Finite Group Databasebeta

← #10614494

Subgroups — #10614494

Order 640.

608 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×C22⋊Q8C211yesmaximal, characteristic
2C2×#4090C211yesmaximal, characteristic, complemented
3C22×#683C211yesmaximal, characteristic
4C2×#4024C211yesmaximal, characteristic
5#65518C211yesmaximal, characteristic, complemented
6#65499C211yesmaximal, characteristic
7#65272C211yesmaximal, characteristic, complemented
8#65269C211yesmaximal, characteristic, complemented
9#65268C211yesmaximal, characteristic, complemented
10#65265C211yesmaximal, characteristic, complemented
11#65261C211yesmaximal, characteristic, complemented
12#65261C211yesmaximal, characteristic, complemented
13#65235C211yesmaximal, characteristic
14#65231C211yesmaximal, characteristic
15C22⋊C4×Dic5C211yesmaximal, characteristic
16C22×C5×Q8C2211yescharacteristic
17C23×C4×C5C2211yescharacteristic
18C23×Dic5C2211yescharacteristic
19C23×Dic5C2211yescharacteristic
20C22×Dic10C2211yescharacteristic
21C5×C22⋊Q8—24no—
22C2×C5×C4⋊C4C2211yescharacteristic
23C2×C5×C4⋊C4C2211yescharacteristic
24C2×C5×C4⋊C4C2211yescharacteristic
25C2×C5×C22⋊C4C2211yescharacteristic
26C2×C5×C22⋊C4C2211yescharacteristic
27C2×C22⋊Dic5C2211yescharacteristic
28C2×C22⋊Dic5C2211yescharacteristic
29#4090—24no—
30C2×C4⋊Dic5C2211yescharacteristic
31C2×#683—24no—
32C2×#683C2211yescharacteristic
33C2×#683C2211yescharacteristic
34C2×#683C2211yescharacteristic
35C2×#683C2211yescharacteristic
36C2×#683C2211yescharacteristic
37C2×#683C2211yescharacteristic
38C2×#683C2211yescharacteristic
39C2×#683C2211yescharacteristic
40C2×#683C2211yescharacteristic
41C2×#683C2211yescharacteristic
42C2×#683C2211yescharacteristic
43C2×#683C2211yescharacteristic
44C2×C4×Dic5C2211yescharacteristic
45C2×C4×Dic5C2211yescharacteristic
46#4024—24no—
47#3963C2211yescharacteristic
48#3963C2211yescharacteristic
49#3963C2211yescharacteristic
50#3963C2211yescharacteristic
51#3963C2211yescharacteristic
52#3963C2211yescharacteristic
53#3963C2211yescharacteristic
54#3963C2211yescharacteristic
55#2440—51nomaximal, Sylow
56C24×C5C2311yescharacteristic
57C2×C5×Q8—22no—
58C2×C5×Q8—22no—
59C2×C5×Q8—41no—
60C22×C4×C5—22no—
61C22×C4×C5—22no—
62C22×C4×C5—41no—
63C22×C4×C5C2311yescharacteristic
64C22×C4×C5C2311yescharacteristic
65C22×C4×C5C2311yescharacteristic
66C22×C4×C5C2311yescharacteristic
67C22×C4×C5C2311yescharacteristic
68C22×C4×C5C2311yescharacteristic
69C22×Dic5—22no—
70C22×Dic5—22no—
71C22×Dic5—22no—
72C22×Dic5—22no—
73C22×Dic5—41no—
74C22×Dic5Q814yescomplemented
75C22×Dic5C2311yescharacteristic
76C22×Dic5C2311yescharacteristic
77C22×Dic5C2311yescharacteristic
78C22×Dic5C2311yescharacteristic
79C22×Dic5C2311yescharacteristic
80C22×Dic5C2311yescharacteristic
81C22×Dic5C2311yescharacteristic
82C22×Dic5C2311yescharacteristic
83C2×Dic10—22no—
84C2×Dic10—22no—
85C2×Dic10—41no—
86C5×C4⋊C4—22no—
87C5×C4⋊C4—22no—
88C5×C4⋊C4—22no—
89C5×C22⋊C4—22no—
90C5×C22⋊C4D414yes—
91C22⋊Dic5—22no—
92C22⋊Dic5Q814yescomplemented
93C4⋊Dic5—22no—
94#683—22no—
95#683—22no—
96#683—22no—
97#683—22no—
98#683—22no—
99#683—22no—
100#683—22no—
101#683—22no—
102#683—22no—
103#683—41no—
104#683—41no—
105#683D414yes—
106C4×Dic5—22no—
107C4×Dic5—22no—
108C2×C4⋊Q8—51no—
109C2×C22⋊Q8—51no—
110C2×C22⋊Q8D511yescharacteristic
111C22×C4⋊C4—51no—
112#398—51no—
113#398—51no—
114#395—51no—
115#395—51no—
116#389—51no—
117#389—51no—
118#385—51no—
119#385—51no—
120#384—51no—
121#380—51no—
122C4×C22⋊C4—51no—
123C23×C5—21no—
124C23×C5—21no—
125C23×C5—21no—
126C23×C5—21no—
127C23×C5C2×D412yes—
128C23×C5C2×Q812yes—
129C23×C5C2×Q812yes—
130C23×C5C2411yescharacteristic
131C5×Q8—44no—
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—21no—
147C2×C4×C5—44no—
148C2×C4×C5C2×D412yes—
149C2×C4×C5C2×D412yes—
150C2×C4×C5C2×D412yes—
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—28no—
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—44no—
170C2×Dic5C2×D412yes—
171C2×Dic5C2×D412yes—
172C2×Dic5C2×Q812yes—
173C2×Dic5C2×Q812yes—
174C2×Dic5C2×Q812yes—
175C2×Dic5C2×Q812yes—
176C2×Dic5#4112yes—
177Dic10—44no—
178C22×Q8—51no—
179C22×Q8C2×D511yescharacteristic
180C23×C4—51no—
181C23×C4—51no—
182C23×C4C2×D511yescharacteristic
183C4⋊Q8—104no—
184C22⋊Q8—24no—
185C22⋊Q8—104no—
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⋊C4—51no—
193C2×C4⋊C4—51no—
194C2×C4⋊C4—51no—
195C2×C4⋊C4—51no—
196C2×C4⋊C4—51no—
197C2×C4⋊C4—51no—
198C2×C4⋊C4—51no—
199C2×C4⋊C4—104no—
200C2×C4⋊C4C2×D511yescharacteristic
201C2×C4⋊C4C2×D511yescharacteristic
202C2×C4⋊C4C2×D511yescharacteristic
203C2×C22⋊C4—51no—
204C2×C22⋊C4—51no—
205C2×C22⋊C4C2×D511yescharacteristic
206C2×C22⋊C4C2×D511yescharacteristic
207C2×C42—51no—
208C2×C42—51no—
209#95—51no—
210#95—51no—
211#95—51no—
212#95—51no—
213#95—51no—
214#95—51no—
215#95—51no—
216#95—51no—
217C22×C5—22no—
218C22×C5—22no—
219C22×C5—22no—
220C22×C5—22no—
221C22×C5—22no—
222C22×C5—22no—
223C22×C5C4⋊Q814yescomplemented
224C22×C5C22×D411yescharacteristic
225C22×C5C22×D411yescharacteristic
226C22×C5C22×Q811yescharacteristic
227C22×C5C22×Q811yescharacteristic
228C22×C5C2×#4111yescharacteristic
229C22×C5H(5,𝔽2)11yescharacteristic
230C22×C52−1+411yescharacteristic
231C4×C5—22no—
232C4×C5—22no—
233C4×C5—22no—
234C4×C5—41no—
235C4×C5—41no—
236C4×C5—41no—
237Dic5—22no—
238Dic5—22no—
239Dic5—22no—
240Dic5—22no—
241Dic5—41no—
242Dic5—41no—
243Dic5—41no—
244Dic5C22⋊Q814yescomplemented
245C24C22×D511yescharacteristic
246C2×Q8—22no—
247C2×Q8—22no—
248C2×Q8—41no—
249C2×Q8—102no—
250C2×Q8—102no—
251C2×Q8—201no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—41no—
255C22×C4—51no—
256C22×C4—54no—
257C22×C4—51no—
258C22×C4—51no—
259C22×C4—51no—
260C22×C4—51no—
261C22×C4—51no—
262C22×C4—51no—
263C22×C4—51no—
264C22×C4—102no—
265C22×C4—102no—
266C22×C4—102no—
267C22×C4—102no—
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—54no—
279C4⋊C4—102no—
280C4⋊C4—102no—
281C4⋊C4—102no—
282C4⋊C4—102no—
283C4⋊C4—102no—
284C4⋊C4—102no—
285C4⋊C4—102no—
286C4⋊C4—102no—
287C4⋊C4—102no—
288C4⋊C4—102no—
289C4⋊C4—201no—
290C4⋊C4—201no—
291C22⋊C4—22no—
292C22⋊C4—54no—
293C22⋊C4—102no—
294C22⋊C4C22⋊D514yes—
295C42—102no—
296C42—102no—
297C2×C5—24no—
298C2×C5C2×C22⋊Q811yescharacteristic
299C2×C5C2×C4⋊Q811yescharacteristic
300C2×C5#54411yescharacteristic
301C2×C5D4211yescharacteristic
302C2×C5#54711yescharacteristic
303C2×C5D4×Q811yescharacteristic
304C2×C5#55411yescharacteristic
305C23—21no—
306C23—21no—
307C23—21no—
308C23—21no—
309C23Q8×D512yes—
310C23Q8×D512yes—
311C23C2×C22⋊D512yes—
312C23C23×D511yescharacteristic
313Q8—44no—
314Q8—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—52no—
333C2×C4—52no—
334C2×C4—52no—
335C2×C4—52no—
336C2×C4—52no—
337C2×C4—52no—
338C2×C4—108no—
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×C4—101no—
353C2×C4—101no—
354C2×C4—101no—
355C2×C4—101no—
356C2×C4—204no—
357C2×C4D4×D512yes—
358C2×C4C2×C22⋊D512yes—
359C2×C4C2×C22⋊D512yes—
360C5#244011yescharacteristic, complemented, Sylow
361C22—22no—
362C22—22no—
363C22—22no—
364C22—22no—
365C22—22no—
366C22—22no—
367C22#409014yescomplemented
368C22C2×D4×D511yescharacteristic
369C22#414411yescharacteristic
370C22C2×Q8×D511yescharacteristic
371C22C2×Q8×D511yescharacteristic
372C22D5×#4111yescharacteristic
373C22#415011yescharacteristic
374C22C22×C22⋊D511yescharacteristic
375C4—22no—
376C4—22no—
377C4—22no—
378C4—41no—
379C4—41no—
380C4—41no—
381C4—54no—
382C4—102no—
383C4—102no—
384C4—102no—
385C4—102no—
386C4—201no—
387C4—201no—
388C4—201no—
389C2—24no—
390C2#6595711yescharacteristic
391C2D5×C22⋊Q811yescharacteristic
392C2#6596611yescharacteristic
393C2#6597011yescharacteristic
394C2D4×C22⋊D511yescharacteristic
395C2C2×#409011yescharacteristic
396C2#6615811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC2×C5×C22⋊Q8C2
prime radicalC24×C5C23
hypercenterC2×C22⋊Q8D5
hypocenterC5#2440
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C2×#4090C2semidirect
5. #65518C2semidirect
7. #65272C2semidirect
8. #65269C2semidirect
9. #65268C2semidirect
10. #65265C2semidirect
11. #65261C2semidirect
12. #65261C2semidirect
74. C22×Dic5Q8semidirect
92. C22⋊Dic5Q8semidirect
223. C22×C5C4⋊Q8semidirect
244. Dic5C22⋊Q8semidirect
360. C5#2440semidirect
367. C22#4090semidirect

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
C22×#6832
C2×#40242
#655182
#654992
#652722
#652692
#652682
#652652
#652612
#652352
#652312
C22⋊C4×Dic52
#24405