Finite Group Databasebeta

← #12010012

Subgroups — #12010012

Order 1280.

616 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 616 conjugacy classes of subgroups — too many to draw. Only its 95 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
1C2×C5×#399C211yesmaximal, characteristic
2#10602335C211yesmaximal, characteristic
3#10602332C211yesmaximal, characteristic
4#10598142C214yesmaximal, complemented
5C22×C5×C4⋊C4—21no—
6C22×C5×C4⋊C4C411yescharacteristic, complemented
7C22×C5×C22⋊C4—21no—
8C22×C5×C22⋊C4C2211yescharacteristic
9C5×#399—21no—
10C5×#399—21no—
11C5×#399C412yescomplemented
12C5×#399C2212yes—
13C2×C5×#95C411yescharacteristic, complemented
14#65536—21no—
15#65496—21no—
16#64746—21no—
17#64746C2212yes—
18#64745—21no—
19#64745C2212yes—
20#11809—51nomaximal, Sylow
21C25×C5D411yescharacteristic
22C23×C4×C5—21no—
23C23×C4×C5—21no—
24C23×C4×C5C2×C411yescharacteristic
25C23×C4×C5D411yescharacteristic
26C2×C5×C4⋊C4—21no—
27C2×C5×C4⋊C4—22no—
28C2×C5×C4⋊C4—22no—
29C2×C5×C4⋊C4—41no—
30C2×C5×C4⋊C4—41no—
31C2×C5×C4⋊C4—44no—
32C2×C5×C4⋊C4C2×C412yes—
33C2×C5×C22⋊C4—22no—
34C2×C5×C22⋊C4—22no—
35C2×C5×C22⋊C4—41no—
36C2×C5×C22⋊C4—41no—
37C2×C5×C22⋊C4—44no—
38C2×C5×C22⋊C4C2×C411yescharacteristic
39C2×C5×C22⋊C4D411yescharacteristic
40C2×C5×C22⋊C4D411yescharacteristic
41C2×C5×C22⋊C4C2311yescharacteristic
42C2×C22⋊Dic5—22no—
43C2×C22⋊Dic5—41no—
44C2×C22⋊Dic5—41no—
45C2×#681—41no—
46C5×#95—21no—
47C5×#95C2×C412yes—
48C22⋊#168—22no—
49C22⋊#168—41no—
50C2×#399D511yescharacteristic
51#1819—51no—
52#1462—51no—
53#1352—54no—
54C24×C5—22no—
55C24×C5—21no—
56C24×C5—21no—
57C24×C5—41no—
58C24×C5—42no—
59C24×C5—42no—
60C24×C5C22⋊C411yescharacteristic
61C24×C5C22⋊C411yescharacteristic
62C24×C5C2×D411yescharacteristic
63C22×C4×C5—24no—
64C22×C4×C5—21no—
65C22×C4×C5—21no—
66C22×C4×C5—21no—
67C22×C4×C5—22no—
68C22×C4×C5—21no—
69C22×C4×C5—22no—
70C22×C4×C5—42no—
71C22×C4×C5—42no—
72C22×C4×C5—42no—
73C22×C4×C5—42no—
74C22×C4×C5—42no—
75C22×C4×C5—42no—
76C22×C4×C5—41no—
77C22×C4×C5—41no—
78C22×C4×C5C22⋊C411yescharacteristic
79C22×C4×C5C22⋊C411yescharacteristic
80C22×C4×C5C22×C411yescharacteristic
81C22×C4×C5C2×D411yescharacteristic
82C22×Dic5—42no—
83C5×C4⋊C4—42no—
84C5×C4⋊C4—44no—
85C5×C4⋊C4—44no—
86C5×C4⋊C4—42no—
87C5×C22⋊C4—41no—
88C5×C22⋊C4—41no—
89C5×C22⋊C4—41no—
90C5×C22⋊C4—41no—
91C5×C22⋊C4—44no—
92C5×C22⋊C4—44no—
93C22⋊Dic5—42no—
94C22⋊Dic5—42no—
95C22⋊Dic5—82no—
96C22⋊Dic5—82no—
97#681—82no—
98C2×#168—42no—
99C22×C4⋊C4—21no—
100C22×C4⋊C4Dic511yescharacteristic, complemented
101C22×C22⋊C4—21no—
102C22×C22⋊C4C2×D511yescharacteristic
103#407—101no—
104#399—21no—
105#399—21no—
106#399Dic512yescomplemented
107#399C2×D512yes—
108#379—101no—
109C2×#95Dic511yescharacteristic, complemented
110#342—52no—
111#342—101no—
112#323—52no—
113#323—101no—
114C23×C5—22no—
115C23×C5—22no—
116C23×C5—21no—
117C23×C5—21no—
118C23×C5—21no—
119C23×C5—21no—
120C23×C5—21no—
121C23×C5—21no—
122C23×C5—21no—
123C23×C5—42no—
124C23×C5—42no—
125C23×C5—41no—
126C23×C5—41no—
127C23×C5—41no—
128C23×C5—41no—
129C23×C5—48no—
130C23×C5—42no—
131C23×C5—42no—
132C23×C5—81no—
133C23×C5—81no—
134C23×C5—84no—
135C23×C5#9911yescharacteristic
136C23×C5#9911yescharacteristic
137C23×C5C2×C22⋊C411yescharacteristic
138C23×C5#13712yes—
139C2×C4×C5—24no—
140C2×C4×C5—21no—
141C2×C4×C5—21no—
142C2×C4×C5—21no—
143C2×C4×C5—21no—
144C2×C4×C5—48no—
145C2×C4×C5—42no—
146C2×C4×C5—42no—
147C2×C4×C5—42no—
148C2×C4×C5—42no—
149C2×C4×C5—42no—
150C2×C4×C5—41no—
151C2×C4×C5—41no—
152C2×C4×C5—41no—
153C2×C4×C5—41no—
154C2×C4×C5—42no—
155C2×C4×C5—42no—
156C2×C4×C5—42no—
157C2×C4×C5—81no—
158C2×C4×C5—82no—
159C2×C4×C5—82no—
160C2×Dic5—82no—
161C2×Dic5—82no—
162C2×Dic5—82no—
163#168—82no—
164C25C22⋊D511yescharacteristic
165C23×C4—21no—
166C23×C4—21no—
167C23×C4C2×Dic511yescharacteristic
168C23×C4C22⋊D511yescharacteristic
169C2×M16—201no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—41no—
174C2×C4⋊C4—41no—
175C2×C4⋊C4—44no—
176C2×C4⋊C4C2×Dic512yes—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—22no—
179C2×C22⋊C4—41no—
180C2×C22⋊C4—41no—
181C2×C22⋊C4—44no—
182C2×C22⋊C4—102no—
183C2×C22⋊C4—201no—
184C2×C22⋊C4—201no—
185C2×C22⋊C4C2×Dic511yescharacteristic
186C2×C22⋊C4C22⋊D511yescharacteristic
187C2×C22⋊C4C22⋊D511yescharacteristic
188C2×C22⋊C4C22×D511yescharacteristic
189C22⋊C8—102no—
190C22⋊C8—201no—
191#95—21no—
192#95C2×Dic512yes—
193C22×C5—24no—
194C22×C5—24no—
195C22×C5—21no—
196C22×C5—21no—
197C22×C5—21no—
198C22×C5—21no—
199C22×C5—21no—
200C22×C5—21no—
201C22×C5—22no—
202C22×C5—42no—
203C22×C5—42no—
204C22×C5—41no—
205C22×C5—41no—
206C22×C5—41no—
207C22×C5—41no—
208C22×C5—41no—
209C22×C5—81no—
210C22×C5—84no—
211C22×C5—84no—
212C22×C5—81no—
213C22×C5#35212yes—
214C22×C5#35312yes—
215C22×C5C2×#9911yescharacteristic
216C22×C5#41911yescharacteristic
217C22×C5#42111yescharacteristic
218C4×C5—44no—
219C4×C5—41no—
220C4×C5—41no—
221C4×C5—41no—
222C4×C5—41no—
223C4×C5—82no—
224Dic5—162no—
225C24—22no—
226C24—21no—
227C24—21no—
228C24—41no—
229C24—42no—
230C24—42no—
231C24C22⋊Dic511yescharacteristic
232C24C22⋊Dic511yescharacteristic
233C24C2×C22⋊D511yescharacteristic
234C22×C4—24no—
235C22×C4—22no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—22no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—202no—
250C22×C4C22⋊Dic511yescharacteristic
251C22×C4C22⋊Dic511yescharacteristic
252C22×C4C22×Dic511yescharacteristic
253C22×C4C2×C22⋊D511yescharacteristic
254M16—402no—
255C2×C8—202no—
256C4⋊C4—42no—
257C4⋊C4—44no—
258C4⋊C4—44no—
259C4⋊C4—42no—
260C22⋊C4—41no—
261C22⋊C4—41no—
262C22⋊C4—41no—
263C22⋊C4—41no—
264C22⋊C4—44no—
265C22⋊C4—44no—
266C22⋊C4—202no—
267C22⋊C4—202no—
268C22⋊C4—402no—
269C22⋊C4—402no—
270C2×C5—21no—
271C2×C5—21no—
272C2×C5—22no—
273C2×C5—22no—
274C2×C5—42no—
275C2×C5—81no—
276C2×C5#151111yescharacteristic
277C2×C5C2×#35211yescharacteristic
278C2×C5C2×#35311yescharacteristic
279C23—21no—
280C23—21no—
281C23—22no—
282C23—22no—
283C23—21no—
284C23—21no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—42no—
289C23—42no—
290C23—41no—
291C23—41no—
292C23—41no—
293C23—41no—
294C23—48no—
295C23—42no—
296C23—42no—
297C23—81no—
298C23—81no—
299C23—84no—
300C23#396611yescharacteristic
301C23#396611yescharacteristic
302C23#408812yes—
303C23C2×C22⋊Dic511yescharacteristic
304C2×C4—24no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—48no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—81no—
323C2×C4—82no—
324C2×C4—82no—
325C2×C4—402no—
326C2×C4—402no—
327C2×C4—402no—
328C8—402no—
329C5#1180911yescharacteristic, complemented, Sylow
330C22—24no—
331C22—24no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—21no—
336C22—21no—
337C22—21no—
338C22—22no—
339C22—42no—
340C22—42no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—81no—
347C22—84no—
348C22—84no—
349C22—81no—
350C22#6475912yes—
351C22#6476512yes—
352C22C2×#396611yescharacteristic
353C22#6551411yescharacteristic
354C22#6552511yescharacteristic
355C4—44no—
356C4—41no—
357C4—41no—
358C4—41no—
359C4—41no—
360C4—82no—
361C4—802no—
362C2—21no—
363C2—21no—
364C2—22no—
365C2—22no—
366C2—42no—
367C2—81no—
368C2#1060248211yescharacteristic
369C2C2×#6475911yescharacteristic
370C2C2×#6476511yescharacteristic
subgroupG/H
centerC22C2×#3966
commutatorC22×C4×C5C22×C4
FrattiniC2×C22⋊C4C22×D5
FittingC2×C5×#399C2
socleC22×C5C2×#99
prime radicalC25×C5D4
cosocleC2×C5×C22⋊C4C23
hypercenterC2×#399D5
hypocenterC5#11809
LCSG ▷ C22×C4×C5 ▷ C23×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ C2×#399

Complements

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

NH ≅ G/Ntype
4. #10598142C2semidirect
6. C22×C5×C4⋊C4C4semidirect
11. C5×#399C4semidirect
13. C2×C5×#95C4semidirect
100. C22×C4⋊C4Dic5semidirect
106. #399Dic5semidirect
109. C2×#95Dic5semidirect
329. C5#11809semidirect

Maximal subgroups

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

subgroupindex
C2×C5×#3992
#106023352
#106023322
#105981422
#118095