Group #43252637 — subgroups

Subgroups — #43252637

Order 28544.

152 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 152 conjugacy classes of subgroups — too many to draw. Only its 77 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	#28383633	C₂	1	1	yes	maximal, characteristic
2	C₂×#20297215	C₂	1	1	yes	maximal, characteristic
3	#431×C₂₂₃	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₂₂₃⋊C₈	C₄	1	2	yes	—
5	C₄⋊(C₂₂₃⋊C₈)	—	2	1	no	—
6	C₂²⋊(C₂₂₃⋊C₈)	C₄	1	2	yes	—
7	#20297215	—	2	1	no	—
8	#20297215	—	2	1	no	—
9	C₄×C₂₂₃⋊C₈	—	2	1	no	—
10	C₂×C₄²×C₂₂₃	C₂²	1	1	yes	characteristic
11	C₄⋊C₈×C₂₂₃	—	2	1	no	—
12	C₂²⋊C₈×C₂₂₃	C₄	1	2	yes	—
13	C₄×C₈×C₂₂₃	—	2	1	no	—
14	C₂×C₂₂₃⋊C₈	—	2	2	no	—
15	C₂×C₂₂₃⋊C₈	—	2	2	no	—
16	C₂×C₂₂₃⋊C₈	—	2	2	no	—
17	C₂×C₂₂₃⋊C₈	C₈	1	4	yes	complemented
18	C₂²×C₄×C₂₂₃	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄×C₂₂₃	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₄×C₂₂₃	C₂×C₄	1	1	yes	characteristic
21	C₂×C₈×C₂₂₃	—	2	2	no	—
22	C₄²×C₂₂₃	D₄	1	1	yes	characteristic
23	C₄²×C₂₂₃	D₄	1	1	yes	characteristic
24	C₄²×C₂₂₃	D₄	1	1	yes	characteristic
25	C₄²×C₂₂₃	Q₈	1	1	yes	characteristic
26	C₂₂₃⋊C₈	—	2	4	no	—
27	C₂₂₃⋊C₈	—	4	2	no	—
28	C₂³×C₂₂₃	C₄²	1	1	yes	characteristic
29	C₂×C₄×C₂₂₃	—	4	1	no	—
30	C₂×C₄×C₂₂₃	—	4	1	no	—
31	C₂×C₄×C₂₂₃	C₂²⋊C₄	1	1	yes	characteristic
32	C₂×C₄×C₂₂₃	C₂²⋊C₄	1	1	yes	characteristic
33	C₂×C₄×C₂₂₃	C₂²⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₂₂₃	C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₄×C₂₂₃	C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₄×C₂₂₃	C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₄×C₂₂₃	C₂×C₈	1	2	yes	—
38	C₂×C₄×C₂₂₃	M₁₆	1	2	yes	—
39	C₈×C₂₂₃	—	4	2	no	—
40	C₂²×C₂₂₃	—	2	1	no	—
41	C₂²×C₂₂₃	—	2	1	no	—
42	C₂²×C₂₂₃	#95	1	1	yes	characteristic
43	C₂²×C₂₂₃	C₄×C₈	1	1	yes	characteristic
44	C₂²×C₂₂₃	#97	1	1	yes	characteristic
45	C₄×C₂₂₃	—	2	1	no	—
46	C₄×C₂₂₃	—	2	1	no	—
47	C₄×C₂₂₃	—	2	1	no	—
48	C₄×C₂₂₃	—	2	1	no	—
49	C₄×C₂₂₃	C₂²⋊C₈	1	2	yes	—
50	C₄×C₂₂₃	C₄⋊C₈	1	2	yes	—
51	C₂×C₂₂₃	—	2	1	no	—
52	C₂×C₂₂₃	—	2	1	no	—
53	C₂×C₂₂₃	#336	1	1	yes	characteristic
54	C₂×C₂₂₃	#344	1	1	yes	characteristic
55	C₂×C₂₂₃	#344	1	1	yes	characteristic
56	C₂₂₃	#1281	1	1	yes	characteristic, complemented, Sylow
57	#1281	—	223	1	no	maximal, Sylow
58	#431	—	223	1	no	—
59	#431	D₂₂₃	1	1	yes	characteristic
60	C₂×#97	—	223	1	no	—
61	C₂²×C₈	—	223	2	no	—
62	C₂×C₄²	C₂×D₂₂₃	1	1	yes	characteristic
63	C₄⋊C₈	—	2	1	no	—
64	C₄⋊C₈	—	446	1	no	—
65	C₂²⋊C₈	—	223	2	no	—
66	C₂²⋊C₈	Dic₂₂₃	1	2	yes	—
67	#97	—	446	1	no	—
68	#97	—	446	1	no	—
69	C₄×C₈	—	2	1	no	—
70	C₄×C₈	—	446	1	no	—
71	C₂²×C₄	C₄×D₂₂₃	1	1	yes	characteristic
72	C₂²×C₄	C₄×D₂₂₃	1	1	yes	characteristic
73	C₂²×C₄	C₂×Dic₂₂₃	1	1	yes	characteristic
74	C₂×C₈	—	2	2	no	—
75	C₂×C₈	—	446	2	no	—
76	C₂×C₈	—	446	2	no	—
77	C₂×C₈	—	446	2	no	—
78	C₂×C₈	—	223	4	no	—
79	C₄²	C₂²⋊D₂₂₃	1	1	yes	characteristic
80	C₄²	C₂²⋊D₂₂₃	1	1	yes	characteristic
81	C₄²	D₈₉₂	1	1	yes	characteristic
82	C₄²	Dic₄₄₆	1	1	yes	characteristic
83	C₂³	C₄×Dic₂₂₃	1	1	yes	characteristic
84	C₂×C₄	—	4	1	no	—
85	C₂×C₄	—	4	1	no	—
86	C₂×C₄	(C₂×D₂₂₃)⋊C₄	1	1	yes	characteristic
87	C₂×C₄	(C₂×D₂₂₃)⋊C₄	1	1	yes	characteristic
88	C₂×C₄	C₂²⋊Dic₂₂₃	1	1	yes	characteristic
89	C₂×C₄	Dic₂₂₃⋊C₄	1	1	yes	characteristic
90	C₂×C₄	Dic₂₂₃⋊C₄	1	1	yes	characteristic
91	C₂×C₄	C₄⋊Dic₂₂₃	1	1	yes	characteristic
92	C₂×C₄	C₈×D₂₂₃	1	2	yes	—
93	C₂×C₄	#16813153	1	2	yes	—
94	C₈	—	4	2	no	—
95	C₈	—	892	2	no	—
96	C₈	—	446	4	no	—
97	C₂²	—	2	1	no	—
98	C₂²	—	2	1	no	—
99	C₂²	C₂₂₃⋊#95	1	1	yes	characteristic
100	C₂²	C₈×Dic₂₂₃	1	1	yes	characteristic
101	C₂²	#20297214	1	1	yes	characteristic
102	C₄	—	2	1	no	—
103	C₄	—	2	1	no	—
104	C₄	—	2	1	no	—
105	C₄	—	2	1	no	—
106	C₄	(C₂×D₂₂₃)⋊C₈	1	2	yes	—
107	C₄	Dic₂₂₃⋊C₈	1	2	yes	—
108	C₂	—	2	1	no	—
109	C₂	—	2	1	no	—
110	C₂	#28383334	1	1	yes	characteristic
111	C₂	#28383350	1	1	yes	characteristic
112	C₂	#28383352	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#28383633

C₂

yes

maximal, characteristic

C₂×#20297215

C₂

yes

maximal, characteristic

#431×C₂₂₃

C₂

yes

maximal, characteristic

C₂²×C₂₂₃⋊C₈

C₄

yes

—

C₄⋊(C₂₂₃⋊C₈)

—

C₂²⋊(C₂₂₃⋊C₈)

C₄

yes

—

#20297215

—

#20297215

—

C₄×C₂₂₃⋊C₈

—

C₂×C₄²×C₂₂₃

C₂²

yes

characteristic

C₄⋊C₈×C₂₂₃

—

C₂²⋊C₈×C₂₂₃

C₄

yes

—

C₄×C₈×C₂₂₃

—

C₂×C₂₂₃⋊C₈

—

C₂×C₂₂₃⋊C₈

—

C₂×C₂₂₃⋊C₈

—

C₂×C₂₂₃⋊C₈

C₈

yes

complemented

C₂²×C₄×C₂₂₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₂₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₂₃

C₂×C₄

yes

characteristic

C₂×C₈×C₂₂₃

—

C₄²×C₂₂₃

D₄

yes

characteristic

C₄²×C₂₂₃

D₄

yes

characteristic

C₄²×C₂₂₃

D₄

yes

characteristic

C₄²×C₂₂₃

Q₈

yes

characteristic

C₂₂₃⋊C₈

—

C₂₂₃⋊C₈

—

C₂³×C₂₂₃

C₄²

yes

characteristic

C₂×C₄×C₂₂₃

—

C₂×C₄×C₂₂₃

—

C₂×C₄×C₂₂₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₂₂₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₂₂₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₂₂₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₂₂₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₂₂₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₂₂₃

C₂×C₈

yes

—

C₂×C₄×C₂₂₃

M₁₆

yes

—

C₈×C₂₂₃

—

C₂²×C₂₂₃

—

C₂²×C₂₂₃

—

C₂²×C₂₂₃

#95

yes

characteristic

C₂²×C₂₂₃

C₄×C₈

yes

characteristic

C₂²×C₂₂₃

#97

yes

characteristic

C₄×C₂₂₃

—

C₄×C₂₂₃

—

C₄×C₂₂₃

—

C₄×C₂₂₃

—

C₄×C₂₂₃

C₂²⋊C₈

yes

—

C₄×C₂₂₃

C₄⋊C₈

yes

—

C₂×C₂₂₃

—

C₂×C₂₂₃

—

C₂×C₂₂₃

yes

C₂×C₂₂₃

yes

C₂×C₂₂₃

yes

C₂₂₃

yes

characteristic, complemented, Sylow

#1281

—

223

maximal, Sylow

#431

—

223

—

#431

D₂₂₃

yes

characteristic

C₂×#97

—

223

—

C₂²×C₈

—

223

—

C₂×C₄²

C₂×D₂₂₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

446

—

C₂²⋊C₈

—

223

—

C₂²⋊C₈

Dic₂₂₃

yes

—

#97

—

446

—

#97

—

446

—

C₄×C₈

—

C₄×C₈

—

446

—

C₂²×C₄

C₄×D₂₂₃

yes

characteristic

C₂²×C₄

C₄×D₂₂₃

yes

characteristic

C₂²×C₄

C₂×Dic₂₂₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

446

—

C₂×C₈

—

446

—

C₂×C₈

—

446

—

C₂×C₈

—

223

—

C₄²

C₂²⋊D₂₂₃

yes

characteristic

C₄²

C₂²⋊D₂₂₃

yes

characteristic

C₄²

D₈₉₂

yes

characteristic

C₄²

Dic₄₄₆

yes

characteristic

C₂³

C₄×Dic₂₂₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

(C₂×D₂₂₃)⋊C₄

yes

characteristic

C₂×C₄

(C₂×D₂₂₃)⋊C₄

yes

characteristic

C₂×C₄

C₂²⋊Dic₂₂₃

yes

characteristic

C₂×C₄

Dic₂₂₃⋊C₄

yes

characteristic

C₂×C₄

Dic₂₂₃⋊C₄

yes

characteristic

C₂×C₄

C₄⋊Dic₂₂₃

yes

characteristic

C₂×C₄

C₈×D₂₂₃

yes

—

C₂×C₄

#16813153

yes

—

C₈

—

C₈

—

892

—

C₈

—

446

—

C₂²

—

C₂²

—

C₂²

C₂₂₃⋊#95

yes

characteristic

100

C₂²

C₈×Dic₂₂₃

yes

characteristic

101

C₂²

#20297214

yes

characteristic

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

—

106

C₄

(C₂×D₂₂₃)⋊C₈

yes

—

107

C₄

Dic₂₂₃⋊C₈

yes

—

108

C₂

—

109

C₂

—

110

C₂

#28383334

yes

characteristic

111

C₂

#28383350

yes

characteristic

112

C₂

#28383352

yes

characteristic

	subgroup	G/H
center	C₂×C₄	(C₂×D₂₂₃)⋊C₄
commutator	C₂²×C₂₂₃	C₄×C₈
Frattini	C₂×C₄²	C₂×D₂₂₃
Fitting	#431×C₂₂₃	C₂
socle	C₂²×C₂₂₃	#95
prime radical	C₂³×C₂₂₃	C₄²
cosocle	C₂×C₄²×C₂₂₃	C₂²
hypercenter	#431	D₂₂₃
hypocenter	C₂₂₃	#1281
LCS	G ▷ C₂²×C₂₂₃ ▷ C₂×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #431

subgroup

G/H

center

C₂×C₄

(C₂×D₂₂₃)⋊C₄

commutator

C₂²×C₂₂₃

C₄×C₈

Frattini

C₂×C₄²

C₂×D₂₂₃

Fitting

#431×C₂₂₃

C₂

socle

C₂²×C₂₂₃

#95

prime radical

C₂³×C₂₂₃

C₄²

cosocle

C₂×C₄²×C₂₂₃

C₂²

D₂₂₃

C₂₂₃

G ▷ C₂²×C₂₂₃ ▷ C₂×C₂₂₃ ▷ C₂₂₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #431

N	H ≅ G/N	type
17. C₂×C₂₂₃⋊C₈	C₈	semidirect
56. C₂₂₃	#1281	semidirect

H ≅ G/N

type

17. C₂×C₂₂₃⋊C₈

C₈

semidirect

56. C₂₂₃

#1281

semidirect

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

subgroup	index
#28383633	2
C₂×#20297215	2
#431×C₂₂₃	2
#1281	223

subgroup

index

#28383633

C₂×#20297215

#431×C₂₂₃

#1281

223

Subgroups — #43252637

Conjugacy classes of subgroups

Complements

Maximal subgroups