Group #37534689 — subgroups

Subgroups — #37534689

Order 23168.

260 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 105 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×#19153808	C₂	1	1	yes	maximal, characteristic
2	C₂×#19153727	C₂	1	1	yes	maximal, characteristic
3	#26041041	C₂	1	4	yes	maximal
4	#384×C₁₈₁	C₂	1	1	yes	maximal, characteristic
5	C₂²×#13089678	C₂²	1	2	yes	—
6	#19153808	—	2	2	no	—
7	#19153808	C₄	1	4	yes	complemented
8	C₂×C₄⋊Dic₁₈₁	C₂²	1	2	yes	—
9	C₂×#15668428	—	2	2	no	—
10	#19153727	—	2	2	no	—
11	C₂×C₄⋊C₄×C₁₈₁	—	2	2	no	—
12	C₂×C₄⋊C₄×C₁₈₁	C₂²	1	2	yes	—
13	C₂×C₄²×C₁₈₁	C₂²	1	1	yes	characteristic
14	C₂²×Dic₁₈₁	—	2	2	no	—
15	C₂×#13089678	—	2	2	no	—
16	C₂×#13089678	—	2	2	no	—
17	C₂×#13089678	Q₈	1	4	yes	—
18	C₄⋊Dic₁₈₁	—	2	2	no	—
19	C₄⋊Dic₁₈₁	C₂×C₄	1	4	yes	—
20	#15668428	—	2	8	no	—
21	C₂²×C₄×C₁₈₁	—	2	2	no	—
22	C₂²×C₄×C₁₈₁	D₄	1	2	yes	—
23	C₂²×C₄×C₁₈₁	C₂³	1	1	yes	characteristic
24	C₄⋊C₄×C₁₈₁	—	2	4	no	—
25	C₄⋊C₄×C₁₈₁	—	4	4	no	—
26	C₄²×C₁₈₁	C₂×C₄	1	2	yes	—
27	C₄²×C₁₈₁	D₄	1	2	yes	—
28	C₂×Dic₁₈₁	—	2	4	no	—
29	C₂×Dic₁₈₁	—	4	2	no	—
30	C₂×Dic₁₈₁	—	4	2	no	—
31	#13089678	—	2	4	no	—
32	C₂³×C₁₈₁	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×C₁₈₁	—	2	2	no	—
34	C₂×C₄×C₁₈₁	—	2	2	no	—
35	C₂×C₄×C₁₈₁	—	4	2	no	—
36	C₂×C₄×C₁₈₁	—	4	4	no	—
37	C₂×C₄×C₁₈₁	C₂²⋊C₄	1	4	yes	—
38	C₂×C₄×C₁₈₁	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₄×C₁₈₁	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₁₈₁	C₂×Q₈	1	2	yes	—
41	C₂×C₄×C₁₈₁	#41	1	2	yes	—
42	C₂²×C₁₈₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
43	C₂²×C₁₈₁	#124	1	1	yes	characteristic
44	C₂²×C₁₈₁	C₄⋊Q₈	1	1	yes	characteristic
45	C₂²×C₁₈₁	AGL(1,ℤ/8)	1	2	yes	—
46	C₂²×C₁₈₁	#137	1	2	yes	—
47	C₄×C₁₈₁	—	2	4	no	—
48	C₄×C₁₈₁	—	8	2	no	—
49	C₄×C₁₈₁	C₄×Q₈	1	2	yes	—
50	C₄×C₁₈₁	C₂²⋊Q₈	1	2	yes	—
51	Dic₁₈₁	—	4	4	no	—
52	C₂×C₁₈₁	#391	1	1	yes	characteristic
53	C₂×C₁₈₁	#417	1	2	yes	—
54	C₂×C₁₈₁	#490	1	1	yes	characteristic
55	C₂×C₁₈₁	#491	1	1	yes	characteristic
56	C₂×C₁₈₁	#501	1	2	yes	—
57	C₁₈₁	#1992	1	1	yes	characteristic, complemented, Sylow
58	#1992	—	181	1	no	maximal, Sylow
59	C₂×#125	—	181	1	no	—
60	C₂×#97	—	181	1	no	—
61	#384	D₁₈₁	1	1	yes	characteristic
62	#340	—	181	4	no	—
63	C₂²×C₈	—	181	2	no	—
64	#125	—	362	2	no	—
65	#125	—	181	4	no	—
66	C₂×C₄⋊C₄	—	2	2	no	—
67	C₂×C₄⋊C₄	—	362	2	no	—
68	C₂×C₄⋊C₄	—	181	2	no	—
69	C₂×C₄⋊C₄	C₂×D₁₈₁	1	2	yes	—
70	C₂×C₄²	C₂×D₁₈₁	1	1	yes	characteristic
71	#97	—	362	2	no	—
72	C₂²×C₄	—	2	2	no	—
73	C₂²×C₄	—	362	2	no	—
74	C₂²×C₄	C₂²⋊D₁₈₁	1	2	yes	—
75	C₂²×C₄	C₂²×D₁₈₁	1	1	yes	characteristic
76	C₂×C₈	—	362	2	no	—
77	C₂×C₈	—	362	2	no	—
78	C₂×C₈	—	181	4	no	—
79	C₄⋊C₄	—	2	4	no	—
80	C₄⋊C₄	—	4	4	no	—
81	C₄⋊C₄	—	362	8	no	—
82	C₄⋊C₄	—	362	2	no	—
83	C₄⋊C₄	—	181	4	no	—
84	C₄²	C₄×D₁₈₁	1	2	yes	—
85	C₄²	D₇₂₄	1	2	yes	—
86	C₂³	C₂×C₂²⋊D₁₈₁	1	1	yes	characteristic
87	C₂×C₄	—	2	2	no	—
88	C₂×C₄	—	2	2	no	—
89	C₂×C₄	—	4	2	no	—
90	C₂×C₄	—	4	4	no	—
91	C₂×C₄	—	362	4	no	—
92	C₂×C₄	—	724	2	no	—
93	C₂×C₄	—	724	2	no	—
94	C₂×C₄	#15668426	1	4	yes	—
95	C₂×C₄	C₂×C₄×D₁₈₁	1	1	yes	characteristic
96	C₂×C₄	C₂×D₇₂₄	1	1	yes	characteristic
97	C₂×C₄	Q₈×D₁₈₁	1	2	yes	—
98	C₂×C₄	#15668448	1	2	yes	—
99	C₈	—	362	4	no	—
100	C₂²	C₂×#15668426	1	1	yes	characteristic
101	C₂²	#19153803	1	1	yes	characteristic
102	C₂²	#19153814	1	1	yes	characteristic
103	C₂²	#19153842	1	2	yes	—
104	C₂²	#19153847	1	2	yes	—
105	C₄	—	2	4	no	—
106	C₄	—	8	2	no	—
107	C₄	—	724	4	no	—
108	C₄	#19153781	1	2	yes	—
109	C₄	#19153796	1	2	yes	—
110	C₂	#26041184	1	1	yes	characteristic
111	C₂	#26041278	1	2	yes	—
112	C₂	#26041643	1	1	yes	characteristic
113	C₂	#26041647	1	1	yes	characteristic
114	C₂	#26041685	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19153808

C₂

yes

maximal, characteristic

C₂×#19153727

C₂

yes

maximal, characteristic

#26041041

C₂

yes

maximal

#384×C₁₈₁

C₂

yes

maximal, characteristic

C₂²×#13089678

C₂²

yes

—

#19153808

—

#19153808

C₄

yes

complemented

C₂×C₄⋊Dic₁₈₁

C₂²

yes

—

C₂×#15668428

—

#19153727

—

C₂×C₄⋊C₄×C₁₈₁

—

C₂×C₄⋊C₄×C₁₈₁

C₂²

yes

—

C₂×C₄²×C₁₈₁

C₂²

yes

characteristic

C₂²×Dic₁₈₁

—

C₂×#13089678

—

C₂×#13089678

—

C₂×#13089678

Q₈

yes

—

C₄⋊Dic₁₈₁

—

C₄⋊Dic₁₈₁

C₂×C₄

yes

—

#15668428

—

C₂²×C₄×C₁₈₁

—

C₂²×C₄×C₁₈₁

D₄

yes

—

C₂²×C₄×C₁₈₁

C₂³

yes

characteristic

C₄⋊C₄×C₁₈₁

—

C₄⋊C₄×C₁₈₁

—

C₄²×C₁₈₁

C₂×C₄

yes

—

C₄²×C₁₈₁

D₄

yes

—

C₂×Dic₁₈₁

—

C₂×Dic₁₈₁

—

C₂×Dic₁₈₁

—

#13089678

—

C₂³×C₁₈₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₈₁

—

C₂×C₄×C₁₈₁

—

C₂×C₄×C₁₈₁

—

C₂×C₄×C₁₈₁

—

C₂×C₄×C₁₈₁

C₂²⋊C₄

yes

—

C₂×C₄×C₁₈₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₈₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₈₁

C₂×Q₈

yes

—

C₂×C₄×C₁₈₁

#41

yes

—

C₂²×C₁₈₁

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₈₁

#124

yes

characteristic

C₂²×C₁₈₁

C₄⋊Q₈

yes

characteristic

C₂²×C₁₈₁

AGL(1,ℤ/8)

yes

—

C₂²×C₁₈₁

#137

yes

—

C₄×C₁₈₁

—

C₄×C₁₈₁

—

C₄×C₁₈₁

C₄×Q₈

yes

—

C₄×C₁₈₁

C₂²⋊Q₈

yes

—

Dic₁₈₁

—

C₂×C₁₈₁

#391

yes

characteristic

C₂×C₁₈₁

#417

yes

—

C₂×C₁₈₁

yes

C₂×C₁₈₁

yes

C₂×C₁₈₁

yes

—

C₁₈₁

#1992

yes

characteristic, complemented, Sylow

#1992

—

181

maximal, Sylow

C₂×#125

—

181

—

C₂×#97

—

181

—

#384

D₁₈₁

yes

characteristic

#340

—

181

—

C₂²×C₈

—

181

—

#125

—

362

—

#125

—

181

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

362

—

C₂×C₄⋊C₄

—

181

—

C₂×C₄⋊C₄

C₂×D₁₈₁

yes

—

C₂×C₄²

C₂×D₁₈₁

yes

characteristic

#97

—

362

—

C₂²×C₄

—

C₂²×C₄

—

362

—

C₂²×C₄

C₂²⋊D₁₈₁

yes

—

C₂²×C₄

C₂²×D₁₈₁

yes

characteristic

C₂×C₈

—

362

—

C₂×C₈

—

362

—

C₂×C₈

—

181

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

362

—

C₄⋊C₄

—

362

—

C₄⋊C₄

—

181

—

C₄²

C₄×D₁₈₁

yes

—

C₄²

D₇₂₄

yes

—

C₂³

C₂×C₂²⋊D₁₈₁

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

362

—

C₂×C₄

—

724

—

C₂×C₄

—

724

—

C₂×C₄

#15668426

yes

—

C₂×C₄

C₂×C₄×D₁₈₁

yes

characteristic

C₂×C₄

C₂×D₇₂₄

yes

characteristic

C₂×C₄

Q₈×D₁₈₁

yes

—

C₂×C₄

#15668448

yes

—

C₈

—

362

—

100

C₂²

C₂×#15668426

yes

characteristic

101

C₂²

#19153803

yes

characteristic

102

C₂²

#19153814

yes

characteristic

103

C₂²

#19153842

yes

—

104

C₂²

#19153847

yes

—

105

C₄

—

106

C₄

—

107

C₄

—

724

—

108

C₄

#19153781

yes

—

109

C₄

#19153796

yes

—

110

C₂

#26041184

yes

characteristic

111

C₂

#26041278

yes

—

112

C₂

#26041643

yes

characteristic

113

C₂

#26041647

yes

characteristic

114

C₂

#26041685

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₈₁
commutator	C₂×C₄×C₁₈₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₈₁
Fitting	#384×C₁₈₁	C₂
socle, prime radical	C₂³×C₁₈₁	C₂×D₄
cosocle	C₂²×C₄×C₁₈₁	C₂³
hypercenter	#384	D₁₈₁
hypocenter	C₁₈₁	#1992
LCS	G ▷ C₂×C₄×C₁₈₁ ▷ C₂×C₁₈₁ ▷ C₁₈₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₈₁

commutator

C₂×C₄×C₁₈₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₈₁

Fitting

#384×C₁₈₁

C₂

socle, prime radical

C₂³×C₁₈₁

C₂×D₄

cosocle

C₂²×C₄×C₁₈₁

C₂³

D₁₈₁

C₁₈₁

G ▷ C₂×C₄×C₁₈₁ ▷ C₂×C₁₈₁ ▷ C₁₈₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

N	H ≅ G/N	type
7. #19153808	C₄	semidirect
57. C₁₈₁	#1992	semidirect

H ≅ G/N

type

7. #19153808

C₄

semidirect

57. C₁₈₁

#1992

semidirect

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

subgroup	index
C₂×#19153808	2
C₂×#19153727	2
#26041041	2
#384×C₁₈₁	2
#1992	181

subgroup

index

#384×C₁₈₁

181

Subgroups — #37534689

Conjugacy classes of subgroups

Complements

Maximal subgroups