Group #37550138 — subgroups

Subgroups — #37550138

Order 23168.

126 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 126 conjugacy classes of subgroups — too many to draw. Only its 64 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	#26042200	C₂	1	1	yes	maximal, characteristic
2	#26042200	C₂	1	1	yes	maximal, characteristic
3	C₄⋊#15668452	C₂	1	1	yes	maximal, characteristic
4	#26042168	C₂	1	1	yes	maximal, characteristic
5	#26042167	C₂	1	1	yes	maximal, characteristic
6	#26042165	C₂	1	1	yes	maximal, characteristic
7	#26041403	C₂	1	1	yes	maximal, characteristic
8	C₂×#15668452	C₂²	1	1	yes	characteristic
9	C₂×#15668452	C₂²	1	1	yes	characteristic
10	C₂×#15668452	C₂²	1	1	yes	characteristic
11	C₂×#15668452	C₂²	1	1	yes	characteristic
12	C₄×Dic₃₆₂	C₄	1	1	yes	characteristic
13	#19153745	C₄	1	1	yes	characteristic
14	#19153745	C₄	1	1	yes	characteristic
15	C₄×#13089678	C₂²	1	1	yes	characteristic
16	C₈×Dic₁₈₁	C₂²	1	1	yes	characteristic
17	C₈×Dic₁₈₁	C₂²	1	1	yes	characteristic
18	C₄⋊C₈×C₁₈₁	C₄	1	1	yes	characteristic
19	#15668452	—	2	1	no	—
20	#15668452	—	2	1	no	—
21	#15668452	—	2	1	no	—
22	#15668452	Q₈	1	2	yes	complemented
23	C₂×Dic₃₆₂	C₈	1	1	yes	characteristic
24	C₂×#13089678	C₂×C₄	1	1	yes	characteristic
25	C₂×#13089678	C₂³	1	1	yes	characteristic
26	C₄⋊Dic₁₈₁	C₈	1	1	yes	characteristic
27	#15668428	C₈	1	1	yes	characteristic
28	#15668428	C₈	1	1	yes	characteristic
29	C₄×Dic₁₈₁	C₂×C₄	1	1	yes	characteristic
30	C₄×Dic₁₈₁	C₂×C₄	1	1	yes	characteristic
31	C₂×C₈×C₁₈₁	C₂×C₄	1	1	yes	characteristic
32	C₂×C₈×C₁₈₁	C₂×C₄	1	1	yes	characteristic
33	C₄²×C₁₈₁	C₂×C₄	1	1	yes	characteristic
34	Dic₃₆₂	—	2	2	no	—
35	C₂×Dic₁₈₁	C₂×C₈	1	1	yes	characteristic
36	C₂×Dic₁₈₁	C₂×C₈	1	1	yes	characteristic
37	C₂×Dic₁₈₁	C₂×C₈	1	1	yes	characteristic
38	C₂×Dic₁₈₁	C₂×C₈	1	1	yes	characteristic
39	#13089678	—	2	1	no	—
40	#13089678	C₂×Q₈	1	1	yes	characteristic
41	#13089678	#41	1	1	yes	characteristic
42	C₂×C₄×C₁₈₁	C₂×C₈	1	1	yes	characteristic
43	C₂×C₄×C₁₈₁	C₂×C₈	1	1	yes	characteristic
44	C₂×C₄×C₁₈₁	C₂²×C₄	1	1	yes	characteristic
45	C₈×C₁₈₁	—	2	1	no	—
46	C₈×C₁₈₁	—	2	1	no	—
47	C₂²×C₁₈₁	C₂²×C₈	1	1	yes	characteristic
48	C₄×C₁₈₁	—	2	1	no	—
49	C₄×C₁₈₁	—	2	1	no	—
50	C₄×C₁₈₁	C₄×Q₈	1	1	yes	characteristic
51	C₄×C₁₈₁	#131	1	1	yes	characteristic
52	Dic₁₈₁	—	2	1	no	—
53	Dic₁₈₁	—	2	1	no	—
54	Dic₁₈₁	—	2	1	no	—
55	Dic₁₈₁	M₃₂	1	2	yes	—
56	C₂×C₁₈₁	C₈×Q₈	1	1	yes	characteristic
57	C₂×C₁₈₁	C₂×M₃₂	1	1	yes	characteristic
58	C₂×C₁₈₁	#504	1	1	yes	characteristic
59	C₁₈₁	#2183	1	1	yes	characteristic, complemented, Sylow
60	#2183	—	181	1	no	maximal, Sylow
61	C₈×Q₈	—	181	1	no	—
62	C₄⋊C₁₆	—	181	1	no	—
63	C₄⋊C₁₆	—	181	1	no	—
64	C₄⋊C₁₆	—	181	1	no	—
65	#346	—	181	1	no	—
66	#346	—	181	1	no	—
67	C₄×C₁₆	—	181	1	no	—
68	C₄×Q₈	—	181	1	no	—
69	C₂×C₁₆	—	181	1	no	—
70	C₂×C₁₆	—	181	1	no	—
71	C₂×C₁₆	—	181	1	no	—
72	C₂×C₁₆	—	181	1	no	—
73	C₄⋊C₈	—	181	1	no	—
74	C₄⋊C₈	—	181	1	no	—
75	C₄⋊C₈	#10624382	1	1	yes	characteristic
76	C₄×C₈	—	181	1	no	—
77	C₄×C₈	—	181	1	no	—
78	C₄×C₈	—	181	1	no	—
79	C₂×Q₈	—	181	1	no	—
80	C₂×C₈	—	181	1	no	—
81	C₂×C₈	—	181	1	no	—
82	C₂×C₈	C₂×#10624382	1	1	yes	characteristic
83	C₂×C₈	C₂×#10624382	1	1	yes	characteristic
84	C₄⋊C₄	—	181	1	no	—
85	C₄⋊C₄	—	181	1	no	—
86	C₄⋊C₄	—	181	1	no	—
87	C₄²	—	181	1	no	—
88	C₄²	—	181	1	no	—
89	C₄²	C₂×#10624382	1	1	yes	characteristic
90	C₁₆	—	362	1	no	—
91	C₁₆	—	362	1	no	—
92	C₁₆	—	362	1	no	—
93	C₁₆	—	181	2	no	—
94	Q₈	—	362	2	no	—
95	C₂×C₄	—	181	1	no	—
96	C₂×C₄	—	181	1	no	—
97	C₂×C₄	—	181	1	no	—
98	C₂×C₄	—	181	1	no	—
99	C₂×C₄	D₁₈₁⋊C₈	1	1	yes	characteristic
100	C₂×C₄	D₁₈₁⋊C₈	1	1	yes	characteristic
101	C₂×C₄	C₂²×#10624382	1	1	yes	characteristic
102	C₈	—	2	1	no	—
103	C₈	—	2	1	no	—
104	C₈	—	362	1	no	—
105	C₈	—	181	1	no	—
106	C₈	—	181	1	no	—
107	C₂²	C₂×D₁₈₁⋊C₈	1	1	yes	characteristic
108	C₄	—	2	1	no	—
109	C₄	—	2	1	no	—
110	C₄	—	362	1	no	—
111	C₄	—	362	1	no	—
112	C₄	—	362	1	no	—
113	C₄	—	181	2	no	—
114	C₄	Q₈×#10624382	1	1	yes	characteristic
115	C₄	#19153903	1	1	yes	characteristic
116	C₂	#26042328	1	1	yes	characteristic
117	C₂	D₁₈₁⋊M₃₂	1	1	yes	characteristic
118	C₂	#26042338	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26042200

C₂

yes

maximal, characteristic

#26042200

C₂

yes

maximal, characteristic

C₄⋊#15668452

C₂

yes

maximal, characteristic

#26042168

C₂

yes

maximal, characteristic

#26042167

C₂

yes

maximal, characteristic

#26042165

C₂

yes

maximal, characteristic

#26041403

C₂

yes

maximal, characteristic

C₂×#15668452

C₂²

yes

characteristic

C₂×#15668452

C₂²

yes

characteristic

C₂×#15668452

C₂²

yes

characteristic

C₂×#15668452

C₂²

yes

characteristic

C₄×Dic₃₆₂

C₄

yes

characteristic

#19153745

C₄

yes

characteristic

#19153745

C₄

yes

characteristic

C₄×#13089678

C₂²

yes

characteristic

C₈×Dic₁₈₁

C₂²

yes

characteristic

C₈×Dic₁₈₁

C₂²

yes

characteristic

C₄⋊C₈×C₁₈₁

C₄

yes

—

—

—

Q₈

yes

complemented

C₂×Dic₃₆₂

C₈

yes

characteristic

C₂×#13089678

C₂×C₄

yes

characteristic

C₂×#13089678

C₂³

yes

characteristic

C₄⋊Dic₁₈₁

C₈

yes

characteristic

#15668428

C₈

yes

characteristic

#15668428

C₈

yes

characteristic

C₄×Dic₁₈₁

C₂×C₄

yes

characteristic

C₄×Dic₁₈₁

C₂×C₄

yes

characteristic

C₂×C₈×C₁₈₁

C₂×C₄

yes

characteristic

C₂×C₈×C₁₈₁

C₂×C₄

yes

characteristic

C₄²×C₁₈₁

C₂×C₄

yes

characteristic

Dic₃₆₂

—

C₂×Dic₁₈₁

C₂×C₈

yes

characteristic

C₂×Dic₁₈₁

C₂×C₈

yes

characteristic

C₂×Dic₁₈₁

C₂×C₈

yes

characteristic

C₂×Dic₁₈₁

C₂×C₈

yes

characteristic

#13089678

—

#13089678

C₂×Q₈

yes

yes

C₂×C₄×C₁₈₁

C₂×C₈

yes

characteristic

C₂×C₄×C₁₈₁

C₂×C₈

yes

characteristic

C₂×C₄×C₁₈₁

C₂²×C₄

yes

characteristic

C₈×C₁₈₁

—

C₈×C₁₈₁

—

C₂²×C₁₈₁

C₂²×C₈

yes

characteristic

C₄×C₁₈₁

—

C₄×C₁₈₁

—

C₄×C₁₈₁

C₄×Q₈

yes

characteristic

C₄×C₁₈₁

#131

yes

characteristic

Dic₁₈₁

—

Dic₁₈₁

—

Dic₁₈₁

—

Dic₁₈₁

M₃₂

yes

—

C₂×C₁₈₁

C₈×Q₈

yes

characteristic

C₂×C₁₈₁

C₂×M₃₂

yes

C₂×C₁₈₁

yes

C₁₈₁

yes

characteristic, complemented, Sylow

#2183

—

181

maximal, Sylow

C₈×Q₈

—

181

—

C₄⋊C₁₆

—

181

—

C₄⋊C₁₆

—

181

—

C₄⋊C₁₆

—

181

—

#346

—

181

—

#346

—

181

—

C₄×C₁₆

—

181

—

C₄×Q₈

—

181

—

C₂×C₁₆

—

181

—

C₂×C₁₆

—

181

—

C₂×C₁₆

—

181

—

C₂×C₁₆

—

181

—

C₄⋊C₈

—

181

—

C₄⋊C₈

—

181

—

C₄⋊C₈

#10624382

yes

characteristic

C₄×C₈

—

181

—

C₄×C₈

—

181

—

C₄×C₈

—

181

—

C₂×Q₈

—

181

—

C₂×C₈

—

181

—

C₂×C₈

—

181

—

C₂×C₈

yes

C₂×C₈

yes

C₄⋊C₄

—

181

—

C₄⋊C₄

—

181

—

C₄⋊C₄

—

181

—

C₄²

—

181

—

C₄²

—

181

—

C₄²

C₂×#10624382

yes

characteristic

C₁₆

—

362

—

C₁₆

—

362

—

C₁₆

—

362

—

C₁₆

—

181

—

Q₈

—

362

—

C₂×C₄

—

181

—

C₂×C₄

—

181

—

C₂×C₄

—

181

—

C₂×C₄

—

181

—

C₂×C₄

D₁₈₁⋊C₈

yes

characteristic

100

C₂×C₄

D₁₈₁⋊C₈

yes

characteristic

101

C₂×C₄

C₂²×#10624382

yes

characteristic

102

C₈

—

103

C₈

—

104

C₈

—

362

—

105

C₈

—

181

—

106

C₈

—

181

—

107

C₂²

C₂×D₁₈₁⋊C₈

yes

characteristic

108

C₄

—

109

C₄

—

110

C₄

—

362

—

111

C₄

—

362

—

112

C₄

—

362

—

113

C₄

—

181

—

114

C₄

Q₈×#10624382

yes

characteristic

115

C₄

#19153903

yes

characteristic

116

C₂

#26042328

yes

characteristic

117

C₂

D₁₈₁⋊M₃₂

yes

characteristic

118

C₂

#26042338

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×#10624382
commutator, socle, prime radical	C₂²×C₁₈₁	C₂²×C₈
Fitting	C₄⋊C₈×C₁₈₁	C₄
cosocle	C₂×#13089678	C₂³
hypercenter	C₄⋊C₈	#10624382
hypocenter	C₁₈₁	#2183
LCS	G ▷ C₂²×C₁₈₁ ▷ C₁₈₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄⋊C₈

subgroup

G/H

center, Frattini

C₂×C₄

C₂²×#10624382

commutator, socle, prime radical

C₂²×C₁₈₁

C₂²×C₈

Fitting

C₄⋊C₈×C₁₈₁

C₄

C₂³

C₄⋊C₈

C₁₈₁

G ▷ C₂²×C₁₈₁ ▷ C₁₈₁

C₁ ◁ C₂×C₄ ◁ C₄⋊C₈

N	H ≅ G/N	type
22. #15668452	Q₈	semidirect
59. C₁₈₁	#2183	semidirect

H ≅ G/N

type

22. #15668452

Q₈

semidirect

59. C₁₈₁

#2183

semidirect

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

subgroup	index
#26042200	2
C₄⋊#15668452	2
#26042168	2
#26042167	2
#26042165	2
#26041403	2
#2183	181

subgroup

index

181

Subgroups — #37550138

Conjugacy classes of subgroups

Complements

Maximal subgroups