Finite Group Databasebeta

← #12252348

Subgroups — #12252348

Order 1280.

658 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#65196C211yesmaximal, characteristic
2#10610910C211yesmaximal, characteristic
3#10610877C211yesmaximal, characteristic
4C5×#2159C211yesmaximal, characteristic, complemented
5#10602221C211yesmaximal, characteristic, complemented
6#10601441C211yesmaximal, characteristic, complemented
7C2×#64710C211yesmaximal, characteristic, complemented
8#10600139C212yesmaximal, complemented
9D5×#367C212yesmaximal, complemented
10#10600119C212yesmaximal, complemented
11#10600118C212yesmaximal, complemented
12C2×#4047C2211yescharacteristic
13#65857—21no—
14#65838—21no—
15C2×C5×M32C2211yescharacteristic
16C5×#427C2211yescharacteristic
17C2×C5×#107C2211yescharacteristic
18#65395C2211yescharacteristic
19C2×#3950C2211yescharacteristic
20C22×#672C2211yescharacteristic
21#65284—21no—
22#65253—21no—
23#65196C418yescomplemented
24C2×#3930C2212yes—
25C2×D5×C16C2212yes—
26#65170—22no—
27#65169C2212yes—
28#65169C2212yes—
29D5×#107C2212yes—
30D5×#107C2212yes—
31#65150—22no—
32C5×#367C2212yes—
33C5×#366C2212yes—
34#64725C2212yes—
35#64724C2212yes—
36#64710C2212yes—
37#64710C2212yes—
38#64710C2211yescharacteristic
39#64710C2211yescharacteristic
40#64709C2211yescharacteristic
41#64709C2211yescharacteristic
42#33411—51nomaximal, Sylow
43C2×#709D411yescharacteristic
44C22×C5×C8C2311yescharacteristic
45C5×#117—21no—
46C2×C5×C4⋊C4—21no—
47C4×C22⋊D5—42no—
48#4072—21no—
49C2×C4⋊Dic5—21no—
50C2×#681D411yescharacteristic
51#4047C2×C414yes—
52#4047Q814yes—
53C2×#676D412yes—
54C2×C8×D5C2312yes—
55#4039—41no—
56#4038—22no—
57#4038—22no—
58D5×C4⋊C4—22no—
59D5×C4⋊C4—22no—
60#4033—41no—
61#4027—42no—
62C4×D20—41no—
63C4×Dic10—41no—
64C5×M32C2×C414yescomplemented
65C2×C5×C16C2312yes—
66C5×#107—21no—
67C5×#107C2312yes—
68C5×#107C2311yescharacteristic
69C5×#107C2311yescharacteristic
70C5×#106—21no—
71#3950—21no—
72#3950C2312yes—
73#3950C2311yescharacteristic
74#3950C2311yescharacteristic
75#3949—21no—
76C2×#672C2×C414yescomplemented
77C2×#672C2311yescharacteristic
78C2×#672C2311yescharacteristic
79#3940—22no—
80#3940—22no—
81#3939—22no—
82#3939—22no—
83#3930C2×C418yescomplemented
84D5×C16C2×C418yescomplemented
85C2×#504—51no—
86#2913—51no—
87#2913—51no—
88#2159—51no—
89#2159—52no—
90#2159—52no—
91#2159D511yescharacteristic, complemented
92#2158—51no—
93#2158—52no—
94C2×#367—51no—
95C2×#367—52no—
96C22×C4×C5—41no—
97C22×C4×C5C2×D411yescharacteristic
98C2×C22⋊D5D812yes—
99C22×Dic5—41no—
100#709C4⋊C414yes—
101#709Q1614yes—
102C2×D20D811yescharacteristic
103C2×C4×D5—42no—
104C2×C4×D5—42no—
105C2×C4×D5C2×D412yes—
106C2×Dic10D811yescharacteristic
107C2×C5×C8C22×C412yes—
108C2×C5×C8C2×D411yescharacteristic
109C2×C5×C8C2×Q812yes—
110C2×C5×C8C2411yescharacteristic
111C5×C4⋊C4—22no—
112C5×C4⋊C4—21no—
113C5×C4⋊C4—21no—
114C5×C4⋊C4—41no—
115C5×C22⋊C4—41no—
116C42×C5—41no—
117C22⋊Dic5—41no—
118#685—42no—
119#685—42no—
120C4⋊Dic5—21no—
121C4⋊Dic5—22no—
122C4⋊Dic5—21no—
123C4⋊Dic5—41no—
124#683—42no—
125#683—42no—
126C4×Dic5—41no—
127C4×Dic5—42no—
128C4×Dic5—42no—
129#681C4⋊C414yes—
130C2×#168C2×D412yes—
131#676C4⋊C418yes—
132C8×D5C22×C414yes—
133C8×D5C2×Q814yes—
134C5×C16C22×C414yes—
135#672C22×C412yes—
136#672C22×C412yes—
137C2×#131—51no—
138#520—101no—
139#520—101no—
140#504—58no—
141C2×M32—52no—
142C2×M32C2×D511yescharacteristic
143C22×C16—51no—
144C22×C16—52no—
145#428—102no—
146#428—102no—
147#428—101no—
148#428—101no—
149#427—51no—
150#427—52no—
151#427—52no—
152#427C2×D511yescharacteristic
153C2×#107—51no—
154C2×#107—52no—
155C2×#107—52no—
156C2×#107C2×D511yescharacteristic
157#367—52no—
158#367—51no—
159#367—52no—
160#367—52no—
161#367—51no—
162#367C2×D512yes—
163#366—52no—
164#366—51no—
165#366—51no—
166#366C2×D512yes—
167C23×C5C2×D811yescharacteristic
168C22×D5C2×D812yes—
169C2×C4×C5—41no—
170C2×C4×C5—42no—
171C2×C4×C5—41no—
172C2×C4×C5—161no—
173C2×C4×C5C2×C4⋊C412yes—
174C2×C4×C5C2×D811yescharacteristic
175C2×C4×C5C2×Q1612yes—
176C2×C4×C5C22×D411yescharacteristic
177C22⋊D5#10718yes—
178C2×Dic5—41no—
179C2×Dic5—41no—
180C2×Dic5—42no—
181C2×Dic5—161no—
182C2×Dic5C2×D812yes—
183D20#10714yes—
184C4×D5—162no—
185C4×D5—162no—
186C4×D5C2×C4⋊C414yes—
187C4×D5C2×Q1614yes—
188Dic10#10714yes—
189C5×C8C2×C4⋊C412yes—
190C5×C8C23×C411yescharacteristic
191C5×C8C22×Q811yescharacteristic
192#168C2×C4⋊C414yes—
193C2×#41—51no—
194#131—54no—
195#131—54no—
196C2×M16—51no—
197C2×M16—52no—
198C22×C8—52no—
199C22×C8C22×D511yescharacteristic
200C4×Q8—201no—
201C4×Q8—201no—
202C4×D4—201no—
203C4×D4—202no—
204C4×D4—201no—
205C4×D4—202no—
206#117—21no—
207#117—102no—
208#117—102no—
209#117—101no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—102no—
212C2×C4⋊C4—102no—
213C2×C4⋊C4—101no—
214M32—58no—
215M32C4×D514yescomplemented
216C2×C16—54no—
217C2×C16—51no—
218C2×C16—51no—
219C2×C16—58no—
220C2×C16C22×D512yes—
221#107—21no—
222#107—51no—
223#107—52no—
224#107—51no—
225#107—102no—
226#107—102no—
227#107—101no—
228#107C22×D511yescharacteristic
229#107C22×D511yescharacteristic
230#107C22×D512yes—
231#106—21no—
232#106—102no—
233#106—102no—
234#106—101no—
235C22×C5—21no—
236C22×C5—21no—
237C22×C5C2×#10712yes—
238C22×C5C22×D811yescharacteristic
239C2×D5—22no—
240C2×D5—22no—
241C2×D5C2×#10714yes—
242C4×C5—81no—
243C4×C5—82no—
244C4×C5—81no—
245C4×C5C2×#10712yes—
246C4×C5C22×C4⋊C411yescharacteristic
247C4×C5C22×Q1611yescharacteristic
248Dic5—81no—
249Dic5—82no—
250Dic5—81no—
251Dic5C2×#10714yes—
252#41—54no—
253#41—54no—
254C2×Q8—51no—
255C2×D4—52no—
256C2×D4—51no—
257C22×C4—41no—
258C22×C4—52no—
259C22×C4—202no—
260C22×C4—202no—
261C22×C4—201no—
262C22×C4D4×D511yescharacteristic
263M16—54no—
264M16—58no—
265C2×C8—52no—
266C2×C8—54no—
267C2×C8—54no—
268C2×C8C2×C4×D512yes—
269C2×C8D4×D511yescharacteristic
270C2×C8Q8×D512yes—
271C2×C8C23×D511yescharacteristic
272C4⋊C4—21no—
273C4⋊C4—21no—
274C4⋊C4—22no—
275C4⋊C4—41no—
276C4⋊C4—101no—
277C4⋊C4—102no—
278C4⋊C4—101no—
279C4⋊C4—202no—
280C4⋊C4—202no—
281C4⋊C4—201no—
282C22⋊C4—41no—
283C22⋊C4—202no—
284C22⋊C4—202no—
285C22⋊C4—201no—
286C42—41no—
287C42—202no—
288C42—202no—
289C42—201no—
290C16—52no—
291C16—52no—
292C16C2×C4×D514yes—
293C2×C5—22no—
294C2×C5C22×#10711yescharacteristic
295C2×C5#341611yescharacteristic
296C2×C5#341611yescharacteristic
297D5—24no—
298C23—52no—
299C23D5×D811yescharacteristic
300Q8—54no—
301D4—58no—
302D4—54no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—42no—
306C2×C4—52no—
307C2×C4—54no—
308C2×C4—54no—
309C2×C4—161no—
310C2×C4—201no—
311C2×C4—201no—
312C2×C4—202no—
313C2×C4—802no—
314C2×C4—802no—
315C2×C4—801no—
316C2×C4D5×C4⋊C412yes—
317C2×C4D5×D811yescharacteristic
318C2×C4D5×Q1612yes—
319C2×C4C2×D4×D511yescharacteristic
320C8—54no—
321C8D5×C4⋊C412yes—
322C8C22×C4×D511yescharacteristic
323C8C2×Q8×D511yescharacteristic
324C5#3341111yescharacteristic, complemented, Sylow
325C22—21no—
326C22—21no—
327C22—54no—
328C22—102no—
329C22—102no—
330C22D5×#10712yes—
331C22C2×D5×D811yescharacteristic
332C4—54no—
333C4—82no—
334C4—81no—
335C4—81no—
336C4—401no—
337C4—402no—
338C4—401no—
339C4D5×#10712yes—
340C4C2×D5×C4⋊C411yescharacteristic
341C4C2×D5×Q1611yescharacteristic
342C2—22no—
343C2—104no—
344C2C2×D5×#10711yescharacteristic
345C2#1061376811yescharacteristic
346C2#1061377111yescharacteristic
subgroupG/H
centerC22C2×D5×D8
commutatorC5×C8C23×C4
FrattiniC2×C8C23×D5
FittingC5×#2159C2
socleC22×C5C22×D8
prime radicalC2×#709D4
cosocleC2×C5×C8C24
hypercenter#2159D5
hypocenterC5#33411
LCSG ▷ C5×C8 ▷ C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2159

Complements

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

NH ≅ G/Ntype
4. C5×#2159C2semidirect
5. #10602221C2semidirect
6. #10601441C2semidirect
7. C2×#64710C2semidirect
8. #10600139C2semidirect
9. D5×#367C2semidirect
10. #10600119C2semidirect
11. #10600118C2semidirect
23. #65196C4semidirect
64. C5×M32C2×C4semidirect
76. C2×#672C2×C4semidirect
83. #3930C2×C4semidirect
84. D5×C16C2×C4semidirect
91. #2159D5semidirect
215. M32C4×D5semidirect
324. C5#33411semidirect

Maximal subgroups

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

subgroupindex
C2×#651962
#106109102
#106108772
C5×#21592
#106022212
#106014412
C2×#647102
#106001392
D5×#3672
#106001192
#106001182
#334115