Group #37529410 — subgroups

Subgroups — #1467×C₁₈₁ #37529410

Order 23168.

236 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 100 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×C₄×Q₈×C₁₈₁	C₂	1	1	yes	maximal, characteristic
2	#432×C₁₈₁	C₂	1	2	yes	maximal
3	#324×C₁₈₁	C₂	1	4	yes	maximal
4	C₂²×Q₈×C₁₈₁	C₄	1	1	yes	characteristic
5	C₄×Q₈×C₁₈₁	—	2	2	no	—
6	C₄×Q₈×C₁₈₁	—	2	2	no	—
7	C₂×C₄⋊C₄×C₁₈₁	C₄	1	1	yes	characteristic
8	C₂×C₄⋊C₄×C₁₈₁	C₂²	1	2	yes	—
9	C₂×C₄²×C₁₈₁	C₄	1	2	yes	—
10	C₂×C₄²×C₁₈₁	C₂²	1	1	yes	characteristic
11	C₄⋊C₈×C₁₈₁	—	2	2	no	—
12	C₂²⋊C₈×C₁₈₁	C₂²	1	4	yes	—
13	#97×C₁₈₁	—	2	2	no	—
14	C₂×Q₈×C₁₈₁	—	2	2	no	—
15	C₂×Q₈×C₁₈₁	—	2	4	no	—
16	C₂²×C₄×C₁₈₁	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₄×C₁₈₁	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₄×C₁₈₁	C₂×C₄	1	2	yes	—
19	C₂²×C₄×C₁₈₁	C₂×C₄	1	2	yes	—
20	C₂²×C₄×C₁₈₁	C₂³	1	1	yes	characteristic
21	C₂×C₈×C₁₈₁	—	2	4	no	—
22	C₄⋊C₄×C₁₈₁	—	2	2	no	—
23	C₄⋊C₄×C₁₈₁	—	2	4	no	—
24	C₄²×C₁₈₁	—	2	2	no	—
25	C₄²×C₁₈₁	—	2	2	no	—
26	C₄²×C₁₈₁	D₄	1	2	yes	—
27	C₄²×C₁₈₁	D₄	1	2	yes	—
28	C₂³×C₁₈₁	C₂²×C₄	1	1	yes	characteristic
29	Q₈×C₁₈₁	—	2	8	no	—
30	C₂×C₄×C₁₈₁	—	2	2	no	—
31	C₂×C₄×C₁₈₁	—	2	2	no	—
32	C₂×C₄×C₁₈₁	—	2	2	no	—
33	C₂×C₄×C₁₈₁	—	2	2	no	—
34	C₂×C₄×C₁₈₁	—	2	2	no	—
35	C₂×C₄×C₁₈₁	—	4	1	no	—
36	C₂×C₄×C₁₈₁	—	4	2	no	—
37	C₂×C₄×C₁₈₁	C₂²⋊C₄	1	1	yes	characteristic
38	C₂×C₄×C₁₈₁	C₂²⋊C₄	1	2	yes	—
39	C₂×C₄×C₁₈₁	C₂²⋊C₄	1	1	yes	characteristic
40	C₂×C₄×C₁₈₁	M₁₆	1	4	yes	—
41	C₂×C₄×C₁₈₁	C₂×D₄	1	2	yes	—
42	C₈×C₁₈₁	—	4	4	no	—
43	C₂²×C₁₈₁	—	2	1	no	—
44	C₂²×C₁₈₁	—	2	1	no	—
45	C₂²×C₁₈₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
46	C₂²×C₁₈₁	C₂×M₁₆	1	2	yes	—
47	C₄×C₁₈₁	—	2	2	no	—
48	C₄×C₁₈₁	—	2	4	no	—
49	C₄×C₁₈₁	—	2	2	no	—
50	C₄×C₁₈₁	—	2	1	no	—
51	C₄×C₁₈₁	—	4	2	no	—
52	C₄×C₁₈₁	#101	1	2	yes	—
53	C₂×C₁₈₁	—	2	2	no	—
54	C₂×C₁₈₁	#407	1	1	yes	characteristic
55	C₂×C₁₈₁	#410	1	1	yes	characteristic
56	C₂×C₁₈₁	C₂×#101	1	1	yes	characteristic
57	C₁₈₁	#1467	1	1	yes	characteristic, direct product factor, Sylow
58	#1467	C₁₈₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
59	C₂×C₄×Q₈	C₂×C₁₈₁	1	1	yes	characteristic
60	#432	C₂×C₁₈₁	1	2	yes	—
61	#324	C₂×C₁₈₁	1	4	yes	—
62	C₂²×Q₈	C₄×C₁₈₁	1	1	yes	characteristic
63	C₄×Q₈	—	2	2	no	—
64	C₄×Q₈	—	2	2	no	—
65	C₂×C₄⋊C₄	C₄×C₁₈₁	1	1	yes	characteristic
66	C₂×C₄⋊C₄	C₂²×C₁₈₁	1	2	yes	—
67	C₂×C₄²	C₄×C₁₈₁	1	2	yes	—
68	C₂×C₄²	C₂²×C₁₈₁	1	1	yes	characteristic
69	C₄⋊C₈	—	2	2	no	—
70	C₂²⋊C₈	C₂²×C₁₈₁	1	4	yes	—
71	#97	—	2	2	no	—
72	C₂×Q₈	—	2	2	no	—
73	C₂×Q₈	—	2	4	no	—
74	C₂²×C₄	C₂×C₄×C₁₈₁	1	1	yes	characteristic
75	C₂²×C₄	C₂×C₄×C₁₈₁	1	1	yes	characteristic
76	C₂²×C₄	C₂×C₄×C₁₈₁	1	2	yes	—
77	C₂²×C₄	C₂×C₄×C₁₈₁	1	2	yes	—
78	C₂²×C₄	C₂³×C₁₈₁	1	1	yes	characteristic
79	C₂×C₈	—	2	4	no	—
80	C₄⋊C₄	—	2	2	no	—
81	C₄⋊C₄	—	2	4	no	—
82	C₄²	—	2	2	no	—
83	C₄²	—	2	2	no	—
84	C₄²	D₄×C₁₈₁	1	2	yes	—
85	C₄²	D₄×C₁₈₁	1	2	yes	—
86	C₂³	C₂²×C₄×C₁₈₁	1	1	yes	characteristic
87	Q₈	—	2	8	no	—
88	C₂×C₄	—	2	2	no	—
89	C₂×C₄	—	2	2	no	—
90	C₂×C₄	—	2	2	no	—
91	C₂×C₄	—	2	2	no	—
92	C₂×C₄	—	2	2	no	—
93	C₂×C₄	—	4	1	no	—
94	C₂×C₄	—	4	2	no	—
95	C₂×C₄	C₂²⋊C₄×C₁₈₁	1	1	yes	characteristic
96	C₂×C₄	C₂²⋊C₄×C₁₈₁	1	2	yes	—
97	C₂×C₄	C₂²⋊C₄×C₁₈₁	1	1	yes	characteristic
98	C₂×C₄	M₁₆×C₁₈₁	1	4	yes	—
99	C₂×C₄	C₂×D₄×C₁₈₁	1	2	yes	—
100	C₈	—	4	4	no	—
101	C₂²	—	2	1	no	—
102	C₂²	—	2	1	no	—
103	C₂²	C₂×C₂²⋊C₄×C₁₈₁	1	1	yes	characteristic
104	C₂²	C₂×M₁₆×C₁₈₁	1	2	yes	—
105	C₄	—	2	2	no	—
106	C₄	—	2	4	no	—
107	C₄	—	2	2	no	—
108	C₄	—	2	1	no	—
109	C₄	—	4	2	no	—
110	C₄	#101×C₁₈₁	1	2	yes	—
111	C₂	—	2	2	no	—
112	C₂	#407×C₁₈₁	1	1	yes	characteristic
113	C₂	#410×C₁₈₁	1	1	yes	characteristic
114	C₂	C₂×#101×C₁₈₁	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×Q₈×C₁₈₁

C₂

yes

maximal, characteristic

#432×C₁₈₁

C₂

yes

maximal

#324×C₁₈₁

C₂

yes

maximal

C₂²×Q₈×C₁₈₁

C₄

yes

characteristic

C₄×Q₈×C₁₈₁

—

C₄×Q₈×C₁₈₁

—

C₂×C₄⋊C₄×C₁₈₁

C₄

yes

characteristic

C₂×C₄⋊C₄×C₁₈₁

C₂²

yes

—

C₂×C₄²×C₁₈₁

C₄

yes

—

C₂×C₄²×C₁₈₁

C₂²

yes

characteristic

C₄⋊C₈×C₁₈₁

—

C₂²⋊C₈×C₁₈₁

C₂²

yes

—

#97×C₁₈₁

—

C₂×Q₈×C₁₈₁

—

C₂×Q₈×C₁₈₁

—

C₂²×C₄×C₁₈₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₈₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₈₁

C₂×C₄

yes

—

C₂²×C₄×C₁₈₁

C₂×C₄

yes

—

C₂²×C₄×C₁₈₁

C₂³

yes

characteristic

C₂×C₈×C₁₈₁

—

C₄⋊C₄×C₁₈₁

—

C₄⋊C₄×C₁₈₁

—

C₄²×C₁₈₁

—

C₄²×C₁₈₁

—

C₄²×C₁₈₁

D₄

yes

—

C₄²×C₁₈₁

D₄

yes

—

C₂³×C₁₈₁

C₂²×C₄

yes

characteristic

Q₈×C₁₈₁

—

C₂×C₄×C₁₈₁

—

C₂×C₄×C₁₈₁

—

C₂×C₄×C₁₈₁

—

C₂×C₄×C₁₈₁

—

C₂×C₄×C₁₈₁

—

C₂×C₄×C₁₈₁

—

C₂×C₄×C₁₈₁

—

C₂×C₄×C₁₈₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₈₁

C₂²⋊C₄

yes

—

C₂×C₄×C₁₈₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₈₁

M₁₆

yes

—

C₂×C₄×C₁₈₁

C₂×D₄

yes

—

C₈×C₁₈₁

—

C₂²×C₁₈₁

—

C₂²×C₁₈₁

—

C₂²×C₁₈₁

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₈₁

C₂×M₁₆

yes

—

C₄×C₁₈₁

—

C₄×C₁₈₁

—

C₄×C₁₈₁

—

C₄×C₁₈₁

—

C₄×C₁₈₁

—

C₄×C₁₈₁

#101

yes

—

C₂×C₁₈₁

—

C₂×C₁₈₁

yes

C₂×C₁₈₁

yes

C₂×C₁₈₁

yes

C₁₈₁

yes

characteristic, direct product factor, Sylow

#1467

C₁₈₁

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₄×Q₈

C₂×C₁₈₁

yes

characteristic

#432

C₂×C₁₈₁

yes

—

#324

C₂×C₁₈₁

yes

—

C₂²×Q₈

C₄×C₁₈₁

yes

characteristic

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

C₄×C₁₈₁

yes

characteristic

C₂×C₄⋊C₄

C₂²×C₁₈₁

yes

—

C₂×C₄²

C₄×C₁₈₁

yes

—

C₂×C₄²

C₂²×C₁₈₁

yes

characteristic

C₄⋊C₈

—

C₂²⋊C₈

C₂²×C₁₈₁

yes

—

#97

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

C₂×C₄×C₁₈₁

yes

characteristic

C₂²×C₄

C₂×C₄×C₁₈₁

yes

characteristic

C₂²×C₄

C₂×C₄×C₁₈₁

yes

—

C₂²×C₄

C₂×C₄×C₁₈₁

yes

—

C₂²×C₄

C₂³×C₁₈₁

yes

characteristic

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₄²

D₄×C₁₈₁

yes

—

C₄²

D₄×C₁₈₁

yes

—

C₂³

C₂²×C₄×C₁₈₁

yes

characteristic

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²⋊C₄×C₁₈₁

yes

characteristic

C₂×C₄

C₂²⋊C₄×C₁₈₁

yes

—

C₂×C₄

C₂²⋊C₄×C₁₈₁

yes

characteristic

C₂×C₄

M₁₆×C₁₈₁

yes

—

C₂×C₄

C₂×D₄×C₁₈₁

yes

—

100

C₈

—

101

C₂²

—

102

C₂²

—

103

C₂²

C₂×C₂²⋊C₄×C₁₈₁

yes

characteristic

104

C₂²

C₂×M₁₆×C₁₈₁

yes

—

105

C₄

—

106

C₄

—

107

C₄

—

108

C₄

—

109

C₄

—

110

C₄

#101×C₁₈₁

yes

—

111

C₂

—

112

C₂

#407×C₁₈₁

yes

characteristic

113

C₂

#410×C₁₈₁

yes

characteristic

114

C₂

C₂×#101×C₁₈₁

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₁₈₁	C₂×C₂²⋊C₄
commutator	C₂³	C₂²×C₄×C₁₈₁
Frattini, cosocle	C₂²×C₄	C₂³×C₁₈₁
prime radical	C₂³×C₁₈₁	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₈₁ ◁ C₂×C₄²×C₁₈₁ ◁ G

subgroup

G/H

center, socle

C₂²×C₁₈₁

C₂×C₂²⋊C₄

commutator

C₂³

C₂²×C₄×C₁₈₁

Frattini, cosocle

C₂²×C₄

C₂³×C₁₈₁

prime radical

C₂³×C₁₈₁

C₂²×C₄

LCS

G ▷ C₂³ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₁₈₁ ◁ C₂×C₄²×C₁₈₁ ◁ G

N	H ≅ G/N	type
57. C₁₈₁	#1467	direct (N × H)
58. #1467	C₁₈₁	direct (N × H)

H ≅ G/N

type

57. C₁₈₁

#1467

direct (N × H)

58. #1467

C₁₈₁

direct (N × H)

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

subgroup	index
C₂×C₄×Q₈×C₁₈₁	2
#432×C₁₈₁	2
#324×C₁₈₁	2
#1467	181

subgroup

index

C₂×C₄×Q₈×C₁₈₁

#432×C₁₈₁

#324×C₁₈₁

#1467

181

Subgroups — #1467×C₁₈₁ #37529410

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1467×C181 #37529410

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1467×C₁₈₁ #37529410