Group #37542330 — subgroups

Subgroups — #37542330

Order 23168.

440 conjugacy classes of subgroups; only the 249 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 249 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	#26042046	C₂	1	1	yes	maximal, characteristic
2	C₂×#19153823	C₂	1	1	yes	maximal, characteristic
3	#26041327	C₂	1	6	yes	maximal, complemented
4	C₂×#19153751	C₂	1	6	yes	maximal, complemented
5	#568×C₁₈₁	C₂	1	1	yes	maximal, characteristic
6	#19153823	—	2	4	no	—
7	#19153823	C₄	1	8	yes	—
8	C₂×#15668433	C₂²	1	6	yes	—
9	C₂×#15668433	C₂²	1	3	yes	—
10	C₂²×#13089678	C₂²	1	3	yes	—
11	#19153751	C₂²	1	4	yes	—
12	#19153751	C₂²	1	12	yes	—
13	C₂×#41×C₁₈₁	C₂²	1	1	yes	characteristic
14	#131×C₁₈₁	—	2	4	no	—
15	C₂×M₁₆×C₁₈₁	C₂²	1	6	yes	—
16	#15668433	—	2	6	no	—
17	#15668433	C₂×C₄	1	12	yes	—
18	#15668433	C₂³	1	4	yes	—
19	C₂×#13089678	—	2	6	no	—
20	C₂×#13089678	C₂×C₄	1	12	yes	—
21	C₂×#13089678	C₂³	1	1	yes	characteristic
22	C₂×#13089678	C₂³	1	3	yes	—
23	#41×C₁₈₁	C₂×C₄	1	4	yes	—
24	#41×C₁₈₁	D₄	1	4	yes	—
25	C₂×Q₈×C₁₈₁	Q₈	1	1	yes	characteristic
26	C₂×D₄×C₁₈₁	Q₈	1	3	yes	—
27	C₂²×C₄×C₁₈₁	C₂³	1	3	yes	—
28	M₁₆×C₁₈₁	—	2	6	no	—
29	M₁₆×C₁₈₁	C₂³	1	4	yes	—
30	C₂×C₈×C₁₈₁	—	2	6	no	—
31	#13089678	—	2	4	no	—
32	#13089678	C₂²×C₄	1	2	yes	—
33	#13089678	C₂²×C₄	1	6	yes	—
34	C₂³×C₁₈₁	C₂×Q₈	1	3	yes	—
35	Q₈×C₁₈₁	C₄⋊C₄	1	4	yes	—
36	D₄×C₁₈₁	C₄⋊C₄	1	12	yes	—
37	C₂×C₄×C₁₈₁	C₂²×C₄	1	6	yes	—
38	C₂×C₄×C₁₈₁	C₂×D₄	1	6	yes	—
39	C₂×C₄×C₁₈₁	C₂×Q₈	1	3	yes	—
40	C₂×C₄×C₁₈₁	C₂⁴	1	1	yes	characteristic
41	C₈×C₁₈₁	—	2	4	no	—
42	C₂²×C₁₈₁	—	4	3	no	—
43	C₂²×C₁₈₁	C₂×C₄⋊C₄	1	6	yes	—
44	C₂²×C₁₈₁	C₂²×Q₈	1	1	yes	characteristic
45	C₄×C₁₈₁	C₂×C₄⋊C₄	1	6	yes	—
46	C₄×C₁₈₁	C₂³×C₄	1	1	yes	characteristic
47	C₄×C₁₈₁	C₂²×D₄	1	1	yes	characteristic
48	C₂×C₁₈₁	—	2	1	no	—
49	C₂×C₁₈₁	—	2	6	no	—
50	C₂×C₁₈₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
51	C₁₈₁	#2916	1	1	yes	characteristic, complemented, Sylow
52	#2916	—	181	1	no	maximal, Sylow
53	#568	—	181	1	no	—
54	#568	D₁₈₁	1	1	yes	characteristic
55	C₂×#131	—	181	1	no	—
56	#430	—	181	6	no	—
57	C₂×#108	—	181	6	no	—
58	C₂×#41	C₂×D₁₈₁	1	1	yes	characteristic
59	#131	—	2	4	no	—
60	#131	—	362	4	no	—
61	#131	—	181	8	no	—
62	C₂×M₁₆	—	181	6	no	—
63	C₂×M₁₆	—	181	3	no	—
64	C₂×M₁₆	C₂×D₁₈₁	1	6	yes	—
65	C₂²×C₈	—	181	3	no	—
66	#108	—	181	4	no	—
67	#108	—	181	12	no	—
68	#41	C₄×D₁₈₁	1	4	yes	—
69	#41	C₂²⋊D₁₈₁	1	4	yes	—
70	C₂×Q₈	Dic₃₆₂	1	1	yes	characteristic
71	C₂×D₄	Dic₃₆₂	1	3	yes	—
72	C₂²×C₄	C₂²×D₁₈₁	1	3	yes	—
73	M₁₆	—	2	6	no	—
74	M₁₆	—	362	6	no	—
75	M₁₆	—	181	4	no	—
76	M₁₆	—	181	12	no	—
77	M₁₆	C₂²×D₁₈₁	1	4	yes	—
78	C₂×C₈	—	2	6	no	—
79	C₂×C₈	—	362	6	no	—
80	C₂×C₈	—	181	12	no	—
81	C₂×C₈	—	181	1	no	—
82	C₂×C₈	—	181	3	no	—
83	C₂³	C₂×Dic₃₆₂	1	3	yes	—
84	Q₈	#15668428	1	4	yes	—
85	D₄	#15668428	1	12	yes	—
86	C₂×C₄	C₂×C₄×D₁₈₁	1	6	yes	—
87	C₂×C₄	C₂×C₂²⋊D₁₈₁	1	6	yes	—
88	C₂×C₄	C₂×Dic₃₆₂	1	3	yes	—
89	C₂×C₄	C₂³×D₁₈₁	1	1	yes	characteristic
90	C₈	—	2	4	no	—
91	C₈	—	362	4	no	—
92	C₈	—	181	2	no	—
93	C₈	—	181	6	no	—
94	C₂²	—	4	3	no	—
95	C₂²	C₂×#15668428	1	6	yes	—
96	C₂²	C₂²×Dic₃₆₂	1	1	yes	characteristic
97	C₄	C₂×#15668428	1	6	yes	—
98	C₄	C₂²×C₄×D₁₈₁	1	1	yes	characteristic
99	C₄	C₂²×C₂²⋊D₁₈₁	1	1	yes	characteristic
100	C₂	—	2	1	no	—
101	C₂	—	2	6	no	—
102	C₂	C₂²×#15668428	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26042046

C₂

yes

maximal, characteristic

C₂×#19153823

C₂

yes

maximal, characteristic

#26041327

C₂

yes

maximal, complemented

C₂×#19153751

C₂

yes

maximal, complemented

#568×C₁₈₁

C₂

yes

maximal, characteristic

#19153823

—

#19153823

C₄

yes

—

C₂×#15668433

C₂²

yes

—

C₂×#15668433

C₂²

yes

—

C₂²×#13089678

C₂²

yes

—

#19153751

C₂²

yes

—

#19153751

C₂²

yes

—

C₂×#41×C₁₈₁

C₂²

yes

characteristic

#131×C₁₈₁

—

C₂×M₁₆×C₁₈₁

C₂²

yes

—

#15668433

—

#15668433

C₂×C₄

yes

—

#15668433

C₂³

yes

—

C₂×#13089678

—

C₂×#13089678

C₂×C₄

yes

—

C₂×#13089678

C₂³

yes

characteristic

C₂×#13089678

C₂³

yes

—

#41×C₁₈₁

C₂×C₄

yes

—

#41×C₁₈₁

D₄

yes

—

C₂×Q₈×C₁₈₁

Q₈

yes

characteristic

C₂×D₄×C₁₈₁

Q₈

yes

—

C₂²×C₄×C₁₈₁

C₂³

yes

—

M₁₆×C₁₈₁

—

M₁₆×C₁₈₁

C₂³

yes

—

C₂×C₈×C₁₈₁

—

#13089678

—

#13089678

C₂²×C₄

yes

—

#13089678

C₂²×C₄

yes

—

C₂³×C₁₈₁

C₂×Q₈

yes

—

Q₈×C₁₈₁

C₄⋊C₄

yes

—

D₄×C₁₈₁

C₄⋊C₄

yes

—

C₂×C₄×C₁₈₁

C₂²×C₄

yes

—

C₂×C₄×C₁₈₁

C₂×D₄

yes

—

C₂×C₄×C₁₈₁

C₂×Q₈

yes

—

C₂×C₄×C₁₈₁

C₂⁴

yes

characteristic

C₈×C₁₈₁

—

C₂²×C₁₈₁

—

C₂²×C₁₈₁

C₂×C₄⋊C₄

yes

—

C₂²×C₁₈₁

C₂²×Q₈

yes

characteristic

C₄×C₁₈₁

C₂×C₄⋊C₄

yes

—

C₄×C₁₈₁

C₂³×C₄

yes

characteristic

C₄×C₁₈₁

C₂²×D₄

yes

characteristic

C₂×C₁₈₁

—

C₂×C₁₈₁

—

C₂×C₁₈₁

C₂²×C₄⋊C₄

yes

characteristic

C₁₈₁

#2916

yes

characteristic, complemented, Sylow

#2916

—

181

maximal, Sylow

#568

—

181

—

#568

D₁₈₁

yes

characteristic

C₂×#131

—

181

—

#430

—

181

—

C₂×#108

—

181

—

C₂×#41

C₂×D₁₈₁

yes

characteristic

#131

—

#131

—

362

—

#131

—

181

—

C₂×M₁₆

—

181

—

C₂×M₁₆

—

181

—

C₂×M₁₆

C₂×D₁₈₁

yes

—

C₂²×C₈

—

181

—

#108

—

181

—

#108

—

181

—

#41

C₄×D₁₈₁

yes

—

#41

C₂²⋊D₁₈₁

yes

—

C₂×Q₈

Dic₃₆₂

yes

characteristic

C₂×D₄

Dic₃₆₂

yes

—

C₂²×C₄

C₂²×D₁₈₁

yes

—

M₁₆

—

M₁₆

—

362

—

M₁₆

—

181

—

M₁₆

—

181

—

M₁₆

C₂²×D₁₈₁

yes

—

C₂×C₈

—

C₂×C₈

—

362

—

C₂×C₈

—

181

—

C₂×C₈

—

181

—

C₂×C₈

—

181

—

C₂³

C₂×Dic₃₆₂

yes

—

Q₈

#15668428

yes

—

D₄

#15668428

yes

—

C₂×C₄

C₂×C₄×D₁₈₁

yes

—

C₂×C₄

C₂×C₂²⋊D₁₈₁

yes

—

C₂×C₄

C₂×Dic₃₆₂

yes

—

C₂×C₄

C₂³×D₁₈₁

yes

characteristic

C₈

—

C₈

—

362

—

C₈

—

181

—

C₈

—

181

—

C₂²

—

C₂²

C₂×#15668428

yes

—

C₂²

C₂²×Dic₃₆₂

yes

characteristic

C₄

C₂×#15668428

yes

—

C₄

C₂²×C₄×D₁₈₁

yes

characteristic

C₄

C₂²×C₂²⋊D₁₈₁

yes

characteristic

100

C₂

—

101

C₂

—

102

C₂

C₂²×#15668428

yes

characteristic

	subgroup	G/H
center	C₄	C₂²×C₂²⋊D₁₈₁
commutator	C₄×C₁₈₁	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₈₁
Fitting	#568×C₁₈₁	C₂
socle	C₂×C₁₈₁	C₂²×C₄⋊C₄
prime radical	C₂×#41×C₁₈₁	C₂²
cosocle	C₂×C₄×C₁₈₁	C₂⁴
hypercenter	#568	D₁₈₁
hypocenter	C₁₈₁	#2916
LCS	G ▷ C₄×C₁₈₁ ▷ C₂×C₁₈₁ ▷ C₁₈₁
UCS	C₁ ◁ C₄ ◁ C₂×#41 ◁ #568

subgroup

G/H

center

C₄

C₂²×C₂²⋊D₁₈₁

commutator

C₄×C₁₈₁

C₂³×C₄

Frattini

C₂×C₄

C₂³×D₁₈₁

Fitting

#568×C₁₈₁

C₂

socle

C₂×C₁₈₁

C₂²×C₄⋊C₄

prime radical

C₂×#41×C₁₈₁

C₂²

cosocle

C₂×C₄×C₁₈₁

C₂⁴

D₁₈₁

C₁₈₁

G ▷ C₄×C₁₈₁ ▷ C₂×C₁₈₁ ▷ C₁₈₁

UCS

C₁ ◁ C₄ ◁ C₂×#41 ◁ #568

N	H ≅ G/N	type
3. #26041327	C₂	semidirect
4. C₂×#19153751	C₂	semidirect
51. C₁₈₁	#2916	semidirect

H ≅ G/N

type

3. #26041327

C₂

semidirect

4. C₂×#19153751

C₂

semidirect

51. C₁₈₁

#2916

semidirect

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

subgroup	index
#26042046	2
C₂×#19153823	2
#26041327	2
C₂×#19153751	2
#568×C₁₈₁	2
#2916	181

subgroup

index

#568×C₁₈₁

181

Subgroups — #37542330

Conjugacy classes of subgroups

Complements

Maximal subgroups