Group #25001672 — subgroups

Subgroups — #25001672

Order 11008.

478 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 141 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	#19135401	C₂	1	3	yes	maximal
2	#19134723	C₂	1	3	yes	maximal, complemented
3	C₄₃×#1273	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#13063322	C₂²	1	1	yes	characteristic
5	C₂×#13063247	—	2	6	no	—
6	#14582769	C₂²	1	3	yes	—
7	#14582662	—	2	6	no	—
8	C₂×C₄×C₈×C₄₃	C₂²	1	3	yes	—
9	C₂²×D₁₇₂	—	2	3	no	—
10	#13063322	—	2	3	no	—
11	#13063322	Q₈	1	2	yes	—
12	C₂×C₄⋊Dic₄₃	—	2	3	no	—
13	C₂×C₄⋊Dic₄₃	—	2	3	no	—
14	C₂×C₄⋊Dic₄₃	—	2	3	no	—
15	#13063247	—	4	12	no	—
16	C₂²×C₈×C₄₃	D₄	1	6	yes	—
17	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
18	C₄×C₈×C₄₃	—	2	6	no	—
19	C₂³×D₄₃	—	4	1	no	—
20	C₂²×C₄×C₄₃	C₂×D₄	1	3	yes	—
21	C₂²×Dic₄₃	—	4	3	no	—
22	C₂×D₁₇₂	—	2	6	no	—
23	C₂×D₁₇₂	—	4	3	no	—
24	C₂×D₁₇₂	—	4	6	no	—
25	C₂×D₁₇₂	—	4	6	no	—
26	C₂×C₈×C₄₃	—	2	6	no	—
27	C₂×C₈×C₄₃	—	2	6	no	—
28	C₂×C₈×C₄₃	—	2	6	no	—
29	C₄²×C₄₃	C₂×Q₈	1	1	yes	characteristic
30	C₄²×C₄₃	#41	1	3	yes	—
31	C₄⋊Dic₄₃	—	4	6	no	—
32	C₄⋊Dic₄₃	—	4	6	no	—
33	C₄⋊Dic₄₃	—	4	6	no	—
34	C₂³×C₄₃	#127	1	1	yes	characteristic
35	C₂²×D₄₃	—	4	2	no	—
36	C₂²×D₄₃	—	8	3	no	—
37	C₂²×D₄₃	—	8	3	no	—
38	C₂×C₄×C₄₃	C₂²⋊Q₈	1	3	yes	—
39	C₂×C₄×C₄₃	#123	1	3	yes	—
40	C₂×C₄×C₄₃	#135	1	6	yes	—
41	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	6	yes	—
42	C₂×Dic₄₃	—	8	3	no	—
43	C₂×Dic₄₃	—	8	3	no	—
44	C₂×Dic₄₃	—	8	3	no	—
45	D₁₇₂	—	4	12	no	—
46	D₁₇₂	—	8	6	no	—
47	C₈×C₄₃	—	4	6	no	—
48	#10678	—	43	1	no	maximal, Sylow
49	C₂²×C₄₃	#399	1	1	yes	characteristic
50	C₂²×C₄₃	#495	1	3	yes	—
51	C₂²×C₄₃	#496	1	3	yes	—
52	C₂×D₄₃	—	8	6	no	—
53	C₂×D₄₃	—	16	1	no	—
54	C₂×D₄₃	—	16	3	no	—
55	C₄×C₄₃	#478	1	6	yes	—
56	C₄×C₄₃	#482	1	6	yes	—
57	Dic₄₃	—	16	3	no	—
58	#1957	—	43	3	no	—
59	#1834	—	43	3	no	—
60	#1273	D₄₃	1	1	yes	characteristic, complemented
61	C₂×C₄₃	#1711	1	1	yes	characteristic
62	C₂×C₄₃	#1717	1	3	yes	—
63	C₂×C₄₃	#2092	1	3	yes	—
64	D₄₃	—	16	2	no	—
65	C₂×#127	—	43	1	no	—
66	C₂×#102	—	86	6	no	—
67	C₂×C₄×C₈	C₂×D₄₃	1	3	yes	—
68	#384	—	43	3	no	—
69	#340	—	86	6	no	—
70	C₄₃	#10678	1	1	yes	characteristic, complemented, Sylow
71	C₂²×D₄	—	86	3	no	—
72	C₂²×C₈	D₁₇₂	1	6	yes	—
73	#127	—	43	2	no	—
74	#127	—	86	3	no	—
75	C₂×C₄⋊C₄	—	86	3	no	—
76	C₂×C₄⋊C₄	—	86	3	no	—
77	C₂×C₄⋊C₄	—	86	3	no	—
78	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
79	#102	—	172	12	no	—
80	C₄×C₈	—	2	6	no	—
81	C₂⁴	—	172	1	no	—
82	C₂×D₄	—	86	6	no	—
83	C₂×D₄	—	172	3	no	—
84	C₂×D₄	—	172	6	no	—
85	C₂×D₄	—	172	6	no	—
86	C₂²×C₄	—	172	3	no	—
87	C₂²×C₄	C₂×D₁₇₂	1	3	yes	—
88	C₂×C₈	—	2	6	no	—
89	C₂×C₈	—	2	6	no	—
90	C₂×C₈	—	2	6	no	—
91	C₄⋊C₄	—	172	6	no	—
92	C₄⋊C₄	—	172	6	no	—
93	C₄⋊C₄	—	172	6	no	—
94	C₄²	#10621900	1	3	yes	—
95	C₄²	Q₈×D₄₃	1	1	yes	characteristic
96	C₂³	—	344	3	no	—
97	C₂³	—	344	3	no	—
98	C₂³	—	172	2	no	—
99	C₂³	#13063322	1	1	yes	characteristic
100	D₄	—	344	6	no	—
101	D₄	—	172	12	no	—
102	C₂×C₄	—	344	3	no	—
103	C₂×C₄	—	344	3	no	—
104	C₂×C₄	—	344	3	no	—
105	C₂×C₄	#13063305	1	3	yes	—
106	C₂×C₄	#13063309	1	3	yes	—
107	C₂×C₄	#13063347	1	6	yes	—
108	C₂×C₄	#13063353	1	6	yes	—
109	C₈	—	4	6	no	—
110	C₂²	—	344	6	no	—
111	C₂²	—	688	3	no	—
112	C₂²	—	688	1	no	—
113	C₂²	#14582836	1	1	yes	characteristic
114	C₂²	#14583283	1	3	yes	—
115	C₂²	#14583288	1	3	yes	—
116	C₄	—	688	3	no	—
117	C₄	#14583200	1	6	yes	—
118	C₄	#14583224	1	6	yes	—
119	C₂	—	688	2	no	—
120	C₂	#19134207	1	1	yes	characteristic
121	C₂	#19134234	1	3	yes	—
122	C₂	#19136115	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19135401

C₂

yes

maximal

#19134723

C₂

yes

maximal, complemented

C₄₃×#1273

C₂

yes

maximal, characteristic, complemented

C₂×#13063322

C₂²

yes

characteristic

C₂×#13063247

—

#14582769

C₂²

yes

—

#14582662

—

C₂×C₄×C₈×C₄₃

C₂²

yes

—

C₂²×D₁₇₂

—

#13063322

—

#13063322

Q₈

yes

—

C₂×C₄⋊Dic₄₃

—

C₂×C₄⋊Dic₄₃

—

C₂×C₄⋊Dic₄₃

—

#13063247

—

C₂²×C₈×C₄₃

D₄

yes

—

C₂×C₄²×C₄₃

C₂³

yes

characteristic

C₄×C₈×C₄₃

—

C₂³×D₄₃

—

C₂²×C₄×C₄₃

C₂×D₄

yes

—

C₂²×Dic₄₃

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₄²×C₄₃

C₂×Q₈

yes

characteristic

C₄²×C₄₃

#41

yes

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₂³×C₄₃

#127

yes

characteristic

C₂²×D₄₃

—

C₂²×D₄₃

—

C₂²×D₄₃

—

C₂×C₄×C₄₃

C₂²⋊Q₈

yes

—

C₂×C₄×C₄₃

#123

yes

—

C₂×C₄×C₄₃

#135

yes

—

C₂×C₄×C₄₃

AGL(1,ℤ/8)

yes

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

D₁₇₂

—

D₁₇₂

—

C₈×C₄₃

—

—

C₂²×C₄₃

yes

C₂²×C₄₃

yes

—

C₂²×C₄₃

#496

yes

—

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₄×C₄₃

#478

yes

—

C₄×C₄₃

#482

yes

—

Dic₄₃

—

#1957

—

#1834

—

#1273

D₄₃

yes

characteristic, complemented

C₂×C₄₃

#1711

yes

characteristic

C₂×C₄₃

#1717

yes

—

C₂×C₄₃

#2092

yes

—

D₄₃

—

C₂×#127

—

C₂×#102

—

C₂×C₄×C₈

C₂×D₄₃

yes

—

#384

—

#340

—

C₄₃

#10678

yes

characteristic, complemented, Sylow

C₂²×D₄

—

C₂²×C₈

D₁₇₂

yes

—

#127

—

#127

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂²×D₄₃

yes

characteristic

#102

—

172

—

C₄×C₈

—

C₂⁴

—

172

—

C₂×D₄

—

C₂×D₄

—

172

—

C₂×D₄

—

172

—

C₂×D₄

—

172

—

C₂²×C₄

—

172

—

C₂²×C₄

C₂×D₁₇₂

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

172

—

C₄⋊C₄

—

172

—

C₄⋊C₄

—

172

—

C₄²

#10621900

yes

—

C₄²

Q₈×D₄₃

yes

characteristic

C₂³

—

344

—

C₂³

—

344

—

C₂³

—

172

—

C₂³

#13063322

yes

characteristic

100

D₄

—

344

—

101

D₄

—

172

—

102

C₂×C₄

—

344

—

103

C₂×C₄

—

344

—

104

C₂×C₄

—

344

—

105

C₂×C₄

#13063305

yes

—

106

C₂×C₄

#13063309

yes

—

107

C₂×C₄

#13063347

yes

—

108

C₂×C₄

#13063353

yes

—

109

C₈

—

110

C₂²

—

344

—

111

C₂²

—

688

—

112

C₂²

—

688

—

113

C₂²

#14582836

yes

characteristic

114

C₂²

#14583283

yes

—

115

C₂²

#14583288

yes

—

116

C₄

—

688

—

117

C₄

#14583200

yes

—

118

C₄

#14583224

yes

—

119

C₂

—

688

—

120

C₂

#19134207

yes

characteristic

121

C₂

#19134234

yes

—

122

C₂

#19136115

yes

—

	subgroup	G/H
center	C₂³	#13063322
commutator, cosocle	C₂×C₄²×C₄₃	C₂³
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₄₃×#1273	C₂
socle	C₂³×C₄₃	#127
prime radical	C₂×#13063322	C₂²
hypercenter	#1273	D₄₃
hypocenter	C₄₃	#10678
LCS	G ▷ C₂×C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1273

subgroup

G/H

center

C₂³

#13063322

commutator, cosocle

C₂×C₄²×C₄₃

C₂³

Frattini

C₂×C₄²

C₂²×D₄₃

Fitting

C₄₃×#1273

C₂

C₂³×C₄₃

C₂²

D₄₃

C₄₃

G ▷ C₂×C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1273

N	H ≅ G/N	type
2. #19134723	C₂	semidirect
3. C₄₃×#1273	C₂	semidirect
60. #1273	D₄₃	semidirect
70. C₄₃	#10678	semidirect

H ≅ G/N

type

2. #19134723

C₂

semidirect

3. C₄₃×#1273

C₂

semidirect

60. #1273

D₄₃

semidirect

70. C₄₃

#10678

semidirect

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

subgroup	index
#19135401	2
#19134723	2
C₄₃×#1273	2
#10678	43

subgroup

index

#19135401

#19134723

C₄₃×#1273

#10678

Subgroups — #25001672

Conjugacy classes of subgroups

Complements

Maximal subgroups