Group #15669647 — subgroups

Subgroups — C₂₃×#2028 #15669647

Order 2944.

246 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 84 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×#544	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#419	C₂	1	2	yes	maximal
3	C₂₃×#391	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#343	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#339	C₂	1	2	yes	maximal
6	C₂²×Q₈×C₂₃	C₂²	1	1	yes	characteristic
7	C₂×M₁₆×C₂₃	C₂²	1	2	yes	—
8	C₂₃×C₄⋊Q₈	—	2	2	no	—
9	C₂₃×#125	—	2	2	no	—
10	C₂₃×C₂²⋊Q₈	—	2	2	no	—
11	C₂₃×#117	C₂²	1	2	yes	—
12	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
13	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
14	C₂₃×#103	—	2	2	no	—
15	C₂₃×#103	—	2	2	no	—
16	C₂₃×#95	—	2	2	no	—
17	C₂×Q₈×C₂₃	—	2	2	no	—
18	C₂×Q₈×C₂₃	—	2	1	no	—
19	C₂×Q₈×C₂₃	—	2	2	no	—
20	C₂×Q₈×C₂₃	—	2	2	no	—
21	C₂×Q₈×C₂₃	Q₈	1	2	yes	complemented
22	C₂²×C₄×C₂₃	—	2	1	no	—
23	C₂²×C₄×C₂₃	—	2	1	no	—
24	C₂²×C₄×C₂₃	D₄	1	2	yes	—
25	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
26	M₁₆×C₂₃	—	2	2	no	—
27	M₁₆×C₂₃	—	2	2	no	—
28	C₂×C₈×C₂₃	—	2	2	no	—
29	C₄⋊C₄×C₂₃	—	2	2	no	—
30	C₄⋊C₄×C₂₃	—	2	4	no	—
31	C₄⋊C₄×C₂₃	—	2	2	no	—
32	C₄⋊C₄×C₂₃	D₄	1	2	yes	—
33	C₄⋊C₄×C₂₃	D₄	1	2	yes	—
34	C₂²⋊C₄×C₂₃	—	2	2	no	—
35	C₄²×C₂₃	—	2	2	no	—
36	C₄²×C₂₃	—	2	2	no	—
37	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
38	Q₈×C₂₃	—	2	4	no	—
39	Q₈×C₂₃	—	4	2	no	—
40	Q₈×C₂₃	—	8	2	no	—
41	C₂×C₄×C₂₃	—	2	2	no	—
42	C₂×C₄×C₂₃	—	2	2	no	—
43	C₂×C₄×C₂₃	—	2	2	no	—
44	C₂×C₄×C₂₃	—	2	2	no	—
45	C₂×C₄×C₂₃	—	2	2	no	—
46	C₂×C₄×C₂₃	—	4	1	no	—
47	C₂×C₄×C₂₃	—	4	1	no	—
48	C₂×C₄×C₂₃	—	4	1	no	—
49	C₂×C₄×C₂₃	—	4	1	no	—
50	C₂×C₄×C₂₃	—	4	2	no	—
51	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
52	C₂×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
53	C₂×C₄×C₂₃	#41	1	2	yes	—
54	C₂×C₄×C₂₃	#41	1	1	yes	characteristic
55	C₈×C₂₃	—	4	2	no	—
56	#2028	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
57	C₂²×C₂₃	—	2	1	no	—
58	C₂²×C₂₃	—	2	1	no	—
59	C₂²×C₂₃	#120	1	1	yes	characteristic
60	C₂²×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
61	C₂²×C₂₃	#124	1	1	yes	characteristic
62	C₄×C₂₃	—	4	2	no	—
63	C₄×C₂₃	—	4	2	no	—
64	C₄×C₂₃	—	4	2	no	—
65	C₄×C₂₃	—	4	2	no	—
66	C₄×C₂₃	—	4	2	no	—
67	C₄×C₂₃	C₂²⋊Q₈	1	2	yes	—
68	C₄×C₂₃	#124	1	2	yes	—
69	#544	C₂×C₂₃	1	1	yes	characteristic
70	#419	C₂×C₂₃	1	2	yes	—
71	#391	C₂×C₂₃	1	1	yes	characteristic
72	#343	C₂×C₂₃	1	1	yes	characteristic
73	#339	C₂×C₂₃	1	2	yes	—
74	C₂×C₂₃	—	2	1	no	—
75	C₂×C₂₃	—	2	1	no	—
76	C₂×C₂₃	#393	1	1	yes	characteristic
77	C₂×C₂₃	#455	1	1	yes	characteristic
78	C₂×C₂₃	#456	1	1	yes	characteristic
79	C₂²×Q₈	C₂²×C₂₃	1	1	yes	characteristic
80	C₂×M₁₆	C₂²×C₂₃	1	2	yes	—
81	C₄⋊Q₈	—	2	2	no	—
82	#125	—	2	2	no	—
83	C₂²⋊Q₈	—	2	2	no	—
84	#117	C₂²×C₂₃	1	2	yes	—
85	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
86	C₂×C₄²	C₂²×C₂₃	1	1	yes	characteristic
87	#103	—	2	2	no	—
88	#103	—	2	2	no	—
89	#95	—	2	2	no	—
90	C₂₃	#2028	1	1	yes	characteristic, direct product factor, Sylow
91	C₂×Q₈	—	2	2	no	—
92	C₂×Q₈	—	2	1	no	—
93	C₂×Q₈	—	2	2	no	—
94	C₂×Q₈	—	2	2	no	—
95	C₂×Q₈	Q₈×C₂₃	1	2	yes	complemented
96	C₂²×C₄	—	2	1	no	—
97	C₂²×C₄	—	2	1	no	—
98	C₂²×C₄	D₄×C₂₃	1	2	yes	—
99	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
100	M₁₆	—	2	2	no	—
101	M₁₆	—	2	2	no	—
102	C₂×C₈	—	2	2	no	—
103	C₄⋊C₄	—	2	2	no	—
104	C₄⋊C₄	—	2	4	no	—
105	C₄⋊C₄	—	2	2	no	—
106	C₄⋊C₄	D₄×C₂₃	1	2	yes	—
107	C₄⋊C₄	D₄×C₂₃	1	2	yes	—
108	C₂²⋊C₄	—	2	2	no	—
109	C₄²	—	2	2	no	—
110	C₄²	—	2	2	no	—
111	C₂³	C₂×D₄×C₂₃	1	1	yes	characteristic
112	Q₈	—	2	4	no	—
113	Q₈	—	4	2	no	—
114	Q₈	—	8	2	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	2	2	no	—
119	C₂×C₄	—	2	2	no	—
120	C₂×C₄	—	4	1	no	—
121	C₂×C₄	—	4	1	no	—
122	C₂×C₄	—	4	1	no	—
123	C₂×C₄	—	4	1	no	—
124	C₂×C₄	—	4	2	no	—
125	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
126	C₂×C₄	C₂×Q₈×C₂₃	1	1	yes	characteristic
127	C₂×C₄	#41×C₂₃	1	2	yes	—
128	C₂×C₄	#41×C₂₃	1	1	yes	characteristic
129	C₈	—	4	2	no	—
130	C₂²	—	2	1	no	—
131	C₂²	—	2	1	no	—
132	C₂²	C₂₃×#120	1	1	yes	characteristic
133	C₂²	C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
134	C₂²	C₂₃×#124	1	1	yes	characteristic
135	C₄	—	4	2	no	—
136	C₄	—	4	2	no	—
137	C₄	—	4	2	no	—
138	C₄	—	4	2	no	—
139	C₄	—	4	2	no	—
140	C₄	C₂₃×C₂²⋊Q₈	1	2	yes	—
141	C₄	C₂₃×#124	1	2	yes	—
142	C₂	—	2	1	no	—
143	C₂	—	2	1	no	—
144	C₂	C₂₃×#393	1	1	yes	characteristic
145	C₂	C₂₃×#455	1	1	yes	characteristic
146	C₂	C₂₃×#456	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂₃×#544

C₂

yes

maximal, characteristic

C₂₃×#419

C₂

yes

maximal

C₂₃×#391

C₂

yes

maximal, characteristic

C₂₃×#343

C₂

yes

maximal, characteristic

C₂₃×#339

C₂

yes

maximal

C₂²×Q₈×C₂₃

C₂²

yes

characteristic

C₂×M₁₆×C₂₃

C₂²

yes

—

C₂₃×C₄⋊Q₈

—

C₂₃×#125

—

C₂₃×C₂²⋊Q₈

—

C₂₃×#117

C₂²

yes

—

C₂×C₄⋊C₄×C₂₃

C₂²

yes

characteristic

C₂×C₄²×C₂₃

C₂²

yes

characteristic

C₂₃×#103

—

C₂₃×#103

—

C₂₃×#95

—

C₂×Q₈×C₂₃

—

C₂×Q₈×C₂₃

—

C₂×Q₈×C₂₃

—

C₂×Q₈×C₂₃

—

C₂×Q₈×C₂₃

Q₈

yes

complemented

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

D₄

yes

—

C₂²×C₄×C₂₃

C₂³

yes

characteristic

M₁₆×C₂₃

—

M₁₆×C₂₃

—

C₂×C₈×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

D₄

yes

—

C₄⋊C₄×C₂₃

D₄

yes

—

C₂²⋊C₄×C₂₃

—

C₄²×C₂₃

—

C₄²×C₂₃

—

C₂³×C₂₃

C₂×D₄

yes

characteristic

Q₈×C₂₃

—

Q₈×C₂₃

—

Q₈×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂×D₄

yes

—

C₂×C₄×C₂₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₃

#41

yes

—

C₂×C₄×C₂₃

#41

yes

characteristic

C₈×C₂₃

—

#2028

C₂₃

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

#120

yes

characteristic

C₂²×C₂₃

C₂²⋊Q₈

yes

characteristic

C₂²×C₂₃

#124

yes

characteristic

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

C₂²⋊Q₈

yes

—

C₄×C₂₃

#124

yes

—

#544

C₂×C₂₃

yes

characteristic

#419

C₂×C₂₃

yes

—

#391

C₂×C₂₃

yes

characteristic

#343

C₂×C₂₃

yes

characteristic

#339

C₂×C₂₃

yes

—

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂²×Q₈

C₂²×C₂₃

yes

characteristic

C₂×M₁₆

C₂²×C₂₃

yes

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

#117

C₂²×C₂₃

yes

—

C₂×C₄⋊C₄

C₂²×C₂₃

yes

characteristic

C₂×C₄²

C₂²×C₂₃

yes

—

—

—

C₂₃

#2028

yes

characteristic, direct product factor, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

Q₈×C₂₃

yes

complemented

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

D₄×C₂₃

yes

—

C₂²×C₄

C₂³×C₂₃

yes

characteristic

100

M₁₆

—

101

M₁₆

—

102

C₂×C₈

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

D₄×C₂₃

yes

—

107

C₄⋊C₄

D₄×C₂₃

yes

—

108

C₂²⋊C₄

—

109

C₄²

—

110

C₄²

—

111

C₂³

C₂×D₄×C₂₃

yes

characteristic

112

Q₈

—

113

Q₈

—

114

Q₈

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

C₂×D₄×C₂₃

yes

—

126

C₂×C₄

C₂×Q₈×C₂₃

yes

characteristic

127

C₂×C₄

#41×C₂₃

yes

—

128

C₂×C₄

#41×C₂₃

yes

characteristic

129

C₈

—

130

C₂²

—

131

C₂²

—

132

C₂²

C₂₃×#120

yes

characteristic

133

C₂²

C₂₃×C₂²⋊Q₈

yes

characteristic

134

C₂²

C₂₃×#124

yes

characteristic

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

C₂₃×C₂²⋊Q₈

yes

—

141

C₄

C₂₃×#124

yes

—

142

C₂

—

143

C₂

—

144

C₂

C₂₃×#393

yes

characteristic

145

C₂

C₂₃×#455

yes

characteristic

146

C₂

C₂₃×#456

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	#120
commutator, Frattini, cosocle	C₂²×C₄	C₂³×C₂₃
prime radical	C₂³×C₂₃	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂²×C₄×C₂₃ ◁ G

subgroup

G/H

center, socle

C₂²×C₂₃

#120

commutator, Frattini, cosocle

C₂²×C₄

C₂³×C₂₃

prime radical

C₂³×C₂₃

C₂×D₄

LCS

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₂₃ ◁ C₂²×C₄×C₂₃ ◁ G

N	H ≅ G/N	type
21. C₂×Q₈×C₂₃	Q₈	semidirect
56. #2028	C₂₃	direct (N × H)
90. C₂₃	#2028	direct (N × H)
95. C₂×Q₈	Q₈×C₂₃	semidirect

H ≅ G/N

type

21. C₂×Q₈×C₂₃

Q₈

semidirect

56. #2028

C₂₃

direct (N × H)

90. C₂₃

#2028

direct (N × H)

95. C₂×Q₈

Q₈×C₂₃

semidirect

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

subgroup	index
C₂₃×#544	2
C₂₃×#419	2
C₂₃×#391	2
C₂₃×#343	2
C₂₃×#339	2
#2028	23

subgroup

index

C₂₃×#544

C₂₃×#419

C₂₃×#391

C₂₃×#343

C₂₃×#339

#2028

Subgroups — C₂₃×#2028 #15669647

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C23×#2028 #15669647

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#2028 #15669647