Finite Group Databasebeta

← #12016690

Subgroups — #12016690

Order 1280.

510 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#10615473C211yesmaximal, characteristic
2C22×#3944C211yesmaximal, characteristic, complemented
3C5×#1807C211yesmaximal, characteristic, complemented
4C2×#64708C212yesmaximal, complemented
5#10598212C212yesmaximal, complemented
6C23×C5×C8C2211yescharacteristic
7#66125—21no—
8C2×#4050C2211yescharacteristic
9C2×C5×#106C2212yes—
10C5×#380—21no—
11#65405C411yescharacteristic, complemented
12#65405C411yescharacteristic, complemented
13#65404C411yescharacteristic, complemented
14#65404C411yescharacteristic, complemented
15#65403—21no—
16#65403—21no—
17#65402—21no—
18#65395C2211yescharacteristic
19C2×#3944—24no—
20C2×#3944—21no—
21C2×#3944C2212yes—
22C22×#672—21no—
23C5×#340—21no—
24#64708—24no—
25#13065—51nomaximal, Sylow
26C23×C4×C5D411yescharacteristic
27C2×#709—21no—
28C22×C5×C8—22no—
29C22×C5×C8—22no—
30C22×C5×C8C2×C411yescharacteristic
31C22×C5×C8C2×C411yescharacteristic
32C22×C5×C8D411yescharacteristic
33C22×C5×C8D411yescharacteristic
34C22×C5×C8D411yescharacteristic
35C22×C5×C8C2311yescharacteristic
36C2×C5×C4⋊C4—22no—
37C2×C5×C22⋊C4—41no—
38#4099—41no—
39#4076—21no—
40#4076—21no—
41#4075—41no—
42C4×C22⋊D5—41no—
43C4×C22⋊D5—41no—
44#4072—21no—
45#4070—21no—
46#4070—21no—
47C2×Dic20C2×C411yescharacteristic
48#4050—22no—
49#4050—22no—
50C2×D40C2×C411yescharacteristic
51C2×#677—21no—
52C5×#106—22no—
53C5×#106—22no—
54C5×#95—41no—
55#3953—21no—
56#3953—21no—
57#3950C2×C411yescharacteristic
58#3950C2×C411yescharacteristic
59#3949—21no—
60#3948—21no—
61#3948—21no—
62#3944—24no—
63#3944—22no—
64#3944D414yescomplemented
65C2×#672—24no—
66C2×#672—22no—
67C22×M32—51no—
68#3050—51no—
69C2×#360—52no—
70#1807D511yescharacteristic, complemented
71#1386—52no—
72C24×C5D811yescharacteristic
73C22×C4×C5—22no—
74C22×C4×C5—22no—
75C22×C4×C5—42no—
76C22×C4×C5C22⋊C411yescharacteristic
77C22×C4×C5C22⋊C411yescharacteristic
78C22×C4×C5D811yescharacteristic
79C22×C4×C5SD1611yescharacteristic
80C22×C4×C5SD1611yescharacteristic
81C22×C4×C5C2×D411yescharacteristic
82C2×C22⋊D5—41no—
83C2×C22⋊D5—41no—
84C2×C22⋊D5—41no—
85#709—42no—
86#709—42no—
87C2×D20—21no—
88C2×C4×D5—41no—
89C2×Dic10—21no—
90C2×C5×C8—22no—
91C2×C5×C8—22no—
92C2×C5×C8—22no—
93C2×C5×C8—21no—
94C2×C5×C8—21no—
95C2×C5×C8—21no—
96C2×C5×C8—21no—
97C2×C5×C8C22⋊C411yescharacteristic
98C2×C5×C8C22⋊C411yescharacteristic
99C2×C5×C8C22×C411yescharacteristic
100C2×C5×C8C2×D411yescharacteristic
101C2×C5×C8C2×D412yes—
102C2×C5×C8#4112yes—
103C5×C4⋊C4—42no—
104C5×C4⋊C4—42no—
105C5×C22⋊C4—82no—
106C22⋊Dic5—41no—
107C22⋊Dic5—41no—
108C22⋊Dic5—41no—
109#685—41no—
110#685—41no—
111C4⋊Dic5—21no—
112C4⋊Dic5—21no—
113#683—41no—
114#683—41no—
115C4×Dic5—41no—
116Dic20—22no—
117D40—22no—
118#677—41no—
119#677—41no—
120#672—24no—
121C2×#135—51no—
122C23×C8C2×D511yescharacteristic
123#525—101no—
124C2×M32—52no—
125C2×M32—104no—
126C2×M32—101no—
127C22×C16—101no—
128#467—51no—
129#467—51no—
130#466—51no—
131#466—51no—
132#465—101no—
133#465—101no—
134#427—51no—
135C2×#106C2×D512yes—
136#416—101no—
137#380—21no—
138#360—104no—
139#340—21no—
140C23×C5—22no—
141C23×C5—22no—
142C23×C5—41no—
143C23×C5#10211yescharacteristic
144C23×C5#10211yescharacteristic
145C23×C5C2×D811yescharacteristic
146C22×D5—41no—
147C2×C4×C5—22no—
148C2×C4×C5—21no—
149C2×C4×C5—21no—
150C2×C4×C5—21no—
151C2×C4×C5—21no—
152C2×C4×C5—22no—
153C2×C4×C5—22no—
154C2×C4×C5—82no—
155C2×C4×C5—82no—
156C2×C4×C5—82no—
157C2×C4×C5#10211yescharacteristic
158C2×C4×C5#10211yescharacteristic
159C2×C4×C5C2×C22⋊C411yescharacteristic
160C2×C4×C5#12112yes—
161C2×C4×C5C2×SD1611yescharacteristic
162C2×C4×C5#13512yes—
163C22⋊D5—82no—
164C22⋊D5—82no—
165C22⋊D5—82no—
166C2×Dic5—41no—
167C2×Dic5—41no—
168C2×Dic5—41no—
169D20—41no—
170D20—41no—
171C4×D5—82no—
172Dic10—41no—
173Dic10—41no—
174C5×C8—21no—
175C5×C8—21no—
176C5×C8C4×D412yes—
177C5×C8#12011yescharacteristic
178C5×C8#12311yescharacteristic
179C2×#41—101no—
180C23×C4C22⋊D511yescharacteristic
181#135—102no—
182#135—102no—
183C2×Q16—51no—
184C2×SD16—101no—
185C2×D8—51no—
186C22×C8—22no—
187C22×C8—22no—
188C22×C8C4×D511yescharacteristic
189C22×C8C4×D511yescharacteristic
190C22×C8D2011yescharacteristic
191C22×C8D2011yescharacteristic
192C22×C8C22⋊D511yescharacteristic
193C22×C8C22×D511yescharacteristic
194#123—201no—
195C22⋊Q8—101no—
196C22⋊Q8—101no—
197#121—101no—
198#121—101no—
199#120—201no—
200C4×D4—201no—
201C4×D4—201no—
202#117—101no—
203C2×C4⋊C4—22no—
204C2×C22⋊C4—41no—
205M32—54no—
206M32—104no—
207M32—102no—
208C2×C16—104no—
209C2×C16—102no—
210#107—51no—
211#107—51no—
212#106—22no—
213#106—22no—
214#106—101no—
215#103—101no—
216#103—101no—
217#102—101no—
218#102—101no—
219#95—41no—
220C22×C5—21no—
221C22×C5—21no—
222C22×C5—21no—
223C22×C5—21no—
224C22×C5—21no—
225C22×C5—21no—
226C22×C5—21no—
227C22×C5—44no—
228C22×C5#36012yescomplemented
229C22×C5C2×#10211yescharacteristic
230C22×C5#46612yes—
231C2×D5—81no—
232C2×D5—81no—
233C2×D5—81no—
234C4×C5—21no—
235C4×C5—21no—
236C4×C5—162no—
237C4×C5#38611yescharacteristic
238C4×C5#41611yescharacteristic
239C4×C5#46512yes—
240Dic5—81no—
241Dic5—81no—
242Dic5—81no—
243C24#68611yescharacteristic
244#41—202no—
245#41—202no—
246C2×Q8—101no—
247C2×D4—101no—
248C2×D4—201no—
249C2×D4—201no—
250C2×D4—201no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—42no—
254C22×C4—201no—
255C22×C4#68511yescharacteristic
256C22×C4#68511yescharacteristic
257C22×C4#68611yescharacteristic
258C22×C4#68811yescharacteristic
259C22×C4#68811yescharacteristic
260C22×C4C2×C22⋊D511yescharacteristic
261Q16—102no—
262SD16—201no—
263SD16—201no—
264D8—102no—
265C2×C8—22no—
266C2×C8—22no—
267C2×C8—22no—
268C2×C8—21no—
269C2×C8—21no—
270C2×C8—21no—
271C2×C8—21no—
272C2×C8#68511yescharacteristic
273C2×C8#68511yescharacteristic
274C2×C8C2×C4×D511yescharacteristic
275C2×C8C2×D2011yescharacteristic
276C2×C8D4×D512yes—
277C2×C8#71112yes—
278C4⋊C4—42no—
279C4⋊C4—42no—
280C4⋊C4—101no—
281C4⋊C4—101no—
282C4⋊C4—201no—
283C4⋊C4—201no—
284C22⋊C4—82no—
285C22⋊C4—201no—
286C22⋊C4—201no—
287C22⋊C4—201no—
288C22⋊C4—201no—
289C22⋊C4—201no—
290C42—201no—
291C16—104no—
292C2×C5—22no—
293C2×C5—22no—
294C2×C5—41no—
295C2×C5#189311yescharacteristic
296C2×C5C2×#36011yescharacteristic
297C2×C5#214511yescharacteristic
298D5—161no—
299C23—22no—
300C23—22no—
301C23—41no—
302C23—201no—
303C23#394111yescharacteristic
304C23#394111yescharacteristic
305C23C2×#68611yescharacteristic
306Q8—201no—
307Q8—201no—
308D4—201no—
309D4—201no—
310D4—402no—
311D4—402no—
312D4—402no—
313C2×C4—22no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—82no—
321C2×C4—82no—
322C2×C4—82no—
323C2×C4—201no—
324C2×C4—201no—
325C2×C4—201no—
326C2×C4—402no—
327C2×C4#394111yescharacteristic
328C2×C4#394111yescharacteristic
329C2×C4C2×#68511yescharacteristic
330C2×C4#408512yes—
331C2×C4C2×#68811yescharacteristic
332C2×C4#409612yes—
333C8—21no—
334C8—21no—
335C8#402712yes—
336C8#402811yescharacteristic
337C8#403211yescharacteristic
338C5#1306511yescharacteristic, complemented, Sylow
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—44no—
347C22—401no—
348C22—401no—
349C22—401no—
350C22#6470812yescomplemented
351C22C2×#394111yescharacteristic
352C22#6532712yes—
353C4—21no—
354C4—21no—
355C4—162no—
356C4—401no—
357C4—401no—
358C4—401no—
359C4#6524711yescharacteristic
360C4#6529111yescharacteristic
361C4#6533712yes—
362C2—22no—
363C2—22no—
364C2—41no—
365C2—801no—
366C2#1060030811yescharacteristic
367C2C2×#6470811yescharacteristic
368C2#1060144911yescharacteristic
subgroupG/H
centerC22C2×#3941
commutatorC2×C5×C8C22×C4
FrattiniC22×C8C22×D5
FittingC5×#1807C2
socleC22×C5C2×#102
prime radical#10615473C2
cosocleC22×C5×C8C23
hypercenter#1807D5
hypocenterC5#13065
LCSG ▷ C2×C5×C8 ▷ C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C23×C4 ◁ C23×C8 ◁ #1807

Complements

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

NH ≅ G/Ntype
2. C22×#3944C2semidirect
3. C5×#1807C2semidirect
4. C2×#64708C2semidirect
5. #10598212C2semidirect
11. #65405C4semidirect
12. #65405C4semidirect
13. #65404C4semidirect
14. #65404C4semidirect
64. #3944D4semidirect
70. #1807D5semidirect
228. C22×C5#360semidirect
338. C5#13065semidirect
350. C22#64708semidirect

Maximal subgroups

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

subgroupindex
#106154732
C22×#39442
C5×#18072
C2×#647082
#105982122
#130655