Group #43251060 — subgroups

Subgroups — #2045×C₂₂₃ #43251060

Order 28544.

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

normal subgroups only

This group has 296 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	C₂²×D₈×C₂₂₃	C₂	1	1	yes	maximal, characteristic
2	#424×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
3	#418×C₂₂₃	C₂	1	2	yes	maximal, complemented
4	C₂×#100×C₂₂₃	C₂	1	2	yes	maximal, complemented
5	#341×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₄×C₂₂₃	C₂²	1	2	yes	—
7	C₂×D₈×C₂₂₃	—	2	4	no	—
8	C₂×D₈×C₂₂₃	—	2	2	no	—
9	C₂×M₁₆×C₂₂₃	C₂²	1	2	yes	—
10	C₂×M₁₆×C₂₂₃	C₂²	1	1	yes	characteristic
11	C₂²×C₈×C₂₂₃	C₂²	1	1	yes	characteristic
12	#117×C₂₂₃	C₂²	1	1	yes	characteristic
13	C₄⋊C₈×C₂₂₃	—	2	2	no	—
14	#102×C₂₂₃	—	2	4	no	—
15	#100×C₂₂₃	—	2	4	no	—
16	C₂⁴×C₂₂₃	—	2	2	no	—
17	C₂×D₄×C₂₂₃	—	2	2	no	—
18	C₂×D₄×C₂₂₃	—	2	2	no	—
19	C₂×D₄×C₂₂₃	—	2	2	no	—
20	C₂×D₄×C₂₂₃	—	2	2	no	—
21	C₂×D₄×C₂₂₃	—	2	2	no	—
22	C₂×D₄×C₂₂₃	D₄	1	4	yes	—
23	C₂²×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
24	D₈×C₂₂₃	—	2	8	no	—
25	M₁₆×C₂₂₃	—	2	2	no	—
26	M₁₆×C₂₂₃	—	2	2	no	—
27	M₁₆×C₂₂₃	—	2	2	no	—
28	C₂×C₈×C₂₂₃	—	2	2	no	—
29	C₂×C₈×C₂₂₃	—	2	2	no	—
30	C₂×C₈×C₂₂₃	—	2	1	no	—
31	C₂×C₈×C₂₂₃	D₄	1	2	yes	—
32	C₄⋊C₄×C₂₂₃	D₄	1	2	yes	—
33	C₂²⋊C₄×C₂₂₃	—	2	1	no	—
34	C₄²×C₂₂₃	—	2	1	no	—
35	C₂³×C₂₂₃	—	2	4	no	—
36	C₂³×C₂₂₃	—	4	2	no	—
37	C₂³×C₂₂₃	—	4	2	no	—
38	C₂³×C₂₂₃	—	8	4	no	—
39	C₂³×C₂₂₃	#41	1	1	yes	characteristic
40	D₄×C₂₂₃	—	2	4	no	—
41	D₄×C₂₂₃	—	2	4	no	—
42	D₄×C₂₂₃	—	4	2	no	—
43	D₄×C₂₂₃	—	4	2	no	—
44	C₂×C₄×C₂₂₃	—	2	2	no	—
45	C₂×C₄×C₂₂₃	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₂₂₃	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₂₂₃	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₂₂₃	#41	1	2	yes	—
49	C₈×C₂₂₃	—	2	2	no	—
50	C₈×C₂₂₃	—	4	1	no	—
51	C₈×C₂₂₃	—	4	2	no	—
52	C₂²×C₂₂₃	—	2	2	no	—
53	C₂²×C₂₂₃	—	4	4	no	—
54	C₂²×C₂₂₃	—	8	4	no	—
55	C₂²×C₂₂₃	—	8	4	no	—
56	C₂²×C₂₂₃	—	8	4	no	—
57	C₂²×C₂₂₃	#121	1	1	yes	characteristic
58	C₂²×C₂₂₃	#123	1	2	yes	—
59	C₄×C₂₂₃	—	4	2	no	—
60	C₄×C₂₂₃	#120	1	1	yes	characteristic
61	C₄×C₂₂₃	#121	1	2	yes	—
62	C₄×C₂₂₃	#124	1	1	yes	characteristic
63	C₂×C₂₂₃	—	2	2	no	—
64	C₂×C₂₂₃	—	8	4	no	—
65	C₂×C₂₂₃	#394	1	1	yes	characteristic
66	C₂×C₂₂₃	#472	1	2	yes	—
67	C₂₂₃	#2045	1	1	yes	characteristic, direct product factor, Sylow
68	#2045	C₂₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
69	C₂²×D₈	C₂×C₂₂₃	1	1	yes	characteristic
70	#424	C₂×C₂₂₃	1	1	yes	characteristic, complemented
71	#418	C₂×C₂₂₃	1	2	yes	complemented
72	C₂×#100	C₂×C₂₂₃	1	2	yes	complemented
73	#341	C₂×C₂₂₃	1	1	yes	characteristic, complemented
74	C₂²×D₄	C₂²×C₂₂₃	1	2	yes	—
75	C₂×D₈	—	2	4	no	—
76	C₂×D₈	—	2	2	no	—
77	C₂×M₁₆	C₂²×C₂₂₃	1	2	yes	—
78	C₂×M₁₆	C₂²×C₂₂₃	1	1	yes	characteristic
79	C₂²×C₈	C₂²×C₂₂₃	1	1	yes	characteristic
80	#117	C₂²×C₂₂₃	1	1	yes	characteristic
81	C₄⋊C₈	—	2	2	no	—
82	#102	—	2	4	no	—
83	#100	—	2	4	no	—
84	C₂⁴	—	2	2	no	—
85	C₂×D₄	—	2	2	no	—
86	C₂×D₄	—	2	2	no	—
87	C₂×D₄	—	2	2	no	—
88	C₂×D₄	—	2	2	no	—
89	C₂×D₄	—	2	2	no	—
90	C₂×D₄	D₄×C₂₂₃	1	4	yes	—
91	C₂²×C₄	C₂³×C₂₂₃	1	1	yes	characteristic
92	D₈	—	2	8	no	—
93	M₁₆	—	2	2	no	—
94	M₁₆	—	2	2	no	—
95	M₁₆	—	2	2	no	—
96	C₂×C₈	—	2	2	no	—
97	C₂×C₈	—	2	2	no	—
98	C₂×C₈	—	2	1	no	—
99	C₂×C₈	D₄×C₂₂₃	1	2	yes	—
100	C₄⋊C₄	D₄×C₂₂₃	1	2	yes	—
101	C₂²⋊C₄	—	2	1	no	—
102	C₄²	—	2	1	no	—
103	C₂³	—	2	4	no	—
104	C₂³	—	4	2	no	—
105	C₂³	—	4	2	no	—
106	C₂³	—	8	4	no	—
107	C₂³	#41×C₂₂₃	1	1	yes	characteristic
108	D₄	—	2	4	no	—
109	D₄	—	2	4	no	—
110	D₄	—	4	2	no	—
111	D₄	—	4	2	no	—
112	C₂×C₄	—	2	2	no	—
113	C₂×C₄	C₂×D₄×C₂₂₃	1	2	yes	—
114	C₂×C₄	C₂×D₄×C₂₂₃	1	1	yes	characteristic
115	C₂×C₄	C₂×D₄×C₂₂₃	1	1	yes	characteristic
116	C₂×C₄	#41×C₂₂₃	1	2	yes	—
117	C₈	—	2	2	no	—
118	C₈	—	4	1	no	—
119	C₈	—	4	2	no	—
120	C₂²	—	2	2	no	—
121	C₂²	—	4	4	no	—
122	C₂²	—	8	4	no	—
123	C₂²	—	8	4	no	—
124	C₂²	—	8	4	no	—
125	C₂²	#121×C₂₂₃	1	1	yes	characteristic
126	C₂²	#123×C₂₂₃	1	2	yes	—
127	C₄	—	4	2	no	—
128	C₄	#120×C₂₂₃	1	1	yes	characteristic
129	C₄	#121×C₂₂₃	1	2	yes	—
130	C₄	#124×C₂₂₃	1	1	yes	characteristic
131	C₂	—	2	2	no	—
132	C₂	—	8	4	no	—
133	C₂	#394×C₂₂₃	1	1	yes	characteristic
134	C₂	#472×C₂₂₃	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×D₈×C₂₂₃

C₂

yes

maximal, characteristic

#424×C₂₂₃

C₂

yes

maximal, characteristic, complemented

#418×C₂₂₃

C₂

yes

maximal, complemented

C₂×#100×C₂₂₃

C₂

yes

maximal, complemented

#341×C₂₂₃

C₂

yes

maximal, characteristic, complemented

C₂²×D₄×C₂₂₃

C₂²

yes

—

C₂×D₈×C₂₂₃

—

C₂×D₈×C₂₂₃

—

C₂×M₁₆×C₂₂₃

C₂²

yes

—

C₂×M₁₆×C₂₂₃

C₂²

yes

characteristic

C₂²×C₈×C₂₂₃

C₂²

yes

characteristic

#117×C₂₂₃

C₂²

yes

characteristic

C₄⋊C₈×C₂₂₃

—

#102×C₂₂₃

—

#100×C₂₂₃

—

C₂⁴×C₂₂₃

—

C₂×D₄×C₂₂₃

—

C₂×D₄×C₂₂₃

—

C₂×D₄×C₂₂₃

—

C₂×D₄×C₂₂₃

—

C₂×D₄×C₂₂₃

—

C₂×D₄×C₂₂₃

D₄

yes

—

C₂²×C₄×C₂₂₃

C₂³

yes

characteristic

D₈×C₂₂₃

—

M₁₆×C₂₂₃

—

M₁₆×C₂₂₃

—

M₁₆×C₂₂₃

—

C₂×C₈×C₂₂₃

—

C₂×C₈×C₂₂₃

—

C₂×C₈×C₂₂₃

—

C₂×C₈×C₂₂₃

D₄

yes

—

C₄⋊C₄×C₂₂₃

D₄

yes

—

C₂²⋊C₄×C₂₂₃

—

C₄²×C₂₂₃

—

C₂³×C₂₂₃

—

C₂³×C₂₂₃

—

C₂³×C₂₂₃

—

C₂³×C₂₂₃

—

C₂³×C₂₂₃

#41

yes

characteristic

D₄×C₂₂₃

—

D₄×C₂₂₃

—

D₄×C₂₂₃

—

D₄×C₂₂₃

—

C₂×C₄×C₂₂₃

—

C₂×C₄×C₂₂₃

C₂×D₄

yes

—

C₂×C₄×C₂₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₂₃

#41

yes

—

C₈×C₂₂₃

—

C₈×C₂₂₃

—

C₈×C₂₂₃

—

C₂²×C₂₂₃

—

C₂²×C₂₂₃

—

C₂²×C₂₂₃

—

C₂²×C₂₂₃

—

C₂²×C₂₂₃

—

C₂²×C₂₂₃

#121

yes

characteristic

C₂²×C₂₂₃

#123

yes

—

C₄×C₂₂₃

—

C₄×C₂₂₃

#120

yes

characteristic

C₄×C₂₂₃

#121

yes

—

C₄×C₂₂₃

#124

yes

characteristic

C₂×C₂₂₃

—

C₂×C₂₂₃

—

C₂×C₂₂₃

#394

yes

characteristic

C₂×C₂₂₃

#472

yes

—

C₂₂₃

#2045

yes

characteristic, direct product factor, Sylow

#2045

C₂₂₃

yes

maximal, characteristic, direct product factor, Sylow

C₂²×D₈

C₂×C₂₂₃

yes

characteristic

#424

C₂×C₂₂₃

yes

characteristic, complemented

#418

C₂×C₂₂₃

yes

complemented

C₂×#100

C₂×C₂₂₃

yes

complemented

#341

C₂×C₂₂₃

yes

characteristic, complemented

C₂²×D₄

C₂²×C₂₂₃

yes

—

C₂×D₈

—

C₂×D₈

—

C₂×M₁₆

C₂²×C₂₂₃

yes

—

C₂×M₁₆

C₂²×C₂₂₃

yes

characteristic

C₂²×C₈

C₂²×C₂₂₃

yes

characteristic

#117

C₂²×C₂₂₃

yes

characteristic

C₄⋊C₈

—

#102

—

#100

—

C₂⁴

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

D₄×C₂₂₃

yes

—

C₂²×C₄

C₂³×C₂₂₃

yes

characteristic

D₈

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

D₄×C₂₂₃

yes

—

100

C₄⋊C₄

D₄×C₂₂₃

yes

—

101

C₂²⋊C₄

—

102

C₄²

—

103

C₂³

—

104

C₂³

—

105

C₂³

—

106

C₂³

—

107

C₂³

#41×C₂₂₃

yes

characteristic

108

D₄

—

109

D₄

—

110

D₄

—

111

D₄

—

112

C₂×C₄

—

113

C₂×C₄

C₂×D₄×C₂₂₃

yes

—

114

C₂×C₄

C₂×D₄×C₂₂₃

yes

characteristic

115

C₂×C₄

C₂×D₄×C₂₂₃

yes

characteristic

116

C₂×C₄

#41×C₂₂₃

yes

—

117

C₈

—

118

C₈

—

119

C₈

—

120

C₂²

—

121

C₂²

—

122

C₂²

—

123

C₂²

—

124

C₂²

—

125

C₂²

#121×C₂₂₃

yes

characteristic

126

C₂²

#123×C₂₂₃

yes

—

127

C₄

—

128

C₄

#120×C₂₂₃

yes

characteristic

129

C₄

#121×C₂₂₃

yes

—

130

C₄

#124×C₂₂₃

yes

characteristic

131

C₂

—

132

C₂

—

133

C₂

#394×C₂₂₃

yes

characteristic

134

C₂

#472×C₂₂₃

yes

—

	subgroup	G/H
center, socle	C₂²×C₂₂₃	#121
commutator, Frattini, cosocle	C₂²×C₄	C₂³×C₂₂₃
prime radical	C₂²×D₈×C₂₂₃	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₂₃ ◁ C₂²×C₄×C₂₂₃ ◁ G

subgroup

G/H

center, socle

C₂²×C₂₂₃

#121

commutator, Frattini, cosocle

C₂²×C₄

C₂³×C₂₂₃

prime radical

C₂²×D₈×C₂₂₃

C₂

LCS

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₂₂₃ ◁ C₂²×C₄×C₂₂₃ ◁ G

N	H ≅ G/N	type
2. #424×C₂₂₃	C₂	semidirect
3. #418×C₂₂₃	C₂	semidirect
4. C₂×#100×C₂₂₃	C₂	semidirect
5. #341×C₂₂₃	C₂	semidirect
67. C₂₂₃	#2045	direct (N × H)
68. #2045	C₂₂₃	direct (N × H)
70. #424	C₂×C₂₂₃	semidirect
71. #418	C₂×C₂₂₃	semidirect
72. C₂×#100	C₂×C₂₂₃	semidirect
73. #341	C₂×C₂₂₃	semidirect

H ≅ G/N

type

2. #424×C₂₂₃

C₂

semidirect

3. #418×C₂₂₃

C₂

semidirect

4. C₂×#100×C₂₂₃

C₂

semidirect

5. #341×C₂₂₃

C₂

semidirect

67. C₂₂₃

#2045

direct (N × H)

68. #2045

C₂₂₃

direct (N × H)

70. #424

C₂×C₂₂₃

semidirect

71. #418

C₂×C₂₂₃

semidirect

72. C₂×#100

C₂×C₂₂₃

semidirect

73. #341

C₂×C₂₂₃

semidirect

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

subgroup	index
C₂²×D₈×C₂₂₃	2
#424×C₂₂₃	2
#418×C₂₂₃	2
C₂×#100×C₂₂₃	2
#341×C₂₂₃	2
#2045	223

subgroup

index

C₂²×D₈×C₂₂₃

#424×C₂₂₃

#418×C₂₂₃

C₂×#100×C₂₂₃

#341×C₂₂₃

#2045

223

Subgroups — #2045×C₂₂₃ #43251060

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2045×C223 #43251060

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2045×C₂₂₃ #43251060