Group #17994644 — subgroups

Subgroups — #17994644

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	C₂×#11924543	C₂	1	1	yes	maximal, characteristic
2	C₂×#11924471	C₂	1	2	yes	maximal, complemented
3	#14557517	C₂	1	1	yes	maximal, characteristic
4	#14557291	C₂	1	2	yes	maximal, complemented
5	C₃₇×#413	C₂	1	1	yes	maximal, characteristic
6	#11924543	—	2	4	no	—
7	C₂×#10596934	C₄	1	2	yes	—
8	C₂×#10596934	C₂²	1	2	yes	—
9	C₂×#10596934	C₂²	1	1	yes	characteristic
10	C₂²×#64436	C₄	1	2	yes	—
11	C₂²×#64436	C₂²	1	1	yes	characteristic
12	#11924471	—	2	4	no	—
13	#11924455	—	2	1	no	—
14	#11924453	—	2	1	no	—
15	C₂×#41×C₃₇	C₂²	1	1	yes	characteristic
16	C₂×M₁₆×C₃₇	C₂²	1	2	yes	—
17	#101×C₃₇	—	2	1	no	—
18	#100×C₃₇	—	2	1	no	—
19	#41×C₃₇	—	2	2	no	—
20	#41×C₃₇	—	2	2	no	—
21	C₂×Q₈×C₃₇	D₄	1	1	yes	characteristic
22	C₂×D₄×C₃₇	D₄	1	1	yes	characteristic
23	C₂×D₄×C₃₇	Q₈	1	2	yes	—
24	C₂²×C₄×C₃₇	C₂×C₄	1	2	yes	—
25	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
26	M₁₆×C₃₇	—	2	2	no	—
27	M₁₆×C₃₇	—	2	2	no	—
28	C₂×C₈×C₃₇	—	2	2	no	—
29	#10596934	—	2	2	no	—
30	#10596934	—	2	4	no	—
31	#10596934	—	2	2	no	—
32	#10596934	—	2	2	no	—
33	C₂×#64436	—	2	4	no	—
34	C₂×#64436	—	2	2	no	—
35	C₂×#64436	C₂×C₄	1	2	yes	—
36	C₂×#64436	C₂×C₄	1	1	yes	characteristic
37	C₂×#64436	C₂×C₄	1	1	yes	characteristic
38	C₂×#64436	D₄	1	4	yes	—
39	C₂³×C₃₇	C₄⋊C₄	1	2	yes	—
40	C₂³×C₃₇	C₂×Q₈	1	1	yes	characteristic
41	Q₈×C₃₇	—	2	2	no	—
42	D₄×C₃₇	—	2	4	no	—
43	D₄×C₃₇	—	2	2	no	—
44	C₂×C₄×C₃₇	—	2	2	no	—
45	C₂×C₄×C₃₇	—	2	2	no	—
46	C₂×C₄×C₃₇	C₄⋊C₄	1	2	yes	—
47	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₄×C₃₇	C₂×D₄	1	2	yes	—
49	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₃₇	#41	1	2	yes	—
51	C₈×C₃₇	—	4	2	no	—
52	#64436	—	2	2	no	—
53	#64436	—	4	2	no	—
54	#64436	C₂²⋊C₄	1	2	yes	—
55	#64436	C₂²⋊C₄	1	2	yes	—
56	C₂²×C₃₇	—	2	2	no	—
57	C₂²×C₃₇	—	4	1	no	—
58	C₂²×C₃₇	—	4	2	no	—
59	C₂²×C₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₂²×C₃₇	C₂²⋊Q₈	1	2	yes	—
61	C₄×C₃₇	—	2	2	no	—
62	C₄×C₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
63	C₄×C₃₇	#117	1	1	yes	characteristic
64	C₄×C₃₇	#121	1	2	yes	—
65	#1813	—	37	1	no	maximal, Sylow
66	C₂×C₃₇	—	2	1	no	—
67	C₂×C₃₇	—	2	2	no	—
68	C₂×C₃₇	—	4	2	no	—
69	C₂×C₃₇	#380	1	1	yes	characteristic
70	C₂×#131	—	37	1	no	—
71	C₂×#108	—	37	2	no	—
72	#413	—	37	1	no	—
73	#413	D₃₇	1	1	yes	characteristic
74	#338	—	37	2	no	—
75	C₃₇	#1813	1	1	yes	characteristic, complemented, Sylow
76	C₂×#41	C₂×D₃₇	1	1	yes	characteristic
77	#131	—	74	4	no	—
78	C₂×M₁₆	—	37	2	no	—
79	C₂×M₁₆	—	37	1	no	—
80	C₂×M₁₆	—	37	2	no	—
81	C₂×M₁₆	C₂×D₃₇	1	2	yes	—
82	C₂²×C₈	—	37	1	no	—
83	C₂²×C₈	—	37	2	no	—
84	#108	—	74	4	no	—
85	#101	—	2	1	no	—
86	#101	—	74	1	no	—
87	#100	—	2	1	no	—
88	#100	—	74	1	no	—
89	#41	—	2	2	no	—
90	#41	—	2	2	no	—
91	C₂×Q₈	C₂²⋊D₃₇	1	1	yes	characteristic
92	C₂×D₄	Dic₇₄	1	2	yes	—
93	C₂×D₄	C₂²⋊D₃₇	1	1	yes	characteristic
94	C₂²×C₄	C₄×D₃₇	1	2	yes	—
95	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
96	M₁₆	—	2	2	no	—
97	M₁₆	—	2	2	no	—
98	M₁₆	—	74	4	no	—
99	M₁₆	—	74	2	no	—
100	M₁₆	—	74	2	no	—
101	M₁₆	—	74	2	no	—
102	C₂×C₈	—	2	2	no	—
103	C₂×C₈	—	37	1	no	—
104	C₂×C₈	—	37	1	no	—
105	C₂×C₈	—	37	4	no	—
106	C₂×C₈	—	37	2	no	—
107	C₂×C₈	—	74	4	no	—
108	C₂×C₈	—	74	2	no	—
109	C₂³	#10596936	1	2	yes	—
110	C₂³	C₂×Dic₇₄	1	1	yes	characteristic
111	Q₈	—	2	2	no	—
112	D₄	—	2	4	no	—
113	D₄	—	2	2	no	—
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	#10596936	1	2	yes	—
117	C₂×C₄	C₂×C₄×D₃₇	1	1	yes	characteristic
118	C₂×C₄	D₄×D₃₇	1	2	yes	—
119	C₂×C₄	#10596964	1	2	yes	—
120	C₂×C₄	C₂×C₂²⋊D₃₇	1	1	yes	characteristic
121	C₈	—	4	2	no	—
122	C₈	—	37	2	no	—
123	C₈	—	37	2	no	—
124	C₈	—	74	2	no	—
125	C₈	—	148	2	no	—
126	C₂²	—	2	2	no	—
127	C₂²	—	4	1	no	—
128	C₂²	—	4	2	no	—
129	C₂²	C₂×#10596936	1	1	yes	characteristic
130	C₂²	#11924513	1	2	yes	—
131	C₄	—	2	2	no	—
132	C₄	C₂²⋊C₄×D₃₇	1	1	yes	characteristic
133	C₄	#11924492	1	1	yes	characteristic
134	C₄	#11924508	1	2	yes	—
135	C₂	—	2	1	no	—
136	C₂	—	2	2	no	—
137	C₂	—	4	2	no	—
138	C₂	#14557391	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11924543

C₂

yes

maximal, characteristic

C₂×#11924471

C₂

yes

maximal, complemented

#14557517

C₂

yes

maximal, characteristic

#14557291

C₂

yes

maximal, complemented

C₃₇×#413

C₂

yes

maximal, characteristic

#11924543

—

C₂×#10596934

C₄

yes

—

C₂×#10596934

C₂²

yes

—

C₂×#10596934

C₂²

yes

characteristic

C₂²×#64436

C₄

yes

—

C₂²×#64436

C₂²

yes

—

—

—

C₂×#41×C₃₇

C₂²

yes

characteristic

C₂×M₁₆×C₃₇

C₂²

yes

—

#101×C₃₇

—

#100×C₃₇

—

#41×C₃₇

—

#41×C₃₇

—

C₂×Q₈×C₃₇

D₄

yes

characteristic

C₂×D₄×C₃₇

D₄

yes

characteristic

C₂×D₄×C₃₇

Q₈

yes

—

C₂²×C₄×C₃₇

C₂×C₄

yes

—

C₂²×C₄×C₃₇

C₂³

yes

characteristic

M₁₆×C₃₇

—

M₁₆×C₃₇

—

C₂×C₈×C₃₇

—

—

—

—

—

—

—

C₂×C₄

yes

—

C₂×#64436

C₂×C₄

yes

characteristic

C₂×#64436

C₂×C₄

yes

characteristic

C₂×#64436

D₄

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⋊C₄

yes

—

#64436

C₂²⋊C₄

yes

—

C₂²×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₃₇

C₂²⋊Q₈

yes

—

C₄×C₃₇

—

C₄×C₃₇

C₂×C₂²⋊C₄

yes

C₄×C₃₇

yes

C₄×C₃₇

yes

—

#1813

—

maximal, Sylow

C₂×C₃₇

—

C₂×C₃₇

—

C₂×C₃₇

—

C₂×C₃₇

yes

—

—

—

D₃₇

yes

characteristic

#338

—

C₃₇

#1813

yes

characteristic, complemented, Sylow

C₂×#41

C₂×D₃₇

yes

characteristic

#131

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₃₇

yes

—

C₂²×C₈

—

C₂²×C₈

—

—

—

—

—

—

—

—

C₂×Q₈

C₂²⋊D₃₇

yes

characteristic

C₂×D₄

Dic₇₄

yes

—

C₂×D₄

C₂²⋊D₃₇

yes

characteristic

C₂²×C₄

C₄×D₃₇

yes

—

C₂²×C₄

C₂²×D₃₇

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

100

M₁₆

—

101

M₁₆

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂³

#10596936

yes

—

110

C₂³

C₂×Dic₇₄

yes

characteristic

111

Q₈

—

112

D₄

—

113

D₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

#10596936

yes

—

117

C₂×C₄

C₂×C₄×D₃₇

yes

characteristic

118

C₂×C₄

D₄×D₃₇

yes

—

119

C₂×C₄

#10596964

yes

—

120

C₂×C₄

C₂×C₂²⋊D₃₇

yes

characteristic

121

C₈

—

122

C₈

—

123

C₈

—

124

C₈

—

125

C₈

—

148

—

126

C₂²

—

127

C₂²

—

128

C₂²

—

129

C₂²

C₂×#10596936

yes

characteristic

130

C₂²

#11924513

yes

—

131

C₄

—

132

C₄

C₂²⋊C₄×D₃₇

yes

characteristic

133

C₄

#11924492

yes

characteristic

134

C₄

#11924508

yes

—

135

C₂

—

136

C₂

—

137

C₂

—

138

C₂

#14557391

yes

characteristic

	subgroup	G/H
center	C₄	#11924492
commutator	C₂×C₄×C₃₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₃₇
Fitting	C₃₇×#413	C₂
socle	C₂×C₃₇	#380
prime radical	C₂×#41×C₃₇	C₂²
cosocle	C₂²×C₄×C₃₇	C₂³
hypercenter	#413	D₃₇
hypocenter	C₃₇	#1813
LCS	G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #413

subgroup

G/H

center

C₄

#11924492

commutator

C₂×C₄×C₃₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₃₇

Fitting

C₃₇×#413

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 ◁ #413

N	H ≅ G/N	type
2. C₂×#11924471	C₂	semidirect
4. #14557291	C₂	semidirect
75. C₃₇	#1813	semidirect

H ≅ G/N

type

2. C₂×#11924471

C₂

semidirect

4. #14557291

C₂

semidirect

75. C₃₇

#1813

semidirect

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

subgroup	index
C₂×#11924543	2
C₂×#11924471	2
#14557517	2
#14557291	2
C₃₇×#413	2
#1813	37

subgroup

index

C₃₇×#413

Subgroups — #17994644

Conjugacy classes of subgroups

Complements

Maximal subgroups