Group #18004931 — subgroups

Subgroups — C₂×#14557898 #18004931

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×#11924533	C₂	1	1	yes	maximal, characteristic
2	C₂×#11924528	C₂	1	1	yes	maximal, characteristic, complemented
3	#14557898	C₂	1	8	yes	maximal, direct product factor
4	C₂×#11924458	C₂	1	4	yes	maximal, complemented
5	C₂×#97×C₃₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₁₄₈	—	2	2	no	—
7	C₂²×D₁₄₈	C₂²	1	2	yes	—
8	#11924533	—	2	2	no	—
9	#11924533	C₂²	1	4	yes	—
10	#11924528	—	2	2	no	—
11	#11924528	C₂²	1	4	yes	complemented
12	C₂×C₄⋊Dic₃₇	C₂²	1	2	yes	—
13	C₂×#10596936	—	2	2	no	—
14	#11924458	C₂²	1	16	yes	complemented
15	C₂²×C₈×C₃₇	C₂²	1	2	yes	—
16	C₂×C₄²×C₃₇	C₂²	1	1	yes	characteristic
17	#97×C₃₇	C₂²	1	4	yes	complemented
18	C₂³×D₃₇	—	2	2	no	—
19	C₂²×C₄×C₃₇	D₄	1	2	yes	—
20	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
21	C₂²×Dic₃₇	—	2	2	no	—
22	C₂×D₁₄₈	—	2	8	no	—
23	C₂×D₁₄₈	—	2	2	no	—
24	C₂×D₁₄₈	—	2	8	no	—
25	C₂×D₁₄₈	—	4	8	no	—
26	C₂×D₁₄₈	C₂³	1	4	yes	—
27	C₂×C₈×C₃₇	—	2	4	no	—
28	C₂×C₈×C₃₇	C₂³	1	4	yes	—
29	C₄²×C₃₇	D₄	1	2	yes	—
30	C₄²×C₃₇	C₂³	1	2	yes	—
31	C₄⋊Dic₃₇	—	2	2	no	—
32	C₄⋊Dic₃₇	C₂³	1	4	yes	—
33	#10596936	—	2	8	no	—
34	C₂³×C₃₇	C₂×D₄	1	1	yes	characteristic
35	C₂²×D₃₇	—	2	4	no	—
36	C₂²×D₃₇	—	4	4	no	—
37	C₂²×D₃₇	—	4	8	no	—
38	C₂×C₄×C₃₇	—	2	4	no	—
39	C₂×C₄×C₃₇	C₂×D₄	1	4	yes	—
40	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₃₇	#41	1	4	yes	—
42	C₂×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
43	C₂×Dic₃₇	—	2	4	no	—
44	C₂×Dic₃₇	—	4	4	no	—
45	D₁₄₈	—	2	8	no	—
46	D₁₄₈	—	4	16	no	—
47	D₁₄₈	—	4	4	no	—
48	C₈×C₃₇	—	2	4	no	—
49	C₂²×C₃₇	#124	1	2	yes	—
50	C₂²×C₃₇	AGL(1,ℤ/8)	1	4	yes	—
51	C₂²×C₃₇	C₂²×D₄	1	1	yes	characteristic
52	C₂×D₃₇	—	4	4	no	—
53	C₂×D₃₇	—	4	8	no	—
54	C₂×D₃₇	—	8	8	no	—
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	#490	1	4	yes	complemented
61	C₂×C₃₇	C₂×#124	1	1	yes	characteristic
62	C₂×C₃₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
63	D₃₇	—	8	4	no	—
64	C₂×#127	—	37	1	no	—
65	C₂×#125	—	37	1	no	—
66	#490	—	37	8	no	—
67	C₂×#102	—	37	4	no	—
68	C₂×#97	D₃₇	1	1	yes	characteristic, complemented
69	C₃₇	C₂×#490	1	1	yes	characteristic, complemented, Sylow
70	C₂²×D₄	—	37	2	no	—
71	C₂²×D₄	—	74	2	no	—
72	C₂²×C₈	C₂×D₃₇	1	2	yes	—
73	#127	—	37	4	no	—
74	#127	—	74	2	no	—
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	C₂×D₃₇	1	4	yes	complemented
82	C₂⁴	—	74	2	no	—
83	C₂×D₄	—	37	4	no	—
84	C₂×D₄	—	74	8	no	—
85	C₂×D₄	—	74	2	no	—
86	C₂×D₄	—	74	8	no	—
87	C₂×D₄	—	148	8	no	—
88	C₂²×C₄	—	74	2	no	—
89	C₂²×C₄	D₁₄₈	1	2	yes	—
90	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
91	C₂×C₈	—	2	4	no	—
92	C₂×C₈	C₂²×D₃₇	1	4	yes	—
93	C₄⋊C₄	—	37	4	no	—
94	C₄⋊C₄	—	74	8	no	—
95	C₄⋊C₄	—	74	2	no	—
96	C₄²	D₁₄₈	1	2	yes	—
97	C₄²	C₂²×D₃₇	1	2	yes	—
98	C₂³	—	74	4	no	—
99	C₂³	—	148	4	no	—
100	C₂³	—	148	8	no	—
101	C₂³	C₂×D₁₄₈	1	1	yes	characteristic
102	D₄	—	74	8	no	—
103	D₄	—	148	16	no	—
104	D₄	—	148	4	no	—
105	C₂×C₄	—	2	4	no	—
106	C₂×C₄	—	74	4	no	—
107	C₂×C₄	—	148	4	no	—
108	C₂×C₄	C₂×D₁₄₈	1	4	yes	—
109	C₂×C₄	C₂×D₁₄₈	1	1	yes	characteristic
110	C₂×C₄	#10596962	1	4	yes	—
111	C₂×C₄	C₂³×D₃₇	1	1	yes	characteristic
112	C₈	—	2	4	no	—
113	C₂²	—	296	8	no	—
114	C₂²	—	148	4	no	—
115	C₂²	—	148	8	no	—
116	C₂²	#11924524	1	2	yes	—
117	C₂²	#11924564	1	4	yes	—
118	C₂²	C₂²×D₁₄₈	1	1	yes	characteristic
119	C₄	—	2	4	no	—
120	C₄	—	148	4	no	—
121	C₄	#11924524	1	2	yes	—
122	C₄	C₂×#10596962	1	2	yes	—
123	C₂	—	296	4	no	—
124	C₂	#14557898	1	4	yes	direct product factor
125	C₂	C₂×#11924524	1	1	yes	characteristic
126	C₂	C₂×#11924564	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11924533

C₂

yes

maximal, characteristic

C₂×#11924528

C₂

yes

maximal, characteristic, complemented

#14557898

C₂

yes

maximal, direct product factor

C₂×#11924458

C₂

yes

maximal, complemented

C₂×#97×C₃₇

C₂

yes

maximal, characteristic, complemented

C₂²×D₁₄₈

—

C₂²×D₁₄₈

C₂²

yes

—

#11924533

—

#11924533

C₂²

yes

—

#11924528

—

#11924528

C₂²

yes

complemented

C₂×C₄⋊Dic₃₇

C₂²

yes

—

C₂×#10596936

—

#11924458

C₂²

yes

complemented

C₂²×C₈×C₃₇

C₂²

yes

—

C₂×C₄²×C₃₇

C₂²

yes

characteristic

#97×C₃₇

C₂²

yes

complemented

C₂³×D₃₇

—

C₂²×C₄×C₃₇

D₄

yes

—

C₂²×C₄×C₃₇

C₂³

yes

characteristic

C₂²×Dic₃₇

—

C₂×D₁₄₈

—

C₂×D₁₄₈

—

C₂×D₁₄₈

—

C₂×D₁₄₈

—

C₂×D₁₄₈

C₂³

yes

—

C₂×C₈×C₃₇

—

C₂×C₈×C₃₇

C₂³

yes

—

C₄²×C₃₇

D₄

yes

—

C₄²×C₃₇

C₂³

yes

—

C₄⋊Dic₃₇

—

C₄⋊Dic₃₇

C₂³

yes

—

#10596936

—

C₂³×C₃₇

C₂×D₄

yes

characteristic

C₂²×D₃₇

—

C₂²×D₃₇

—

C₂²×D₃₇

—

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

—

D₁₄₈

—

D₁₄₈

—

D₁₄₈

—

C₈×C₃₇

—

C₂²×C₃₇

#124

yes

—

C₂²×C₃₇

AGL(1,ℤ/8)

yes

—

C₂²×C₃₇

C₂²×D₄

yes

characteristic

C₂×D₃₇

—

C₂×D₃₇

—

C₂×D₃₇

—

C₄×C₃₇

—

C₄×C₃₇

#124

yes

—

C₄×C₃₇

C₂×#41

yes

—

Dic₃₇

—

—

C₂×C₃₇

yes

C₂×C₃₇

yes

C₂×C₃₇

yes

—

D₃₇

—

—

—

—

—

C₂×#97

D₃₇

yes

characteristic, complemented

C₃₇

C₂×#490

yes

characteristic, complemented, Sylow

C₂²×D₄

—

C₂²×D₄

—

C₂²×C₈

C₂×D₃₇

yes

—

—

—

—

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₃₇

yes

characteristic

#102

—

#97

C₂×D₃₇

yes

complemented

C₂⁴

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

148

—

C₂²×C₄

—

C₂²×C₄

D₁₄₈

yes

—

C₂²×C₄

C₂²×D₃₇

yes

characteristic

C₂×C₈

—

C₂×C₈

C₂²×D₃₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

D₁₄₈

yes

—

C₄²

C₂²×D₃₇

yes

—

C₂³

—

C₂³

—

148

—

100

C₂³

—

148

—

101

C₂³

C₂×D₁₄₈

yes

characteristic

102

D₄

—

103

D₄

—

148

—

104

D₄

—

148

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

148

—

108

C₂×C₄

C₂×D₁₄₈

yes

—

109

C₂×C₄

C₂×D₁₄₈

yes

characteristic

110

C₂×C₄

#10596962

yes

—

111

C₂×C₄

C₂³×D₃₇

yes

characteristic

112

C₈

—

113

C₂²

—

296

—

114

C₂²

—

148

—

115

C₂²

—

148

—

116

C₂²

#11924524

yes

—

117

C₂²

#11924564

yes

—

118

C₂²

C₂²×D₁₄₈

yes

characteristic

119

C₄

—

120

C₄

—

148

—

121

C₄

#11924524

yes

—

122

C₄

C₂×#10596962

yes

—

123

C₂

—

296

—

124

C₂

#14557898

yes

direct product factor

125

C₂

C₂×#11924524

yes

characteristic

126

C₂

C₂×#11924564

yes

—

	subgroup	G/H
center	C₂³	C₂×D₁₄₈
commutator, cosocle	C₂×C₄×C₃₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₃₇
Fitting	C₂×#97×C₃₇	C₂
socle	C₂³×C₃₇	C₂×D₄
prime radical	C₂×#11924533	C₂
hypercenter	C₂×#97	D₃₇
hypocenter	C₃₇	C₂×#490
LCS	G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

subgroup

G/H

center

C₂³

C₂×D₁₄₈

commutator, cosocle

C₂×C₄×C₃₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₃₇

Fitting

C₂×#97×C₃₇

C₂

socle

C₂³×C₃₇

C₂×D₄

prime radical

C₂×#11924533

C₂

hypercenter

C₂×#97

D₃₇

hypocenter

C₃₇

C₂×#490

LCS

G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

N	H ≅ G/N	type
2. C₂×#11924528	C₂	semidirect
3. #14557898	C₂	direct (N × H)
4. C₂×#11924458	C₂	semidirect
5. C₂×#97×C₃₇	C₂	semidirect
11. #11924528	C₂²	semidirect
14. #11924458	C₂²	semidirect
17. #97×C₃₇	C₂²	semidirect
60. C₂×C₃₇	#490	semidirect
68. C₂×#97	D₃₇	semidirect
69. C₃₇	C₂×#490	semidirect
81. #97	C₂×D₃₇	semidirect
124. C₂	#14557898	direct (N × H)

H ≅ G/N

type

2. C₂×#11924528

C₂

semidirect

3. #14557898

C₂

direct (N × H)

4. C₂×#11924458

C₂

semidirect

5. C₂×#97×C₃₇

C₂

semidirect

11. #11924528

C₂²

semidirect

14. #11924458

C₂²

semidirect

17. #97×C₃₇

C₂²

semidirect

60. C₂×C₃₇

#490

semidirect

68. C₂×#97

D₃₇

semidirect

69. C₃₇

C₂×#490

semidirect

81. #97

C₂×D₃₇

semidirect

124. C₂

#14557898

direct (N × H)

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

subgroup	index
C₂×#11924533	2
C₂×#11924528	2
#14557898	2
C₂×#11924458	2
C₂×#97×C₃₇	2
C₂×#490	37

subgroup

index

C₂×#97×C₃₇

Subgroups — C₂×#14557898 #18004931

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#14557898 #18004931

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#14557898 #18004931