Finite Group Databasebeta

← C5×#13224 #11935944

Subgroups — C5×#13224 #11935944

Order 1280.

552 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 126 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
1C5×#2547C211yesmaximal, characteristic
2C2×C5×#354C214yesmaximal
3C5×#1799C211yesmaximal, characteristic
4C4×C5×#99C211yesmaximal, characteristic
5C2×C5×C4⋊Q8C411yescharacteristic, complemented
6C2×C5×#124C2212yes—
7C2×C4×C5×D4C2211yescharacteristic
8C2×C5×#99C2212yes—
9C2×C5×#99C2212yes—
10C5×#397—21no—
11C5×#397—21no—
12C5×#391C412yescomplemented
13C5×#380—21no—
14C5×#380—21no—
15C5×#380—21no—
16C4×C5×C4⋊C4C411yescharacteristic, complemented
17C4×C5×C22⋊C4—21no—
18C5×#354—28no—
19C5×#342—21no—
20C5×#342—21no—
21#13224C511yesmaximal, characteristic, direct product factor, Sylow
22C22×C5×Q8C2×C412yes—
23C22×C5×D4C2311yescharacteristic
24C23×C4×C5D411yescharacteristic
25C23×C4×C5D411yescharacteristic
26C5×C4⋊Q8—41no—
27C5×C4⋊Q8—41no—
28C5×#124—24no—
29C5×#124—24no—
30C4×C5×D4—41no—
31C4×C5×D4—41no—
32C2×C5×C4⋊C4—21no—
33C2×C5×C4⋊C4—21no—
34C2×C5×C4⋊C4—21no—
35C2×C5×C4⋊C4—41no—
36C2×C5×C4⋊C4C2×C411yescharacteristic
37C2×C5×C22⋊C4—22no—
38C2×C5×C22⋊C4—22no—
39C2×C5×C22⋊C4—22no—
40C2×C5×C22⋊C4—22no—
41C2×C5×C22⋊C4D411yescharacteristic
42C2×C5×C22⋊C4D411yescharacteristic
43C2×C42×C5—41no—
44C2×C42×C5C2×C412yes—
45C2×C42×C5C2×C411yescharacteristic
46C5×#99—22no—
47C5×#99—24no—
48C5×#99D414yes—
49C5×#95—21no—
50C5×#95—22no—
51C5×#95—22no—
52C5×#95—41no—
53C5×#95—41no—
54#2547C2×C511yescharacteristic
55C2×#354C2×C514yes—
56#1799C2×C511yescharacteristic
57C4×#99C2×C511yescharacteristic
58C24×C5C2×D411yescharacteristic
59C24×C5C2×D411yescharacteristic
60C2×C5×Q8—24no—
61C2×C5×Q8—42no—
62C2×C5×Q8—42no—
63C2×C5×D4—41no—
64C2×C5×D4—41no—
65C2×C5×D4C2×D412yes—
66C2×C5×D4#4112yes—
67C22×C4×C5—21no—
68C22×C4×C5—21no—
69C22×C4×C5—21no—
70C22×C4×C5—21no—
71C22×C4×C5—22no—
72C22×C4×C5—22no—
73C22×C4×C5—42no—
74C22×C4×C5—42no—
75C22×C4×C5—42no—
76C22×C4×C5—42no—
77C22×C4×C5C22⋊C411yescharacteristic
78C22×C4×C5C22⋊C411yescharacteristic
79C22×C4×C5C22⋊C411yescharacteristic
80C22×C4×C5C22⋊C411yescharacteristic
81C22×C4×C5C22×C411yescharacteristic
82C5×C4⋊C4—41no—
83C5×C4⋊C4—42no—
84C5×C4⋊C4—42no—
85C5×C4⋊C4—41no—
86C5×C4⋊C4—41no—
87C5×C4⋊C4—82no—
88C5×C22⋊C4—24no—
89C5×C22⋊C4—41no—
90C5×C22⋊C4—41no—
91C5×C22⋊C4—42no—
92C5×C22⋊C4—44no—
93C5×C22⋊C4—42no—
94C5×C22⋊C4—42no—
95C5×C22⋊C4—42no—
96C5×C22⋊C4—42no—
97C42×C5—21no—
98C42×C5—21no—
99C42×C5—24no—
100C42×C5—82no—
101C2×C4⋊Q8C4×C511yescharacteristic, complemented
102C2×#124C22×C512yes—
103C2×C4×D4C22×C511yescharacteristic
104C2×#99C22×C512yes—
105C2×#99C22×C512yes—
106#397—21no—
107#397—21no—
108#391C4×C512yescomplemented
109#380—21no—
110#380—21no—
111#380—21no—
112C4×C4⋊C4C4×C511yescharacteristic, complemented
113C4×C22⋊C4—21no—
114#354—28no—
115#342—21no—
116#342—21no—
117C23×C5—21no—
118C23×C5—21no—
119C23×C5—21no—
120C23×C5—21no—
121C23×C5—41no—
122C23×C5—41no—
123C23×C5—41no—
124C23×C5—41no—
125C23×C5C2×C22⋊C411yescharacteristic
126C23×C5#12112yes—
127C23×C5#12411yescharacteristic
128C23×C5#12711yescharacteristic
129C5×Q8—84no—
130C5×D4—42no—
131C5×D4—42no—
132C2×C4×C5—21no—
133C2×C4×C5—21no—
134C2×C4×C5—22no—
135C2×C4×C5—21no—
136C2×C4×C5—21no—
137C2×C4×C5—21no—
138C2×C4×C5—42no—
139C2×C4×C5—42no—
140C2×C4×C5—41no—
141C2×C4×C5—41no—
142C2×C4×C5—41no—
143C2×C4×C5—44no—
144C2×C4×C5—42no—
145C2×C4×C5—42no—
146C2×C4×C5—42no—
147C2×C4×C5—42no—
148C2×C4×C5—42no—
149C2×C4×C5—42no—
150C2×C4×C5—81no—
151C2×C4×C5—81no—
152C2×C4×C5—82no—
153C2×C4×C5—82no—
154C2×C4×C5—82no—
155C2×C4×C5—82no—
156C2×C4×C5—82no—
157C2×C4×C5#9911yescharacteristic
158C2×C4×C5#9911yescharacteristic
159C2×C4×C5C4×D412yes—
160C22×Q8C2×C4×C512yes—
161C22×D4C23×C511yescharacteristic
162C23×C4C5×D411yescharacteristic
163C23×C4C5×D411yescharacteristic
164C4⋊Q8—41no—
165C4⋊Q8—41no—
166#124—24no—
167#124—24no—
168C4×D4—41no—
169C4×D4—41no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—41no—
174C2×C4⋊C4C2×C4×C511yescharacteristic
175C2×C22⋊C4—22no—
176C2×C22⋊C4—22no—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—22no—
179C2×C22⋊C4C5×D411yescharacteristic
180C2×C22⋊C4C5×D411yescharacteristic
181C2×C42—41no—
182C2×C42C2×C4×C512yes—
183C2×C42C2×C4×C511yescharacteristic
184#99—22no—
185#99—24no—
186#99C5×D414yes—
187#95—21no—
188#95—22no—
189#95—22no—
190#95—41no—
191#95—41no—
192C22×C5—21no—
193C22×C5—21no—
194C22×C5—22no—
195C22×C5—21no—
196C22×C5—21no—
197C22×C5—42no—
198C22×C5—41no—
199C22×C5—41no—
200C22×C5—41no—
201C22×C5—41no—
202C22×C5—41no—
203C22×C5—41no—
204C22×C5—81no—
205C22×C5—81no—
206C22×C5#39011yescharacteristic
207C22×C5C2×#9911yescharacteristic
208C22×C5#41011yescharacteristic
209C4×C5—21no—
210C4×C5—22no—
211C4×C5—41no—
212C4×C5—41no—
213C4×C5—41no—
214C4×C5—82no—
215C4×C5—84no—
216C4×C5—82no—
217C4×C5—162no—
218C4×C5#35412yescomplemented
219C24C2×C5×D411yescharacteristic
220C24C2×C5×D411yescharacteristic
221C2×Q8—24no—
222C2×Q8—42no—
223C2×Q8—42no—
224C2×D4—41no—
225C2×D4—41no—
226C2×D4C2×C5×D412yes—
227C2×D4C5×#4112yes—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4C5×C22⋊C411yescharacteristic
239C22×C4C5×C22⋊C411yescharacteristic
240C22×C4C5×C22⋊C411yescharacteristic
241C22×C4C5×C22⋊C411yescharacteristic
242C22×C4C22×C4×C511yescharacteristic
243C4⋊C4—41no—
244C4⋊C4—42no—
245C4⋊C4—42no—
246C4⋊C4—41no—
247C4⋊C4—41no—
248C4⋊C4—82no—
249C22⋊C4—24no—
250C22⋊C4—41no—
251C22⋊C4—41no—
252C22⋊C4—42no—
253C22⋊C4—44no—
254C22⋊C4—42no—
255C22⋊C4—42no—
256C22⋊C4—42no—
257C22⋊C4—42no—
258C42—21no—
259C42—21no—
260C42—24no—
261C42—82no—
262C2×C5—21no—
263C2×C5—21no—
264C2×C5—42no—
265C2×C5—41no—
266C2×C5—41no—
267C2×C5#196411yescharacteristic
268C2×C5C2×#35411yescharacteristic
269C2×C5#212911yescharacteristic
270C23—21no—
271C23—21no—
272C23—21no—
273C23—21no—
274C23—41no—
275C23—41no—
276C23—41no—
277C23—41no—
278C23C2×C5×C22⋊C411yescharacteristic
279C23C5×#12112yes—
280C23C5×#12411yescharacteristic
281C23C5×#12711yescharacteristic
282Q8—84no—
283D4—42no—
284D4—42no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—22no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—44no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—81no—
304C2×C4—81no—
305C2×C4—82no—
306C2×C4—82no—
307C2×C4—82no—
308C2×C4—82no—
309C2×C4—82no—
310C2×C4C5×#9911yescharacteristic
311C2×C4C5×#9911yescharacteristic
312C2×C4C4×C5×D412yes—
313C5#1322411yescharacteristic, direct product factor, Sylow
314C22—21no—
315C22—21no—
316C22—22no—
317C22—21no—
318C22—21no—
319C22—42no—
320C22—41no—
321C22—41no—
322C22—41no—
323C22—41no—
324C22—41no—
325C22—41no—
326C22—81no—
327C22—81no—
328C22C5×#39011yescharacteristic
329C22C2×C5×#9911yescharacteristic
330C22C5×#41011yescharacteristic
331C4—21no—
332C4—22no—
333C4—41no—
334C4—41no—
335C4—41no—
336C4—82no—
337C4—84no—
338C4—82no—
339C4—162no—
340C4C5×#35412yescomplemented
341C2—21no—
342C2—21no—
343C2—42no—
344C2—41no—
345C2—41no—
346C2C5×#196411yescharacteristic
347C2C2×C5×#35411yescharacteristic
348C2C5×#212911yescharacteristic
subgroupG/H
center, socleC22×C5C2×#99
commutatorC22×C4C22×C4×C5
Frattini, cosocleC22×D4C23×C5
prime radicalC22×C5×D4C23
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C5 ◁ C22×C4×C5 ◁ C2×C4×C5×D4 ◁ G

Complements

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

NH ≅ G/Ntype
5. C2×C5×C4⋊Q8C4semidirect
12. C5×#391C4semidirect
16. C4×C5×C4⋊C4C4semidirect
21. #13224C5direct (N × H)
101. C2×C4⋊Q8C4×C5semidirect
108. #391C4×C5semidirect
112. C4×C4⋊C4C4×C5semidirect
218. C4×C5#354semidirect
313. C5#13224direct (N × H)
340. C4C5×#354semidirect

Maximal subgroups

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

subgroupindex
C5×#25472
C2×C5×#3542
C5×#17992
C4×C5×#992
#132245