Finite Group Databasebeta

← #12010214

Subgroups — #12010214

Order 1280.

396 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 83 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#10613907C211yesmaximal, characteristic
2C2×C5×#324C211yesmaximal, characteristic
3#10602355C211yesmaximal, characteristic
4#10597950C212yesmaximal, complemented
5#10597949C212yesmaximal, complemented
6C22×C5×C4⋊C4C2211yescharacteristic
7C2×#4072C411yescharacteristic
8C2×C5×C22⋊C8—21no—
9#65500—21no—
10#65500C411yescharacteristic
11#65499—21no—
12#65496—21no—
13#65275C411yescharacteristic
14#65275C2211yescharacteristic
15#65272C411yescharacteristic
16#65272C2211yescharacteristic
17#65267—21no—
18#65267—21no—
19C5×#324—21no—
20C5×#324C2212yes—
21#64750—21no—
22#64750C2212yes—
23#11850—51nomaximal, Sylow
24C23×C4×C5C2×C411yescharacteristic
25C23×C4×C5D411yescharacteristic
26C23×C4×C5D411yescharacteristic
27C22×C5×C8—41no—
28C2×C5×C4⋊C4—41no—
29C2×C5×C4⋊C4—41no—
30C2×C5×C4⋊C4C2×C411yescharacteristic
31C2×C5×C4⋊C4D411yescharacteristic
32C2×C5×C4⋊C4D411yescharacteristic
33C2×C5×C4⋊C4C2311yescharacteristic
34C2×C22⋊Dic5—21no—
35C2×C22⋊Dic5—21no—
36#4072—41no—
37#4072—41no—
38C2×C4⋊Dic5C2×C411yescharacteristic
39C2×C4⋊Dic5C2×C411yescharacteristic
40C2×#683—21no—
41C2×#683—21no—
42C2×C4×Dic5—21no—
43C2×#681—22no—
44C5×C22⋊C8—22no—
45C5×C22⋊C8—41no—
46#3963—21no—
47#3963—21no—
48#3963—21no—
49#3963—21no—
50#3963—21no—
51#3963—21no—
52#3963—21no—
53#3963C2×C411yescharacteristic
54#3963C2×C411yescharacteristic
55C22⋊#168—41no—
56C22⋊#168—41no—
57#2484—51no—
58#1821—51no—
59C2×#324D511yescharacteristic
60#1353—52no—
61#1346—52no—
62C24×C5C22⋊C411yescharacteristic
63C22×C4×C5—21no—
64C22×C4×C5—21no—
65C22×C4×C5—21no—
66C22×C4×C5—21no—
67C22×C4×C5—21no—
68C22×C4×C5—21no—
69C22×C4×C5—22no—
70C22×C4×C5—41no—
71C22×C4×C5—42no—
72C22×C4×C5—42no—
73C22×C4×C5C22⋊C411yescharacteristic
74C22×C4×C5C22⋊C411yescharacteristic
75C22×C4×C5C22⋊C411yescharacteristic
76C22×C4×C5C22×C411yescharacteristic
77C22×C4×C5C2×D411yescharacteristic
78C22×C4×C5C2×D411yescharacteristic
79C22×Dic5—21no—
80C22×Dic5—21no—
81C22×Dic5—21no—
82C22×Dic5—21no—
83C2×C5×C8—42no—
84C2×C5×C8—82no—
85C5×C4⋊C4—41no—
86C5×C4⋊C4—41no—
87C5×C4⋊C4—41no—
88C5×C4⋊C4—41no—
89C22⋊Dic5—42no—
90C22⋊Dic5—41no—
91C22⋊Dic5—41no—
92C4⋊Dic5—41no—
93C4⋊Dic5—41no—
94#683—42no—
95#683—42no—
96C4×Dic5—42no—
97#681—42no—
98#681—42no—
99C2×#168—42no—
100C2×#117—51no—
101C22×C4⋊C4C2×D511yescharacteristic
102#407—101no—
103C2×C22⋊C8—21no—
104#400—51no—
105#400—51no—
106#398—51no—
107#398—51no—
108#387—101no—
109#387—101no—
110#385—101no—
111#381—51no—
112#381—101no—
113#343—52no—
114#343—101no—
115#324—21no—
116#324C2×D512yes—
117C23×C5—22no—
118C23×C5—21no—
119C23×C5—21no—
120C23×C5—41no—
121C23×C5#9911yescharacteristic
122C23×C5#9911yescharacteristic
123C23×C5C2×C22⋊C411yescharacteristic
124C2×C4×C5—21no—
125C2×C4×C5—21no—
126C2×C4×C5—21no—
127C2×C4×C5—21no—
128C2×C4×C5—21no—
129C2×C4×C5—21no—
130C2×C4×C5—21no—
131C2×C4×C5—21no—
132C2×C4×C5—21no—
133C2×C4×C5—42no—
134C2×C4×C5—42no—
135C2×C4×C5—42no—
136C2×C4×C5—42no—
137C2×C4×C5—42no—
138C2×C4×C5—41no—
139C2×C4×C5—41no—
140C2×C4×C5—41no—
141C2×C4×C5—41no—
142C2×C4×C5—81no—
143C2×C4×C5#13512yes—
144C2×Dic5—41no—
145C2×Dic5—41no—
146C2×Dic5—41no—
147C2×Dic5—41no—
148C2×Dic5—42no—
149C2×Dic5—41no—
150C2×Dic5—41no—
151C2×Dic5—41no—
152C2×Dic5—41no—
153C2×Dic5—41no—
154C2×Dic5—41no—
155C5×C8—82no—
156#168—82no—
157C23×C4C4×D511yescharacteristic
158C23×C4D2011yescharacteristic
159C23×C4C22⋊D511yescharacteristic
160C2×M16—102no—
161C22×C8—41no—
162#117—201no—
163#117—201no—
164C2×C4⋊C4—41no—
165C2×C4⋊C4—41no—
166C2×C4⋊C4—51no—
167C2×C4⋊C4—51no—
168C2×C4⋊C4—101no—
169C2×C4⋊C4—101no—
170C2×C4⋊C4C4×D511yescharacteristic
171C2×C4⋊C4D2011yescharacteristic
172C2×C4⋊C4C22⋊D511yescharacteristic
173C2×C4⋊C4C22×D511yescharacteristic
174C2×C22⋊C4—101no—
175C2×C22⋊C4—101no—
176C2×C42—101no—
177C22⋊C8—22no—
178C22⋊C8—41no—
179C22⋊C8—201no—
180C22⋊C8—201no—
181#95—51no—
182#95—51no—
183#95—101no—
184#95—101no—
185#95—101no—
186#95—101no—
187#95—101no—
188#95—101no—
189#95—101no—
190C22×C5—21no—
191C22×C5—21no—
192C22×C5—21no—
193C22×C5—21no—
194C22×C5—21no—
195C22×C5—21no—
196C22×C5—21no—
197C22×C5—42no—
198C22×C5—42no—
199C22×C5#35212yes—
200C22×C5C2×#9911yescharacteristic
201C22×C5#41611yescharacteristic
202C22×C5#42111yescharacteristic
203C4×C5—22no—
204C4×C5—41no—
205C4×C5—41no—
206C4×C5—41no—
207C4×C5—41no—
208C4×C5—41no—
209Dic5—81no—
210Dic5—81no—
211Dic5—81no—
212Dic5—81no—
213C24#68511yescharacteristic
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—22no—
221C22×C4—41no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—101no—
225C22×C4—101no—
226C22×C4—101no—
227C22×C4—101no—
228C22×C4#68511yescharacteristic
229C22×C4#68511yescharacteristic
230C22×C4#68511yescharacteristic
231C22×C4C2×C4×D511yescharacteristic
232C22×C4C2×D2011yescharacteristic
233C22×C4C2×C22⋊D511yescharacteristic
234M16—202no—
235M16—202no—
236C2×C8—42no—
237C2×C8—82no—
238C2×C8—202no—
239C4⋊C4—41no—
240C4⋊C4—41no—
241C4⋊C4—41no—
242C4⋊C4—41no—
243C4⋊C4—201no—
244C4⋊C4—201no—
245C4⋊C4—202no—
246C4⋊C4—202no—
247C22⋊C4—202no—
248C22⋊C4—201no—
249C22⋊C4—201no—
250C42—202no—
251C2×C5—21no—
252C2×C5—21no—
253C2×C5—22no—
254C2×C5—41no—
255C2×C5#151611yescharacteristic
256C2×C5C2×#35211yescharacteristic
257C2×C5#213311yescharacteristic
258C23—21no—
259C23—21no—
260C23—22no—
261C23—41no—
262C23#393811yescharacteristic
263C23#393811yescharacteristic
264C23C2×#68511yescharacteristic
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—81no—
284C2×C4—201no—
285C2×C4—201no—
286C2×C4—201no—
287C2×C4—201no—
288C2×C4—202no—
289C2×C4—201no—
290C2×C4—201no—
291C2×C4—201no—
292C2×C4—201no—
293C2×C4—201no—
294C2×C4—201no—
295C2×C4#405012yes—
296C8—82no—
297C8—402no—
298C5#1185011yescharacteristic, complemented, Sylow
299C22—21no—
300C22—21no—
301C22—21no—
302C22—21no—
303C22—21no—
304C22—21no—
305C22—21no—
306C22—42no—
307C22—42no—
308C22#6469212yes—
309C22C2×#393811yescharacteristic
310C22#6540211yescharacteristic
311C22#6542811yescharacteristic
312C4—22no—
313C4—41no—
314C4—41no—
315C4—41no—
316C4—41no—
317C4—41no—
318C4—401no—
319C4—401no—
320C4—401no—
321C4—401no—
322C2—21no—
323C2—21no—
324C2—22no—
325C2—41no—
326C2#1060073611yescharacteristic
327C2C2×#6469211yescharacteristic
328C2#1060092111yescharacteristic
subgroupG/H
centerC22C2×#3938
commutatorC22×C4×C5C22×C4
FrattiniC2×C4⋊C4C22×D5
FittingC2×C5×#324C2
socleC22×C5C2×#99
prime radicalC24×C5C22⋊C4
cosocleC2×C5×C4⋊C4C23
hypercenterC2×#324D5
hypocenterC5#11850
LCSG ▷ C22×C4×C5 ▷ C23×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ C2×#324

Complements

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

NH ≅ G/Ntype
4. #10597950C2semidirect
5. #10597949C2semidirect
298. C5#11850semidirect

Maximal subgroups

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

subgroupindex
#106139072
C2×C5×#3242
#106023552
#105979502
#105979492
#118505