Group #17994632 — subgroups

Subgroups — #17994632

Order 4736.

292 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 101 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	C₂³×#64436	C₂	1	1	yes	maximal, characteristic
2	C₂×#11924471	C₂	1	2	yes	maximal, complemented
3	#14557497	C₂	1	1	yes	maximal, characteristic
4	#14557297	C₂	1	2	yes	maximal, complemented
5	C₃₇×#407	C₂	1	1	yes	maximal, characteristic
6	C₂×#10596934	C₂²	1	2	yes	—
7	C₂²×#64436	—	2	4	no	—
8	C₂²×#64436	C₄	1	4	yes	—
9	C₂²×#64436	C₂²	1	1	yes	characteristic
10	C₂²×#64436	C₂²	1	1	yes	characteristic
11	#11924471	—	2	4	no	—
12	C₂²⋊#64436	—	2	2	no	—
13	C₂³×C₄×C₃₇	C₂²	1	1	yes	characteristic
14	C₂×M₁₆×C₃₇	C₂²	1	2	yes	—
15	C₂²⋊C₈×C₃₇	—	2	2	no	—
16	C₂⁴×C₃₇	Q₈	1	1	yes	characteristic
17	C₂²×C₄×C₃₇	—	2	2	no	—
18	C₂²×C₄×C₃₇	—	2	2	no	—
19	C₂²×C₄×C₃₇	C₂×C₄	1	2	yes	—
20	C₂²×C₄×C₃₇	D₄	1	2	yes	—
21	C₂²×C₄×C₃₇	Q₈	1	1	yes	characteristic
22	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
23	M₁₆×C₃₇	—	2	2	no	—
24	M₁₆×C₃₇	—	2	2	no	—
25	C₂×C₈×C₃₇	—	2	2	no	—
26	#10596934	—	2	2	no	—
27	#10596934	—	2	2	no	—
28	C₂×#64436	—	2	2	no	—
29	C₂×#64436	—	2	8	no	—
30	C₂×#64436	—	2	2	no	—
31	C₂×#64436	C₂×C₄	1	2	yes	—
32	C₂×#64436	C₂×C₄	1	2	yes	—
33	C₂×#64436	D₄	1	4	yes	—
34	C₂³×C₃₇	—	2	4	no	—
35	C₂³×C₃₇	—	2	2	no	—
36	C₂³×C₃₇	C₄⋊C₄	1	2	yes	—
37	C₂³×C₃₇	C₂×Q₈	1	1	yes	characteristic
38	C₂×C₄×C₃₇	—	2	2	no	—
39	C₂×C₄×C₃₇	—	2	2	no	—
40	C₂×C₄×C₃₇	—	2	2	no	—
41	C₂×C₄×C₃₇	—	2	4	no	—
42	C₂×C₄×C₃₇	C₄⋊C₄	1	2	yes	—
43	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₄×C₃₇	C₂×D₄	1	2	yes	—
45	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₃₇	#41	1	2	yes	—
47	C₈×C₃₇	—	4	2	no	—
48	#64436	—	2	2	no	—
49	#64436	—	4	2	no	—
50	#64436	C₂²⋊C₄	1	4	yes	—
51	C₂²×C₃₇	—	2	8	no	—
52	C₂²×C₃₇	—	2	2	no	—
53	C₂²×C₃₇	—	2	2	no	—
54	C₂²×C₃₇	—	2	2	no	—
55	C₂²×C₃₇	#108	1	4	yes	complemented
56	C₂²×C₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₂²×C₃₇	C₂²⋊Q₈	1	2	yes	—
58	C₄×C₃₇	—	2	2	no	—
59	C₄×C₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
60	C₄×C₃₇	#117	1	1	yes	characteristic
61	C₄×C₃₇	#121	1	2	yes	—
62	#1810	—	37	1	no	maximal, Sylow
63	C₂×C₃₇	—	2	4	no	—
64	C₂×C₃₇	—	2	2	no	—
65	C₂×C₃₇	#380	1	1	yes	characteristic
66	C₂×C₃₇	C₂×#108	1	2	yes	—
67	C₂³×C₈	—	37	1	no	—
68	C₂×#108	—	37	2	no	—
69	#407	—	37	1	no	—
70	#407	D₃₇	1	1	yes	characteristic
71	#341	—	37	2	no	—
72	C₃₇	#1810	1	1	yes	characteristic, complemented, Sylow
73	C₂³×C₄	C₂×D₃₇	1	1	yes	characteristic
74	C₂×M₁₆	—	37	2	no	—
75	C₂×M₁₆	C₂×D₃₇	1	2	yes	—
76	C₂²×C₈	—	37	1	no	—
77	C₂²×C₈	—	37	4	no	—
78	C₂²×C₈	—	37	1	no	—
79	C₂²×C₈	—	74	4	no	—
80	#108	—	74	4	no	—
81	C₂²⋊C₈	—	2	2	no	—
82	C₂²⋊C₈	—	74	2	no	—
83	C₂⁴	Dic₇₄	1	1	yes	characteristic
84	C₂²×C₄	—	2	2	no	—
85	C₂²×C₄	—	2	2	no	—
86	C₂²×C₄	Dic₇₄	1	1	yes	characteristic
87	C₂²×C₄	C₄×D₃₇	1	2	yes	—
88	C₂²×C₄	C₂²⋊D₃₇	1	2	yes	—
89	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
90	M₁₆	—	2	2	no	—
91	M₁₆	—	2	2	no	—
92	M₁₆	—	74	2	no	—
93	M₁₆	—	74	2	no	—
94	C₂×C₈	—	2	2	no	—
95	C₂×C₈	—	37	2	no	—
96	C₂×C₈	—	37	4	no	—
97	C₂×C₈	—	37	2	no	—
98	C₂×C₈	—	74	8	no	—
99	C₂×C₈	—	74	2	no	—
100	C₂×C₈	—	74	2	no	—
101	C₂³	—	2	2	no	—
102	C₂³	—	2	4	no	—
103	C₂³	#10596936	1	2	yes	—
104	C₂³	C₂×Dic₇₄	1	1	yes	characteristic
105	C₂×C₄	—	2	2	no	—
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	—	2	2	no	—
108	C₂×C₄	—	2	4	no	—
109	C₂×C₄	#10596936	1	2	yes	—
110	C₂×C₄	C₂×C₄×D₃₇	1	1	yes	characteristic
111	C₂×C₄	D₄×D₃₇	1	2	yes	—
112	C₂×C₄	#10596964	1	2	yes	—
113	C₂×C₄	C₂×C₂²⋊D₃₇	1	1	yes	characteristic
114	C₈	—	4	2	no	—
115	C₈	—	37	4	no	—
116	C₈	—	74	2	no	—
117	C₈	—	148	2	no	—
118	C₂²	—	2	8	no	—
119	C₂²	—	2	2	no	—
120	C₂²	—	2	2	no	—
121	C₂²	—	2	2	no	—
122	C₂²	#11924471	1	4	yes	complemented
123	C₂²	C₂×#10596936	1	1	yes	characteristic
124	C₂²	#11924513	1	2	yes	—
125	C₄	—	2	2	no	—
126	C₄	C₂²⋊C₄×D₃₇	1	1	yes	characteristic
127	C₄	#11924492	1	1	yes	characteristic
128	C₄	#11924508	1	2	yes	—
129	C₂	—	2	2	no	—
130	C₂	—	2	4	no	—
131	C₂	#14557391	1	1	yes	characteristic
132	C₂	C₂×#11924471	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂³×#64436

C₂

yes

maximal, characteristic

C₂×#11924471

C₂

yes

maximal, complemented

#14557497

C₂

yes

maximal, characteristic

#14557297

C₂

yes

maximal, complemented

C₃₇×#407

C₂

yes

maximal, characteristic

C₂×#10596934

C₂²

yes

—

C₂²×#64436

—

C₂²×#64436

C₄

yes

—

C₂²×#64436

C₂²

yes

characteristic

C₂²×#64436

C₂²

yes

characteristic

#11924471

—

C₂²⋊#64436

—

C₂³×C₄×C₃₇

C₂²

yes

characteristic

C₂×M₁₆×C₃₇

C₂²

yes

—

C₂²⋊C₈×C₃₇

—

C₂⁴×C₃₇

Q₈

yes

characteristic

C₂²×C₄×C₃₇

—

C₂²×C₄×C₃₇

—

C₂²×C₄×C₃₇

C₂×C₄

yes

—

C₂²×C₄×C₃₇

D₄

yes

—

C₂²×C₄×C₃₇

Q₈

yes

characteristic

C₂²×C₄×C₃₇

C₂³

yes

characteristic

M₁₆×C₃₇

—

M₁₆×C₃₇

—

C₂×C₈×C₃₇

—

—

—

—

—

—

C₂×C₄

yes

—

C₂×#64436

C₂×C₄

yes

—

C₂×#64436

D₄

yes

—

C₂³×C₃₇

—

C₂³×C₃₇

—

C₂³×C₃₇

C₄⋊C₄

yes

—

C₂³×C₃₇

C₂×Q₈

yes

characteristic

C₂×C₄×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×D₄

yes

characteristic

C₂×C₄×C₃₇

#41

yes

—

C₈×C₃₇

—

#64436

—

#64436

—

#64436

C₂²⋊C₄

yes

—

C₂²×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

#108

yes

complemented

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

—

#1810

—

maximal, Sylow

C₂×C₃₇

—

C₂×C₃₇

—

C₂×C₃₇

#380

yes

characteristic

C₂×C₃₇

C₂×#108

yes

—

C₂³×C₈

—

C₂×#108

—

#407

—

#407

D₃₇

yes

characteristic

#341

—

C₃₇

#1810

yes

characteristic, complemented, Sylow

C₂³×C₄

C₂×D₃₇

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

C₂×D₃₇

yes

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

#108

—

C₂²⋊C₈

—

C₂²⋊C₈

—

C₂⁴

Dic₇₄

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

Dic₇₄

yes

characteristic

C₂²×C₄

C₄×D₃₇

yes

—

C₂²×C₄

C₂²⋊D₃₇

yes

—

C₂²×C₄

C₂²×D₃₇

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂³

—

102

C₂³

—

103

C₂³

#10596936

yes

—

104

C₂³

C₂×Dic₇₄

yes

characteristic

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

#10596936

yes

—

110

C₂×C₄

C₂×C₄×D₃₇

yes

characteristic

111

C₂×C₄

D₄×D₃₇

yes

—

112

C₂×C₄

#10596964

yes

—

113

C₂×C₄

C₂×C₂²⋊D₃₇

yes

characteristic

114

C₈

—

115

C₈

—

116

C₈

—

117

C₈

—

148

—

118

C₂²

—

119

C₂²

—

120

C₂²

—

121

C₂²

—

122

C₂²

#11924471

yes

complemented

123

C₂²

C₂×#10596936

yes

characteristic

124

C₂²

#11924513

yes

—

125

C₄

—

126

C₄

C₂²⋊C₄×D₃₇

yes

characteristic

127

C₄

#11924492

yes

characteristic

128

C₄

#11924508

yes

—

129

C₂

—

130

C₂

—

131

C₂

#14557391

yes

characteristic

132

C₂

C₂×#11924471

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₃₇
commutator	C₂×C₄×C₃₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₃₇
Fitting	C₃₇×#407	C₂
socle	C₂²×C₃₇	C₂×C₄⋊C₄
prime radical	C₂⁴×C₃₇	Q₈
cosocle	C₂²×C₄×C₃₇	C₂³
hypercenter	#407	D₃₇
hypocenter	C₃₇	#1810
LCS	G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #407

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₃₇

commutator

C₂×C₄×C₃₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₃₇

Fitting

C₃₇×#407

C₂

socle

C₂²×C₃₇

C₂×C₄⋊C₄

prime radical

C₂⁴×C₃₇

Q₈

cosocle

C₂²×C₄×C₃₇

C₂³

D₃₇

C₃₇

G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #407

N	H ≅ G/N	type
2. C₂×#11924471	C₂	semidirect
4. #14557297	C₂	semidirect
55. C₂²×C₃₇	#108	semidirect
72. C₃₇	#1810	semidirect
122. C₂²	#11924471	semidirect

H ≅ G/N

type

2. C₂×#11924471

C₂

semidirect

4. #14557297

C₂

semidirect

55. C₂²×C₃₇

#108

semidirect

72. C₃₇

#1810

semidirect

122. C₂²

#11924471

semidirect

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

subgroup	index
C₂³×#64436	2
C₂×#11924471	2
#14557497	2
#14557297	2
C₃₇×#407	2
#1810	37

subgroup

index

C₂³×#64436

C₃₇×#407

Subgroups — #17994632

Conjugacy classes of subgroups

Complements

Maximal subgroups