Finite Group Databasebeta

← #12009581

Subgroups — #12009581

Order 1280.

468 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 107 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#10613976C211yesmaximal, characteristic
2C5×#1458C211yesmaximal, characteristic
3#10602385C211yesmaximal, characteristic
4#10597942C212yesmaximal
5#10597940C212yesmaximal
6C2×C5×#117C2211yescharacteristic
7C2×#4072—21no—
8C2×C4×Dic10C411yescharacteristic, complemented
9C5×#431—21no—
10C4×C22⋊Dic5—21no—
11#65275C411yescharacteristic, complemented
12#65273—21no—
13#65268C411yescharacteristic, complemented
14#65238C2211yescharacteristic
15#65237—21no—
16#65236C2211yescharacteristic
17C4×C4⋊Dic5C411yescharacteristic, complemented
18C4×#683—21no—
19C5×#324—21no—
20C5×#323C2212yes—
21#64746—21no—
22#64746C2212yes—
23#11715—51nomaximal, Sylow
24C23×C4×C5D411yescharacteristic
25C22×Dic10C2×C411yescharacteristic
26C5×#117—41no—
27C5×#117—41no—
28C2×C5×C4⋊C4C2×C411yescharacteristic
29C2×C5×C4⋊C4D411yescharacteristic
30C2×C5×C22⋊C4D411yescharacteristic
31C2×C5×C22⋊C4C2311yescharacteristic
32C2×C42×C5C2×C411yescharacteristic
33C2×C42×C5D411yescharacteristic
34C2×C22⋊Dic5—22no—
35C2×C22⋊Dic5—21no—
36C2×C22⋊Dic5—21no—
37#4072—24no—
38#4072—42no—
39C2×C4⋊Dic5—21no—
40C2×C4⋊Dic5—41no—
41C2×C4⋊Dic5C2×C411yescharacteristic
42C2×#683—21no—
43C2×#683—21no—
44C2×#683—21no—
45C2×C4×Dic5—21no—
46C2×C4×Dic5—21no—
47C2×C4×Dic5—41no—
48C4×Dic10—41no—
49C4×Dic10—41no—
50C5×C4⋊C8—41no—
51C5×C22⋊C8—22no—
52C4×C5×C8—41no—
53#3963—21no—
54#3963—21no—
55#3963—21no—
56#3963—21no—
57#3963C2×C411yescharacteristic
58#3963C2×C411yescharacteristic
59#2396—51no—
60#1740—51no—
61#1458D511yescharacteristic
62#1352—52no—
63#1351—52no—
64C24×C5C2×D411yescharacteristic
65C22×C4×C5—21no—
66C22×C4×C5—21no—
67C22×C4×C5—22no—
68C22×C4×C5C22⋊C411yescharacteristic
69C22×C4×C5C22⋊C411yescharacteristic
70C22×C4×C5C22⋊C411yescharacteristic
71C22×C4×C5C22⋊C411yescharacteristic
72C22×C4×C5SD1612yes—
73C22×C4×C5Q1612yes—
74C22×C4×C5C22×C411yescharacteristic
75C22×C4×C5C2×D411yescharacteristic
76C22×Dic5—21no—
77C22×Dic5—21no—
78C22×Dic5—21no—
79C22×Dic5—21no—
80C22×Dic5—42no—
81C2×Dic10—41no—
82C2×Dic10—42no—
83C2×C5×C8—42no—
84C5×C4⋊C4—41no—
85C5×C4⋊C4—41no—
86C5×C22⋊C4—41no—
87C5×C22⋊C4—41no—
88C42×C5—21no—
89C42×C5—21no—
90C42×C5—21no—
91C42×C5—21no—
92C22⋊Dic5—24no—
93C22⋊Dic5—42no—
94C22⋊Dic5—42no—
95C22⋊Dic5—42no—
96C4⋊Dic5—42no—
97C4⋊Dic5—44no—
98C4⋊Dic5—41no—
99#683—42no—
100#683—42no—
101#683—42no—
102C4×Dic5—44no—
103C4×Dic5—41no—
104C4×Dic5—41no—
105C4×Dic5—42no—
106C2×C4×Q8—51no—
107C2×#117—101no—
108C2×#117C2×D511yescharacteristic
109#431—21no—
110#400—51no—
111#399—51no—
112#398—101no—
113#396—101no—
114#395—51no—
115#393—51no—
116C4×C4⋊C4—51no—
117C4×C4⋊C4—101no—
118C4×C22⋊C4—101no—
119#342—52no—
120#342—101no—
121#324—21no—
122#323C2×D512yes—
123C23×C5—21no—
124C23×C5—21no—
125C23×C5—41no—
126C23×C5—41no—
127C23×C5C2×C22⋊C411yescharacteristic
128C23×C5C2×SD1611yescharacteristic
129C23×C5C2×Q1611yescharacteristic
130C2×C4×C5—21no—
131C2×C4×C5—21no—
132C2×C4×C5—22no—
133C2×C4×C5—22no—
134C2×C4×C5—22no—
135C2×C4×C5—22no—
136C2×C4×C5—21no—
137C2×C4×C5—21no—
138C2×C4×C5—21no—
139C2×C4×C5—21no—
140C2×C4×C5—41no—
141C2×C4×C5—41no—
142C2×C4×C5—41no—
143C2×C4×C5—41no—
144C2×C4×C5#9911yescharacteristic
145C2×C4×C5#9911yescharacteristic
146C2×C4×C5#10312yes—
147C2×C4×C5#10312yes—
148C2×C4×C5#10412yes—
149C2×Dic5—22no—
150C2×Dic5—22no—
151C2×Dic5—41no—
152C2×Dic5—41no—
153C2×Dic5—41no—
154C2×Dic5—41no—
155C2×Dic5—44no—
156C2×Dic5—41no—
157C2×Dic5—41no—
158C2×Dic5—41no—
159C2×Dic5—41no—
160C2×Dic5—41no—
161C2×Dic5—41no—
162C2×Dic5—82no—
163C2×Dic5—82no—
164Dic10—42no—
165Dic10—42no—
166C5×C8—82no—
167C22×Q8—51no—
168C23×C4D2011yescharacteristic
169C4×Q8—201no—
170C4×Q8—201no—
171#117—41no—
172#117—41no—
173#117—104no—
174#117—202no—
175C2×C4⋊C4—51no—
176C2×C4⋊C4—101no—
177C2×C4⋊C4—101no—
178C2×C4⋊C4—101no—
179C2×C4⋊C4—101no—
180C2×C4⋊C4—201no—
181C2×C4⋊C4C4×D511yescharacteristic
182C2×C4⋊C4C22⋊D511yescharacteristic
183C2×C22⋊C4—102no—
184C2×C22⋊C4—101no—
185C2×C22⋊C4—101no—
186C2×C22⋊C4D2011yescharacteristic
187C2×C22⋊C4C22×D511yescharacteristic
188C2×C42—101no—
189C2×C42—101no—
190C2×C42—201no—
191C2×C42C4×D511yescharacteristic
192C2×C42C22⋊D511yescharacteristic
193C4⋊C8—41no—
194C22⋊C8—22no—
195C4×C8—41no—
196#95—51no—
197#95—51no—
198#95—101no—
199#95—101no—
200#95—101no—
201#95—101no—
202C22×C5—21no—
203C22×C5—21no—
204C22×C5—21no—
205C22×C5—21no—
206C22×C5—41no—
207C22×C5—41no—
208C22×C5—41no—
209C22×C5—41no—
210C22×C5—81no—
211C22×C5C2×#9911yescharacteristic
212C22×C5C2×#10311yescharacteristic
213C22×C5C2×#10411yescharacteristic
214C4×C5—22no—
215C4×C5—41no—
216C4×C5—41no—
217C4×C5—41no—
218C4×C5—41no—
219C4×C5#32812yes—
220C4×C5#32812yes—
221Dic5—42no—
222Dic5—42no—
223Dic5—81no—
224Dic5—84no—
225Dic5—81no—
226C24C2×D2011yescharacteristic
227C2×Q8—201no—
228C2×Q8—202no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—22no—
232C22×C4—101no—
233C22×C4—101no—
234C22×C4—101no—
235C22×C4—101no—
236C22×C4—202no—
237C22×C4#67712yes—
238C22×C4Dic2012yes—
239C22×C4#68511yescharacteristic
240C22×C4#68511yescharacteristic
241C22×C4#68511yescharacteristic
242C22×C4#68511yescharacteristic
243C22×C4C2×C4×D511yescharacteristic
244C22×C4C2×C22⋊D511yescharacteristic
245C2×C8—42no—
246C4⋊C4—41no—
247C4⋊C4—41no—
248C4⋊C4—201no—
249C4⋊C4—202no—
250C4⋊C4—204no—
251C4⋊C4—202no—
252C4⋊C4—202no—
253C4⋊C4—202no—
254C22⋊C4—41no—
255C22⋊C4—41no—
256C22⋊C4—104no—
257C22⋊C4—202no—
258C22⋊C4—202no—
259C22⋊C4—202no—
260C42—21no—
261C42—21no—
262C42—21no—
263C42—21no—
264C42—204no—
265C42—201no—
266C42—201no—
267C42—202no—
268C2×C5—21no—
269C2×C5—21no—
270C2×C5—41no—
271C2×C5—41no—
272C2×C5C2×#32811yescharacteristic
273C2×C5#212011yescharacteristic
274C2×C5#212611yescharacteristic
275C23—21no—
276C23—21no—
277C23—41no—
278C23—41no—
279C23C2×#67711yescharacteristic
280C23C2×Dic2011yescharacteristic
281C23C2×#68511yescharacteristic
282Q8—202no—
283Q8—202no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—102no—
299C2×C4—102no—
300C2×C4—201no—
301C2×C4—201no—
302C2×C4—201no—
303C2×C4—201no—
304C2×C4—204no—
305C2×C4—201no—
306C2×C4—201no—
307C2×C4—201no—
308C2×C4—201no—
309C2×C4—201no—
310C2×C4—201no—
311C2×C4—402no—
312C2×C4—402no—
313C2×C4#393811yescharacteristic
314C2×C4#393811yescharacteristic
315C2×C4#394812yes—
316C2×C4#394812yes—
317C2×C4#395712yes—
318C8—82no—
319C5#1171511yescharacteristic, complemented, Sylow
320C22—21no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—41no—
325C22—41no—
326C22—41no—
327C22—41no—
328C22—81no—
329C22C2×#393811yescharacteristic
330C22C2×#394811yescharacteristic
331C22C2×#395711yescharacteristic
332C4—22no—
333C4—41no—
334C4—41no—
335C4—41no—
336C4—41no—
337C4—202no—
338C4—202no—
339C4—401no—
340C4—404no—
341C4—401no—
342C4#6468712yes—
343C4#6468712yes—
344C2—21no—
345C2—21no—
346C2—41no—
347C2—41no—
348C2C2×#6468711yescharacteristic
349C2#1060085311yescharacteristic
350C2#1060093811yescharacteristic
subgroupG/H
centerC2×C4#3938
commutatorC22×C4×C5C22×C4
FrattiniC2×C22⋊C4C22×D5
FittingC5×#1458C2
socleC22×C5C2×#99
prime radicalC24×C5C2×D4
cosocleC2×C5×C22⋊C4C23
hypercenter#1458D5
hypocenterC5#11715
LCSG ▷ C22×C4×C5 ▷ C23×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×#117 ◁ #1458

Complements

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

NH ≅ G/Ntype
8. C2×C4×Dic10C4semidirect
11. #65275C4semidirect
13. #65268C4semidirect
17. C4×C4⋊Dic5C4semidirect
319. C5#11715semidirect

Maximal subgroups

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

subgroupindex
#106139762
C5×#14582
#106023852
#105979422
#105979402
#117155