Finite Group Databasebeta

← #10618923

Subgroups — #10618923

Order 640.

1012 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#557C211yesmaximal, characteristic, complemented
2#66067C213yesmaximal, complemented
3#66064C213yesmaximal, complemented
4#66031C213yesmaximal, complemented
5#66029C213yesmaximal, complemented
6#66028C213yesmaximal, complemented
7#66025C213yesmaximal, complemented
8#65918C213yesmaximal, complemented
9#65914C213yesmaximal, complemented
10#65910C211yesmaximal, characteristic, complemented
11#65910C211yesmaximal, characteristic, complemented
12Q8×D20C211yesmaximal, characteristic, complemented
13#65908C211yesmaximal, characteristic, complemented
14#65906C211yesmaximal, characteristic, complemented
15C4×Q8×D5C211yesmaximal, characteristic, complemented
16C2×#713C2211yescharacteristic
17C2×Q8×D5C2211yescharacteristic
18C2×Q8×D5C2211yescharacteristic
19C5×C4⋊Q8C2213yes—
20C5×#125C2213yes—
21C5×#125C2213yes—
22C4×C5×Q8C2211yescharacteristic
23C4×C5×Q8C2211yescharacteristic
24C4×C5×Q8C2211yescharacteristic
25C4×C5×Q8C2213yes—
26#4093C2213yes—
27#4092C2213yes—
28#4092C2216yes—
29Q8×Dic5C2211yescharacteristic
30#4044C2216yes—
31#4044C2213yes—
32#4044C2213yes—
33#4044C2216yes—
34#4043C2213yes—
35#4043C2213yes—
36#4043C2213yes—
37#4043C2213yes—
38#4041C2213yes—
39#4041C2213yes—
40#4041C2213yes—
41#4040C2216yes—
42#4040C2211yescharacteristic
43#4040C2213yes—
44#4039C2216yes—
45#4039C2213yes—
46#4038C2213yes—
47#4038C2216yes—
48D5×C4⋊C4C2213yes—
49D5×C4⋊C4C2213yes—
50#4036C2216yes—
51#4036C2213yes—
52#4033C2213yes—
53#4033C2213yes—
54#4021C2213yes—
55#4021C2213yes—
56C4×D20C2213yes—
57C4×D20C2213yes—
58C4×D20C2213yes—
59C4×D20C2213yes—
60#4018C2213yes—
61#4018C2213yes—
62C42×D5C2213yes—
63#4015C2213yes—
64C4×Dic10C2213yes—
65C4×Dic10C2213yes—
66#3516—51nomaximal, Sylow
67C2×C5×Q8C2313yes—
68C2×C5×Q8C2311yescharacteristic
69#713—24no—
70Q8×D5—24no—
71Q8×D5—24no—
72C2×D20C2313yes—
73C2×D20C2311yescharacteristic
74C2×D20C2313yes—
75C2×D20C2313yes—
76C2×C4×D5C2313yes—
77C2×C4×D5C2313yes—
78C2×C4×D5C2313yes—
79C2×C4×D5C2311yescharacteristic
80C2×C4×D5C2313yes—
81C2×C4×D5C2316yes—
82C2×Dic10C2313yes—
83C2×Dic10C2313yes—
84C5×C4⋊C4C2311yescharacteristic
85C5×C4⋊C4C2313yes—
86C5×C4⋊C4C2313yes—
87C5×C4⋊C4C2313yes—
88C5×C4⋊C4C2313yes—
89C5×C4⋊C4C2313yes—
90C5×C4⋊C4C2316yes—
91C42×C5C2313yes—
92C42×C5C2313yes—
93C42×C5C2313yes—
94#685C2311yescharacteristic
95#685C2313yes—
96#685C2313yes—
97#685C2313yes—
98#685C2311yescharacteristic
99#685C2311yescharacteristic
100#685C2313yes—
101#685C2313yes—
102#685C2313yes—
103#685C2313yes—
104#685C2316yes—
105#685C2316yes—
106C4⋊Dic5C2313yes—
107C4⋊Dic5C2313yes—
108C4⋊Dic5C2316yes—
109C4⋊Dic5C2313yes—
110C4⋊Dic5C2313yes—
111#683C2313yes—
112#683C2316yes—
113#683C2311yescharacteristic
114#683C2313yes—
115#683C2313yes—
116C4×Dic5C2313yes—
117C4×Dic5C2316yes—
118C4×Dic5C2313yes—
119C4×Dic5C2313yes—
120#557D511yescharacteristic, complemented
121#556—53no—
122#554—53no—
123#553—53no—
124#552—53no—
125D4×Q8—51no—
126#548—51no—
127#548—51no—
128#542—53no—
129#541—53no—
130#529—53no—
131#529—53no—
132#520—51no—
133#519—51no—
134C2×C4×Q8—51no—
135C22×D5C2411yescharacteristic
136C22×D5C2413yes—
137C22×D5C2411yescharacteristic
138C5×Q8—43no—
139C5×Q8#4114yes—
140C2×C4×C5C2411yescharacteristic
141C2×C4×C5C2413yes—
142C2×C4×C5C2413yes—
143C2×C4×C5C2411yescharacteristic
144C2×C4×C5C2411yescharacteristic
145C2×C4×C5C2413yes—
146C2×C4×C5C2413yes—
147C2×Dic5C2411yescharacteristic
148C2×Dic5C2413yes—
149C2×Dic5C2413yes—
150C2×Dic5C2413yes—
151C2×Dic5C2411yescharacteristic
152D20—22no—
153D20—26no—
154D20—26no—
155D20—43no—
156C4×D5—22no—
157C4×D5—26no—
158C4×D5—26no—
159C4×D5—26no—
160C4×D5—46no—
161C4×D5—43no—
162Dic10—26no—
163Dic10—26no—
164C2×#41—51no—
165C22×Q8—51no—
166C22×Q8—51no—
167C4⋊Q8—53no—
168C4⋊Q8C2×D513yes—
169#126—56no—
170#126—53no—
171#126—53no—
172#126—56no—
173#125—53no—
174#125—56no—
175#125C2×D513yes—
176#125C2×D513yes—
177#124—53no—
178#124—53no—
179#124—53no—
180#123—51no—
181#123—53no—
182#123—56no—
183C22⋊Q8—53no—
184C22⋊Q8—56no—
185C22⋊Q8—53no—
186C22⋊Q8—53no—
187C22⋊Q8—53no—
188C22⋊Q8—53no—
189#121—53no—
190#121—53no—
191#121—53no—
192C4×Q8—51no—
193C4×Q8—53no—
194C4×Q8—53no—
195C4×Q8—53no—
196C4×Q8—53no—
197C4×Q8C2×D511yescharacteristic
198C4×Q8C2×D511yescharacteristic
199C4×Q8C2×D511yescharacteristic
200C4×Q8C2×D513yes—
201C4×D4—53no—
202C4×D4—56no—
203C4×D4—53no—
204C4×D4—53no—
205C4×D4—53no—
206C4×D4—53no—
207#117—53no—
208#117—56no—
209#117—53no—
210#117—53no—
211C2×C4⋊C4—53no—
212C2×C4⋊C4—53no—
213C2×C42—53no—
214C22×C5C2511yescharacteristic
215C2×D5—21no—
216C2×D5—21no—
217C2×D5—21no—
218C2×D5—23no—
219C2×D5—21no—
220C2×D5—23no—
221C2×D5—23no—
222C2×D5—21no—
223C2×D52−1+412yes—
224C4×C5—21no—
225C4×C5—23no—
226C4×C5—21no—
227C4×C5—23no—
228C4×C5—23no—
229C4×C5—21no—
230C4×C5C2×#4116yes—
231Dic5—23no—
232Dic5—23no—
233Dic5—21no—
234Dic5—21no—
235Dic5—23no—
236#41—104no—
237C2×Q8—53no—
238C2×Q8—53no—
239C2×Q8—104no—
240C2×Q8—104no—
241C2×Q8C22×D513yes—
242C2×Q8C22×D511yescharacteristic
243C2×D4—53no—
244C2×D4—51no—
245C2×D4—53no—
246C2×D4—53no—
247C22×C4—51no—
248C22×C4—53no—
249C22×C4—53no—
250C22×C4—53no—
251C22×C4—53no—
252C22×C4—56no—
253C4⋊C4—53no—
254C4⋊C4—53no—
255C4⋊C4—56no—
256C4⋊C4—51no—
257C4⋊C4—53no—
258C4⋊C4—53no—
259C4⋊C4—56no—
260C4⋊C4—53no—
261C4⋊C4—53no—
262C4⋊C4—53no—
263C4⋊C4C22×D513yes—
264C4⋊C4C22×D513yes—
265C4⋊C4C22×D513yes—
266C4⋊C4C22×D513yes—
267C4⋊C4C22×D516yes—
268C4⋊C4C22×D511yescharacteristic
269C4⋊C4C22×D513yes—
270C22⋊C4—53no—
271C22⋊C4—51no—
272C22⋊C4—53no—
273C22⋊C4—53no—
274C22⋊C4—53no—
275C22⋊C4—56no—
276C22⋊C4—53no—
277C22⋊C4—51no—
278C22⋊C4—51no—
279C22⋊C4—56no—
280C22⋊C4—53no—
281C22⋊C4—53no—
282C42—53no—
283C42—53no—
284C42—53no—
285C42—56no—
286C42C22×D513yes—
287C42C22×D513yes—
288C42C22×D513yes—
289C2×C5C22×#4111yescharacteristic
290C2×C5C2×2−1+411yescharacteristic
291C2×C5#58511yescharacteristic
292D5—22no—
293D5—41no—
294D5—43no—
295C23—51no—
296C23—51no—
297C23—53no—
298Q8—43no—
299Q8—106no—
300Q8—106no—
301Q8#71314yes—
302D4—106no—
303D4—102no—
304D4—106no—
305D4—203no—
306C2×C4—51no—
307C2×C4—53no—
308C2×C4—53no—
309C2×C4—53no—
310C2×C4—51no—
311C2×C4—102no—
312C2×C4—106no—
313C2×C4—106no—
314C2×C4—106no—
315C2×C4—206no—
316C2×C4—203no—
317C2×C4C23×D511yescharacteristic
318C2×C4C23×D513yes—
319C2×C4C23×D513yes—
320C2×C4C23×D511yescharacteristic
321C2×C4C23×D511yescharacteristic
322C2×C4C23×D513yes—
323C2×C4C23×D513yes—
324C5#351611yescharacteristic, complemented, Sylow
325C22—52no—
326C22—101no—
327C22—101no—
328C22—101no—
329C22—103no—
330C22—103no—
331C22—101no—
332C22—101no—
333C22—103no—
334C22C24×D511yescharacteristic
335C4—23no—
336C4—23no—
337C4—21no—
338C4—21no—
339C4—23no—
340C4—21no—
341C4—101no—
342C4—103no—
343C4—103no—
344C4—103no—
345C4—101no—
346C4C2×#71316yes—
347C2—102no—
348C2—201no—
349C2—203no—
350C2C22×#71311yescharacteristic
351C2#6628311yescharacteristic
352C2D5×2−1+411yescharacteristic
subgroupG/H
center, FrattiniC22C24×D5
commutator, socle, cosocleC22×C5C25
FittingC5×#557C2
hypercenter#557D5
hypocenterC5#3516
LCSG ▷ C22×C5 ▷ C5
UCSC1 ◁ C22 ◁ #557

Complements

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

NH ≅ G/Ntype
1. C5×#557C2semidirect
2. #66067C2semidirect
3. #66064C2semidirect
4. #66031C2semidirect
5. #66029C2semidirect
6. #66028C2semidirect
7. #66025C2semidirect
8. #65918C2semidirect
9. #65914C2semidirect
10. #65910C2semidirect
11. #65910C2semidirect
12. Q8×D20C2semidirect
13. #65908C2semidirect
14. #65906C2semidirect
15. C4×Q8×D5C2semidirect
120. #557D5semidirect
324. C5#3516semidirect

Maximal subgroups

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

subgroupindex
C5×#5572
#660672
#660642
#660312
#660292
#660282
#660252
#659182
#659142
#659102
Q8×D202
#659082
#659062
C4×Q8×D52
#35165