Group #25017628 — subgroups

Subgroups — #25017628

Order 11008.

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

normal subgroups only

This group has 150 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	#19136874	C₂	1	1	yes	maximal, characteristic
2	#19136870	C₂	1	1	yes	maximal, characteristic
3	#19132797	C₂	1	2	yes	maximal
4	#19132795	C₂	1	2	yes	maximal
5	C₄₃×#1397	C₂	1	1	yes	maximal, characteristic
6	#14583304	—	2	1	no	—
7	#14583301	—	2	1	no	—
8	#14582727	—	2	1	no	—
9	#14582725	—	2	1	no	—
10	#14582725	C₂²	1	2	yes	—
11	#14582725	C₂²	1	1	yes	characteristic
12	#14582725	C₂²	1	1	yes	characteristic
13	C₂×C₃₂×C₄₃	C₂²	1	2	yes	—
14	C₄×C₁₆×C₄₃	C₂²	1	1	yes	characteristic
15	#13063325	—	4	1	no	—
16	#13063317	—	4	1	no	—
17	#13063259	—	2	1	no	—
18	#13063259	—	2	1	no	—
19	#13063259	—	2	2	no	—
20	#13063259	—	4	1	no	—
21	#13063257	—	4	1	no	—
22	C₂×C₁₆×C₄₃	D₄	1	1	yes	characteristic
23	C₂×C₁₆×C₄₃	C₂³	1	1	yes	characteristic
24	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
25	C₃₂×C₄₃	Q₈	1	4	yes	complemented
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	1	no	—
31	C₄⋊Dic₄₃	—	4	1	no	—
32	C₄⋊Dic₄₃	—	4	2	no	—
33	C₄⋊Dic₄₃	—	8	1	no	—
34	Dic₄₃⋊C₄	—	8	1	no	—
35	Dic₄₃⋊C₄	—	8	1	no	—
36	C₁₆×C₄₃	Q₁₆	1	2	yes	—
37	C₁₆×C₄₃	C₂×Q₈	1	2	yes	—
38	C₂×C₄×C₄₃	D₁₆	1	1	yes	characteristic
39	C₂×C₄×C₄₃	D₁₆	1	1	yes	characteristic
40	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
41	C₂×Dic₄₃	—	8	1	no	—
42	C₂×Dic₄₃	—	8	1	no	—
43	C₂×Dic₄₃	—	8	2	no	—
44	Dic₈₆	—	16	2	no	—
45	C₈×C₄₃	Q₃₂	1	2	yes	—
46	C₈×C₄₃	C₄⋊Q₈	1	1	yes	characteristic
47	C₈×C₄₃	C₂×Q₁₆	1	1	yes	characteristic
48	#13734	—	43	1	no	maximal, Sylow
49	C₂²×C₄₃	C₂×D₁₆	1	1	yes	characteristic
50	C₄×C₄₃	—	2	1	no	—
51	C₄×C₄₃	—	2	1	no	—
52	C₄×C₄₃	#500	1	1	yes	characteristic
53	C₄×C₄₃	C₂×Q₃₂	1	1	yes	characteristic
54	Dic₄₃	—	16	2	no	—
55	Dic₄₃	—	16	1	no	—
56	Dic₄₃	—	16	1	no	—
57	#2253	—	43	1	no	—
58	#2252	—	43	1	no	—
59	#1424	—	43	2	no	—
60	#1423	—	43	2	no	—
61	#1397	D₄₃	1	1	yes	characteristic
62	C₂×C₄₃	#2252	1	1	yes	characteristic
63	C₂×C₄₃	#2263	1	1	yes	characteristic
64	C₂×C₄₃	#2264	1	1	yes	characteristic
65	#500	—	86	1	no	—
66	#499	—	86	1	no	—
67	C₂×C₃₂	C₂×D₄₃	1	2	yes	—
68	#367	—	86	1	no	—
69	#366	—	43	1	no	—
70	#366	—	43	2	no	—
71	#366	—	43	1	no	—
72	#366	—	86	1	no	—
73	C₄×C₁₆	C₂×D₄₃	1	1	yes	characteristic
74	C₄₃	#13734	1	1	yes	characteristic, complemented, Sylow
75	C₄⋊Q₈	—	172	1	no	—
76	#125	—	172	1	no	—
77	C₂×C₁₆	D₁₇₂	1	1	yes	characteristic
78	C₂×C₁₆	C₂²×D₄₃	1	1	yes	characteristic
79	#107	—	86	2	no	—
80	#107	—	86	1	no	—
81	#107	—	86	1	no	—
82	#107	—	172	1	no	—
83	#106	—	172	1	no	—
84	C₄×C₈	D₁₇₂	1	1	yes	characteristic
85	C₃₂	Dic₈₆	1	4	yes	complemented
86	C₂×Q₈	—	344	1	no	—
87	C₂×C₈	D₃₄₄	1	1	yes	characteristic
88	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
89	C₄⋊C₄	—	344	1	no	—
90	C₄⋊C₄	—	344	1	no	—
91	C₄⋊C₄	—	344	1	no	—
92	C₄⋊C₄	—	172	1	no	—
93	C₄⋊C₄	—	172	1	no	—
94	C₄⋊C₄	—	172	2	no	—
95	C₄²	D₃₄₄	1	1	yes	characteristic
96	C₁₆	Dic₁₇₂	1	2	yes	—
97	C₁₆	C₂×Dic₈₆	1	2	yes	—
98	Q₈	—	688	2	no	—
99	C₂×C₄	—	344	1	no	—
100	C₂×C₄	—	344	1	no	—
101	C₂×C₄	—	344	2	no	—
102	C₂×C₄	D₆₈₈	1	1	yes	characteristic
103	C₂×C₄	D₆₈₈	1	1	yes	characteristic
104	C₂×C₄	C₂×D₃₄₄	1	1	yes	characteristic
105	C₈	Dic₃₄₄	1	2	yes	—
106	C₈	#13063325	1	1	yes	characteristic
107	C₈	C₂×Dic₁₇₂	1	1	yes	characteristic
108	C₂²	C₂×D₆₈₈	1	1	yes	characteristic
109	C₄	—	2	1	no	—
110	C₄	—	2	1	no	—
111	C₄	—	688	2	no	—
112	C₄	—	688	1	no	—
113	C₄	—	688	1	no	—
114	C₄	#14583304	1	1	yes	characteristic
115	C₄	C₂×Dic₃₄₄	1	1	yes	characteristic
116	C₂	#19136870	1	1	yes	characteristic
117	C₂	#19136910	1	1	yes	characteristic
118	C₂	#19136915	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19136874

C₂

yes

maximal, characteristic

#19136870

C₂

yes

maximal, characteristic

#19132797

C₂

yes

maximal

#19132795

C₂

yes

maximal

C₄₃×#1397

C₂

yes

maximal, characteristic

—

—

—

—

C₂²

yes

—

#14582725

C₂²

yes

characteristic

#14582725

C₂²

yes

characteristic

C₂×C₃₂×C₄₃

C₂²

yes

—

C₄×C₁₆×C₄₃

C₂²

yes

—

—

—

—

—

—

—

C₂×C₁₆×C₄₃

D₄

yes

characteristic

C₂×C₁₆×C₄₃

C₂³

yes

characteristic

C₄×C₈×C₄₃

D₄

yes

characteristic

C₃₂×C₄₃

Q₈

yes

complemented

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

—

Dic₄₃⋊C₄

—

Dic₄₃⋊C₄

—

C₁₆×C₄₃

Q₁₆

yes

—

C₁₆×C₄₃

C₂×Q₈

yes

—

C₂×C₄×C₄₃

D₁₆

yes

characteristic

C₂×C₄×C₄₃

D₁₆

yes

characteristic

C₂×C₄×C₄₃

C₂×D₈

yes

characteristic

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

Dic₈₆

—

C₈×C₄₃

Q₃₂

yes

—

C₈×C₄₃

C₄⋊Q₈

yes

characteristic

C₈×C₄₃

C₂×Q₁₆

yes

characteristic

#13734

—

maximal, Sylow

C₂²×C₄₃

C₂×D₁₆

yes

characteristic

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

#500

yes

characteristic

C₄×C₄₃

C₂×Q₃₂

yes

characteristic

Dic₄₃

—

Dic₄₃

—

Dic₄₃

—

—

—

—

—

D₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

—

—

C₂×C₃₂

C₂×D₄₃

yes

—

—

—

—

—

—

C₄×C₁₆

C₂×D₄₃

yes

characteristic

C₄₃

#13734

yes

characteristic, complemented, Sylow

C₄⋊Q₈

—

172

—

#125

—

172

—

C₂×C₁₆

D₁₇₂

yes

characteristic

C₂×C₁₆

C₂²×D₄₃

yes

—

—

—

—

172

—

#106

—

172

—

C₄×C₈

D₁₇₂

yes

characteristic

C₃₂

Dic₈₆

yes

complemented

C₂×Q₈

—

344

—

C₂×C₈

D₃₄₄

yes

characteristic

C₂×C₈

C₂×D₁₇₂

yes

characteristic

C₄⋊C₄

—

344

—

C₄⋊C₄

—

344

—

C₄⋊C₄

—

344

—

C₄⋊C₄

—

172

—

C₄⋊C₄

—

172

—

C₄⋊C₄

—

172

—

C₄²

D₃₄₄

yes

characteristic

C₁₆

Dic₁₇₂

yes

—

C₁₆

C₂×Dic₈₆

yes

—

Q₈

—

688

—

C₂×C₄

—

344

—

100

C₂×C₄

—

344

—

101

C₂×C₄

—

344

—

102

C₂×C₄

D₆₈₈

yes

characteristic

103

C₂×C₄

D₆₈₈

yes

characteristic

104

C₂×C₄

C₂×D₃₄₄

yes

characteristic

105

C₈

Dic₃₄₄

yes

—

106

C₈

#13063325

yes

characteristic

107

C₈

C₂×Dic₁₇₂

yes

characteristic

108

C₂²

C₂×D₆₈₈

yes

characteristic

109

C₄

—

110

C₄

—

111

C₄

—

688

—

112

C₄

—

688

—

113

C₄

—

688

—

114

C₄

#14583304

yes

characteristic

115

C₄

C₂×Dic₃₄₄

yes

characteristic

116

C₂

#19136870

yes

characteristic

117

C₂

#19136910

yes

characteristic

118

C₂

#19136915

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₆₈₈
commutator, cosocle	C₂×C₁₆×C₄₃	C₂³
Frattini	C₂×C₁₆	C₂²×D₄₃
Fitting	C₄₃×#1397	C₂
socle, prime radical	C₂²×C₄₃	C₂×D₁₆
hypercenter	#1397	D₄₃
hypocenter	C₄₃	#13734
LCS	G ▷ C₂×C₁₆×C₄₃ ▷ C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ #1397

subgroup

G/H

center

C₂²

C₂×D₆₈₈

commutator, cosocle

C₂×C₁₆×C₄₃

C₂³

Frattini

C₂×C₁₆

C₂²×D₄₃

Fitting

C₄₃×#1397

C₂

socle, prime radical

C₂²×C₄₃

C₂×D₁₆

D₄₃

C₄₃

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ #1397

N	H ≅ G/N	type
25. C₃₂×C₄₃	Q₈	semidirect
74. C₄₃	#13734	semidirect
85. C₃₂	Dic₈₆	semidirect

H ≅ G/N

type

25. C₃₂×C₄₃

Q₈

semidirect

74. C₄₃

#13734

semidirect

85. C₃₂

Dic₈₆

semidirect

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

subgroup	index
#19136874	2
#19136870	2
#19132797	2
#19132795	2
C₄₃×#1397	2
#13734	43

subgroup

index

C₄₃×#1397

Subgroups — #25017628

Conjugacy classes of subgroups

Complements

Maximal subgroups