Group #17994640 — subgroups

Subgroups — #17994640

Order 4736.

260 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 89 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×#11924543	C₂	1	1	yes	maximal, characteristic
2	#14557569	C₂	1	2	yes	maximal, complemented
3	#14557507	C₂	1	1	yes	maximal, characteristic
4	#14557289	C₂	1	2	yes	maximal, complemented
5	C₃₇×#410	C₂	1	1	yes	maximal, characteristic
6	#11924543	—	2	2	no	—
7	#11924543	—	2	2	no	—
8	C₂×#10596934	C₄	1	2	yes	complemented
9	C₂×#10596934	C₂²	1	1	yes	characteristic
10	C₂²×#64436	C₄	1	2	yes	complemented
11	C₂²×#64436	C₂²	1	1	yes	characteristic
12	#11924495	C₂²	1	2	yes	—
13	#11924469	—	2	2	no	—
14	#11924467	—	2	2	no	—
15	#11924451	—	2	2	no	—
16	C₂×#41×C₃₇	C₂²	1	1	yes	characteristic
17	#117×C₃₇	C₂²	1	2	yes	—
18	#99×C₃₇	—	2	2	no	—
19	#41×C₃₇	—	2	2	no	—
20	#41×C₃₇	—	2	2	no	—
21	C₂×Q₈×C₃₇	Q₈	1	1	yes	characteristic
22	C₂×D₄×C₃₇	D₄	1	2	yes	—
23	C₂×D₄×C₃₇	Q₈	1	1	yes	characteristic
24	C₂²×C₄×C₃₇	C₂×C₄	1	2	yes	—
25	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
26	C₄⋊C₄×C₃₇	—	2	2	no	—
27	C₂²⋊C₄×C₃₇	—	2	2	no	—
28	C₄²×C₃₇	—	2	2	no	—
29	C₂²⋊Dic₃₇	—	2	2	no	—
30	C₄⋊Dic₃₇	—	2	2	no	—
31	C₄×Dic₃₇	—	2	2	no	—
32	#10596934	—	2	2	no	—
33	#10596934	—	2	4	no	—
34	C₂×#64436	—	2	2	no	—
35	C₂×#64436	—	2	2	no	—
36	C₂×#64436	C₂×C₄	1	2	yes	—
37	C₂×#64436	C₂×C₄	1	1	yes	characteristic
38	C₂×#64436	C₂×C₄	1	1	yes	characteristic
39	C₂×#64436	D₄	1	2	yes	—
40	C₂×#64436	D₄	1	2	yes	—
41	C₂³×C₃₇	C₄⋊C₄	1	2	yes	—
42	C₂³×C₃₇	C₂×D₄	1	1	yes	characteristic
43	Q₈×C₃₇	—	2	2	no	—
44	D₄×C₃₇	—	2	4	no	—
45	D₄×C₃₇	—	2	2	no	—
46	C₂×C₄×C₃₇	—	2	2	no	—
47	C₂×C₄×C₃₇	—	2	2	no	—
48	C₂×C₄×C₃₇	—	4	2	no	—
49	C₂×C₄×C₃₇	C₄⋊C₄	1	2	yes	—
50	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₄×C₃₇	C₂×D₄	1	2	yes	—
52	C₂×C₄×C₃₇	C₂×Q₈	1	1	yes	characteristic
53	C₂×C₄×C₃₇	#41	1	2	yes	—
54	C₂×Dic₃₇	—	4	2	no	—
55	#64436	—	2	2	no	—
56	#64436	C₂²⋊C₄	1	1	yes	characteristic
57	#64436	C₂²⋊C₄	1	1	yes	characteristic
58	#64436	C₂²⋊C₄	1	2	yes	—
59	C₂²×C₃₇	—	2	2	no	—
60	C₂²×C₃₇	—	4	1	no	—
61	C₂²×C₃₇	—	4	2	no	—
62	C₂²×C₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
63	C₂²×C₃₇	#121	1	2	yes	—
64	C₄×C₃₇	—	2	2	no	—
65	C₄×C₃₇	—	8	2	no	—
66	C₄×C₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
67	C₄×C₃₇	#117	1	1	yes	characteristic
68	C₄×C₃₇	C₂²⋊Q₈	1	2	yes	—
69	Dic₃₇	—	8	2	no	—
70	#1812	—	37	1	no	maximal, Sylow
71	C₂×C₃₇	—	2	2	no	—
72	C₂×C₃₇	—	2	1	no	—
73	C₂×C₃₇	—	4	2	no	—
74	C₂×C₃₇	#380	1	1	yes	characteristic
75	C₂×#131	—	37	1	no	—
76	#427	—	37	2	no	—
77	#410	—	37	1	no	—
78	#410	D₃₇	1	1	yes	characteristic
79	H(3,ℤ/4)	—	37	2	no	—
80	C₃₇	#1812	1	1	yes	characteristic, complemented, Sylow
81	C₂×#41	C₂×D₃₇	1	1	yes	characteristic
82	#131	—	74	2	no	—
83	#131	—	74	2	no	—
84	C₂×M₁₆	—	37	2	no	—
85	C₂×M₁₆	—	37	1	no	—
86	C₂²×C₈	—	37	1	no	—
87	C₂²×C₈	—	37	2	no	—
88	#117	—	37	2	no	—
89	#117	C₂×D₃₇	1	2	yes	—
90	#107	—	74	2	no	—
91	#106	—	74	2	no	—
92	#99	—	2	2	no	—
93	#99	—	74	2	no	—
94	#41	—	2	2	no	—
95	#41	—	2	2	no	—
96	C₂×Q₈	Dic₇₄	1	1	yes	characteristic
97	C₂×D₄	Dic₇₄	1	1	yes	characteristic
98	C₂×D₄	C₂²⋊D₃₇	1	2	yes	—
99	C₂²×C₄	C₄×D₃₇	1	2	yes	—
100	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
101	M₁₆	—	74	4	no	—
102	M₁₆	—	74	2	no	—
103	C₂×C₈	—	37	1	no	—
104	C₂×C₈	—	37	1	no	—
105	C₂×C₈	—	37	2	no	—
106	C₂×C₈	—	37	2	no	—
107	C₂×C₈	—	37	2	no	—
108	C₂×C₈	—	74	2	no	—
109	C₂×C₈	—	74	2	no	—
110	C₄⋊C₄	—	2	2	no	—
111	C₄⋊C₄	—	74	2	no	—
112	C₂²⋊C₄	—	2	2	no	—
113	C₂²⋊C₄	—	74	2	no	—
114	C₄²	—	2	2	no	—
115	C₄²	—	74	2	no	—
116	C₂³	#10596936	1	2	yes	—
117	C₂³	C₂×C₂²⋊D₃₇	1	1	yes	characteristic
118	Q₈	—	2	2	no	—
119	D₄	—	2	4	no	—
120	D₄	—	2	2	no	—
121	C₂×C₄	—	2	2	no	—
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	4	2	no	—
124	C₂×C₄	—	148	2	no	—
125	C₂×C₄	#10596936	1	2	yes	—
126	C₂×C₄	C₂×Dic₇₄	1	1	yes	characteristic
127	C₂×C₄	C₂×C₄×D₃₇	1	1	yes	characteristic
128	C₂×C₄	D₄×D₃₇	1	2	yes	—
129	C₂×C₄	#10596964	1	2	yes	—
130	C₈	—	37	1	no	—
131	C₈	—	37	1	no	—
132	C₈	—	37	2	no	—
133	C₈	—	74	2	no	—
134	C₂²	—	2	2	no	—
135	C₂²	—	4	1	no	—
136	C₂²	—	4	2	no	—
137	C₂²	C₂×#10596936	1	1	yes	characteristic
138	C₂²	#11924508	1	2	yes	—
139	C₄	—	2	2	no	—
140	C₄	—	8	2	no	—
141	C₄	—	296	2	no	—
142	C₄	C₂²⋊C₄×D₃₇	1	1	yes	characteristic
143	C₄	#11924492	1	1	yes	characteristic
144	C₄	#11924513	1	2	yes	—
145	C₂	—	2	2	no	—
146	C₂	—	2	1	no	—
147	C₂	—	4	2	no	—
148	C₂	#14557391	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11924543

C₂

yes

maximal, characteristic

#14557569

C₂

yes

maximal, complemented

#14557507

C₂

yes

maximal, characteristic

#14557289

C₂

yes

maximal, complemented

C₃₇×#410

C₂

yes

maximal, characteristic

#11924543

—

#11924543

—

C₂×#10596934

C₄

yes

complemented

C₂×#10596934

C₂²

yes

characteristic

C₂²×#64436

C₄

yes

complemented

C₂²×#64436

C₂²

yes

characteristic

#11924495

C₂²

yes

—

#11924469

—

#11924467

—

#11924451

—

C₂×#41×C₃₇

C₂²

yes

characteristic

#117×C₃₇

C₂²

yes

—

#99×C₃₇

—

#41×C₃₇

—

#41×C₃₇

—

C₂×Q₈×C₃₇

Q₈

yes

characteristic

C₂×D₄×C₃₇

D₄

yes

—

C₂×D₄×C₃₇

Q₈

yes

characteristic

C₂²×C₄×C₃₇

C₂×C₄

yes

—

C₂²×C₄×C₃₇

C₂³

yes

characteristic

C₄⋊C₄×C₃₇

—

C₂²⋊C₄×C₃₇

—

C₄²×C₃₇

—

C₂²⋊Dic₃₇

—

C₄⋊Dic₃₇

—

C₄×Dic₃₇

—

—

—

—

—

C₂×C₄

yes

—

C₂×#64436

C₂×C₄

yes

characteristic

C₂×#64436

C₂×C₄

yes

characteristic

C₂×#64436

D₄

yes

—

C₂×#64436

D₄

yes

—

C₂³×C₃₇

C₄⋊C₄

yes

—

C₂³×C₃₇

C₂×D₄

yes

characteristic

Q₈×C₃₇

—

D₄×C₃₇

—

D₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₄⋊C₄

yes

—

C₂×C₄×C₃₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₇

C₂×D₄

yes

—

C₂×C₄×C₃₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₃₇

#41

yes

—

C₂×Dic₃₇

—

#64436

—

#64436

C₂²⋊C₄

yes

characteristic

#64436

C₂²⋊C₄

yes

characteristic

#64436

C₂²⋊C₄

yes

—

C₂²×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₃₇

#121

yes

—

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₃₇

#117

yes

characteristic

C₄×C₃₇

C₂²⋊Q₈

yes

—

Dic₃₇

—

#1812

—

maximal, Sylow

C₂×C₃₇

—

C₂×C₃₇

—

C₂×C₃₇

—

C₂×C₃₇

yes

—

—

—

D₃₇

yes

characteristic

H(3,ℤ/4)

—

C₃₇

#1812

yes

characteristic, complemented, Sylow

C₂×#41

C₂×D₃₇

yes

characteristic

#131

—

#131

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

#117

—

#117

C₂×D₃₇

yes

—

—

—

—

—

—

—

C₂×Q₈

Dic₇₄

yes

characteristic

C₂×D₄

Dic₇₄

yes

characteristic

C₂×D₄

C₂²⋊D₃₇

yes

—

C₂²×C₄

C₄×D₃₇

yes

—

100

C₂²×C₄

C₂²×D₃₇

yes

characteristic

101

M₁₆

—

102

M₁₆

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₂²⋊C₄

—

113

C₂²⋊C₄

—

114

C₄²

—

115

C₄²

—

116

C₂³

#10596936

yes

—

117

C₂³

C₂×C₂²⋊D₃₇

yes

characteristic

118

Q₈

—

119

D₄

—

120

D₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

148

—

125

C₂×C₄

#10596936

yes

—

126

C₂×C₄

C₂×Dic₇₄

yes

characteristic

127

C₂×C₄

C₂×C₄×D₃₇

yes

characteristic

128

C₂×C₄

D₄×D₃₇

yes

—

129

C₂×C₄

#10596964

yes

—

130

C₈

—

131

C₈

—

132

C₈

—

133

C₈

—

134

C₂²

—

135

C₂²

—

136

C₂²

—

137

C₂²

C₂×#10596936

yes

characteristic

138

C₂²

#11924508

yes

—

139

C₄

—

140

C₄

—

141

C₄

—

296

—

142

C₄

C₂²⋊C₄×D₃₇

yes

characteristic

143

C₄

#11924492

yes

characteristic

144

C₄

#11924513

yes

—

145

C₂

—

146

C₂

—

147

C₂

—

148

C₂

#14557391

yes

characteristic

	subgroup	G/H
center	C₄	#11924492
commutator	C₂×C₄×C₃₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₃₇
Fitting	C₃₇×#410	C₂
socle	C₂×C₃₇	#380
prime radical	C₂×#41×C₃₇	C₂²
cosocle	C₂²×C₄×C₃₇	C₂³
hypercenter	#410	D₃₇
hypocenter	C₃₇	#1812
LCS	G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #410

subgroup

G/H

center

C₄

#11924492

commutator

C₂×C₄×C₃₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₃₇

Fitting

C₃₇×#410

C₂

socle

C₂×C₃₇

#380

prime radical

C₂×#41×C₃₇

C₂²

cosocle

C₂²×C₄×C₃₇

C₂³

D₃₇

C₃₇

G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #410

N	H ≅ G/N	type
2. #14557569	C₂	semidirect
4. #14557289	C₂	semidirect
8. C₂×#10596934	C₄	semidirect
10. C₂²×#64436	C₄	semidirect
80. C₃₇	#1812	semidirect

H ≅ G/N

type

2. #14557569

C₂

semidirect

4. #14557289

C₂

semidirect

8. C₂×#10596934

C₄

semidirect

10. C₂²×#64436

C₄

semidirect

80. C₃₇

#1812

semidirect

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

subgroup	index
C₂×#11924543	2
#14557569	2
#14557507	2
#14557289	2
C₃₇×#410	2
#1812	37

subgroup

index

C₃₇×#410

Subgroups — #17994640

Conjugacy classes of subgroups

Complements

Maximal subgroups