Finite Group Databasebeta

← #13023198

Subgroups — #13023198

Order 1280.

434 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 434 conjugacy classes of subgroups — too many to draw. Only its 104 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#10613888C211yesmaximal, characteristic
2#10604877C211yesmaximal, characteristic
3#10604877C211yesmaximal, characteristic
4#10598820C214yesmaximal
5C22×C5×C4⋊C4C411yescharacteristic
6C2×#4072C411yescharacteristic
7C2×#4072C2211yescharacteristic
8#65748—21no—
9#65748—21no—
10#65499C412yes—
11#65499C2212yes—
12#65266—21no—
13#65265—21no—
14#65262—21no—
15#65262—21no—
16#64933C2212yes—
17#64933C2212yes—
18#64913—21no—
19#64913—21no—
20#11454—51nomaximal, Sylow
21C23×C4×C5C2×C412yes—
22C23×C4×C5C2×C411yescharacteristic
23C2×C5×C4⋊C4—21no—
24C2×C5×C4⋊C4—21no—
25C2×C5×C4⋊C4—41no—
26C2×C5×C4⋊C4—41no—
27C2×C22⋊Dic5C2×C412yes—
28C2×C22⋊Dic5C2×C411yescharacteristic
29C2×C22⋊Dic5C2311yescharacteristic
30#4072—22no—
31#4072—41no—
32#4072—41no—
33#4072C814yescomplemented
34C2×C4⋊Dic5—21no—
35C2×C4⋊Dic5D411yescharacteristic
36C2×C4⋊Dic5D411yescharacteristic
37C2×#683—22no—
38C2×#683—22no—
39C2×C4×Dic5—21no—
40C2×C4×Dic5D411yescharacteristic
41C2×C4×Dic5D411yescharacteristic
42C22⋊#170—22no—
43C22⋊#170—22no—
44C4⋊#170—41no—
45C4⋊#170—41no—
46C4×#170—41no—
47C4×#170—41no—
48#3963—22no—
49#3963—22no—
50#2317—51no—
51#1458—51no—
52#1458—51no—
53#1326—54no—
54C24×C5C22×C411yescharacteristic
55C22×C4×C5—21no—
56C22×C4×C5—21no—
57C22×C4×C5—22no—
58C22×C4×C5—22no—
59C22×C4×C5—42no—
60C22×C4×C5—82no—
61C22×C4×C5C22⋊C411yescharacteristic
62C22×C4×C5C22⋊C411yescharacteristic
63C22×C4×C5C2×C812yes—
64C22×C4×C5M1612yes—
65C22×Dic5—22no—
66C22×Dic5C22⋊C411yescharacteristic
67C22×Dic5C22⋊C411yescharacteristic
68C22×Dic5C2×D411yescharacteristic
69C22×Dic5C2×D411yescharacteristic
70C2×#170—42no—
71C2×#170—42no—
72C5×C4⋊C4—42no—
73C5×C4⋊C4—41no—
74C5×C4⋊C4—41no—
75C22⋊Dic5—22no—
76C22⋊Dic5—41no—
77C22⋊Dic5—41no—
78C22⋊Dic5C2×C814yes—
79C4⋊Dic5—24no—
80C4⋊Dic5—41no—
81C4⋊Dic5—41no—
82#683—82no—
83#683—82no—
84C4×Dic5—24no—
85C4×Dic5—21no—
86C4×Dic5—21no—
87C4×Dic5—21no—
88C4×Dic5—21no—
89C2×#117—51no—
90C2×#117—51no—
91C22×C4⋊C4AGL(1,𝔽5)11yescharacteristic
92#431—101no—
93#431—101no—
94#389—101no—
95#389—101no—
96#385—52no—
97#385—52no—
98#384—101no—
99#382—101no—
100#324—101no—
101#324—101no—
102#323—52no—
103#323—52no—
104C23×C5—21no—
105C23×C5—21no—
106C23×C5—41no—
107C23×C5—41no—
108C23×C5C2×C22⋊C411yescharacteristic
109C23×C5C22×C811yescharacteristic
110C23×C5C2×M1611yescharacteristic
111C2×C4×C5—21no—
112C2×C4×C5—21no—
113C2×C4×C5—22no—
114C2×C4×C5—22no—
115C2×C4×C5—22no—
116C2×C4×C5—22no—
117C2×C4×C5—42no—
118C2×C4×C5—42no—
119C2×C4×C5—42no—
120C2×C4×C5—84no—
121C2×C4×C5C22⋊C812yes—
122C2×C4×C5C22⋊C812yes—
123C2×C4×C5#9911yescharacteristic
124C2×C4×C5#9911yescharacteristic
125C2×C4×C5#10112yes—
126C2×Dic5—24no—
127C2×Dic5—21no—
128C2×Dic5—21no—
129C2×Dic5—21no—
130C2×Dic5—21no—
131C2×Dic5—42no—
132C2×Dic5—42no—
133C2×Dic5—41no—
134C2×Dic5—41no—
135C2×Dic5—41no—
136C2×Dic5—41no—
137#170—82no—
138#170—82no—
139C23×C4C2×AGL(1,𝔽5)12yes—
140C23×C4C2×AGL(1,𝔽5)11yescharacteristic
141#117—54no—
142#117—102no—
143#117—201no—
144#117—201no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—41no—
148C2×C4⋊C4—41no—
149C2×C4⋊C4—51no—
150C2×C4⋊C4—51no—
151C2×C4⋊C4—101no—
152C2×C4⋊C4—102no—
153C2×C4⋊C4—102no—
154C2×C22⋊C4—52no—
155C2×C22⋊C4—51no—
156C2×C22⋊C4—51no—
157C2×C42—51no—
158C2×C42—51no—
159C2×C42—101no—
160C4⋊C8—201no—
161C4⋊C8—201no—
162C22⋊C8—102no—
163C22⋊C8—102no—
164C4×C8—201no—
165C4×C8—201no—
166#95—102no—
167#95—102no—
168C22×C5—21no—
169C22×C5—21no—
170C22×C5—21no—
171C22×C5—21no—
172C22×C5—41no—
173C22×C5—41no—
174C22×C5—42no—
175C22×C5—41no—
176C22×C5—41no—
177C22×C5C2×C22⋊C811yescharacteristic
178C22×C5C2×#9911yescharacteristic
179C22×C5C2×#10111yescharacteristic
180C4×C5—22no—
181C4×C5—82no—
182C4×C5#32412yes—
183C4×C5#32412yes—
184Dic5—44no—
185Dic5—41no—
186Dic5—41no—
187Dic5—41no—
188Dic5—41no—
189C24C22×AGL(1,𝔽5)11yescharacteristic
190C22×C4—22no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—22no—
194C22×C4—42no—
195C22×C4—51no—
196C22×C4—51no—
197C22×C4—51no—
198C22×C4—51no—
199C22×C4—82no—
200C22×C4—102no—
201C22×C4D5⋊C812yes—
202C22×C4#70012yes—
203C22×C4C22⋊AGL(1,𝔽5)11yescharacteristic
204C22×C4C22⋊AGL(1,𝔽5)11yescharacteristic
205C2×C8—202no—
206C2×C8—202no—
207C4⋊C4—42no—
208C4⋊C4—41no—
209C4⋊C4—41no—
210C4⋊C4—104no—
211C4⋊C4—201no—
212C4⋊C4—201no—
213C4⋊C4—402no—
214C4⋊C4—402no—
215C22⋊C4—54no—
216C22⋊C4—102no—
217C22⋊C4—201no—
218C22⋊C4—201no—
219C42—104no—
220C42—101no—
221C42—101no—
222C42—101no—
223C42—101no—
224C2×C5—21no—
225C2×C5—21no—
226C2×C5—41no—
227C2×C5—41no—
228C2×C5C2×#32411yescharacteristic
229C2×C5#212211yescharacteristic
230C2×C5#212311yescharacteristic
231C23—21no—
232C23—21no—
233C23—41no—
234C23—41no—
235C23C2×D5⋊C811yescharacteristic
236C23C2×#70011yescharacteristic
237C23C2×C22⋊AGL(1,𝔽5)11yescharacteristic
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—84no—
248C2×C4—101no—
249C2×C4—101no—
250C2×C4—101no—
251C2×C4—101no—
252C2×C4—104no—
253C2×C4—202no—
254C2×C4—202no—
255C2×C4—201no—
256C2×C4—201no—
257C2×C4—201no—
258C2×C4—201no—
259C2×C4#400312yes—
260C2×C4#400312yes—
261C2×C4#400512yes—
262C2×C4#401111yescharacteristic
263C2×C4#401111yescharacteristic
264C8—402no—
265C8—402no—
266C5#1145411yescharacteristic, complemented, Sylow
267C22—21no—
268C22—21no—
269C22—21no—
270C22—21no—
271C22—41no—
272C22—41no—
273C22—42no—
274C22—41no—
275C22—41no—
276C22C2×#400311yescharacteristic
277C22C2×#400511yescharacteristic
278C22C2×#401111yescharacteristic
279C4—22no—
280C4—82no—
281C4—204no—
282C4—201no—
283C4—201no—
284C4—201no—
285C4—201no—
286C4#6491412yes—
287C4#6491412yes—
288C2—21no—
289C2—21no—
290C2—41no—
291C2—41no—
292C2C2×#6491411yescharacteristic
293C2#1060494011yescharacteristic
294C2#1060498011yescharacteristic
subgroupG/H
centerC22C2×#4011
commutatorC23×C5C22×C8
FrattiniC24C22×AGL(1,𝔽5)
FittingC22×C5×C4⋊C4C4
socleC22×C5C2×#99
prime radicalC24×C5C22×C4
cosocleC2×C22⋊Dic5C23
hypercenterC22×C4⋊C4AGL(1,𝔽5)
hypocenterC5#11454
LCSG ▷ C23×C5 ▷ C22×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
33. #4072C8semidirect
266. C5#11454semidirect

Maximal subgroups

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

subgroupindex
#106138882
#106048772
#105988202
#114545