Finite Group Databasebeta

← #10614521

Subgroups — #10614521

Order 640.

562 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×C5×C22⋊Q8C211yesmaximal, characteristic, complemented
2C2×#4090C211yesmaximal, characteristic, complemented
3C2×#4075C211yesmaximal, characteristic
4C2×#4031C211yesmaximal, characteristic
5#65520C211yesmaximal, characteristic, complemented
6#65499C211yesmaximal, characteristic, complemented
7#65283C211yesmaximal, characteristic, complemented
8#65281C211yesmaximal, characteristic, complemented
9#65279C211yesmaximal, characteristic, complemented
10#65268C211yesmaximal, characteristic, complemented
11#65267C211yesmaximal, characteristic, complemented
12#65262C211yesmaximal, characteristic, complemented
13#65248C211yesmaximal, characteristic
14#65241C211yesmaximal, characteristic, complemented
15#65240C211yesmaximal, characteristic, complemented
16C22×C5×Q8C2211yescharacteristic
17C23×C4×C5C2211yescharacteristic
18C22×C22⋊D5C2211yescharacteristic
19C22×Dic10C2211yescharacteristic
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#4090—24no—
30#4075—24no—
31C2×#685C2211yescharacteristic
32C2×#685C2211yescharacteristic
33C2×#685C2211yescharacteristic
34C2×#685C2211yescharacteristic
35C2×#685C2211yescharacteristic
36C2×#685C2211yescharacteristic
37C2×C4⋊Dic5C2211yescharacteristic
38C2×#683C2211yescharacteristic
39C2×#683C2211yescharacteristic
40C2×#683C2211yescharacteristic
41C2×#683C2211yescharacteristic
42C2×#683C2211yescharacteristic
43C2×#683C2211yescharacteristic
44C2×C4×Dic5C2211yescharacteristic
45C2×C4×Dic5C2211yescharacteristic
46#4031—24no—
47#3963C2211yescharacteristic
48#3963C2211yescharacteristic
49#3963C2211yescharacteristic
50#3963C2211yescharacteristic
51#3963C2211yescharacteristic
52#3963C2211yescharacteristic
53#3963C2211yescharacteristic
54#3963C2211yescharacteristic
55#2674—51nomaximal, Sylow
56C24×C5C2311yescharacteristic
57C23×D5C2311yescharacteristic
58C2×C5×Q8—22no—
59C2×C5×Q8—22no—
60C2×C5×Q8—41no—
61C22×C4×C5—22no—
62C22×C4×C5—22no—
63C22×C4×C5—41no—
64C22×C4×C5C2311yescharacteristic
65C22×C4×C5C2311yescharacteristic
66C22×C4×C5C2311yescharacteristic
67C22×C4×C5C2311yescharacteristic
68C22×C4×C5C2311yescharacteristic
69C22×C4×C5C2311yescharacteristic
70C2×C22⋊D5—22no—
71C2×C22⋊D5—22no—
72C2×C22⋊D5—41no—
73C22×Dic5C2311yescharacteristic
74C22×Dic5C2311yescharacteristic
75C22×Dic5C2311yescharacteristic
76C22×Dic5C2311yescharacteristic
77C22×Dic5C2311yescharacteristic
78C22×Dic5C2311yescharacteristic
79C22×Dic5C2311yescharacteristic
80C2×Dic10—22no—
81C2×Dic10—22no—
82C2×Dic10—41no—
83C5×C4⋊C4—22no—
84C5×C4⋊C4—22no—
85C5×C4⋊C4—22no—
86C5×C22⋊C4—22no—
87C5×C22⋊C4D414yes—
88C22⋊Dic5—22no—
89C22⋊Dic5—22no—
90C22⋊Dic5—22no—
91#685—22no—
92#685—22no—
93#685—22no—
94#685—22no—
95#685—22no—
96#685—41no—
97C4⋊Dic5—22no—
98#683—22no—
99#683—22no—
100#683—22no—
101#683—22no—
102#683—41no—
103#683D414yes—
104C4×Dic5—22no—
105C4×Dic5—22no—
106C2×C4⋊Q8—51no—
107C2×#124—51no—
108C2×#123—51no—
109C2×C22⋊Q8D511yescharacteristic, complemented
110#399—51no—
111#397—51no—
112#396—51no—
113#395—51no—
114#394—51no—
115#393—51no—
116#389—51no—
117#388—51no—
118#388—51no—
119#387—51no—
120#385—51no—
121C23×C5—21no—
122C23×C5—21no—
123C23×C5—21no—
124C23×C5—21no—
125C23×C5—21no—
126C23×C5—21no—
127C23×C5C2×D412yes—
128C23×C5C2411yescharacteristic
129C22×D5—21no—
130C22×D5—21no—
131C22×D5—21no—
132C22×D5—21no—
133C22×D5—21no—
134C22×D5—21no—
135C22×D5—21no—
136C5×Q8—44no—
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—44no—
153C2×C4×C5C2×D412yes—
154C2×C4×C5C2×D412yes—
155C2×C4×C5C2×D412yes—
156C22⋊D5—44no—
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×Dic5—21no—
175C2×Dic5C2×D412yes—
176C2×Dic5C2×D412yes—
177C2×Dic5#4112yes—
178Dic10—44no—
179C22×Q8—51no—
180C22×Q8C2×D511yescharacteristic
181C22×D4—51no—
182C23×C4C2×D511yescharacteristic
183C4⋊Q8—104no—
184#124—104no—
185#123—104no—
186C22⋊Q8—24no—
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⋊C4—51no—
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⋊C4C2×D511yescharacteristic
207C2×C22⋊C4C2×D511yescharacteristic
208C2×C42—51no—
209C2×C42—51no—
210#95—51no—
211#95—51no—
212#95—51no—
213#95—51no—
214#95—51no—
215#95—51no—
216#95—51no—
217#95—51no—
218C22×C5—22no—
219C22×C5—22no—
220C22×C5—22no—
221C22×C5—41no—
222C22×C5—41no—
223C22×C5—41no—
224C22×C5—41no—
225C22×C5C22×D411yescharacteristic
226C22×C5C22×D411yescharacteristic
227C22×C5C2×#4111yescharacteristic
228C22×C5H(5,𝔽2)11yescharacteristic
229C22×C52−1+411yescharacteristic
230C22×C52−1+411yescharacteristic
231C22×C52−1+411yescharacteristic
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—
245Dic5—22no—
246Dic5—22no—
247Dic5—22no—
248Dic5—41no—
249Dic5—41no—
250Dic5—41no—
251Dic5—41no—
252C24—51no—
253C24C22×D511yescharacteristic
254C2×Q8—22no—
255C2×Q8—22no—
256C2×Q8—41no—
257C2×Q8—102no—
258C2×Q8—102no—
259C2×Q8—201no—
260C2×D4—102no—
261C2×D4—102no—
262C2×D4—201no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—41no—
266C22×C4—51no—
267C22×C4—51no—
268C22×C4—51no—
269C22×C4—51no—
270C22×C4—51no—
271C22×C4—51no—
272C22×C4—51no—
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—54no—
283C4⋊C4—102no—
284C4⋊C4—102no—
285C4⋊C4—102no—
286C4⋊C4—102no—
287C4⋊C4—102no—
288C4⋊C4—201no—
289C22⋊C4—22no—
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—201no—
299C22⋊C4C22⋊D514yes—
300C42—102no—
301C42—102no—
302C2×C5—42no—
303C2×C5#53611yescharacteristic
304C2×C5#53611yescharacteristic
305C2×C5#54211yescharacteristic
306C2×C5D4211yescharacteristic
307C2×C5#54711yescharacteristic
308C2×C5#54811yescharacteristic
309C2×C5#56311yescharacteristic
310D5—81no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—21no—
317C23—101no—
318C23—101no—
319C23—101no—
320C23—101no—
321C23—101no—
322C23—101no—
323C23—101no—
324C23C2×C22⋊D512yes—
325C23C23×D511yescharacteristic
326Q8—44no—
327Q8—204no—
328D4—204no—
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—44no—
345C2×C4—52no—
346C2×C4—52no—
347C2×C4—52no—
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—101no—
362C2×C4—101no—
363C2×C4—101no—
364C2×C4—101no—
365C2×C4—101no—
366C2×C4D4×D512yes—
367C2×C4C2×C22⋊D512yes—
368C2×C4C2×C22⋊D512yes—
369C5#267411yescharacteristic, complemented, Sylow
370C22—22no—
371C22—22no—
372C22—22no—
373C22—41no—
374C22—41no—
375C22—41no—
376C22—41no—
377C22—201no—
378C22—201no—
379C22—201no—
380C22—201no—
381C22—201no—
382C22—201no—
383C22—201no—
384C22C2×D4×D511yescharacteristic
385C22#414411yescharacteristic
386C22#414711yescharacteristic
387C22#414711yescharacteristic
388C22D5×#4111yescharacteristic
389C22#415011yescharacteristic
390C22C22×C22⋊D511yescharacteristic
391C4—22no—
392C4—22no—
393C4—22no—
394C4—41no—
395C4—41no—
396C4—41no—
397C4—102no—
398C4—102no—
399C4—102no—
400C4—201no—
401C4—201no—
402C4—201no—
403C4—201no—
404C2—42no—
405C2—401no—
406C2#6596211yescharacteristic
407C2#6596711yescharacteristic
408C2#6597411yescharacteristic
409C2#6598011yescharacteristic
410C2D4×C22⋊D511yescharacteristic
411C2#6615011yescharacteristic
412C2#6615811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC2×C5×C22⋊Q8C2
prime radicalC22×C22⋊D5C22
hypercenterC2×C22⋊Q8D5
hypocenterC5#2674
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×#4090C2semidirect
5. #65520C2semidirect
6. #65499C2semidirect
7. #65283C2semidirect
8. #65281C2semidirect
9. #65279C2semidirect
10. #65268C2semidirect
11. #65267C2semidirect
12. #65262C2semidirect
14. #65241C2semidirect
15. #65240C2semidirect
109. C2×C22⋊Q8D5semidirect
369. C5#2674semidirect

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×#40902
C2×#40752
C2×#40312
#655202
#654992
#652832
#652812
#652792
#652682
#652672
#652622
#652482
#652412
#652402
#26745