Group #43251156 — subgroups

Subgroups — #2141×C₂₂₃ #43251156

Order 28544.

220 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 104 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	C₂²×Q₁₆×C₂₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂×M₃₂×C₂₂₃	C₂	1	1	yes	maximal, characteristic
3	#427×C₂₂₃	C₂	1	1	yes	maximal, characteristic
4	#358×C₂₂₃	C₂	1	4	yes	maximal, complemented
5	C₂²×Q₈×C₂₂₃	—	2	1	no	—
6	C₂×Q₁₆×C₂₂₃	—	2	1	no	—
7	C₂×Q₁₆×C₂₂₃	—	2	2	no	—
8	C₂×Q₁₆×C₂₂₃	C₄	1	4	yes	complemented
9	C₂×Q₁₆×C₂₂₃	C₂²	1	2	yes	—
10	C₂²×C₈×C₂₂₃	C₂²	1	1	yes	characteristic
11	#117×C₂₂₃	—	2	1	no	—
12	M₃₂×C₂₂₃	—	2	2	no	—
13	C₂×C₁₆×C₂₂₃	C₂²	1	2	yes	—
14	#107×C₂₂₃	C₂²	1	2	yes	—
15	#106×C₂₂₃	—	2	1	no	—
16	C₂×Q₈×C₂₂₃	—	2	4	no	—
17	C₂×Q₈×C₂₂₃	—	2	2	no	—
18	C₂×Q₈×C₂₂₃	—	4	1	no	—
19	C₂×Q₈×C₂₂₃	—	4	2	no	—
20	C₂²×C₄×C₂₂₃	—	4	1	no	—
21	C₂²×C₄×C₂₂₃	D₄	1	1	yes	characteristic
22	Q₁₆×C₂₂₃	—	2	4	no	—
23	Q₁₆×C₂₂₃	—	2	2	no	—
24	Q₁₆×C₂₂₃	C₂×C₄	1	4	yes	—
25	C₂×C₈×C₂₂₃	C₂×C₄	1	2	yes	—
26	C₂×C₈×C₂₂₃	D₄	1	1	yes	characteristic
27	C₂×C₈×C₂₂₃	D₄	1	2	yes	—
28	C₂×C₈×C₂₂₃	C₂³	1	1	yes	characteristic
29	C₄⋊C₄×C₂₂₃	—	2	2	no	—
30	C₂²⋊C₄×C₂₂₃	—	4	1	no	—
31	C₄²×C₂₂₃	—	4	1	no	—
32	C₁₆×C₂₂₃	—	2	2	no	—
33	C₂³×C₂₂₃	D₈	1	1	yes	characteristic
34	Q₈×C₂₂₃	—	2	4	no	—
35	Q₈×C₂₂₃	—	4	4	no	—
36	Q₈×C₂₂₃	—	4	2	no	—
37	C₂×C₄×C₂₂₃	—	4	4	no	—
38	C₂×C₄×C₂₂₃	—	4	2	no	—
39	C₂×C₄×C₂₂₃	—	4	2	no	—
40	C₂×C₄×C₂₂₃	C₂²⋊C₄	1	2	yes	—
41	C₂×C₄×C₂₂₃	D₈	1	1	yes	characteristic
42	C₂×C₄×C₂₂₃	SD₁₆	1	2	yes	—
43	C₂×C₄×C₂₂₃	C₂×D₄	1	1	yes	characteristic
44	C₈×C₂₂₃	C₂²⋊C₄	1	2	yes	—
45	C₈×C₂₂₃	C₂²×C₄	1	1	yes	characteristic
46	C₈×C₂₂₃	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₂₂₃	—	2	2	no	—
48	C₂²×C₂₂₃	#102	1	2	yes	—
49	C₂²×C₂₂₃	C₂×D₈	1	1	yes	characteristic
50	C₄×C₂₂₃	—	4	4	no	—
51	C₄×C₂₂₃	—	8	2	no	—
52	C₄×C₂₂₃	#102	1	2	yes	—
53	C₄×C₂₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
54	C₄×C₂₂₃	C₂×SD₁₆	1	1	yes	characteristic
55	C₂×C₂₂₃	—	2	2	no	—
56	C₂×C₂₂₃	C₂×#102	1	1	yes	characteristic
57	C₂×C₂₂₃	#510	1	2	yes	—
58	C₂₂₃	#2141	1	1	yes	characteristic, direct product factor, Sylow
59	#2141	C₂₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
60	C₂²×Q₁₆	C₂×C₂₂₃	1	1	yes	characteristic
61	C₂×M₃₂	C₂×C₂₂₃	1	1	yes	characteristic
62	#427	C₂×C₂₂₃	1	1	yes	characteristic
63	#358	C₂×C₂₂₃	1	4	yes	complemented
64	C₂²×Q₈	—	2	1	no	—
65	C₂×Q₁₆	—	2	1	no	—
66	C₂×Q₁₆	—	2	2	no	—
67	C₂×Q₁₆	C₄×C₂₂₃	1	4	yes	complemented
68	C₂×Q₁₆	C₂²×C₂₂₃	1	2	yes	—
69	C₂²×C₈	C₂²×C₂₂₃	1	1	yes	characteristic
70	#117	—	2	1	no	—
71	M₃₂	—	2	2	no	—
72	C₂×C₁₆	C₂²×C₂₂₃	1	2	yes	—
73	#107	C₂²×C₂₂₃	1	2	yes	—
74	#106	—	2	1	no	—
75	C₂×Q₈	—	2	4	no	—
76	C₂×Q₈	—	2	2	no	—
77	C₂×Q₈	—	4	1	no	—
78	C₂×Q₈	—	4	2	no	—
79	C₂²×C₄	—	4	1	no	—
80	C₂²×C₄	D₄×C₂₂₃	1	1	yes	characteristic
81	Q₁₆	—	2	4	no	—
82	Q₁₆	—	2	2	no	—
83	Q₁₆	C₂×C₄×C₂₂₃	1	4	yes	—
84	C₂×C₈	C₂×C₄×C₂₂₃	1	2	yes	—
85	C₂×C₈	D₄×C₂₂₃	1	1	yes	characteristic
86	C₂×C₈	D₄×C₂₂₃	1	2	yes	—
87	C₂×C₈	C₂³×C₂₂₃	1	1	yes	characteristic
88	C₄⋊C₄	—	2	2	no	—
89	C₂²⋊C₄	—	4	1	no	—
90	C₄²	—	4	1	no	—
91	C₁₆	—	2	2	no	—
92	C₂³	D₈×C₂₂₃	1	1	yes	characteristic
93	Q₈	—	2	4	no	—
94	Q₈	—	4	4	no	—
95	Q₈	—	4	2	no	—
96	C₂×C₄	—	4	4	no	—
97	C₂×C₄	—	4	2	no	—
98	C₂×C₄	—	4	2	no	—
99	C₂×C₄	C₂²⋊C₄×C₂₂₃	1	2	yes	—
100	C₂×C₄	D₈×C₂₂₃	1	1	yes	characteristic
101	C₂×C₄	SD₁₆×C₂₂₃	1	2	yes	—
102	C₂×C₄	C₂×D₄×C₂₂₃	1	1	yes	characteristic
103	C₈	C₂²⋊C₄×C₂₂₃	1	2	yes	—
104	C₈	C₂²×C₄×C₂₂₃	1	1	yes	characteristic
105	C₈	C₂×D₄×C₂₂₃	1	1	yes	characteristic
106	C₂²	—	2	2	no	—
107	C₂²	#102×C₂₂₃	1	2	yes	—
108	C₂²	C₂×D₈×C₂₂₃	1	1	yes	characteristic
109	C₄	—	4	4	no	—
110	C₄	—	8	2	no	—
111	C₄	#102×C₂₂₃	1	2	yes	—
112	C₄	C₂×C₂²⋊C₄×C₂₂₃	1	1	yes	characteristic
113	C₄	C₂×SD₁₆×C₂₂₃	1	1	yes	characteristic
114	C₂	—	2	2	no	—
115	C₂	C₂×#102×C₂₂₃	1	1	yes	characteristic
116	C₂	#510×C₂₂₃	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Q₁₆×C₂₂₃

C₂

yes

maximal, characteristic

C₂×M₃₂×C₂₂₃

C₂

yes

maximal, characteristic

#427×C₂₂₃

C₂

yes

maximal, characteristic

#358×C₂₂₃

C₂

yes

maximal, complemented

C₂²×Q₈×C₂₂₃

—

C₂×Q₁₆×C₂₂₃

—

C₂×Q₁₆×C₂₂₃

—

C₂×Q₁₆×C₂₂₃

C₄

yes

complemented

C₂×Q₁₆×C₂₂₃

C₂²

yes

—

C₂²×C₈×C₂₂₃

C₂²

yes

characteristic

#117×C₂₂₃

—

M₃₂×C₂₂₃

—

C₂×C₁₆×C₂₂₃

C₂²

yes

—

#107×C₂₂₃

C₂²

yes

—

#106×C₂₂₃

—

C₂×Q₈×C₂₂₃

—

C₂×Q₈×C₂₂₃

—

C₂×Q₈×C₂₂₃

—

C₂×Q₈×C₂₂₃

—

C₂²×C₄×C₂₂₃

—

C₂²×C₄×C₂₂₃

D₄

yes

characteristic

Q₁₆×C₂₂₃

—

Q₁₆×C₂₂₃

—

Q₁₆×C₂₂₃

C₂×C₄

yes

—

C₂×C₈×C₂₂₃

C₂×C₄

yes

—

C₂×C₈×C₂₂₃

D₄

yes

characteristic

C₂×C₈×C₂₂₃

D₄

yes

—

C₂×C₈×C₂₂₃

C₂³

yes

characteristic

C₄⋊C₄×C₂₂₃

—

C₂²⋊C₄×C₂₂₃

—

C₄²×C₂₂₃

—

C₁₆×C₂₂₃

—

C₂³×C₂₂₃

D₈

yes

characteristic

Q₈×C₂₂₃

—

Q₈×C₂₂₃

—

Q₈×C₂₂₃

—

C₂×C₄×C₂₂₃

—

C₂×C₄×C₂₂₃

—

C₂×C₄×C₂₂₃

—

C₂×C₄×C₂₂₃

C₂²⋊C₄

yes

—

C₂×C₄×C₂₂₃

D₈

yes

characteristic

C₂×C₄×C₂₂₃

SD₁₆

yes

—

C₂×C₄×C₂₂₃

C₂×D₄

yes

characteristic

C₈×C₂₂₃

C₂²⋊C₄

yes

—

C₈×C₂₂₃

C₂²×C₄

yes

characteristic

C₈×C₂₂₃

C₂×D₄

yes

characteristic

C₂²×C₂₂₃

—

C₂²×C₂₂₃

#102

yes

—

C₂²×C₂₂₃

C₂×D₈

yes

characteristic

C₄×C₂₂₃

—

C₄×C₂₂₃

—

C₄×C₂₂₃

#102

yes

—

C₄×C₂₂₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₂₂₃

C₂×SD₁₆

yes

characteristic

C₂×C₂₂₃

—

C₂×C₂₂₃

C₂×#102

yes

characteristic

C₂×C₂₂₃

#510

yes

—

C₂₂₃

#2141

yes

characteristic, direct product factor, Sylow

#2141

C₂₂₃

yes

maximal, characteristic, direct product factor, Sylow

C₂²×Q₁₆

C₂×C₂₂₃

yes

characteristic

C₂×M₃₂

C₂×C₂₂₃

yes

characteristic

#427

C₂×C₂₂₃

yes

characteristic

#358

C₂×C₂₂₃

yes

complemented

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

C₄×C₂₂₃

yes

complemented

C₂×Q₁₆

C₂²×C₂₂₃

yes

—

C₂²×C₈

C₂²×C₂₂₃

yes

characteristic

#117

—

M₃₂

—

C₂×C₁₆

C₂²×C₂₂₃

yes

—

#107

C₂²×C₂₂₃

yes

—

#106

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

D₄×C₂₂₃

yes

characteristic

Q₁₆

—

Q₁₆

—

Q₁₆

C₂×C₄×C₂₂₃

yes

—

C₂×C₈

C₂×C₄×C₂₂₃

yes

—

C₂×C₈

D₄×C₂₂₃

yes

characteristic

C₂×C₈

D₄×C₂₂₃

yes

—

C₂×C₈

C₂³×C₂₂₃

yes

characteristic

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₁₆

—

C₂³

D₈×C₂₂₃

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²⋊C₄×C₂₂₃

yes

—

100

C₂×C₄

D₈×C₂₂₃

yes

characteristic

101

C₂×C₄

SD₁₆×C₂₂₃

yes

—

102

C₂×C₄

C₂×D₄×C₂₂₃

yes

characteristic

103

C₈

C₂²⋊C₄×C₂₂₃

yes

—

104

C₈

C₂²×C₄×C₂₂₃

yes

characteristic

105

C₈

C₂×D₄×C₂₂₃

yes

characteristic

106

C₂²

—

107

C₂²

#102×C₂₂₃

yes

—

108

C₂²

C₂×D₈×C₂₂₃

yes

characteristic

109

C₄

—

110

C₄

—

111

C₄

#102×C₂₂₃

yes

—

112

C₄

C₂×C₂²⋊C₄×C₂₂₃

yes

characteristic

113

C₄

C₂×SD₁₆×C₂₂₃

yes

characteristic

114

C₂

—

115

C₂

C₂×#102×C₂₂₃

yes

characteristic

116

C₂

#510×C₂₂₃

yes

—

	subgroup	G/H
center, socle	C₂²×C₂₂₃	C₂×D₈
commutator	C₈	C₂²×C₄×C₂₂₃
Frattini, cosocle	C₂×C₈	C₂³×C₂₂₃
prime radical	C₂³×C₂₂₃	D₈
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₂₃ ◁ C₂²×C₄×C₂₂₃ ◁ C₂²×C₈×C₂₂₃ ◁ G

subgroup

G/H

center, socle

C₂²×C₂₂₃

C₂×D₈

commutator

C₈

C₂²×C₄×C₂₂₃

Frattini, cosocle

C₂×C₈

C₂³×C₂₂₃

prime radical

C₂³×C₂₂₃

D₈

LCS

G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₂₂₃ ◁ C₂²×C₄×C₂₂₃ ◁ C₂²×C₈×C₂₂₃ ◁ G

N	H ≅ G/N	type
4. #358×C₂₂₃	C₂	semidirect
8. C₂×Q₁₆×C₂₂₃	C₄	semidirect
58. C₂₂₃	#2141	direct (N × H)
59. #2141	C₂₂₃	direct (N × H)
63. #358	C₂×C₂₂₃	semidirect
67. C₂×Q₁₆	C₄×C₂₂₃	semidirect

H ≅ G/N

type

4. #358×C₂₂₃

C₂

semidirect

8. C₂×Q₁₆×C₂₂₃

C₄

semidirect

58. C₂₂₃

#2141

direct (N × H)

59. #2141

C₂₂₃

direct (N × H)

63. #358

C₂×C₂₂₃

semidirect

67. C₂×Q₁₆

C₄×C₂₂₃

semidirect

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

subgroup	index
C₂²×Q₁₆×C₂₂₃	2
C₂×M₃₂×C₂₂₃	2
#427×C₂₂₃	2
#358×C₂₂₃	2
#2141	223

subgroup

index

C₂²×Q₁₆×C₂₂₃

C₂×M₃₂×C₂₂₃

#427×C₂₂₃

#358×C₂₂₃

#2141

223

Subgroups — #2141×C₂₂₃ #43251156

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2141×C223 #43251156

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2141×C₂₂₃ #43251156