Group #43252684 — subgroups

Subgroups — #43252684

Order 28544.

150 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 150 conjugacy classes of subgroups — too many to draw. Only its 65 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	#28383604	C₂	1	1	yes	maximal, characteristic
2	C₄×#16813154	C₂	1	1	yes	maximal, characteristic
3	#431×C₂₂₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#16813154	—	2	1	no	—
5	C₂×#16813154	C₄	1	2	yes	—
6	C₄⋊(C₂₂₃⋊C₈)	—	2	1	no	—
7	C₄⋊(C₂₂₃⋊C₈)	—	2	1	no	—
8	#20297215	C₄	1	2	yes	—
9	C₄×C₂₂₃⋊C₈	—	2	1	no	—
10	C₂×C₄²×C₂₂₃	C₂²	1	1	yes	characteristic
11	C₄⋊C₈×C₂₂₃	C₄	1	2	yes	—
12	C₂²⋊C₈×C₂₂₃	—	2	1	no	—
13	C₄×C₈×C₂₂₃	—	2	1	no	—
14	#16813154	—	2	4	no	—
15	#16813154	—	4	2	no	—
16	C₂×C₂₂₃⋊C₈	—	2	2	no	—
17	C₂×C₂₂₃⋊C₈	—	2	2	no	—
18	C₂²×C₄×C₂₂₃	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄×C₂₂₃	D₄	1	1	yes	characteristic
20	C₂²×C₄×C₂₂₃	Q₈	1	1	yes	characteristic
21	C₂×C₈×C₂₂₃	—	2	2	no	—
22	C₄²×C₂₂₃	C₂×C₄	1	1	yes	characteristic
23	C₄²×C₂₂₃	C₂×C₄	1	1	yes	characteristic
24	C₄²×C₂₂₃	D₄	1	1	yes	characteristic
25	C₄²×C₂₂₃	D₄	1	1	yes	characteristic
26	C₂₂₃⋊C₈	—	2	4	no	—
27	C₂₂₃⋊C₈	—	4	2	no	—
28	C₂³×C₂₂₃	C₄⋊C₄	1	1	yes	characteristic
29	C₂×C₄×C₂₂₃	—	2	1	no	—
30	C₂×C₄×C₂₂₃	—	2	1	no	—
31	C₂×C₄×C₂₂₃	—	2	2	no	—
32	C₂×C₄×C₂₂₃	—	2	2	no	—
33	C₂×C₄×C₂₂₃	C₄²	1	1	yes	characteristic
34	C₂×C₄×C₂₂₃	C₂²⋊C₄	1	1	yes	characteristic
35	C₂×C₄×C₂₂₃	C₂²⋊C₄	1	1	yes	characteristic
36	C₂×C₄×C₂₂₃	C₂²⋊C₄	1	1	yes	characteristic
37	C₂×C₄×C₂₂₃	C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₄×C₂₂₃	C₄⋊C₄	1	1	yes	characteristic
39	C₈×C₂₂₃	—	4	2	no	—
40	C₂²×C₂₂₃	—	2	1	no	—
41	C₂²×C₂₂₃	—	2	1	no	—
42	C₂²×C₂₂₃	—	2	1	no	—
43	C₂²×C₂₂₃	#95	1	1	yes	characteristic
44	C₄×C₂₂₃	—	2	1	no	—
45	C₄×C₂₂₃	—	2	1	no	—
46	C₄×C₂₂₃	—	2	1	no	—
47	C₄×C₂₂₃	#100	1	1	yes	characteristic
48	C₄×C₂₂₃	#101	1	1	yes	characteristic
49	C₄×C₂₂₃	#104	1	2	yes	—
50	C₄×C₂₂₃	#108	1	2	yes	—
51	C₂×C₂₂₃	—	4	1	no	—
52	C₂×C₂₂₃	#339	1	1	yes	characteristic
53	C₂×C₂₂₃	#341	1	1	yes	characteristic
54	C₂×C₂₂₃	#343	1	1	yes	characteristic
55	C₂₂₃	#1299	1	1	yes	characteristic, complemented, Sylow
56	#1299	—	223	1	no	maximal, Sylow
57	#431	D₂₂₃	1	1	yes	characteristic
58	#423	—	223	1	no	—
59	C₄×M₁₆	—	223	1	no	—
60	C₂×M₁₆	—	446	1	no	—
61	C₂×M₁₆	—	223	2	no	—
62	C₂×C₄²	C₂×D₂₂₃	1	1	yes	characteristic
63	C₄⋊C₈	—	446	1	no	—
64	C₄⋊C₈	—	446	1	no	—
65	C₄⋊C₈	Dic₂₂₃	1	2	yes	—
66	C₂²⋊C₈	—	2	1	no	—
67	#97	—	223	2	no	—
68	C₄×C₈	—	2	1	no	—
69	C₄×C₈	—	446	1	no	—
70	C₂²×C₄	C₄×D₂₂₃	1	1	yes	characteristic
71	C₂²×C₄	C₂²⋊D₂₂₃	1	1	yes	characteristic
72	C₂²×C₄	Dic₄₄₆	1	1	yes	characteristic
73	M₁₆	—	892	2	no	—
74	M₁₆	—	446	4	no	—
75	C₂×C₈	—	2	2	no	—
76	C₂×C₈	—	446	2	no	—
77	C₂×C₈	—	446	2	no	—
78	C₄²	C₄×D₂₂₃	1	1	yes	characteristic
79	C₄²	C₂×Dic₂₂₃	1	1	yes	characteristic
80	C₄²	C₂²⋊D₂₂₃	1	1	yes	characteristic
81	C₄²	D₈₉₂	1	1	yes	characteristic
82	C₂³	Dic₂₂₃⋊C₄	1	1	yes	characteristic
83	C₂×C₄	—	2	1	no	—
84	C₂×C₄	—	2	1	no	—
85	C₂×C₄	—	2	2	no	—
86	C₂×C₄	—	2	2	no	—
87	C₂×C₄	C₄×Dic₂₂₃	1	1	yes	characteristic
88	C₂×C₄	(C₂×D₂₂₃)⋊C₄	1	1	yes	characteristic
89	C₂×C₄	(C₂×D₂₂₃)⋊C₄	1	1	yes	characteristic
90	C₂×C₄	C₂²⋊Dic₂₂₃	1	1	yes	characteristic
91	C₂×C₄	Dic₂₂₃⋊C₄	1	1	yes	characteristic
92	C₂×C₄	C₄⋊Dic₂₂₃	1	1	yes	characteristic
93	C₈	—	4	2	no	—
94	C₈	—	892	2	no	—
95	C₈	—	446	4	no	—
96	C₂²	—	2	1	no	—
97	C₂²	—	2	1	no	—
98	C₂²	—	2	1	no	—
99	C₂²	C₂₂₃⋊#95	1	1	yes	characteristic
100	C₄	—	2	1	no	—
101	C₄	—	2	1	no	—
102	C₄	—	2	1	no	—
103	C₄	#20297220	1	1	yes	characteristic
104	C₄	#20297222	1	1	yes	characteristic
105	C₄	#20297231	1	2	yes	—
106	C₄	#20297240	1	2	yes	—
107	C₂	—	4	1	no	—
108	C₂	#28383339	1	1	yes	characteristic
109	C₂	#28383345	1	1	yes	characteristic
110	C₂	#28383348	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#28383604

C₂

yes

maximal, characteristic

C₄×#16813154

C₂

yes

maximal, characteristic

#431×C₂₂₃

C₂

yes

maximal, characteristic

C₂×#16813154

—

C₂×#16813154

C₄

yes

—

C₄⋊(C₂₂₃⋊C₈)

—

C₄⋊(C₂₂₃⋊C₈)

—

#20297215

C₄

yes

—

C₄×C₂₂₃⋊C₈

—

C₂×C₄²×C₂₂₃

C₂²

yes

characteristic

C₄⋊C₈×C₂₂₃

C₄

yes

—

C₂²⋊C₈×C₂₂₃

—

C₄×C₈×C₂₂₃

—

#16813154

—

#16813154

—

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

—

C₄²×C₂₂₃

C₂×C₄

yes

characteristic

C₄²×C₂₂₃

C₂×C₄

yes

characteristic

C₄²×C₂₂₃

D₄

yes

characteristic

C₄²×C₂₂₃

D₄

yes

characteristic

C₂₂₃⋊C₈

—

C₂₂₃⋊C₈

—

C₂³×C₂₂₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₂₂₃

—

C₂×C₄×C₂₂₃

—

C₂×C₄×C₂₂₃

—

C₂×C₄×C₂₂₃

—

C₂×C₄×C₂₂₃

C₄²

yes

characteristic

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

C₈×C₂₂₃

—

C₂²×C₂₂₃

—

C₂²×C₂₂₃

—

C₂²×C₂₂₃

—

C₂²×C₂₂₃

#95

yes

characteristic

C₄×C₂₂₃

—

C₄×C₂₂₃

—

C₄×C₂₂₃

—

C₄×C₂₂₃

yes

C₄×C₂₂₃

yes

C₄×C₂₂₃

yes

—

C₄×C₂₂₃

#108

yes

—

C₂×C₂₂₃

—

C₂×C₂₂₃

yes

C₂×C₂₂₃

yes

C₂×C₂₂₃

yes

C₂₂₃

yes

characteristic, complemented, Sylow

#1299

—

223

maximal, Sylow

#431

D₂₂₃

yes

characteristic

#423

—

223

—

C₄×M₁₆

—

223

—

C₂×M₁₆

—

446

—

C₂×M₁₆

—

223

—

C₂×C₄²

C₂×D₂₂₃

yes

characteristic

C₄⋊C₈

—

446

—

C₄⋊C₈

—

446

—

C₄⋊C₈

Dic₂₂₃

yes

—

C₂²⋊C₈

—

#97

—

223

—

C₄×C₈

—

C₄×C₈

—

446

—

C₂²×C₄

C₄×D₂₂₃

yes

characteristic

C₂²×C₄

C₂²⋊D₂₂₃

yes

characteristic

C₂²×C₄

Dic₄₄₆

yes

characteristic

M₁₆

—

892

—

M₁₆

—

446

—

C₂×C₈

—

C₂×C₈

—

446

—

C₂×C₈

—

446

—

C₄²

C₄×D₂₂₃

yes

characteristic

C₄²

C₂×Dic₂₂₃

yes

characteristic

C₄²

C₂²⋊D₂₂₃

yes

characteristic

C₄²

D₈₉₂

yes

characteristic

C₂³

Dic₂₂₃⋊C₄

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄×Dic₂₂₃

yes

characteristic

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

C₄⋊Dic₂₂₃

yes

characteristic

C₈

—

C₈

—

892

—

C₈

—

446

—

C₂²

—

C₂²

—

C₂²

—

C₂²

C₂₂₃⋊#95

yes

characteristic

100

C₄

—

101

C₄

—

102

C₄

—

103

C₄

#20297220

yes

characteristic

104

C₄

#20297222

yes

characteristic

105

C₄

#20297231

yes

—

106

C₄

#20297240

yes

—

107

C₂

—

108

C₂

#28383339

yes

characteristic

109

C₂

#28383345

yes

characteristic

110

C₂

#28383348

yes

characteristic

	subgroup	G/H
center	C₂×C₄	(C₂×D₂₂₃)⋊C₄
commutator	C₂×C₄×C₂₂₃	C₄²
Frattini	C₂×C₄²	C₂×D₂₂₃
Fitting	#431×C₂₂₃	C₂
socle	C₂²×C₂₂₃	#95
prime radical	C₂³×C₂₂₃	C₄⋊C₄
cosocle	C₂×C₄²×C₂₂₃	C₂²
hypercenter	#431	D₂₂₃
hypocenter	C₂₂₃	#1299
LCS	G ▷ C₂×C₄×C₂₂₃ ▷ C₂²×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #431

subgroup

G/H

center

C₂×C₄

(C₂×D₂₂₃)⋊C₄

commutator

C₂×C₄×C₂₂₃

C₄²

Frattini

C₂×C₄²

C₂×D₂₂₃

Fitting

#431×C₂₂₃

C₂

socle

C₂²×C₂₂₃

#95

prime radical

C₂³×C₂₂₃

C₄⋊C₄

cosocle

C₂×C₄²×C₂₂₃

C₂²

D₂₂₃

C₂₂₃

G ▷ C₂×C₄×C₂₂₃ ▷ C₂²×C₂₂₃ ▷ C₂₂₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #431

N	H ≅ G/N	type
55. C₂₂₃	#1299	semidirect

H ≅ G/N

type

55. C₂₂₃

#1299

semidirect

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

subgroup	index
#28383604	2
C₄×#16813154	2
#431×C₂₂₃	2
#1299	223

subgroup

index

#28383604

C₄×#16813154

#431×C₂₂₃

#1299

223

Subgroups — #43252684

Conjugacy classes of subgroups

Complements

Maximal subgroups