Finite Group Databasebeta

← #12017522

Subgroups — #12017522

Order 1280.

514 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#65480C211yesmaximal, characteristic
2C2×C5×#358C211yesmaximal, characteristic, complemented
3C2×#64784C211yesmaximal, characteristic, complemented
4C2×#64729C211yesmaximal, characteristic
5C2×C16×Dic5C211yesmaximal, characteristic, complemented
6#10601400C211yesmaximal, characteristic, complemented
7C2×#64711C211yesmaximal, characteristic
8C22×C5×Q16C2211yescharacteristic
9C2×#4093—21no—
10C22×#688—21no—
11C22×D40C2211yescharacteristic
12C22×C5×C16C2211yescharacteristic
13C2×C5×#107C2211yescharacteristic
14#65480—22no—
15#65480C414yescomplemented
16C2×#3950C2211yescharacteristic
17C2×C8×Dic5C2211yescharacteristic
18C22×#672C2211yescharacteristic
19#65266—21no—
20C2×#3940—21no—
21C5×#358—22no—
22#64784—22no—
23#64729—22no—
24C16×Dic5—22no—
25#64711—22no—
26#13204—51nomaximal, Sylow
27C22×C5×Q8—21no—
28C22×D20—21no—
29C2×C5×Q16—21no—
30C2×C5×Q16—21no—
31C2×C5×Q16—21no—
32C2×C5×Q16—21no—
33C2×C5×Q16—21no—
34C2×C5×Q16C2×C412yes—
35C22×C5×C8C2311yescharacteristic
36C2×C5×C4⋊C4—21no—
37#4093—24no—
38#4093—42no—
39C2×#688—24no—
40C2×#688—42no—
41C2×#688—42no—
42C2×#685—42no—
43C2×C4⋊Dic5—21no—
44C2×C4×Dic5D411yescharacteristic
45C22×#168D411yescharacteristic
46C2×D40—21no—
47C2×D40—21no—
48C2×D40—21no—
49C2×D40—21no—
50C2×D40—21no—
51C2×D40C2×C412yes—
52C2×C5×C16—21no—
53C2×C5×C16—21no—
54C2×C5×C16D412yes—
55C5×#107—21no—
56C5×#107—21no—
57#3963—42no—
58#3950—21no—
59#3950—21no—
60C8×Dic5C2×C412yes—
61C8×Dic5D412yes—
62C2×#672—21no—
63C2×#672—21no—
64C2×#672D412yes—
65#3940—42no—
66C2×#495—51no—
67C2×#358—51no—
68C2×#358D511yescharacteristic, complemented
69C2×#357—51no—
70C2×#357—51no—
71C2×C4×C16—51no—
72#1846—51no—
73C23×D5—41no—
74C2×C5×Q8—22no—
75C2×C5×Q8—41no—
76C2×C5×Q8—41no—
77C2×C5×Q8—41no—
78C2×C5×Q8—41no—
79C2×C5×Q8—41no—
80C22×C4×C5—41no—
81C22×C4×C5—41no—
82C22×C4×C5C2×D411yescharacteristic
83C22×Dic5—41no—
84C22×Dic5D812yes—
85C2×D20—22no—
86C2×D20—41no—
87C2×D20—41no—
88C2×D20—41no—
89C2×D20—41no—
90C2×D20—41no—
91C5×Q16—22no—
92C5×Q16—22no—
93C5×Q16—41no—
94C5×Q16—41no—
95C2×C5×C8C22×C411yescharacteristic
96C2×C5×C8C2×D411yescharacteristic
97C2×C5×C8C2×D411yescharacteristic
98C2×C5×C8C2×D411yescharacteristic
99C2×C5×C8#4111yescharacteristic
100C2×C5×C8#4111yescharacteristic
101C5×C4⋊C4—41no—
102C5×C4⋊C4—41no—
103#688—48no—
104#685—48no—
105C4⋊Dic5—41no—
106C4⋊Dic5—41no—
107C4×Dic5C22⋊C412yes—
108C4×Dic5SD1612yes—
109C2×#168C22⋊C412yes—
110C2×#168D812yes—
111C2×#168SD1612yes—
112D40—22no—
113D40—22no—
114D40—41no—
115D40—41no—
116C5×C16—22no—
117#672—22no—
118C22×Q16C2×D511yescharacteristic
119C22×SD16—101no—
120C22×D8—51no—
121C2×#124—101no—
122C22×C16—51no—
123C22×C16C2×D511yescharacteristic
124#495—54no—
125#495—102no—
126C2×#107—51no—
127C2×#107C2×D511yescharacteristic
128C2×#106—101no—
129C2×C4×C8—51no—
130#382—101no—
131#358—22no—
132#358—102no—
133#357—102no—
134#357—102no—
135C4×C16—102no—
136C23×C5C2×D811yescharacteristic
137C22×D5—42no—
138C22×D5—81no—
139C22×D5—81no—
140C22×D5—82no—
141C22×D5—82no—
142C5×Q8—42no—
143C5×Q8—42no—
144C5×Q8—81no—
145C5×Q8—81no—
146C2×C4×C5—42no—
147C2×C4×C5—81no—
148C2×C4×C5—81no—
149C2×C4×C5—81no—
150C2×C4×C5—82no—
151C2×C4×C5C2×C22⋊C411yescharacteristic
152C2×C4×C5#12411yescharacteristic
153C2×C4×C5#12711yescharacteristic
154C2×C4×C5C2×D811yescharacteristic
155C2×C4×C5C2×SD1611yescharacteristic
156C2×C4×C5C2×SD1611yescharacteristic
157C2×Dic5—22no—
158C2×Dic5—22no—
159C2×Dic5—81no—
160C2×Dic5—82no—
161C2×Dic5#10214yes—
162D20—42no—
163D20—42no—
164D20—81no—
165D20—81no—
166C5×C8C4×D411yescharacteristic
167C5×C8C4×D411yescharacteristic
168C5×C8#12111yescharacteristic
169C5×C8#12111yescharacteristic
170#168#10214yes—
171C22×Q8—21no—
172C22×D4—101no—
173C2×Q16—21no—
174C2×Q16—21no—
175C2×Q16—21no—
176C2×Q16—21no—
177C2×Q16—21no—
178C2×Q16C4×D512yes—
179C2×SD16—104no—
180C2×SD16—202no—
181C2×SD16—202no—
182C2×D8—52no—
183C2×D8—101no—
184C2×D8—101no—
185C2×D8—101no—
186C2×D8—101no—
187C2×D8—101no—
188C22×C8—51no—
189C22×C8C22×D511yescharacteristic
190#124—104no—
191#124—202no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—101no—
194C2×C22⋊C4—202no—
195C2×C42—51no—
196C2×C16—21no—
197C2×C16—21no—
198C2×C16—52no—
199C2×C16—101no—
200C2×C16—101no—
201C2×C16C22⋊D512yes—
202#107—21no—
203#107—21no—
204#107—101no—
205#107—101no—
206#106—202no—
207C4×C8—52no—
208C4×C8—52no—
209#95—202no—
210C22×C5C2×#10211yescharacteristic
211C22×C5#48611yescharacteristic
212C22×C5#49311yescharacteristic
213C22×C5#50812yes—
214C22×C5#50812yes—
215C2×D5—84no—
216C2×D5—82no—
217C2×D5—162no—
218C2×D5—161no—
219C2×D5—161no—
220C4×C5—82no—
221C4×C5—161no—
222C4×C5#39011yescharacteristic
223C4×C5C2×#10211yescharacteristic
224C4×C5#48611yescharacteristic
225C4×C5#49211yescharacteristic
226Dic5—24no—
227Dic5—161no—
228C24—201no—
229C2×Q8—22no—
230C2×Q8—41no—
231C2×Q8—41no—
232C2×Q8—41no—
233C2×Q8—41no—
234C2×Q8—41no—
235C2×D4—102no—
236C2×D4—201no—
237C2×D4—201no—
238C2×D4—201no—
239C2×D4—201no—
240C2×D4—201no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—52no—
244C22×C4—201no—
245C22×C4D4×D511yescharacteristic
246Q16—22no—
247Q16—22no—
248Q16—41no—
249Q16—41no—
250SD16—208no—
251D8—102no—
252D8—102no—
253D8—201no—
254D8—201no—
255C2×C8—52no—
256C2×C8—52no—
257C2×C8—52no—
258C2×C8C2×C4×D511yescharacteristic
259C2×C8#70911yescharacteristic
260C2×C8D4×D511yescharacteristic
261C2×C8D4×D511yescharacteristic
262C2×C8#71111yescharacteristic
263C2×C8C2×C22⋊D511yescharacteristic
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—201no—
267C4⋊C4—201no—
268C22⋊C4—208no—
269C42—52no—
270C42—52no—
271C16—22no—
272C16—102no—
273C2×C5#195711yescharacteristic
274C2×C5#213812yes—
275C2×C5#224212yes—
276C2×C5#224911yescharacteristic
277C2×C5#224911yescharacteristic
278D5—162no—
279C23—202no—
280C23—401no—
281C23—401no—
282C23—402no—
283C23—402no—
284C23D5×D811yescharacteristic
285Q8—42no—
286Q8—42no—
287Q8—81no—
288Q8—81no—
289D4—202no—
290D4—202no—
291D4—401no—
292D4—401no—
293C2×C4—42no—
294C2×C4—54no—
295C2×C4—81no—
296C2×C4—81no—
297C2×C4—81no—
298C2×C4—82no—
299C2×C4—102no—
300C2×C4—102no—
301C2×C4—401no—
302C2×C4—402no—
303C2×C4D5×C22⋊C411yescharacteristic
304C2×C4#403111yescharacteristic
305C2×C4D5×D811yescharacteristic
306C2×C4D5×SD1611yescharacteristic
307C2×C4D5×SD1611yescharacteristic
308C2×C4#408611yescharacteristic
309C8—54no—
310C8#402711yescharacteristic
311C8#403011yescharacteristic
312C8C4×C22⋊D511yescharacteristic
313C8#408511yescharacteristic
314C5#1320411yescharacteristic, complemented, Sylow
315C22—404no—
316C22—402no—
317C22—801no—
318C22—801no—
319C22—802no—
320C22#6504911yescharacteristic
321C22D5×#10211yescharacteristic
322C22#6520512yes—
323C22#6520812yes—
324C22#6544011yescharacteristic
325C4—82no—
326C4—104no—
327C4—161no—
328C4—801no—
329C4#6504911yescharacteristic
330C4D5×#10211yescharacteristic
331C4#6524611yescharacteristic
332C4#6546411yescharacteristic
333C2—802no—
334C2#1060000912yes—
335C2#1060001312yes—
336C2#1060095011yescharacteristic
337C2#1060227311yescharacteristic
338C2#1060229311yescharacteristic
subgroupG/H
centerC23D5×D8
commutatorC2×C5×C8C22×C4
FrattiniC22×C8C22×D5
FittingC2×C5×#358C2
socleC23×C5C2×D8
prime radicalC22×D40C22
cosocleC22×C5×C8C23
hypercenterC2×#358D5
hypocenterC5#13204
LCSG ▷ C2×C5×C8 ▷ C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×C8 ◁ C2×#358

Complements

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

NH ≅ G/Ntype
2. C2×C5×#358C2semidirect
3. C2×#64784C2semidirect
5. C2×C16×Dic5C2semidirect
6. #10601400C2semidirect
15. #65480C4semidirect
68. C2×#358D5semidirect
314. C5#13204semidirect

Maximal subgroups

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

subgroupindex
C2×#654802
C2×C5×#3582
C2×#647842
C2×#647292
C2×C16×Dic52
#106014002
C2×#647112
#132045