Group #17994816 — subgroups

Subgroups — #17994816

Order 4736.

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

normal subgroups only

This group has 252 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	#14557581	C₂	1	2	yes	maximal, complemented
2	#14557517	C₂	1	1	yes	maximal, characteristic
3	#14557500	C₂	1	1	yes	maximal, characteristic
4	#14557297	C₂	1	2	yes	maximal, complemented
5	C₃₇×#568	C₂	1	1	yes	maximal, characteristic
6	C₂×#10596934	C₂²	1	1	yes	characteristic
7	C₂×#10596934	C₂²	1	2	yes	—
8	C₂²×#64436	C₂²	1	1	yes	characteristic
9	#11924471	—	2	4	no	—
10	#11924455	C₄	1	2	yes	—
11	#11924453	C₄	1	2	yes	—
12	C₂²⋊#64436	—	2	2	no	—
13	C₂×#41×C₃₇	C₂²	1	1	yes	characteristic
14	#131×C₃₇	—	2	4	no	—
15	C₂×M₁₆×C₃₇	—	2	2	no	—
16	C₂×M₁₆×C₃₇	C₂²	1	2	yes	—
17	#41×C₃₇	—	2	2	no	—
18	#41×C₃₇	—	2	2	no	—
19	C₂×Q₈×C₃₇	C₂×C₄	1	1	yes	characteristic
20	C₂×D₄×C₃₇	C₂×C₄	1	1	yes	characteristic
21	C₂×D₄×C₃₇	Q₈	1	2	yes	—
22	C₂²×C₄×C₃₇	D₄	1	2	yes	—
23	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
24	M₁₆×C₃₇	—	2	4	no	—
25	M₁₆×C₃₇	—	2	2	no	—
26	M₁₆×C₃₇	D₄	1	4	yes	—
27	C₂×C₈×C₃₇	—	2	4	no	—
28	C₂×C₈×C₃₇	—	2	2	no	—
29	#10596934	—	2	2	no	—
30	#10596934	—	2	2	no	—
31	#10596934	C₂×C₄	1	4	yes	—
32	C₂×#64436	—	2	2	no	—
33	C₂×#64436	—	2	1	no	—
34	C₂×#64436	—	2	1	no	—
35	C₂×#64436	—	2	2	no	—
36	C₂³×C₃₇	C₄⋊C₄	1	2	yes	—
37	C₂³×C₃₇	C₂×Q₈	1	1	yes	characteristic
38	Q₈×C₃₇	—	2	2	no	—
39	D₄×C₃₇	—	2	4	no	—
40	D₄×C₃₇	—	2	2	no	—
41	C₂×C₄×C₃₇	—	2	2	no	—
42	C₂×C₄×C₃₇	—	2	2	no	—
43	C₂×C₄×C₃₇	C₄⋊C₄	1	2	yes	—
44	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₃₇	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₃₇	#41	1	2	yes	—
48	C₈×C₃₇	—	2	4	no	—
49	#64436	—	2	4	no	—
50	#64436	—	4	1	no	—
51	#64436	—	4	1	no	—
52	C₂²×C₃₇	—	2	2	no	—
53	C₂²×C₃₇	—	4	1	no	—
54	C₂²×C₃₇	—	4	2	no	—
55	C₂²×C₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₃₇	C₂²⋊Q₈	1	2	yes	—
57	C₄×C₃₇	—	2	2	no	—
58	C₄×C₃₇	C₄×D₄	1	2	yes	—
59	C₄×C₃₇	#120	1	1	yes	characteristic
60	C₄×C₃₇	#123	1	1	yes	characteristic
61	#1858	—	37	1	no	maximal, Sylow
62	C₂×C₃₇	—	2	1	no	—
63	C₂×C₃₇	—	2	2	no	—
64	C₂×C₃₇	—	4	2	no	—
65	C₂×C₃₇	#385	1	1	yes	characteristic
66	#568	D₃₇	1	1	yes	characteristic
67	#430	—	37	2	no	—
68	#413	—	37	1	no	—
69	#408	—	37	1	no	—
70	#341	—	37	2	no	—
71	C₃₇	#1858	1	1	yes	characteristic, complemented, Sylow
72	C₂×#41	C₂×D₃₇	1	1	yes	characteristic
73	#131	—	2	4	no	—
74	C₂×M₁₆	—	2	2	no	—
75	C₂×M₁₆	—	37	2	no	—
76	C₂×M₁₆	—	37	1	no	—
77	C₂×M₁₆	C₂×D₃₇	1	2	yes	—
78	C₂²×C₈	—	37	1	no	—
79	#108	—	74	4	no	—
80	#101	—	37	2	no	—
81	#100	—	37	2	no	—
82	C₂²⋊C₈	—	74	2	no	—
83	#41	—	2	2	no	—
84	#41	—	2	2	no	—
85	C₂×Q₈	C₄×D₃₇	1	1	yes	characteristic
86	C₂×D₄	Dic₇₄	1	2	yes	—
87	C₂×D₄	C₄×D₃₇	1	1	yes	characteristic
88	C₂²×C₄	C₂²⋊D₃₇	1	2	yes	—
89	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
90	M₁₆	—	2	4	no	—
91	M₁₆	—	2	2	no	—
92	M₁₆	—	37	4	no	—
93	M₁₆	—	74	2	no	—
94	M₁₆	—	74	2	no	—
95	M₁₆	C₂²⋊D₃₇	1	4	yes	—
96	C₂×C₈	—	2	4	no	—
97	C₂×C₈	—	2	2	no	—
98	C₂×C₈	—	74	2	no	—
99	C₂×C₈	—	74	2	no	—
100	C₂×C₈	—	74	1	no	—
101	C₂×C₈	—	74	1	no	—
102	C₂³	#10596936	1	2	yes	—
103	C₂³	C₂×Dic₇₄	1	1	yes	characteristic
104	Q₈	—	2	2	no	—
105	D₄	—	2	4	no	—
106	D₄	—	2	2	no	—
107	C₂×C₄	—	2	2	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	#10596936	1	2	yes	—
110	C₂×C₄	C₂×C₄×D₃₇	1	1	yes	characteristic
111	C₂×C₄	#10596962	1	2	yes	—
112	C₂×C₄	C₂×C₂²⋊D₃₇	1	2	yes	—
113	C₂×C₄	C₂×C₂²⋊D₃₇	1	1	yes	characteristic
114	C₈	—	2	4	no	—
115	C₈	—	74	4	no	—
116	C₈	—	148	1	no	—
117	C₈	—	148	1	no	—
118	C₂²	—	2	2	no	—
119	C₂²	—	4	1	no	—
120	C₂²	—	4	2	no	—
121	C₂²	C₂×#10596936	1	1	yes	characteristic
122	C₂²	#11924515	1	2	yes	—
123	C₄	—	2	2	no	—
124	C₄	C₄×C₂²⋊D₃₇	1	2	yes	—
125	C₄	#11924505	1	1	yes	characteristic
126	C₄	#11924519	1	1	yes	characteristic
127	C₂	—	2	1	no	—
128	C₂	—	2	2	no	—
129	C₂	—	4	2	no	—
130	C₂	#14557406	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14557581

C₂

yes

maximal, complemented

#14557517

C₂

yes

maximal, characteristic

#14557500

C₂

yes

maximal, characteristic

#14557297

C₂

yes

maximal, complemented

C₃₇×#568

C₂

yes

maximal, characteristic

C₂×#10596934

C₂²

yes

characteristic

C₂×#10596934

C₂²

yes

—

C₂²×#64436

C₂²

yes

characteristic

#11924471

—

#11924455

C₄

yes

—

#11924453

C₄

yes

—

C₂²⋊#64436

—

C₂×#41×C₃₇

C₂²

yes

characteristic

#131×C₃₇

—

C₂×M₁₆×C₃₇

—

C₂×M₁₆×C₃₇

C₂²

yes

—

#41×C₃₇

—

#41×C₃₇

—

C₂×Q₈×C₃₇

C₂×C₄

yes

characteristic

C₂×D₄×C₃₇

C₂×C₄

yes

characteristic

C₂×D₄×C₃₇

Q₈

yes

—

C₂²×C₄×C₃₇

D₄

yes

—

C₂²×C₄×C₃₇

C₂³

yes

characteristic

M₁₆×C₃₇

—

M₁₆×C₃₇

—

M₁₆×C₃₇

D₄

yes

—

C₂×C₈×C₃₇

—

C₂×C₈×C₃₇

—

#10596934

—

#10596934

—

#10596934

C₂×C₄

yes

—

—

—

—

—

C₂³×C₃₇

C₄⋊C₄

yes

—

C₂³×C₃₇

C₂×Q₈

yes

characteristic

Q₈×C₃₇

—

D₄×C₃₇

—

D₄×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×D₄

yes

characteristic

C₂×C₄×C₃₇

#41

yes

—

C₈×C₃₇

—

#64436

—

#64436

—

#64436

—

C₂²×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₃₇

C₂²⋊Q₈

yes

—

C₄×C₃₇

—

C₄×C₃₇

C₄×D₄

yes

—

C₄×C₃₇

yes

C₄×C₃₇

yes

—

C₂×C₃₇

—

C₂×C₃₇

—

C₂×C₃₇

—

C₂×C₃₇

#385

yes

characteristic

#568

D₃₇

yes

—

—

—

—

C₃₇

#1858

yes

characteristic, complemented, Sylow

C₂×#41

C₂×D₃₇

yes

characteristic

#131

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₃₇

yes

—

C₂²×C₈

—

#108

—

#101

—

#100

—

C₂²⋊C₈

—

#41

—

#41

—

C₂×Q₈

C₄×D₃₇

yes

characteristic

C₂×D₄

Dic₇₄

yes

—

C₂×D₄

C₄×D₃₇

yes

characteristic

C₂²×C₄

C₂²⋊D₃₇

yes

—

C₂²×C₄

C₂²×D₃₇

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

C₂²⋊D₃₇

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂³

#10596936

yes

—

103

C₂³

C₂×Dic₇₄

yes

characteristic

104

Q₈

—

105

D₄

—

106

D₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

#10596936

yes

—

110

C₂×C₄

C₂×C₄×D₃₇

yes

characteristic

111

C₂×C₄

#10596962

yes

—

112

C₂×C₄

C₂×C₂²⋊D₃₇

yes

—

113

C₂×C₄

C₂×C₂²⋊D₃₇

yes

characteristic

114

C₈

—

115

C₈

—

116

C₈

—

148

—

117

C₈

—

148

—

118

C₂²

—

119

C₂²

—

120

C₂²

—

121

C₂²

C₂×#10596936

yes

characteristic

122

C₂²

#11924515

yes

—

123

C₄

—

124

C₄

C₄×C₂²⋊D₃₇

yes

—

125

C₄

#11924505

yes

characteristic

126

C₄

#11924519

yes

characteristic

127

C₂

—

128

C₂

—

129

C₂

—

130

C₂

#14557406

yes

characteristic

	subgroup	G/H
center	C₄	#11924505
commutator	C₂×C₄×C₃₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₃₇
Fitting	C₃₇×#568	C₂
socle	C₂×C₃₇	#385
prime radical	C₂×#41×C₃₇	C₂²
cosocle	C₂²×C₄×C₃₇	C₂³
hypercenter	#568	D₃₇
hypocenter	C₃₇	#1858
LCS	G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #568

subgroup

G/H

center

C₄

#11924505

commutator

C₂×C₄×C₃₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₃₇

Fitting

C₃₇×#568

C₂

socle

C₂×C₃₇

#385

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 ◁ #568

N	H ≅ G/N	type
1. #14557581	C₂	semidirect
4. #14557297	C₂	semidirect
71. C₃₇	#1858	semidirect

H ≅ G/N

type

1. #14557581

C₂

semidirect

4. #14557297

C₂

semidirect

71. C₃₇

#1858

semidirect

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

subgroup	index
#14557581	2
#14557517	2
#14557500	2
#14557297	2
C₃₇×#568	2
#1858	37

subgroup

index

C₃₇×#568

Subgroups — #17994816

Conjugacy classes of subgroups

Complements

Maximal subgroups