Group #17996209 — subgroups

Subgroups — #17996209

Order 4736.

228 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 93 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×#11924542	C₂	1	1	yes	maximal, characteristic
2	C₂×#11924476	C₂	1	1	yes	maximal, characteristic, complemented
3	#14557312	C₂	1	4	yes	maximal, complemented
4	C₂²×C₁₆×C₃₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#10596962	C₄	1	1	yes	characteristic
6	#11924542	—	2	2	no	—
7	#11924542	—	2	2	no	—
8	C₂×#10596934	C₄	1	1	yes	characteristic
9	C₂×#10596929	C₄	1	2	yes	—
10	C₂×C₈×D₃₇	C₂²	1	2	yes	—
11	#11924476	—	2	2	no	—
12	C₂×#10596925	C₂²	1	2	yes	—
13	C₂²×C₈×C₃₇	C₂²	1	1	yes	characteristic
14	C₂×C₁₆×C₃₇	—	2	2	no	—
15	C₂×C₁₆×C₃₇	C₂²	1	2	yes	—
16	C₂²×C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
17	C₂×C₂²⋊D₃₇	C₈	1	2	yes	—
18	#10596962	—	2	2	no	—
19	#10596962	—	2	2	no	—
20	C₂×D₁₄₈	C₈	1	1	yes	characteristic
21	C₂×C₄×D₃₇	C₂×C₄	1	2	yes	—
22	C₂×Dic₇₄	C₈	1	1	yes	characteristic
23	C₂×C₈×C₃₇	C₂×C₄	1	1	yes	characteristic
24	C₂×C₈×C₃₇	D₄	1	2	yes	—
25	C₂×C₈×C₃₇	D₄	1	2	yes	—
26	C₂×C₈×C₃₇	C₂³	1	1	yes	characteristic
27	#10596934	—	2	2	no	—
28	C₂×#64436	C₂×C₄	1	2	yes	—
29	#10596929	—	2	4	no	—
30	C₈×D₃₇	—	2	2	no	—
31	C₈×D₃₇	—	2	2	no	—
32	C₁₆×C₃₇	—	2	2	no	—
33	#10596925	—	2	2	no	—
34	C₂³×C₃₇	C₂×C₈	1	1	yes	characteristic
35	C₂²×D₃₇	C₂×C₈	1	2	yes	—
36	C₂×C₄×C₃₇	C₂²⋊C₄	1	2	yes	—
37	C₂×C₄×C₃₇	C₂×C₈	1	1	yes	characteristic
38	C₂×C₄×C₃₇	M₁₆	1	2	yes	—
39	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
40	C₂²⋊D₃₇	—	2	4	no	—
41	C₂×Dic₃₇	C₂×C₈	1	2	yes	—
42	D₁₄₈	—	2	2	no	—
43	C₄×D₃₇	—	2	2	no	—
44	C₄×D₃₇	—	2	2	no	—
45	Dic₇₄	—	2	2	no	—
46	C₈×C₃₇	C₂²⋊C₄	1	2	yes	—
47	C₈×C₃₇	C₂×D₄	1	1	yes	characteristic
48	C₈×C₃₇	C₂×D₄	1	1	yes	characteristic
49	#64436	—	2	2	no	—
50	C₂²×C₃₇	—	2	2	no	—
51	C₂²×C₃₇	C₂²⋊C₈	1	2	yes	—
52	C₂²×C₃₇	C₂²×C₈	1	1	yes	characteristic
53	C₂×D₃₇	—	2	2	no	—
54	C₂×D₃₇	—	2	4	no	—
55	C₄×C₃₇	C₂²⋊C₈	1	2	yes	—
56	C₄×C₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
57	C₄×C₃₇	C₂×M₁₆	1	1	yes	characteristic
58	Dic₃₇	—	2	2	no	—
59	#2113	—	37	1	no	maximal, Sylow
60	C₂×C₃₇	—	2	2	no	—
61	C₂×C₃₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
62	C₂×C₃₇	#504	1	2	yes	—
63	D₃₇	—	4	2	no	—
64	C₂×#131	—	37	1	no	—
65	C₂×M₃₂	—	37	1	no	—
66	C₂²×C₁₆	D₃₇	1	1	yes	characteristic, complemented
67	C₂²⋊C₁₆	—	37	4	no	—
68	C₃₇	#2113	1	1	yes	characteristic, complemented, Sylow
69	C₂×#41	—	37	1	no	—
70	#131	—	74	2	no	—
71	#131	—	74	2	no	—
72	C₂×M₁₆	—	37	1	no	—
73	C₂×M₁₆	—	37	2	no	—
74	C₂²×C₈	—	37	2	no	—
75	C₂²×C₈	C₂×D₃₇	1	1	yes	characteristic
76	M₃₂	—	74	2	no	—
77	C₂×C₁₆	—	2	2	no	—
78	C₂×C₁₆	—	37	2	no	—
79	C₂×C₁₆	C₂×D₃₇	1	2	yes	—
80	#41	—	74	2	no	—
81	#41	—	74	2	no	—
82	C₂×Q₈	—	37	1	no	—
83	C₂×D₄	—	37	2	no	—
84	C₂×D₄	—	37	1	no	—
85	C₂²×C₄	—	37	2	no	—
86	C₂²×C₄	C₄×D₃₇	1	1	yes	characteristic
87	M₁₆	—	74	2	no	—
88	M₁₆	—	74	4	no	—
89	C₂×C₈	—	37	2	no	—
90	C₂×C₈	—	74	2	no	—
91	C₂×C₈	—	74	2	no	—
92	C₂×C₈	C₄×D₃₇	1	1	yes	characteristic
93	C₂×C₈	D₁₄₈	1	2	yes	—
94	C₂×C₈	C₂²⋊D₃₇	1	2	yes	—
95	C₂×C₈	C₂²×D₃₇	1	1	yes	characteristic
96	C₁₆	—	2	2	no	—
97	C₁₆	—	74	2	no	—
98	C₂³	—	37	2	no	—
99	C₂³	C₈×D₃₇	1	1	yes	characteristic
100	Q₈	—	74	2	no	—
101	D₄	—	74	4	no	—
102	D₄	—	74	2	no	—
103	C₂×C₄	—	37	2	no	—
104	C₂×C₄	—	74	2	no	—
105	C₂×C₄	—	74	2	no	—
106	C₂×C₄	C₈×D₃₇	1	1	yes	characteristic
107	C₂×C₄	#10596929	1	2	yes	—
108	C₂×C₄	#10596938	1	2	yes	—
109	C₂×C₄	C₂×C₄×D₃₇	1	1	yes	characteristic
110	C₈	—	74	2	no	—
111	C₈	#10596938	1	2	yes	—
112	C₈	C₂×D₁₄₈	1	1	yes	characteristic
113	C₈	C₂×C₂²⋊D₃₇	1	1	yes	characteristic
114	C₂²	—	2	2	no	—
115	C₂²	—	74	2	no	—
116	C₂²	—	74	4	no	—
117	C₂²	#11924448	1	2	yes	—
118	C₂²	C₂×C₈×D₃₇	1	1	yes	characteristic
119	C₄	—	74	2	no	—
120	C₄	#11924448	1	2	yes	—
121	C₄	C₂×#10596938	1	1	yes	characteristic
122	C₄	C₂×#10596929	1	1	yes	characteristic
123	C₂	—	2	2	no	—
124	C₂	—	148	2	no	—
125	C₂	C₂×#11924448	1	1	yes	characteristic
126	C₂	#14557949	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11924542

C₂

yes

maximal, characteristic

C₂×#11924476

C₂

yes

maximal, characteristic, complemented

#14557312

C₂

yes

maximal, complemented

C₂²×C₁₆×C₃₇

C₂

yes

maximal, characteristic, complemented

C₂×#10596962

C₄

yes

—

—

C₄

yes

characteristic

C₂×#10596929

C₄

yes

—

C₂×C₈×D₃₇

C₂²

yes

—

#11924476

—

C₂×#10596925

C₂²

yes

—

C₂²×C₈×C₃₇

C₂²

yes

characteristic

C₂×C₁₆×C₃₇

—

C₂×C₁₆×C₃₇

C₂²

yes

—

C₂²×C₄×C₃₇

C₂×C₄

yes

characteristic

C₂×C₂²⋊D₃₇

C₈

yes

—

#10596962

—

#10596962

—

C₂×D₁₄₈

C₈

yes

characteristic

C₂×C₄×D₃₇

C₂×C₄

yes

—

C₂×Dic₇₄

C₈

yes

characteristic

C₂×C₈×C₃₇

C₂×C₄

yes

characteristic

C₂×C₈×C₃₇

D₄

yes

—

C₂×C₈×C₃₇

D₄

yes

—

C₂×C₈×C₃₇

C₂³

yes

characteristic

#10596934

—

C₂×#64436

C₂×C₄

yes

—

#10596929

—

C₈×D₃₇

—

C₈×D₃₇

—

C₁₆×C₃₇

—

#10596925

—

C₂³×C₃₇

C₂×C₈

yes

characteristic

C₂²×D₃₇

C₂×C₈

yes

—

C₂×C₄×C₃₇

C₂²⋊C₄

yes

—

C₂×C₄×C₃₇

C₂×C₈

yes

characteristic

C₂×C₄×C₃₇

M₁₆

yes

—

C₂×C₄×C₃₇

C₂²×C₄

yes

characteristic

C₂²⋊D₃₇

—

C₂×Dic₃₇

C₂×C₈

yes

—

D₁₄₈

—

C₄×D₃₇

—

C₄×D₃₇

—

Dic₇₄

—

C₈×C₃₇

C₂²⋊C₄

yes

—

C₈×C₃₇

C₂×D₄

yes

characteristic

C₈×C₃₇

C₂×D₄

yes

characteristic

#64436

—

C₂²×C₃₇

—

C₂²×C₃₇

C₂²⋊C₈

yes

—

C₂²×C₃₇

C₂²×C₈

yes

characteristic

C₂×D₃₇

—

C₂×D₃₇

—

C₄×C₃₇

C₂²⋊C₈

yes

—

C₄×C₃₇

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₃₇

C₂×M₁₆

yes

characteristic

Dic₃₇

—

#2113

—

maximal, Sylow

C₂×C₃₇

—

C₂×C₃₇

C₂×C₂²⋊C₈

yes

characteristic

C₂×C₃₇

#504

yes

—

D₃₇

—

C₂×#131

—

C₂×M₃₂

—

C₂²×C₁₆

D₃₇

yes

characteristic, complemented

C₂²⋊C₁₆

—

C₃₇

#2113

yes

characteristic, complemented, Sylow

C₂×#41

—

#131

—

#131

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

C₂×D₃₇

yes

characteristic

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₃₇

yes

—

#41

—

#41

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

C₄×D₃₇

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄×D₃₇

yes

characteristic

C₂×C₈

D₁₄₈

yes

—

C₂×C₈

C₂²⋊D₃₇

yes

—

C₂×C₈

C₂²×D₃₇

yes

characteristic

C₁₆

—

C₁₆

—

C₂³

—

C₂³

C₈×D₃₇

yes

characteristic

100

Q₈

—

101

D₄

—

102

D₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

C₈×D₃₇

yes

characteristic

107

C₂×C₄

#10596929

yes

—

108

C₂×C₄

#10596938

yes

—

109

C₂×C₄

C₂×C₄×D₃₇

yes

characteristic

110

C₈

—

111

C₈

#10596938

yes

—

112

C₈

C₂×D₁₄₈

yes

characteristic

113

C₈

C₂×C₂²⋊D₃₇

yes

characteristic

114

C₂²

—

115

C₂²

—

116

C₂²

—

117

C₂²

#11924448

yes

—

118

C₂²

C₂×C₈×D₃₇

yes

characteristic

119

C₄

—

120

C₄

#11924448

yes

—

121

C₄

C₂×#10596938

yes

characteristic

122

C₄

C₂×#10596929

yes

characteristic

123

C₂

—

124

C₂

—

148

—

125

C₂

C₂×#11924448

yes

characteristic

126

C₂

#14557949

yes

—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₃₇
commutator, socle	C₂²×C₃₇	C₂²×C₈
Fitting	C₂²×C₁₆×C₃₇	C₂
prime radical	C₂×#10596962	C₄
cosocle	C₂×C₈×C₃₇	C₂³
hypercenter	C₂²×C₁₆	D₃₇
hypocenter	C₃₇	#2113
LCS	G ▷ C₂²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₁₆

subgroup

G/H

center, Frattini

C₂×C₈

C₂²×D₃₇

commutator, socle

C₂²×C₃₇

C₂²×C₈

Fitting

C₂²×C₁₆×C₃₇

C₂

prime radical

C₂×#10596962

C₄

cosocle

C₂×C₈×C₃₇

C₂³

hypercenter

C₂²×C₁₆

D₃₇

C₃₇

G ▷ C₂²×C₃₇ ▷ C₃₇

C₁ ◁ C₂×C₈ ◁ C₂²×C₁₆

N	H ≅ G/N	type
2. C₂×#11924476	C₂	semidirect
3. #14557312	C₂	semidirect
4. C₂²×C₁₆×C₃₇	C₂	semidirect
66. C₂²×C₁₆	D₃₇	semidirect
68. C₃₇	#2113	semidirect

H ≅ G/N

type

2. C₂×#11924476

C₂

semidirect

3. #14557312

C₂

semidirect

4. C₂²×C₁₆×C₃₇

C₂

semidirect

66. C₂²×C₁₆

D₃₇

semidirect

68. C₃₇

#2113

semidirect

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

subgroup	index
C₂×#11924542	2
C₂×#11924476	2
#14557312	2
C₂²×C₁₆×C₃₇	2
#2113	37

subgroup

index

C₂²×C₁₆×C₃₇

Subgroups — #17996209

Conjugacy classes of subgroups

Complements

Maximal subgroups