂×Dic₃₇	—	8	4	no	—
50	C₈×C₃₇	—	4	2	no	—
51	#64436	—	4	2	no	—
52	C₂²×C₃₇	—	2	4	no	—
53	C₂²×C₃₇	—	4	1	no	—
54	C₂²×C₃₇	—	4	4	no	—
55	C₂²×C₃₇	—	8	2	no	—
56	C₂²×C₃₇	—	8	2	no	—
57	C₂²×C₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₃₇	#117	1	1	yes	characteristic
59	C₂²×C₃₇	#117	1	1	yes	characteristic
60	C₄×C₃₇	#123	1	2	yes	—
61	C₄×C₃₇	#123	1	2	yes	—
62	Dic₃₇	—	4	2	no	—
63	Dic₃₇	—	4	2	no	—
64	#1829	—	37	1	no	maximal, Sylow
65	C₂×C₃₇	—	2	1	no	—
66	C₂×C₃₇	—	2	1	no	—
67	C₂×C₃₇	—	2	1	no	—
68	C₂×C₃₇	—	4	4	no	—
69	C₂×C₃₇	#381	1	1	yes	characteristic
70	#518	—	37	1	no	—
71	C₂×#100	—	37	1	no	—
72	C₂×#100	D₃₇	1	1	yes	characteristic
73	#344	—	37	2	no	—
74	#344	—	37	2	no	—
75	C₃₇	#1829	1	1	yes	characteristic, complemented, Sylow
76	C₂²×D₄	C₂×D₃₇	1	1	yes	characteristic
77	C₂×M₁₆	—	37	2	no	—
78	C₂×M₁₆	C₂×D₃₇	1	2	yes	—
79	C₄×D₄	—	74	2	no	—
80	C₄×D₄	—	74	2	no	—
81	#117	—	37	1	no	—
82	#117	—	37	1	no	—
83	C₂×C₂²⋊C₄	—	37	4	no	—
84	#100	—	2	2	no	—
85	#100	—	74	2	no	—
86	C₂⁴	C₄×D₃₇	1	2	yes	—
87	C₂×D₄	—	2	2	no	—
88	C₂×D₄	—	2	2	no	—
89	C₂×D₄	D₁₄₈	1	2	yes	—
90	C₂×D₄	C₂²⋊D₃₇	1	2	yes	—
91	C₂²×C₄	—	74	4	no	—
92	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
93	M₁₆	—	2	2	no	—
94	M₁₆	—	2	2	no	—
95	M₁₆	—	74	2	no	—
96	M₁₆	—	74	2	no	—
97	C₂×C₈	—	2	2	no	—
98	C₂×C₈	—	74	2	no	—
99	C₄⋊C₄	—	74	1	no	—
100	C₄⋊C₄	—	74	1	no	—
101	C₂²⋊C₄	—	37	2	no	—
102	C₂²⋊C₄	—	37	2	no	—
103	C₂²⋊C₄	—	74	4	no	—
104	C₄²	—	74	1	no	—
105	C₄²	—	74	1	no	—
106	C₂³	—	2	2	no	—
107	C₂³	—	2	2	no	—
108	C₂³	—	8	2	no	—
109	C₂³	#10596938	1	4	yes	—
110	C₂³	C₂×C₄×D₃₇	1	1	yes	characteristic
111	D₄	—	2	4	no	—
112	D₄	—	2	4	no	—
113	C₂×C₄	—	296	4	no	—
114	C₂×C₄	—	74	2	no	—
115	C₂×C₄	—	74	2	no	—
116	C₂×C₄	C₂×D₁₄₈	1	1	yes	characteristic
117	C₂×C₄	#10596964	1	2	yes	—
118	C₂×C₄	#10596964	1	2	yes	—
119	C₂×C₄	C₂×C₂²⋊D₃₇	1	1	yes	characteristic
120	C₈	—	4	2	no	—
121	C₈	—	148	2	no	—
122	C₂²	—	2	4	no	—
123	C₂²	—	4	1	no	—
124	C₂²	—	4	4	no	—
125	C₂²	—	8	2	no	—
126	C₂²	—	8	2	no	—
127	C₂²	C₂×#10596938	1	1	yes	characteristic
128	C₂²	#11924492	1	1	yes	characteristic
129	C₂²	#11924492	1	1	yes	characteristic
130	C₄	—	148	2	no	—
131	C₄	—	148	2	no	—
132	C₄	#11924517	1	2	yes	—
133	C₄	#11924520	1	2	yes	—
134	C₂	—	2	1	no	—
135	C₂	—	2	1	no	—
136	C₂	—	2	1	no	—
137	C₂	—	4	4	no	—
138	C₂	#14557394	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14558008

C₂

yes

maximal, characteristic

C₂×#11924453

C₂

yes

maximal, characteristic

#14557305

C₂

yes

maximal, complemented

#14557304

C₂

yes

maximal, complemented

C₂×#100×C₃₇

C₂

yes

maximal, characteristic

C₂×#10596934

C₂²

yes

—

D₄×Dic₃₇

—

D₄×Dic₃₇

—

#11924495

C₂²

yes

characteristic

#11924495

C₂²

yes

characteristic

C₂×C₂²⋊Dic₃₇

C₄

yes

—

#11924453

—

C₂²×D₄×C₃₇

C₂²

yes

characteristic

C₂×M₁₆×C₃₇

C₂²

yes

—

#100×C₃₇

—

C₂⁴×C₃₇

C₂×C₄

yes

—

C₂×D₄×C₃₇

—

C₂×D₄×C₃₇

—

C₂×D₄×C₃₇

D₄

yes

—

C₂×D₄×C₃₇

D₄

yes

—

C₂²×C₄×C₃₇

C₂³

yes

characteristic

C₂²×Dic₃₇

—

M₁₆×C₃₇

—

M₁₆×C₃₇

—

C₂×C₈×C₃₇

—

C₂²⋊Dic₃₇

—

C₂²⋊Dic₃₇

C₂×C₄

yes

—

C₂²⋊Dic₃₇

C₂×C₄

yes

—

C₄⋊Dic₃₇

—

C₄⋊Dic₃₇

—

C₄×Dic₃₇

—

C₄×Dic₃₇

—

#10596934

—

#10596934

—

C₂×#64436

—

C₂³×C₃₇

—

C₂³×C₃₇

—

C₂³×C₃₇

—

C₂³×C₃₇

C₂²⋊C₄

yes

—

C₂³×C₃₇

C₂²×C₄

yes

characteristic

D₄×C₃₇

—

D₄×C₃₇

—

C₂×C₄×C₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₃₇

#41

yes

—

C₂×C₄×C₃₇

#41

yes

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₈×C₃₇

—

#64436

—

C₂²×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

C₂×C₂²⋊C₄

yes

C₂²×C₃₇

yes

C₂²×C₃₇

yes

C₄×C₃₇

yes

—

C₄×C₃₇

#123

yes

—

Dic₃₇

—

Dic₃₇

—

#1829

—

maximal, Sylow

C₂×C₃₇

—

C₂×C₃₇

—

C₂×C₃₇

—

C₂×C₃₇

—

C₂×C₃₇

yes

—

—

D₃₇

yes

characteristic

#344

—

#344

—

C₃₇

#1829

yes

characteristic, complemented, Sylow

C₂²×D₄

C₂×D₃₇

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

C₂×D₃₇

yes

—

C₄×D₄

—

C₄×D₄

—

#117

—

#117

—

C₂×C₂²⋊C₄

—

#100

—

#100

—

C₂⁴

C₄×D₃₇

yes

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

D₁₄₈

yes

—

C₂×D₄

C₂²⋊D₃₇

yes

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₃₇

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₂²⋊C₄

—

102

C₂²⋊C₄

—

103

C₂²⋊C₄

—

104

C₄²

—

105

C₄²

—

106

C₂³

—

107

C₂³

—

108

C₂³

—

109

C₂³

#10596938

yes

—

110

C₂³

C₂×C₄×D₃₇

yes

characteristic

111

D₄

—

112

D₄

—

113

C₂×C₄

—

296

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

C₂×D₁₄₈

yes

characteristic

117

C₂×C₄

#10596964

yes

—

118

C₂×C₄

#10596964

yes

—

119

C₂×C₄

C₂×C₂²⋊D₃₇

yes

characteristic

120

C₈

—

121

C₈

—

148

—

122

C₂²

—

123

C₂²

—

124

C₂²

—

125

C₂²

—

126

C₂²

—

127

C₂²

C₂×#10596938

yes

characteristic

128

C₂²

#11924492

yes

characteristic

129

C₂²

#11924492

yes

characteristic

130

C₄

—

148

—

131

C₄

—

148

—

132

C₄

#11924517

yes

—

133

C₄

#11924520

yes

—

134

C₂

—

135

C₂

—

136

C₂

—

137

C₂

—

138

C₂

#14557394

yes

characteristic

	subgroup	G/H
center	C₂	#14557394
commutator	C₂³×C₃₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₃₇
Fitting	C₂×#100×C₃₇	C₂
socle	C₂×C₃₇	#381
prime radical	C₂²×D₄×C₃₇	C₂²
cosocle	C₂²×C₄×C₃₇	C₂³
hypercenter	C₂×#100	D₃₇
hypocenter	C₃₇	#1829
LCS	G ▷ C₂³×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#100

subgroup

G/H

center

C₂

#14557394

commutator

C₂³×C₃₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₃₇

Fitting

C₂×#100×C₃₇

C₂

socle

C₂×C₃₇

#381

prime radical

C₂²×D₄×C₃₇

C₂²

cosocle

C₂²×C₄×C₃₇

C₂³

D₃₇

C₃₇

G ▷ C₂³×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#100

N	H ≅ G/N	type
3. #14557305	C₂	semidirect
4. #14557304	C₂	semidirect
75. C₃₇	#1829	semidirect

H ≅ G/N

type

3. #14557305

C₂

semidirect

4. #14557304

C₂

semidirect

75. C₃₇

#1829

semidirect

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

subgroup	index
#14558008	2
C₂×#11924453	2
#14557305	2
#14557304	2
C₂×#100×C₃₇	2
#1829	37

subgroup

index

C₂×#100×C₃₇

Subgroups — #17994705

Conjugacy classes of subgroups

Complements

Maximal subgroups