Finite Group Databasebeta

← #12010351

Subgroups — #12010351

Order 1280.

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

normal subgroups only

This group has 460 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×#1459C211yesmaximal, characteristic
3#10602410C211yesmaximal, characteristic
4#10597949C214yesmaximal, complemented
5C2×C4×C5×D4C2211yescharacteristic
6C2×#4081—21no—
7C22×C4⋊Dic5C411yescharacteristic
8C2×C5×C22⋊C8—21no—
9C2×#3965—21no—
10#65501—21no—
11#65501C411yescharacteristic
12#65272C2212yes—
13#65242C412yes—
14#65232—21no—
15#65217—21no—
16#65213—21no—
17C5×#324C2212yes—
18C5×#323—21no—
19#64750C2212yes—
20#11881—51nomaximal, Sylow
21C22×C5×D4D411yescharacteristic
22C23×C4×C5C2×C411yescharacteristic
23C23×C4×C5D411yescharacteristic
24C23×Dic5—21no—
25C22×C5×C8—41no—
26C4×C5×D4—41no—
27C4×C5×D4—41no—
28C2×C5×C4⋊C4C2311yescharacteristic
29C2×C5×C22⋊C4C2×C411yescharacteristic
30C2×C5×C22⋊C4D411yescharacteristic
31C2×C42×C5D411yescharacteristic
32C2×C22⋊Dic5—21no—
33C2×C22⋊Dic5—22no—
34#4081—42no—
35#4081—42no—
36C2×C4⋊Dic5—21no—
37C2×C4⋊Dic5—21no—
38C2×C4⋊Dic5—41no—
39C2×C4⋊Dic5—41no—
40C2×C4⋊Dic5C2×C412yes—
41C2×#683—22no—
42C2×#681—22no—
43C5×C22⋊C8—22no—
44C5×C22⋊C8—41no—
45#3965—42no—
46#3963—22no—
47#3963—22no—
48#3963C2×C412yes—
49C4⋊#168—41no—
50C4⋊#168—41no—
51#2486—51no—
52#1800—51no—
53#1459D511yescharacteristic
54#1346—54no—
55C24×C5C22⋊C411yescharacteristic
56C24×C5C22⋊C411yescharacteristic
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×C5C22⋊C411yescharacteristic
68C22×C4×C5C22⋊C411yescharacteristic
69C22×C4×C5C22×C411yescharacteristic
70C22×C4×C5C2×D411yescharacteristic
71C22×C4×C5C2×D411yescharacteristic
72C22×Dic5—22no—
73C22×Dic5—22no—
74C22×Dic5—42no—
75C22×Dic5—42no—
76C22×Dic5—42no—
77C2×C5×C8—42no—
78C2×C5×C8—82no—
79C5×C4⋊C4—41no—
80C5×C22⋊C4—41no—
81C5×C22⋊C4—41no—
82C42×C5—21no—
83C42×C5—21no—
84C22⋊Dic5—42no—
85C22⋊Dic5—44no—
86C4⋊Dic5—42no—
87C4⋊Dic5—41no—
88C4⋊Dic5—41no—
89C4⋊Dic5—42no—
90#683—42no—
91#683—42no—
92#681—42no—
93#681—42no—
94C2×#168—42no—
95C2×#123—101no—
96C2×C4×D4C2×D511yescharacteristic
97C22×C4⋊C4—51no—
98#423—101no—
99C2×#100—101no—
100C2×C22⋊C8—21no—
101#398—52no—
102#397—52no—
103#389—101no—
104#381—101no—
105#380—51no—
106#380—101no—
107#343—52no—
108#324C2×D512yes—
109#323—21no—
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×C5#10212yes—
120C23×C5C2×C22⋊C411yescharacteristic
121C5×D4—42no—
122C5×D4—42no—
123C2×C4×C5—21no—
124C2×C4×C5—21no—
125C2×C4×C5—22no—
126C2×C4×C5—21no—
127C2×C4×C5—21no—
128C2×C4×C5—21no—
129C2×C4×C5—42no—
130C2×C4×C5—42no—
131C2×C4×C5—42no—
132C2×C4×C5—41no—
133C2×C4×C5—41no—
134C2×C4×C5—41no—
135C2×C4×C5—81no—
136C2×C4×C5#9911yescharacteristic
137C2×C4×C5#9911yescharacteristic
138C2×C4×C5C2×D811yescharacteristic
139C2×C4×C5C2×SD1611yescharacteristic
140C2×Dic5—42no—
141C2×Dic5—42no—
142C2×Dic5—42no—
143C2×Dic5—42no—
144C2×Dic5—42no—
145C2×Dic5—42no—
146C2×Dic5—84no—
147C5×C8—82no—
148#168—82no—
149C22×D4D2011yescharacteristic
150C23×C4—101no—
151C23×C4C4×D511yescharacteristic
152C23×C4C22⋊D511yescharacteristic
153C2×M16—102no—
154C22×C8—41no—
155#123—202no—
156#123—202no—
157C4×D4—41no—
158C4×D4—41no—
159C2×C4⋊C4—52no—
160C2×C4⋊C4—101no—
161C2×C4⋊C4—101no—
162C2×C4⋊C4—102no—
163C2×C4⋊C4—201no—
164C2×C4⋊C4—201no—
165C2×C4⋊C4C22×D511yescharacteristic
166C2×C22⋊C4—101no—
167C2×C22⋊C4—102no—
168C2×C22⋊C4C4×D511yescharacteristic
169C2×C22⋊C4C22⋊D511yescharacteristic
170C2×C42D2011yescharacteristic
171C4⋊C8—201no—
172C4⋊C8—201no—
173#100—202no—
174C22⋊C8—22no—
175C22⋊C8—41no—
176#95—52no—
177#95—102no—
178#95—102no—
179C22×C5—21no—
180C22×C5—21no—
181C22×C5—22no—
182C22×C5—22no—
183C22×C5—42no—
184C22×C5—42no—
185C22×C5—42no—
186C22×C5—41no—
187C22×C5—41no—
188C22×C5—41no—
189C22×C5—41no—
190C22×C5—81no—
191C22×C5C2×#9911yescharacteristic
192C22×C5C2×#10211yescharacteristic
193C22×C5#42111yescharacteristic
194C4×C5—21no—
195C4×C5—21no—
196C4×C5—21no—
197C4×C5—41no—
198C4×C5—41no—
199C4×C5—41no—
200C4×C5#35212yes—
201Dic5—82no—
202Dic5—82no—
203C24#68511yescharacteristic
204C24#68511yescharacteristic
205C2×D4—41no—
206C2×D4—41no—
207C2×D4#67712yes—
208C2×D4D4012yes—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—102no—
216C22×C4—102no—
217C22×C4—202no—
218C22×C4—202no—
219C22×C4—202no—
220C22×C4#68511yescharacteristic
221C22×C4#68511yescharacteristic
222C22×C4C2×C4×D511yescharacteristic
223C22×C4C2×D2011yescharacteristic
224C22×C4C2×C22⋊D511yescharacteristic
225M16—202no—
226M16—202no—
227C2×C8—42no—
228C2×C8—82no—
229C2×C8—202no—
230C4⋊C4—41no—
231C4⋊C4—202no—
232C4⋊C4—202no—
233C4⋊C4—201no—
234C4⋊C4—201no—
235C4⋊C4—202no—
236C4⋊C4—202no—
237C22⋊C4—41no—
238C22⋊C4—41no—
239C22⋊C4—202no—
240C22⋊C4—204no—
241C42—21no—
242C42—21no—
243C2×C5—21no—
244C2×C5—21no—
245C2×C5—42no—
246C2×C5—42no—
247C2×C5#151911yescharacteristic
248C2×C5C2×#35211yescharacteristic
249C2×C5#213511yescharacteristic
250C23—21no—
251C23—21no—
252C23—21no—
253C23—21no—
254C23—41no—
255C23—41no—
256C23—41no—
257C23—41no—
258C23#395312yes—
259C23#395312yes—
260C23C2×#68511yescharacteristic
261D4—42no—
262D4—42no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—22no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—81no—
276C2×C4—202no—
277C2×C4—202no—
278C2×C4—202no—
279C2×C4—202no—
280C2×C4—202no—
281C2×C4—202no—
282C2×C4—404no—
283C2×C4#393811yescharacteristic
284C2×C4#393811yescharacteristic
285C2×C4C2×#67711yescharacteristic
286C2×C4C2×D4011yescharacteristic
287C8—82no—
288C8—402no—
289C5#1188111yescharacteristic, complemented, Sylow
290C22—21no—
291C22—21no—
292C22—22no—
293C22—22no—
294C22—42no—
295C22—42no—
296C22—42no—
297C22—41no—
298C22—41no—
299C22—41no—
300C22—41no—
301C22—81no—
302C22C2×#393811yescharacteristic
303C22C2×#395311yescharacteristic
304C22#6542811yescharacteristic
305C4—21no—
306C4—21no—
307C4—21no—
308C4—41no—
309C4—41no—
310C4—41no—
311C4—402no—
312C4—402no—
313C4#6469212yes—
314C2—21no—
315C2—21no—
316C2—42no—
317C2—42no—
318C2#1060073711yescharacteristic
319C2C2×#6469211yescharacteristic
320C2#1060094311yescharacteristic
subgroupG/H
centerC22C2×#3938
commutatorC22×C4×C5C22×C4
FrattiniC2×C4⋊C4C22×D5
FittingC5×#1459C2
socleC22×C5C2×#99
prime radicalC22×C5×D4D4
cosocleC2×C5×C4⋊C4C23
hypercenter#1459D5
hypocenterC5#11881
LCSG ▷ C22×C4×C5 ▷ C23×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1459

Complements

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

NH ≅ G/Ntype
4. #10597949C2semidirect
289. C5#11881semidirect

Maximal subgroups

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

subgroupindex
#106141042
C5×#14592
#106024102
#105979492
#118815