Finite Group Databasebeta

← C5×#85302 #14511040

Subgroups — C5×#85302 #14511040

Order 1920.

638 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 638 conjugacy classes of subgroups — too many to draw. Only its 108 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×A4⋊Q8C211yesmaximal, characteristic
2C5×A4×C4⋊C4C211yesmaximal, characteristic
3C5×SL(2,ℤ/4)⋊C4C212yesmaximal
4C4×C5×SL(2,ℤ/4)C211yesmaximal, characteristic
5C4×C5×SL(2,ℤ/4)C212yesmaximal
6C5×#2360—31nomaximal, Hall
7C2×C4×C5×A4C2212yes—
8C2×C4×C5×A4C2211yescharacteristic
9C2×C5×SL(2,ℤ/4)C2211yescharacteristic
10C2×C5×SL(2,ℤ/4)C2212yes—
11C2×C5×SL(2,ℤ/4)C2211yescharacteristic
12C5×A4⋊Q8C414yescomplemented
13C5×#887—41nomaximal
14#85302C511yesmaximal, characteristic, direct product factor, Hall
15C2×C5×C22⋊Q8—31no—
16C2×C4×C5×Q8—31no—
17C22×C5×C4⋊C4D311yescharacteristic
18C5×#391—31no—
19C5×#389—31no—
20C5×#389—32no—
21C5×#387—32no—
22C5×#385—32no—
23C4×C5×C4⋊C4—31no—
24C4×C5×C22⋊C4—31no—
25C4×C5×C22⋊C4—32no—
26C22×C5×A4C2311yescharacteristic
27C2×C5×Dic6—41no—
28C4×C5×A4—22no—
29C4×C5×A4C2×C412yes—
30C5×SL(2,ℤ/4)—21no—
31C5×SL(2,ℤ/4)C2×C414yes—
32C5×SL(2,ℤ/4)Q812yescomplemented
33C3×C5×C4⋊C4—41no—
34C5×Dic3⋊C4—42no—
35C4×C5×Dic3—42no—
36C4×C5×Dic3—41no—
37C2×A4⋊Q8C2×C511yescharacteristic
38A4×C4⋊C4C2×C511yescharacteristic
39SL(2,ℤ/4)⋊C4C2×C512yes—
40C4×SL(2,ℤ/4)C2×C511yescharacteristic
41C4×SL(2,ℤ/4)C2×C512yes—
42C22×C5×Q8—31no—
43C23×C4×C5C2×D312yes—
44C23×C4×C5C2×D311yescharacteristic
45C5×C22⋊Q8—34no—
46C5×C22⋊Q8—34no—
47C4×C5×Q8—61no—
48C4×C5×Q8—62no—
49C4×C5×Q8—61no—
50C2×C5×C4⋊C4—31no—
51C2×C5×C4⋊C4—32no—
52C2×C5×C4⋊C4—31no—
53C2×C5×C4⋊C4—32no—
54C2×C5×C4⋊C4—31no—
55C2×C5×C4⋊C4—32no—
56C2×C5×C4⋊C4—32no—
57C2×C5×C22⋊C4—31no—
58C2×C5×C22⋊C4—32no—
59C2×C5×C22⋊C4—31no—
60C2×C42×C5—31no—
61C2×C42×C5—32no—
62C2×C42×C5—31no—
63C2×C42×C5—32no—
64C5×#95—31no—
65C5×#95—32no—
66C5×#95—31no—
67C5×#95—32no—
68C5×#95—32no—
69C5×#95—32no—
70#2360—31noSylow
71C2×C5×A4C22×C411yescharacteristic
72C2×C5×A4C2×Q811yescharacteristic
73C2×C5×A4#4111yescharacteristic
74C2×C3×C4×C5—41no—
75C2×C3×C4×C5—42no—
76C2×C5×Dic3—41no—
77C2×C5×Dic3—42no—
78C2×C5×Dic3—41no—
79C5×Dic6—44no—
80C2×C4×A4C22×C512yes—
81C2×C4×A4C22×C511yescharacteristic
82C2×SL(2,ℤ/4)C22×C511yescharacteristic
83C2×SL(2,ℤ/4)C22×C512yes—
84C2×SL(2,ℤ/4)C22×C511yescharacteristic
85A4⋊Q8C4×C514yescomplemented
86#887—41no—
87C24×C5C22×D311yescharacteristic
88C2×C5×Q8—34no—
89C2×C5×Q8—61no—
90C2×C5×Q8—62no—
91C2×C5×Q8—61no—
92C22×C4×C5—22no—
93C22×C4×C5—31no—
94C22×C4×C5—31no—
95C22×C4×C5—32no—
96C22×C4×C5—31no—
97C22×C4×C5—31no—
98C22×C4×C5—32no—
99C22×C4×C5—32no—
100C22×C4×C5—32no—
101C22×C4×C5—32no—
102C22×C4×C5—31no—
103C22×C4×C5—31no—
104C22×C4×C5—62no—
105C22×C4×C5C4×D312yes—
106C5×C4⋊C4—34no—
107C5×C4⋊C4—38no—
108C5×C4⋊C4—32no—
109C5×C4⋊C4—31no—
110C5×C4⋊C4—61no—
111C5×C4⋊C4—62no—
112C5×C4⋊C4—62no—
113C5×C4⋊C4—122no—
114C5×C4⋊C4S411yescharacteristic
115C5×C22⋊C4—32no—
116C5×C22⋊C4—34no—
117C5×C22⋊C4—34no—
118C5×C22⋊C4—32no—
119C5×C22⋊C4—61no—
120C5×C22⋊C4—61no—
121C42×C5—62no—
122C42×C5—61no—
123C42×C5—62no—
124C42×C5—62no—
125C42×C5—61no—
126C42×C5—122no—
127C2×C22⋊Q8—31no—
128C2×C4×Q8—31no—
129C22×C4⋊C4C5×D311yescharacteristic
130#391—31no—
131#389—31no—
132#389—32no—
133#387—32no—
134#385—32no—
135C4×C4⋊C4—31no—
136C4×C22⋊C4—31no—
137C4×C22⋊C4—32no—
138C22×C3×C5—41no—
139C5×A4C4×Q811yescharacteristic, complemented
140C3×C4×C5—42no—
141C3×C4×C5—82no—
142C5×Dic3—44no—
143C5×Dic3—42no—
144C5×Dic3—81no—
145C22×A4C23×C511yescharacteristic
146C2×Dic6—41no—
147C4×A4—22no—
148C4×A4C2×C4×C512yes—
149SL(2,ℤ/4)—21no—
150SL(2,ℤ/4)C2×C4×C514yes—
151SL(2,ℤ/4)C5×Q812yescomplemented
152C3×C4⋊C4—41no—
153Dic3⋊C4—42no—
154C4×Dic3—42no—
155C4×Dic3—41no—
156C23×C5—31no—
157C23×C5—31no—
158C23×C5—31no—
159C23×C5—31no—
160C23×C5C2×C4×D311yescharacteristic
161C23×C5#23711yescharacteristic
162C23×C5D3×Q811yescharacteristic
163C5×Q8—64no—
164C5×Q8—64no—
165C2×C4×C5—31no—
166C2×C4×C5—32no—
167C2×C4×C5—32no—
168C2×C4×C5—32no—
169C2×C4×C5—34no—
170C2×C4×C5—34no—
171C2×C4×C5—32no—
172C2×C4×C5—32no—
173C2×C4×C5—32no—
174C2×C4×C5—62no—
175C2×C4×C5—62no—
176C2×C4×C5—61no—
177C2×C4×C5—61no—
178C2×C4×C5—61no—
179C2×C4×C5—62no—
180C2×C4×C5—62no—
181C2×C4×C5—61no—
182C2×C4×C5—62no—
183C2×C4×C5—61no—
184C2×C4×C5—62no—
185C2×C4×C5—62no—
186C2×C4×C5—62no—
187C2×C4×C5—61no—
188C2×C4×C5—61no—
189C2×C4×C5—61no—
190C2×C4×C5—61no—
191C2×C4×C5—61no—
192C2×C4×C5—61no—
193C2×C4×C5C2×S412yes—
194C2×C4×C5C2×S411yescharacteristic
195C22×Q8—31no—
196C23×C4C2×C5×D312yes—
197C23×C4C2×C5×D311yescharacteristic
198C22⋊Q8—34no—
199C22⋊Q8—34no—
200C4×Q8—61no—
201C4×Q8—62no—
202C4×Q8—61no—
203C2×C4⋊C4—31no—
204C2×C4⋊C4—32no—
205C2×C4⋊C4—31no—
206C2×C4⋊C4—32no—
207C2×C4⋊C4—31no—
208C2×C4⋊C4—32no—
209C2×C4⋊C4—32no—
210C2×C22⋊C4—32no—
211C2×C22⋊C4—31no—
212C2×C22⋊C4—31no—
213C2×C42—32no—
214C2×C42—31no—
215C2×C42—31no—
216C2×C42—32no—
217#95—31no—
218#95—32no—
219#95—31no—
220#95—32no—
221#95—32no—
222#95—32no—
223C2×C3×C5—41no—
224C2×C3×C5—41no—
225C2×C3×C5—41no—
226C2×A4C22×C4×C511yescharacteristic
227C2×A4C2×C5×Q811yescharacteristic
228C2×A4C5×#4111yescharacteristic
229C2×C3×C4—41no—
230C2×C3×C4—42no—
231C2×Dic3—41no—
232C2×Dic3—42no—
233C2×Dic3—41no—
234Dic6—44no—
235C22×C5—31no—
236C22×C5—31no—
237C22×C5—31no—
238C22×C5—31no—
239C22×C5—31no—
240C22×C5—31no—
241C22×C5—31no—
242C22×C5—31no—
243C22×C5—31no—
244C22×C5—61no—
245C22×C5#88711yescharacteristic, complemented
246C22×C5C22×S411yescharacteristic
247C4×C5—22no—
248C4×C5—32no—
249C4×C5—64no—
250C4×C5—62no—
251C4×C5—64no—
252C4×C5—62no—
253C4×C5—62no—
254C4×C5—121no—
255C4×C5—121no—
256C4×C5C4×S412yes—
257C24C22×C5×D311yescharacteristic
258C2×Q8—34no—
259C2×Q8—61no—
260C2×Q8—62no—
261C2×Q8—61no—
262C22×C4—22no—
263C22×C4—31no—
264C22×C4—31no—
265C22×C4—32no—
266C22×C4—31no—
267C22×C4—31no—
268C22×C4—32no—
269C22×C4—32no—
270C22×C4—32no—
271C22×C4—32no—
272C22×C4—31no—
273C22×C4—31no—
274C22×C4—62no—
275C22×C4C4×C5×D312yes—
276C4⋊C4—34no—
277C4⋊C4—38no—
278C4⋊C4—32no—
279C4⋊C4—31no—
280C4⋊C4—61no—
281C4⋊C4—62no—
282C4⋊C4—62no—
283C4⋊C4—122no—
284C4⋊C4C5×S411yescharacteristic
285C22⋊C4—34no—
286C22⋊C4—32no—
287C22⋊C4—34no—
288C22⋊C4—32no—
289C22⋊C4—61no—
290C22⋊C4—61no—
291C42—61no—
292C42—62no—
293C42—62no—
294C42—62no—
295C42—61no—
296C42—122no—
297C3×C5—41noHall
298C22×C3—41no—
299A4C4×C5×Q811yescharacteristic, complemented
300C3×C4—42no—
301C3×C4—82no—
302Dic3—44no—
303Dic3—42no—
304Dic3—81no—
305C2×C5—31no—
306C2×C5—31no—
307C2×C5—31no—
308C2×C5—31no—
309C2×C5C2×C4×S411yescharacteristic
310C2×C5#591411yescharacteristic
311C2×C5Q8×S411yescharacteristic
312C23—31no—
313C23—31no—
314C23—31no—
315C23—31no—
316C23C2×C4×C5×D311yescharacteristic
317C23C5×#23711yescharacteristic
318C23C5×D3×Q811yescharacteristic
319Q8—64no—
320Q8—64no—
321C2×C4—31no—
322C2×C4—32no—
323C2×C4—32no—
324C2×C4—32no—
325C2×C4—34no—
326C2×C4—34no—
327C2×C4—32no—
328C2×C4—32no—
329C2×C4—32no—
330C2×C4—62no—
331C2×C4—62no—
332C2×C4—61no—
333C2×C4—61no—
334C2×C4—61no—
335C2×C4—62no—
336C2×C4—62no—
337C2×C4—62no—
338C2×C4—61no—
339C2×C4—61no—
340C2×C4—62no—
341C2×C4—62no—
342C2×C4—62no—
343C2×C4—61no—
344C2×C4—61no—
345C2×C4—61no—
346C2×C4—61no—
347C2×C4—61no—
348C2×C4—61no—
349C2×C4C2×C5×S412yes—
350C2×C4C2×C5×S411yescharacteristic
351C2×C3—41no—
352C2×C3—41no—
353C2×C3—41no—
354C5#8530211yescharacteristic, direct product factor, Sylow
355C22—31no—
356C22—31no—
357C22—31no—
358C22—31no—
359C22—31no—
360C22—31no—
361C22—31no—
362C22—31no—
363C22—31no—
364C22—61no—
365C22C5×#88711yescharacteristic, complemented
366C22C22×C5×S411yescharacteristic
367C4—22no—
368C4—32no—
369C4—64no—
370C4—62no—
371C4—64no—
372C4—62no—
373C4—62no—
374C4—121no—
375C4—121no—
376C4C4×C5×S412yes—
377C3—41noSylow
378C2—31no—
379C2—31no—
380C2—31no—
381C2—31no—
382C2C2×C4×C5×S411yescharacteristic
383C2C5×#591411yescharacteristic
384C2C5×Q8×S411yescharacteristic
subgroupG/H
centerC22×C5C22×S4
commutatorC2×A4C22×C4×C5
FrattiniC22C22×C5×S4
FittingC22×C5×C4⋊C4D3
socleC24×C5C22×D3
prime radicalC22×C5×A4C23
cosocleC22×A4C23×C5
hypercenterC5×C4⋊C4S4
hypocenterA4C4×C5×Q8
LCSG ▷ C2×A4 ▷ A4
UCSC1 ◁ C22×C5 ◁ C5×C4⋊C4

Complements

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

NH ≅ G/Ntype
12. C5×A4⋊Q8C4semidirect
14. #85302C5direct (N × H)
32. C5×SL(2,ℤ/4)Q8semidirect
85. A4⋊Q8C4×C5semidirect
139. C5×A4C4×Q8semidirect
151. SL(2,ℤ/4)C5×Q8semidirect
245. C22×C5#887semidirect
299. A4C4×C5×Q8semidirect
354. C5#85302direct (N × H)
365. C22C5×#887semidirect

Maximal subgroups

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

subgroupindex
C2×C5×A4⋊Q82
C5×A4×C4⋊C42
C5×SL(2,ℤ/4)⋊C42
C4×C5×SL(2,ℤ/4)2
C5×#23603
C5×#8874
#853025