Finite Group Databasebeta

← #10615196

Subgroups — #10615196

Order 640.

358 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#542C211yesmaximal, characteristic, complemented
2#65869C211yesmaximal, characteristic, complemented
3#65376C211yesmaximal, characteristic, complemented
4#65373C211yesmaximal, characteristic, complemented
5#65346C211yesmaximal, characteristic, complemented
6#65346C211yesmaximal, characteristic, complemented
7#65338C211yesmaximal, characteristic, complemented
8#65337C211yesmaximal, characteristic, complemented
9#65328C211yesmaximal, characteristic, complemented
10#65327C211yesmaximal, characteristic, complemented
11#65316C211yesmaximal, characteristic, complemented
12#65311C211yesmaximal, characteristic, complemented
13#65304C211yesmaximal, characteristic, complemented
14#65298C211yesmaximal, characteristic, complemented
15#65288C211yesmaximal, characteristic, complemented
16C5×C4⋊Q8C2211yescharacteristic
17C5×#126—21no—
18C5×#124—21no—
19C5×#124C2211yescharacteristic
20C5×#123—21no—
21C5×C22⋊Q8—21no—
22C5×C22⋊Q8C2211yescharacteristic
23C5×#121C2211yescharacteristic
24C4×C5×Q8C2211yescharacteristic
25C4×C5×D4C2211yescharacteristic
26C5×#117C2211yescharacteristic
27C2×#688C2211yescharacteristic
28C2×#688C2211yescharacteristic
29C2×#686C2211yescharacteristic
30C2×#686C2211yescharacteristic
31#4076C2211yescharacteristic
32#4076C2211yescharacteristic
33C2×#681C2211yescharacteristic
34C22×#168C2211yescharacteristic
35#4040—21no—
36#4040—21no—
37#4030—21no—
38#4030—21no—
39#4020C2211yescharacteristic
40C4×D20C2211yescharacteristic
41C4×D20C2211yescharacteristic
42#4016C2211yescharacteristic
43#3967C2211yescharacteristic
44#3967C2211yescharacteristic
45#3964C2211yescharacteristic
46#3964C2211yescharacteristic
47#3941C2211yescharacteristic
48#3941C2211yescharacteristic
49#3941C2211yescharacteristic
50#3941C2211yescharacteristic
51#3940C2211yescharacteristic
52#3940C2211yescharacteristic
53#3939C2211yescharacteristic
54#3939C2211yescharacteristic
55C4⋊#168C2211yescharacteristic
56C4⋊#168C2211yescharacteristic
57C4⋊#168C2211yescharacteristic
58C4⋊#168C2211yescharacteristic
59#3257—51nomaximal, Sylow
60C2×C5×Q8—21no—
61C2×C5×Q8C2311yescharacteristic
62C2×C5×D4—21no—
63C2×C5×D4C2311yescharacteristic
64C22×C4×C5—21no—
65C22×C4×C5C2311yescharacteristic
66C2×C22⋊D5—21no—
67C2×C22⋊D5—21no—
68C2×D20—21no—
69C2×D20—21no—
70C2×D20C2311yescharacteristic
71C2×D20C2311yescharacteristic
72C2×C4×D5—21no—
73C2×C4×D5—21no—
74C5×C4⋊C4—21no—
75C5×C4⋊C4—21no—
76C5×C4⋊C4—21no—
77C5×C4⋊C4D411yescharacteristic
78C5×C4⋊C4D411yescharacteristic
79C5×C4⋊C4C2311yescharacteristic
80C5×C4⋊C4C2311yescharacteristic
81C5×C22⋊C4—21no—
82C5×C22⋊C4—21no—
83C5×C22⋊C4—21no—
84C5×C22⋊C4—21no—
85C5×C22⋊C4—21no—
86C5×C22⋊C4D411yescharacteristic
87C5×C22⋊C4D411yescharacteristic
88C42×C5—21no—
89C42×C5C2311yescharacteristic
90C42×C5C2311yescharacteristic
91#688—41no—
92#688—41no—
93#686—41no—
94#686—41no—
95#685—21no—
96#685—21no—
97#685—21no—
98#685—21no—
99C4⋊Dic5C2311yescharacteristic
100C4⋊Dic5C2311yescharacteristic
101#683—21no—
102#683—21no—
103#681—41no—
104C2×#168—41no—
105C2×#168C2311yescharacteristic
106C2×#168C2311yescharacteristic
107C2×#168C2311yescharacteristic
108C2×#168C2311yescharacteristic
109#542D511yescharacteristic, complemented
110#540—51no—
111#479—51no—
112#478—51no—
113#476—51no—
114#474—51no—
115#469—51no—
116#468—51no—
117#466—51no—
118#465—51no—
119#463—51no—
120#463—51no—
121#460—51no—
122#459—51no—
123#424—51no—
124C23×C5—21no—
125C23×C5C2×D411yescharacteristic
126C22×D5—21no—
127C22×D5—21no—
128C5×Q8—21no—
129C5×Q8—21no—
130C5×Q8—81no—
131C5×D4—21no—
132C5×D4—21no—
133C5×D4—81no—
134C2×C4×C5—21no—
135C2×C4×C5—21no—
136C2×C4×C5—21no—
137C2×C4×C5—21no—
138C2×C4×C5—21no—
139C2×C4×C5—81no—
140C2×C4×C5C2×D411yescharacteristic
141C2×C4×C5C2×D411yescharacteristic
142C2×C4×C5C2×D411yescharacteristic
143C2×C4×C5C2×D411yescharacteristic
144C2×C4×C5C2×D411yescharacteristic
145C2×C4×C5C2411yescharacteristic
146C22⋊D5—81no—
147C22⋊D5—81no—
148C2×Dic5—21no—
149C2×Dic5—21no—
150D20—21no—
151D20—21no—
152D20—21no—
153D20—21no—
154D20—81no—
155D20—81no—
156C4×D5—81no—
157C4×D5—81no—
158#168—21no—
159#168—21no—
160#168—21no—
161#168—21no—
162C2×SD16—51no—
163C2×SD16—51no—
164C2×D8—51no—
165C2×D8—51no—
166C2×M16—51no—
167C22×C8—51no—
168C4⋊Q8C2×D511yescharacteristic
169#127—51no—
170#126—21no—
171#125—51no—
172#124—21no—
173#124C2×D511yescharacteristic
174#123—21no—
175#123—101no—
176#123—101no—
177C22⋊Q8—21no—
178C22⋊Q8C2×D511yescharacteristic
179#121—51no—
180#121—51no—
181#121—101no—
182#121—101no—
183#121C2×D511yescharacteristic
184C4×Q8C2×D511yescharacteristic
185C4×D4—51no—
186C4×D4—51no—
187C4×D4C2×D511yescharacteristic
188#117C2×D511yescharacteristic
189#107—51no—
190#107—51no—
191#106—51no—
192#106—51no—
193C4⋊C8—51no—
194C4⋊C8—51no—
195C4⋊C8—51no—
196C4⋊C8—51no—
197#103—51no—
198#103—51no—
199#102—51no—
200#102—51no—
201#102—51no—
202#102—51no—
203#102—51no—
204#102—51no—
205C22×C5—21no—
206C22×C5—21no—
207C22×C5—21no—
208C22×C5—41no—
209C22×C5—41no—
210C22×C5—41no—
211C22×C5C22×D411yescharacteristic
212C2×D5—41no—
213C2×D5—41no—
214C2×D5—41no—
215C2×D5—41no—
216C2×D5—41no—
217C2×D5—41no—
218C4×C5—21no—
219C4×C5—21no—
220C4×C5—21no—
221C4×C5—21no—
222C4×C5—21no—
223C4×C5—41no—
224C4×C5—41no—
225C4×C5—41no—
226C4×C5H(5,𝔽2)11yescharacteristic
227C4×C52−1+411yescharacteristic
228Dic5—41no—
229Dic5—41no—
230C2×Q8—21no—
231C2×Q8C22×D511yescharacteristic
232C2×D4—21no—
233C2×D4—51no—
234C2×D4—51no—
235C2×D4—101no—
236C2×D4—101no—
237C2×D4—101no—
238C2×D4—101no—
239C2×D4C22×D511yescharacteristic
240C22×C4—21no—
241C22×C4—101no—
242C22×C4—101no—
243C22×C4C22×D511yescharacteristic
244SD16—201no—
245SD16—201no—
246D8—201no—
247D8—201no—
248M16—201no—
249C2×C8—51no—
250C2×C8—51no—
251C2×C8—51no—
252C2×C8—51no—
253C2×C8—201no—
254C4⋊C4—21no—
255C4⋊C4—21no—
256C4⋊C4—21no—
257C4⋊C4—51no—
258C4⋊C4—51no—
259C4⋊C4—101no—
260C4⋊C4—101no—
261C4⋊C4C22⋊D511yescharacteristic
262C4⋊C4C22⋊D511yescharacteristic
263C4⋊C4C22×D511yescharacteristic
264C4⋊C4C22×D511yescharacteristic
265C22⋊C4—21no—
266C22⋊C4—21no—
267C22⋊C4—21no—
268C22⋊C4—21no—
269C22⋊C4—21no—
270C22⋊C4—101no—
271C22⋊C4—101no—
272C22⋊C4—101no—
273C22⋊C4—101no—
274C22⋊C4C22⋊D511yescharacteristic
275C22⋊C4C22⋊D511yescharacteristic
276C42—21no—
277C42C22×D511yescharacteristic
278C42C22×D511yescharacteristic
279C2×C5—41no—
280C2×C5—81no—
281C2×C5#53711yescharacteristic
282C2×C5#57611yescharacteristic
283C2×C5#57711yescharacteristic
284D5—81no—
285D5—81no—
286C23—21no—
287C23—101no—
288C23—101no—
289C23C2×C22⋊D511yescharacteristic
290Q8—21no—
291Q8—21no—
292Q8—81no—
293D4—21no—
294D4—21no—
295D4—81no—
296D4—101no—
297D4—101no—
298D4—101no—
299D4—101no—
300D4—401no—
301D4—401no—
302D4—401no—
303D4—401no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—81no—
310C2×C4—101no—
311C2×C4—101no—
312C2×C4—401no—
313C2×C4—401no—
314C2×C4C2×C22⋊D511yescharacteristic
315C2×C4C2×C22⋊D511yescharacteristic
316C2×C4C2×C22⋊D511yescharacteristic
317C2×C4C2×C22⋊D511yescharacteristic
318C2×C4C2×C22⋊D511yescharacteristic
319C2×C4C23×D511yescharacteristic
320C8—101no—
321C8—101no—
322C8—101no—
323C8—101no—
324C5#325711yescharacteristic, complemented, Sylow
325C22—21no—
326C22—21no—
327C22—21no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22—201no—
332C22—201no—
333C22—201no—
334C22—201no—
335C22—201no—
336C22—201no—
337C22C22×C22⋊D511yescharacteristic
338C4—21no—
339C4—21no—
340C4—21no—
341C4—21no—
342C4—21no—
343C4—41no—
344C4—41no—
345C4—41no—
346C4—201no—
347C4—201no—
348C4#414911yescharacteristic
349C4#415011yescharacteristic
350C2—41no—
351C2—81no—
352C2—401no—
353C2—401no—
354C2#6616711yescharacteristic
355C2#6616911yescharacteristic
356C2#6617111yescharacteristic
subgroupG/H
centerC22C22×C22⋊D5
commutator, cosocleC2×C4×C5C24
FrattiniC2×C4C23×D5
FittingC5×#542C2
socleC22×C5C22×D4
hypercenter#542D5
hypocenterC5#3257
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ #117 ◁ #542

Complements

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

NH ≅ G/Ntype
1. C5×#542C2semidirect
2. #65869C2semidirect
3. #65376C2semidirect
4. #65373C2semidirect
5. #65346C2semidirect
6. #65346C2semidirect
7. #65338C2semidirect
8. #65337C2semidirect
9. #65328C2semidirect
10. #65327C2semidirect
11. #65316C2semidirect
12. #65311C2semidirect
13. #65304C2semidirect
14. #65298C2semidirect
15. #65288C2semidirect
109. #542D5semidirect
324. C5#3257semidirect

Maximal subgroups

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

subgroupindex
C5×#5422
#658692
#653762
#653732
#653462
#653382
#653372
#653282
#653272
#653162
#653112
#653042
#652982
#652882
#32575