Finite Group Databasebeta

← C5×#85303 #14511041

Subgroups — C5×#85303 #14511041

Order 1920.

622 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 622 conjugacy classes of subgroups — too many to draw. Only its 96 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⋊Q8C212yesmaximal
2C5×A4×C4⋊C4C211yesmaximal, characteristic
3C5×C4⋊SL(2,ℤ/4)C211yesmaximal, characteristic
4C5×SL(2,ℤ/4)⋊C4C212yesmaximal
5C4×C5×SL(2,ℤ/4)C211yesmaximal, characteristic
6C5×#2440—31nomaximal, Hall
7C2×C4×C5×A4C2212yes—
8C2×C4×C5×A4C2211yescharacteristic
9C2×C5×SL(2,ℤ/4)C2212yes—
10C2×C5×SL(2,ℤ/4)C2211yescharacteristic
11C2×C5×SL(2,ℤ/4)C2211yescharacteristic
12C5×A4⋊Q8—24no—
13C5×#888—41nomaximal
14#85303C511yesmaximal, characteristic, direct product factor, Hall
15C2×C5×C4⋊Q8—31no—
16C2×C5×C22⋊Q8—32no—
17C22×C5×C4⋊C4D311yescharacteristic
18C5×#398—32no—
19C5×#395—32no—
20C5×#389—31no—
21C5×#389—31no—
22C5×#385—32no—
23C5×#384—31no—
24C5×#380—31no—
25C4×C5×C22⋊C4—31no—
26C22×C5×A4C2311yescharacteristic
27C2×C5×Dic6—42no—
28C4×C5×A4—22no—
29C4×C5×A4Q812yescomplemented
30C5×SL(2,ℤ/4)—22no—
31C5×SL(2,ℤ/4)D412yes—
32C5×SL(2,ℤ/4)Q812yescomplemented
33C3×C5×C4⋊C4—41no—
34C5×C4⋊Dic3—41no—
35C5×Dic3⋊C4—42no—
36C4×C5×Dic3—41no—
37C2×A4⋊Q8C2×C512yes—
38A4×C4⋊C4C2×C511yescharacteristic
39C4⋊SL(2,ℤ/4)C2×C511yescharacteristic
40SL(2,ℤ/4)⋊C4C2×C512yes—
41C4×SL(2,ℤ/4)C2×C511yescharacteristic
42C22×C5×Q8—32no—
43C23×C4×C5C2×D312yes—
44C23×C4×C5C2×D311yescharacteristic
45C5×C4⋊Q8—62no—
46C5×C4⋊Q8—61no—
47C5×C4⋊Q8—61no—
48C5×C22⋊Q8—64no—
49C5×C22⋊Q8—64no—
50C2×C5×C4⋊C4—31no—
51C2×C5×C4⋊C4—31no—
52C2×C5×C4⋊C4—32no—
53C2×C5×C4⋊C4—31no—
54C2×C5×C4⋊C4—31no—
55C2×C5×C4⋊C4—32no—
56C2×C5×C4⋊C4—32no—
57C2×C5×C4⋊C4—32no—
58C2×C5×C4⋊C4—32no—
59C2×C5×C4⋊C4—32no—
60C2×C5×C22⋊C4—31no—
61C2×C5×C22⋊C4—31no—
62C2×C5×C22⋊C4—32no—
63C2×C42×C5—31no—
64C2×C42×C5—31no—
65C5×#95—31no—
66C5×#95—31no—
67C5×#95—32no—
68C5×#95—32no—
69C5×#95—32no—
70#2440—31noSylow
71C2×C5×A4C2×D411yescharacteristic
72C2×C5×A4C2×Q811yescharacteristic
73C2×C5×A4C2×Q811yescharacteristic
74C2×C3×C4×C5—41no—
75C2×C3×C4×C5—42no—
76C2×C5×Dic3—41no—
77C2×C5×Dic3—41no—
78C2×C5×Dic3—42no—
79C5×Dic6—84no—
80C2×C4×A4C22×C512yes—
81C2×C4×A4C22×C511yescharacteristic
82C2×SL(2,ℤ/4)C22×C512yes—
83C2×SL(2,ℤ/4)C22×C511yescharacteristic
84C2×SL(2,ℤ/4)C22×C511yescharacteristic
85A4⋊Q8—24no—
86#888—41no—
87C24×C5C22×D311yescharacteristic
88C2×C5×Q8—64no—
89C2×C5×Q8—62no—
90C2×C5×Q8—62no—
91C2×C5×Q8—122no—
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—31no—
99C22×C4×C5—31no—
100C22×C4×C5—32no—
101C22×C4×C5—32no—
102C22×C4×C5—32no—
103C22×C4×C5—32no—
104C22×C4×C5—62no—
105C22×C4×C5Dic612yescomplemented
106C5×C4⋊C4—32no—
107C5×C4⋊C4—31no—
108C5×C4⋊C4—61no—
109C5×C4⋊C4—62no—
110C5×C4⋊C4—64no—
111C5×C4⋊C4—64no—
112C5×C4⋊C4—64no—
113C5×C4⋊C4—61no—
114C5×C4⋊C4—62no—
115C5×C4⋊C4—62no—
116C5×C4⋊C4—61no—
117C5×C4⋊C4—122no—
118C5×C4⋊C4S411yescharacteristic
119C5×C22⋊C4—32no—
120C5×C22⋊C4—32no—
121C5×C22⋊C4—32no—
122C5×C22⋊C4—32no—
123C5×C22⋊C4—62no—
124C5×C22⋊C4—62no—
125C42×C5—62no—
126C42×C5—61no—
127C42×C5—61no—
128C2×C4⋊Q8—31no—
129C2×C22⋊Q8—32no—
130C22×C4⋊C4C5×D311yescharacteristic
131#398—32no—
132#395—32no—
133#389—31no—
134#389—31no—
135#385—32no—
136#384—31no—
137#380—31no—
138C4×C22⋊C4—31no—
139C22×C3×C5—41no—
140C5×A4C4⋊Q811yescharacteristic, complemented
141C3×C4×C5—42no—
142C3×C4×C5—82no—
143C5×Dic3—42no—
144C5×Dic3—42no—
145C5×Dic3—82no—
146C22×A4C23×C511yescharacteristic
147C2×Dic6—42no—
148C4×A4—22no—
149C4×A4C5×Q812yescomplemented
150SL(2,ℤ/4)—22no—
151SL(2,ℤ/4)C5×D412yes—
152SL(2,ℤ/4)C5×Q812yescomplemented
153C3×C4⋊C4—41no—
154C4⋊Dic3—41no—
155Dic3⋊C4—42no—
156C4×Dic3—41no—
157C23×C5—31no—
158C23×C5—31no—
159C23×C5—31no—
160C23×C5—31no—
161C23×C5C2×Dic611yescharacteristic
162C23×C5D3×D411yescharacteristic
163C23×C5D3×Q811yescharacteristic
164C5×Q8—124no—
165C5×Q8—124no—
166C2×C4×C5—31no—
167C2×C4×C5—32no—
168C2×C4×C5—32no—
169C2×C4×C5—32no—
170C2×C4×C5—32no—
171C2×C4×C5—32no—
172C2×C4×C5—32no—
173C2×C4×C5—32no—
174C2×C4×C5—32no—
175C2×C4×C5—62no—
176C2×C4×C5—62no—
177C2×C4×C5—61no—
178C2×C4×C5—61no—
179C2×C4×C5—61no—
180C2×C4×C5—61no—
181C2×C4×C5—61no—
182C2×C4×C5—61no—
183C2×C4×C5—61no—
184C2×C4×C5—61no—
185C2×C4×C5—61no—
186C2×C4×C5—62no—
187C2×C4×C5—62no—
188C2×C4×C5—62no—
189C2×C4×C5—62no—
190C2×C4×C5—62no—
191C2×C4×C5—62no—
192C2×C4×C5—62no—
193C2×C4×C5—62no—
194C2×C4×C5C2×S412yes—
195C2×C4×C5C2×S411yescharacteristic
196C22×Q8—32no—
197C23×C4C2×C5×D312yes—
198C23×C4C2×C5×D311yescharacteristic
199C4⋊Q8—62no—
200C4⋊Q8—61no—
201C4⋊Q8—61no—
202C22⋊Q8—64no—
203C22⋊Q8—64no—
204C2×C4⋊C4—31no—
205C2×C4⋊C4—31no—
206C2×C4⋊C4—32no—
207C2×C4⋊C4—32no—
208C2×C4⋊C4—32no—
209C2×C4⋊C4—32no—
210C2×C4⋊C4—31no—
211C2×C4⋊C4—32no—
212C2×C4⋊C4—31no—
213C2×C4⋊C4—32no—
214C2×C22⋊C4—31no—
215C2×C22⋊C4—31no—
216C2×C22⋊C4—32no—
217C2×C42—31no—
218C2×C42—31no—
219#95—31no—
220#95—31no—
221#95—32no—
222#95—32no—
223#95—32no—
224C2×C3×C5—41no—
225C2×C3×C5—41no—
226C2×C3×C5—41no—
227C2×A4C2×C5×D411yescharacteristic
228C2×A4C2×C5×Q811yescharacteristic
229C2×A4C2×C5×Q811yescharacteristic
230C2×C3×C4—41no—
231C2×C3×C4—42no—
232C2×Dic3—41no—
233C2×Dic3—41no—
234C2×Dic3—42no—
235Dic6—84no—
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—31no—
245C22×C5—61no—
246C22×C5#88811yescharacteristic, complemented
247C22×C5C22×S411yescharacteristic
248C4×C5—22no—
249C4×C5—32no—
250C4×C5—62no—
251C4×C5—62no—
252C4×C5—62no—
253C4×C5—62no—
254C4×C5—62no—
255C4×C5—122no—
256C4×C5—122no—
257C4×C5A4⋊Q812yescomplemented
258C24C22×C5×D311yescharacteristic
259C2×Q8—64no—
260C2×Q8—62no—
261C2×Q8—62no—
262C2×Q8—122no—
263C22×C4—22no—
264C22×C4—31no—
265C22×C4—31no—
266C22×C4—32no—
267C22×C4—31no—
268C22×C4—31no—
269C22×C4—31no—
270C22×C4—31no—
271C22×C4—32no—
272C22×C4—32no—
273C22×C4—32no—
274C22×C4—32no—
275C22×C4—62no—
276C22×C4C5×Dic612yescomplemented
277C4⋊C4—32no—
278C4⋊C4—31no—
279C4⋊C4—61no—
280C4⋊C4—64no—
281C4⋊C4—64no—
282C4⋊C4—64no—
283C4⋊C4—61no—
284C4⋊C4—62no—
285C4⋊C4—62no—
286C4⋊C4—62no—
287C4⋊C4—61no—
288C4⋊C4—122no—
289C4⋊C4C5×S411yescharacteristic
290C22⋊C4—32no—
291C22⋊C4—32no—
292C22⋊C4—32no—
293C22⋊C4—32no—
294C22⋊C4—62no—
295C22⋊C4—62no—
296C42—61no—
297C42—62no—
298C42—61no—
299C3×C5—41noHall
300C22×C3—41no—
301A4C5×C4⋊Q811yescharacteristic, complemented
302C3×C4—42no—
303C3×C4—82no—
304Dic3—42no—
305Dic3—42no—
306Dic3—82no—
307C2×C5—31no—
308C2×C5—31no—
309C2×C5—31no—
310C2×C5—31no—
311C2×C5C2×A4⋊Q811yescharacteristic
312C2×C5D4×S411yescharacteristic
313C2×C5Q8×S411yescharacteristic
314C23—31no—
315C23—31no—
316C23—31no—
317C23—31no—
318C23C2×C5×Dic611yescharacteristic
319C23C5×D3×D411yescharacteristic
320C23C5×D3×Q811yescharacteristic
321Q8—124no—
322Q8—124no—
323C2×C4—31no—
324C2×C4—32no—
325C2×C4—32no—
326C2×C4—32no—
327C2×C4—32no—
328C2×C4—32no—
329C2×C4—32no—
330C2×C4—32no—
331C2×C4—32no—
332C2×C4—62no—
333C2×C4—62no—
334C2×C4—61no—
335C2×C4—61no—
336C2×C4—61no—
337C2×C4—61no—
338C2×C4—61no—
339C2×C4—61no—
340C2×C4—61no—
341C2×C4—61no—
342C2×C4—61no—
343C2×C4—62no—
344C2×C4—62no—
345C2×C4—62no—
346C2×C4—62no—
347C2×C4—62no—
348C2×C4—62no—
349C2×C4—62no—
350C2×C4—62no—
351C2×C4C2×C5×S412yes—
352C2×C4C2×C5×S411yescharacteristic
353C2×C3—41no—
354C2×C3—41no—
355C2×C3—41no—
356C5#8530311yescharacteristic, direct product factor, Sylow
357C22—31no—
358C22—31no—
359C22—31no—
360C22—31no—
361C22—31no—
362C22—31no—
363C22—31no—
364C22—31no—
365C22—31no—
366C22—61no—
367C22C5×#88811yescharacteristic, complemented
368C22C22×C5×S411yescharacteristic
369C4—22no—
370C4—32no—
371C4—62no—
372C4—62no—
373C4—62no—
374C4—62no—
375C4—62no—
376C4—122no—
377C4—122no—
378C4C5×A4⋊Q812yescomplemented
379C3—41noSylow
380C2—31no—
381C2—31no—
382C2—31no—
383C2—31no—
384C2C2×C5×A4⋊Q811yescharacteristic
385C2C5×D4×S411yescharacteristic
386C2C5×Q8×S411yescharacteristic
subgroupG/H
centerC22×C5C22×S4
commutator, cosocleC22×A4C23×C5
FrattiniC22C22×C5×S4
FittingC22×C5×C4⋊C4D3
socleC24×C5C22×D3
prime radicalC22×C5×A4C23
hypercenterC5×C4⋊C4S4
hypocenterA4C5×C4⋊Q8
LCSG ▷ C22×A4 ▷ A4
UCSC1 ◁ C22×C5 ◁ C5×C4⋊C4

Complements

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

NH ≅ G/Ntype
14. #85303C5direct (N × H)
29. C4×C5×A4Q8semidirect
32. C5×SL(2,ℤ/4)Q8semidirect
105. C22×C4×C5Dic6semidirect
140. C5×A4C4⋊Q8semidirect
149. C4×A4C5×Q8semidirect
152. SL(2,ℤ/4)C5×Q8semidirect
246. C22×C5#888semidirect
257. C4×C5A4⋊Q8semidirect
276. C22×C4C5×Dic6semidirect
301. A4C5×C4⋊Q8semidirect
356. C5#85303direct (N × H)
367. C22C5×#888semidirect
378. C4C5×A4⋊Q8semidirect

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×C4⋊SL(2,ℤ/4)2
C5×SL(2,ℤ/4)⋊C42
C4×C5×SL(2,ℤ/4)2
C5×#24403
C5×#8884
#853035