Group #43252276 — subgroups

Subgroups — #3261×C₂₂₃ #43252276

Order 28544.

322 conjugacy classes of subgroups; only the 168 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 168 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	#541×C₂₂₃	C₂	1	2	yes	maximal
2	#479×C₂₂₃	C₂	1	4	yes	maximal, complemented
3	#470×C₂₂₃	C₂	1	2	yes	maximal
4	#467×C₂₂₃	C₂	1	2	yes	maximal
5	#462×C₂₂₃	C₂	1	4	yes	maximal, complemented
6	#424×C₂₂₃	C₂	1	1	yes	maximal, characteristic
7	C₂×Q₁₆×C₂₂₃	C₂²	1	2	yes	—
8	C₂×Q₁₆×C₂₂₃	C₂²	1	2	yes	—
9	C₂×M₁₆×C₂₂₃	C₂²	1	1	yes	characteristic
10	C₂²×C₈×C₂₂₃	C₂²	1	1	yes	characteristic
11	C₄⋊Q₈×C₂₂₃	C₂²	1	2	yes	—
12	#126×C₂₂₃	—	2	4	no	—
13	#125×C₂₂₃	—	2	4	no	—
14	#125×C₂₂₃	C₂²	1	2	yes	—
15	C₂²⋊Q₈×C₂₂₃	C₂²	1	4	yes	—
16	C₄×Q₈×C₂₂₃	C₂²	1	4	yes	—
17	#117×C₂₂₃	C₂²	1	1	yes	characteristic
18	#107×C₂₂₃	C₂²	1	2	yes	—
19	#106×C₂₂₃	C₂²	1	2	yes	—
20	C₄⋊C₈×C₂₂₃	C₂²	1	4	yes	—
21	#103×C₂₂₃	C₂²	1	4	yes	—
22	#103×C₂₂₃	C₂²	1	4	yes	—
23	C₂×Q₈×C₂₂₃	C₂³	1	4	yes	—
24	C₂²×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
25	Q₁₆×C₂₂₃	—	4	2	no	—
26	Q₁₆×C₂₂₃	—	4	2	no	—
27	M₁₆×C₂₂₃	—	4	1	no	—
28	C₂×C₈×C₂₂₃	—	4	1	no	—
29	C₂×C₈×C₂₂₃	C₂³	1	2	yes	—
30	C₂×C₈×C₂₂₃	C₂³	1	2	yes	—
31	C₄⋊C₄×C₂₂₃	—	2	4	no	—
32	C₄⋊C₄×C₂₂₃	—	2	4	no	—
33	C₄⋊C₄×C₂₂₃	—	2	4	no	—
34	C₄⋊C₄×C₂₂₃	—	2	4	no	—
35	C₄⋊C₄×C₂₂₃	D₄	1	2	yes	—
36	C₄⋊C₄×C₂₂₃	C₂³	1	4	yes	—
37	C₂²⋊C₄×C₂₂₃	—	2	4	no	—
38	C₂²⋊C₄×C₂₂₃	D₄	1	2	yes	—
39	C₄²×C₂₂₃	—	2	4	no	—
40	C₄²×C₂₂₃	C₂³	1	2	yes	—
41	C₂³×C₂₂₃	C₂×D₄	1	1	yes	characteristic
42	Q₈×C₂₂₃	—	2	4	no	—
43	Q₈×C₂₂₃	—	2	4	no	—
44	C₂×C₄×C₂₂₃	—	2	1	no	—
45	C₂×C₄×C₂₂₃	—	2	1	no	—
46	C₂×C₄×C₂₂₃	—	2	4	no	—
47	C₂×C₄×C₂₂₃	—	2	4	no	—
48	C₂×C₄×C₂₂₃	C₂×D₄	1	4	yes	—
49	C₂×C₄×C₂₂₃	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₂₂₃	C₂⁴	1	1	yes	characteristic
51	C₈×C₂₂₃	—	2	2	no	—
52	C₈×C₂₂₃	—	2	2	no	—
53	C₂²×C₂₂₃	—	2	1	no	—
54	C₂²×C₂₂₃	—	2	2	no	—
55	C₂²×C₂₂₃	C₂²×D₄	1	1	yes	characteristic
56	C₄×C₂₂₃	—	2	1	no	—
57	C₄×C₂₂₃	—	2	4	no	—
58	C₄×C₂₂₃	—	4	4	no	—
59	C₄×C₂₂₃	—	4	4	no	—
60	C₄×C₂₂₃	H(5,𝔽₂)	1	1	yes	characteristic
61	C₄×C₂₂₃	2₋¹⁺⁴	1	1	yes	characteristic
62	C₂×C₂₂₃	—	4	1	no	—
63	C₂×C₂₂₃	#537	1	1	yes	characteristic
64	C₂×C₂₂₃	#578	1	2	yes	—
65	C₂₂₃	#3261	1	1	yes	characteristic, direct product factor, Sylow
66	#3261	C₂₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
67	#541	C₂×C₂₂₃	1	2	yes	—
68	#479	C₂×C₂₂₃	1	4	yes	complemented
69	#470	C₂×C₂₂₃	1	2	yes	—
70	#467	C₂×C₂₂₃	1	2	yes	—
71	#462	C₂×C₂₂₃	1	4	yes	complemented
72	#424	C₂×C₂₂₃	1	1	yes	characteristic
73	C₂×Q₁₆	C₂²×C₂₂₃	1	2	yes	—
74	C₂×Q₁₆	C₂²×C₂₂₃	1	2	yes	—
75	C₂×M₁₆	C₂²×C₂₂₃	1	1	yes	characteristic
76	C₂²×C₈	C₂²×C₂₂₃	1	1	yes	characteristic
77	C₄⋊Q₈	C₂²×C₂₂₃	1	2	yes	—
78	#126	—	2	4	no	—
79	#125	—	2	4	no	—
80	#125	C₂²×C₂₂₃	1	2	yes	—
81	C₂²⋊Q₈	C₂²×C₂₂₃	1	4	yes	—
82	C₄×Q₈	C₂²×C₂₂₃	1	4	yes	—
83	#117	C₂²×C₂₂₃	1	1	yes	characteristic
84	#107	C₂²×C₂₂₃	1	2	yes	—
85	#106	C₂²×C₂₂₃	1	2	yes	—
86	C₄⋊C₈	C₂²×C₂₂₃	1	4	yes	—
87	#103	C₂²×C₂₂₃	1	4	yes	—
88	#103	C₂²×C₂₂₃	1	4	yes	—
89	C₂×Q₈	C₂³×C₂₂₃	1	4	yes	—
90	C₂²×C₄	C₂³×C₂₂₃	1	1	yes	characteristic
91	Q₁₆	—	4	2	no	—
92	Q₁₆	—	4	2	no	—
93	M₁₆	—	4	1	no	—
94	C₂×C₈	—	4	1	no	—
95	C₂×C₈	C₂³×C₂₂₃	1	2	yes	—
96	C₂×C₈	C₂³×C₂₂₃	1	2	yes	—
97	C₄⋊C₄	—	2	4	no	—
98	C₄⋊C₄	—	2	4	no	—
99	C₄⋊C₄	—	2	4	no	—
100	C₄⋊C₄	—	2	4	no	—
101	C₄⋊C₄	D₄×C₂₂₃	1	2	yes	—
102	C₄⋊C₄	C₂³×C₂₂₃	1	4	yes	—
103	C₂²⋊C₄	—	2	4	no	—
104	C₂²⋊C₄	D₄×C₂₂₃	1	2	yes	—
105	C₄²	—	2	4	no	—
106	C₄²	C₂³×C₂₂₃	1	2	yes	—
107	C₂³	C₂×D₄×C₂₂₃	1	1	yes	characteristic
108	Q₈	—	2	4	no	—
109	Q₈	—	2	4	no	—
110	C₂×C₄	—	2	1	no	—
111	C₂×C₄	—	2	1	no	—
112	C₂×C₄	—	2	4	no	—
113	C₂×C₄	—	2	4	no	—
114	C₂×C₄	C₂×D₄×C₂₂₃	1	4	yes	—
115	C₂×C₄	C₂×D₄×C₂₂₃	1	1	yes	characteristic
116	C₂×C₄	C₂⁴×C₂₂₃	1	1	yes	characteristic
117	C₈	—	2	2	no	—
118	C₈	—	2	2	no	—
119	C₂²	—	2	1	no	—
120	C₂²	—	2	2	no	—
121	C₂²	C₂²×D₄×C₂₂₃	1	1	yes	characteristic
122	C₄	—	2	1	no	—
123	C₄	—	2	4	no	—
124	C₄	—	4	4	no	—
125	C₄	—	4	4	no	—
126	C₄	H(5,𝔽₂)×C₂₂₃	1	1	yes	characteristic
127	C₄	2₋¹⁺⁴×C₂₂₃	1	1	yes	characteristic
128	C₂	—	4	1	no	—
129	C₂	#537×C₂₂₃	1	1	yes	characteristic
130	C₂	#578×C₂₂₃	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#541×C₂₂₃

C₂

yes

maximal

#479×C₂₂₃

C₂

yes

maximal, complemented

#470×C₂₂₃

C₂

yes

maximal

#467×C₂₂₃

C₂

yes

maximal

#462×C₂₂₃

C₂

yes

maximal, complemented

#424×C₂₂₃

C₂

yes

maximal, characteristic

C₂×Q₁₆×C₂₂₃

C₂²

yes

—

C₂×Q₁₆×C₂₂₃

C₂²

yes

—

C₂×M₁₆×C₂₂₃

C₂²

yes

characteristic

C₂²×C₈×C₂₂₃

C₂²

yes

characteristic

C₄⋊Q₈×C₂₂₃

C₂²

yes

—

#126×C₂₂₃

—

#125×C₂₂₃

—

#125×C₂₂₃

C₂²

yes

—

C₂²⋊Q₈×C₂₂₃

C₂²

yes

—

C₄×Q₈×C₂₂₃

C₂²

yes

—

#117×C₂₂₃

C₂²

yes

characteristic

#107×C₂₂₃

C₂²

yes

—

#106×C₂₂₃

C₂²

yes

—

C₄⋊C₈×C₂₂₃

C₂²

yes

—

#103×C₂₂₃

C₂²

yes

—

#103×C₂₂₃

C₂²

yes

—

C₂×Q₈×C₂₂₃

C₂³

yes

—

C₂²×C₄×C₂₂₃

C₂³

yes

characteristic

Q₁₆×C₂₂₃

—

Q₁₆×C₂₂₃

—

M₁₆×C₂₂₃

—

C₂×C₈×C₂₂₃

—

C₂×C₈×C₂₂₃

C₂³

yes

—

C₂×C₈×C₂₂₃

C₂³

yes

—

C₄⋊C₄×C₂₂₃

—

C₄⋊C₄×C₂₂₃

—

C₄⋊C₄×C₂₂₃

—

C₄⋊C₄×C₂₂₃

—

C₄⋊C₄×C₂₂₃

D₄

yes

—

C₄⋊C₄×C₂₂₃

C₂³

yes

—

C₂²⋊C₄×C₂₂₃

—

C₂²⋊C₄×C₂₂₃

D₄

yes

—

C₄²×C₂₂₃

—

C₄²×C₂₂₃

C₂³

yes

—

C₂³×C₂₂₃

C₂×D₄

yes

characteristic

Q₈×C₂₂₃

—

Q₈×C₂₂₃

—

C₂×C₄×C₂₂₃

—

C₂×C₄×C₂₂₃

—

C₂×C₄×C₂₂₃

—

C₂×C₄×C₂₂₃

—

C₂×C₄×C₂₂₃

C₂×D₄

yes

—

C₂×C₄×C₂₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₂₃

C₂⁴

yes

characteristic

C₈×C₂₂₃

—

C₈×C₂₂₃

—

C₂²×C₂₂₃

—

C₂²×C₂₂₃

—

C₂²×C₂₂₃

C₂²×D₄

yes

characteristic

C₄×C₂₂₃

—

C₄×C₂₂₃

—

C₄×C₂₂₃

—

C₄×C₂₂₃

—

C₄×C₂₂₃

H(5,𝔽₂)

yes

characteristic

C₄×C₂₂₃

2₋¹⁺⁴

yes

characteristic

C₂×C₂₂₃

—

C₂×C₂₂₃

#537

yes

characteristic

C₂×C₂₂₃

#578

yes

—

C₂₂₃

#3261

yes

characteristic, direct product factor, Sylow

#3261

C₂₂₃

yes

maximal, characteristic, direct product factor, Sylow

#541

C₂×C₂₂₃

yes

—

#479

C₂×C₂₂₃

yes

complemented

#470

C₂×C₂₂₃

yes

—

#467

C₂×C₂₂₃

yes

—

#462

C₂×C₂₂₃

yes

complemented

#424

C₂×C₂₂₃

yes

characteristic

C₂×Q₁₆

C₂²×C₂₂₃

yes

—

C₂×Q₁₆

C₂²×C₂₂₃

yes

—

C₂×M₁₆

C₂²×C₂₂₃

yes

characteristic

C₂²×C₈

C₂²×C₂₂₃

yes

characteristic

C₄⋊Q₈

C₂²×C₂₂₃

yes

—

#126

—

#125

—

#125

C₂²×C₂₂₃

yes

—

C₂²⋊Q₈

C₂²×C₂₂₃

yes

—

C₄×Q₈

C₂²×C₂₂₃

yes

—

#117

C₂²×C₂₂₃

yes

characteristic

#107

C₂²×C₂₂₃

yes

—

#106

C₂²×C₂₂₃

yes

—

C₄⋊C₈

C₂²×C₂₂₃

yes

—

#103

C₂²×C₂₂₃

yes

—

#103

C₂²×C₂₂₃

yes

—

C₂×Q₈

C₂³×C₂₂₃

yes

—

C₂²×C₄

C₂³×C₂₂₃

yes

characteristic

Q₁₆

—

Q₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

C₂³×C₂₂₃

yes

—

C₂×C₈

C₂³×C₂₂₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

D₄×C₂₂₃

yes

—

102

C₄⋊C₄

C₂³×C₂₂₃

yes

—

103

C₂²⋊C₄

—

104

C₂²⋊C₄

D₄×C₂₂₃

yes

—

105

C₄²

—

106

C₄²

C₂³×C₂₂₃

yes

—

107

C₂³

C₂×D₄×C₂₂₃

yes

characteristic

108

Q₈

—

109

Q₈

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

C₂×D₄×C₂₂₃

yes

—

115

C₂×C₄

C₂×D₄×C₂₂₃

yes

characteristic

116

C₂×C₄

C₂⁴×C₂₂₃

yes

characteristic

117

C₈

—

118

C₈

—

119

C₂²

—

120

C₂²

—

121

C₂²

C₂²×D₄×C₂₂₃

yes

characteristic

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

H(5,𝔽₂)×C₂₂₃

yes

characteristic

127

C₄

2₋¹⁺⁴×C₂₂₃

yes

characteristic

128

C₂

—

129

C₂

#537×C₂₂₃

yes

characteristic

130

C₂

#578×C₂₂₃

yes

—

	subgroup	G/H
center, socle	C₂²×C₂₂₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₂₂₃
prime radical	C₂³×C₂₂₃	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₂₃ ◁ #117×C₂₂₃ ◁ G

subgroup

G/H

center, socle

C₂²×C₂₂₃

C₂²×D₄

commutator, Frattini, cosocle

C₂×C₄

C₂⁴×C₂₂₃

prime radical

C₂³×C₂₂₃

C₂×D₄

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₂₂₃ ◁ #117×C₂₂₃ ◁ G

N	H ≅ G/N	type
2. #479×C₂₂₃	C₂	semidirect
5. #462×C₂₂₃	C₂	semidirect
65. C₂₂₃	#3261	direct (N × H)
66. #3261	C₂₂₃	direct (N × H)
68. #479	C₂×C₂₂₃	semidirect
71. #462	C₂×C₂₂₃	semidirect

H ≅ G/N

type

2. #479×C₂₂₃

C₂

semidirect

5. #462×C₂₂₃

C₂

semidirect

65. C₂₂₃

#3261

direct (N × H)

66. #3261

C₂₂₃

direct (N × H)

68. #479

C₂×C₂₂₃

semidirect

71. #462

C₂×C₂₂₃

semidirect

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

subgroup	index
#541×C₂₂₃	2
#479×C₂₂₃	2
#470×C₂₂₃	2
#467×C₂₂₃	2
#462×C₂₂₃	2
#424×C₂₂₃	2
#3261	223

subgroup

index

#541×C₂₂₃

#479×C₂₂₃

#470×C₂₂₃

#467×C₂₂₃

#462×C₂₂₃

#424×C₂₂₃

#3261

223

Subgroups — #3261×C₂₂₃ #43252276

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #3261×C223 #43252276

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #3261×C₂₂₃ #43252276