Finite Group Databasebeta

← #11997120

Subgroups — #11997120

Order 1280.

838 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 838 conjugacy classes of subgroups — too many to draw. Only its 107 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×#2404C211yesmaximal, characteristic
2#10613824C211yesmaximal, characteristic, complemented
3#10602931C212yesmaximal, complemented
4#10602332C211yesmaximal, characteristic, complemented
5#10598119C212yesmaximal, complemented
6C22×C5×#41C2211yescharacteristic
7C2×C5×#120—21no—
8C22×C5×C22⋊C4C2211yescharacteristic
9C5×#393C412yescomplemented
10C5×#392C412yescomplemented
11C5×#386—21no—
12C5×#386—22no—
13C5×#381C2211yescharacteristic
14#65536—21no—
15C2×#3969—24no—
16#65501—21no—
17C4×C22⋊Dic5C2212yes—
18#65496C2212yes—
19#65240—24no—
20#64745—22no—
21#9479—51nomaximal, Sylow
22C25×C5D411yescharacteristic
23C2×C5×#41—22no—
24C2×C5×#41—22no—
25C2×C5×#41—44no—
26C22×C5×Q8C2×C411yescharacteristic
27C22×C5×D4—21no—
28C22×C5×D4—22no—
29C22×C5×D4C2×C411yescharacteristic
30C23×C4×C5—21no—
31C23×C4×C5D411yescharacteristic
32C23×C4×C5C2311yescharacteristic
33C5×#120—44no—
34C2×C5×C22⋊C4—22no—
35C2×C5×C22⋊C4—22no—
36C2×C5×C22⋊C4—22no—
37C2×C5×C22⋊C4—21no—
38C2×C5×C22⋊C4—22no—
39C2×C5×C22⋊C4C2×C412yes—
40C2×C5×C22⋊C4D412yes—
41C2×C22⋊Dic5—22no—
42C2×C22⋊Dic5—22no—
43C2×C22⋊Dic5—22no—
44C2×C4⋊Dic5—22no—
45C2×C4×Dic5D414yescomplemented
46C2×#681—22no—
47C5×#95—22no—
48C5×#95—21no—
49C5×#95C2×C412yes—
50#3969—48no—
51#3963—22no—
52#3963—22no—
53C22⋊#168—22no—
54C22⋊#168—22no—
55#2404D511yescharacteristic
56#2312—51no—
57#1789—52no—
58#1462—51no—
59#1284—52no—
60C24×C5—21no—
61C24×C5—21no—
62C24×C5—21no—
63C24×C5—42no—
64C24×C5—42no—
65C24×C5—81no—
66C24×C5C22⋊C412yes—
67C24×C5C2×D411yescharacteristic
68C5×#41—28no—
69C5×#41—44no—
70C5×#41—44no—
71C5×#41—44no—
72C2×C5×Q8—22no—
73C2×C5×Q8—42no—
74C2×C5×D4—24no—
75C2×C5×D4—22no—
76C2×C5×D4—42no—
77C2×C5×D4—42no—
78C2×C5×D4—41no—
79C2×C5×D4—41no—
80C2×C5×D4—44no—
81C2×C5×D4—44no—
82C2×C5×D4—82no—
83C22×C4×C5—21no—
84C22×C4×C5—21no—
85C22×C4×C5—21no—
86C22×C4×C5—22no—
87C22×C4×C5—42no—
88C22×C4×C5—42no—
89C22×C4×C5—42no—
90C22×C4×C5—42no—
91C22×C4×C5—81no—
92C22×C4×C5C22⋊C412yes—
93C22×C4×C5C22×C411yescharacteristic
94C22×C4×C5C2×D412yes—
95C22×C4×C5C2×D411yescharacteristic
96C22×C4×C5#4112yes—
97C22×Dic5—24no—
98C5×C22⋊C4—24no—
99C5×C22⋊C4—24no—
100C5×C22⋊C4—44no—
101C5×C22⋊C4—42no—
102C5×C22⋊C4—82no—
103C22⋊Dic5—44no—
104C22⋊Dic5—82no—
105C22⋊Dic5—82no—
106C4⋊Dic5—82no—
107C4×Dic5—44no—
108#681—44no—
109C2×#168—42no—
110C22×#41C2×D511yescharacteristic
111C2×#120—21no—
112C22×C22⋊C4C2×D511yescharacteristic
113C2×#104—104no—
114#407—52no—
115#393Dic512yescomplemented
116#392Dic512yescomplemented
117#388—104no—
118#386—21no—
119#386—22no—
120#381C2×D511yescharacteristic
121#380—101no—
122#379—101no—
123C4×C22⋊C4—52no—
124#323—102no—
125C23×C5—22no—
126C23×C5—22no—
127C23×C5—22no—
128C23×C5—22no—
129C23×C5—22no—
130C23×C5—22no—
131C23×C5—41no—
132C23×C5—42no—
133C23×C5—42no—
134C23×C5—42no—
135C23×C5—42no—
136C23×C5—41no—
137C23×C5—42no—
138C23×C5—42no—
139C23×C5—41no—
140C23×C5—42no—
141C23×C5—42no—
142C23×C5—81no—
143C23×C5—81no—
144C23×C5—88no—
145C23×C5C2×C22⋊C411yescharacteristic
146C23×C5#12011yescharacteristic
147C23×C5#12311yescharacteristic
148C5×Q8—24no—
149C5×Q8—42no—
150C5×D4—24no—
151C5×D4—42no—
152C5×D4—44no—
153C5×D4—44no—
154C5×D4—88no—
155C2×C4×C5—24no—
156C2×C4×C5—22no—
157C2×C4×C5—22no—
158C2×C4×C5—41no—
159C2×C4×C5—41no—
160C2×C4×C5—42no—
161C2×C4×C5—48no—
162C2×C4×C5—42no—
163C2×C4×C5—41no—
164C2×C4×C5—41no—
165C2×C4×C5—42no—
166C2×C4×C5—42no—
167C2×C4×C5—42no—
168C2×C4×C5—42no—
169C2×C4×C5—81no—
170C2×C4×C5C4×D412yes—
171C2×C4×C5#12112yes—
172C2×Dic5—44no—
173C2×Dic5—44no—
174C2×Dic5—44no—
175#168—82no—
176C25C22⋊D511yescharacteristic
177C2×#41—22no—
178C2×#41—22no—
179C2×#41—44no—
180C22×Q8C2×Dic511yescharacteristic
181C22×D4—21no—
182C22×D4—22no—
183C22×D4C2×Dic511yescharacteristic
184C23×C4—21no—
185C23×C4C22⋊D511yescharacteristic
186C23×C4C22×D511yescharacteristic
187C2×M16—102no—
188#120—44no—
189C2×C4⋊C4—102no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—21no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—102no—
196C2×C22⋊C4—102no—
197C2×C22⋊C4—102no—
198C2×C22⋊C4C2×Dic512yes—
199C2×C22⋊C4C22⋊D512yes—
200C2×C42—54no—
201#104—208no—
202C22⋊C8—102no—
203C22⋊C8—102no—
204#95—22no—
205#95—21no—
206#95—102no—
207#95—102no—
208#95C2×Dic512yes—
209C22×C5—21no—
210C22×C5—21no—
211C22×C5—21no—
212C22×C5—21no—
213C22×C5—21no—
214C22×C5—21no—
215C22×C5—21no—
216C22×C5—44no—
217C22×C5—44no—
218C22×C5—44no—
219C22×C5—42no—
220C22×C5—42no—
221C22×C5—42no—
222C22×C5—42no—
223C22×C5—82no—
224C22×C5—82no—
225C22×C5—84no—
226C22×C5—84no—
227C22×C5—81no—
228C22×C5—81no—
229C22×C5#38611yescharacteristic
230C22×C5#42112yes—
231C22×C5B(4,𝔽2)12yes—
232C4×C5—22no—
233C4×C5—22no—
234C4×C5—42no—
235C4×C5—42no—
236C4×C5—82no—
237Dic5—84no—
238C24—21no—
239C24—21no—
240C24—21no—
241C24—42no—
242C24—42no—
243C24—81no—
244C24C22⋊Dic512yes—
245C24C2×C22⋊D511yescharacteristic
246#41—28no—
247#41—44no—
248#41—44no—
249#41—44no—
250C2×Q8—22no—
251C2×Q8—42no—
252C2×D4—24no—
253C2×D4—22no—
254C2×D4—42no—
255C2×D4—42no—
256C2×D4—41no—
257C2×D4—41no—
258C2×D4—44no—
259C2×D4—44no—
260C2×D4—82no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—22no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—81no—
270C22×C4—104no—
271C22×C4C22⋊Dic512yes—
272C22×C4D4×D512yes—
273C22×C4#71112yes—
274C22×C4C22×Dic511yescharacteristic
275C22×C4C2×C22⋊D511yescharacteristic
276M16—204no—
277C2×C8—202no—
278C4⋊C4—402no—
279C22⋊C4—24no—
280C22⋊C4—24no—
281C22⋊C4—44no—
282C22⋊C4—42no—
283C22⋊C4—82no—
284C22⋊C4—204no—
285C22⋊C4—402no—
286C22⋊C4—402no—
287C42—204no—
288C2×C5—22no—
289C2×C5—22no—
290C2×C5—41no—
291C2×C5—82no—
292C2×C5—82no—
293C2×C5#188911yescharacteristic
294C2×C5#189711yescharacteristic
295C2×C5#191111yescharacteristic
296C23—22no—
297C23—22no—
298C23—22no—
299C23—22no—
300C23—22no—
301C23—22no—
302C23—41no—
303C23—42no—
304C23—42no—
305C23—42no—
306C23—42no—
307C23—41no—
308C23—42no—
309C23—42no—
310C23—42no—
311C23—41no—
312C23—42no—
313C23—81no—
314C23—81no—
315C23—88no—
316C23#408111yescharacteristic
317C23#408311yescharacteristic
318C23C2×C22⋊Dic511yescharacteristic
319Q8—24no—
320Q8—42no—
321D4—24no—
322D4—42no—
323D4—44no—
324D4—44no—
325D4—88no—
326C2×C4—24no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—41no—
330C2×C4—42no—
331C2×C4—41no—
332C2×C4—48no—
333C2×C4—42no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—81no—
341C2×C4—204no—
342C2×C4—204no—
343C2×C4—204no—
344C2×C4D4×Dic512yes—
345C2×C4#408512yes—
346C8—402no—
347C5#947911yescharacteristic, complemented, Sylow
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—44no—
356C22—44no—
357C22—44no—
358C22—42no—
359C22—42no—
360C22—42no—
361C22—42no—
362C22—82no—
363C22—84no—
364C22—82no—
365C22—84no—
366C22—81no—
367C22—81no—
368C22#6532112yes—
369C22#6550911yescharacteristic
370C22#6552512yes—
371C4—22no—
372C4—22no—
373C4—42no—
374C4—42no—
375C4—82no—
376C4—404no—
377C2—22no—
378C2—22no—
379C2—41no—
380C2—82no—
381C2—82no—
382C2#1060246211yescharacteristic
383C2#1060250511yescharacteristic
384C2#1060264611yescharacteristic
subgroupG/H
centerC22#65509
commutatorC22×C4×C5C22×C4
FrattiniC23×C4C22×D5
Fitting, prime radicalC5×#2404C2
socleC22×C5#386
cosocleC23×C4×C5C23
hypercenter#2404D5
hypocenterC5#9479
LCSG ▷ C22×C4×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2404

Complements

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

NH ≅ G/Ntype
2. #10613824C2semidirect
3. #10602931C2semidirect
4. #10602332C2semidirect
5. #10598119C2semidirect
9. C5×#393C4semidirect
10. C5×#392C4semidirect
45. C2×C4×Dic5D4semidirect
115. #393Dic5semidirect
116. #392Dic5semidirect
347. C5#9479semidirect

Maximal subgroups

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

subgroupindex
C5×#24042
#106138242
#106029312
#106023322
#105981192
#94795