Group #43251186 — subgroups

Subgroups — #2171×C₂₂₃ #43251186

Order 28544.

144 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 144 conjugacy classes of subgroups — too many to draw. Only its 92 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	#504×C₂₂₃	C₂	1	2	yes	maximal, complemented
2	#443×C₂₂₃	C₂	1	1	yes	maximal, characteristic
3	#364×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
4	#350×C₂₂₃	C₂	1	2	yes	maximal, complemented
5	#346×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
6	#135×C₂₂₃	C₄	1	1	yes	characteristic
7	#131×C₂₂₃	C₂²	1	2	yes	—
8	M₃₂×C₂₂₃	—	2	2	no	—
9	M₃₂×C₂₂₃	C₂²	1	2	yes	—
10	C₂×C₁₆×C₂₂₃	—	2	2	no	—
11	C₂×C₁₆×C₂₂₃	C₂²	1	1	yes	characteristic
12	C₂×C₁₆×C₂₂₃	C₂²	1	1	yes	characteristic
13	#108×C₂₂₃	C₄	1	1	yes	characteristic
14	#104×C₂₂₃	C₄	1	2	yes	—
15	C₄×C₈×C₂₂₃	C₂²	1	1	yes	characteristic
16	#41×C₂₂₃	C₂×C₄	1	2	yes	—
17	Q₁₆×C₂₂₃	C₈	1	1	yes	characteristic
18	SD₁₆×C₂₂₃	C₈	1	2	yes	—
19	D₈×C₂₂₃	C₈	1	1	yes	characteristic
20	M₁₆×C₂₂₃	—	2	2	no	—
21	M₁₆×C₂₂₃	C₂×C₄	1	2	yes	—
22	C₂×C₈×C₂₂₃	—	2	2	no	—
23	C₂×C₈×C₂₂₃	C₂×C₄	1	1	yes	characteristic
24	C₂×C₈×C₂₂₃	C₂³	1	1	yes	characteristic
25	C₄²×C₂₂₃	C₂×C₄	1	1	yes	characteristic
26	C₁₆×C₂₂₃	—	2	2	no	—
27	C₁₆×C₂₂₃	—	2	1	no	—
28	C₁₆×C₂₂₃	D₄	1	2	yes	—
29	Q₈×C₂₂₃	C₂×C₈	1	2	yes	—
30	D₄×C₂₂₃	—	2	2	no	—
31	D₄×C₂₂₃	C₂×C₈	1	2	yes	—
32	C₂×C₄×C₂₂₃	—	2	2	no	—
33	C₂×C₄×C₂₂₃	—	2	1	no	—
34	C₂×C₄×C₂₂₃	C₂²×C₄	1	1	yes	characteristic
35	C₈×C₂₂₃	—	2	2	no	—
36	C₈×C₂₂₃	C₂×C₈	1	1	yes	characteristic
37	C₈×C₂₂₃	C₂×C₈	1	1	yes	characteristic
38	C₈×C₂₂₃	C₂×D₄	1	1	yes	characteristic
39	C₈×C₂₂₃	#41	1	1	yes	characteristic
40	C₂²×C₂₂₃	—	2	2	no	—
41	C₂²×C₂₂₃	#131	1	1	yes	characteristic
42	C₄×C₂₂₃	—	2	2	no	—
43	C₄×C₂₂₃	—	4	1	no	—
44	C₄×C₂₂₃	C₄×D₄	1	1	yes	characteristic
45	C₄×C₂₂₃	C₂²×C₈	1	1	yes	characteristic
46	C₂×C₂₂₃	—	2	1	no	—
47	C₂×C₂₂₃	—	4	2	no	—
48	C₂×C₂₂₃	C₈×D₄	1	1	yes	characteristic
49	C₂₂₃	#2171	1	1	yes	characteristic, direct product factor, Sylow
50	#2171	C₂₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
51	#504	C₂×C₂₂₃	1	2	yes	complemented
52	#443	C₂×C₂₂₃	1	1	yes	characteristic
53	#364	C₂×C₂₂₃	1	1	yes	characteristic, complemented
54	#350	C₂×C₂₂₃	1	2	yes	complemented
55	#346	C₂×C₂₂₃	1	1	yes	characteristic, complemented
56	#135	C₄×C₂₂₃	1	1	yes	characteristic
57	#131	C₂²×C₂₂₃	1	2	yes	—
58	M₃₂	—	2	2	no	—
59	M₃₂	C₂²×C₂₂₃	1	2	yes	—
60	C₂×C₁₆	—	2	2	no	—
61	C₂×C₁₆	C₂²×C₂₂₃	1	1	yes	characteristic
62	C₂×C₁₆	C₂²×C₂₂₃	1	1	yes	characteristic
63	#108	C₄×C₂₂₃	1	1	yes	characteristic
64	#104	C₄×C₂₂₃	1	2	yes	—
65	C₄×C₈	C₂²×C₂₂₃	1	1	yes	characteristic
66	#41	C₂×C₄×C₂₂₃	1	2	yes	—
67	Q₁₆	C₈×C₂₂₃	1	1	yes	characteristic
68	SD₁₆	C₈×C₂₂₃	1	2	yes	—
69	D₈	C₈×C₂₂₃	1	1	yes	characteristic
70	M₁₆	—	2	2	no	—
71	M₁₆	C₂×C₄×C₂₂₃	1	2	yes	—
72	C₂×C₈	—	2	2	no	—
73	C₂×C₈	C₂×C₄×C₂₂₃	1	1	yes	characteristic
74	C₂×C₈	C₂³×C₂₂₃	1	1	yes	characteristic
75	C₄²	C₂×C₄×C₂₂₃	1	1	yes	characteristic
76	C₁₆	—	2	2	no	—
77	C₁₆	—	2	1	no	—
78	C₁₆	D₄×C₂₂₃	1	2	yes	—
79	Q₈	C₂×C₈×C₂₂₃	1	2	yes	—
80	D₄	—	2	2	no	—
81	D₄	C₂×C₈×C₂₂₃	1	2	yes	—
82	C₂×C₄	—	2	2	no	—
83	C₂×C₄	—	2	1	no	—
84	C₂×C₄	C₂²×C₄×C₂₂₃	1	1	yes	characteristic
85	C₈	—	2	2	no	—
86	C₈	C₂×C₈×C₂₂₃	1	1	yes	characteristic
87	C₈	C₂×C₈×C₂₂₃	1	1	yes	characteristic
88	C₈	C₂×D₄×C₂₂₃	1	1	yes	characteristic
89	C₈	#41×C₂₂₃	1	1	yes	characteristic
90	C₂²	—	2	2	no	—
91	C₂²	#131×C₂₂₃	1	1	yes	characteristic
92	C₄	—	2	2	no	—
93	C₄	—	4	1	no	—
94	C₄	C₄×D₄×C₂₂₃	1	1	yes	characteristic
95	C₄	C₂²×C₈×C₂₂₃	1	1	yes	characteristic
96	C₂	—	2	1	no	—
97	C₂	—	4	2	no	—
98	C₂	C₈×D₄×C₂₂₃	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#504×C₂₂₃

C₂

yes

maximal, complemented

#443×C₂₂₃

C₂

yes

maximal, characteristic

#364×C₂₂₃

C₂

yes

maximal, characteristic, complemented

#350×C₂₂₃

C₂

yes

maximal, complemented

#346×C₂₂₃

C₂

yes

maximal, characteristic, complemented

#135×C₂₂₃

C₄

yes

characteristic

#131×C₂₂₃

C₂²

yes

—

M₃₂×C₂₂₃

—

M₃₂×C₂₂₃

C₂²

yes

—

C₂×C₁₆×C₂₂₃

—

C₂×C₁₆×C₂₂₃

C₂²

yes

characteristic

C₂×C₁₆×C₂₂₃

C₂²

yes

characteristic

#108×C₂₂₃

C₄

yes

characteristic

#104×C₂₂₃

C₄

yes

—

C₄×C₈×C₂₂₃

C₂²

yes

characteristic

#41×C₂₂₃

C₂×C₄

yes

—

Q₁₆×C₂₂₃

C₈

yes

characteristic

SD₁₆×C₂₂₃

C₈

yes

—

D₈×C₂₂₃

C₈

yes

characteristic

M₁₆×C₂₂₃

—

M₁₆×C₂₂₃

C₂×C₄

yes

—

C₂×C₈×C₂₂₃

—

C₂×C₈×C₂₂₃

C₂×C₄

yes

characteristic

C₂×C₈×C₂₂₃

C₂³

yes

characteristic

C₄²×C₂₂₃

C₂×C₄

yes

characteristic

C₁₆×C₂₂₃

—

C₁₆×C₂₂₃

—

C₁₆×C₂₂₃

D₄

yes

—

Q₈×C₂₂₃

C₂×C₈

yes

—

D₄×C₂₂₃

—

D₄×C₂₂₃

C₂×C₈

yes

—

C₂×C₄×C₂₂₃

—

C₂×C₄×C₂₂₃

—

C₂×C₄×C₂₂₃

C₂²×C₄

yes

characteristic

C₈×C₂₂₃

—

C₈×C₂₂₃

C₂×C₈

yes

characteristic

C₈×C₂₂₃

C₂×C₈

yes

characteristic

C₈×C₂₂₃

C₂×D₄

yes

characteristic

C₈×C₂₂₃

#41

yes

characteristic

C₂²×C₂₂₃

—

C₂²×C₂₂₃

#131

yes

characteristic

C₄×C₂₂₃

—

C₄×C₂₂₃

—

C₄×C₂₂₃

C₄×D₄

yes

characteristic

C₄×C₂₂₃

C₂²×C₈

yes

characteristic

C₂×C₂₂₃

—

C₂×C₂₂₃

—

C₂×C₂₂₃

C₈×D₄

yes

characteristic

C₂₂₃

#2171

yes

characteristic, direct product factor, Sylow

#2171

C₂₂₃

yes

maximal, characteristic, direct product factor, Sylow

#504

C₂×C₂₂₃

yes

complemented

#443

C₂×C₂₂₃

yes

characteristic

#364

C₂×C₂₂₃

yes

characteristic, complemented

#350

C₂×C₂₂₃

yes

complemented

#346

C₂×C₂₂₃

yes

characteristic, complemented

#135

C₄×C₂₂₃

yes

characteristic

#131

C₂²×C₂₂₃

yes

—

M₃₂

—

M₃₂

C₂²×C₂₂₃

yes

—

C₂×C₁₆

—

C₂×C₁₆

C₂²×C₂₂₃

yes

characteristic

C₂×C₁₆

C₂²×C₂₂₃

yes

characteristic

#108

C₄×C₂₂₃

yes

characteristic

#104

C₄×C₂₂₃

yes

—

C₄×C₈

C₂²×C₂₂₃

yes

characteristic

#41

C₂×C₄×C₂₂₃

yes

—

Q₁₆

C₈×C₂₂₃

yes

characteristic

SD₁₆

C₈×C₂₂₃

yes

—

D₈

C₈×C₂₂₃

yes

characteristic

M₁₆

—

M₁₆

C₂×C₄×C₂₂₃

yes

—

C₂×C₈

—

C₂×C₈

C₂×C₄×C₂₂₃

yes

characteristic

C₂×C₈

C₂³×C₂₂₃

yes

characteristic

C₄²

C₂×C₄×C₂₂₃

yes

characteristic

C₁₆

—

C₁₆

—

C₁₆

D₄×C₂₂₃

yes

—

Q₈

C₂×C₈×C₂₂₃

yes

—

D₄

—

D₄

C₂×C₈×C₂₂₃

yes

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²×C₄×C₂₂₃

yes

characteristic

C₈

—

C₈

C₂×C₈×C₂₂₃

yes

characteristic

C₈

C₂×C₈×C₂₂₃

yes

characteristic

C₈

C₂×D₄×C₂₂₃

yes

characteristic

C₈

#41×C₂₂₃

yes

characteristic

C₂²

—

C₂²

#131×C₂₂₃

yes

characteristic

C₄

—

C₄

—

C₄

C₄×D₄×C₂₂₃

yes

characteristic

C₄

C₂²×C₈×C₂₂₃

yes

characteristic

C₂

—

C₂

—

C₂

C₈×D₄×C₂₂₃

yes

characteristic

	subgroup	G/H
center	C₈×C₂₂₃	C₂×D₄
commutator	C₄	C₂²×C₈×C₂₂₃
Frattini, cosocle	C₂×C₈	C₂³×C₂₂₃
socle	C₂×C₂₂₃	C₈×D₄
prime radical	#135×C₂₂₃	C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×C₂₂₃ ◁ C₂×C₁₆×C₂₂₃ ◁ G

subgroup

G/H

center

C₈×C₂₂₃

C₂×D₄

commutator

C₄

C₂²×C₈×C₂₂₃

Frattini, cosocle

C₂×C₈

C₂³×C₂₂₃

socle

C₂×C₂₂₃

C₈×D₄

prime radical

#135×C₂₂₃

C₄

LCS

G ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₈×C₂₂₃ ◁ C₂×C₁₆×C₂₂₃ ◁ G

N	H ≅ G/N	type
1. #504×C₂₂₃	C₂	semidirect
3. #364×C₂₂₃	C₂	semidirect
4. #350×C₂₂₃	C₂	semidirect
5. #346×C₂₂₃	C₂	semidirect
49. C₂₂₃	#2171	direct (N × H)
50. #2171	C₂₂₃	direct (N × H)
51. #504	C₂×C₂₂₃	semidirect
53. #364	C₂×C₂₂₃	semidirect
54. #350	C₂×C₂₂₃	semidirect
55. #346	C₂×C₂₂₃	semidirect

H ≅ G/N

type

1. #504×C₂₂₃

C₂

semidirect

3. #364×C₂₂₃

C₂

semidirect

4. #350×C₂₂₃

C₂

semidirect

5. #346×C₂₂₃

C₂

semidirect

49. C₂₂₃

#2171

direct (N × H)

50. #2171

C₂₂₃

direct (N × H)

51. #504

C₂×C₂₂₃

semidirect

53. #364

C₂×C₂₂₃

semidirect

54. #350

C₂×C₂₂₃

semidirect

55. #346

C₂×C₂₂₃

semidirect

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

subgroup	index
#504×C₂₂₃	2
#443×C₂₂₃	2
#364×C₂₂₃	2
#350×C₂₂₃	2
#346×C₂₂₃	2
#2171	223

subgroup

index

#504×C₂₂₃

#443×C₂₂₃

#364×C₂₂₃

#350×C₂₂₃

#346×C₂₂₃

#2171

223

Subgroups — #2171×C₂₂₃ #43251186

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2171×C223 #43251186

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2171×C₂₂₃ #43251186