Finite Group Databasebeta

← #12009973

Subgroups — #12009973

Order 1280.

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

normal subgroups only

This group has 502 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#10614017C211yesmaximal, characteristic
2C5×#1820C211yesmaximal, characteristic, complemented
3#10602365C211yesmaximal, characteristic, complemented
4#10597980C214yesmaximal, complemented
5C2×C5×#117C2211yescharacteristic
6C2×#4076—21no—
7C2×#4020C411yescharacteristic, complemented
8C5×#381—21no—
9#65280—21no—
10#65280C412yescomplemented
11#65249—21no—
12#65249C2212yes—
13#65228—21no—
14#65225C411yescharacteristic, complemented
15#65215—21no—
16C5×#342—21no—
17C5×#342C2212yes—
18#64748—21no—
19#64745C2212yes—
20#11800—51nomaximal, Sylow
21C23×C4×C5D411yescharacteristic
22C22×C22⋊D5—22no—
23C22×D20—21no—
24C22×D20—21no—
25C22×D20C2×C412yes—
26C5×#117—41no—
27C5×#117—41no—
28C2×C5×C4⋊C4C2×C411yescharacteristic
29C2×C5×C4⋊C4D411yescharacteristic
30C2×C5×C22⋊C4—22no—
31C2×C5×C22⋊C4D411yescharacteristic
32C2×C5×C22⋊C4C2311yescharacteristic
33C2×C42×C5C2×C411yescharacteristic
34C2×C42×C5D411yescharacteristic
35#4076—44no—
36C2×#685—22no—
37C2×#685—22no—
38C2×#685—22no—
39C2×C4⋊Dic5—21no—
40#4020—41no—
41#4020—41no—
42C5×#95—41no—
43C5×#95—41no—
44#3963—21no—
45#3963C2×C412yes—
46C22⋊#168—22no—
47C4⋊#168—41no—
48#3935—41no—
49#2619—51no—
50#1820D511yescharacteristic, complemented
51#1464—51no—
52#1343—54no—
53C24×C5C2×D411yescharacteristic
54C23×D5—22no—
55C22×C4×C5—22no—
56C22×C4×C5—21no—
57C22×C4×C5—21no—
58C22×C4×C5—41no—
59C22×C4×C5—42no—
60C22×C4×C5C22⋊C411yescharacteristic
61C22×C4×C5C22⋊C411yescharacteristic
62C22×C4×C5C22⋊C411yescharacteristic
63C22×C4×C5C22⋊C411yescharacteristic
64C22×C4×C5C22×C411yescharacteristic
65C22×C4×C5C2×D411yescharacteristic
66C2×C22⋊D5—44no—
67C2×C22⋊D5—82no—
68C2×C22⋊D5—82no—
69C22×Dic5—22no—
70C2×D20—42no—
71C2×D20—42no—
72C2×D20—42no—
73C2×D20—42no—
74C2×D20—41no—
75C2×D20—41no—
76C2×D20—82no—
77C2×D20—82no—
78C5×C4⋊C4—41no—
79C5×C4⋊C4—41no—
80C5×C22⋊C4—41no—
81C5×C22⋊C4—41no—
82C5×C22⋊C4—42no—
83C5×C22⋊C4—42no—
84C42×C5—21no—
85C42×C5—21no—
86C42×C5—21no—
87C42×C5—21no—
88#685—44no—
89#685—82no—
90#685—82no—
91C4⋊Dic5—42no—
92C2×#168—42no—
93C2×#127—51no—
94C2×#121—101no—
95C2×#117C2×D511yescharacteristic
96#432—101no—
97#394—52no—
98#394—101no—
99#392—52no—
100#392—101no—
101#390—101no—
102#382—51no—
103#381—21no—
104#342—21no—
105#342C2×D512yes—
106#324—101no—
107#323—52no—
108C23×C5—21no—
109C23×C5—21no—
110C23×C5—41no—
111C23×C5—41no—
112C23×C5C2×C22⋊C411yescharacteristic
113C23×C5AGL(1,ℤ/8)12yes—
114C22×D5—42no—
115C22×D5—42no—
116C22×D5—42no—
117C22×D5—42no—
118C22×D5—42no—
119C22×D5—42no—
120C22×D5—42no—
121C2×C4×C5—21no—
122C2×C4×C5—21no—
123C2×C4×C5—21no—
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—42no—
131C2×C4×C5—42no—
132C2×C4×C5—41no—
133C2×C4×C5—41no—
134C2×C4×C5—41no—
135C2×C4×C5—41no—
136C2×C4×C5—82no—
137C2×C4×C5—82no—
138C2×C4×C5—82no—
139C2×C4×C5#9911yescharacteristic
140C2×C4×C5#9911yescharacteristic
141C22⋊D5—88no—
142C2×Dic5—42no—
143C2×Dic5—42no—
144C2×Dic5—42no—
145D20—82no—
146D20—82no—
147D20—84no—
148D20—84no—
149#168—82no—
150C22×D4—52no—
151C22×D4—102no—
152C22×D4—101no—
153C22×D4—101no—
154C23×C4C22⋊D511yescharacteristic
155#127—201no—
156#127—201no—
157#121—204no—
158#117—41no—
159#117—41no—
160C2×C4⋊C4—101no—
161C2×C4⋊C4C4×D511yescharacteristic
162C2×C4⋊C4D2011yescharacteristic
163C2×C22⋊C4—22no—
164C2×C22⋊C4—102no—
165C2×C22⋊C4—102no—
166C2×C22⋊C4—102no—
167C2×C22⋊C4C22⋊D511yescharacteristic
168C2×C22⋊C4C22×D511yescharacteristic
169C2×C42C4×D511yescharacteristic
170C2×C42D2011yescharacteristic
171C4⋊C8—201no—
172C22⋊C8—102no—
173#97—201no—
174#95—41no—
175#95—41no—
176#95—52no—
177#95—101no—
178C22×C5—21no—
179C22×C5—21no—
180C22×C5—22no—
181C22×C5—42no—
182C22×C5—41no—
183C22×C5—41no—
184C22×C5—81no—
185C22×C5C2×#9911yescharacteristic
186C22×C5#41811yescharacteristic
187C22×C5#42111yescharacteristic
188C2×D5—82no—
189C2×D5—82no—
190C2×D5—82no—
191C2×D5—82no—
192C2×D5—82no—
193C2×D5—82no—
194C2×D5—82no—
195C4×C5—21no—
196C4×C5—21no—
197C4×C5—41no—
198C4×C5—41no—
199C4×C5—41no—
200C4×C5—41no—
201C4×C5—41no—
202C4×C5—162no—
203Dic5—82no—
204C24—102no—
205C24C2×C22⋊D511yescharacteristic
206C2×D4—202no—
207C2×D4—202no—
208C2×D4—202no—
209C2×D4—202no—
210C2×D4—204no—
211C2×D4—201no—
212C2×D4—201no—
213C2×D4—402no—
214C2×D4—402no—
215C2×D4—402no—
216C2×D4—402no—
217C22×C4—22no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—41no—
221C22×C4—42no—
222C22×C4—102no—
223C22×C4#68511yescharacteristic
224C22×C4#68511yescharacteristic
225C22×C4#68511yescharacteristic
226C22×C4#68511yescharacteristic
227C22×C4C2×C4×D511yescharacteristic
228C22×C4C2×D2011yescharacteristic
229C2×C8—202no—
230C4⋊C4—41no—
231C4⋊C4—41no—
232C4⋊C4—202no—
233C22⋊C4—41no—
234C22⋊C4—41no—
235C22⋊C4—42no—
236C22⋊C4—42no—
237C22⋊C4—204no—
238C22⋊C4—402no—
239C22⋊C4—402no—
240C42—21no—
241C42—21no—
242C42—21no—
243C42—21no—
244C2×C5—21no—
245C2×C5—21no—
246C2×C5—42no—
247C2×C5#151011yescharacteristic
248C2×C5#212211yescharacteristic
249C2×C5#212811yescharacteristic
250D5—162no—
251C23—21no—
252C23—21no—
253C23—41no—
254C23—41no—
255C23—202no—
256C23—202no—
257C23—202no—
258C23—202no—
259C23—202no—
260C23—202no—
261C23—202no—
262C23C2×#68511yescharacteristic
263C23#409512yes—
264D4—402no—
265D4—402no—
266D4—404no—
267D4—404no—
268D4—408no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—82no—
285C2×C4—82no—
286C2×C4—82no—
287C2×C4—202no—
288C2×C4—202no—
289C2×C4—202no—
290C2×C4#393811yescharacteristic
291C2×C4#393811yescharacteristic
292C8—402no—
293C5#1180011yescharacteristic, complemented, Sylow
294C22—21no—
295C22—21no—
296C22—22no—
297C22—42no—
298C22—41no—
299C22—41no—
300C22—81no—
301C22—402no—
302C22—402no—
303C22—402no—
304C22—402no—
305C22—402no—
306C22—402no—
307C22—402no—
308C22C2×#393811yescharacteristic
309C22#6529011yescharacteristic
310C22#6529411yescharacteristic
311C4—21no—
312C4—21no—
313C4—41no—
314C4—41no—
315C4—41no—
316C4—41no—
317C4—41no—
318C4—162no—
319C4—402no—
320C2—21no—
321C2—21no—
322C2—42no—
323C2—802no—
324C2#1060032411yescharacteristic
325C2#1060085011yescharacteristic
326C2#1060085411yescharacteristic
subgroupG/H
centerC22C2×#3938
commutatorC22×C4×C5C22×C4
FrattiniC2×C22⋊C4C22×D5
FittingC5×#1820C2
socleC22×C5C2×#99
prime radical#10614017C2
cosocleC2×C5×C22⋊C4C23
hypercenter#1820D5
hypocenterC5#11800
LCSG ▷ C22×C4×C5 ▷ C23×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1820

Complements

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

NH ≅ G/Ntype
2. C5×#1820C2semidirect
3. #10602365C2semidirect
4. #10597980C2semidirect
7. C2×#4020C4semidirect
10. #65280C4semidirect
14. #65225C4semidirect
50. #1820D5semidirect
293. C5#11800semidirect

Maximal subgroups

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

subgroupindex
#106140172
C5×#18202
#106023652
#105979802
#118005