Group #37571428 — subgroups

Subgroups — #37571428

Order 24448.

148 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 148 conjugacy classes of subgroups — too many to draw. Only its 63 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	#27132790	C₂	1	2	yes	maximal
2	C₄×Dic₇₆₄	C₂	1	1	yes	maximal, characteristic
3	#27132393	C₂	1	2	yes	maximal
4	#27132384	C₂	1	1	yes	maximal, characteristic
5	#334×C₁₉₁	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₇₆₄	C₄	1	1	yes	characteristic
7	C₄×Dic₃₈₂	C₂²	1	2	yes	—
8	#20259256	C₄	1	1	yes	characteristic
9	Dic₁₉₁⋊C₈	—	2	2	no	—
10	#20259247	C₄	1	2	yes	—
11	#20259232	—	2	2	no	—
12	C₄×C₁₉₁⋊C₈	C₂²	1	1	yes	characteristic
13	C₄×C₁₉₁⋊C₈	C₂²	1	1	yes	characteristic
14	C₄⋊C₈×C₁₉₁	C₂²	1	2	yes	—
15	C₄×C₈×C₁₉₁	C₂²	1	1	yes	characteristic
16	C₂×Dic₃₈₂	C₂×C₄	1	2	yes	—
17	Dic₇₆₄	—	2	2	no	—
18	C₂×C₁₉₁⋊C₈	—	2	1	no	—
19	C₂×C₁₉₁⋊C₈	D₄	1	2	yes	—
20	C₄⋊Dic₁₉₁	C₂×C₄	1	2	yes	—
21	Dic₁₉₁⋊C₄	—	2	2	no	—
22	C₄×Dic₁₉₁	—	2	2	no	—
23	C₂×C₈×C₁₉₁	—	2	2	no	—
24	C₂×C₈×C₁₉₁	C₂×C₄	1	1	yes	characteristic
25	C₂×C₈×C₁₉₁	C₂×C₄	1	1	yes	characteristic
26	C₄²×C₁₉₁	C₂³	1	1	yes	characteristic
27	Dic₃₈₂	—	2	2	no	—
28	Dic₃₈₂	—	2	2	no	—
29	C₂×Dic₁₉₁	—	2	2	no	—
30	C₂×Dic₁₉₁	—	2	2	no	—
31	C₁₉₁⋊C₈	—	2	2	no	—
32	C₁₉₁⋊C₈	—	4	1	no	—
33	C₂×C₄×C₁₉₁	C₂²×C₄	1	1	yes	characteristic
34	C₂×C₄×C₁₉₁	C₂×D₄	1	1	yes	characteristic
35	C₂×C₄×C₁₉₁	#41	1	1	yes	characteristic
36	C₈×C₁₉₁	—	2	1	no	—
37	C₈×C₁₉₁	—	4	2	no	—
38	C₈×C₁₉₁	M₁₆	1	2	yes	—
39	C₂²×C₁₉₁	C₄×D₄	1	1	yes	characteristic
40	C₄×C₁₉₁	—	2	1	no	—
41	C₄×C₁₉₁	C₂×M₁₆	1	1	yes	characteristic
42	C₄×C₁₉₁	#131	1	1	yes	characteristic
43	C₄×C₁₉₁	#137	1	2	yes	—
44	Dic₁₉₁	—	4	2	no	—
45	Dic₁₉₁	—	4	2	no	—
46	C₂×C₁₉₁	#436	1	1	yes	characteristic
47	C₂×C₁₉₁	#441	1	1	yes	characteristic
48	C₂×C₁₉₁	#444	1	1	yes	characteristic
49	C₁₉₁	#1593	1	1	yes	characteristic, complemented, Sylow
50	#1593	—	191	1	no	maximal, Sylow
51	#446	—	191	2	no	—
52	C₄×Q₁₆	—	191	1	no	—
53	#334	D₁₉₁	1	1	yes	characteristic
54	Q₈⋊C₈	—	191	2	no	—
55	#322	—	191	1	no	—
56	C₂×Q₁₆	—	191	1	no	—
57	C₄×Q₈	—	191	2	no	—
58	#107	—	191	1	no	—
59	C₄⋊C₈	—	382	2	no	—
60	C₄⋊C₈	C₂×D₁₉₁	1	2	yes	—
61	#103	—	191	2	no	—
62	#97	—	382	2	no	—
63	C₄×C₈	—	191	1	no	—
64	C₄×C₈	—	191	1	no	—
65	C₄×C₈	C₂×D₁₉₁	1	1	yes	characteristic
66	C₂×Q₈	—	191	2	no	—
67	Q₁₆	—	382	2	no	—
68	C₂×C₈	—	2	2	no	—
69	C₂×C₈	—	382	1	no	—
70	C₂×C₈	—	191	2	no	—
71	C₂×C₈	C₄×D₁₉₁	1	1	yes	characteristic
72	C₂×C₈	C₄×D₁₉₁	1	1	yes	characteristic
73	C₄⋊C₄	—	382	2	no	—
74	C₄⋊C₄	—	191	2	no	—
75	C₄²	—	382	2	no	—
76	C₄²	C₂²×D₁₉₁	1	1	yes	characteristic
77	Q₈	—	382	2	no	—
78	Q₈	—	382	2	no	—
79	C₂×C₄	—	382	2	no	—
80	C₂×C₄	—	382	2	no	—
81	C₂×C₄	C₂×C₄×D₁₉₁	1	1	yes	characteristic
82	C₂×C₄	D₄×D₁₉₁	1	1	yes	characteristic
83	C₂×C₄	#15690288	1	1	yes	characteristic
84	C₈	—	2	1	no	—
85	C₈	—	4	2	no	—
86	C₈	—	382	2	no	—
87	C₈	—	764	1	no	—
88	C₈	#15690270	1	2	yes	—
89	C₂²	#20259284	1	1	yes	characteristic
90	C₄	—	2	1	no	—
91	C₄	—	764	2	no	—
92	C₄	—	764	2	no	—
93	C₄	C₂×#15690270	1	1	yes	characteristic
94	C₄	#20259330	1	1	yes	characteristic
95	C₄	#20259351	1	2	yes	—
96	C₂	#27132740	1	1	yes	characteristic
97	C₂	#27132769	1	1	yes	characteristic
98	C₂	#27132784	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#27132790

C₂

yes

maximal

C₄×Dic₇₆₄

C₂

yes

maximal, characteristic

#27132393

C₂

yes

maximal

#27132384

C₂

yes

maximal, characteristic

#334×C₁₉₁

C₂

yes

maximal, characteristic

C₂×Dic₇₆₄

C₄

yes

characteristic

C₄×Dic₃₈₂

C₂²

yes

—

#20259256

C₄

yes

characteristic

Dic₁₉₁⋊C₈

—

#20259247

C₄

yes

—

#20259232

—

C₄×C₁₉₁⋊C₈

C₂²

yes

characteristic

C₄×C₁₉₁⋊C₈

C₂²

yes

characteristic

C₄⋊C₈×C₁₉₁

C₂²

yes

—

C₄×C₈×C₁₉₁

C₂²

yes

characteristic

C₂×Dic₃₈₂

C₂×C₄

yes

—

Dic₇₆₄

—

C₂×C₁₉₁⋊C₈

—

C₂×C₁₉₁⋊C₈

D₄

yes

—

C₄⋊Dic₁₉₁

C₂×C₄

yes

—

Dic₁₉₁⋊C₄

—

C₄×Dic₁₉₁

—

C₂×C₈×C₁₉₁

—

C₂×C₈×C₁₉₁

C₂×C₄

yes

characteristic

C₂×C₈×C₁₉₁

C₂×C₄

yes

characteristic

C₄²×C₁₉₁

C₂³

yes

characteristic

Dic₃₈₂

—

Dic₃₈₂

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

—

C₁₉₁⋊C₈

—

C₁₉₁⋊C₈

—

C₂×C₄×C₁₉₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₉₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₁

#41

yes

characteristic

C₈×C₁₉₁

—

C₈×C₁₉₁

—

C₈×C₁₉₁

M₁₆

yes

—

C₂²×C₁₉₁

C₄×D₄

yes

characteristic

C₄×C₁₉₁

—

C₄×C₁₉₁

C₂×M₁₆

yes

C₄×C₁₉₁

yes

C₄×C₁₉₁

yes

—

Dic₁₉₁

—

Dic₁₉₁

—

C₂×C₁₉₁

yes

C₂×C₁₉₁

yes

C₂×C₁₉₁

yes

C₁₉₁

yes

characteristic, complemented, Sylow

#1593

—

191

maximal, Sylow

#446

—

191

—

C₄×Q₁₆

—

191

—

#334

D₁₉₁

yes

characteristic

Q₈⋊C₈

—

191

—

#322

—

191

—

C₂×Q₁₆

—

191

—

C₄×Q₈

—

191

—

#107

—

191

—

C₄⋊C₈

—

382

—

C₄⋊C₈

C₂×D₁₉₁

yes

—

#103

—

191

—

#97

—

382

—

C₄×C₈

—

191

—

C₄×C₈

—

191

—

C₄×C₈

C₂×D₁₉₁

yes

characteristic

C₂×Q₈

—

191

—

Q₁₆

—

382

—

C₂×C₈

—

C₂×C₈

—

382

—

C₂×C₈

—

191

—

C₂×C₈

C₄×D₁₉₁

yes

characteristic

C₂×C₈

C₄×D₁₉₁

yes

characteristic

C₄⋊C₄

—

382

—

C₄⋊C₄

—

191

—

C₄²

—

382

—

C₄²

C₂²×D₁₉₁

yes

characteristic

Q₈

—

382

—

Q₈

—

382

—

C₂×C₄

—

382

—

C₂×C₄

—

382

—

C₂×C₄

C₂×C₄×D₁₉₁

yes

characteristic

C₂×C₄

D₄×D₁₉₁

yes

characteristic

C₂×C₄

#15690288

yes

characteristic

C₈

—

C₈

—

C₈

—

382

—

C₈

—

764

—

C₈

#15690270

yes

—

C₂²

#20259284

yes

characteristic

C₄

—

C₄

—

764

—

C₄

—

764

—

C₄

yes

C₄

yes

C₄

yes

—

C₂

yes

C₂

yes

C₂

yes

	subgroup	G/H
center	C₂×C₄	D₄×D₁₉₁
commutator	C₂×C₄×C₁₉₁	C₂²×C₄
Frattini	C₄²	C₂²×D₁₉₁
Fitting	#334×C₁₉₁	C₂
socle, prime radical	C₂²×C₁₉₁	C₄×D₄
cosocle	C₄²×C₁₉₁	C₂³
hypercenter	#334	D₁₉₁
hypocenter	C₁₉₁	#1593
LCS	G ▷ C₂×C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ #334

subgroup

G/H

center

C₂×C₄

D₄×D₁₉₁

commutator

C₂×C₄×C₁₉₁

C₂²×C₄

Frattini

C₄²

C₂²×D₁₉₁

Fitting

#334×C₁₉₁

C₂

socle, prime radical

C₂²×C₁₉₁

C₄×D₄

cosocle

C₄²×C₁₉₁

C₂³

D₁₉₁

C₁₉₁

G ▷ C₂×C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ #334

N	H ≅ G/N	type
49. C₁₉₁	#1593	semidirect

H ≅ G/N

type

49. C₁₉₁

#1593

semidirect

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

subgroup	index
#27132790	2
C₄×Dic₇₆₄	2
#27132393	2
#27132384	2
#334×C₁₉₁	2
#1593	191

subgroup

index

C₄×Dic₇₆₄

#334×C₁₉₁

191

Subgroups — #37571428

Conjugacy classes of subgroups

Complements

Maximal subgroups