Finite Group Databasebeta

← C3×C5×#1854 #14274888

Subgroups — C3×C5×#1854 #14274888

Order 1920.

632 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C5×#117C211yesmaximal, characteristic
2C2×C3×C5×#107C211yesmaximal, characteristic, complemented
3C2×C3×C5×#106C211yesmaximal, characteristic, complemented
4C2×C3×C5×C22⋊C8C211yesmaximal, characteristic
5C3×C5×#380C211yesmaximal, characteristic
6C3×C5×#340C211yesmaximal, characteristic, complemented
7C3×C5×#340C211yesmaximal, characteristic, complemented
8C5×#1854C311yesmaximal, characteristic, direct product factor, Hall
9C23×C3×C4×C5C2211yescharacteristic
10C22×C3×C5×C8C2211yescharacteristic
11C22×C3×C5×C8C2211yescharacteristic
12C3×C5×#117—22no—
13C3×C5×#117—22no—
14C2×C3×C5×C4⋊C4C2211yescharacteristic
15C2×C3×C5×C4⋊C4C2211yescharacteristic
16C2×C3×C5×C4⋊C4C2211yescharacteristic
17C2×C3×C5×C4⋊C4C2211yescharacteristic
18C2×C3×C5×C22⋊C4—21no—
19C2×C3×C5×C22⋊C4—21no—
20C2×C3×C42×C5—21no—
21C3×C5×#107—22no—
22C3×C5×#106—22no—
23C3×C5×C22⋊C8C414yescomplemented
24C3×C5×#95—21no—
25C3×#1854C511yesmaximal, characteristic, direct product factor, Hall
26C2×C5×#117C2×C311yescharacteristic
27C2×C5×#107C2×C311yescharacteristic, complemented
28C2×C5×#106C2×C311yescharacteristic, complemented
29C2×C5×C22⋊C8C2×C311yescharacteristic
30C5×#380C2×C311yescharacteristic
31C5×#340C2×C311yescharacteristic, complemented
32C5×#340C2×C311yescharacteristic, complemented
33C24×C3×C5D411yescharacteristic
34C22×C3×C4×C5—21no—
35C22×C3×C4×C5—21no—
36C22×C3×C4×C5—21no—
37C22×C3×C4×C5—21no—
38C22×C3×C4×C5—21no—
39C22×C3×C4×C5—21no—
40C22×C3×C4×C5—21no—
41C22×C3×C4×C5—21no—
42C22×C3×C4×C5C2×C412yes—
43C22×C3×C4×C5D411yescharacteristic
44C22×C3×C4×C5Q812yes—
45C22×C3×C4×C5C2311yescharacteristic
46C2×C3×C5×C8—21no—
47C2×C3×C5×C8—21no—
48C2×C3×C5×C8—21no—
49C2×C3×C5×C8—21no—
50C2×C3×C5×C8C2×C412yes—
51C2×C3×C5×C8C2×C412yes—
52C3×C5×C4⋊C4—21no—
53C3×C5×C4⋊C4—21no—
54C3×C5×C4⋊C4—21no—
55C3×C5×C4⋊C4—21no—
56C3×C5×C4⋊C4—21no—
57C3×C5×C4⋊C4—21no—
58C3×C5×C4⋊C4D412yes—
59C3×C5×C4⋊C4D412yes—
60C3×C5×C22⋊C4—24no—
61C3×C5×C22⋊C4—42no—
62C3×C42×C5—24no—
63C2×C3×#117C2×C511yescharacteristic
64C2×C3×#107C2×C511yescharacteristic, complemented
65C2×C3×#106C2×C511yescharacteristic, complemented
66C2×C3×C22⋊C8C2×C511yescharacteristic
67C3×#380C2×C511yescharacteristic
68C3×#340C2×C511yescharacteristic, complemented
69C3×#340C2×C511yescharacteristic, complemented
70C23×C4×C5C22×C311yescharacteristic
71C22×C5×C8C22×C311yescharacteristic
72C22×C5×C8C22×C311yescharacteristic
73C5×#117—22no—
74C5×#117—22no—
75C2×C5×C4⋊C4C22×C311yescharacteristic
76C2×C5×C4⋊C4C22×C311yescharacteristic
77C2×C5×C4⋊C4C22×C311yescharacteristic
78C2×C5×C4⋊C4C22×C311yescharacteristic
79C2×C5×C22⋊C4—21no—
80C2×C5×C22⋊C4—21no—
81C2×C42×C5—21no—
82C5×#107—22no—
83C5×#106—22no—
84C5×C22⋊C8C3×C414yescomplemented
85C5×#95—21no—
86#1854C3×C511yescharacteristic, direct product factor, Sylow
87C23×C3×C5—21no—
88C23×C3×C5—21no—
89C23×C3×C5—21no—
90C23×C3×C5—21no—
91C23×C3×C5—21no—
92C23×C3×C5—21no—
93C23×C3×C5C4⋊C412yes—
94C23×C3×C5C2×D411yescharacteristic
95C2×C3×C4×C5—21no—
96C2×C3×C4×C5—21no—
97C2×C3×C4×C5—22no—
98C2×C3×C4×C5—22no—
99C2×C3×C4×C5—22no—
100C2×C3×C4×C5—22no—
101C2×C3×C4×C5—22no—
102C2×C3×C4×C5—22no—
103C2×C3×C4×C5—41no—
104C2×C3×C4×C5—41no—
105C2×C3×C4×C5—41no—
106C2×C3×C4×C5—41no—
107C2×C3×C4×C5—41no—
108C2×C3×C4×C5—41no—
109C2×C3×C4×C5—41no—
110C2×C3×C4×C5—41no—
111C2×C3×C4×C5—41no—
112C2×C3×C4×C5—41no—
113C2×C3×C4×C5C4⋊C412yes—
114C2×C3×C4×C5C22×C411yescharacteristic
115C2×C3×C4×C5C2×D411yescharacteristic
116C2×C3×C4×C5C2×D411yescharacteristic
117C2×C3×C4×C5C2×Q811yescharacteristic
118C2×C3×C4×C5#4111yescharacteristic
119C2×C3×C4×C5#4111yescharacteristic
120C3×C5×C8—22no—
121C3×C5×C8—22no—
122C23×C3×C4C22×C511yescharacteristic
123C22×C3×C8C22×C511yescharacteristic
124C22×C3×C8C22×C511yescharacteristic
125C3×#117—22no—
126C3×#117—22no—
127C2×C3×C4⋊C4C22×C511yescharacteristic
128C2×C3×C4⋊C4C22×C511yescharacteristic
129C2×C3×C4⋊C4C22×C511yescharacteristic
130C2×C3×C4⋊C4C22×C511yescharacteristic
131C2×C3×C22⋊C4—21no—
132C2×C3×C22⋊C4—21no—
133C2×C3×C42—21no—
134C3×#107—22no—
135C3×#106—22no—
136C3×C22⋊C8C4×C514yescomplemented
137C3×#95—21no—
138C24×C5C3×D411yescharacteristic
139C22×C4×C5—21no—
140C22×C4×C5—21no—
141C22×C4×C5—21no—
142C22×C4×C5—21no—
143C22×C4×C5—21no—
144C22×C4×C5—21no—
145C22×C4×C5—21no—
146C22×C4×C5—21no—
147C22×C4×C5C2×C3×C412yes—
148C22×C4×C5C3×D411yescharacteristic
149C22×C4×C5C3×Q812yes—
150C22×C4×C5C23×C311yescharacteristic
151C2×C5×C8—21no—
152C2×C5×C8—21no—
153C2×C5×C8—21no—
154C2×C5×C8—21no—
155C2×C5×C8C2×C3×C412yes—
156C2×C5×C8C2×C3×C412yes—
157C5×C4⋊C4—21no—
158C5×C4⋊C4—21no—
159C5×C4⋊C4—21no—
160C5×C4⋊C4—21no—
161C5×C4⋊C4—21no—
162C5×C4⋊C4—21no—
163C5×C4⋊C4C3×D412yes—
164C5×C4⋊C4C3×D412yes—
165C5×C22⋊C4—24no—
166C5×C22⋊C4—42no—
167C42×C5—24no—
168C2×#117C2×C3×C511yescharacteristic
169C2×#107C2×C3×C511yescharacteristic, complemented
170C2×#106C2×C3×C511yescharacteristic, complemented
171C2×C22⋊C8C2×C3×C511yescharacteristic
172#380C2×C3×C511yescharacteristic
173#340C2×C3×C511yescharacteristic, complemented
174#340C2×C3×C511yescharacteristic, complemented
175C22×C3×C5—22no—
176C22×C3×C5—22no—
177C22×C3×C5—22no—
178C22×C3×C5—41no—
179C22×C3×C5—41no—
180C22×C3×C5—41no—
181C22×C3×C5—41no—
182C22×C3×C5C2×C4⋊C411yescharacteristic
183C22×C3×C5#12011yescharacteristic
184C22×C3×C5#12311yescharacteristic
185C22×C3×C5#13511yescharacteristic
186C22×C3×C5#13511yescharacteristic
187C22×C3×C5AGL(1,ℤ/8)11yescharacteristic
188C22×C3×C5#13711yescharacteristic
189C3×C4×C5—22no—
190C3×C4×C5—42no—
191C3×C4×C5—42no—
192C3×C4×C5—81no—
193C3×C4×C5—81no—
194C3×C4×C5C4×D411yescharacteristic
195C3×C4×C5C4×D411yescharacteristic
196C3×C4×C5C22⋊Q811yescharacteristic
197C3×C4×C5C22⋊Q811yescharacteristic
198C24×C3C5×D411yescharacteristic
199C22×C3×C4—21no—
200C22×C3×C4—21no—
201C22×C3×C4—21no—
202C22×C3×C4—21no—
203C22×C3×C4—21no—
204C22×C3×C4—21no—
205C22×C3×C4—21no—
206C22×C3×C4—21no—
207C22×C3×C4C2×C4×C512yes—
208C22×C3×C4C5×D411yescharacteristic
209C22×C3×C4C5×Q812yes—
210C22×C3×C4C23×C511yescharacteristic
211C2×C3×C8—21no—
212C2×C3×C8—21no—
213C2×C3×C8—21no—
214C2×C3×C8—21no—
215C2×C3×C8C2×C4×C512yes—
216C2×C3×C8C2×C4×C512yes—
217C3×C4⋊C4—21no—
218C3×C4⋊C4—21no—
219C3×C4⋊C4—21no—
220C3×C4⋊C4—21no—
221C3×C4⋊C4—21no—
222C3×C4⋊C4—21no—
223C3×C4⋊C4C5×D412yes—
224C3×C4⋊C4C5×D412yes—
225C3×C22⋊C4—24no—
226C3×C22⋊C4—42no—
227C3×C42—24no—
228C23×C5—21no—
229C23×C5—21no—
230C23×C5—21no—
231C23×C5—21no—
232C23×C5—21no—
233C23×C5—21no—
234C23×C5C3×C4⋊C412yes—
235C23×C5C2×C3×D411yescharacteristic
236C2×C4×C5—21no—
237C2×C4×C5—21no—
238C2×C4×C5—22no—
239C2×C4×C5—22no—
240C2×C4×C5—22no—
241C2×C4×C5—22no—
242C2×C4×C5—22no—
243C2×C4×C5—22no—
244C2×C4×C5—41no—
245C2×C4×C5—41no—
246C2×C4×C5—41no—
247C2×C4×C5—41no—
248C2×C4×C5—41no—
249C2×C4×C5—41no—
250C2×C4×C5—41no—
251C2×C4×C5—41no—
252C2×C4×C5—41no—
253C2×C4×C5—41no—
254C2×C4×C5C3×C4⋊C412yes—
255C2×C4×C5C22×C3×C411yescharacteristic
256C2×C4×C5C2×C3×D411yescharacteristic
257C2×C4×C5C2×C3×D411yescharacteristic
258C2×C4×C5C2×C3×Q811yescharacteristic
259C2×C4×C5C3×#4111yescharacteristic
260C2×C4×C5C3×#4111yescharacteristic
261C5×C8—22no—
262C5×C8—22no—
263C23×C4C22×C3×C511yescharacteristic
264C22×C8C22×C3×C511yescharacteristic
265C22×C8C22×C3×C511yescharacteristic
266#117—22no—
267#117—22no—
268C2×C4⋊C4C22×C3×C511yescharacteristic
269C2×C4⋊C4C22×C3×C511yescharacteristic
270C2×C4⋊C4C22×C3×C511yescharacteristic
271C2×C4⋊C4C22×C3×C511yescharacteristic
272C2×C22⋊C4—21no—
273C2×C22⋊C4—21no—
274C2×C42—21no—
275#107—22no—
276#106—22no—
277C22⋊C8C3×C4×C514yescomplemented
278#95—21no—
279C2×C3×C5—42no—
280C2×C3×C5#38511yescharacteristic
281C2×C3×C5#42711yescharacteristic
282C2×C3×C5#42811yescharacteristic
283C2×C3×C5#44911yescharacteristic
284C2×C3×C5#45211yescharacteristic
285C2×C3×C5#48211yescharacteristic
286C2×C3×C5#48511yescharacteristic
287C23×C3—21no—
288C23×C3—21no—
289C23×C3—21no—
290C23×C3—21no—
291C23×C3—21no—
292C23×C3—21no—
293C23×C3C5×C4⋊C412yes—
294C23×C3C2×C5×D411yescharacteristic
295C2×C3×C4—21no—
296C2×C3×C4—21no—
297C2×C3×C4—22no—
298C2×C3×C4—22no—
299C2×C3×C4—22no—
300C2×C3×C4—22no—
301C2×C3×C4—22no—
302C2×C3×C4—22no—
303C2×C3×C4—41no—
304C2×C3×C4—41no—
305C2×C3×C4—41no—
306C2×C3×C4—41no—
307C2×C3×C4—41no—
308C2×C3×C4—41no—
309C2×C3×C4—41no—
310C2×C3×C4—41no—
311C2×C3×C4—41no—
312C2×C3×C4—41no—
313C2×C3×C4C5×C4⋊C412yes—
314C2×C3×C4C22×C4×C511yescharacteristic
315C2×C3×C4C2×C5×D411yescharacteristic
316C2×C3×C4C2×C5×D411yescharacteristic
317C2×C3×C4C2×C5×Q811yescharacteristic
318C2×C3×C4C5×#4111yescharacteristic
319C2×C3×C4C5×#4111yescharacteristic
320C3×C8—22no—
321C3×C8—22no—
322C22×C5—22no—
323C22×C5—22no—
324C22×C5—22no—
325C22×C5—41no—
326C22×C5—41no—
327C22×C5—41no—
328C22×C5—41no—
329C22×C5C2×C3×C4⋊C411yescharacteristic
330C22×C5C3×#12011yescharacteristic
331C22×C5C3×#12311yescharacteristic
332C22×C5C3×#13511yescharacteristic
333C22×C5C3×#13511yescharacteristic
334C22×C5C3×AGL(1,ℤ/8)11yescharacteristic
335C22×C5C3×#13711yescharacteristic
336C4×C5—22no—
337C4×C5—42no—
338C4×C5—42no—
339C4×C5—81no—
340C4×C5—81no—
341C4×C5C3×C4×D411yescharacteristic
342C4×C5C3×C4×D411yescharacteristic
343C4×C5C3×C22⋊Q811yescharacteristic
344C4×C5C3×C22⋊Q811yescharacteristic
345C24C3×C5×D411yescharacteristic
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4C2×C3×C4×C512yes—
355C22×C4C3×C5×D411yescharacteristic
356C22×C4C3×C5×Q812yes—
357C22×C4C23×C3×C511yescharacteristic
358C2×C8—21no—
359C2×C8—21no—
360C2×C8—21no—
361C2×C8—21no—
362C2×C8C2×C3×C4×C512yes—
363C2×C8C2×C3×C4×C512yes—
364C4⋊C4—21no—
365C4⋊C4—21no—
366C4⋊C4—21no—
367C4⋊C4—21no—
368C4⋊C4—21no—
369C4⋊C4—21no—
370C4⋊C4C3×C5×D412yes—
371C4⋊C4C3×C5×D412yes—
372C22⋊C4—24no—
373C22⋊C4—42no—
374C42—24no—
375C3×C5#185411yescharacteristic, direct product factor, Hall
376C22×C3—22no—
377C22×C3—22no—
378C22×C3—22no—
379C22×C3—41no—
380C22×C3—41no—
381C22×C3—41no—
382C22×C3—41no—
383C22×C3C2×C5×C4⋊C411yescharacteristic
384C22×C3C5×#12011yescharacteristic
385C22×C3C5×#12311yescharacteristic
386C22×C3C5×#13511yescharacteristic
387C22×C3C5×#13511yescharacteristic
388C22×C3C5×AGL(1,ℤ/8)11yescharacteristic
389C22×C3C5×#13711yescharacteristic
390C3×C4—22no—
391C3×C4—42no—
392C3×C4—42no—
393C3×C4—81no—
394C3×C4—81no—
395C3×C4C4×C5×D411yescharacteristic
396C3×C4C4×C5×D411yescharacteristic
397C3×C4C5×C22⋊Q811yescharacteristic
398C3×C4C5×C22⋊Q811yescharacteristic
399C2×C5—42no—
400C2×C5C3×#38511yescharacteristic
401C2×C5C3×#42711yescharacteristic
402C2×C5C3×#42811yescharacteristic
403C2×C5C3×#44911yescharacteristic
404C2×C5C3×#45211yescharacteristic
405C2×C5C3×#48211yescharacteristic
406C2×C5C3×#48511yescharacteristic
407C23—21no—
408C23—21no—
409C23—21no—
410C23—21no—
411C23—21no—
412C23—21no—
413C23C3×C5×C4⋊C412yes—
414C23C2×C3×C5×D411yescharacteristic
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4C3×C5×C4⋊C412yes—
434C2×C4C22×C3×C4×C511yescharacteristic
435C2×C4C2×C3×C5×D411yescharacteristic
436C2×C4C2×C3×C5×D411yescharacteristic
437C2×C4C2×C3×C5×Q811yescharacteristic
438C2×C4C3×C5×#4111yescharacteristic
439C2×C4C3×C5×#4111yescharacteristic
440C8—22no—
441C8—22no—
442C2×C3—42no—
443C2×C3C5×#38511yescharacteristic
444C2×C3C5×#42711yescharacteristic
445C2×C3C5×#42811yescharacteristic
446C2×C3C5×#44911yescharacteristic
447C2×C3C5×#45211yescharacteristic
448C2×C3C5×#48211yescharacteristic
449C2×C3C5×#48511yescharacteristic
450C5C3×#185411yescharacteristic, direct product factor, Sylow
451C22—22no—
452C22—22no—
453C22—22no—
454C22—41no—
455C22—41no—
456C22—41no—
457C22—41no—
458C22C2×C3×C5×C4⋊C411yescharacteristic
459C22C3×C5×#12011yescharacteristic
460C22C3×C5×#12311yescharacteristic
461C22C3×C5×#13511yescharacteristic
462C22C3×C5×#13511yescharacteristic
463C22C3×C5×AGL(1,ℤ/8)11yescharacteristic
464C22C3×C5×#13711yescharacteristic
465C4—22no—
466C4—42no—
467C4—42no—
468C4—81no—
469C4—81no—
470C4C3×C4×C5×D411yescharacteristic
471C4C3×C4×C5×D411yescharacteristic
472C4C3×C5×C22⋊Q811yescharacteristic
473C4C3×C5×C22⋊Q811yescharacteristic
474C3C5×#185411yescharacteristic, direct product factor, Sylow
475C2—42no—
476C2C3×C5×#38511yescharacteristic
477C2C3×C5×#42711yescharacteristic
478C2C3×C5×#42811yescharacteristic
479C2C3×C5×#44911yescharacteristic
480C2C3×C5×#45211yescharacteristic
481C2C3×C5×#48211yescharacteristic
482C2C3×C5×#48511yescharacteristic
subgroupG/H
center, socleC23×C3×C5C2×D4
commutatorC2×C4C22×C3×C4×C5
Frattini, cosocleC22×C4C23×C3×C5
prime radicalC24×C3×C5D4
LCSG ▷ C2×C4 ▷ C2 ▷ 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×#107C2semidirect
3. C2×C3×C5×#106C2semidirect
6. C3×C5×#340C2semidirect
7. C3×C5×#340C2semidirect
8. C5×#1854C3direct (N × H)
23. C3×C5×C22⋊C8C4semidirect
25. C3×#1854C5direct (N × H)
27. C2×C5×#107C2×C3semidirect
28. C2×C5×#106C2×C3semidirect
31. C5×#340C2×C3semidirect
32. C5×#340C2×C3semidirect
64. C2×C3×#107C2×C5semidirect
65. C2×C3×#106C2×C5semidirect
68. C3×#340C2×C5semidirect
69. C3×#340C2×C5semidirect
84. C5×C22⋊C8C3×C4semidirect
86. #1854C3×C5direct (N × H)
136. C3×C22⋊C8C4×C5semidirect
169. C2×#107C2×C3×C5semidirect
170. C2×#106C2×C3×C5semidirect
173. #340C2×C3×C5semidirect
174. #340C2×C3×C5semidirect
277. C22⋊C8C3×C4×C5semidirect
375. C3×C5#1854direct (N × H)
450. C5C3×#1854direct (N × H)
474. C3C5×#1854direct (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×#1172
C2×C3×C5×#1072
C2×C3×C5×#1062
C2×C3×C5×C22⋊C82
C3×C5×#3802
C3×C5×#3402
C5×#18543
C3×#18545