Group #25011315 — subgroups

Subgroups — #25011315

Order 11008.

354 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 115 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	#19136851	C₂	1	4	yes	maximal, complemented
2	#19134213	C₂	1	1	yes	maximal, characteristic, complemented
3	#19134210	C₂	1	1	yes	maximal, characteristic
4	C₈×C₁₆×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#13063270	—	2	8	no	—
6	#14583304	—	2	2	no	—
7	#14583279	C₂²	1	2	yes	—
8	#14583276	—	2	2	no	—
9	#14583276	C₂²	1	2	yes	—
10	C₄×C₁₆×C₄₃	C₂²	1	2	yes	—
11	C₈²×C₄₃	C₂²	1	1	yes	characteristic
12	C₂×Dic₁₇₂	—	2	4	no	—
13	C₂×D₃₄₄	—	2	4	no	—
14	C₂×D₃₄₄	—	4	4	no	—
15	#13063325	—	2	2	no	—
16	#13063322	—	2	2	no	—
17	#13063270	—	4	16	no	—
18	#13063259	—	4	4	no	—
19	C₂×C₁₆×C₄₃	D₄	1	4	yes	—
20	C₄×C₈×C₄₃	D₄	1	2	yes	—
21	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
22	C₂×D₁₇₂	—	4	2	no	—
23	C₂×D₁₇₂	—	4	4	no	—
24	C₂×Dic₈₆	—	4	2	no	—
25	C₂×C₈×C₄₃	D₈	1	4	yes	—
26	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
27	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
28	C₄⋊Dic₄₃	—	4	4	no	—
29	Dic₁₇₂	—	4	8	no	—
30	D₃₄₄	—	4	8	no	—
31	D₃₄₄	—	8	8	no	—
32	C₁₆×C₄₃	D₈	1	8	yes	complemented
33	C₂²×D₄₃	—	8	2	no	—
34	C₂×C₄×C₄₃	#127	1	1	yes	characteristic
35	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
36	C₂×Dic₄₃	—	8	2	no	—
37	D₁₇₂	—	8	4	no	—
38	D₁₇₂	—	8	8	no	—
39	Dic₈₆	—	8	4	no	—
40	C₈×C₄₃	SD₃₂	1	8	yes	complemented
41	C₈×C₄₃	C₂×D₈	1	4	yes	—
42	#12517	—	43	1	no	maximal, Sylow
43	C₂²×C₄₃	#493	1	1	yes	characteristic
44	C₂×D₄₃	—	16	2	no	—
45	C₂×D₄₃	—	16	4	no	—
46	C₄×C₄₃	#493	1	2	yes	—
47	C₄×C₄₃	C₂×SD₃₂	1	4	yes	—
48	Dic₄₃	—	16	2	no	—
49	#2248	—	43	4	no	—
50	#1713	—	43	1	no	—
51	#1712	—	43	1	no	—
52	C₈×C₁₆	D₄₃	1	1	yes	characteristic, complemented
53	C₂×C₄₃	#1712	1	1	yes	characteristic
54	C₂×C₄₃	#2248	1	2	yes	—
55	D₄₃	—	32	2	no	—
56	C₂×SD₃₂	—	86	8	no	—
57	#500	—	86	2	no	—
58	#494	—	43	2	no	—
59	#493	—	43	2	no	—
60	#493	—	86	2	no	—
61	C₄×C₁₆	C₂×D₄₃	1	2	yes	—
62	C₈²	C₂×D₄₃	1	1	yes	characteristic
63	C₄₃	#12517	1	1	yes	characteristic, complemented, Sylow
64	C₂×Q₁₆	—	86	4	no	—
65	C₂×D₈	—	86	4	no	—
66	C₂×D₈	—	172	4	no	—
67	C₄⋊Q₈	—	86	2	no	—
68	#127	—	86	2	no	—
69	SD₃₂	—	172	16	no	—
70	C₂×C₁₆	D₁₇₂	1	4	yes	—
71	#107	—	172	4	no	—
72	C₄×C₈	D₁₇₂	1	2	yes	—
73	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
74	C₂×Q₈	—	172	2	no	—
75	C₂×D₄	—	172	2	no	—
76	C₂×D₄	—	172	4	no	—
77	Q₁₆	—	172	8	no	—
78	D₈	—	344	8	no	—
79	D₈	—	172	8	no	—
80	C₂×C₈	D₃₄₄	1	4	yes	—
81	C₂×C₈	C₂×D₁₇₂	1	2	yes	—
82	C₄⋊C₄	—	172	4	no	—
83	C₄²	C₂×D₁₇₂	1	1	yes	characteristic
84	C₁₆	D₃₄₄	1	8	yes	complemented
85	C₂³	—	344	2	no	—
86	Q₈	—	344	4	no	—
87	D₄	—	344	8	no	—
88	D₄	—	344	4	no	—
89	C₂×C₄	—	344	2	no	—
90	C₂×C₄	#13063322	1	1	yes	characteristic
91	C₂×C₄	C₂×D₃₄₄	1	2	yes	—
92	C₈	#13063270	1	8	yes	complemented
93	C₈	C₂×D₃₄₄	1	4	yes	—
94	C₂²	—	688	2	no	—
95	C₂²	—	688	4	no	—
96	C₂²	#14583276	1	1	yes	characteristic
97	C₄	—	688	2	no	—
98	C₄	#14583276	1	2	yes	—
99	C₄	C₂×#13063270	1	4	yes	—
100	C₂	—	1376	2	no	—
101	C₂	#19134210	1	1	yes	characteristic
102	C₂	#19136851	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19136851

C₂

yes

maximal, complemented

#19134213

C₂

yes

maximal, characteristic, complemented

#19134210

C₂

yes

maximal, characteristic

C₈×C₁₆×C₄₃

C₂

yes

maximal, characteristic, complemented

C₂×#13063270

—

#14583304

—

#14583279

C₂²

yes

—

#14583276

—

#14583276

C₂²

yes

—

C₄×C₁₆×C₄₃

C₂²

yes

—

C₈²×C₄₃

C₂²

yes

characteristic

C₂×Dic₁₇₂

—

C₂×D₃₄₄

—

C₂×D₃₄₄

—

—

—

—

—

C₂×C₁₆×C₄₃

D₄

yes

—

C₄×C₈×C₄₃

D₄

yes

—

C₄×C₈×C₄₃

C₂³

yes

characteristic

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×Dic₈₆

—

C₂×C₈×C₄₃

D₈

yes

—

C₂×C₈×C₄₃

C₂×D₄

yes

—

C₄²×C₄₃

C₂×D₄

yes

characteristic

C₄⋊Dic₄₃

—

Dic₁₇₂

—

D₃₄₄

—

D₃₄₄

—

C₁₆×C₄₃

D₈

yes

complemented

C₂²×D₄₃

—

C₂×C₄×C₄₃

#127

yes

characteristic

C₂×C₄×C₄₃

C₂×D₈

yes

—

C₂×Dic₄₃

—

D₁₇₂

—

D₁₇₂

—

Dic₈₆

—

C₈×C₄₃

SD₃₂

yes

complemented

C₈×C₄₃

C₂×D₈

yes

—

—

C₂²×C₄₃

yes

C₂×D₄₃

—

C₂×D₄₃

—

C₄×C₄₃

#493

yes

—

C₄×C₄₃

C₂×SD₃₂

yes

—

Dic₄₃

—

#2248

—

#1713

—

#1712

—

C₈×C₁₆

D₄₃

yes

characteristic, complemented

C₂×C₄₃

#1712

yes

characteristic

C₂×C₄₃

#2248

yes

—

D₄₃

—

C₂×SD₃₂

—

—

—

—

—

C₄×C₁₆

C₂×D₄₃

yes

—

C₈²

C₂×D₄₃

yes

characteristic

C₄₃

#12517

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂×D₈

—

C₂×D₈

—

172

—

C₄⋊Q₈

—

#127

—

SD₃₂

—

172

—

C₂×C₁₆

D₁₇₂

yes

—

#107

—

172

—

C₄×C₈

D₁₇₂

yes

—

C₄×C₈

C₂²×D₄₃

yes

characteristic

C₂×Q₈

—

172

—

C₂×D₄

—

172

—

C₂×D₄

—

172

—

Q₁₆

—

172

—

D₈

—

344

—

D₈

—

172

—

C₂×C₈

D₃₄₄

yes

—

C₂×C₈

C₂×D₁₇₂

yes

—

C₄⋊C₄

—

172

—

C₄²

C₂×D₁₇₂

yes

characteristic

C₁₆

D₃₄₄

yes

complemented

C₂³

—

344

—

Q₈

—

344

—

D₄

—

344

—

D₄

—

344

—

C₂×C₄

—

344

—

C₂×C₄

#13063322

yes

characteristic

C₂×C₄

C₂×D₃₄₄

yes

—

C₈

#13063270

yes

complemented

C₈

C₂×D₃₄₄

yes

—

C₂²

—

688

—

C₂²

—

688

—

C₂²

#14583276

yes

characteristic

C₄

—

688

—

C₄

#14583276

yes

—

C₄

C₂×#13063270

yes

—

100

C₂

—

1376

—

101

C₂

#19134210

yes

characteristic

102

C₂

#19136851

yes

—

	subgroup	G/H
center	C₂²	#14583276
commutator, cosocle	C₄×C₈×C₄₃	C₂³
Frattini	C₄×C₈	C₂²×D₄₃
Fitting	C₈×C₁₆×C₄₃	C₂
socle	C₂²×C₄₃	#493
prime radical	#19134210	C₂
hypercenter	C₈×C₁₆	D₄₃
hypocenter	C₄₃	#12517
LCS	G ▷ C₄×C₈×C₄₃ ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ C₈×C₁₆

subgroup

G/H

center

C₂²

#14583276

commutator, cosocle

C₄×C₈×C₄₃

C₂³

Frattini

C₄×C₈

C₂²×D₄₃

Fitting

C₈×C₁₆×C₄₃

C₂

C₂²×C₄₃

C₂

C₈×C₁₆

D₄₃

hypocenter

C₄₃

#12517

LCS

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ C₈×C₁₆

N	H ≅ G/N	type
1. #19136851	C₂	semidirect
2. #19134213	C₂	semidirect
4. C₈×C₁₆×C₄₃	C₂	semidirect
32. C₁₆×C₄₃	D₈	semidirect
40. C₈×C₄₃	SD₃₂	semidirect
52. C₈×C₁₆	D₄₃	semidirect
63. C₄₃	#12517	semidirect
84. C₁₆	D₃₄₄	semidirect
92. C₈	#13063270	semidirect

H ≅ G/N

type

1. #19136851

C₂

semidirect

2. #19134213

C₂

semidirect

4. C₈×C₁₆×C₄₃

C₂

semidirect

32. C₁₆×C₄₃

D₈

semidirect

40. C₈×C₄₃

SD₃₂

semidirect

52. C₈×C₁₆

D₄₃

semidirect

63. C₄₃

#12517

semidirect

84. C₁₆

D₃₄₄

semidirect

92. C₈

#13063270

semidirect

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

subgroup	index
#19136851	2
#19134213	2
#19134210	2
C₈×C₁₆×C₄₃	2
#12517	43

subgroup

index

C₈×C₁₆×C₄₃

Subgroups — #25011315

Conjugacy classes of subgroups

Complements

Maximal subgroups