Finite Group Databasebeta

← #12969642

Subgroups — #12969642

Order 1280.

1830 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#10618105C212yesmaximal
2C2×C5×#494C211yesmaximal, characteristic, complemented
3#10615618C214yesmaximal
4#10615605C214yesmaximal, complemented
5#10614929C212yesmaximal, complemented
6#10613525C214yesmaximal, complemented
7#10613524C218yesmaximal, complemented
8#10613523C214yesmaximal, complemented
9C4×#4047C211yesmaximal, characteristic
10#10610173C211yesmaximal, characteristic, complemented
11C22×C5×Q16C2214yes—
12C2×C5×C4⋊Q8C2212yes—
13#66150C2214yes—
14#66146C2214yes—
15C2×#4047C2212yes—
16#66066—24no—
17#66065C2218yes—
18#66062C2218yes—
19#66055—24no—
20#65847—28no—
21C4×#709C2211yescharacteristic
22#65804C2212yes—
23C5×#494C2212yes—
24C5×#494C2214yes—
25C5×#494C2212yes—
26C2×C4×C5×C8C2211yescharacteristic
27#65514C2218yes—
28#65478C22116yes—
29#65477C22116yes—
30#65473C22116yes—
31Q16×Dic5C22116yes—
32#65395C2214yes—
33#65390C2212yes—
34#65371C2218yes—
35#65370C2218yes—
36C4×#681C2211yescharacteristic
37#64978C2214yes—
38C4×#676C2212yes—
39C4×C8×D5C2212yes—
40#64972C2214yes—
41#64971C2212yes—
42#64969C2212yes—
43#60715—51nomaximal, Sylow
44C22×C5×Q8C2314yes—
45C2×#709C2311yescharacteristic
46C2×C5×Q16—216no—
47C2×C5×Q16C2318yes—
48C2×C5×Q16C2318yes—
49C22×C5×C8C2312yes—
50C5×C4⋊Q8—24no—
51C5×C4⋊Q8C2314yes—
52C5×C4⋊Q8C2314yes—
53C2×C5×C4⋊C4—24no—
54C2×C42×C5C2311yescharacteristic
55#4093—24no—
56#4092C23116yes—
57Q8×Dic5—28no—
58Q8×Dic5C23116yes—
59#4090—24no—
60#4075—28no—
61C4×C22⋊D5D414yes—
62#4072—28no—
63#4072C2314yes—
64#4072C2311yescharacteristic
65#4070—24no—
66C2×#681C2312yes—
67#4047—28no—
68C2×#676C2314yes—
69C2×C8×D5C2314yes—
70#4044—216no—
71#4043—28no—
72#4038—28no—
73#4036—28no—
74#4035—216no—
75#4033—28no—
76C4×D20D412yes—
77#4018C2312yes—
78C42×D5C2312yes—
79#4016C2314yes—
80#4015C2314yes—
81C4×Dic10—24no—
82C4×Dic10D412yes—
83C4×C5×C8C2311yescharacteristic
84C4×C5×C8C2311yescharacteristic
85C4×C5×C8C2312yes—
86#3967C23132yes—
87#3950C2318yes—
88#3949C2318yes—
89#3947C2314yes—
90C8×Dic5C2314yes—
91#3935C2312yes—
92C4×#168C2312yes—
93#3509—52no—
94#3185—54no—
95#3177—58no—
96#3170—54no—
97#3160—51no—
98C2×#494D511yescharacteristic, complemented
99#3139—52no—
100#3059—54no—
101#2950—54no—
102C4×#131—51no—
103C2×C5×Q8—24no—
104C2×C5×Q8—216no—
105C2×C5×Q8C2418yes—
106C22×C4×C5—24no—
107C22×C4×C5C2×D412yes—
108C22×C4×C5C2411yescharacteristic
109C2×C22⋊D5C2×D412yes—
110#709—24no—
111C2×D20C2×D411yescharacteristic
112C2×C4×D5C2×D414yes—
113C2×C4×D5C2412yes—
114C2×Dic10—24no—
115C2×Dic10C2×D411yescharacteristic
116C5×Q16—216no—
117C5×Q16—216no—
118C2×C5×C8#4118yes—
119C2×C5×C8C2412yes—
120C2×C5×C8C2412yes—
121C5×C4⋊C4—28no—
122C5×C4⋊C4—28no—
123C42×C5C2×D412yes—
124C42×C5C2411yescharacteristic
125C42×C5C2411yescharacteristic
126C22⋊Dic5—24no—
127C22⋊Dic5C2×D412yes—
128#685—216no—
129#685C2×D414yes—
130C4⋊Dic5—24no—
131C4⋊Dic5—216no—
132C4⋊Dic5C2×D412yes—
133C4⋊Dic5C2418yes—
134#683—28no—
135#683—216no—
136#683C2×D414yes—
137C4×Dic5—24no—
138C4×Dic5—216no—
139C4×Dic5C2×D414yes—
140C4×Dic5C2412yes—
141#681—24no—
142C2×#168C2414yes—
143#676—28no—
144C8×D5—28no—
145C22×Q16C2×D514yes—
146C2×#131—52no—
147#557—104no—
148#556—104no—
149#552—108no—
150#541—58no—
151#536—54no—
152#533—58no—
153#533—52no—
154C2×C4⋊Q8C2×D512yes—
155#519—54no—
156C4×#41—51no—
157#500—52no—
158#499—54no—
159#498—52no—
160#494C2×D512yes—
161#494C2×D514yes—
162#494C2×D512yes—
163#491—58no—
164#487—58no—
165#470—516no—
166#467—516no—
167#441—516no—
168C4×Q16—516no—
169#427—54no—
170#419—58no—
171#405—52no—
172#405—54no—
173C4×M16—51no—
174C4×M16—52no—
175C2×C4×C8—52no—
176C2×C4×C8C2×D511yescharacteristic
177C23×C5C22×D411yescharacteristic
178C22×D5C22×D412yes—
179C5×Q8—216no—
180C5×Q8—48no—
181C2×C4×C5—24no—
182C2×C4×C5—28no—
183C2×C4×C5—48no—
184C2×C4×C5C22×D412yes—
185C2×C4×C5C22×D411yescharacteristic
186C2×C4×C5C22×D412yes—
187C2×C4×C5C2×#4114yes—
188C2×C4×C5C2511yescharacteristic
189C22⋊D5—24no—
190C2×Dic5—28no—
191C2×Dic5C22×D414yes—
192C2×Dic5C22×D412yes—
193D20—22no—
194C4×D5—28no—
195C4×D5—24no—
196Dic10—22no—
197Dic10—48no—
198C5×C8C2×#4114yes—
199C5×C8C2×#4114yes—
200#168—24no—
201C2×#41—51no—
202C22×Q8C22×D514yes—
203C2×Q16—216no—
204C2×Q16C22×D518yes—
205C2×Q16C22×D518yes—
206#131—108no—
207C2×M16—52no—
208C2×M16—54no—
209C22×C8—54no—
210C22×C8C22×D512yes—
211C4⋊Q8—24no—
212C4⋊Q8—54no—
213C4⋊Q8—104no—
214C4⋊Q8C22×D514yes—
215C4⋊Q8C22×D514yes—
216#126—1016no—
217#125—54no—
218#125—108no—
219#125—1016no—
220#124—104no—
221#123—108no—
222C22⋊Q8—516no—
223C22⋊Q8—108no—
224C22⋊Q8—104no—
225C4×Q8—52no—
226C4×Q8—516no—
227C4×Q8—108no—
228C4×Q8—108no—
229C4×Q8—104no—
230C4×D4—54no—
231C4×D4—52no—
232#117—51no—
233#117—52no—
234#117—54no—
235#117—108no—
236#117—108no—
237C2×C4⋊C4—24no—
238C2×C42—52no—
239C2×C42C22×D511yescharacteristic
240#107—58no—
241#106—58no—
242#103—532no—
243#97—54no—
244#97—52no—
245C4×C8—54no—
246C4×C8—52no—
247C4×C8C22×D511yescharacteristic
248C4×C8C22×D511yescharacteristic
249C4×C8C22×D512yes—
250C22×C5—22no—
251C22×C5#53212yes—
252C22×C5C23×D411yescharacteristic
253C2×D5—22no—
254C2×D5—24no—
255C4×C5—22no—
256C4×C5—48no—
257C4×C5#53212yes—
258C4×C5#57814yes—
259C4×C5C22×#4112yes—
260Dic5—22no—
261Dic5—24no—
262Dic5—48no—
263#41—104no—
264C2×Q8—24no—
265C2×Q8—216no—
266C2×Q8—51no—
267C2×Q8—104no—
268C2×Q8C23×D518yes—
269C2×D4—52no—
270C2×D4—51no—
271C22×C4—24no—
272C22×C4—54no—
273C22×C4—52no—
274C22×C4D4×D512yes—
275C22×C4C23×D511yescharacteristic
276Q16—216no—
277Q16—216no—
278M16—104no—
279M16—108no—
280C2×C8—54no—
281C2×C8—108no—
282C2×C8#71118yes—
283C2×C8C23×D512yes—
284C2×C8C23×D512yes—
285C4⋊C4—28no—
286C4⋊C4—28no—
287C4⋊C4—52no—
288C4⋊C4—54no—
289C4⋊C4—58no—
290C4⋊C4—1016no—
291C4⋊C4—1016no—
292C4⋊C4—104no—
293C4⋊C4—108no—
294C22⋊C4—52no—
295C22⋊C4—54no—
296C22⋊C4—1016no—
297C22⋊C4—104no—
298C42—54no—
299C42—52no—
300C42—1016no—
301C42—104no—
302C42D4×D512yes—
303C42C23×D511yescharacteristic
304C42C23×D511yescharacteristic
305C2×C5—22no—
306C2×C5C2×#53211yescharacteristic
307C2×C5C2×#57812yes—
308D5—42no—
309C23—52no—
310C23C2×D4×D511yescharacteristic
311Q8—216no—
312Q8—48no—
313Q8—102no—
314Q8—208no—
315D4—104no—
316D4—102no—
317C2×C4—24no—
318C2×C4—28no—
319C2×C4—48no—
320C2×C4—54no—
321C2×C4—52no—
322C2×C4—108no—
323C2×C4—104no—
324C2×C4—108no—
325C2×C4C2×D4×D512yes—
326C2×C4C2×D4×D511yescharacteristic
327C2×C4C2×D4×D512yes—
328C2×C4C2×#71114yes—
329C2×C4C24×D511yescharacteristic
330C8—104no—
331C8C2×#71114yes—
332C8C2×#71114yes—
333C5#6071511yescharacteristic, complemented, Sylow
334C22—22no—
335C22—104no—
336C22—102no—
337C22#6605012yes—
338C22C22×D4×D511yescharacteristic
339C4—22no—
340C4—48no—
341C4—104no—
342C4—102no—
343C4—208no—
344C4#6605012yes—
345C4#6610014yes—
346C4C22×#71112yes—
347C2—22no—
348C2—202no—
349C2C2×#6605011yescharacteristic
350C2C2×#6610012yes—
subgroupG/H
centerC22C22×D4×D5
commutator, cosocleC2×C4×C5C25
FrattiniC2×C4C24×D5
FittingC2×C5×#494C2
socleC22×C5C23×D4
prime radicalC2×#709C23
hypercenterC2×#494D5
hypocenterC5#60715
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#494

Complements

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

NH ≅ G/Ntype
2. C2×C5×#494C2semidirect
4. #10615605C2semidirect
5. #10614929C2semidirect
6. #10613525C2semidirect
7. #10613524C2semidirect
8. #10613523C2semidirect
10. #10610173C2semidirect
98. C2×#494D5semidirect
333. C5#60715semidirect

Maximal subgroups

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

subgroupindex
#106181052
C2×C5×#4942
#106156182
#106156052
#106149292
#106135252
#106135242
#106135232
C4×#40472
#106101732
#607155