Finite Group Databasebeta

← #12254648

Subgroups — #12254648

Order 1280.

570 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#3164C211yesmaximal, characteristic, complemented
2#10610307C211yesmaximal, characteristic
3#10602176C212yesmaximal, complemented
4#10602175C212yesmaximal, complemented
5#10602146C212yesmaximal, complemented
6#10602144C212yesmaximal, complemented
7#10601447C211yesmaximal, characteristic, complemented
8#10601444C211yesmaximal, characteristic
9C2×#64712C211yesmaximal, characteristic, complemented
10#10601391C211yesmaximal, characteristic
11#10600674C211yesmaximal, characteristic, complemented
12C5×#544—21no—
13C2×#4055—21no—
14C22×Dic20C2211yescharacteristic
15C2×#4050C2211yescharacteristic
16#65860—21no—
17C5×#501—22no—
18C5×#500C2212yes—
19C5×#499C2212yes—
20C5×#428—21no—
21C5×#427C2211yescharacteristic
22C2×C5×#107C2211yescharacteristic
23C5×#405C2211yescharacteristic
24C2×#3944C2211yescharacteristic
25C22×#672C2211yescharacteristic
26#65004—22no—
27#64988C2212yes—
28#64985C2212yes—
29#64712C2212yes—
30#64712C2212yes—
31#64712C2211yescharacteristic
32#64712C2212yes—
33#64712C2212yes—
34#64712C2211yescharacteristic
35#64712C2212yes—
36#64711C2211yescharacteristic
37#64711C2212yes—
38#64711C2211yescharacteristic
39#64681C2212yes—
40C4×#672C2212yes—
41#33703—51nomaximal, Sylow
42C2×#709—21no—
43C22×Dic10—21no—
44C2×C5×M16D411yescharacteristic
45C22×C5×C8C2311yescharacteristic
46C5×C4⋊Q8—22no—
47C5×C4⋊Q8—41no—
48C5×#125—22no—
49C5×#125—41no—
50C5×C22⋊Q8—42no—
51C5×#117—21no—
52C5×#117D411yescharacteristic
53C2×C5×C4⋊C4—21no—
54#4055—44no—
55C2×Dic20—22no—
56C2×Dic20—22no—
57C2×Dic20—21no—
58C2×Dic20—22no—
59C2×Dic20C2311yescharacteristic
60C2×Dic20C2312yes—
61#4050—22no—
62#4050—22no—
63C2×D40C2311yescharacteristic
64C2×#677—21no—
65C2×#677—22no—
66#4043—41no—
67#4043—41no—
68#4032—42no—
69#4021—22no—
70#4015—22no—
71C5×#107—21no—
72C5×#107—22no—
73C5×#107C2312yes—
74C5×#107C2311yescharacteristic
75C5×#107C2311yescharacteristic
76C5×#106—21no—
77C5×#106—22no—
78C5×#97D412yes—
79C4×C5×C8C2312yes—
80#3944—22no—
81C2×#672—22no—
82C2×#672C2311yescharacteristic
83C2×#672C2311yescharacteristic
84C2×#672C2312yes—
85#3164D511yescharacteristic, complemented
86#3156—51no—
87#2245—52no—
88#2243—52no—
89#2242—52no—
90#2241—52no—
91#2141—51no—
92#2139—51no—
93#2138—51no—
94C2×#358—51no—
95#2108—51no—
96C2×C5×Q8—42no—
97C22×C4×C5—41no—
98C22×C4×C5C2×D411yescharacteristic
99C2×C22⋊D5—41no—
100C22×Dic5—41no—
101#709—42no—
102#709—42no—
103C2×D20—21no—
104C2×C4×D5—41no—
105C2×Dic10—22no—
106C2×Dic10—21no—
107C2×Dic10—42no—
108C2×Dic10—42no—
109C2×Dic10—41no—
110C5×M16SD1614yes—
111C2×C5×C8C2×D411yescharacteristic
112C2×C5×C8C2×D412yes—
113C2×C5×C8#4112yes—
114C2×C5×C8#4112yes—
115C2×C5×C8C2411yescharacteristic
116C5×C4⋊C4—22no—
117C5×C4⋊C4—21no—
118C5×C4⋊C4—21no—
119C5×C4⋊C4—41no—
120C5×C4⋊C4—42no—
121C5×C4⋊C4—42no—
122C5×C4⋊C4—42no—
123C5×C4⋊C4D811yescharacteristic
124C5×C4⋊C4D811yescharacteristic
125C5×C22⋊C4—41no—
126C5×C22⋊C4D812yes—
127C42×C5—41no—
128C42×C5C2×D412yes—
129#685—42no—
130#685—42no—
131C4⋊Dic5—42no—
132C4⋊Dic5—42no—
133Dic20—21no—
134Dic20—21no—
135Dic20—22no—
136Dic20—22no—
137Dic20—42no—
138Dic20—44no—
139D40—21no—
140D40—21no—
141#677—42no—
142#677—44no—
143#672—21no—
144#672—21no—
145#672—22no—
146C2×#137—101no—
147C2×#135—51no—
148C22×Q16—51no—
149#544—21no—
150#536—101no—
151C2×M32—51no—
152C22×C16—51no—
153#501—22no—
154#500C2×D512yes—
155#499C2×D512yes—
156#497—102no—
157#495—52no—
158#494—52no—
159#428—21no—
160#427C2×D511yescharacteristic
161C2×#107C2×D511yescharacteristic
162#405C2×D511yescharacteristic
163#358—52no—
164#358—52no—
165#358—51no—
166#358—51no—
167#358—52no—
168#358—52no—
169#358—52no—
170#357—51no—
171#357—51no—
172#357—52no—
173#346—52no—
174C4×C16—52no—
175C23×C5C2×D811yescharacteristic
176C22×D5—41no—
177C5×Q8—162no—
178C2×C4×C5—41no—
179C2×C4×C5—41no—
180C2×C4×C5—42no—
181C2×C4×C5—161no—
182C2×C4×C5#12412yes—
183C2×C4×C5C2×D812yes—
184C2×C4×C5C2×D812yes—
185C2×C4×C5C2×D811yescharacteristic
186C2×C4×C5C2×SD1612yes—
187C2×C4×C5C22×D411yescharacteristic
188C22⋊D5—82no—
189C2×Dic5—41no—
190C2×Dic5—42no—
191C2×Dic5—82no—
192D20—41no—
193D20—41no—
194C4×D5—82no—
195Dic10—41no—
196Dic10—42no—
197Dic10—42no—
198Dic10—41no—
199Dic10—82no—
200C5×C8#12412yes—
201C5×C8C2×SD1614yes—
202C5×C8C2×#4111yescharacteristic
203C5×C8C2×#4111yescharacteristic
204C2×#41—101no—
205C22×Q8—101no—
206#137—204no—
207#135—102no—
208#135—102no—
209C2×Q16—52no—
210C2×Q16—51no—
211C2×Q16—102no—
212C2×Q16—102no—
213C2×Q16—101no—
214C2×Q16—102no—
215C2×SD16—101no—
216C2×SD16—102no—
217C2×D8—51no—
218C2×M16C22⋊D511yescharacteristic
219C22×C8C22×D511yescharacteristic
220C4⋊Q8—22no—
221C4⋊Q8—41no—
222C4⋊Q8—102no—
223#125—22no—
224#125—41no—
225#124—102no—
226#123—202no—
227C22⋊Q8—42no—
228C22⋊Q8—201no—
229C22⋊Q8—201no—
230#117—21no—
231#117C22⋊D511yescharacteristic
232C2×C4⋊C4—21no—
233M32—102no—
234C2×C16—51no—
235C2×C16—52no—
236C2×C16—51no—
237C2×C16—102no—
238#107—21no—
239#107—22no—
240#107C22×D512yes—
241#107C22×D511yescharacteristic
242#107C22×D511yescharacteristic
243#106—21no—
244#106—22no—
245#97C22⋊D512yes—
246C4×C8C22×D512yes—
247C22×C5—21no—
248C22×C5—21no—
249C22×C5#48612yes—
250C22×C5C22×D811yescharacteristic
251C2×D5—81no—
252C2×D5—81no—
253C2×D5—81no—
254C4×C5—22no—
255C4×C5—22no—
256C4×C5—82no—
257C4×C5—81no—
258C4×C5—81no—
259C4×C5#48612yes—
260C4×C5C2×#12411yescharacteristic
261C4×C5C22×SD1611yescharacteristic
262Dic5—82no—
263Dic5—81no—
264#41—202no—
265#41—202no—
266C2×Q8—42no—
267C2×Q8—102no—
268C2×Q8—101no—
269C2×Q8—201no—
270C2×Q8—202no—
271C2×Q8—202no—
272C2×D4—101no—
273C2×D4—201no—
274C22×C4—41no—
275C22×C4—201no—
276C22×C4—201no—
277C22×C4C2×C22⋊D511yescharacteristic
278Q16—102no—
279Q16—102no—
280Q16—101no—
281Q16—101no—
282Q16—204no—
283Q16—202no—
284SD16—204no—
285SD16—202no—
286D8—101no—
287D8—101no—
288M16#68814yes—
289C2×C8#71312yes—
290C2×C8#71312yes—
291C2×C8C2×C22⋊D512yes—
292C2×C8C2×C22⋊D511yescharacteristic
293C2×C8C23×D511yescharacteristic
294C4⋊C4—21no—
295C4⋊C4—22no—
296C4⋊C4—21no—
297C4⋊C4—41no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C4⋊C4—202no—
302C4⋊C4—202no—
303C4⋊C4#68611yescharacteristic
304C4⋊C4#68611yescharacteristic
305C22⋊C4—41no—
306C22⋊C4—202no—
307C22⋊C4—202no—
308C22⋊C4#68612yes—
309C42—41no—
310C42C2×C22⋊D512yes—
311C16—101no—
312C16—101no—
313C16—102no—
314C2×C5—22no—
315C2×C5C2×#48611yescharacteristic
316C2×C5#341611yescharacteristic
317C2×C5#341711yescharacteristic
318D5—161no—
319C23—201no—
320C23C2×#68611yescharacteristic
321Q8—162no—
322Q8—202no—
323Q8—201no—
324Q8—201no—
325Q8—202no—
326Q8—402no—
327D4—201no—
328D4—201no—
329D4—402no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—42no—
333C2×C4—161no—
334C2×C4—202no—
335C2×C4—201no—
336C2×C4—402no—
337C2×C4—402no—
338C2×C4C2×#68612yes—
339C2×C4C2×#68612yes—
340C2×C4C2×#68611yescharacteristic
341C2×C4C2×#68812yes—
342C2×C4#409312yes—
343C2×C4C22×C22⋊D511yescharacteristic
344C8C2×#68814yes—
345C8#409312yes—
346C8C2×#71311yescharacteristic
347C8C2×#71311yescharacteristic
348C5#3370311yescharacteristic, complemented, Sylow
349C22—21no—
350C22—21no—
351C22—401no—
352C22—401no—
353C22—401no—
354C22#6537712yes—
355C22C22×#68611yescharacteristic
356C4—22no—
357C4—22no—
358C4—82no—
359C4—81no—
360C4—81no—
361C4—401no—
362C4—402no—
363C4#6537712yes—
364C4C22×#68811yescharacteristic
365C4C2×#409311yescharacteristic
366C2—22no—
367C2—801no—
368C2C2×#6537711yescharacteristic
369C2#1061573911yescharacteristic
370C2#1061574811yescharacteristic
subgroupG/H
centerC22C22×#686
commutator, cosocleC2×C5×C8C24
FrattiniC2×C8C23×D5
FittingC5×#3164C2
socleC22×C5C22×D8
prime radicalC2×#4050C22
hypercenter#3164D5
hypocenterC5#33703
LCSG ▷ C2×C5×C8 ▷ C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ #117 ◁ #405 ◁ #3164

Complements

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

NH ≅ G/Ntype
1. C5×#3164C2semidirect
3. #10602176C2semidirect
4. #10602175C2semidirect
5. #10602146C2semidirect
6. #10602144C2semidirect
7. #10601447C2semidirect
9. C2×#64712C2semidirect
11. #10600674C2semidirect
85. #3164D5semidirect
348. C5#33703semidirect

Maximal subgroups

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

subgroupindex
C5×#31642
#106103072
#106021762
#106021752
#106021462
#106021442
#106014472
#106014442
C2×#647122
#106013912
#106006742
#337035