Group #25011302 — subgroups

Subgroups — #25011302

Order 11008.

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

normal subgroups only

This group has 290 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	#19136820	C₂	1	4	yes	maximal, complemented
2	#19134210	C₂	1	1	yes	maximal, characteristic
3	#19132512	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄₃×#1710	C₂	1	1	yes	maximal, characteristic, complemented
5	#14583276	—	2	2	no	—
6	#14583276	C₂²	1	2	yes	—
7	#14582699	—	2	8	no	—
8	C₄×C₄₃⋊C₁₆	C₂²	1	2	yes	—
9	C₄₃×#500	C₂²	1	2	yes	—
10	C₄₃×#487	—	2	2	no	—
11	C₈²×C₄₃	C₂²	1	1	yes	characteristic
12	C₂×D₃₄₄	—	2	4	no	—
13	C₂×D₃₄₄	—	4	4	no	—
14	#13063322	—	2	2	no	—
15	C₂×C₄₃⋊C₁₆	D₄	1	4	yes	—
16	C₄⋊Q₈×C₄₃	—	2	2	no	—
17	#107×C₄₃	—	2	4	no	—
18	#103×C₄₃	—	4	4	no	—
19	C₄×C₈×C₄₃	D₄	1	2	yes	—
20	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
21	C₂×Q₈×C₄₃	—	4	2	no	—
22	C₂×D₁₇₂	—	4	2	no	—
23	C₂×D₁₇₂	—	4	4	no	—
24	C₂×C₈×C₄₃	D₈	1	4	yes	—
25	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
26	C₄⋊C₄×C₄₃	—	4	2	no	—
27	C₄⋊C₄×C₄₃	—	4	2	no	—
28	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
29	D₃₄₄	—	4	8	no	—
30	D₃₄₄	—	8	8	no	—
31	C₄₃⋊C₁₆	—	2	4	no	—
32	C₂²×D₄₃	—	8	2	no	—
33	Q₈×C₄₃	—	8	4	no	—
34	C₂×C₄×C₄₃	—	8	2	no	—
35	C₂×C₄×C₄₃	#127	1	1	yes	characteristic
36	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
37	D₁₇₂	—	8	4	no	—
38	D₁₇₂	—	8	8	no	—
39	C₈×C₄₃	—	2	4	no	—
40	C₈×C₄₃	#135	1	4	yes	—
41	#12513	—	43	1	no	maximal, Sylow
42	C₂²×C₄₃	#493	1	1	yes	characteristic
43	C₂×D₄₃	—	16	2	no	—
44	C₂×D₄₃	—	16	4	no	—
45	C₄×C₄₃	—	16	2	no	—
46	C₄×C₄₃	#495	1	2	yes	—
47	C₄×C₄₃	#509	1	4	yes	—
48	#2240	—	43	4	no	—
49	#1712	—	43	1	no	—
50	#1710	D₄₃	1	1	yes	characteristic, complemented
51	#1311	—	43	1	no	—
52	C₂×C₄₃	#1709	1	1	yes	characteristic
53	C₂×C₄₃	#2250	1	2	yes	—
54	D₄₃	—	32	2	no	—
55	#500	C₂×D₄₃	1	2	yes	—
56	#493	—	43	2	no	—
57	#493	—	86	2	no	—
58	#487	—	2	2	no	—
59	#357	—	86	8	no	—
60	C₄×C₁₆	—	43	2	no	—
61	C₈²	C₂×D₄₃	1	1	yes	characteristic
62	C₄₃	#12513	1	1	yes	characteristic, complemented, Sylow
63	C₂×D₈	—	86	4	no	—
64	C₂×D₈	—	172	4	no	—
65	C₄⋊Q₈	—	2	2	no	—
66	#127	—	86	2	no	—
67	C₂×C₁₆	—	43	4	no	—
68	#107	—	2	4	no	—
69	#103	—	4	4	no	—
70	C₄×C₈	C₂²⋊D₄₃	1	2	yes	—
71	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
72	C₂×Q₈	—	4	2	no	—
73	C₂×D₄	—	172	2	no	—
74	C₂×D₄	—	172	4	no	—
75	D₈	—	344	8	no	—
76	D₈	—	172	8	no	—
77	C₂×C₈	#10621882	1	4	yes	—
78	C₂×C₈	D₄×D₄₃	1	2	yes	—
79	C₄⋊C₄	—	4	2	no	—
80	C₄⋊C₄	—	4	2	no	—
81	C₄²	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
82	C₁₆	—	86	4	no	—
83	C₂³	—	344	2	no	—
84	Q₈	—	8	4	no	—
85	D₄	—	344	8	no	—
86	D₄	—	344	4	no	—
87	C₂×C₄	—	8	2	no	—
88	C₂×C₄	#13063321	1	1	yes	characteristic
89	C₂×C₄	C₂×#10621882	1	2	yes	—
90	C₈	—	2	4	no	—
91	C₈	#13063348	1	4	yes	—
92	C₂²	—	688	4	no	—
93	C₂²	—	688	2	no	—
94	C₂²	#14583275	1	1	yes	characteristic
95	C₄	—	16	2	no	—
96	C₄	#14583284	1	2	yes	—
97	C₄	#14583334	1	4	yes	—
98	C₂	—	1376	2	no	—
99	C₂	#19134202	1	1	yes	characteristic
100	C₂	#19136862	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19136820

C₂

yes

maximal, complemented

#19134210

C₂

yes

maximal, characteristic

#19132512

C₂

yes

maximal, characteristic, complemented

C₄₃×#1710

C₂

yes

maximal, characteristic, complemented

#14583276

—

#14583276

C₂²

yes

—

#14582699

—

C₄×C₄₃⋊C₁₆

C₂²

yes

—

C₄₃×#500

C₂²

yes

—

C₄₃×#487

—

C₈²×C₄₃

C₂²

yes

characteristic

C₂×D₃₄₄

—

C₂×D₃₄₄

—

#13063322

—

C₂×C₄₃⋊C₁₆

D₄

yes

—

C₄⋊Q₈×C₄₃

—

#107×C₄₃

—

#103×C₄₃

—

C₄×C₈×C₄₃

D₄

yes

—

C₄×C₈×C₄₃

C₂³

yes

characteristic

C₂×Q₈×C₄₃

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×C₈×C₄₃

D₈

yes

—

C₂×C₈×C₄₃

C₂×D₄

yes

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄²×C₄₃

C₂×D₄

yes

characteristic

D₃₄₄

—

D₃₄₄

—

C₄₃⋊C₁₆

—

C₂²×D₄₃

—

Q₈×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

#127

yes

characteristic

C₂×C₄×C₄₃

C₂×D₈

yes

—

D₁₇₂

—

D₁₇₂

—

C₈×C₄₃

—

C₈×C₄₃

#135

yes

—

—

C₂²×C₄₃

yes

C₂×D₄₃

—

C₂×D₄₃

—

C₄×C₄₃

—

C₄×C₄₃

#495

yes

—

C₄×C₄₃

#509

yes

—

#2240

—

#1712

—

#1710

D₄₃

yes

characteristic, complemented

#1311

—

C₂×C₄₃

#1709

yes

characteristic

C₂×C₄₃

#2250

yes

—

D₄₃

—

#500

C₂×D₄₃

yes

—

—

—

—

—

C₄×C₁₆

—

C₈²

C₂×D₄₃

yes

characteristic

C₄₃

#12513

yes

characteristic, complemented, Sylow

C₂×D₈

—

C₂×D₈

—

172

—

C₄⋊Q₈

—

#127

—

C₂×C₁₆

—

#107

—

#103

—

C₄×C₈

C₂²⋊D₄₃

yes

—

C₄×C₈

C₂²×D₄₃

yes

characteristic

C₂×Q₈

—

C₂×D₄

—

172

—

C₂×D₄

—

172

—

D₈

—

344

—

D₈

—

172

—

C₂×C₈

#10621882

yes

—

C₂×C₈

D₄×D₄₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂×C₂²⋊D₄₃

yes

characteristic

C₁₆

—

C₂³

—

344

—

Q₈

—

D₄

—

344

—

D₄

—

344

—

C₂×C₄

—

C₂×C₄

#13063321

yes

characteristic

C₂×C₄

C₂×#10621882

yes

—

C₈

—

C₈

#13063348

yes

—

C₂²

—

688

—

C₂²

—

688

—

C₂²

#14583275

yes

characteristic

C₄

—

C₄

#14583284

yes

—

C₄

#14583334

yes

—

C₂

—

1376

—

C₂

#19134202

yes

characteristic

100

C₂

#19136862

yes

—

	subgroup	G/H
center	C₂²	#14583275
commutator, cosocle	C₄×C₈×C₄₃	C₂³
Frattini	C₄×C₈	C₂²×D₄₃
Fitting	C₄₃×#1710	C₂
socle	C₂²×C₄₃	#493
prime radical	#19134210	C₂
hypercenter	#1710	D₄₃
hypocenter	C₄₃	#12513
LCS	G ▷ C₄×C₈×C₄₃ ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1710

subgroup

G/H

center

C₂²

#14583275

commutator, cosocle

C₄×C₈×C₄₃

C₂³

Frattini

C₄×C₈

C₂²×D₄₃

Fitting

C₄₃×#1710

C₂

C₂²×C₄₃

C₂

D₄₃

C₄₃

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1710

N	H ≅ G/N	type
1. #19136820	C₂	semidirect
3. #19132512	C₂	semidirect
4. C₄₃×#1710	C₂	semidirect
50. #1710	D₄₃	semidirect
62. C₄₃	#12513	semidirect

H ≅ G/N

type

1. #19136820

C₂

semidirect

3. #19132512

C₂

semidirect

4. C₄₃×#1710

C₂

semidirect

50. #1710

D₄₃

semidirect

62. C₄₃

#12513

semidirect

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

subgroup	index
#19136820	2
#19134210	2
#19132512	2
C₄₃×#1710	2
#12513	43

subgroup

index

C₄₃×#1710

Subgroups — #25011302

Conjugacy classes of subgroups

Complements

Maximal subgroups