Finite Group Databasebeta

← #12017948

Subgroups — #12017948

Order 1280.

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

normal subgroups only

This group has 612 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#10614080C211yesmaximal, characteristic
2C4×C5×#100C211yesmaximal, characteristic
3#10602412C211yesmaximal, characteristic
4#10598189C214yesmaximal
5C23×Dic10C411yescharacteristic, complemented
6C2×C4×C5×D4C2211yescharacteristic
7C2×#4070—21no—
8C2×C5×#100C2212yes—
9C4×C5×M16—21no—
10#65509C2212yes—
11C2×#3966C2212yes—
12#65501—21no—
13#65501C411yescharacteristic, complemented
14#65268—21no—
15#65236—21no—
16#65235C412yescomplemented
17#65218—21no—
18C5×#343—21no—
19#64746—21no—
20#13293—51nomaximal, Sylow
21C22×C5×D4C2311yescharacteristic
22C23×C4×C5C2×C411yescharacteristic
23C23×C4×C5D411yescharacteristic
24C23×Dic5C2×C412yes—
25C22×Dic10—21no—
26C22×Dic10—21no—
27C22×Dic10—44no—
28C22×Dic10—41no—
29C22×Dic10—41no—
30C2×C5×M16—22no—
31C4×C5×D4—41no—
32C4×C5×D4—41no—
33C2×C5×C4⋊C4D411yescharacteristic
34C2×C5×C22⋊C4C2×C411yescharacteristic
35C2×C5×C22⋊C4D411yescharacteristic
36C2×C42×C5D411yescharacteristic
37C2×C22⋊Dic5—22no—
38C2×C22⋊Dic5—22no—
39C2×C22⋊Dic5C2×C412yes—
40C2×C4⋊Dic5—21no—
41C2×C4⋊Dic5—41no—
42#4070—44no—
43C2×#683—22no—
44C5×#100—22no—
45C5×#100Q814yescomplemented
46C5×#97—41no—
47C4×C5×C8—41no—
48#3966—24no—
49#3963—22no—
50#3963—21no—
51#3963—22no—
52#3963—41no—
53#2409—51no—
54#1799—51no—
55C4×#100D511yescharacteristic
56#1390—54no—
57C24×C5C22×C411yescharacteristic
58C24×C5C2×D411yescharacteristic
59C2×C5×D4—41no—
60C2×C5×D4—41no—
61C2×C5×D4C2×Q812yes—
62C2×C5×D4#4112yes—
63C22×C4×C5—21no—
64C22×C4×C5—21no—
65C22×C4×C5—21no—
66C22×C4×C5—21no—
67C22×C4×C5—42no—
68C22×C4×C5—42no—
69C22×C4×C5C22⋊C411yescharacteristic
70C22×C4×C5C22⋊C411yescharacteristic
71C22×C4×C5C22⋊C411yescharacteristic
72C22×C4×C5C22⋊C411yescharacteristic
73C22×C4×C5C2×D411yescharacteristic
74C22×Dic5—22no—
75C22×Dic5—22no—
76C22×Dic5—44no—
77C22×Dic5—42no—
78C22×Dic5—42no—
79C2×Dic10—24no—
80C2×Dic10—44no—
81C2×Dic10—42no—
82C2×Dic10—42no—
83C2×Dic10—48no—
84C2×Dic10—44no—
85C2×Dic10—41no—
86C2×Dic10—41no—
87C2×Dic10—44no—
88C5×M16—24no—
89C5×M16—42no—
90C2×C5×C8—42no—
91C5×C4⋊C4—41no—
92C5×C22⋊C4—41no—
93C5×C22⋊C4—41no—
94C42×C5—21no—
95C42×C5—21no—
96C22⋊Dic5—44no—
97C22⋊Dic5—42no—
98C22⋊Dic5—42no—
99C22⋊Dic5—42no—
100C4⋊Dic5—42no—
101C4⋊Dic5—82no—
102#683—44no—
103C23×Q8—51no—
104C2×C22⋊Q8—101no—
105C2×C4×D4C2×D511yescharacteristic
106C2×#100C2×D512yes—
107C2×#99—52no—
108C4×M16—21no—
109#395—101no—
110#393—101no—
111#391—101no—
112#386—52no—
113#385—52no—
114#380—51no—
115#380—101no—
116#343—21no—
117#342—101no—
118C23×C5—21no—
119C23×C5—21no—
120C23×C5—21no—
121C23×C5—21no—
122C23×C5—41no—
123C23×C5—41no—
124C23×C5—41no—
125C23×C5—41no—
126C23×C5C2×C22⋊C411yescharacteristic
127C23×C5C4×Q812yes—
128C23×C5#12411yescharacteristic
129C23×C5C4⋊Q811yescharacteristic
130C5×D4—42no—
131C5×D4—42no—
132C2×C4×C5—21no—
133C2×C4×C5—21no—
134C2×C4×C5—22no—
135C2×C4×C5—21no—
136C2×C4×C5—21no—
137C2×C4×C5—21no—
138C2×C4×C5—42no—
139C2×C4×C5—42no—
140C2×C4×C5—42no—
141C2×C4×C5—41no—
142C2×C4×C5—41no—
143C2×C4×C5—41no—
144C2×C4×C5—81no—
145C2×C4×C5#9911yescharacteristic
146C2×C4×C5#9911yescharacteristic
147C2×C4×C5C22⋊Q812yes—
148C2×Dic5—24no—
149C2×Dic5—42no—
150C2×Dic5—44no—
151C2×Dic5—42no—
152C2×Dic5—44no—
153C2×Dic5—42no—
154C2×Dic5—42no—
155C2×Dic5—42no—
156C2×Dic5—82no—
157C2×Dic5—82no—
158C2×Dic5—82no—
159Dic10—44no—
160Dic10—48no—
161Dic10—44no—
162C5×C8—44no—
163C22×Q8—101no—
164C22×Q8—101no—
165C22×Q8—201no—
166C22×Q8—201no—
167C22×Q8—204no—
168C22×D4C22×D511yescharacteristic
169C23×C4—52no—
170C23×C4C4×D511yescharacteristic
171C23×C4D2011yescharacteristic
172C2×M16—22no—
173C22⋊Q8—204no—
174C4×D4—41no—
175C4×D4—41no—
176C2×C4⋊C4—102no—
177C2×C4⋊C4—101no—
178C2×C4⋊C4—201no—
179C2×C4⋊C4C22⋊D511yescharacteristic
180C2×C22⋊C4—52no—
181C2×C22⋊C4—102no—
182C2×C22⋊C4—102no—
183C2×C22⋊C4C4×D511yescharacteristic
184C2×C22⋊C4D2011yescharacteristic
185C2×C42C22⋊D511yescharacteristic
186#100—22no—
187#100Dic1014yescomplemented
188#99—104no—
189#97—41no—
190C4×C8—41no—
191#95—101no—
192#95—102no—
193#95—102no—
194#95—201no—
195C22×C5—21no—
196C22×C5—21no—
197C22×C5—21no—
198C22×C5—22no—
199C22×C5—21no—
200C22×C5—42no—
201C22×C5—41no—
202C22×C5—41no—
203C22×C5—42no—
204C22×C5—41no—
205C22×C5—41no—
206C22×C5—41no—
207C22×C5—41no—
208C22×C5—81no—
209C22×C5#39111yescharacteristic
210C22×C5C2×#9911yescharacteristic
211C22×C5#41011yescharacteristic
212C4×C5—21no—
213C4×C5—22no—
214C4×C5—41no—
215C4×C5—41no—
216C4×C5—41no—
217C4×C5AΣL(1,𝔽16)12yes—
218Dic5—44no—
219Dic5—82no—
220Dic5—162no—
221C24C2×C4×D511yescharacteristic
222C24C2×D2011yescharacteristic
223C2×Q8—104no—
224C2×Q8—202no—
225C2×Q8—202no—
226C2×Q8—204no—
227C2×Q8—204no—
228C2×Q8—208no—
229C2×Q8—204no—
230C2×Q8—201no—
231C2×Q8—201no—
232C2×D4—41no—
233C2×D4—41no—
234C2×D4C2×Dic1012yes—
235C2×D4#70912yes—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—102no—
243C22×C4—102no—
244C22×C4—204no—
245C22×C4—202no—
246C22×C4—202no—
247C22×C4#68511yescharacteristic
248C22×C4#68511yescharacteristic
249C22×C4#68511yescharacteristic
250C22×C4#68511yescharacteristic
251C22×C4C2×C22⋊D511yescharacteristic
252M16—24no—
253M16—42no—
254C2×C8—42no—
255C4⋊C4—41no—
256C4⋊C4—202no—
257C4⋊C4—204no—
258C4⋊C4—402no—
259C22⋊C4—41no—
260C22⋊C4—41no—
261C22⋊C4—204no—
262C22⋊C4—202no—
263C22⋊C4—202no—
264C22⋊C4—202no—
265C42—21no—
266C42—21no—
267C2×C5—21no—
268C2×C5—21no—
269C2×C5—41no—
270C2×C5—42no—
271C2×C5—41no—
272C2×C5#199611yescharacteristic
273C2×C5C2×AΣL(1,𝔽16)11yescharacteristic
274C2×C5#212311yescharacteristic
275C23—21no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—41no—
280C23—41no—
281C23—41no—
282C23—41no—
283C23C4×Dic1012yes—
284C23#401511yescharacteristic
285C23#402111yescharacteristic
286C23C2×#68511yescharacteristic
287Q8—204no—
288Q8—208no—
289Q8—204no—
290D4—42no—
291D4—42no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—22no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—81no—
305C2×C4—104no—
306C2×C4—202no—
307C2×C4—204no—
308C2×C4—202no—
309C2×C4—204no—
310C2×C4—202no—
311C2×C4—202no—
312C2×C4—202no—
313C2×C4—402no—
314C2×C4—402no—
315C2×C4—402no—
316C2×C4#393811yescharacteristic
317C2×C4#393811yescharacteristic
318C2×C4#407012yes—
319C8—44no—
320C5#1329311yescharacteristic, complemented, Sylow
321C22—21no—
322C22—21no—
323C22—21no—
324C22—22no—
325C22—21no—
326C22—42no—
327C22—41no—
328C22—41no—
329C22—42no—
330C22—41no—
331C22—41no—
332C22—41no—
333C22—41no—
334C22—81no—
335C22#6521811yescharacteristic
336C22C2×#393811yescharacteristic
337C22#6529511yescharacteristic
338C4—22no—
339C4—21no—
340C4—41no—
341C4—41no—
342C4—41no—
343C4—204no—
344C4—402no—
345C4—802no—
346C4#6469512yes—
347C2—21no—
348C2—21no—
349C2—41no—
350C2—42no—
351C2—41no—
352C2#1060025011yescharacteristic
353C2C2×#6469511yescharacteristic
354C2#1060094011yescharacteristic
subgroupG/H
centerC22C2×#3938
commutatorC24×C5C22×C4
FrattiniC22×D4C22×D5
FittingC4×C5×#100C2
socleC22×C5C2×#99
prime radical, cosocleC22×C5×D4C23
hypercenterC4×#100D5
hypocenterC5#13293
LCSG ▷ C24×C5 ▷ C22×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ C4×#100

Complements

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

NH ≅ G/Ntype
5. C23×Dic10C4semidirect
13. #65501C4semidirect
16. #65235C4semidirect
45. C5×#100Q8semidirect
187. #100Dic10semidirect
320. C5#13293semidirect

Maximal subgroups

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

subgroupindex
#106140802
C4×C5×#1002
#106024122
#105981892
#132935