Group #19154896 — subgroups

Subgroups — C₂₃×#7270 #19154896

Order 5888.

208 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 114 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×#364	C₂	1	2	yes	maximal
2	C₂₃×#1848	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂₃×M₃₂	—	2	2	no	—
4	C₂×C₂₃×#108	—	2	2	no	—
5	C₂₃×#423	C₄	1	2	yes	—
6	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
7	C₂₃×#364	—	2	2	no	—
8	C₂₃×#364	C₄	1	4	yes	—
9	C₂×M₁₆×C₂₃	—	2	2	no	—
10	C₂²×C₈×C₂₃	D₄	1	2	yes	—
11	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
12	C₂₃×M₃₂	—	2	4	no	—
13	C₂₃×M₃₂	—	4	2	no	—
14	C₂×C₁₆×C₂₃	—	4	2	no	—
15	C₂₃×#108	—	4	2	no	—
16	C₂₃×#108	—	4	2	no	—
17	C₂₃×C₄⋊C₈	—	4	2	no	—
18	C₄×C₈×C₂₃	C₂×C₄	1	2	yes	—
19	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
20	C₄×C₈×C₂₃	Q₈	1	1	yes	characteristic
21	C₂²×C₄×C₂₃	—	2	1	no	—
22	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
23	M₁₆×C₂₃	—	4	4	no	—
24	C₂×C₈×C₂₃	—	4	2	no	—
25	C₂×C₈×C₂₃	C₂²⋊C₄	1	2	yes	—
26	C₂×C₈×C₂₃	C₄⋊C₄	1	2	yes	—
27	C₂×C₈×C₂₃	D₈	1	2	yes	—
28	C₂×C₈×C₂₃	SD₁₆	1	2	yes	—
29	C₂×C₈×C₂₃	SD₁₆	1	2	yes	—
30	C₂×C₈×C₂₃	Q₁₆	1	2	yes	—
31	C₄²×C₂₃	—	2	1	no	—
32	C₄²×C₂₃	C₄²	1	1	yes	characteristic
33	C₄²×C₂₃	C₄⋊C₄	1	1	yes	characteristic
34	C₁₆×C₂₃	—	4	4	no	—
35	#7270	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
36	C₂³×C₂₃	#101	1	1	yes	characteristic
37	C₂×C₄×C₂₃	—	2	1	no	—
38	C₂×C₄×C₂₃	—	2	1	no	—
39	C₂×C₄×C₂₃	—	4	2	no	—
40	C₂×C₄×C₂₃	#95	1	1	yes	characteristic
41	C₂×C₄×C₂₃	#100	1	1	yes	characteristic
42	C₂×C₄×C₂₃	#102	1	2	yes	—
43	C₂×C₄×C₂₃	#103	1	2	yes	—
44	C₈×C₂₃	—	8	2	no	—
45	C₈×C₂₃	#102	1	2	yes	—
46	C₈×C₂₃	#103	1	2	yes	—
47	C₈×C₂₃	#106	1	2	yes	—
48	C₈×C₂₃	#107	1	2	yes	—
49	C₂×#364	C₂×C₂₃	1	2	yes	—
50	#1848	C₂×C₂₃	1	1	yes	characteristic
51	C₂²×C₂₃	—	4	1	no	—
52	C₂²×C₂₃	#332	1	2	yes	—
53	C₂²×C₂₃	#343	1	1	yes	characteristic
54	C₄×C₂₃	—	2	2	no	—
55	C₄×C₂₃	—	4	1	no	—
56	C₄×C₂₃	#331	1	1	yes	characteristic
57	C₄×C₂₃	#333	1	1	yes	characteristic
58	C₄×C₂₃	#340	1	2	yes	—
59	C₂×M₃₂	—	2	2	no	—
60	C₂×#108	—	2	2	no	—
61	#423	C₄×C₂₃	1	2	yes	—
62	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
63	#364	—	2	2	no	—
64	#364	C₄×C₂₃	1	4	yes	—
65	C₂×C₂₃	—	2	1	no	—
66	C₂×C₂₃	—	2	2	no	—
67	C₂×C₂₃	#1296	1	1	yes	characteristic
68	C₂×M₁₆	—	2	2	no	—
69	C₂²×C₈	D₄×C₂₃	1	2	yes	—
70	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
71	M₃₂	—	2	4	no	—
72	M₃₂	—	4	2	no	—
73	C₂×C₁₆	—	4	2	no	—
74	#108	—	4	2	no	—
75	#108	—	4	2	no	—
76	C₄⋊C₈	—	4	2	no	—
77	C₄×C₈	C₂×C₄×C₂₃	1	2	yes	—
78	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
79	C₄×C₈	Q₈×C₂₃	1	1	yes	characteristic
80	C₂₃	#7270	1	1	yes	characteristic, direct product factor, Sylow
81	C₂²×C₄	—	2	1	no	—
82	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
83	M₁₆	—	4	4	no	—
84	C₂×C₈	—	4	2	no	—
85	C₂×C₈	C₂²⋊C₄×C₂₃	1	2	yes	—
86	C₂×C₈	C₄⋊C₄×C₂₃	1	2	yes	—
87	C₂×C₈	D₈×C₂₃	1	2	yes	—
88	C₂×C₈	SD₁₆×C₂₃	1	2	yes	—
89	C₂×C₈	SD₁₆×C₂₃	1	2	yes	—
90	C₂×C₈	Q₁₆×C₂₃	1	2	yes	—
91	C₄²	—	2	1	no	—
92	C₄²	C₄²×C₂₃	1	1	yes	characteristic
93	C₄²	C₄⋊C₄×C₂₃	1	1	yes	characteristic
94	C₁₆	—	4	4	no	—
95	C₂³	C₂₃×#101	1	1	yes	characteristic
96	C₂×C₄	—	2	1	no	—
97	C₂×C₄	—	2	1	no	—
98	C₂×C₄	—	4	2	no	—
99	C₂×C₄	C₂₃×#95	1	1	yes	characteristic
100	C₂×C₄	C₂₃×#100	1	1	yes	characteristic
101	C₂×C₄	C₂₃×#102	1	2	yes	—
102	C₂×C₄	C₂₃×#103	1	2	yes	—
103	C₈	—	8	2	no	—
104	C₈	C₂₃×#102	1	2	yes	—
105	C₈	C₂₃×#103	1	2	yes	—
106	C₈	C₂₃×#106	1	2	yes	—
107	C₈	C₂₃×#107	1	2	yes	—
108	C₂²	—	4	1	no	—
109	C₂²	C₂₃×#332	1	2	yes	—
110	C₂²	C₂₃×#343	1	1	yes	characteristic
111	C₄	—	2	2	no	—
112	C₄	—	4	1	no	—
113	C₄	C₂₃×#331	1	1	yes	characteristic
114	C₄	C₂₃×#333	1	1	yes	characteristic
115	C₄	C₂₃×#340	1	2	yes	—
116	C₂	—	2	1	no	—
117	C₂	—	2	2	no	—
118	C₂	C₂₃×#1296	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₂₃×#364

C₂

yes

maximal

C₂₃×#1848

C₂

yes

maximal, characteristic

C₂×C₂₃×M₃₂

—

C₂×C₂₃×#108

—

C₂₃×#423

C₄

yes

—

C₂×C₄×C₈×C₂₃

C₂²

yes

characteristic

C₂₃×#364

—

C₂₃×#364

C₄

yes

—

C₂×M₁₆×C₂₃

—

C₂²×C₈×C₂₃

D₄

yes

—

C₂×C₄²×C₂₃

C₂×C₄

yes

characteristic

C₂₃×M₃₂

—

C₂₃×M₃₂

—

C₂×C₁₆×C₂₃

—

C₂₃×#108

—

C₂₃×#108

—

C₂₃×C₄⋊C₈

—

C₄×C₈×C₂₃

C₂×C₄

yes

—

C₄×C₈×C₂₃

D₄

yes

characteristic

C₄×C₈×C₂₃

Q₈

yes

characteristic

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

C₂²⋊C₄

yes

characteristic

M₁₆×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

C₂²⋊C₄

yes

—

C₂×C₈×C₂₃

C₄⋊C₄

yes

—

C₂×C₈×C₂₃

D₈

yes

—

C₂×C₈×C₂₃

SD₁₆

yes

—

C₂×C₈×C₂₃

SD₁₆

yes

—

C₂×C₈×C₂₃

Q₁₆

yes

—

C₄²×C₂₃

—

C₄²×C₂₃

C₄²

yes

characteristic

C₄²×C₂₃

C₄⋊C₄

yes

characteristic

C₁₆×C₂₃

—

#7270

C₂₃

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₂₃

#101

yes

characteristic

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

yes

C₂×C₄×C₂₃

yes

C₂×C₄×C₂₃

yes

—

C₂×C₄×C₂₃

#103

yes

—

C₈×C₂₃

—

C₈×C₂₃

#102

yes

—

C₈×C₂₃

#103

yes

—

C₈×C₂₃

#106

yes

—

C₈×C₂₃

#107

yes

—

C₂×#364

C₂×C₂₃

yes

—

#1848

C₂×C₂₃

yes

characteristic

C₂²×C₂₃

—

C₂²×C₂₃

#332

yes

—

C₂²×C₂₃

#343

yes

characteristic

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

yes

C₄×C₂₃

yes

C₄×C₂₃

yes

—

C₂×M₃₂

—

C₂×#108

—

#423

C₄×C₂₃

yes

—

C₂×C₄×C₈

C₂²×C₂₃

yes

characteristic

#364

—

#364

C₄×C₂₃

yes

—

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

#1296

yes

characteristic

C₂×M₁₆

—

C₂²×C₈

D₄×C₂₃

yes

—

C₂×C₄²

C₂×C₄×C₂₃

yes

characteristic

M₃₂

—

M₃₂

—

C₂×C₁₆

—

#108

—

#108

—

C₄⋊C₈

—

C₄×C₈

C₂×C₄×C₂₃

yes

—

C₄×C₈

D₄×C₂₃

yes

characteristic

C₄×C₈

Q₈×C₂₃

yes

characteristic

C₂₃

#7270

yes

characteristic, direct product factor, Sylow

C₂²×C₄

—

C₂²×C₄

C₂²⋊C₄×C₂₃

yes

characteristic

M₁₆

—

C₂×C₈

—

C₂×C₈

C₂²⋊C₄×C₂₃

yes

—

C₂×C₈

C₄⋊C₄×C₂₃

yes

—

C₂×C₈

D₈×C₂₃

yes

—

C₂×C₈

SD₁₆×C₂₃

yes

—

C₂×C₈

SD₁₆×C₂₃

yes

—

C₂×C₈

Q₁₆×C₂₃

yes

—

C₄²

—

C₄²

C₄²×C₂₃

yes

characteristic

C₄²

C₄⋊C₄×C₂₃

yes

characteristic

C₁₆

—

C₂³

C₂₃×#101

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂₃×#95

yes

characteristic

100

C₂×C₄

C₂₃×#100

yes

characteristic

101

C₂×C₄

C₂₃×#102

yes

—

102

C₂×C₄

C₂₃×#103

yes

—

103

C₈

—

104

C₈

C₂₃×#102

yes

—

105

C₈

C₂₃×#103

yes

—

106

C₈

C₂₃×#106

yes

—

107

C₈

C₂₃×#107

yes

—

108

C₂²

—

109

C₂²

C₂₃×#332

yes

—

110

C₂²

C₂₃×#343

yes

characteristic

111

C₄

—

112

C₄

—

113

C₄

C₂₃×#331

yes

characteristic

114

C₄

C₂₃×#333

yes

characteristic

115

C₄

C₂₃×#340

yes

—

116

C₂

—

117

C₂

—

118

C₂

C₂₃×#1296

yes

characteristic

	subgroup	G/H
center	C₄×C₂₃	#331
commutator	C₄²	C₄²×C₂₃
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₂₃
socle	C₂×C₂₃	#1296
prime radical	C₂³×C₂₃	#101
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₂₃ ◁ C₂²×C₄×C₂₃ ◁ C₂×C₄×C₈×C₂₃ ◁ G

subgroup

G/H

center

C₄×C₂₃

#331

commutator

C₄²

C₄²×C₂₃

Frattini, cosocle

C₂×C₄×C₈

C₂²×C₂₃

C₂×C₂₃

C₂³×C₂₃

G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

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

N	H ≅ G/N	type
35. #7270	C₂₃	direct (N × H)
80. C₂₃	#7270	direct (N × H)

H ≅ G/N

type

35. #7270

C₂₃

direct (N × H)

80. C₂₃

#7270

direct (N × H)

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

subgroup	index
C₂×C₂₃×#364	2
C₂₃×#1848	2
#7270	23

subgroup

index

C₂×C₂₃×#364

C₂₃×#1848

#7270

Subgroups — C₂₃×#7270 #19154896

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C23×#7270 #19154896

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#7270 #19154896