Finite Group Databasebeta

← C5×#70368 #14309024

Subgroups — C5×#70368 #14309024

Order 1920.

488 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#5470C211yesmaximal, characteristic, complemented
2C2×C3×C5×#104C211yesmaximal, characteristic
3C5×#5240C211yesmaximal, characteristic
4C5×#5236C211yesmaximal, characteristic
5C5×#4559C211yesmaximal, characteristic, complemented
6C5×#4557C211yesmaximal, characteristic, complemented
7C5×#4523C211yesmaximal, characteristic, complemented
8C5×#2031—31nomaximal, Hall
9C2×C3×C5×#41C2211yescharacteristic
10C2×C3×C5×M16C2211yescharacteristic
11C2×C3×C42×C5C2211yescharacteristic
12C5×#926C2211yescharacteristic
13C5×#926C2211yescharacteristic
14C5×#926C2211yescharacteristic
15C5×#924—22no—
16C2×C5×#208C2211yescharacteristic
17C5×#870—21no—
18C5×#869—21no—
19C4×C5×Dic6—22no—
20C3×C5×#104—21no—
21C3×C5×#104—21no—
22C5×#837—22no—
23C5×#834—22no—
24#70368C511yesmaximal, characteristic, direct product factor, Hall
25C5×#533—31no—
26C5×#421—31no—
27C2×C5×#104D311yescharacteristic
28C5×#410—31no—
29C5×#344—31no—
30C5×#339—31no—
31C5×H(3,ℤ/4)—31no—
32C3×C5×#41—21no—
33C3×C5×#41—21no—
34C3×C5×#41—21no—
35C3×C5×#41—21no—
36C2×C3×C5×Q8Q811yescharacteristic, complemented
37C2×C3×C5×D4—21no—
38C2×C3×C5×D4Q811yescharacteristic, complemented
39C22×C3×C4×C5—21no—
40C22×C3×C4×C5—21no—
41C22×C3×C4×C5C2311yescharacteristic
42C2×C5×Dic6—22no—
43C3×C5×M16—21no—
44C3×C5×M16—21no—
45C2×C3×C5×C8—21no—
46C3×C42×C5—21no—
47C3×C42×C5D411yescharacteristic
48C3×C42×C5D411yescharacteristic
49C5×#217—21no—
50C5×#217—21no—
51C5×#217D412yes—
52C5×C4⋊Dic3—22no—
53C5×C4⋊Dic3—21no—
54C5×C4⋊Dic3—21no—
55C5×C4⋊Dic3—21no—
56C5×Dic3⋊C4—24no—
57C4×C5×Dic3—21no—
58C4×C5×Dic3—21no—
59C4×C5×Dic3D412yes—
60C5×#208—21no—
61C5×#208—21no—
62C2×C5×C3⋊C8—21no—
63#5470C2×C511yescharacteristic, complemented
64C2×C3×#104C2×C511yescharacteristic
65#5240C2×C511yescharacteristic
66#5236C2×C511yescharacteristic
67#4559C2×C511yescharacteristic, complemented
68#4557C2×C511yescharacteristic, complemented
69#4523C2×C511yescharacteristic, complemented
70C2×C5×#41C2×D311yescharacteristic
71C2×C5×M16—31no—
72C2×C5×M16C2×D311yescharacteristic
73C5×C4⋊Q8—61no—
74C5×#125—61no—
75C5×C22⋊Q8—62no—
76C4×C5×Q8—62no—
77C5×#117—31no—
78C5×#117—31no—
79C5×#117—31no—
80C2×C42×C5C2×D311yescharacteristic
81C5×#104—21no—
82C5×#104—21no—
83C5×#104—62no—
84C5×#99—62no—
85#2031—31noSylow
86C23×C3×C5—21no—
87C23×C3×C5C2×D411yescharacteristic
88C3×C5×Q8—21no—
89C3×C5×Q8—21no—
90C3×C5×D4—21no—
91C3×C5×D4—21no—
92C3×C5×D4—41no—
93C3×C5×D4—41no—
94C2×C3×C4×C5—21no—
95C2×C3×C4×C5—21no—
96C2×C3×C4×C5—21no—
97C2×C3×C4×C5—42no—
98C2×C3×C4×C5—41no—
99C2×C3×C4×C5—41no—
100C2×C3×C4×C5C2×D411yescharacteristic
101C2×C3×C4×C5C2×D411yescharacteristic
102C2×C3×C4×C5C2×Q811yescharacteristic
103C2×C3×C4×C5#4111yescharacteristic
104C2×C3×C4×C5#4111yescharacteristic
105C2×C3×C4×C5#4111yescharacteristic
106C2×C5×Dic3—22no—
107C2×C5×Dic3—22no—
108C2×C5×Dic3—41no—
109C5×Dic6—44no—
110C3×C5×C8—41no—
111C5×C3⋊C8—41no—
112C2×C3×#41C22×C511yescharacteristic
113C2×C3×M16C22×C511yescharacteristic
114C2×C3×C42C22×C511yescharacteristic
115#926C22×C511yescharacteristic
116#926C22×C511yescharacteristic
117#926C22×C511yescharacteristic
118#924—22no—
119C2×#208C22×C511yescharacteristic
120#870—21no—
121#869—21no—
122C4×Dic6—22no—
123C3×#104—21no—
124C3×#104—21no—
125#837—22no—
126#834—22no—
127C5×#41—21no—
128C5×#41—21no—
129C5×#41—21no—
130C5×#41—21no—
131C2×C5×Q8—62no—
132C2×C5×Q8Dic611yescharacteristic, complemented
133C2×C5×D4—21no—
134C2×C5×D4Dic611yescharacteristic, complemented
135C22×C4×C5—21no—
136C22×C4×C5—21no—
137C22×C4×C5C22×D311yescharacteristic
138C5×M16—21no—
139C5×M16—21no—
140C5×M16—61no—
141C5×M16—61no—
142C2×C5×C8—21no—
143C2×C5×C8—61no—
144C5×C4⋊C4—64no—
145C5×C4⋊C4—62no—
146C5×C4⋊C4—61no—
147C5×C4⋊C4—61no—
148C5×C4⋊C4—61no—
149C5×C22⋊C4—32no—
150C5×C22⋊C4—61no—
151C5×C22⋊C4—61no—
152C42×C5—21no—
153C42×C5—32no—
154C42×C5—61no—
155C42×C5—61no—
156C42×C5#6711yescharacteristic
157C42×C5#6711yescharacteristic
158#533—31no—
159#421—31no—
160C2×#104C5×D311yescharacteristic
161#410—31no—
162#344—31no—
163#339—31no—
164H(3,ℤ/4)—31no—
165C22×C3×C5—41no—
166C22×C3×C5—41no—
167C22×C3×C5—81no—
168C22×C3×C5#12011yescharacteristic
169C22×C3×C5C22⋊Q811yescharacteristic
170C22×C3×C5#12411yescharacteristic
171C3×C4×C5—22no—
172C3×C4×C5—41no—
173C3×C4×C5—41no—
174C3×C4×C5C22⋊Q811yescharacteristic
175C3×C4×C5C22⋊Q811yescharacteristic
176C3×C4×C5#12411yescharacteristic
177C3×C4×C5#12411yescharacteristic
178C5×Dic3—42no—
179C5×Dic3—42no—
180C5×Dic3—81no—
181C3×#41—21no—
182C3×#41—21no—
183C3×#41—21no—
184C3×#41—21no—
185C2×C3×Q8C5×Q811yescharacteristic, complemented
186C2×C3×D4—21no—
187C2×C3×D4C5×Q811yescharacteristic, complemented
188C22×C3×C4—21no—
189C22×C3×C4—21no—
190C22×C3×C4C23×C511yescharacteristic
191C2×Dic6—22no—
192C3×M16—21no—
193C3×M16—21no—
194C2×C3×C8—21no—
195C3×C42—21no—
196C3×C42C5×D411yescharacteristic
197C3×C42C5×D411yescharacteristic
198#217—21no—
199#217—21no—
200#217C5×D412yes—
201C4⋊Dic3—22no—
202C4⋊Dic3—21no—
203C4⋊Dic3—21no—
204C4⋊Dic3—21no—
205Dic3⋊C4—24no—
206C4×Dic3—21no—
207C4×Dic3—21no—
208C4×Dic3C5×D412yes—
209#208—21no—
210#208—21no—
211C2×C3⋊C8—21no—
212C23×C5—21no—
213C23×C5D3×D411yescharacteristic
214C5×Q8—21no—
215C5×Q8—21no—
216C5×Q8—124no—
217C5×D4—21no—
218C5×D4—21no—
219C5×D4—41no—
220C5×D4—41no—
221C2×C4×C5—21no—
222C2×C4×C5—21no—
223C2×C4×C5—21no—
224C2×C4×C5—42no—
225C2×C4×C5—41no—
226C2×C4×C5—41no—
227C2×C4×C5—62no—
228C2×C4×C5—62no—
229C2×C4×C5—121no—
230C2×C4×C5C2×Dic611yescharacteristic
231C2×C4×C5#23511yescharacteristic
232C2×C4×C5D3×D411yescharacteristic
233C2×C4×C5#23711yescharacteristic
234C2×C4×C5#23711yescharacteristic
235C2×C4×C5C2×#6711yescharacteristic
236C5×C8—41no—
237C5×C8—121no—
238C2×#41C2×C5×D311yescharacteristic
239C2×M16—31no—
240C2×M16C2×C5×D311yescharacteristic
241C4⋊Q8—61no—
242#125—61no—
243C22⋊Q8—62no—
244C4×Q8—62no—
245#117—31no—
246#117—31no—
247#117—31no—
248C2×C42C2×C5×D311yescharacteristic
249#104—21no—
250#104—21no—
251#104—62no—
252#99—62no—
253C2×C3×C5—21no—
254C2×C3×C5—21no—
255C2×C3×C5—21no—
256C2×C3×C5—81no—
257C2×C3×C5#39311yescharacteristic
258C23×C3—21no—
259C23×C3C2×C5×D411yescharacteristic
260C3×Q8—21no—
261C3×Q8—21no—
262C3×D4—21no—
263C3×D4—21no—
264C3×D4—41no—
265C3×D4—41no—
266C2×C3×C4—21no—
267C2×C3×C4—21no—
268C2×C3×C4—21no—
269C2×C3×C4—42no—
270C2×C3×C4—41no—
271C2×C3×C4—41no—
272C2×C3×C4C2×C5×D411yescharacteristic
273C2×C3×C4C2×C5×D411yescharacteristic
274C2×C3×C4C2×C5×Q811yescharacteristic
275C2×C3×C4C5×#4111yescharacteristic
276C2×C3×C4C5×#4111yescharacteristic
277C2×C3×C4C5×#4111yescharacteristic
278C2×Dic3—22no—
279C2×Dic3—22no—
280C2×Dic3—41no—
281Dic6—44no—
282C3×C8—41no—
283C3⋊C8—41no—
284C22×C5—41no—
285C22×C5—41no—
286C22×C5—81no—
287C22×C5#87811yescharacteristic
288C22×C5#88511yescharacteristic
289C22×C5#93711yescharacteristic
290C4×C5—22no—
291C4×C5—41no—
292C4×C5—41no—
293C4×C5—122no—
294C4×C5—122no—
295C4×C5—241no—
296C4×C5#87811yescharacteristic
297C4×C5#88511yescharacteristic
298C4×C5#92411yescharacteristic
299C4×C5#93611yescharacteristic
300#41—21no—
301#41—21no—
302#41—21no—
303#41—21no—
304C2×Q8—62no—
305C2×Q8C5×Dic611yescharacteristic, complemented
306C2×D4—21no—
307C2×D4C5×Dic611yescharacteristic, complemented
308C22×C4—21no—
309C22×C4—21no—
310C22×C4C22×C5×D311yescharacteristic
311M16—21no—
312M16—21no—
313M16—61no—
314M16—61no—
315C2×C8—21no—
316C2×C8—61no—
317C4⋊C4—64no—
318C4⋊C4—62no—
319C4⋊C4—61no—
320C4⋊C4—61no—
321C4⋊C4—61no—
322C22⋊C4—32no—
323C22⋊C4—61no—
324C22⋊C4—61no—
325C42—21no—
326C42—32no—
327C42—61no—
328C42—61no—
329C42C5×#6711yescharacteristic
330C42C5×#6711yescharacteristic
331C3×C5#203111yescharacteristic, complemented, Hall
332C22×C3—41no—
333C22×C3—41no—
334C22×C3—81no—
335C22×C3C5×#12011yescharacteristic
336C22×C3C5×C22⋊Q811yescharacteristic
337C22×C3C5×#12411yescharacteristic
338C3×C4—22no—
339C3×C4—41no—
340C3×C4—41no—
341C3×C4C5×C22⋊Q811yescharacteristic
342C3×C4C5×C22⋊Q811yescharacteristic
343C3×C4C5×#12411yescharacteristic
344C3×C4C5×#12411yescharacteristic
345Dic3—42no—
346Dic3—42no—
347Dic3—81no—
348C2×C5—21no—
349C2×C5—21no—
350C2×C5—21no—
351C2×C5—81no—
352C2×C5#494711yescharacteristic
353C23—21no—
354C23C5×D3×D411yescharacteristic
355Q8—21no—
356Q8—21no—
357Q8—124no—
358D4—21no—
359D4—21no—
360D4—41no—
361D4—41no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—42no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—62no—
369C2×C4—62no—
370C2×C4—121no—
371C2×C4C2×C5×Dic611yescharacteristic
372C2×C4C5×#23511yescharacteristic
373C2×C4C5×D3×D411yescharacteristic
374C2×C4C5×#23711yescharacteristic
375C2×C4C5×#23711yescharacteristic
376C2×C4C2×C5×#6711yescharacteristic
377C8—41no—
378C8—121no—
379C2×C3—21no—
380C2×C3—21no—
381C2×C3—21no—
382C2×C3—81no—
383C2×C3C5×#39311yescharacteristic
384C5#7036811yescharacteristic, direct product factor, Sylow
385C22—41no—
386C22—41no—
387C22—81no—
388C22C5×#87811yescharacteristic
389C22C5×#88511yescharacteristic
390C22C5×#93711yescharacteristic
391C4—22no—
392C4—41no—
393C4—41no—
394C4—122no—
395C4—122no—
396C4—241no—
397C4C5×#87811yescharacteristic
398C4C5×#88511yescharacteristic
399C4C5×#92411yescharacteristic
400C4C5×#93611yescharacteristic
401C3C5×#203111yescharacteristic, complemented, Sylow
402C2—21no—
403C2—21no—
404C2—21no—
405C2—81no—
406C2C5×#494711yescharacteristic
subgroupG/H
centerC4×C5#885
commutator, cosocleC22×C3×C4C23×C5
FrattiniC22×C4C22×C5×D3
FittingC2×C3×C5×#104C2
socleC2×C3×C5#393
prime radicalC2×C3×C5×#41C22
hypercenterC2×C5×#104D3
hypocenterC3C5×#2031
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4×C5 ◁ C22×C4×C5 ◁ C2×C5×#104

Complements

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

NH ≅ G/Ntype
1. C5×#5470C2semidirect
5. C5×#4559C2semidirect
6. C5×#4557C2semidirect
7. C5×#4523C2semidirect
24. #70368C5direct (N × H)
36. C2×C3×C5×Q8Q8semidirect
38. C2×C3×C5×D4Q8semidirect
63. #5470C2×C5semidirect
67. #4559C2×C5semidirect
68. #4557C2×C5semidirect
69. #4523C2×C5semidirect
132. C2×C5×Q8Dic6semidirect
134. C2×C5×D4Dic6semidirect
185. C2×C3×Q8C5×Q8semidirect
187. C2×C3×D4C5×Q8semidirect
305. C2×Q8C5×Dic6semidirect
307. C2×D4C5×Dic6semidirect
331. C3×C5#2031semidirect
384. C5#70368direct (N × H)
401. C3C5×#2031semidirect

Maximal subgroups

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

subgroupindex
C5×#54702
C2×C3×C5×#1042
C5×#52402
C5×#52362
C5×#45592
C5×#45572
C5×#45232
C5×#20313
#703685