Finite Group Databasebeta

← #12009580

Subgroups — #12009580

Order 1280.

468 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 115 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×#2396C211yesmaximal, characteristic
2#10602385C211yesmaximal, characteristic
3#10602364C211yesmaximal, characteristic
4#10598142C212yesmaximal
5#10598137C212yesmaximal
6C2×C4×C5×Q8C411yescharacteristic, complemented
7C2×C5×#117C2211yescharacteristic
8C2×#4072—21no—
9C5×#400C411yescharacteristic, complemented
10C5×#399C2211yescharacteristic
11C5×#398—21no—
12C5×#396—21no—
13C5×#395C411yescharacteristic, complemented
14C5×#393C2211yescharacteristic
15C4×C5×C4⋊C4—21no—
16C4×C5×C4⋊C4C411yescharacteristic, complemented
17C4×C5×C22⋊C4—21no—
18#65214—21no—
19#64748—21no—
20#64746—21no—
21#64746C2212yes—
22#64745C2212yes—
23#11715—51nomaximal, Sylow
24C22×C5×Q8C2×C411yescharacteristic
25C23×C4×C5D411yescharacteristic
26C4×C5×Q8—41no—
27C4×C5×Q8—41no—
28C5×#117—41no—
29C5×#117—41no—
30C2×C5×C4⋊C4—21no—
31C2×C5×C4⋊C4—21no—
32C2×C5×C4⋊C4—21no—
33C2×C5×C4⋊C4—21no—
34C2×C5×C4⋊C4C2×C411yescharacteristic
35C2×C5×C4⋊C4C2×C411yescharacteristic
36C2×C5×C4⋊C4D411yescharacteristic
37C2×C5×C22⋊C4—21no—
38C2×C5×C22⋊C4—21no—
39C2×C5×C22⋊C4D411yescharacteristic
40C2×C5×C22⋊C4C2311yescharacteristic
41C2×C42×C5—21no—
42C2×C42×C5—21no—
43C2×C42×C5C2×C411yescharacteristic
44C2×C42×C5D411yescharacteristic
45C2×C22⋊Dic5—22no—
46#4072—24no—
47#4072—42no—
48C2×C4⋊Dic5—41no—
49C2×C4×Dic5—41no—
50C5×#95—21no—
51C5×#95—21no—
52C5×#95—21no—
53C5×#95—21no—
54C5×#95C2×C411yescharacteristic
55C5×#95C2×C411yescharacteristic
56C22⋊#168—22no—
57C4⋊#168—41no—
58C4×#168—41no—
59#2396D511yescharacteristic
60#1740—51no—
61#1458—51no—
62#1352—52no—
63#1351—52no—
64C24×C5C2×D411yescharacteristic
65C2×C5×Q8—42no—
66C2×C5×Q8—41no—
67C22×C4×C5—21no—
68C22×C4×C5—21no—
69C22×C4×C5—22no—
70C22×C4×C5—21no—
71C22×C4×C5—21no—
72C22×C4×C5—21no—
73C22×C4×C5—21no—
74C22×C4×C5C22⋊C411yescharacteristic
75C22×C4×C5C22⋊C411yescharacteristic
76C22×C4×C5C22⋊C411yescharacteristic
77C22×C4×C5C22⋊C411yescharacteristic
78C22×C4×C5SD1612yes—
79C22×C4×C5Q1612yes—
80C22×C4×C5C22×C411yescharacteristic
81C22×C4×C5C2×D411yescharacteristic
82C22×Dic5—42no—
83C5×C4⋊C4—41no—
84C5×C4⋊C4—41no—
85C5×C4⋊C4—41no—
86C5×C4⋊C4—42no—
87C5×C4⋊C4—42no—
88C5×C4⋊C4—42no—
89C5×C4⋊C4—42no—
90C5×C22⋊C4—41no—
91C5×C22⋊C4—41no—
92C5×C22⋊C4—42no—
93C5×C22⋊C4—42no—
94C42×C5—21no—
95C42×C5—21no—
96C42×C5—21no—
97C42×C5—21no—
98C42×C5—41no—
99C42×C5—41no—
100C42×C5—42no—
101C22⋊Dic5—24no—
102C22⋊Dic5—42no—
103C4⋊Dic5—44no—
104C4×Dic5—44no—
105C2×#168—42no—
106C2×C4×Q8Dic511yescharacteristic, complemented
107C2×#117—101no—
108C2×#117C2×D511yescharacteristic
109#431—101no—
110#400Dic511yescharacteristic, complemented
111#399C2×D511yescharacteristic
112#398—21no—
113#396—21no—
114#395Dic511yescharacteristic, complemented
115#393C2×D511yescharacteristic
116C4×C4⋊C4—21no—
117C4×C4⋊C4Dic511yescharacteristic, complemented
118C4×C22⋊C4—21no—
119#342—52no—
120#342—101no—
121#324—101no—
122#323—52no—
123C23×C5—21no—
124C23×C5—21no—
125C23×C5—41no—
126C23×C5—41no—
127C23×C5C2×C22⋊C411yescharacteristic
128C23×C5C2×SD1611yescharacteristic
129C23×C5C2×Q1611yescharacteristic
130C5×Q8—42no—
131C5×Q8—42no—
132C2×C4×C5—21no—
133C2×C4×C5—21no—
134C2×C4×C5—22no—
135C2×C4×C5—22no—
136C2×C4×C5—22no—
137C2×C4×C5—22no—
138C2×C4×C5—22no—
139C2×C4×C5—22no—
140C2×C4×C5—21no—
141C2×C4×C5—21no—
142C2×C4×C5—21no—
143C2×C4×C5—21no—
144C2×C4×C5—41no—
145C2×C4×C5—41no—
146C2×C4×C5—41no—
147C2×C4×C5—41no—
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—41no—
155C2×C4×C5—41no—
156C2×C4×C5—41no—
157C2×C4×C5—41no—
158C2×C4×C5#9911yescharacteristic
159C2×C4×C5#9911yescharacteristic
160C2×C4×C5#10312yes—
161C2×C4×C5#10312yes—
162C2×C4×C5#10412yes—
163C2×Dic5—44no—
164C2×Dic5—82no—
165C2×Dic5—82no—
166#168—82no—
167C22×Q8C2×Dic511yescharacteristic
168C23×C4C22⋊D511yescharacteristic
169C4×Q8—41no—
170C4×Q8—41no—
171#117—41no—
172#117—41no—
173#117—104no—
174#117—202no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—201no—
180C2×C4⋊C4C2×Dic511yescharacteristic
181C2×C4⋊C4C2×Dic511yescharacteristic
182C2×C4⋊C4C22⋊D511yescharacteristic
183C2×C22⋊C4—21no—
184C2×C22⋊C4—21no—
185C2×C22⋊C4—102no—
186C2×C22⋊C4C22⋊D511yescharacteristic
187C2×C22⋊C4C22×D511yescharacteristic
188C2×C42—21no—
189C2×C42—21no—
190C2×C42—201no—
191C2×C42C2×Dic511yescharacteristic
192C2×C42C22⋊D511yescharacteristic
193C4⋊C8—201no—
194C22⋊C8—102no—
195C4×C8—201no—
196#95—21no—
197#95—21no—
198#95—21no—
199#95—21no—
200#95C2×Dic511yescharacteristic
201#95C2×Dic511yescharacteristic
202C22×C5—21no—
203C22×C5—21no—
204C22×C5—21no—
205C22×C5—21no—
206C22×C5—41no—
207C22×C5—41no—
208C22×C5—41no—
209C22×C5—41no—
210C22×C5—81no—
211C22×C5C2×#9911yescharacteristic
212C22×C5C2×#10311yescharacteristic
213C22×C5C2×#10411yescharacteristic
214C4×C5—22no—
215C4×C5—42no—
216C4×C5—41no—
217C4×C5—42no—
218C4×C5—41no—
219C4×C5—41no—
220C4×C5—41no—
221C4×C5—81no—
222C4×C5—81no—
223C4×C5#32812yes—
224C4×C5#32812yes—
225Dic5—84no—
226C24C2×C22⋊D511yescharacteristic
227C2×Q8—42no—
228C2×Q8—41no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—22no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—202no—
237C22×C4#68812yes—
238C22×C4#68912yes—
239C22×C4C22⋊Dic511yescharacteristic
240C22×C4C22⋊Dic511yescharacteristic
241C22×C4C22⋊Dic511yescharacteristic
242C22×C4C22⋊Dic511yescharacteristic
243C22×C4C22×Dic511yescharacteristic
244C22×C4C2×C22⋊D511yescharacteristic
245C2×C8—202no—
246C4⋊C4—41no—
247C4⋊C4—41no—
248C4⋊C4—41no—
249C4⋊C4—42no—
250C4⋊C4—42no—
251C4⋊C4—42no—
252C4⋊C4—42no—
253C4⋊C4—204no—
254C22⋊C4—41no—
255C22⋊C4—41no—
256C22⋊C4—42no—
257C22⋊C4—42no—
258C22⋊C4—104no—
259C22⋊C4—202no—
260C42—21no—
261C42—21no—
262C42—21no—
263C42—21no—
264C42—41no—
265C42—41no—
266C42—42no—
267C42—204no—
268C2×C5—21no—
269C2×C5—21no—
270C2×C5—41no—
271C2×C5—41no—
272C2×C5C2×#32811yescharacteristic
273C2×C5#212011yescharacteristic
274C2×C5#212611yescharacteristic
275C23—21no—
276C23—21no—
277C23—41no—
278C23—41no—
279C23C2×#68811yescharacteristic
280C23C2×#68911yescharacteristic
281C23C2×C22⋊Dic511yescharacteristic
282Q8—42no—
283Q8—42no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—204no—
311C2×C4—402no—
312C2×C4—402no—
313C2×C4#396611yescharacteristic
314C2×C4#396611yescharacteristic
315C2×C4#396712yes—
316C2×C4#396712yes—
317C2×C4#396912yes—
318C8—402no—
319C5#1171511yescharacteristic, complemented, Sylow
320C22—21no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—41no—
325C22—41no—
326C22—41no—
327C22—41no—
328C22—81no—
329C22C2×#396611yescharacteristic
330C22C2×#396711yescharacteristic
331C22C2×#396911yescharacteristic
332C4—22no—
333C4—42no—
334C4—42no—
335C4—41no—
336C4—41no—
337C4—41no—
338C4—41no—
339C4—81no—
340C4—81no—
341C4—404no—
342C4#6475812yes—
343C4#6475812yes—
344C2—21no—
345C2—21no—
346C2—41no—
347C2—41no—
348C2C2×#6475811yescharacteristic
349C2#1060250911yescharacteristic
350C2#1060261211yescharacteristic
subgroupG/H
centerC2×C4#3966
commutatorC22×C4×C5C22×C4
FrattiniC2×C22⋊C4C22×D5
FittingC5×#2396C2
socleC22×C5C2×#99
prime radicalC24×C5C2×D4
cosocleC2×C5×C22⋊C4C23
hypercenter#2396D5
hypocenterC5#11715
LCSG ▷ C22×C4×C5 ▷ C23×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×#117 ◁ #2396

Complements

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

NH ≅ G/Ntype
6. C2×C4×C5×Q8C4semidirect
9. C5×#400C4semidirect
13. C5×#395C4semidirect
16. C4×C5×C4⋊C4C4semidirect
106. C2×C4×Q8Dic5semidirect
110. #400Dic5semidirect
114. #395Dic5semidirect
117. C4×C4⋊C4Dic5semidirect
319. C5#11715semidirect

Maximal subgroups

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

subgroupindex
C5×#23962
#106023852
#106023642
#105981422
#105981372
#117155