Group #19160515 — subgroups

Subgroups — C₂₃×#12889 #19160515

Order 5888.

310 conjugacy classes of subgroups; only the 174 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 174 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×C₄×C₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#1846	C₂	1	2	yes	maximal
3	C₂₃×#1380	C₂	1	4	yes	maximal
4	C₂²×C₁₆×C₂₃	C₂²	1	2	yes	—
5	C₂×C₂₃×#107	C₂²	1	4	yes	—
6	C₂×C₂₃×#106	—	2	2	no	—
7	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
8	C₂₃×#382	—	2	2	no	—
9	C₄×C₁₆×C₂₃	C₄	1	4	yes	complemented
10	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
11	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
12	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
13	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
14	C₂×C₁₆×C₂₃	—	2	4	no	—
15	C₂×C₁₆×C₂₃	C₂×C₄	1	4	yes	—
16	C₂₃×#107	—	2	8	no	—
17	C₂₃×#106	—	4	4	no	—
18	C₄×C₈×C₂₃	C₂×C₄	1	2	yes	—
19	C₄×C₈×C₂₃	Q₈	1	2	yes	—
20	C₂₃×#95	—	4	4	no	—
21	C₂²×C₄×C₂₃	—	4	4	no	—
22	C₂²×C₄×C₂₃	D₈	1	2	yes	—
23	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
24	C₂×C₈×C₂₃	C₄⋊C₄	1	2	yes	—
25	C₂×C₈×C₂₃	SD₁₆	1	4	yes	—
26	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
27	C₂×C₈×C₂₃	C₂×Q₈	1	1	yes	characteristic
28	C₂×C₈×C₂₃	#41	1	4	yes	—
29	C₄⋊C₄×C₂₃	—	4	4	no	—
30	C₄⋊C₄×C₂₃	—	4	4	no	—
31	C₄²×C₂₃	C₄⋊C₄	1	2	yes	—
32	C₄²×C₂₃	Q₁₆	1	2	yes	—
33	C₁₆×C₂₃	—	2	4	no	—
34	#12889	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
35	C₂³×C₂₃	C₂×D₈	1	1	yes	characteristic
36	C₂×C₄×C₂₃	—	2	4	no	—
37	C₂×C₄×C₂₃	—	8	4	no	—
38	C₂×C₄×C₂₃	—	8	8	no	—
39	C₂×C₄×C₂₃	#107	1	4	yes	—
40	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₄×C₂₃	#124	1	2	yes	—
42	C₂×C₄×C₂₃	C₂×SD₁₆	1	2	yes	—
43	C₂×C₄×C₂₃	C₂×Q₁₆	1	1	yes	characteristic
44	C₈×C₂₃	#106	1	4	yes	—
45	C₈×C₂₃	#117	1	2	yes	—
46	C₈×C₂₃	#125	1	2	yes	—
47	C₂×C₄×C₁₆	C₂×C₂₃	1	1	yes	characteristic
48	#1846	C₂×C₂₃	1	2	yes	—
49	#1380	C₂×C₂₃	1	4	yes	—
50	C₂²×C₂₃	C₂×#107	1	1	yes	characteristic
51	C₂²×C₂₃	#486	1	2	yes	—
52	C₂²×C₂₃	#508	1	4	yes	—
53	C₄×C₂₃	—	2	4	no	—
54	C₄×C₂₃	—	16	4	no	—
55	C₄×C₂₃	#382	1	1	yes	characteristic
56	C₄×C₂₃	C₂×#106	1	1	yes	characteristic
57	C₄×C₂₃	#487	1	2	yes	—
58	C₂²×C₁₆	C₂²×C₂₃	1	2	yes	—
59	C₂×#107	C₂²×C₂₃	1	4	yes	—
60	C₂×#106	—	2	2	no	—
61	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
62	#382	—	2	2	no	—
63	C₄×C₁₆	C₄×C₂₃	1	4	yes	complemented
64	C₂×C₂₃	#1834	1	1	yes	characteristic
65	C₂×C₂₃	#2158	1	2	yes	—
66	C₂×C₂₃	#2242	1	4	yes	—
67	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
68	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
69	C₂×C₄⋊C₄	—	2	4	no	—
70	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
71	C₂×C₁₆	—	2	4	no	—
72	C₂×C₁₆	C₂×C₄×C₂₃	1	4	yes	—
73	#107	—	2	8	no	—
74	#106	—	4	4	no	—
75	C₄×C₈	C₂×C₄×C₂₃	1	2	yes	—
76	C₄×C₈	Q₈×C₂₃	1	2	yes	—
77	#95	—	4	4	no	—
78	C₂₃	#12889	1	1	yes	characteristic, direct product factor, Sylow
79	C₂²×C₄	—	4	4	no	—
80	C₂²×C₄	D₈×C₂₃	1	2	yes	—
81	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
82	C₂×C₈	C₄⋊C₄×C₂₃	1	2	yes	—
83	C₂×C₈	SD₁₆×C₂₃	1	4	yes	—
84	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
85	C₂×C₈	C₂×Q₈×C₂₃	1	1	yes	characteristic
86	C₂×C₈	#41×C₂₃	1	4	yes	—
87	C₄⋊C₄	—	4	4	no	—
88	C₄⋊C₄	—	4	4	no	—
89	C₄²	C₄⋊C₄×C₂₃	1	2	yes	—
90	C₄²	Q₁₆×C₂₃	1	2	yes	—
91	C₁₆	—	2	4	no	—
92	C₂³	C₂×D₈×C₂₃	1	1	yes	characteristic
93	C₂×C₄	—	2	4	no	—
94	C₂×C₄	—	8	4	no	—
95	C₂×C₄	—	8	8	no	—
96	C₂×C₄	C₂₃×#107	1	4	yes	—
97	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
98	C₂×C₄	C₂₃×#124	1	2	yes	—
99	C₂×C₄	C₂×SD₁₆×C₂₃	1	2	yes	—
100	C₂×C₄	C₂×Q₁₆×C₂₃	1	1	yes	characteristic
101	C₈	C₂₃×#106	1	4	yes	—
102	C₈	C₂₃×#117	1	2	yes	—
103	C₈	C₂₃×#125	1	2	yes	—
104	C₂²	C₂×C₂₃×#107	1	1	yes	characteristic
105	C₂²	C₂₃×#486	1	2	yes	—
106	C₂²	C₂₃×#508	1	4	yes	—
107	C₄	—	2	4	no	—
108	C₄	—	16	4	no	—
109	C₄	C₂₃×#382	1	1	yes	characteristic
110	C₄	C₂×C₂₃×#106	1	1	yes	characteristic
111	C₄	C₂₃×#487	1	2	yes	—
112	C₂	C₂₃×#1834	1	1	yes	characteristic
113	C₂	C₂₃×#2158	1	2	yes	—
114	C₂	C₂₃×#2242	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×C₁₆×C₂₃

C₂

yes

maximal, characteristic

C₂₃×#1846

C₂

yes

maximal

C₂₃×#1380

C₂

yes

maximal

C₂²×C₁₆×C₂₃

C₂²

yes

—

C₂×C₂₃×#107

C₂²

yes

—

C₂×C₂₃×#106

—

C₂×C₄×C₈×C₂₃

C₂²

yes

characteristic

C₂₃×#382

—

C₄×C₁₆×C₂₃

C₄

yes

complemented

C₂²×C₈×C₂₃

D₄

yes

characteristic

C₂²×C₈×C₂₃

C₂³

yes

characteristic

C₂×C₄⋊C₄×C₂₃

—

C₂×C₄²×C₂₃

D₄

yes

characteristic

C₂×C₁₆×C₂₃

—

C₂×C₁₆×C₂₃

C₂×C₄

yes

—

C₂₃×#107

—

C₂₃×#106

—

C₄×C₈×C₂₃

C₂×C₄

yes

—

C₄×C₈×C₂₃

Q₈

yes

—

C₂₃×#95

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

D₈

yes

—

C₂²×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₈×C₂₃

C₄⋊C₄

yes

—

C₂×C₈×C₂₃

SD₁₆

yes

—

C₂×C₈×C₂₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₂₃

C₂×Q₈

yes

characteristic

C₂×C₈×C₂₃

#41

yes

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄²×C₂₃

C₄⋊C₄

yes

—

C₄²×C₂₃

Q₁₆

yes

—

C₁₆×C₂₃

—

#12889

C₂₃

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₂₃

C₂×D₈

yes

characteristic

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

#107

yes

—

C₂×C₄×C₂₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₂₃

#124

yes

—

C₂×C₄×C₂₃

C₂×SD₁₆

yes

—

C₂×C₄×C₂₃

C₂×Q₁₆

yes

characteristic

C₈×C₂₃

#106

yes

—

C₈×C₂₃

#117

yes

—

C₈×C₂₃

#125

yes

—

C₂×C₄×C₁₆

C₂×C₂₃

yes

characteristic

#1846

C₂×C₂₃

yes

—

#1380

C₂×C₂₃

yes

—

C₂²×C₂₃

C₂×#107

yes

characteristic

C₂²×C₂₃

#486

yes

—

C₂²×C₂₃

#508

yes

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

yes

C₄×C₂₃

yes

C₄×C₂₃

yes

—

C₂²×C₁₆

C₂²×C₂₃

yes

—

C₂×#107

C₂²×C₂₃

yes

—

C₂×#106

—

C₂×C₄×C₈

C₂²×C₂₃

yes

characteristic

#382

—

C₄×C₁₆

C₄×C₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

—

C₂×C₂₃

#2242

yes

—

C₂²×C₈

D₄×C₂₃

yes

characteristic

C₂²×C₈

C₂³×C₂₃

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄²

D₄×C₂₃

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

C₂×C₄×C₂₃

yes

—

#107

—

#106

—

C₄×C₈

C₂×C₄×C₂₃

yes

—

C₄×C₈

Q₈×C₂₃

yes

—

#95

—

C₂₃

#12889

yes

characteristic, direct product factor, Sylow

C₂²×C₄

—

C₂²×C₄

D₈×C₂₃

yes

—

C₂²×C₄

C₂×D₄×C₂₃

yes

characteristic

C₂×C₈

C₄⋊C₄×C₂₃

yes

—

C₂×C₈

SD₁₆×C₂₃

yes

—

C₂×C₈

C₂²×C₄×C₂₃

yes

characteristic

C₂×C₈

C₂×Q₈×C₂₃

yes

characteristic

C₂×C₈

#41×C₂₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₄⋊C₄×C₂₃

yes

—

C₄²

Q₁₆×C₂₃

yes

—

C₁₆

—

C₂³

C₂×D₈×C₂₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂₃×#107

yes

—

C₂×C₄

C₂×C₄⋊C₄×C₂₃

yes

characteristic

C₂×C₄

C₂₃×#124

yes

—

C₂×C₄

C₂×SD₁₆×C₂₃

yes

—

100

C₂×C₄

C₂×Q₁₆×C₂₃

yes

characteristic

101

C₈

C₂₃×#106

yes

—

102

C₈

C₂₃×#117

yes

—

103

C₈

C₂₃×#125

yes

—

104

C₂²

C₂×C₂₃×#107

yes

characteristic

105

C₂²

C₂₃×#486

yes

—

106

C₂²

C₂₃×#508

yes

—

107

C₄

—

108

C₄

—

109

C₄

C₂₃×#382

yes

characteristic

110

C₄

C₂×C₂₃×#106

yes

characteristic

111

C₄

C₂₃×#487

yes

—

112

C₂

C₂₃×#1834

yes

characteristic

113

C₂

C₂₃×#2158

yes

—

114

C₂

C₂₃×#2242

yes

—

	subgroup	G/H
center, socle, prime radical	C₂³×C₂₃	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₂₃
Frattini, cosocle	C₂²×C₈	C₂³×C₂₃
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ C₂×C₄×C₈×C₂₃ ◁ G

subgroup

G/H

center, socle, prime radical

C₂³×C₂₃

C₂×D₈

commutator

C₂×C₈

C₂²×C₄×C₂₃

Frattini, cosocle

C₂²×C₈

C₂³×C₂₃

LCS

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ C₂×C₄×C₈×C₂₃ ◁ G

N	H ≅ G/N	type
9. C₄×C₁₆×C₂₃	C₄	semidirect
34. #12889	C₂₃	direct (N × H)
63. C₄×C₁₆	C₄×C₂₃	semidirect
78. C₂₃	#12889	direct (N × H)

H ≅ G/N

type

9. C₄×C₁₆×C₂₃

C₄

semidirect

34. #12889

C₂₃

direct (N × H)

63. C₄×C₁₆

C₄×C₂₃

semidirect

78. C₂₃

#12889

direct (N × H)

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

subgroup	index
C₂×C₄×C₁₆×C₂₃	2
C₂₃×#1846	2
C₂₃×#1380	2
#12889	23

subgroup

index

C₂×C₄×C₁₆×C₂₃

C₂₃×#1846

C₂₃×#1380

#12889

Subgroups — C₂₃×#12889 #19160515

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C23×#12889 #19160515

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#12889 #19160515