Finite Group Databasebeta

← C2×C3×C5×#327 #14274532

Subgroups — C2×C3×C5×#327 #14274532

Order 1920.

776 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C5×#121C211yesmaximal, characteristic
2C2×C3×C5×C22⋊C8C211yesmaximal, characteristic, complemented
3C2×C3×C5×#95C211yesmaximal, characteristic, complemented
4C3×C5×#327C214yesmaximal, direct product factor
5C2×C5×#327C311yesmaximal, characteristic, direct product factor, Hall
6C22×C3×C5×D4—21no—
7C22×C3×C5×D4C411yescharacteristic, complemented
8C23×C3×C4×C5—21no—
9C23×C3×C4×C5C2211yescharacteristic
10C22×C3×C5×C8—21no—
11C3×C5×#121—21no—
12C3×C5×#121—21no—
13C3×C5×#121C412yescomplemented
14C3×C5×#121C2212yes—
15C2×C3×C5×C4⋊C4C411yescharacteristic, complemented
16C2×C3×C5×C22⋊C4—21no—
17C3×C5×C22⋊C8—21no—
18C3×C5×C22⋊C8C2212yescomplemented
19C3×C5×#95—21no—
20C3×C5×#95C2212yescomplemented
21C2×C3×#327C511yesmaximal, characteristic, direct product factor, Hall
22C2×C5×#121C2×C311yescharacteristic
23C2×C5×C22⋊C8C2×C311yescharacteristic, complemented
24C2×C5×#95C2×C311yescharacteristic, complemented
25C5×#327C2×C314yesdirect product factor
26C24×C3×C5—21no—
27C24×C3×C5D411yescharacteristic
28C2×C3×C5×D4—22no—
29C2×C3×C5×D4—22no—
30C2×C3×C5×D4—21no—
31C2×C3×C5×D4—42no—
32C2×C3×C5×D4—42no—
33C2×C3×C5×D4—42no—
34C2×C3×C5×D4C2×C412yescomplemented
35C22×C3×C4×C5—24no—
36C22×C3×C4×C5—21no—
37C22×C3×C4×C5—22no—
38C22×C3×C4×C5—42no—
39C22×C3×C4×C5—42no—
40C22×C3×C4×C5—42no—
41C22×C3×C4×C5C2×C411yescharacteristic
42C22×C3×C4×C5C2×C411yescharacteristic
43C22×C3×C4×C5D411yescharacteristic
44C22×C3×C4×C5D411yescharacteristic
45C22×C3×C4×C5D411yescharacteristic
46C22×C3×C4×C5C2311yescharacteristic
47C2×C3×C5×C8—22no—
48C2×C3×C5×C8—42no—
49C3×C5×C4⋊C4—21no—
50C3×C5×C4⋊C4C2×C412yescomplemented
51C3×C5×C22⋊C4—22no—
52C3×C5×C22⋊C4—22no—
53C2×C3×#121C2×C511yescharacteristic
54C2×C3×C22⋊C8C2×C511yescharacteristic, complemented
55C2×C3×#95C2×C511yescharacteristic, complemented
56C3×#327C2×C514yesdirect product factor
57C22×C5×D4—21no—
58C22×C5×D4C3×C411yescharacteristic, complemented
59C23×C4×C5—21no—
60C23×C4×C5C22×C311yescharacteristic
61C22×C5×C8—21no—
62C5×#121—21no—
63C5×#121—21no—
64C5×#121C3×C412yescomplemented
65C5×#121C22×C312yes—
66C2×C5×C4⋊C4C3×C411yescharacteristic, complemented
67C2×C5×C22⋊C4—21no—
68C5×C22⋊C8—21no—
69C5×C22⋊C8C22×C312yescomplemented
70C5×#95—21no—
71C5×#95C22×C312yescomplemented
72C2×#327C3×C511yescharacteristic, direct product factor, Sylow
73C23×C3×C5—22no—
74C23×C3×C5—22no—
75C23×C3×C5—22no—
76C23×C3×C5—42no—
77C23×C3×C5—42no—
78C23×C3×C5—42no—
79C23×C3×C5C22⋊C411yescharacteristic
80C23×C3×C5C22⋊C411yescharacteristic
81C23×C3×C5D812yes—
82C23×C3×C5SD1612yes—
83C23×C3×C5C2×D411yescharacteristic
84C3×C5×D4—42no—
85C3×C5×D4—42no—
86C3×C5×D4—44no—
87C3×C5×D4—42no—
88C3×C5×D4—42no—
89C2×C3×C4×C5—22no—
90C2×C3×C4×C5—22no—
91C2×C3×C4×C5—24no—
92C2×C3×C4×C5—22no—
93C2×C3×C4×C5—41no—
94C2×C3×C4×C5—41no—
95C2×C3×C4×C5—41no—
96C2×C3×C4×C5—41no—
97C2×C3×C4×C5—48no—
98C2×C3×C4×C5—42no—
99C2×C3×C4×C5—42no—
100C2×C3×C4×C5—42no—
101C2×C3×C4×C5C22⋊C411yescharacteristic
102C2×C3×C4×C5C22⋊C411yescharacteristic
103C2×C3×C4×C5C22×C411yescharacteristic
104C2×C3×C4×C5C2×D411yescharacteristic
105C3×C5×C8—42no—
106C22×C3×D4—21no—
107C22×C3×D4C4×C511yescharacteristic, complemented
108C23×C3×C4—21no—
109C23×C3×C4C22×C511yescharacteristic
110C22×C3×C8—21no—
111C3×#121—21no—
112C3×#121—21no—
113C3×#121C4×C512yescomplemented
114C3×#121C22×C512yes—
115C2×C3×C4⋊C4C4×C511yescharacteristic, complemented
116C2×C3×C22⋊C4—21no—
117C3×C22⋊C8—21no—
118C3×C22⋊C8C22×C512yescomplemented
119C3×#95—21no—
120C3×#95C22×C512yescomplemented
121C24×C5—21no—
122C24×C5C3×D411yescharacteristic
123C2×C5×D4—22no—
124C2×C5×D4—22no—
125C2×C5×D4—21no—
126C2×C5×D4—42no—
127C2×C5×D4—42no—
128C2×C5×D4—42no—
129C2×C5×D4C2×C3×C412yescomplemented
130C22×C4×C5—24no—
131C22×C4×C5—21no—
132C22×C4×C5—22no—
133C22×C4×C5—42no—
134C22×C4×C5—42no—
135C22×C4×C5—42no—
136C22×C4×C5C2×C3×C411yescharacteristic
137C22×C4×C5C2×C3×C411yescharacteristic
138C22×C4×C5C3×D411yescharacteristic
139C22×C4×C5C3×D411yescharacteristic
140C22×C4×C5C3×D411yescharacteristic
141C22×C4×C5C23×C311yescharacteristic
142C2×C5×C8—22no—
143C2×C5×C8—42no—
144C5×C4⋊C4—21no—
145C5×C4⋊C4C2×C3×C412yescomplemented
146C5×C22⋊C4—22no—
147C5×C22⋊C4—22no—
148C2×#121C2×C3×C511yescharacteristic
149C2×C22⋊C8C2×C3×C511yescharacteristic, complemented
150C2×#95C2×C3×C511yescharacteristic, complemented
151#327C2×C3×C514yesdirect product factor
152C22×C3×C5—24no—
153C22×C3×C5—24no—
154C22×C3×C5—21no—
155C22×C3×C5—21no—
156C22×C3×C5—21no—
157C22×C3×C5—21no—
158C22×C3×C5—42no—
159C22×C3×C5—42no—
160C22×C3×C5—42no—
161C22×C3×C5—84no—
162C22×C3×C5#9912yes—
163C22×C3×C5#10212yes—
164C22×C3×C5#10212yes—
165C22×C3×C5#10412yes—
166C22×C3×C5C2×C22⋊C411yescharacteristic
167C22×C3×C5C2×D811yescharacteristic
168C22×C3×C5C2×SD1611yescharacteristic
169C3×C4×C5—21no—
170C3×C4×C5—21no—
171C3×C4×C5—21no—
172C3×C4×C5—21no—
173C3×C4×C5—44no—
174C3×C4×C5—42no—
175C24×C3—21no—
176C24×C3C5×D411yescharacteristic
177C2×C3×D4—22no—
178C2×C3×D4—22no—
179C2×C3×D4—21no—
180C2×C3×D4—42no—
181C2×C3×D4—42no—
182C2×C3×D4—42no—
183C2×C3×D4C2×C4×C512yescomplemented
184C22×C3×C4—24no—
185C22×C3×C4—21no—
186C22×C3×C4—22no—
187C22×C3×C4—42no—
188C22×C3×C4—42no—
189C22×C3×C4—42no—
190C22×C3×C4C2×C4×C511yescharacteristic
191C22×C3×C4C2×C4×C511yescharacteristic
192C22×C3×C4C5×D411yescharacteristic
193C22×C3×C4C5×D411yescharacteristic
194C22×C3×C4C5×D411yescharacteristic
195C22×C3×C4C23×C511yescharacteristic
196C2×C3×C8—22no—
197C2×C3×C8—42no—
198C3×C4⋊C4—21no—
199C3×C4⋊C4C2×C4×C512yescomplemented
200C3×C22⋊C4—22no—
201C3×C22⋊C4—22no—
202C23×C5—22no—
203C23×C5—22no—
204C23×C5—22no—
205C23×C5—42no—
206C23×C5—42no—
207C23×C5—42no—
208C23×C5C3×C22⋊C411yescharacteristic
209C23×C5C3×C22⋊C411yescharacteristic
210C23×C5C3×D812yes—
211C23×C5C3×SD1612yes—
212C23×C5C2×C3×D411yescharacteristic
213C5×D4—42no—
214C5×D4—42no—
215C5×D4—44no—
216C5×D4—42no—
217C5×D4—42no—
218C2×C4×C5—22no—
219C2×C4×C5—22no—
220C2×C4×C5—24no—
221C2×C4×C5—22no—
222C2×C4×C5—41no—
223C2×C4×C5—41no—
224C2×C4×C5—41no—
225C2×C4×C5—41no—
226C2×C4×C5—42no—
227C2×C4×C5—48no—
228C2×C4×C5—42no—
229C2×C4×C5—42no—
230C2×C4×C5C3×C22⋊C411yescharacteristic
231C2×C4×C5C3×C22⋊C411yescharacteristic
232C2×C4×C5C22×C3×C411yescharacteristic
233C2×C4×C5C2×C3×D411yescharacteristic
234C5×C8—42no—
235C22×D4—21no—
236C22×D4C3×C4×C511yescharacteristic, complemented
237C23×C4—21no—
238C23×C4C22×C3×C511yescharacteristic
239C22×C8—21no—
240#121—21no—
241#121—21no—
242#121C3×C4×C512yescomplemented
243#121C22×C3×C512yes—
244C2×C4⋊C4C3×C4×C511yescharacteristic, complemented
245C2×C22⋊C4—21no—
246C22⋊C8—21no—
247C22⋊C8C22×C3×C512yescomplemented
248#95—21no—
249#95C22×C3×C512yescomplemented
250C2×C3×C5—21no—
251C2×C3×C5—22no—
252C2×C3×C5—21no—
253C2×C3×C5—82no—
254C2×C3×C5#32714yesdirect product factor
255C2×C3×C5C2×#9911yescharacteristic
256C2×C3×C5C2×#10211yescharacteristic
257C2×C3×C5C2×#10411yescharacteristic
258C23×C3—22no—
259C23×C3—22no—
260C23×C3—22no—
261C23×C3—42no—
262C23×C3—42no—
263C23×C3—42no—
264C23×C3C5×C22⋊C411yescharacteristic
265C23×C3C5×C22⋊C411yescharacteristic
266C23×C3C5×D812yes—
267C23×C3C5×SD1612yes—
268C23×C3C2×C5×D411yescharacteristic
269C3×D4—42no—
270C3×D4—42no—
271C3×D4—42no—
272C3×D4—42no—
273C3×D4—44no—
274C2×C3×C4—22no—
275C2×C3×C4—22no—
276C2×C3×C4—24no—
277C2×C3×C4—22no—
278C2×C3×C4—41no—
279C2×C3×C4—41no—
280C2×C3×C4—41no—
281C2×C3×C4—41no—
282C2×C3×C4—48no—
283C2×C3×C4—42no—
284C2×C3×C4—42no—
285C2×C3×C4—42no—
286C2×C3×C4C5×C22⋊C411yescharacteristic
287C2×C3×C4C5×C22⋊C411yescharacteristic
288C2×C3×C4C22×C4×C511yescharacteristic
289C2×C3×C4C2×C5×D411yescharacteristic
290C3×C8—42no—
291C22×C5—21no—
292C22×C5—21no—
293C22×C5—21no—
294C22×C5—21no—
295C22×C5—24no—
296C22×C5—24no—
297C22×C5—42no—
298C22×C5—42no—
299C22×C5—42no—
300C22×C5—84no—
301C22×C5C3×#9912yes—
302C22×C5C3×#10212yes—
303C22×C5C3×#10212yes—
304C22×C5C3×#10412yes—
305C22×C5C2×C3×C22⋊C411yescharacteristic
306C22×C5C2×C3×D811yescharacteristic
307C22×C5C2×C3×SD1611yescharacteristic
308C4×C5—21no—
309C4×C5—21no—
310C4×C5—21no—
311C4×C5—21no—
312C4×C5—44no—
313C4×C5—42no—
314C24—21no—
315C24C3×C5×D411yescharacteristic
316C2×D4—22no—
317C2×D4—22no—
318C2×D4—21no—
319C2×D4—42no—
320C2×D4—42no—
321C2×D4—42no—
322C2×D4C2×C3×C4×C512yescomplemented
323C22×C4—24no—
324C22×C4—21no—
325C22×C4—22no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4C2×C3×C4×C511yescharacteristic
330C22×C4C2×C3×C4×C511yescharacteristic
331C22×C4C3×C5×D411yescharacteristic
332C22×C4C3×C5×D411yescharacteristic
333C22×C4C3×C5×D411yescharacteristic
334C22×C4C23×C3×C511yescharacteristic
335C2×C8—22no—
336C2×C8—42no—
337C4⋊C4—21no—
338C4⋊C4C2×C3×C4×C512yescomplemented
339C22⋊C4—22no—
340C22⋊C4—22no—
341C3×C5C2×#32711yescharacteristic, direct product factor, Hall
342C22×C3—24no—
343C22×C3—24no—
344C22×C3—21no—
345C22×C3—21no—
346C22×C3—21no—
347C22×C3—21no—
348C22×C3—42no—
349C22×C3—42no—
350C22×C3—42no—
351C22×C3—84no—
352C22×C3C5×#9912yes—
353C22×C3C5×#10212yes—
354C22×C3C5×#10212yes—
355C22×C3C5×#10412yes—
356C22×C3C2×C5×C22⋊C411yescharacteristic
357C22×C3C2×C5×D811yescharacteristic
358C22×C3C2×C5×SD1611yescharacteristic
359C3×C4—21no—
360C3×C4—21no—
361C3×C4—21no—
362C3×C4—21no—
363C3×C4—44no—
364C3×C4—42no—
365C2×C5—21no—
366C2×C5—22no—
367C2×C5—21no—
368C2×C5—82no—
369C2×C5C3×#32714yesdirect product factor
370C2×C5C2×C3×#9911yescharacteristic
371C2×C5C2×C3×#10211yescharacteristic
372C2×C5C2×C3×#10411yescharacteristic
373C23—22no—
374C23—22no—
375C23—22no—
376C23—42no—
377C23—42no—
378C23—42no—
379C23C3×C5×C22⋊C411yescharacteristic
380C23C3×C5×C22⋊C411yescharacteristic
381C23C3×C5×D812yes—
382C23C3×C5×SD1612yes—
383C23C2×C3×C5×D411yescharacteristic
384D4—42no—
385D4—42no—
386D4—42no—
387D4—42no—
388D4—44no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—24no—
392C2×C4—22no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—48no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4C3×C5×C22⋊C411yescharacteristic
402C2×C4C3×C5×C22⋊C411yescharacteristic
403C2×C4C22×C3×C4×C511yescharacteristic
404C2×C4C2×C3×C5×D411yescharacteristic
405C8—42no—
406C2×C3—21no—
407C2×C3—21no—
408C2×C3—22no—
409C2×C3—82no—
410C2×C3C5×#32714yesdirect product factor
411C2×C3C2×C5×#9911yescharacteristic
412C2×C3C2×C5×#10211yescharacteristic
413C2×C3C2×C5×#10411yescharacteristic
414C5C2×C3×#32711yescharacteristic, direct product factor, Sylow
415C22—24no—
416C22—24no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—42no—
422C22—42no—
423C22—42no—
424C22—84no—
425C22C3×C5×#9912yes—
426C22C3×C5×#10212yes—
427C22C3×C5×#10212yes—
428C22C3×C5×#10412yes—
429C22C2×C3×C5×C22⋊C411yescharacteristic
430C22C2×C3×C5×D811yescharacteristic
431C22C2×C3×C5×SD1611yescharacteristic
432C4—21no—
433C4—21no—
434C4—21no—
435C4—21no—
436C4—44no—
437C4—42no—
438C3C2×C5×#32711yescharacteristic, direct product factor, Sylow
439C2—21no—
440C2—21no—
441C2—22no—
442C2—82no—
443C2C3×C5×#32714yesdirect product factor
444C2C2×C3×C5×#9911yescharacteristic
445C2C2×C3×C5×#10211yescharacteristic
446C2C2×C3×C5×#10411yescharacteristic
subgroupG/H
center, socleC23×C3×C5C22⋊C4
commutatorC2×C4C22×C3×C4×C5
Frattini, cosocleC22×C4C23×C3×C5
prime radicalC2×C3×C5×#121C2
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C3×C5 ◁ C23×C3×C4×C5 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C3×C5×C22⋊C8C2semidirect
3. C2×C3×C5×#95C2semidirect
4. C3×C5×#327C2direct (N × H)
5. C2×C5×#327C3direct (N × H)
7. C22×C3×C5×D4C4semidirect
13. C3×C5×#121C4semidirect
15. C2×C3×C5×C4⋊C4C4semidirect
18. C3×C5×C22⋊C8C22semidirect
20. C3×C5×#95C22semidirect
21. C2×C3×#327C5direct (N × H)
23. C2×C5×C22⋊C8C2×C3semidirect
24. C2×C5×#95C2×C3semidirect
25. C5×#327C2×C3direct (N × H)
34. C2×C3×C5×D4C2×C4semidirect
50. C3×C5×C4⋊C4C2×C4semidirect
54. C2×C3×C22⋊C8C2×C5semidirect
55. C2×C3×#95C2×C5semidirect
56. C3×#327C2×C5direct (N × H)
58. C22×C5×D4C3×C4semidirect
64. C5×#121C3×C4semidirect
66. C2×C5×C4⋊C4C3×C4semidirect
69. C5×C22⋊C8C22×C3semidirect
71. C5×#95C22×C3semidirect
72. C2×#327C3×C5direct (N × H)
107. C22×C3×D4C4×C5semidirect
113. C3×#121C4×C5semidirect
115. C2×C3×C4⋊C4C4×C5semidirect
118. C3×C22⋊C8C22×C5semidirect
120. C3×#95C22×C5semidirect
129. C2×C5×D4C2×C3×C4semidirect
145. C5×C4⋊C4C2×C3×C4semidirect
149. C2×C22⋊C8C2×C3×C5semidirect
150. C2×#95C2×C3×C5semidirect
151. #327C2×C3×C5direct (N × H)
183. C2×C3×D4C2×C4×C5semidirect
199. C3×C4⋊C4C2×C4×C5semidirect
236. C22×D4C3×C4×C5semidirect
242. #121C3×C4×C5semidirect
244. C2×C4⋊C4C3×C4×C5semidirect
247. C22⋊C8C22×C3×C5semidirect
249. #95C22×C3×C5semidirect
254. C2×C3×C5#327direct (N × H)
322. C2×D4C2×C3×C4×C5semidirect
338. C4⋊C4C2×C3×C4×C5semidirect
341. C3×C5C2×#327direct (N × H)
369. C2×C5C3×#327direct (N × H)
410. C2×C3C5×#327direct (N × H)
414. C5C2×C3×#327direct (N × H)
438. C3C2×C5×#327direct (N × H)
443. C2C3×C5×#327direct (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×#1212
C2×C3×C5×C22⋊C82
C2×C3×C5×#952
C3×C5×#3272
C2×C5×#3273
C2×C3×#3275