Finite Group Databasebeta

← C5×#68714 #14309021

Subgroups — C5×#68714 #14309021

Order 1920.

488 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#5517C211yesmaximal, characteristic, complemented
2C2×C5×#825C211yesmaximal, characteristic
3C5×#5006C211yesmaximal, characteristic
4C5×#5005C211yesmaximal, characteristic
5C3×C5×H(3,ℤ/4)C211yesmaximal, characteristic, complemented
6C5×#4559C211yesmaximal, characteristic, complemented
7C5×#4532C211yesmaximal, characteristic, complemented
8C5×#2031—31nomaximal, Hall
9C2×C5×#235C2211yescharacteristic
10C2×C3×C5×M16C2211yescharacteristic
11C3×C5×#117C2211yescharacteristic
12C3×C5×#117C2211yescharacteristic
13C5×#926C2211yescharacteristic
14C2×C4×C5×Dic3C2211yescharacteristic
15C2×C5×#208C2211yescharacteristic
16C5×#890—21no—
17C5×#888—21no—
18C5×#878—22no—
19C4×C5×Dic6—22no—
20C5×#825—21no—
21C5×#825—21no—
22C5×#806—22no—
23C5×#805—22no—
24#68714C511yesmaximal, characteristic, direct product factor, Hall
25C5×#533—31no—
26C5×#421—31no—
27C2×C5×#104—31no—
28C5×#410—31no—
29C5×#344—31no—
30C5×#339—31no—
31C5×H(3,ℤ/4)D311yescharacteristic, complemented
32C22×C3×C4×C5C2311yescharacteristic
33C2×C5×#67—21no—
34C22×C5×Dic3—21no—
35C5×#235—21no—
36C5×#235—21no—
37C5×#235—21no—
38C5×#235—21no—
39C2×C5×D12Q811yescharacteristic, complemented
40C2×C4×C5×D3—21no—
41C2×C5×Dic6—22no—
42C2×C5×Dic6Q811yescharacteristic, complemented
43C3×C5×M16—21no—
44C3×C5×M16—21no—
45C2×C3×C5×C8—21no—
46C3×C5×C4⋊C4—21no—
47C3×C5×C4⋊C4—21no—
48C3×C5×C22⋊C4—21no—
49C3×C5×C22⋊C4—21no—
50C3×C42×C5—21no—
51C3×C42×C5D412yes—
52C5×#217D412yes—
53C5×C4⋊Dic3—22no—
54C5×C4⋊Dic3—21no—
55C5×Dic3⋊C4—22no—
56C5×Dic3⋊C4—22no—
57C4×C5×Dic3—21no—
58C4×C5×Dic3—21no—
59C4×C5×Dic3D411yescharacteristic
60C4×C5×Dic3D411yescharacteristic
61C5×#208—21no—
62C5×#208—21no—
63C2×C5×C3⋊C8—21no—
64#5517C2×C511yescharacteristic, complemented
65C2×#825C2×C511yescharacteristic
66#5006C2×C511yescharacteristic
67#5005C2×C511yescharacteristic
68C3×H(3,ℤ/4)C2×C511yescharacteristic, complemented
69#4559C2×C511yescharacteristic, complemented
70#4532C2×C511yescharacteristic, complemented
71C2×C5×#41—31no—
72C2×C5×M16—31no—
73C2×C5×M16C2×D311yescharacteristic
74C5×C4⋊Q8—61no—
75C5×#125—61no—
76C5×C22⋊Q8—62no—
77C4×C5×Q8—62no—
78C5×#117—31no—
79C5×#117C2×D311yescharacteristic
80C5×#117C2×D311yescharacteristic
81C2×C42×C5—31no—
82C5×#104—61no—
83C5×#104—61no—
84C5×#104—62no—
85C5×#99—62no—
86#2031—31noSylow
87C23×C3×C5C2×D411yescharacteristic
88C22×C5×D3—21no—
89C2×C3×C4×C5—22no—
90C2×C3×C4×C5—41no—
91C2×C3×C4×C5C2×D411yescharacteristic
92C2×C3×C4×C5C2×D411yescharacteristic
93C2×C3×C4×C5C2×Q811yescharacteristic
94C2×C3×C4×C5#4111yescharacteristic
95C2×C3×C4×C5#4111yescharacteristic
96C2×C3×C4×C5#4111yescharacteristic
97C5×#67—41no—
98C5×#67—41no—
99C2×C5×Dic3—21no—
100C2×C5×Dic3—21no—
101C2×C5×Dic3—22no—
102C2×C5×Dic3—21no—
103C2×C5×Dic3—42no—
104C5×D12—21no—
105C5×D12—21no—
106C4×C5×D3—41no—
107C4×C5×D3—41no—
108C5×Dic6—21no—
109C5×Dic6—21no—
110C5×Dic6—44no—
111C3×C5×C8—41no—
112C5×C3⋊C8—41no—
113C2×#235C22×C511yescharacteristic
114C2×C3×M16C22×C511yescharacteristic
115C3×#117C22×C511yescharacteristic
116C3×#117C22×C511yescharacteristic
117#926C22×C511yescharacteristic
118C2×C4×Dic3C22×C511yescharacteristic
119C2×#208C22×C511yescharacteristic
120#890—21no—
121#888—21no—
122#878—22no—
123C4×Dic6—22no—
124#825—21no—
125#825—21no—
126#806—22no—
127#805—22no—
128C5×#41—61no—
129C5×#41—61no—
130C5×#41—61no—
131C5×#41—61no—
132C2×C5×Q8—31no—
133C2×C5×Q8—62no—
134C2×C5×D4—31no—
135C2×C5×D4—61no—
136C22×C4×C5—61no—
137C22×C4×C5—61no—
138C22×C4×C5C22×D311yescharacteristic
139C5×M16—21no—
140C5×M16—21no—
141C5×M16—61no—
142C5×M16—61no—
143C2×C5×C8—21no—
144C2×C5×C8—61no—
145C5×C4⋊C4—21no—
146C5×C4⋊C4—21no—
147C5×C4⋊C4—62no—
148C5×C4⋊C4—62no—
149C5×C4⋊C4—62no—
150C5×C4⋊C4—61no—
151C5×C22⋊C4—21no—
152C5×C22⋊C4—21no—
153C5×C22⋊C4—32no—
154C42×C5—21no—
155C42×C5—31no—
156C42×C5—31no—
157C42×C5—61no—
158C42×C5—61no—
159C42×C5D1212yes—
160#533—31no—
161#421—31no—
162C2×#104—31no—
163#410—31no—
164#344—31no—
165#339—31no—
166H(3,ℤ/4)C5×D311yescharacteristic, complemented
167C22×C3×C5—41no—
168C22×C3×C5#12011yescharacteristic
169C22×C3×C5C22⋊Q811yescharacteristic
170C22×C3×C5#12411yescharacteristic
171C2×C5×D3—41no—
172C2×C5×D3—81no—
173C3×C4×C5—42no—
174C3×C4×C5—81no—
175C3×C4×C5C22⋊Q811yescharacteristic
176C3×C4×C5C22⋊Q811yescharacteristic
177C3×C4×C5#12411yescharacteristic
178C3×C4×C5#12411yescharacteristic
179C5×Dic3—22no—
180C5×Dic3—42no—
181C5×Dic3—41no—
182C5×Dic3—41no—
183C22×C3×C4C23×C511yescharacteristic
184C2×#67—21no—
185C22×Dic3—21no—
186#235—21no—
187#235—21no—
188#235—21no—
189#235—21no—
190C2×D12C5×Q811yescharacteristic, complemented
191C2×C4×D3—21no—
192C2×Dic6—22no—
193C2×Dic6C5×Q811yescharacteristic, complemented
194C3×M16—21no—
195C3×M16—21no—
196C2×C3×C8—21no—
197C3×C4⋊C4—21no—
198C3×C4⋊C4—21no—
199C3×C22⋊C4—21no—
200C3×C22⋊C4—21no—
201C3×C42—21no—
202C3×C42C5×D412yes—
203#217C5×D412yes—
204C4⋊Dic3—22no—
205C4⋊Dic3—21no—
206Dic3⋊C4—22no—
207Dic3⋊C4—22no—
208C4×Dic3—21no—
209C4×Dic3—21no—
210C4×Dic3C5×D411yescharacteristic
211C4×Dic3C5×D411yescharacteristic
212#208—21no—
213#208—21no—
214C2×C3⋊C8—21no—
215C23×C5—61no—
216C23×C5D3×D411yescharacteristic
217C5×Q8—61no—
218C5×Q8—61no—
219C5×Q8—124no—
220C5×D4—61no—
221C5×D4—61no—
222C5×D4—121no—
223C5×D4—121no—
224C2×C4×C5—22no—
225C2×C4×C5—41no—
226C2×C4×C5—61no—
227C2×C4×C5—61no—
228C2×C4×C5—62no—
229C2×C4×C5—61no—
230C2×C4×C5—122no—
231C2×C4×C5—121no—
232C2×C4×C5—121no—
233C2×C4×C5C2×D1211yescharacteristic
234C2×C4×C5#23511yescharacteristic
235C2×C4×C5#23511yescharacteristic
236C2×C4×C5D3×D411yescharacteristic
237C2×C4×C5#23711yescharacteristic
238C2×C4×C5D3×Q811yescharacteristic
239C5×C8—41no—
240C5×C8—121no—
241C2×#41—31no—
242C2×M16—31no—
243C2×M16C2×C5×D311yescharacteristic
244C4⋊Q8—61no—
245#125—61no—
246C22⋊Q8—62no—
247C4×Q8—62no—
248#117—31no—
249#117C2×C5×D311yescharacteristic
250#117C2×C5×D311yescharacteristic
251C2×C42—31no—
252#104—61no—
253#104—61no—
254#104—62no—
255#99—62no—
256C2×C3×C5—21no—
257C2×C3×C5—21no—
258C2×C3×C5—21no—
259C2×C3×C5#39311yescharacteristic
260C5×D3—81no—
261C23×C3C2×C5×D411yescharacteristic
262C22×D3—21no—
263C2×C3×C4—22no—
264C2×C3×C4—41no—
265C2×C3×C4C2×C5×D411yescharacteristic
266C2×C3×C4C2×C5×D411yescharacteristic
267C2×C3×C4C2×C5×Q811yescharacteristic
268C2×C3×C4C5×#4111yescharacteristic
269C2×C3×C4C5×#4111yescharacteristic
270C2×C3×C4C5×#4111yescharacteristic
271#67—41no—
272#67—41no—
273C2×Dic3—21no—
274C2×Dic3—21no—
275C2×Dic3—22no—
276C2×Dic3—21no—
277C2×Dic3—42no—
278D12—21no—
279D12—21no—
280C4×D3—41no—
281C4×D3—41no—
282Dic6—21no—
283Dic6—21no—
284Dic6—44no—
285C3×C8—41no—
286C3⋊C8—41no—
287C22×C5—41no—
288C22×C5—121no—
289C22×C5—241no—
290C22×C5#88211yescharacteristic
291C22×C5#88511yescharacteristic
292C22×C5#89611yescharacteristic
293C4×C5—42no—
294C4×C5—62no—
295C4×C5—81no—
296C4×C5—121no—
297C4×C5—122no—
298C4×C5—121no—
299C4×C5#87511yescharacteristic
300C4×C5#88511yescharacteristic
301C4×C5#89611yescharacteristic
302C4×C5#89711yescharacteristic
303#41—61no—
304#41—61no—
305#41—61no—
306#41—61no—
307C2×Q8—31no—
308C2×Q8—62no—
309C2×D4—31no—
310C2×D4—61no—
311C22×C4—61no—
312C22×C4—61no—
313C22×C4C22×C5×D311yescharacteristic
314M16—21no—
315M16—21no—
316M16—61no—
317M16—61no—
318C2×C8—21no—
319C2×C8—61no—
320C4⋊C4—21no—
321C4⋊C4—21no—
322C4⋊C4—62no—
323C4⋊C4—62no—
324C4⋊C4—62no—
325C4⋊C4—61no—
326C22⋊C4—21no—
327C22⋊C4—21no—
328C22⋊C4—32no—
329C42—21no—
330C42—31no—
331C42—31no—
332C42—61no—
333C42—61no—
334C42C5×D1212yes—
335C3×C5#203111yescharacteristic, complemented, Hall
336C22×C3—41no—
337C22×C3C5×#12011yescharacteristic
338C22×C3C5×C22⋊Q811yescharacteristic
339C22×C3C5×#12411yescharacteristic
340C2×D3—41no—
341C2×D3—81no—
342C3×C4—42no—
343C3×C4—81no—
344C3×C4C5×C22⋊Q811yescharacteristic
345C3×C4C5×C22⋊Q811yescharacteristic
346C3×C4C5×#12411yescharacteristic
347C3×C4C5×#12411yescharacteristic
348Dic3—22no—
349Dic3—42no—
350Dic3—41no—
351Dic3—41no—
352C2×C5—21no—
353C2×C5—21no—
354C2×C5—21no—
355C2×C5—241no—
356C2×C5#467311yescharacteristic
357C23—61no—
358C23C5×D3×D411yescharacteristic
359Q8—61no—
360Q8—61no—
361Q8—124no—
362D4—61no—
363D4—61no—
364D4—121no—
365D4—121no—
366C2×C4—22no—
367C2×C4—41no—
368C2×C4—61no—
369C2×C4—61no—
370C2×C4—62no—
371C2×C4—61no—
372C2×C4—122no—
373C2×C4—121no—
374C2×C4—121no—
375C2×C4C2×C5×D1211yescharacteristic
376C2×C4C5×#23511yescharacteristic
377C2×C4C5×#23511yescharacteristic
378C2×C4C5×D3×D411yescharacteristic
379C2×C4C5×#23711yescharacteristic
380C2×C4C5×D3×Q811yescharacteristic
381C8—41no—
382C8—121no—
383C2×C3—21no—
384C2×C3—21no—
385C2×C3—21no—
386C2×C3C5×#39311yescharacteristic
387D3—81no—
388C5#6871411yescharacteristic, direct product factor, Sylow
389C22—41no—
390C22—121no—
391C22—241no—
392C22C5×#88211yescharacteristic
393C22C5×#88511yescharacteristic
394C22C5×#89611yescharacteristic
395C4—42no—
396C4—62no—
397C4—81no—
398C4—121no—
399C4—122no—
400C4—121no—
401C4C5×#87511yescharacteristic
402C4C5×#88511yescharacteristic
403C4C5×#89611yescharacteristic
404C4C5×#89711yescharacteristic
405C3C5×#203111yescharacteristic, complemented, Sylow
406C2—21no—
407C2—21no—
408C2—21no—
409C2—241no—
410C2C5×#467311yescharacteristic
subgroupG/H
centerC4×C5#875
commutator, cosocleC22×C3×C4C23×C5
FrattiniC22×C4C22×C5×D3
FittingC3×C5×H(3,ℤ/4)C2
socleC2×C3×C5#393
prime radicalC2×C5×#235C22
hypercenterC5×H(3,ℤ/4)D3
hypocenterC3C5×#2031
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4×C5 ◁ C22×C4×C5 ◁ C5×H(3,ℤ/4)

Complements

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

NH ≅ G/Ntype
1. C5×#5517C2semidirect
5. C3×C5×H(3,ℤ/4)C2semidirect
6. C5×#4559C2semidirect
7. C5×#4532C2semidirect
24. #68714C5direct (N × H)
31. C5×H(3,ℤ/4)D3semidirect
39. C2×C5×D12Q8semidirect
42. C2×C5×Dic6Q8semidirect
64. #5517C2×C5semidirect
68. C3×H(3,ℤ/4)C2×C5semidirect
69. #4559C2×C5semidirect
70. #4532C2×C5semidirect
166. H(3,ℤ/4)C5×D3semidirect
190. C2×D12C5×Q8semidirect
193. C2×Dic6C5×Q8semidirect
335. C3×C5#2031semidirect
388. C5#68714direct (N × H)
405. C3C5×#2031semidirect

Maximal subgroups

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

subgroupindex
C5×#55172
C2×C5×#8252
C5×#50062
C5×#50052
C3×C5×H(3,ℤ/4)2
C5×#45592
C5×#45322
C5×#20313
#687145