Finite Group Databasebeta

← C2×C3×#65354 #14439904

Subgroups — C2×C3×#65354 #14439904

Order 1920.

768 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C5×#125C211yesmaximal, characteristic
2C2×C3×#4015C211yesmaximal, characteristic
3C3×#65354C218yesmaximal, direct product factor
4C2×C3×#3940C212yesmaximal
5C2×C3×#3939C212yesmaximal
6C2×C3×#3935C211yesmaximal, characteristic
7C2×#65354C311yesmaximal, characteristic, direct product factor, Hall
8C22×C3×Dic10—21no—
9C3×C5×#125—22no—
10C3×C5×#125C2214yes—
11C2×C3×C5×C4⋊C4—21no—
12C2×C3×C5×C4⋊C4—21no—
13C2×C3×C5×C4⋊C4C2211yescharacteristic
14C2×C3×C5×C4⋊C4C2211yescharacteristic
15C2×C3×C42×C5C2211yescharacteristic
16C2×C3×C4⋊Dic5—21no—
17C2×C3×C4⋊Dic5C2212yes—
18C22×C3×#168C2212yes—
19C3×#4015—22no—
20C3×#4015C2214yes—
21C3×#3940C2218yes—
22C3×#3939C2218yes—
23C3×#3935C2214yes—
24C2×C3×#501—51nomaximal, Hall
25C2×C5×#125C2×C311yescharacteristic
26C2×#4015C2×C311yescharacteristic
27#65354C2×C318yesdirect product factor
28C2×#3940C2×C312yes—
29C2×#3939C2×C312yes—
30C2×#3935C2×C311yescharacteristic
31C22×C3×C4×C5—21no—
32C22×C3×C4×C5—21no—
33C22×C3×C4×C5D411yescharacteristic
34C22×C3×C4×C5D411yescharacteristic
35C22×C3×C4×C5C2311yescharacteristic
36C22×C3×Dic5—22no—
37C2×C3×Dic10—24no—
38C2×C3×Dic10—44no—
39C3×C5×C4⋊C4—24no—
40C3×C5×C4⋊C4—24no—
41C3×C5×C4⋊C4—21no—
42C3×C5×C4⋊C4—21no—
43C3×C5×C4⋊C4C2312yes—
44C3×C5×C4⋊C4C2312yes—
45C3×C42×C5D412yes—
46C3×C42×C5C2312yes—
47C3×C4⋊Dic5—22no—
48C3×C4⋊Dic5—24no—
49C3×C4⋊Dic5C2314yes—
50C2×C3×#168Q818yescomplemented
51C2×C3×#168C2314yes—
52C2×C3×C4⋊Q8—51no—
53C2×C3×#125D511yescharacteristic
54C3×#501—58no—
55C2×C3×#107—52no—
56C2×C3×#106—52no—
57C2×C3×#97—51no—
58C22×Dic10—21no—
59C5×#125—22no—
60C5×#125C22×C314yes—
61C2×C5×C4⋊C4—21no—
62C2×C5×C4⋊C4—21no—
63C2×C5×C4⋊C4C22×C311yescharacteristic
64C2×C5×C4⋊C4C22×C311yescharacteristic
65C2×C42×C5C22×C311yescharacteristic
66C2×C4⋊Dic5—21no—
67C2×C4⋊Dic5C22×C312yes—
68C22×#168C22×C312yes—
69#4015—22no—
70#4015C22×C314yes—
71#3940C22×C318yes—
72#3939C22×C318yes—
73#3935C22×C314yes—
74C2×#501—51noSylow
75C23×C3×C5C2×D411yescharacteristic
76C2×C3×C4×C5—22no—
77C2×C3×C4×C5—22no—
78C2×C3×C4×C5—22no—
79C2×C3×C4×C5—22no—
80C2×C3×C4×C5—42no—
81C2×C3×C4×C5—42no—
82C2×C3×C4×C5C2×D412yes—
83C2×C3×C4×C5C2×D412yes—
84C2×C3×C4×C5C2×D411yescharacteristic
85C2×C3×C4×C5C2×Q814yes—
86C2×C3×C4×C5C2411yescharacteristic
87C2×C3×Dic5—24no—
88C2×C3×Dic5—44no—
89C3×Dic10—48no—
90C3×#168C2×Q818yescomplemented
91C22×C3×Q8—101no—
92C22×C3×C8—52no—
93C3×C4⋊Q8—54no—
94C3×C4⋊Q8—102no—
95C3×#125—22no—
96C3×#125C2×D514yes—
97C2×C3×C4⋊C4—21no—
98C2×C3×C4⋊C4—21no—
99C2×C3×C4⋊C4—52no—
100C2×C3×C4⋊C4—101no—
101C2×C3×C4⋊C4C2×D511yescharacteristic
102C2×C3×C4⋊C4C2×D511yescharacteristic
103C2×C3×C42C2×D511yescharacteristic
104C3×#107—58no—
105C3×#106—58no—
106C3×#97—54no—
107C22×C4×C5—21no—
108C22×C4×C5—21no—
109C22×C4×C5C3×D411yescharacteristic
110C22×C4×C5C3×D411yescharacteristic
111C22×C4×C5C23×C311yescharacteristic
112C22×Dic5—22no—
113C2×Dic10—24no—
114C2×Dic10—44no—
115C5×C4⋊C4—24no—
116C5×C4⋊C4—24no—
117C5×C4⋊C4—21no—
118C5×C4⋊C4—21no—
119C5×C4⋊C4C23×C312yes—
120C5×C4⋊C4C23×C312yes—
121C42×C5C3×D412yes—
122C42×C5C23×C312yes—
123C4⋊Dic5—22no—
124C4⋊Dic5—24no—
125C4⋊Dic5C23×C314yes—
126C2×#168C3×Q818yescomplemented
127C2×#168C23×C314yes—
128C2×C4⋊Q8—51no—
129C2×#125C3×D511yescharacteristic
130#501—58no—
131C2×#107—52no—
132C2×#106—52no—
133C2×#97—51no—
134C22×C3×C5C4⋊Q812yes—
135C22×C3×C5AGL(1,ℤ/8)12yes—
136C22×C3×C5#13712yes—
137C22×C3×C5C22×D411yescharacteristic
138C3×C4×C5—22no—
139C3×C4×C5—22no—
140C3×C4×C5—42no—
141C3×C4×C5—42no—
142C3×C4×C5C4⋊Q812yes—
143C3×C4×C5C22×Q812yes—
144C3×Dic5—44no—
145C2×C3×Q8—104no—
146C2×C3×Q8—204no—
147C22×C3×C4—21no—
148C22×C3×C4—21no—
149C22×C3×C4—102no—
150C22×C3×C4C22⋊D511yescharacteristic
151C22×C3×C4C22⋊D511yescharacteristic
152C22×C3×C4C22×D511yescharacteristic
153C2×C3×C8—58no—
154C2×C3×C8—54no—
155C3×C4⋊C4—24no—
156C3×C4⋊C4—24no—
157C3×C4⋊C4—21no—
158C3×C4⋊C4—21no—
159C3×C4⋊C4—54no—
160C3×C4⋊C4—102no—
161C3×C4⋊C4—104no—
162C3×C4⋊C4C22×D512yes—
163C3×C4⋊C4C22×D512yes—
164C3×C42C22⋊D512yes—
165C3×C42C22×D512yes—
166C23×C5C2×C3×D411yescharacteristic
167C2×C4×C5—22no—
168C2×C4×C5—22no—
169C2×C4×C5—22no—
170C2×C4×C5—22no—
171C2×C4×C5—42no—
172C2×C4×C5—42no—
173C2×C4×C5C2×C3×D412yes—
174C2×C4×C5C2×C3×D412yes—
175C2×C4×C5C2×C3×D411yescharacteristic
176C2×C4×C5C2×C3×Q814yes—
177C2×C4×C5C24×C311yescharacteristic
178C2×Dic5—24no—
179C2×Dic5—44no—
180Dic10—48no—
181#168C2×C3×Q818yescomplemented
182C22×Q8—101no—
183C22×C8—52no—
184C4⋊Q8—54no—
185C4⋊Q8—102no—
186#125—22no—
187#125C2×C3×D514yes—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—52no—
191C2×C4⋊C4—101no—
192C2×C4⋊C4C2×C3×D511yescharacteristic
193C2×C4⋊C4C2×C3×D511yescharacteristic
194C2×C42C2×C3×D511yescharacteristic
195#107—58no—
196#106—58no—
197#97—54no—
198C2×C3×C5#50114yescomplemented
199C2×C3×C5C2×C4⋊Q811yescharacteristic
200C2×C3×C5C2×AGL(1,ℤ/8)11yescharacteristic
201C2×C3×C5C2×#13711yescharacteristic
202C23×C3C2×C22⋊D511yescharacteristic
203C3×Q8—208no—
204C2×C3×C4—22no—
205C2×C3×C4—22no—
206C2×C3×C4—22no—
207C2×C3×C4—22no—
208C2×C3×C4—42no—
209C2×C3×C4—42no—
210C2×C3×C4—104no—
211C2×C3×C4—204no—
212C2×C3×C4Q8×D514yes—
213C2×C3×C4C2×C22⋊D512yes—
214C2×C3×C4C2×C22⋊D512yes—
215C2×C3×C4C2×C22⋊D511yescharacteristic
216C2×C3×C4C23×D511yescharacteristic
217C3×C8—58no—
218C22×C5C3×C4⋊Q812yes—
219C22×C5C3×AGL(1,ℤ/8)12yes—
220C22×C5C3×#13712yes—
221C22×C5C22×C3×D411yescharacteristic
222C4×C5—22no—
223C4×C5—22no—
224C4×C5—42no—
225C4×C5—42no—
226C4×C5C3×C4⋊Q812yes—
227C4×C5C22×C3×Q812yes—
228Dic5—44no—
229C2×Q8—104no—
230C2×Q8—204no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—102no—
234C22×C4C3×C22⋊D511yescharacteristic
235C22×C4C3×C22⋊D511yescharacteristic
236C22×C4C22×C3×D511yescharacteristic
237C2×C8—58no—
238C2×C8—54no—
239C4⋊C4—24no—
240C4⋊C4—24no—
241C4⋊C4—21no—
242C4⋊C4—21no—
243C4⋊C4—54no—
244C4⋊C4—102no—
245C4⋊C4—104no—
246C4⋊C4C22×C3×D512yes—
247C4⋊C4C22×C3×D512yes—
248C42C3×C22⋊D512yes—
249C42C22×C3×D512yes—
250C3×C5C2×#50111yescharacteristic, complemented, Hall
251C22×C3#409012yes—
252C22×C3#409512yes—
253C22×C3#409712yes—
254C22×C3C22×C22⋊D511yescharacteristic
255C3×C4—22no—
256C3×C4—22no—
257C3×C4—42no—
258C3×C4—42no—
259C3×C4—204no—
260C3×C4#409012yes—
261C3×C4C2×Q8×D512yes—
262C2×C5C3×#50114yescomplemented
263C2×C5C2×C3×C4⋊Q811yescharacteristic
264C2×C5C2×C3×AGL(1,ℤ/8)11yescharacteristic
265C2×C5C2×C3×#13711yescharacteristic
266C23C2×C3×C22⋊D511yescharacteristic
267Q8—208no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—104no—
275C2×C4—204no—
276C2×C4C3×Q8×D514yes—
277C2×C4C2×C3×C22⋊D512yes—
278C2×C4C2×C3×C22⋊D512yes—
279C2×C4C2×C3×C22⋊D511yescharacteristic
280C2×C4C23×C3×D511yescharacteristic
281C8—58no—
282C2×C3#6535414yesdirect product factor
283C2×C3C2×#409011yescharacteristic
284C2×C3C2×#409511yescharacteristic
285C2×C3C2×#409711yescharacteristic
286C5C2×C3×#50111yescharacteristic, complemented, Sylow
287C22C3×#409012yes—
288C22C3×#409512yes—
289C22C3×#409712yes—
290C22C22×C3×C22⋊D511yescharacteristic
291C4—22no—
292C4—22no—
293C4—42no—
294C4—42no—
295C4—204no—
296C4C3×#409012yes—
297C4C2×C3×Q8×D512yes—
298C3C2×#6535411yescharacteristic, direct product factor, Sylow
299C2C3×#6535414yesdirect product factor
300C2C2×C3×#409011yescharacteristic
301C2C2×C3×#409511yescharacteristic
302C2C2×C3×#409711yescharacteristic
subgroupG/H
centerC23×C3C2×C22⋊D5
commutator, cosocleC2×C4×C5C24×C3
FrattiniC2×C4C23×C3×D5
FittingC2×C3×C5×#125C2
socle, prime radicalC23×C3×C5C2×D4
hypercenterC2×C3×#125D5
hypocenterC5C2×C3×#501
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23×C3 ◁ C2×C3×C42 ◁ C2×C3×#125

Complements

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

NH ≅ G/Ntype
3. C3×#65354C2direct (N × H)
7. C2×#65354C3direct (N × H)
27. #65354C2×C3direct (N × H)
50. C2×C3×#168Q8semidirect
90. C3×#168C2×Q8semidirect
126. C2×#168C3×Q8semidirect
181. #168C2×C3×Q8semidirect
198. C2×C3×C5#501semidirect
250. C3×C5C2×#501semidirect
262. C2×C5C3×#501semidirect
282. C2×C3#65354direct (N × H)
286. C5C2×C3×#501semidirect
298. C3C2×#65354direct (N × H)
299. C2C3×#65354direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C5×#1252
C2×C3×#40152
C3×#653542
C2×C3×#39402
C2×C3×#39392
C2×C3×#39352
C2×#653543
C2×C3×#5015