Finite Group Databasebeta

← #12010231

Subgroups — #12010231

Order 1280.

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

normal subgroups only

This group has 412 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#10613912C211yesmaximal, characteristic
2C5×#1821C211yesmaximal, characteristic
3C2×#64748C211yesmaximal, characteristic
4#10597986C212yesmaximal, complemented
5#10597985C212yesmaximal, complemented
6C22×C5×C4⋊C4C2211yescharacteristic
7C2×#4070C411yescharacteristic
8C5×#407—21no—
9#65499—21no—
10#65499—21no—
11C4×C22⋊Dic5C411yescharacteristic
12C2×C22⋊#168—21no—
13#65272C411yescharacteristic
14#65272C2211yescharacteristic
15#65268C411yescharacteristic
16#65268C2211yescharacteristic
17#65261—21no—
18#65261—21no—
19C5×#343—21no—
20C5×#343C2212yes—
21#64748—21no—
22#64748C2212yes—
23#11854—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⋊Dic5C2×C411yescharacteristic
38#4070—41no—
39#4070—41no—
40C2×#683—21no—
41C2×#683—21no—
42C2×#683—21no—
43C2×#683—21no—
44C2×#683—21no—
45C2×C4×Dic5—21no—
46C22×#168—41no—
47C5×C22⋊C8—41no—
48C5×C22⋊C8—41no—
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#2583—51no—
58#1821D511yescharacteristic
59C2×#324—51no—
60#1354—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—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—24no—
93C22⋊Dic5—41no—
94C22⋊Dic5—41no—
95C4⋊Dic5—41no—
96#683—42no—
97#683—42no—
98#683—42no—
99#683—81no—
100#683—81no—
101C4×Dic5—81no—
102C2×#168—42no—
103C2×#168—82no—
104C2×C22⋊Q8—51no—
105C22×C4⋊C4C2×D511yescharacteristic
106#407—21no—
107C2×C22⋊C8—101no—
108#398—51no—
109#398—51no—
110#395—51no—
111#395—51no—
112#389—101no—
113#389—101no—
114#385—101no—
115#385—101no—
116C4×C22⋊C4—51no—
117#343—21no—
118#343C2×D512yes—
119#324—52no—
120#324—101no—
121C23×C5—22no—
122C23×C5—21no—
123C23×C5—21no—
124C23×C5—41no—
125C23×C5#9911yescharacteristic
126C23×C5#9911yescharacteristic
127C23×C5C2×C22⋊C411yescharacteristic
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—21no—
136C2×C4×C5—21no—
137C2×C4×C5—42no—
138C2×C4×C5—42no—
139C2×C4×C5—42no—
140C2×C4×C5—41no—
141C2×C4×C5—41no—
142C2×C4×C5—41no—
143C2×C4×C5—41no—
144C2×C4×C5—81no—
145C2×C4×C5—81no—
146C2×C4×C5—81no—
147C2×C4×C5#13512yes—
148C2×Dic5—22no—
149C2×Dic5—22no—
150C2×Dic5—41no—
151C2×Dic5—41no—
152C2×Dic5—41no—
153C2×Dic5—41no—
154C2×Dic5—41no—
155C2×Dic5—41no—
156C2×Dic5—41no—
157C2×Dic5—41no—
158C2×Dic5—41no—
159C2×Dic5—41no—
160Dic10—82no—
161C5×C8—82no—
162#168—82no—
163C22×Q8—51no—
164C23×C4C4×D511yescharacteristic
165C23×C4D2011yescharacteristic
166C23×C4C22⋊D511yescharacteristic
167C2×M16—22no—
168C22×C8—201no—
169C22⋊Q8—201no—
170C22⋊Q8—201no—
171C2×C4⋊C4—41no—
172C2×C4⋊C4—41no—
173C2×C4⋊C4—51no—
174C2×C4⋊C4—101no—
175C2×C4⋊C4—101no—
176C2×C4⋊C4—101no—
177C2×C4⋊C4—101no—
178C2×C4⋊C4—101no—
179C2×C4⋊C4C4×D511yescharacteristic
180C2×C4⋊C4D2011yescharacteristic
181C2×C4⋊C4C22⋊D511yescharacteristic
182C2×C4⋊C4C22×D511yescharacteristic
183C2×C22⋊C4—101no—
184C2×C22⋊C4—101no—
185C2×C42—101no—
186C22⋊C8—41no—
187C22⋊C8—41no—
188C22⋊C8—102no—
189C22⋊C8—201no—
190#95—51no—
191#95—51no—
192#95—101no—
193#95—101no—
194#95—101no—
195#95—101no—
196C22×C5—21no—
197C22×C5—21no—
198C22×C5—21no—
199C22×C5—21no—
200C22×C5—21no—
201C22×C5—21no—
202C22×C5—21no—
203C22×C5—42no—
204C22×C5—42no—
205C22×C5#35612yes—
206C22×C5C2×#9911yescharacteristic
207C22×C5#41611yescharacteristic
208C22×C5#42111yescharacteristic
209C4×C5—22no—
210C4×C5—41no—
211C4×C5—41no—
212C4×C5—41no—
213C4×C5—41no—
214C4×C5—41no—
215Dic5—42no—
216Dic5—42no—
217Dic5—81no—
218Dic5—81no—
219C24#68511yescharacteristic
220C2×Q8—201no—
221C2×Q8—401no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—22no—
229C22×C4—41no—
230C22×C4—81no—
231C22×C4—81no—
232C22×C4—101no—
233C22×C4—101no—
234C22×C4—101no—
235C22×C4—101no—
236C22×C4#68511yescharacteristic
237C22×C4#68511yescharacteristic
238C22×C4#68511yescharacteristic
239C22×C4C2×C4×D511yescharacteristic
240C22×C4C2×D2011yescharacteristic
241C22×C4C2×C22⋊D511yescharacteristic
242M16—42no—
243M16—42no—
244C2×C8—42no—
245C2×C8—202no—
246C2×C8—402no—
247C4⋊C4—41no—
248C4⋊C4—41no—
249C4⋊C4—41no—
250C4⋊C4—41no—
251C4⋊C4—201no—
252C4⋊C4—202no—
253C4⋊C4—202no—
254C4⋊C4—202no—
255C4⋊C4—401no—
256C4⋊C4—401no—
257C22⋊C4—104no—
258C22⋊C4—201no—
259C22⋊C4—201no—
260C42—401no—
261C2×C5—21no—
262C2×C5—21no—
263C2×C5—22no—
264C2×C5—41no—
265C2×C5#151611yescharacteristic
266C2×C5#212111yescharacteristic
267C2×C5C2×#35611yescharacteristic
268C23—21no—
269C23—21no—
270C23—22no—
271C23—41no—
272C23#393811yescharacteristic
273C23#393811yescharacteristic
274C23C2×#68511yescharacteristic
275Q8—402no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—81no—
293C2×C4—81no—
294C2×C4—81no—
295C2×C4—102no—
296C2×C4—102no—
297C2×C4—201no—
298C2×C4—201no—
299C2×C4—201no—
300C2×C4—201no—
301C2×C4—201no—
302C2×C4—201no—
303C2×C4—201no—
304C2×C4—201no—
305C2×C4—201no—
306C2×C4—201no—
307C2×C4#409612yes—
308C8—82no—
309C8—402no—
310C5#1185411yescharacteristic, complemented, Sylow
311C22—21no—
312C22—21no—
313C22—21no—
314C22—21no—
315C22—21no—
316C22—21no—
317C22—21no—
318C22—42no—
319C22—42no—
320C22#6469812yes—
321C22C2×#393811yescharacteristic
322C22#6529111yescharacteristic
323C22#6529411yescharacteristic
324C4—22no—
325C4—41no—
326C4—41no—
327C4—41no—
328C4—41no—
329C4—41no—
330C4—202no—
331C4—202no—
332C4—401no—
333C4—401no—
334C2—21no—
335C2—21no—
336C2—22no—
337C2—41no—
338C2#1060032511yescharacteristic
339C2#1060088711yescharacteristic
340C2C2×#6469811yescharacteristic
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#11854
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. #10597986C2semidirect
5. #10597985C2semidirect
310. C5#11854semidirect

Maximal subgroups

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

subgroupindex
#106139122
C5×#18212
C2×#647482
#105979862
#105979852
#118545