Finite Group Databasebeta

← C2×C3×#64915 #14280745

Subgroups — C2×C3×#64915 #14280745

Order 1920.

912 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#685C211yesmaximal, characteristic
2C2×C3×C22⋊#170C212yesmaximal, complemented
3C3×#64915C214yesmaximal, direct product factor
4C2×#64915C311yesmaximal, characteristic, direct product factor, Hall
5C24×C3×D5C411yescharacteristic
6C23×C3×C4×C5C411yescharacteristic
7C23×C3×Dic5C2211yescharacteristic
8C22×C3×#170—22no—
9C2×C3×#685—22no—
10C2×C3×#685—22no—
11C2×C3×#685C412yes—
12C2×C3×#685C2212yes—
13C3×C22⋊#170—22no—
14C3×C22⋊#170C2214yescomplemented
15C2×C3×#323—51nomaximal, Hall
16C22×#685C2×C311yescharacteristic
17C2×C22⋊#170C2×C312yescomplemented
18#64915C2×C314yesdirect product factor
19C24×C3×C5C2×C411yescharacteristic
20C23×C3×D5—21no—
21C23×C3×D5—21no—
22C23×C3×D5—22no—
23C23×C3×D5—42no—
24C23×C3×D5—42no—
25C23×C3×D5C814yescomplemented
26C23×C3×D5C2×C412yes—
27C22×C3×C4×C5—21no—
28C22×C3×C4×C5—21no—
29C22×C3×C4×C5—42no—
30C22×C3×C4×C5C2×C412yes—
31C22×C3×Dic5—42no—
32C22×C3×Dic5C2×C411yescharacteristic
33C22×C3×Dic5D412yes—
34C22×C3×Dic5D412yes—
35C22×C3×Dic5C2311yescharacteristic
36C2×C3×#170—24no—
37C2×C3×#170—44no—
38C3×#685—24no—
39C3×#685—24no—
40C22×C3×C22⋊C4—51no—
41C2×C3×C22⋊C8—52no—
42C3×#323—54no—
43C24×D5C3×C411yescharacteristic
44C23×C4×C5C3×C411yescharacteristic
45C23×Dic5C22×C311yescharacteristic
46C22×#170—22no—
47C2×#685—22no—
48C2×#685—22no—
49C2×#685C3×C412yes—
50C2×#685C22×C312yes—
51C22⋊#170—22no—
52C22⋊#170C22×C314yescomplemented
53C2×#323—51noSylow
54C23×C3×C5—22no—
55C23×C3×C5—22no—
56C23×C3×C5C22⋊C411yescharacteristic
57C23×C3×C5C22⋊C411yescharacteristic
58C23×C3×C5C2×C812yes—
59C23×C3×C5M1612yes—
60C23×C3×C5C22×C411yescharacteristic
61C22×C3×D5—24no—
62C22×C3×D5—24no—
63C22×C3×D5—22no—
64C22×C3×D5—22no—
65C22×C3×D5—48no—
66C22×C3×D5—42no—
67C22×C3×D5—42no—
68C22×C3×D5—42no—
69C22×C3×D5—42no—
70C22×C3×D5—48no—
71C22×C3×D5—42no—
72C22×C3×D5—42no—
73C22×C3×D5C2×C814yescomplemented
74C2×C3×C4×C5—22no—
75C2×C3×C4×C5—42no—
76C2×C3×C4×C5—42no—
77C2×C3×C4×C5—42no—
78C2×C3×Dic5—24no—
79C2×C3×Dic5—41no—
80C2×C3×Dic5—42no—
81C2×C3×Dic5—41no—
82C2×C3×Dic5C22⋊C412yes—
83C2×C3×Dic5C2×D412yes—
84C3×#170—44no—
85C25×C3—51no—
86C23×C3×C4—51no—
87C23×C3×C4AGL(1,𝔽5)11yescharacteristic
88C22×C3×C8—102no—
89C2×C3×C22⋊C4—52no—
90C2×C3×C22⋊C4—52no—
91C2×C3×C22⋊C4—102no—
92C2×C3×C22⋊C4—102no—
93C3×C22⋊C8—54no—
94C3×C22⋊C8—102no—
95C24×C5C2×C3×C411yescharacteristic
96C23×D5—21no—
97C23×D5—21no—
98C23×D5—22no—
99C23×D5—42no—
100C23×D5—42no—
101C23×D5C3×C814yescomplemented
102C23×D5C2×C3×C412yes—
103C22×C4×C5—21no—
104C22×C4×C5—21no—
105C22×C4×C5—42no—
106C22×C4×C5C2×C3×C412yes—
107C22×Dic5—42no—
108C22×Dic5C2×C3×C411yescharacteristic
109C22×Dic5C3×D412yes—
110C22×Dic5C3×D412yes—
111C22×Dic5C23×C311yescharacteristic
112C2×#170—24no—
113C2×#170—44no—
114#685—24no—
115#685—24no—
116C22×C22⋊C4—51no—
117C2×C22⋊C8—52no—
118#323—54no—
119C22×C3×C5—24no—
120C22×C3×C5—24no—
121C22×C3×C5—21no—
122C22×C3×C5—21no—
123C22×C3×C5—21no—
124C22×C3×C5—21no—
125C22×C3×C5C22⋊C812yes—
126C22×C3×C5C22⋊C812yes—
127C22×C3×C5#9912yes—
128C22×C3×C5#10012yes—
129C22×C3×C5C2×C22⋊C411yescharacteristic
130C22×C3×C5C22×C811yescharacteristic
131C22×C3×C5C2×M1611yescharacteristic
132C2×C3×D5—24no—
133C2×C3×D5—48no—
134C2×C3×D5—44no—
135C2×C3×D5—48no—
136C2×C3×D5—42no—
137C2×C3×D5—42no—
138C2×C3×D5—44no—
139C3×C4×C5—42no—
140C3×Dic5—22no—
141C3×Dic5—22no—
142C24×C3—54no—
143C24×C3—52no—
144C24×C3—102no—
145C24×C3—101no—
146C24×C3—101no—
147C24×C3—202no—
148C24×C3—202no—
149C24×C3C2×AGL(1,𝔽5)11yescharacteristic
150C22×C3×C4—21no—
151C22×C3×C4—21no—
152C22×C3×C4—42no—
153C22×C3×C4—52no—
154C22×C3×C4—52no—
155C22×C3×C4—51no—
156C22×C3×C4—51no—
157C22×C3×C4—202no—
158C22×C3×C4C2×AGL(1,𝔽5)12yes—
159C2×C3×C8—104no—
160C2×C3×C8—204no—
161C3×C22⋊C4—104no—
162C3×C22⋊C4—104no—
163C23×C5—22no—
164C23×C5—22no—
165C23×C5C3×C22⋊C411yescharacteristic
166C23×C5C3×C22⋊C411yescharacteristic
167C23×C5C2×C3×C812yes—
168C23×C5C3×M1612yes—
169C23×C5C22×C3×C411yescharacteristic
170C22×D5—24no—
171C22×D5—24no—
172C22×D5—22no—
173C22×D5—22no—
174C22×D5—42no—
175C22×D5—42no—
176C22×D5—48no—
177C22×D5—42no—
178C22×D5—42no—
179C22×D5—48no—
180C22×D5—42no—
181C22×D5—42no—
182C22×D5C2×C3×C814yescomplemented
183C2×C4×C5—22no—
184C2×C4×C5—42no—
185C2×C4×C5—42no—
186C2×C4×C5—42no—
187C2×Dic5—24no—
188C2×Dic5—41no—
189C2×Dic5—42no—
190C2×Dic5—41no—
191C2×Dic5C3×C22⋊C412yes—
192C2×Dic5C2×C3×D412yes—
193#170—44no—
194C25—51no—
195C23×C4—51no—
196C23×C4C3×AGL(1,𝔽5)11yescharacteristic
197C22×C8—102no—
198C2×C22⋊C4—52no—
199C2×C22⋊C4—52no—
200C2×C22⋊C4—102no—
201C2×C22⋊C4—102no—
202C22⋊C8—54no—
203C22⋊C8—102no—
204C2×C3×C5—22no—
205C2×C3×C5—21no—
206C2×C3×C5—21no—
207C2×C3×C5#32314yescomplemented
208C2×C3×C5C2×C22⋊C811yescharacteristic
209C2×C3×C5C2×#9911yescharacteristic
210C2×C3×C5C2×#10011yescharacteristic
211C3×D5—44no—
212C23×C3—22no—
213C23×C3—22no—
214C23×C3—54no—
215C23×C3—102no—
216C23×C3—102no—
217C23×C3—104no—
218C23×C3—104no—
219C23×C3—202no—
220C23×C3—208no—
221C23×C3—202no—
222C23×C3—202no—
223C23×C3—208no—
224C23×C3—202no—
225C23×C3—202no—
226C23×C3—202no—
227C23×C3D5⋊C812yes—
228C23×C3#70012yes—
229C23×C3C22⋊AGL(1,𝔽5)11yescharacteristic
230C23×C3C22⋊AGL(1,𝔽5)11yescharacteristic
231C23×C3C22×AGL(1,𝔽5)11yescharacteristic
232C2×C3×C4—22no—
233C2×C3×C4—42no—
234C2×C3×C4—42no—
235C2×C3×C4—42no—
236C2×C3×C4—52no—
237C2×C3×C4—52no—
238C2×C3×C4—104no—
239C2×C3×C4—201no—
240C2×C3×C4—202no—
241C2×C3×C4—201no—
242C3×C8—204no—
243C22×C5—21no—
244C22×C5—21no—
245C22×C5—21no—
246C22×C5—21no—
247C22×C5—24no—
248C22×C5—24no—
249C22×C5C3×C22⋊C812yes—
250C22×C5C3×C22⋊C812yes—
251C22×C5C3×#9912yes—
252C22×C5C3×#10012yes—
253C22×C5C2×C3×C22⋊C411yescharacteristic
254C22×C5C22×C3×C811yescharacteristic
255C22×C5C2×C3×M1611yescharacteristic
256C2×D5—24no—
257C2×D5—48no—
258C2×D5—44no—
259C2×D5—42no—
260C2×D5—42no—
261C2×D5—48no—
262C2×D5—44no—
263C4×C5—42no—
264Dic5—22no—
265Dic5—22no—
266C24—54no—
267C24—52no—
268C24—102no—
269C24—101no—
270C24—101no—
271C24—202no—
272C24—202no—
273C24C2×C3×AGL(1,𝔽5)11yescharacteristic
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—42no—
277C22×C4—52no—
278C22×C4—52no—
279C22×C4—51no—
280C22×C4—51no—
281C22×C4—202no—
282C22×C4C2×C3×AGL(1,𝔽5)12yes—
283C2×C8—104no—
284C2×C8—204no—
285C22⋊C4—104no—
286C22⋊C4—104no—
287C3×C5C2×#32311yescharacteristic, complemented, Hall
288C22×C3—21no—
289C22×C3—21no—
290C22×C3—24no—
291C22×C3—24no—
292C22×C3—21no—
293C22×C3—21no—
294C22×C3—104no—
295C22×C3—204no—
296C22×C3—204no—
297C22×C3—208no—
298C22×C3—202no—
299C22×C3—208no—
300C22×C3—202no—
301C22×C3#399912yes—
302C22×C3#400312yes—
303C22×C3#400312yes—
304C22×C3#401312yes—
305C22×C3C2×D5⋊C811yescharacteristic
306C22×C3C2×#70011yescharacteristic
307C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
308C3×C4—42no—
309C3×C4—102no—
310C3×C4—102no—
311C2×C5—22no—
312C2×C5—21no—
313C2×C5—21no—
314C2×C5C3×#32314yescomplemented
315C2×C5C2×C3×C22⋊C811yescharacteristic
316C2×C5C2×C3×#9911yescharacteristic
317C2×C5C2×C3×#10011yescharacteristic
318D5—44no—
319C23—22no—
320C23—22no—
321C23—54no—
322C23—102no—
323C23—104no—
324C23—104no—
325C23—102no—
326C23—202no—
327C23—208no—
328C23—202no—
329C23—202no—
330C23—208no—
331C23—202no—
332C23—202no—
333C23—202no—
334C23C3×D5⋊C812yes—
335C23C3×#70012yes—
336C23C3×C22⋊AGL(1,𝔽5)11yescharacteristic
337C23C3×C22⋊AGL(1,𝔽5)11yescharacteristic
338C23C22×C3×AGL(1,𝔽5)11yescharacteristic
339C2×C4—22no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—52no—
344C2×C4—52no—
345C2×C4—104no—
346C2×C4—201no—
347C2×C4—202no—
348C2×C4—201no—
349C8—204no—
350C2×C3—21no—
351C2×C3—21no—
352C2×C3—22no—
353C2×C3—204no—
354C2×C3#6491514yesdirect product factor
355C2×C3C2×#399911yescharacteristic
356C2×C3C2×#400311yescharacteristic
357C2×C3C2×#401311yescharacteristic
358C5C2×C3×#32311yescharacteristic, complemented, Sylow
359C22—21no—
360C22—21no—
361C22—24no—
362C22—24no—
363C22—21no—
364C22—21no—
365C22—104no—
366C22—204no—
367C22—204no—
368C22—208no—
369C22—202no—
370C22—208no—
371C22—202no—
372C22C3×#399912yes—
373C22C3×#400312yes—
374C22C3×#400312yes—
375C22C3×#401312yes—
376C22C2×C3×D5⋊C811yescharacteristic
377C22C2×C3×#70011yescharacteristic
378C22C2×C3×C22⋊AGL(1,𝔽5)11yescharacteristic
379C4—42no—
380C4—102no—
381C4—102no—
382C3C2×#6491511yescharacteristic, direct product factor, Sylow
383C2—21no—
384C2—21no—
385C2—22no—
386C2—204no—
387C2C3×#6491514yesdirect product factor
388C2C2×C3×#399911yescharacteristic
389C2C2×C3×#400311yescharacteristic
390C2C2×C3×#401311yescharacteristic
subgroupG/H
centerC23×C3C22⋊AGL(1,𝔽5)
commutatorC22×C5C22×C3×C8
FrattiniC23C22×C3×AGL(1,𝔽5)
FittingC23×C3×C4×C5C4
socleC23×C3×C5C22⋊C4
prime radicalC24×C3×D5C4
cosocleC22×Dic5C23×C3
hypercenterC23×C3×C4AGL(1,𝔽5)
hypocenterC5C2×C3×#323
LCSG ▷ C22×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23×C3 ◁ C24×C3 ◁ C23×C3×C4

Complements

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

NH ≅ G/Ntype
2. C2×C3×C22⋊#170C2semidirect
3. C3×#64915C2direct (N × H)
4. C2×#64915C3direct (N × H)
14. C3×C22⋊#170C22semidirect
17. C2×C22⋊#170C2×C3semidirect
18. #64915C2×C3direct (N × H)
25. C23×C3×D5C8semidirect
52. C22⋊#170C22×C3semidirect
73. C22×C3×D5C2×C8semidirect
101. C23×D5C3×C8semidirect
182. C22×D5C2×C3×C8semidirect
207. C2×C3×C5#323semidirect
287. C3×C5C2×#323semidirect
314. C2×C5C3×#323semidirect
354. C2×C3#64915direct (N × H)
358. C5C2×C3×#323semidirect
382. C3C2×#64915direct (N × H)
387. C2C3×#64915direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#6852
C2×C3×C22⋊#1702
C3×#649152
C2×#649153
C2×C3×#3235