Finite Group Databasebeta

← #13105208

Subgroups — #13105208

Order 1600.

502 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11717198C211yesmaximal, characteristic, complemented
2C5×#4032C211yesmaximal, characteristic
3C5×#4023C211yesmaximal, characteristic, complemented
4#11717091C211yesmaximal, characteristic
5C2×#87781C211yesmaximal, characteristic, complemented
6#11717079C211yesmaximal, characteristic
7#11717073C211yesmaximal, characteristic, complemented
8#11717015C211yesmaximal, characteristic, complemented
9#11716995C211yesmaximal, characteristic, complemented
10#11716988C211yesmaximal, characteristic, complemented
11#11716978C211yesmaximal, characteristic, complemented
12#11716977C211yesmaximal, characteristic, complemented
13#11716965C211yesmaximal, characteristic, complemented
14#11716965C211yesmaximal, characteristic, complemented
15#11716962C211yesmaximal, characteristic, complemented
16C22×#1039C2211yescharacteristic
17C2×#6045C2211yescharacteristic
18C2×C5×C22⋊D5C2211yescharacteristic
19C22×C5×Dic5C2211yescharacteristic
20C22×C5×Dic5C2211yescharacteristic
21C2×D5×Dic5C2211yescharacteristic
22C52×C22⋊C4C2211yescharacteristic
23C22⋊#1039C2211yescharacteristic
24C4⋊#1039C2211yescharacteristic
25#87806C2211yescharacteristic
26#87806C2211yescharacteristic
27C5×C22⋊Dic5C2211yescharacteristic
28C5×#685C2211yescharacteristic
29C5×#685C2211yescharacteristic
30C5×C4⋊Dic5C2211yescharacteristic
31C5×C4⋊Dic5C2211yescharacteristic
32C5×#683C2211yescharacteristic
33C5×#683C2211yescharacteristic
34C4×C5×Dic5C2211yescharacteristic
35C4×C5×Dic5C2211yescharacteristic
36#87781C2211yescharacteristic
37#87781C2211yescharacteristic
38#87781C2211yescharacteristic
39#87781C2211yescharacteristic
40#87781C2211yescharacteristic
41#87781C2211yescharacteristic
42#87780C2211yescharacteristic
43#87780C2211yescharacteristic
44#87780C2211yescharacteristic
45#87778C2211yescharacteristic
46#87778C2211yescharacteristic
47#87778C2211yescharacteristic
48Dic52C2211yescharacteristic
49Dic52C2211yescharacteristic
50Dic52C2211yescharacteristic
51#65984—51nomaximal
52#65865—51nomaximal
53C23×C52C2311yescharacteristic
54C22×C5×D5C2311yescharacteristic
55C2×C4×C52C2311yescharacteristic
56C2×C4×C52C2311yescharacteristic
57C2×#1039—22no—
58C2×#1039C2311yescharacteristic
59C2×#1039C2311yescharacteristic
60C2×#1039C2311yescharacteristic
61C2×#1039C2311yescharacteristic
62#6045—22no—
63C5×C22⋊D5—22no—
64C2×C5×Dic5—22no—
65C2×C5×Dic5—22no—
66C2×C5×Dic5C2311yescharacteristic
67C2×C5×Dic5C2311yescharacteristic
68C2×C5×Dic5C2311yescharacteristic
69C2×C5×Dic5C2311yescharacteristic
70C2×C5×Dic5C2311yescharacteristic
71C2×C5×Dic5C2311yescharacteristic
72C2×C5×Dic5C2311yescharacteristic
73D5×Dic5—22no—
74C5×#123—51no—
75C5×#117—51no—
76#4081—51no—
77D4×Dic5—51no—
78#4075—51no—
79#4072—51no—
80C2×#683—51no—
81C2×#683—51no—
82#4044—51no—
83#4042—51no—
84#4036—51no—
85#4035—51no—
86#4035—51no—
87#4035—51no—
88#4035—51no—
89#4033—51no—
90#4032D511yescharacteristic
91#4027—51no—
92#4025—51no—
93#4025—51no—
94#4024—51no—
95#4024—51no—
96#4024—51no—
97#4024—102no—
98#4023—51no—
99#4023—51no—
100#4023D511yescharacteristic, complemented
101#4022—51no—
102#4018—51no—
103#4016—51no—
104C4×Dic10—51no—
105C22×C52—21no—
106C22×C52—21no—
107C22×C52#4112yes—
108C22×C52C2411yescharacteristic
109C2×C5×D5—21no—
110C2×C5×D5—21no—
111C2×C5×D5—21no—
112C4×C52—21no—
113C4×C52—21no—
114#1039—21no—
115#1039—21no—
116#1039—21no—
117#1039#4112yes—
118C5×Dic5—21no—
119C5×Dic5—21no—
120C5×Dic5—21no—
121C5×Dic5—21no—
122C5×Dic5—21no—
123C5×Dic5—21no—
124C5×Dic5—21no—
125C2×C5×D4—51no—
126C22×C4×C5—51no—
127C22×C4×C5—51no—
128C2×C22⋊D5C2×D511yescharacteristic
129C22×Dic5—51no—
130C22×Dic5—51no—
131C22×Dic5—51no—
132C22×Dic5—102no—
133C22×Dic5C2×D511yescharacteristic
134C22×Dic5C2×D511yescharacteristic
135C2×C4×D5—51no—
136C2×Dic10—51no—
137C2×Dic10—51no—
138C2×Dic10—102no—
139C5×C4⋊C4—51no—
140C5×C4⋊C4—51no—
141C5×C4⋊C4—51no—
142C5×C4⋊C4—51no—
143C5×C22⋊C4—22no—
144C5×C22⋊C4—51no—
145C5×C22⋊C4—51no—
146C5×C22⋊C4—51no—
147C5×C22⋊C4C2×D511yescharacteristic
148C5×C22⋊C4C2×D511yescharacteristic
149C42×C5—51no—
150C42×C5—51no—
151C22⋊Dic5—51no—
152C22⋊Dic5—51no—
153C22⋊Dic5—51no—
154C22⋊Dic5—51no—
155C22⋊Dic5—51no—
156C22⋊Dic5—102no—
157C22⋊Dic5C2×D511yescharacteristic
158#685—51no—
159#685—51no—
160#685—51no—
161#685C2×D511yescharacteristic
162#685C2×D511yescharacteristic
163C4⋊Dic5—51no—
164C4⋊Dic5—51no—
165C4⋊Dic5—51no—
166C4⋊Dic5—51no—
167C4⋊Dic5—51no—
168C4⋊Dic5—102no—
169C4⋊Dic5C2×D511yescharacteristic
170C4⋊Dic5C2×D511yescharacteristic
171#683—51no—
172#683—51no—
173#683—51no—
174#683—51no—
175#683—51no—
176#683—51no—
177#683—51no—
178#683—51no—
179#683—51no—
180#683—51no—
181#683—51no—
182#683—51no—
183#683—51no—
184#683—51no—
185#683—51no—
186#683—51no—
187#683—51no—
188#683—51no—
189#683—51no—
190#683—102no—
191#683—102no—
192#683C2×D511yescharacteristic
193#683C2×D511yescharacteristic
194C4×Dic5—51no—
195C4×Dic5—51no—
196C4×Dic5—51no—
197C4×Dic5—51no—
198C4×Dic5—51no—
199C4×Dic5—51no—
200C4×Dic5C2×D511yescharacteristic
201C4×Dic5C2×D511yescharacteristic
202#552—251noSylow
203C2×C52—22no—
204C2×C52C2×#4111yescharacteristic
205C2×C52C2×#4111yescharacteristic
206C2×C522−1+411yescharacteristic
207C5×D5—41no—
208C23×C5—22no—
209C23×C5—51no—
210C23×C5C22×D511yescharacteristic
211C23×C5C22×D511yescharacteristic
212C22×D5C22×D511yescharacteristic
213C5×D4—102no—
214C2×C4×C5—22no—
215C2×C4×C5—22no—
216C2×C4×C5—51no—
217C2×C4×C5—51no—
218C2×C4×C5—51no—
219C2×C4×C5—51no—
220C2×C4×C5—51no—
221C2×C4×C5—51no—
222C2×C4×C5—51no—
223C2×C4×C5—102no—
224C2×C4×C5—102no—
225C2×C4×C5C22×D511yescharacteristic
226C2×C4×C5C22×D511yescharacteristic
227C2×C4×C5C22×D511yescharacteristic
228C2×C4×C5C22×D511yescharacteristic
229C22⋊D5—22no—
230C2×Dic5—22no—
231C2×Dic5—22no—
232C2×Dic5—51no—
233C2×Dic5—51no—
234C2×Dic5—51no—
235C2×Dic5—51no—
236C2×Dic5—51no—
237C2×Dic5—51no—
238C2×Dic5—51no—
239C2×Dic5—51no—
240C2×Dic5—102no—
241C2×Dic5—102no—
242C2×Dic5—102no—
243C2×Dic5—102no—
244C2×Dic5—102no—
245C2×Dic5—102no—
246C2×Dic5—102no—
247C2×Dic5—204no—
248C2×Dic5C22×D511yescharacteristic
249C2×Dic5C22×D511yescharacteristic
250C2×Dic5C22×D511yescharacteristic
251C2×Dic5C22×D511yescharacteristic
252C2×Dic5C22×D511yescharacteristic
253C2×Dic5C22×D511yescharacteristic
254C2×Dic5C22×D511yescharacteristic
255C4×D5—102no—
256Dic10—102no—
257Dic10—102no—
258Dic10—204no—
259#126—251no—
260#126—251no—
261#125—251no—
262#125—251no—
263#125—251no—
264#123—51no—
265#123—251no—
266C22⋊Q8—251no—
267C22⋊Q8—251no—
268C4×Q8—251no—
269C4×D4—251no—
270#117—51no—
271#117—251no—
272#117—251no—
273C2×C4⋊C4—251no—
274C52#55211yescharacteristic, complemented, Sylow
275C22×C5—21no—
276C22×C5—21no—
277C22×C5—21no—
278C22×C5—21no—
279C22×C5—24no—
280C22×C5—22no—
281C22×C5—42no—
282C22×C5—42no—
283C22×C5—101no—
284C22×C5—101no—
285C22×C5—101no—
286C22×C5#70912yes—
287C22×C5#71112yes—
288C22×C5C23×D511yescharacteristic
289C22×C5C23×D511yescharacteristic
290C2×D5—21no—
291C2×D5—21no—
292C2×D5—21no—
293C4×C5—21no—
294C4×C5—21no—
295C4×C5—21no—
296C4×C5—21no—
297C4×C5—42no—
298C4×C5—42no—
299C4×C5—101no—
300C4×C5—101no—
301C4×C5—101no—
302C4×C5—101no—
303C4×C5—101no—
304C4×C5—101no—
305C4×C5—101no—
306Dic5—21no—
307Dic5—21no—
308Dic5—21no—
309Dic5—21no—
310Dic5—21no—
311Dic5—21no—
312Dic5—21no—
313Dic5—52no—
314Dic5—52no—
315Dic5—104no—
316Dic5—101no—
317Dic5—101no—
318Dic5—101no—
319Dic5—101no—
320Dic5—101no—
321Dic5—101no—
322Dic5—202no—
323Dic5—202no—
324Dic5—202no—
325C2×Q8—251no—
326C2×D4—51no—
327C22×C4—51no—
328C22×C4—51no—
329C22×C4—251no—
330C22×C4—251no—
331C4⋊C4—51no—
332C4⋊C4—51no—
333C4⋊C4—51no—
334C4⋊C4—51no—
335C4⋊C4—251no—
336C4⋊C4—251no—
337C4⋊C4—251no—
338C4⋊C4—251no—
339C4⋊C4—251no—
340C4⋊C4—251no—
341C4⋊C4—251no—
342C4⋊C4—251no—
343C4⋊C4—251no—
344C4⋊C4—251no—
345C4⋊C4—251no—
346C4⋊C4—251no—
347C22⋊C4—51no—
348C22⋊C4—51no—
349C22⋊C4—51no—
350C22⋊C4—251no—
351C22⋊C4—251no—
352C22⋊C4—251no—
353C22⋊C4—251no—
354C22⋊C4D5211yescharacteristic
355C42—51no—
356C42—51no—
357C42—251no—
358C42—251no—
359C42—251no—
360C2×C5—22no—
361C2×C5—22no—
362C2×C5—22no—
363C2×C5—22no—
364C2×C5—22no—
365C2×C5—44no—
366C2×C5—201no—
367C2×C5C2×#70911yescharacteristic
368C2×C5C2×#71111yescharacteristic
369C2×C5D5×#4111yescharacteristic
370C2×C5D5×#4111yescharacteristic
371C2×C5#415011yescharacteristic
372C2×C5#415011yescharacteristic
373D5—41no—
374C23—51no—
375C23C2×D5211yescharacteristic
376Q8—502no—
377D4—102no—
378C2×C4—51no—
379C2×C4—51no—
380C2×C4—51no—
381C2×C4—51no—
382C2×C4—51no—
383C2×C4—51no—
384C2×C4—51no—
385C2×C4—102no—
386C2×C4—102no—
387C2×C4—251no—
388C2×C4—251no—
389C2×C4—251no—
390C2×C4—251no—
391C2×C4—502no—
392C2×C4—502no—
393C2×C4C2×D5211yescharacteristic
394C2×C4C2×D5211yescharacteristic
395C5—22no—
396C5#6586511yescharacteristic, complemented
397C5#6598411yescharacteristic, complemented
398C22—21no—
399C22—21no—
400C22—101no—
401C22—101no—
402C22—101no—
403C22#8787012yes—
404C22C22×D5211yescharacteristic
405C4—21no—
406C4—21no—
407C4—101no—
408C4—101no—
409C4—101no—
410C4—101no—
411C4—101no—
412C4—101no—
413C4—101no—
414C4—252no—
415C4—501no—
416C4—501no—
417C4—501no—
418C2—22no—
419C2—201no—
420C2C2×#8787011yescharacteristic
421C2#1171747411yescharacteristic
422C2#1171747711yescharacteristic
subgroupG/H
center, FrattiniC22C22×D52
commutator, socle, cosocleC22×C52C24
FittingC52×C22⋊C4C22
prime radicalC2×C5×C22⋊D5C22
hypercenterC22⋊C4D52
hypocenterC52#552
LCSG ▷ C22×C52 ▷ C52
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
1. #11717198C2semidirect
3. C5×#4023C2semidirect
5. C2×#87781C2semidirect
7. #11717073C2semidirect
8. #11717015C2semidirect
9. #11716995C2semidirect
10. #11716988C2semidirect
11. #11716978C2semidirect
12. #11716977C2semidirect
13. #11716965C2semidirect
14. #11716965C2semidirect
15. #11716962C2semidirect
100. #4023D5semidirect
274. C52#552semidirect
396. C5#65865semidirect
397. C5#65984semidirect

Maximal subgroups

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

subgroupindex
#117171982
C5×#40322
C5×#40232
#117170912
C2×#877812
#117170792
#117170732
#117170152
#117169952
#117169882
#117169782
#117169772
#117169652
#117169622
#659845
#658655