Finite Group Databasebeta

← C2×C3×C5×#352 #14275153

Subgroups — C2×C3×C5×#352 #14275153

Order 1920.

556 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C5×#123C211yesmaximal, characteristic
2C2×C3×C5×#100C211yesmaximal, characteristic
3C2×C3×C5×#99C211yesmaximal, characteristic
4C3×C5×#352C214yesmaximal, direct product factor
5C2×C5×#352C311yesmaximal, characteristic, direct product factor, Hall
6C22×C3×C5×D4C2211yescharacteristic
7C23×C3×C4×C5C411yescharacteristic, complemented
8C2×C3×C5×M16—21no—
9C3×C5×#123—21no—
10C3×C5×#123—21no—
11C3×C5×#123C412yescomplemented
12C3×C5×#123C2212yes—
13C2×C3×C5×C4⋊C4—21no—
14C2×C3×C5×C22⋊C4—21no—
15C2×C3×C5×C22⋊C4—21no—
16C2×C3×C5×C22⋊C4C411yescharacteristic, complemented
17C3×C5×#100—21no—
18C3×C5×#100C2212yes—
19C3×C5×#99—21no—
20C3×C5×#99C2212yes—
21C2×C3×#352C511yesmaximal, characteristic, direct product factor, Hall
22C2×C5×#123C2×C311yescharacteristic
23C2×C5×#100C2×C311yescharacteristic
24C2×C5×#99C2×C311yescharacteristic
25C5×#352C2×C314yesdirect product factor
26C24×C3×C5C2×C411yescharacteristic
27C24×C3×C5D411yescharacteristic
28C2×C3×C5×D4—41no—
29C2×C3×C5×D4—41no—
30C2×C3×C5×D4C2×C411yescharacteristic
31C2×C3×C5×D4D411yescharacteristic
32C2×C3×C5×D4D411yescharacteristic
33C2×C3×C5×D4C2311yescharacteristic
34C22×C3×C4×C5—21no—
35C22×C3×C4×C5—21no—
36C22×C3×C4×C5—21no—
37C22×C3×C4×C5—41no—
38C22×C3×C4×C5—41no—
39C22×C3×C4×C5—41no—
40C22×C3×C4×C5C2×C412yescomplemented
41C22×C3×C4×C5D411yescharacteristic
42C3×C5×M16—22no—
43C3×C5×M16—41no—
44C2×C3×C5×C8—41no—
45C3×C5×C4⋊C4—22no—
46C3×C5×C4⋊C4—22no—
47C3×C5×C22⋊C4—22no—
48C3×C5×C22⋊C4—21no—
49C3×C5×C22⋊C4—22no—
50C3×C5×C22⋊C4—22no—
51C3×C5×C22⋊C4—41no—
52C3×C5×C22⋊C4C2×C412yescomplemented
53C2×C3×#123C2×C511yescharacteristic
54C2×C3×#100C2×C511yescharacteristic
55C2×C3×#99C2×C511yescharacteristic
56C3×#352C2×C514yesdirect product factor
57C22×C5×D4C22×C311yescharacteristic
58C23×C4×C5C3×C411yescharacteristic, complemented
59C2×C5×M16—21no—
60C5×#123—21no—
61C5×#123—21no—
62C5×#123C3×C412yescomplemented
63C5×#123C22×C312yes—
64C2×C5×C4⋊C4—21no—
65C2×C5×C22⋊C4—21no—
66C2×C5×C22⋊C4—21no—
67C2×C5×C22⋊C4C3×C411yescharacteristic, complemented
68C5×#100—21no—
69C5×#100C22×C312yes—
70C5×#99—21no—
71C5×#99C22×C312yes—
72C2×#352C3×C511yescharacteristic, direct product factor, Sylow
73C23×C3×C5—21no—
74C23×C3×C5—21no—
75C23×C3×C5—21no—
76C23×C3×C5—21no—
77C23×C3×C5—42no—
78C23×C3×C5—42no—
79C23×C3×C5C22⋊C411yescharacteristic
80C23×C3×C5C22⋊C411yescharacteristic
81C23×C3×C5C22⋊C411yescharacteristic
82C23×C3×C5C22×C411yescharacteristic
83C23×C3×C5C2×D411yescharacteristic
84C3×C5×D4—41no—
85C3×C5×D4—41no—
86C3×C5×D4—41no—
87C3×C5×D4—41no—
88C2×C3×C4×C5—21no—
89C2×C3×C4×C5—21no—
90C2×C3×C4×C5—22no—
91C2×C3×C4×C5—22no—
92C2×C3×C4×C5—42no—
93C2×C3×C4×C5—42no—
94C2×C3×C4×C5—41no—
95C2×C3×C4×C5—41no—
96C2×C3×C4×C5—42no—
97C2×C3×C4×C5—42no—
98C2×C3×C4×C5—82no—
99C2×C3×C4×C5C22⋊C411yescharacteristic
100C2×C3×C4×C5C2×D411yescharacteristic
101C3×C5×C8—42no—
102C22×C3×D4C22×C511yescharacteristic
103C23×C3×C4C4×C511yescharacteristic, complemented
104C2×C3×M16—21no—
105C3×#123—21no—
106C3×#123—21no—
107C3×#123C4×C512yescomplemented
108C3×#123C22×C512yes—
109C2×C3×C4⋊C4—21no—
110C2×C3×C22⋊C4—21no—
111C2×C3×C22⋊C4—21no—
112C2×C3×C22⋊C4C4×C511yescharacteristic, complemented
113C3×#100—21no—
114C3×#100C22×C512yes—
115C3×#99—21no—
116C3×#99C22×C512yes—
117C24×C5C2×C3×C411yescharacteristic
118C24×C5C3×D411yescharacteristic
119C2×C5×D4—41no—
120C2×C5×D4—41no—
121C2×C5×D4C2×C3×C411yescharacteristic
122C2×C5×D4C3×D411yescharacteristic
123C2×C5×D4C3×D411yescharacteristic
124C2×C5×D4C23×C311yescharacteristic
125C22×C4×C5—21no—
126C22×C4×C5—21no—
127C22×C4×C5—21no—
128C22×C4×C5—41no—
129C22×C4×C5—41no—
130C22×C4×C5—41no—
131C22×C4×C5C2×C3×C412yescomplemented
132C22×C4×C5C3×D411yescharacteristic
133C5×M16—22no—
134C5×M16—41no—
135C2×C5×C8—41no—
136C5×C4⋊C4—22no—
137C5×C4⋊C4—22no—
138C5×C22⋊C4—22no—
139C5×C22⋊C4—21no—
140C5×C22⋊C4—22no—
141C5×C22⋊C4—22no—
142C5×C22⋊C4—41no—
143C5×C22⋊C4C2×C3×C412yescomplemented
144C2×#123C2×C3×C511yescharacteristic
145C2×#100C2×C3×C511yescharacteristic
146C2×#99C2×C3×C511yescharacteristic
147#352C2×C3×C514yesdirect product factor
148C22×C3×C5—22no—
149C22×C3×C5—21no—
150C22×C3×C5—21no—
151C22×C3×C5—21no—
152C22×C3×C5—21no—
153C22×C3×C5—41no—
154C22×C3×C5—41no—
155C22×C3×C5—41no—
156C22×C3×C5—41no—
157C22×C3×C5—42no—
158C22×C3×C5—42no—
159C22×C3×C5—42no—
160C22×C3×C5—81no—
161C22×C3×C5#9911yescharacteristic
162C22×C3×C5#9911yescharacteristic
163C22×C3×C5C2×C22⋊C411yescharacteristic
164C3×C4×C5—21no—
165C3×C4×C5—21no—
166C3×C4×C5—42no—
167C3×C4×C5—42no—
168C3×C4×C5—82no—
169C24×C3C2×C4×C511yescharacteristic
170C24×C3C5×D411yescharacteristic
171C2×C3×D4—41no—
172C2×C3×D4—41no—
173C2×C3×D4C2×C4×C511yescharacteristic
174C2×C3×D4C5×D411yescharacteristic
175C2×C3×D4C5×D411yescharacteristic
176C2×C3×D4C23×C511yescharacteristic
177C22×C3×C4—21no—
178C22×C3×C4—21no—
179C22×C3×C4—21no—
180C22×C3×C4—41no—
181C22×C3×C4—41no—
182C22×C3×C4—41no—
183C22×C3×C4C2×C4×C512yescomplemented
184C22×C3×C4C5×D411yescharacteristic
185C3×M16—22no—
186C3×M16—41no—
187C2×C3×C8—41no—
188C3×C4⋊C4—22no—
189C3×C4⋊C4—22no—
190C3×C22⋊C4—22no—
191C3×C22⋊C4—21no—
192C3×C22⋊C4—22no—
193C3×C22⋊C4—22no—
194C3×C22⋊C4—41no—
195C3×C22⋊C4C2×C4×C512yescomplemented
196C23×C5—21no—
197C23×C5—21no—
198C23×C5—21no—
199C23×C5—21no—
200C23×C5—42no—
201C23×C5—42no—
202C23×C5C3×C22⋊C411yescharacteristic
203C23×C5C3×C22⋊C411yescharacteristic
204C23×C5C3×C22⋊C411yescharacteristic
205C23×C5C22×C3×C411yescharacteristic
206C23×C5C2×C3×D411yescharacteristic
207C5×D4—41no—
208C5×D4—41no—
209C5×D4—41no—
210C5×D4—41no—
211C2×C4×C5—21no—
212C2×C4×C5—21no—
213C2×C4×C5—22no—
214C2×C4×C5—22no—
215C2×C4×C5—42no—
216C2×C4×C5—42no—
217C2×C4×C5—41no—
218C2×C4×C5—41no—
219C2×C4×C5—42no—
220C2×C4×C5—42no—
221C2×C4×C5—82no—
222C2×C4×C5C3×C22⋊C411yescharacteristic
223C2×C4×C5C2×C3×D411yescharacteristic
224C5×C8—42no—
225C22×D4C22×C3×C511yescharacteristic
226C23×C4C3×C4×C511yescharacteristic, complemented
227C2×M16—21no—
228#123—21no—
229#123—21no—
230#123C3×C4×C512yescomplemented
231#123C22×C3×C512yes—
232C2×C4⋊C4—21no—
233C2×C22⋊C4—21no—
234C2×C22⋊C4—21no—
235C2×C22⋊C4C3×C4×C511yescharacteristic, complemented
236#100—21no—
237#100C22×C3×C512yes—
238#99—21no—
239#99C22×C3×C512yes—
240C2×C3×C5—21no—
241C2×C3×C5—21no—
242C2×C3×C5—41no—
243C2×C3×C5—41no—
244C2×C3×C5—41no—
245C2×C3×C5—41no—
246C2×C3×C5#35212yesdirect product factor
247C2×C3×C5C2×#9911yescharacteristic
248C23×C3—21no—
249C23×C3—21no—
250C23×C3—21no—
251C23×C3—21no—
252C23×C3—42no—
253C23×C3—42no—
254C23×C3C5×C22⋊C411yescharacteristic
255C23×C3C5×C22⋊C411yescharacteristic
256C23×C3C5×C22⋊C411yescharacteristic
257C23×C3C22×C4×C511yescharacteristic
258C23×C3C2×C5×D411yescharacteristic
259C3×D4—41no—
260C3×D4—41no—
261C3×D4—41no—
262C3×D4—41no—
263C2×C3×C4—21no—
264C2×C3×C4—21no—
265C2×C3×C4—22no—
266C2×C3×C4—22no—
267C2×C3×C4—42no—
268C2×C3×C4—42no—
269C2×C3×C4—41no—
270C2×C3×C4—41no—
271C2×C3×C4—42no—
272C2×C3×C4—42no—
273C2×C3×C4—82no—
274C2×C3×C4C5×C22⋊C411yescharacteristic
275C2×C3×C4C2×C5×D411yescharacteristic
276C3×C8—42no—
277C22×C5—22no—
278C22×C5—21no—
279C22×C5—21no—
280C22×C5—21no—
281C22×C5—21no—
282C22×C5—41no—
283C22×C5—41no—
284C22×C5—41no—
285C22×C5—41no—
286C22×C5—42no—
287C22×C5—42no—
288C22×C5—42no—
289C22×C5—81no—
290C22×C5C3×#9911yescharacteristic
291C22×C5C3×#9911yescharacteristic
292C22×C5C2×C3×C22⋊C411yescharacteristic
293C4×C5—21no—
294C4×C5—21no—
295C4×C5—42no—
296C4×C5—42no—
297C4×C5—82no—
298C24C2×C3×C4×C511yescharacteristic
299C24C3×C5×D411yescharacteristic
300C2×D4—41no—
301C2×D4—41no—
302C2×D4C2×C3×C4×C511yescharacteristic
303C2×D4C3×C5×D411yescharacteristic
304C2×D4C3×C5×D411yescharacteristic
305C2×D4C23×C3×C511yescharacteristic
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4C2×C3×C4×C512yescomplemented
313C22×C4C3×C5×D411yescharacteristic
314M16—22no—
315M16—41no—
316C2×C8—41no—
317C4⋊C4—22no—
318C4⋊C4—22no—
319C22⋊C4—22no—
320C22⋊C4—21no—
321C22⋊C4—22no—
322C22⋊C4—22no—
323C22⋊C4—41no—
324C22⋊C4C2×C3×C4×C512yescomplemented
325C3×C5C2×#35211yescharacteristic, direct product factor, Hall
326C22×C3—22no—
327C22×C3—21no—
328C22×C3—21no—
329C22×C3—21no—
330C22×C3—21no—
331C22×C3—41no—
332C22×C3—41no—
333C22×C3—41no—
334C22×C3—41no—
335C22×C3—42no—
336C22×C3—42no—
337C22×C3—42no—
338C22×C3—81no—
339C22×C3C5×#9911yescharacteristic
340C22×C3C5×#9911yescharacteristic
341C22×C3C2×C5×C22⋊C411yescharacteristic
342C3×C4—21no—
343C3×C4—21no—
344C3×C4—42no—
345C3×C4—42no—
346C3×C4—82no—
347C2×C5—21no—
348C2×C5—21no—
349C2×C5—41no—
350C2×C5—41no—
351C2×C5—41no—
352C2×C5—41no—
353C2×C5C3×#35212yesdirect product factor
354C2×C5C2×C3×#9911yescharacteristic
355C23—21no—
356C23—21no—
357C23—21no—
358C23—21no—
359C23—42no—
360C23—42no—
361C23C3×C5×C22⋊C411yescharacteristic
362C23C3×C5×C22⋊C411yescharacteristic
363C23C3×C5×C22⋊C411yescharacteristic
364C23C22×C3×C4×C511yescharacteristic
365C23C2×C3×C5×D411yescharacteristic
366D4—41no—
367D4—41no—
368D4—41no—
369D4—41no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—82no—
381C2×C4C3×C5×C22⋊C411yescharacteristic
382C2×C4C2×C3×C5×D411yescharacteristic
383C8—42no—
384C2×C3—21no—
385C2×C3—21no—
386C2×C3—41no—
387C2×C3—41no—
388C2×C3—41no—
389C2×C3—41no—
390C2×C3C5×#35212yesdirect product factor
391C2×C3C2×C5×#9911yescharacteristic
392C5C2×C3×#35211yescharacteristic, direct product factor, Sylow
393C22—22no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—42no—
403C22—42no—
404C22—42no—
405C22—81no—
406C22C3×C5×#9911yescharacteristic
407C22C3×C5×#9911yescharacteristic
408C22C2×C3×C5×C22⋊C411yescharacteristic
409C4—21no—
410C4—21no—
411C4—42no—
412C4—42no—
413C4—82no—
414C3C2×C5×#35211yescharacteristic, direct product factor, Sylow
415C2—21no—
416C2—21no—
417C2—41no—
418C2—41no—
419C2—41no—
420C2—41no—
421C2C3×C5×#35212yesdirect product factor
422C2C2×C3×C5×#9911yescharacteristic
subgroupG/H
center, socleC22×C3×C5#99
commutatorC23C22×C3×C4×C5
Frattini, cosocleC2×D4C23×C3×C5
prime radicalC22×C3×C5×D4C22
LCSG ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3×C5 ◁ C23×C3×C5 ◁ C22×C3×C5×D4 ◁ G

Complements

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

NH ≅ G/Ntype
4. C3×C5×#352C2direct (N × H)
5. C2×C5×#352C3direct (N × H)
7. C23×C3×C4×C5C4semidirect
11. C3×C5×#123C4semidirect
16. C2×C3×C5×C22⋊C4C4semidirect
21. C2×C3×#352C5direct (N × H)
25. C5×#352C2×C3direct (N × H)
40. C22×C3×C4×C5C2×C4semidirect
52. C3×C5×C22⋊C4C2×C4semidirect
56. C3×#352C2×C5direct (N × H)
58. C23×C4×C5C3×C4semidirect
62. C5×#123C3×C4semidirect
67. C2×C5×C22⋊C4C3×C4semidirect
72. C2×#352C3×C5direct (N × H)
103. C23×C3×C4C4×C5semidirect
107. C3×#123C4×C5semidirect
112. C2×C3×C22⋊C4C4×C5semidirect
131. C22×C4×C5C2×C3×C4semidirect
143. C5×C22⋊C4C2×C3×C4semidirect
147. #352C2×C3×C5direct (N × H)
183. C22×C3×C4C2×C4×C5semidirect
195. C3×C22⋊C4C2×C4×C5semidirect
226. C23×C4C3×C4×C5semidirect
230. #123C3×C4×C5semidirect
235. C2×C22⋊C4C3×C4×C5semidirect
246. C2×C3×C5#352direct (N × H)
312. C22×C4C2×C3×C4×C5semidirect
324. C22⋊C4C2×C3×C4×C5semidirect
325. C3×C5C2×#352direct (N × H)
353. C2×C5C3×#352direct (N × H)
390. C2×C3C5×#352direct (N × H)
392. C5C2×C3×#352direct (N × H)
414. C3C2×C5×#352direct (N × H)
421. C2C3×C5×#352direct (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×#1232
C2×C3×C5×#1002
C2×C3×C5×#992
C3×C5×#3522
C2×C5×#3523
C2×C3×#3525