Group #37544223 — subgroups

Subgroups — C₂×#26041679 #37544223

Order 23168.

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

normal subgroups only

This group has 360 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×#19153807	C₂	1	2	yes	maximal
2	#26041679	C₂	1	8	yes	maximal, direct product factor
3	C₂×#19153749	C₂	1	2	yes	maximal
4	C₂×#19153748	C₂	1	1	yes	maximal, characteristic
5	C₂×C₈×Dic₁₈₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#106×C₁₈₁	C₂	1	1	yes	maximal, characteristic
7	C₂²×#13089678	C₂²	1	1	yes	characteristic
8	#19153807	—	2	4	no	—
9	#19153807	C₂²	1	8	yes	—
10	C₂×C₄⋊Dic₁₈₁	—	2	2	no	—
11	C₂×C₄⋊Dic₁₈₁	C₂²	1	2	yes	—
12	C₂×#15668428	—	2	2	no	—
13	C₂×C₄×Dic₁₈₁	C₂²	1	1	yes	characteristic
14	#19153749	C₂²	1	8	yes	—
15	#19153748	C₂²	1	4	yes	—
16	C₈×Dic₁₈₁	C₂²	1	4	yes	—
17	C₂²×C₈×C₁₈₁	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₄×C₁₈₁	C₂²	1	2	yes	—
19	#106×C₁₈₁	C₂²	1	4	yes	—
20	C₂²×Dic₁₈₁	—	2	2	no	—
21	C₂²×Dic₁₈₁	D₄	1	2	yes	—
22	C₂×#13089678	Q₈	1	4	yes	—
23	C₂×#13089678	C₂³	1	2	yes	—
24	C₄⋊Dic₁₈₁	—	2	8	no	—
25	C₄⋊Dic₁₈₁	—	2	2	no	—
26	C₄⋊Dic₁₈₁	C₂³	1	4	yes	—
27	#15668428	—	2	8	no	—
28	C₄×Dic₁₈₁	D₄	1	2	yes	—
29	C₄×Dic₁₈₁	C₂³	1	2	yes	—
30	C₂²×C₄×C₁₈₁	—	2	2	no	—
31	C₂²×C₄×C₁₈₁	C₂³	1	1	yes	characteristic
32	C₂×C₈×C₁₈₁	Q₈	1	4	yes	—
33	C₂×C₈×C₁₈₁	C₂³	1	2	yes	—
34	C₄⋊C₄×C₁₈₁	—	2	2	no	—
35	C₄⋊C₄×C₁₈₁	C₂³	1	4	yes	—
36	C₂×Dic₁₈₁	—	2	4	no	—
37	C₂×Dic₁₈₁	—	2	4	no	—
38	C₂×Dic₁₈₁	—	4	4	no	—
39	C₂×Dic₁₈₁	C₂×D₄	1	4	yes	—
40	#13089678	C₂×Q₈	1	4	yes	—
41	C₂³×C₁₈₁	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₁₈₁	—	2	4	no	—
43	C₂×C₄×C₁₈₁	—	4	4	no	—
44	C₂×C₄×C₁₈₁	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₁₈₁	C₂×Q₈	1	2	yes	—
46	C₂×C₄×C₁₈₁	C₂×Q₈	1	2	yes	—
47	C₂×C₄×C₁₈₁	C₂⁴	1	1	yes	characteristic
48	C₈×C₁₈₁	C₂×Q₈	1	4	yes	—
49	C₂²×C₁₈₁	C₄⋊Q₈	1	2	yes	—
50	C₂²×C₁₈₁	#135	1	4	yes	—
51	C₂²×C₁₈₁	C₂²×D₄	1	1	yes	characteristic
52	C₄×C₁₈₁	—	4	4	no	—
53	C₄×C₁₈₁	C₄⋊Q₈	1	2	yes	—
54	C₄×C₁₈₁	C₂²×Q₈	1	1	yes	characteristic
55	C₄×C₁₈₁	C₂²×Q₈	1	1	yes	characteristic
56	Dic₁₈₁	—	2	4	no	—
57	Dic₁₈₁	—	4	4	no	—
58	C₂×C₁₈₁	#499	1	4	yes	complemented
59	C₂×C₁₈₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
60	C₂×C₁₈₁	C₂×#135	1	2	yes	—
61	C₁₈₁	C₂×#499	1	1	yes	characteristic, complemented, Sylow
62	C₂×#499	—	181	1	no	maximal, Sylow
63	C₂×#125	—	181	2	no	—
64	#499	—	181	8	no	—
65	C₂×#107	—	181	2	no	—
66	C₂×#106	—	181	1	no	—
67	C₂×#106	D₁₈₁	1	1	yes	characteristic
68	C₂×C₄×C₈	—	181	1	no	—
69	C₂²×C₈	—	181	1	no	—
70	C₂²×C₈	C₂×D₁₈₁	1	1	yes	characteristic
71	#125	—	362	4	no	—
72	#125	—	181	8	no	—
73	C₂×C₄⋊C₄	—	362	2	no	—
74	C₂×C₄⋊C₄	—	362	2	no	—
75	C₂×C₄⋊C₄	—	181	2	no	—
76	C₂×C₄⋊C₄	C₂×D₁₈₁	1	2	yes	—
77	C₂×C₄²	—	181	1	no	—
78	#107	—	181	8	no	—
79	#106	—	181	4	no	—
80	#106	C₂×D₁₈₁	1	4	yes	—
81	C₄×C₈	—	181	4	no	—
82	C₂²×C₄	—	2	2	no	—
83	C₂²×C₄	—	362	2	no	—
84	C₂²×C₄	—	181	2	no	—
85	C₂²×C₄	C₂²×D₁₈₁	1	1	yes	characteristic
86	C₂×C₈	—	181	4	no	—
87	C₂×C₈	—	181	2	no	—
88	C₂×C₈	Dic₃₆₂	1	4	yes	—
89	C₂×C₈	C₂²×D₁₈₁	1	2	yes	—
90	C₄⋊C₄	—	2	2	no	—
91	C₄⋊C₄	—	362	8	no	—
92	C₄⋊C₄	—	362	8	no	—
93	C₄⋊C₄	—	362	2	no	—
94	C₄⋊C₄	—	181	4	no	—
95	C₄⋊C₄	C₂²×D₁₈₁	1	4	yes	—
96	C₄²	—	181	2	no	—
97	C₄²	—	181	2	no	—
98	C₂³	D₄×D₁₈₁	1	1	yes	characteristic
99	C₂×C₄	—	2	4	no	—
100	C₂×C₄	—	4	4	no	—
101	C₂×C₄	—	362	4	no	—
102	C₂×C₄	—	362	4	no	—
103	C₂×C₄	—	181	4	no	—
104	C₂×C₄	—	724	4	no	—
105	C₂×C₄	D₄×D₁₈₁	1	1	yes	characteristic
106	C₂×C₄	Q₈×D₁₈₁	1	2	yes	—
107	C₂×C₄	C₂×Dic₃₆₂	1	2	yes	—
108	C₂×C₄	C₂³×D₁₈₁	1	1	yes	characteristic
109	C₈	—	181	4	no	—
110	C₈	C₂×Dic₃₆₂	1	4	yes	—
111	C₂²	#19153815	1	2	yes	—
112	C₂²	#19153837	1	4	yes	—
113	C₂²	C₂×D₄×D₁₈₁	1	1	yes	characteristic
114	C₄	—	4	4	no	—
115	C₄	—	362	4	no	—
116	C₄	—	724	4	no	—
117	C₄	#19153815	1	2	yes	—
118	C₄	C₂×Q₈×D₁₈₁	1	1	yes	characteristic
119	C₄	C₂²×Dic₃₆₂	1	1	yes	characteristic
120	C₂	#26041679	1	4	yes	direct product factor
121	C₂	C₂×#19153815	1	1	yes	characteristic
122	C₂	C₂×#19153837	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19153807

C₂

yes

maximal

#26041679

C₂

yes

maximal, direct product factor

C₂×#19153749

C₂

yes

maximal

C₂×#19153748

C₂

yes

maximal, characteristic

C₂×C₈×Dic₁₈₁

C₂

yes

maximal, characteristic

C₂×#106×C₁₈₁

C₂

yes

maximal, characteristic

C₂²×#13089678

C₂²

yes

characteristic

#19153807

—

#19153807

C₂²

yes

—

C₂×C₄⋊Dic₁₈₁

—

C₂×C₄⋊Dic₁₈₁

C₂²

yes

—

C₂×#15668428

—

C₂×C₄×Dic₁₈₁

C₂²

yes

characteristic

#19153749

C₂²

yes

—

#19153748

C₂²

yes

—

C₈×Dic₁₈₁

C₂²

yes

—

C₂²×C₈×C₁₈₁

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₁₈₁

C₂²

yes

—

#106×C₁₈₁

C₂²

yes

—

C₂²×Dic₁₈₁

—

C₂²×Dic₁₈₁

D₄

yes

—

C₂×#13089678

Q₈

yes

—

C₂×#13089678

C₂³

yes

—

C₄⋊Dic₁₈₁

—

C₄⋊Dic₁₈₁

—

C₄⋊Dic₁₈₁

C₂³

yes

—

#15668428

—

C₄×Dic₁₈₁

D₄

yes

—

C₄×Dic₁₈₁

C₂³

yes

—

C₂²×C₄×C₁₈₁

—

C₂²×C₄×C₁₈₁

C₂³

yes

characteristic

C₂×C₈×C₁₈₁

Q₈

yes

—

C₂×C₈×C₁₈₁

C₂³

yes

—

C₄⋊C₄×C₁₈₁

—

C₄⋊C₄×C₁₈₁

C₂³

yes

—

C₂×Dic₁₈₁

—

C₂×Dic₁₈₁

—

C₂×Dic₁₈₁

—

C₂×Dic₁₈₁

C₂×D₄

yes

—

#13089678

C₂×Q₈

yes

—

C₂³×C₁₈₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₈₁

—

C₂×C₄×C₁₈₁

—

C₂×C₄×C₁₈₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₈₁

C₂×Q₈

yes

—

C₂×C₄×C₁₈₁

C₂×Q₈

yes

—

C₂×C₄×C₁₈₁

C₂⁴

yes

characteristic

C₈×C₁₈₁

C₂×Q₈

yes

—

C₂²×C₁₈₁

C₄⋊Q₈

yes

—

C₂²×C₁₈₁

#135

yes

—

C₂²×C₁₈₁

C₂²×D₄

yes

characteristic

C₄×C₁₈₁

—

C₄×C₁₈₁

C₄⋊Q₈

yes

—

C₄×C₁₈₁

C₂²×Q₈

yes

characteristic

C₄×C₁₈₁

C₂²×Q₈

yes

characteristic

Dic₁₈₁

—

Dic₁₈₁

—

C₂×C₁₈₁

#499

yes

complemented

C₂×C₁₈₁

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₁₈₁

C₂×#135

yes

—

C₁₈₁

C₂×#499

yes

characteristic, complemented, Sylow

C₂×#499

—

181

maximal, Sylow

C₂×#125

—

181

—

#499

—

181

—

C₂×#107

—

181

—

C₂×#106

—

181

—

C₂×#106

D₁₈₁

yes

characteristic

C₂×C₄×C₈

—

181

—

C₂²×C₈

—

181

—

C₂²×C₈

C₂×D₁₈₁

yes

characteristic

#125

—

362

—

#125

—

181

—

C₂×C₄⋊C₄

—

362

—

C₂×C₄⋊C₄

—

362

—

C₂×C₄⋊C₄

—

181

—

C₂×C₄⋊C₄

C₂×D₁₈₁

yes

—

C₂×C₄²

—

181

—

#107

—

181

—

#106

—

181

—

#106

C₂×D₁₈₁

yes

—

C₄×C₈

—

181

—

C₂²×C₄

—

C₂²×C₄

—

362

—

C₂²×C₄

—

181

—

C₂²×C₄

C₂²×D₁₈₁

yes

characteristic

C₂×C₈

—

181

—

C₂×C₈

—

181

—

C₂×C₈

Dic₃₆₂

yes

—

C₂×C₈

C₂²×D₁₈₁

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

362

—

C₄⋊C₄

—

362

—

C₄⋊C₄

—

362

—

C₄⋊C₄

—

181

—

C₄⋊C₄

C₂²×D₁₈₁

yes

—

C₄²

—

181

—

C₄²

—

181

—

C₂³

D₄×D₁₈₁

yes

characteristic

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

362

—

102

C₂×C₄

—

362

—

103

C₂×C₄

—

181

—

104

C₂×C₄

—

724

—

105

C₂×C₄

D₄×D₁₈₁

yes

characteristic

106

C₂×C₄

Q₈×D₁₈₁

yes

—

107

C₂×C₄

C₂×Dic₃₆₂

yes

—

108

C₂×C₄

C₂³×D₁₈₁

yes

characteristic

109

C₈

—

181

—

110

C₈

C₂×Dic₃₆₂

yes

—

111

C₂²

#19153815

yes

—

112

C₂²

#19153837

yes

—

113

C₂²

C₂×D₄×D₁₈₁

yes

characteristic

114

C₄

—

115

C₄

—

362

—

116

C₄

—

724

—

117

C₄

#19153815

yes

—

118

C₄

C₂×Q₈×D₁₈₁

yes

characteristic

119

C₄

C₂²×Dic₃₆₂

yes

characteristic

120

C₂

#26041679

yes

direct product factor

121

C₂

C₂×#19153815

yes

characteristic

122

C₂

C₂×#19153837

yes

—

	subgroup	G/H
center	C₂³	D₄×D₁₈₁
commutator, cosocle	C₂×C₄×C₁₈₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₈₁
Fitting	C₂×#106×C₁₈₁	C₂
socle, prime radical	C₂³×C₁₈₁	C₂×D₄
hypercenter	C₂×#106	D₁₈₁
hypocenter	C₁₈₁	C₂×#499
LCS	G ▷ C₂×C₄×C₁₈₁ ▷ C₂×C₁₈₁ ▷ C₁₈₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#106

subgroup

G/H

center

C₂³

D₄×D₁₈₁

commutator, cosocle

C₂×C₄×C₁₈₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₈₁

Fitting

C₂×#106×C₁₈₁

C₂

socle, prime radical

C₂³×C₁₈₁

C₂×D₄

D₁₈₁

C₁₈₁

G ▷ C₂×C₄×C₁₈₁ ▷ C₂×C₁₈₁ ▷ C₁₈₁

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#106

N	H ≅ G/N	type
2. #26041679	C₂	direct (N × H)
58. C₂×C₁₈₁	#499	semidirect
61. C₁₈₁	C₂×#499	semidirect
120. C₂	#26041679	direct (N × H)

H ≅ G/N

type

2. #26041679

C₂

direct (N × H)

58. C₂×C₁₈₁

#499

semidirect

61. C₁₈₁

C₂×#499

semidirect

120. C₂

#26041679

direct (N × H)

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

subgroup	index
C₂×#19153807	2
#26041679	2
C₂×#19153749	2
C₂×#19153748	2
C₂×C₈×Dic₁₈₁	2
C₂×#106×C₁₈₁	2
C₂×#499	181

subgroup

index

C₂×C₈×Dic₁₈₁

C₂×#106×C₁₈₁

C₂×#499

181

Subgroups — C₂×#26041679 #37544223

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#26041679 #37544223

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26041679 #37544223