Finite Group Databasebeta

← #10607726

Subgroups — #10607726

Order 640.

604 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4042C211yesmaximal, characteristic, complemented
2C2×#4042C211yesmaximal, characteristic, complemented
3C2×#4039C211yesmaximal, characteristic
4C2×#4018C211yesmaximal, characteristic, complemented
5C5×#389C211yesmaximal, characteristic, complemented
6#65282C211yesmaximal, characteristic, complemented
7#65280C211yesmaximal, characteristic
8#65279C211yesmaximal, characteristic, complemented
9#65261C211yesmaximal, characteristic, complemented
10#65261C211yesmaximal, characteristic, complemented
11#65229C211yesmaximal, characteristic, complemented
12#64964C211yesmaximal, characteristic, complemented
13#64961C211yesmaximal, characteristic
14#64956C211yesmaximal, characteristic, complemented
15#64935C211yesmaximal, characteristic, complemented
16C22×D20C2211yescharacteristic
17C22×C4×D5C2211yescharacteristic
18C22×C4×D5C2211yescharacteristic
19C22×Dic10C2211yescharacteristic
20C2×C5×C4⋊C4C2211yescharacteristic
21C2×C5×C4⋊C4C2211yescharacteristic
22C2×C5×C4⋊C4C2211yescharacteristic
23C2×C5×C4⋊C4C2211yescharacteristic
24C2×C42×C5C2211yescharacteristic
25C2×#685C2211yescharacteristic
26C2×#685C2211yescharacteristic
27C2×#685C2211yescharacteristic
28C2×#685C2211yescharacteristic
29C2×#685C2211yescharacteristic
30C2×#685C2211yescharacteristic
31C2×#685C2211yescharacteristic
32C2×#685C2211yescharacteristic
33C2×#685C2211yescharacteristic
34C2×#685C2211yescharacteristic
35C2×#683C2211yescharacteristic
36C2×#683C2211yescharacteristic
37C2×#683C2211yescharacteristic
38C2×#683C2211yescharacteristic
39C2×#683C2211yescharacteristic
40C2×#683C2211yescharacteristic
41C2×C4×Dic5C2211yescharacteristic
42C2×C4×Dic5C2211yescharacteristic
43#4042—24no—
44#4042—24no—
45#4039—24no—
46#4018—24no—
47C5×#95C2211yescharacteristic
48C5×#95C2211yescharacteristic
49#3963C2211yescharacteristic
50#3963C2211yescharacteristic
51#3963C2211yescharacteristic
52#3963C2211yescharacteristic
53#3963C2211yescharacteristic
54#3963C2211yescharacteristic
55#2415—51nomaximal, Sylow
56C23×D5C2311yescharacteristic
57C23×D5C2311yescharacteristic
58C22×C4×C5C2311yescharacteristic
59C22×C4×C5C2311yescharacteristic
60C22×C4×C5C2311yescharacteristic
61C22×C4×C5C2311yescharacteristic
62C22×C4×C5C2311yescharacteristic
63C22×C4×C5C2311yescharacteristic
64C22×C4×C5C2311yescharacteristic
65C22×Dic5C2311yescharacteristic
66C22×Dic5C2311yescharacteristic
67C22×Dic5C2311yescharacteristic
68C22×Dic5C2311yescharacteristic
69C22×Dic5C2311yescharacteristic
70C22×Dic5C2311yescharacteristic
71C2×D20—22no—
72C2×D20—22no—
73C2×D20—22no—
74C2×C4×D5—22no—
75C2×C4×D5—22no—
76C2×C4×D5—22no—
77C2×C4×D5—22no—
78C2×C4×D5—22no—
79C2×C4×D5D414yes—
80C2×Dic10—22no—
81C2×Dic10—22no—
82C2×Dic10—22no—
83C5×C4⋊C4—22no—
84C5×C4⋊C4—22no—
85C5×C4⋊C4—22no—
86C5×C4⋊C4—22no—
87C42×C5—22no—
88#685—22no—
89#685—22no—
90#685—22no—
91#685—22no—
92#685—22no—
93#685—22no—
94#685—22no—
95#685—22no—
96#685—22no—
97#685—22no—
98#683—22no—
99#683—22no—
100#683—22no—
101#683—22no—
102#683—22no—
103#683D414yescomplemented
104C4×Dic5—22no—
105C4×Dic5—22no—
106C2×C22⋊Q8—51no—
107C2×C22⋊Q8—51no—
108C2×C4×D4—51no—
109C2×#117—51no—
110#397—51no—
111#397—51no—
112#394—51no—
113#394—51no—
114#391—51no—
115#389—51no—
116#389—51no—
117#389D511yescharacteristic, complemented
118#388—51no—
119#388—51no—
120#380—51no—
121C23×C5C2411yescharacteristic
122C22×D5—21no—
123C22×D5—21no—
124C22×D5—21no—
125C22×D5—21no—
126C22×D5—21no—
127C22×D5—21no—
128C22×D5—21no—
129C22×D5—21no—
130C22×D5—21no—
131C22×D5—21no—
132C22×D5—21no—
133C22×D5—21no—
134C22×D5C2×D412yes—
135C22×D5#4112yes—
136C2×C4×C5—21no—
137C2×C4×C5—21no—
138C2×C4×C5—21no—
139C2×C4×C5—21no—
140C2×C4×C5—21no—
141C2×C4×C5—21no—
142C2×C4×C5—21no—
143C2×C4×C5—21no—
144C2×C4×C5—21no—
145C2×C4×C5—21no—
146C2×C4×C5—21no—
147C2×C4×C5—21no—
148C2×C4×C5—21no—
149C2×C4×C5—21no—
150C2×C4×C5—21no—
151C2×C4×C5—21no—
152C2×C4×C5—21no—
153C2×C4×C5—21no—
154C2×C4×C5C2×D412yes—
155C2×C4×C5C2×D412yes—
156C2×C4×C5#4112yes—
157C2×Dic5—21no—
158C2×Dic5—21no—
159C2×Dic5—21no—
160C2×Dic5—21no—
161C2×Dic5—21no—
162C2×Dic5—21no—
163C2×Dic5—21no—
164C2×Dic5—21no—
165C2×Dic5—21no—
166C2×Dic5—21no—
167C2×Dic5—21no—
168C2×Dic5—21no—
169C2×Dic5—21no—
170C2×Dic5—21no—
171C2×Dic5C2×D412yes—
172C2×Dic5C2×D412yes—
173C2×Dic5C2×D412yes—
174C2×Dic5#4112yes—
175D20—28no—
176C4×D5—28no—
177C4×D5—44no—
178Dic10—44no—
179C22×Q8—51no—
180C22×D4—51no—
181C23×C4—51no—
182C23×C4—51no—
183C22⋊Q8—104no—
184C22⋊Q8—104no—
185C4×D4—104no—
186#117—104no—
187C2×C4⋊C4—51no—
188C2×C4⋊C4—51no—
189C2×C4⋊C4—51no—
190C2×C4⋊C4—51no—
191C2×C4⋊C4—51no—
192C2×C4⋊C4—51no—
193C2×C4⋊C4C2×D511yescharacteristic
194C2×C4⋊C4C2×D511yescharacteristic
195C2×C4⋊C4C2×D511yescharacteristic
196C2×C4⋊C4C2×D511yescharacteristic
197C2×C22⋊C4—51no—
198C2×C22⋊C4—51no—
199C2×C22⋊C4—51no—
200C2×C22⋊C4—51no—
201C2×C22⋊C4—51no—
202C2×C22⋊C4—51no—
203C2×C22⋊C4—51no—
204C2×C22⋊C4—51no—
205C2×C22⋊C4—51no—
206C2×C22⋊C4—51no—
207C2×C42—51no—
208C2×C42—51no—
209C2×C42C2×D511yescharacteristic
210#95—51no—
211#95—51no—
212#95—51no—
213#95—51no—
214#95—51no—
215#95—51no—
216#95C2×D511yescharacteristic
217#95C2×D511yescharacteristic
218C22×C5C22×D411yescharacteristic
219C22×C5C22×D411yescharacteristic
220C22×C5C2×#4111yescharacteristic
221C22×C5C2×#4111yescharacteristic
222C22×C5C2×#4111yescharacteristic
223C22×C52−1+411yescharacteristic
224C22×C52−1+411yescharacteristic
225C2×D5—22no—
226C2×D5—22no—
227C2×D5—22no—
228C2×D5—22no—
229C2×D5—22no—
230C2×D5—22no—
231C2×D5—41no—
232C2×D5—41no—
233C2×D5—41no—
234C2×D5—41no—
235C2×D5—41no—
236C2×D5—41no—
237C2×D5—41no—
238C2×D5—41no—
239C4×C5—22no—
240C4×C5—22no—
241C4×C5—22no—
242C4×C5—41no—
243C4×C5—41no—
244C4×C5—41no—
245C4×C5—41no—
246Dic5—22no—
247Dic5—41no—
248Dic5—41no—
249Dic5—41no—
250Dic5—41no—
251Dic5#12114yes—
252C24—51no—
253C24—51no—
254C2×Q8—102no—
255C2×Q8—102no—
256C2×Q8—102no—
257C2×D4—102no—
258C2×D4—102no—
259C2×D4—102no—
260C22×C4—51no—
261C22×C4—51no—
262C22×C4—54no—
263C22×C4—51no—
264C22×C4—51no—
265C22×C4—51no—
266C22×C4—51no—
267C22×C4—102no—
268C22×C4—102no—
269C22×C4—102no—
270C22×C4—102no—
271C22×C4—102no—
272C22×C4C22×D511yescharacteristic
273C22×C4C22×D511yescharacteristic
274C22×C4C22×D511yescharacteristic
275C22×C4C22×D511yescharacteristic
276C22×C4C22×D511yescharacteristic
277C22×C4C22×D511yescharacteristic
278C22×C4C22×D511yescharacteristic
279C4⋊C4—22no—
280C4⋊C4—22no—
281C4⋊C4—22no—
282C4⋊C4—22no—
283C4⋊C4—54no—
284C4⋊C4—102no—
285C4⋊C4—102no—
286C4⋊C4—102no—
287C4⋊C4—102no—
288C4⋊C4—102no—
289C22⋊C4—102no—
290C22⋊C4—102no—
291C22⋊C4—102no—
292C22⋊C4—102no—
293C22⋊C4—102no—
294C22⋊C4—102no—
295C22⋊C4—102no—
296C22⋊C4—102no—
297C22⋊C4—102no—
298C22⋊C4—102no—
299C42—22no—
300C42—102no—
301C42—102no—
302C2×C5C2×#12111yescharacteristic
303C2×C5#52511yescharacteristic
304C2×C5#53611yescharacteristic
305C2×C5#54711yescharacteristic
306C2×C5#54811yescharacteristic
307C2×C5#55211yescharacteristic
308C2×C5#55611yescharacteristic
309D5—42no—
310D5—42no—
311C23—52no—
312C23—52no—
313C23—101no—
314C23—101no—
315C23—101no—
316C23—101no—
317C23—101no—
318C23—101no—
319C23—101no—
320C23—101no—
321C23—101no—
322C23—101no—
323C23—101no—
324C23—101no—
325C23C23×D511yescharacteristic
326Q8—204no—
327D4—108no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—52no—
347C2×C4—52no—
348C2×C4—52no—
349C2×C4—52no—
350C2×C4—108no—
351C2×C4—101no—
352C2×C4—101no—
353C2×C4—101no—
354C2×C4—101no—
355C2×C4—101no—
356C2×C4—101no—
357C2×C4—101no—
358C2×C4—101no—
359C2×C4—101no—
360C2×C4—101no—
361C2×C4—101no—
362C2×C4—101no—
363C2×C4—101no—
364C2×C4—101no—
365C2×C4—204no—
366C2×C4#70912yes—
367C2×C4D4×D512yes—
368C2×C4D4×D512yes—
369C5#241511yescharacteristic, complemented, Sylow
370C22—102no—
371C22—102no—
372C22—102no—
373C22—102no—
374C22—102no—
375C22—102no—
376C22—201no—
377C22—201no—
378C22—201no—
379C22—201no—
380C22—201no—
381C22—201no—
382C22—201no—
383C22—201no—
384C22C2×#70911yescharacteristic
385C22C2×D4×D511yescharacteristic
386C22C2×D4×D511yescharacteristic
387C22#414711yescharacteristic
388C22#414711yescharacteristic
389C22D5×#4111yescharacteristic
390C22D5×#4111yescharacteristic
391C4—22no—
392C4—22no—
393C4—22no—
394C4—41no—
395C4—41no—
396C4—41no—
397C4—41no—
398C4—54no—
399C4—102no—
400C4—201no—
401C4—201no—
402C4—201no—
403C4—201no—
404C2—202no—
405C2—202no—
406C2#6584111yescharacteristic
407C2#6588111yescharacteristic
408C2#6590211yescharacteristic
409C2D5×#12111yescharacteristic
410C2#6596711yescharacteristic
411C2#6602711yescharacteristic
412C2#6605811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#389C2
prime radicalC22×D20C22
hypercenter#389D5
hypocenterC5#2415
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #389

Complements

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

NH ≅ G/Ntype
1. C2×#4042C2semidirect
2. C2×#4042C2semidirect
4. C2×#4018C2semidirect
5. C5×#389C2semidirect
6. #65282C2semidirect
8. #65279C2semidirect
9. #65261C2semidirect
10. #65261C2semidirect
11. #65229C2semidirect
12. #64964C2semidirect
14. #64956C2semidirect
15. #64935C2semidirect
103. #683D4semidirect
117. #389D5semidirect
369. C5#2415semidirect

Maximal subgroups

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

subgroupindex
C2×#40422
C2×#40392
C2×#40182
C5×#3892
#652822
#652802
#652792
#652612
#652292
#649642
#649612
#649562
#649352
#24155