Finite Group Databasebeta

← C5×#8550 #11931270

Subgroups — C5×#8550 #11931270

Order 1280.

1134 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 1134 conjugacy classes of subgroups — too many to draw. Only its 166 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
1C22×C5×C22⋊Q8C211yesmaximal, characteristic
2C5×#1784C212yesmaximal, complemented
3C5×#1463C212yesmaximal, complemented
4C5×#1462C212yesmaximal
5C23×C5×Q8C2211yescharacteristic
6C24×C4×C5C411yescharacteristic
7C2×C5×C22⋊Q8—24no—
8C2×C5×C22⋊Q8—24no—
9C2×C5×C22⋊Q8—24no—
10C22×C5×C4⋊C4C411yescharacteristic
11C22×C5×C4⋊C4C411yescharacteristic
12C22×C5×C4⋊C4C2211yescharacteristic
13C22×C5×C22⋊C4C411yescharacteristic
14C22×C5×C22⋊C4C2211yescharacteristic
15C2×C5×#101—24no—
16C5×#407C2214yes—
17C5×#324—24no—
18C5×#323—24no—
19#8550C511yesmaximal, characteristic, direct product factor, Sylow
20C25×C5C2×C411yescharacteristic
21C22×C5×Q8—22no—
22C22×C5×Q8—22no—
23C22×C5×Q8—44no—
24C22×C5×Q8D414yes—
25C23×C4×C5—22no—
26C23×C4×C5—22no—
27C23×C4×C5—22no—
28C23×C4×C5—44no—
29C23×C4×C5C2×C411yescharacteristic
30C23×C4×C5C2×C411yescharacteristic
31C23×C4×C5C2×C411yescharacteristic
32C23×C4×C5C2×C411yescharacteristic
33C23×C4×C5C2×C411yescharacteristic
34C23×C4×C5C2311yescharacteristic
35C2×C5×M16—24no—
36C5×C22⋊Q8—232no—
37C2×C5×C4⋊C4—22no—
38C2×C5×C4⋊C4—22no—
39C2×C5×C4⋊C4—22no—
40C2×C5×C4⋊C4—22no—
41C2×C5×C4⋊C4—22no—
42C2×C5×C4⋊C4—22no—
43C2×C5×C4⋊C4—22no—
44C2×C5×C4⋊C4—22no—
45C2×C5×C4⋊C4D414yes—
46C2×C5×C22⋊C4—22no—
47C2×C5×C22⋊C4—22no—
48C2×C5×C22⋊C4—22no—
49C2×C5×C22⋊C4—22no—
50C2×C5×C22⋊C4—22no—
51C2×C5×C22⋊C4D414yes—
52C5×#101—48no—
53C5×C22⋊C8—24no—
54C5×C22⋊C8—24no—
55C22×C22⋊Q8C2×C511yescharacteristic
56#1784C2×C512yescomplemented
57#1463C2×C512yescomplemented
58#1462C2×C512yes—
59C24×C5—24no—
60C24×C5—21no—
61C24×C5—21no—
62C24×C5—42no—
63C24×C5—42no—
64C24×C5C22⋊C412yes—
65C24×C5C22×C411yescharacteristic
66C2×C5×Q8—28no—
67C2×C5×Q8—28no—
68C2×C5×Q8—44no—
69C2×C5×Q8—48no—
70C2×C5×Q8—48no—
71C22×C4×C5—21no—
72C22×C4×C5—21no—
73C22×C4×C5—24no—
74C22×C4×C5—28no—
75C22×C4×C5—21no—
76C22×C4×C5—21no—
77C22×C4×C5—21no—
78C22×C4×C5—21no—
79C22×C4×C5—21no—
80C22×C4×C5—21no—
81C22×C4×C5—21no—
82C22×C4×C5—21no—
83C22×C4×C5—42no—
84C22×C4×C5—42no—
85C22×C4×C5—42no—
86C22×C4×C5—48no—
87C22×C4×C5—48no—
88C22×C4×C5—44no—
89C22×C4×C5—44no—
90C22×C4×C5—42no—
91C22×C4×C5—42no—
92C22×C4×C5C22⋊C412yes—
93C22×C4×C5C22⋊C412yes—
94C22×C4×C5C22⋊C412yes—
95C22×C4×C5C22⋊C412yes—
96C22×C4×C5C22⋊C412yes—
97C22×C4×C5C2×D412yes—
98C22×C4×C5C2×D412yes—
99C22×C4×C5C2×D412yes—
100C5×M16—48no—
101C2×C5×C8—44no—
102C5×C4⋊C4—28no—
103C5×C4⋊C4—28no—
104C5×C4⋊C4—28no—
105C5×C22⋊C4—28no—
106C5×C22⋊C4—28no—
107C23×Q8C22×C511yescharacteristic
108C24×C4C4×C511yescharacteristic
109C2×C22⋊Q8—24no—
110C2×C22⋊Q8—24no—
111C2×C22⋊Q8—24no—
112C22×C4⋊C4C4×C511yescharacteristic
113C22×C4⋊C4C4×C511yescharacteristic
114C22×C4⋊C4C22×C511yescharacteristic
115C22×C22⋊C4C4×C511yescharacteristic
116C22×C22⋊C4C22×C511yescharacteristic
117C2×#101—24no—
118#407C22×C514yes—
119#324—24no—
120#323—24no—
121C23×C5—22no—
122C23×C5—22no—
123C23×C5—22no—
124C23×C5—24no—
125C23×C5—24no—
126C23×C5—22no—
127C23×C5—22no—
128C23×C5—42no—
129C23×C5—42no—
130C23×C5—42no—
131C23×C5—416no—
132C23×C5—42no—
133C23×C5—42no—
134C23×C5—42no—
135C23×C5#10114yescomplemented
136C23×C5C2×C22⋊C411yescharacteristic
137C23×C5C2×C22⋊C411yescharacteristic
138C23×C5C2×C22⋊C411yescharacteristic
139C5×Q8—416no—
140C2×C4×C5—24no—
141C2×C4×C5—24no—
142C2×C4×C5—22no—
143C2×C4×C5—24no—
144C2×C4×C5—22no—
145C2×C4×C5—22no—
146C2×C4×C5—22no—
147C2×C4×C5—22no—
148C2×C4×C5—22no—
149C2×C4×C5—24no—
150C2×C4×C5—24no—
151C2×C4×C5—42no—
152C2×C4×C5—42no—
153C2×C4×C5—416no—
154C2×C4×C5—42no—
155C2×C4×C5—42no—
156C2×C4×C5—42no—
157C2×C4×C5—42no—
158C2×C4×C5—42no—
159C2×C4×C5—42no—
160C2×C4×C5—42no—
161C2×C4×C5—42no—
162C2×C4×C5—42no—
163C2×C4×C5—42no—
164C2×C4×C5#9914yes—
165C2×C4×C5#12014yes—
166C5×C8—84no—
167C25C2×C4×C511yescharacteristic
168C22×Q8—22no—
169C22×Q8—22no—
170C22×Q8—44no—
171C22×Q8C5×D414yes—
172C23×C4—22no—
173C23×C4—22no—
174C23×C4—22no—
175C23×C4—44no—
176C23×C4C2×C4×C511yescharacteristic
177C23×C4C2×C4×C511yescharacteristic
178C23×C4C2×C4×C511yescharacteristic
179C23×C4C2×C4×C511yescharacteristic
180C23×C4C2×C4×C511yescharacteristic
181C23×C4C23×C511yescharacteristic
182C2×M16—24no—
183C22⋊Q8—232no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4C5×D414yes—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4C5×D414yes—
199#101—48no—
200C22⋊C8—24no—
201C22⋊C8—24no—
202C22×C5—28no—
203C22×C5—21no—
204C22×C5—21no—
205C22×C5—21no—
206C22×C5—21no—
207C22×C5—21no—
208C22×C5—21no—
209C22×C5—24no—
210C22×C5—44no—
211C22×C5—48no—
212C22×C5—44no—
213C22×C5—48no—
214C22×C5—42no—
215C22×C5—42no—
216C22×C5#37911yescharacteristic
217C22×C5C2×#9912yes—
218C22×C5C2×#10112yes—
219C22×C5#41312yes—
220C4×C5—24no—
221C4×C5—24no—
222C4×C5—42no—
223C4×C5—42no—
224C4×C5—42no—
225C4×C5—42no—
226C24—21no—
227C24—21no—
228C24—24no—
229C24—42no—
230C24—42no—
231C24C5×C22⋊C412yes—
232C24C22×C4×C511yescharacteristic
233C2×Q8—28no—
234C2×Q8—28no—
235C2×Q8—44no—
236C2×Q8—48no—
237C2×Q8—48no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—24no—
241C22×C4—28no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—48no—
254C22×C4—48no—
255C22×C4—44no—
256C22×C4—44no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4C5×C22⋊C412yes—
260C22×C4C5×C22⋊C412yes—
261C22×C4C5×C22⋊C412yes—
262C22×C4C5×C22⋊C412yes—
263C22×C4C5×C22⋊C412yes—
264C22×C4C2×C5×D412yes—
265C22×C4C2×C5×D412yes—
266C22×C4C2×C5×D412yes—
267M16—48no—
268C2×C8—44no—
269C4⋊C4—28no—
270C4⋊C4—28no—
271C4⋊C4—28no—
272C22⋊C4—28no—
273C22⋊C4—28no—
274C2×C5—22no—
275C2×C5—22no—
276C2×C5—22no—
277C2×C5—44no—
278C2×C5#178111yescharacteristic
279C2×C5#178411yescharacteristic
280C2×C5#178511yescharacteristic
281C23—22no—
282C23—22no—
283C23—22no—
284C23—24no—
285C23—24no—
286C23—22no—
287C23—22no—
288C23—42no—
289C23—42no—
290C23—42no—
291C23—416no—
292C23—42no—
293C23—42no—
294C23—42no—
295C23C5×#10114yescomplemented
296C23C2×C5×C22⋊C411yescharacteristic
297C23C2×C5×C22⋊C411yescharacteristic
298C23C2×C5×C22⋊C411yescharacteristic
299Q8—416no—
300C2×C4—24no—
301C2×C4—24no—
302C2×C4—22no—
303C2×C4—24no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—24no—
310C2×C4—24no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—416no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4C5×#9914yes—
325C2×C4C5×#12014yes—
326C8—84no—
327C5#855011yescharacteristic, direct product factor, Sylow
328C22—28no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—24no—
336C22—44no—
337C22—48no—
338C22—44no—
339C22—48no—
340C22—42no—
341C22—42no—
342C22C5×#37911yescharacteristic
343C22C2×C5×#9912yes—
344C22C2×C5×#10112yes—
345C22C5×#41312yes—
346C4—24no—
347C4—24no—
348C4—42no—
349C4—42no—
350C4—42no—
351C4—42no—
352C2—22no—
353C2—22no—
354C2—22no—
355C2—44no—
356C2C5×#178111yescharacteristic
357C2C5×#178411yescharacteristic
358C2C5×#178511yescharacteristic
subgroupG/H
center, socleC22×C5#379
commutatorC24C22×C4×C5
Frattini, cosocleC23×C4C23×C5
prime radicalC25×C5C2×C4
LCSG ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C22×C5 ◁ C23×C4×C5 ◁ G

Complements

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

NH ≅ G/Ntype
2. C5×#1784C2semidirect
3. C5×#1463C2semidirect
19. #8550C5direct (N × H)
56. #1784C2×C5semidirect
57. #1463C2×C5semidirect
135. C23×C5#101semidirect
295. C23C5×#101semidirect
327. C5#8550direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C5×C22⋊Q82
C5×#17842
C5×#14632
C5×#14622
#85505