Group #18005042 — subgroups

Subgroups — #18005042

Order 4736.

564 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 165 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×#11924558	C₂	1	4	yes	maximal, complemented
2	#14558083	C₂	1	2	yes	maximal, complemented
3	#14557927	C₂	1	4	yes	maximal, complemented
4	#14557922	C₂	1	4	yes	maximal, complemented
5	C₂×#97×C₃₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#10596962	C₂²	1	4	yes	—
7	#11924558	—	2	16	no	—
8	C₂×Dic₁₄₈	C₂²	1	4	yes	—
9	C₂×#10596930	C₂²	1	8	yes	—
10	C₂×D₂₉₆	C₂²	1	4	yes	—
11	#11924536	C₂²	1	2	yes	—
12	#11924533	C₂²	1	2	yes	—
13	#11924524	C₂²	1	4	yes	—
14	#11924510	—	2	4	no	—
15	C₄×D₁₄₈	—	2	4	no	—
16	C₂²×C₈×C₃₇	C₂²	1	2	yes	—
17	C₂×C₄²×C₃₇	C₂²	1	1	yes	characteristic
18	#97×C₃₇	C₂²	1	4	yes	—
19	C₂²×C₄×C₃₇	D₄	1	2	yes	—
20	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
21	C₂×C₂²⋊D₃₇	—	2	4	no	—
22	#10596962	—	2	16	no	—
23	C₂×D₁₄₈	—	2	4	no	—
24	C₂×D₁₄₈	C₂³	1	4	yes	—
25	C₂×C₄×D₃₇	—	2	4	no	—
26	C₂×Dic₇₄	C₂³	1	4	yes	—
27	C₂×C₈×C₃₇	D₄	1	8	yes	complemented
28	C₂×C₈×C₃₇	C₂³	1	4	yes	—
29	C₄²×C₃₇	D₄	1	2	yes	—
30	C₄²×C₃₇	C₂³	1	2	yes	—
31	#10596938	—	2	8	no	—
32	C₄⋊Dic₃₇	—	2	4	no	—
33	Dic₁₄₈	—	2	8	no	—
34	D₂₉₆	—	2	8	no	—
35	#10596930	—	2	16	no	—
36	C₂³×C₃₇	C₂×D₄	1	1	yes	characteristic
37	C₂²×D₃₇	—	2	4	no	—
38	C₂×C₄×C₃₇	—	4	2	no	—
39	C₂×C₄×C₃₇	C₂×D₄	1	4	yes	—
40	C₂×C₄×C₃₇	C₂×D₄	1	4	yes	—
41	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
43	C₂²⋊D₃₇	—	4	8	no	—
44	C₂×Dic₃₇	—	2	4	no	—
45	D₁₄₈	—	2	8	no	—
46	D₁₄₈	—	4	8	no	—
47	C₄×D₃₇	—	4	8	no	—
48	Dic₇₄	—	2	8	no	—
49	C₈×C₃₇	C₂×D₄	1	8	yes	—
50	C₂²×C₃₇	—	2	2	no	—
51	C₂²×C₃₇	#127	1	2	yes	—
52	C₂²×C₃₇	C₂²×D₄	1	1	yes	characteristic
53	C₂×D₃₇	—	4	4	no	—
54	C₂×D₃₇	—	4	8	no	—
55	C₄×C₃₇	—	2	4	no	—
56	C₄×C₃₇	#127	1	2	yes	—
57	C₄×C₃₇	C₂²×D₄	1	2	yes	—
58	Dic₃₇	—	4	4	no	—
59	#3150	—	37	1	no	maximal, Sylow
60	C₂×C₃₇	—	2	2	no	—
61	C₂×C₃₇	C₂×#127	1	1	yes	characteristic
62	C₂×C₃₇	#575	1	2	yes	—
63	D₃₇	—	8	4	no	—
64	C₂×#135	—	37	4	no	—
65	#532	—	37	2	no	—
66	#497	—	37	4	no	—
67	#496	—	37	4	no	—
68	C₂×#97	D₃₇	1	1	yes	characteristic, complemented
69	C₃₇	#3150	1	1	yes	characteristic, complemented, Sylow
70	C₂×#41	—	37	4	no	—
71	#135	—	74	16	no	—
72	C₂×Q₁₆	—	37	4	no	—
73	C₂×SD₁₆	—	37	8	no	—
74	C₂×D₈	—	37	4	no	—
75	C₂²×C₈	C₂×D₃₇	1	2	yes	—
76	C₄⋊Q₈	—	37	2	no	—
77	#127	—	37	2	no	—
78	#124	—	37	4	no	—
79	#121	—	74	4	no	—
80	C₄×D₄	—	74	4	no	—
81	C₂×C₄²	C₂×D₃₇	1	1	yes	characteristic
82	#97	C₂×D₃₇	1	4	yes	—
83	#41	—	74	16	no	—
84	C₂×Q₈	—	37	4	no	—
85	C₂×D₄	—	37	4	no	—
86	C₂×D₄	—	74	4	no	—
87	C₂×D₄	—	74	4	no	—
88	C₂²×C₄	—	74	4	no	—
89	C₂²×C₄	D₁₄₈	1	2	yes	—
90	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
91	Q₁₆	—	74	8	no	—
92	SD₁₆	—	74	16	no	—
93	D₈	—	74	8	no	—
94	C₂×C₈	D₁₄₈	1	8	yes	complemented
95	C₂×C₈	C₂²×D₃₇	1	4	yes	—
96	C₄⋊C₄	—	74	4	no	—
97	C₂²⋊C₄	—	74	8	no	—
98	C₄²	D₁₄₈	1	2	yes	—
99	C₄²	C₂²×D₃₇	1	2	yes	—
100	C₂³	—	74	4	no	—
101	C₂³	C₂×D₁₄₈	1	1	yes	characteristic
102	Q₈	—	74	8	no	—
103	D₄	—	74	8	no	—
104	D₄	—	148	8	no	—
105	D₄	—	148	8	no	—
106	C₂×C₄	—	4	2	no	—
107	C₂×C₄	—	74	4	no	—
108	C₂×C₄	—	148	8	no	—
109	C₂×C₄	C₂×D₁₄₈	1	4	yes	—
110	C₂×C₄	C₂×D₁₄₈	1	4	yes	—
111	C₂×C₄	C₂×D₁₄₈	1	1	yes	characteristic
112	C₂×C₄	C₂³×D₃₇	1	1	yes	characteristic
113	C₈	C₂×D₁₄₈	1	8	yes	—
114	C₂²	—	2	2	no	—
115	C₂²	—	148	4	no	—
116	C₂²	—	148	8	no	—
117	C₂²	#11924533	1	2	yes	—
118	C₂²	C₂²×D₁₄₈	1	1	yes	characteristic
119	C₄	—	2	4	no	—
120	C₄	—	148	4	no	—
121	C₄	#11924533	1	2	yes	—
122	C₄	C₂²×D₁₄₈	1	2	yes	—
123	C₂	—	2	2	no	—
124	C₂	—	296	4	no	—
125	C₂	C₂×#11924533	1	1	yes	characteristic
126	C₂	#14558333	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11924558

C₂

yes

maximal, complemented

#14558083

C₂

yes

maximal, complemented

#14557927

C₂

yes

maximal, complemented

#14557922

C₂

yes

maximal, complemented

C₂×#97×C₃₇

C₂

yes

maximal, characteristic, complemented

C₂×#10596962

C₂²

yes

—

#11924558

—

C₂×Dic₁₄₈

C₂²

yes

—

C₂×#10596930

C₂²

yes

—

C₂×D₂₉₆

C₂²

yes

—

#11924536

C₂²

yes

—

#11924533

C₂²

yes

—

#11924524

C₂²

yes

—

#11924510

—

C₄×D₁₄₈

—

C₂²×C₈×C₃₇

C₂²

yes

—

C₂×C₄²×C₃₇

C₂²

yes

characteristic

#97×C₃₇

C₂²

yes

—

C₂²×C₄×C₃₇

D₄

yes

—

C₂²×C₄×C₃₇

C₂³

yes

characteristic

C₂×C₂²⋊D₃₇

—

#10596962

—

C₂×D₁₄₈

—

C₂×D₁₄₈

C₂³

yes

—

C₂×C₄×D₃₇

—

C₂×Dic₇₄

C₂³

yes

—

C₂×C₈×C₃₇

D₄

yes

complemented

C₂×C₈×C₃₇

C₂³

yes

—

C₄²×C₃₇

D₄

yes

—

C₄²×C₃₇

C₂³

yes

—

#10596938

—

C₄⋊Dic₃₇

—

Dic₁₄₈

—

D₂₉₆

—

#10596930

—

C₂³×C₃₇

C₂×D₄

yes

characteristic

C₂²×D₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₂×D₄

yes

—

C₂×C₄×C₃₇

C₂×D₄

yes

—

C₂×C₄×C₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₃₇

C₂⁴

yes

characteristic

C₂²⋊D₃₇

—

C₂×Dic₃₇

—

D₁₄₈

—

D₁₄₈

—

C₄×D₃₇

—

Dic₇₄

—

C₈×C₃₇

C₂×D₄

yes

—

C₂²×C₃₇

—

C₂²×C₃₇

#127

yes

—

C₂²×C₃₇

C₂²×D₄

yes

characteristic

C₂×D₃₇

—

C₂×D₃₇

—

C₄×C₃₇

—

C₄×C₃₇

#127

yes

—

C₄×C₃₇

C₂²×D₄

yes

—

Dic₃₇

—

#3150

—

maximal, Sylow

C₂×C₃₇

—

C₂×C₃₇

C₂×#127

yes

characteristic

C₂×C₃₇

#575

yes

—

D₃₇

—

—

—

—

—

C₂×#97

D₃₇

yes

characteristic, complemented

C₃₇

#3150

yes

characteristic, complemented, Sylow

C₂×#41

—

#135

—

C₂×Q₁₆

—

C₂×SD₁₆

—

C₂×D₈

—

C₂²×C₈

C₂×D₃₇

yes

—

C₄⋊Q₈

—

#127

—

#124

—

#121

—

C₄×D₄

—

C₂×C₄²

C₂×D₃₇

yes

characteristic

#97

C₂×D₃₇

yes

—

#41

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

D₁₄₈

yes

—

C₂²×C₄

C₂²×D₃₇

yes

characteristic

Q₁₆

—

SD₁₆

—

D₈

—

C₂×C₈

D₁₄₈

yes

complemented

C₂×C₈

C₂²×D₃₇

yes

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

D₁₄₈

yes

—

C₄²

C₂²×D₃₇

yes

—

100

C₂³

—

101

C₂³

C₂×D₁₄₈

yes

characteristic

102

Q₈

—

103

D₄

—

104

D₄

—

148

—

105

D₄

—

148

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

148

—

109

C₂×C₄

C₂×D₁₄₈

yes

—

110

C₂×C₄

C₂×D₁₄₈

yes

—

111

C₂×C₄

C₂×D₁₄₈

yes

characteristic

112

C₂×C₄

C₂³×D₃₇

yes

characteristic

113

C₈

C₂×D₁₄₈

yes

—

114

C₂²

—

115

C₂²

—

148

—

116

C₂²

—

148

—

117

C₂²

#11924533

yes

—

118

C₂²

C₂²×D₁₄₈

yes

characteristic

119

C₄

—

120

C₄

—

148

—

121

C₄

#11924533

yes

—

122

C₄

C₂²×D₁₄₈

yes

—

123

C₂

—

124

C₂

—

296

—

125

C₂

C₂×#11924533

yes

characteristic

126

C₂

#14558333

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×D₁₄₈
commutator, cosocle	C₂×C₄×C₃₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₃₇
Fitting	C₂×#97×C₃₇	C₂
socle	C₂²×C₃₇	C₂²×D₄
hypercenter	C₂×#97	D₃₇
hypocenter	C₃₇	#3150
LCS	G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#97

subgroup

G/H

center

C₂×C₄

C₂×D₁₄₈

commutator, cosocle

C₂×C₄×C₃₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₃₇

Fitting

C₂×#97×C₃₇

C₂

socle

C₂²×C₃₇

C₂²×D₄

hypercenter

C₂×#97

D₃₇

hypocenter

C₃₇

#3150

LCS

G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#97

N	H ≅ G/N	type
1. C₂×#11924558	C₂	semidirect
2. #14558083	C₂	semidirect
3. #14557927	C₂	semidirect
4. #14557922	C₂	semidirect
5. C₂×#97×C₃₇	C₂	semidirect
27. C₂×C₈×C₃₇	D₄	semidirect
68. C₂×#97	D₃₇	semidirect
69. C₃₇	#3150	semidirect
94. C₂×C₈	D₁₄₈	semidirect

H ≅ G/N

type

1. C₂×#11924558

C₂

semidirect

2. #14558083

C₂

semidirect

3. #14557927

C₂

semidirect

4. #14557922

C₂

semidirect

5. C₂×#97×C₃₇

C₂

semidirect

27. C₂×C₈×C₃₇

D₄

semidirect

68. C₂×#97

D₃₇

semidirect

69. C₃₇

#3150

semidirect

94. C₂×C₈

D₁₄₈

semidirect

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

subgroup	index
C₂×#11924558	2
#14558083	2
#14557927	2
#14557922	2
C₂×#97×C₃₇	2
#3150	37

subgroup

index

C₂×#97×C₃₇

Subgroups — #18005042

Conjugacy classes of subgroups

Complements

Maximal subgroups