Group #43252628 — subgroups

Subgroups — #43252628

Order 28544.

180 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 113 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	C₂×C₄⋊(C₂₂₃⋊C₈)	C₂	1	2	yes	maximal
2	C₂×C₄×C₈×C₂₂₃	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₂₂₃⋊C₈	—	2	2	no	—
4	C₄⋊(C₂₂₃⋊C₈)	—	2	2	no	—
5	C₄⋊(C₂₂₃⋊C₈)	C₄	1	4	yes	—
6	C₂²×C₈×C₂₂₃	C₄	1	2	yes	—
7	C₂×C₄²×C₂₂₃	C₂²	1	1	yes	characteristic
8	C₄×C₈×C₂₂₃	—	2	2	no	—
9	C₂×C₂₂₃⋊C₈	—	2	4	no	—
10	C₂×C₂₂₃⋊C₈	—	4	2	no	—
11	C₂×C₂₂₃⋊C₈	—	4	2	no	—
12	C₂²×C₄×C₂₂₃	C₂×C₄	1	1	yes	characteristic
13	C₂²×C₄×C₂₂₃	D₄	1	1	yes	characteristic
14	C₂²×C₄×C₂₂₃	Q₈	1	1	yes	characteristic
15	C₂×C₈×C₂₂₃	—	2	2	no	—
16	C₂×C₈×C₂₂₃	—	2	2	no	—
17	C₂×C₈×C₂₂₃	C₈	1	4	yes	complemented
18	C₄²×C₂₂₃	C₂×C₄	1	2	yes	—
19	C₄²×C₂₂₃	D₄	1	2	yes	—
20	C₂₂₃⋊C₈	—	4	4	no	—
21	C₂³×C₂₂₃	C₄⋊C₄	1	1	yes	characteristic
22	C₂×C₄×C₂₂₃	—	2	2	no	—
23	C₂×C₄×C₂₂₃	C₄²	1	1	yes	characteristic
24	C₂×C₄×C₂₂₃	C₂²⋊C₄	1	2	yes	—
25	C₂×C₄×C₂₂₃	C₂²⋊C₄	1	1	yes	characteristic
26	C₂×C₄×C₂₂₃	C₄⋊C₄	1	2	yes	—
27	C₂×C₄×C₂₂₃	C₂×C₈	1	2	yes	—
28	C₂×C₄×C₂₂₃	M₁₆	1	2	yes	—
29	C₂×C₄×C₂₂₃	D₈	1	1	yes	characteristic
30	C₂×C₄×C₂₂₃	SD₁₆	1	2	yes	—
31	C₂×C₄×C₂₂₃	Q₁₆	1	1	yes	characteristic
32	C₈×C₂₂₃	—	2	4	no	—
33	C₂²×C₂₂₃	#95	1	1	yes	characteristic
34	C₂²×C₂₂₃	C₄⋊C₈	1	2	yes	—
35	C₂²×C₂₂₃	#106	1	1	yes	characteristic
36	C₂²×C₂₂₃	#107	1	1	yes	characteristic
37	C₂²×C₂₂₃	#108	1	2	yes	—
38	C₄×C₂₂₃	—	2	2	no	—
39	C₄×C₂₂₃	C₄×C₈	1	1	yes	characteristic
40	C₄×C₂₂₃	#97	1	1	yes	characteristic
41	C₄×C₂₂₃	C₂²⋊C₈	1	2	yes	—
42	C₄×C₂₂₃	#102	1	2	yes	—
43	C₄×C₂₂₃	#103	1	2	yes	—
44	C₂×C₂₂₃	#334	1	2	yes	—
45	C₂×C₂₂₃	#335	1	2	yes	—
46	C₂×C₂₂₃	#336	1	1	yes	characteristic
47	C₂×C₂₂₃	#340	1	1	yes	characteristic
48	C₂×C₂₂₃	#341	1	1	yes	characteristic
49	C₂₂₃	#1277	1	1	yes	characteristic, complemented, Sylow
50	#1277	—	223	1	no	maximal, Sylow
51	C₂×C₄⋊C₈	—	223	2	no	—
52	C₂×C₄×C₈	D₂₂₃	1	1	yes	characteristic
53	C₂²×C₈	—	446	2	no	—
54	C₂²×C₈	Dic₂₂₃	1	2	yes	—
55	C₂×C₄²	C₂×D₂₂₃	1	1	yes	characteristic
56	C₄⋊C₈	—	446	2	no	—
57	C₄⋊C₈	—	223	4	no	—
58	C₄×C₈	—	2	2	no	—
59	C₂²×C₄	C₂×Dic₂₂₃	1	1	yes	characteristic
60	C₂²×C₄	D₈₉₂	1	1	yes	characteristic
61	C₂²×C₄	Dic₄₄₆	1	1	yes	characteristic
62	C₂×C₈	—	2	2	no	—
63	C₂×C₈	—	2	2	no	—
64	C₂×C₈	—	892	2	no	—
65	C₂×C₈	—	892	2	no	—
66	C₂×C₈	—	446	4	no	—
67	C₂×C₈	C₂₂₃⋊C₈	1	4	yes	complemented
68	C₄²	C₄×D₂₂₃	1	2	yes	—
69	C₄²	C₂²⋊D₂₂₃	1	2	yes	—
70	C₂³	C₄⋊Dic₂₂₃	1	1	yes	characteristic
71	C₂×C₄	—	2	2	no	—
72	C₂×C₄	C₄×Dic₂₂₃	1	1	yes	characteristic
73	C₂×C₄	(C₂×D₂₂₃)⋊C₄	1	2	yes	—
74	C₂×C₄	C₂²⋊Dic₂₂₃	1	1	yes	characteristic
75	C₂×C₄	Dic₂₂₃⋊C₄	1	2	yes	—
76	C₂×C₄	C₂×C₂₂₃⋊C₈	1	2	yes	—
77	C₂×C₄	#16813154	1	2	yes	—
78	C₂×C₄	D₁₇₈₄	1	1	yes	characteristic
79	C₂×C₄	#16813159	1	2	yes	—
80	C₂×C₄	Dic₈₉₂	1	1	yes	characteristic
81	C₈	—	2	4	no	—
82	C₈	—	892	4	no	—
83	C₂²	C₂₂₃⋊#95	1	1	yes	characteristic
84	C₂²	C₄⋊(C₂₂₃⋊C₈)	1	2	yes	—
85	C₂²	#20297236	1	1	yes	characteristic
86	C₂²	#20297238	1	1	yes	characteristic
87	C₂²	#20297240	1	2	yes	—
88	C₄	—	2	2	no	—
89	C₄	C₄×C₂₂₃⋊C₈	1	1	yes	characteristic
90	C₄	#20297215	1	1	yes	characteristic
91	C₄	C₂²⋊(C₂₂₃⋊C₈)	1	2	yes	—
92	C₄	#20297226	1	2	yes	—
93	C₄	#20297229	1	2	yes	—
94	C₂	#28383331	1	2	yes	—
95	C₂	#28383333	1	2	yes	—
96	C₂	#28383335	1	1	yes	characteristic
97	C₂	#28383343	1	1	yes	characteristic
98	C₂	#28383345	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊(C₂₂₃⋊C₈)

C₂

yes

maximal

C₂×C₄×C₈×C₂₂₃

C₂

yes

maximal, characteristic

C₂²×C₂₂₃⋊C₈

—

C₄⋊(C₂₂₃⋊C₈)

—

C₄⋊(C₂₂₃⋊C₈)

C₄

yes

—

C₂²×C₈×C₂₂₃

C₄

yes

—

C₂×C₄²×C₂₂₃

C₂²

yes

characteristic

C₄×C₈×C₂₂₃

—

C₂×C₂₂₃⋊C₈

—

C₂×C₂₂₃⋊C₈

—

C₂×C₂₂₃⋊C₈

—

C₂²×C₄×C₂₂₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₂₃

D₄

yes

characteristic

C₂²×C₄×C₂₂₃

Q₈

yes

characteristic

C₂×C₈×C₂₂₃

—

C₂×C₈×C₂₂₃

—

C₂×C₈×C₂₂₃

C₈

yes

complemented

C₄²×C₂₂₃

C₂×C₄

yes

—

C₄²×C₂₂₃

D₄

yes

—

C₂₂₃⋊C₈

—

C₂³×C₂₂₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₂₂₃

—

C₂×C₄×C₂₂₃

C₄²

yes

characteristic

C₂×C₄×C₂₂₃

C₂²⋊C₄

yes

—

C₂×C₄×C₂₂₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₂₂₃

C₄⋊C₄

yes

—

C₂×C₄×C₂₂₃

C₂×C₈

yes

—

C₂×C₄×C₂₂₃

M₁₆

yes

—

C₂×C₄×C₂₂₃

D₈

yes

characteristic

C₂×C₄×C₂₂₃

SD₁₆

yes

—

C₂×C₄×C₂₂₃

Q₁₆

yes

characteristic

C₈×C₂₂₃

—

C₂²×C₂₂₃

#95

yes

characteristic

C₂²×C₂₂₃

C₄⋊C₈

yes

—

C₂²×C₂₂₃

yes

C₂²×C₂₂₃

yes

C₂²×C₂₂₃

yes

—

C₄×C₂₂₃

—

C₄×C₂₂₃

C₄×C₈

yes

characteristic

C₄×C₂₂₃

#97

yes

characteristic

C₄×C₂₂₃

C₂²⋊C₈

yes

—

C₄×C₂₂₃

#102

yes

—

C₄×C₂₂₃

#103

yes

—

C₂×C₂₂₃

#334

yes

—

C₂×C₂₂₃

#335

yes

—

C₂×C₂₂₃

yes

C₂×C₂₂₃

yes

C₂×C₂₂₃

yes

C₂₂₃

yes

characteristic, complemented, Sylow

#1277

—

223

maximal, Sylow

C₂×C₄⋊C₈

—

223

—

C₂×C₄×C₈

D₂₂₃

yes

characteristic

C₂²×C₈

—

446

—

C₂²×C₈

Dic₂₂₃

yes

—

C₂×C₄²

C₂×D₂₂₃

yes

characteristic

C₄⋊C₈

—

446

—

C₄⋊C₈

—

223

—

C₄×C₈

—

C₂²×C₄

C₂×Dic₂₂₃

yes

characteristic

C₂²×C₄

D₈₉₂

yes

characteristic

C₂²×C₄

Dic₄₄₆

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

892

—

C₂×C₈

—

892

—

C₂×C₈

—

446

—

C₂×C₈

C₂₂₃⋊C₈

yes

complemented

C₄²

C₄×D₂₂₃

yes

—

C₄²

C₂²⋊D₂₂₃

yes

—

C₂³

C₄⋊Dic₂₂₃

yes

characteristic

C₂×C₄

—

C₂×C₄

C₄×Dic₂₂₃

yes

characteristic

C₂×C₄

(C₂×D₂₂₃)⋊C₄

yes

—

C₂×C₄

C₂²⋊Dic₂₂₃

yes

characteristic

C₂×C₄

Dic₂₂₃⋊C₄

yes

—

C₂×C₄

C₂×C₂₂₃⋊C₈

yes

—

C₂×C₄

#16813154

yes

—

C₂×C₄

D₁₇₈₄

yes

characteristic

C₂×C₄

#16813159

yes

—

C₂×C₄

Dic₈₉₂

yes

characteristic

C₈

—

C₈

—

892

—

C₂²

C₂₂₃⋊#95

yes

characteristic

C₂²

C₄⋊(C₂₂₃⋊C₈)

yes

—

C₂²

yes

C₂²

yes

C₂²

yes

—

C₄

—

C₄

C₄×C₂₂₃⋊C₈

yes

characteristic

C₄

#20297215

yes

characteristic

C₄

C₂²⋊(C₂₂₃⋊C₈)

yes

—

C₄

#20297226

yes

—

C₄

#20297229

yes

—

C₂

#28383331

yes

—

C₂

#28383333

yes

—

C₂

yes

C₂

yes

C₂

yes

	subgroup	G/H
center	C₂²×C₄	D₈₉₂
commutator	C₄×C₂₂₃	C₄×C₈
Frattini	C₂×C₄²	C₂×D₂₂₃
Fitting	C₂×C₄×C₈×C₂₂₃	C₂
socle, prime radical	C₂³×C₂₂₃	C₄⋊C₄
cosocle	C₂×C₄²×C₂₂₃	C₂²
hypercenter	C₂×C₄×C₈	D₂₂₃
hypocenter	C₂₂₃	#1277
LCS	G ▷ C₄×C₂₂₃ ▷ C₂×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂²×C₄

D₈₉₂

commutator

C₄×C₂₂₃

C₄×C₈

Frattini

C₂×C₄²

C₂×D₂₂₃

Fitting

C₂×C₄×C₈×C₂₂₃

C₂

socle, prime radical

C₂³×C₂₂₃

C₄⋊C₄

cosocle

C₂×C₄²×C₂₂₃

C₂²

hypercenter

C₂×C₄×C₈

D₂₂₃

hypocenter

C₂₂₃

#1277

LCS

G ▷ C₄×C₂₂₃ ▷ C₂×C₂₂₃ ▷ C₂₂₃

UCS

C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
17. C₂×C₈×C₂₂₃	C₈	semidirect
49. C₂₂₃	#1277	semidirect
67. C₂×C₈	C₂₂₃⋊C₈	semidirect

H ≅ G/N

type

17. C₂×C₈×C₂₂₃

C₈

semidirect

49. C₂₂₃

#1277

semidirect

67. C₂×C₈

C₂₂₃⋊C₈

semidirect

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

subgroup	index
C₂×C₄⋊(C₂₂₃⋊C₈)	2
C₂×C₄×C₈×C₂₂₃	2
#1277	223

subgroup

index

C₂×C₄⋊(C₂₂₃⋊C₈)

C₂×C₄×C₈×C₂₂₃

#1277

223

Subgroups — #43252628

Conjugacy classes of subgroups

Complements

Maximal subgroups