Finite Group Databasebeta

← #10614535

Subgroups — #10614535

Order 640.

498 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×C22⋊Q8C211yesmaximal, characteristic, complemented
2C2×#4044C211yesmaximal, characteristic, complemented
3#65520C211yesmaximal, characteristic, complemented
4#65518C211yesmaximal, characteristic, complemented
5#65502C211yesmaximal, characteristic
6#65499C211yesmaximal, characteristic, complemented
7#65282C211yesmaximal, characteristic, complemented
8#65281C211yesmaximal, characteristic, complemented
9#65270C211yesmaximal, characteristic, complemented
10#65266C211yesmaximal, characteristic, complemented
11#65262C211yesmaximal, characteristic, complemented
12#65250C211yesmaximal, characteristic
13#65246C211yesmaximal, characteristic
14#65242C211yesmaximal, characteristic, complemented
15#65233C211yesmaximal, characteristic, complemented
16C22×C5×Q8C2211yescharacteristic
17C23×C4×C5C2211yescharacteristic
18C22×C22⋊D5C2211yescharacteristic
19C5×C22⋊Q8—24no—
20C2×C5×C4⋊C4C2211yescharacteristic
21C2×C5×C4⋊C4C2211yescharacteristic
22C2×C5×C4⋊C4C2211yescharacteristic
23C2×C5×C22⋊C4C2211yescharacteristic
24C2×C5×C22⋊C4C2211yescharacteristic
25C2×C22⋊Dic5C2211yescharacteristic
26C2×C22⋊Dic5C2211yescharacteristic
27C2×C22⋊Dic5C2211yescharacteristic
28C2×#685C2211yescharacteristic
29C2×#685C2211yescharacteristic
30C2×#685C2211yescharacteristic
31C2×#685C2211yescharacteristic
32C2×#685C2211yescharacteristic
33C2×#685C2211yescharacteristic
34C2×C4⋊Dic5C2211yescharacteristic
35C2×C4⋊Dic5C2211yescharacteristic
36C2×#683C2211yescharacteristic
37C2×#683C2211yescharacteristic
38C2×#683C2211yescharacteristic
39C2×#683C2211yescharacteristic
40C2×C4×Dic5C2211yescharacteristic
41C2×C4×Dic5C2211yescharacteristic
42#4044—24no—
43#3963C2211yescharacteristic
44#3963C2211yescharacteristic
45#3963C2211yescharacteristic
46#3963C2211yescharacteristic
47#3963C2211yescharacteristic
48#3963C2211yescharacteristic
49#3963C2211yescharacteristic
50#3963C2211yescharacteristic
51#3963C2211yescharacteristic
52#3963C2211yescharacteristic
53#2723—51nomaximal, Sylow
54C24×C5C2311yescharacteristic
55C23×D5C2311yescharacteristic
56C2×C5×Q8—22no—
57C2×C5×Q8—41no—
58C2×C5×Q8—41no—
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—22no—
69C2×C22⋊D5—22no—
70C2×C22⋊D5—41no—
71C22×Dic5C2311yescharacteristic
72C22×Dic5C2311yescharacteristic
73C22×Dic5C2311yescharacteristic
74C22×Dic5C2311yescharacteristic
75C22×Dic5C2311yescharacteristic
76C22×Dic5C2311yescharacteristic
77C22×Dic5C2311yescharacteristic
78C5×C4⋊C4—22no—
79C5×C4⋊C4—22no—
80C5×C4⋊C4D414yescomplemented
81C5×C22⋊C4—22no—
82C5×C22⋊C4—22no—
83C22⋊Dic5—22no—
84C22⋊Dic5—22no—
85C22⋊Dic5—41no—
86#685—22no—
87#685—22no—
88#685—22no—
89#685—41no—
90#685—41no—
91#685—41no—
92C4⋊Dic5—22no—
93C4⋊Dic5—22no—
94#683—22no—
95#683—41no—
96#683—41no—
97#683—41no—
98C4×Dic5—22no—
99C4×Dic5—22no—
100C2×#126—51no—
101C2×C22⋊Q8D511yescharacteristic, complemented
102#399—51no—
103#397—51no—
104#397—51no—
105#395—51no—
106#394—51no—
107#393—51no—
108#390—51no—
109#389—51no—
110#388—51no—
111#387—51no—
112#386—51no—
113#385—51no—
114#382—51no—
115C23×C5—21no—
116C23×C5—21no—
117C23×C5—21no—
118C23×C5—21no—
119C23×C5—21no—
120C23×C5—21no—
121C23×C5#4112yes—
122C23×C5C2411yescharacteristic
123C22×D5—21no—
124C22×D5—21no—
125C22×D5—21no—
126C22×D5—21no—
127C22×D5—21no—
128C22×D5—21no—
129C22×D5—21no—
130C5×Q8—44no—
131C2×C4×C5—21no—
132C2×C4×C5—21no—
133C2×C4×C5—21no—
134C2×C4×C5—21no—
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—44no—
147C2×C4×C5C2×D412yes—
148C2×C4×C5C2×D412yes—
149C2×C4×C5C2×D412yes—
150C22⋊D5—44no—
151C2×Dic5—21no—
152C2×Dic5—21no—
153C2×Dic5—21no—
154C2×Dic5—21no—
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#4112yes—
172C22×Q8C2×D511yescharacteristic
173C22×D4—51no—
174C23×C4C2×D511yescharacteristic
175#126—104no—
176C22⋊Q8—24no—
177C2×C4⋊C4—51no—
178C2×C4⋊C4—51no—
179C2×C4⋊C4—51no—
180C2×C4⋊C4—51no—
181C2×C4⋊C4—51no—
182C2×C4⋊C4—51no—
183C2×C4⋊C4C2×D511yescharacteristic
184C2×C4⋊C4C2×D511yescharacteristic
185C2×C4⋊C4C2×D511yescharacteristic
186C2×C22⋊C4—51no—
187C2×C22⋊C4—51no—
188C2×C22⋊C4—51no—
189C2×C22⋊C4—51no—
190C2×C22⋊C4—51no—
191C2×C22⋊C4—51no—
192C2×C22⋊C4—51no—
193C2×C22⋊C4—51no—
194C2×C22⋊C4—51no—
195C2×C22⋊C4C2×D511yescharacteristic
196C2×C22⋊C4C2×D511yescharacteristic
197C2×C42—51no—
198C2×C42—51no—
199#95—51no—
200#95—51no—
201#95—51no—
202#95—51no—
203#95—51no—
204#95—51no—
205#95—51no—
206#95—51no—
207#95—51no—
208#95—51no—
209C22×C5—22no—
210C22×C5—22no—
211C22×C5—22no—
212C22×C5—41no—
213C22×C5—41no—
214C22×C5—41no—
215C22×C5—41no—
216C22×C5C22×D411yescharacteristic
217C22×C5C2×#4111yescharacteristic
218C22×C5C2×#4111yescharacteristic
219C22×C5H(5,𝔽2)11yescharacteristic
220C22×C5H(5,𝔽2)11yescharacteristic
221C22×C52−1+411yescharacteristic
222C22×C52−1+411yescharacteristic
223C2×D5—41no—
224C2×D5—41no—
225C2×D5—41no—
226C2×D5—41no—
227C2×D5—41no—
228C2×D5—41no—
229C2×D5—41no—
230C4×C5—22no—
231C4×C5—22no—
232C4×C5—22no—
233C4×C5—41no—
234C4×C5—41no—
235C4×C5—41no—
236Dic5—22no—
237Dic5—41no—
238Dic5—41no—
239Dic5—41no—
240Dic5—41no—
241Dic5—41no—
242Dic5—41no—
243C24—51no—
244C24C22×D511yescharacteristic
245C2×Q8—22no—
246C2×Q8—41no—
247C2×Q8—41no—
248C2×D4—102no—
249C2×D4—102no—
250C2×D4—201no—
251C22×C4—22no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—51no—
255C22×C4—51no—
256C22×C4—51no—
257C22×C4—51no—
258C22×C4—51no—
259C22×C4—51no—
260C22×C4—51no—
261C22×C4C22×D511yescharacteristic
262C22×C4C22×D511yescharacteristic
263C22×C4C22×D511yescharacteristic
264C22×C4C22×D511yescharacteristic
265C22×C4C22×D511yescharacteristic
266C22×C4C22×D511yescharacteristic
267C4⋊C4—22no—
268C4⋊C4—22no—
269C4⋊C4—102no—
270C4⋊C4—102no—
271C4⋊C4—102no—
272C4⋊C4—201no—
273C4⋊C4—201no—
274C4⋊C4—201no—
275C4⋊C4C22⋊D514yescomplemented
276C22⋊C4—22no—
277C22⋊C4—22no—
278C22⋊C4—102no—
279C22⋊C4—102no—
280C22⋊C4—102no—
281C22⋊C4—102no—
282C22⋊C4—102no—
283C22⋊C4—201no—
284C22⋊C4—201no—
285C22⋊C4—201no—
286C22⋊C4—201no—
287C42—102no—
288C42—102no—
289C2×C5—42no—
290C2×C5#53511yescharacteristic
291C2×C5#53911yescharacteristic
292C2×C5#54211yescharacteristic
293C2×C5#54711yescharacteristic
294C2×C5#54811yescharacteristic
295C2×C5#55111yescharacteristic
296C2×C5#56311yescharacteristic
297D5—81no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—101no—
305C23—101no—
306C23—101no—
307C23—101no—
308C23—101no—
309C23—101no—
310C23—101no—
311C23#71312yes—
312C23C23×D511yescharacteristic
313Q8—44no—
314D4—204no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—44no—
331C2×C4—52no—
332C2×C4—101no—
333C2×C4—101no—
334C2×C4—101no—
335C2×C4—101no—
336C2×C4—101no—
337C2×C4—101no—
338C2×C4—101no—
339C2×C4—101no—
340C2×C4—101no—
341C2×C4—101no—
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×C4C2×C22⋊D512yes—
353C2×C4C2×C22⋊D512yes—
354C2×C4C2×C22⋊D512yes—
355C5#272311yescharacteristic, complemented, Sylow
356C22—22no—
357C22—22no—
358C22—22no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—201no—
364C22—201no—
365C22—201no—
366C22—201no—
367C22—201no—
368C22—201no—
369C22—201no—
370C22#414411yescharacteristic
371C22#414411yescharacteristic
372C22C2×#71311yescharacteristic
373C22#414711yescharacteristic
374C22D5×#4111yescharacteristic
375C22#415011yescharacteristic
376C22C22×C22⋊D511yescharacteristic
377C4—22no—
378C4—22no—
379C4—22no—
380C4—41no—
381C4—41no—
382C4—41no—
383C4—102no—
384C4—201no—
385C4—201no—
386C4—201no—
387C4—201no—
388C4—201no—
389C4—201no—
390C2—42no—
391C2—401no—
392C2#6597111yescharacteristic
393C2#6597411yescharacteristic
394C2#6597911yescharacteristic
395C2#6598011yescharacteristic
396C2#6613911yescharacteristic
397C2#6615111yescharacteristic
398C2#6615811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC2×C5×C22⋊Q8C2
prime radicalC22×C22⋊D5C22
hypercenterC2×C22⋊Q8D5
hypocenterC5#2723
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×#4044C2semidirect
3. #65520C2semidirect
4. #65518C2semidirect
6. #65499C2semidirect
7. #65282C2semidirect
8. #65281C2semidirect
9. #65270C2semidirect
10. #65266C2semidirect
11. #65262C2semidirect
14. #65242C2semidirect
15. #65233C2semidirect
80. C5×C4⋊C4D4semidirect
101. C2×C22⋊Q8D5semidirect
275. C4⋊C4C22⋊D5semidirect
355. C5#2723semidirect

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×#40442
#655202
#655182
#655022
#654992
#652822
#652812
#652702
#652662
#652622
#652502
#652462
#652422
#652332
#27235