Finite Group Databasebeta

← #12010436

Subgroups — #12010436

Order 1280.

508 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 508 conjugacy classes of subgroups — too many to draw. Only its 107 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
3#10602393C211yesmaximal, characteristic
4#10598142C214yesmaximal, complemented
5C2×C5×C4⋊Q8C411yescharacteristic, complemented
6C2×C5×C22⋊Q8—21no—
7C2×C5×C22⋊Q8—21no—
8C2×C4×C5×D4C2211yescharacteristic
9C5×#399C412yescomplemented
10C5×#399C2212yes—
11C5×#390—21no—
12C5×#384—21no—
13C4×C5×C4⋊C4C411yescharacteristic, complemented
14C2×#3966—21no—
15#65501—21no—
16C2×C22⋊#168—21no—
17#64748—21no—
18#64746C2212yes—
19#64745C2212yes—
20#11901—51nomaximal, Sylow
21C22×C5×Q8—21no—
22C22×C5×D4D411yescharacteristic
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—21no—
33C2×C5×C4⋊C4—22no—
34C2×C5×C4⋊C4—22no—
35C2×C5×C4⋊C4C2×C412yes—
36C2×C5×C4⋊C4D411yescharacteristic
37C2×C5×C22⋊C4—22no—
38C2×C5×C22⋊C4—22no—
39C2×C5×C22⋊C4C2×C411yescharacteristic
40C2×C5×C22⋊C4C2311yescharacteristic
41C2×C42×C5—21no—
42C2×C42×C5C2×C411yescharacteristic
43C2×C22⋊Dic5—22no—
44C2×C4⋊Dic5—41no—
45C22×#168—41no—
46C5×#95C2×C412yes—
47#3966—42no—
48#3963—41no—
49C22⋊#168—22no—
50C22⋊#168—41no—
51#2542D511yescharacteristic
52#1799—51no—
53#1459—51no—
54#1352—54no—
55C24×C5C22⋊C411yescharacteristic
56C24×C5C2×D411yescharacteristic
57C2×C5×Q8—42no—
58C2×C5×Q8—42no—
59C2×C5×Q8—41no—
60C2×C5×Q8—41no—
61C2×C5×D4—41no—
62C2×C5×D4—41no—
63C2×C5×D4SD1612yes—
64C2×C5×D4Q1612yes—
65C22×C4×C5—22no—
66C22×C4×C5—21no—
67C22×C4×C5—21no—
68C22×C4×C5—21no—
69C22×C4×C5—21no—
70C22×C4×C5—22no—
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×Dic5—42no—
79C5×C4⋊C4—24no—
80C5×C4⋊C4—41no—
81C5×C4⋊C4—42no—
82C5×C4⋊C4—44no—
83C5×C4⋊C4—44no—
84C5×C4⋊C4—42no—
85C5×C22⋊C4—41no—
86C5×C22⋊C4—41no—
87C5×C22⋊C4—44no—
88C5×C22⋊C4—44no—
89C42×C5—21no—
90C42×C5—21no—
91C42×C5—42no—
92C22⋊Dic5—42no—
93C22⋊Dic5—42no—
94C4⋊Dic5—82no—
95C2×#168—42no—
96C2×#168—82no—
97C2×C4⋊Q8Dic511yescharacteristic, complemented
98C2×C22⋊Q8—21no—
99C2×C22⋊Q8—21no—
100C2×C4×D4C2×D511yescharacteristic
101C2×#99—101no—
102C2×C22⋊C8—101no—
103#399Dic512yescomplemented
104#399C2×D512yes—
105#390—21no—
106#384—21no—
107#380—101no—
108C4×C4⋊C4Dic511yescharacteristic, complemented
109#342—52no—
110#324—101no—
111#323—52no—
112C23×C5—21no—
113C23×C5—21no—
114C23×C5—21no—
115C23×C5—21no—
116C23×C5—41no—
117C23×C5—41no—
118C23×C5—41no—
119C23×C5—41no—
120C23×C5#10312yes—
121C23×C5C2×C22⋊C411yescharacteristic
122C23×C5C2×SD1611yescharacteristic
123C23×C5C2×Q1611yescharacteristic
124C5×Q8—82no—
125C5×Q8—82no—
126C5×D4—42no—
127C5×D4—42no—
128C2×C4×C5—21no—
129C2×C4×C5—21no—
130C2×C4×C5—22no—
131C2×C4×C5—24no—
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—42no—
139C2×C4×C5—41no—
140C2×C4×C5—41no—
141C2×C4×C5—41no—
142C2×C4×C5—42no—
143C2×C4×C5—42no—
144C2×C4×C5—42no—
145C2×C4×C5—81no—
146C2×C4×C5—81no—
147C2×C4×C5#9911yescharacteristic
148C2×C4×C5#9911yescharacteristic
149C2×C4×C5#10312yes—
150C2×Dic5—82no—
151C2×Dic5—82no—
152C2×Dic5—82no—
153#168—82no—
154C22×Q8—21no—
155C22×D4C22⋊D511yescharacteristic
156C23×C4C22⋊D511yescharacteristic
157C23×C4C22⋊D511yescharacteristic
158C22×C8—201no—
159C4⋊Q8—41no—
160C4⋊Q8—41no—
161C22⋊Q8—44no—
162C22⋊Q8—44no—
163C4×D4—41no—
164C4×D4—41no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—201no—
170C2×C4⋊C4C2×Dic512yes—
171C2×C4⋊C4C22⋊D511yescharacteristic
172C2×C22⋊C4—22no—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—102no—
175C2×C22⋊C4C2×Dic511yescharacteristic
176C2×C22⋊C4C22×D511yescharacteristic
177C2×C42—21no—
178C2×C42C2×Dic511yescharacteristic
179#99—202no—
180C22⋊C8—102no—
181C22⋊C8—201no—
182#95—201no—
183#95C2×Dic512yes—
184C22×C5—21no—
185C22×C5—21no—
186C22×C5—21no—
187C22×C5—22no—
188C22×C5—21no—
189C22×C5—42no—
190C22×C5—41no—
191C22×C5—41no—
192C22×C5—41no—
193C22×C5—41no—
194C22×C5—41no—
195C22×C5—41no—
196C22×C5—81no—
197C22×C5—81no—
198C22×C5C2×#9911yescharacteristic
199C22×C5C2×#10311yescharacteristic
200C22×C5#42111yescharacteristic
201C4×C5—21no—
202C4×C5—22no—
203C4×C5—44no—
204C4×C5—41no—
205C4×C5—41no—
206C4×C5—41no—
207C4×C5—82no—
208C4×C5#35312yes—
209Dic5—162no—
210C24C22⋊Dic511yescharacteristic
211C24C2×C22⋊D511yescharacteristic
212C2×Q8—42no—
213C2×Q8—42no—
214C2×Q8—41no—
215C2×Q8—41no—
216C2×D4—41no—
217C2×D4—41no—
218C2×D4#68812yes—
219C2×D4#68912yes—
220C22×C4—22no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—22no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—202no—
229C22×C4C22⋊Dic511yescharacteristic
230C22×C4C22⋊Dic511yescharacteristic
231C22×C4C22⋊Dic511yescharacteristic
232C22×C4C22×Dic511yescharacteristic
233C22×C4C2×C22⋊D511yescharacteristic
234C2×C8—202no—
235C2×C8—402no—
236C4⋊C4—24no—
237C4⋊C4—41no—
238C4⋊C4—42no—
239C4⋊C4—44no—
240C4⋊C4—44no—
241C4⋊C4—42no—
242C4⋊C4—402no—
243C22⋊C4—41no—
244C22⋊C4—41no—
245C22⋊C4—44no—
246C22⋊C4—44no—
247C22⋊C4—202no—
248C22⋊C4—202no—
249C42—21no—
250C42—21no—
251C42—42no—
252C2×C5—21no—
253C2×C5—21no—
254C2×C5—41no—
255C2×C5—42no—
256C2×C5—41no—
257C2×C5#152011yescharacteristic
258C2×C5#212311yescharacteristic
259C2×C5C2×#35311yescharacteristic
260C23—21no—
261C23—21no—
262C23—21no—
263C23—21no—
264C23—41no—
265C23—41no—
266C23—41no—
267C23—41no—
268C23#396712yes—
269C23C2×#68811yescharacteristic
270C23C2×#68911yescharacteristic
271C23C2×C22⋊Dic511yescharacteristic
272Q8—82no—
273Q8—82no—
274D4—42no—
275D4—42no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—22no—
279C2×C4—24no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—81no—
294C2×C4—81no—
295C2×C4—402no—
296C2×C4—402no—
297C2×C4—402no—
298C2×C4#396611yescharacteristic
299C2×C4#396611yescharacteristic
300C2×C4#396712yes—
301C8—402no—
302C5#1190111yescharacteristic, complemented, Sylow
303C22—21no—
304C22—21no—
305C22—21no—
306C22—22no—
307C22—21no—
308C22—42no—
309C22—41no—
310C22—41no—
311C22—41no—
312C22—41no—
313C22—41no—
314C22—41no—
315C22—81no—
316C22—81no—
317C22C2×#396611yescharacteristic
318C22C2×#396711yescharacteristic
319C22#6552511yescharacteristic
320C4—21no—
321C4—22no—
322C4—44no—
323C4—41no—
324C4—41no—
325C4—41no—
326C4—82no—
327C4—802no—
328C4#6476512yes—
329C2—21no—
330C2—21no—
331C2—41no—
332C2—42no—
333C2—41no—
334C2#1060248311yescharacteristic
335C2C2×#6476511yescharacteristic
336C2#1060266311yescharacteristic
subgroupG/H
centerC22C2×#3966
commutatorC22×C4×C5C22×C4
FrattiniC2×C22⋊C4C22×D5
FittingC5×#2542C2
socleC22×C5C2×#99
prime radicalC22×C5×D4D4
cosocleC2×C5×C22⋊C4C23
hypercenter#2542D5
hypocenterC5#11901
LCSG ▷ C22×C4×C5 ▷ C23×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
4. #10598142C2semidirect
5. C2×C5×C4⋊Q8C4semidirect
9. C5×#399C4semidirect
13. C4×C5×C4⋊C4C4semidirect
97. C2×C4⋊Q8Dic5semidirect
103. #399Dic5semidirect
108. C4×C4⋊C4Dic5semidirect
302. C5#11901semidirect

Maximal subgroups

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

subgroupindex
C5×#25422
#106024122
#106023932
#105981422
#119015