Group #19154871 — subgroups

Subgroups — C₂₃×#7245 #19154871

Order 5888.

200 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 98 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×#2150	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#1567	C₂	1	2	yes	maximal
3	C₂×C₂₃×M₃₂	C₄	1	2	yes	—
4	C₂₃×#423	—	2	2	no	—
5	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
6	C₂₃×C₄⋊C₁₆	—	2	2	no	—
7	C₂₃×#335	—	2	2	no	—
8	C₂₃×#335	C₄	1	4	yes	—
9	C₂×M₁₆×C₂₃	—	4	2	no	—
10	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
11	C₂²×C₈×C₂₃	Q₈	1	1	yes	characteristic
12	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
13	C₂₃×M₃₂	—	4	2	no	—
14	C₂×C₁₆×C₂₃	—	2	2	no	—
15	C₂₃×C₄⋊C₈	—	2	4	no	—
16	C₂₃×C₄⋊C₈	—	4	2	no	—
17	C₄×C₈×C₂₃	C₂×C₄	1	2	yes	—
18	C₄×C₈×C₂₃	D₄	1	2	yes	—
19	C₂²×C₄×C₂₃	C₄⋊C₄	1	1	yes	characteristic
20	C₂²×C₄×C₂₃	SD₁₆	1	2	yes	—
21	M₁₆×C₂₃	—	8	4	no	—
22	C₂×C₈×C₂₃	—	2	1	no	—
23	C₂×C₈×C₂₃	—	2	1	no	—
24	C₂×C₈×C₂₃	—	4	1	no	—
25	C₂×C₈×C₂₃	—	4	4	no	—
26	C₂×C₈×C₂₃	C₄²	1	1	yes	characteristic
27	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
28	C₂×C₈×C₂₃	C₄⋊C₄	1	2	yes	—
29	C₄²×C₂₃	C₂²⋊C₄	1	2	yes	—
30	C₄²×C₂₃	D₈	1	1	yes	characteristic
31	C₄²×C₂₃	Q₁₆	1	1	yes	characteristic
32	C₁₆×C₂₃	—	4	2	no	—
33	#7245	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
34	C₂³×C₂₃	#106	1	1	yes	characteristic
35	C₂×C₄×C₂₃	—	2	1	no	—
36	C₂×C₄×C₂₃	—	2	1	no	—
37	C₂×C₄×C₂₃	—	2	2	no	—
38	C₂×C₄×C₂₃	—	2	2	no	—
39	C₂×C₄×C₂₃	#95	1	1	yes	characteristic
40	C₂×C₄×C₂₃	#102	1	2	yes	—
41	C₂×C₄×C₂₃	#103	1	2	yes	—
42	C₂×C₄×C₂₃	#107	1	1	yes	characteristic
43	C₈×C₂₃	—	2	2	no	—
44	C₈×C₂₃	—	2	1	no	—
45	C₈×C₂₃	—	8	4	no	—
46	C₈×C₂₃	#100	1	1	yes	characteristic
47	C₈×C₂₃	#101	1	1	yes	characteristic
48	#2150	C₂×C₂₃	1	1	yes	characteristic
49	#1567	C₂×C₂₃	1	2	yes	—
50	C₂²×C₂₃	—	2	1	no	—
51	C₂²×C₂₃	—	2	2	no	—
52	C₂²×C₂₃	#340	1	1	yes	characteristic
53	C₄×C₂₃	—	2	1	no	—
54	C₄×C₂₃	—	2	2	no	—
55	C₄×C₂₃	#338	1	1	yes	characteristic
56	C₄×C₂₃	#343	1	1	yes	characteristic
57	C₄×C₂₃	#361	1	2	yes	—
58	C₄×C₂₃	#362	1	2	yes	—
59	C₂×M₃₂	C₄×C₂₃	1	2	yes	—
60	#423	—	2	2	no	—
61	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
62	C₄⋊C₁₆	—	2	2	no	—
63	#335	—	2	2	no	—
64	#335	C₄×C₂₃	1	4	yes	—
65	C₂×C₂₃	—	4	1	no	—
66	C₂×C₂₃	#1292	1	1	yes	characteristic
67	C₂×C₂₃	#1387	1	2	yes	—
68	C₂×M₁₆	—	4	2	no	—
69	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
70	C₂²×C₈	Q₈×C₂₃	1	1	yes	characteristic
71	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
72	M₃₂	—	4	2	no	—
73	C₂×C₁₆	—	2	2	no	—
74	C₄⋊C₈	—	2	4	no	—
75	C₄⋊C₈	—	4	2	no	—
76	C₄×C₈	C₂×C₄×C₂₃	1	2	yes	—
77	C₄×C₈	D₄×C₂₃	1	2	yes	—
78	C₂₃	#7245	1	1	yes	characteristic, direct product factor, Sylow
79	C₂²×C₄	C₄⋊C₄×C₂₃	1	1	yes	characteristic
80	C₂²×C₄	SD₁₆×C₂₃	1	2	yes	—
81	M₁₆	—	8	4	no	—
82	C₂×C₈	—	2	1	no	—
83	C₂×C₈	—	2	1	no	—
84	C₂×C₈	—	4	1	no	—
85	C₂×C₈	—	4	4	no	—
86	C₂×C₈	C₄²×C₂₃	1	1	yes	characteristic
87	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
88	C₂×C₈	C₄⋊C₄×C₂₃	1	2	yes	—
89	C₄²	C₂²⋊C₄×C₂₃	1	2	yes	—
90	C₄²	D₈×C₂₃	1	1	yes	characteristic
91	C₄²	Q₁₆×C₂₃	1	1	yes	characteristic
92	C₁₆	—	4	2	no	—
93	C₂³	C₂₃×#106	1	1	yes	characteristic
94	C₂×C₄	—	2	1	no	—
95	C₂×C₄	—	2	1	no	—
96	C₂×C₄	—	2	2	no	—
97	C₂×C₄	—	2	2	no	—
98	C₂×C₄	C₂₃×#95	1	1	yes	characteristic
99	C₂×C₄	C₂₃×#102	1	2	yes	—
100	C₂×C₄	C₂₃×#103	1	2	yes	—
101	C₂×C₄	C₂₃×#107	1	1	yes	characteristic
102	C₈	—	2	2	no	—
103	C₈	—	2	1	no	—
104	C₈	—	8	4	no	—
105	C₈	C₂₃×#100	1	1	yes	characteristic
106	C₈	C₂₃×#101	1	1	yes	characteristic
107	C₂²	—	2	2	no	—
108	C₂²	—	2	1	no	—
109	C₂²	C₂₃×#340	1	1	yes	characteristic
110	C₄	—	2	2	no	—
111	C₄	—	2	1	no	—
112	C₄	C₂₃×#338	1	1	yes	characteristic
113	C₄	C₂₃×#343	1	1	yes	characteristic
114	C₄	C₂₃×#361	1	2	yes	—
115	C₄	C₂₃×#362	1	2	yes	—
116	C₂	—	4	1	no	—
117	C₂	C₂₃×#1292	1	1	yes	characteristic
118	C₂	C₂₃×#1387	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂₃×#2150

C₂

yes

maximal, characteristic

C₂₃×#1567

C₂

yes

maximal

C₂×C₂₃×M₃₂

C₄

yes

—

C₂₃×#423

—

C₂×C₄×C₈×C₂₃

C₂²

yes

characteristic

C₂₃×C₄⋊C₁₆

—

C₂₃×#335

—

C₂₃×#335

C₄

yes

—

C₂×M₁₆×C₂₃

—

C₂²×C₈×C₂₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₂₃

Q₈

yes

characteristic

C₂×C₄²×C₂₃

D₄

yes

characteristic

C₂₃×M₃₂

—

C₂×C₁₆×C₂₃

—

C₂₃×C₄⋊C₈

—

C₂₃×C₄⋊C₈

—

C₄×C₈×C₂₃

C₂×C₄

yes

—

C₄×C₈×C₂₃

D₄

yes

—

C₂²×C₄×C₂₃

C₄⋊C₄

yes

characteristic

C₂²×C₄×C₂₃

SD₁₆

yes

—

M₁₆×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

C₄²

yes

characteristic

C₂×C₈×C₂₃

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₂₃

C₄⋊C₄

yes

—

C₄²×C₂₃

C₂²⋊C₄

yes

—

C₄²×C₂₃

D₈

yes

characteristic

C₄²×C₂₃

Q₁₆

yes

characteristic

C₁₆×C₂₃

—

#7245

C₂₃

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₂₃

#106

yes

characteristic

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

#95

yes

characteristic

C₂×C₄×C₂₃

#102

yes

—

C₂×C₄×C₂₃

#103

yes

—

C₂×C₄×C₂₃

#107

yes

characteristic

C₈×C₂₃

—

C₈×C₂₃

—

C₈×C₂₃

—

C₈×C₂₃

yes

C₈×C₂₃

yes

C₂×C₂₃

yes

characteristic

#1567

C₂×C₂₃

yes

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

#340

yes

characteristic

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

yes

C₄×C₂₃

yes

C₄×C₂₃

yes

—

C₄×C₂₃

#362

yes

—

C₂×M₃₂

C₄×C₂₃

yes

—

#423

—

C₂×C₄×C₈

C₂²×C₂₃

yes

characteristic

C₄⋊C₁₆

—

#335

—

#335

C₄×C₂₃

yes

—

C₂×C₂₃

—

C₂×C₂₃

#1292

yes

characteristic

C₂×C₂₃

#1387

yes

—

C₂×M₁₆

—

C₂²×C₈

C₂×C₄×C₂₃

yes

characteristic

C₂²×C₈

Q₈×C₂₃

yes

characteristic

C₂×C₄²

D₄×C₂₃

yes

characteristic

M₃₂

—

C₂×C₁₆

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

C₂×C₄×C₂₃

yes

—

C₄×C₈

D₄×C₂₃

yes

—

C₂₃

#7245

yes

characteristic, direct product factor, Sylow

C₂²×C₄

C₄⋊C₄×C₂₃

yes

characteristic

C₂²×C₄

SD₁₆×C₂₃

yes

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄²×C₂₃

yes

characteristic

C₂×C₈

C₂²⋊C₄×C₂₃

yes

characteristic

C₂×C₈

C₄⋊C₄×C₂₃

yes

—

C₄²

C₂²⋊C₄×C₂₃

yes

—

C₄²

D₈×C₂₃

yes

characteristic

C₄²

Q₁₆×C₂₃

yes

characteristic

C₁₆

—

C₂³

C₂₃×#106

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂₃×#95

yes

characteristic

C₂×C₄

C₂₃×#102

yes

—

100

C₂×C₄

C₂₃×#103

yes

—

101

C₂×C₄

C₂₃×#107

yes

characteristic

102

C₈

—

103

C₈

—

104

C₈

—

105

C₈

C₂₃×#100

yes

characteristic

106

C₈

C₂₃×#101

yes

characteristic

107

C₂²

—

108

C₂²

—

109

C₂²

C₂₃×#340

yes

characteristic

110

C₄

—

111

C₄

—

112

C₄

C₂₃×#338

yes

characteristic

113

C₄

C₂₃×#343

yes

characteristic

114

C₄

C₂₃×#361

yes

—

115

C₄

C₂₃×#362

yes

—

116

C₂

—

117

C₂

C₂₃×#1292

yes

characteristic

118

C₂

C₂₃×#1387

yes

—

	subgroup	G/H
center, socle	C₂²×C₂₃	#340
commutator	C₂×C₈	C₄²×C₂₃
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₂₃
prime radical	C₂³×C₂₃	#106
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ C₂×C₄×C₈×C₂₃ ◁ G

subgroup

G/H

center, socle

C₂²×C₂₃

#340

commutator

C₂×C₈

C₄²×C₂₃

Frattini, cosocle

C₂×C₄×C₈

C₂²×C₂₃

prime radical

C₂³×C₂₃

#106

LCS

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ C₂×C₄×C₈×C₂₃ ◁ G

N	H ≅ G/N	type
33. #7245	C₂₃	direct (N × H)
78. C₂₃	#7245	direct (N × H)

H ≅ G/N

type

33. #7245

C₂₃

direct (N × H)

78. C₂₃

#7245

direct (N × H)

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

subgroup	index
C₂₃×#2150	2
C₂₃×#1567	2
#7245	23

subgroup

index

C₂₃×#2150

C₂₃×#1567

#7245

Subgroups — C₂₃×#7245 #19154871

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C23×#7245 #19154871

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#7245 #19154871