Finite Group Databasebeta

← #10614538

Subgroups — #10614538

Order 640.

528 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×C22⋊Q8C211yesmaximal, characteristic, complemented
2C2×#4092C211yesmaximal, characteristic, complemented
3#65519C211yesmaximal, characteristic, complemented
4#65502C211yesmaximal, characteristic
5#65499C211yesmaximal, characteristic, complemented
6#65283C211yesmaximal, characteristic, complemented
7#65281C211yesmaximal, characteristic, complemented
8#65278C211yesmaximal, characteristic, complemented
9#65272C211yesmaximal, characteristic, complemented
10#65269C211yesmaximal, characteristic, complemented
11#65261C211yesmaximal, characteristic, complemented
12#65246C211yesmaximal, characteristic
13#65245C211yesmaximal, characteristic
14#65238C211yesmaximal, characteristic, complemented
15#65237C211yesmaximal, characteristic, complemented
16C22×C5×Q8C2211yescharacteristic
17C23×C4×C5C2211yescharacteristic
18C22×C22⋊D5C2211yescharacteristic
19C22×C4×D5C2211yescharacteristic
20C5×C22⋊Q8—24no—
21C2×C5×C4⋊C4C2211yescharacteristic
22C2×C5×C4⋊C4C2211yescharacteristic
23C2×C5×C4⋊C4C2211yescharacteristic
24C2×C5×C22⋊C4C2211yescharacteristic
25C2×C5×C22⋊C4C2211yescharacteristic
26C2×C22⋊Dic5C2211yescharacteristic
27C2×C22⋊Dic5C2211yescharacteristic
28C2×C22⋊Dic5C2211yescharacteristic
29#4092—24no—
30C2×#685C2211yescharacteristic
31C2×#685C2211yescharacteristic
32C2×#685C2211yescharacteristic
33C2×#685C2211yescharacteristic
34C2×#685C2211yescharacteristic
35C2×C4⋊Dic5C2211yescharacteristic
36C2×C4⋊Dic5C2211yescharacteristic
37C2×#683C2211yescharacteristic
38C2×#683C2211yescharacteristic
39C2×#683C2211yescharacteristic
40C2×#683C2211yescharacteristic
41C2×#683C2211yescharacteristic
42C2×#683C2211yescharacteristic
43C2×#683C2211yescharacteristic
44C2×C4×Dic5C2211yescharacteristic
45#3963C2211yescharacteristic
46#3963C2211yescharacteristic
47#3963C2211yescharacteristic
48#3963C2211yescharacteristic
49#3963C2211yescharacteristic
50#3963C2211yescharacteristic
51#3963C2211yescharacteristic
52#3963C2211yescharacteristic
53#2732—51nomaximal, Sylow
54C24×C5C2311yescharacteristic
55C23×D5C2311yescharacteristic
56C2×C5×Q8—41no—
57C2×C5×Q8—41no—
58C2×C5×Q8D414yescomplemented
59C22×C4×C5—22no—
60C22×C4×C5—41no—
61C22×C4×C5—41no—
62C22×C4×C5C2311yescharacteristic
63C22×C4×C5C2311yescharacteristic
64C22×C4×C5C2311yescharacteristic
65C22×C4×C5C2311yescharacteristic
66C22×C4×C5C2311yescharacteristic
67C22×C4×C5C2311yescharacteristic
68C2×C22⋊D5—41no—
69C2×C22⋊D5—41no—
70C2×C22⋊D5Q814yescomplemented
71C22×Dic5C2311yescharacteristic
72C22×Dic5C2311yescharacteristic
73C22×Dic5C2311yescharacteristic
74C22×Dic5C2311yescharacteristic
75C22×Dic5C2311yescharacteristic
76C22×Dic5C2311yescharacteristic
77C22×Dic5C2311yescharacteristic
78C2×C4×D5—22no—
79C2×C4×D5—41no—
80C2×C4×D5—41no—
81C5×C4⋊C4—22no—
82C5×C4⋊C4—22no—
83C5×C4⋊C4—22no—
84C5×C22⋊C4—22no—
85C5×C22⋊C4—22no—
86C22⋊Dic5—22no—
87C22⋊Dic5—41no—
88C22⋊Dic5—41no—
89#685—22no—
90#685—22no—
91#685—22no—
92#685—41no—
93#685—41no—
94C4⋊Dic5—22no—
95C4⋊Dic5—41no—
96#683—22no—
97#683—22no—
98#683—22no—
99#683—41no—
100#683—41no—
101#683—41no—
102#683—41no—
103C4×Dic5—41no—
104C2×C22⋊Q8—51no—
105C2×C22⋊Q8D511yescharacteristic, complemented
106#399—51no—
107#399—51no—
108#398—51no—
109#398—51no—
110#396—51no—
111#396—51no—
112#391—51no—
113#390—51no—
114#389—51no—
115#386—51no—
116#386—51no—
117#385—51no—
118#385—51no—
119C23×C5—21no—
120C23×C5—21no—
121C23×C5—21no—
122C23×C5—21no—
123C23×C5—21no—
124C23×C5—21no—
125C23×C5C2×Q812yes—
126C23×C5C2411yescharacteristic
127C22×D5—21no—
128C22×D5—21no—
129C22×D5—21no—
130C22×D5—21no—
131C22×D5—21no—
132C22×D5—21no—
133C22×D5C2×Q812yes—
134C5×Q8—44no—
135C2×C4×C5—21no—
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—44no—
151C2×C4×C5C2×D412yes—
152C2×C4×C5C2×D412yes—
153C2×C4×C5C2×D412yes—
154C22⋊D5—44no—
155C2×Dic5—21no—
156C2×Dic5—21no—
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×Dic5—21no—
172C2×Dic5—21no—
173C2×Dic5—21no—
174C2×Dic5C2×Q812yes—
175C2×Dic5#4112yes—
176C4×D5—44no—
177C22×Q8C2×D511yescharacteristic
178C22×D4—51no—
179C23×C4—51no—
180C23×C4C2×D511yescharacteristic
181C22⋊Q8—24no—
182C22⋊Q8—104no—
183C2×C4⋊C4—51no—
184C2×C4⋊C4—51no—
185C2×C4⋊C4—51no—
186C2×C4⋊C4—51no—
187C2×C4⋊C4—51no—
188C2×C4⋊C4—51no—
189C2×C4⋊C4—51no—
190C2×C4⋊C4—51no—
191C2×C4⋊C4—51no—
192C2×C4⋊C4C2×D511yescharacteristic
193C2×C4⋊C4C2×D511yescharacteristic
194C2×C4⋊C4C2×D511yescharacteristic
195C2×C22⋊C4—51no—
196C2×C22⋊C4—51no—
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⋊C4C2×D511yescharacteristic
204C2×C22⋊C4C2×D511yescharacteristic
205C2×C42—51no—
206#95—51no—
207#95—51no—
208#95—51no—
209#95—51no—
210#95—51no—
211#95—51no—
212#95—51no—
213#95—51no—
214C22×C5—22no—
215C22×C5—22no—
216C22×C5—22no—
217C22×C5—41no—
218C22×C5—41no—
219C22×C5—41no—
220C22×C5—41no—
221C22×C5C22×D411yescharacteristic
222C22×C5C22×Q811yescharacteristic
223C22×C5C2×#4111yescharacteristic
224C22×C5H(5,𝔽2)11yescharacteristic
225C22×C5H(5,𝔽2)11yescharacteristic
226C22×C5H(5,𝔽2)11yescharacteristic
227C22×C52−1+411yescharacteristic
228C2×D5—22no—
229C2×D5—22no—
230C2×D5—22no—
231C2×D5—41no—
232C2×D5—41no—
233C2×D5—41no—
234C2×D5—41no—
235C4×C5—22no—
236C4×C5—22no—
237C4×C5—22no—
238C4×C5—41no—
239C4×C5—41no—
240C4×C5—41no—
241Dic5—22no—
242Dic5—22no—
243Dic5—41no—
244Dic5—41no—
245Dic5—41no—
246Dic5—41no—
247Dic5—41no—
248C24—51no—
249C24C22×D511yescharacteristic
250C2×Q8—41no—
251C2×Q8—41no—
252C2×Q8C22⋊D514yescomplemented
253C2×D4—54no—
254C2×D4—201no—
255C2×D4—201no—
256C22×C4—22no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—51no—
260C22×C4—51no—
261C22×C4—51no—
262C22×C4—51no—
263C22×C4—51no—
264C22×C4—51no—
265C22×C4—51no—
266C22×C4—102no—
267C22×C4—201no—
268C22×C4—201no—
269C22×C4C22×D511yescharacteristic
270C22×C4C22×D511yescharacteristic
271C22×C4C22×D511yescharacteristic
272C22×C4C22×D511yescharacteristic
273C22×C4C22×D511yescharacteristic
274C22×C4C22×D511yescharacteristic
275C4⋊C4—22no—
276C4⋊C4—22no—
277C4⋊C4—22no—
278C4⋊C4—102no—
279C4⋊C4—102no—
280C4⋊C4—102no—
281C4⋊C4—102no—
282C4⋊C4—201no—
283C4⋊C4—201no—
284C4⋊C4—201no—
285C4⋊C4—201no—
286C4⋊C4—201no—
287C22⋊C4—22no—
288C22⋊C4—22no—
289C22⋊C4—102no—
290C22⋊C4—102no—
291C22⋊C4—102no—
292C22⋊C4—102no—
293C22⋊C4—201no—
294C22⋊C4—201no—
295C22⋊C4—201no—
296C22⋊C4—201no—
297C42—201no—
298C2×C5—42no—
299C2×C5#53411yescharacteristic
300C2×C5#53911yescharacteristic
301C2×C5#54311yescharacteristic
302C2×C5D4×Q811yescharacteristic
303C2×C5#55011yescharacteristic
304C2×C5#55411yescharacteristic
305C2×C5#56211yescharacteristic
306D5—42no—
307C23—21no—
308C23—21no—
309C23—21no—
310C23—21no—
311C23—21no—
312C23—21no—
313C23—52no—
314C23—101no—
315C23—101no—
316C23—101no—
317C23—101no—
318C23—101no—
319C23—101no—
320C23Q8×D512yes—
321C23C23×D511yescharacteristic
322Q8—44no—
323D4—204no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
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—44no—
340C2×C4—52no—
341C2×C4—52no—
342C2×C4—101no—
343C2×C4—101no—
344C2×C4—101no—
345C2×C4—101no—
346C2×C4—101no—
347C2×C4—101no—
348C2×C4—101no—
349C2×C4—101no—
350C2×C4—101no—
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—204no—
362C2×C4C2×C22⋊D512yes—
363C2×C4C2×C22⋊D512yes—
364C2×C4C2×C22⋊D512yes—
365C5#273211yescharacteristic, complemented, Sylow
366C22—22no—
367C22—22no—
368C22—22no—
369C22—41no—
370C22—41no—
371C22—41no—
372C22—41no—
373C22—102no—
374C22—102no—
375C22—102no—
376C22—201no—
377C22—201no—
378C22—201no—
379C22—201no—
380C22#414411yescharacteristic
381C22#414411yescharacteristic
382C22C2×Q8×D511yescharacteristic
383C22#414711yescharacteristic
384C22D5×#4111yescharacteristic
385C22#414911yescharacteristic
386C22C22×C22⋊D511yescharacteristic
387C4—22no—
388C4—22no—
389C4—22no—
390C4—41no—
391C4—41no—
392C4—41no—
393C4—102no—
394C4—102no—
395C4—201no—
396C4—201no—
397C4—201no—
398C4—201no—
399C4—201no—
400C2—42no—
401C2—202no—
402C2#6596811yescharacteristic
403C2#6597011yescharacteristic
404C2#6597211yescharacteristic
405C2#6598111yescharacteristic
406C2#6613811yescharacteristic
407C2Q8×C22⋊D511yescharacteristic
408C2#6616811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC2×C5×C22⋊Q8C2
prime radicalC22×C22⋊D5C22
hypercenterC2×C22⋊Q8D5
hypocenterC5#2732
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×C5×C22⋊Q8C2semidirect
2. C2×#4092C2semidirect
3. #65519C2semidirect
5. #65499C2semidirect
6. #65283C2semidirect
7. #65281C2semidirect
8. #65278C2semidirect
9. #65272C2semidirect
10. #65269C2semidirect
11. #65261C2semidirect
14. #65238C2semidirect
15. #65237C2semidirect
58. C2×C5×Q8D4semidirect
70. C2×C22⋊D5Q8semidirect
105. C2×C22⋊Q8D5semidirect
252. C2×Q8C22⋊D5semidirect
365. C5#2732semidirect

Maximal subgroups

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

subgroupindex
C2×C5×C22⋊Q82
C2×#40922
#655192
#655022
#654992
#652832
#652812
#652782
#652722
#652692
#652612
#652462
#652452
#652382
#652372
#27325