Finite Group Databasebeta

← #11995992

Subgroups — #11995992

Order 1280.

490 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×#2961C211yesmaximal, characteristic, complemented
2#10615086C211yesmaximal, characteristic
3#10602631C211yesmaximal, characteristic, complemented
4#10601295C211yesmaximal, characteristic
5#10600616C211yesmaximal, characteristic
6#10598204C211yesmaximal, characteristic, complemented
7#10598151C211yesmaximal, characteristic, complemented
8C5×#533C2211yescharacteristic
9C2×#4020C2211yescharacteristic
10C5×#446—22no—
11C5×#446—22no—
12C5×#424—21no—
13C5×#423C2211yescharacteristic
14C5×#405—21no—
15C2×C5×#97C2211yescharacteristic
16C2×#3953—21no—
17C2×#3953—21no—
18#65377C412yescomplemented
19#65374C412yescomplemented
20#65335—22no—
21#65316—22no—
22#65225C2211yescharacteristic
23#65214C2211yescharacteristic
24#65214C2211yescharacteristic
25#64758—22no—
26#64705—22no—
27#9294—51nomaximal, Sylow
28C22×D20—21no—
29C22×D20—21no—
30C22×D20—21no—
31C2×C5×M16—21no—
32C22×C5×C8D411yescharacteristic
33C22×C5×C8D411yescharacteristic
34C5×C4⋊Q8C2×C411yescharacteristic
35C5×C4⋊Q8C2×C411yescharacteristic
36C5×#125—21no—
37C5×C22⋊Q8—21no—
38C5×C22⋊Q8D412yes—
39C4×C5×Q8—21no—
40C4×C5×Q8D412yes—
41C5×#117—21no—
42C2×C42×C5C2311yescharacteristic
43C2×#688—24no—
44C2×C4⋊Dic5—21no—
45#4020—21no—
46#4020—21no—
47#4020—21no—
48#4020C2×C412yes—
49C5×C4⋊C8—24no—
50C5×C4⋊C8—21no—
51C5×C4⋊C8D412yes—
52C5×#97—22no—
53C5×#97—21no—
54C5×#97—21no—
55C4×C5×C8—22no—
56#3963—21no—
57#3963—21no—
58C22⋊#168—21no—
59C22⋊#168—21no—
60#3953—42no—
61#3953—42no—
62#3941—24no—
63C4⋊#168—21no—
64C4⋊#168—21no—
65C4×#168—21no—
66C4×#168C2×C412yes—
67#3208—51no—
68#2961D511yescharacteristic, complemented
69#1961—51no—
70#1545—51no—
71#1518—51no—
72#1286—51no—
73#1275—51no—
74C23×D5—41no—
75C2×C5×Q8—21no—
76C2×C5×Q8C22⋊C412yes—
77C22×C4×C5C2×D411yescharacteristic
78C22×C4×C5C2×D411yescharacteristic
79C22×C4×C5#4111yescharacteristic
80C22×Dic5—41no—
81C2×D20—22no—
82C2×D20—22no—
83C2×D20—22no—
84C2×D20—41no—
85C2×D20—41no—
86C2×D20—41no—
87C2×D20—41no—
88C2×D20—41no—
89C2×D20—41no—
90C2×D20—41no—
91C2×D20—82no—
92C2×D20—82no—
93C5×M16—41no—
94C5×M16—41no—
95C2×C5×C8—21no—
96C2×C5×C8—21no—
97C2×C5×C8—21no—
98C2×C5×C8—22no—
99C2×C5×C8—21no—
100C2×C5×C8—21no—
101C2×C5×C8SD1612yescomplemented
102C5×C4⋊C4—21no—
103C5×C4⋊C4—21no—
104C5×C4⋊C4—21no—
105C5×C4⋊C4—22no—
106C5×C4⋊C4—22no—
107C5×C4⋊C4C22⋊C412yes—
108C5×C22⋊C4—22no—
109C42×C5—22no—
110C42×C5C22×C411yescharacteristic
111C42×C5C2×D411yescharacteristic
112C42×C5C2×D411yescharacteristic
113C42×C5#4111yescharacteristic
114#688—84no—
115C4⋊Dic5—41no—
116C4⋊Dic5—41no—
117C2×#168—22no—
118C2×#168—41no—
119#533C2×D511yescharacteristic
120C2×#127—51no—
121#492—52no—
122#486—52no—
123#460—102no—
124#450—102no—
125#446—22no—
126#446—22no—
127#431—51no—
128#431—51no—
129#424—21no—
130#423C2×D511yescharacteristic
131C2×#102—101no—
132C2×#102—101no—
133#405—21no—
134C2×#97C2×D511yescharacteristic
135#382—51no—
136#331—102no—
137#328—102no—
138C23×C5#12111yescharacteristic
139C22×D5—42no—
140C22×D5—82no—
141C22×D5—82no—
142C22×D5—81no—
143C22×D5—81no—
144C5×Q8—42no—
145C5×Q8—81no—
146C2×C4×C5—22no—
147C2×C4×C5—22no—
148C2×C4×C5—41no—
149C2×C4×C5—41no—
150C2×C4×C5C2×C22⋊C411yescharacteristic
151C2×C4×C5C4×D411yescharacteristic
152C2×C4×C5C4×D411yescharacteristic
153C2×C4×C5#12011yescharacteristic
154C2×C4×C5#12111yescharacteristic
155C2×C4×C5#12311yescharacteristic
156C2×C4×C5C2×SD1611yescharacteristic
157C2×C4×C5#13511yescharacteristic
158C2×C4×C5AGL(1,ℤ/8)11yescharacteristic
159C2×C4×C5#13711yescharacteristic
160C2×Dic5—81no—
161C2×Dic5—81no—
162C2×Dic5—81no—
163D20—44no—
164D20—42no—
165D20—42no—
166D20—82no—
167D20—82no—
168D20—82no—
169D20—81no—
170D20—81no—
171C5×C8—22no—
172C5×C8—41no—
173C5×C8—42no—
174#168—42no—
175#168—81no—
176C22×D4—101no—
177C22×D4—101no—
178C22×D4—101no—
179C2×SD16—104no—
180C2×M16—21no—
181C22×C8C22⋊D511yescharacteristic
182C22×C8C22⋊D511yescharacteristic
183C4⋊Q8C4×D511yescharacteristic
184C4⋊Q8C4×D511yescharacteristic
185#127—52no—
186#127—101no—
187#127—101no—
188#127—101no—
189#125—21no—
190C22⋊Q8—21no—
191C22⋊Q8D2012yes—
192C4×Q8—21no—
193C4×Q8C22⋊D512yes—
194#117—21no—
195C2×C4⋊C4—101no—
196C2×C42C22×D511yescharacteristic
197C4⋊C8—24no—
198C4⋊C8—21no—
199C4⋊C8—101no—
200C4⋊C8—101no—
201C4⋊C8C22⋊D512yes—
202#102—104no—
203#102—202no—
204#102—202no—
205C22⋊C8—101no—
206C22⋊C8—101no—
207#97—22no—
208#97—21no—
209#97—21no—
210C4×C8—22no—
211C4×C8—52no—
212C4×C8—101no—
213#95—101no—
214#95—101no—
215C22×C5—21no—
216C22×C5—21no—
217C22×C5#38611yescharacteristic
218C22×C5#46011yescharacteristic
219C22×C5#46411yescharacteristic
220C2×D5—82no—
221C2×D5—82no—
222C2×D5—82no—
223C2×D5—162no—
224C2×D5—162no—
225C4×C5—21no—
226C4×C5—21no—
227C4×C5—21no—
228C4×C5—21no—
229C4×C5—42no—
230C4×C5—42no—
231C4×C5C4×SD1611yescharacteristic
232C4×C5#44011yescharacteristic
233C4×C5#44811yescharacteristic
234C4×C5#44911yescharacteristic
235Dic5—161no—
236C24—201no—
237C2×Q8—21no—
238C2×Q8#68512yes—
239C2×D4—102no—
240C2×D4—102no—
241C2×D4—102no—
242C2×D4—201no—
243C2×D4—201no—
244C2×D4—201no—
245C2×D4—201no—
246C2×D4—201no—
247C2×D4—201no—
248C2×D4—201no—
249C2×D4—402no—
250C2×D4—402no—
251C22×C4—201no—
252C22×C4C2×D2011yescharacteristic
253C22×C4#70911yescharacteristic
254C22×C4C2×C22⋊D511yescharacteristic
255SD16—404no—
256M16—41no—
257M16—41no—
258C2×C8—21no—
259C2×C8—21no—
260C2×C8—21no—
261C2×C8—22no—
262C2×C8—21no—
263C2×C8—21no—
264C2×C8—102no—
265C2×C8—201no—
266C2×C8#68812yescomplemented
267C4⋊C4—21no—
268C4⋊C4—21no—
269C4⋊C4—21no—
270C4⋊C4—22no—
271C4⋊C4—22no—
272C4⋊C4—201no—
273C4⋊C4—201no—
274C4⋊C4#68512yes—
275C22⋊C4—22no—
276C42—22no—
277C42C2×C4×D511yescharacteristic
278C42#70911yescharacteristic
279C42C2×C22⋊D511yescharacteristic
280C42C2×C22⋊D511yescharacteristic
281C2×C5—21no—
282C2×C5—21no—
283C2×C5#188011yescharacteristic
284C2×C5#190311yescharacteristic
285C2×C5#191011yescharacteristic
286D5—162no—
287C23—202no—
288C23—402no—
289C23—402no—
290C23—401no—
291C23—401no—
292C23#407611yescharacteristic
293Q8—42no—
294Q8—81no—
295D4—204no—
296D4—202no—
297D4—202no—
298D4—402no—
299D4—402no—
300D4—402no—
301D4—401no—
302D4—401no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—401no—
308C2×C4—401no—
309C2×C4—401no—
310C2×C4C2×#68511yescharacteristic
311C2×C4C4×C22⋊D511yescharacteristic
312C2×C4C4×C22⋊D511yescharacteristic
313C2×C4#407511yescharacteristic
314C2×C4#407611yescharacteristic
315C2×C4C2×#68811yescharacteristic
316C2×C4#408811yescharacteristic
317C2×C4#409511yescharacteristic
318C2×C4#409611yescharacteristic
319C2×C4#409911yescharacteristic
320C8—22no—
321C8—41no—
322C8—42no—
323C8—202no—
324C8—401no—
325C5#929411yescharacteristic, complemented, Sylow
326C22—21no—
327C22—21no—
328C22—402no—
329C22—402no—
330C22—402no—
331C22—802no—
332C22—802no—
333C22#6531611yescharacteristic
334C22#6531711yescharacteristic
335C22#6550211yescharacteristic
336C4—21no—
337C4—21no—
338C4—21no—
339C4—21no—
340C4—42no—
341C4—42no—
342C4—801no—
343C4C4×#68811yescharacteristic
344C4#6531511yescharacteristic
345C4#6551611yescharacteristic
346C4#6552711yescharacteristic
347C2—21no—
348C2—21no—
349C2—802no—
350C2#1060242611yescharacteristic
351C2#1060281611yescharacteristic
352C2#1060281811yescharacteristic
subgroupG/H
centerC22#65502
commutatorC42×C5C22×C4
FrattiniC2×C42C22×D5
FittingC5×#2961C2
socleC22×C5#386
prime radicalC2×#4020C22
cosocleC2×C42×C5C23
hypercenter#2961D5
hypocenterC5#9294
LCSG ▷ C42×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2961

Complements

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

NH ≅ G/Ntype
1. C5×#2961C2semidirect
3. #10602631C2semidirect
6. #10598204C2semidirect
7. #10598151C2semidirect
18. #65377C4semidirect
19. #65374C4semidirect
68. #2961D5semidirect
101. C2×C5×C8SD16semidirect
266. C2×C8#688semidirect
325. C5#9294semidirect

Maximal subgroups

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

subgroupindex
C5×#29612
#106150862
#106026312
#106012952
#106006162
#105982042
#105981512
#92945