Group #19160509 — subgroups

Subgroups — C₂₃×#12883 #19160509

Order 5888.

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

normal subgroups only

This group has 342 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	C₂₃×#2150	C₂	1	2	yes	maximal
2	C₄²×C₈×C₂₃	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1376	C₂	1	4	yes	maximal
4	C₂×C₂₃×M₃₂	C₂²	1	4	yes	—
5	C₂×C₄×C₈×C₂₃	C₄	1	2	yes	—
6	C₂×C₄×C₈×C₂₃	C₄	1	1	yes	characteristic
7	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
8	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
9	C₄³×C₂₃	C₄	1	1	yes	characteristic
10	C₂₃×C₄⋊C₁₆	—	2	4	no	—
11	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
12	C₂²×C₈×C₂₃	C₂×C₄	1	2	yes	—
13	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
14	C₂×C₄²×C₂₃	—	2	2	no	—
15	C₂×C₄²×C₂₃	C₂×C₄	1	2	yes	—
16	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
17	C₂₃×M₃₂	—	2	8	no	—
18	C₂×C₁₆×C₂₃	—	2	4	no	—
19	C₄×C₈×C₂₃	—	2	4	no	—
20	C₄×C₈×C₂₃	—	2	4	no	—
21	C₄×C₈×C₂₃	C₈	1	2	yes	—
22	C₄×C₈×C₂₃	C₈	1	2	yes	—
23	C₄×C₈×C₂₃	D₄	1	2	yes	—
24	C₄×C₈×C₂₃	Q₈	1	2	yes	—
25	C₂²×C₄×C₂₃	—	2	2	no	—
26	C₂²×C₄×C₂₃	M₁₆	1	2	yes	—
27	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
28	C₂×C₈×C₂₃	—	2	2	no	—
29	C₂×C₈×C₂₃	—	2	2	no	—
30	C₂×C₈×C₂₃	—	2	2	no	—
31	C₂×C₈×C₂₃	C₄⋊C₄	1	2	yes	—
32	C₂×C₈×C₂₃	C₂×C₈	1	4	yes	—
33	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
34	C₂×C₈×C₂₃	C₂×Q₈	1	1	yes	characteristic
35	C₂×C₈×C₂₃	#41	1	4	yes	—
36	C₄²×C₂₃	—	2	8	no	—
37	C₄²×C₂₃	—	2	2	no	—
38	C₄²×C₂₃	C₄⋊C₄	1	2	yes	—
39	C₄²×C₂₃	C₂×C₈	1	1	yes	characteristic
40	C₄²×C₂₃	C₂×C₈	1	1	yes	characteristic
41	C₄²×C₂₃	M₁₆	1	4	yes	—
42	C₁₆×C₂₃	—	4	4	no	—
43	#12883	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
44	C₂³×C₂₃	C₂×M₁₆	1	1	yes	characteristic
45	C₂×C₄×C₂₃	—	2	8	no	—
46	C₂×C₄×C₂₃	—	2	4	no	—
47	C₂×C₄×C₂₃	—	2	4	no	—
48	C₂×C₄×C₂₃	C₄⋊C₈	1	2	yes	—
49	C₂×C₄×C₂₃	C₄⋊C₈	1	2	yes	—
50	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
51	C₂×C₄×C₂₃	#117	1	2	yes	—
52	C₂×C₄×C₂₃	C₂²×C₈	1	1	yes	characteristic
53	C₂×C₄×C₂₃	C₂×M₁₆	1	2	yes	—
54	C₈×C₂₃	—	2	2	no	—
55	C₈×C₂₃	—	2	4	no	—
56	C₈×C₂₃	#124	1	2	yes	—
57	C₈×C₂₃	#125	1	2	yes	—
58	#2150	C₂×C₂₃	1	2	yes	—
59	C₄²×C₈	C₂×C₂₃	1	1	yes	characteristic
60	#1376	C₂×C₂₃	1	4	yes	—
61	C₂²×C₂₃	—	2	2	no	—
62	C₂²×C₂₃	C₂×C₄⋊C₈	1	1	yes	characteristic
63	C₂²×C₂₃	#432	1	2	yes	—
64	C₄×C₂₃	—	2	8	no	—
65	C₄×C₂₃	—	2	2	no	—
66	C₄×C₂₃	#364	1	4	yes	—
67	C₄×C₂₃	#382	1	1	yes	characteristic
68	C₄×C₂₃	#423	1	1	yes	characteristic
69	C₄×C₂₃	#431	1	2	yes	—
70	C₂×M₃₂	C₂²×C₂₃	1	4	yes	—
71	C₂×C₄×C₈	C₄×C₂₃	1	2	yes	—
72	C₂×C₄×C₈	C₄×C₂₃	1	1	yes	characteristic
73	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
74	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
75	C₄³	C₄×C₂₃	1	1	yes	characteristic
76	C₄⋊C₁₆	—	2	4	no	—
77	C₂×C₂₃	—	2	2	no	—
78	C₂×C₂₃	#1831	1	1	yes	characteristic
79	C₂×C₂₃	C₂×#364	1	2	yes	—
80	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
81	C₂²×C₈	C₂×C₄×C₂₃	1	2	yes	—
82	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
83	C₂×C₄²	—	2	2	no	—
84	C₂×C₄²	C₂×C₄×C₂₃	1	2	yes	—
85	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
86	M₃₂	—	2	8	no	—
87	C₂×C₁₆	—	2	4	no	—
88	C₄×C₈	—	2	4	no	—
89	C₄×C₈	—	2	4	no	—
90	C₄×C₈	C₈×C₂₃	1	2	yes	—
91	C₄×C₈	C₈×C₂₃	1	2	yes	—
92	C₄×C₈	D₄×C₂₃	1	2	yes	—
93	C₄×C₈	Q₈×C₂₃	1	2	yes	—
94	C₂₃	#12883	1	1	yes	characteristic, direct product factor, Sylow
95	C₂²×C₄	—	2	2	no	—
96	C₂²×C₄	M₁₆×C₂₃	1	2	yes	—
97	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
98	C₂×C₈	—	2	2	no	—
99	C₂×C₈	—	2	2	no	—
100	C₂×C₈	—	2	2	no	—
101	C₂×C₈	C₄⋊C₄×C₂₃	1	2	yes	—
102	C₂×C₈	C₂×C₈×C₂₃	1	4	yes	—
103	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
104	C₂×C₈	C₂×Q₈×C₂₃	1	1	yes	characteristic
105	C₂×C₈	#41×C₂₃	1	4	yes	—
106	C₄²	—	2	8	no	—
107	C₄²	—	2	2	no	—
108	C₄²	C₄⋊C₄×C₂₃	1	2	yes	—
109	C₄²	C₂×C₈×C₂₃	1	1	yes	characteristic
110	C₄²	C₂×C₈×C₂₃	1	1	yes	characteristic
111	C₄²	M₁₆×C₂₃	1	4	yes	—
112	C₁₆	—	4	4	no	—
113	C₂³	C₂×M₁₆×C₂₃	1	1	yes	characteristic
114	C₂×C₄	—	2	8	no	—
115	C₂×C₄	—	2	4	no	—
116	C₂×C₄	—	2	4	no	—
117	C₂×C₄	C₂₃×C₄⋊C₈	1	2	yes	—
118	C₂×C₄	C₂₃×C₄⋊C₈	1	2	yes	—
119	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
120	C₂×C₄	C₂₃×#117	1	2	yes	—
121	C₂×C₄	C₂²×C₈×C₂₃	1	1	yes	characteristic
122	C₂×C₄	C₂×M₁₆×C₂₃	1	2	yes	—
123	C₈	—	2	2	no	—
124	C₈	—	2	4	no	—
125	C₈	C₂₃×#124	1	2	yes	—
126	C₈	C₂₃×#125	1	2	yes	—
127	C₂²	—	2	2	no	—
128	C₂²	C₂×C₂₃×C₄⋊C₈	1	1	yes	characteristic
129	C₂²	C₂₃×#432	1	2	yes	—
130	C₄	—	2	8	no	—
131	C₄	—	2	2	no	—
132	C₄	C₂₃×#364	1	4	yes	—
133	C₄	C₂₃×#382	1	1	yes	characteristic
134	C₄	C₂₃×#423	1	1	yes	characteristic
135	C₄	C₂₃×#431	1	2	yes	—
136	C₂	—	2	2	no	—
137	C₂	C₂₃×#1831	1	1	yes	characteristic
138	C₂	C₂×C₂₃×#364	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂₃×#2150

C₂

yes

maximal

C₄²×C₈×C₂₃

C₂

yes

maximal, characteristic

C₂₃×#1376

C₂

yes

maximal

C₂×C₂₃×M₃₂

C₂²

yes

—

C₂×C₄×C₈×C₂₃

C₄

yes

—

C₂×C₄×C₈×C₂₃

C₄

yes

characteristic

C₂×C₄×C₈×C₂₃

C₂²

yes

—

C₂×C₄×C₈×C₂₃

C₂²

yes

characteristic

C₄³×C₂₃

C₄

yes

characteristic

C₂₃×C₄⋊C₁₆

—

C₂²×C₈×C₂₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₂₃

C₂×C₄

yes

—

C₂²×C₈×C₂₃

C₂³

yes

characteristic

C₂×C₄²×C₂₃

—

C₂×C₄²×C₂₃

C₂×C₄

yes

—

C₂×C₄²×C₂₃

C₂×C₄

yes

characteristic

C₂₃×M₃₂

—

C₂×C₁₆×C₂₃

—

C₄×C₈×C₂₃

—

C₄×C₈×C₂₃

—

C₄×C₈×C₂₃

C₈

yes

—

C₄×C₈×C₂₃

C₈

yes

—

C₄×C₈×C₂₃

D₄

yes

—

C₄×C₈×C₂₃

Q₈

yes

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

M₁₆

yes

—

C₂²×C₄×C₂₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

C₄⋊C₄

yes

—

C₂×C₈×C₂₃

C₂×C₈

yes

—

C₂×C₈×C₂₃

C₂×D₄

yes

characteristic

C₂×C₈×C₂₃

C₂×Q₈

yes

characteristic

C₂×C₈×C₂₃

#41

yes

—

C₄²×C₂₃

—

C₄²×C₂₃

—

C₄²×C₂₃

C₄⋊C₄

yes

—

C₄²×C₂₃

C₂×C₈

yes

characteristic

C₄²×C₂₃

C₂×C₈

yes

characteristic

C₄²×C₂₃

M₁₆

yes

—

C₁₆×C₂₃

—

#12883

C₂₃

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₂₃

C₂×M₁₆

yes

characteristic

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₄⋊C₈

yes

—

C₂×C₄×C₂₃

C₄⋊C₈

yes

—

C₂×C₄×C₂₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₂₃

#117

yes

—

C₂×C₄×C₂₃

C₂²×C₈

yes

characteristic

C₂×C₄×C₂₃

C₂×M₁₆

yes

—

C₈×C₂₃

—

C₈×C₂₃

—

C₈×C₂₃

#124

yes

—

C₈×C₂₃

#125

yes

—

#2150

C₂×C₂₃

yes

—

C₄²×C₈

C₂×C₂₃

yes

characteristic

#1376

C₂×C₂₃

yes

—

C₂²×C₂₃

—

C₂²×C₂₃

C₂×C₄⋊C₈

yes

characteristic

C₂²×C₂₃

#432

yes

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

#364

yes

—

C₄×C₂₃

yes

C₄×C₂₃

yes

C₄×C₂₃

yes

—

C₂×M₃₂

C₂²×C₂₃

yes

—

C₂×C₄×C₈

C₄×C₂₃

yes

—

C₂×C₄×C₈

C₄×C₂₃

yes

characteristic

C₂×C₄×C₈

C₂²×C₂₃

yes

—

C₂×C₄×C₈

C₂²×C₂₃

yes

characteristic

C₄³

C₄×C₂₃

yes

characteristic

C₄⋊C₁₆

—

C₂×C₂₃

—

C₂×C₂₃

#1831

yes

characteristic

C₂×C₂₃

C₂×#364

yes

—

C₂²×C₈

C₂×C₄×C₂₃

yes

characteristic

C₂²×C₈

C₂×C₄×C₂₃

yes

—

C₂²×C₈

C₂³×C₂₃

yes

characteristic

C₂×C₄²

—

C₂×C₄²

C₂×C₄×C₂₃

yes

—

C₂×C₄²

C₂×C₄×C₂₃

yes

characteristic

M₃₂

—

C₂×C₁₆

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₈×C₂₃

yes

—

C₄×C₈

C₈×C₂₃

yes

—

C₄×C₈

D₄×C₂₃

yes

—

C₄×C₈

Q₈×C₂₃

yes

—

C₂₃

#12883

yes

characteristic, direct product factor, Sylow

C₂²×C₄

—

C₂²×C₄

M₁₆×C₂₃

yes

—

C₂²×C₄

C₂²×C₄×C₂₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

C₄⋊C₄×C₂₃

yes

—

102

C₂×C₈

C₂×C₈×C₂₃

yes

—

103

C₂×C₈

C₂×D₄×C₂₃

yes

characteristic

104

C₂×C₈

C₂×Q₈×C₂₃

yes

characteristic

105

C₂×C₈

#41×C₂₃

yes

—

106

C₄²

—

107

C₄²

—

108

C₄²

C₄⋊C₄×C₂₃

yes

—

109

C₄²

C₂×C₈×C₂₃

yes

characteristic

110

C₄²

C₂×C₈×C₂₃

yes

characteristic

111

C₄²

M₁₆×C₂₃

yes

—

112

C₁₆

—

113

C₂³

C₂×M₁₆×C₂₃

yes

characteristic

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

C₂₃×C₄⋊C₈

yes

—

118

C₂×C₄

C₂₃×C₄⋊C₈

yes

—

119

C₂×C₄

C₂×C₄⋊C₄×C₂₃

yes

characteristic

120

C₂×C₄

C₂₃×#117

yes

—

121

C₂×C₄

C₂²×C₈×C₂₃

yes

characteristic

122

C₂×C₄

C₂×M₁₆×C₂₃

yes

—

123

C₈

—

124

C₈

—

125

C₈

C₂₃×#124

yes

—

126

C₈

C₂₃×#125

yes

—

127

C₂²

—

128

C₂²

C₂×C₂₃×C₄⋊C₈

yes

characteristic

129

C₂²

C₂₃×#432

yes

—

130

C₄

—

131

C₄

—

132

C₄

C₂₃×#364

yes

—

133

C₄

C₂₃×#382

yes

characteristic

134

C₄

C₂₃×#423

yes

characteristic

135

C₄

C₂₃×#431

yes

—

136

C₂

—

137

C₂

C₂₃×#1831

yes

characteristic

138

C₂

C₂×C₂₃×#364

yes

—

	subgroup	G/H
center	C₂×C₈×C₂₃	C₂×D₄
commutator	C₂×C₄	C₂²×C₈×C₂₃
Frattini, cosocle	C₂²×C₈	C₂³×C₂₃
socle	C₂²×C₂₃	C₂×C₄⋊C₈
prime radical	C₂³×C₂₃	C₂×M₁₆
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₂₃ ◁ C₂×C₄×C₈×C₂₃ ◁ G

subgroup

G/H

center

C₂×C₈×C₂₃

C₂×D₄

commutator

C₂×C₄

C₂²×C₈×C₂₃

Frattini, cosocle

C₂²×C₈

C₂³×C₂₃

socle

C₂²×C₂₃

C₂×C₄⋊C₈

prime radical

C₂³×C₂₃

C₂×M₁₆

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₈×C₂₃ ◁ C₂×C₄×C₈×C₂₃ ◁ G

N	H ≅ G/N	type
43. #12883	C₂₃	direct (N × H)
94. C₂₃	#12883	direct (N × H)

H ≅ G/N

type

43. #12883

C₂₃

direct (N × H)

94. C₂₃

#12883

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
C₄²×C₈×C₂₃	2
C₂₃×#1376	2
#12883	23

subgroup

index

C₂₃×#2150

C₄²×C₈×C₂₃

C₂₃×#1376

#12883

Subgroups — C₂₃×#12883 #19160509

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C23×#12883 #19160509

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#12883 #19160509