Finite Group Databasebeta

← #12017971

Subgroups — #12017971

Order 1280.

556 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 111 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
1C5×#2542C211yesmaximal, characteristic
2#10602412C211yesmaximal, characteristic
3C4×#3966C211yesmaximal, characteristic
4#10598132C214yesmaximal
5C2×C5×C4⋊Q8C411yescharacteristic, complemented
6C2×C5×C22⋊Q8—21no—
7C2×C5×C22⋊Q8—21no—
8C2×C4×C5×D4C2211yescharacteristic
9C5×#399—21no—
10C5×#399—21no—
11C5×#390C2212yes—
12C5×#384C412yescomplemented
13C4×C5×C4⋊C4C411yescharacteristic, complemented
14C2×#3966C2212yes—
15C2×#3966C2212yes—
16#65501—21no—
17C4×C22⋊Dic5—21no—
18#64746—21no—
19#64746—21no—
20#13298—51nomaximal, Sylow
21C22×C5×Q8—21no—
22C22×C5×D4C2311yescharacteristic
23C23×C4×C5D411yescharacteristic
24C23×C4×C5D411yescharacteristic
25C5×C4⋊Q8—41no—
26C5×C4⋊Q8—41no—
27C5×C22⋊Q8—44no—
28C5×C22⋊Q8—44no—
29C4×C5×D4—41no—
30C4×C5×D4—41no—
31C2×C5×C4⋊C4—21no—
32C2×C5×C4⋊C4—22no—
33C2×C5×C4⋊C4—22no—
34C2×C5×C4⋊C4—21no—
35C2×C5×C4⋊C4C2×C412yes—
36C2×C5×C4⋊C4C2×C411yescharacteristic
37C2×C5×C22⋊C4—22no—
38C2×C5×C22⋊C4—22no—
39C2×C5×C22⋊C4D411yescharacteristic
40C2×C5×C22⋊C4D411yescharacteristic
41C2×C42×C5C2×C411yescharacteristic
42C2×C42×C5C2×C412yes—
43C2×C22⋊Dic5—22no—
44C2×C22⋊Dic5—22no—
45C2×C4⋊Dic5—41no—
46C2×C4×Dic5—41no—
47C5×#95—21no—
48#3966—22no—
49#3966—24no—
50#3966Q814yescomplemented
51#3963—41no—
52#3963—41no—
53#2542D511yescharacteristic
54#1799—51no—
55C4×#99—51no—
56#1393—54no—
57C24×C5C2×D411yescharacteristic
58C24×C5C2×D411yescharacteristic
59C2×C5×Q8—42no—
60C2×C5×Q8—42no—
61C2×C5×Q8—41no—
62C2×C5×Q8—41no—
63C2×C5×D4—41no—
64C2×C5×D4—41no—
65C2×C5×D4C2×Q812yes—
66C2×C5×D4#4112yes—
67C22×C4×C5—22no—
68C22×C4×C5—21no—
69C22×C4×C5—21no—
70C22×C4×C5—21no—
71C22×C4×C5—21no—
72C22×C4×C5—22no—
73C22×C4×C5—42no—
74C22×C4×C5—42no—
75C22×C4×C5C22⋊C411yescharacteristic
76C22×C4×C5C22⋊C411yescharacteristic
77C22×C4×C5C22⋊C411yescharacteristic
78C22×C4×C5C22⋊C411yescharacteristic
79C22×C4×C5C22×C411yescharacteristic
80C22×Dic5—42no—
81C22×Dic5—42no—
82C5×C4⋊C4—24no—
83C5×C4⋊C4—41no—
84C5×C4⋊C4—42no—
85C5×C4⋊C4—44no—
86C5×C4⋊C4—44no—
87C5×C4⋊C4—42no—
88C5×C22⋊C4—41no—
89C5×C22⋊C4—41no—
90C5×C22⋊C4—44no—
91C5×C22⋊C4—44no—
92C42×C5—21no—
93C42×C5—21no—
94C42×C5—42no—
95C22⋊Dic5—24no—
96C22⋊Dic5—42no—
97C22⋊Dic5—42no—
98C22⋊Dic5—42no—
99C4⋊Dic5—82no—
100C4×Dic5—82no—
101C2×C4⋊Q8Dic511yescharacteristic, complemented
102C2×C22⋊Q8—21no—
103C2×C22⋊Q8—21no—
104C2×C4×D4C2×D511yescharacteristic
105C2×#99—52no—
106C2×#99—52no—
107#399—21no—
108#399—21no—
109#390C2×D512yes—
110#384Dic512yescomplemented
111#380—101no—
112C4×C4⋊C4Dic511yescharacteristic, complemented
113C4×C22⋊C4—101no—
114#342—101no—
115#342—101no—
116C23×C5—21no—
117C23×C5—21no—
118C23×C5—21no—
119C23×C5—21no—
120C23×C5—41no—
121C23×C5—41no—
122C23×C5—41no—
123C23×C5—41no—
124C23×C5C2×C22⋊C411yescharacteristic
125C23×C5C22⋊Q812yes—
126C23×C5#12411yescharacteristic
127C23×C5C4⋊Q811yescharacteristic
128C5×Q8—82no—
129C5×Q8—82no—
130C5×D4—42no—
131C5×D4—42no—
132C2×C4×C5—21no—
133C2×C4×C5—21no—
134C2×C4×C5—22no—
135C2×C4×C5—24no—
136C2×C4×C5—21no—
137C2×C4×C5—21no—
138C2×C4×C5—21no—
139C2×C4×C5—42no—
140C2×C4×C5—42no—
141C2×C4×C5—42no—
142C2×C4×C5—42no—
143C2×C4×C5—41no—
144C2×C4×C5—41no—
145C2×C4×C5—41no—
146C2×C4×C5—42no—
147C2×C4×C5—42no—
148C2×C4×C5—42no—
149C2×C4×C5—81no—
150C2×C4×C5—81no—
151C2×C4×C5#9911yescharacteristic
152C2×C4×C5#9911yescharacteristic
153C2×C4×C5C4×Q812yes—
154C2×Dic5—44no—
155C2×Dic5—82no—
156C2×Dic5—82no—
157C2×Dic5—82no—
158C2×Dic5—82no—
159C2×Dic5—82no—
160C22×Q8—21no—
161C22×D4C22×D511yescharacteristic
162C23×C4C22⋊D511yescharacteristic
163C23×C4C22⋊D511yescharacteristic
164C4⋊Q8—41no—
165C4⋊Q8—41no—
166C22⋊Q8—44no—
167C22⋊Q8—44no—
168C4×D4—41no—
169C4×D4—41no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—201no—
175C2×C4⋊C4C2×Dic512yes—
176C2×C4⋊C4C2×Dic511yescharacteristic
177C2×C22⋊C4—22no—
178C2×C22⋊C4—22no—
179C2×C22⋊C4—102no—
180C2×C22⋊C4—102no—
181C2×C22⋊C4C22⋊D511yescharacteristic
182C2×C22⋊C4C22⋊D511yescharacteristic
183C2×C42—201no—
184C2×C42C2×Dic511yescharacteristic
185C2×C42C2×Dic512yes—
186#99—54no—
187#99—102no—
188#99—104no—
189#95—21no—
190#95—201no—
191#95—201no—
192C22×C5—21no—
193C22×C5—21no—
194C22×C5—22no—
195C22×C5—21no—
196C22×C5—21no—
197C22×C5—42no—
198C22×C5—41no—
199C22×C5—41no—
200C22×C5—41no—
201C22×C5—41no—
202C22×C5—41no—
203C22×C5—41no—
204C22×C5—81no—
205C22×C5—81no—
206C22×C5#39111yescharacteristic
207C22×C5C2×#9911yescharacteristic
208C22×C5#41011yescharacteristic
209C4×C5—21no—
210C4×C5—22no—
211C4×C5—44no—
212C4×C5—41no—
213C4×C5—41no—
214C4×C5—41no—
215C4×C5—82no—
216C4×C5#35312yes—
217Dic5—84no—
218Dic5—162no—
219C24C2×C22⋊D511yescharacteristic
220C24C2×C22⋊D511yescharacteristic
221C2×Q8—42no—
222C2×Q8—42no—
223C2×Q8—41no—
224C2×Q8—41no—
225C2×D4—41no—
226C2×D4—41no—
227C2×D4Q8×D512yes—
228C2×D4#71312yes—
229C22×C4—22no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—22no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—202no—
238C22×C4—202no—
239C22×C4C22⋊Dic511yescharacteristic
240C22×C4C22⋊Dic511yescharacteristic
241C22×C4C22⋊Dic511yescharacteristic
242C22×C4C22⋊Dic511yescharacteristic
243C22×C4C22×Dic511yescharacteristic
244C4⋊C4—24no—
245C4⋊C4—41no—
246C4⋊C4—42no—
247C4⋊C4—44no—
248C4⋊C4—44no—
249C4⋊C4—42no—
250C4⋊C4—402no—
251C22⋊C4—41no—
252C22⋊C4—41no—
253C22⋊C4—44no—
254C22⋊C4—44no—
255C22⋊C4—104no—
256C22⋊C4—202no—
257C22⋊C4—202no—
258C22⋊C4—202no—
259C42—21no—
260C42—21no—
261C42—42no—
262C42—402no—
263C2×C5—21no—
264C2×C5—21no—
265C2×C5—42no—
266C2×C5—41no—
267C2×C5—41no—
268C2×C5#199611yescharacteristic
269C2×C5C2×#35311yescharacteristic
270C2×C5#212911yescharacteristic
271C23—21no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23—41no—
276C23—41no—
277C23—41no—
278C23—41no—
279C23#409011yescharacteristic
280C23#409212yes—
281C23#409311yescharacteristic
282C23C2×C22⋊Dic511yescharacteristic
283Q8—82no—
284Q8—82no—
285D4—42no—
286D4—42no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—22no—
290C2×C4—24no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—81no—
305C2×C4—81no—
306C2×C4—204no—
307C2×C4—402no—
308C2×C4—402no—
309C2×C4—402no—
310C2×C4—402no—
311C2×C4—402no—
312C2×C4#396611yescharacteristic
313C2×C4#396611yescharacteristic
314C2×C4Q8×Dic512yes—
315C5#1329811yescharacteristic, complemented, Sylow
316C22—21no—
317C22—21no—
318C22—22no—
319C22—21no—
320C22—21no—
321C22—41no—
322C22—42no—
323C22—41no—
324C22—41no—
325C22—41no—
326C22—41no—
327C22—41no—
328C22—81no—
329C22—81no—
330C22C2×#396611yescharacteristic
331C22#6551911yescharacteristic
332C22#6552911yescharacteristic
333C4—21no—
334C4—22no—
335C4—44no—
336C4—41no—
337C4—41no—
338C4—41no—
339C4—82no—
340C4—404no—
341C4—802no—
342C4#6476512yes—
343C2—21no—
344C2—21no—
345C2—42no—
346C2—41no—
347C2—41no—
348C2#1060249411yescharacteristic
349C2C2×#6476511yescharacteristic
350C2#1060266411yescharacteristic
subgroupG/H
centerC22C2×#3966
commutatorC22×C4×C5C22×C4
FrattiniC22×D4C22×D5
FittingC5×#2542C2
socleC22×C5C2×#99
prime radical, cosocleC22×C5×D4C23
hypercenter#2542D5
hypocenterC5#13298
LCSG ▷ C22×C4×C5 ▷ C22×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2542

Complements

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

NH ≅ G/Ntype
5. C2×C5×C4⋊Q8C4semidirect
12. C5×#384C4semidirect
13. C4×C5×C4⋊C4C4semidirect
50. #3966Q8semidirect
101. C2×C4⋊Q8Dic5semidirect
110. #384Dic5semidirect
112. C4×C4⋊C4Dic5semidirect
315. C5#13298semidirect

Maximal subgroups

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

subgroupindex
C5×#25422
#106024122
C4×#39662
#105981322
#132985