Group #17995421 — subgroups

Subgroups — #17995421

Order 4736.

412 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 113 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×#11924457	C₂	1	4	yes	maximal, complemented
2	C₂×#11924447	C₂	1	1	yes	maximal, characteristic, complemented
3	#14557399	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#127×C₃₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×#64436	C₂²	1	2	yes	—
6	C₂×C₄⋊Dic₃₇	C₂²	1	2	yes	—
7	#11924457	—	2	8	no	—
8	#11924447	—	2	2	no	—
9	#11924443	—	2	2	no	—
10	C₂²×D₄×C₃₇	—	2	2	no	—
11	C₂²×D₄×C₃₇	C₂²	1	2	yes	—
12	#127×C₃₇	—	2	2	no	—
13	#127×C₃₇	C₄	1	4	yes	complemented
14	C₂×C₄²×C₃₇	C₂²	1	1	yes	characteristic
15	C₂⁴×C₃₇	—	2	2	no	—
16	C₂×D₄×C₃₇	—	2	8	no	—
17	C₂×D₄×C₃₇	—	2	4	no	—
18	C₂×D₄×C₃₇	—	2	2	no	—
19	C₂×D₄×C₃₇	—	2	4	no	—
20	C₂×D₄×C₃₇	—	4	8	no	—
21	C₂×D₄×C₃₇	C₂×C₄	1	4	yes	—
22	C₂²×C₄×C₃₇	D₄	1	2	yes	—
23	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
24	C₂²×Dic₃₇	—	2	2	no	—
25	C₄²×C₃₇	C₂×C₄	1	2	yes	—
26	C₄²×C₃₇	D₄	1	2	yes	—
27	C₄⋊Dic₃₇	—	2	4	no	—
28	C₂×#64436	—	2	2	no	—
29	C₂×#64436	—	2	2	no	—
30	C₂×#64436	D₄	1	4	yes	complemented
31	C₂³×C₃₇	—	2	4	no	—
32	C₂³×C₃₇	—	4	2	no	—
33	C₂³×C₃₇	—	4	2	no	—
34	C₂³×C₃₇	—	4	8	no	—
35	C₂³×C₃₇	C₂×D₄	1	1	yes	characteristic
36	D₄×C₃₇	—	2	8	no	—
37	D₄×C₃₇	—	4	16	no	—
38	D₄×C₃₇	—	4	4	no	—
39	C₂×C₄×C₃₇	—	2	4	no	—
40	C₂×C₄×C₃₇	C₂²⋊C₄	1	4	yes	—
41	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₄×C₃₇	C₂×D₄	1	2	yes	—
43	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₃₇	#41	1	2	yes	—
45	C₂×Dic₃₇	—	4	2	no	—
46	C₂×Dic₃₇	—	4	4	no	—
47	#64436	—	2	4	no	—
48	C₂²×C₃₇	—	4	4	no	—
49	C₂²×C₃₇	—	4	8	no	—
50	C₂²×C₃₇	—	8	4	no	—
51	C₂²×C₃₇	—	8	4	no	—
52	C₂²×C₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₃₇	#124	1	1	yes	characteristic
54	C₂²×C₃₇	#127	1	1	yes	characteristic
55	C₂²×C₃₇	AGL(1,ℤ/8)	1	4	yes	—
56	C₄×C₃₇	—	2	4	no	—
57	C₄×C₃₇	C₄×D₄	1	2	yes	—
58	C₄×C₃₇	#121	1	2	yes	—
59	Dic₃₇	—	8	2	no	—
60	#1961	—	37	1	no	maximal, Sylow
61	C₂×C₃₇	—	8	4	no	—
62	C₂×C₃₇	#390	1	1	yes	characteristic
63	C₂×C₃₇	#418	1	2	yes	—
64	C₂×C₃₇	#490	1	2	yes	—
65	C₂×C₃₇	#496	1	2	yes	—
66	C₂×#127	D₃₇	1	1	yes	characteristic
67	C₂×#102	—	37	4	no	—
68	C₂×#97	—	37	1	no	—
69	#382	—	37	1	no	—
70	C₃₇	#1961	1	1	yes	characteristic, complemented, Sylow
71	C₂²×D₄	—	2	2	no	—
72	C₂²×D₄	C₂×D₃₇	1	2	yes	—
73	C₂²×C₈	—	37	2	no	—
74	#127	—	2	2	no	—
75	#127	Dic₃₇	1	4	yes	complemented
76	C₂×C₄⋊C₄	—	37	2	no	—
77	C₂×C₄²	C₂×D₃₇	1	1	yes	characteristic
78	#102	—	74	8	no	—
79	#97	—	74	2	no	—
80	#95	—	74	2	no	—
81	C₂⁴	—	2	2	no	—
82	C₂×D₄	—	2	8	no	—
83	C₂×D₄	—	2	4	no	—
84	C₂×D₄	—	2	4	no	—
85	C₂×D₄	—	2	2	no	—
86	C₂×D₄	—	4	8	no	—
87	C₂×D₄	C₂×Dic₃₇	1	4	yes	—
88	C₂²×C₄	—	74	2	no	—
89	C₂²×C₄	C₂²⋊D₃₇	1	2	yes	—
90	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
91	C₂×C₈	—	37	4	no	—
92	C₂×C₈	—	74	2	no	—
93	C₂×C₈	—	74	2	no	—
94	C₄⋊C₄	—	74	4	no	—
95	C₄²	C₂×Dic₃₇	1	2	yes	—
96	C₄²	C₂²⋊D₃₇	1	2	yes	—
97	C₂³	—	2	4	no	—
98	C₂³	—	4	2	no	—
99	C₂³	—	4	2	no	—
100	C₂³	—	4	8	no	—
101	C₂³	C₂×C₂²⋊D₃₇	1	1	yes	characteristic
102	D₄	—	2	8	no	—
103	D₄	—	4	16	no	—
104	D₄	—	4	4	no	—
105	C₂×C₄	—	2	4	no	—
106	C₂×C₄	—	148	2	no	—
107	C₂×C₄	—	148	4	no	—
108	C₂×C₄	C₂²⋊Dic₃₇	1	4	yes	—
109	C₂×C₄	D₄×D₃₇	1	2	yes	—
110	C₂×C₄	#10596964	1	2	yes	—
111	C₂×C₄	C₂²×Dic₃₇	1	1	yes	characteristic
112	C₂×C₄	C₂×C₂²⋊D₃₇	1	1	yes	characteristic
113	C₈	—	74	4	no	—
114	C₂²	—	4	8	no	—
115	C₂²	—	4	4	no	—
116	C₂²	—	8	4	no	—
117	C₂²	—	8	4	no	—
118	C₂²	C₂×C₂²⋊Dic₃₇	1	1	yes	characteristic
119	C₂²	#11924523	1	1	yes	characteristic
120	C₂²	#11924532	1	1	yes	characteristic
121	C₂²	#11924562	1	4	yes	—
122	C₄	—	2	4	no	—
123	C₄	—	296	2	no	—
124	C₄	D₄×Dic₃₇	1	2	yes	—
125	C₄	#11924507	1	2	yes	—
126	C₂	—	8	4	no	—
127	C₂	#14557436	1	1	yes	characteristic
128	C₂	#14557538	1	2	yes	—
129	C₂	#14557897	1	2	yes	—
130	C₂	#14557920	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11924457

C₂

yes

maximal, complemented

C₂×#11924447

C₂

yes

maximal, characteristic, complemented

#14557399

C₂

yes

maximal, characteristic, complemented

C₂×#127×C₃₇

C₂

yes

maximal, characteristic

C₂²×#64436

C₂²

yes

—

C₂×C₄⋊Dic₃₇

C₂²

yes

—

#11924457

—

#11924447

—

#11924443

—

C₂²×D₄×C₃₇

—

C₂²×D₄×C₃₇

C₂²

yes

—

#127×C₃₇

—

#127×C₃₇

C₄

yes

complemented

C₂×C₄²×C₃₇

C₂²

yes

characteristic

C₂⁴×C₃₇

—

C₂×D₄×C₃₇

—

C₂×D₄×C₃₇

—

C₂×D₄×C₃₇

—

C₂×D₄×C₃₇

—

C₂×D₄×C₃₇

—

C₂×D₄×C₃₇

C₂×C₄

yes

—

C₂²×C₄×C₃₇

D₄

yes

—

C₂²×C₄×C₃₇

C₂³

yes

characteristic

C₂²×Dic₃₇

—

C₄²×C₃₇

C₂×C₄

yes

—

C₄²×C₃₇

D₄

yes

—

C₄⋊Dic₃₇

—

C₂×#64436

—

C₂×#64436

—

C₂×#64436

D₄

yes

complemented

C₂³×C₃₇

—

C₂³×C₃₇

—

C₂³×C₃₇

—

C₂³×C₃₇

—

C₂³×C₃₇

C₂×D₄

yes

characteristic

D₄×C₃₇

—

D₄×C₃₇

—

D₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₂²⋊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

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

#64436

—

C₂²×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

C₂×C₂²⋊C₄

yes

C₂²×C₃₇

yes

C₂²×C₃₇

yes

C₂²×C₃₇

yes

—

C₄×C₃₇

—

C₄×C₃₇

C₄×D₄

yes

—

C₄×C₃₇

#121

yes

—

Dic₃₇

—

#1961

—

maximal, Sylow

C₂×C₃₇

—

C₂×C₃₇

#390

yes

characteristic

C₂×C₃₇

#418

yes

—

C₂×C₃₇

#490

yes

—

C₂×C₃₇

#496

yes

—

C₂×#127

D₃₇

yes

characteristic

C₂×#102

—

C₂×#97

—

#382

—

C₃₇

#1961

yes

characteristic, complemented, Sylow

C₂²×D₄

—

C₂²×D₄

C₂×D₃₇

yes

—

C₂²×C₈

—

#127

—

#127

Dic₃₇

yes

complemented

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₃₇

yes

—

—

—

C₂⁴

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×Dic₃₇

yes

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₃₇

yes

—

C₂²×C₄

C₂²×D₃₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄²

C₂×Dic₃₇

yes

—

C₄²

C₂²⋊D₃₇

yes

—

C₂³

—

C₂³

—

C₂³

—

100

C₂³

—

101

C₂³

C₂×C₂²⋊D₃₇

yes

characteristic

102

D₄

—

103

D₄

—

104

D₄

—

105

C₂×C₄

—

106

C₂×C₄

—

148

—

107

C₂×C₄

—

148

—

108

C₂×C₄

C₂²⋊Dic₃₇

yes

—

109

C₂×C₄

D₄×D₃₇

yes

—

110

C₂×C₄

#10596964

yes

—

111

C₂×C₄

C₂²×Dic₃₇

yes

characteristic

112

C₂×C₄

C₂×C₂²⋊D₃₇

yes

characteristic

113

C₈

—

114

C₂²

—

115

C₂²

—

116

C₂²

—

117

C₂²

—

118

C₂²

C₂×C₂²⋊Dic₃₇

yes

characteristic

119

C₂²

#11924523

yes

characteristic

120

C₂²

#11924532

yes

characteristic

121

C₂²

#11924562

yes

—

122

C₄

—

123

C₄

—

296

—

124

C₄

D₄×Dic₃₇

yes

—

125

C₄

#11924507

yes

—

126

C₂

—

127

C₂

#14557436

yes

characteristic

128

C₂

#14557538

yes

—

129

C₂

#14557897

yes

—

130

C₂

#14557920

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₃₇
commutator	C₂×C₄×C₃₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₃₇
Fitting, prime radical	C₂×#127×C₃₇	C₂
socle	C₂³×C₃₇	C₂×D₄
cosocle	C₂²×C₄×C₃₇	C₂³
hypercenter	C₂×#127	D₃₇
hypocenter	C₃₇	#1961
LCS	G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#127

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₃₇

commutator

C₂×C₄×C₃₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₃₇

Fitting, prime radical

C₂×#127×C₃₇

C₂

socle

C₂³×C₃₇

C₂×D₄

cosocle

C₂²×C₄×C₃₇

C₂³

D₃₇

C₃₇

G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#127

N	H ≅ G/N	type
1. C₂×#11924457	C₂	semidirect
2. C₂×#11924447	C₂	semidirect
3. #14557399	C₂	semidirect
13. #127×C₃₇	C₄	semidirect
30. C₂×#64436	D₄	semidirect
70. C₃₇	#1961	semidirect
75. #127	Dic₃₇	semidirect

H ≅ G/N

type

1. C₂×#11924457

C₂

semidirect

2. C₂×#11924447

C₂

semidirect

3. #14557399

C₂

semidirect

13. #127×C₃₇

C₄

semidirect

30. C₂×#64436

D₄

semidirect

70. C₃₇

#1961

semidirect

75. #127

Dic₃₇

semidirect

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

subgroup	index
C₂×#11924457	2
C₂×#11924447	2
#14557399	2
C₂×#127×C₃₇	2
#1961	37

subgroup

index

C₂×#127×C₃₇

Subgroups — #17995421

Conjugacy classes of subgroups

Complements

Maximal subgroups