Finite Group Databasebeta

← #12010022

Subgroups — #12010022

Order 1280.

496 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 87 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×#2626C211yesmaximal, characteristic
2#10602335C211yesmaximal, characteristic
3#10602332C211yesmaximal, characteristic
4#10598142C212yesmaximal, complemented
5#10598137C212yesmaximal, complemented
6C2×C5×C22⋊Q8C411yescharacteristic, complemented
7C22×C5×C22⋊C4C2211yescharacteristic
8C5×#399C411yescharacteristic, complemented
9C5×#399C2211yescharacteristic
10C5×#396—21no—
11C5×#396—21no—
12C5×#393C411yescharacteristic, complemented
13C5×#393C2211yescharacteristic
14C5×#385—21no—
15C5×#380C411yescharacteristic, complemented
16C5×#379—21no—
17#65536—21no—
18#65496—21no—
19#64746—21no—
20#64746C2212yes—
21#64745—21no—
22#64745C2212yes—
23#11811—51nomaximal, Sylow
24C25×C5D411yescharacteristic
25C22×C5×Q8C2×C411yescharacteristic
26C23×C4×C5C2×C411yescharacteristic
27C23×C4×C5D411yescharacteristic
28C5×C22⋊Q8—41no—
29C5×C22⋊Q8—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×C22⋊C4—21no—
36C2×C5×C22⋊C4—21no—
37C2×C5×C22⋊C4—21no—
38C2×C5×C22⋊C4—21no—
39C2×C5×C22⋊C4—21no—
40C2×C5×C22⋊C4—21no—
41C2×C5×C22⋊C4—41no—
42C2×C5×C22⋊C4—41no—
43C2×C5×C22⋊C4C2×C411yescharacteristic
44C2×C5×C22⋊C4D411yescharacteristic
45C2×C5×C22⋊C4D411yescharacteristic
46C2×C5×C22⋊C4C2311yescharacteristic
47C2×C22⋊Dic5—22no—
48C2×C22⋊Dic5—41no—
49C2×C22⋊Dic5—41no—
50C2×#681—41no—
51C5×#95—21no—
52C5×#95—21no—
53C5×#95C2×C411yescharacteristic
54C5×#95C2×C411yescharacteristic
55C22⋊#168—22no—
56C22⋊#168—41no—
57#2626D511yescharacteristic
58#1819—51no—
59#1462—51no—
60#1352—52no—
61#1351—52no—
62C24×C5—21no—
63C24×C5—21no—
64C24×C5—22no—
65C24×C5—41no—
66C24×C5—81no—
67C24×C5—81no—
68C24×C5C22⋊C411yescharacteristic
69C24×C5C22⋊C411yescharacteristic
70C24×C5C2×D411yescharacteristic
71C2×C5×Q8—41no—
72C2×C5×Q8—81no—
73C22×C4×C5—21no—
74C22×C4×C5—21no—
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—81no—
82C22×C4×C5—81no—
83C22×C4×C5C22⋊C411yescharacteristic
84C22×C4×C5C22⋊C411yescharacteristic
85C22×C4×C5C22×C411yescharacteristic
86C22×C4×C5C2×D411yescharacteristic
87C22×Dic5—42no—
88C5×C4⋊C4—41no—
89C5×C4⋊C4—42no—
90C5×C4⋊C4—81no—
91C5×C4⋊C4—81no—
92C5×C4⋊C4—81no—
93C5×C22⋊C4—41no—
94C5×C22⋊C4—41no—
95C5×C22⋊C4—41no—
96C5×C22⋊C4—41no—
97C5×C22⋊C4—42no—
98C5×C22⋊C4—41no—
99C5×C22⋊C4—41no—
100C5×C22⋊C4—81no—
101C5×C22⋊C4—81no—
102C5×C22⋊C4—81no—
103C5×C22⋊C4—81no—
104C22⋊Dic5—42no—
105C22⋊Dic5—42no—
106C22⋊Dic5—82no—
107C22⋊Dic5—82no—
108#681—82no—
109C2×#168—42no—
110C2×C22⋊Q8Dic511yescharacteristic, complemented
111C22×C22⋊C4C2×D511yescharacteristic
112#407—101no—
113#399Dic511yescharacteristic, complemented
114#399C2×D511yescharacteristic
115#396—21no—
116#396—21no—
117#393Dic511yescharacteristic, complemented
118#393C2×D511yescharacteristic
119#385—21no—
120#380Dic511yescharacteristic, complemented
121#379—21no—
122#379—101no—
123#342—52no—
124#342—101no—
125#323—52no—
126#323—101no—
127C23×C5—21no—
128C23×C5—21no—
129C23×C5—21no—
130C23×C5—21no—
131C23×C5—21no—
132C23×C5—21no—
133C23×C5—21no—
134C23×C5—41no—
135C23×C5—41no—
136C23×C5—42no—
137C23×C5—42no—
138C23×C5—41no—
139C23×C5—41no—
140C23×C5—41no—
141C23×C5—41no—
142C23×C5—42no—
143C23×C5—42no—
144C23×C5—81no—
145C23×C5—81no—
146C23×C5—84no—
147C23×C5—84no—
148C23×C5#9911yescharacteristic
149C23×C5#9911yescharacteristic
150C23×C5C2×C22⋊C411yescharacteristic
151C23×C5#13712yes—
152C5×Q8—82no—
153C2×C4×C5—21no—
154C2×C4×C5—21no—
155C2×C4×C5—21no—
156C2×C4×C5—21no—
157C2×C4×C5—42no—
158C2×C4×C5—41no—
159C2×C4×C5—41no—
160C2×C4×C5—41no—
161C2×C4×C5—41no—
162C2×C4×C5—41no—
163C2×C4×C5—41no—
164C2×C4×C5—41no—
165C2×C4×C5—41no—
166C2×C4×C5—41no—
167C2×C4×C5—41no—
168C2×C4×C5—41no—
169C2×C4×C5—41no—
170C2×C4×C5—41no—
171C2×C4×C5—41no—
172C2×C4×C5—41no—
173C2×C4×C5—41no—
174C2×C4×C5—81no—
175C2×C4×C5—81no—
176C2×C4×C5—81no—
177C2×Dic5—82no—
178C2×Dic5—82no—
179C2×Dic5—82no—
180#168—82no—
181C25C22⋊D511yescharacteristic
182C22×Q8C2×Dic511yescharacteristic
183C23×C4C2×Dic511yescharacteristic
184C23×C4C22⋊D511yescharacteristic
185C2×M16—201no—
186C22⋊Q8—41no—
187C22⋊Q8—41no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4C2×Dic511yescharacteristic
193C2×C22⋊C4—21no—
194C2×C22⋊C4—21no—
195C2×C22⋊C4—21no—
196C2×C22⋊C4—21no—
197C2×C22⋊C4—21no—
198C2×C22⋊C4—21no—
199C2×C22⋊C4—41no—
200C2×C22⋊C4—41no—
201C2×C22⋊C4—102no—
202C2×C22⋊C4—201no—
203C2×C22⋊C4—201no—
204C2×C22⋊C4C2×Dic511yescharacteristic
205C2×C22⋊C4C22⋊D511yescharacteristic
206C2×C22⋊C4C22⋊D511yescharacteristic
207C2×C22⋊C4C22×D511yescharacteristic
208C22⋊C8—102no—
209C22⋊C8—201no—
210#95—21no—
211#95—21no—
212#95C2×Dic511yescharacteristic
213#95C2×Dic511yescharacteristic
214C22×C5—21no—
215C22×C5—21no—
216C22×C5—21no—
217C22×C5—21no—
218C22×C5—21no—
219C22×C5—21no—
220C22×C5—22no—
221C22×C5—21no—
222C22×C5—21no—
223C22×C5—42no—
224C22×C5—42no—
225C22×C5—42no—
226C22×C5—42no—
227C22×C5—41no—
228C22×C5—41no—
229C22×C5—41no—
230C22×C5—41no—
231C22×C5—41no—
232C22×C5—81no—
233C22×C5—81no—
234C22×C5—84no—
235C22×C5—84no—
236C22×C5C2×#9911yescharacteristic
237C22×C5#41911yescharacteristic
238C22×C5#42111yescharacteristic
239C4×C5—41no—
240C4×C5—41no—
241C4×C5—41no—
242C4×C5—41no—
243C4×C5—81no—
244C4×C5—81no—
245C4×C5—81no—
246C4×C5—81no—
247Dic5—162no—
248C24—21no—
249C24—21no—
250C24—22no—
251C24—41no—
252C24—81no—
253C24—81no—
254C24C22⋊Dic511yescharacteristic
255C24C22⋊Dic511yescharacteristic
256C24C2×C22⋊D511yescharacteristic
257C2×Q8—41no—
258C2×Q8—81no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—81no—
268C22×C4—81no—
269C22×C4—202no—
270C22×C4C22⋊Dic511yescharacteristic
271C22×C4C22⋊Dic511yescharacteristic
272C22×C4C22×Dic511yescharacteristic
273C22×C4C2×C22⋊D511yescharacteristic
274M16—402no—
275C2×C8—202no—
276C4⋊C4—41no—
277C4⋊C4—42no—
278C4⋊C4—81no—
279C4⋊C4—81no—
280C4⋊C4—81no—
281C22⋊C4—41no—
282C22⋊C4—41no—
283C22⋊C4—41no—
284C22⋊C4—41no—
285C22⋊C4—42no—
286C22⋊C4—41no—
287C22⋊C4—41no—
288C22⋊C4—81no—
289C22⋊C4—81no—
290C22⋊C4—81no—
291C22⋊C4—81no—
292C22⋊C4—202no—
293C22⋊C4—202no—
294C22⋊C4—402no—
295C22⋊C4—402no—
296C2×C5—21no—
297C2×C5—21no—
298C2×C5—42no—
299C2×C5—41no—
300C2×C5—41no—
301C2×C5—81no—
302C2×C5#151111yescharacteristic
303C2×C5#212111yescharacteristic
304C2×C5#212711yescharacteristic
305C23—21no—
306C23—21no—
307C23—21no—
308C23—21no—
309C23—21no—
310C23—21no—
311C23—21no—
312C23—41no—
313C23—41no—
314C23—42no—
315C23—42no—
316C23—41no—
317C23—41no—
318C23—41no—
319C23—41no—
320C23—42no—
321C23—42no—
322C23—81no—
323C23—81no—
324C23—84no—
325C23—84no—
326C23#396611yescharacteristic
327C23#396611yescharacteristic
328C23#408812yes—
329C23C2×C22⋊Dic511yescharacteristic
330Q8—82no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—42no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—81no—
353C2×C4—81no—
354C2×C4—81no—
355C2×C4—402no—
356C2×C4—402no—
357C2×C4—402no—
358C8—402no—
359C5#1181111yescharacteristic, complemented, Sylow
360C22—21no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—22no—
367C22—21no—
368C22—21no—
369C22—42no—
370C22—42no—
371C22—42no—
372C22—42no—
373C22—41no—
374C22—41no—
375C22—41no—
376C22—41no—
377C22—41no—
378C22—81no—
379C22—81no—
380C22—84no—
381C22—84no—
382C22C2×#396611yescharacteristic
383C22#6551411yescharacteristic
384C22#6552511yescharacteristic
385C4—41no—
386C4—41no—
387C4—41no—
388C4—41no—
389C4—81no—
390C4—81no—
391C4—81no—
392C4—81no—
393C4—802no—
394C2—21no—
395C2—21no—
396C2—42no—
397C2—41no—
398C2—41no—
399C2—81no—
400C2#1060248211yescharacteristic
401C2#1060264011yescharacteristic
402C2#1060265311yescharacteristic
subgroupG/H
centerC22C2×#3966
commutatorC22×C4×C5C22×C4
FrattiniC2×C22⋊C4C22×D5
FittingC5×#2626C2
socleC22×C5C2×#99
prime radicalC25×C5D4
cosocleC2×C5×C22⋊C4C23
hypercenter#2626D5
hypocenterC5#11811
LCSG ▷ C22×C4×C5 ▷ C23×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ #2626

Complements

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

NH ≅ G/Ntype
4. #10598142C2semidirect
5. #10598137C2semidirect
6. C2×C5×C22⋊Q8C4semidirect
8. C5×#399C4semidirect
12. C5×#393C4semidirect
15. C5×#380C4semidirect
110. C2×C22⋊Q8Dic5semidirect
113. #399Dic5semidirect
117. #393Dic5semidirect
120. #380Dic5semidirect
359. C5#11811semidirect

Maximal subgroups

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

subgroupindex
C5×#26262
#106023352
#106023322
#105981422
#105981372
#118115