Group #15670605 — subgroups

Subgroups — C₂₃×#2986 #15670605

Order 2944.

330 conjugacy classes of subgroups; only the 256 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 256 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×#533	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#446	C₂	1	4	yes	maximal, complemented
3	C₈×Q₈×C₂₃	C₂	1	4	yes	maximal, complemented
4	C₂₃×#433	C₂	1	4	yes	maximal
5	C₂₃×#431	C₂	1	2	yes	maximal
6	C₂₃×C₄⋊Q₈	C₄	1	2	yes	—
7	C₂₃×#125	C₄	1	2	yes	—
8	C₂₃×C₂²⋊Q₈	C₄	1	4	yes	—
9	C₄×Q₈×C₂₃	C₂²	1	4	yes	—
10	C₂₃×#117	C₂²	1	2	yes	—
11	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
12	C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
13	C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
14	C₂₃×C₄⋊C₈	C₂²	1	8	yes	—
15	C₂₃×C₂²⋊C₈	C₂²	1	4	yes	—
16	C₂₃×#97	C₂²	1	4	yes	—
17	C₄×C₈×C₂₃	C₂²	1	4	yes	—
18	C₄×C₈×C₂₃	C₂²	1	4	yes	—
19	C₂×Q₈×C₂₃	C₂×C₄	1	4	yes	—
20	C₂²×C₄×C₂₃	C₂×C₄	1	2	yes	—
21	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
22	C₂×C₈×C₂₃	C₂³	1	4	yes	—
23	C₂×C₈×C₂₃	C₂³	1	4	yes	—
24	C₄⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
25	C₄⋊C₄×C₂₃	C₂×C₄	1	8	yes	—
26	C₄⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
27	C₂²⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
28	C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
29	C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
30	C₄²×C₂₃	C₂³	1	2	yes	—
31	C₄²×C₂₃	C₂³	1	4	yes	—
32	C₂³×C₂₃	C₂²×C₄	1	1	yes	characteristic
33	Q₈×C₂₃	—	2	8	no	—
34	C₂×C₄×C₂₃	—	2	4	no	—
35	C₂×C₄×C₂₃	—	2	2	no	—
36	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
38	C₂×C₄×C₂₃	C₂²×C₄	1	8	yes	—
39	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
40	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
41	C₈×C₂₃	—	2	4	no	—
42	C₈×C₂₃	—	2	4	no	—
43	#2986	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
44	C₂²×C₂₃	—	2	1	no	—
45	C₂²×C₂₃	—	2	2	no	—
46	C₂²×C₂₃	C₂³×C₄	1	1	yes	characteristic
47	C₄×C₂₃	—	2	1	no	—
48	C₄×C₂₃	—	2	2	no	—
49	C₄×C₂₃	—	2	8	no	—
50	C₄×C₂₃	#131	1	4	yes	—
51	C₄×C₂₃	2₋¹⁺⁴	1	2	yes	—
52	#533	C₂×C₂₃	1	1	yes	characteristic
53	#446	C₂×C₂₃	1	4	yes	complemented
54	C₈×Q₈	C₂×C₂₃	1	4	yes	complemented
55	#433	C₂×C₂₃	1	4	yes	—
56	#431	C₂×C₂₃	1	2	yes	—
57	C₂×C₂₃	—	4	1	no	—
58	C₂×C₂₃	#519	1	1	yes	characteristic
59	C₂×C₂₃	C₂×#131	1	2	yes	—
60	C₄⋊Q₈	C₄×C₂₃	1	2	yes	—
61	#125	C₄×C₂₃	1	2	yes	—
62	C₂²⋊Q₈	C₄×C₂₃	1	4	yes	—
63	C₄×Q₈	C₂²×C₂₃	1	4	yes	—
64	#117	C₂²×C₂₃	1	2	yes	—
65	C₂×C₄²	C₂²×C₂₃	1	1	yes	characteristic
66	C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
67	C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
68	C₄⋊C₈	C₂²×C₂₃	1	8	yes	—
69	C₂²⋊C₈	C₂²×C₂₃	1	4	yes	—
70	#97	C₂²×C₂₃	1	4	yes	—
71	C₄×C₈	C₂²×C₂₃	1	4	yes	—
72	C₄×C₈	C₂²×C₂₃	1	4	yes	—
73	C₂₃	#2986	1	1	yes	characteristic, direct product factor, Sylow
74	C₂×Q₈	C₂×C₄×C₂₃	1	4	yes	—
75	C₂²×C₄	C₂×C₄×C₂₃	1	2	yes	—
76	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
77	C₂×C₈	C₂³×C₂₃	1	4	yes	—
78	C₂×C₈	C₂³×C₂₃	1	4	yes	—
79	C₄⋊C₄	C₂×C₄×C₂₃	1	4	yes	—
80	C₄⋊C₄	C₂×C₄×C₂₃	1	8	yes	—
81	C₄⋊C₄	C₂×C₄×C₂₃	1	4	yes	—
82	C₂²⋊C₄	C₂×C₄×C₂₃	1	4	yes	—
83	C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
84	C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
85	C₄²	C₂³×C₂₃	1	2	yes	—
86	C₄²	C₂³×C₂₃	1	4	yes	—
87	C₂³	C₂²×C₄×C₂₃	1	1	yes	characteristic
88	Q₈	—	2	8	no	—
89	C₂×C₄	—	2	4	no	—
90	C₂×C₄	—	2	2	no	—
91	C₂×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
92	C₂×C₄	C₂²×C₄×C₂₃	1	2	yes	—
93	C₂×C₄	C₂²×C₄×C₂₃	1	8	yes	—
94	C₂×C₄	C₂²×C₄×C₂₃	1	2	yes	—
95	C₂×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
96	C₈	—	2	4	no	—
97	C₈	—	2	4	no	—
98	C₂²	—	2	2	no	—
99	C₂²	—	2	1	no	—
100	C₂²	C₂³×C₄×C₂₃	1	1	yes	characteristic
101	C₄	—	2	2	no	—
102	C₄	—	2	8	no	—
103	C₄	—	2	1	no	—
104	C₄	C₂₃×#131	1	4	yes	—
105	C₄	C₂₃×2₋¹⁺⁴	1	2	yes	—
106	C₂	—	4	1	no	—
107	C₂	C₂₃×#519	1	1	yes	characteristic
108	C₂	C₂×C₂₃×#131	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂₃×#533

C₂

yes

maximal, characteristic

C₂₃×#446

C₂

yes

maximal, complemented

C₈×Q₈×C₂₃

C₂

yes

maximal, complemented

C₂₃×#433

C₂

yes

maximal

C₂₃×#431

C₂

yes

maximal

C₂₃×C₄⋊Q₈

C₄

yes

—

C₂₃×#125

C₄

yes

—

C₂₃×C₂²⋊Q₈

C₄

yes

—

C₄×Q₈×C₂₃

C₂²

yes

—

C₂₃×#117

C₂²

yes

—

C₂×C₄²×C₂₃

C₂²

yes

characteristic

C₂₃×C₄⋊C₈

C₂²

yes

—

C₂₃×C₄⋊C₈

C₂²

yes

—

C₂₃×C₄⋊C₈

C₂²

yes

—

C₂₃×C₂²⋊C₈

C₂²

yes

—

C₂₃×#97

C₂²

yes

—

C₄×C₈×C₂₃

C₂²

yes

—

C₄×C₈×C₂₃

C₂²

yes

—

C₂×Q₈×C₂₃

C₂×C₄

yes

—

C₂²×C₄×C₂₃

C₂×C₄

yes

—

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₂×C₈×C₂₃

C₂³

yes

—

C₂×C₈×C₂₃

C₂³

yes

—

C₄⋊C₄×C₂₃

C₂×C₄

yes

—

C₄⋊C₄×C₂₃

C₂×C₄

yes

—

C₄⋊C₄×C₂₃

C₂×C₄

yes

—

C₂²⋊C₄×C₂₃

C₂×C₄

yes

—

C₄²×C₂₃

C₂×C₄

yes

characteristic

C₄²×C₂₃

C₂×C₄

yes

characteristic

C₄²×C₂₃

C₂³

yes

—

C₄²×C₂₃

C₂³

yes

—

C₂³×C₂₃

C₂²×C₄

yes

characteristic

Q₈×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₃

C₂²×C₄

yes

—

C₂×C₄×C₂₃

C₂²×C₄

yes

—

C₂×C₄×C₂₃

C₂²×C₄

yes

—

C₂×C₄×C₂₃

C₂⁴

yes

characteristic

C₈×C₂₃

—

C₈×C₂₃

—

#2986

C₂₃

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

C₂³×C₄

yes

characteristic

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

#131

yes

—

C₄×C₂₃

2₋¹⁺⁴

yes

—

#533

C₂×C₂₃

yes

characteristic

#446

C₂×C₂₃

yes

complemented

C₈×Q₈

C₂×C₂₃

yes

complemented

#433

C₂×C₂₃

yes

—

#431

C₂×C₂₃

yes

—

C₂×C₂₃

—

C₂×C₂₃

#519

yes

characteristic

C₂×C₂₃

C₂×#131

yes

—

C₄⋊Q₈

C₄×C₂₃

yes

—

#125

C₄×C₂₃

yes

—

C₂²⋊Q₈

C₄×C₂₃

yes

—

C₄×Q₈

C₂²×C₂₃

yes

—

#117

C₂²×C₂₃

yes

—

C₂×C₄²

C₂²×C₂₃

yes

characteristic

C₄⋊C₈

C₂²×C₂₃

yes

—

C₄⋊C₈

C₂²×C₂₃

yes

—

C₄⋊C₈

C₂²×C₂₃

yes

—

C₂²⋊C₈

C₂²×C₂₃

yes

—

#97

C₂²×C₂₃

yes

—

C₄×C₈

C₂²×C₂₃

yes

—

C₄×C₈

C₂²×C₂₃

yes

—

C₂₃

#2986

yes

characteristic, direct product factor, Sylow

C₂×Q₈

C₂×C₄×C₂₃

yes

—

C₂²×C₄

C₂×C₄×C₂₃

yes

—

C₂²×C₄

C₂³×C₂₃

yes

characteristic

C₂×C₈

C₂³×C₂₃

yes

—

C₂×C₈

C₂³×C₂₃

yes

—

C₄⋊C₄

C₂×C₄×C₂₃

yes

—

C₄⋊C₄

C₂×C₄×C₂₃

yes

—

C₄⋊C₄

C₂×C₄×C₂₃

yes

—

C₂²⋊C₄

C₂×C₄×C₂₃

yes

—

C₄²

C₂×C₄×C₂₃

yes

characteristic

C₄²

C₂×C₄×C₂₃

yes

characteristic

C₄²

C₂³×C₂₃

yes

—

C₄²

C₂³×C₂₃

yes

—

C₂³

C₂²×C₄×C₂₃

yes

characteristic

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²×C₄×C₂₃

yes

characteristic

C₂×C₄

C₂²×C₄×C₂₃

yes

—

C₂×C₄

C₂²×C₄×C₂₃

yes

—

C₂×C₄

C₂²×C₄×C₂₃

yes

—

C₂×C₄

C₂⁴×C₂₃

yes

characteristic

C₈

—

C₈

—

C₂²

—

C₂²

—

100

C₂²

C₂³×C₄×C₂₃

yes

characteristic

101

C₄

—

102

C₄

—

103

C₄

—

104

C₄

C₂₃×#131

yes

—

105

C₄

C₂₃×2₋¹⁺⁴

yes

—

106

C₂

—

107

C₂

C₂₃×#519

yes

characteristic

108

C₂

C₂×C₂₃×#131

yes

—

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

subgroup

G/H

center

C₂×C₄×C₂₃

C₂⁴

commutator

C₂²

C₂³×C₄×C₂₃

Frattini, cosocle

C₂×C₄

C₂⁴×C₂₃

socle

C₂²×C₂₃

C₂³×C₄

prime radical

C₂³×C₂₃

C₂²×C₄

LCS

G ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂×C₄×C₂₃ ◁ G

N	H ≅ G/N	type
2. C₂₃×#446	C₂	semidirect
3. C₈×Q₈×C₂₃	C₂	semidirect
43. #2986	C₂₃	direct (N × H)
53. #446	C₂×C₂₃	semidirect
54. C₈×Q₈	C₂×C₂₃	semidirect
73. C₂₃	#2986	direct (N × H)

H ≅ G/N

type

2. C₂₃×#446

C₂

semidirect

3. C₈×Q₈×C₂₃

C₂

semidirect

43. #2986

C₂₃

direct (N × H)

53. #446

C₂×C₂₃

semidirect

54. C₈×Q₈

C₂×C₂₃

semidirect

73. C₂₃

#2986

direct (N × H)

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

subgroup	index
C₂₃×#533	2
C₂₃×#446	2
C₈×Q₈×C₂₃	2
C₂₃×#433	2
C₂₃×#431	2
#2986	23

subgroup

index

C₂₃×#533

C₂₃×#446

C₈×Q₈×C₂₃

C₂₃×#433

C₂₃×#431

#2986

Subgroups — C₂₃×#2986 #15670605

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C23×#2986 #15670605

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#2986 #15670605