Group #25001282 — subgroups

Subgroups — #25001282

Order 11008.

328 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 328 conjugacy classes of subgroups — too many to draw. Only its 79 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	#19133386	C₂	1	3	yes	maximal, complemented
2	#19133264	C₂	1	3	yes	maximal
3	C₄₃×#1309	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#13063322	C₂²	1	1	yes	characteristic
5	#14582952	C₂²	1	3	yes	—
6	#14582651	—	2	3	no	—
7	#14582649	—	2	3	no	—
8	#14582642	—	2	3	no	—
9	#14582642	—	2	3	no	—
10	C₄₃×#423	C₂²	1	3	yes	—
11	C₂²×D₁₇₂	—	2	3	no	—
12	#13063322	—	2	3	no	—
13	#13063322	Q₈	1	2	yes	—
14	C₄⋊(C₄₃⋊C₈)	D₄	1	3	yes	—
15	C₄⋊(C₄₃⋊C₈)	D₄	1	3	yes	—
16	C₂²⋊(C₄₃⋊C₈)	—	2	3	no	—
17	C₄×C₄₃⋊C₈	—	2	3	no	—
18	C₂×M₁₆×C₄₃	—	2	3	no	—
19	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
20	C₄⋊C₈×C₄₃	—	2	3	no	—
21	C₄⋊C₈×C₄₃	—	2	3	no	—
22	C₂³×D₄₃	—	4	1	no	—
23	C₂²×C₄×C₄₃	#41	1	3	yes	—
24	C₂×D₁₇₂	—	2	6	no	—
25	C₂×D₁₇₂	—	4	3	no	—
26	C₂×D₁₇₂	—	8	3	no	—
27	C₂×D₁₇₂	—	8	3	no	—
28	M₁₆×C₄₃	—	8	3	no	—
29	C₂×C₈×C₄₃	—	4	3	no	—
30	C₄²×C₄₃	C₂×D₄	1	3	yes	—
31	C₄²×C₄₃	C₂×Q₈	1	1	yes	characteristic
32	C₂×C₄₃⋊C₈	—	2	3	no	—
33	C₂×C₄₃⋊C₈	—	2	3	no	—
34	C₂³×C₄₃	#126	1	1	yes	characteristic
35	C₂²×D₄₃	—	4	2	no	—
36	C₂²×D₄₃	—	8	6	no	—
37	C₂×C₄×C₄₃	—	2	3	no	—
38	C₂×C₄×C₄₃	—	2	3	no	—
39	C₂×C₄×C₄₃	#121	1	3	yes	—
40	C₂×C₄×C₄₃	C₂²⋊Q₈	1	3	yes	—
41	D₁₇₂	—	8	6	no	—
42	D₁₇₂	—	8	3	no	—
43	D₁₇₂	—	8	3	no	—
44	C₈×C₄₃	—	8	3	no	—
45	C₄₃⋊C₈	—	4	3	no	—
46	C₄₃⋊C₈	—	4	3	no	—
47	#10612	—	43	1	no	maximal, Sylow
48	C₂²×C₄₃	—	2	3	no	—
49	C₂²×C₄₃	#396	1	1	yes	characteristic
50	C₂×D₄₃	—	8	6	no	—
51	C₂×D₄₃	—	16	4	no	—
52	C₄×C₄₃	—	2	3	no	—
53	C₄×C₄₃	#471	1	3	yes	—
54	C₄×C₄₃	#472	1	3	yes	—
55	#1574	—	43	3	no	—
56	#1545	—	43	3	no	—
57	#1309	D₄₃	1	1	yes	characteristic, complemented
58	C₂×C₄₃	—	4	1	no	—
59	C₂×C₄₃	#2063	1	3	yes	—
60	D₄₃	—	16	2	no	—
61	C₂×#127	—	43	1	no	—
62	#431	—	43	3	no	—
63	#423	C₂×D₄₃	1	3	yes	—
64	#335	—	86	3	no	—
65	#334	—	86	3	no	—
66	#331	—	86	3	no	—
67	#331	—	86	3	no	—
68	C₄₃	#10612	1	1	yes	characteristic, complemented, Sylow
69	C₂²×D₄	—	86	3	no	—
70	C₂×M₁₆	—	2	3	no	—
71	#127	—	43	2	no	—
72	#127	—	86	3	no	—
73	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
74	C₄⋊C₈	—	2	3	no	—
75	C₄⋊C₈	—	2	3	no	—
76	C₄⋊C₈	—	43	3	no	—
77	C₄⋊C₈	—	43	3	no	—
78	C₂²⋊C₈	—	86	3	no	—
79	C₄×C₈	—	86	3	no	—
80	C₂⁴	—	172	1	no	—
81	C₂×D₄	—	86	6	no	—
82	C₂×D₄	—	344	3	no	—
83	C₂×D₄	—	344	3	no	—
84	C₂×D₄	—	172	3	no	—
85	C₂²×C₄	#10621898	1	3	yes	—
86	M₁₆	—	8	3	no	—
87	C₂×C₈	—	4	3	no	—
88	C₂×C₈	—	86	3	no	—
89	C₂×C₈	—	86	3	no	—
90	C₄²	D₄×D₄₃	1	3	yes	—
91	C₄²	Q₈×D₄₃	1	1	yes	characteristic
92	C₂³	—	344	6	no	—
93	C₂³	—	172	2	no	—
94	C₂³	#13063319	1	1	yes	characteristic
95	D₄	—	344	6	no	—
96	D₄	—	344	3	no	—
97	D₄	—	344	3	no	—
98	C₂×C₄	—	2	3	no	—
99	C₂×C₄	—	2	3	no	—
100	C₂×C₄	#13063298	1	3	yes	—
101	C₂×C₄	#13063303	1	3	yes	—
102	C₈	—	8	3	no	—
103	C₈	—	172	3	no	—
104	C₈	—	172	3	no	—
105	C₂²	—	2	3	no	—
106	C₂²	—	344	6	no	—
107	C₂²	—	688	4	no	—
108	C₂²	#14582823	1	1	yes	characteristic
109	C₄	—	2	3	no	—
110	C₄	#14583165	1	3	yes	—
111	C₄	#14583168	1	3	yes	—
112	C₂	—	4	1	no	—
113	C₂	—	688	2	no	—
114	C₂	#19135956	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19133386

C₂

yes

maximal, complemented

#19133264

C₂

yes

maximal

C₄₃×#1309

C₂

yes

maximal, characteristic, complemented

C₂×#13063322

C₂²

yes

characteristic

#14582952

C₂²

yes

—

—

—

—

—

C₄₃×#423

C₂²

yes

—

C₂²×D₁₇₂

—

#13063322

—

#13063322

Q₈

yes

—

C₄⋊(C₄₃⋊C₈)

D₄

yes

—

C₄⋊(C₄₃⋊C₈)

D₄

yes

—

C₂²⋊(C₄₃⋊C₈)

—

C₄×C₄₃⋊C₈

—

C₂×M₁₆×C₄₃

—

C₂×C₄²×C₄₃

C₂³

yes

characteristic

C₄⋊C₈×C₄₃

—

C₄⋊C₈×C₄₃

—

C₂³×D₄₃

—

C₂²×C₄×C₄₃

#41

yes

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

M₁₆×C₄₃

—

C₂×C₈×C₄₃

—

C₄²×C₄₃

C₂×D₄

yes

—

C₄²×C₄₃

C₂×Q₈

yes

characteristic

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂³×C₄₃

#126

yes

characteristic

C₂²×D₄₃

—

C₂²×D₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

#121

yes

—

C₂×C₄×C₄₃

C₂²⋊Q₈

yes

—

D₁₇₂

—

D₁₇₂

—

D₁₇₂

—

C₈×C₄₃

—

C₄₃⋊C₈

—

C₄₃⋊C₈

—

#10612

—

maximal, Sylow

C₂²×C₄₃

—

C₂²×C₄₃

#396

yes

characteristic

C₂×D₄₃

—

C₂×D₄₃

—

C₄×C₄₃

—

C₄×C₄₃

#471

yes

—

C₄×C₄₃

#472

yes

—

#1574

—

#1545

—

#1309

D₄₃

yes

characteristic, complemented

C₂×C₄₃

—

C₂×C₄₃

#2063

yes

—

D₄₃

—

C₂×#127

—

#431

—

#423

C₂×D₄₃

yes

—

—

—

—

—

C₄₃

#10612

yes

characteristic, complemented, Sylow

C₂²×D₄

—

C₂×M₁₆

—

#127

—

#127

—

C₂×C₄²

C₂²×D₄₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₂⁴

—

172

—

C₂×D₄

—

C₂×D₄

—

344

—

C₂×D₄

—

344

—

C₂×D₄

—

172

—

C₂²×C₄

#10621898

yes

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄²

D₄×D₄₃

yes

—

C₄²

Q₈×D₄₃

yes

characteristic

C₂³

—

344

—

C₂³

—

172

—

C₂³

#13063319

yes

characteristic

D₄

—

344

—

D₄

—

344

—

D₄

—

344

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

#13063298

yes

—

101

C₂×C₄

#13063303

yes

—

102

C₈

—

103

C₈

—

172

—

104

C₈

—

172

—

105

C₂²

—

106

C₂²

—

344

—

107

C₂²

—

688

—

108

C₂²

#14582823

yes

characteristic

109

C₄

—

110

C₄

#14583165

yes

—

111

C₄

#14583168

yes

—

112

C₂

—

113

C₂

—

688

—

114

C₂

#19135956

yes

—

	subgroup	G/H
center	C₂²	#14582823
commutator, cosocle	C₂×C₄²×C₄₃	C₂³
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₄₃×#1309	C₂
socle	C₂²×C₄₃	#396
prime radical	C₂×#13063322	C₂²
hypercenter	#1309	D₄₃
hypocenter	C₄₃	#10612
LCS	G ▷ C₂×C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1309

subgroup

G/H

center

C₂²

#14582823

commutator, cosocle

C₂×C₄²×C₄₃

C₂³

Frattini

C₂×C₄²

C₂²×D₄₃

Fitting

C₄₃×#1309

C₂

C₂²×C₄₃

C₂²

D₄₃

C₄₃

G ▷ C₂×C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1309

N	H ≅ G/N	type
1. #19133386	C₂	semidirect
3. C₄₃×#1309	C₂	semidirect
57. #1309	D₄₃	semidirect
68. C₄₃	#10612	semidirect

H ≅ G/N

type

1. #19133386

C₂

semidirect

3. C₄₃×#1309

C₂

semidirect

57. #1309

D₄₃

semidirect

68. C₄₃

#10612

semidirect

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

subgroup	index
#19133386	2
#19133264	2
C₄₃×#1309	2
#10612	43

subgroup

index

#19133386

#19133264

C₄₃×#1309

#10612

Subgroups — #25001282

Conjugacy classes of subgroups

Complements

Maximal subgroups