Group #15669455 — subgroups

Subgroups — C₂₃×#1836 #15669455

Order 2944.

224 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 120 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×#97	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#384	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#382	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#340	C₂	1	4	yes	maximal
5	C₂²×C₈×C₂₃	C₂²	1	2	yes	—
6	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
7	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
8	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
9	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
10	C₂₃×#97	C₄	1	4	yes	complemented
11	C₂₃×#95	—	2	2	no	—
12	C₂²×C₄×C₂₃	—	2	2	no	—
13	C₂²×C₄×C₂₃	—	2	2	no	—
14	C₂²×C₄×C₂₃	D₄	1	2	yes	—
15	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
16	C₂×C₈×C₂₃	—	2	2	no	—
17	C₂×C₈×C₂₃	—	2	2	no	—
18	C₂×C₈×C₂₃	C₂×C₄	1	4	yes	—
19	C₄⋊C₄×C₂₃	—	2	4	no	—
20	C₄⋊C₄×C₂₃	—	2	4	no	—
21	C₄⋊C₄×C₂₃	—	4	4	no	—
22	C₄²×C₂₃	C₂×C₄	1	2	yes	—
23	C₄²×C₂₃	Q₈	1	2	yes	—
24	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
25	C₂×C₄×C₂₃	—	2	2	no	—
26	C₂×C₄×C₂₃	—	2	2	no	—
27	C₂×C₄×C₂₃	—	4	2	no	—
28	C₂×C₄×C₂₃	—	4	2	no	—
29	C₂×C₄×C₂₃	—	4	4	no	—
30	C₂×C₄×C₂₃	—	4	2	no	—
31	C₂×C₄×C₂₃	—	4	2	no	—
32	C₂×C₄×C₂₃	C₄⋊C₄	1	4	yes	—
33	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
34	C₂×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
35	C₂×C₄×C₂₃	#41	1	2	yes	—
36	C₂×C₄×C₂₃	#41	1	2	yes	—
37	C₈×C₂₃	—	2	4	no	—
38	#1836	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
39	C₂²×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
40	C₂²×C₂₃	#124	1	1	yes	characteristic
41	C₂²×C₂₃	#124	1	1	yes	characteristic
42	C₂²×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
43	C₂²×C₂₃	#137	1	2	yes	—
44	C₄×C₂₃	—	2	4	no	—
45	C₄×C₂₃	—	8	2	no	—
46	C₄×C₂₃	—	8	2	no	—
47	C₄×C₂₃	#117	1	2	yes	—
48	C₄×C₂₃	#125	1	2	yes	—
49	C₂×#97	C₂×C₂₃	1	1	yes	characteristic
50	#384	C₂×C₂₃	1	1	yes	characteristic
51	#382	C₂×C₂₃	1	1	yes	characteristic
52	#340	C₂×C₂₃	1	4	yes	—
53	C₂×C₂₃	#382	1	1	yes	characteristic
54	C₂×C₂₃	#428	1	2	yes	—
55	C₂×C₂₃	#489	1	2	yes	—
56	C₂×C₂₃	#490	1	1	yes	characteristic
57	C₂×C₂₃	#491	1	1	yes	characteristic
58	C₂²×C₈	C₂²×C₂₃	1	2	yes	—
59	C₂×C₄⋊C₄	—	2	2	no	—
60	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
61	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
62	C₂×C₄²	C₂²×C₂₃	1	1	yes	characteristic
63	#97	C₄×C₂₃	1	4	yes	complemented
64	#95	—	2	2	no	—
65	C₂₃	#1836	1	1	yes	characteristic, direct product factor, Sylow
66	C₂²×C₄	—	2	2	no	—
67	C₂²×C₄	—	2	2	no	—
68	C₂²×C₄	D₄×C₂₃	1	2	yes	—
69	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
70	C₂×C₈	—	2	2	no	—
71	C₂×C₈	—	2	2	no	—
72	C₂×C₈	C₂×C₄×C₂₃	1	4	yes	—
73	C₄⋊C₄	—	2	4	no	—
74	C₄⋊C₄	—	2	4	no	—
75	C₄⋊C₄	—	4	4	no	—
76	C₄²	C₂×C₄×C₂₃	1	2	yes	—
77	C₄²	Q₈×C₂₃	1	2	yes	—
78	C₂³	C₂×D₄×C₂₃	1	1	yes	characteristic
79	C₂×C₄	—	2	2	no	—
80	C₂×C₄	—	2	2	no	—
81	C₂×C₄	—	4	2	no	—
82	C₂×C₄	—	4	2	no	—
83	C₂×C₄	—	4	4	no	—
84	C₂×C₄	—	4	2	no	—
85	C₂×C₄	—	4	2	no	—
86	C₂×C₄	C₄⋊C₄×C₂₃	1	4	yes	—
87	C₂×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
88	C₂×C₄	C₂×Q₈×C₂₃	1	1	yes	characteristic
89	C₂×C₄	#41×C₂₃	1	2	yes	—
90	C₂×C₄	#41×C₂₃	1	2	yes	—
91	C₈	—	2	4	no	—
92	C₂²	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
93	C₂²	C₂₃×#124	1	1	yes	characteristic
94	C₂²	C₂₃×#124	1	1	yes	characteristic
95	C₂²	C₂₃×AGL(1,ℤ/8)	1	2	yes	—
96	C₂²	C₂₃×#137	1	2	yes	—
97	C₄	—	2	4	no	—
98	C₄	—	8	2	no	—
99	C₄	—	8	2	no	—
100	C₄	C₂₃×#117	1	2	yes	—
101	C₄	C₂₃×#125	1	2	yes	—
102	C₂	C₂₃×#382	1	1	yes	characteristic
103	C₂	C₂₃×#428	1	2	yes	—
104	C₂	C₂₃×#489	1	2	yes	—
105	C₂	C₂₃×#490	1	1	yes	characteristic
106	C₂	C₂₃×#491	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₂₃×#97

C₂

yes

maximal, characteristic

C₂₃×#384

C₂

yes

maximal, characteristic

C₂₃×#382

C₂

yes

maximal, characteristic

C₂₃×#340

C₂

yes

maximal

C₂²×C₈×C₂₃

C₂²

yes

—

C₂×C₄⋊C₄×C₂₃

—

C₂×C₄⋊C₄×C₂₃

C₂²

yes

—

C₂×C₄⋊C₄×C₂₃

C₂²

yes

—

C₂×C₄²×C₂₃

C₂²

yes

characteristic

C₂₃×#97

C₄

yes

complemented

C₂₃×#95

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

D₄

yes

—

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

C₂×C₄

yes

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄²×C₂₃

C₂×C₄

yes

—

C₄²×C₂₃

Q₈

yes

—

C₂³×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₄⋊C₄

yes

—

C₂×C₄×C₂₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₃

#41

yes

—

C₂×C₄×C₂₃

#41

yes

—

C₈×C₂₃

—

#1836

C₂₃

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₂₃

C₂×C₄⋊C₄

yes

C₂²×C₂₃

yes

C₂²×C₂₃

yes

C₂²×C₂₃

yes

—

C₂²×C₂₃

#137

yes

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

#117

yes

—

C₄×C₂₃

#125

yes

—

C₂×#97

C₂×C₂₃

yes

characteristic

#384

C₂×C₂₃

yes

characteristic

#382

C₂×C₂₃

yes

characteristic

#340

C₂×C₂₃

yes

—

C₂×C₂₃

#382

yes

characteristic

C₂×C₂₃

#428

yes

—

C₂×C₂₃

#489

yes

—

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂²×C₈

C₂²×C₂₃

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂²×C₂₃

yes

—

C₂×C₄⋊C₄

C₂²×C₂₃

yes

—

C₂×C₄²

C₂²×C₂₃

yes

characteristic

#97

C₄×C₂₃

yes

complemented

#95

—

C₂₃

#1836

yes

characteristic, direct product factor, Sylow

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

D₄×C₂₃

yes

—

C₂²×C₄

C₂³×C₂₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×C₄×C₂₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂×C₄×C₂₃

yes

—

C₄²

Q₈×C₂₃

yes

—

C₂³

C₂×D₄×C₂₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄⋊C₄×C₂₃

yes

—

C₂×C₄

C₂²×C₄×C₂₃

yes

characteristic

C₂×C₄

C₂×Q₈×C₂₃

yes

characteristic

C₂×C₄

#41×C₂₃

yes

—

C₂×C₄

#41×C₂₃

yes

—

C₈

—

C₂²

C₂×C₄⋊C₄×C₂₃

yes

characteristic

C₂²

C₂₃×#124

yes

characteristic

C₂²

C₂₃×#124

yes

characteristic

C₂²

C₂₃×AGL(1,ℤ/8)

yes

—

C₂²

C₂₃×#137

yes

—

C₄

—

C₄

—

C₄

—

100

C₄

C₂₃×#117

yes

—

101

C₄

C₂₃×#125

yes

—

102

C₂

C₂₃×#382

yes

characteristic

103

C₂

C₂₃×#428

yes

—

104

C₂

C₂₃×#489

yes

—

105

C₂

C₂₃×#490

yes

characteristic

106

C₂

C₂₃×#491

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₂₃	C₂×D₄
commutator	C₂×C₄	C₂²×C₄×C₂₃
Frattini, cosocle	C₂²×C₄	C₂³×C₂₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

subgroup

G/H

center, socle, prime radical

C₂³×C₂₃

C₂×D₄

commutator

C₂×C₄

C₂²×C₄×C₂₃

Frattini, cosocle

C₂²×C₄

C₂³×C₂₃

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

N	H ≅ G/N	type
10. C₂₃×#97	C₄	semidirect
38. #1836	C₂₃	direct (N × H)
63. #97	C₄×C₂₃	semidirect
65. C₂₃	#1836	direct (N × H)

H ≅ G/N

type

10. C₂₃×#97

C₄

semidirect

38. #1836

C₂₃

direct (N × H)

63. #97

C₄×C₂₃

semidirect

65. C₂₃

#1836

direct (N × H)

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

subgroup	index
C₂×C₂₃×#97	2
C₂₃×#384	2
C₂₃×#382	2
C₂₃×#340	2
#1836	23

subgroup

index

C₂×C₂₃×#97

C₂₃×#384

C₂₃×#382

C₂₃×#340

#1836

Subgroups — C₂₃×#1836 #15669455

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C23×#1836 #15669455

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#1836 #15669455