Finite Group Databasebeta

← #12010046

Subgroups — #12010046

Order 1280.

386 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 79 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#10613914C211yesmaximal, characteristic
2C5×#1821C211yesmaximal, characteristic
3#10602345C211yesmaximal, characteristic
4#10597987C212yesmaximal, complemented
5#10597985C212yesmaximal, complemented
6C22×C5×C4⋊C4C2211yescharacteristic
7C2×#4070C411yescharacteristic
8C5×#407—21no—
9#65501C411yescharacteristic
10#65500—21no—
11#65499—21no—
12#65496—21no—
13#65275C411yescharacteristic
14#65275C2211yescharacteristic
15#65273—21no—
16#65273—21no—
17#65268C411yescharacteristic
18#65268C2211yescharacteristic
19C5×#343—21no—
20C5×#343C2212yes—
21#64748—21no—
22#64748C2212yes—
23#11817—51nomaximal, Sylow
24C23×C4×C5C2×C411yescharacteristic
25C23×C4×C5D411yescharacteristic
26C23×C4×C5D411yescharacteristic
27C22×Dic10C2×C411yescharacteristic
28C2×C5×M16—22no—
29C2×C5×C4⋊C4—41no—
30C2×C5×C4⋊C4—41no—
31C2×C5×C4⋊C4C2×C411yescharacteristic
32C2×C5×C4⋊C4D411yescharacteristic
33C2×C5×C4⋊C4D411yescharacteristic
34C2×C5×C4⋊C4C2311yescharacteristic
35C2×C22⋊Dic5—21no—
36C2×C22⋊Dic5—21no—
37C2×C4⋊Dic5—21no—
38C2×C4⋊Dic5C2×C411yescharacteristic
39#4070—41no—
40#4070—41no—
41C2×#683—21no—
42C2×#683—21no—
43C2×#683—21no—
44C2×#681—41no—
45C5×C22⋊C8—41no—
46C5×C22⋊C8—41no—
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—22no—
56C22⋊#168—41no—
57#2627—51no—
58#1821D511yescharacteristic
59#1463—51no—
60#1354—52no—
61#1353—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—81no—
72C22×C4×C5—81no—
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×Dic10—41no—
84C2×Dic10—81no—
85C5×M16—42no—
86C5×M16—42no—
87C2×C5×C8—42no—
88C5×C4⋊C4—41no—
89C5×C4⋊C4—41no—
90C5×C4⋊C4—41no—
91C5×C4⋊C4—41no—
92C22⋊Dic5—42no—
93C22⋊Dic5—41no—
94C22⋊Dic5—41no—
95C4⋊Dic5—41no—
96C4⋊Dic5—81no—
97#683—42no—
98#683—81no—
99#683—81no—
100#681—82no—
101C2×#168—42no—
102C2×C22⋊Q8—51no—
103C22×C4⋊C4C2×D511yescharacteristic
104#407—21no—
105#407—101no—
106#400—51no—
107#400—51no—
108#398—101no—
109#398—101no—
110#395—51no—
111#395—51no—
112#385—101no—
113#381—101no—
114#380—51no—
115#343—21no—
116#343C2×D512yes—
117#324—52no—
118#324—101no—
119C23×C5—21no—
120C23×C5—21no—
121C23×C5—41no—
122C23×C5—41no—
123C23×C5#9911yescharacteristic
124C23×C5#9911yescharacteristic
125C23×C5C2×C22⋊C411yescharacteristic
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—21no—
134C2×C4×C5—21no—
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—81no—
144C2×C4×C5—81no—
145C2×C4×C5#13712yes—
146C2×Dic5—41no—
147C2×Dic5—41no—
148C2×Dic5—41no—
149C2×Dic5—41no—
150C2×Dic5—41no—
151C2×Dic5—41no—
152C2×Dic5—42no—
153C2×Dic5—41no—
154C2×Dic5—41no—
155C2×Dic5—41no—
156C2×Dic5—41no—
157Dic10—82no—
158C5×C8—82no—
159#168—82no—
160C22×Q8—51no—
161C23×C4C4×D511yescharacteristic
162C23×C4D2011yescharacteristic
163C23×C4C22⋊D511yescharacteristic
164C2×M16—22no—
165C2×M16—201no—
166C22⋊Q8—201no—
167C22⋊Q8—201no—
168C2×C4⋊C4—41no—
169C2×C4⋊C4—41no—
170C2×C4⋊C4—51no—
171C2×C4⋊C4—101no—
172C2×C4⋊C4—101no—
173C2×C4⋊C4—101no—
174C2×C4⋊C4—101no—
175C2×C4⋊C4C4×D511yescharacteristic
176C2×C4⋊C4D2011yescharacteristic
177C2×C4⋊C4C22⋊D511yescharacteristic
178C2×C4⋊C4C22×D511yescharacteristic
179C2×C22⋊C4—101no—
180C2×C22⋊C4—101no—
181C22⋊C8—41no—
182C22⋊C8—41no—
183C22⋊C8—102no—
184C22⋊C8—201no—
185#95—51no—
186#95—51no—
187#95—101no—
188#95—101no—
189#95—101no—
190#95—101no—
191#95—101no—
192#95—101no—
193C22×C5—21no—
194C22×C5—21no—
195C22×C5—21no—
196C22×C5—21no—
197C22×C5—21no—
198C22×C5—21no—
199C22×C5—21no—
200C22×C5—21no—
201C22×C5—42no—
202C22×C5—42no—
203C22×C5C2×#9911yescharacteristic
204C22×C5#41911yescharacteristic
205C22×C5#42111yescharacteristic
206C4×C5—22no—
207C4×C5—41no—
208C4×C5—41no—
209C4×C5—41no—
210C4×C5—41no—
211C4×C5—41no—
212Dic5—81no—
213Dic5—81no—
214Dic5—81no—
215Dic5—81no—
216C24#68511yescharacteristic
217C2×Q8—201no—
218C2×Q8—401no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—22no—
226C22×C4—41no—
227C22×C4—81no—
228C22×C4—81no—
229C22×C4—101no—
230C22×C4—101no—
231C22×C4—101no—
232C22×C4—101no—
233C22×C4#68511yescharacteristic
234C22×C4#68511yescharacteristic
235C22×C4#68511yescharacteristic
236C22×C4C2×C4×D511yescharacteristic
237C22×C4C2×D2011yescharacteristic
238C22×C4C2×C22⋊D511yescharacteristic
239M16—42no—
240M16—42no—
241M16—402no—
242C2×C8—42no—
243C2×C8—202no—
244C4⋊C4—41no—
245C4⋊C4—41no—
246C4⋊C4—41no—
247C4⋊C4—41no—
248C4⋊C4—201no—
249C4⋊C4—202no—
250C4⋊C4—401no—
251C4⋊C4—401no—
252C4⋊C4—401no—
253C22⋊C4—202no—
254C22⋊C4—201no—
255C22⋊C4—201no—
256C2×C5—21no—
257C2×C5—21no—
258C2×C5—41no—
259C2×C5—41no—
260C2×C5#151111yescharacteristic
261C2×C5#212111yescharacteristic
262C2×C5#213311yescharacteristic
263C23—21no—
264C23—21no—
265C23—41no—
266C23—41no—
267C23#393811yescharacteristic
268C23#393811yescharacteristic
269C23C2×#68511yescharacteristic
270Q8—402no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—81no—
288C2×C4—81no—
289C2×C4—81no—
290C2×C4—201no—
291C2×C4—201no—
292C2×C4—201no—
293C2×C4—201no—
294C2×C4—201no—
295C2×C4—201no—
296C2×C4—202no—
297C2×C4—201no—
298C2×C4—201no—
299C2×C4—201no—
300C2×C4—201no—
301C2×C4#409712yes—
302C8—82no—
303C8—402no—
304C5#1181711yescharacteristic, complemented, Sylow
305C22—21no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—21no—
312C22—21no—
313C22—42no—
314C22—42no—
315C22C2×#393811yescharacteristic
316C22#6529311yescharacteristic
317C22#6529411yescharacteristic
318C4—22no—
319C4—41no—
320C4—41no—
321C4—41no—
322C4—41no—
323C4—41no—
324C4—401no—
325C4—401no—
326C4—401no—
327C4—401no—
328C2—21no—
329C2—21no—
330C2—41no—
331C2—41no—
332C2#1060032611yescharacteristic
333C2#1060088711yescharacteristic
334C2#1060092111yescharacteristic
subgroupG/H
centerC22C2×#3938
commutatorC22×C4×C5C22×C4
FrattiniC2×C4⋊C4C22×D5
FittingC5×#1821C2
socleC22×C5C2×#99
prime radicalC24×C5C22⋊C4
cosocleC2×C5×C4⋊C4C23
hypercenter#1821D5
hypocenterC5#11817
LCSG ▷ C22×C4×C5 ▷ C23×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ #1821

Complements

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

NH ≅ G/Ntype
4. #10597987C2semidirect
5. #10597985C2semidirect
304. C5#11817semidirect

Maximal subgroups

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

subgroupindex
#106139142
C5×#18212
#106023452
#105979872
#105979852
#118175