Finite Group Databasebeta

← #12968690

Subgroups — #12968690

Order 1280.

1998 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#3499C211yesmaximal, characteristic
2#10617730C211yesmaximal, characteristic, complemented
3#10615842C214yesmaximal, complemented
4#10615132C2112yesmaximal, complemented
5#10615015C213yesmaximal, complemented
6#10615013C213yesmaximal, complemented
7#10614929C213yesmaximal, complemented
8C2×#65371C213yesmaximal, complemented
9#10614242C211yesmaximal, characteristic, complemented
10C2×C5×2−1+4C2212yes—
11C5×D4×Q8—22no—
12C5×#547—26no—
13C5×#536C2216yes—
14C5×#532C2213yes—
15C2×C5×C4⋊Q8C2213yes—
16C4×C5×#41C2211yescharacteristic
17#66161C2212yes—
18C2×#4094C2212yes—
19#65903—22no—
20#65873—26no—
21#65853C2216yes—
22#65804C2213yes—
23C2×#4016C2213yes—
24#65522C22112yes—
25#65521C22112yes—
26#65514C22112yes—
27C2×#3967C22112yes—
28#65371C22112yes—
29#65371C2213yes—
30#65371C2216yes—
31#65371C2213yes—
32#65370C22112yes—
33#65360C2211yescharacteristic
34#65360C2213yes—
35#65345C22112yes—
36#65344C22112yes—
37#65286C2216yes—
38C4×#681C2213yes—
39C2×#3935C2213yes—
40#60627—51nomaximal, Sylow
41C5×2−1+4—216no—
42C2×C5×#41—26no—
43C2×C5×#41C2316yes—
44C2×C5×#41C2311yescharacteristic
45C22×C5×Q8—22no—
46C22×C5×Q8C2316yes—
47C5×C4⋊Q8—26no—
48C5×C4⋊Q8C2316yes—
49C5×C4⋊Q8C2313yes—
50C5×#127C2311yescharacteristic
51C5×#124C2316yes—
52C5×#123—212no—
53C5×C22⋊Q8—212no—
54C5×#121—26no—
55C4×C5×Q8D412yes—
56C4×C5×D4—26no—
57C4×C5×D4D416yes—
58C5×#117C2313yes—
59C2×C5×C4⋊C4—26no—
60C2×C42×C5C2313yes—
61#4094—28no—
62Q8×Dic5—22no—
63D4×Dic5—26no—
64#4072C2316yes—
65C2×C4⋊Dic5C2316yes—
66#4070—26no—
67C2×#683—26no—
68C2×#681C2316yes—
69C22×#168C2316yes—
70#4036—26no—
71#4034—26no—
72#4024—212no—
73#4016—26no—
74#4016C2313yes—
75#4016C2316yes—
76#4016C2311yescharacteristic
77#4015C2316yes—
78C4×Dic10—26no—
79#3967C23112yes—
80#3967C23112yes—
81#3967C23124yes—
82#3964C2314yes—
83#3964C23112yes—
84#3935C2316yes—
85#3935C2311yescharacteristic
86#3935C2313yes—
87C4×#168C2316yes—
88#3504—51no—
89#3499D511yescharacteristic
90#3142—53no—
91#3139—53no—
92#3137—512no—
93#3135—53no—
94C2×#491—53no—
95#2898—54no—
96#2875—51no—
97C5×#41—224no—
98C5×#41—424no—
99C5×#41#4118yes—
100C2×C5×Q8—26no—
101C2×C5×Q8—224no—
102C2×C5×Q8—48no—
103C2×C5×Q8C2×D411yescharacteristic
104C2×C5×Q8C2416yes—
105C2×C5×D4—26no—
106C2×C5×D4—22no—
107C2×C5×D4C2×D413yes—
108C2×C5×D4C2412yes—
109C22×C4×C5—26no—
110C22×C4×C5—26no—
111C22×C4×C5C2×D416yes—
112C22×C4×C5C2413yes—
113C22×Dic5—26no—
114C2×Dic10—26no—
115C5×C4⋊C4—26no—
116C5×C4⋊C4—212no—
117C5×C4⋊C4C2×D416yes—
118C5×C22⋊C4—212no—
119C5×C22⋊C4C2×D416yes—
120C42×C5C2×D416yes—
121C42×C5C2411yescharacteristic
122C42×C5C2413yes—
123C22⋊Dic5—26no—
124C4⋊Dic5—26no—
125C4⋊Dic5—26no—
126C4⋊Dic5C2412yes—
127C4⋊Dic5C2416yes—
128#683—22no—
129#683—26no—
130#683—212no—
131C4×Dic5—26no—
132#681—212no—
133C2×#168—212no—
134C2×#168C2416yes—
135C2×#168C2412yes—
136C2×2−1+4C2×D512yes—
137C2×#131—52no—
138#557—102no—
139#554—106no—
140D4×Q8—22no—
141#547—26no—
142#544—56no—
143#536C2×D516yes—
144#533—53no—
145#532C2×D513yes—
146C2×C4⋊Q8C2×D513yes—
147C2×#125—53no—
148#520—52no—
149C4×#41C2×D511yescharacteristic
150#491—53no—
151#491—512no—
152#491—56no—
153#491—53no—
154#490—51no—
155#490—53no—
156#489—512no—
157#488—512no—
158#487—512no—
159#419—512no—
160#417—512no—
161#416—512no—
162C2×#103—512no—
163#405—56no—
164C4×M16—53no—
165C2×#97—53no—
166C23×C5—22no—
167C23×C5C22×D413yes—
168C5×Q8—212no—
169C5×Q8—412no—
170C5×Q8#12414yes—
171C5×D4—24no—
172C5×D4—44no—
173C5×D4—412no—
174C5×D4#124112yes—
175C2×C4×C5—26no—
176C2×C4×C5—46no—
177C2×C4×C5—412no—
178C2×C4×C5—412no—
179C2×C4×C5C22×D412yes—
180C2×C4×C5C22×D416yes—
181C2×C4×C5C22×D413yes—
182C2×C4×C5C2×#41112yes—
183C2×C4×C5C2511yescharacteristic
184C2×Dic5—22no—
185C2×Dic5—26no—
186C2×Dic5—86no—
187Dic10—86no—
188#168—22no—
189#168—26no—
1902−1+4—216no—
191C2×#41—26no—
192C2×#41C22×D516yes—
193C2×#41C22×D511yescharacteristic
194C22×Q8—22no—
195C22×Q8C22×D516yes—
196#131—108no—
197C2×M16—56no—
198C22×C8—56no—
199C4⋊Q8—26no—
200C4⋊Q8—56no—
201C4⋊Q8—106no—
202C4⋊Q8C22×D513yes—
203C4⋊Q8C22×D516yes—
204#127C22×D511yescharacteristic
205#125—53no—
206#125—56no—
207#125—51no—
208#125—106no—
209#125—106no—
210#124C22×D516yes—
211#123—212no—
212C22⋊Q8—212no—
213C22⋊Q8—106no—
214C22⋊Q8—1012no—
215#121—26no—
216C4×Q8—102no—
217C4×Q8—106no—
218C4×Q8C22⋊D512yes—
219C4×D4—26no—
220C4×D4—106no—
221C4×D4C22⋊D516yes—
222#117—56no—
223#117C22×D513yes—
224C2×C4⋊C4—26no—
225C2×C4⋊C4—56no—
226C2×C4⋊C4—106no—
227C2×C42C22×D513yes—
228#103—512no—
229#103—524no—
230#103—512no—
231#102—54no—
232#102—512no—
233#97—51no—
234#97—53no—
235#97—56no—
236C4×C8—56no—
237C22×C5—26no—
238C22×C5—44no—
239C22×C5—42no—
240C22×C5C2×#12416yes—
241C22×C5C23×D411yescharacteristic
242C4×C5—26no—
243C4×C5—22no—
244C4×C5—46no—
245C4×C5C2×#12416yes—
246C4×C5C22×#4112yes—
247Dic5—42no—
248Dic5—46no—
249#41—224no—
250#41—424no—
251#41#71118yes—
252C2×Q8—26no—
253C2×Q8—224no—
254C2×Q8—48no—
255C2×Q8—106no—
256C2×Q8C2×C22⋊D511yescharacteristic
257C2×Q8C23×D516yes—
258C2×D4—26no—
259C2×D4—22no—
260C2×D4C2×C22⋊D513yes—
261C2×D4C23×D512yes—
262C22×C4—26no—
263C22×C4—26no—
264C22×C4—106no—
265C22×C4C2×C22⋊D516yes—
266C22×C4C23×D513yes—
267M16—1012no—
268C2×C8—56no—
269C2×C8—52no—
270C2×C8—1012no—
271C4⋊C4—26no—
272C4⋊C4—212no—
273C4⋊C4—56no—
274C4⋊C4—52no—
275C4⋊C4—106no—
276C4⋊C4—106no—
277C4⋊C4—1012no—
278C4⋊C4—102no—
279C4⋊C4—106no—
280C4⋊C4C2×C22⋊D516yes—
281C22⋊C4—212no—
282C22⋊C4—106no—
283C22⋊C4C2×C22⋊D516yes—
284C42—106no—
285C42C2×C22⋊D516yes—
286C42C23×D513yes—
287C42C23×D511yescharacteristic
288C2×C5—26no—
289C2×C5—82no—
290C2×C5C22×#12411yescharacteristic
291C2×C5#358612yes—
292C23—22no—
293C23C22×C22⋊D513yes—
294Q8—212no—
295Q8—412no—
296Q8—406no—
297Q8#408214yes—
298D4—24no—
299D4—44no—
300D4—412no—
301D4#4082112yes—
302C2×C4—26no—
303C2×C4—46no—
304C2×C4—412no—
305C2×C4—412no—
306C2×C4—102no—
307C2×C4—106no—
308C2×C4—406no—
309C2×C4C2×#711112yes—
310C2×C4C22×C22⋊D512yes—
311C2×C4C22×C22⋊D516yes—
312C2×C4C22×C22⋊D513yes—
313C2×C4C24×D511yescharacteristic
314C8—102no—
315C8—106no—
316C5#6062711yescharacteristic, complemented, Sylow
317C22—26no—
318C22—44no—
319C22—42no—
320C22C2×#408216yes—
321C22C23×C22⋊D511yescharacteristic
322C4—22no—
323C4—26no—
324C4—46no—
325C4—206no—
326C4—202no—
327C4C2×#408216yes—
328C4C22×#71112yes—
329C2—26no—
330C2—82no—
331C2C22×#408211yescharacteristic
332C2#1061918812yes—
subgroupG/H
centerC22C23×C22⋊D5
commutator, cosocleC2×C4×C5C25
FrattiniC2×C4C24×D5
Fitting, prime radicalC5×#3499C2
socleC22×C5C23×D4
hypercenter#3499D5
hypocenterC5#60627
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C4×#41 ◁ #3499

Complements

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

NH ≅ G/Ntype
2. #10617730C2semidirect
3. #10615842C2semidirect
4. #10615132C2semidirect
5. #10615015C2semidirect
6. #10615013C2semidirect
7. #10614929C2semidirect
8. C2×#65371C2semidirect
9. #10614242C2semidirect
316. C5#60627semidirect

Maximal subgroups

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

subgroupindex
C5×#34992
#106177302
#106158422
#106151322
#106150152
#106150132
#106149292
C2×#653712
#106142422
#606275