Finite Group Databasebeta

← C23×#6715 #14509641

Subgroups — C23×#6715 #14509641

Order 1920.

3952 conjugacy classes of subgroups; only the 983 normal subgroups are shown.

normal subgroups only

This group has 3952 conjugacy classes of subgroups — too many to draw. Only its 983 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C3×C5×D4C211yesmaximal, characteristic, complemented
2C23×D60C211yesmaximal, characteristic, complemented
3C22×#6715C2128yesmaximal, direct product factor
4C23×C3⋊#168C211yesmaximal, characteristic, complemented
5C23×#686—31nomaximal, Hall
6C25×C3×C5—21no—
7C24×D15—21no—
8C22×C3×C5×D4—27no—
9C22×C3×C5×D4C22114yescomplemented
10C23×C3×C4×C5C2211yescharacteristic
11C22×D60—27no—
12C22×D60C22114yescomplemented
13C2×#6715C221112yesdirect product factor
14C22×C3⋊#168C22114yescomplemented
15C23×#213—51nomaximal, Hall
16C23×C5×D4D311yescharacteristic, complemented
17C23×D20—31no—
18C22×#686—328no—
19C23×#168—31no—
20C24×C3×C5—214no—
21C24×C3×C5—48no—
22C24×C3×C5D411yescharacteristic
23C23×D15—214no—
24C23×D15—48no—
25C2×C3×C5×D4—242no—
26C2×C3×C5×D4C23128yescomplemented
27C22×C3×C4×C5D417yes—
28C22×C3×C4×C5C2317yes—
29C2×D60—242no—
30C2×D60C23128yescomplemented
31#6715C23164yesdirect product factor
32C2×C3⋊#168C23128yescomplemented
33C23×C3×D4D511yescharacteristic, complemented
34C23×D12—51no—
35C22×#213—528no—
36C23×C3⋊C8—51no—
37C25×C5—21no—
38C24×D5—61no—
39C22×C5×D4—27no—
40C22×C5×D4C2×D3114yescomplemented
41C23×C4×C5C2×D311yescharacteristic
42C22×D20—314no—
43C22×D20—67no—
44C2×#686—3112no—
45C22×#168—314no—
46C23×D8—151noSylow
47C23×C3×C5—228no—
48C23×C3×C5—456no—
49C23×C3×C5D818yescomplemented
50C23×C3×C5C2×D417yes—
51C22×D15—228no—
52C22×D15—456no—
53C3×C5×D4—228no—
54C3×C5×D4C2418yescomplemented
55C2×C3×C4×C5C2×D4121yes—
56C2×C3×C4×C5C2417yes—
57D60—228no—
58D60C2418yescomplemented
59C3⋊#168C2418yescomplemented
60C25×C3—21no—
61C24×D3—101no—
62C22×C3×D4—27no—
63C22×C3×D4C2×D5114yescomplemented
64C23×C3×C4C2×D511yescharacteristic
65C22×D12—514no—
66C22×D12—107no—
67C2×#213—5112no—
68C22×C3⋊C8—514no—
69C24×C5—214no—
70C24×C5—48no—
71C24×C5#6711yescharacteristic
72C23×D5—614no—
73C23×D5—128no—
74C2×C5×D4—242no—
75C2×C5×D4C22×D3128yescomplemented
76C22×C4×C5#6717yes—
77C22×C4×C5C22×D317yes—
78C2×D20—328no—
79C2×D20—642no—
80#686—364no—
81C2×#168—328no—
82C23×D4—151no—
83C23×D4D1511yescharacteristic, complemented
84C22×D8—1528no—
85C23×C8—151no—
86C22×C3×C5—28no—
87C22×C3×C5—456no—
88C22×C3×C5C2×D8128yescomplemented
89C22×C3×C5C22×D417yes—
90C2×D15—28no—
91C2×D15—456no—
92C3×C4×C5C22×D417yes—
93C3×C4×C5C2511yescharacteristic
94C24×C3—214no—
95C24×C3—48no—
96C24×C3C22⋊D511yescharacteristic
97C23×D3—1014no—
98C23×D3—208no—
99C2×C3×D4—242no—
100C2×C3×D4C22×D5128yescomplemented
101C22×C3×C4C22⋊D517yes—
102C22×C3×C4C22×D517yes—
103C2×D12—528no—
104C2×D12—1042no—
105#213—564no—
106C2×C3⋊C8—528no—
107C23×C5—228no—
108C23×C5—456no—
109C23×C5#21318yescomplemented
110C23×C5C2×#6717yes—
111C22×D5—628no—
112C22×D5—1256no—
113C5×D4—228no—
114C5×D4C23×D318yescomplemented
115C2×C4×C5C2×#67121yes—
116C2×C4×C5C23×D317yes—
117D20—38no—
118D20—628no—
119#168—38no—
120C25—21no—
121C25—301no—
122C22×D4—27no—
123C22×D4—1514no—
124C22×D4—307no—
125C22×D4C2×D15114yescomplemented
126C23×C4C2×D1511yescharacteristic
127C2×D8—15112no—
128C22×C8—1514no—
129C2×C3×C5—48no—
130C2×C3×C5C22×D8114yescomplemented
131C2×C3×C5C23×D411yescharacteristic
132D15—48no—
133C23×C3—228no—
134C23×C3—456no—
135C23×C3#68618yescomplemented
136C23×C3C2×C22⋊D517yes—
137C22×D3—1028no—
138C22×D3—2056no—
139C3×D4—228no—
140C3×D4C23×D518yescomplemented
141C2×C3×C4C2×C22⋊D5121yes—
142C2×C3×C4C23×D517yes—
143D12—58no—
144D12—1028no—
145C3⋊C8—58no—
146C22×C5—28no—
147C22×C5—456no—
148C22×C5C2×#213128yescomplemented
149C22×C5C22×#6717yes—
150C2×D5—68no—
151C2×D5—1256no—
152C4×C5C22×#6717yes—
153C4×C5C24×D311yescharacteristic
154C24—214no—
155C24—48no—
156C24—3014no—
157C24—608no—
158C24#122011yescharacteristic
159C2×D4—242no—
160C2×D4—1528no—
161C2×D4—3042no—
162C2×D4C22×D15128yescomplemented
163C22×C4#122017yes—
164C22×C4C22×D1517yes—
165D8—1564no—
166C2×C8—1528no—
167C3×C5C23×D811yescharacteristic, complemented, Hall
168C22×C3—28no—
169C22×C3—456no—
170C22×C3C2×#686128yescomplemented
171C22×C3C22×C22⋊D517yes—
172C2×D3—108no—
173C2×D3—2056no—
174C3×C4C22×C22⋊D517yes—
175C3×C4C24×D511yescharacteristic
176C2×C5—48no—
177C2×C5C22×#213114yescomplemented
178C2×C5C23×#6711yescharacteristic
179D5—128no—
180C23—228no—
181C23—456no—
182C23—3028no—
183C23—6056no—
184C23#671518yesdirect product factor
185C23C2×#122017yes—
186D4—228no—
187D4—158no—
188D4—3028no—
189D4C23×D1518yescomplemented
190C2×C4C2×#1220121yes—
191C2×C4C23×D1517yes—
192C8—158no—
193C2×C3—48no—
194C2×C3C22×#686114yescomplemented
195C2×C3C23×C22⋊D511yescharacteristic
196D3—208no—
197C5C23×#21311yescharacteristic, complemented, Sylow
198C22—28no—
199C22—456no—
200C22—308no—
201C22—6056no—
202C22C2×#6715128yesdirect product factor
203C22C22×#122017yes—
204C4C22×#122017yes—
205C4C24×D1511yescharacteristic
206C3C23×#68611yescharacteristic, complemented, Sylow
207C2—48no—
208C2—608no—
209C2C22×#6715114yesdirect product factor
210C2C23×#122011yescharacteristic
subgroupG/H
centerC24#1220
commutator, cosocleC3×C4×C5C25
FrattiniC4C24×D15
FittingC23×C3×C5×D4C2
socleC24×C3×C5D4
hypercenterC23×D4D15
hypocenterC3×C5C23×D8
LCSG ▷ C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C24 ◁ C23×C4 ◁ C23×D4

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C23×C3×C5×D4C2semidirect
2. C23×D60C2semidirect
3. C22×#6715C2direct (N × H)
4. C23×C3⋊#168C2semidirect
9. C22×C3×C5×D4C22semidirect
12. C22×D60C22semidirect
13. C2×#6715C22direct (N × H)
14. C22×C3⋊#168C22semidirect
16. C23×C5×D4D3semidirect
26. C2×C3×C5×D4C23semidirect
30. C2×D60C23semidirect
31. #6715C23direct (N × H)
32. C2×C3⋊#168C23semidirect
33. C23×C3×D4D5semidirect
40. C22×C5×D4C2×D3semidirect
49. C23×C3×C5D8semidirect
54. C3×C5×D4C24semidirect
58. D60C24semidirect
59. C3⋊#168C24semidirect
63. C22×C3×D4C2×D5semidirect
75. C2×C5×D4C22×D3semidirect
83. C23×D4D15semidirect
88. C22×C3×C5C2×D8semidirect
100. C2×C3×D4C22×D5semidirect
109. C23×C5#213semidirect
114. C5×D4C23×D3semidirect
125. C22×D4C2×D15semidirect
130. C2×C3×C5C22×D8semidirect
135. C23×C3#686semidirect
140. C3×D4C23×D5semidirect
148. C22×C5C2×#213semidirect
162. C2×D4C22×D15semidirect
167. C3×C5C23×D8semidirect
170. C22×C3C2×#686semidirect
177. C2×C5C22×#213semidirect
184. C23#6715direct (N × H)
189. D4C23×D15semidirect
194. C2×C3C22×#686semidirect
197. C5C23×#213semidirect
202. C22C2×#6715direct (N × H)
206. C3C23×#686semidirect
209. C2C22×#6715direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C3×C5×D42
C23×D602
C22×#67152
C23×C3⋊#1682
C23×#6863
C23×#2135