Finite Group Databasebeta

← #12015300

Subgroups — #12015300

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
1C4×#4090C211yesmaximal, characteristic
2C5×#1801C211yesmaximal, characteristic
3#10602430C211yesmaximal, characteristic
4C2×#64698C212yesmaximal
5#10598190C212yesmaximal
6C2×C4×C5×Q8C2211yescharacteristic
7C2×#4090C2211yescharacteristic
8C2×C4×Dic10C411yescharacteristic
9C5×#423—21no—
10C2×C5×#101C2212yes—
11#65519C2211yescharacteristic
12C2×#3968C2212yes—
13#65265C411yescharacteristic
14#65263C411yescharacteristic
15#65261—21no—
16#65261—21no—
17C4×#683—21no—
18C4×#683—21no—
19C42×Dic5C411yescharacteristic
20#65213—21no—
21C5×#343—21no—
22#64750—21no—
23#64698—24no—
24#12802—51nomaximal, Sylow
25C22×C5×Q8C2311yescharacteristic
26C22×Dic10C2×C411yescharacteristic
27C2×C5×M16—22no—
28C4×C5×Q8—41no—
29C4×C5×Q8—41no—
30C2×C5×C4⋊C4C2×C411yescharacteristic
31C2×C5×C4⋊C4D411yescharacteristic
32C2×C5×C4⋊C4D411yescharacteristic
33C2×C42×C5C2×C411yescharacteristic
34C2×C42×C5D411yescharacteristic
35C2×C42×C5D411yescharacteristic
36#4090—21no—
37#4090—21no—
38#4090D412yes—
39#4090Q812yes—
40C2×C4⋊Dic5C2×C411yescharacteristic
41C2×#683—21no—
42C2×#683—21no—
43C2×#683—21no—
44C2×#683—21no—
45C2×#683—21no—
46C2×C4×Dic5—21no—
47C2×C4×Dic5—21no—
48C2×C4×Dic5—21no—
49C2×C4×Dic5C2×C411yescharacteristic
50C2×C4×Dic5C2×C411yescharacteristic
51C2×#681—22no—
52C4×Dic10—41no—
53C4×Dic10—41no—
54C5×C4⋊C8—41no—
55C5×C4⋊C8—41no—
56C5×#101—22no—
57C5×#101—22no—
58#3968—22no—
59#3968—22no—
60#3963—21no—
61#3963—21no—
62#3963—21no—
63#3963—21no—
64C4⋊#168—41no—
65C4⋊#168—41no—
66C4×C4⋊Q8—51no—
67C2×#356—52no—
68#1801—51no—
69#1801D511yescharacteristic
70#1394—52no—
71C2×C5×Q8—41no—
72C2×C5×Q8—41no—
73C2×C5×Q8C2×D411yescharacteristic
74C2×C5×Q8C2×Q811yescharacteristic
75C2×C5×Q8#4111yescharacteristic
76C2×C5×Q8#4111yescharacteristic
77C22×C4×C5C22⋊C411yescharacteristic
78C22×C4×C5C22⋊C411yescharacteristic
79C22×C4×C5C22⋊C411yescharacteristic
80C22×C4×C5C22⋊C411yescharacteristic
81C22×C4×C5C22×C411yescharacteristic
82C22×C4×C5C2×D411yescharacteristic
83C22×C4×C5C2×D411yescharacteristic
84C22×Dic5—21no—
85C22×Dic5—21no—
86C22×Dic5—21no—
87C22×Dic5—21no—
88C2×Dic10—21no—
89C2×Dic10—41no—
90C2×Dic10—41no—
91C2×Dic10C4⋊C412yes—
92C5×M16—42no—
93C5×M16—42no—
94C2×C5×C8—42no—
95C5×C4⋊C4—41no—
96C5×C4⋊C4—41no—
97C5×C4⋊C4—41no—
98C42×C5—21no—
99C42×C5—21no—
100C42×C5—21no—
101C42×C5—21no—
102C42×C5—21no—
103C42×C5—21no—
104C4⋊Dic5—41no—
105#683—22no—
106#683—22no—
107#683—22no—
108#683—22no—
109#683—42no—
110#683—42no—
111#683—42no—
112C4×Dic5—24no—
113C4×Dic5—21no—
114C4×Dic5—42no—
115C4×Dic5—41no—
116C4×Dic5—41no—
117C4×Dic5—41no—
118C4×Dic5C4⋊C412yes—
119#681—42no—
120#681—42no—
121C2×#168—42no—
122C2×C4⋊Q8—51no—
123C2×C4×Q8—51no—
124C2×C4×Q8C2×D511yescharacteristic
125#423—21no—
126#423—101no—
127C2×#101—52no—
128C2×#101C2×D512yes—
129#391—51no—
130#391—51no—
131#389—101no—
132#389—101no—
133#384—51no—
134C4×C4⋊C4—101no—
135C4×C4⋊C4—101no—
136C43—51no—
137#356—104no—
138#343—21no—
139#343—101no—
140C23×C5C2×C22⋊C411yescharacteristic
141C5×Q8—41no—
142C5×Q8—41no—
143C5×Q8—41no—
144C5×Q8—41no—
145C2×C4×C5—21no—
146C2×C4×C5—21no—
147C2×C4×C5—21no—
148C2×C4×C5—21no—
149C2×C4×C5—21no—
150C2×C4×C5—21no—
151C2×C4×C5—22no—
152C2×C4×C5—21no—
153C2×C4×C5—21no—
154C2×C4×C5—21no—
155C2×C4×C5—41no—
156C2×C4×C5—41no—
157C2×C4×C5—41no—
158C2×C4×C5#9911yescharacteristic
159C2×C4×C5#9911yescharacteristic
160C2×C4×C5C2×C4⋊C411yescharacteristic
161C2×C4×C5#11711yescharacteristic
162C2×C4×C5#12111yescharacteristic
163C2×C4×C5#12111yescharacteristic
164C2×C4×C5C22⋊Q811yescharacteristic
165C2×C4×C5C22⋊Q811yescharacteristic
166C2×Dic5—22no—
167C2×Dic5—22no—
168C2×Dic5—22no—
169C2×Dic5—22no—
170C2×Dic5—22no—
171C2×Dic5—41no—
172C2×Dic5—41no—
173C2×Dic5—41no—
174C2×Dic5—41no—
175C2×Dic5—41no—
176C2×Dic5—41no—
177C2×Dic5—41no—
178Dic10—42no—
179Dic10—42no—
180C5×C8—82no—
181#168—82no—
182C22×Q8—51no—
183C22×Q8C22×D511yescharacteristic
184C2×M16—22no—
185C2×M16—102no—
186C4⋊Q8—52no—
187C4⋊Q8—52no—
188C4⋊Q8—101no—
189C4⋊Q8—101no—
190C4×Q8—41no—
191C4×Q8—41no—
192C4×Q8—201no—
193C4×Q8—201no—
194C2×C4⋊C4—51no—
195C2×C4⋊C4—101no—
196C2×C4⋊C4—101no—
197C2×C4⋊C4—101no—
198C2×C4⋊C4—101no—
199C2×C4⋊C4—101no—
200C2×C4⋊C4C4×D511yescharacteristic
201C2×C4⋊C4D2011yescharacteristic
202C2×C4⋊C4C22⋊D511yescharacteristic
203C2×C42—51no—
204C2×C42—51no—
205C2×C42—101no—
206C2×C42—101no—
207C2×C42—101no—
208C2×C42C4×D511yescharacteristic
209C2×C42D2011yescharacteristic
210C2×C42C22⋊D511yescharacteristic
211C4⋊C8—41no—
212C4⋊C8—41no—
213C4⋊C8—201no—
214C4⋊C8—201no—
215#101—22no—
216#101—22no—
217#101—102no—
218#101—102no—
219#95—101no—
220#95—101no—
221#95—101no—
222#95—101no—
223C22×C5—21no—
224C22×C5—21no—
225C22×C5#38011yescharacteristic
226C22×C5C2×#9911yescharacteristic
227C22×C5#41011yescharacteristic
228C4×C5—21no—
229C4×C5—21no—
230C4×C5—21no—
231C4×C5—21no—
232C4×C5—21no—
233C4×C5—21no—
234C4×C5—21no—
235C4×C5—41no—
236C4×C5—41no—
237C4×C5—41no—
238C4×C5#35612yescomplemented
239Dic5—24no—
240Dic5—42no—
241Dic5—42no—
242Dic5—81no—
243C2×Q8—41no—
244C2×Q8—41no—
245C2×Q8—52no—
246C2×Q8—101no—
247C2×Q8—201no—
248C2×Q8—201no—
249C2×Q8D4×D511yescharacteristic
250C2×Q8#71111yescharacteristic
251C2×Q8Q8×D511yescharacteristic
252C2×Q8#71311yescharacteristic
253C22×C4—101no—
254C22×C4—101no—
255C22×C4—101no—
256C22×C4—101no—
257C22×C4#68511yescharacteristic
258C22×C4#68511yescharacteristic
259C22×C4#68511yescharacteristic
260C22×C4#68511yescharacteristic
261C22×C4C2×C4×D511yescharacteristic
262C22×C4C2×D2011yescharacteristic
263C22×C4C2×C22⋊D511yescharacteristic
264M16—42no—
265M16—42no—
266M16—202no—
267M16—202no—
268C2×C8—42no—
269C2×C8—202no—
270C4⋊C4—41no—
271C4⋊C4—41no—
272C4⋊C4—41no—
273C4⋊C4—102no—
274C4⋊C4—102no—
275C4⋊C4—102no—
276C4⋊C4—102no—
277C4⋊C4—201no—
278C4⋊C4—202no—
279C4⋊C4—202no—
280C4⋊C4—202no—
281C42—21no—
282C42—21no—
283C42—21no—
284C42—21no—
285C42—21no—
286C42—21no—
287C42—52no—
288C42—104no—
289C42—101no—
290C42—201no—
291C42—202no—
292C42—201no—
293C42—201no—
294C2×C5—21no—
295C2×C5—21no—
296C2×C5#179911yescharacteristic
297C2×C5C2×#35611yescharacteristic
298C2×C5#213211yescharacteristic
299C23C2×#68511yescharacteristic
300Q8—41no—
301Q8—41no—
302Q8—41no—
303Q8—41no—
304Q8—202no—
305Q8—202no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—22no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—102no—
320C2×C4—102no—
321C2×C4—102no—
322C2×C4—102no—
323C2×C4—102no—
324C2×C4—201no—
325C2×C4—201no—
326C2×C4—201no—
327C2×C4—201no—
328C2×C4—201no—
329C2×C4—201no—
330C2×C4—201no—
331C2×C4#393811yescharacteristic
332C2×C4#393811yescharacteristic
333C2×C4D5×C4⋊C411yescharacteristic
334C2×C4#403811yescharacteristic
335C2×C4#404111yescharacteristic
336C2×C4#404311yescharacteristic
337C2×C4#408411yescharacteristic
338C2×C4#409211yescharacteristic
339C8—82no—
340C8—402no—
341C5#1280211yescharacteristic, complemented, Sylow
342C22—21no—
343C22—21no—
344C22C2×#393811yescharacteristic
345C22#6527611yescharacteristic
346C22#6529511yescharacteristic
347C4—21no—
348C4—21no—
349C4—21no—
350C4—21no—
351C4—21no—
352C4—21no—
353C4—21no—
354C4—41no—
355C4—41no—
356C4—41no—
357C4—104no—
358C4—202no—
359C4—202no—
360C4—401no—
361C4#6469812yescomplemented
362C2—21no—
363C2—21no—
364C2#1060032311yescharacteristic
365C2C2×#6469811yescharacteristic
366C2#1060094111yescharacteristic
subgroupG/H
centerC22C2×#3938
commutatorC22×C4×C5C22×C4
FrattiniC22×Q8C22×D5
FittingC5×#1801C2
socleC22×C5C2×#99
prime radicalC23×C5C2×C22⋊C4
cosocleC22×C5×Q8C23
hypercenter#1801D5
hypocenterC5#12802
LCSG ▷ C22×C4×C5 ▷ C22×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #1801

Complements

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

NH ≅ G/Ntype
238. C4×C5#356semidirect
341. C5#12802semidirect
361. C4#64698semidirect

Maximal subgroups

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

subgroupindex
C4×#40902
C5×#18012
#106024302
C2×#646982
#105981902
#128025