Group #17994645 — subgroups

Subgroups — #17994645

Order 4736.

252 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 81 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×#11924544	C₂	1	1	yes	maximal, characteristic
2	C₂×#11924471	C₂	1	2	yes	maximal, complemented
3	#14557515	C₂	1	2	yes	maximal
4	#14557291	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₇×#338	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#10596962	C₂²	1	1	yes	characteristic
7	#11924544	—	2	4	no	—
8	C₂×#10596934	C₂²	1	2	yes	—
9	C₂×#10596929	C₄	1	2	yes	—
10	C₂²×#64436	C₂²	1	1	yes	characteristic
11	C₂×C₈×D₃₇	C₄	1	2	yes	—
12	#11924471	—	2	4	no	—
13	#11924454	—	2	2	no	—
14	#11924452	—	2	2	no	—
15	C₂×M₁₆×C₃₇	C₂²	1	2	yes	—
16	C₂×M₁₆×C₃₇	C₂²	1	1	yes	characteristic
17	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
18	C₂×C₂²⋊D₃₇	Q₈	1	2	yes	—
19	#10596962	—	2	2	no	—
20	#10596962	—	2	2	no	—
21	C₂×D₁₄₈	D₄	1	1	yes	characteristic
22	C₂×C₄×D₃₇	C₂×C₄	1	2	yes	—
23	C₂×Dic₇₄	D₄	1	1	yes	characteristic
24	M₁₆×C₃₇	—	2	2	no	—
25	M₁₆×C₃₇	—	2	2	no	—
26	M₁₆×C₃₇	—	2	2	no	—
27	C₂×C₈×C₃₇	—	2	2	no	—
28	C₂×C₈×C₃₇	C₂×C₄	1	2	yes	—
29	#10596934	—	2	2	no	—
30	#10596934	—	2	2	no	—
31	C₂×#64436	—	2	2	no	—
32	C₂×#64436	C₂×C₄	1	1	yes	characteristic
33	C₂×#64436	C₂×C₄	1	1	yes	characteristic
34	C₂×#64436	D₄	1	4	yes	—
35	#10596929	—	2	4	no	—
36	C₈×D₃₇	—	2	4	no	—
37	C₂³×C₃₇	C₂×Q₈	1	1	yes	characteristic
38	C₂²×D₃₇	C₄⋊C₄	1	2	yes	—
39	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₄×C₃₇	C₂×D₄	1	2	yes	—
41	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₃₇	#41	1	2	yes	—
43	C₂²⋊D₃₇	—	2	4	no	—
44	C₂×Dic₃₇	C₄⋊C₄	1	2	yes	—
45	D₁₄₈	—	2	2	no	—
46	C₄×D₃₇	—	2	2	no	—
47	C₄×D₃₇	—	2	2	no	—
48	Dic₇₄	—	2	2	no	—
49	C₈×C₃₇	—	2	2	no	—
50	C₈×C₃₇	—	4	2	no	—
51	#64436	—	4	2	no	—
52	#64436	C₂²⋊C₄	1	2	yes	—
53	#64436	C₂²⋊C₄	1	2	yes	—
54	C₂²×C₃₇	—	4	1	no	—
55	C₂²×C₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₃₇	C₂²⋊Q₈	1	2	yes	—
57	C₂×D₃₇	—	2	2	no	—
58	C₂×D₃₇	—	4	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	Dic₃₇	—	2	2	no	—
63	#1813	—	37	1	no	maximal, Sylow
64	C₂×C₃₇	—	2	1	no	—
65	C₂×C₃₇	—	2	2	no	—
66	C₂×C₃₇	#380	1	1	yes	characteristic
67	D₃₇	—	4	2	no	—
68	C₂×#131	—	37	1	no	—
69	C₂×#108	—	37	2	no	—
70	#413	—	37	2	no	—
71	#338	—	37	1	no	—
72	#338	D₃₇	1	1	yes	characteristic, complemented
73	C₃₇	#1813	1	1	yes	characteristic, complemented, Sylow
74	C₂×#41	—	37	1	no	—
75	#131	—	74	4	no	—
76	C₂×M₁₆	—	37	2	no	—
77	C₂×M₁₆	—	37	2	no	—
78	C₂×M₁₆	C₂×D₃₇	1	2	yes	—
79	C₂×M₁₆	C₂×D₃₇	1	1	yes	characteristic
80	C₂²×C₈	—	37	1	no	—
81	C₂²×C₈	—	37	2	no	—
82	#108	—	74	4	no	—
83	#101	—	74	2	no	—
84	#100	—	74	2	no	—
85	#41	—	74	2	no	—
86	#41	—	74	2	no	—
87	C₂×Q₈	—	37	1	no	—
88	C₂×D₄	—	37	2	no	—
89	C₂×D₄	—	37	1	no	—
90	C₂²×C₄	—	37	2	no	—
91	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
92	M₁₆	—	2	2	no	—
93	M₁₆	—	2	2	no	—
94	M₁₆	—	2	2	no	—
95	M₁₆	—	74	4	no	—
96	M₁₆	—	74	2	no	—
97	M₁₆	—	74	2	no	—
98	C₂×C₈	—	2	2	no	—
99	C₂×C₈	—	37	1	no	—
100	C₂×C₈	—	37	1	no	—
101	C₂×C₈	—	37	4	no	—
102	C₂×C₈	—	74	4	no	—
103	C₂×C₈	—	74	2	no	—
104	C₂×C₈	C₄×D₃₇	1	2	yes	—
105	C₂³	—	37	2	no	—
106	C₂³	Q₈×D₃₇	1	1	yes	characteristic
107	Q₈	—	74	2	no	—
108	D₄	—	74	4	no	—
109	D₄	—	74	2	no	—
110	C₂×C₄	—	37	2	no	—
111	C₂×C₄	—	74	2	no	—
112	C₂×C₄	—	74	2	no	—
113	C₂×C₄	C₂×C₄×D₃₇	1	1	yes	characteristic
114	C₂×C₄	#10596962	1	2	yes	—
115	C₂×C₄	D₄×D₃₇	1	2	yes	—
116	C₂×C₄	D₄×D₃₇	1	1	yes	characteristic
117	C₈	—	2	2	no	—
118	C₈	—	4	2	no	—
119	C₈	—	37	2	no	—
120	C₈	—	37	2	no	—
121	C₈	—	148	2	no	—
122	C₂²	—	4	1	no	—
123	C₂²	—	74	2	no	—
124	C₂²	—	148	2	no	—
125	C₂²	C₄⋊C₄×D₃₇	1	1	yes	characteristic
126	C₂²	#11924514	1	2	yes	—
127	C₄	—	74	2	no	—
128	C₄	C₂²⋊C₄×D₃₇	1	1	yes	characteristic
129	C₄	#11924493	1	1	yes	characteristic
130	C₄	#11924509	1	2	yes	—
131	C₂	—	2	1	no	—
132	C₂	—	2	2	no	—
133	C₂	—	148	2	no	—
134	C₂	#14557392	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11924544

C₂

yes

maximal, characteristic

C₂×#11924471

C₂

yes

maximal, complemented

#14557515

C₂

yes

maximal

#14557291

C₂

yes

maximal, characteristic, complemented

C₃₇×#338

C₂

yes

maximal, characteristic, complemented

C₂×#10596962

C₂²

yes

characteristic

#11924544

—

C₂×#10596934

C₂²

yes

—

C₂×#10596929

C₄

yes

—

C₂²×#64436

C₂²

yes

characteristic

C₂×C₈×D₃₇

C₄

yes

—

#11924471

—

#11924454

—

#11924452

—

C₂×M₁₆×C₃₇

C₂²

yes

—

C₂×M₁₆×C₃₇

C₂²

yes

characteristic

C₂²×C₄×C₃₇

C₂³

yes

characteristic

C₂×C₂²⋊D₃₇

Q₈

yes

—

#10596962

—

#10596962

—

C₂×D₁₄₈

D₄

yes

characteristic

C₂×C₄×D₃₇

C₂×C₄

yes

—

C₂×Dic₇₄

D₄

yes

characteristic

M₁₆×C₃₇

—

M₁₆×C₃₇

—

M₁₆×C₃₇

—

C₂×C₈×C₃₇

—

C₂×C₈×C₃₇

C₂×C₄

yes

—

—

—

—

C₂×C₄

yes

characteristic

C₂×#64436

C₂×C₄

yes

characteristic

C₂×#64436

D₄

yes

—

#10596929

—

C₈×D₃₇

—

C₂³×C₃₇

C₂×Q₈

yes

characteristic

C₂²×D₃₇

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

—

C₂²⋊D₃₇

—

C₂×Dic₃₇

C₄⋊C₄

yes

—

D₁₄₈

—

C₄×D₃₇

—

C₄×D₃₇

—

Dic₇₄

—

C₈×C₃₇

—

C₈×C₃₇

—

#64436

—

#64436

C₂²⋊C₄

yes

—

#64436

C₂²⋊C₄

yes

—

C₂²×C₃₇

—

C₂²×C₃₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₃₇

C₂²⋊Q₈

yes

—

C₂×D₃₇

—

C₂×D₃₇

—

C₄×C₃₇

C₂×C₂²⋊C₄

yes

C₄×C₃₇

yes

C₄×C₃₇

yes

—

Dic₃₇

—

#1813

—

maximal, Sylow

C₂×C₃₇

—

C₂×C₃₇

—

C₂×C₃₇

#380

yes

characteristic

D₃₇

—

—

—

—

—

D₃₇

yes

characteristic, complemented

C₃₇

#1813

yes

characteristic, complemented, Sylow

C₂×#41

—

#131

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₃₇

yes

—

C₂×M₁₆

C₂×D₃₇

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

—

—

—

—

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₃₇

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

C₄×D₃₇

yes

—

105

C₂³

—

106

C₂³

Q₈×D₃₇

yes

characteristic

107

Q₈

—

108

D₄

—

109

D₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

C₂×C₄×D₃₇

yes

characteristic

114

C₂×C₄

#10596962

yes

—

115

C₂×C₄

D₄×D₃₇

yes

—

116

C₂×C₄

D₄×D₃₇

yes

characteristic

117

C₈

—

118

C₈

—

119

C₈

—

120

C₈

—

121

C₈

—

148

—

122

C₂²

—

123

C₂²

—

124

C₂²

—

148

—

125

C₂²

C₄⋊C₄×D₃₇

yes

characteristic

126

C₂²

#11924514

yes

—

127

C₄

—

128

C₄

C₂²⋊C₄×D₃₇

yes

characteristic

129

C₄

#11924493

yes

characteristic

130

C₄

#11924509

yes

—

131

C₂

—

132

C₂

—

133

C₂

—

148

—

134

C₂

#14557392

yes

characteristic

	subgroup	G/H
center	C₄	#11924493
commutator	C₂×C₄×C₃₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₃₇
Fitting	C₃₇×#338	C₂
socle	C₂×C₃₇	#380
prime radical	C₂×#10596962	C₂²
cosocle	C₂²×C₄×C₃₇	C₂³
hypercenter	#338	D₃₇
hypocenter	C₃₇	#1813
LCS	G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ #338

subgroup

G/H

center

C₄

#11924493

commutator

C₂×C₄×C₃₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₃₇

Fitting

C₃₇×#338

C₂

C₂×C₃₇

C₂²

C₂²×C₄×C₃₇

C₂³

D₃₇

C₃₇

G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ #338

N	H ≅ G/N	type
2. C₂×#11924471	C₂	semidirect
4. #14557291	C₂	semidirect
5. C₃₇×#338	C₂	semidirect
72. #338	D₃₇	semidirect
73. C₃₇	#1813	semidirect

H ≅ G/N

type

2. C₂×#11924471

C₂

semidirect

4. #14557291

C₂

semidirect

5. C₃₇×#338

C₂

semidirect

72. #338

D₃₇

semidirect

73. C₃₇

#1813

semidirect

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

subgroup	index
C₂×#11924544	2
C₂×#11924471	2
#14557515	2
#14557291	2
C₃₇×#338	2
#1813	37

subgroup

index

C₃₇×#338

Subgroups — #17994645

Conjugacy classes of subgroups

Complements

Maximal subgroups