Finite Group Databasebeta

← #12254135

Subgroups — #12254135

Order 1280.

580 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#65405C211yesmaximal, characteristic
2C2×#65395C211yesmaximal, characteristic
3C5×#2112C211yesmaximal, characteristic
4#10602238C212yesmaximal, complemented
5#10602221C212yesmaximal, complemented
6#10599893C214yesmaximal, complemented
7#10599892C212yesmaximal, complemented
8#10599891C212yesmaximal, complemented
9C23×C5×C8C2211yescharacteristic
10C2×#4072—21no—
11C2×#4070—21no—
12C22×Dic20C2211yescharacteristic
13C2×C5×M32C2212yes—
14#65405—21no—
15#65405—21no—
16#65405C2212yes—
17#65405C2212yes—
18#65395—21no—
19#65395—21no—
20#65395C2212yes—
21#65395C2212yes—
22C2×#3950C2211yescharacteristic
23C2×#3950C2211yescharacteristic
24C2×#3950C2211yescharacteristic
25C2×#3949—21no—
26C2×#3948—21no—
27C5×C22⋊C16C2212yes—
28C5×C22⋊C16C2212yes—
29#64725C2214yescomplemented
30#64724C2214yescomplemented
31#64723C2214yescomplemented
32#64723C2214yescomplemented
33#33649—51nomaximal, Sylow
34C23×C4×C5D411yescharacteristic
35C22×Dic10—21no—
36C22×C5×C8—24no—
37C22×C5×C8D411yescharacteristic
38C22×C5×C8D411yescharacteristic
39C22×C5×C8D411yescharacteristic
40C22×C5×C8C2311yescharacteristic
41C22×C5×C8C2311yescharacteristic
42C22×C5×C8C2311yescharacteristic
43C2×C22⋊Dic5—41no—
44C2×C22⋊Dic5—41no—
45#4072—22no—
46#4072—22no—
47#4072—41no—
48#4072—41no—
49C2×C4⋊Dic5—21no—
50C2×C4⋊Dic5—21no—
51C2×C4⋊Dic5—21no—
52#4070—22no—
53#4070—22no—
54#4070—41no—
55#4070—41no—
56C2×#683—41no—
57C2×C4×Dic5—41no—
58C2×Dic20—24no—
59C2×Dic20—21no—
60C2×Dic20C2312yes—
61C5×M32—24no—
62C2×C5×C16C2314yes—
63#3950—21no—
64#3950—21no—
65#3950—21no—
66#3950C2312yes—
67#3950C2312yes—
68#3950C2312yes—
69#3949—22no—
70#3949—22no—
71#3948—22no—
72#3948—22no—
73C2×#467—51no—
74C2×#427—51no—
75#2237—54no—
76#2236—52no—
77#2235—52no—
78#2159—52no—
79#2141—52no—
80#2112D511yescharacteristic
81C24×C5D811yescharacteristic
82C22×C4×C5—22no—
83C22×C4×C5—22no—
84C22×C4×C5D811yescharacteristic
85C22×C4×C5D811yescharacteristic
86C22×C4×C5D811yescharacteristic
87C22×C4×C5C2×D411yescharacteristic
88C22×C4×C5C2×D411yescharacteristic
89C22×C4×C5C2×D411yescharacteristic
90C22×Dic5—41no—
91C22×Dic5—41no—
92C22×Dic5—41no—
93C22×Dic5—41no—
94C2×Dic10—22no—
95C2×Dic10—42no—
96C2×Dic10—42no—
97C2×Dic10—41no—
98C2×C5×C8—22no—
99C2×C5×C8—22no—
100C2×C5×C8—22no—
101C2×C5×C8—22no—
102C2×C5×C8—22no—
103C2×C5×C8C2×D411yescharacteristic
104C2×C5×C8C2×D411yescharacteristic
105C2×C5×C8C2×D411yescharacteristic
106C2×C5×C8#4114yes—
107C2×C5×C8C2411yescharacteristic
108C22⋊Dic5—42no—
109C22⋊Dic5—42no—
110C22⋊Dic5—42no—
111C22⋊Dic5—42no—
112C4⋊Dic5—22no—
113C4⋊Dic5—22no—
114C4⋊Dic5—22no—
115C4⋊Dic5—41no—
116C4⋊Dic5—41no—
117C4⋊Dic5—41no—
118#683—42no—
119#683—42no—
120C4×Dic5—42no—
121C4×Dic5—42no—
122Dic20—24no—
123Dic20—42no—
124C5×C16—24no—
125C22×Q16—51no—
126C23×C8C2×D511yescharacteristic
127C2×C22⋊Q8—101no—
128C2×#117—101no—
129C2×M32C2×D512yes—
130#467—52no—
131#467—52no—
132#467—101no—
133#467—101no—
134#427—52no—
135#427—52no—
136#427—101no—
137#427—101no—
138C2×#107—51no—
139C2×#107—51no—
140C2×#107—51no—
141C2×#106—101no—
142C2×#103—101no—
143#367—54no—
144#366—54no—
145#358—54no—
146#358—54no—
147C22⋊C16C2×D512yes—
148C22⋊C16C2×D512yes—
149C23×C5—22no—
150C23×C5—22no—
151C23×C5—41no—
152C23×C5C2×D811yescharacteristic
153C23×C5C2×D811yescharacteristic
154C23×C5C2×D811yescharacteristic
155C2×C4×C5—22no—
156C2×C4×C5—21no—
157C2×C4×C5—21no—
158C2×C4×C5—21no—
159C2×C4×C5—21no—
160C2×C4×C5—22no—
161C2×C4×C5—22no—
162C2×C4×C5#12312yes—
163C2×C4×C5C2×D811yescharacteristic
164C2×C4×C5C2×D811yescharacteristic
165C2×C4×C5C2×D811yescharacteristic
166C2×C4×C5#13712yes—
167C2×C4×C5C22×D411yescharacteristic
168C2×Dic5—42no—
169C2×Dic5—42no—
170C2×Dic5—42no—
171C2×Dic5—42no—
172C2×Dic5—82no—
173C2×Dic5—82no—
174C2×Dic5—82no—
175C2×Dic5—82no—
176Dic10—42no—
177Dic10—42no—
178Dic10—82no—
179C5×C8—21no—
180C5×C8—21no—
181C5×C8#12312yes—
182C5×C8C2×#4112yes—
183C22×Q8—101no—
184C23×C4D2011yescharacteristic
185C2×Q16—52no—
186C2×Q16—101no—
187C2×Q16—104no—
188C22×C8—24no—
189C22×C8D2011yescharacteristic
190C22×C8D2011yescharacteristic
191C22×C8D2011yescharacteristic
192C22×C8C22×D511yescharacteristic
193C22×C8C22×D511yescharacteristic
194C22×C8C22×D511yescharacteristic
195C22⋊Q8—102no—
196C22⋊Q8—102no—
197C22⋊Q8—201no—
198C22⋊Q8—201no—
199#117—102no—
200#117—102no—
201#117—201no—
202#117—201no—
203C2×C4⋊C4—101no—
204C2×C4⋊C4—101no—
205C2×C4⋊C4—101no—
206C2×C4⋊C4—201no—
207C2×C22⋊C4—201no—
208C2×C22⋊C4—201no—
209C2×C42—201no—
210M32—24no—
211C2×C16C22×D514yes—
212#107—52no—
213#107—52no—
214#107—52no—
215#107—101no—
216#107—101no—
217#107—101no—
218#106—102no—
219#106—102no—
220#103—102no—
221#103—102no—
222C22×C5—21no—
223C22×C5—21no—
224C22×C5—21no—
225C22×C5—21no—
226C22×C5—21no—
227C22×C5—21no—
228C22×C5—21no—
229C22×C5—44no—
230C22×C5#48012yes—
231C22×C5#51012yes—
232C22×C5C22×D811yescharacteristic
233C4×C5—21no—
234C4×C5—21no—
235C4×C5#48012yes—
236C4×C5C2×#12311yescharacteristic
237C4×C5C2×#13711yescharacteristic
238Dic5—82no—
239Dic5—82no—
240Dic5—82no—
241Dic5—82no—
242C24D4011yescharacteristic
243C2×Q8—102no—
244C2×Q8—201no—
245C2×Q8—202no—
246C2×Q8—202no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—201no—
250C22×C4—201no—
251C22×C4—201no—
252C22×C4—201no—
253C22×C4D4011yescharacteristic
254C22×C4D4011yescharacteristic
255C22×C4D4011yescharacteristic
256C22×C4C2×D2011yescharacteristic
257C22×C4C2×D2011yescharacteristic
258C22×C4C2×D2011yescharacteristic
259Q16—104no—
260Q16—202no—
261C2×C8—22no—
262C2×C8—22no—
263C2×C8—22no—
264C2×C8—22no—
265C2×C8—22no—
266C2×C8C2×D2011yescharacteristic
267C2×C8C2×D2011yescharacteristic
268C2×C8C2×D2011yescharacteristic
269C2×C8#71114yes—
270C2×C8C23×D511yescharacteristic
271C4⋊C4—102no—
272C4⋊C4—102no—
273C4⋊C4—102no—
274C4⋊C4—201no—
275C4⋊C4—201no—
276C4⋊C4—201no—
277C4⋊C4—202no—
278C4⋊C4—202no—
279C22⋊C4—202no—
280C22⋊C4—202no—
281C22⋊C4—202no—
282C22⋊C4—202no—
283C42—202no—
284C42—202no—
285C16—24no—
286C2×C5—22no—
287C2×C5—22no—
288C2×C5—41no—
289C2×C5C2×#48011yescharacteristic
290C2×C5C2×#51011yescharacteristic
291C2×C5#341411yescharacteristic
292C23—22no—
293C23—22no—
294C23—41no—
295C23C2×D4011yescharacteristic
296C23C2×D4011yescharacteristic
297C23C2×D4011yescharacteristic
298Q8—202no—
299Q8—202no—
300Q8—402no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—202no—
309C2×C4—202no—
310C2×C4—202no—
311C2×C4—202no—
312C2×C4—402no—
313C2×C4—402no—
314C2×C4—402no—
315C2×C4—402no—
316C2×C4#403212yes—
317C2×C4C2×D4011yescharacteristic
318C2×C4C2×D4011yescharacteristic
319C2×C4C2×D4011yescharacteristic
320C2×C4#405512yes—
321C2×C4C22×D2011yescharacteristic
322C8—21no—
323C8—21no—
324C8#403212yes—
325C8C2×#71112yes—
326C5#3364911yescharacteristic, complemented, Sylow
327C22—21no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—44no—
335C22#6502512yes—
336C22#6519812yes—
337C22C22×D4011yescharacteristic
338C4—21no—
339C4—21no—
340C4—402no—
341C4—402no—
342C4—402no—
343C4—402no—
344C4#6502512yes—
345C4C2×#403211yescharacteristic
346C4C2×#405511yescharacteristic
347C2—22no—
348C2—22no—
349C2—41no—
350C2C2×#6502511yescharacteristic
351C2#1061374011yescharacteristic
352C2C2×#6519811yescharacteristic
subgroupG/H
centerC22C22×D40
commutator, cosocleC2×C5×C8C24
FrattiniC2×C8C23×D5
FittingC5×#2112C2
socleC22×C5C22×D8
prime radicalC24×C5D8
hypercenter#2112D5
hypocenterC5#33649
LCSG ▷ C2×C5×C8 ▷ C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C23×C4 ◁ C23×C8 ◁ #2112

Complements

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

NH ≅ G/Ntype
4. #10602238C2semidirect
5. #10602221C2semidirect
6. #10599893C2semidirect
7. #10599892C2semidirect
8. #10599891C2semidirect
29. #64725C22semidirect
30. #64724C22semidirect
31. #64723C22semidirect
32. #64723C22semidirect
326. C5#33649semidirect

Maximal subgroups

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

subgroupindex
C2×#654052
C2×#653952
C5×#21122
#106022382
#106022212
#105998932
#105998922
#105998912
#336495