Finite Group Databasebeta

← #12010318

Subgroups — #12010318

Order 1280.

472 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 99 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#10614104C211yesmaximal, characteristic
2C5×#1799C211yesmaximal, characteristic
3#10602394C211yesmaximal, characteristic
4#10597982C214yesmaximal, complemented
5C2×C4×C5×D4C2211yescharacteristic
6C2×#4081—21no—
7C22×C4⋊Dic5C411yescharacteristic, complemented
8C2×C5×#99—21no—
9C5×#380—21no—
10#65501—21no—
11#65501C411yescharacteristic, complemented
12#65272C412yescomplemented
13#65242C2212yes—
14#65232—21no—
15#65217—21no—
16#65214—21no—
17C5×#342C2212yes—
18#64745—21no—
19#64745C2212yes—
20#11874—51nomaximal, Sylow
21C22×C5×D4D411yescharacteristic
22C23×C4×C5C2×C411yescharacteristic
23C23×C4×C5D411yescharacteristic
24C23×Dic5—21no—
25C4×C5×D4—41no—
26C4×C5×D4—41no—
27C2×C5×C4⋊C4—41no—
28C2×C5×C4⋊C4C2×C411yescharacteristic
29C2×C5×C22⋊C4—22no—
30C2×C5×C22⋊C4D411yescharacteristic
31C2×C5×C22⋊C4C2311yescharacteristic
32C2×C42×C5D411yescharacteristic
33C2×C22⋊Dic5—21no—
34C2×C22⋊Dic5—22no—
35#4081—42no—
36#4081—42no—
37C2×C4⋊Dic5—21no—
38C2×C4⋊Dic5—21no—
39C2×C4⋊Dic5—41no—
40C2×C4⋊Dic5—41no—
41C2×C4⋊Dic5C2×C412yes—
42C2×#683—22no—
43C5×#99—42no—
44C5×#95—41no—
45#3963—22no—
46#3963—22no—
47#3963C2×C412yes—
48C22⋊#168—22no—
49C4⋊#168—41no—
50C4×#168—41no—
51#2486—51no—
52#1799D511yescharacteristic
53#1460—51no—
54#1344—54no—
55C24×C5C22⋊C411yescharacteristic
56C24×C5C2×D411yescharacteristic
57C2×C5×D4—41no—
58C2×C5×D4—41no—
59C2×C5×D4D812yes—
60C2×C5×D4SD1612yes—
61C22×C4×C5—21no—
62C22×C4×C5—21no—
63C22×C4×C5—21no—
64C22×C4×C5—21no—
65C22×C4×C5—42no—
66C22×C4×C5—42no—
67C22×C4×C5—42no—
68C22×C4×C5C22⋊C411yescharacteristic
69C22×C4×C5C22⋊C411yescharacteristic
70C22×C4×C5C22⋊C411yescharacteristic
71C22×C4×C5C22×C411yescharacteristic
72C22×C4×C5C2×D411yescharacteristic
73C22×Dic5—22no—
74C22×Dic5—22no—
75C22×Dic5—42no—
76C22×Dic5—42no—
77C22×Dic5—42no—
78C5×C4⋊C4—41no—
79C5×C4⋊C4—82no—
80C5×C22⋊C4—41no—
81C5×C22⋊C4—41no—
82C5×C22⋊C4—42no—
83C5×C22⋊C4—42no—
84C42×C5—21no—
85C42×C5—21no—
86C22⋊Dic5—42no—
87C22⋊Dic5—42no—
88C22⋊Dic5—42no—
89C4⋊Dic5—42no—
90C4⋊Dic5—41no—
91C4⋊Dic5—41no—
92C4⋊Dic5—42no—
93#683—44no—
94C2×#168—42no—
95C2×#123—101no—
96C2×C4×D4C2×D511yescharacteristic
97C22×C4⋊C4—51no—
98#431—101no—
99C2×#99—21no—
100#398—52no—
101#397—52no—
102#389—101no—
103#381—101no—
104#380—21no—
105#380—51no—
106#380—101no—
107#342C2×D512yes—
108#323—52no—
109#323—101no—
110C23×C5—21no—
111C23×C5—21no—
112C23×C5—21no—
113C23×C5—21no—
114C23×C5—41no—
115C23×C5—41no—
116C23×C5—41no—
117C23×C5—41no—
118C23×C5#10212yes—
119C23×C5C2×C22⋊C411yescharacteristic
120C23×C5C2×D811yescharacteristic
121C23×C5C2×SD1611yescharacteristic
122C5×D4—42no—
123C5×D4—42no—
124C2×C4×C5—21no—
125C2×C4×C5—21no—
126C2×C4×C5—22no—
127C2×C4×C5—21no—
128C2×C4×C5—21no—
129C2×C4×C5—21no—
130C2×C4×C5—42no—
131C2×C4×C5—42no—
132C2×C4×C5—42no—
133C2×C4×C5—41no—
134C2×C4×C5—41no—
135C2×C4×C5—41no—
136C2×C4×C5—81no—
137C2×C4×C5—82no—
138C2×C4×C5—82no—
139C2×C4×C5—82no—
140C2×C4×C5#9911yescharacteristic
141C2×C4×C5#9911yescharacteristic
142C2×C4×C5#10212yes—
143C2×Dic5—42no—
144C2×Dic5—42no—
145C2×Dic5—42no—
146C2×Dic5—42no—
147C2×Dic5—42no—
148C2×Dic5—42no—
149C2×Dic5—84no—
150#168—82no—
151C22×D4C22⋊D511yescharacteristic
152C23×C4—101no—
153C23×C4C4×D511yescharacteristic
154C23×C4C22⋊D511yescharacteristic
155#123—202no—
156#123—202no—
157C4×D4—41no—
158C4×D4—41no—
159C2×C4⋊C4—41no—
160C2×C4⋊C4—52no—
161C2×C4⋊C4—102no—
162C2×C4⋊C4—101no—
163C2×C4⋊C4—101no—
164C2×C4⋊C4—201no—
165C2×C4⋊C4—201no—
166C2×C4⋊C4C4×D511yescharacteristic
167C2×C22⋊C4—22no—
168C2×C22⋊C4—101no—
169C2×C22⋊C4—102no—
170C2×C22⋊C4D2011yescharacteristic
171C2×C22⋊C4C22×D511yescharacteristic
172C2×C42D2011yescharacteristic
173C4⋊C8—201no—
174#99—42no—
175C22⋊C8—102no—
176C4×C8—201no—
177#95—41no—
178#95—52no—
179#95—102no—
180#95—102no—
181C22×C5—21no—
182C22×C5—21no—
183C22×C5—22no—
184C22×C5—21no—
185C22×C5—21no—
186C22×C5—42no—
187C22×C5—41no—
188C22×C5—41no—
189C22×C5—42no—
190C22×C5—41no—
191C22×C5—41no—
192C22×C5—41no—
193C22×C5—41no—
194C22×C5—81no—
195C22×C5C2×#9911yescharacteristic
196C22×C5C2×#10211yescharacteristic
197C22×C5#42111yescharacteristic
198C4×C5—21no—
199C4×C5—22no—
200C4×C5—41no—
201C4×C5—41no—
202C4×C5—41no—
203C4×C5—162no—
204C4×C5#35212yes—
205Dic5—82no—
206Dic5—82no—
207C24#68511yescharacteristic
208C24C2×C22⋊D511yescharacteristic
209C2×D4—41no—
210C2×D4—41no—
211C2×D4#68612yes—
212C2×D4#68812yes—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—102no—
221C22×C4—102no—
222C22×C4—202no—
223C22×C4—202no—
224C22×C4—202no—
225C22×C4#68511yescharacteristic
226C22×C4#68511yescharacteristic
227C22×C4#68511yescharacteristic
228C22×C4C2×C4×D511yescharacteristic
229C22×C4C2×D2011yescharacteristic
230C2×C8—202no—
231C4⋊C4—41no—
232C4⋊C4—82no—
233C4⋊C4—202no—
234C4⋊C4—202no—
235C4⋊C4—201no—
236C4⋊C4—201no—
237C4⋊C4—204no—
238C22⋊C4—41no—
239C22⋊C4—41no—
240C22⋊C4—42no—
241C22⋊C4—42no—
242C22⋊C4—202no—
243C22⋊C4—202no—
244C22⋊C4—202no—
245C42—21no—
246C42—21no—
247C2×C5—21no—
248C2×C5—21no—
249C2×C5—41no—
250C2×C5—42no—
251C2×C5—41no—
252C2×C5#151911yescharacteristic
253C2×C5C2×#35211yescharacteristic
254C2×C5#212911yescharacteristic
255C23—21no—
256C23—21no—
257C23—21no—
258C23—21no—
259C23—41no—
260C23—41no—
261C23—41no—
262C23—41no—
263C23#394112yes—
264C23C2×#68511yescharacteristic
265C23C2×#68611yescharacteristic
266C23C2×#68811yescharacteristic
267D4—42no—
268D4—42no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—22no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—81no—
282C2×C4—82no—
283C2×C4—82no—
284C2×C4—82no—
285C2×C4—202no—
286C2×C4—202no—
287C2×C4—202no—
288C2×C4—202no—
289C2×C4—202no—
290C2×C4—202no—
291C2×C4—404no—
292C2×C4#393811yescharacteristic
293C2×C4#393811yescharacteristic
294C2×C4#394112yes—
295C8—402no—
296C5#1187411yescharacteristic, complemented, Sylow
297C22—21no—
298C22—21no—
299C22—21no—
300C22—22no—
301C22—21no—
302C22—42no—
303C22—41no—
304C22—41no—
305C22—42no—
306C22—41no—
307C22—41no—
308C22—41no—
309C22—41no—
310C22—81no—
311C22C2×#393811yescharacteristic
312C22C2×#394111yescharacteristic
313C22#6529411yescharacteristic
314C4—21no—
315C4—22no—
316C4—41no—
317C4—41no—
318C4—41no—
319C4—162no—
320C4—402no—
321C4—402no—
322C4#6469212yes—
323C2—21no—
324C2—21no—
325C2—41no—
326C2—42no—
327C2—41no—
328C2#1060030211yescharacteristic
329C2C2×#6469211yescharacteristic
330C2#1060085211yescharacteristic
subgroupG/H
centerC22C2×#3938
commutatorC22×C4×C5C22×C4
FrattiniC2×C22⋊C4C22×D5
FittingC5×#1799C2
socleC22×C5C2×#99
prime radicalC22×C5×D4D4
cosocleC2×C5×C22⋊C4C23
hypercenter#1799D5
hypocenterC5#11874
LCSG ▷ C22×C4×C5 ▷ C23×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1799

Complements

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

NH ≅ G/Ntype
4. #10597982C2semidirect
7. C22×C4⋊Dic5C4semidirect
11. #65501C4semidirect
12. #65272C4semidirect
296. C5#11874semidirect

Maximal subgroups

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

subgroupindex
#106141042
C5×#17992
#106023942
#105979822
#118745