Finite Group Databasebeta

← #10617780

Subgroups — #10617780

Order 640.

1030 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#519C211yesmaximal, characteristic, complemented
2#66151C211yesmaximal, characteristic, complemented
3#66151C211yesmaximal, characteristic, complemented
4#65919C213yesmaximal, complemented
5#65916C213yesmaximal, complemented
6#65915C213yesmaximal, complemented
7#65914C213yesmaximal, complemented
8#65911C211yesmaximal, characteristic, complemented
9#65908C211yesmaximal, characteristic, complemented
10C4×#713C211yesmaximal, characteristic, complemented
11#65903C211yesmaximal, characteristic, complemented
12#65869C213yesmaximal, complemented
13#65866C213yesmaximal, complemented
14#65864C213yesmaximal, complemented
15#65862C213yesmaximal, complemented
16C22×C5×Q8C2211yescharacteristic
17C2×#713C2211yescharacteristic
18C2×#713C2211yescharacteristic
19C4×C5×Q8C2211yescharacteristic
20C4×C5×Q8C2211yescharacteristic
21C4×C5×Q8C2213yes—
22C4×C5×Q8C2213yes—
23C5×#117C2213yes—
24C5×#117C2213yes—
25#4093C2213yes—
26#4093C2213yes—
27#4092C2213yes—
28#4092C2213yes—
29Q8×Dic5C2211yescharacteristic
30Q8×Dic5C2211yescharacteristic
31#4076C2213yes—
32#4076C2213yes—
33C4×C22⋊D5C2213yes—
34C4×C22⋊D5C2213yes—
35#4044C2213yes—
36#4042C2213yes—
37#4041C2213yes—
38#4041C2213yes—
39#4040C2213yes—
40#4040C2216yes—
41#4040C2216yes—
42#4039C2213yes—
43#4039C2213yes—
44#4038C2213yes—
45D5×C4⋊C4C2213yes—
46#4036C2213yes—
47#4035C2213yes—
48#4034C2213yes—
49#4031C2213yes—
50#4030C2213yes—
51#4030C2216yes—
52#4029C2213yes—
53#4025C2211yescharacteristic
54#4022C2216yes—
55#4021C2216yes—
56#4020C2213yes—
57C4×D20C2213yes—
58C4×D20C2216yes—
59C4×D20C2213yes—
60C4×D20C2213yes—
61#4018C2213yes—
62#4018C2216yes—
63C42×D5C2213yes—
64#4016C2213yes—
65C4×Dic10C2213yes—
66#3524—51nomaximal, Sylow
67C2×C5×Q8—24no—
68C2×C5×Q8C2313yes—
69C2×C5×Q8C2311yescharacteristic
70C22×C4×C5C2313yes—
71C2×C22⋊D5C2313yes—
72C2×C22⋊D5C2313yes—
73#713—24no—
74#713—24no—
75C2×D20C2313yes—
76C2×D20C2313yes—
77C2×D20C2313yes—
78C2×D20C2316yes—
79C2×C4×D5C2313yes—
80C2×C4×D5C2313yes—
81C2×C4×D5C2313yes—
82C2×C4×D5C2313yes—
83C2×C4×D5C2316yes—
84C2×Dic10C2313yes—
85C5×C4⋊C4C2313yes—
86C5×C4⋊C4C2313yes—
87C5×C4⋊C4C2313yes—
88C5×C4⋊C4C2313yes—
89C5×C22⋊C4C2311yescharacteristic
90C5×C22⋊C4C2313yes—
91C42×C5C2313yes—
92C42×C5C2313yes—
93C42×C5C2316yes—
94C22⋊Dic5C2311yescharacteristic
95C22⋊Dic5C2311yescharacteristic
96#685C2313yes—
97#685C2313yes—
98#685C2313yes—
99#685C2316yes—
100#685C2313yes—
101#685C2316yes—
102#685C2313yes—
103#685C2316yes—
104#685C2313yes—
105C4⋊Dic5C2313yes—
106C4⋊Dic5C2313yes—
107C4⋊Dic5C2313yes—
108C4⋊Dic5C2311yescharacteristic
109#683C2311yescharacteristic
110#683C2313yes—
111#683C2316yes—
112#683C2311yescharacteristic
113#683C2313yes—
114#683C2313yes—
115#683C2313yes—
116C4×Dic5C2313yes—
117C4×Dic5C2313yes—
118C4×Dic5C2311yescharacteristic
119C4×Dic5C2313yes—
120#559—53no—
121#557—51no—
122#555—53no—
123#553—53no—
124#552—53no—
125#550—51no—
126#548—51no—
127#548—51no—
128#542—53no—
129#540—53no—
130#529—53no—
131#529—53no—
132#520—51no—
133#519D511yescharacteristic, complemented
134C4×#41—51no—
135C23×C5C2411yescharacteristic
136C22×D5C2413yes—
137C22×D5C2413yes—
138C5×Q8—26no—
139C5×Q8#4114yes—
140C2×C4×C5—26no—
141C2×C4×C5C2411yescharacteristic
142C2×C4×C5C2413yes—
143C2×C4×C5C2411yescharacteristic
144C2×C4×C5C2413yes—
145C2×C4×C5C2413yes—
146C2×C4×C5C2413yes—
147C22⋊D5—43no—
148C22⋊D5—43no—
149C2×Dic5C2411yescharacteristic
150C2×Dic5C2411yescharacteristic
151C2×Dic5C2411yescharacteristic
152C2×Dic5C2413yes—
153C2×Dic5C2413yes—
154C2×Dic5C2411yescharacteristic
155D20—26no—
156D20—26no—
157D20—46no—
158D20—43no—
159C4×D5—26no—
160C4×D5—26no—
161C4×D5—43no—
162C4×D5—43no—
163C4×D5—46no—
164Dic10—43no—
165C2×#41—51no—
166C2×#41—51no—
167C22×Q8C2×D511yescharacteristic
168C4⋊Q8—53no—
169#127—53no—
170#126—53no—
171#126—51no—
172#126—56no—
173#125—53no—
174#125—53no—
175#125—53no—
176#124—53no—
177#124—53no—
178#124—53no—
179#124—56no—
180#123—56no—
181#123—53no—
182#123—56no—
183#123—53no—
184C22⋊Q8—53no—
185C22⋊Q8—53no—
186C22⋊Q8—53no—
187#121—53no—
188#121—53no—
189#121—53no—
190#121—53no—
191#121—53no—
192#121—56no—
193C4×Q8—51no—
194C4×Q8—51no—
195C4×Q8—53no—
196C4×Q8C2×D511yescharacteristic
197C4×Q8C2×D513yes—
198C4×Q8C2×D511yescharacteristic
199C4×Q8C2×D513yes—
200C4×D4—53no—
201C4×D4—53no—
202C4×D4—53no—
203C4×D4—53no—
204C4×D4—53no—
205C4×D4—53no—
206C4×D4—53no—
207C4×D4—56no—
208#117—53no—
209#117—53no—
210#117—56no—
211#117C2×D513yes—
212#117C2×D513yes—
213C2×C4⋊C4—53no—
214C2×C42—53no—
215C22×C5—21no—
216C22×C5—21no—
217C22×C5—21no—
218C22×C5C2511yescharacteristic
219C2×D5—23no—
220C2×D5—23no—
221C2×D5—23no—
222C2×D5—23no—
223C2×D5—23no—
224C2×D5—23no—
225C4×C5—23no—
226C4×C5—23no—
227C4×C5—21no—
228C4×C5—23no—
229C4×C5—21no—
230C4×C5C2×#4116yes—
231Dic5—21no—
232Dic5—21no—
233Dic5—21no—
234Dic5—23no—
235Dic5—21no—
236Dic5—23no—
237#41—104no—
238#41—104no—
239C2×Q8—24no—
240C2×Q8—53no—
241C2×Q8C22×D513yes—
242C2×Q8C22×D511yescharacteristic
243C2×D4—53no—
244C2×D4—53no—
245C2×D4—53no—
246C2×D4—53no—
247C2×D4—56no—
248C2×D4—53no—
249C22×C4—53no—
250C22×C4—53no—
251C22×C4—53no—
252C22×C4—53no—
253C22×C4—56no—
254C22×C4C22×D513yes—
255C4⋊C4—53no—
256C4⋊C4—53no—
257C4⋊C4—51no—
258C4⋊C4—53no—
259C4⋊C4—53no—
260C4⋊C4—56no—
261C4⋊C4—51no—
262C4⋊C4—53no—
263C4⋊C4—53no—
264C4⋊C4—51no—
265C4⋊C4—53no—
266C4⋊C4C22×D513yes—
267C4⋊C4C22×D513yes—
268C4⋊C4C22×D513yes—
269C4⋊C4C22×D513yes—
270C22⋊C4—51no—
271C22⋊C4—51no—
272C22⋊C4—53no—
273C22⋊C4—53no—
274C22⋊C4—56no—
275C22⋊C4—53no—
276C22⋊C4—56no—
277C22⋊C4—53no—
278C22⋊C4—53no—
279C22⋊C4—56no—
280C22⋊C4—53no—
281C22⋊C4C22×D511yescharacteristic
282C22⋊C4C22×D513yes—
283C42—51no—
284C42—53no—
285C42—53no—
286C42—53no—
287C42C22×D513yes—
288C42C22×D516yes—
289C42C22×D513yes—
290C2×C5—41no—
291C2×C5C22×#4111yescharacteristic
292C2×C5#58511yescharacteristic
293C2×C5#58511yescharacteristic
294D5—43no—
295D5—43no—
296C23—53no—
297C23—53no—
298C23C23×D511yescharacteristic
299Q8—26no—
300Q8—203no—
301Q8#70914yes—
302D4—106no—
303D4—106no—
304D4—203no—
305D4—206no—
306D4—203no—
307D4—203no—
308C2×C4—26no—
309C2×C4—51no—
310C2×C4—51no—
311C2×C4—51no—
312C2×C4—53no—
313C2×C4—53no—
314C2×C4—51no—
315C2×C4—106no—
316C2×C4—106no—
317C2×C4—203no—
318C2×C4—206no—
319C2×C4—203no—
320C2×C4C23×D511yescharacteristic
321C2×C4C23×D513yes—
322C2×C4C23×D513yes—
323C2×C4C23×D511yescharacteristic
324C2×C4C23×D513yes—
325C2×C4C23×D513yes—
326C5#352411yescharacteristic, complemented, Sylow
327C22—21no—
328C22—21no—
329C22—21no—
330C22—103no—
331C22—103no—
332C22—103no—
333C22—103no—
334C22—103no—
335C22—103no—
336C22C24×D511yescharacteristic
337C4—23no—
338C4—23no—
339C4—21no—
340C4—21no—
341C4—23no—
342C4—101no—
343C4—101no—
344C4—101no—
345C4—103no—
346C4—103no—
347C4—101no—
348C4C2×#70916yes—
349C2—41no—
350C2—203no—
351C2—203no—
352C2C22×#70911yescharacteristic
353C2#6628711yescharacteristic
354C2#6628711yescharacteristic
subgroupG/H
center, FrattiniC22C24×D5
commutator, socle, cosocleC22×C5C25
FittingC5×#519C2
hypercenter#519D5
hypocenterC5#3524
LCSG ▷ C22×C5 ▷ C5
UCSC1 ◁ C22 ◁ #519

Complements

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

NH ≅ G/Ntype
1. C5×#519C2semidirect
2. #66151C2semidirect
3. #66151C2semidirect
4. #65919C2semidirect
5. #65916C2semidirect
6. #65915C2semidirect
7. #65914C2semidirect
8. #65911C2semidirect
9. #65908C2semidirect
10. C4×#713C2semidirect
11. #65903C2semidirect
12. #65869C2semidirect
13. #65866C2semidirect
14. #65864C2semidirect
15. #65862C2semidirect
133. #519D5semidirect
326. C5#3524semidirect

Maximal subgroups

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

subgroupindex
C5×#5192
#661512
#659192
#659162
#659152
#659142
#659112
#659082
C4×#7132
#659032
#658692
#658662
#658642
#658622
#35245