Group #24990271 — subgroups

Subgroups — #24990271

Order 11008.

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	#19134712	C₂	1	2	yes	maximal
2	#19132430	C₂	1	4	yes	maximal
3	C₄₃×#1744	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	4	yes	—
5	#14582654	—	2	4	no	—
6	C₂×C₄×C₈×C₄₃	C₂²	1	2	yes	—
7	C₄³×C₄₃	C₂²	1	1	yes	characteristic
8	C₄₃×#336	C₄	1	4	yes	—
9	C₂²×C₄₃⋊C₈	—	2	4	no	—
10	C₄⋊(C₄₃⋊C₈)	—	2	8	no	—
11	C₂²×C₈×C₄₃	C₂×C₄	1	4	yes	—
12	C₂×C₄²×C₄₃	C₂×C₄	1	2	yes	—
13	C₂×C₄²×C₄₃	D₄	1	2	yes	—
14	C₂×C₄²×C₄₃	Q₈	1	2	yes	—
15	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
16	C₄×C₈×C₄₃	—	2	4	no	—
17	C₂²×C₄×C₄₃	C₄⋊C₄	1	4	yes	—
18	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
19	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
20	C₂²×C₄×C₄₃	C₂×Q₈	1	1	yes	characteristic
21	C₂×C₈×C₄₃	—	2	4	no	—
22	C₂×C₈×C₄₃	—	2	8	no	—
23	C₄²×C₄₃	—	2	4	no	—
24	C₄²×C₄₃	—	2	4	no	—
25	C₄²×C₄₃	—	4	2	no	—
26	C₄²×C₄₃	#41	1	4	yes	—
27	C₂×C₄₃⋊C₈	—	4	4	no	—
28	C₂×C₄₃⋊C₈	—	4	8	no	—
29	C₂³×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
30	C₂×C₄×C₄₃	—	2	2	no	—
31	C₂×C₄×C₄₃	—	2	4	no	—
32	C₂×C₄×C₄₃	—	2	4	no	—
33	C₂×C₄×C₄₃	—	2	4	no	—
34	C₂×C₄×C₄₃	#117	1	2	yes	—
35	C₂×C₄×C₄₃	#124	1	2	yes	—
36	C₂×C₄×C₄₃	#125	1	2	yes	—
37	C₂×C₄×C₄₃	#131	1	4	yes	—
38	C₂×C₄×C₄₃	#135	1	4	yes	—
39	C₈×C₄₃	—	4	4	no	—
40	C₄₃⋊C₈	—	8	4	no	—
41	#8788	—	43	1	no	maximal, Sylow
42	C₂²×C₄₃	#382	1	1	yes	characteristic
43	C₂²×C₄₃	#424	1	2	yes	—
44	C₂²×C₄₃	#427	1	2	yes	—
45	C₂²×C₄₃	#430	1	2	yes	—
46	C₄×C₄₃	—	2	2	no	—
47	C₄×C₄₃	—	4	4	no	—
48	C₄×C₄₃	#433	1	4	yes	—
49	C₄×C₄₃	#488	1	4	yes	—
50	#1831	—	43	2	no	—
51	#1744	D₄₃	1	1	yes	characteristic
52	#1277	—	43	4	no	—
53	C₂×C₄₃	#1572	1	4	yes	—
54	C₂×C₄₃	#1832	1	1	yes	characteristic
55	C₂×C₄₃	#1835	1	1	yes	characteristic
56	C₂×C₄₃	#1838	1	1	yes	characteristic
57	C₂×C₄⋊C₈	—	43	4	no	—
58	C₂×C₄×C₈	C₂×D₄₃	1	2	yes	—
59	C₄³	C₂×D₄₃	1	1	yes	characteristic
60	#336	—	86	4	no	—
61	#336	Dic₄₃	1	4	yes	—
62	C₄₃	#8788	1	1	yes	characteristic, complemented, Sylow
63	C₂²×C₈	—	86	4	no	—
64	C₂²×C₈	C₂×Dic₄₃	1	4	yes	—
65	C₂×C₄²	Dic₈₆	1	2	yes	—
66	C₂×C₄²	D₁₇₂	1	2	yes	—
67	C₂×C₄²	C₂×Dic₄₃	1	2	yes	—
68	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
69	C₄⋊C₈	—	86	8	no	—
70	C₄×C₈	—	2	4	no	—
71	C₂²×C₄	C₄⋊Dic₄₃	1	4	yes	—
72	C₂²×C₄	C₂×Dic₈₆	1	1	yes	characteristic
73	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
74	C₂²×C₄	C₂²×Dic₄₃	1	1	yes	characteristic
75	C₂×C₈	—	2	4	no	—
76	C₂×C₈	—	2	8	no	—
77	C₂×C₈	—	172	4	no	—
78	C₂×C₈	—	172	8	no	—
79	C₄²	—	2	4	no	—
80	C₄²	—	2	4	no	—
81	C₄²	—	4	2	no	—
82	C₄²	#10621898	1	4	yes	—
83	C₂³	C₂×C₄⋊Dic₄₃	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	#13063284	1	2	yes	—
89	C₂×C₄	#13063313	1	2	yes	—
90	C₂×C₄	#13063317	1	2	yes	—
91	C₂×C₄	#13063332	1	4	yes	—
92	C₂×C₄	#13063347	1	4	yes	—
93	C₈	—	4	4	no	—
94	C₈	—	344	4	no	—
95	C₂²	#14582765	1	1	yes	characteristic
96	C₂²	#14582926	1	2	yes	—
97	C₂²	#14582937	1	2	yes	—
98	C₂²	#14582948	1	2	yes	—
99	C₄	—	2	2	no	—
100	C₄	—	4	4	no	—
101	C₄	#14582960	1	4	yes	—
102	C₄	#14583254	1	4	yes	—
103	C₂	#19133375	1	4	yes	—
104	C₂	#19134715	1	1	yes	characteristic
105	C₂	#19134726	1	1	yes	characteristic
106	C₂	#19134736	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19134712

C₂

yes

maximal

#19132430

C₂

yes

maximal

C₄₃×#1744

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₄₃⋊C₈)

C₂²

yes

—

#14582654

—

C₂×C₄×C₈×C₄₃

C₂²

yes

—

C₄³×C₄₃

C₂²

yes

characteristic

C₄₃×#336

C₄

yes

—

C₂²×C₄₃⋊C₈

—

C₄⋊(C₄₃⋊C₈)

—

C₂²×C₈×C₄₃

C₂×C₄

yes

—

C₂×C₄²×C₄₃

C₂×C₄

yes

—

C₂×C₄²×C₄₃

D₄

yes

—

C₂×C₄²×C₄₃

Q₈

yes

—

C₂×C₄²×C₄₃

C₂³

yes

characteristic

C₄×C₈×C₄₃

—

C₂²×C₄×C₄₃

C₄⋊C₄

yes

—

C₂²×C₄×C₄₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₄₃

C₂×Q₈

yes

characteristic

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₄²×C₄₃

—

C₄²×C₄₃

—

C₄²×C₄₃

—

C₄²×C₄₃

#41

yes

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂³×C₄₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

#117

yes

—

C₂×C₄×C₄₃

#124

yes

—

C₂×C₄×C₄₃

#125

yes

—

C₂×C₄×C₄₃

#131

yes

—

C₂×C₄×C₄₃

#135

yes

—

C₈×C₄₃

—

C₄₃⋊C₈

—

—

C₂²×C₄₃

yes

C₂²×C₄₃

yes

—

C₂²×C₄₃

#427

yes

—

C₂²×C₄₃

#430

yes

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

#433

yes

—

C₄×C₄₃

#488

yes

—

#1831

—

#1744

D₄₃

yes

characteristic

#1277

—

C₂×C₄₃

#1572

yes

—

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄⋊C₈

—

C₂×C₄×C₈

C₂×D₄₃

yes

—

C₄³

C₂×D₄₃

yes

characteristic

#336

—

#336

Dic₄₃

yes

—

C₄₃

#8788

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

C₂×Dic₄₃

yes

—

C₂×C₄²

Dic₈₆

yes

—

C₂×C₄²

D₁₇₂

yes

—

C₂×C₄²

C₂×Dic₄₃

yes

—

C₂×C₄²

C₂²×D₄₃

yes

characteristic

C₄⋊C₈

—

C₄×C₈

—

C₂²×C₄

C₄⋊Dic₄₃

yes

—

C₂²×C₄

C₂×Dic₈₆

yes

characteristic

C₂²×C₄

C₂×D₁₇₂

yes

characteristic

C₂²×C₄

C₂²×Dic₄₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

172

—

C₂×C₈

—

172

—

C₄²

—

C₄²

—

C₄²

—

C₄²

#10621898

yes

—

C₂³

C₂×C₄⋊Dic₄₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#13063284

yes

—

C₂×C₄

#13063313

yes

—

C₂×C₄

#13063317

yes

—

C₂×C₄

#13063332

yes

—

C₂×C₄

#13063347

yes

—

C₈

—

C₈

—

344

—

C₂²

#14582765

yes

characteristic

C₂²

#14582926

yes

—

C₂²

#14582937

yes

—

C₂²

#14582948

yes

—

C₄

—

100

C₄

—

101

C₄

#14582960

yes

—

102

C₄

#14583254

yes

—

103

C₂

#19133375

yes

—

104

C₂

#19134715

yes

characteristic

105

C₂

#19134726

yes

characteristic

106

C₂

#19134736

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₁₇₂
commutator	C₂²×C₄×C₄₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₄₃×#1744	C₂
socle, prime radical	C₂³×C₄₃	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₄₃	C₂³
hypercenter	#1744	D₄₃
hypocenter	C₄₃	#8788
LCS	G ▷ C₂²×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1744

subgroup

G/H

center

C₂²×C₄

C₂×D₁₇₂

commutator

C₂²×C₄×C₄₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₄₃

Fitting

C₄₃×#1744

C₂

socle, prime radical

C₂³×C₄₃

C₂×C₄⋊C₄

cosocle

C₂×C₄²×C₄₃

C₂³

D₄₃

C₄₃

G ▷ C₂²×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1744

N	H ≅ G/N	type
62. C₄₃	#8788	semidirect

H ≅ G/N

type

62. C₄₃

#8788

semidirect

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

subgroup	index
#19134712	2
#19132430	2
C₄₃×#1744	2
#8788	43

subgroup

index

#19134712

#19132430

C₄₃×#1744

#8788

Subgroups — #24990271

Conjugacy classes of subgroups

Complements

Maximal subgroups