Group #37533399 — subgroups

Subgroups — #37533399

Order 23168.

176 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 65 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	#26041682	C₂	1	1	yes	maximal, characteristic
2	#26041677	C₂	1	1	yes	maximal, characteristic
3	#26041658	C₂	1	1	yes	maximal, characteristic
4	#26041631	C₂	1	1	yes	maximal, characteristic
5	#26041629	C₂	1	1	yes	maximal, characteristic
6	#26041000	C₂	1	1	yes	maximal, characteristic
7	#500×C₁₈₁	C₂	1	1	yes	maximal, characteristic
8	C₂×#15668439	—	2	2	no	—
9	#19153816	C₂²	1	1	yes	characteristic
10	#19153816	C₂²	1	1	yes	characteristic
11	#19153749	—	2	2	no	—
12	#19153748	—	2	2	no	—
13	#19153740	—	2	2	no	—
14	#19153739	—	2	2	no	—
15	C₄×#13089678	C₂²	1	1	yes	characteristic
16	C₄×#13089678	C₂²	1	1	yes	characteristic
17	C₄⋊Q₈×C₁₈₁	C₂²	1	1	yes	characteristic
18	C₄⋊Q₈×C₁₈₁	C₂²	1	1	yes	characteristic
19	#107×C₁₈₁	—	2	2	no	—
20	C₄×C₈×C₁₈₁	C₂²	1	1	yes	characteristic
21	C₂×Dic₃₆₂	—	2	1	no	—
22	C₂×Dic₃₆₂	—	2	1	no	—
23	#15668439	—	4	4	no	—
24	C₂×#13089678	D₄	1	2	yes	—
25	C₂×#13089678	D₄	1	2	yes	—
26	C₄⋊Dic₁₈₁	—	2	1	no	—
27	C₄⋊Dic₁₈₁	—	2	1	no	—
28	C₄⋊Dic₁₈₁	—	2	1	no	—
29	C₄⋊Dic₁₈₁	—	2	1	no	—
30	C₂×Q₈×C₁₈₁	—	2	1	no	—
31	C₂×Q₈×C₁₈₁	—	2	1	no	—
32	C₂×C₈×C₁₈₁	D₄	1	2	yes	—
33	C₄⋊C₄×C₁₈₁	—	2	1	no	—
34	C₄⋊C₄×C₁₈₁	—	2	1	no	—
35	C₄⋊C₄×C₁₈₁	—	2	1	no	—
36	C₄⋊C₄×C₁₈₁	—	2	1	no	—
37	C₄²×C₁₈₁	C₂³	1	1	yes	characteristic
38	Dic₃₆₂	—	4	2	no	—
39	Dic₃₆₂	—	4	2	no	—
40	C₂×Dic₁₈₁	—	4	1	no	—
41	C₂×Dic₁₈₁	—	4	1	no	—
42	#13089678	—	2	2	no	—
43	#13089678	—	2	2	no	—
44	Q₈×C₁₈₁	—	4	2	no	—
45	Q₈×C₁₈₁	—	4	2	no	—
46	C₂×C₄×C₁₈₁	—	4	1	no	—
47	C₂×C₄×C₁₈₁	—	4	1	no	—
48	C₂×C₄×C₁₈₁	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₁₈₁	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₁₈₁	C₂×D₄	1	1	yes	characteristic
51	C₈×C₁₈₁	Q₁₆	1	4	yes	complemented
52	C₂²×C₁₈₁	#127	1	1	yes	characteristic
53	C₄×C₁₈₁	—	8	1	no	—
54	C₄×C₁₈₁	—	8	1	no	—
55	C₄×C₁₈₁	C₂×Q₁₆	1	2	yes	—
56	C₄×C₁₈₁	AGL(1,ℤ/8)	1	2	yes	—
57	C₄×C₁₈₁	#137	1	2	yes	—
58	Dic₁₈₁	—	8	1	no	—
59	Dic₁₈₁	—	8	1	no	—
60	C₂×C₁₈₁	#494	1	1	yes	characteristic
61	C₂×C₁₈₁	#496	1	1	yes	characteristic
62	C₂×C₁₈₁	#497	1	1	yes	characteristic
63	C₁₈₁	#1723	1	1	yes	characteristic, complemented, Sylow
64	#1723	—	181	1	no	maximal, Sylow
65	#500	—	181	1	no	—
66	#500	D₁₈₁	1	1	yes	characteristic
67	#498	—	181	1	no	—
68	#494	—	181	1	no	—
69	#487	—	181	1	no	—
70	#487	—	181	1	no	—
71	#322	—	181	1	no	—
72	C₂×Q₁₆	—	362	2	no	—
73	C₄⋊Q₈	—	181	1	no	—
74	C₄⋊Q₈	—	181	1	no	—
75	C₄⋊Q₈	C₂×D₁₈₁	1	1	yes	characteristic
76	C₄⋊Q₈	C₂×D₁₈₁	1	1	yes	characteristic
77	#107	—	2	2	no	—
78	#107	—	362	2	no	—
79	#106	—	362	2	no	—
80	#103	—	362	2	no	—
81	#103	—	362	2	no	—
82	C₄×C₈	—	181	1	no	—
83	C₄×C₈	—	181	1	no	—
84	C₄×C₈	C₂×D₁₈₁	1	1	yes	characteristic
85	C₂×Q₈	—	2	1	no	—
86	C₂×Q₈	—	2	1	no	—
87	C₂×Q₈	—	362	1	no	—
88	C₂×Q₈	—	362	1	no	—
89	Q₁₆	—	724	4	no	—
90	C₂×C₈	—	181	2	no	—
91	C₂×C₈	—	181	2	no	—
92	C₂×C₈	C₂²⋊D₁₈₁	1	2	yes	—
93	C₄⋊C₄	—	2	1	no	—
94	C₄⋊C₄	—	2	1	no	—
95	C₄⋊C₄	—	2	1	no	—
96	C₄⋊C₄	—	2	1	no	—
97	C₄⋊C₄	—	362	1	no	—
98	C₄⋊C₄	—	362	1	no	—
99	C₄⋊C₄	—	362	1	no	—
100	C₄⋊C₄	—	362	1	no	—
101	C₄²	C₂²×D₁₈₁	1	1	yes	characteristic
102	Q₈	—	4	2	no	—
103	Q₈	—	4	2	no	—
104	Q₈	—	724	2	no	—
105	Q₈	—	724	2	no	—
106	C₂×C₄	—	4	1	no	—
107	C₂×C₄	—	4	1	no	—
108	C₂×C₄	—	724	1	no	—
109	C₂×C₄	—	724	1	no	—
110	C₂×C₄	D₄×D₁₈₁	1	1	yes	characteristic
111	C₂×C₄	D₄×D₁₈₁	1	1	yes	characteristic
112	C₂×C₄	C₂×C₂²⋊D₁₈₁	1	1	yes	characteristic
113	C₈	—	362	2	no	—
114	C₈	—	362	2	no	—
115	C₈	#15668439	1	4	yes	complemented
116	C₂²	#19153812	1	1	yes	characteristic
117	C₄	—	8	1	no	—
118	C₄	—	8	1	no	—
119	C₄	—	1448	1	no	—
120	C₄	—	1448	1	no	—
121	C₄	C₂×#15668439	1	2	yes	—
122	C₄	#19153840	1	2	yes	—
123	C₄	#19153848	1	2	yes	—
124	C₂	#26041658	1	1	yes	characteristic
125	C₂	#26041667	1	1	yes	characteristic
126	C₂	#26041672	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26041682

C₂

yes

maximal, characteristic

#26041677

C₂

yes

maximal, characteristic

#26041658

C₂

yes

maximal, characteristic

#26041631

C₂

yes

maximal, characteristic

#26041629

C₂

yes

maximal, characteristic

#26041000

C₂

yes

maximal, characteristic

#500×C₁₈₁

C₂

yes

maximal, characteristic

C₂×#15668439

—

#19153816

C₂²

yes

characteristic

#19153816

C₂²

yes

—

—

—

—

C₂²

yes

characteristic

C₄×#13089678

C₂²

yes

characteristic

C₄⋊Q₈×C₁₈₁

C₂²

yes

characteristic

C₄⋊Q₈×C₁₈₁

C₂²

yes

characteristic

#107×C₁₈₁

—

C₄×C₈×C₁₈₁

C₂²

yes

characteristic

C₂×Dic₃₆₂

—

C₂×Dic₃₆₂

—

#15668439

—

C₂×#13089678

D₄

yes

—

C₂×#13089678

D₄

yes

—

C₄⋊Dic₁₈₁

—

C₄⋊Dic₁₈₁

—

C₄⋊Dic₁₈₁

—

C₄⋊Dic₁₈₁

—

C₂×Q₈×C₁₈₁

—

C₂×Q₈×C₁₈₁

—

C₂×C₈×C₁₈₁

D₄

yes

—

C₄⋊C₄×C₁₈₁

—

C₄⋊C₄×C₁₈₁

—

C₄⋊C₄×C₁₈₁

—

C₄⋊C₄×C₁₈₁

—

C₄²×C₁₈₁

C₂³

yes

characteristic

Dic₃₆₂

—

Dic₃₆₂

—

C₂×Dic₁₈₁

—

C₂×Dic₁₈₁

—

#13089678

—

#13089678

—

Q₈×C₁₈₁

—

Q₈×C₁₈₁

—

C₂×C₄×C₁₈₁

—

C₂×C₄×C₁₈₁

—

C₂×C₄×C₁₈₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₈₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₈₁

C₂×D₄

yes

characteristic

C₈×C₁₈₁

Q₁₆

yes

complemented

C₂²×C₁₈₁

#127

yes

characteristic

C₄×C₁₈₁

—

C₄×C₁₈₁

—

C₄×C₁₈₁

C₂×Q₁₆

yes

—

C₄×C₁₈₁

AGL(1,ℤ/8)

yes

—

C₄×C₁₈₁

#137

yes

—

Dic₁₈₁

—

Dic₁₈₁

—

C₂×C₁₈₁

yes

C₂×C₁₈₁

yes

C₂×C₁₈₁

yes

C₁₈₁

yes

characteristic, complemented, Sylow

#1723

—

181

maximal, Sylow

#500

—

181

—

#500

D₁₈₁

yes

characteristic

#498

—

181

—

#494

—

181

—

#487

—

181

—

#487

—

181

—

#322

—

181

—

C₂×Q₁₆

—

362

—

C₄⋊Q₈

—

181

—

C₄⋊Q₈

—

181

—

C₄⋊Q₈

C₂×D₁₈₁

yes

characteristic

C₄⋊Q₈

C₂×D₁₈₁

yes

characteristic

#107

—

#107

—

362

—

#106

—

362

—

#103

—

362

—

#103

—

362

—

C₄×C₈

—

181

—

C₄×C₈

—

181

—

C₄×C₈

C₂×D₁₈₁

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

362

—

C₂×Q₈

—

362

—

Q₁₆

—

724

—

C₂×C₈

—

181

—

C₂×C₈

—

181

—

C₂×C₈

C₂²⋊D₁₈₁

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

362

—

C₄⋊C₄

—

362

—

C₄⋊C₄

—

362

—

100

C₄⋊C₄

—

362

—

101

C₄²

C₂²×D₁₈₁

yes

characteristic

102

Q₈

—

103

Q₈

—

104

Q₈

—

724

—

105

Q₈

—

724

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

724

—

109

C₂×C₄

—

724

—

110

C₂×C₄

D₄×D₁₈₁

yes

characteristic

111

C₂×C₄

D₄×D₁₈₁

yes

characteristic

112

C₂×C₄

C₂×C₂²⋊D₁₈₁

yes

characteristic

113

C₈

—

362

—

114

C₈

—

362

—

115

C₈

#15668439

yes

complemented

116

C₂²

#19153812

yes

characteristic

117

C₄

—

118

C₄

—

119

C₄

—

1448

—

120

C₄

—

1448

—

121

C₄

C₂×#15668439

yes

—

122

C₄

#19153840

yes

—

123

C₄

#19153848

yes

—

124

C₂

#26041658

yes

characteristic

125

C₂

#26041667

yes

characteristic

126

C₂

#26041672

yes

characteristic

	subgroup	G/H
center	C₂²	#19153812
commutator, cosocle	C₄²×C₁₈₁	C₂³
Frattini	C₄²	C₂²×D₁₈₁
Fitting	#500×C₁₈₁	C₂
socle, prime radical	C₂²×C₁₈₁	#127
hypercenter	#500	D₁₈₁
hypocenter	C₁₈₁	#1723
LCS	G ▷ C₄²×C₁₈₁ ▷ C₂²×C₁₈₁ ▷ C₁₈₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #500

subgroup

G/H

center

C₂²

#19153812

commutator, cosocle

C₄²×C₁₈₁

C₂³

Frattini

C₄²

C₂²×D₁₈₁

Fitting

#500×C₁₈₁

C₂

C₂²×C₁₈₁

D₁₈₁

C₁₈₁

G ▷ C₄²×C₁₈₁ ▷ C₂²×C₁₈₁ ▷ C₁₈₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #500

N	H ≅ G/N	type
51. C₈×C₁₈₁	Q₁₆	semidirect
63. C₁₈₁	#1723	semidirect
115. C₈	#15668439	semidirect

H ≅ G/N

type

51. C₈×C₁₈₁

Q₁₆

semidirect

63. C₁₈₁

#1723

semidirect

115. C₈

#15668439

semidirect

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

subgroup	index
#26041682	2
#26041677	2
#26041658	2
#26041631	2
#26041629	2
#26041000	2
#500×C₁₈₁	2
#1723	181

subgroup

index

#500×C₁₈₁

181

Subgroups — #37533399

Conjugacy classes of subgroups

Complements

Maximal subgroups