Group #43250635 — subgroups

Subgroups — #1620×C₂₂₃ #43250635

Order 28544.

240 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 68 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	#559×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
2	#493×C₂₂₃	C₂	1	1	yes	maximal, characteristic
3	#460×C₂₂₃	C₂	1	2	yes	maximal, complemented
4	#331×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
5	Q₈⋊C₈×C₂₂₃	C₂	1	2	yes	maximal, complemented
6	C₂×SD₁₆×C₂₂₃	—	2	2	no	—
7	C₂×D₈×C₂₂₃	—	2	2	no	—
8	#127×C₂₂₃	C₂²	1	1	yes	characteristic
9	#127×C₂₂₃	C₂²	1	1	yes	characteristic
10	#124×C₂₂₃	—	2	2	no	—
11	#123×C₂₂₃	—	2	2	no	—
12	C₄×Q₈×C₂₂₃	C₂²	1	2	yes	—
13	C₄×D₄×C₂₂₃	—	2	2	no	—
14	C₄⋊C₈×C₂₂₃	C₂²	1	2	yes	—
15	#102×C₂₂₃	—	2	2	no	—
16	C₄×C₈×C₂₂₃	C₂²	1	1	yes	characteristic
17	C₂×Q₈×C₂₂₃	D₄	1	2	yes	—
18	C₂×D₄×C₂₂₃	—	2	2	no	—
19	C₂×D₄×C₂₂₃	—	2	2	no	—
20	C₂×D₄×C₂₂₃	—	2	1	no	—
21	C₂×D₄×C₂₂₃	D₄	1	2	yes	—
22	C₂²×C₄×C₂₂₃	—	2	2	no	—
23	SD₁₆×C₂₂₃	—	4	4	no	—
24	D₈×C₂₂₃	—	4	4	no	—
25	C₂×C₈×C₂₂₃	—	2	1	no	—
26	C₂×C₈×C₂₂₃	—	2	2	no	—
27	C₄⋊C₄×C₂₂₃	—	2	2	no	—
28	C₄⋊C₄×C₂₂₃	D₄	1	2	yes	—
29	C₂²⋊C₄×C₂₂₃	—	2	4	no	—
30	C₂²⋊C₄×C₂₂₃	—	2	2	no	—
31	C₄²×C₂₂₃	—	2	2	no	—
32	C₄²×C₂₂₃	C₂³	1	1	yes	characteristic
33	C₂³×C₂₂₃	—	2	2	no	—
34	C₂³×C₂₂₃	—	4	1	no	—
35	Q₈×C₂₂₃	—	2	2	no	—
36	Q₈×C₂₂₃	—	2	2	no	—
37	D₄×C₂₂₃	—	4	2	no	—
38	D₄×C₂₂₃	—	4	2	no	—
39	D₄×C₂₂₃	—	4	2	no	—
40	D₄×C₂₂₃	—	4	2	no	—
41	D₄×C₂₂₃	—	4	2	no	—
42	D₄×C₂₂₃	—	4	2	no	—
43	C₂×C₄×C₂₂₃	—	2	2	no	—
44	C₂×C₄×C₂₂₃	—	2	2	no	—
45	C₂×C₄×C₂₂₃	—	8	2	no	—
46	C₂×C₄×C₂₂₃	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₂₂₃	C₂×D₄	1	2	yes	—
48	C₈×C₂₂₃	—	2	2	no	—
49	C₈×C₂₂₃	—	4	2	no	—
50	C₂²×C₂₂₃	—	4	4	no	—
51	C₂²×C₂₂₃	—	4	2	no	—
52	C₂²×C₂₂₃	—	8	2	no	—
53	C₂²×C₂₂₃	—	8	1	no	—
54	C₂²×C₂₂₃	#120	1	1	yes	characteristic
55	C₄×C₂₂₃	—	2	1	no	—
56	C₄×C₂₂₃	—	4	2	no	—
57	C₄×C₂₂₃	—	4	2	no	—
58	C₄×C₂₂₃	#135	1	2	yes	—
59	C₄×C₂₂₃	AGL(1,ℤ/8)	1	2	yes	—
60	C₂×C₂₂₃	—	8	2	no	—
61	C₂×C₂₂₃	—	16	1	no	—
62	C₂×C₂₂₃	#449	1	2	yes	—
63	C₂×C₂₂₃	#453	1	1	yes	characteristic
64	C₂₂₃	#1620	1	1	yes	characteristic, direct product factor, Sylow
65	#1620	C₂₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
66	#559	C₂×C₂₂₃	1	1	yes	characteristic, complemented
67	#493	C₂×C₂₂₃	1	1	yes	characteristic
68	#460	C₂×C₂₂₃	1	2	yes	complemented
69	#331	C₂×C₂₂₃	1	1	yes	characteristic, complemented
70	Q₈⋊C₈	C₂×C₂₂₃	1	2	yes	complemented
71	C₂×SD₁₆	—	2	2	no	—
72	C₂×D₈	—	2	2	no	—
73	#127	C₂²×C₂₂₃	1	1	yes	characteristic
74	#127	C₂²×C₂₂₃	1	1	yes	characteristic
75	#124	—	2	2	no	—
76	#123	—	2	2	no	—
77	C₄×Q₈	C₂²×C₂₂₃	1	2	yes	—
78	C₄×D₄	—	2	2	no	—
79	C₄⋊C₈	C₂²×C₂₂₃	1	2	yes	—
80	#102	—	2	2	no	—
81	C₄×C₈	C₂²×C₂₂₃	1	1	yes	characteristic
82	C₂×Q₈	D₄×C₂₂₃	1	2	yes	—
83	C₂×D₄	—	2	2	no	—
84	C₂×D₄	—	2	2	no	—
85	C₂×D₄	—	2	1	no	—
86	C₂×D₄	D₄×C₂₂₃	1	2	yes	—
87	C₂²×C₄	—	2	2	no	—
88	SD₁₆	—	4	4	no	—
89	D₈	—	4	4	no	—
90	C₂×C₈	—	2	1	no	—
91	C₂×C₈	—	2	2	no	—
92	C₄⋊C₄	—	2	2	no	—
93	C₄⋊C₄	D₄×C₂₂₃	1	2	yes	—
94	C₂²⋊C₄	—	2	4	no	—
95	C₂²⋊C₄	—	2	2	no	—
96	C₄²	—	2	2	no	—
97	C₄²	C₂³×C₂₂₃	1	1	yes	characteristic
98	C₂³	—	2	2	no	—
99	C₂³	—	4	1	no	—
100	Q₈	—	2	2	no	—
101	Q₈	—	2	2	no	—
102	D₄	—	4	2	no	—
103	D₄	—	4	2	no	—
104	D₄	—	4	2	no	—
105	D₄	—	4	2	no	—
106	D₄	—	4	2	no	—
107	D₄	—	4	2	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	2	2	no	—
110	C₂×C₄	—	8	2	no	—
111	C₂×C₄	C₂×D₄×C₂₂₃	1	1	yes	characteristic
112	C₂×C₄	C₂×D₄×C₂₂₃	1	2	yes	—
113	C₈	—	2	2	no	—
114	C₈	—	4	2	no	—
115	C₂²	—	4	4	no	—
116	C₂²	—	4	2	no	—
117	C₂²	—	8	2	no	—
118	C₂²	—	8	1	no	—
119	C₂²	#120×C₂₂₃	1	1	yes	characteristic
120	C₄	—	2	1	no	—
121	C₄	—	4	2	no	—
122	C₄	—	4	2	no	—
123	C₄	#135×C₂₂₃	1	2	yes	—
124	C₄	AGL(1,ℤ/8)×C₂₂₃	1	2	yes	—
125	C₂	—	8	2	no	—
126	C₂	—	16	1	no	—
127	C₂	#449×C₂₂₃	1	2	yes	—
128	C₂	#453×C₂₂₃	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#559×C₂₂₃

C₂

yes

maximal, characteristic, complemented

#493×C₂₂₃

C₂

yes

maximal, characteristic

#460×C₂₂₃

C₂

yes

maximal, complemented

#331×C₂₂₃

C₂

yes

maximal, characteristic, complemented

Q₈⋊C₈×C₂₂₃

C₂

yes

maximal, complemented

C₂×SD₁₆×C₂₂₃

—

C₂×D₈×C₂₂₃

—

#127×C₂₂₃

C₂²

yes

characteristic

#127×C₂₂₃

C₂²

yes

characteristic

#124×C₂₂₃

—

#123×C₂₂₃

—

C₄×Q₈×C₂₂₃

C₂²

yes

—

C₄×D₄×C₂₂₃

—

C₄⋊C₈×C₂₂₃

C₂²

yes

—

#102×C₂₂₃

—

C₄×C₈×C₂₂₃

C₂²

yes

characteristic

C₂×Q₈×C₂₂₃

D₄

yes

—

C₂×D₄×C₂₂₃

—

C₂×D₄×C₂₂₃

—

C₂×D₄×C₂₂₃

—

C₂×D₄×C₂₂₃

D₄

yes

—

C₂²×C₄×C₂₂₃

—

SD₁₆×C₂₂₃

—

D₈×C₂₂₃

—

C₂×C₈×C₂₂₃

—

C₂×C₈×C₂₂₃

—

C₄⋊C₄×C₂₂₃

—

C₄⋊C₄×C₂₂₃

D₄

yes

—

C₂²⋊C₄×C₂₂₃

—

C₂²⋊C₄×C₂₂₃

—

C₄²×C₂₂₃

—

C₄²×C₂₂₃

C₂³

yes

characteristic

C₂³×C₂₂₃

—

C₂³×C₂₂₃

—

Q₈×C₂₂₃

—

Q₈×C₂₂₃

—

D₄×C₂₂₃

—

D₄×C₂₂₃

—

D₄×C₂₂₃

—

D₄×C₂₂₃

—

D₄×C₂₂₃

—

D₄×C₂₂₃

—

C₂×C₄×C₂₂₃

—

C₂×C₄×C₂₂₃

—

C₂×C₄×C₂₂₃

—

C₂×C₄×C₂₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₂₃

C₂×D₄

yes

—

C₈×C₂₂₃

—

C₈×C₂₂₃

—

C₂²×C₂₂₃

—

C₂²×C₂₂₃

—

C₂²×C₂₂₃

—

C₂²×C₂₂₃

—

C₂²×C₂₂₃

#120

yes

characteristic

C₄×C₂₂₃

—

C₄×C₂₂₃

—

C₄×C₂₂₃

—

C₄×C₂₂₃

#135

yes

—

C₄×C₂₂₃

AGL(1,ℤ/8)

yes

—

C₂×C₂₂₃

—

C₂×C₂₂₃

—

C₂×C₂₂₃

#449

yes

—

C₂×C₂₂₃

#453

yes

characteristic

C₂₂₃

#1620

yes

characteristic, direct product factor, Sylow

#1620

C₂₂₃

yes

maximal, characteristic, direct product factor, Sylow

#559

C₂×C₂₂₃

yes

characteristic, complemented

#493

C₂×C₂₂₃

yes

characteristic

#460

C₂×C₂₂₃

yes

complemented

#331

C₂×C₂₂₃

yes

characteristic, complemented

Q₈⋊C₈

C₂×C₂₂₃

yes

complemented

C₂×SD₁₆

—

C₂×D₈

—

#127

C₂²×C₂₂₃

yes

characteristic

#127

C₂²×C₂₂₃

yes

characteristic

#124

—

#123

—

C₄×Q₈

C₂²×C₂₂₃

yes

—

C₄×D₄

—

C₄⋊C₈

C₂²×C₂₂₃

yes

—

#102

—

C₄×C₈

C₂²×C₂₂₃

yes

characteristic

C₂×Q₈

D₄×C₂₂₃

yes

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

D₄×C₂₂₃

yes

—

C₂²×C₄

—

SD₁₆

—

D₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

D₄×C₂₂₃

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₂³×C₂₂₃

yes

characteristic

C₂³

—

C₂³

—

100

Q₈

—

101

Q₈

—

102

D₄

—

103

D₄

—

104

D₄

—

105

D₄

—

106

D₄

—

107

D₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

C₂×D₄×C₂₂₃

yes

characteristic

112

C₂×C₄

C₂×D₄×C₂₂₃

yes

—

113

C₈

—

114

C₈

—

115

C₂²

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

119

C₂²

#120×C₂₂₃

yes

characteristic

120

C₄

—

121

C₄

—

122

C₄

—

123

C₄

#135×C₂₂₃

yes

—

124

C₄

AGL(1,ℤ/8)×C₂₂₃

yes

—

125

C₂

—

126

C₂

—

127

C₂

#449×C₂₂₃

yes

—

128

C₂

#453×C₂₂₃

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₂₂₃	#120
commutator, Frattini, cosocle	C₄²	C₂³×C₂₂₃
prime radical	#493×C₂₂₃	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₂₃ ◁ C₄²×C₂₂₃ ◁ G

subgroup

G/H

center, socle

C₂²×C₂₂₃

#120

commutator, Frattini, cosocle

C₄²

C₂³×C₂₂₃

prime radical

#493×C₂₂₃

C₂

LCS

G ▷ C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₂₂₃ ◁ C₄²×C₂₂₃ ◁ G

N	H ≅ G/N	type
1. #559×C₂₂₃	C₂	semidirect
3. #460×C₂₂₃	C₂	semidirect
4. #331×C₂₂₃	C₂	semidirect
5. Q₈⋊C₈×C₂₂₃	C₂	semidirect
64. C₂₂₃	#1620	direct (N × H)
65. #1620	C₂₂₃	direct (N × H)
66. #559	C₂×C₂₂₃	semidirect
68. #460	C₂×C₂₂₃	semidirect
69. #331	C₂×C₂₂₃	semidirect
70. Q₈⋊C₈	C₂×C₂₂₃	semidirect

H ≅ G/N

type

1. #559×C₂₂₃

C₂

semidirect

3. #460×C₂₂₃

C₂

semidirect

4. #331×C₂₂₃

C₂

semidirect

5. Q₈⋊C₈×C₂₂₃

C₂

semidirect

64. C₂₂₃

#1620

direct (N × H)

65. #1620

C₂₂₃

direct (N × H)

66. #559

C₂×C₂₂₃

semidirect

68. #460

C₂×C₂₂₃

semidirect

69. #331

C₂×C₂₂₃

semidirect

70. Q₈⋊C₈

C₂×C₂₂₃

semidirect

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

subgroup	index
#559×C₂₂₃	2
#493×C₂₂₃	2
#460×C₂₂₃	2
#331×C₂₂₃	2
Q₈⋊C₈×C₂₂₃	2
#1620	223

subgroup

index

#559×C₂₂₃

#493×C₂₂₃

#460×C₂₂₃

#331×C₂₂₃

Q₈⋊C₈×C₂₂₃

#1620

223

Subgroups — #1620×C₂₂₃ #43250635

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1620×C223 #43250635

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1620×C₂₂₃ #43250635