Finite Group Databasebeta

← C3×C5×#2370 #14275404

Subgroups — C3×C5×#2370 #14275404

Order 1920.

928 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C5×#125C211yesmaximal, characteristic
2C3×C5×#389C211yesmaximal, characteristic
3C3×C5×#389C211yesmaximal, characteristic
4C3×C5×#389C212yesmaximal
5C3×C5×#387C212yesmaximal
6C3×C5×#387C212yesmaximal
7C3×C4×C5×C4⋊C4C211yesmaximal, characteristic
8C3×C4×C5×C4⋊C4C212yesmaximal
9C3×C4×C5×C4⋊C4C211yesmaximal, characteristic
10C3×C4×C5×C4⋊C4C211yesmaximal, characteristic
11C3×C4×C5×C4⋊C4C211yesmaximal, characteristic
12C5×#2370C311yesmaximal, characteristic, direct product factor, Hall
13C3×C5×#125C418yescomplemented
14C2×C3×C5×C4⋊C4C2211yescharacteristic
15C2×C3×C5×C4⋊C4C2211yescharacteristic
16C2×C3×C5×C4⋊C4C2211yescharacteristic
17C2×C3×C5×C4⋊C4C2212yes—
18C2×C3×C5×C4⋊C4C2212yes—
19C2×C3×C5×C4⋊C4C2212yes—
20C2×C3×C5×C4⋊C4C2211yescharacteristic
21C2×C3×C5×C4⋊C4C2211yescharacteristic
22C2×C3×C5×C4⋊C4C2212yes—
23C2×C3×C5×C4⋊C4C2211yescharacteristic
24C2×C3×C42×C5C2212yes—
25C2×C3×C42×C5C2211yescharacteristic
26C2×C3×C42×C5C2211yescharacteristic
27C2×C3×C42×C5C2212yes—
28C2×C3×C42×C5C2212yes—
29C2×C3×C42×C5C2211yescharacteristic
30C3×C5×#95C2211yescharacteristic
31C3×C5×#95C2211yescharacteristic
32C3×C5×#95C2212yes—
33C3×C5×#95C2212yes—
34C3×C5×#95C2212yes—
35C3×C5×#95C2212yes—
36C3×C5×#95C2212yes—
37C3×#2370C511yesmaximal, characteristic, direct product factor, Hall
38C2×C5×#125C2×C311yescharacteristic
39C5×#389C2×C311yescharacteristic
40C5×#389C2×C311yescharacteristic
41C5×#389C2×C312yes—
42C5×#387C2×C312yes—
43C5×#387C2×C312yes—
44C4×C5×C4⋊C4C2×C311yescharacteristic
45C4×C5×C4⋊C4C2×C312yes—
46C4×C5×C4⋊C4C2×C311yescharacteristic
47C4×C5×C4⋊C4C2×C311yescharacteristic
48C4×C5×C4⋊C4C2×C311yescharacteristic
49C22×C3×C4×C5C2311yescharacteristic
50C22×C3×C4×C5C2312yes—
51C22×C3×C4×C5C2311yescharacteristic
52C22×C3×C4×C5C2311yescharacteristic
53C22×C3×C4×C5C2312yes—
54C22×C3×C4×C5C2311yescharacteristic
55C22×C3×C4×C5C2312yes—
56C22×C3×C4×C5C2312yes—
57C22×C3×C4×C5C2311yescharacteristic
58C22×C3×C4×C5C2311yescharacteristic
59C22×C3×C4×C5C2311yescharacteristic
60C3×C5×C4⋊C4—22no—
61C3×C5×C4⋊C4—24no—
62C3×C5×C4⋊C4—22no—
63C3×C5×C4⋊C4—22no—
64C3×C5×C4⋊C4—42no—
65C3×C5×C4⋊C4C2×C418yes—
66C3×C5×C4⋊C4C2×C418yes—
67C3×C5×C4⋊C4C2×C414yes—
68C3×C5×C4⋊C4C2×C414yes—
69C3×C5×C4⋊C4Q814yes—
70C3×C42×C5—24no—
71C3×C42×C5—22no—
72C3×C42×C5—22no—
73C3×C42×C5—42no—
74C3×C42×C5—42no—
75C3×C42×C5C2×C414yes—
76C2×C3×#125C2×C511yescharacteristic
77C3×#389C2×C511yescharacteristic
78C3×#389C2×C512yes—
79C3×#389C2×C511yescharacteristic
80C3×#387C2×C512yes—
81C3×#387C2×C512yes—
82C3×C4×C4⋊C4C2×C511yescharacteristic
83C3×C4×C4⋊C4C2×C512yes—
84C3×C4×C4⋊C4C2×C511yescharacteristic
85C3×C4×C4⋊C4C2×C511yescharacteristic
86C3×C4×C4⋊C4C2×C511yescharacteristic
87C5×#125C3×C418yescomplemented
88C2×C5×C4⋊C4C22×C311yescharacteristic
89C2×C5×C4⋊C4C22×C311yescharacteristic
90C2×C5×C4⋊C4C22×C311yescharacteristic
91C2×C5×C4⋊C4C22×C312yes—
92C2×C5×C4⋊C4C22×C312yes—
93C2×C5×C4⋊C4C22×C312yes—
94C2×C5×C4⋊C4C22×C311yescharacteristic
95C2×C5×C4⋊C4C22×C311yescharacteristic
96C2×C5×C4⋊C4C22×C312yes—
97C2×C5×C4⋊C4C22×C311yescharacteristic
98C2×C42×C5C22×C312yes—
99C2×C42×C5C22×C311yescharacteristic
100C2×C42×C5C22×C311yescharacteristic
101C2×C42×C5C22×C312yes—
102C2×C42×C5C22×C312yes—
103C2×C42×C5C22×C311yescharacteristic
104C5×#95C22×C311yescharacteristic
105C5×#95C22×C311yescharacteristic
106C5×#95C22×C312yes—
107C5×#95C22×C312yes—
108C5×#95C22×C312yes—
109C5×#95C22×C312yes—
110C5×#95C22×C312yes—
111#2370C3×C511yescharacteristic, direct product factor, Sylow
112C23×C3×C5C2411yescharacteristic
113C2×C3×C4×C5—22no—
114C2×C3×C4×C5—21no—
115C2×C3×C4×C5—21no—
116C2×C3×C4×C5—22no—
117C2×C3×C4×C5—21no—
118C2×C3×C4×C5—22no—
119C2×C3×C4×C5—21no—
120C2×C3×C4×C5—21no—
121C2×C3×C4×C5—21no—
122C2×C3×C4×C5—22no—
123C2×C3×C4×C5—22no—
124C2×C3×C4×C5—21no—
125C2×C3×C4×C5—22no—
126C2×C3×C4×C5—21no—
127C2×C3×C4×C5—21no—
128C2×C3×C4×C5—21no—
129C2×C3×C4×C5—21no—
130C2×C3×C4×C5—22no—
131C2×C3×C4×C5—21no—
132C2×C3×C4×C5—22no—
133C2×C3×C4×C5—22no—
134C2×C3×C4×C5—21no—
135C2×C3×C4×C5—21no—
136C2×C3×C4×C5C22×C412yes—
137C2×C3×C4×C5C22×C412yes—
138C2×C3×C4×C5C22×C414yes—
139C2×C3×C4×C5C22×C414yes—
140C2×C3×C4×C5C22×C412yes—
141C2×C3×C4×C5C2×Q814yes—
142C2×C3×C4×C5C2×Q812yes—
143C2×C3×C4×C5#4112yes—
144C2×C3×C4×C5#4112yes—
145C2×C3×C4×C5#4112yes—
146C3×#125C4×C518yescomplemented
147C2×C3×C4⋊C4C22×C511yescharacteristic
148C2×C3×C4⋊C4C22×C511yescharacteristic
149C2×C3×C4⋊C4C22×C511yescharacteristic
150C2×C3×C4⋊C4C22×C512yes—
151C2×C3×C4⋊C4C22×C512yes—
152C2×C3×C4⋊C4C22×C512yes—
153C2×C3×C4⋊C4C22×C511yescharacteristic
154C2×C3×C4⋊C4C22×C511yescharacteristic
155C2×C3×C4⋊C4C22×C512yes—
156C2×C3×C4⋊C4C22×C511yescharacteristic
157C2×C3×C42C22×C512yes—
158C2×C3×C42C22×C511yescharacteristic
159C2×C3×C42C22×C511yescharacteristic
160C2×C3×C42C22×C512yes—
161C2×C3×C42C22×C512yes—
162C2×C3×C42C22×C511yescharacteristic
163C3×#95C22×C511yescharacteristic
164C3×#95C22×C511yescharacteristic
165C3×#95C22×C512yes—
166C3×#95C22×C512yes—
167C3×#95C22×C512yes—
168C3×#95C22×C512yes—
169C3×#95C22×C512yes—
170C22×C4×C5C23×C311yescharacteristic
171C22×C4×C5C23×C312yes—
172C22×C4×C5C23×C311yescharacteristic
173C22×C4×C5C23×C311yescharacteristic
174C22×C4×C5C23×C312yes—
175C22×C4×C5C23×C311yescharacteristic
176C22×C4×C5C23×C312yes—
177C22×C4×C5C23×C312yes—
178C22×C4×C5C23×C311yescharacteristic
179C22×C4×C5C23×C311yescharacteristic
180C22×C4×C5C23×C311yescharacteristic
181C5×C4⋊C4—22no—
182C5×C4⋊C4—24no—
183C5×C4⋊C4—22no—
184C5×C4⋊C4—22no—
185C5×C4⋊C4—42no—
186C5×C4⋊C4C2×C3×C418yes—
187C5×C4⋊C4C2×C3×C418yes—
188C5×C4⋊C4C2×C3×C414yes—
189C5×C4⋊C4C2×C3×C414yes—
190C5×C4⋊C4C3×Q814yes—
191C42×C5—24no—
192C42×C5—22no—
193C42×C5—22no—
194C42×C5—42no—
195C42×C5—42no—
196C42×C5C2×C3×C414yes—
197C2×#125C2×C3×C511yescharacteristic
198#389C2×C3×C511yescharacteristic
199#389C2×C3×C512yes—
200#389C2×C3×C511yescharacteristic
201#387C2×C3×C512yes—
202#387C2×C3×C512yes—
203C4×C4⋊C4C2×C3×C511yescharacteristic
204C4×C4⋊C4C2×C3×C512yes—
205C4×C4⋊C4C2×C3×C511yescharacteristic
206C4×C4⋊C4C2×C3×C511yescharacteristic
207C4×C4⋊C4C2×C3×C511yescharacteristic
208C22×C3×C5C23×C411yescharacteristic
209C22×C3×C5C22×Q811yescharacteristic
210C22×C3×C5C2×#4111yescharacteristic
211C22×C3×C5C2×#4111yescharacteristic
212C22×C3×C5C2×#4111yescharacteristic
213C22×C3×C5H(5,𝔽2)11yescharacteristic
214C22×C3×C52−1+411yescharacteristic
215C3×C4×C5—24no—
216C3×C4×C5—22no—
217C3×C4×C5—22no—
218C3×C4×C5—24no—
219C3×C4×C5—22no—
220C3×C4×C5—22no—
221C3×C4×C5—24no—
222C3×C4×C5—41no—
223C3×C4×C5—41no—
224C3×C4×C5—42no—
225C3×C4×C5C4×Q814yes—
226C22×C3×C4C23×C511yescharacteristic
227C22×C3×C4C23×C511yescharacteristic
228C22×C3×C4C23×C512yes—
229C22×C3×C4C23×C511yescharacteristic
230C22×C3×C4C23×C512yes—
231C22×C3×C4C23×C511yescharacteristic
232C22×C3×C4C23×C512yes—
233C22×C3×C4C23×C512yes—
234C22×C3×C4C23×C511yescharacteristic
235C22×C3×C4C23×C511yescharacteristic
236C22×C3×C4C23×C511yescharacteristic
237C3×C4⋊C4—22no—
238C3×C4⋊C4—24no—
239C3×C4⋊C4—22no—
240C3×C4⋊C4—22no—
241C3×C4⋊C4—42no—
242C3×C4⋊C4C2×C4×C518yes—
243C3×C4⋊C4C2×C4×C518yes—
244C3×C4⋊C4C2×C4×C514yes—
245C3×C4⋊C4C2×C4×C514yes—
246C3×C4⋊C4C5×Q814yes—
247C3×C42—24no—
248C3×C42—22no—
249C3×C42—22no—
250C3×C42—42no—
251C3×C42—42no—
252C3×C42C2×C4×C514yes—
253C23×C5C24×C311yescharacteristic
254C2×C4×C5—22no—
255C2×C4×C5—21no—
256C2×C4×C5—21no—
257C2×C4×C5—22no—
258C2×C4×C5—21no—
259C2×C4×C5—22no—
260C2×C4×C5—21no—
261C2×C4×C5—21no—
262C2×C4×C5—21no—
263C2×C4×C5—22no—
264C2×C4×C5—22no—
265C2×C4×C5—21no—
266C2×C4×C5—22no—
267C2×C4×C5—21no—
268C2×C4×C5—21no—
269C2×C4×C5—21no—
270C2×C4×C5—21no—
271C2×C4×C5—22no—
272C2×C4×C5—21no—
273C2×C4×C5—22no—
274C2×C4×C5—22no—
275C2×C4×C5—21no—
276C2×C4×C5—21no—
277C2×C4×C5C22×C3×C412yes—
278C2×C4×C5C22×C3×C412yes—
279C2×C4×C5C22×C3×C414yes—
280C2×C4×C5C22×C3×C414yes—
281C2×C4×C5C22×C3×C412yes—
282C2×C4×C5C2×C3×Q814yes—
283C2×C4×C5C2×C3×Q812yes—
284C2×C4×C5C3×#4112yes—
285C2×C4×C5C3×#4112yes—
286C2×C4×C5C3×#4112yes—
287#125C3×C4×C518yescomplemented
288C2×C4⋊C4C22×C3×C511yescharacteristic
289C2×C4⋊C4C22×C3×C511yescharacteristic
290C2×C4⋊C4C22×C3×C511yescharacteristic
291C2×C4⋊C4C22×C3×C512yes—
292C2×C4⋊C4C22×C3×C512yes—
293C2×C4⋊C4C22×C3×C512yes—
294C2×C4⋊C4C22×C3×C511yescharacteristic
295C2×C4⋊C4C22×C3×C511yescharacteristic
296C2×C4⋊C4C22×C3×C512yes—
297C2×C4⋊C4C22×C3×C511yescharacteristic
298C2×C42C22×C3×C512yes—
299C2×C42C22×C3×C511yescharacteristic
300C2×C42C22×C3×C511yescharacteristic
301C2×C42C22×C3×C512yes—
302C2×C42C22×C3×C512yes—
303C2×C42C22×C3×C511yescharacteristic
304#95C22×C3×C511yescharacteristic
305#95C22×C3×C511yescharacteristic
306#95C22×C3×C512yes—
307#95C22×C3×C512yes—
308#95C22×C3×C512yes—
309#95C22×C3×C512yes—
310#95C22×C3×C512yes—
311C2×C3×C5C2×C4×Q811yescharacteristic
312C2×C3×C5C4×#4111yescharacteristic
313C2×C3×C5#52011yescharacteristic
314C2×C3×C5#55211yescharacteristic
315C2×C3×C5#55311yescharacteristic
316C2×C3×C5#55411yescharacteristic
317C2×C3×C5#55711yescharacteristic
318C23×C3C24×C511yescharacteristic
319C2×C3×C4—22no—
320C2×C3×C4—21no—
321C2×C3×C4—21no—
322C2×C3×C4—22no—
323C2×C3×C4—21no—
324C2×C3×C4—22no—
325C2×C3×C4—21no—
326C2×C3×C4—21no—
327C2×C3×C4—21no—
328C2×C3×C4—22no—
329C2×C3×C4—22no—
330C2×C3×C4—21no—
331C2×C3×C4—22no—
332C2×C3×C4—21no—
333C2×C3×C4—21no—
334C2×C3×C4—21no—
335C2×C3×C4—21no—
336C2×C3×C4—22no—
337C2×C3×C4—21no—
338C2×C3×C4—22no—
339C2×C3×C4—22no—
340C2×C3×C4—21no—
341C2×C3×C4—21no—
342C2×C3×C4C22×C4×C512yes—
343C2×C3×C4C22×C4×C512yes—
344C2×C3×C4C22×C4×C514yes—
345C2×C3×C4C22×C4×C514yes—
346C2×C3×C4C22×C4×C512yes—
347C2×C3×C4C2×C5×Q814yes—
348C2×C3×C4C2×C5×Q812yes—
349C2×C3×C4C5×#4112yes—
350C2×C3×C4C5×#4112yes—
351C2×C3×C4C5×#4112yes—
352C22×C5C23×C3×C411yescharacteristic
353C22×C5C22×C3×Q811yescharacteristic
354C22×C5C2×C3×#4111yescharacteristic
355C22×C5C2×C3×#4111yescharacteristic
356C22×C5C2×C3×#4111yescharacteristic
357C22×C5C3×H(5,𝔽2)11yescharacteristic
358C22×C5C3×2−1+411yescharacteristic
359C4×C5—24no—
360C4×C5—22no—
361C4×C5—22no—
362C4×C5—24no—
363C4×C5—22no—
364C4×C5—22no—
365C4×C5—24no—
366C4×C5—41no—
367C4×C5—41no—
368C4×C5—42no—
369C4×C5C3×C4×Q814yes—
370C22×C4C23×C3×C511yescharacteristic
371C22×C4C23×C3×C511yescharacteristic
372C22×C4C23×C3×C512yes—
373C22×C4C23×C3×C511yescharacteristic
374C22×C4C23×C3×C512yes—
375C22×C4C23×C3×C511yescharacteristic
376C22×C4C23×C3×C512yes—
377C22×C4C23×C3×C512yes—
378C22×C4C23×C3×C511yescharacteristic
379C22×C4C23×C3×C511yescharacteristic
380C22×C4C23×C3×C511yescharacteristic
381C4⋊C4—22no—
382C4⋊C4—24no—
383C4⋊C4—22no—
384C4⋊C4—22no—
385C4⋊C4—42no—
386C4⋊C4C2×C3×C4×C518yes—
387C4⋊C4C2×C3×C4×C518yes—
388C4⋊C4C2×C3×C4×C514yes—
389C4⋊C4C2×C3×C4×C514yes—
390C4⋊C4C3×C5×Q814yes—
391C42—24no—
392C42—22no—
393C42—22no—
394C42—42no—
395C42—42no—
396C42C2×C3×C4×C514yes—
397C3×C5#237011yescharacteristic, direct product factor, Hall
398C22×C3C23×C4×C511yescharacteristic
399C22×C3C22×C5×Q811yescharacteristic
400C22×C3C2×C5×#4111yescharacteristic
401C22×C3C2×C5×#4111yescharacteristic
402C22×C3C2×C5×#4111yescharacteristic
403C22×C3C5×H(5,𝔽2)11yescharacteristic
404C22×C3C5×2−1+411yescharacteristic
405C3×C4—24no—
406C3×C4—22no—
407C3×C4—22no—
408C3×C4—24no—
409C3×C4—22no—
410C3×C4—22no—
411C3×C4—24no—
412C3×C4—41no—
413C3×C4—41no—
414C3×C4—42no—
415C3×C4C4×C5×Q814yes—
416C2×C5C2×C3×C4×Q811yescharacteristic
417C2×C5C3×C4×#4111yescharacteristic
418C2×C5C3×#52011yescharacteristic
419C2×C5C3×#55211yescharacteristic
420C2×C5C3×#55311yescharacteristic
421C2×C5C3×#55411yescharacteristic
422C2×C5C3×#55711yescharacteristic
423C23C24×C3×C511yescharacteristic
424C2×C4—22no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—22no—
428C2×C4—21no—
429C2×C4—22no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—21no—
436C2×C4—22no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—22no—
442C2×C4—21no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4C22×C3×C4×C512yes—
448C2×C4C22×C3×C4×C512yes—
449C2×C4C22×C3×C4×C514yes—
450C2×C4C22×C3×C4×C514yes—
451C2×C4C22×C3×C4×C512yes—
452C2×C4C2×C3×C5×Q814yes—
453C2×C4C2×C3×C5×Q812yes—
454C2×C4C3×C5×#4112yes—
455C2×C4C3×C5×#4112yes—
456C2×C4C3×C5×#4112yes—
457C2×C3C2×C4×C5×Q811yescharacteristic
458C2×C3C4×C5×#4111yescharacteristic
459C2×C3C5×#52011yescharacteristic
460C2×C3C5×#55211yescharacteristic
461C2×C3C5×#55311yescharacteristic
462C2×C3C5×#55411yescharacteristic
463C2×C3C5×#55711yescharacteristic
464C5C3×#237011yescharacteristic, direct product factor, Sylow
465C22C23×C3×C4×C511yescharacteristic
466C22C22×C3×C5×Q811yescharacteristic
467C22C2×C3×C5×#4111yescharacteristic
468C22C2×C3×C5×#4111yescharacteristic
469C22C2×C3×C5×#4111yescharacteristic
470C22C3×C5×H(5,𝔽2)11yescharacteristic
471C22C3×C5×2−1+411yescharacteristic
472C4—24no—
473C4—22no—
474C4—22no—
475C4—24no—
476C4—22no—
477C4—22no—
478C4—24no—
479C4—41no—
480C4—41no—
481C4—42no—
482C4C3×C4×C5×Q814yes—
483C3C5×#237011yescharacteristic, direct product factor, Sylow
484C2C2×C3×C4×C5×Q811yescharacteristic
485C2C3×C4×C5×#4111yescharacteristic
486C2C3×C5×#52011yescharacteristic
487C2C3×C5×#55211yescharacteristic
488C2C3×C5×#55311yescharacteristic
489C2C3×C5×#55411yescharacteristic
490C2C3×C5×#55711yescharacteristic
subgroupG/H
center, socle, prime radicalC23×C3×C5C24
commutatorC22C23×C3×C4×C5
Frattini, cosocleC23C24×C3×C5
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C23×C3×C5 ◁ G

Complements

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

NH ≅ G/Ntype
12. C5×#2370C3direct (N × H)
13. C3×C5×#125C4semidirect
37. C3×#2370C5direct (N × H)
87. C5×#125C3×C4semidirect
111. #2370C3×C5direct (N × H)
146. C3×#125C4×C5semidirect
287. #125C3×C4×C5semidirect
397. C3×C5#2370direct (N × H)
464. C5C3×#2370direct (N × H)
483. C3C5×#2370direct (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
C3×C5×#3892
C3×C5×#3872
C3×C4×C5×C4⋊C42
C5×#23703
C3×#23705