Group #17994723 — subgroups

Subgroups — #17994723

Order 4736.

248 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 133 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×C₄×#64436	C₂	1	1	yes	maximal, characteristic
2	#14557403	C₂	1	1	yes	maximal, characteristic
3	#14557295	C₂	1	4	yes	maximal
4	C₃₇×#384	C₂	1	1	yes	maximal, characteristic
5	C₂²×#64436	C₂²	1	2	yes	—
6	C₂×C₄⋊Dic₃₇	—	2	1	no	—
7	C₂×C₄⋊Dic₃₇	—	2	1	no	—
8	C₂×C₄⋊Dic₃₇	C₂²	1	2	yes	—
9	C₄×#64436	C₄	1	4	yes	complemented
10	C₂×C₄⋊C₄×C₃₇	—	2	1	no	—
11	C₂×C₄⋊C₄×C₃₇	—	2	1	no	—
12	C₂×C₄⋊C₄×C₃₇	C₂²	1	2	yes	—
13	C₂×C₄²×C₃₇	C₂²	1	1	yes	characteristic
14	C₂²×C₄×C₃₇	—	2	2	no	—
15	C₂²×C₄×C₃₇	D₄	1	1	yes	characteristic
16	C₂²×C₄×C₃₇	D₄	1	1	yes	characteristic
17	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
18	C₂²×Dic₃₇	—	2	2	no	—
19	C₄⋊C₄×C₃₇	—	2	4	no	—
20	C₄⋊C₄×C₃₇	—	4	2	no	—
21	C₄⋊C₄×C₃₇	—	4	2	no	—
22	C₄²×C₃₇	C₂×C₄	1	2	yes	—
23	C₄²×C₃₇	Q₈	1	2	yes	—
24	C₄⋊Dic₃₇	—	2	4	no	—
25	C₄⋊Dic₃₇	—	4	2	no	—
26	C₄⋊Dic₃₇	—	4	2	no	—
27	C₂×#64436	—	2	2	no	—
28	C₂×#64436	—	2	2	no	—
29	C₂×#64436	C₂×C₄	1	4	yes	—
30	C₂³×C₃₇	C₂×D₄	1	1	yes	characteristic
31	C₂×C₄×C₃₇	—	4	2	no	—
32	C₂×C₄×C₃₇	—	4	2	no	—
33	C₂×C₄×C₃₇	—	4	2	no	—
34	C₂×C₄×C₃₇	C₄⋊C₄	1	2	yes	—
35	C₂×C₄×C₃₇	C₄⋊C₄	1	2	yes	—
36	C₂×C₄×C₃₇	D₈	1	2	yes	—
37	C₂×C₄×C₃₇	SD₁₆	1	2	yes	—
38	C₂×C₄×C₃₇	SD₁₆	1	2	yes	—
39	C₂×C₄×C₃₇	Q₁₆	1	2	yes	—
40	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₃₇	C₂×Q₈	1	1	yes	characteristic
42	C₂×C₄×C₃₇	#41	1	2	yes	—
43	C₂×C₄×C₃₇	#41	1	2	yes	—
44	C₂×Dic₃₇	—	4	2	no	—
45	C₂×Dic₃₇	—	4	2	no	—
46	C₂×Dic₃₇	—	4	2	no	—
47	#64436	—	2	4	no	—
48	C₂²×C₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
49	C₂²×C₃₇	#124	1	1	yes	characteristic
50	C₂²×C₃₇	#124	1	1	yes	characteristic
51	C₂²×C₃₇	C₂×D₈	1	1	yes	characteristic
52	C₂²×C₃₇	C₂×SD₁₆	1	1	yes	characteristic
53	C₂²×C₃₇	C₂×SD₁₆	1	1	yes	characteristic
54	C₂²×C₃₇	C₂×Q₁₆	1	1	yes	characteristic
55	C₄×C₃₇	—	8	2	no	—
56	C₄×C₃₇	#106	1	4	yes	—
57	C₄×C₃₇	#107	1	4	yes	—
58	C₄×C₃₇	#117	1	2	yes	—
59	C₄×C₃₇	#125	1	2	yes	—
60	Dic₃₇	—	8	2	no	—
61	#1834	—	37	1	no	maximal, Sylow
62	C₂×C₃₇	#382	1	1	yes	characteristic
63	C₂×C₃₇	C₂×#106	1	1	yes	characteristic
64	C₂×C₃₇	C₂×#107	1	1	yes	characteristic
65	C₂×C₃₇	#486	1	1	yes	characteristic
66	C₂×C₃₇	#486	1	1	yes	characteristic
67	C₂×C₃₇	#487	1	1	yes	characteristic
68	C₂×C₃₇	#487	1	1	yes	characteristic
69	C₂×C₄×C₈	—	37	1	no	—
70	#384	—	37	1	no	—
71	#384	D₃₇	1	1	yes	characteristic
72	#340	—	37	4	no	—
73	C₃₇	#1834	1	1	yes	characteristic, complemented, Sylow
74	C₂²×C₈	—	37	2	no	—
75	C₂×C₄⋊C₄	—	2	1	no	—
76	C₂×C₄⋊C₄	—	2	1	no	—
77	C₂×C₄⋊C₄	—	37	2	no	—
78	C₂×C₄⋊C₄	—	74	1	no	—
79	C₂×C₄⋊C₄	—	74	1	no	—
80	C₂×C₄⋊C₄	C₂×D₃₇	1	2	yes	—
81	C₂×C₄²	C₂×D₃₇	1	1	yes	characteristic
82	C₄×C₈	—	37	4	no	—
83	C₂²×C₄	—	2	2	no	—
84	C₂²×C₄	—	74	2	no	—
85	C₂²×C₄	C₂²⋊D₃₇	1	1	yes	characteristic
86	C₂²×C₄	C₂²⋊D₃₇	1	1	yes	characteristic
87	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
88	C₂×C₈	—	37	4	no	—
89	C₂×C₈	—	74	2	no	—
90	C₂×C₈	—	74	2	no	—
91	C₄⋊C₄	—	2	4	no	—
92	C₄⋊C₄	—	4	2	no	—
93	C₄⋊C₄	—	4	2	no	—
94	C₄⋊C₄	—	74	4	no	—
95	C₄⋊C₄	—	148	2	no	—
96	C₄⋊C₄	—	148	2	no	—
97	C₄²	Dic₇₄	1	2	yes	—
98	C₄²	C₄×D₃₇	1	2	yes	—
99	C₂³	C₂×C₂²⋊D₃₇	1	1	yes	characteristic
100	C₂×C₄	—	4	2	no	—
101	C₂×C₄	—	4	2	no	—
102	C₂×C₄	—	4	2	no	—
103	C₂×C₄	—	148	2	no	—
104	C₂×C₄	—	148	2	no	—
105	C₂×C₄	—	148	2	no	—
106	C₂×C₄	#10596936	1	2	yes	—
107	C₂×C₄	#10596936	1	2	yes	—
108	C₂×C₄	#10596939	1	2	yes	—
109	C₂×C₄	#10596940	1	2	yes	—
110	C₂×C₄	#10596941	1	2	yes	—
111	C₂×C₄	#10596942	1	2	yes	—
112	C₂×C₄	C₂×Dic₇₄	1	1	yes	characteristic
113	C₂×C₄	C₂×C₄×D₃₇	1	1	yes	characteristic
114	C₂×C₄	#10596964	1	2	yes	—
115	C₂×C₄	#10596966	1	2	yes	—
116	C₈	—	74	4	no	—
117	C₂²	C₂×#10596936	1	1	yes	characteristic
118	C₂²	#11924523	1	1	yes	characteristic
119	C₂²	#11924525	1	1	yes	characteristic
120	C₂²	C₂×#10596939	1	1	yes	characteristic
121	C₂²	C₂×#10596941	1	1	yes	characteristic
122	C₂²	C₂×#10596940	1	1	yes	characteristic
123	C₂²	C₂×#10596942	1	1	yes	characteristic
124	C₄	—	8	2	no	—
125	C₄	—	296	2	no	—
126	C₄	#11924467	1	4	yes	—
127	C₄	#11924469	1	4	yes	—
128	C₄	#11924494	1	2	yes	—
129	C₄	#11924527	1	2	yes	—
130	C₂	#14557398	1	1	yes	characteristic
131	C₂	C₂×#11924467	1	1	yes	characteristic
132	C₂	C₂×#11924469	1	1	yes	characteristic
133	C₂	#14557879	1	1	yes	characteristic
134	C₂	#14557881	1	1	yes	characteristic
135	C₂	#14557883	1	1	yes	characteristic
136	C₂	#14557885	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×#64436

C₂

yes

maximal, characteristic

#14557403

C₂

yes

maximal, characteristic

#14557295

C₂

yes

maximal

C₃₇×#384

C₂

yes

maximal, characteristic

C₂²×#64436

C₂²

yes

—

C₂×C₄⋊Dic₃₇

—

C₂×C₄⋊Dic₃₇

—

C₂×C₄⋊Dic₃₇

C₂²

yes

—

C₄×#64436

C₄

yes

complemented

C₂×C₄⋊C₄×C₃₇

—

C₂×C₄⋊C₄×C₃₇

—

C₂×C₄⋊C₄×C₃₇

C₂²

yes

—

C₂×C₄²×C₃₇

C₂²

yes

characteristic

C₂²×C₄×C₃₇

—

C₂²×C₄×C₃₇

D₄

yes

characteristic

C₂²×C₄×C₃₇

D₄

yes

characteristic

C₂²×C₄×C₃₇

C₂³

yes

characteristic

C₂²×Dic₃₇

—

C₄⋊C₄×C₃₇

—

C₄⋊C₄×C₃₇

—

C₄⋊C₄×C₃₇

—

C₄²×C₃₇

C₂×C₄

yes

—

C₄²×C₃₇

Q₈

yes

—

C₄⋊Dic₃₇

—

C₄⋊Dic₃₇

—

C₄⋊Dic₃₇

—

C₂×#64436

—

C₂×#64436

—

C₂×#64436

C₂×C₄

yes

—

C₂³×C₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₄⋊C₄

yes

—

C₂×C₄×C₃₇

C₄⋊C₄

yes

—

C₂×C₄×C₃₇

D₈

yes

—

C₂×C₄×C₃₇

SD₁₆

yes

—

C₂×C₄×C₃₇

SD₁₆

yes

—

C₂×C₄×C₃₇

Q₁₆

yes

—

C₂×C₄×C₃₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₃₇

#41

yes

—

C₂×C₄×C₃₇

#41

yes

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

#64436

—

C₂²×C₃₇

C₂×C₄⋊C₄

yes

C₂²×C₃₇

yes

C₂²×C₃₇

yes

C₂²×C₃₇

C₂×D₈

yes

characteristic

C₂²×C₃₇

C₂×SD₁₆

yes

characteristic

C₂²×C₃₇

C₂×SD₁₆

yes

characteristic

C₂²×C₃₇

C₂×Q₁₆

yes

characteristic

C₄×C₃₇

—

C₄×C₃₇

#106

yes

—

C₄×C₃₇

#107

yes

—

C₄×C₃₇

#117

yes

—

C₄×C₃₇

#125

yes

—

Dic₃₇

—

—

C₂×C₃₇

yes

C₂×C₃₇

yes

C₂×C₃₇

yes

C₂×C₃₇

yes

C₂×C₃₇

yes

C₂×C₃₇

yes

C₂×C₃₇

yes

C₂×C₄×C₈

—

#384

—

#384

D₃₇

yes

characteristic

#340

—

C₃₇

#1834

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₃₇

yes

—

C₂×C₄²

C₂×D₃₇

yes

characteristic

C₄×C₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₃₇

yes

characteristic

C₂²×C₄

C₂²⋊D₃₇

yes

characteristic

C₂²×C₄

C₂²×D₃₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

148

—

C₄⋊C₄

—

148

—

C₄²

Dic₇₄

yes

—

C₄²

C₄×D₃₇

yes

—

C₂³

C₂×C₂²⋊D₃₇

yes

characteristic

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

148

—

104

C₂×C₄

—

148

—

105

C₂×C₄

—

148

—

106

C₂×C₄

#10596936

yes

—

107

C₂×C₄

#10596936

yes

—

108

C₂×C₄

#10596939

yes

—

109

C₂×C₄

#10596940

yes

—

110

C₂×C₄

#10596941

yes

—

111

C₂×C₄

#10596942

yes

—

112

C₂×C₄

C₂×Dic₇₄

yes

characteristic

113

C₂×C₄

C₂×C₄×D₃₇

yes

characteristic

114

C₂×C₄

#10596964

yes

—

115

C₂×C₄

#10596966

yes

—

116

C₈

—

117

C₂²

C₂×#10596936

yes

characteristic

118

C₂²

#11924523

yes

characteristic

119

C₂²

#11924525

yes

characteristic

120

C₂²

C₂×#10596939

yes

characteristic

121

C₂²

C₂×#10596941

yes

characteristic

122

C₂²

C₂×#10596940

yes

characteristic

123

C₂²

C₂×#10596942

yes

characteristic

124

C₄

—

125

C₄

—

296

—

126

C₄

#11924467

yes

—

127

C₄

#11924469

yes

—

128

C₄

#11924494

yes

—

129

C₄

#11924527

yes

—

130

C₂

#14557398

yes

characteristic

131

C₂

C₂×#11924467

yes

characteristic

132

C₂

C₂×#11924469

yes

characteristic

133

C₂

#14557879

yes

characteristic

134

C₂

#14557881

yes

characteristic

135

C₂

#14557883

yes

characteristic

136

C₂

#14557885

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₃₇
commutator	C₂×C₄×C₃₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₃₇
Fitting	C₃₇×#384	C₂
socle, prime radical	C₂³×C₃₇	C₂×D₄
cosocle	C₂²×C₄×C₃₇	C₂³
hypercenter	#384	D₃₇
hypocenter	C₃₇	#1834
LCS	G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₃₇

commutator

C₂×C₄×C₃₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₃₇

Fitting

C₃₇×#384

C₂

socle, prime radical

C₂³×C₃₇

C₂×D₄

cosocle

C₂²×C₄×C₃₇

C₂³

D₃₇

C₃₇

G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

N	H ≅ G/N	type
9. C₄×#64436	C₄	semidirect
73. C₃₇	#1834	semidirect

H ≅ G/N

type

9. C₄×#64436

C₄

semidirect

73. C₃₇

#1834

semidirect

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

subgroup	index
C₂×C₄×#64436	2
#14557403	2
#14557295	2
C₃₇×#384	2
#1834	37

subgroup

index

C₂×C₄×#64436

#14557403

#14557295

C₃₇×#384

#1834

Subgroups — #17994723

Conjugacy classes of subgroups

Complements

Maximal subgroups