Group #19158017 — subgroups

Subgroups — C₂₃×#10391 #19158017

Order 5888.

542 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 90 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×#2840	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#1819	C₂	1	3	yes	maximal
3	C₂₃×#1304	C₂	1	3	yes	maximal
4	C₂²×C₂²⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
5	C₂₃×#393	—	2	6	no	—
6	C₂₃×#379	C₂²	1	3	yes	—
7	C₂₃×#379	C₂²	1	3	yes	—
8	C₂₃×#342	—	2	6	no	—
9	C₂⁵×C₂₃	C₂³	1	1	yes	characteristic
10	C₂²×Q₈×C₂₃	—	2	2	no	—
11	C₂³×C₄×C₂₃	Q₈	1	2	yes	complemented
12	C₂×C₂²⋊C₄×C₂₃	—	2	3	no	—
13	C₂×C₂²⋊C₄×C₂₃	—	2	3	no	—
14	C₂×C₂²⋊C₄×C₂₃	—	2	3	no	—
15	C₂×C₂²⋊C₄×C₂₃	—	2	6	no	—
16	C₂×C₂²⋊C₄×C₂₃	—	2	6	no	—
17	C₂×C₂²⋊C₄×C₂₃	D₄	1	6	yes	—
18	C₂₃×#95	—	2	6	no	—
19	C₂⁴×C₂₃	—	4	2	no	—
20	C₂⁴×C₂₃	—	8	2	no	—
21	C₂⁴×C₂₃	C₂×D₄	1	3	yes	—
22	C₂⁴×C₂₃	C₂×Q₈	1	1	yes	characteristic
23	C₂⁴×C₂₃	#41	1	3	yes	—
24	C₂×Q₈×C₂₃	—	8	4	no	—
25	C₂×Q₈×C₂₃	—	8	2	no	—
26	C₂²×C₄×C₂₃	—	2	6	no	—
27	C₂²×C₄×C₂₃	—	2	6	no	—
28	C₂²×C₄×C₂₃	—	4	3	no	—
29	C₂²×C₄×C₂₃	—	8	2	no	—
30	C₂²⋊C₄×C₂₃	—	4	1	no	—
31	C₂²⋊C₄×C₂₃	—	4	3	no	—
32	C₂²⋊C₄×C₂₃	—	4	6	no	—
33	C₂²⋊C₄×C₂₃	—	4	6	no	—
34	C₂²⋊C₄×C₂₃	—	8	6	no	—
35	C₂²⋊C₄×C₂₃	—	8	6	no	—
36	#10391	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
37	C₂³×C₂₃	—	2	6	no	—
38	C₂³×C₂₃	—	4	2	no	—
39	C₂³×C₂₃	—	4	1	no	—
40	C₂³×C₂₃	—	4	6	no	—
41	C₂³×C₂₃	—	4	6	no	—
42	C₂³×C₂₃	—	4	3	no	—
43	C₂³×C₂₃	—	8	8	no	—
44	C₂³×C₂₃	#120	1	1	yes	characteristic
45	C₂³×C₂₃	C₂²⋊Q₈	1	3	yes	—
46	C₂³×C₂₃	#124	1	3	yes	—
47	Q₈×C₂₃	—	8	8	no	—
48	C₂×C₄×C₂₃	—	4	12	no	—
49	C₂×C₄×C₂₃	—	4	2	no	—
50	C₂×C₄×C₂₃	—	4	6	no	—
51	C₂×C₄×C₂₃	—	8	6	no	—
52	C₂×C₄×C₂₃	—	8	3	no	—
53	C₂×C₄×C₂₃	—	8	6	no	—
54	#2840	C₂×C₂₃	1	1	yes	characteristic
55	#1819	C₂×C₂₃	1	3	yes	—
56	#1304	C₂×C₂₃	1	3	yes	—
57	C₂²×C₂₃	—	2	2	no	—
58	C₂²×C₂₃	—	2	3	no	—
59	C₂²×C₂₃	—	2	3	no	—
60	C₂²×C₂₃	—	2	6	no	—
61	C₂²×C₂₃	—	2	6	no	—
62	C₂²×C₂₃	—	4	4	no	—
63	C₂²×C₂₃	—	8	6	no	—
64	C₂²×C₂₃	—	8	6	no	—
65	C₂²×C₂₃	#393	1	1	yes	characteristic
66	C₂²×C₂₃	#458	1	2	yes	—
67	C₄×C₂₃	—	8	6	no	—
68	C₄×C₂₃	—	8	2	no	—
69	C₄×C₂₃	—	16	3	no	—
70	C₂²×C₂²⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
71	#393	—	2	6	no	—
72	#379	C₂²×C₂₃	1	3	yes	—
73	#379	C₂²×C₂₃	1	3	yes	—
74	#342	—	2	6	no	—
75	C₂×C₂₃	—	2	2	no	—
76	C₂×C₂₃	—	4	3	no	—
77	C₂×C₂₃	—	4	3	no	—
78	C₂×C₂₃	#2030	1	1	yes	characteristic
79	C₂×C₂₃	#2032	1	2	yes	—
80	C₂⁵	C₂³×C₂₃	1	1	yes	characteristic
81	C₂²×Q₈	—	2	2	no	—
82	C₂³×C₄	Q₈×C₂₃	1	2	yes	complemented
83	C₂×C₂²⋊C₄	—	2	3	no	—
84	C₂×C₂²⋊C₄	—	2	3	no	—
85	C₂×C₂²⋊C₄	—	2	3	no	—
86	C₂×C₂²⋊C₄	—	2	6	no	—
87	C₂×C₂²⋊C₄	—	2	6	no	—
88	C₂×C₂²⋊C₄	D₄×C₂₃	1	6	yes	—
89	#95	—	2	6	no	—
90	C₂₃	#10391	1	1	yes	characteristic, direct product factor, Sylow
91	C₂⁴	—	4	2	no	—
92	C₂⁴	—	8	2	no	—
93	C₂⁴	C₂×D₄×C₂₃	1	3	yes	—
94	C₂⁴	C₂×Q₈×C₂₃	1	1	yes	characteristic
95	C₂⁴	#41×C₂₃	1	3	yes	—
96	C₂×Q₈	—	8	4	no	—
97	C₂×Q₈	—	8	2	no	—
98	C₂²×C₄	—	2	6	no	—
99	C₂²×C₄	—	2	6	no	—
100	C₂²×C₄	—	4	3	no	—
101	C₂²×C₄	—	8	2	no	—
102	C₂²⋊C₄	—	4	1	no	—
103	C₂²⋊C₄	—	4	3	no	—
104	C₂²⋊C₄	—	4	6	no	—
105	C₂²⋊C₄	—	4	6	no	—
106	C₂²⋊C₄	—	8	6	no	—
107	C₂²⋊C₄	—	8	6	no	—
108	C₂³	—	2	6	no	—
109	C₂³	—	4	2	no	—
110	C₂³	—	4	1	no	—
111	C₂³	—	4	6	no	—
112	C₂³	—	4	6	no	—
113	C₂³	—	4	3	no	—
114	C₂³	—	8	8	no	—
115	C₂³	C₂₃×#120	1	1	yes	characteristic
116	C₂³	C₂₃×C₂²⋊Q₈	1	3	yes	—
117	C₂³	C₂₃×#124	1	3	yes	—
118	Q₈	—	8	8	no	—
119	C₂×C₄	—	4	12	no	—
120	C₂×C₄	—	4	2	no	—
121	C₂×C₄	—	4	6	no	—
122	C₂×C₄	—	8	6	no	—
123	C₂×C₄	—	8	3	no	—
124	C₂×C₄	—	8	6	no	—
125	C₂²	—	2	2	no	—
126	C₂²	—	2	3	no	—
127	C₂²	—	2	3	no	—
128	C₂²	—	2	6	no	—
129	C₂²	—	2	6	no	—
130	C₂²	—	4	4	no	—
131	C₂²	—	8	6	no	—
132	C₂²	—	8	6	no	—
133	C₂²	C₂₃×#393	1	1	yes	characteristic
134	C₂²	C₂₃×#458	1	2	yes	—
135	C₄	—	8	6	no	—
136	C₄	—	8	2	no	—
137	C₄	—	16	3	no	—
138	C₂	—	2	2	no	—
139	C₂	—	4	3	no	—
140	C₂	—	4	3	no	—
141	C₂	C₂₃×#2030	1	1	yes	characteristic
142	C₂	C₂₃×#2032	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂₃×#2840

C₂

yes

maximal, characteristic

C₂₃×#1819

C₂

yes

maximal

C₂₃×#1304

C₂

yes

maximal

C₂²×C₂²⋊C₄×C₂₃

C₂²

yes

characteristic

C₂₃×#393

—

C₂₃×#379

C₂²

yes

—

C₂₃×#379

C₂²

yes

—

C₂₃×#342

—

C₂⁵×C₂₃

C₂³

yes

characteristic

C₂²×Q₈×C₂₃

—

C₂³×C₄×C₂₃

Q₈

yes

complemented

C₂×C₂²⋊C₄×C₂₃

—

C₂×C₂²⋊C₄×C₂₃

—

C₂×C₂²⋊C₄×C₂₃

—

C₂×C₂²⋊C₄×C₂₃

—

C₂×C₂²⋊C₄×C₂₃

—

C₂×C₂²⋊C₄×C₂₃

D₄

yes

—

C₂₃×#95

—

C₂⁴×C₂₃

—

C₂⁴×C₂₃

—

C₂⁴×C₂₃

C₂×D₄

yes

—

C₂⁴×C₂₃

C₂×Q₈

yes

characteristic

C₂⁴×C₂₃

#41

yes

—

C₂×Q₈×C₂₃

—

C₂×Q₈×C₂₃

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

—

C₂²⋊C₄×C₂₃

—

C₂²⋊C₄×C₂₃

—

C₂²⋊C₄×C₂₃

—

C₂²⋊C₄×C₂₃

—

C₂²⋊C₄×C₂₃

—

C₂²⋊C₄×C₂₃

—

#10391

C₂₃

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₂₃

—

C₂³×C₂₃

—

C₂³×C₂₃

—

C₂³×C₂₃

—

C₂³×C₂₃

—

C₂³×C₂₃

—

C₂³×C₂₃

—

C₂³×C₂₃

#120

yes

characteristic

C₂³×C₂₃

C₂²⋊Q₈

yes

—

C₂³×C₂₃

#124

yes

—

Q₈×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

#2840

C₂×C₂₃

yes

characteristic

#1819

C₂×C₂₃

yes

—

#1304

C₂×C₂₃

yes

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

#393

yes

characteristic

C₂²×C₂₃

#458

yes

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₂²×C₂²⋊C₄

C₂²×C₂₃

yes

characteristic

#393

—

#379

C₂²×C₂₃

yes

—

#379

C₂²×C₂₃

yes

—

#342

—

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

#2030

yes

characteristic

C₂×C₂₃

#2032

yes

—

C₂⁵

C₂³×C₂₃

yes

characteristic

C₂²×Q₈

—

C₂³×C₄

Q₈×C₂₃

yes

complemented

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

D₄×C₂₃

yes

—

#95

—

C₂₃

#10391

yes

characteristic, direct product factor, Sylow

C₂⁴

—

C₂⁴

—

C₂⁴

C₂×D₄×C₂₃

yes

—

C₂⁴

C₂×Q₈×C₂₃

yes

characteristic

C₂⁴

#41×C₂₃

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

100

C₂²×C₄

—

101

C₂²×C₄

—

102

C₂²⋊C₄

—

103

C₂²⋊C₄

—

104

C₂²⋊C₄

—

105

C₂²⋊C₄

—

106

C₂²⋊C₄

—

107

C₂²⋊C₄

—

108

C₂³

—

109

C₂³

—

110

C₂³

—

111

C₂³

—

112

C₂³

—

113

C₂³

—

114

C₂³

—

115

C₂³

C₂₃×#120

yes

characteristic

116

C₂³

C₂₃×C₂²⋊Q₈

yes

—

117

C₂³

C₂₃×#124

yes

—

118

Q₈

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂²

—

126

C₂²

—

127

C₂²

—

128

C₂²

—

129

C₂²

—

130

C₂²

—

131

C₂²

—

132

C₂²

—

133

C₂²

C₂₃×#393

yes

characteristic

134

C₂²

C₂₃×#458

yes

—

135

C₄

—

136

C₄

—

137

C₄

—

138

C₂

—

139

C₂

—

140

C₂

—

141

C₂

C₂₃×#2030

yes

characteristic

142

C₂

C₂₃×#2032

yes

—

	subgroup	G/H
center, socle	C₂²×C₂₃	#393
commutator, Frattini, cosocle	C₂⁵	C₂³×C₂₃
prime radical	C₂⁵×C₂₃	C₂³
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂⁵×C₂₃ ◁ G

subgroup

G/H

center, socle

C₂²×C₂₃

#393

commutator, Frattini, cosocle

C₂⁵

C₂³×C₂₃

prime radical

C₂⁵×C₂₃

C₂³

LCS

G ▷ C₂⁵ ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₂₃ ◁ C₂⁵×C₂₃ ◁ G

N	H ≅ G/N	type
11. C₂³×C₄×C₂₃	Q₈	semidirect
36. #10391	C₂₃	direct (N × H)
82. C₂³×C₄	Q₈×C₂₃	semidirect
90. C₂₃	#10391	direct (N × H)

H ≅ G/N

type

11. C₂³×C₄×C₂₃

Q₈

semidirect

36. #10391

C₂₃

direct (N × H)

82. C₂³×C₄

Q₈×C₂₃

semidirect

90. C₂₃

#10391

direct (N × H)

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

subgroup	index
C₂₃×#2840	2
C₂₃×#1819	2
C₂₃×#1304	2
#10391	23

subgroup

index

C₂₃×#2840

C₂₃×#1819

C₂₃×#1304

#10391

Subgroups — C₂₃×#10391 #19158017

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C23×#10391 #19158017

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#10391 #19158017