Group #37572425 — subgroups

Subgroups — #37572425

Order 24448.

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

normal subgroups only

This group has 220 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×#20259257	C₂	1	2	yes	maximal
2	M₁₆×Dic₁₉₁	C₂	1	2	yes	maximal
3	C₂×#20259232	C₂	1	1	yes	maximal, characteristic
4	#27132428	C₂	1	1	yes	maximal, characteristic
5	#341×C₁₉₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#15690271	C₂²	1	2	yes	—
7	C₂²×C₁₉₁⋊C₈	C₂²	1	1	yes	characteristic
8	C₂×C₄×Dic₁₉₁	C₂²	1	1	yes	characteristic
9	#20259257	C₄	1	8	yes	complemented
10	#20259232	—	2	2	no	—
11	#20259232	C₄	1	4	yes	—
12	C₈×Dic₁₉₁	—	2	2	no	—
13	C₂×M₁₆×C₁₉₁	C₂²	1	2	yes	—
14	C₂²×C₈×C₁₉₁	C₂²	1	1	yes	characteristic
15	C₂²×Dic₁₉₁	Q₈	1	2	yes	—
16	#15690271	—	2	2	no	—
17	#15690271	C₂×C₄	1	4	yes	—
18	C₂×C₁₉₁⋊C₈	—	2	2	no	—
19	C₂×C₁₉₁⋊C₈	C₂×C₄	1	2	yes	—
20	C₂×C₁₉₁⋊C₈	C₂×C₄	1	4	yes	—
21	C₄×Dic₁₉₁	C₂×C₄	1	2	yes	—
22	C₄×Dic₁₉₁	D₄	1	2	yes	—
23	C₂²×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
24	M₁₆×C₁₉₁	—	2	2	no	—
25	M₁₆×C₁₉₁	C₂×C₄	1	4	yes	—
26	C₂×C₈×C₁₉₁	—	2	2	no	—
27	C₂×C₈×C₁₉₁	—	2	2	no	—
28	C₂×C₈×C₁₉₁	C₂×C₄	1	2	yes	—
29	C₂×Dic₁₉₁	—	4	2	no	—
30	C₂×Dic₁₉₁	C₄⋊C₄	1	4	yes	—
31	C₁₉₁⋊C₈	—	2	4	no	—
32	C₁₉₁⋊C₈	C₄²	1	4	yes	—
33	C₂³×C₁₉₁	C₂×Q₈	1	1	yes	characteristic
34	C₂×C₄×C₁₉₁	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₄×C₁₉₁	C₂²×C₄	1	2	yes	—
36	C₂×C₄×C₁₉₁	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₁₉₁	#41	1	2	yes	—
38	C₈×C₁₉₁	—	2	2	no	—
39	C₈×C₁₉₁	—	2	4	no	—
40	C₂²×C₁₉₁	—	2	2	no	—
41	C₂²×C₁₉₁	C₂×C₄⋊C₄	1	1	yes	characteristic
42	C₂²×C₁₉₁	C₄×Q₈	1	2	yes	—
43	C₄×C₁₉₁	C₂×C₄²	1	1	yes	characteristic
44	C₄×C₁₉₁	#117	1	1	yes	characteristic
45	C₄×C₁₉₁	C₄×D₄	1	2	yes	—
46	Dic₁₉₁	—	2	4	no	—
47	C₂×C₁₉₁	—	2	2	no	—
48	C₂×C₁₉₁	C₄×C₄⋊C₄	1	1	yes	characteristic
49	C₂×C₁₉₁	#430	1	2	yes	—
50	C₁₉₁	#1778	1	1	yes	characteristic, complemented, Sylow
51	#1778	—	191	1	no	maximal, Sylow
52	C₂×#108	—	191	2	no	—
53	C₄×M₁₆	—	191	2	no	—
54	C₂×#97	—	191	1	no	—
55	#341	—	191	1	no	—
56	#341	D₁₉₁	1	1	yes	characteristic
57	C₂×M₁₆	—	191	2	no	—
58	C₂×M₁₆	C₂×D₁₉₁	1	2	yes	—
59	C₂²×C₈	—	191	1	no	—
60	C₂²×C₈	C₂×D₁₉₁	1	1	yes	characteristic
61	C₂×C₄²	—	191	1	no	—
62	#108	—	191	8	no	—
63	#97	—	382	2	no	—
64	#97	—	191	4	no	—
65	C₄×C₈	—	382	2	no	—
66	C₂²×C₄	—	191	2	no	—
67	C₂²×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
68	M₁₆	—	2	2	no	—
69	M₁₆	—	382	2	no	—
70	M₁₆	—	191	4	no	—
71	M₁₆	C₄×D₁₉₁	1	4	yes	—
72	C₂×C₈	—	2	2	no	—
73	C₂×C₈	—	2	2	no	—
74	C₂×C₈	—	382	2	no	—
75	C₂×C₈	—	191	2	no	—
76	C₂×C₈	—	191	4	no	—
77	C₂×C₈	C₄×D₁₉₁	1	2	yes	—
78	C₄²	—	191	2	no	—
79	C₄²	—	191	2	no	—
80	C₂³	Q₈×D₁₉₁	1	1	yes	characteristic
81	C₂×C₄	—	191	4	no	—
82	C₂×C₄	—	764	2	no	—
83	C₂×C₄	C₂×C₄×D₁₉₁	1	1	yes	characteristic
84	C₂×C₄	C₂×C₄×D₁₉₁	1	2	yes	—
85	C₂×C₄	D₄×D₁₉₁	1	1	yes	characteristic
86	C₂×C₄	#15690286	1	2	yes	—
87	C₈	—	2	2	no	—
88	C₈	—	2	4	no	—
89	C₈	—	382	4	no	—
90	C₈	—	191	4	no	—
91	C₂²	—	2	2	no	—
92	C₂²	C₄⋊C₄×D₁₉₁	1	1	yes	characteristic
93	C₂²	#20259287	1	2	yes	—
94	C₄	—	382	4	no	—
95	C₄	C₄²×D₁₉₁	1	1	yes	characteristic
96	C₄	#20259278	1	1	yes	characteristic
97	C₄	(C₂²⋊D₁₉₁)⋊C₄	1	2	yes	—
98	C₂	—	2	2	no	—
99	C₂	#27132514	1	1	yes	characteristic
100	C₂	#27132710	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#20259257

C₂

yes

maximal

M₁₆×Dic₁₉₁

C₂

yes

maximal

C₂×#20259232

C₂

yes

maximal, characteristic

#27132428

C₂

yes

maximal, characteristic

#341×C₁₉₁

C₂

yes

maximal, characteristic

C₂×#15690271

C₂²

yes

—

C₂²×C₁₉₁⋊C₈

C₂²

yes

characteristic

C₂×C₄×Dic₁₉₁

C₂²

yes

characteristic

#20259257

C₄

yes

complemented

#20259232

—

#20259232

C₄

yes

—

C₈×Dic₁₉₁

—

C₂×M₁₆×C₁₉₁

C₂²

yes

—

C₂²×C₈×C₁₉₁

C₂²

yes

characteristic

C₂²×Dic₁₉₁

Q₈

yes

—

#15690271

—

#15690271

C₂×C₄

yes

—

C₂×C₁₉₁⋊C₈

—

C₂×C₁₉₁⋊C₈

C₂×C₄

yes

—

C₂×C₁₉₁⋊C₈

C₂×C₄

yes

—

C₄×Dic₁₉₁

C₂×C₄

yes

—

C₄×Dic₁₉₁

D₄

yes

—

C₂²×C₄×C₁₉₁

C₂³

yes

characteristic

M₁₆×C₁₉₁

—

M₁₆×C₁₉₁

C₂×C₄

yes

—

C₂×C₈×C₁₉₁

—

C₂×C₈×C₁₉₁

—

C₂×C₈×C₁₉₁

C₂×C₄

yes

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

C₄⋊C₄

yes

—

C₁₉₁⋊C₈

—

C₁₉₁⋊C₈

C₄²

yes

—

C₂³×C₁₉₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₉₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₉₁

C₂²×C₄

yes

—

C₂×C₄×C₁₉₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₁

#41

yes

—

C₈×C₁₉₁

—

C₈×C₁₉₁

—

C₂²×C₁₉₁

—

C₂²×C₁₉₁

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₉₁

C₄×Q₈

yes

—

C₄×C₁₉₁

C₂×C₄²

yes

characteristic

C₄×C₁₉₁

#117

yes

characteristic

C₄×C₁₉₁

C₄×D₄

yes

—

Dic₁₉₁

—

C₂×C₁₉₁

—

C₂×C₁₉₁

C₄×C₄⋊C₄

yes

characteristic

C₂×C₁₉₁

#430

yes

—

C₁₉₁

#1778

yes

characteristic, complemented, Sylow

#1778

—

191

maximal, Sylow

C₂×#108

—

191

—

C₄×M₁₆

—

191

—

C₂×#97

—

191

—

#341

—

191

—

#341

D₁₉₁

yes

characteristic

C₂×M₁₆

—

191

—

C₂×M₁₆

C₂×D₁₉₁

yes

—

C₂²×C₈

—

191

—

C₂²×C₈

C₂×D₁₉₁

yes

characteristic

C₂×C₄²

—

191

—

#108

—

191

—

#97

—

382

—

#97

—

191

—

C₄×C₈

—

382

—

C₂²×C₄

—

191

—

C₂²×C₄

C₂²×D₁₉₁

yes

characteristic

M₁₆

—

M₁₆

—

382

—

M₁₆

—

191

—

M₁₆

C₄×D₁₉₁

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

382

—

C₂×C₈

—

191

—

C₂×C₈

—

191

—

C₂×C₈

C₄×D₁₉₁

yes

—

C₄²

—

191

—

C₄²

—

191

—

C₂³

Q₈×D₁₉₁

yes

characteristic

C₂×C₄

—

191

—

C₂×C₄

—

764

—

C₂×C₄

C₂×C₄×D₁₉₁

yes

characteristic

C₂×C₄

C₂×C₄×D₁₉₁

yes

—

C₂×C₄

D₄×D₁₉₁

yes

characteristic

C₂×C₄

#15690286

yes

—

C₈

—

C₈

—

C₈

—

382

—

C₈

—

191

—

C₂²

—

C₂²

C₄⋊C₄×D₁₉₁

yes

characteristic

C₂²

#20259287

yes

—

C₄

—

382

—

C₄

C₄²×D₁₉₁

yes

characteristic

C₄

#20259278

yes

characteristic

C₄

(C₂²⋊D₁₉₁)⋊C₄

yes

—

C₂

—

C₂

#27132514

yes

characteristic

100

C₂

#27132710

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₁₉₁
commutator	C₄×C₁₉₁	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₁₉₁
Fitting	#341×C₁₉₁	C₂
socle	C₂²×C₁₉₁	C₂×C₄⋊C₄
prime radical	C₂³×C₁₉₁	C₂×Q₈
cosocle	C₂²×C₄×C₁₉₁	C₂³
hypercenter	#341	D₁₉₁
hypocenter	C₁₉₁	#1778
LCS	G ▷ C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #341

subgroup

G/H

center

C₂×C₄

D₄×D₁₉₁

commutator

C₄×C₁₉₁

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₁₉₁

Fitting

#341×C₁₉₁

C₂

socle

C₂²×C₁₉₁

C₂×C₄⋊C₄

prime radical

C₂³×C₁₉₁

C₂×Q₈

cosocle

C₂²×C₄×C₁₉₁

C₂³

D₁₉₁

C₁₉₁

G ▷ C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #341

N	H ≅ G/N	type
9. #20259257	C₄	semidirect
50. C₁₉₁	#1778	semidirect

H ≅ G/N

type

9. #20259257

C₄

semidirect

50. C₁₉₁

#1778

semidirect

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

subgroup	index
C₂×#20259257	2
M₁₆×Dic₁₉₁	2
C₂×#20259232	2
#27132428	2
#341×C₁₉₁	2
#1778	191

subgroup

index

M₁₆×Dic₁₉₁

#341×C₁₉₁

191

Subgroups — #37572425

Conjugacy classes of subgroups

Complements

Maximal subgroups