Group #15670404 — subgroups

Subgroups — C₂₃×#2785 #15670404

Order 2944.

428 conjugacy classes of subgroups; only the 176 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 176 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×#395	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂₃×#394	C₂	1	3	yes	maximal
3	C₂₃×#393	C₂	1	2	yes	maximal, complemented
4	C₂₃×#388	C₂	1	6	yes	maximal, complemented
5	C₂₃×#382	C₂	1	3	yes	maximal, complemented
6	C₂²×Q₈×C₂₃	C₂²	1	1	yes	characteristic
7	C₂²×D₄×C₂₃	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊C₄×C₂₃	C₂²	1	3	yes	—
9	C₂×C₄⋊C₄×C₂₃	C₂²	1	3	yes	—
10	C₂×C₂²⋊C₄×C₂₃	C₂²	1	6	yes	—
11	C₂×C₂²⋊C₄×C₂₃	C₂²	1	6	yes	—
12	C₂×C₄²×C₂₃	C₂²	1	3	yes	—
13	C₂₃×#95	C₂²	1	3	yes	—
14	C₂₃×#95	C₂²	1	6	yes	—
15	C₂₃×#95	C₂²	1	3	yes	—
16	C₂⁴×C₂₃	C₂³	1	2	yes	—
17	C₂×Q₈×C₂₃	—	4	3	no	—
18	C₂×D₄×C₂₃	—	4	3	no	—
19	C₂²×C₄×C₂₃	C₂³	1	6	yes	—
20	C₂²×C₄×C₂₃	C₂³	1	3	yes	—
21	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
22	C₂²×C₄×C₂₃	C₂³	1	3	yes	—
23	C₄⋊C₄×C₂₃	—	4	3	no	—
24	C₄⋊C₄×C₂₃	—	4	3	no	—
25	C₂²⋊C₄×C₂₃	—	2	12	no	—
26	C₂²⋊C₄×C₂₃	—	4	6	no	—
27	C₄²×C₂₃	—	2	6	no	—
28	C₂³×C₂₃	—	2	6	no	—
29	C₂³×C₂₃	—	2	6	no	—
30	C₂³×C₂₃	—	2	2	no	—
31	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
32	Q₈×C₂₃	—	4	4	no	—
33	D₄×C₂₃	—	4	4	no	—
34	C₂×C₄×C₂₃	—	2	3	no	—
35	C₂×C₄×C₂₃	—	2	3	no	—
36	C₂×C₄×C₂₃	—	2	6	no	—
37	C₂×C₄×C₂₃	—	2	6	no	—
38	C₂×C₄×C₂₃	—	2	6	no	—
39	C₂×C₄×C₂₃	—	2	6	no	—
40	C₂×C₄×C₂₃	—	2	3	no	—
41	C₂×C₄×C₂₃	—	2	3	no	—
42	C₂×C₄×C₂₃	#41	1	6	yes	—
43	#2785	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
44	C₂²×C₂₃	—	4	6	no	—
45	C₂²×C₂₃	—	4	6	no	—
46	C₂²×C₂₃	—	4	2	no	—
47	C₂²×C₂₃	C₂×#41	1	3	yes	—
48	C₂²×C₂₃	H(5,𝔽₂)	1	3	yes	—
49	C₂²×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
50	C₄×C₂₃	—	2	6	no	—
51	C₄×C₂₃	—	4	6	no	—
52	C₄×C₂₃	—	4	1	no	—
53	C₄×C₂₃	—	4	3	no	—
54	#395	C₂×C₂₃	1	1	yes	characteristic, complemented
55	#394	C₂×C₂₃	1	3	yes	—
56	#393	C₂×C₂₃	1	2	yes	complemented
57	#388	C₂×C₂₃	1	6	yes	complemented
58	#382	C₂×C₂₃	1	3	yes	complemented
59	C₂×C₂₃	—	8	2	no	—
60	C₂×C₂₃	#539	1	3	yes	—
61	C₂×C₂₃	#555	1	3	yes	—
62	C₂×C₂₃	H(3,𝔽₄)	1	1	yes	characteristic
63	C₂²×Q₈	C₂²×C₂₃	1	1	yes	characteristic
64	C₂²×D₄	C₂²×C₂₃	1	1	yes	characteristic
65	C₂×C₄⋊C₄	C₂²×C₂₃	1	3	yes	—
66	C₂×C₄⋊C₄	C₂²×C₂₃	1	3	yes	—
67	C₂×C₂²⋊C₄	C₂²×C₂₃	1	6	yes	—
68	C₂×C₂²⋊C₄	C₂²×C₂₃	1	6	yes	—
69	C₂×C₄²	C₂²×C₂₃	1	3	yes	—
70	#95	C₂²×C₂₃	1	3	yes	—
71	#95	C₂²×C₂₃	1	6	yes	—
72	#95	C₂²×C₂₃	1	3	yes	—
73	C₂₃	#2785	1	1	yes	characteristic, direct product factor, Sylow
74	C₂⁴	C₂³×C₂₃	1	2	yes	—
75	C₂×Q₈	—	4	3	no	—
76	C₂×D₄	—	4	3	no	—
77	C₂²×C₄	C₂³×C₂₃	1	6	yes	—
78	C₂²×C₄	C₂³×C₂₃	1	3	yes	—
79	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
80	C₂²×C₄	C₂³×C₂₃	1	3	yes	—
81	C₄⋊C₄	—	4	3	no	—
82	C₄⋊C₄	—	4	3	no	—
83	C₂²⋊C₄	—	2	12	no	—
84	C₂²⋊C₄	—	4	6	no	—
85	C₄²	—	2	6	no	—
86	C₂³	—	2	6	no	—
87	C₂³	—	2	6	no	—
88	C₂³	—	2	2	no	—
89	C₂³	C₂⁴×C₂₃	1	1	yes	characteristic
90	Q₈	—	4	4	no	—
91	D₄	—	4	4	no	—
92	C₂×C₄	—	2	3	no	—
93	C₂×C₄	—	2	3	no	—
94	C₂×C₄	—	2	6	no	—
95	C₂×C₄	—	2	6	no	—
96	C₂×C₄	—	2	6	no	—
97	C₂×C₄	—	2	6	no	—
98	C₂×C₄	—	2	3	no	—
99	C₂×C₄	—	2	3	no	—
100	C₂×C₄	#41×C₂₃	1	6	yes	—
101	C₂²	—	4	6	no	—
102	C₂²	—	4	2	no	—
103	C₂²	—	4	6	no	—
104	C₂²	C₂×#41×C₂₃	1	3	yes	—
105	C₂²	C₂₃×H(5,𝔽₂)	1	3	yes	—
106	C₂²	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
107	C₄	—	2	6	no	—
108	C₄	—	4	6	no	—
109	C₄	—	4	1	no	—
110	C₄	—	4	3	no	—
111	C₂	—	8	2	no	—
112	C₂	C₂₃×#539	1	3	yes	—
113	C₂	C₂₃×#555	1	3	yes	—
114	C₂	C₂₃×H(3,𝔽₄)	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂₃×#395

C₂

yes

maximal, characteristic, complemented

C₂₃×#394

C₂

yes

maximal

C₂₃×#393

C₂

yes

maximal, complemented

C₂₃×#388

C₂

yes

maximal, complemented

C₂₃×#382

C₂

yes

maximal, complemented

C₂²×Q₈×C₂₃

C₂²

yes

characteristic

C₂²×D₄×C₂₃

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₂₃

C₂²

yes

—

C₂×C₄⋊C₄×C₂₃

C₂²

yes

—

C₂×C₂²⋊C₄×C₂₃

C₂²

yes

—

C₂×C₂²⋊C₄×C₂₃

C₂²

yes

—

C₂×C₄²×C₂₃

C₂²

yes

—

C₂₃×#95

C₂²

yes

—

C₂₃×#95

C₂²

yes

—

C₂₃×#95

C₂²

yes

—

C₂⁴×C₂₃

C₂³

yes

—

C₂×Q₈×C₂₃

—

C₂×D₄×C₂₃

—

C₂²×C₄×C₂₃

C₂³

yes

—

C₂²×C₄×C₂₃

C₂³

yes

—

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₂²×C₄×C₂₃

C₂³

yes

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₂²⋊C₄×C₂₃

—

C₂²⋊C₄×C₂₃

—

C₄²×C₂₃

—

C₂³×C₂₃

—

C₂³×C₂₃

—

C₂³×C₂₃

—

C₂³×C₂₃

C₂⁴

yes

characteristic

Q₈×C₂₃

—

D₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

#41

yes

—

#2785

C₂₃

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

C₂×#41

yes

—

C₂²×C₂₃

H(5,𝔽₂)

yes

—

C₂²×C₂₃

2₋¹⁺⁴

yes

characteristic

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

#395

C₂×C₂₃

yes

characteristic, complemented

#394

C₂×C₂₃

yes

—

#393

C₂×C₂₃

yes

complemented

#388

C₂×C₂₃

yes

complemented

#382

C₂×C₂₃

yes

complemented

C₂×C₂₃

—

C₂×C₂₃

#539

yes

—

C₂×C₂₃

#555

yes

—

C₂×C₂₃

H(3,𝔽₄)

yes

characteristic

C₂²×Q₈

C₂²×C₂₃

yes

characteristic

C₂²×D₄

C₂²×C₂₃

yes

characteristic

C₂×C₄⋊C₄

C₂²×C₂₃

yes

—

C₂×C₄⋊C₄

C₂²×C₂₃

yes

—

C₂×C₂²⋊C₄

C₂²×C₂₃

yes

—

C₂×C₂²⋊C₄

C₂²×C₂₃

yes

—

C₂×C₄²

C₂²×C₂₃

yes

—

#95

C₂²×C₂₃

yes

—

#95

C₂²×C₂₃

yes

—

#95

C₂²×C₂₃

yes

—

C₂₃

#2785

yes

characteristic, direct product factor, Sylow

C₂⁴

C₂³×C₂₃

yes

—

C₂×Q₈

—

C₂×D₄

—

C₂²×C₄

C₂³×C₂₃

yes

—

C₂²×C₄

C₂³×C₂₃

yes

—

C₂²×C₄

C₂³×C₂₃

yes

characteristic

C₂²×C₄

C₂³×C₂₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₂³

—

C₂³

—

C₂³

—

C₂³

C₂⁴×C₂₃

yes

characteristic

Q₈

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

#41×C₂₃

yes

—

101

C₂²

—

102

C₂²

—

103

C₂²

—

104

C₂²

C₂×#41×C₂₃

yes

—

105

C₂²

C₂₃×H(5,𝔽₂)

yes

—

106

C₂²

C₂₃×2₋¹⁺⁴

yes

characteristic

107

C₄

—

108

C₄

—

109

C₄

—

110

C₄

—

111

C₂

—

112

C₂

C₂₃×#539

yes

—

113

C₂

C₂₃×#555

yes

—

114

C₂

C₂₃×H(3,𝔽₄)

yes

characteristic

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

subgroup

G/H

center, socle

C₂³×C₂₃

C₂⁴

commutator, Frattini, cosocle

C₂³

C₂⁴×C₂₃

prime radical

C₂²×D₄×C₂₃

C₂²

LCS

G ▷ C₂³ ▷ C₁

UCS

C₁ ◁ C₂³×C₂₃ ◁ G

N	H ≅ G/N	type
1. C₂₃×#395	C₂	semidirect
3. C₂₃×#393	C₂	semidirect
4. C₂₃×#388	C₂	semidirect
5. C₂₃×#382	C₂	semidirect
43. #2785	C₂₃	direct (N × H)
54. #395	C₂×C₂₃	semidirect
56. #393	C₂×C₂₃	semidirect
57. #388	C₂×C₂₃	semidirect
58. #382	C₂×C₂₃	semidirect
73. C₂₃	#2785	direct (N × H)

H ≅ G/N

type

1. C₂₃×#395

C₂

semidirect

3. C₂₃×#393

C₂

semidirect

4. C₂₃×#388

C₂

semidirect

5. C₂₃×#382

C₂

semidirect

43. #2785

C₂₃

direct (N × H)

54. #395

C₂×C₂₃

semidirect

56. #393

C₂×C₂₃

semidirect

57. #388

C₂×C₂₃

semidirect

58. #382

C₂×C₂₃

semidirect

73. C₂₃

#2785

direct (N × H)

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

subgroup	index
C₂₃×#395	2
C₂₃×#394	2
C₂₃×#393	2
C₂₃×#388	2
C₂₃×#382	2
#2785	23

subgroup

index

C₂₃×#395

C₂₃×#394

C₂₃×#393

C₂₃×#388

C₂₃×#382

#2785

Subgroups — C₂₃×#2785 #15670404

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C23×#2785 #15670404

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#2785 #15670404