Group #37548587 — subgroups

Subgroups — #37548587

Order 23168.

138 conjugacy classes of subgroups; only the 44 normal subgroups are shown.

normal subgroups only

This group has 138 conjugacy classes of subgroups — too many to draw. Only its 44 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	#26042301	C₂	1	1	yes	maximal, characteristic
2	C₄×#15668459	C₂	1	1	yes	maximal, characteristic
3	#26041335	C₂	1	1	yes	maximal, characteristic
4	C₂×#15668459	—	2	1	no	—
5	C₄⋊#13089685	—	2	1	no	—
6	#19153873	C₄	1	2	yes	—
7	#19153871	—	2	1	no	—
8	#19153871	—	2	1	no	—
9	C₄×#13089685	C₄	1	2	yes	—
10	C₄²×D₁₈₁	C₂²	1	1	yes	characteristic
11	C₄⋊#13089678	C₄	1	1	yes	characteristic
12	#19153728	—	2	1	no	—
13	#19153726	—	2	1	no	—
14	C₄⋊C₈×C₁₈₁	C₄	1	1	yes	characteristic
15	#15668459	—	2	2	no	—
16	#15668459	—	2	2	no	—
17	C₂×#13089685	—	2	2	no	—
18	C₂×#13089685	—	2	2	no	—
19	C₂×C₄×D₁₈₁	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×D₁₈₁	D₄	1	1	yes	characteristic
21	C₂×C₄×D₁₈₁	Q₈	1	1	yes	characteristic
22	C₂×#13089678	—	2	1	no	—
23	C₄×Dic₁₈₁	C₂×C₄	1	1	yes	characteristic
24	C₄×Dic₁₈₁	D₄	1	1	yes	characteristic
25	C₄×Dic₁₈₁	D₄	1	1	yes	characteristic
26	C₂×C₈×C₁₈₁	—	2	1	no	—
27	C₄²×C₁₈₁	C₂×C₄	1	1	yes	characteristic
28	#13089685	—	2	4	no	—
29	#13089685	—	4	2	no	—
30	C₂²×D₁₈₁	C₄⋊C₄	1	1	yes	characteristic
31	C₂×Dic₁₈₁	C₂²⋊C₄	1	1	yes	characteristic
32	C₂×Dic₁₈₁	C₂²⋊C₄	1	1	yes	characteristic
33	C₂×Dic₁₈₁	C₄⋊C₄	1	1	yes	characteristic
34	C₄×D₁₈₁	—	2	1	no	—
35	C₄×D₁₈₁	—	2	1	no	—
36	C₄×D₁₈₁	—	4	1	no	—
37	C₄×D₁₈₁	—	4	1	no	—
38	#13089678	—	4	1	no	—
39	C₂×C₄×C₁₈₁	C₄²	1	1	yes	characteristic
40	C₂×C₄×C₁₈₁	C₂²⋊C₄	1	1	yes	characteristic
41	C₂×C₄×C₁₈₁	C₄⋊C₄	1	1	yes	characteristic
42	C₈×C₁₈₁	—	4	1	no	—
43	C₂²×C₁₈₁	#95	1	1	yes	characteristic
44	C₂×D₁₈₁	—	2	1	no	—
45	C₂×D₁₈₁	—	2	1	no	—
46	C₂×D₁₈₁	—	2	1	no	—
47	C₄×C₁₈₁	—	2	1	no	—
48	C₄×C₁₈₁	—	2	1	no	—
49	C₄×C₁₈₁	#104	1	1	yes	characteristic
50	C₄×C₁₈₁	#104	1	1	yes	characteristic
51	Dic₁₈₁	—	2	1	no	—
52	Dic₁₈₁	—	2	1	no	—
53	Dic₁₈₁	—	2	1	no	—
54	C₂×C₁₈₁	#338	1	1	yes	characteristic
55	C₂×C₁₈₁	#339	1	1	yes	characteristic
56	C₂×C₁₈₁	#344	1	1	yes	characteristic
57	D₁₈₁	—	4	1	no	—
58	C₁₈₁	#1291	1	1	yes	characteristic, complemented, Sylow
59	#1291	—	181	1	no	maximal, Sylow
60	#432	—	181	1	no	—
61	#432	—	181	1	no	—
62	C₄×M₁₆	—	181	1	no	—
63	C₂×M₁₆	—	362	1	no	—
64	C₂×C₄²	—	181	1	no	—
65	C₄⋊C₈	—	362	1	no	—
66	C₄⋊C₈	—	181	1	no	—
67	C₄⋊C₈	#10624382	1	1	yes	characteristic
68	C₂²⋊C₈	—	362	1	no	—
69	C₂²⋊C₈	—	181	2	no	—
70	#97	—	362	1	no	—
71	#97	—	362	1	no	—
72	#97	—	362	1	no	—
73	C₄×C₈	—	181	2	no	—
74	C₂²×C₄	—	181	1	no	—
75	C₂²×C₄	—	181	1	no	—
76	C₂²×C₄	—	181	1	no	—
77	M₁₆	—	362	2	no	—
78	M₁₆	—	362	2	no	—
79	C₂×C₈	—	2	1	no	—
80	C₂×C₈	—	362	2	no	—
81	C₂×C₈	—	362	2	no	—
82	C₂×C₈	—	362	1	no	—
83	C₄²	—	181	1	no	—
84	C₄²	—	181	1	no	—
85	C₄²	—	181	1	no	—
86	C₄²	C₂×#10624382	1	1	yes	characteristic
87	C₂³	—	181	1	no	—
88	C₂×C₄	—	362	1	no	—
89	C₂×C₄	—	362	1	no	—
90	C₂×C₄	—	181	1	no	—
91	C₂×C₄	—	181	1	no	—
92	C₂×C₄	—	181	1	no	—
93	C₂×C₄	—	724	1	no	—
94	C₂×C₄	—	724	1	no	—
95	C₂×C₄	C₄×#10624382	1	1	yes	characteristic
96	C₂×C₄	C₂²⋊#10624382	1	1	yes	characteristic
97	C₂×C₄	C₄⋊#10624382	1	1	yes	characteristic
98	C₈	—	4	1	no	—
99	C₈	—	362	4	no	—
100	C₈	—	724	2	no	—
101	C₈	—	724	1	no	—
102	C₂²	—	362	1	no	—
103	C₂²	—	362	1	no	—
104	C₂²	—	362	1	no	—
105	C₂²	(C₂×#10624382)⋊C₄	1	1	yes	characteristic
106	C₄	—	2	1	no	—
107	C₄	—	2	1	no	—
108	C₄	—	362	1	no	—
109	C₄	—	362	1	no	—
110	C₄	—	362	1	no	—
111	C₄	#19153880	1	1	yes	characteristic
112	C₄	#19153881	1	1	yes	characteristic
113	C₂	—	724	1	no	—
114	C₂	#26042148	1	1	yes	characteristic
115	C₂	#26042149	1	1	yes	characteristic
116	C₂	#26042162	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26042301

C₂

yes

maximal, characteristic

C₄×#15668459

C₂

yes

maximal, characteristic

#26041335

C₂

yes

maximal, characteristic

C₂×#15668459

—

C₄⋊#13089685

—

#19153873

C₄

yes

—

#19153871

—

#19153871

—

C₄×#13089685

C₄

yes

—

C₄²×D₁₈₁

C₂²

yes

characteristic

C₄⋊#13089678

C₄

yes

characteristic

#19153728

—

#19153726

—

C₄⋊C₈×C₁₈₁

C₄

yes

—

—

—

—

C₂×C₄×D₁₈₁

C₂×C₄

yes

characteristic

C₂×C₄×D₁₈₁

D₄

yes

characteristic

C₂×C₄×D₁₈₁

Q₈

yes

characteristic

C₂×#13089678

—

C₄×Dic₁₈₁

C₂×C₄

yes

characteristic

C₄×Dic₁₈₁

D₄

yes

characteristic

C₄×Dic₁₈₁

D₄

yes

characteristic

C₂×C₈×C₁₈₁

—

C₄²×C₁₈₁

C₂×C₄

yes

characteristic

#13089685

—

#13089685

—

C₂²×D₁₈₁

C₄⋊C₄

yes

characteristic

C₂×Dic₁₈₁

C₂²⋊C₄

yes

characteristic

C₂×Dic₁₈₁

C₂²⋊C₄

yes

characteristic

C₂×Dic₁₈₁

C₄⋊C₄

yes

characteristic

C₄×D₁₈₁

—

C₄×D₁₈₁

—

C₄×D₁₈₁

—

C₄×D₁₈₁

—

#13089678

—

C₂×C₄×C₁₈₁

C₄²

yes

characteristic

C₂×C₄×C₁₈₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₈₁

C₄⋊C₄

yes

characteristic

C₈×C₁₈₁

—

C₂²×C₁₈₁

#95

yes

characteristic

C₂×D₁₈₁

—

C₂×D₁₈₁

—

C₂×D₁₈₁

—

C₄×C₁₈₁

—

C₄×C₁₈₁

—

C₄×C₁₈₁

yes

C₄×C₁₈₁

yes

Dic₁₈₁

—

Dic₁₈₁

—

Dic₁₈₁

—

C₂×C₁₈₁

yes

C₂×C₁₈₁

yes

C₂×C₁₈₁

yes

D₁₈₁

—

C₁₈₁

#1291

yes

characteristic, complemented, Sylow

#1291

—

181

maximal, Sylow

#432

—

181

—

#432

—

181

—

C₄×M₁₆

—

181

—

C₂×M₁₆

—

362

—

C₂×C₄²

—

181

—

C₄⋊C₈

—

362

—

C₄⋊C₈

—

181

—

C₄⋊C₈

#10624382

yes

characteristic

C₂²⋊C₈

—

362

—

C₂²⋊C₈

—

181

—

#97

—

362

—

#97

—

362

—

#97

—

362

—

C₄×C₈

—

181

—

C₂²×C₄

—

181

—

C₂²×C₄

—

181

—

C₂²×C₄

—

181

—

M₁₆

—

362

—

M₁₆

—

362

—

C₂×C₈

—

C₂×C₈

—

362

—

C₂×C₈

—

362

—

C₂×C₈

—

362

—

C₄²

—

181

—

C₄²

—

181

—

C₄²

—

181

—

C₄²

C₂×#10624382

yes

characteristic

C₂³

—

181

—

C₂×C₄

—

362

—

C₂×C₄

—

362

—

C₂×C₄

—

181

—

C₂×C₄

—

181

—

C₂×C₄

—

181

—

C₂×C₄

—

724

—

C₂×C₄

—

724

—

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₈

—

C₈

—

362

—

100

C₈

—

724

—

101

C₈

—

724

—

102

C₂²

—

362

—

103

C₂²

—

362

—

104

C₂²

—

362

—

105

C₂²

(C₂×#10624382)⋊C₄

yes

characteristic

106

C₄

—

107

C₄

—

108

C₄

—

362

—

109

C₄

—

362

—

110

C₄

—

362

—

111

C₄

#19153880

yes

characteristic

112

C₄

#19153881

yes

characteristic

113

C₂

—

724

—

114

C₂

#26042148

yes

characteristic

115

C₂

#26042149

yes

characteristic

116

C₂

#26042162

yes

characteristic

	subgroup	G/H
center	C₂²	(C₂×#10624382)⋊C₄
commutator	C₂×C₄×C₁₈₁	C₄²
Frattini	C₄²	C₂×#10624382
Fitting	C₄⋊C₈×C₁₈₁	C₄
socle	C₂²×C₁₈₁	#95
prime radical	C₂²×D₁₈₁	C₄⋊C₄
cosocle	C₄²×D₁₈₁	C₂²
hypercenter	C₄⋊C₈	#10624382
hypocenter	C₁₈₁	#1291
LCS	G ▷ C₂×C₄×C₁₈₁ ▷ C₂×C₁₈₁ ▷ C₁₈₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊C₈

subgroup

G/H

center

C₂²

(C₂×#10624382)⋊C₄

commutator

C₂×C₄×C₁₈₁

C₄²

Frattini

C₄²

C₂×#10624382

Fitting

C₄⋊C₈×C₁₈₁

C₄

socle

C₂²×C₁₈₁

#95

prime radical

C₂²×D₁₈₁

C₄⋊C₄

cosocle

C₄²×D₁₈₁

C₂²

C₄⋊C₈

C₁₈₁

G ▷ C₂×C₄×C₁₈₁ ▷ C₂×C₁₈₁ ▷ C₁₈₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊C₈

N	H ≅ G/N	type
58. C₁₈₁	#1291	semidirect

H ≅ G/N

type

58. C₁₈₁

#1291

semidirect

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

subgroup	index
#26042301	2
C₄×#15668459	2
#26041335	2
#1291	181

subgroup

index

181

Subgroups — #37548587

Conjugacy classes of subgroups

Complements

Maximal subgroups