Finite Group Databasebeta

← C3×C5×#1867 #14274901

Subgroups — C3×C5×#1867 #14274901

Order 1920.

748 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C5×#518C211yesmaximal, characteristic
2C2×C3×C5×#99C211yesmaximal, characteristic
3C3×C5×#386C211yesmaximal, characteristic
4C3×C5×#385C212yesmaximal, complemented
5C3×C5×#342C212yesmaximal, complemented
6C5×#1867C311yesmaximal, characteristic, direct product factor, Hall
7C22×C3×C5×D4C2211yescharacteristic
8C23×C3×C4×C5C2211yescharacteristic
9C3×C4×C5×D4—22no—
10C3×C4×C5×D4—22no—
11C3×C5×#117—21no—
12C2×C3×C5×C4⋊C4—22no—
13C2×C3×C5×C22⋊C4—21no—
14C2×C3×C5×C22⋊C4—21no—
15C2×C3×C5×C22⋊C4C2211yescharacteristic
16C2×C3×C5×C22⋊C4C2212yes—
17C2×C3×C5×C22⋊C4C2212yes—
18C3×C5×#99C414yescomplemented
19C3×C5×#95—22no—
20C3×C5×#95—21no—
21C3×#1867C511yesmaximal, characteristic, direct product factor, Hall
22C5×#518C2×C311yescharacteristic
23C2×C5×#99C2×C311yescharacteristic
24C5×#386C2×C311yescharacteristic
25C5×#385C2×C312yescomplemented
26C5×#342C2×C312yescomplemented
27C24×C3×C5D411yescharacteristic
28C24×C3×C5C2311yescharacteristic
29C2×C3×C5×D4—22no—
30C2×C3×C5×D4—22no—
31C2×C3×C5×D4C2×C412yes—
32C2×C3×C5×D4Q812yes—
33C22×C3×C4×C5—21no—
34C22×C3×C4×C5—22no—
35C22×C3×C4×C5—22no—
36C22×C3×C4×C5—21no—
37C22×C3×C4×C5—21no—
38C22×C3×C4×C5—21no—
39C22×C3×C4×C5—42no—
40C22×C3×C4×C5D411yescharacteristic
41C22×C3×C4×C5D412yes—
42C22×C3×C4×C5D412yes—
43C3×C5×C4⋊C4—21no—
44C3×C5×C4⋊C4—21no—
45C3×C5×C4⋊C4—44no—
46C3×C5×C22⋊C4—22no—
47C3×C5×C22⋊C4—22no—
48C3×C5×C22⋊C4—22no—
49C3×C5×C22⋊C4—22no—
50C3×C5×C22⋊C4—22no—
51C3×C5×C22⋊C4—42no—
52C3×C5×C22⋊C4C2×C414yescomplemented
53C3×C42×C5—22no—
54C3×#518C2×C511yescharacteristic
55C2×C3×#99C2×C511yescharacteristic
56C3×#386C2×C511yescharacteristic
57C3×#385C2×C512yescomplemented
58C3×#342C2×C512yescomplemented
59C22×C5×D4C22×C311yescharacteristic
60C23×C4×C5C22×C311yescharacteristic
61C4×C5×D4—22no—
62C4×C5×D4—22no—
63C5×#117—21no—
64C2×C5×C4⋊C4—22no—
65C2×C5×C22⋊C4—21no—
66C2×C5×C22⋊C4—21no—
67C2×C5×C22⋊C4C22×C311yescharacteristic
68C2×C5×C22⋊C4C22×C312yes—
69C2×C5×C22⋊C4C22×C312yes—
70C5×#99C3×C414yescomplemented
71C5×#95—22no—
72C5×#95—21no—
73#1867C3×C511yescharacteristic, direct product factor, Sylow
74C23×C3×C5—21no—
75C23×C3×C5—21no—
76C23×C3×C5—41no—
77C23×C3×C5—41no—
78C23×C3×C5—41no—
79C23×C3×C5—41no—
80C23×C3×C5C4⋊C412yescomplemented
81C23×C3×C5C22×C411yescharacteristic
82C23×C3×C5C2×D411yescharacteristic
83C23×C3×C5C2×D412yes—
84C23×C3×C5C2×Q811yescharacteristic
85C23×C3×C5#4112yes—
86C3×C5×D4—24no—
87C3×C5×D4—24no—
88C2×C3×C4×C5—21no—
89C2×C3×C4×C5—21no—
90C2×C3×C4×C5—22no—
91C2×C3×C4×C5—24no—
92C2×C3×C4×C5—24no—
93C2×C3×C4×C5—22no—
94C2×C3×C4×C5—22no—
95C2×C3×C4×C5—42no—
96C2×C3×C4×C5—42no—
97C2×C3×C4×C5—42no—
98C2×C3×C4×C5—41no—
99C2×C3×C4×C5—41no—
100C2×C3×C4×C5—42no—
101C2×C3×C4×C5—42no—
102C2×C3×C4×C5—41no—
103C2×C3×C4×C5—41no—
104C2×C3×C4×C5—41no—
105C2×C3×C4×C5—81no—
106C2×C3×C4×C5—81no—
107C2×C3×C4×C5C4⋊C412yescomplemented
108C22×C3×D4C22×C511yescharacteristic
109C23×C3×C4C22×C511yescharacteristic
110C3×C4×D4—22no—
111C3×C4×D4—22no—
112C3×#117—21no—
113C2×C3×C4⋊C4—22no—
114C2×C3×C22⋊C4—21no—
115C2×C3×C22⋊C4—21no—
116C2×C3×C22⋊C4C22×C511yescharacteristic
117C2×C3×C22⋊C4C22×C512yes—
118C2×C3×C22⋊C4C22×C512yes—
119C3×#99C4×C514yescomplemented
120C3×#95—22no—
121C3×#95—21no—
122C24×C5C3×D411yescharacteristic
123C24×C5C23×C311yescharacteristic
124C2×C5×D4—22no—
125C2×C5×D4—22no—
126C2×C5×D4C2×C3×C412yes—
127C2×C5×D4C3×Q812yes—
128C22×C4×C5—21no—
129C22×C4×C5—22no—
130C22×C4×C5—22no—
131C22×C4×C5—21no—
132C22×C4×C5—21no—
133C22×C4×C5—21no—
134C22×C4×C5—42no—
135C22×C4×C5C3×D411yescharacteristic
136C22×C4×C5C3×D412yes—
137C22×C4×C5C3×D412yes—
138C5×C4⋊C4—21no—
139C5×C4⋊C4—21no—
140C5×C4⋊C4—44no—
141C5×C22⋊C4—22no—
142C5×C22⋊C4—22no—
143C5×C22⋊C4—22no—
144C5×C22⋊C4—22no—
145C5×C22⋊C4—22no—
146C5×C22⋊C4—42no—
147C5×C22⋊C4C2×C3×C414yescomplemented
148C42×C5—22no—
149#518C2×C3×C511yescharacteristic
150C2×#99C2×C3×C511yescharacteristic
151#386C2×C3×C511yescharacteristic
152#385C2×C3×C512yescomplemented
153#342C2×C3×C512yescomplemented
154C22×C3×C5—22no—
155C22×C3×C5—21no—
156C22×C3×C5—22no—
157C22×C3×C5—21no—
158C22×C3×C5—22no—
159C22×C3×C5—41no—
160C22×C3×C5—41no—
161C22×C3×C5—42no—
162C22×C3×C5—42no—
163C22×C3×C5—41no—
164C22×C3×C5—41no—
165C22×C3×C5—81no—
166C22×C3×C5C2×C4⋊C411yescharacteristic
167C22×C3×C5C4×D412yes—
168C22×C3×C5#12011yescharacteristic
169C22×C3×C5C22⋊Q812yes—
170C22×C3×C5#12311yescharacteristic
171C3×C4×C5—22no—
172C3×C4×C5—44no—
173C3×C4×C5—42no—
174C3×C4×C5—42no—
175C3×C4×C5—42no—
176C3×C4×C5—81no—
177C24×C3C5×D411yescharacteristic
178C24×C3C23×C511yescharacteristic
179C2×C3×D4—22no—
180C2×C3×D4—22no—
181C2×C3×D4C2×C4×C512yes—
182C2×C3×D4C5×Q812yes—
183C22×C3×C4—21no—
184C22×C3×C4—22no—
185C22×C3×C4—22no—
186C22×C3×C4—21no—
187C22×C3×C4—21no—
188C22×C3×C4—21no—
189C22×C3×C4—42no—
190C22×C3×C4C5×D411yescharacteristic
191C22×C3×C4C5×D412yes—
192C22×C3×C4C5×D412yes—
193C3×C4⋊C4—21no—
194C3×C4⋊C4—21no—
195C3×C4⋊C4—44no—
196C3×C22⋊C4—22no—
197C3×C22⋊C4—22no—
198C3×C22⋊C4—22no—
199C3×C22⋊C4—22no—
200C3×C22⋊C4—22no—
201C3×C22⋊C4—42no—
202C3×C22⋊C4C2×C4×C514yescomplemented
203C3×C42—22no—
204C23×C5—21no—
205C23×C5—21no—
206C23×C5—41no—
207C23×C5—41no—
208C23×C5—41no—
209C23×C5—41no—
210C23×C5C3×C4⋊C412yescomplemented
211C23×C5C22×C3×C411yescharacteristic
212C23×C5C2×C3×D411yescharacteristic
213C23×C5C2×C3×D412yes—
214C23×C5C2×C3×Q811yescharacteristic
215C23×C5C3×#4112yes—
216C5×D4—24no—
217C5×D4—24no—
218C2×C4×C5—21no—
219C2×C4×C5—21no—
220C2×C4×C5—22no—
221C2×C4×C5—24no—
222C2×C4×C5—24no—
223C2×C4×C5—22no—
224C2×C4×C5—22no—
225C2×C4×C5—42no—
226C2×C4×C5—42no—
227C2×C4×C5—42no—
228C2×C4×C5—41no—
229C2×C4×C5—41no—
230C2×C4×C5—42no—
231C2×C4×C5—42no—
232C2×C4×C5—41no—
233C2×C4×C5—41no—
234C2×C4×C5—41no—
235C2×C4×C5—81no—
236C2×C4×C5—81no—
237C2×C4×C5C3×C4⋊C412yescomplemented
238C22×D4C22×C3×C511yescharacteristic
239C23×C4C22×C3×C511yescharacteristic
240C4×D4—22no—
241C4×D4—22no—
242#117—21no—
243C2×C4⋊C4—22no—
244C2×C22⋊C4—21no—
245C2×C22⋊C4—21no—
246C2×C22⋊C4C22×C3×C511yescharacteristic
247C2×C22⋊C4C22×C3×C512yes—
248C2×C22⋊C4C22×C3×C512yes—
249#99C3×C4×C514yescomplemented
250#95—22no—
251#95—21no—
252C2×C3×C5—22no—
253C2×C3×C5—21no—
254C2×C3×C5—21no—
255C2×C3×C5—22no—
256C2×C3×C5—42no—
257C2×C3×C5#38511yescharacteristic
258C2×C3×C5B(4,𝔽2)11yescharacteristic
259C2×C3×C5#45811yescharacteristic
260C23×C3—21no—
261C23×C3—21no—
262C23×C3—41no—
263C23×C3—41no—
264C23×C3—41no—
265C23×C3—41no—
266C23×C3C5×C4⋊C412yescomplemented
267C23×C3C22×C4×C511yescharacteristic
268C23×C3C2×C5×D411yescharacteristic
269C23×C3C2×C5×D412yes—
270C23×C3C2×C5×Q811yescharacteristic
271C23×C3C5×#4112yes—
272C3×D4—24no—
273C3×D4—24no—
274C2×C3×C4—21no—
275C2×C3×C4—21no—
276C2×C3×C4—22no—
277C2×C3×C4—24no—
278C2×C3×C4—24no—
279C2×C3×C4—22no—
280C2×C3×C4—22no—
281C2×C3×C4—42no—
282C2×C3×C4—42no—
283C2×C3×C4—42no—
284C2×C3×C4—41no—
285C2×C3×C4—41no—
286C2×C3×C4—42no—
287C2×C3×C4—42no—
288C2×C3×C4—41no—
289C2×C3×C4—41no—
290C2×C3×C4—41no—
291C2×C3×C4—81no—
292C2×C3×C4—81no—
293C2×C3×C4C5×C4⋊C412yescomplemented
294C22×C5—22no—
295C22×C5—21no—
296C22×C5—22no—
297C22×C5—21no—
298C22×C5—22no—
299C22×C5—41no—
300C22×C5—41no—
301C22×C5—42no—
302C22×C5—42no—
303C22×C5—41no—
304C22×C5—41no—
305C22×C5—81no—
306C22×C5C2×C3×C4⋊C411yescharacteristic
307C22×C5C3×C4×D412yes—
308C22×C5C3×#12011yescharacteristic
309C22×C5C3×C22⋊Q812yes—
310C22×C5C3×#12311yescharacteristic
311C4×C5—22no—
312C4×C5—44no—
313C4×C5—42no—
314C4×C5—42no—
315C4×C5—42no—
316C4×C5—81no—
317C24C3×C5×D411yescharacteristic
318C24C23×C3×C511yescharacteristic
319C2×D4—22no—
320C2×D4—22no—
321C2×D4C2×C3×C4×C512yes—
322C2×D4C3×C5×Q812yes—
323C22×C4—21no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—42no—
330C22×C4C3×C5×D411yescharacteristic
331C22×C4C3×C5×D412yes—
332C22×C4C3×C5×D412yes—
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—44no—
336C22⋊C4—22no—
337C22⋊C4—22no—
338C22⋊C4—22no—
339C22⋊C4—22no—
340C22⋊C4—22no—
341C22⋊C4—42no—
342C22⋊C4C2×C3×C4×C514yescomplemented
343C42—22no—
344C3×C5#186711yescharacteristic, direct product factor, Hall
345C22×C3—22no—
346C22×C3—21no—
347C22×C3—22no—
348C22×C3—21no—
349C22×C3—22no—
350C22×C3—41no—
351C22×C3—41no—
352C22×C3—42no—
353C22×C3—42no—
354C22×C3—41no—
355C22×C3—41no—
356C22×C3—81no—
357C22×C3C2×C5×C4⋊C411yescharacteristic
358C22×C3C4×C5×D412yes—
359C22×C3C5×#12011yescharacteristic
360C22×C3C5×C22⋊Q812yes—
361C22×C3C5×#12311yescharacteristic
362C3×C4—22no—
363C3×C4—44no—
364C3×C4—42no—
365C3×C4—42no—
366C3×C4—42no—
367C3×C4—81no—
368C2×C5—22no—
369C2×C5—21no—
370C2×C5—21no—
371C2×C5—22no—
372C2×C5—42no—
373C2×C5C3×#38511yescharacteristic
374C2×C5C3×B(4,𝔽2)11yescharacteristic
375C2×C5C3×#45811yescharacteristic
376C23—21no—
377C23—21no—
378C23—41no—
379C23—41no—
380C23—41no—
381C23—41no—
382C23C3×C5×C4⋊C412yescomplemented
383C23C22×C3×C4×C511yescharacteristic
384C23C2×C3×C5×D411yescharacteristic
385C23C2×C3×C5×D412yes—
386C23C2×C3×C5×Q811yescharacteristic
387C23C3×C5×#4112yes—
388D4—24no—
389D4—24no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—22no—
393C2×C4—24no—
394C2×C4—24no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—41no—
400C2×C4—42no—
401C2×C4—41no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—81no—
408C2×C4—81no—
409C2×C4C3×C5×C4⋊C412yescomplemented
410C2×C3—22no—
411C2×C3—21no—
412C2×C3—21no—
413C2×C3—22no—
414C2×C3—42no—
415C2×C3C5×#38511yescharacteristic
416C2×C3C5×B(4,𝔽2)11yescharacteristic
417C2×C3C5×#45811yescharacteristic
418C5C3×#186711yescharacteristic, direct product factor, Sylow
419C22—22no—
420C22—21no—
421C22—22no—
422C22—21no—
423C22—22no—
424C22—41no—
425C22—41no—
426C22—42no—
427C22—42no—
428C22—41no—
429C22—41no—
430C22—81no—
431C22C2×C3×C5×C4⋊C411yescharacteristic
432C22C3×C4×C5×D412yes—
433C22C3×C5×#12011yescharacteristic
434C22C3×C5×C22⋊Q812yes—
435C22C3×C5×#12311yescharacteristic
436C4—22no—
437C4—44no—
438C4—42no—
439C4—42no—
440C4—42no—
441C4—81no—
442C3C5×#186711yescharacteristic, direct product factor, Sylow
443C2—22no—
444C2—21no—
445C2—21no—
446C2—22no—
447C2—42no—
448C2C3×C5×#38511yescharacteristic
449C2C3×C5×B(4,𝔽2)11yescharacteristic
450C2C3×C5×#45811yescharacteristic
subgroupG/H
center, socleC22×C3×C5#120
commutatorC23C22×C3×C4×C5
Frattini, cosocleC24C23×C3×C5
prime radicalC22×C3×C5×D4C22
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3×C5 ◁ C24×C3×C5 ◁ G

Complements

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

NH ≅ G/Ntype
4. C3×C5×#385C2semidirect
5. C3×C5×#342C2semidirect
6. C5×#1867C3direct (N × H)
18. C3×C5×#99C4semidirect
21. C3×#1867C5direct (N × H)
25. C5×#385C2×C3semidirect
26. C5×#342C2×C3semidirect
52. C3×C5×C22⋊C4C2×C4semidirect
57. C3×#385C2×C5semidirect
58. C3×#342C2×C5semidirect
70. C5×#99C3×C4semidirect
73. #1867C3×C5direct (N × H)
80. C23×C3×C5C4⋊C4semidirect
107. C2×C3×C4×C5C4⋊C4semidirect
119. C3×#99C4×C5semidirect
147. C5×C22⋊C4C2×C3×C4semidirect
152. #385C2×C3×C5semidirect
153. #342C2×C3×C5semidirect
202. C3×C22⋊C4C2×C4×C5semidirect
210. C23×C5C3×C4⋊C4semidirect
237. C2×C4×C5C3×C4⋊C4semidirect
249. #99C3×C4×C5semidirect
266. C23×C3C5×C4⋊C4semidirect
293. C2×C3×C4C5×C4⋊C4semidirect
342. C22⋊C4C2×C3×C4×C5semidirect
344. C3×C5#1867direct (N × H)
382. C23C3×C5×C4⋊C4semidirect
409. C2×C4C3×C5×C4⋊C4semidirect
418. C5C3×#1867direct (N × H)
442. C3C5×#1867direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C5×#5182
C2×C3×C5×#992
C3×C5×#3862
C3×C5×#3852
C3×C5×#3422
C5×#18673
C3×#18675