Group #26109952 — subgroups

Subgroups — #26109952

Order 12032.

322 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 147 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	#20243505	C₂	1	2	yes	maximal
2	#20241223	C₂	1	4	yes	maximal
3	C₄₇×#1744	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	4	yes	—
5	#15688878	—	2	4	no	—
6	C₂×C₄×C₈×C₄₇	C₂²	1	2	yes	—
7	C₄³×C₄₇	C₂²	1	1	yes	characteristic
8	C₄₇×#336	C₄	1	4	yes	—
9	C₂²×C₄₇⋊C₈	—	2	4	no	—
10	C₄⋊(C₄₇⋊C₈)	—	2	8	no	—
11	C₂²×C₈×C₄₇	C₂×C₄	1	4	yes	—
12	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
13	C₂×C₄²×C₄₇	D₄	1	2	yes	—
14	C₂×C₄²×C₄₇	Q₈	1	2	yes	—
15	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
16	C₄×C₈×C₄₇	—	2	4	no	—
17	C₂²×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
18	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
19	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
20	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
21	C₂×C₈×C₄₇	—	2	4	no	—
22	C₂×C₈×C₄₇	—	2	8	no	—
23	C₄²×C₄₇	—	2	4	no	—
24	C₄²×C₄₇	—	2	4	no	—
25	C₄²×C₄₇	—	4	2	no	—
26	C₄²×C₄₇	#41	1	4	yes	—
27	C₂×C₄₇⋊C₈	—	4	4	no	—
28	C₂×C₄₇⋊C₈	—	4	8	no	—
29	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
30	C₂×C₄×C₄₇	—	2	2	no	—
31	C₂×C₄×C₄₇	—	2	4	no	—
32	C₂×C₄×C₄₇	—	2	4	no	—
33	C₂×C₄×C₄₇	—	2	4	no	—
34	C₂×C₄×C₄₇	#117	1	2	yes	—
35	C₂×C₄×C₄₇	#124	1	2	yes	—
36	C₂×C₄×C₄₇	#125	1	2	yes	—
37	C₂×C₄×C₄₇	#131	1	4	yes	—
38	C₂×C₄×C₄₇	#135	1	4	yes	—
39	C₈×C₄₇	—	4	4	no	—
40	C₄₇⋊C₈	—	8	4	no	—
41	#8788	—	47	1	no	maximal, Sylow
42	C₂²×C₄₇	#382	1	1	yes	characteristic
43	C₂²×C₄₇	#424	1	2	yes	—
44	C₂²×C₄₇	#427	1	2	yes	—
45	C₂²×C₄₇	#430	1	2	yes	—
46	C₄×C₄₇	—	2	2	no	—
47	C₄×C₄₇	—	4	4	no	—
48	C₄×C₄₇	#433	1	4	yes	—
49	C₄×C₄₇	#488	1	4	yes	—
50	#1831	—	47	2	no	—
51	#1744	D₄₇	1	1	yes	characteristic
52	#1277	—	47	4	no	—
53	C₂×C₄₇	#1572	1	4	yes	—
54	C₂×C₄₇	#1832	1	1	yes	characteristic
55	C₂×C₄₇	#1835	1	1	yes	characteristic
56	C₂×C₄₇	#1838	1	1	yes	characteristic
57	C₂×C₄⋊C₈	—	47	4	no	—
58	C₂×C₄×C₈	C₂×D₄₇	1	2	yes	—
59	C₄³	C₂×D₄₇	1	1	yes	characteristic
60	#336	—	94	4	no	—
61	#336	Dic₄₇	1	4	yes	—
62	C₄₇	#8788	1	1	yes	characteristic, complemented, Sylow
63	C₂²×C₈	—	94	4	no	—
64	C₂²×C₈	C₂×Dic₄₇	1	4	yes	—
65	C₂×C₄²	Dic₉₄	1	2	yes	—
66	C₂×C₄²	D₁₈₈	1	2	yes	—
67	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
68	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
69	C₄⋊C₈	—	94	8	no	—
70	C₄×C₈	—	2	4	no	—
71	C₂²×C₄	C₄⋊Dic₄₇	1	4	yes	—
72	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
73	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
74	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
75	C₂×C₈	—	2	4	no	—
76	C₂×C₈	—	2	8	no	—
77	C₂×C₈	—	188	4	no	—
78	C₂×C₈	—	188	8	no	—
79	C₄²	—	2	4	no	—
80	C₄²	—	2	4	no	—
81	C₄²	—	4	2	no	—
82	C₄²	#10625350	1	4	yes	—
83	C₂³	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
84	C₂×C₄	—	2	2	no	—
85	C₂×C₄	—	2	4	no	—
86	C₂×C₄	—	2	4	no	—
87	C₂×C₄	—	2	4	no	—
88	C₂×C₄	#13093983	1	2	yes	—
89	C₂×C₄	#13094012	1	2	yes	—
90	C₂×C₄	#13094016	1	2	yes	—
91	C₂×C₄	#13094031	1	4	yes	—
92	C₂×C₄	#13094046	1	4	yes	—
93	C₈	—	4	4	no	—
94	C₈	—	376	4	no	—
95	C₂²	#15688989	1	1	yes	characteristic
96	C₂²	#15689150	1	2	yes	—
97	C₂²	#15689161	1	2	yes	—
98	C₂²	#15689172	1	2	yes	—
99	C₄	—	2	2	no	—
100	C₄	—	4	4	no	—
101	C₄	#15689184	1	4	yes	—
102	C₄	#15689478	1	4	yes	—
103	C₂	#20242168	1	4	yes	—
104	C₂	#20243508	1	1	yes	characteristic
105	C₂	#20243519	1	1	yes	characteristic
106	C₂	#20243529	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20243505

C₂

yes

maximal

#20241223

C₂

yes

maximal

C₄₇×#1744

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₄₇⋊C₈)

C₂²

yes

—

#15688878

—

C₂×C₄×C₈×C₄₇

C₂²

yes

—

C₄³×C₄₇

C₂²

yes

characteristic

C₄₇×#336

C₄

yes

—

C₂²×C₄₇⋊C₈

—

C₄⋊(C₄₇⋊C₈)

—

C₂²×C₈×C₄₇

C₂×C₄

yes

—

C₂×C₄²×C₄₇

C₂×C₄

yes

—

C₂×C₄²×C₄₇

D₄

yes

—

C₂×C₄²×C₄₇

Q₈

yes

—

C₂×C₄²×C₄₇

C₂³

yes

characteristic

C₄×C₈×C₄₇

—

C₂²×C₄×C₄₇

C₄⋊C₄

yes

—

C₂²×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₄₇

C₂×Q₈

yes

characteristic

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

#41

yes

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

#117

yes

—

C₂×C₄×C₄₇

#124

yes

—

C₂×C₄×C₄₇

#125

yes

—

C₂×C₄×C₄₇

#131

yes

—

C₂×C₄×C₄₇

#135

yes

—

C₈×C₄₇

—

C₄₇⋊C₈

—

—

C₂²×C₄₇

yes

C₂²×C₄₇

yes

—

C₂²×C₄₇

#427

yes

—

C₂²×C₄₇

#430

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#433

yes

—

C₄×C₄₇

#488

yes

—

#1831

—

#1744

D₄₇

yes

characteristic

#1277

—

C₂×C₄₇

#1572

yes

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄⋊C₈

—

C₂×C₄×C₈

C₂×D₄₇

yes

—

C₄³

C₂×D₄₇

yes

characteristic

#336

—

#336

Dic₄₇

yes

—

C₄₇

#8788

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

C₂×Dic₄₇

yes

—

C₂×C₄²

Dic₉₄

yes

—

C₂×C₄²

D₁₈₈

yes

—

C₂×C₄²

C₂×Dic₄₇

yes

—

C₂×C₄²

C₂²×D₄₇

yes

characteristic

C₄⋊C₈

—

C₄×C₈

—

C₂²×C₄

C₄⋊Dic₄₇

yes

—

C₂²×C₄

C₂×Dic₉₄

yes

characteristic

C₂²×C₄

C₂×D₁₈₈

yes

characteristic

C₂²×C₄

C₂²×Dic₄₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

188

—

C₂×C₈

—

188

—

C₄²

—

C₄²

—

C₄²

—

C₄²

#10625350

yes

—

C₂³

C₂×C₄⋊Dic₄₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#13093983

yes

—

C₂×C₄

#13094012

yes

—

C₂×C₄

#13094016

yes

—

C₂×C₄

#13094031

yes

—

C₂×C₄

#13094046

yes

—

C₈

—

C₈

—

376

—

C₂²

#15688989

yes

characteristic

C₂²

#15689150

yes

—

C₂²

#15689161

yes

—

C₂²

#15689172

yes

—

C₄

—

100

C₄

—

101

C₄

#15689184

yes

—

102

C₄

#15689478

yes

—

103

C₂

#20242168

yes

—

104

C₂

#20243508

yes

characteristic

105

C₂

#20243519

yes

characteristic

106

C₂

#20243529

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₁₈₈
commutator	C₂²×C₄×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1744	C₂
socle, prime radical	C₂³×C₄₇	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#1744	D₄₇
hypocenter	C₄₇	#8788
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1744

subgroup

G/H

center

C₂²×C₄

C₂×D₁₈₈

commutator

C₂²×C₄×C₄₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₄₇

Fitting

C₄₇×#1744

C₂

socle, prime radical

C₂³×C₄₇

C₂×C₄⋊C₄

cosocle

C₂×C₄²×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1744

N	H ≅ G/N	type
62. C₄₇	#8788	semidirect

H ≅ G/N

type

62. C₄₇

#8788

semidirect

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

subgroup	index
#20243505	2
#20241223	2
C₄₇×#1744	2
#8788	47

subgroup

index

#20243505

#20241223

C₄₇×#1744

#8788

Subgroups — #26109952

Conjugacy classes of subgroups

Complements

Maximal subgroups