Group #37581129 — subgroups

Subgroups — #37581129

Order 24448.

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	#27133430	C₂	1	1	yes	maximal, characteristic
2	C₂×#20259329	C₂	1	1	yes	maximal, characteristic
3	#27132711	C₂	1	6	yes	maximal, complemented
4	C₂×#20259257	C₂	1	6	yes	maximal, complemented
5	#568×C₁₉₁	C₂	1	1	yes	maximal, characteristic
6	#20259329	—	2	4	no	—
7	#20259329	C₄	1	8	yes	—
8	C₂×#15690271	C₂²	1	6	yes	—
9	C₂×#15690271	C₂²	1	3	yes	—
10	C₂²×C₁₉₁⋊C₈	C₂²	1	3	yes	—
11	#20259257	C₂²	1	4	yes	—
12	#20259257	C₂²	1	12	yes	—
13	C₂×#41×C₁₉₁	C₂²	1	1	yes	characteristic
14	#131×C₁₉₁	—	2	4	no	—
15	C₂×M₁₆×C₁₉₁	C₂²	1	6	yes	—
16	#15690271	—	2	6	no	—
17	#15690271	C₂×C₄	1	12	yes	—
18	#15690271	C₂³	1	4	yes	—
19	C₂×C₁₉₁⋊C₈	—	2	6	no	—
20	C₂×C₁₉₁⋊C₈	C₂×C₄	1	12	yes	—
21	C₂×C₁₉₁⋊C₈	C₂³	1	1	yes	characteristic
22	C₂×C₁₉₁⋊C₈	C₂³	1	3	yes	—
23	#41×C₁₉₁	C₂×C₄	1	4	yes	—
24	#41×C₁₉₁	D₄	1	4	yes	—
25	C₂×Q₈×C₁₉₁	Q₈	1	1	yes	characteristic
26	C₂×D₄×C₁₉₁	Q₈	1	3	yes	—
27	C₂²×C₄×C₁₉₁	C₂³	1	3	yes	—
28	M₁₆×C₁₉₁	—	2	6	no	—
29	M₁₆×C₁₉₁	C₂³	1	4	yes	—
30	C₂×C₈×C₁₉₁	—	2	6	no	—
31	C₁₉₁⋊C₈	—	2	4	no	—
32	C₁₉₁⋊C₈	C₂²×C₄	1	2	yes	—
33	C₁₉₁⋊C₈	C₂²×C₄	1	6	yes	—
34	C₂³×C₁₉₁	C₂×Q₈	1	3	yes	—
35	Q₈×C₁₉₁	C₄⋊C₄	1	4	yes	—
36	D₄×C₁₉₁	C₄⋊C₄	1	12	yes	—
37	C₂×C₄×C₁₉₁	C₂²×C₄	1	6	yes	—
38	C₂×C₄×C₁₉₁	C₂×D₄	1	6	yes	—
39	C₂×C₄×C₁₉₁	C₂×Q₈	1	3	yes	—
40	C₂×C₄×C₁₉₁	C₂⁴	1	1	yes	characteristic
41	C₈×C₁₉₁	—	2	4	no	—
42	C₂²×C₁₉₁	—	4	3	no	—
43	C₂²×C₁₉₁	C₂×C₄⋊C₄	1	6	yes	—
44	C₂²×C₁₉₁	C₂²×Q₈	1	1	yes	characteristic
45	C₄×C₁₉₁	C₂×C₄⋊C₄	1	6	yes	—
46	C₄×C₁₉₁	C₂³×C₄	1	1	yes	characteristic
47	C₄×C₁₉₁	C₂²×D₄	1	1	yes	characteristic
48	C₂×C₁₉₁	—	2	1	no	—
49	C₂×C₁₉₁	—	2	6	no	—
50	C₂×C₁₉₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
51	C₁₉₁	#2916	1	1	yes	characteristic, complemented, Sylow
52	#2916	—	191	1	no	maximal, Sylow
53	#568	—	191	1	no	—
54	#568	D₁₉₁	1	1	yes	characteristic
55	C₂×#131	—	191	1	no	—
56	#430	—	191	6	no	—
57	C₂×#108	—	191	6	no	—
58	C₂×#41	C₂×D₁₉₁	1	1	yes	characteristic
59	#131	—	2	4	no	—
60	#131	—	382	4	no	—
61	#131	—	191	8	no	—
62	C₂×M₁₆	—	191	6	no	—
63	C₂×M₁₆	—	191	3	no	—
64	C₂×M₁₆	C₂×D₁₉₁	1	6	yes	—
65	C₂²×C₈	—	191	3	no	—
66	#108	—	191	4	no	—
67	#108	—	191	12	no	—
68	#41	C₄×D₁₉₁	1	4	yes	—
69	#41	C₂²⋊D₁₉₁	1	4	yes	—
70	C₂×Q₈	Dic₃₈₂	1	1	yes	characteristic
71	C₂×D₄	Dic₃₈₂	1	3	yes	—
72	C₂²×C₄	C₂²×D₁₉₁	1	3	yes	—
73	M₁₆	—	2	6	no	—
74	M₁₆	—	382	6	no	—
75	M₁₆	—	191	4	no	—
76	M₁₆	—	191	12	no	—
77	M₁₆	C₂²×D₁₉₁	1	4	yes	—
78	C₂×C₈	—	2	6	no	—
79	C₂×C₈	—	382	6	no	—
80	C₂×C₈	—	191	12	no	—
81	C₂×C₈	—	191	1	no	—
82	C₂×C₈	—	191	3	no	—
83	C₂³	C₂×Dic₃₈₂	1	3	yes	—
84	Q₈	Dic₁₉₁⋊C₄	1	4	yes	—
85	D₄	Dic₁₉₁⋊C₄	1	12	yes	—
86	C₂×C₄	C₂×C₄×D₁₉₁	1	6	yes	—
87	C₂×C₄	C₂×C₂²⋊D₁₉₁	1	6	yes	—
88	C₂×C₄	C₂×Dic₃₈₂	1	3	yes	—
89	C₂×C₄	C₂³×D₁₉₁	1	1	yes	characteristic
90	C₈	—	2	4	no	—
91	C₈	—	382	4	no	—
92	C₈	—	191	2	no	—
93	C₈	—	191	6	no	—
94	C₂²	—	4	3	no	—
95	C₂²	C₂×Dic₁₉₁⋊C₄	1	6	yes	—
96	C₂²	C₂²×Dic₃₈₂	1	1	yes	characteristic
97	C₄	C₂×Dic₁₉₁⋊C₄	1	6	yes	—
98	C₄	C₂²×C₄×D₁₉₁	1	1	yes	characteristic
99	C₄	C₂²×C₂²⋊D₁₉₁	1	1	yes	characteristic
100	C₂	—	2	1	no	—
101	C₂	—	2	6	no	—
102	C₂	C₂²×Dic₁₉₁⋊C₄	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#27133430

C₂

yes

maximal, characteristic

C₂×#20259329

C₂

yes

maximal, characteristic

#27132711

C₂

yes

maximal, complemented

C₂×#20259257

C₂

yes

maximal, complemented

#568×C₁₉₁

C₂

yes

maximal, characteristic

#20259329

—

#20259329

C₄

yes

—

C₂×#15690271

C₂²

yes

—

C₂×#15690271

C₂²

yes

—

C₂²×C₁₉₁⋊C₈

C₂²

yes

—

#20259257

C₂²

yes

—

#20259257

C₂²

yes

—

C₂×#41×C₁₉₁

C₂²

yes

characteristic

#131×C₁₉₁

—

C₂×M₁₆×C₁₉₁

C₂²

yes

—

#15690271

—

#15690271

C₂×C₄

yes

—

#15690271

C₂³

yes

—

C₂×C₁₉₁⋊C₈

—

C₂×C₁₉₁⋊C₈

C₂×C₄

yes

—

C₂×C₁₉₁⋊C₈

C₂³

yes

characteristic

C₂×C₁₉₁⋊C₈

C₂³

yes

—

#41×C₁₉₁

C₂×C₄

yes

—

#41×C₁₉₁

D₄

yes

—

C₂×Q₈×C₁₉₁

Q₈

yes

characteristic

C₂×D₄×C₁₉₁

Q₈

yes

—

C₂²×C₄×C₁₉₁

C₂³

yes

—

M₁₆×C₁₉₁

—

M₁₆×C₁₉₁

C₂³

yes

—

C₂×C₈×C₁₉₁

—

C₁₉₁⋊C₈

—

C₁₉₁⋊C₈

C₂²×C₄

yes

—

C₁₉₁⋊C₈

C₂²×C₄

yes

—

C₂³×C₁₉₁

C₂×Q₈

yes

—

Q₈×C₁₉₁

C₄⋊C₄

yes

—

D₄×C₁₉₁

C₄⋊C₄

yes

—

C₂×C₄×C₁₉₁

C₂²×C₄

yes

—

C₂×C₄×C₁₉₁

C₂×D₄

yes

—

C₂×C₄×C₁₉₁

C₂×Q₈

yes

—

C₂×C₄×C₁₉₁

C₂⁴

yes

characteristic

C₈×C₁₉₁

—

C₂²×C₁₉₁

—

C₂²×C₁₉₁

C₂×C₄⋊C₄

yes

—

C₂²×C₁₉₁

C₂²×Q₈

yes

characteristic

C₄×C₁₉₁

C₂×C₄⋊C₄

yes

—

C₄×C₁₉₁

C₂³×C₄

yes

characteristic

C₄×C₁₉₁

C₂²×D₄

yes

characteristic

C₂×C₁₉₁

—

C₂×C₁₉₁

—

C₂×C₁₉₁

C₂²×C₄⋊C₄

yes

characteristic

C₁₉₁

#2916

yes

characteristic, complemented, Sylow

#2916

—

191

maximal, Sylow

#568

—

191

—

#568

D₁₉₁

yes

characteristic

C₂×#131

—

191

—

#430

—

191

—

C₂×#108

—

191

—

C₂×#41

C₂×D₁₉₁

yes

characteristic

#131

—

#131

—

382

—

#131

—

191

—

C₂×M₁₆

—

191

—

C₂×M₁₆

—

191

—

C₂×M₁₆

C₂×D₁₉₁

yes

—

C₂²×C₈

—

191

—

#108

—

191

—

#108

—

191

—

#41

C₄×D₁₉₁

yes

—

#41

C₂²⋊D₁₉₁

yes

—

C₂×Q₈

Dic₃₈₂

yes

characteristic

C₂×D₄

Dic₃₈₂

yes

—

C₂²×C₄

C₂²×D₁₉₁

yes

—

M₁₆

—

M₁₆

—

382

—

M₁₆

—

191

—

M₁₆

—

191

—

M₁₆

C₂²×D₁₉₁

yes

—

C₂×C₈

—

C₂×C₈

—

382

—

C₂×C₈

—

191

—

C₂×C₈

—

191

—

C₂×C₈

—

191

—

C₂³

C₂×Dic₃₈₂

yes

—

Q₈

Dic₁₉₁⋊C₄

yes

—

D₄

Dic₁₉₁⋊C₄

yes

—

C₂×C₄

C₂×C₄×D₁₉₁

yes

—

C₂×C₄

C₂×C₂²⋊D₁₉₁

yes

—

C₂×C₄

C₂×Dic₃₈₂

yes

—

C₂×C₄

C₂³×D₁₉₁

yes

characteristic

C₈

—

C₈

—

382

—

C₈

—

191

—

C₈

—

191

—

C₂²

—

C₂²

C₂×Dic₁₉₁⋊C₄

yes

—

C₂²

C₂²×Dic₃₈₂

yes

characteristic

C₄

C₂×Dic₁₉₁⋊C₄

yes

—

C₄

C₂²×C₄×D₁₉₁

yes

characteristic

C₄

C₂²×C₂²⋊D₁₉₁

yes

characteristic

100

C₂

—

101

C₂

—

102

C₂

C₂²×Dic₁₉₁⋊C₄

yes

characteristic

	subgroup	G/H
center	C₄	C₂²×C₂²⋊D₁₉₁
commutator	C₄×C₁₉₁	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₉₁
Fitting	#568×C₁₉₁	C₂
socle	C₂×C₁₉₁	C₂²×C₄⋊C₄
prime radical	C₂×#41×C₁₉₁	C₂²
cosocle	C₂×C₄×C₁₉₁	C₂⁴
hypercenter	#568	D₁₉₁
hypocenter	C₁₉₁	#2916
LCS	G ▷ C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₄ ◁ C₂×#41 ◁ #568

subgroup

G/H

center

C₄

C₂²×C₂²⋊D₁₉₁

commutator

C₄×C₁₉₁

C₂³×C₄

Frattini

C₂×C₄

C₂³×D₁₉₁

Fitting

#568×C₁₉₁

C₂

socle

C₂×C₁₉₁

C₂²×C₄⋊C₄

prime radical

C₂×#41×C₁₉₁

C₂²

cosocle

C₂×C₄×C₁₉₁

C₂⁴

D₁₉₁

C₁₉₁

G ▷ C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁

UCS

C₁ ◁ C₄ ◁ C₂×#41 ◁ #568

N	H ≅ G/N	type
3. #27132711	C₂	semidirect
4. C₂×#20259257	C₂	semidirect
51. C₁₉₁	#2916	semidirect

H ≅ G/N

type

3. #27132711

C₂

semidirect

4. C₂×#20259257

C₂

semidirect

51. C₁₉₁

#2916

semidirect

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

subgroup	index
#27133430	2
C₂×#20259329	2
#27132711	2
C₂×#20259257	2
#568×C₁₉₁	2
#2916	191

subgroup

index

#568×C₁₉₁

191

Subgroups — #37581129

Conjugacy classes of subgroups

Complements

Maximal subgroups