Group #18004930 — subgroups

Subgroups — C₂×#14557897 #18004930

Order 4736.

488 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 149 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×#11924528	C₂	1	1	yes	maximal, characteristic, complemented
2	#14557897	C₂	1	8	yes	maximal, direct product factor
3	C₂×#11924457	C₂	1	4	yes	maximal, complemented
4	C₂×#11924447	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#127×C₃₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×#64436	C₂²	1	2	yes	—
7	#11924528	—	2	2	no	—
8	#11924528	C₂²	1	4	yes	complemented
9	C₂×C₄⋊Dic₃₇	C₂²	1	2	yes	—
10	C₂×#10596936	—	2	2	no	—
11	#11924457	C₂²	1	16	yes	complemented
12	#11924447	C₂²	1	4	yes	complemented
13	C₂²×D₄×C₃₇	—	2	2	no	—
14	C₂²×D₄×C₃₇	C₂²	1	2	yes	—
15	#127×C₃₇	—	2	2	no	—
16	#127×C₃₇	C₂²	1	4	yes	—
17	C₂×C₄²×C₃₇	C₂²	1	1	yes	characteristic
18	C₂⁴×C₃₇	—	2	2	no	—
19	C₂×D₄×C₃₇	—	2	2	no	—
20	C₂×D₄×C₃₇	—	2	8	no	—
21	C₂×D₄×C₃₇	—	2	8	no	—
22	C₂×D₄×C₃₇	—	4	8	no	—
23	C₂×D₄×C₃₇	C₂³	1	4	yes	—
24	C₂²×C₄×C₃₇	D₄	1	2	yes	—
25	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
26	C₂²×Dic₃₇	—	2	2	no	—
27	C₄²×C₃₇	D₄	1	2	yes	—
28	C₄²×C₃₇	C₂³	1	2	yes	—
29	C₄⋊Dic₃₇	—	2	2	no	—
30	C₄⋊Dic₃₇	C₂³	1	4	yes	—
31	#10596936	—	2	8	no	—
32	C₂×#64436	—	2	4	no	—
33	C₂×#64436	C₂³	1	4	yes	—
34	C₂³×C₃₇	—	2	4	no	—
35	C₂³×C₃₇	—	4	4	no	—
36	C₂³×C₃₇	—	4	8	no	—
37	C₂³×C₃₇	C₂×D₄	1	1	yes	characteristic
38	D₄×C₃₇	—	2	8	no	—
39	D₄×C₃₇	—	4	16	no	—
40	D₄×C₃₇	—	4	4	no	—
41	C₂×C₄×C₃₇	—	2	4	no	—
42	C₂×C₄×C₃₇	C₂×D₄	1	4	yes	—
43	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₃₇	#41	1	4	yes	—
45	C₂×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
46	C₂×Dic₃₇	—	2	4	no	—
47	C₂×Dic₃₇	—	4	4	no	—
48	#64436	—	2	4	no	—
49	C₂²×C₃₇	—	4	4	no	—
50	C₂²×C₃₇	—	4	8	no	—
51	C₂²×C₃₇	—	8	8	no	—
52	C₂²×C₃₇	#124	1	2	yes	—
53	C₂²×C₃₇	AGL(1,ℤ/8)	1	4	yes	—
54	C₂²×C₃₇	C₂²×D₄	1	1	yes	characteristic
55	C₄×C₃₇	—	2	4	no	—
56	C₄×C₃₇	#124	1	2	yes	—
57	C₄×C₃₇	C₂×#41	1	2	yes	—
58	Dic₃₇	—	4	4	no	—
59	C₂×#490	—	37	1	no	maximal, Sylow
60	C₂×C₃₇	—	8	4	no	—
61	C₂×C₃₇	#490	1	4	yes	complemented
62	C₂×C₃₇	C₂×#124	1	1	yes	characteristic
63	C₂×C₃₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
64	C₂×#127	D₃₇	1	1	yes	characteristic
65	C₂×#125	—	37	1	no	—
66	#490	—	37	8	no	—
67	C₂×#102	—	37	4	no	—
68	C₂×#97	—	37	1	no	—
69	C₃₇	C₂×#490	1	1	yes	characteristic, complemented, Sylow
70	C₂²×D₄	—	2	2	no	—
71	C₂²×D₄	C₂×D₃₇	1	2	yes	—
72	C₂²×C₈	—	37	2	no	—
73	#127	—	2	2	no	—
74	#127	C₂×D₃₇	1	4	yes	—
75	#125	—	37	4	no	—
76	#125	—	74	2	no	—
77	C₂×C₄⋊C₄	—	37	2	no	—
78	C₂×C₄⋊C₄	—	74	2	no	—
79	C₂×C₄²	C₂×D₃₇	1	1	yes	characteristic
80	#102	—	37	16	no	—
81	#97	—	37	4	no	—
82	C₂⁴	—	2	2	no	—
83	C₂×D₄	—	2	8	no	—
84	C₂×D₄	—	2	2	no	—
85	C₂×D₄	—	2	8	no	—
86	C₂×D₄	—	4	8	no	—
87	C₂×D₄	C₂²×D₃₇	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	4	no	—
93	C₄⋊C₄	—	37	4	no	—
94	C₄⋊C₄	—	74	8	no	—
95	C₄⋊C₄	—	74	2	no	—
96	C₄²	C₂²⋊D₃₇	1	2	yes	—
97	C₄²	C₂²×D₃₇	1	2	yes	—
98	C₂³	—	2	4	no	—
99	C₂³	—	4	4	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	—	74	4	no	—
107	C₂×C₄	—	148	4	no	—
108	C₂×C₄	#10596964	1	4	yes	—
109	C₂×C₄	C₂×C₂²⋊D₃₇	1	4	yes	—
110	C₂×C₄	C₂×C₂²⋊D₃₇	1	1	yes	characteristic
111	C₂×C₄	C₂³×D₃₇	1	1	yes	characteristic
112	C₈	—	74	4	no	—
113	C₂²	—	4	4	no	—
114	C₂²	—	4	8	no	—
115	C₂²	—	8	8	no	—
116	C₂²	#11924523	1	2	yes	—
117	C₂²	#11924562	1	4	yes	—
118	C₂²	C₂²×C₂²⋊D₃₇	1	1	yes	characteristic
119	C₄	—	2	4	no	—
120	C₄	—	148	4	no	—
121	C₄	#11924523	1	2	yes	—
122	C₄	C₂×#10596964	1	2	yes	—
123	C₂	—	8	4	no	—
124	C₂	#14557897	1	4	yes	direct product factor
125	C₂	C₂×#11924523	1	1	yes	characteristic
126	C₂	C₂×#11924562	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11924528

C₂

yes

maximal, characteristic, complemented

#14557897

C₂

yes

maximal, direct product factor

C₂×#11924457

C₂

yes

maximal, complemented

C₂×#11924447

C₂

yes

maximal, characteristic, complemented

C₂×#127×C₃₇

C₂

yes

maximal, characteristic

C₂²×#64436

C₂²

yes

—

#11924528

—

#11924528

C₂²

yes

complemented

C₂×C₄⋊Dic₃₇

C₂²

yes

—

C₂×#10596936

—

#11924457

C₂²

yes

complemented

#11924447

C₂²

yes

complemented

C₂²×D₄×C₃₇

—

C₂²×D₄×C₃₇

C₂²

yes

—

#127×C₃₇

—

#127×C₃₇

C₂²

yes

—

C₂×C₄²×C₃₇

C₂²

yes

characteristic

C₂⁴×C₃₇

—

C₂×D₄×C₃₇

—

C₂×D₄×C₃₇

—

C₂×D₄×C₃₇

—

C₂×D₄×C₃₇

—

C₂×D₄×C₃₇

C₂³

yes

—

C₂²×C₄×C₃₇

D₄

yes

—

C₂²×C₄×C₃₇

C₂³

yes

characteristic

C₂²×Dic₃₇

—

C₄²×C₃₇

D₄

yes

—

C₄²×C₃₇

C₂³

yes

—

C₄⋊Dic₃₇

—

C₄⋊Dic₃₇

C₂³

yes

—

#10596936

—

C₂×#64436

—

C₂×#64436

C₂³

yes

—

C₂³×C₃₇

—

C₂³×C₃₇

—

C₂³×C₃₇

—

C₂³×C₃₇

C₂×D₄

yes

characteristic

D₄×C₃₇

—

D₄×C₃₇

—

D₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₂×D₄

yes

—

C₂×C₄×C₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₃₇

#41

yes

—

C₂×C₄×C₃₇

C₂⁴

yes

characteristic

C₂×Dic₃₇

—

C₂×Dic₃₇

—

#64436

—

C₂²×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

#124

yes

—

C₂²×C₃₇

AGL(1,ℤ/8)

yes

—

C₂²×C₃₇

C₂²×D₄

yes

characteristic

C₄×C₃₇

—

C₄×C₃₇

#124

yes

—

C₄×C₃₇

C₂×#41

yes

—

Dic₃₇

—

C₂×#490

—

maximal, Sylow

C₂×C₃₇

—

C₂×C₃₇

yes

C₂×C₃₇

yes

C₂×C₃₇

yes

—

C₂×#127

D₃₇

yes

—

—

—

C₂×#97

—

C₃₇

C₂×#490

yes

characteristic, complemented, Sylow

C₂²×D₄

—

C₂²×D₄

C₂×D₃₇

yes

—

C₂²×C₈

—

#127

—

#127

C₂×D₃₇

yes

—

#125

—

#125

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₃₇

yes

characteristic

#102

—

#97

—

C₂⁴

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂²×D₃₇

yes

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₃₇

yes

—

C₂²×C₄

C₂²×D₃₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂²⋊D₃₇

yes

—

C₄²

C₂²×D₃₇

yes

—

C₂³

—

C₂³

—

100

C₂³

—

101

C₂³

C₂×C₂²⋊D₃₇

yes

characteristic

102

D₄

—

103

D₄

—

104

D₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

148

—

108

C₂×C₄

#10596964

yes

—

109

C₂×C₄

C₂×C₂²⋊D₃₇

yes

—

110

C₂×C₄

C₂×C₂²⋊D₃₇

yes

characteristic

111

C₂×C₄

C₂³×D₃₇

yes

characteristic

112

C₈

—

113

C₂²

—

114

C₂²

—

115

C₂²

—

116

C₂²

#11924523

yes

—

117

C₂²

#11924562

yes

—

118

C₂²

C₂²×C₂²⋊D₃₇

yes

characteristic

119

C₄

—

120

C₄

—

148

—

121

C₄

#11924523

yes

—

122

C₄

C₂×#10596964

yes

—

123

C₂

—

124

C₂

#14557897

yes

direct product factor

125

C₂

C₂×#11924523

yes

characteristic

126

C₂

C₂×#11924562

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₃₇
commutator, cosocle	C₂×C₄×C₃₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₃₇
Fitting, prime radical	C₂×#127×C₃₇	C₂
socle	C₂³×C₃₇	C₂×D₄
hypercenter	C₂×#127	D₃₇
hypocenter	C₃₇	C₂×#490
LCS	G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#127

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₃₇

commutator, cosocle

C₂×C₄×C₃₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₃₇

Fitting, prime radical

C₂×#127×C₃₇

C₂

socle

C₂³×C₃₇

C₂×D₄

D₃₇

C₃₇

G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#127

N	H ≅ G/N	type
1. C₂×#11924528	C₂	semidirect
2. #14557897	C₂	direct (N × H)
3. C₂×#11924457	C₂	semidirect
4. C₂×#11924447	C₂	semidirect
8. #11924528	C₂²	semidirect
11. #11924457	C₂²	semidirect
12. #11924447	C₂²	semidirect
61. C₂×C₃₇	#490	semidirect
69. C₃₇	C₂×#490	semidirect
124. C₂	#14557897	direct (N × H)

H ≅ G/N

type

1. C₂×#11924528

C₂

semidirect

2. #14557897

C₂

direct (N × H)

3. C₂×#11924457

C₂

semidirect

4. C₂×#11924447

C₂

semidirect

8. #11924528

C₂²

semidirect

11. #11924457

C₂²

semidirect

12. #11924447

C₂²

semidirect

61. C₂×C₃₇

#490

semidirect

69. C₃₇

C₂×#490

semidirect

124. C₂

#14557897

direct (N × H)

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

subgroup	index
C₂×#11924528	2
#14557897	2
C₂×#11924457	2
C₂×#11924447	2
C₂×#127×C₃₇	2
C₂×#490	37

subgroup

index

C₂×#127×C₃₇

Subgroups — C₂×#14557897 #18004930

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#14557897 #18004930

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#14557897 #18004930