Finite Group Databasebeta

← #12009592

Subgroups — #12009592

Order 1280.

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

normal subgroups only

This group has 448 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
1#10613995C211yesmaximal, characteristic
2C5×#1740C211yesmaximal, characteristic
3#10602364C211yesmaximal, characteristic
4#10597983C214yesmaximal
5C2×C5×#117—21no—
6C2×C5×#117C2211yescharacteristic
7C2×#4015C411yescharacteristic, complemented
8#65501—21no—
9#65275C412yescomplemented
10#65268—21no—
11#65238C2212yes—
12#65236—21no—
13#65225C411yescharacteristic, complemented
14#65217—21no—
15#65214—21no—
16C5×#342—21no—
17C5×#342C2212yes—
18#64748—21no—
19#64745C2212yes—
20#11718—51nomaximal, Sylow
21C23×C4×C5D411yescharacteristic
22C22×Dic10—21no—
23C5×#117—24no—
24C5×#117—41no—
25C5×#117—41no—
26C5×#117—42no—
27C2×C5×C4⋊C4—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×C5—41no—
34C2×C42×C5C2×C411yescharacteristic
35C2×C42×C5D411yescharacteristic
36C2×C22⋊Dic5—22no—
37C2×C4⋊Dic5—21no—
38C2×C4⋊Dic5—21no—
39C2×C4⋊Dic5C2×C412yes—
40C2×#683—22no—
41#4015—41no—
42#4015—41no—
43#3963—22no—
44#3963—21no—
45#3963—22no—
46#3963C2×C412yes—
47C22⋊#168—22no—
48C4⋊#168—41no—
49C4×#168—41no—
50#2628—51no—
51#1740D511yescharacteristic
52#1458—51no—
53#1352—54no—
54C24×C5C2×D411yescharacteristic
55C22×C4×C5—21no—
56C22×C4×C5—21no—
57C22×C4×C5—22no—
58C22×C4×C5—42no—
59C22×C4×C5C22⋊C411yescharacteristic
60C22×C4×C5C22⋊C411yescharacteristic
61C22×C4×C5C22⋊C411yescharacteristic
62C22×C4×C5C22⋊C411yescharacteristic
63C22×C4×C5SD1612yes—
64C22×C4×C5Q1612yes—
65C22×C4×C5C22×C411yescharacteristic
66C22×C4×C5C2×D411yescharacteristic
67C22×Dic5—22no—
68C22×Dic5—22no—
69C2×Dic10—42no—
70C2×Dic10—82no—
71C5×C4⋊C4—41no—
72C5×C4⋊C4—41no—
73C5×C4⋊C4—44no—
74C5×C22⋊C4—24no—
75C5×C22⋊C4—41no—
76C5×C22⋊C4—41no—
77C5×C22⋊C4—42no—
78C42×C5—21no—
79C42×C5—21no—
80C42×C5—21no—
81C42×C5—21no—
82C42×C5—44no—
83C22⋊Dic5—82no—
84C4⋊Dic5—41no—
85C4⋊Dic5—41no—
86C4⋊Dic5—42no—
87C4⋊Dic5—42no—
88#683—82no—
89C2×#168—42no—
90C2×C4⋊Q8—51no—
91C2×#117—21no—
92C2×#117C2×D511yescharacteristic
93#431—101no—
94#400—52no—
95#399—52no—
96#395—101no—
97#393—101no—
98#389—101no—
99#382—51no—
100#380—101no—
101#342—21no—
102#342C2×D512yes—
103#324—101no—
104#323—52no—
105C23×C5—21no—
106C23×C5—21no—
107C23×C5—41no—
108C23×C5—41no—
109C23×C5C2×C22⋊C411yescharacteristic
110C23×C5C2×SD1611yescharacteristic
111C23×C5C2×Q1611yescharacteristic
112C2×C4×C5—21no—
113C2×C4×C5—21no—
114C2×C4×C5—22no—
115C2×C4×C5—22no—
116C2×C4×C5—22no—
117C2×C4×C5—22no—
118C2×C4×C5—21no—
119C2×C4×C5—21no—
120C2×C4×C5—21no—
121C2×C4×C5—21no—
122C2×C4×C5—41no—
123C2×C4×C5—41no—
124C2×C4×C5—41no—
125C2×C4×C5—41no—
126C2×C4×C5—44no—
127C2×C4×C5—82no—
128C2×C4×C5—82no—
129C2×C4×C5#9911yescharacteristic
130C2×C4×C5#9911yescharacteristic
131C2×C4×C5#10312yes—
132C2×C4×C5#10312yes—
133C2×C4×C5#10412yes—
134C2×Dic5—42no—
135C2×Dic5—42no—
136C2×Dic5—42no—
137C2×Dic5—42no—
138C2×Dic5—42no—
139C2×Dic5—42no—
140Dic10—84no—
141#168—82no—
142C22×Q8—101no—
143C23×C4C22⋊D511yescharacteristic
144C4⋊Q8—201no—
145C4⋊Q8—201no—
146#117—24no—
147#117—41no—
148#117—41no—
149#117—42no—
150C2×C4⋊C4—41no—
151C2×C4⋊C4—52no—
152C2×C4⋊C4—101no—
153C2×C4⋊C4—101no—
154C2×C4⋊C4—102no—
155C2×C4⋊C4C4×D511yescharacteristic
156C2×C4⋊C4D2011yescharacteristic
157C2×C22⋊C4—22no—
158C2×C22⋊C4—102no—
159C2×C22⋊C4C22⋊D511yescharacteristic
160C2×C22⋊C4C22×D511yescharacteristic
161C2×C42—41no—
162C2×C42C4×D511yescharacteristic
163C2×C42D2011yescharacteristic
164C4⋊C8—201no—
165C22⋊C8—102no—
166C4×C8—201no—
167#95—52no—
168#95—102no—
169#95—101no—
170#95—102no—
171C22×C5—21no—
172C22×C5—21no—
173C22×C5—21no—
174C22×C5—21no—
175C22×C5—41no—
176C22×C5—41no—
177C22×C5—41no—
178C22×C5—41no—
179C22×C5—81no—
180C22×C5C2×#9911yescharacteristic
181C22×C5C2×#10311yescharacteristic
182C22×C5C2×#10411yescharacteristic
183C4×C5—22no—
184C4×C5—41no—
185C4×C5—41no—
186C4×C5—41no—
187C4×C5—41no—
188C4×C5—84no—
189C4×C5#32812yes—
190C4×C5#32812yes—
191Dic5—82no—
192Dic5—82no—
193C24C2×C22⋊D511yescharacteristic
194C2×Q8—202no—
195C2×Q8—402no—
196C22×C4—22no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—42no—
200C22×C4—102no—
201C22×C4—102no—
202C22×C4#68511yescharacteristic
203C22×C4#68511yescharacteristic
204C22×C4#68511yescharacteristic
205C22×C4#68511yescharacteristic
206C22×C4#68712yes—
207C22×C4#68912yes—
208C22×C4C2×C4×D511yescharacteristic
209C22×C4C2×D2011yescharacteristic
210C2×C8—202no—
211C4⋊C4—41no—
212C4⋊C4—41no—
213C4⋊C4—44no—
214C4⋊C4—202no—
215C4⋊C4—202no—
216C4⋊C4—201no—
217C4⋊C4—201no—
218C4⋊C4—402no—
219C22⋊C4—24no—
220C22⋊C4—41no—
221C22⋊C4—41no—
222C22⋊C4—42no—
223C22⋊C4—402no—
224C42—21no—
225C42—21no—
226C42—21no—
227C42—21no—
228C42—44no—
229C2×C5—21no—
230C2×C5—21no—
231C2×C5—41no—
232C2×C5—41no—
233C2×C5C2×#32811yescharacteristic
234C2×C5#212311yescharacteristic
235C2×C5#212811yescharacteristic
236C23—21no—
237C23—21no—
238C23—41no—
239C23—41no—
240C23C2×#68511yescharacteristic
241C23C2×#68711yescharacteristic
242C23C2×#68911yescharacteristic
243Q8—404no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—44no—
259C2×C4—82no—
260C2×C4—82no—
261C2×C4—202no—
262C2×C4—202no—
263C2×C4—202no—
264C2×C4—202no—
265C2×C4—202no—
266C2×C4—202no—
267C2×C4#393712yes—
268C2×C4#393811yescharacteristic
269C2×C4#393811yescharacteristic
270C2×C4#394212yes—
271C2×C4#394212yes—
272C8—402no—
273C5#1171811yescharacteristic, complemented, Sylow
274C22—21no—
275C22—21no—
276C22—21no—
277C22—21no—
278C22—41no—
279C22—41no—
280C22—41no—
281C22—41no—
282C22—81no—
283C22C2×#393711yescharacteristic
284C22C2×#393811yescharacteristic
285C22C2×#394211yescharacteristic
286C4—22no—
287C4—41no—
288C4—41no—
289C4—41no—
290C4—41no—
291C4—84no—
292C4—402no—
293C4—402no—
294C4#6467212yes—
295C4#6467212yes—
296C2—21no—
297C2—21no—
298C2—41no—
299C2—41no—
300C2C2×#6467211yescharacteristic
301C2#1060085111yescharacteristic
302C2#1060085411yescharacteristic
subgroupG/H
centerC22C2×#3938
commutatorC22×C4×C5C22×C4
FrattiniC2×C22⋊C4C22×D5
FittingC5×#1740C2
socleC22×C5C2×#99
prime radicalC24×C5C2×D4
cosocleC2×C5×C22⋊C4C23
hypercenter#1740D5
hypocenterC5#11718
LCSG ▷ C22×C4×C5 ▷ C23×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1740

Complements

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

NH ≅ G/Ntype
7. C2×#4015C4semidirect
9. #65275C4semidirect
13. #65225C4semidirect
273. C5#11718semidirect

Maximal subgroups

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

subgroupindex
#106139952
C5×#17402
#106023642
#105979832
#117185