Group #37544225 — subgroups

Subgroups — C₂×#26041681 #37544225

Order 23168.

360 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 181 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	2	yes	maximal
2	#26041681	C₂	1	8	yes	maximal, direct product factor
3	C₂×#19153750	C₂	1	2	yes	maximal
4	C₂×#19153748	C₂	1	2	yes	maximal
5	C₂×C₄×C₈×C₁₈₁	C₂	1	1	yes	maximal, characteristic
6	#19153808	—	2	4	no	—
7	#19153808	C₂²	1	8	yes	—
8	C₂×C₄⋊Dic₁₈₁	C₂²	1	4	yes	—
9	C₂×#15668428	—	2	4	no	—
10	#19153750	C₂²	1	8	yes	—
11	#19153748	C₂²	1	8	yes	—
12	C₂²×C₈×C₁₈₁	C₂²	1	1	yes	characteristic
13	C₂²×C₈×C₁₈₁	C₂²	1	1	yes	characteristic
14	C₂×C₄²×C₁₈₁	C₂²	1	1	yes	characteristic
15	C₄×C₈×C₁₈₁	C₂²	1	4	yes	—
16	C₂²×Dic₁₈₁	—	2	4	no	—
17	C₄⋊Dic₁₈₁	—	2	4	no	—
18	C₄⋊Dic₁₈₁	C₂³	1	8	yes	—
19	#15668428	—	2	16	no	—
20	C₂²×C₄×C₁₈₁	D₄	1	2	yes	—
21	C₂²×C₄×C₁₈₁	C₂³	1	1	yes	characteristic
22	C₂×C₈×C₁₈₁	Q₈	1	4	yes	—
23	C₂×C₈×C₁₈₁	Q₈	1	4	yes	—
24	C₂×C₈×C₁₈₁	C₂³	1	2	yes	—
25	C₂×C₈×C₁₈₁	C₂³	1	2	yes	—
26	C₄²×C₁₈₁	D₄	1	2	yes	—
27	C₄²×C₁₈₁	C₂³	1	2	yes	—
28	C₂×Dic₁₈₁	—	2	8	no	—
29	C₂×Dic₁₈₁	—	4	8	no	—
30	C₂³×C₁₈₁	C₂×D₄	1	1	yes	characteristic
31	C₂×C₄×C₁₈₁	—	2	4	no	—
32	C₂×C₄×C₁₈₁	C₂×D₄	1	4	yes	—
33	C₂×C₄×C₁₈₁	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₁₈₁	C₂×Q₈	1	2	yes	—
35	C₂×C₄×C₁₈₁	C₂×Q₈	1	2	yes	—
36	C₂×C₄×C₁₈₁	C₂⁴	1	1	yes	characteristic
37	C₈×C₁₈₁	C₂×Q₈	1	4	yes	—
38	C₈×C₁₈₁	C₂×Q₈	1	4	yes	—
39	C₂²×C₁₈₁	C₄⋊Q₈	1	2	yes	—
40	C₂²×C₁₈₁	#135	1	4	yes	—
41	C₂²×C₁₈₁	C₂²×D₄	1	1	yes	characteristic
42	C₄×C₁₈₁	—	2	4	no	—
43	C₄×C₁₈₁	C₄⋊Q₈	1	2	yes	—
44	C₄×C₁₈₁	C₂²×Q₈	1	1	yes	characteristic
45	C₄×C₁₈₁	C₂²×Q₈	1	1	yes	characteristic
46	Dic₁₈₁	—	4	8	no	—
47	C₂×C₁₈₁	#499	1	4	yes	complemented
48	C₂×C₁₈₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
49	C₂×C₁₈₁	C₂×#135	1	2	yes	—
50	C₁₈₁	C₂×#499	1	1	yes	characteristic, complemented, Sylow
51	C₂×#499	—	181	1	no	maximal, Sylow
52	C₂×#125	—	181	2	no	—
53	#499	—	181	8	no	—
54	C₂×#107	—	181	2	no	—
55	C₂×#106	—	181	2	no	—
56	C₂×C₄×C₈	D₁₈₁	1	1	yes	characteristic
57	C₂²×C₈	C₂×D₁₈₁	1	1	yes	characteristic
58	C₂²×C₈	C₂×D₁₈₁	1	1	yes	characteristic
59	#125	—	362	4	no	—
60	#125	—	181	8	no	—
61	C₂×C₄⋊C₄	—	362	4	no	—
62	C₂×C₄⋊C₄	—	181	4	no	—
63	C₂×C₄²	C₂×D₁₈₁	1	1	yes	characteristic
64	#107	—	181	8	no	—
65	#106	—	181	8	no	—
66	C₄×C₈	C₂×D₁₈₁	1	4	yes	—
67	C₂²×C₄	—	362	4	no	—
68	C₂²×C₄	D₇₂₄	1	2	yes	—
69	C₂²×C₄	C₂²×D₁₈₁	1	1	yes	characteristic
70	C₂×C₈	Dic₃₆₂	1	4	yes	—
71	C₂×C₈	Dic₃₆₂	1	4	yes	—
72	C₂×C₈	C₂²×D₁₈₁	1	2	yes	—
73	C₂×C₈	C₂²×D₁₈₁	1	2	yes	—
74	C₄⋊C₄	—	362	16	no	—
75	C₄⋊C₄	—	362	4	no	—
76	C₄⋊C₄	—	181	8	no	—
77	C₄²	D₇₂₄	1	2	yes	—
78	C₄²	C₂²×D₁₈₁	1	2	yes	—
79	C₂³	C₂×D₇₂₄	1	1	yes	characteristic
80	C₂×C₄	—	2	4	no	—
81	C₂×C₄	—	362	8	no	—
82	C₂×C₄	—	724	8	no	—
83	C₂×C₄	C₂×D₇₂₄	1	4	yes	—
84	C₂×C₄	C₂×D₇₂₄	1	1	yes	characteristic
85	C₂×C₄	C₂×Dic₃₆₂	1	2	yes	—
86	C₂×C₄	C₂×Dic₃₆₂	1	2	yes	—
87	C₂×C₄	C₂³×D₁₈₁	1	1	yes	characteristic
88	C₈	C₂×Dic₃₆₂	1	4	yes	—
89	C₈	C₂×Dic₃₆₂	1	4	yes	—
90	C₂²	#19153816	1	2	yes	—
91	C₂²	#19153838	1	4	yes	—
92	C₂²	C₂²×D₇₂₄	1	1	yes	characteristic
93	C₄	—	2	4	no	—
94	C₄	—	724	8	no	—
95	C₄	#19153816	1	2	yes	—
96	C₄	C₂²×Dic₃₆₂	1	1	yes	characteristic
97	C₄	C₂²×Dic₃₆₂	1	1	yes	characteristic
98	C₂	#26041681	1	4	yes	direct product factor
99	C₂	C₂×#19153816	1	1	yes	characteristic
100	C₂	C₂×#19153838	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19153808

C₂

yes

maximal

#26041681

C₂

yes

maximal, direct product factor

C₂×#19153750

C₂

yes

maximal

C₂×#19153748

C₂

yes

maximal

C₂×C₄×C₈×C₁₈₁

C₂

yes

maximal, characteristic

#19153808

—

#19153808

C₂²

yes

—

C₂×C₄⋊Dic₁₈₁

C₂²

yes

—

C₂×#15668428

—

#19153750

C₂²

yes

—

#19153748

C₂²

yes

—

C₂²×C₈×C₁₈₁

C₂²

yes

characteristic

C₂²×C₈×C₁₈₁

C₂²

yes

characteristic

C₂×C₄²×C₁₈₁

C₂²

yes

characteristic

C₄×C₈×C₁₈₁

C₂²

yes

—

C₂²×Dic₁₈₁

—

C₄⋊Dic₁₈₁

—

C₄⋊Dic₁₈₁

C₂³

yes

—

#15668428

—

C₂²×C₄×C₁₈₁

D₄

yes

—

C₂²×C₄×C₁₈₁

C₂³

yes

characteristic

C₂×C₈×C₁₈₁

Q₈

yes

—

C₂×C₈×C₁₈₁

Q₈

yes

—

C₂×C₈×C₁₈₁

C₂³

yes

—

C₂×C₈×C₁₈₁

C₂³

yes

—

C₄²×C₁₈₁

D₄

yes

—

C₄²×C₁₈₁

C₂³

yes

—

C₂×Dic₁₈₁

—

C₂×Dic₁₈₁

—

C₂³×C₁₈₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₈₁

—

C₂×C₄×C₁₈₁

C₂×D₄

yes

—

C₂×C₄×C₁₈₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₈₁

C₂×Q₈

yes

—

C₂×C₄×C₁₈₁

C₂×Q₈

yes

—

C₂×C₄×C₁₈₁

C₂⁴

yes

characteristic

C₈×C₁₈₁

C₂×Q₈

yes

—

C₈×C₁₈₁

C₂×Q₈

yes

—

C₂²×C₁₈₁

C₄⋊Q₈

yes

—

C₂²×C₁₈₁

#135

yes

—

C₂²×C₁₈₁

C₂²×D₄

yes

characteristic

C₄×C₁₈₁

—

C₄×C₁₈₁

C₄⋊Q₈

yes

—

C₄×C₁₈₁

C₂²×Q₈

yes

characteristic

C₄×C₁₈₁

C₂²×Q₈

yes

characteristic

Dic₁₈₁

—

C₂×C₁₈₁

#499

yes

complemented

C₂×C₁₈₁

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₁₈₁

C₂×#135

yes

—

C₁₈₁

C₂×#499

yes

characteristic, complemented, Sylow

C₂×#499

—

181

maximal, Sylow

C₂×#125

—

181

—

#499

—

181

—

C₂×#107

—

181

—

C₂×#106

—

181

—

C₂×C₄×C₈

D₁₈₁

yes

characteristic

C₂²×C₈

C₂×D₁₈₁

yes

characteristic

C₂²×C₈

C₂×D₁₈₁

yes

characteristic

#125

—

362

—

#125

—

181

—

C₂×C₄⋊C₄

—

362

—

C₂×C₄⋊C₄

—

181

—

C₂×C₄²

C₂×D₁₈₁

yes

characteristic

#107

—

181

—

#106

—

181

—

C₄×C₈

C₂×D₁₈₁

yes

—

C₂²×C₄

—

362

—

C₂²×C₄

D₇₂₄

yes

—

C₂²×C₄

C₂²×D₁₈₁

yes

characteristic

C₂×C₈

Dic₃₆₂

yes

—

C₂×C₈

Dic₃₆₂

yes

—

C₂×C₈

C₂²×D₁₈₁

yes

—

C₂×C₈

C₂²×D₁₈₁

yes

—

C₄⋊C₄

—

362

—

C₄⋊C₄

—

362

—

C₄⋊C₄

—

181

—

C₄²

D₇₂₄

yes

—

C₄²

C₂²×D₁₈₁

yes

—

C₂³

C₂×D₇₂₄

yes

characteristic

C₂×C₄

—

C₂×C₄

—

362

—

C₂×C₄

—

724

—

C₂×C₄

C₂×D₇₂₄

yes

—

C₂×C₄

C₂×D₇₂₄

yes

characteristic

C₂×C₄

C₂×Dic₃₆₂

yes

—

C₂×C₄

C₂×Dic₃₆₂

yes

—

C₂×C₄

C₂³×D₁₈₁

yes

characteristic

C₈

C₂×Dic₃₆₂

yes

—

C₈

C₂×Dic₃₆₂

yes

—

C₂²

#19153816

yes

—

C₂²

#19153838

yes

—

C₂²

C₂²×D₇₂₄

yes

characteristic

C₄

—

C₄

—

724

—

C₄

#19153816

yes

—

C₄

C₂²×Dic₃₆₂

yes

characteristic

C₄

C₂²×Dic₃₆₂

yes

characteristic

C₂

#26041681

yes

direct product factor

C₂

C₂×#19153816

yes

characteristic

100

C₂

C₂×#19153838

yes

—

	subgroup	G/H
center	C₂³	C₂×D₇₂₄
commutator, cosocle	C₂×C₄×C₁₈₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₈₁
Fitting	C₂×C₄×C₈×C₁₈₁	C₂
socle, prime radical	C₂³×C₁₈₁	C₂×D₄
hypercenter	C₂×C₄×C₈	D₁₈₁
hypocenter	C₁₈₁	C₂×#499
LCS	G ▷ C₂×C₄×C₁₈₁ ▷ C₂×C₁₈₁ ▷ C₁₈₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂³

C₂×D₇₂₄

commutator, cosocle

C₂×C₄×C₁₈₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₈₁

Fitting

C₂×C₄×C₈×C₁₈₁

C₂

socle, prime radical

C₂³×C₁₈₁

C₂×D₄

hypercenter

C₂×C₄×C₈

D₁₈₁

hypocenter

C₁₈₁

C₂×#499

LCS

G ▷ C₂×C₄×C₁₈₁ ▷ C₂×C₁₈₁ ▷ C₁₈₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
2. #26041681	C₂	direct (N × H)
47. C₂×C₁₈₁	#499	semidirect
50. C₁₈₁	C₂×#499	semidirect
98. C₂	#26041681	direct (N × H)

H ≅ G/N

type

2. #26041681

C₂

direct (N × H)

47. C₂×C₁₈₁

#499

semidirect

50. C₁₈₁

C₂×#499

semidirect

98. C₂

#26041681

direct (N × H)

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

subgroup	index
C₂×#19153808	2
#26041681	2
C₂×#19153750	2
C₂×#19153748	2
C₂×C₄×C₈×C₁₈₁	2
C₂×#499	181

subgroup

index

C₂×C₄×C₈×C₁₈₁

181

Subgroups — C₂×#26041681 #37544225

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#26041681 #37544225

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26041681 #37544225