Group #43251126 — subgroups

Subgroups — C₂×C₂²⋊C₁₆×C₂₂₃ #43251126

Order 28544.

272 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 152 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	C₂³×C₈×C₂₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₁₆×C₂₂₃	C₂	1	2	yes	maximal, complemented
3	C₂²⋊C₁₆×C₂₂₃	C₂	1	4	yes	maximal, direct product factor
4	C₂³×C₄×C₂₂₃	C₄	1	1	yes	characteristic
5	C₂²×C₈×C₂₂₃	—	2	4	no	—
6	C₂²×C₈×C₂₂₃	C₄	1	1	yes	characteristic
7	C₂²×C₈×C₂₂₃	C₄	1	2	yes	—
8	C₂²×C₈×C₂₂₃	C₂²	1	2	yes	—
9	C₂²×C₈×C₂₂₃	C₂²	1	1	yes	characteristic
10	C₂×C₁₆×C₂₂₃	—	2	4	no	—
11	C₂×C₁₆×C₂₂₃	C₂²	1	4	yes	complemented
12	C₂⁴×C₂₂₃	C₈	1	1	yes	characteristic
13	C₂²×C₄×C₂₂₃	—	2	2	no	—
14	C₂²×C₄×C₂₂₃	—	2	2	no	—
15	C₂²×C₄×C₂₂₃	C₈	1	1	yes	characteristic
16	C₂²×C₄×C₂₂₃	C₈	1	2	yes	—
17	C₂²×C₄×C₂₂₃	C₂×C₄	1	2	yes	—
18	C₂²×C₄×C₂₂₃	C₂×C₄	1	1	yes	characteristic
19	C₂×C₈×C₂₂₃	—	2	2	no	—
20	C₂×C₈×C₂₂₃	—	2	4	no	—
21	C₂×C₈×C₂₂₃	—	2	4	no	—
22	C₂×C₈×C₂₂₃	C₂×C₄	1	2	yes	—
23	C₂×C₈×C₂₂₃	C₂×C₄	1	1	yes	characteristic
24	C₂×C₈×C₂₂₃	D₄	1	4	yes	—
25	C₂×C₈×C₂₂₃	C₂³	1	1	yes	characteristic
26	C₁₆×C₂₂₃	—	2	4	no	—
27	C₂³×C₂₂₃	—	2	2	no	—
28	C₂³×C₂₂₃	—	2	2	no	—
29	C₂³×C₂₂₃	C₁₆	1	4	yes	complemented
30	C₂³×C₂₂₃	C₂×C₈	1	2	yes	—
31	C₂³×C₂₂₃	C₂×C₈	1	1	yes	characteristic
32	C₂×C₄×C₂₂₃	—	2	2	no	—
33	C₂×C₄×C₂₂₃	—	2	4	no	—
34	C₂×C₄×C₂₂₃	—	2	2	no	—
35	C₂×C₄×C₂₂₃	—	2	2	no	—
36	C₂×C₄×C₂₂₃	C₂²⋊C₄	1	2	yes	—
37	C₂×C₄×C₂₂₃	C₂×C₈	1	2	yes	—
38	C₂×C₄×C₂₂₃	C₂×C₈	1	1	yes	characteristic
39	C₂×C₄×C₂₂₃	M₁₆	1	2	yes	—
40	C₂×C₄×C₂₂₃	C₂²×C₄	1	1	yes	characteristic
41	C₈×C₂₂₃	—	2	2	no	—
42	C₈×C₂₂₃	C₂²⋊C₄	1	2	yes	—
43	C₈×C₂₂₃	C₂×D₄	1	2	yes	—
44	C₂²×C₂₂₃	—	2	8	no	—
45	C₂²×C₂₂₃	—	2	2	no	—
46	C₂²×C₂₂₃	—	2	2	no	—
47	C₂²×C₂₂₃	C₂²⋊C₈	1	2	yes	—
48	C₂²×C₂₂₃	C₂×C₁₆	1	2	yes	—
49	C₂²×C₂₂₃	C₂×C₁₆	1	4	yes	complemented
50	C₂²×C₂₂₃	M₃₂	1	2	yes	—
51	C₂²×C₂₂₃	C₂²×C₈	1	1	yes	characteristic
52	C₄×C₂₂₃	—	2	2	no	—
53	C₄×C₂₂₃	C₂²⋊C₈	1	2	yes	—
54	C₄×C₂₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
55	C₄×C₂₂₃	C₂×M₁₆	1	1	yes	characteristic
56	C₂×C₂₂₃	—	2	4	no	—
57	C₂×C₂₂₃	C₂²⋊C₁₆	1	4	yes	direct product factor
58	C₂×C₂₂₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
59	C₂×C₂₂₃	C₂²×C₁₆	1	1	yes	characteristic
60	C₂×C₂₂₃	C₂×M₃₂	1	1	yes	characteristic
61	C₂₂₃	C₂×C₂²⋊C₁₆	1	1	yes	characteristic, direct product factor, Sylow
62	C₂×C₂²⋊C₁₆	C₂₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
63	C₂³×C₈	C₂×C₂₂₃	1	1	yes	characteristic
64	C₂²×C₁₆	C₂×C₂₂₃	1	2	yes	complemented
65	C₂²⋊C₁₆	C₂×C₂₂₃	1	4	yes	direct product factor
66	C₂³×C₄	C₄×C₂₂₃	1	1	yes	characteristic
67	C₂²×C₈	—	2	4	no	—
68	C₂²×C₈	C₄×C₂₂₃	1	1	yes	characteristic
69	C₂²×C₈	C₄×C₂₂₃	1	2	yes	—
70	C₂²×C₈	C₂²×C₂₂₃	1	2	yes	—
71	C₂²×C₈	C₂²×C₂₂₃	1	1	yes	characteristic
72	C₂×C₁₆	—	2	4	no	—
73	C₂×C₁₆	C₂²×C₂₂₃	1	4	yes	complemented
74	C₂⁴	C₈×C₂₂₃	1	1	yes	characteristic
75	C₂²×C₄	—	2	2	no	—
76	C₂²×C₄	—	2	2	no	—
77	C₂²×C₄	C₈×C₂₂₃	1	1	yes	characteristic
78	C₂²×C₄	C₈×C₂₂₃	1	2	yes	—
79	C₂²×C₄	C₂×C₄×C₂₂₃	1	2	yes	—
80	C₂²×C₄	C₂×C₄×C₂₂₃	1	1	yes	characteristic
81	C₂×C₈	—	2	2	no	—
82	C₂×C₈	—	2	4	no	—
83	C₂×C₈	—	2	4	no	—
84	C₂×C₈	C₂×C₄×C₂₂₃	1	2	yes	—
85	C₂×C₈	C₂×C₄×C₂₂₃	1	1	yes	characteristic
86	C₂×C₈	D₄×C₂₂₃	1	4	yes	—
87	C₂×C₈	C₂³×C₂₂₃	1	1	yes	characteristic
88	C₁₆	—	2	4	no	—
89	C₂³	—	2	2	no	—
90	C₂³	—	2	2	no	—
91	C₂³	C₁₆×C₂₂₃	1	4	yes	complemented
92	C₂³	C₂×C₈×C₂₂₃	1	2	yes	—
93	C₂³	C₂×C₈×C₂₂₃	1	1	yes	characteristic
94	C₂×C₄	—	2	2	no	—
95	C₂×C₄	—	2	4	no	—
96	C₂×C₄	—	2	2	no	—
97	C₂×C₄	—	2	2	no	—
98	C₂×C₄	C₂²⋊C₄×C₂₂₃	1	2	yes	—
99	C₂×C₄	C₂×C₈×C₂₂₃	1	2	yes	—
100	C₂×C₄	C₂×C₈×C₂₂₃	1	1	yes	characteristic
101	C₂×C₄	M₁₆×C₂₂₃	1	2	yes	—
102	C₂×C₄	C₂²×C₄×C₂₂₃	1	1	yes	characteristic
103	C₈	—	2	2	no	—
104	C₈	C₂²⋊C₄×C₂₂₃	1	2	yes	—
105	C₈	C₂×D₄×C₂₂₃	1	2	yes	—
106	C₂²	—	2	8	no	—
107	C₂²	—	2	2	no	—
108	C₂²	—	2	2	no	—
109	C₂²	C₂²⋊C₈×C₂₂₃	1	2	yes	—
110	C₂²	C₂×C₁₆×C₂₂₃	1	2	yes	—
111	C₂²	C₂×C₁₆×C₂₂₃	1	4	yes	complemented
112	C₂²	M₃₂×C₂₂₃	1	2	yes	—
113	C₂²	C₂²×C₈×C₂₂₃	1	1	yes	characteristic
114	C₄	—	2	2	no	—
115	C₄	C₂²⋊C₈×C₂₂₃	1	2	yes	—
116	C₄	C₂×C₂²⋊C₄×C₂₂₃	1	1	yes	characteristic
117	C₄	C₂×M₁₆×C₂₂₃	1	1	yes	characteristic
118	C₂	—	2	4	no	—
119	C₂	C₂²⋊C₁₆×C₂₂₃	1	4	yes	direct product factor
120	C₂	C₂×C₂²⋊C₈×C₂₂₃	1	1	yes	characteristic
121	C₂	C₂²×C₁₆×C₂₂₃	1	1	yes	characteristic
122	C₂	C₂×M₃₂×C₂₂₃	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂³×C₈×C₂₂₃

C₂

yes

maximal, characteristic

C₂²×C₁₆×C₂₂₃

C₂

yes

maximal, complemented

C₂²⋊C₁₆×C₂₂₃

C₂

yes

maximal, direct product factor

C₂³×C₄×C₂₂₃

C₄

yes

characteristic

C₂²×C₈×C₂₂₃

—

C₂²×C₈×C₂₂₃

C₄

yes

characteristic

C₂²×C₈×C₂₂₃

C₄

yes

—

C₂²×C₈×C₂₂₃

C₂²

yes

—

C₂²×C₈×C₂₂₃

C₂²

yes

characteristic

C₂×C₁₆×C₂₂₃

—

C₂×C₁₆×C₂₂₃

C₂²

yes

complemented

C₂⁴×C₂₂₃

C₈

yes

characteristic

C₂²×C₄×C₂₂₃

—

C₂²×C₄×C₂₂₃

—

C₂²×C₄×C₂₂₃

C₈

yes

characteristic

C₂²×C₄×C₂₂₃

C₈

yes

—

C₂²×C₄×C₂₂₃

C₂×C₄

yes

—

C₂²×C₄×C₂₂₃

C₂×C₄

yes

characteristic

C₂×C₈×C₂₂₃

—

C₂×C₈×C₂₂₃

—

C₂×C₈×C₂₂₃

—

C₂×C₈×C₂₂₃

C₂×C₄

yes

—

C₂×C₈×C₂₂₃

C₂×C₄

yes

characteristic

C₂×C₈×C₂₂₃

D₄

yes

—

C₂×C₈×C₂₂₃

C₂³

yes

characteristic

C₁₆×C₂₂₃

—

C₂³×C₂₂₃

—

C₂³×C₂₂₃

—

C₂³×C₂₂₃

C₁₆

yes

complemented

C₂³×C₂₂₃

C₂×C₈

yes

—

C₂³×C₂₂₃

C₂×C₈

yes

characteristic

C₂×C₄×C₂₂₃

—

C₂×C₄×C₂₂₃

—

C₂×C₄×C₂₂₃

—

C₂×C₄×C₂₂₃

—

C₂×C₄×C₂₂₃

C₂²⋊C₄

yes

—

C₂×C₄×C₂₂₃

C₂×C₈

yes

—

C₂×C₄×C₂₂₃

C₂×C₈

yes

characteristic

C₂×C₄×C₂₂₃

M₁₆

yes

—

C₂×C₄×C₂₂₃

C₂²×C₄

yes

characteristic

C₈×C₂₂₃

—

C₈×C₂₂₃

C₂²⋊C₄

yes

—

C₈×C₂₂₃

C₂×D₄

yes

—

C₂²×C₂₂₃

—

C₂²×C₂₂₃

—

C₂²×C₂₂₃

—

C₂²×C₂₂₃

C₂²⋊C₈

yes

—

C₂²×C₂₂₃

C₂×C₁₆

yes

—

C₂²×C₂₂₃

C₂×C₁₆

yes

complemented

C₂²×C₂₂₃

M₃₂

yes

—

C₂²×C₂₂₃

C₂²×C₈

yes

characteristic

C₄×C₂₂₃

—

C₄×C₂₂₃

C₂²⋊C₈

yes

—

C₄×C₂₂₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₂₂₃

C₂×M₁₆

yes

characteristic

C₂×C₂₂₃

—

C₂×C₂₂₃

C₂²⋊C₁₆

yes

direct product factor

C₂×C₂₂₃

C₂×C₂²⋊C₈

yes

characteristic

C₂×C₂₂₃

C₂²×C₁₆

yes

characteristic

C₂×C₂₂₃

C₂×M₃₂

yes

characteristic

C₂₂₃

C₂×C₂²⋊C₁₆

yes

characteristic, direct product factor, Sylow

C₂×C₂²⋊C₁₆

C₂₂₃

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₈

C₂×C₂₂₃

yes

characteristic

C₂²×C₁₆

C₂×C₂₂₃

yes

complemented

C₂²⋊C₁₆

C₂×C₂₂₃

yes

direct product factor

C₂³×C₄

C₄×C₂₂₃

yes

characteristic

C₂²×C₈

—

C₂²×C₈

C₄×C₂₂₃

yes

characteristic

C₂²×C₈

C₄×C₂₂₃

yes

—

C₂²×C₈

C₂²×C₂₂₃

yes

—

C₂²×C₈

C₂²×C₂₂₃

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

C₂²×C₂₂₃

yes

complemented

C₂⁴

C₈×C₂₂₃

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₈×C₂₂₃

yes

characteristic

C₂²×C₄

C₈×C₂₂₃

yes

—

C₂²×C₄

C₂×C₄×C₂₂₃

yes

—

C₂²×C₄

C₂×C₄×C₂₂₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×C₄×C₂₂₃

yes

—

C₂×C₈

C₂×C₄×C₂₂₃

yes

characteristic

C₂×C₈

D₄×C₂₂₃

yes

—

C₂×C₈

C₂³×C₂₂₃

yes

characteristic

C₁₆

—

C₂³

—

C₂³

—

C₂³

C₁₆×C₂₂₃

yes

complemented

C₂³

C₂×C₈×C₂₂₃

yes

—

C₂³

C₂×C₈×C₂₂₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²⋊C₄×C₂₂₃

yes

—

C₂×C₄

C₂×C₈×C₂₂₃

yes

—

100

C₂×C₄

C₂×C₈×C₂₂₃

yes

characteristic

101

C₂×C₄

M₁₆×C₂₂₃

yes

—

102

C₂×C₄

C₂²×C₄×C₂₂₃

yes

characteristic

103

C₈

—

104

C₈

C₂²⋊C₄×C₂₂₃

yes

—

105

C₈

C₂×D₄×C₂₂₃

yes

—

106

C₂²

—

107

C₂²

—

108

C₂²

—

109

C₂²

C₂²⋊C₈×C₂₂₃

yes

—

110

C₂²

C₂×C₁₆×C₂₂₃

yes

—

111

C₂²

C₂×C₁₆×C₂₂₃

yes

complemented

112

C₂²

M₃₂×C₂₂₃

yes

—

113

C₂²

C₂²×C₈×C₂₂₃

yes

characteristic

114

C₄

—

115

C₄

C₂²⋊C₈×C₂₂₃

yes

—

116

C₄

C₂×C₂²⋊C₄×C₂₂₃

yes

characteristic

117

C₄

C₂×M₁₆×C₂₂₃

yes

characteristic

118

C₂

—

119

C₂

C₂²⋊C₁₆×C₂₂₃

yes

direct product factor

120

C₂

C₂×C₂²⋊C₈×C₂₂₃

yes

characteristic

121

C₂

C₂²×C₁₆×C₂₂₃

yes

characteristic

122

C₂

C₂×M₃₂×C₂₂₃

yes

characteristic

	subgroup	G/H
center	C₂²×C₈×C₂₂₃	C₂²
commutator	C₂	C₂²×C₁₆×C₂₂₃
Frattini, cosocle	C₂×C₈	C₂³×C₂₂₃
socle	C₂³×C₂₂₃	C₂×C₈
prime radical	C₂⁴×C₂₂₃	C₈
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈×C₂₂₃ ◁ G

subgroup

G/H

center

C₂²×C₈×C₂₂₃

C₂²

commutator

C₂

C₂²×C₁₆×C₂₂₃

Frattini, cosocle

C₂×C₈

C₂³×C₂₂₃

socle

C₂³×C₂₂₃

C₂×C₈

prime radical

C₂⁴×C₂₂₃

C₈

LCS

G ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₈×C₂₂₃ ◁ G

N	H ≅ G/N	type
2. C₂²×C₁₆×C₂₂₃	C₂	semidirect
3. C₂²⋊C₁₆×C₂₂₃	C₂	direct (N × H)
11. C₂×C₁₆×C₂₂₃	C₂²	semidirect
29. C₂³×C₂₂₃	C₁₆	semidirect
49. C₂²×C₂₂₃	C₂×C₁₆	semidirect
57. C₂×C₂₂₃	C₂²⋊C₁₆	direct (N × H)
61. C₂₂₃	C₂×C₂²⋊C₁₆	direct (N × H)
62. C₂×C₂²⋊C₁₆	C₂₂₃	direct (N × H)
64. C₂²×C₁₆	C₂×C₂₂₃	semidirect
65. C₂²⋊C₁₆	C₂×C₂₂₃	direct (N × H)
73. C₂×C₁₆	C₂²×C₂₂₃	semidirect
91. C₂³	C₁₆×C₂₂₃	semidirect
111. C₂²	C₂×C₁₆×C₂₂₃	semidirect
119. C₂	C₂²⋊C₁₆×C₂₂₃	direct (N × H)

H ≅ G/N

type

2. C₂²×C₁₆×C₂₂₃

C₂

semidirect

3. C₂²⋊C₁₆×C₂₂₃

C₂

direct (N × H)

11. C₂×C₁₆×C₂₂₃

C₂²

semidirect

29. C₂³×C₂₂₃

C₁₆

semidirect

49. C₂²×C₂₂₃

C₂×C₁₆

semidirect

57. C₂×C₂₂₃

C₂²⋊C₁₆

direct (N × H)

61. C₂₂₃

C₂×C₂²⋊C₁₆

direct (N × H)

62. C₂×C₂²⋊C₁₆

C₂₂₃

direct (N × H)

64. C₂²×C₁₆

C₂×C₂₂₃

semidirect

65. C₂²⋊C₁₆

C₂×C₂₂₃

direct (N × H)

73. C₂×C₁₆

C₂²×C₂₂₃

semidirect

91. C₂³

C₁₆×C₂₂₃

semidirect

111. C₂²

C₂×C₁₆×C₂₂₃

semidirect

119. C₂

C₂²⋊C₁₆×C₂₂₃

direct (N × H)

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

subgroup	index
C₂³×C₈×C₂₂₃	2
C₂²×C₁₆×C₂₂₃	2
C₂²⋊C₁₆×C₂₂₃	2
C₂×C₂²⋊C₁₆	223

subgroup

index

C₂³×C₈×C₂₂₃

C₂²×C₁₆×C₂₂₃

C₂²⋊C₁₆×C₂₂₃

C₂×C₂²⋊C₁₆

223

Subgroups — C₂×C₂²⋊C₁₆×C₂₂₃ #43251126

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×C22⋊C16×C223 #43251126

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂²⋊C₁₆×C₂₂₃ #43251126