Finite Group Databasebeta

← #13032956

Subgroups — #13032956

Order 1280.

578 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#10617263C212yesmaximal
2#10612472C211yesmaximal, characteristic
3#10604840C212yesmaximal, complemented
4C2×#64883C211yesmaximal, characteristic
5#10604688C211yesmaximal, characteristic
6#10604509C212yesmaximal
7#10604066C212yesmaximal, complemented
8#10603982C212yesmaximal, complemented
9#10603970C212yesmaximal, complemented
10#65956C2212yes—
11#65788C2212yes—
12Q8×#170C2212yes—
13#65754C2211yescharacteristic
14C4×#704C2211yescharacteristic
15C2×C4⋊#170C2212yes—
16#65709—22no—
17#65709C2212yes—
18#65707—22no—
19#65703C2212yes—
20#65696—22no—
21#65691—22no—
22C5×#467C411yescharacteristic
23Q16×Dic5C2211yescharacteristic
24#65470C411yescharacteristic
25#65405C411yescharacteristic
26C2×C8×Dic5C2211yescharacteristic
27#65340C412yes—
28#65163C411yescharacteristic
29#65162C2211yescharacteristic
30#65081C412yes—
31#65079C2212yes—
32Q8⋊#170C2212yes—
33Q8⋊#170C2212yes—
34#64883C2212yes—
35#64883C2211yescharacteristic
36#64883C2211yescharacteristic
37#170⋊C8C2212yes—
38#64879C2212yes—
39#64872C2212yes—
40#64872C2212yes—
41#18072—51nomaximal, Sylow
42C2×#704D412yes—
43C22×#170—22no—
44C2×C5×Q16C2×C411yescharacteristic
45C22×C5×C8C2×C411yescharacteristic
46C5×C22⋊Q8C2×C412yes—
47Q8×Dic5C2312yes—
48#4090C2×C412yes—
49C2×#689C2×C412yes—
50#4070C2×C412yes—
51C2×C4×Dic5C2311yescharacteristic
52C22×#168C2×C411yescharacteristic
53C2×Dic20C2×C411yescharacteristic
54#4035—22no—
55#4034C2×C412yes—
56#4033—22no—
57#4033C2312yes—
58#4024—22no—
59#4023—22no—
60C22⋊#170—21no—
61#4004—22no—
62#4004—22no—
63#4004—21no—
64#4002—22no—
65#4002—22no—
66#4002D411yescharacteristic
67#4002D411yescharacteristic
68C4⋊#170—22no—
69C4⋊#170—22no—
70C4⋊#170—22no—
71C4⋊#170C2312yes—
72C4⋊#170C2312yes—
73C4×#170—22no—
74C4×#170—22no—
75C4×#170C2312yes—
76C4×#170C2312yes—
77C5×#107C2×C411yescharacteristic
78C5×#103C2×C412yes—
79#3967C2×C412yes—
80#3950C2×C411yescharacteristic
81#3948C2×C412yes—
82C8×Dic5C2×C411yescharacteristic
83C8×Dic5C2×C411yescharacteristic
84C8×Dic5C2311yescharacteristic
85C8×Dic5C2311yescharacteristic
86#3942C2×C412yes—
87#3939C2×C412yes—
88#3170—51no—
89#2962—52no—
90#1593—52no—
91#1587—52no—
92#1587—52no—
93#1579—52no—
94C2×#334—51no—
95#1481—52no—
96#1451—51no—
97C2×C5×Q8C22×C412yes—
98C22×C4×C5C22×C411yescharacteristic
99C22×Dic5C2×D411yescharacteristic
100C22×Dic5#4111yescharacteristic
101C2×Dic10—22no—
102C2×Dic10C22×C412yes—
103#704—24no—
104C2×#170—22no—
105C2×#170—22no—
106C2×#170—22no—
107C2×#170—44no—
108C2×#170C2×D412yes—
109C2×#170C2×D412yes—
110C5×Q16—41no—
111C2×C5×C8—41no—
112C2×C5×C8C22×C411yescharacteristic
113C2×C5×C8C22×C411yescharacteristic
114C5×C4⋊C4—22no—
115C5×C4⋊C4C22×C412yes—
116C5×C22⋊C4—22no—
117C22⋊Dic5—22no—
118#689—24no—
119C4⋊Dic5—22no—
120C4⋊Dic5C22×C412yes—
121#683—22no—
122#683—22no—
123#683—22no—
124#683—22no—
125#683—22no—
126C4×Dic5—22no—
127C4×Dic5—22no—
128C4×Dic5C22×C411yescharacteristic
129C4×Dic5C2×D411yescharacteristic
130C4×Dic5#4111yescharacteristic
131C4×Dic5C2411yescharacteristic
132C2×#168—22no—
133C2×#168C22×C411yescharacteristic
134C2×#168C22×C411yescharacteristic
135Dic20—41no—
136#533—52no—
137#500—51no—
138#494—51no—
139#487—52no—
140#467—51no—
141#467—52no—
142#467AGL(1,𝔽5)11yescharacteristic
143#446—52no—
144#446—52no—
145#446—102no—
146#446—102no—
147C8×Q8—52no—
148C8×Q8—52no—
149C4×Q16—51no—
150C4×Q16—51no—
151C4×Q16—52no—
152#431—51no—
153#424—102no—
154C2×C4⋊C8—52no—
155#405—102no—
156C4×M16—51no—
157C2×C4×C8—51no—
158#334—51no—
159#334—52no—
160#334—51no—
161Q8⋊C8—52no—
162Q8⋊C8—52no—
163Q8⋊C8—52no—
164Q8⋊C8—52no—
165#322—52no—
166#322—52no—
167C23×C5C4×D411yescharacteristic
168C5×Q8—22no—
169C5×Q8—22no—
170C2×C4×C5—21no—
171C2×C4×C5—21no—
172C2×C4×C5—22no—
173C2×C4×C5—22no—
174C2×C4×C5C4×D411yescharacteristic
175C2×C4×C5C23×C411yescharacteristic
176C2×Dic5—22no—
177C2×Dic5—22no—
178C2×Dic5—41no—
179C2×Dic5C4×D411yescharacteristic
180C2×Dic5C4×D411yescharacteristic
181C2×Dic5#13714yes—
182C2×Dic5C22×D411yescharacteristic
183C2×Dic5C2×#4111yescharacteristic
184Dic10—22no—
185Dic10—22no—
186Dic10—82no—
187#170—22no—
188#170—22no—
189#170—42no—
190#170—42no—
191#170—42no—
192C5×C8—21no—
193C5×C8—21no—
194#168—21no—
195#168#13112yes—
196C2×Q16—52no—
197C2×Q16—51no—
198C2×Q16C2×AGL(1,𝔽5)11yescharacteristic
199C2×M16—52no—
200C22×C8—51no—
201C22×C8—102no—
202C22×C8C2×AGL(1,𝔽5)11yescharacteristic
203C4⋊Q8—52no—
204C4⋊Q8—52no—
205#125—102no—
206C22⋊Q8—52no—
207C22⋊Q8—102no—
208C22⋊Q8C2×AGL(1,𝔽5)12yes—
209C4×Q8—52no—
210C4×Q8—52no—
211C4×Q8—102no—
212#117—102no—
213C2×C42—51no—
214#107—51no—
215#107—52no—
216#107C2×AGL(1,𝔽5)11yescharacteristic
217C4⋊C8—52no—
218C4⋊C8—52no—
219C4⋊C8—102no—
220C4⋊C8—101no—
221C4⋊C8—102no—
222C4⋊C8—102no—
223C4⋊C8—102no—
224C4⋊C8—102no—
225#103—52no—
226#103—52no—
227#103—52no—
228#103C2×AGL(1,𝔽5)12yes—
229C22⋊C8—101no—
230#97—51no—
231#97—51no—
232#97—102no—
233#97—102no—
234C4×C8—52no—
235C4×C8—51no—
236C4×C8—51no—
237C4×C8—51no—
238C4×C8—51no—
239C4×C8—52no—
240C4×C8—102no—
241C4×C8—102no—
242C22×C5—21no—
243C22×C5—21no—
244C22×C5#44112yes—
245C22×C5C2×C4×D411yescharacteristic
246C4×C5—21no—
247C4×C5—42no—
248C4×C5—42no—
249C4×C5C2×#13111yescharacteristic
250C4×C5#56811yescharacteristic
251Dic5—21no—
252Dic5—21no—
253Dic5—42no—
254Dic5—42no—
255Dic5#44112yes—
256Dic5C2×#13712yes—
257C2×Q8—52no—
258C2×Q8—102no—
259C2×Q8C22×AGL(1,𝔽5)12yes—
260C22×C4—51no—
261C22×C4—51no—
262C22×C4C22×AGL(1,𝔽5)11yescharacteristic
263Q16—41no—
264Q16—104no—
265Q16—201no—
266M16—104no—
267C2×C8—41no—
268C2×C8—51no—
269C2×C8—52no—
270C2×C8—52no—
271C2×C8—51no—
272C2×C8—102no—
273C2×C8—102no—
274C2×C8—102no—
275C2×C8—102no—
276C2×C8—204no—
277C2×C8C22×AGL(1,𝔽5)11yescharacteristic
278C2×C8C22×AGL(1,𝔽5)11yescharacteristic
279C4⋊C4—22no—
280C4⋊C4—52no—
281C4⋊C4—102no—
282C4⋊C4—102no—
283C4⋊C4—102no—
284C4⋊C4—102no—
285C4⋊C4—102no—
286C4⋊C4—102no—
287C4⋊C4C22×AGL(1,𝔽5)12yes—
288C22⋊C4—22no—
289C22⋊C4—102no—
290C42—51no—
291C42—51no—
292C42—51no—
293C42—51no—
294C42—102no—
295C42—102no—
296C2×C5—22no—
297C2×C5#293511yescharacteristic
298C2×C5C2×#44111yescharacteristic
299C2×C5#295711yescharacteristic
300C23D4×AGL(1,𝔽5)11yescharacteristic
301Q8—22no—
302Q8—22no—
303Q8—102no—
304Q8—102no—
305Q8—402no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—54no—
311C2×C4—51no—
312C2×C4—51no—
313C2×C4—51no—
314C2×C4—51no—
315C2×C4—102no—
316C2×C4—102no—
317C2×C4—201no—
318C2×C4D4×AGL(1,𝔽5)11yescharacteristic
319C2×C4C23×AGL(1,𝔽5)11yescharacteristic
320C8—21no—
321C8—21no—
322C8—52no—
323C8—102no—
324C8—102no—
325C8—101no—
326C8—202no—
327C8—202no—
328C8—202no—
329C5#1807211yescharacteristic, complemented, Sylow
330C22—21no—
331C22—21no—
332C22#6573912yes—
333C22C2×D4×AGL(1,𝔽5)11yescharacteristic
334C4—21no—
335C4—42no—
336C4—42no—
337C4—52no—
338C4—52no—
339C4—101no—
340C4—101no—
341C4—202no—
342C4—202no—
343C4#6625611yescharacteristic
344C4D5⋊#13111yescharacteristic
345C2—22no—
346C2#1061723111yescharacteristic
347C2C2×#6573911yescharacteristic
348C2#1061744511yescharacteristic
subgroupG/H
centerC22C2×D4×AGL(1,𝔽5)
commutatorC2×C4×C5C23×C4
FrattiniC2×C4C23×AGL(1,𝔽5)
FittingC5×#467C4
socleC22×C5C2×C4×D4
prime radicalC23×C5C4×D4
cosocleC4×Dic5C24
hypercenter#467AGL(1,𝔽5)
hypocenterC5#18072
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ #467

Complements

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

NH ≅ G/Ntype
3. #10604840C2semidirect
7. #10604066C2semidirect
8. #10603982C2semidirect
9. #10603970C2semidirect
329. C5#18072semidirect

Maximal subgroups

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

subgroupindex
#106172632
#106124722
#106048402
C2×#648832
#106046882
#106045092
#106040662
#106039822
#106039702
#180725