Finite Group Databasebeta

← #10614622

Subgroups — #10614622

Order 640.

688 conjugacy classes of subgroups; only the 169 normal subgroups are shown.

normal subgroups only

This group has 688 conjugacy classes of subgroups — too many to draw. Only its 169 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×#123C211yesmaximal, characteristic, complemented
2C2×#4085C211yesmaximal, characteristic
3C2×#4075C211yesmaximal, characteristic, complemented
4C22×C4×Dic5C211yesmaximal, characteristic, complemented
5#65509C211yesmaximal, characteristic, complemented
6#65279C212yesmaximal, complemented
7#65261C212yesmaximal, complemented
8#65247C211yesmaximal, characteristic, complemented
9#65246C212yesmaximal, complemented
10#65234C212yesmaximal, complemented
11#65231C211yesmaximal, characteristic, complemented
12C22×C5×D4C2211yescharacteristic
13C23×C4×C5C2211yescharacteristic
14C22×C22⋊D5C2211yescharacteristic
15C22×C22⋊D5C2211yescharacteristic
16C23×Dic5C2211yescharacteristic
17C23×Dic5C2211yescharacteristic
18C5×#123—24no—
19C2×C5×C4⋊C4C2212yes—
20C2×C5×C22⋊C4C2211yescharacteristic
21C2×C5×C22⋊C4C2212yes—
22C2×C22⋊Dic5C2211yescharacteristic
23C2×C22⋊Dic5C2211yescharacteristic
24C2×C22⋊Dic5C2212yes—
25#4085—24no—
26#4075—24no—
27C2×#685C2212yes—
28C2×#685C2211yescharacteristic
29C2×#685C2212yes—
30C2×#683C2212yes—
31C2×#683C2211yescharacteristic
32C2×#683C2212yes—
33C2×C4×Dic5—24no—
34C2×C4×Dic5C2212yes—
35C2×C4×Dic5C2212yes—
36#3963C2212yes—
37#3963C2212yes—
38#3963C2212yes—
39#2854—51nomaximal, Sylow
40C24×C5C2311yescharacteristic
41C24×C5C2311yescharacteristic
42C23×D5C2311yescharacteristic
43C2×C5×D4—22no—
44C2×C5×D4—22no—
45C2×C5×D4—22no—
46C22×C4×C5—22no—
47C22×C4×C5—22no—
48C22×C4×C5D414yescomplemented
49C22×C4×C5C2312yes—
50C22×C4×C5C2311yescharacteristic
51C22×C4×C5C2312yes—
52C2×C22⋊D5—22no—
53C2×C22⋊D5—22no—
54C2×C22⋊D5—22no—
55C2×C22⋊D5—22no—
56C2×C22⋊D5—24no—
57C22×Dic5—22no—
58C22×Dic5—24no—
59C22×Dic5—24no—
60C22×Dic5D414yes—
61C22×Dic5C2312yes—
62C22×Dic5C2311yescharacteristic
63C22×Dic5C2311yescharacteristic
64C22×Dic5C2312yes—
65C22×Dic5C2311yescharacteristic
66C5×C4⋊C4—24no—
67C5×C22⋊C4—22no—
68C5×C22⋊C4—24no—
69C22⋊Dic5—22no—
70C22⋊Dic5—22no—
71C22⋊Dic5—24no—
72#685—24no—
73#685—22no—
74#685—24no—
75#683—24no—
76#683—22no—
77#683—24no—
78C4×Dic5—24no—
79C4×Dic5—24no—
80C2×#123—51no—
81C2×#123D511yescharacteristic, complemented
82C2×#121—51no—
83C22×C42—51no—
84#390—52no—
85#389—52no—
86#388—52no—
87#388—52no—
88#386—51no—
89#386—51no—
90#380—51no—
91C23×C5—22no—
92C23×C5—22no—
93C23×C5—21no—
94C23×C5—21no—
95C23×C5—22no—
96C23×C5—22no—
97C23×C5—21no—
98C23×C5C2×D412yes—
99C23×C5C2×D412yes—
100C23×C5#4112yes—
101C23×C5C2411yescharacteristic
102C22×D5—21no—
103C22×D5—21no—
104C22×D5—22no—
105C22×D5—21no—
106C22×D5—22no—
107C5×D4—44no—
108C2×C4×C5—28no—
109C2×C4×C5—22no—
110C2×C4×C5—22no—
111C2×C4×C5—21no—
112C2×C4×C5—22no—
113C2×C4×C5—21no—
114C2×C4×C5—22no—
115C2×C4×C5—22no—
116C2×C4×C5—21no—
117C2×C4×C5C2×D414yes—
118C22⋊D5—44no—
119C22⋊D5—44no—
120C2×Dic5—28no—
121C2×Dic5—22no—
122C2×Dic5—22no—
123C2×Dic5—22no—
124C2×Dic5—28no—
125C2×Dic5—21no—
126C2×Dic5—22no—
127C2×Dic5—21no—
128C2×Dic5—22no—
129C2×Dic5—22no—
130C2×Dic5—21no—
131C2×Dic5C2×D412yes—
132C2×Dic5C2×D412yes—
133C2×Dic5#4114yes—
134C2×Dic5#4114yes—
135C22×D4—51no—
136C22×D4—51no—
137C22×D4C2×D511yescharacteristic
138C23×C4—51no—
139C23×C4—51no—
140C23×C4C2×D511yescharacteristic
141#123—24no—
142#123—104no—
143#121—104no—
144C2×C4⋊C4—51no—
145C2×C4⋊C4—52no—
146C2×C4⋊C4—52no—
147C2×C4⋊C4C2×D512yes—
148C2×C22⋊C4—51no—
149C2×C22⋊C4—51no—
150C2×C22⋊C4—52no—
151C2×C22⋊C4—51no—
152C2×C22⋊C4—52no—
153C2×C22⋊C4—52no—
154C2×C22⋊C4C2×D511yescharacteristic
155C2×C22⋊C4C2×D512yes—
156C2×C42—52no—
157C2×C42—52no—
158C2×C42—104no—
159#95—52no—
160#95—52no—
161#95—52no—
162C22×C5—22no—
163C22×C5—22no—
164C22×C5—24no—
165C22×C5—24no—
166C22×C5—41no—
167C22×C5—41no—
168C22×C5—41no—
169C22×C5—42no—
170C22×C5—42no—
171C22×C5#12114yes—
172C22×C5C22×D411yescharacteristic
173C22×C5C22×D411yescharacteristic
174C22×C5C2×#4111yescharacteristic
175C22×C5C2×#4112yes—
176C22×C5C2×#4112yes—
177C2×D5—41no—
178C2×D5—41no—
179C2×D5—41no—
180C2×D5—42no—
181C2×D5—42no—
182C4×C5—24no—
183C4×C5—41no—
184C4×C5—42no—
185Dic5—22no—
186Dic5—24no—
187Dic5—41no—
188Dic5—42no—
189Dic5#12314yescomplemented
190C24—51no—
191C24C22×D511yescharacteristic
192C24C22×D511yescharacteristic
193C2×D4—22no—
194C2×D4—22no—
195C2×D4—22no—
196C2×D4—102no—
197C2×D4—104no—
198C2×D4—102no—
199C2×D4—102no—
200C2×D4—102no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—51no—
204C22×C4—54no—
205C22×C4—51no—
206C22×C4—52no—
207C22×C4—52no—
208C22×C4—51no—
209C22×C4—104no—
210C22×C4—104no—
211C22×C4—102no—
212C22×C4C22⋊D514yescomplemented
213C22×C4C22×D512yes—
214C22×C4C22×D511yescharacteristic
215C22×C4C22×D512yes—
216C4⋊C4—24no—
217C4⋊C4—102no—
218C4⋊C4—104no—
219C4⋊C4—104no—
220C22⋊C4—22no—
221C22⋊C4—24no—
222C22⋊C4—102no—
223C22⋊C4—104no—
224C22⋊C4—102no—
225C22⋊C4—104no—
226C22⋊C4—102no—
227C22⋊C4—104no—
228C42—104no—
229C42—104no—
230C2×C5—24no—
231C2×C5—81no—
232C2×C5C2×#12111yescharacteristic
233C2×C5C2×#12311yescharacteristic
234C2×C5#52511yescharacteristic
235C2×C5#52912yes—
236C2×C5#53212yes—
237D5—81no—
238C23—22no—
239C23—22no—
240C23—21no—
241C23—21no—
242C23—22no—
243C23—22no—
244C23—21no—
245C23—101no—
246C23—101no—
247C23—102no—
248C23—102no—
249C23—101no—
250C23D4×D512yes—
251C23#71112yes—
252C23C2×C22⋊D512yes—
253C23C23×D511yescharacteristic
254D4—44no—
255D4—204no—
256D4—204no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—28no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—21no—
266C2×C4—52no—
267C2×C4—54no—
268C2×C4—52no—
269C2×C4—54no—
270C2×C4—108no—
271C2×C4—108no—
272C2×C4—102no—
273C2×C4—102no—
274C2×C4—102no—
275C2×C4—101no—
276C2×C4—101no—
277C2×C4—102no—
278C2×C4—102no—
279C2×C4—102no—
280C2×C4—101no—
281C2×C4C2×C22⋊D514yes—
282C5#285411yescharacteristic, complemented, Sylow
283C22—24no—
284C22—24no—
285C22—22no—
286C22—22no—
287C22—41no—
288C22—42no—
289C22—42no—
290C22—41no—
291C22—41no—
292C22—201no—
293C22—202no—
294C22—202no—
295C22—201no—
296C22—201no—
297C22#408514yes—
298C22C2×D4×D511yescharacteristic
299C22C2×#71111yescharacteristic
300C22D5×#4112yes—
301C22D5×#4112yes—
302C22C22×C22⋊D511yescharacteristic
303C4—24no—
304C4—41no—
305C4—42no—
306C4—54no—
307C4—102no—
308C4—104no—
309C4—201no—
310C4—202no—
311C2—24no—
312C2—81no—
313C2—401no—
314C2#6598312yes—
315C2D5×#12311yescharacteristic
316C2#6599011yescharacteristic
317C2C2×#408511yescharacteristic
318C2#6616612yes—
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC2×C5×#123C2
prime radicalC2×#4085C2
hypercenterC2×#123D5
hypocenterC5#2854
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
1. C2×C5×#123C2semidirect
3. C2×#4075C2semidirect
4. C22×C4×Dic5C2semidirect
5. #65509C2semidirect
6. #65279C2semidirect
7. #65261C2semidirect
8. #65247C2semidirect
9. #65246C2semidirect
10. #65234C2semidirect
11. #65231C2semidirect
48. C22×C4×C5D4semidirect
81. C2×#123D5semidirect
189. Dic5#123semidirect
212. C22×C4C22⋊D5semidirect
282. C5#2854semidirect

Maximal subgroups

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

subgroupindex
C2×C5×#1232
C2×#40852
C2×#40752
C22×C4×Dic52
#655092
#652792
#652612
#652472
#652462
#652342
#652312
#28545