Group #43250608 — subgroups

Subgroups — #1593×C₂₂₃ #43250608

Order 28544.

148 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 148 conjugacy classes of subgroups — too many to draw. Only its 84 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	#446×C₂₂₃	C₂	1	2	yes	maximal
2	C₄×Q₁₆×C₂₂₃	C₂	1	1	yes	maximal, characteristic
3	#334×C₂₂₃	C₂	1	1	yes	maximal, characteristic
4	Q₈⋊C₈×C₂₂₃	C₂	1	2	yes	maximal
5	#322×C₂₂₃	C₂	1	1	yes	maximal, characteristic
6	C₂×Q₁₆×C₂₂₃	C₄	1	1	yes	characteristic
7	C₄×Q₈×C₂₂₃	C₂²	1	2	yes	—
8	#107×C₂₂₃	C₄	1	1	yes	characteristic
9	C₄⋊C₈×C₂₂₃	—	2	2	no	—
10	C₄⋊C₈×C₂₂₃	C₂²	1	2	yes	—
11	#103×C₂₂₃	C₄	1	2	yes	—
12	#97×C₂₂₃	—	2	2	no	—
13	C₄×C₈×C₂₂₃	C₂²	1	1	yes	characteristic
14	C₄×C₈×C₂₂₃	C₂²	1	1	yes	characteristic
15	C₄×C₈×C₂₂₃	C₂²	1	1	yes	characteristic
16	C₂×Q₈×C₂₂₃	C₂×C₄	1	2	yes	—
17	Q₁₆×C₂₂₃	—	2	2	no	—
18	C₂×C₈×C₂₂₃	—	2	2	no	—
19	C₂×C₈×C₂₂₃	—	2	1	no	—
20	C₂×C₈×C₂₂₃	C₂×C₄	1	1	yes	characteristic
21	C₂×C₈×C₂₂₃	C₂×C₄	1	1	yes	characteristic
22	C₂×C₈×C₂₂₃	D₄	1	2	yes	—
23	C₄⋊C₄×C₂₂₃	—	2	2	no	—
24	C₄⋊C₄×C₂₂₃	C₂×C₄	1	2	yes	—
25	C₄²×C₂₂₃	—	2	2	no	—
26	C₄²×C₂₂₃	C₂³	1	1	yes	characteristic
27	Q₈×C₂₂₃	—	2	2	no	—
28	Q₈×C₂₂₃	—	2	2	no	—
29	C₂×C₄×C₂₂₃	—	2	2	no	—
30	C₂×C₄×C₂₂₃	—	2	2	no	—
31	C₂×C₄×C₂₂₃	C₂²×C₄	1	1	yes	characteristic
32	C₂×C₄×C₂₂₃	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×C₂₂₃	#41	1	1	yes	characteristic
34	C₈×C₂₂₃	—	2	1	no	—
35	C₈×C₂₂₃	—	2	2	no	—
36	C₈×C₂₂₃	—	4	1	no	—
37	C₈×C₂₂₃	—	4	2	no	—
38	C₈×C₂₂₃	M₁₆	1	2	yes	—
39	C₂²×C₂₂₃	C₄×D₄	1	1	yes	characteristic
40	C₄×C₂₂₃	—	2	1	no	—
41	C₄×C₂₂₃	—	4	2	no	—
42	C₄×C₂₂₃	—	4	2	no	—
43	C₄×C₂₂₃	C₂×M₁₆	1	1	yes	characteristic
44	C₄×C₂₂₃	#131	1	1	yes	characteristic
45	C₄×C₂₂₃	#137	1	2	yes	—
46	C₂×C₂₂₃	#436	1	1	yes	characteristic
47	C₂×C₂₂₃	#441	1	1	yes	characteristic
48	C₂×C₂₂₃	#444	1	1	yes	characteristic
49	C₂₂₃	#1593	1	1	yes	characteristic, direct product factor, Sylow
50	#1593	C₂₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
51	#446	C₂×C₂₂₃	1	2	yes	—
52	C₄×Q₁₆	C₂×C₂₂₃	1	1	yes	characteristic
53	#334	C₂×C₂₂₃	1	1	yes	characteristic
54	Q₈⋊C₈	C₂×C₂₂₃	1	2	yes	—
55	#322	C₂×C₂₂₃	1	1	yes	characteristic
56	C₂×Q₁₆	C₄×C₂₂₃	1	1	yes	characteristic
57	C₄×Q₈	C₂²×C₂₂₃	1	2	yes	—
58	#107	C₄×C₂₂₃	1	1	yes	characteristic
59	C₄⋊C₈	—	2	2	no	—
60	C₄⋊C₈	C₂²×C₂₂₃	1	2	yes	—
61	#103	C₄×C₂₂₃	1	2	yes	—
62	#97	—	2	2	no	—
63	C₄×C₈	C₂²×C₂₂₃	1	1	yes	characteristic
64	C₄×C₈	C₂²×C₂₂₃	1	1	yes	characteristic
65	C₄×C₈	C₂²×C₂₂₃	1	1	yes	characteristic
66	C₂×Q₈	C₂×C₄×C₂₂₃	1	2	yes	—
67	Q₁₆	—	2	2	no	—
68	C₂×C₈	—	2	2	no	—
69	C₂×C₈	—	2	1	no	—
70	C₂×C₈	C₂×C₄×C₂₂₃	1	1	yes	characteristic
71	C₂×C₈	C₂×C₄×C₂₂₃	1	1	yes	characteristic
72	C₂×C₈	D₄×C₂₂₃	1	2	yes	—
73	C₄⋊C₄	—	2	2	no	—
74	C₄⋊C₄	C₂×C₄×C₂₂₃	1	2	yes	—
75	C₄²	—	2	2	no	—
76	C₄²	C₂³×C₂₂₃	1	1	yes	characteristic
77	Q₈	—	2	2	no	—
78	Q₈	—	2	2	no	—
79	C₂×C₄	—	2	2	no	—
80	C₂×C₄	—	2	2	no	—
81	C₂×C₄	C₂²×C₄×C₂₂₃	1	1	yes	characteristic
82	C₂×C₄	C₂×D₄×C₂₂₃	1	1	yes	characteristic
83	C₂×C₄	#41×C₂₂₃	1	1	yes	characteristic
84	C₈	—	2	1	no	—
85	C₈	—	2	2	no	—
86	C₈	—	4	1	no	—
87	C₈	—	4	2	no	—
88	C₈	M₁₆×C₂₂₃	1	2	yes	—
89	C₂²	C₄×D₄×C₂₂₃	1	1	yes	characteristic
90	C₄	—	2	1	no	—
91	C₄	—	4	2	no	—
92	C₄	—	4	2	no	—
93	C₄	C₂×M₁₆×C₂₂₃	1	1	yes	characteristic
94	C₄	#131×C₂₂₃	1	1	yes	characteristic
95	C₄	#137×C₂₂₃	1	2	yes	—
96	C₂	#436×C₂₂₃	1	1	yes	characteristic
97	C₂	#441×C₂₂₃	1	1	yes	characteristic
98	C₂	#444×C₂₂₃	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#446×C₂₂₃

C₂

yes

maximal

C₄×Q₁₆×C₂₂₃

C₂

yes

maximal, characteristic

#334×C₂₂₃

C₂

yes

maximal, characteristic

Q₈⋊C₈×C₂₂₃

C₂

yes

maximal

#322×C₂₂₃

C₂

yes

maximal, characteristic

C₂×Q₁₆×C₂₂₃

C₄

yes

characteristic

C₄×Q₈×C₂₂₃

C₂²

yes

—

#107×C₂₂₃

C₄

yes

characteristic

C₄⋊C₈×C₂₂₃

—

C₄⋊C₈×C₂₂₃

C₂²

yes

—

#103×C₂₂₃

C₄

yes

—

#97×C₂₂₃

—

C₄×C₈×C₂₂₃

C₂²

yes

characteristic

C₄×C₈×C₂₂₃

C₂²

yes

characteristic

C₄×C₈×C₂₂₃

C₂²

yes

characteristic

C₂×Q₈×C₂₂₃

C₂×C₄

yes

—

Q₁₆×C₂₂₃

—

C₂×C₈×C₂₂₃

—

C₂×C₈×C₂₂₃

—

C₂×C₈×C₂₂₃

C₂×C₄

yes

characteristic

C₂×C₈×C₂₂₃

C₂×C₄

yes

characteristic

C₂×C₈×C₂₂₃

D₄

yes

—

C₄⋊C₄×C₂₂₃

—

C₄⋊C₄×C₂₂₃

C₂×C₄

yes

—

C₄²×C₂₂₃

—

C₄²×C₂₂₃

C₂³

yes

characteristic

Q₈×C₂₂₃

—

Q₈×C₂₂₃

—

C₂×C₄×C₂₂₃

—

C₂×C₄×C₂₂₃

—

C₂×C₄×C₂₂₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₂₃

#41

yes

characteristic

C₈×C₂₂₃

—

C₈×C₂₂₃

—

C₈×C₂₂₃

—

C₈×C₂₂₃

—

C₈×C₂₂₃

M₁₆

yes

—

C₂²×C₂₂₃

C₄×D₄

yes

characteristic

C₄×C₂₂₃

—

C₄×C₂₂₃

—

C₄×C₂₂₃

—

C₄×C₂₂₃

C₂×M₁₆

yes

C₄×C₂₂₃

yes

C₄×C₂₂₃

yes

—

C₂×C₂₂₃

yes

C₂×C₂₂₃

yes

C₂×C₂₂₃

yes

C₂₂₃

yes

characteristic, direct product factor, Sylow

#1593

C₂₂₃

yes

maximal, characteristic, direct product factor, Sylow

#446

C₂×C₂₂₃

yes

—

C₄×Q₁₆

C₂×C₂₂₃

yes

characteristic

#334

C₂×C₂₂₃

yes

characteristic

Q₈⋊C₈

C₂×C₂₂₃

yes

—

#322

C₂×C₂₂₃

yes

characteristic

C₂×Q₁₆

C₄×C₂₂₃

yes

characteristic

C₄×Q₈

C₂²×C₂₂₃

yes

—

#107

C₄×C₂₂₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

C₂²×C₂₂₃

yes

—

#103

C₄×C₂₂₃

yes

—

#97

—

C₄×C₈

C₂²×C₂₂₃

yes

characteristic

C₄×C₈

C₂²×C₂₂₃

yes

characteristic

C₄×C₈

C₂²×C₂₂₃

yes

characteristic

C₂×Q₈

C₂×C₄×C₂₂₃

yes

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×C₄×C₂₂₃

yes

characteristic

C₂×C₈

C₂×C₄×C₂₂₃

yes

characteristic

C₂×C₈

D₄×C₂₂₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

C₂×C₄×C₂₂₃

yes

—

C₄²

—

C₄²

C₂³×C₂₂₃

yes

characteristic

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²×C₄×C₂₂₃

yes

characteristic

C₂×C₄

C₂×D₄×C₂₂₃

yes

characteristic

C₂×C₄

#41×C₂₂₃

yes

characteristic

C₈

—

C₈

—

C₈

—

C₈

—

C₈

M₁₆×C₂₂₃

yes

—

C₂²

C₄×D₄×C₂₂₃

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

C₂×M₁₆×C₂₂₃

yes

characteristic

C₄

#131×C₂₂₃

yes

characteristic

C₄

#137×C₂₂₃

yes

—

C₂

#436×C₂₂₃

yes

characteristic

C₂

#441×C₂₂₃

yes

characteristic

C₂

#444×C₂₂₃

yes

characteristic

	subgroup	G/H
center	C₂×C₄×C₂₂₃	C₂×D₄
commutator	C₂×C₄	C₂²×C₄×C₂₂₃
Frattini, cosocle	C₄²	C₂³×C₂₂₃
socle, prime radical	C₂²×C₂₂₃	C₄×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₂₃ ◁ C₄×C₈×C₂₂₃ ◁ G

subgroup

G/H

center

C₂×C₄×C₂₂₃

C₂×D₄

commutator

C₂×C₄

C₂²×C₄×C₂₂₃

Frattini, cosocle

C₄²

C₂³×C₂₂₃

socle, prime radical

C₂²×C₂₂₃

C₄×D₄

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₄×C₂₂₃ ◁ C₄×C₈×C₂₂₃ ◁ G

N	H ≅ G/N	type
49. C₂₂₃	#1593	direct (N × H)
50. #1593	C₂₂₃	direct (N × H)

H ≅ G/N

type

49. C₂₂₃

#1593

direct (N × H)

50. #1593

C₂₂₃

direct (N × H)

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

subgroup	index
#446×C₂₂₃	2
C₄×Q₁₆×C₂₂₃	2
#334×C₂₂₃	2
Q₈⋊C₈×C₂₂₃	2
#322×C₂₂₃	2
#1593	223

subgroup

index

#446×C₂₂₃

C₄×Q₁₆×C₂₂₃

#334×C₂₂₃

Q₈⋊C₈×C₂₂₃

#322×C₂₂₃

#1593

223

Subgroups — #1593×C₂₂₃ #43250608

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1593×C223 #43250608

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1593×C₂₂₃ #43250608