Finite Group Databasebeta

← C5×#11551 #11934271

Subgroups — C5×#11551 #11934271

Order 1280.

398 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 102 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×#2503C211yesmaximal, characteristic
2C5×#1463C211yesmaximal, characteristic
3C2×C5×#324C211yesmaximal, characteristic
4C5×#1324C212yesmaximal, complemented
5C5×#1319C212yesmaximal, complemented
6C2×C5×C22⋊Q8C411yescharacteristic
7C22×C5×C4⋊C4C2211yescharacteristic
8C5×#407—21no—
9C2×C5×C22⋊C8—21no—
10C5×#400—21no—
11C5×#400—21no—
12C5×#391C411yescharacteristic
13C5×#391C2211yescharacteristic
14C5×#382C411yescharacteristic
15C5×#382C2211yescharacteristic
16C5×#381—21no—
17C5×#381—21no—
18C4×C5×C22⋊C4C411yescharacteristic
19C5×#324—21no—
20C5×#324—21no—
21C5×#324C2212yes—
22C5×#324C2212yes—
23#11551C511yesmaximal, characteristic, direct product factor, Sylow
24C22×C5×Q8C2×C411yescharacteristic
25C23×C4×C5C2×C411yescharacteristic
26C23×C4×C5D411yescharacteristic
27C23×C4×C5D411yescharacteristic
28C2×C5×M16—41no—
29C22×C5×C8—41no—
30C5×C22⋊Q8—41no—
31C5×C22⋊Q8—41no—
32C2×C5×C4⋊C4—21no—
33C2×C5×C4⋊C4—41no—
34C2×C5×C4⋊C4—41no—
35C2×C5×C4⋊C4C2×C411yescharacteristic
36C2×C5×C4⋊C4C2×C411yescharacteristic
37C2×C5×C4⋊C4D411yescharacteristic
38C2×C5×C4⋊C4D411yescharacteristic
39C2×C5×C4⋊C4C2311yescharacteristic
40C2×C5×C22⋊C4—21no—
41C2×C5×C22⋊C4—21no—
42C2×C42×C5C2×C411yescharacteristic
43C2×C42×C5C2×C411yescharacteristic
44C5×C22⋊C8—22no—
45C5×C22⋊C8—22no—
46C5×C22⋊C8—41no—
47C5×C22⋊C8—41no—
48C5×#95—21no—
49C5×#95—21no—
50C5×#95—21no—
51C5×#95—21no—
52C5×#95—21no—
53C5×#95—21no—
54C5×#95—21no—
55C5×#95—21no—
56C5×#95—21no—
57#2503C2×C511yescharacteristic
58#1463C2×C511yescharacteristic
59C2×#324C2×C511yescharacteristic
60#1324C2×C512yescomplemented
61#1319C2×C512yescomplemented
62C24×C5C22⋊C411yescharacteristic
63C2×C5×Q8—22no—
64C2×C5×Q8—41no—
65C2×C5×Q8—41no—
66C22×C4×C5—21no—
67C22×C4×C5—21no—
68C22×C4×C5—21no—
69C22×C4×C5—21no—
70C22×C4×C5—21no—
71C22×C4×C5—21no—
72C22×C4×C5—22no—
73C22×C4×C5—21no—
74C22×C4×C5—21no—
75C22×C4×C5—21no—
76C22×C4×C5—21no—
77C22×C4×C5—41no—
78C22×C4×C5—42no—
79C22×C4×C5—42no—
80C22×C4×C5C22⋊C411yescharacteristic
81C22×C4×C5C22⋊C411yescharacteristic
82C22×C4×C5C22⋊C411yescharacteristic
83C22×C4×C5C22×C411yescharacteristic
84C22×C4×C5C2×D411yescharacteristic
85C22×C4×C5C2×D411yescharacteristic
86C5×M16—82no—
87C2×C5×C8—42no—
88C2×C5×C8—42no—
89C2×C5×C8—82no—
90C5×C4⋊C4—41no—
91C5×C4⋊C4—41no—
92C5×C4⋊C4—41no—
93C5×C4⋊C4—41no—
94C5×C4⋊C4—41no—
95C5×C4⋊C4—42no—
96C5×C22⋊C4—42no—
97C5×C22⋊C4—41no—
98C5×C22⋊C4—41no—
99C42×C5—22no—
100C42×C5—41no—
101C2×C22⋊Q8C4×C511yescharacteristic
102C22×C4⋊C4C22×C511yescharacteristic
103#407—21no—
104C2×C22⋊C8—21no—
105#400—21no—
106#400—21no—
107#391C4×C511yescharacteristic
108#391C22×C511yescharacteristic
109#382C4×C511yescharacteristic
110#382C22×C511yescharacteristic
111#381—21no—
112#381—21no—
113C4×C22⋊C4C4×C511yescharacteristic
114#324—21no—
115#324—21no—
116#324C22×C512yes—
117#324C22×C512yes—
118C23×C5—22no—
119C23×C5—21no—
120C23×C5—21no—
121C23×C5—41no—
122C23×C5#9911yescharacteristic
123C23×C5#9911yescharacteristic
124C23×C5C2×C22⋊C411yescharacteristic
125C5×Q8—82no—
126C2×C4×C5—21no—
127C2×C4×C5—21no—
128C2×C4×C5—21no—
129C2×C4×C5—21no—
130C2×C4×C5—21no—
131C2×C4×C5—21no—
132C2×C4×C5—21no—
133C2×C4×C5—21no—
134C2×C4×C5—21no—
135C2×C4×C5—42no—
136C2×C4×C5—42no—
137C2×C4×C5—42no—
138C2×C4×C5—42no—
139C2×C4×C5—41no—
140C2×C4×C5—41no—
141C2×C4×C5—41no—
142C2×C4×C5—41no—
143C2×C4×C5—41no—
144C2×C4×C5—41no—
145C2×C4×C5—41no—
146C2×C4×C5—41no—
147C2×C4×C5—42no—
148C2×C4×C5—41no—
149C2×C4×C5—41no—
150C2×C4×C5—41no—
151C2×C4×C5—41no—
152C2×C4×C5—41no—
153C2×C4×C5—41no—
154C2×C4×C5—81no—
155C2×C4×C5—81no—
156C2×C4×C5#13112yes—
157C5×C8—82no—
158C5×C8—82no—
159C22×Q8C2×C4×C511yescharacteristic
160C23×C4C2×C4×C511yescharacteristic
161C23×C4C5×D411yescharacteristic
162C23×C4C5×D411yescharacteristic
163C2×M16—41no—
164C22×C8—41no—
165C22⋊Q8—41no—
166C22⋊Q8—41no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—41no—
169C2×C4⋊C4—41no—
170C2×C4⋊C4C2×C4×C511yescharacteristic
171C2×C4⋊C4C2×C4×C511yescharacteristic
172C2×C4⋊C4C5×D411yescharacteristic
173C2×C4⋊C4C5×D411yescharacteristic
174C2×C4⋊C4C23×C511yescharacteristic
175C2×C22⋊C4—21no—
176C2×C22⋊C4—21no—
177C2×C42C2×C4×C511yescharacteristic
178C2×C42C2×C4×C511yescharacteristic
179C22⋊C8—22no—
180C22⋊C8—22no—
181C22⋊C8—41no—
182C22⋊C8—41no—
183#95—21no—
184#95—21no—
185#95—21no—
186#95—21no—
187#95—21no—
188#95—21no—
189#95—21no—
190#95—21no—
191#95—21no—
192C22×C5—21no—
193C22×C5—21no—
194C22×C5—21no—
195C22×C5—21no—
196C22×C5—21no—
197C22×C5—21no—
198C22×C5—21no—
199C22×C5—42no—
200C22×C5—42no—
201C22×C5#35412yes—
202C22×C5#40811yescharacteristic
203C22×C5C2×#9911yescharacteristic
204C22×C5#41311yescharacteristic
205C4×C5—22no—
206C4×C5—41no—
207C4×C5—41no—
208C4×C5—41no—
209C4×C5—41no—
210C4×C5—41no—
211C4×C5—81no—
212C4×C5—81no—
213C4×C5—81no—
214C4×C5—81no—
215C24C5×C22⋊C411yescharacteristic
216C2×Q8—22no—
217C2×Q8—41no—
218C2×Q8—41no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—22no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—41no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4C5×C22⋊C411yescharacteristic
234C22×C4C5×C22⋊C411yescharacteristic
235C22×C4C5×C22⋊C411yescharacteristic
236C22×C4C22×C4×C511yescharacteristic
237C22×C4C2×C5×D411yescharacteristic
238C22×C4C2×C5×D411yescharacteristic
239M16—82no—
240C2×C8—42no—
241C2×C8—42no—
242C2×C8—82no—
243C4⋊C4—41no—
244C4⋊C4—41no—
245C4⋊C4—41no—
246C4⋊C4—41no—
247C4⋊C4—41no—
248C4⋊C4—42no—
249C22⋊C4—42no—
250C22⋊C4—41no—
251C22⋊C4—41no—
252C42—22no—
253C42—41no—
254C2×C5—21no—
255C2×C5—21no—
256C2×C5—22no—
257C2×C5—41no—
258C2×C5#146811yescharacteristic
259C2×C5C2×#35411yescharacteristic
260C2×C5#213311yescharacteristic
261C23—21no—
262C23—21no—
263C23—22no—
264C23—41no—
265C23C5×#9911yescharacteristic
266C23C5×#9911yescharacteristic
267C23C2×C5×C22⋊C411yescharacteristic
268Q8—82no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—42no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—81no—
298C2×C4—81no—
299C2×C4C5×#13112yes—
300C8—82no—
301C8—82no—
302C5#1155111yescharacteristic, direct product factor, Sylow
303C22—21no—
304C22—21no—
305C22—21no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—42no—
311C22—42no—
312C22C5×#35412yes—
313C22C5×#40811yescharacteristic
314C22C2×C5×#9911yescharacteristic
315C22C5×#41311yescharacteristic
316C4—22no—
317C4—41no—
318C4—41no—
319C4—41no—
320C4—41no—
321C4—41no—
322C4—81no—
323C4—81no—
324C4—81no—
325C4—81no—
326C2—21no—
327C2—21no—
328C2—22no—
329C2—41no—
330C2C5×#146811yescharacteristic
331C2C2×C5×#35411yescharacteristic
332C2C5×#213311yescharacteristic
subgroupG/H
center, socleC22×C5C2×#99
commutatorC22×C4C22×C4×C5
Frattini, cosocleC2×C4⋊C4C23×C5
prime radicalC24×C5C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C5 ◁ C24×C5 ◁ C22×C5×C4⋊C4 ◁ G

Complements

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

NH ≅ G/Ntype
4. C5×#1324C2semidirect
5. C5×#1319C2semidirect
23. #11551C5direct (N × H)
60. #1324C2×C5semidirect
61. #1319C2×C5semidirect
302. C5#11551direct (N × H)

Maximal subgroups

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

subgroupindex
C5×#25032
C5×#14632
C2×C5×#3242
C5×#13242
C5×#13192
#115515