Group #25017570 — subgroups

Subgroups — #25017570

Order 11008.

182 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 182 conjugacy classes of subgroups — too many to draw. Only its 75 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	#19136870	C₂	1	1	yes	maximal, characteristic
2	#19136848	C₂	1	1	yes	maximal, characteristic
3	#19132777	C₂	1	4	yes	maximal
4	C₄×C₃₂×C₄₃	C₂	1	1	yes	maximal, characteristic
5	C₂×Dic₃₄₄	—	2	1	no	—
6	C₂×Dic₃₄₄	C₂²	1	2	yes	—
7	#14583304	—	2	1	no	—
8	#14583279	—	2	1	no	—
9	#14582725	—	2	1	no	—
10	#14582725	C₂²	1	2	yes	—
11	C₂×C₃₂×C₄₃	C₂²	1	2	yes	—
12	C₄×C₁₆×C₄₃	C₂²	1	1	yes	characteristic
13	C₂×Dic₁₇₂	—	2	2	no	—
14	C₂×Dic₁₇₂	—	4	1	no	—
15	#13063325	—	4	1	no	—
16	#13063325	—	4	1	no	—
17	Dic₃₄₄	—	2	4	no	—
18	Dic₃₄₄	—	4	2	no	—
19	#13063259	—	2	2	no	—
20	#13063259	—	4	1	no	—
21	C₂×C₁₆×C₄₃	D₄	1	1	yes	characteristic
22	C₂×C₁₆×C₄₃	C₂³	1	1	yes	characteristic
23	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
24	C₃₂×C₄₃	—	2	2	no	—
25	C₂×Dic₈₆	—	4	2	no	—
26	C₂×Dic₈₆	—	8	1	no	—
27	C₂×C₈×C₄₃	D₈	1	1	yes	characteristic
28	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
29	C₄²×C₄₃	D₈	1	1	yes	characteristic
30	C₄⋊Dic₄₃	—	4	2	no	—
31	C₄⋊Dic₄₃	—	8	1	no	—
32	C₄⋊Dic₄₃	—	8	1	no	—
33	C₄⋊Dic₄₃	—	8	1	no	—
34	Dic₁₇₂	—	4	2	no	—
35	Dic₁₇₂	—	4	2	no	—
36	Dic₁₇₂	—	8	2	no	—
37	C₁₆×C₄₃	SD₁₆	1	2	yes	—
38	C₁₆×C₄₃	#41	1	2	yes	—
39	C₂×C₄×C₄₃	D₁₆	1	1	yes	characteristic
40	C₂×C₄×C₄₃	D₁₆	1	1	yes	characteristic
41	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
42	C₂×Dic₄₃	—	8	2	no	—
43	C₂×Dic₄₃	—	8	2	no	—
44	Dic₈₆	—	8	2	no	—
45	Dic₈₆	—	8	2	no	—
46	Dic₈₆	—	16	2	no	—
47	C₈×C₄₃	SD₃₂	1	2	yes	—
48	C₈×C₄₃	#124	1	1	yes	characteristic
49	C₈×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
50	#13720	—	43	1	no	maximal, Sylow
51	C₂²×C₄₃	C₂×D₁₆	1	1	yes	characteristic
52	C₄×C₄₃	SD₆₄	1	2	yes	—
53	C₄×C₄₃	Q₆₄	1	2	yes	—
54	C₄×C₄₃	#486	1	1	yes	characteristic
55	C₄×C₄₃	C₂×SD₃₂	1	1	yes	characteristic
56	Dic₄₃	—	16	2	no	—
57	Dic₄₃	—	16	2	no	—
58	#2252	—	43	1	no	—
59	#2247	—	43	1	no	—
60	#1416	—	43	4	no	—
61	C₄×C₃₂	D₄₃	1	1	yes	characteristic
62	C₂×C₄₃	#2240	1	1	yes	characteristic
63	C₂×C₄₃	C₂×SD₆₄	1	1	yes	characteristic
64	C₂×C₄₃	C₂×Q₆₄	1	1	yes	characteristic
65	C₂×Q₃₂	—	43	2	no	—
66	C₂×Q₃₂	—	86	1	no	—
67	#500	—	86	1	no	—
68	#494	—	86	1	no	—
69	C₂×C₃₂	C₂×D₄₃	1	2	yes	—
70	#366	—	43	2	no	—
71	#366	—	86	1	no	—
72	C₄×C₁₆	C₂×D₄₃	1	1	yes	characteristic
73	C₄₃	#13720	1	1	yes	characteristic, complemented, Sylow
74	C₂×Q₁₆	—	86	2	no	—
75	C₂×Q₁₆	—	172	1	no	—
76	C₄⋊Q₈	—	172	1	no	—
77	C₄⋊Q₈	—	172	1	no	—
78	Q₃₂	—	86	4	no	—
79	Q₃₂	—	172	2	no	—
80	C₂×C₁₆	D₁₇₂	1	1	yes	characteristic
81	C₂×C₁₆	C₂²×D₄₃	1	1	yes	characteristic
82	#107	—	86	2	no	—
83	#107	—	172	1	no	—
84	C₄×C₈	D₁₇₂	1	1	yes	characteristic
85	C₃₂	—	2	2	no	—
86	C₂×Q₈	—	344	1	no	—
87	C₂×Q₈	—	172	2	no	—
88	Q₁₆	—	344	2	no	—
89	Q₁₆	—	172	2	no	—
90	Q₁₆	—	172	2	no	—
91	C₂×C₈	D₃₄₄	1	1	yes	characteristic
92	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
93	C₄⋊C₄	—	344	1	no	—
94	C₄⋊C₄	—	344	1	no	—
95	C₄⋊C₄	—	344	1	no	—
96	C₄⋊C₄	—	172	2	no	—
97	C₄²	D₃₄₄	1	1	yes	characteristic
98	C₁₆	#10621873	1	2	yes	—
99	C₁₆	#10621898	1	2	yes	—
100	Q₈	—	344	2	no	—
101	Q₈	—	344	2	no	—
102	Q₈	—	688	2	no	—
103	C₂×C₄	—	344	2	no	—
104	C₂×C₄	—	344	2	no	—
105	C₂×C₄	D₆₈₈	1	1	yes	characteristic
106	C₂×C₄	D₆₈₈	1	1	yes	characteristic
107	C₂×C₄	C₂×D₃₄₄	1	1	yes	characteristic
108	C₈	#13063270	1	2	yes	—
109	C₈	#13063313	1	1	yes	characteristic
110	C₈	C₂×#10621873	1	1	yes	characteristic
111	C₂²	C₂×D₆₈₈	1	1	yes	characteristic
112	C₄	—	688	2	no	—
113	C₄	—	688	2	no	—
114	C₄	#14582738	1	2	yes	—
115	C₄	Dic₆₈₈	1	2	yes	—
116	C₄	#14583246	1	1	yes	characteristic
117	C₄	C₂×#13063270	1	1	yes	characteristic
118	C₂	#19136819	1	1	yes	characteristic
119	C₂	C₂×#14582738	1	1	yes	characteristic
120	C₂	C₂×Dic₆₈₈	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19136870

C₂

yes

maximal, characteristic

#19136848

C₂

yes

maximal, characteristic

#19132777

C₂

yes

maximal

C₄×C₃₂×C₄₃

C₂

yes

maximal, characteristic

C₂×Dic₃₄₄

—

C₂×Dic₃₄₄

C₂²

yes

—

—

—

—

C₂²

yes

—

C₂×C₃₂×C₄₃

C₂²

yes

—

C₄×C₁₆×C₄₃

C₂²

yes

characteristic

C₂×Dic₁₇₂

—

C₂×Dic₁₇₂

—

#13063325

—

#13063325

—

Dic₃₄₄

—

Dic₃₄₄

—

#13063259

—

#13063259

—

C₂×C₁₆×C₄₃

D₄

yes

characteristic

C₂×C₁₆×C₄₃

C₂³

yes

characteristic

C₄×C₈×C₄₃

D₄

yes

characteristic

C₃₂×C₄₃

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×C₈×C₄₃

D₈

yes

characteristic

C₂×C₈×C₄₃

C₂×D₄

yes

characteristic

C₄²×C₄₃

D₈

yes

characteristic

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

Dic₁₇₂

—

Dic₁₇₂

—

Dic₁₇₂

—

C₁₆×C₄₃

SD₁₆

yes

—

C₁₆×C₄₃

#41

yes

—

C₂×C₄×C₄₃

D₁₆

yes

characteristic

C₂×C₄×C₄₃

D₁₆

yes

characteristic

C₂×C₄×C₄₃

C₂×D₈

yes

characteristic

C₂×Dic₄₃

—

C₂×Dic₄₃

—

Dic₈₆

—

Dic₈₆

—

Dic₈₆

—

C₈×C₄₃

SD₃₂

yes

—

C₈×C₄₃

#124

yes

characteristic

C₈×C₄₃

C₂×SD₁₆

yes

characteristic

#13720

—

maximal, Sylow

C₂²×C₄₃

C₂×D₁₆

yes

characteristic

C₄×C₄₃

SD₆₄

yes

—

C₄×C₄₃

Q₆₄

yes

—

C₄×C₄₃

#486

yes

characteristic

C₄×C₄₃

C₂×SD₃₂

yes

characteristic

Dic₄₃

—

Dic₄₃

—

#2252

—

#2247

—

#1416

—

C₄×C₃₂

D₄₃

yes

characteristic

C₂×C₄₃

#2240

yes

characteristic

C₂×C₄₃

C₂×SD₆₄

yes

characteristic

C₂×C₄₃

C₂×Q₆₄

yes

characteristic

C₂×Q₃₂

—

C₂×Q₃₂

—

#500

—

#494

—

C₂×C₃₂

C₂×D₄₃

yes

—

#366

—

#366

—

C₄×C₁₆

C₂×D₄₃

yes

characteristic

C₄₃

#13720

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂×Q₁₆

—

172

—

C₄⋊Q₈

—

172

—

C₄⋊Q₈

—

172

—

Q₃₂

—

Q₃₂

—

172

—

C₂×C₁₆

D₁₇₂

yes

characteristic

C₂×C₁₆

C₂²×D₄₃

yes

characteristic

#107

—

#107

—

172

—

C₄×C₈

D₁₇₂

yes

characteristic

C₃₂

—

C₂×Q₈

—

344

—

C₂×Q₈

—

172

—

Q₁₆

—

344

—

Q₁₆

—

172

—

Q₁₆

—

172

—

C₂×C₈

D₃₄₄

yes

characteristic

C₂×C₈

C₂×D₁₇₂

yes

characteristic

C₄⋊C₄

—

344

—

C₄⋊C₄

—

344

—

C₄⋊C₄

—

344

—

C₄⋊C₄

—

172

—

C₄²

D₃₄₄

yes

characteristic

C₁₆

#10621873

yes

—

C₁₆

#10621898

yes

—

100

Q₈

—

344

—

101

Q₈

—

344

—

102

Q₈

—

688

—

103

C₂×C₄

—

344

—

104

C₂×C₄

—

344

—

105

C₂×C₄

D₆₈₈

yes

characteristic

106

C₂×C₄

D₆₈₈

yes

characteristic

107

C₂×C₄

C₂×D₃₄₄

yes

characteristic

108

C₈

#13063270

yes

—

109

C₈

#13063313

yes

characteristic

110

C₈

C₂×#10621873

yes

characteristic

111

C₂²

C₂×D₆₈₈

yes

characteristic

112

C₄

—

688

—

113

C₄

—

688

—

114

C₄

#14582738

yes

—

115

C₄

Dic₆₈₈

yes

—

116

C₄

#14583246

yes

characteristic

117

C₄

C₂×#13063270

yes

characteristic

118

C₂

#19136819

yes

characteristic

119

C₂

C₂×#14582738

yes

characteristic

120

C₂

C₂×Dic₆₈₈

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₆₈₈
commutator, cosocle	C₂×C₁₆×C₄₃	C₂³
Frattini	C₂×C₁₆	C₂²×D₄₃
Fitting	C₄×C₃₂×C₄₃	C₂
socle, prime radical	C₂²×C₄₃	C₂×D₁₆
hypercenter	C₄×C₃₂	D₄₃
hypocenter	C₄₃	#13720
LCS	G ▷ C₂×C₁₆×C₄₃ ▷ C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ C₄×C₃₂

subgroup

G/H

center

C₂²

C₂×D₆₈₈

commutator, cosocle

C₂×C₁₆×C₄₃

C₂³

Frattini

C₂×C₁₆

C₂²×D₄₃

Fitting

C₄×C₃₂×C₄₃

C₂

socle, prime radical

C₂²×C₄₃

C₂×D₁₆

hypercenter

C₄×C₃₂

D₄₃

hypocenter

C₄₃

#13720

LCS

G ▷ C₂×C₁₆×C₄₃ ▷ C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ C₄×C₃₂

N	H ≅ G/N	type
73. C₄₃	#13720	semidirect

H ≅ G/N

type

73. C₄₃

#13720

semidirect

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

subgroup	index
#19136870	2
#19136848	2
#19132777	2
C₄×C₃₂×C₄₃	2
#13720	43

subgroup

index

C₄×C₃₂×C₄₃

Subgroups — #25017570

Conjugacy classes of subgroups

Complements

Maximal subgroups