Group #19157834 — subgroups

Subgroups — C₂₃×#10208 #19157834

Order 5888.

508 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 508 conjugacy classes of subgroups — too many to draw. Only its 94 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×#3214	C₂	1	3	yes	maximal, complemented
2	C₂₃×#1514	C₂	1	3	yes	maximal, complemented
3	C₂₃×#1308	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#532	C₂²	1	3	yes	—
5	C₂₃×#496	—	2	3	no	—
6	C₂₃×#489	—	2	3	no	—
7	C₂₃×#461	—	2	3	no	—
8	C₂₃×#459	—	2	3	no	—
9	C₂₃×#449	—	2	6	no	—
10	C₂₃×#432	C₂²	1	3	yes	—
11	C₂₃×#383	C₂²	1	1	yes	characteristic
12	C₂₃×#327	—	2	6	no	—
13	C₂×#41×C₂₃	—	2	3	no	—
14	C₂×SD₁₆×C₂₃	—	2	6	no	—
15	C₂×D₈×C₂₃	—	2	6	no	—
16	C₂₃×C₄⋊Q₈	D₄	1	3	yes	—
17	C₂₃×#127	D₄	1	3	yes	—
18	C₂₃×#124	—	2	3	no	—
19	C₂₃×#121	—	2	3	no	—
20	C₂₃×#121	—	2	3	no	—
21	C₄×D₄×C₂₃	—	2	3	no	—
22	C₄×D₄×C₂₃	—	2	3	no	—
23	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
24	C₂₃×C₄⋊C₈	—	2	3	no	—
25	C₂₃×#103	—	2	6	no	—
26	C₂₃×#102	—	2	6	no	—
27	C₂₃×C₂²⋊C₈	D₄	1	6	yes	—
28	C₂₃×#97	—	2	3	no	—
29	C₂₃×#95	—	2	3	no	—
30	#41×C₂₃	—	8	6	no	—
31	C₂×Q₈×C₂₃	—	2	3	no	—
32	C₂×D₄×C₂₃	—	2	3	no	—
33	C₂×D₄×C₂₃	—	2	3	no	—
34	C₂×D₄×C₂₃	—	2	3	no	—
35	C₂×D₄×C₂₃	—	4	3	no	—
36	C₂²×C₄×C₂₃	—	4	3	no	—
37	C₂²×C₄×C₂₃	—	4	1	no	—
38	C₂²×C₄×C₂₃	C₂×D₄	1	3	yes	—
39	SD₁₆×C₂₃	—	8	6	no	—
40	D₈×C₂₃	—	8	6	no	—
41	C₂×C₈×C₂₃	—	2	6	no	—
42	C₄⋊C₄×C₂₃	—	2	3	no	—
43	C₄⋊C₄×C₂₃	—	2	3	no	—
44	C₂²⋊C₄×C₂₃	—	4	3	no	—
45	C₂²⋊C₄×C₂₃	—	4	3	no	—
46	C₄²×C₂₃	C₂×D₄	1	3	yes	—
47	C₄²×C₂₃	#41	1	1	yes	characteristic
48	#10208	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
49	C₂³×C₂₃	—	4	3	no	—
50	C₂³×C₂₃	#127	1	1	yes	characteristic
51	Q₈×C₂₃	—	4	6	no	—
52	D₄×C₂₃	—	4	3	no	—
53	D₄×C₂₃	—	4	6	no	—
54	D₄×C₂₃	—	4	3	no	—
55	D₄×C₂₃	—	8	3	no	—
56	D₄×C₂₃	—	8	6	no	—
57	C₂×C₄×C₂₃	—	4	3	no	—
58	C₂×C₄×C₂₃	—	4	3	no	—
59	C₂×C₄×C₂₃	—	8	6	no	—
60	C₂×C₄×C₂₃	—	8	3	no	—
61	C₂×C₄×C₂₃	#120	1	3	yes	—
62	C₂×C₄×C₂₃	#121	1	3	yes	—
63	C₈×C₂₃	—	4	6	no	—
64	#3214	C₂×C₂₃	1	3	yes	complemented
65	#1514	C₂×C₂₃	1	3	yes	complemented
66	#1308	C₂×C₂₃	1	1	yes	characteristic, complemented
67	C₂²×C₂₃	—	2	3	no	—
68	C₂²×C₂₃	—	8	3	no	—
69	C₂²×C₂₃	—	8	3	no	—
70	C₂²×C₂₃	—	8	3	no	—
71	C₂²×C₂₃	#392	1	1	yes	characteristic
72	C₄×C₂₃	—	2	3	no	—
73	C₄×C₂₃	—	2	3	no	—
74	C₄×C₂₃	—	8	3	no	—
75	C₄×C₂₃	—	16	1	no	—
76	#532	C₂²×C₂₃	1	3	yes	—
77	#496	—	2	3	no	—
78	#489	—	2	3	no	—
79	#461	—	2	3	no	—
80	#459	—	2	3	no	—
81	#449	—	2	6	no	—
82	#432	C₂²×C₂₃	1	3	yes	—
83	#383	C₂²×C₂₃	1	1	yes	characteristic
84	#327	—	2	6	no	—
85	C₂×C₂₃	—	4	1	no	—
86	C₂×C₂₃	—	16	3	no	—
87	C₂×C₂₃	#2008	1	3	yes	—
88	C₂×#41	—	2	3	no	—
89	C₂×SD₁₆	—	2	6	no	—
90	C₂×D₈	—	2	6	no	—
91	C₄⋊Q₈	D₄×C₂₃	1	3	yes	—
92	#127	D₄×C₂₃	1	3	yes	—
93	#124	—	2	3	no	—
94	#121	—	2	3	no	—
95	#121	—	2	3	no	—
96	C₄×D₄	—	2	3	no	—
97	C₄×D₄	—	2	3	no	—
98	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
99	C₄⋊C₈	—	2	3	no	—
100	#103	—	2	6	no	—
101	#102	—	2	6	no	—
102	C₂²⋊C₈	D₄×C₂₃	1	6	yes	—
103	#97	—	2	3	no	—
104	#95	—	2	3	no	—
105	C₂₃	#10208	1	1	yes	characteristic, direct product factor, Sylow
106	#41	—	8	6	no	—
107	C₂×Q₈	—	2	3	no	—
108	C₂×D₄	—	2	3	no	—
109	C₂×D₄	—	2	3	no	—
110	C₂×D₄	—	2	3	no	—
111	C₂×D₄	—	4	3	no	—
112	C₂²×C₄	—	4	3	no	—
113	C₂²×C₄	—	4	1	no	—
114	C₂²×C₄	C₂×D₄×C₂₃	1	3	yes	—
115	SD₁₆	—	8	6	no	—
116	D₈	—	8	6	no	—
117	C₂×C₈	—	2	6	no	—
118	C₄⋊C₄	—	2	3	no	—
119	C₄⋊C₄	—	2	3	no	—
120	C₂²⋊C₄	—	4	3	no	—
121	C₂²⋊C₄	—	4	3	no	—
122	C₄²	C₂×D₄×C₂₃	1	3	yes	—
123	C₄²	#41×C₂₃	1	1	yes	characteristic
124	C₂³	—	4	3	no	—
125	C₂³	C₂₃×#127	1	1	yes	characteristic
126	Q₈	—	4	6	no	—
127	D₄	—	4	3	no	—
128	D₄	—	4	6	no	—
129	D₄	—	4	3	no	—
130	D₄	—	8	3	no	—
131	D₄	—	8	6	no	—
132	C₂×C₄	—	4	3	no	—
133	C₂×C₄	—	4	3	no	—
134	C₂×C₄	—	8	6	no	—
135	C₂×C₄	—	8	3	no	—
136	C₂×C₄	C₂₃×#120	1	3	yes	—
137	C₂×C₄	C₂₃×#121	1	3	yes	—
138	C₈	—	4	6	no	—
139	C₂²	—	2	3	no	—
140	C₂²	—	8	3	no	—
141	C₂²	—	8	3	no	—
142	C₂²	—	8	3	no	—
143	C₂²	C₂₃×#392	1	1	yes	characteristic
144	C₄	—	2	3	no	—
145	C₄	—	2	3	no	—
146	C₄	—	8	3	no	—
147	C₄	—	16	1	no	—
148	C₂	—	4	1	no	—
149	C₂	—	16	3	no	—
150	C₂	C₂₃×#2008	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂₃×#3214

C₂

yes

maximal, complemented

C₂₃×#1514

C₂

yes

maximal, complemented

C₂₃×#1308

C₂

yes

maximal, characteristic, complemented

C₂₃×#532

C₂²

yes

—

C₂₃×#496

—

C₂₃×#489

—

C₂₃×#461

—

C₂₃×#459

—

C₂₃×#449

—

C₂₃×#432

C₂²

yes

—

C₂₃×#383

C₂²

yes

characteristic

C₂₃×#327

—

C₂×#41×C₂₃

—

C₂×SD₁₆×C₂₃

—

C₂×D₈×C₂₃

—

C₂₃×C₄⋊Q₈

D₄

yes

—

C₂₃×#127

D₄

yes

—

C₂₃×#124

—

C₂₃×#121

—

C₂₃×#121

—

C₄×D₄×C₂₃

—

C₄×D₄×C₂₃

—

C₂×C₄²×C₂₃

C₂³

yes

characteristic

C₂₃×C₄⋊C₈

—

C₂₃×#103

—

C₂₃×#102

—

C₂₃×C₂²⋊C₈

D₄

yes

—

C₂₃×#97

—

C₂₃×#95

—

#41×C₂₃

—

C₂×Q₈×C₂₃

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

C₂×D₄

yes

—

SD₁₆×C₂₃

—

D₈×C₂₃

—

C₂×C₈×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₂²⋊C₄×C₂₃

—

C₂²⋊C₄×C₂₃

—

C₄²×C₂₃

C₂×D₄

yes

—

C₄²×C₂₃

#41

yes

characteristic

#10208

C₂₃

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₂₃

—

C₂³×C₂₃

#127

yes

characteristic

Q₈×C₂₃

—

D₄×C₂₃

—

D₄×C₂₃

—

D₄×C₂₃

—

D₄×C₂₃

—

D₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

#120

yes

—

C₂×C₄×C₂₃

#121

yes

—

C₈×C₂₃

—

#3214

C₂×C₂₃

yes

complemented

#1514

C₂×C₂₃

yes

complemented

#1308

C₂×C₂₃

yes

characteristic, complemented

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

#392

yes

characteristic

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

#532

C₂²×C₂₃

yes

—

—

—

—

—

—

C₂²×C₂₃

yes

—

#383

C₂²×C₂₃

yes

characteristic

#327

—

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

#2008

yes

—

C₂×#41

—

C₂×SD₁₆

—

C₂×D₈

—

C₄⋊Q₈

D₄×C₂₃

yes

—

#127

D₄×C₂₃

yes

—

#124

—

#121

—

#121

—

C₄×D₄

—

C₄×D₄

—

C₂×C₄²

C₂³×C₂₃

yes

characteristic

C₄⋊C₈

—

100

#103

—

101

#102

—

102

C₂²⋊C₈

D₄×C₂₃

yes

—

103

#97

—

104

#95

—

105

C₂₃

#10208

yes

characteristic, direct product factor, Sylow

106

#41

—

107

C₂×Q₈

—

108

C₂×D₄

—

109

C₂×D₄

—

110

C₂×D₄

—

111

C₂×D₄

—

112

C₂²×C₄

—

113

C₂²×C₄

—

114

C₂²×C₄

C₂×D₄×C₂₃

yes

—

115

SD₁₆

—

116

D₈

—

117

C₂×C₈

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

120

C₂²⋊C₄

—

121

C₂²⋊C₄

—

122

C₄²

C₂×D₄×C₂₃

yes

—

123

C₄²

#41×C₂₃

yes

characteristic

124

C₂³

—

125

C₂³

C₂₃×#127

yes

characteristic

126

Q₈

—

127

D₄

—

128

D₄

—

129

D₄

—

130

D₄

—

131

D₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

C₂₃×#120

yes

—

137

C₂×C₄

C₂₃×#121

yes

—

138

C₈

—

139

C₂²

—

140

C₂²

—

141

C₂²

—

142

C₂²

—

143

C₂²

C₂₃×#392

yes

characteristic

144

C₄

—

145

C₄

—

146

C₄

—

147

C₄

—

148

C₂

—

149

C₂

—

150

C₂

C₂₃×#2008

yes

—

	subgroup	G/H
center, socle	C₂²×C₂₃	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₂₃
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

subgroup

G/H

center, socle

C₂²×C₂₃

#392

commutator, Frattini, cosocle

C₂×C₄²

C₂³×C₂₃

LCS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

N	H ≅ G/N	type
1. C₂₃×#3214	C₂	semidirect
2. C₂₃×#1514	C₂	semidirect
3. C₂₃×#1308	C₂	semidirect
48. #10208	C₂₃	direct (N × H)
64. #3214	C₂×C₂₃	semidirect
65. #1514	C₂×C₂₃	semidirect
66. #1308	C₂×C₂₃	semidirect
105. C₂₃	#10208	direct (N × H)

H ≅ G/N

type

1. C₂₃×#3214

C₂

semidirect

2. C₂₃×#1514

C₂

semidirect

3. C₂₃×#1308

C₂

semidirect

48. #10208

C₂₃

direct (N × H)

64. #3214

C₂×C₂₃

semidirect

65. #1514

C₂×C₂₃

semidirect

66. #1308

C₂×C₂₃

semidirect

105. C₂₃

#10208

direct (N × H)

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

subgroup	index
C₂₃×#3214	2
C₂₃×#1514	2
C₂₃×#1308	2
#10208	23

subgroup

index

C₂₃×#3214

C₂₃×#1514

C₂₃×#1308

#10208

Subgroups — C₂₃×#10208 #19157834

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C23×#10208 #19157834

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#10208 #19157834