Group #25008477 — subgroups

Subgroups — #25008477

Order 11008.

214 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 93 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	#19133414	C₂	1	2	yes	maximal
2	#19132661	C₂	1	1	yes	maximal, characteristic
3	#19132565	C₂	1	2	yes	maximal
4	#19132515	C₂	1	1	yes	maximal, characteristic
5	C₄₃×#2176	C₂	1	1	yes	maximal, characteristic
6	Q₈×C₄₃⋊C₈	—	2	2	no	—
7	#14582675	C₂²	1	1	yes	characteristic
8	#14582650	C₂²	1	2	yes	—
9	Q₈⋊(C₄₃⋊C₈)	—	2	2	no	—
10	#14582620	C₂²	1	1	yes	characteristic
11	C₂×Q₃₂×C₄₃	C₄	1	1	yes	characteristic
12	C₄×Q₁₆×C₄₃	C₂²	1	2	yes	—
13	C₄₃×#367	C₄	1	1	yes	characteristic
14	C₄₃×#358	C₄	1	2	yes	—
15	C₄₃×#346	C₂²	1	1	yes	characteristic
16	C₂×C₄₃⋊C₁₆	—	2	1	no	—
17	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
18	C₄⋊(C₄₃⋊C₈)	—	4	2	no	—
19	C₄×C₄₃⋊C₈	—	4	2	no	—
20	C₄×C₄₃⋊C₈	D₄	1	2	yes	—
21	C₂×Q₁₆×C₄₃	C₂×C₄	1	2	yes	—
22	C₄×Q₈×C₄₃	—	2	2	no	—
23	Q₃₂×C₄₃	C₈	1	4	yes	complemented
24	C₂×C₁₆×C₄₃	C₂×C₄	1	1	yes	characteristic
25	C₂×C₁₆×C₄₃	C₂×C₄	1	1	yes	characteristic
26	#107×C₄₃	C₂×C₄	1	2	yes	—
27	#103×C₄₃	—	2	2	no	—
28	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
29	C₂×Q₈×C₄₃	—	2	2	no	—
30	Q₁₆×C₄₃	—	2	2	no	—
31	Q₁₆×C₄₃	C₂×C₈	1	4	yes	—
32	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
33	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
34	C₄⋊C₄×C₄₃	—	2	2	no	—
35	C₄⋊C₄×C₄₃	—	4	2	no	—
36	C₄²×C₄₃	—	4	2	no	—
37	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄₃⋊C₈	—	2	2	no	—
39	C₂×C₄₃⋊C₈	—	4	2	no	—
40	C₁₆×C₄₃	—	2	1	no	—
41	C₁₆×C₄₃	C₂×C₈	1	2	yes	—
42	C₄₃⋊C₁₆	—	4	1	no	—
43	Q₈×C₄₃	—	2	4	no	—
44	Q₈×C₄₃	—	4	2	no	—
45	C₂×C₄×C₄₃	—	4	2	no	—
46	C₂×C₄×C₄₃	—	4	2	no	—
47	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
48	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	2	yes	—
49	C₈×C₄₃	—	2	1	no	—
50	C₈×C₄₃	C₂²×C₈	1	1	yes	characteristic
51	C₈×C₄₃	#131	1	1	yes	characteristic
52	C₄₃⋊C₈	—	4	2	no	—
53	C₄₃⋊C₈	—	8	2	no	—
54	#12066	—	43	1	no	maximal, Sylow
55	C₂²×C₄₃	#442	1	1	yes	characteristic
56	C₄×C₄₃	—	2	2	no	—
57	C₄×C₄₃	—	4	4	no	—
58	C₄×C₄₃	—	8	2	no	—
59	C₄×C₄₃	C₈×D₄	1	1	yes	characteristic
60	C₄×C₄₃	#444	1	1	yes	characteristic
61	#2176	D₄₃	1	1	yes	characteristic
62	#1579	—	43	2	no	—
63	#1368	—	43	1	no	—
64	#1334	—	43	2	no	—
65	#1313	—	43	1	no	—
66	C₂×C₄₃	#1580	1	1	yes	characteristic
67	C₂×C₄₃	#2180	1	2	yes	—
68	C₂×Q₃₂	Dic₄₃	1	1	yes	characteristic
69	C₈×Q₈	—	86	2	no	—
70	C₄×Q₁₆	C₂×D₄₃	1	2	yes	—
71	#367	Dic₄₃	1	1	yes	characteristic
72	#358	Dic₄₃	1	2	yes	—
73	#346	—	43	1	no	—
74	#346	C₂×D₄₃	1	1	yes	characteristic
75	#334	—	43	2	no	—
76	Q₈⋊C₈	—	86	2	no	—
77	#322	—	43	1	no	—
78	C₄₃	#12066	1	1	yes	characteristic, complemented, Sylow
79	C₂×Q₁₆	C₂×Dic₄₃	1	2	yes	—
80	C₄×Q₈	—	2	2	no	—
81	Q₃₂	C₄₃⋊C₈	1	4	yes	complemented
82	C₂×C₁₆	—	86	1	no	—
83	C₂×C₁₆	C₂×Dic₄₃	1	1	yes	characteristic
84	C₂×C₁₆	C₂×Dic₄₃	1	1	yes	characteristic
85	#107	C₂×Dic₄₃	1	2	yes	—
86	C₄⋊C₈	—	86	2	no	—
87	C₄⋊C₈	—	172	2	no	—
88	#103	—	2	2	no	—
89	C₄×C₈	—	43	2	no	—
90	C₄×C₈	—	172	2	no	—
91	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
92	C₂×Q₈	—	2	2	no	—
93	Q₁₆	—	2	2	no	—
94	Q₁₆	C₂×C₄₃⋊C₈	1	4	yes	—
95	C₂×C₈	—	86	2	no	—
96	C₂×C₈	—	172	2	no	—
97	C₂×C₈	#10621900	1	1	yes	characteristic
98	C₂×C₈	C₂²×Dic₄₃	1	1	yes	characteristic
99	C₄⋊C₄	—	2	2	no	—
100	C₄⋊C₄	—	4	2	no	—
101	C₄²	—	4	2	no	—
102	C₄²	D₄×D₄₃	1	1	yes	characteristic
103	C₁₆	—	2	1	no	—
104	C₁₆	—	172	1	no	—
105	C₁₆	C₂×C₄₃⋊C₈	1	2	yes	—
106	Q₈	—	2	4	no	—
107	Q₈	—	4	2	no	—
108	C₂×C₄	—	4	2	no	—
109	C₂×C₄	—	4	2	no	—
110	C₂×C₄	D₄×Dic₄₃	1	1	yes	characteristic
111	C₂×C₄	#13063349	1	2	yes	—
112	C₈	—	2	1	no	—
113	C₈	—	344	2	no	—
114	C₈	—	172	2	no	—
115	C₈	C₂²×C₄₃⋊C₈	1	1	yes	characteristic
116	C₈	#13063332	1	1	yes	characteristic
117	C₂²	#14583011	1	1	yes	characteristic
118	C₄	—	2	2	no	—
119	C₄	—	4	4	no	—
120	C₄	—	8	2	no	—
121	C₄	D₄×C₄₃⋊C₈	1	1	yes	characteristic
122	C₄	#14583021	1	1	yes	characteristic
123	C₂	#19133419	1	1	yes	characteristic
124	C₂	#19136487	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19133414

C₂

yes

maximal

#19132661

C₂

yes

maximal, characteristic

#19132565

C₂

yes

maximal

#19132515

C₂

yes

maximal, characteristic

C₄₃×#2176

C₂

yes

maximal, characteristic

Q₈×C₄₃⋊C₈

—

#14582675

C₂²

yes

characteristic

#14582650

C₂²

yes

—

Q₈⋊(C₄₃⋊C₈)

—

#14582620

C₂²

yes

characteristic

C₂×Q₃₂×C₄₃

C₄

yes

characteristic

C₄×Q₁₆×C₄₃

C₂²

yes

—

C₄₃×#367

C₄

yes

characteristic

C₄₃×#358

C₄

yes

—

C₄₃×#346

C₂²

yes

characteristic

C₂×C₄₃⋊C₁₆

—

C₄⋊(C₄₃⋊C₈)

—

C₄⋊(C₄₃⋊C₈)

—

C₄×C₄₃⋊C₈

—

C₄×C₄₃⋊C₈

D₄

yes

—

C₂×Q₁₆×C₄₃

C₂×C₄

yes

—

C₄×Q₈×C₄₃

—

Q₃₂×C₄₃

C₈

yes

complemented

C₂×C₁₆×C₄₃

C₂×C₄

yes

characteristic

C₂×C₁₆×C₄₃

C₂×C₄

yes

characteristic

#107×C₄₃

C₂×C₄

yes

—

#103×C₄₃

—

C₄×C₈×C₄₃

C₂³

yes

characteristic

C₂×Q₈×C₄₃

—

Q₁₆×C₄₃

—

Q₁₆×C₄₃

C₂×C₈

yes

—

C₂×C₈×C₄₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₄₃

#41

yes

characteristic

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄²×C₄₃

—

C₄²×C₄₃

C₂×D₄

yes

characteristic

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₁₆×C₄₃

—

C₁₆×C₄₃

C₂×C₈

yes

—

C₄₃⋊C₁₆

—

Q₈×C₄₃

—

Q₈×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₄×D₄

yes

characteristic

C₂×C₄×C₄₃

AGL(1,ℤ/8)

yes

—

C₈×C₄₃

—

C₈×C₄₃

C₂²×C₈

yes

characteristic

C₈×C₄₃

#131

yes

characteristic

C₄₃⋊C₈

—

C₄₃⋊C₈

—

—

C₂²×C₄₃

yes

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

C₈×D₄

yes

C₄×C₄₃

yes

D₄₃

yes

—

—

—

—

C₂×C₄₃

#1580

yes

characteristic

C₂×C₄₃

#2180

yes

—

C₂×Q₃₂

Dic₄₃

yes

characteristic

C₈×Q₈

—

C₄×Q₁₆

C₂×D₄₃

yes

—

#367

Dic₄₃

yes

characteristic

#358

Dic₄₃

yes

—

#346

—

#346

C₂×D₄₃

yes

characteristic

#334

—

Q₈⋊C₈

—

#322

—

C₄₃

#12066

yes

characteristic, complemented, Sylow

C₂×Q₁₆

C₂×Dic₄₃

yes

—

C₄×Q₈

—

Q₃₂

C₄₃⋊C₈

yes

complemented

C₂×C₁₆

—

C₂×C₁₆

C₂×Dic₄₃

yes

characteristic

C₂×C₁₆

C₂×Dic₄₃

yes

characteristic

#107

C₂×Dic₄₃

yes

—

C₄⋊C₈

—

C₄⋊C₈

—

172

—

#103

—

C₄×C₈

—

C₄×C₈

—

172

—

C₄×C₈

C₂²×D₄₃

yes

characteristic

C₂×Q₈

—

Q₁₆

—

Q₁₆

C₂×C₄₃⋊C₈

yes

—

C₂×C₈

—

C₂×C₈

—

172

—

C₂×C₈

#10621900

yes

characteristic

C₂×C₈

C₂²×Dic₄₃

yes

characteristic

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄²

—

102

C₄²

D₄×D₄₃

yes

characteristic

103

C₁₆

—

104

C₁₆

—

172

—

105

C₁₆

C₂×C₄₃⋊C₈

yes

—

106

Q₈

—

107

Q₈

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

D₄×Dic₄₃

yes

characteristic

111

C₂×C₄

#13063349

yes

—

112

C₈

—

113

C₈

—

344

—

114

C₈

—

172

—

115

C₈

C₂²×C₄₃⋊C₈

yes

characteristic

116

C₈

#13063332

yes

characteristic

117

C₂²

#14583011

yes

characteristic

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

D₄×C₄₃⋊C₈

yes

characteristic

122

C₄

#14583021

yes

characteristic

123

C₂

#19133419

yes

characteristic

124

C₂

#19136487

yes

—

	subgroup	G/H
center	C₂²	#14583011
commutator	C₈×C₄₃	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₄₃
Fitting	C₄₃×#2176	C₂
socle, prime radical	C₂²×C₄₃	#442
cosocle	C₄×C₈×C₄₃	C₂³
hypercenter	#2176	D₄₃
hypocenter	C₄₃	#12066
LCS	G ▷ C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2176

subgroup

G/H

center

C₂²

#14583011

commutator

C₈×C₄₃

C₂²×C₈

Frattini

C₄×C₈

C₂²×D₄₃

Fitting

C₄₃×#2176

C₂

socle, prime radical

C₂²×C₄₃

#442

cosocle

C₄×C₈×C₄₃

C₂³

D₄₃

C₄₃

G ▷ C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2176

N	H ≅ G/N	type
23. Q₃₂×C₄₃	C₈	semidirect
78. C₄₃	#12066	semidirect
81. Q₃₂	C₄₃⋊C₈	semidirect

H ≅ G/N

type

23. Q₃₂×C₄₃

C₈

semidirect

78. C₄₃

#12066

semidirect

81. Q₃₂

C₄₃⋊C₈

semidirect

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

subgroup	index
#19133414	2
#19132661	2
#19132565	2
#19132515	2
C₄₃×#2176	2
#12066	43

subgroup

index

C₄₃×#2176

Subgroups — #25008477

Conjugacy classes of subgroups

Complements

Maximal subgroups