Finite Group Databasebeta

← #10614529

Subgroups — #10614529

Order 640.

562 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×C22⋊Q8C211yesmaximal, characteristic, complemented
2C2×#4040C211yesmaximal, characteristic, complemented
3#65520C211yesmaximal, characteristic, complemented
4#65520C211yesmaximal, characteristic, complemented
5#65502C211yesmaximal, characteristic, complemented
6#65502C211yesmaximal, characteristic, complemented
7#65282C211yesmaximal, characteristic, complemented
8#65277C211yesmaximal, characteristic, complemented
9#65276C211yesmaximal, characteristic, complemented
10#65271C211yesmaximal, characteristic, complemented
11#65270C211yesmaximal, characteristic, complemented
12#65250C211yesmaximal, characteristic, complemented
13#65249C211yesmaximal, characteristic
14#65245C211yesmaximal, characteristic, complemented
15#65234C211yesmaximal, characteristic, complemented
16C22×C5×Q8C2211yescharacteristic
17C23×C4×C5C2211yescharacteristic
18C22×C22⋊D5C2211yescharacteristic
19C22×C22⋊D5C2211yescharacteristic
20C22×D20C2211yescharacteristic
21C22×C4×D5C2211yescharacteristic
22C5×C22⋊Q8—24no—
23C2×C5×C4⋊C4C2211yescharacteristic
24C2×C5×C4⋊C4C2211yescharacteristic
25C2×C5×C4⋊C4C2211yescharacteristic
26C2×C5×C22⋊C4C2211yescharacteristic
27C2×C5×C22⋊C4C2211yescharacteristic
28C2×C22⋊Dic5C2211yescharacteristic
29C2×C22⋊Dic5C2211yescharacteristic
30C2×#685C2211yescharacteristic
31C2×#685C2211yescharacteristic
32C2×#685C2211yescharacteristic
33C2×#685C2211yescharacteristic
34C2×#685C2211yescharacteristic
35C2×#685C2211yescharacteristic
36C2×#685C2211yescharacteristic
37C2×#685C2211yescharacteristic
38C2×#685C2211yescharacteristic
39C2×C4⋊Dic5C2211yescharacteristic
40C2×#683C2211yescharacteristic
41C2×C4×Dic5C2211yescharacteristic
42#4040—24no—
43#3963C2211yescharacteristic
44#3963C2211yescharacteristic
45#3963C2211yescharacteristic
46#3963C2211yescharacteristic
47#3963C2211yescharacteristic
48#3963C2211yescharacteristic
49#3963C2211yescharacteristic
50#3963C2211yescharacteristic
51#3963C2211yescharacteristic
52#3963C2211yescharacteristic
53#2696—51nomaximal, Sylow
54C24×C5C2311yescharacteristic
55C23×D5C2311yescharacteristic
56C23×D5C2311yescharacteristic
57C2×C5×Q8—22no—
58C2×C5×Q8—41no—
59C2×C5×Q8—41no—
60C22×C4×C5—22no—
61C22×C4×C5—41no—
62C22×C4×C5—41no—
63C22×C4×C5C2311yescharacteristic
64C22×C4×C5C2311yescharacteristic
65C22×C4×C5C2311yescharacteristic
66C22×C4×C5C2311yescharacteristic
67C22×C4×C5C2311yescharacteristic
68C22×C4×C5C2311yescharacteristic
69C2×C22⋊D5—22no—
70C2×C22⋊D5—22no—
71C2×C22⋊D5—22no—
72C2×C22⋊D5—41no—
73C2×C22⋊D5—41no—
74C2×C22⋊D5—41no—
75C22×Dic5C2311yescharacteristic
76C22×Dic5C2311yescharacteristic
77C22×Dic5C2311yescharacteristic
78C22×Dic5C2311yescharacteristic
79C22×Dic5C2311yescharacteristic
80C22×Dic5C2311yescharacteristic
81C2×D20—22no—
82C2×D20—22no—
83C2×D20—41no—
84C2×C4×D5—22no—
85C2×C4×D5—22no—
86C2×C4×D5—41no—
87C5×C4⋊C4—22no—
88C5×C4⋊C4—22no—
89C5×C4⋊C4D414yescomplemented
90C5×C22⋊C4—22no—
91C5×C22⋊C4—22no—
92C22⋊Dic5—22no—
93C22⋊Dic5—41no—
94#685—22no—
95#685—22no—
96#685—22no—
97#685—41no—
98#685—41no—
99#685—41no—
100#685—41no—
101#685—41no—
102#685—41no—
103C4⋊Dic5—41no—
104#683—22no—
105C4×Dic5—41no—
106C2×#123—51no—
107C2×C22⋊Q8D511yescharacteristic, complemented
108#400—51no—
109#397—51no—
110#394—51no—
111#393—51no—
112#393—51no—
113#392—51no—
114#390—51no—
115#388—51no—
116#387—51no—
117#386—51no—
118#386—51no—
119#386—51no—
120#380—51no—
121C23×C5—21no—
122C23×C5—21no—
123C23×C5—21no—
124C23×C5—21no—
125C23×C5—21no—
126C23×C5—21no—
127C23×C5#4112yes—
128C23×C5C2411yescharacteristic
129C22×D5—21no—
130C22×D5—21no—
131C22×D5—21no—
132C22×D5—21no—
133C22×D5—21no—
134C22×D5—21no—
135C22×D5—21no—
136C22×D5—21no—
137C22×D5—21no—
138C22×D5—21no—
139C22×D5—21no—
140C22×D5—21no—
141C22×D5—21no—
142C22×D5#4112yes—
143C5×Q8—44no—
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×C5—21no—
155C2×C4×C5—21no—
156C2×C4×C5—21no—
157C2×C4×C5—21no—
158C2×C4×C5—21no—
159C2×C4×C5—44no—
160C2×C4×C5C2×D412yes—
161C2×C4×C5C2×D412yes—
162C2×C4×C5C2×D412yes—
163C22⋊D5—44no—
164C22⋊D5—44no—
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×Dic5—21no—
176C2×Dic5—21no—
177C2×Dic5—21no—
178C2×Dic5—21no—
179C2×Dic5—21no—
180C2×Dic5—21no—
181C2×Dic5—21no—
182C2×Dic5—21no—
183D20—44no—
184C4×D5—44no—
185C22×Q8C2×D511yescharacteristic
186C22×D4—51no—
187C22×D4—51no—
188C22×D4—51no—
189C23×C4—51no—
190C23×C4C2×D511yescharacteristic
191#123—104no—
192C22⋊Q8—24no—
193C2×C4⋊C4—51no—
194C2×C4⋊C4—51no—
195C2×C4⋊C4C2×D511yescharacteristic
196C2×C4⋊C4C2×D511yescharacteristic
197C2×C4⋊C4C2×D511yescharacteristic
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×C22⋊C4—51no—
208C2×C22⋊C4—51no—
209C2×C22⋊C4C2×D511yescharacteristic
210C2×C22⋊C4C2×D511yescharacteristic
211C2×C42—51no—
212#95—51no—
213#95—51no—
214#95—51no—
215#95—51no—
216#95—51no—
217#95—51no—
218#95—51no—
219#95—51no—
220#95—51no—
221#95—51no—
222C22×C5—22no—
223C22×C5—22no—
224C22×C5—22no—
225C22×C5—41no—
226C22×C5—41no—
227C22×C5—41no—
228C22×C5—41no—
229C22×C5C22×D411yescharacteristic
230C22×C5C2×#4111yescharacteristic
231C22×C5C2×#4111yescharacteristic
232C22×C5H(5,𝔽2)11yescharacteristic
233C22×C5H(5,𝔽2)11yescharacteristic
234C22×C5H(5,𝔽2)11yescharacteristic
235C22×C52−1+411yescharacteristic
236C2×D5—22no—
237C2×D5—22no—
238C2×D5—22no—
239C2×D5—41no—
240C2×D5—41no—
241C2×D5—41no—
242C2×D5—41no—
243C2×D5—41no—
244C2×D5—41no—
245C2×D5—41no—
246C2×D5—41no—
247C2×D5—41no—
248C2×D5—41no—
249C2×D5—41no—
250C4×C5—22no—
251C4×C5—22no—
252C4×C5—22no—
253C4×C5—41no—
254C4×C5—41no—
255C4×C5—41no—
256Dic5—41no—
257Dic5—41no—
258Dic5—41no—
259Dic5—41no—
260Dic5—41no—
261Dic5—41no—
262C24—51no—
263C24—51no—
264C24C22×D511yescharacteristic
265C2×Q8—22no—
266C2×Q8—41no—
267C2×Q8—41no—
268C2×D4—102no—
269C2×D4—102no—
270C2×D4—102no—
271C2×D4—102no—
272C2×D4—102no—
273C2×D4—201no—
274C2×D4—201no—
275C2×D4—201no—
276C2×D4—201no—
277C22×C4—22no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—51no—
281C22×C4—51no—
282C22×C4—51no—
283C22×C4—51no—
284C22×C4—51no—
285C22×C4—51no—
286C22×C4—102no—
287C22×C4—102no—
288C22×C4—201no—
289C22×C4C22×D511yescharacteristic
290C22×C4C22×D511yescharacteristic
291C22×C4C22×D511yescharacteristic
292C22×C4C22×D511yescharacteristic
293C22×C4C22×D511yescharacteristic
294C22×C4C22×D511yescharacteristic
295C4⋊C4—22no—
296C4⋊C4—22no—
297C4⋊C4—102no—
298C4⋊C4—201no—
299C4⋊C4C22⋊D514yescomplemented
300C22⋊C4—22no—
301C22⋊C4—22no—
302C22⋊C4—102no—
303C22⋊C4—102no—
304C22⋊C4—102no—
305C22⋊C4—102no—
306C22⋊C4—201no—
307C22⋊C4—201no—
308C22⋊C4—201no—
309C22⋊C4—201no—
310C22⋊C4—201no—
311C22⋊C4—201no—
312C22⋊C4—201no—
313C42—201no—
314C2×C5—42no—
315C2×C5#53511yescharacteristic
316C2×C5#53811yescharacteristic
317C2×C5#53911yescharacteristic
318C2×C5#54611yescharacteristic
319C2×C5#54811yescharacteristic
320C2×C5#55111yescharacteristic
321C2×C5#56211yescharacteristic
322D5—42no—
323D5—81no—
324C23—21no—
325C23—21no—
326C23—21no—
327C23—21no—
328C23—21no—
329C23—21no—
330C23—52no—
331C23—101no—
332C23—101no—
333C23—101no—
334C23—101no—
335C23—101no—
336C23—101no—
337C23—101no—
338C23—101no—
339C23—101no—
340C23—101no—
341C23—101no—
342C23—101no—
343C23—101no—
344C23#71312yes—
345C23C23×D511yescharacteristic
346Q8—44no—
347D4—204no—
348D4—204no—
349D4—204no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—44no—
366C2×C4—101no—
367C2×C4—101no—
368C2×C4—101no—
369C2×C4—101no—
370C2×C4—101no—
371C2×C4—101no—
372C2×C4—101no—
373C2×C4—101no—
374C2×C4—101no—
375C2×C4—101no—
376C2×C4—101no—
377C2×C4—101no—
378C2×C4—101no—
379C2×C4—101no—
380C2×C4—101no—
381C2×C4—101no—
382C2×C4—101no—
383C2×C4—101no—
384C2×C4—204no—
385C2×C4C2×C22⋊D512yes—
386C2×C4C2×C22⋊D512yes—
387C2×C4C2×C22⋊D512yes—
388C5#269611yescharacteristic, complemented, Sylow
389C22—22no—
390C22—22no—
391C22—22no—
392C22—41no—
393C22—41no—
394C22—41no—
395C22—41no—
396C22—102no—
397C22—102no—
398C22—102no—
399C22—201no—
400C22—201no—
401C22—201no—
402C22—201no—
403C22—201no—
404C22—201no—
405C22—201no—
406C22—201no—
407C22—201no—
408C22—201no—
409C22—201no—
410C22#414411yescharacteristic
411C22#414411yescharacteristic
412C22C2×#71311yescharacteristic
413C22#414711yescharacteristic
414C22D5×#4111yescharacteristic
415C22#414911yescharacteristic
416C22C22×C22⋊D511yescharacteristic
417C4—22no—
418C4—22no—
419C4—22no—
420C4—41no—
421C4—41no—
422C4—41no—
423C4—201no—
424C4—201no—
425C4—201no—
426C4—201no—
427C4—201no—
428C4—201no—
429C2—42no—
430C2—202no—
431C2—401no—
432C2#6597111yescharacteristic
433C2#6597211yescharacteristic
434C2#6597811yescharacteristic
435C2#6598111yescharacteristic
436C2#6613911yescharacteristic
437C2#6615111yescharacteristic
438C2#6616311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC2×C5×C22⋊Q8C2
prime radical#65249C2
hypercenterC2×C22⋊Q8D5
hypocenterC5#2696
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×#4040C2semidirect
3. #65520C2semidirect
4. #65520C2semidirect
5. #65502C2semidirect
6. #65502C2semidirect
7. #65282C2semidirect
8. #65277C2semidirect
9. #65276C2semidirect
10. #65271C2semidirect
11. #65270C2semidirect
12. #65250C2semidirect
14. #65245C2semidirect
15. #65234C2semidirect
89. C5×C4⋊C4D4semidirect
107. C2×C22⋊Q8D5semidirect
299. C4⋊C4C22⋊D5semidirect
388. C5#2696semidirect

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×#40402
#655202
#655022
#652822
#652772
#652762
#652712
#652702
#652502
#652492
#652452
#652342
#26965