Finite Group Databasebeta

← #12968694

Subgroups — #12968694

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×#3504C211yesmaximal, characteristic, complemented
2#10617728C211yesmaximal, characteristic
3#10615202C2112yesmaximal, complemented
4#10615068C213yesmaximal, complemented
5#10615066C213yesmaximal, complemented
6#10614743C213yesmaximal, complemented
7C2×#65358C213yesmaximal, complemented
8#10614301C214yesmaximal, complemented
9#10614242C211yesmaximal, characteristic, complemented
10C2×#4150C2212yes—
11C5×#557—22no—
12C5×#554—26no—
13C5×#544C2216yes—
14C5×#533C2213yes—
15C2×C5×#125C2213yes—
16C5×#520C2212yes—
17C4×C5×#41C2211yescharacteristic
18C2×#4094C2212yes—
19Q8×D20—22no—
20#65889—26no—
21#65860C2216yes—
22#65811C2213yes—
23C2×#4015C2213yes—
24#65382C22112yes—
25#65378C22112yes—
26#65358C2213yes—
27#65358C2213yes—
28#65358C2216yes—
29#65358C22112yes—
30#65357C22112yes—
31#65356C2211yescharacteristic
32#65356C2213yes—
33#65293C22112yes—
34#65291C22112yes—
35#65286C2216yes—
36C2×#3942C22112yes—
37#65255C22112yes—
38C4×#681C2213yes—
39C2×#3935C2213yes—
40#60627—51nomaximal, Sylow
41C2×C5×#41C2311yescharacteristic
42#4150—216no—
43C2×Q8×D5—22no—
44C2×#711—26no—
45C2×#709C2316yes—
46C22×Dic10C2316yes—
47C5×C4⋊Q8—26no—
48C5×C4⋊Q8C2316yes—
49C5×#125—26no—
50C5×#125—26no—
51C5×#125C2313yes—
52C5×#125C2311yescharacteristic
53C5×#125C2316yes—
54C5×C22⋊Q8—26no—
55C5×C22⋊Q8—212no—
56C4×C5×Q8—22no—
57C4×C5×Q8—26no—
58C4×C5×Q8D412yes—
59C4×C5×D4—26no—
60C4×C5×D4D416yes—
61C5×#117C2316yes—
62C5×#117C2313yes—
63C2×C5×C4⋊C4—26no—
64C2×C5×C4⋊C4C2316yes—
65C2×C42×C5C2313yes—
66#4094—28no—
67#4076—26no—
68C2×C4⋊Dic5—26no—
69C2×#681C2316yes—
70C22×#168C2316yes—
71#4043—212no—
72#4032—212no—
73#4021C2316yes—
74#4020C2311yescharacteristic
75C4×D20—26no—
76#4015—26no—
77#4015C2313yes—
78#4015C2316yes—
79#3942C23112yes—
80#3942C23112yes—
81#3942C23124yes—
82#3941C2314yes—
83#3941C23112yes—
84#3935C2316yes—
85#3935C2311yescharacteristic
86#3935C2313yes—
87C4×#168C2316yes—
88#3504D511yescharacteristic, complemented
89#3499—51no—
90#3142—53no—
91#3139—53no—
92#3137—512no—
93#3135—53no—
94C2×#491—53no—
95#2898—54no—
96#2875—51no—
97C5×#41#4118yes—
98C2×C5×Q8—26no—
99C2×C5×Q8C2×D411yescharacteristic
100C2×C5×D4C2×D413yes—
101C22×C4×C5—26no—
102C22×C4×C5C2×D416yes—
103C22×C4×C5C2413yes—
104C2×C22⋊D5—26no—
105C22×Dic5—26no—
106Q8×D5—48no—
107#711—424no—
108#709—224no—
109C2×D20—22no—
110C2×D20C2412yes—
111C2×C4×D5—26no—
112C2×Dic10—224no—
113C2×Dic10—26no—
114C2×Dic10C2416yes—
115C5×C4⋊C4—26no—
116C5×C4⋊C4—212no—
117C5×C4⋊C4—26no—
118C5×C4⋊C4—22no—
119C5×C4⋊C4—26no—
120C5×C4⋊C4C2×D416yes—
121C5×C4⋊C4C2416yes—
122C5×C4⋊C4C2412yes—
123C5×C22⋊C4—26no—
124C5×C22⋊C4C2×D416yes—
125C42×C5—26no—
126C42×C5C2×D416yes—
127C42×C5C2411yescharacteristic
128C42×C5C2413yes—
129#685—212no—
130C4⋊Dic5—26no—
131C4⋊Dic5—212no—
132#681—212no—
133C2×#168—212no—
134C2×#168C2416yes—
135C2×#168C2412yes—
136C2×2−1+4—52no—
137C2×#131—52no—
138#557—22no—
139#554—26no—
140D4×Q8—102no—
141#547—106no—
142#544C2×D516yes—
143#536—56no—
144#533C2×D513yes—
145#532—53no—
146C2×C4⋊Q8—53no—
147C2×#125C2×D513yes—
148#520C2×D512yes—
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×C5C22×D413yes—
167C22×D5—22no—
168C5×Q8—86no—
169C5×Q8#12414yes—
170C5×D4#124112yes—
171C2×C4×C5—22no—
172C2×C4×C5—26no—
173C2×C4×C5—46no—
174C2×C4×C5—86no—
175C2×C4×C5C22×D412yes—
176C2×C4×C5C22×D416yes—
177C2×C4×C5C22×D413yes—
178C2×C4×C5C2×#41112yes—
179C2×C4×C5C2511yescharacteristic
180C22⋊D5—412no—
181C2×Dic5—26no—
182C2×Dic5—412no—
183D20—24no—
184D20—44no—
185C4×D5—412no—
186Dic10—212no—
187Dic10—412no—
188#168—22no—
189#168—26no—
1902−1+4—1016no—
191C2×#41—56no—
192C2×#41—106no—
193C2×#41C22×D511yescharacteristic
194C22×Q8—56no—
195C22×Q8—102no—
196#131—108no—
197C2×M16—56no—
198C22×C8—56no—
199C4⋊Q8—26no—
200C4⋊Q8—53no—
201C4⋊Q8—56no—
202C4⋊Q8—106no—
203C4⋊Q8C22×D516yes—
204#127—51no—
205#125—26no—
206#125—26no—
207#125C22×D513yes—
208#125C22×D516yes—
209#125C22×D511yescharacteristic
210#124—56no—
211#123—1012no—
212C22⋊Q8—26no—
213C22⋊Q8—212no—
214C22⋊Q8—1012no—
215#121—106no—
216C4×Q8—22no—
217C4×Q8—26no—
218C4×Q8C22⋊D512yes—
219C4×D4—26no—
220C4×D4—106no—
221C4×D4C22⋊D516yes—
222#117C22×D516yes—
223#117C22×D513yes—
224C2×C4⋊C4—26no—
225C2×C4⋊C4—106no—
226C2×C4⋊C4C22×D516yes—
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×C5C2×#12416yes—
239C22×C5C23×D411yescharacteristic
240C2×D5—44no—
241C2×D5—42no—
242C4×C5—22no—
243C4×C5—26no—
244C4×C5—46no—
245C4×C5—42no—
246C4×C5C2×#12416yes—
247C4×C5C22×#4112yes—
248Dic5—46no—
249#41—1024no—
250#41—2024no—
251#41#71318yes—
252C2×Q8—26no—
253C2×Q8—56no—
254C2×Q8—106no—
255C2×Q8—1024no—
256C2×Q8—208no—
257C2×Q8C2×C22⋊D511yescharacteristic
258C2×D4—52no—
259C2×D4—106no—
260C2×D4—102no—
261C2×D4C2×C22⋊D513yes—
262C22×C4—26no—
263C22×C4—106no—
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—26no—
274C4⋊C4—22no—
275C4⋊C4—26no—
276C4⋊C4—106no—
277C4⋊C4—1012no—
278C4⋊C4C2×C22⋊D516yes—
279C4⋊C4C23×D516yes—
280C4⋊C4C23×D512yes—
281C22⋊C4—26no—
282C22⋊C4—1012no—
283C22⋊C4C2×C22⋊D516yes—
284C42—26no—
285C42C2×C22⋊D516yes—
286C42C23×D513yes—
287C42C23×D511yescharacteristic
288C2×C5—26no—
289C2×C5C22×#12411yescharacteristic
290C2×C5#358612yes—
291D5—82no—
292C23—102no—
293C23C22×C22⋊D513yes—
294Q8—86no—
295Q8—1012no—
296Q8—2012no—
297Q8#409314yes—
298D4—104no—
299D4—204no—
300D4—2012no—
301D4#4093112yes—
302C2×C4—22no—
303C2×C4—26no—
304C2×C4—46no—
305C2×C4—86no—
306C2×C4—106no—
307C2×C4—2012no—
308C2×C4—2012no—
309C2×C4C2×#713112yes—
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—202no—
319C22—204no—
320C22C2×#409316yes—
321C22C23×C22⋊D511yescharacteristic
322C4—22no—
323C4—26no—
324C4—46no—
325C4—42no—
326C4—206no—
327C4C2×#409316yes—
328C4C22×#71312yes—
329C2—26no—
330C2—402no—
331C2C22×#409311yescharacteristic
332C2#1061920012yes—
subgroupG/H
centerC22C23×C22⋊D5
commutator, cosocleC2×C4×C5C25
FrattiniC2×C4C24×D5
FittingC5×#3504C2
socleC22×C5C23×D4
prime radical#10617728C2
hypercenter#3504D5
hypocenterC5#60627
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C4×#41 ◁ #3504

Complements

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

NH ≅ G/Ntype
1. C5×#3504C2semidirect
3. #10615202C2semidirect
4. #10615068C2semidirect
5. #10615066C2semidirect
6. #10614743C2semidirect
7. C2×#65358C2semidirect
8. #10614301C2semidirect
9. #10614242C2semidirect
88. #3504D5semidirect
316. C5#60627semidirect

Maximal subgroups

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

subgroupindex
C5×#35042
#106177282
#106152022
#106150682
#106150662
#106147432
C2×#653582
#106143012
#106142422
#606275