Group #15670508 — subgroups

Subgroups — C₂₃×#2889 #15670508

Order 2944.

704 conjugacy classes of subgroups; only the 340 normal subgroups are shown.

normal subgroups only

This group has 704 conjugacy classes of subgroups — too many to draw. Only its 340 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	C₂×C₂₃×2₋¹⁺⁴	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#568	C₂	1	2	yes	maximal, complemented
3	C₂₃×#413	C₂	1	6	yes	maximal, complemented
4	C₂×C₂₃×#101	C₂	1	6	yes	maximal, complemented
5	C₂₃×2₋¹⁺⁴	—	2	8	no	—
6	C₂×#41×C₂₃	C₄	1	6	yes	—
7	C₂×#41×C₂₃	C₂²	1	3	yes	—
8	C₂×#41×C₂₃	C₂²	1	1	yes	characteristic
9	C₂²×Q₈×C₂₃	C₄	1	2	yes	—
10	C₂²×Q₈×C₂₃	C₂²	1	3	yes	—
11	C₂₃×#131	—	2	8	no	—
12	C₂×M₁₆×C₂₃	C₂²	1	12	yes	—
13	C₂₃×#101	C₂²	1	12	yes	—
14	C₂₃×#100	C₂²	1	4	yes	—
15	#41×C₂₃	—	2	24	no	—
16	#41×C₂₃	—	2	12	no	—
17	#41×C₂₃	D₄	1	8	yes	—
18	C₂×Q₈×C₂₃	—	2	8	no	—
19	C₂×Q₈×C₂₃	—	2	12	no	—
20	C₂×Q₈×C₂₃	C₂×C₄	1	6	yes	—
21	C₂×Q₈×C₂₃	C₂×C₄	1	1	yes	characteristic
22	C₂×Q₈×C₂₃	C₂³	1	3	yes	—
23	C₂×D₄×C₂₃	C₂×C₄	1	6	yes	—
24	C₂×D₄×C₂₃	C₂×C₄	1	3	yes	—
25	C₂×D₄×C₂₃	C₂³	1	1	yes	characteristic
26	C₂²×C₄×C₂₃	C₂×C₄	1	6	yes	—
27	C₂²×C₄×C₂₃	C₂×C₄	1	6	yes	—
28	C₂²×C₄×C₂₃	C₂³	1	3	yes	—
29	M₁₆×C₂₃	—	2	12	no	—
30	M₁₆×C₂₃	C₂³	1	8	yes	—
31	C₂×C₈×C₂₃	—	2	12	no	—
32	C₂³×C₂₃	C₂²×C₄	1	2	yes	—
33	C₂³×C₂₃	C₂²×C₄	1	3	yes	—
34	Q₈×C₂₃	—	2	12	no	—
35	Q₈×C₂₃	—	2	6	no	—
36	Q₈×C₂₃	C₂²⋊C₄	1	4	yes	—
37	D₄×C₂₃	—	2	12	no	—
38	D₄×C₂₃	—	2	2	no	—
39	D₄×C₂₃	C₂²⋊C₄	1	12	yes	—
40	C₂×C₄×C₂₃	—	2	12	no	—
41	C₂×C₄×C₂₃	—	2	12	no	—
42	C₂×C₄×C₂₃	C₂²×C₄	1	6	yes	—
43	C₂×C₄×C₂₃	C₂²×C₄	1	3	yes	—
44	C₂×C₄×C₂₃	C₂×D₄	1	12	yes	—
45	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
46	C₈×C₂₃	—	2	8	no	—
47	#2889	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
48	C₂²×C₂₃	—	2	2	no	—
49	C₂²×C₂₃	—	4	3	no	—
50	C₂²×C₂₃	—	4	2	no	—
51	C₂²×C₂₃	C₂×C₂²⋊C₄	1	6	yes	—
52	C₂²×C₂₃	C₂³×C₄	1	1	yes	characteristic
53	C₄×C₂₃	—	2	6	no	—
54	C₄×C₂₃	C₂×C₂²⋊C₄	1	6	yes	—
55	C₄×C₂₃	C₂²×D₄	1	2	yes	—
56	C₂×2₋¹⁺⁴	C₂×C₂₃	1	1	yes	characteristic
57	#568	C₂×C₂₃	1	2	yes	complemented
58	#413	C₂×C₂₃	1	6	yes	complemented
59	C₂×#101	C₂×C₂₃	1	6	yes	complemented
60	C₂×C₂₃	—	2	1	no	—
61	C₂×C₂₃	—	2	6	no	—
62	C₂×C₂₃	—	4	2	no	—
63	C₂×C₂₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
64	2₋¹⁺⁴	—	2	8	no	—
65	C₂×#41	C₄×C₂₃	1	6	yes	—
66	C₂×#41	C₂²×C₂₃	1	3	yes	—
67	C₂×#41	C₂²×C₂₃	1	1	yes	characteristic
68	C₂²×Q₈	C₄×C₂₃	1	2	yes	—
69	C₂²×Q₈	C₂²×C₂₃	1	3	yes	—
70	#131	—	2	8	no	—
71	C₂×M₁₆	C₂²×C₂₃	1	12	yes	—
72	#101	C₂²×C₂₃	1	12	yes	—
73	#100	C₂²×C₂₃	1	4	yes	—
74	C₂₃	#2889	1	1	yes	characteristic, direct product factor, Sylow
75	#41	—	2	24	no	—
76	#41	—	2	12	no	—
77	#41	D₄×C₂₃	1	8	yes	—
78	C₂×Q₈	—	2	8	no	—
79	C₂×Q₈	—	2	12	no	—
80	C₂×Q₈	C₂×C₄×C₂₃	1	6	yes	—
81	C₂×Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
82	C₂×Q₈	C₂³×C₂₃	1	3	yes	—
83	C₂×D₄	C₂×C₄×C₂₃	1	6	yes	—
84	C₂×D₄	C₂×C₄×C₂₃	1	3	yes	—
85	C₂×D₄	C₂³×C₂₃	1	1	yes	characteristic
86	C₂²×C₄	C₂×C₄×C₂₃	1	6	yes	—
87	C₂²×C₄	C₂×C₄×C₂₃	1	6	yes	—
88	C₂²×C₄	C₂³×C₂₃	1	3	yes	—
89	M₁₆	—	2	12	no	—
90	M₁₆	C₂³×C₂₃	1	8	yes	—
91	C₂×C₈	—	2	12	no	—
92	C₂³	C₂²×C₄×C₂₃	1	2	yes	—
93	C₂³	C₂²×C₄×C₂₃	1	3	yes	—
94	Q₈	—	2	12	no	—
95	Q₈	—	2	6	no	—
96	Q₈	C₂²⋊C₄×C₂₃	1	4	yes	—
97	D₄	—	2	12	no	—
98	D₄	—	2	2	no	—
99	D₄	C₂²⋊C₄×C₂₃	1	12	yes	—
100	C₂×C₄	—	2	12	no	—
101	C₂×C₄	—	2	12	no	—
102	C₂×C₄	C₂²×C₄×C₂₃	1	6	yes	—
103	C₂×C₄	C₂²×C₄×C₂₃	1	3	yes	—
104	C₂×C₄	C₂×D₄×C₂₃	1	12	yes	—
105	C₂×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
106	C₈	—	2	8	no	—
107	C₂²	—	2	2	no	—
108	C₂²	—	4	3	no	—
109	C₂²	—	4	2	no	—
110	C₂²	C₂×C₂²⋊C₄×C₂₃	1	6	yes	—
111	C₂²	C₂³×C₄×C₂₃	1	1	yes	characteristic
112	C₄	—	2	6	no	—
113	C₄	C₂×C₂²⋊C₄×C₂₃	1	6	yes	—
114	C₄	C₂²×D₄×C₂₃	1	2	yes	—
115	C₂	—	2	1	no	—
116	C₂	—	2	6	no	—
117	C₂	—	4	2	no	—
118	C₂	C₂²×C₂²⋊C₄×C₂₃	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₂₃×2₋¹⁺⁴

C₂

yes

maximal, characteristic

C₂₃×#568

C₂

yes

maximal, complemented

C₂₃×#413

C₂

yes

maximal, complemented

C₂×C₂₃×#101

C₂

yes

maximal, complemented

C₂₃×2₋¹⁺⁴

—

C₂×#41×C₂₃

C₄

yes

—

C₂×#41×C₂₃

C₂²

yes

—

C₂×#41×C₂₃

C₂²

yes

characteristic

C₂²×Q₈×C₂₃

C₄

yes

—

C₂²×Q₈×C₂₃

C₂²

yes

—

C₂₃×#131

—

C₂×M₁₆×C₂₃

C₂²

yes

—

C₂₃×#101

C₂²

yes

—

C₂₃×#100

C₂²

yes

—

#41×C₂₃

—

#41×C₂₃

—

#41×C₂₃

D₄

yes

—

C₂×Q₈×C₂₃

—

C₂×Q₈×C₂₃

—

C₂×Q₈×C₂₃

C₂×C₄

yes

—

C₂×Q₈×C₂₃

C₂×C₄

yes

characteristic

C₂×Q₈×C₂₃

C₂³

yes

—

C₂×D₄×C₂₃

C₂×C₄

yes

—

C₂×D₄×C₂₃

C₂×C₄

yes

—

C₂×D₄×C₂₃

C₂³

yes

characteristic

C₂²×C₄×C₂₃

C₂×C₄

yes

—

C₂²×C₄×C₂₃

C₂×C₄

yes

—

C₂²×C₄×C₂₃

C₂³

yes

—

M₁₆×C₂₃

—

M₁₆×C₂₃

C₂³

yes

—

C₂×C₈×C₂₃

—

C₂³×C₂₃

C₂²×C₄

yes

—

C₂³×C₂₃

C₂²×C₄

yes

—

Q₈×C₂₃

—

Q₈×C₂₃

—

Q₈×C₂₃

C₂²⋊C₄

yes

—

D₄×C₂₃

—

D₄×C₂₃

—

D₄×C₂₃

C₂²⋊C₄

yes

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂²×C₄

yes

—

C₂×C₄×C₂₃

C₂²×C₄

yes

—

C₂×C₄×C₂₃

C₂×D₄

yes

—

C₂×C₄×C₂₃

C₂⁴

yes

characteristic

C₈×C₂₃

—

#2889

C₂₃

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

C₂×C₂²⋊C₄

yes

—

C₂²×C₂₃

C₂³×C₄

yes

characteristic

C₄×C₂₃

—

C₄×C₂₃

C₂×C₂²⋊C₄

yes

—

C₄×C₂₃

C₂²×D₄

yes

—

C₂×2₋¹⁺⁴

C₂×C₂₃

yes

characteristic

#568

C₂×C₂₃

yes

complemented

#413

C₂×C₂₃

yes

complemented

C₂×#101

C₂×C₂₃

yes

complemented

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

C₂²×C₂²⋊C₄

yes

characteristic

2₋¹⁺⁴

—

C₂×#41

C₄×C₂₃

yes

—

C₂×#41

C₂²×C₂₃

yes

—

C₂×#41

C₂²×C₂₃

yes

characteristic

C₂²×Q₈

C₄×C₂₃

yes

—

C₂²×Q₈

C₂²×C₂₃

yes

—

#131

—

C₂×M₁₆

C₂²×C₂₃

yes

—

#101

C₂²×C₂₃

yes

—

#100

C₂²×C₂₃

yes

—

C₂₃

#2889

yes

characteristic, direct product factor, Sylow

#41

—

#41

—

#41

D₄×C₂₃

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×C₄×C₂₃

yes

—

C₂×Q₈

C₂×C₄×C₂₃

yes

characteristic

C₂×Q₈

C₂³×C₂₃

yes

—

C₂×D₄

C₂×C₄×C₂₃

yes

—

C₂×D₄

C₂×C₄×C₂₃

yes

—

C₂×D₄

C₂³×C₂₃

yes

characteristic

C₂²×C₄

C₂×C₄×C₂₃

yes

—

C₂²×C₄

C₂×C₄×C₂₃

yes

—

C₂²×C₄

C₂³×C₂₃

yes

—

M₁₆

—

M₁₆

C₂³×C₂₃

yes

—

C₂×C₈

—

C₂³

C₂²×C₄×C₂₃

yes

—

C₂³

C₂²×C₄×C₂₃

yes

—

Q₈

—

Q₈

—

Q₈

C₂²⋊C₄×C₂₃

yes

—

D₄

—

D₄

—

D₄

C₂²⋊C₄×C₂₃

yes

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

C₂²×C₄×C₂₃

yes

—

103

C₂×C₄

C₂²×C₄×C₂₃

yes

—

104

C₂×C₄

C₂×D₄×C₂₃

yes

—

105

C₂×C₄

C₂⁴×C₂₃

yes

characteristic

106

C₈

—

107

C₂²

—

108

C₂²

—

109

C₂²

—

110

C₂²

C₂×C₂²⋊C₄×C₂₃

yes

—

111

C₂²

C₂³×C₄×C₂₃

yes

characteristic

112

C₄

—

113

C₄

C₂×C₂²⋊C₄×C₂₃

yes

—

114

C₄

C₂²×D₄×C₂₃

yes

—

115

C₂

—

116

C₂

—

117

C₂

—

118

C₂

C₂²×C₂²⋊C₄×C₂₃

yes

characteristic

	subgroup	G/H
center, socle	C₂×C₂₃	C₂²×C₂²⋊C₄
commutator	C₂²	C₂³×C₄×C₂₃
Frattini, cosocle	C₂×C₄	C₂⁴×C₂₃
prime radical	C₂×C₂₃×2₋¹⁺⁴	C₂
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₂₃ ◁ C₂×#41×C₂₃ ◁ G

subgroup

G/H

center, socle

C₂×C₂₃

C₂²×C₂²⋊C₄

commutator

C₂²

C₂³×C₄×C₂₃

Frattini, cosocle

C₂×C₄

C₂⁴×C₂₃

prime radical

C₂×C₂₃×2₋¹⁺⁴

C₂

LCS

G ▷ C₂² ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₂₃ ◁ C₂×#41×C₂₃ ◁ G

N	H ≅ G/N	type
2. C₂₃×#568	C₂	semidirect
3. C₂₃×#413	C₂	semidirect
4. C₂×C₂₃×#101	C₂	semidirect
47. #2889	C₂₃	direct (N × H)
57. #568	C₂×C₂₃	semidirect
58. #413	C₂×C₂₃	semidirect
59. C₂×#101	C₂×C₂₃	semidirect
74. C₂₃	#2889	direct (N × H)

H ≅ G/N

type

2. C₂₃×#568

C₂

semidirect

3. C₂₃×#413

C₂

semidirect

4. C₂×C₂₃×#101

C₂

semidirect

47. #2889

C₂₃

direct (N × H)

57. #568

C₂×C₂₃

semidirect

58. #413

C₂×C₂₃

semidirect

59. C₂×#101

C₂×C₂₃

semidirect

74. C₂₃

#2889

direct (N × H)

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

subgroup	index
C₂×C₂₃×2₋¹⁺⁴	2
C₂₃×#568	2
C₂₃×#413	2
C₂×C₂₃×#101	2
#2889	23

subgroup

index

C₂×C₂₃×2₋¹⁺⁴

C₂₃×#568

C₂₃×#413

C₂×C₂₃×#101

#2889

Subgroups — C₂₃×#2889 #15670508

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C23×#2889 #15670508

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#2889 #15670508