Finite Group Databasebeta

← #10617779

Subgroups — #10617779

Order 640.

1010 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×#519C211yesmaximal, characteristic, complemented
2#66151C211yesmaximal, characteristic, complemented
3Q8×C22⋊D5C211yesmaximal, characteristic, complemented
4#65918C213yesmaximal, complemented
5#65916C213yesmaximal, complemented
6#65912C213yesmaximal, complemented
7#65910C211yesmaximal, characteristic, complemented
8C4×#713C211yesmaximal, characteristic, complemented
9#65906C211yesmaximal, characteristic, complemented
10#65902C213yesmaximal, complemented
11Q8×Dic10C211yesmaximal, characteristic, complemented
12#65868C213yesmaximal, complemented
13#65867C213yesmaximal, complemented
14#65866C213yesmaximal, complemented
15#65862C213yesmaximal, complemented
16C22×C5×Q8C2211yescharacteristic
17C2×#713C2211yescharacteristic
18C2×Q8×D5C2211yescharacteristic
19C4×C5×Q8C2211yescharacteristic
20C4×C5×Q8C2213yes—
21C4×C5×Q8C2211yescharacteristic
22C4×C5×Q8C2213yes—
23C5×#117C2213yes—
24C5×#117C2213yes—
25#4093C2213yes—
26#4092C2213yes—
27#4092C2213yes—
28Q8×Dic5C2211yescharacteristic
29Q8×Dic5C2211yescharacteristic
30#4090C2213yes—
31#4076C2213yes—
32C4×C22⋊D5C2213yes—
33C4×C22⋊D5C2213yes—
34#4070C2213yes—
35#4044C2213yes—
36#4044C2216yes—
37#4043C2213yes—
38#4042C2213yes—
39#4042C2216yes—
40#4041C2213yes—
41#4040C2213yes—
42#4039C2213yes—
43#4038C2213yes—
44#4038C2213yes—
45#4035C2213yes—
46#4034C2213yes—
47#4034C2213yes—
48#4033C2213yes—
49#4031C2211yescharacteristic
50#4031C2216yes—
51#4030C2213yes—
52#4029C2213yes—
53#4025C2213yes—
54#4022C2216yes—
55#4021C2216yes—
56#4021C2213yes—
57C4×D20C2213yes—
58C4×D20C2213yes—
59#4018C2213yes—
60#4018C2216yes—
61C42×D5C2213yes—
62#4015C2213yes—
63C4×Dic10C2213yes—
64C4×Dic10C2216yes—
65C4×Dic10C2213yes—
66#3518—51nomaximal, Sylow
67C2×C5×Q8—24no—
68C2×C5×Q8C2313yes—
69C2×C5×Q8C2311yescharacteristic
70C22×C4×C5C2313yes—
71C2×C22⋊D5C2311yescharacteristic
72C2×C22⋊D5C2313yes—
73#713—24no—
74Q8×D5—24no—
75C2×D20C2313yes—
76C2×D20C2313yes—
77C2×C4×D5C2313yes—
78C2×C4×D5C2313yes—
79C2×C4×D5C2313yes—
80C2×C4×D5C2313yes—
81C2×Dic10C2313yes—
82C2×Dic10C2311yescharacteristic
83C2×Dic10C2316yes—
84C5×C4⋊C4C2313yes—
85C5×C4⋊C4C2313yes—
86C5×C4⋊C4C2313yes—
87C5×C4⋊C4C2313yes—
88C5×C22⋊C4C2313yes—
89C5×C22⋊C4C2311yescharacteristic
90C42×C5C2313yes—
91C42×C5C2313yes—
92C42×C5C2316yes—
93C22⋊Dic5C2311yescharacteristic
94C22⋊Dic5C2313yes—
95#685C2311yescharacteristic
96#685C2313yes—
97#685C2316yes—
98#685C2316yes—
99#685C2311yescharacteristic
100#685C2313yes—
101#685C2316yes—
102#685C2313yes—
103#685C2313yes—
104C4⋊Dic5C2313yes—
105C4⋊Dic5C2313yes—
106C4⋊Dic5C2313yes—
107C4⋊Dic5C2313yes—
108#683C2313yes—
109#683C2313yes—
110#683C2313yes—
111#683C2313yes—
112#683C2316yes—
113#683C2313yes—
114#683C2313yes—
115C4×Dic5C2313yes—
116C4×Dic5C2316yes—
117C4×Dic5C2313yes—
118C4×Dic5C2313yes—
119C4×Dic5C2311yescharacteristic
120Q82—51no—
121#556—53no—
122#556—53no—
123#556—53no—
124#555—53no—
125D4×Q8—51no—
126#548—51no—
127#548—51no—
128#542—53no—
129#542—53no—
130#533—53no—
131#529—53no—
132#519—51no—
133#519D511yescharacteristic, complemented
134C4×#41—51no—
135C23×C5C2411yescharacteristic
136C22×D5C2411yescharacteristic
137C22×D5C2413yes—
138C5×Q8—26no—
139C5×Q8#4114yes—
140C2×C4×C5—26no—
141C2×C4×C5C2413yes—
142C2×C4×C5C2411yescharacteristic
143C2×C4×C5C2413yes—
144C2×C4×C5C2413yes—
145C2×C4×C5C2411yescharacteristic
146C2×C4×C5C2413yes—
147C22⋊D5—22no—
148C22⋊D5—43no—
149C2×Dic5C2411yescharacteristic
150C2×Dic5C2411yescharacteristic
151C2×Dic5C2413yes—
152C2×Dic5C2413yes—
153C2×Dic5C2413yes—
154C2×Dic5C2411yescharacteristic
155D20—26no—
156D20—43no—
157C4×D5—26no—
158C4×D5—26no—
159C4×D5—26no—
160C4×D5—43no—
161Dic10—26no—
162Dic10—22no—
163Dic10—46no—
164C2×#41—51no—
165C22×Q8—51no—
166C22×Q8C2×D511yescharacteristic
167C4⋊Q8—53no—
168C4⋊Q8—53no—
169C4⋊Q8—53no—
170C4⋊Q8—53no—
171#126—53no—
172#126—56no—
173#126—53no—
174#126—56no—
175#125—53no—
176#124—53no—
177#124—51no—
178#124—56no—
179#124—53no—
180#124—56no—
181#123—53no—
182#123—53no—
183C22⋊Q8—53no—
184C22⋊Q8—53no—
185C22⋊Q8—53no—
186C22⋊Q8—53no—
187C22⋊Q8—56no—
188C22⋊Q8—53no—
189#121—53no—
190#121—53no—
191#121—53no—
192C4×Q8—51no—
193C4×Q8—51no—
194C4×Q8—53no—
195C4×Q8—53no—
196C4×Q8—53no—
197C4×Q8—56no—
198C4×Q8C2×D511yescharacteristic
199C4×Q8C2×D513yes—
200C4×Q8C2×D511yescharacteristic
201C4×Q8C2×D513yes—
202C4×D4—53no—
203C4×D4—53no—
204C4×D4—53no—
205C4×D4—53no—
206C4×D4—53no—
207#117—53no—
208#117—53no—
209#117—53no—
210#117—56no—
211#117C2×D513yes—
212#117C2×D513yes—
213C2×C42—53no—
214C22×C5—21no—
215C22×C5—21no—
216C22×C5—21no—
217C22×C5C2511yescharacteristic
218C2×D5—21no—
219C2×D5—23no—
220C2×D5—23no—
221C2×D5—21no—
222C2×D5—23no—
223C2×D5—21no—
224C4×C5—23no—
225C4×C5—21no—
226C4×C5—23no—
227C4×C5—23no—
228C4×C5—21no—
229C4×C5C2×#4116yes—
230Dic5—23no—
231Dic5—21no—
232Dic5—21no—
233Dic5—23no—
234Dic5—23no—
235Dic52−1+412yes—
236#41—104no—
237C2×Q8—24no—
238C2×Q8—51no—
239C2×Q8—53no—
240C2×Q8—56no—
241C2×Q8—104no—
242C2×Q8C22×D513yes—
243C2×Q8C22×D511yescharacteristic
244C2×D4—51no—
245C2×D4—53no—
246C2×D4—53no—
247C2×D4—53no—
248C22×C4—53no—
249C22×C4—53no—
250C22×C4—53no—
251C22×C4—53no—
252C22×C4C22×D513yes—
253C4⋊C4—53no—
254C4⋊C4—53no—
255C4⋊C4—53no—
256C4⋊C4—53no—
257C4⋊C4—53no—
258C4⋊C4—53no—
259C4⋊C4—53no—
260C4⋊C4—53no—
261C4⋊C4—56no—
262C4⋊C4—53no—
263C4⋊C4—53no—
264C4⋊C4C22×D513yes—
265C4⋊C4C22×D513yes—
266C4⋊C4C22×D513yes—
267C4⋊C4C22×D513yes—
268C22⋊C4—51no—
269C22⋊C4—53no—
270C22⋊C4—53no—
271C22⋊C4—51no—
272C22⋊C4—56no—
273C22⋊C4—56no—
274C22⋊C4—56no—
275C22⋊C4—51no—
276C22⋊C4—53no—
277C22⋊C4—53no—
278C22⋊C4—53no—
279C22⋊C4C22×D513yes—
280C22⋊C4C22×D511yescharacteristic
281C42—51no—
282C42—53no—
283C42—53no—
284C42—56no—
285C42—53no—
286C42C22×D513yes—
287C42C22×D516yes—
288C42C22×D513yes—
289C2×C5—41no—
290C2×C5C22×#4111yescharacteristic
291C2×C5C2×2−1+411yescharacteristic
292C2×C5#58511yescharacteristic
293D5—41no—
294D5—43no—
295C23—51no—
296C23—53no—
297C23C23×D511yescharacteristic
298Q8—26no—
299Q8—102no—
300Q8—106no—
301Q8—206no—
302Q8#70914yes—
303D4—102no—
304D4—106no—
305D4—203no—
306D4—203no—
307C2×C4—26no—
308C2×C4—51no—
309C2×C4—51no—
310C2×C4—51no—
311C2×C4—53no—
312C2×C4—53no—
313C2×C4—53no—
314C2×C4—106no—
315C2×C4—106no—
316C2×C4—106no—
317C2×C4—203no—
318C2×C4C23×D513yes—
319C2×C4C23×D511yescharacteristic
320C2×C4C23×D511yescharacteristic
321C2×C4C23×D513yes—
322C2×C4C23×D513yes—
323C2×C4C23×D513yes—
324C5#351811yescharacteristic, complemented, Sylow
325C22—21no—
326C22—21no—
327C22—21no—
328C22—101no—
329C22—103no—
330C22—103no—
331C22—101no—
332C22—103no—
333C22—101no—
334C22C24×D511yescharacteristic
335C4—23no—
336C4—23no—
337C4—21no—
338C4—21no—
339C4—23no—
340C4—52no—
341C4—101no—
342C4—103no—
343C4—101no—
344C4—103no—
345C4—103no—
346C4C2×#70916yes—
347C2—41no—
348C2—201no—
349C2—203no—
350C2C22×#70911yescharacteristic
351C2D5×2−1+411yescharacteristic
352C2#6628711yescharacteristic
subgroupG/H
center, FrattiniC22C24×D5
commutator, socle, cosocleC22×C5C25
FittingC5×#519C2
hypercenter#519D5
hypocenterC5#3518
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. Q8×C22⋊D5C2semidirect
4. #65918C2semidirect
5. #65916C2semidirect
6. #65912C2semidirect
7. #65910C2semidirect
8. C4×#713C2semidirect
9. #65906C2semidirect
10. #65902C2semidirect
11. Q8×Dic10C2semidirect
12. #65868C2semidirect
13. #65867C2semidirect
14. #65866C2semidirect
15. #65862C2semidirect
133. #519D5semidirect
324. C5#3518semidirect

Maximal subgroups

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

subgroupindex
C5×#5192
#661512
Q8×C22⋊D52
#659182
#659162
#659122
#659102
C4×#7132
#659062
#659022
Q8×Dic102
#658682
#658672
#658662
#658622
#35185