Group #37574739 — subgroups

Subgroups — #37574739

Order 24448.

124 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 124 conjugacy classes of subgroups — too many to draw. Only its 45 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	#27132921	C₂	1	1	yes	maximal, characteristic
2	#27132487	C₂	1	2	yes	maximal, complemented
3	#27132478	C₂	1	2	yes	maximal, complemented
4	#27132445	C₂	1	1	yes	maximal, characteristic
5	#430×C₁₉₁	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₇₆₄	C₂²	1	1	yes	characteristic
7	#20259301	—	2	1	no	—
8	#20259262	C₂²	1	2	yes	—
9	#20259254	C₂²	1	1	yes	characteristic
10	#20259247	—	2	1	no	—
11	C₂×M₁₆×C₁₉₁	—	2	1	no	—
12	C₂×M₁₆×C₁₉₁	C₂²	1	1	yes	characteristic
13	#108×C₁₉₁	—	2	1	no	—
14	#108×C₁₉₁	C₂²	1	2	yes	—
15	C₂×Dic₃₈₂	—	2	1	no	—
16	Dic₇₆₄	—	2	2	no	—
17	C₄⋊Dic₁₉₁	—	2	1	no	—
18	Dic₁₉₁⋊C₄	—	4	1	no	—
19	C₂²⋊Dic₁₉₁	—	4	1	no	—
20	C₁₉₁⋊C₁₆	—	2	2	no	—
21	C₂²×C₄×C₁₉₁	D₄	1	1	yes	characteristic
22	M₁₆×C₁₉₁	—	2	2	no	—
23	M₁₆×C₁₉₁	—	2	2	no	—
24	M₁₆×C₁₉₁	—	4	1	no	—
25	C₂×C₈×C₁₉₁	—	4	1	no	—
26	C₂×C₈×C₁₉₁	D₄	1	1	yes	characteristic
27	C₂×C₈×C₁₉₁	C₂³	1	1	yes	characteristic
28	Dic₃₈₂	—	4	1	no	—
29	Dic₃₈₂	—	4	1	no	—
30	C₂×Dic₁₉₁	—	4	1	no	—
31	C₂×Dic₁₉₁	—	4	1	no	—
32	C₂³×C₁₉₁	SD₁₆	1	1	yes	characteristic
33	C₂×C₄×C₁₉₁	—	2	1	no	—
34	C₂×C₄×C₁₉₁	—	2	1	no	—
35	C₂×C₄×C₁₉₁	SD₁₆	1	1	yes	characteristic
36	C₂×C₄×C₁₉₁	C₂×D₄	1	1	yes	characteristic
37	C₈×C₁₉₁	—	2	1	no	—
38	C₈×C₁₉₁	—	4	2	no	—
39	C₈×C₁₉₁	#41	1	2	yes	—
40	C₂²×C₁₉₁	—	2	1	no	—
41	C₂²×C₁₉₁	—	4	1	no	—
42	C₂²×C₁₉₁	C₂×SD₁₆	1	1	yes	characteristic
43	C₄×C₁₉₁	—	2	1	no	—
44	C₄×C₁₉₁	#123	1	1	yes	characteristic
45	C₄×C₁₉₁	AGL(1,ℤ/8)	1	1	yes	characteristic
46	Dic₁₉₁	—	8	1	no	—
47	Dic₁₉₁	—	8	1	no	—
48	C₂×C₁₉₁	—	2	1	no	—
49	C₂×C₁₉₁	—	4	1	no	—
50	C₂×C₁₉₁	#481	1	1	yes	characteristic
51	C₁₉₁	#2239	1	1	yes	characteristic, complemented, Sylow
52	#2239	—	191	1	no	maximal, Sylow
53	#470	—	191	1	no	—
54	#430	D₁₉₁	1	1	yes	characteristic
55	#365	—	191	2	no	—
56	#362	—	191	2	no	—
57	#349	—	191	1	no	—
58	C₂×Q₁₆	—	191	1	no	—
59	C₂×M₁₆	—	2	1	no	—
60	C₂×M₁₆	C₂×D₁₉₁	1	1	yes	characteristic
61	C₂²⋊Q₈	—	382	1	no	—
62	M₃₂	—	191	2	no	—
63	#108	—	2	1	no	—
64	#108	C₂×D₁₉₁	1	2	yes	—
65	#106	—	191	1	no	—
66	#103	—	382	1	no	—
67	C₂×Q₈	—	382	1	no	—
68	C₂²×C₄	C₂²⋊D₁₉₁	1	1	yes	characteristic
69	Q₁₆	—	382	2	no	—
70	M₁₆	—	2	2	no	—
71	M₁₆	—	2	2	no	—
72	M₁₆	—	4	1	no	—
73	C₂×C₈	—	4	1	no	—
74	C₂×C₈	C₂²⋊D₁₉₁	1	1	yes	characteristic
75	C₂×C₈	C₂²×D₁₉₁	1	1	yes	characteristic
76	C₄⋊C₄	—	382	1	no	—
77	C₄⋊C₄	—	764	1	no	—
78	C₂²⋊C₄	—	764	1	no	—
79	C₁₆	—	382	2	no	—
80	C₂³	#15690274	1	1	yes	characteristic
81	Q₈	—	764	1	no	—
82	Q₈	—	764	1	no	—
83	C₂×C₄	—	2	1	no	—
84	C₂×C₄	—	2	1	no	—
85	C₂×C₄	—	764	1	no	—
86	C₂×C₄	—	764	1	no	—
87	C₂×C₄	#15690274	1	1	yes	characteristic
88	C₂×C₄	C₂×C₂²⋊D₁₉₁	1	1	yes	characteristic
89	C₈	—	2	1	no	—
90	C₈	—	4	2	no	—
91	C₈	#15690287	1	2	yes	—
92	C₂²	—	2	1	no	—
93	C₂²	—	4	1	no	—
94	C₂²	C₂×#15690274	1	1	yes	characteristic
95	C₄	—	2	1	no	—
96	C₄	—	1528	1	no	—
97	C₄	—	1528	1	no	—
98	C₄	#20259305	1	1	yes	characteristic
99	C₄	#20259348	1	1	yes	characteristic
100	C₂	—	2	1	no	—
101	C₂	—	4	1	no	—
102	C₂	#27132981	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#27132921

C₂

yes

maximal, characteristic

#27132487

C₂

yes

maximal, complemented

#27132478

C₂

yes

maximal, complemented

#27132445

C₂

yes

maximal, characteristic

#430×C₁₉₁

C₂

yes

maximal, characteristic

C₂×Dic₇₆₄

C₂²

yes

characteristic

#20259301

—

#20259262

C₂²

yes

—

#20259254

C₂²

yes

characteristic

#20259247

—

C₂×M₁₆×C₁₉₁

—

C₂×M₁₆×C₁₉₁

C₂²

yes

characteristic

#108×C₁₉₁

—

#108×C₁₉₁

C₂²

yes

—

C₂×Dic₃₈₂

—

Dic₇₆₄

—

C₄⋊Dic₁₉₁

—

Dic₁₉₁⋊C₄

—

C₂²⋊Dic₁₉₁

—

C₁₉₁⋊C₁₆

—

C₂²×C₄×C₁₉₁

D₄

yes

characteristic

M₁₆×C₁₉₁

—

M₁₆×C₁₉₁

—

M₁₆×C₁₉₁

—

C₂×C₈×C₁₉₁

—

C₂×C₈×C₁₉₁

D₄

yes

characteristic

C₂×C₈×C₁₉₁

C₂³

yes

characteristic

Dic₃₈₂

—

Dic₃₈₂

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

—

C₂³×C₁₉₁

SD₁₆

yes

characteristic

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

SD₁₆

yes

characteristic

C₂×C₄×C₁₉₁

C₂×D₄

yes

characteristic

C₈×C₁₉₁

—

C₈×C₁₉₁

—

C₈×C₁₉₁

#41

yes

—

C₂²×C₁₉₁

—

C₂²×C₁₉₁

—

C₂²×C₁₉₁

C₂×SD₁₆

yes

characteristic

C₄×C₁₉₁

—

C₄×C₁₉₁

yes

C₄×C₁₉₁

yes

Dic₁₉₁

—

Dic₁₉₁

—

C₂×C₁₉₁

—

C₂×C₁₉₁

—

C₂×C₁₉₁

#481

yes

characteristic

C₁₉₁

#2239

yes

characteristic, complemented, Sylow

#2239

—

191

maximal, Sylow

#470

—

191

—

#430

D₁₉₁

yes

characteristic

#365

—

191

—

#362

—

191

—

#349

—

191

—

C₂×Q₁₆

—

191

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₁₉₁

yes

characteristic

C₂²⋊Q₈

—

382

—

M₃₂

—

191

—

#108

—

#108

C₂×D₁₉₁

yes

—

#106

—

191

—

#103

—

382

—

C₂×Q₈

—

382

—

C₂²×C₄

C₂²⋊D₁₉₁

yes

characteristic

Q₁₆

—

382

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

C₂²⋊D₁₉₁

yes

characteristic

C₂×C₈

C₂²×D₁₉₁

yes

characteristic

C₄⋊C₄

—

382

—

C₄⋊C₄

—

764

—

C₂²⋊C₄

—

764

—

C₁₆

—

382

—

C₂³

#15690274

yes

characteristic

Q₈

—

764

—

Q₈

—

764

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

764

—

C₂×C₄

—

764

—

C₂×C₄

#15690274

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₁₉₁

yes

characteristic

C₈

—

C₈

—

C₈

#15690287

yes

—

C₂²

—

C₂²

—

C₂²

C₂×#15690274

yes

characteristic

C₄

—

C₄

—

1528

—

C₄

—

1528

—

C₄

yes

C₄

yes

100

C₂

—

101

C₂

—

102

C₂

#27132981

yes

characteristic

	subgroup	G/H
center	C₂	#27132981
commutator, cosocle	C₂×C₈×C₁₉₁	C₂³
Frattini	C₂×C₈	C₂²×D₁₉₁
Fitting	#430×C₁₉₁	C₂
socle	C₂×C₁₉₁	#481
prime radical	C₂³×C₁₉₁	SD₁₆
hypercenter	#430	D₁₉₁
hypocenter	C₁₉₁	#2239
LCS	G ▷ C₂×C₈×C₁₉₁ ▷ C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×M₁₆ ◁ #430

subgroup

G/H

center

C₂

#27132981

commutator, cosocle

C₂×C₈×C₁₉₁

C₂³

Frattini

C₂×C₈

C₂²×D₁₉₁

Fitting

#430×C₁₉₁

C₂

socle

C₂×C₁₉₁

#481

prime radical

C₂³×C₁₉₁

SD₁₆

D₁₉₁

C₁₉₁

G ▷ C₂×C₈×C₁₉₁ ▷ C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×M₁₆ ◁ #430

N	H ≅ G/N	type
2. #27132487	C₂	semidirect
3. #27132478	C₂	semidirect
51. C₁₉₁	#2239	semidirect

H ≅ G/N

type

2. #27132487

C₂

semidirect

3. #27132478

C₂

semidirect

51. C₁₉₁

#2239

semidirect

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

subgroup	index
#27132921	2
#27132487	2
#27132478	2
#27132445	2
#430×C₁₉₁	2
#2239	191

subgroup

index

#430×C₁₉₁

191

Subgroups — #37574739

Conjugacy classes of subgroups

Complements

Maximal subgroups