Finite Group Databasebeta

← #10609767

Subgroups — #10609767

Order 640.

760 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 760 conjugacy classes of subgroups — too many to draw. Only its 157 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×#4042C212yesmaximal, complemented
2C2×#4039C211yesmaximal, characteristic, complemented
3C2×#4031C211yesmaximal, characteristic, complemented
4C2×#4030C212yesmaximal, complemented
5C5×#399C211yesmaximal, characteristic, complemented
6#65281C211yesmaximal, characteristic, complemented
7#65277C212yesmaximal, complemented
8#65249C211yesmaximal, characteristic, complemented
9#65246C212yesmaximal, complemented
10#64960C211yesmaximal, characteristic, complemented
11#64938C211yesmaximal, characteristic, complemented
12C22×C22⋊D5C2211yescharacteristic
13C22×C22⋊D5C2212yes—
14C22×D20C2211yescharacteristic
15C22×D20C2212yes—
16C22×C4×D5C2212yes—
17C22×Dic10C2211yescharacteristic
18C2×C5×C4⋊C4C2211yescharacteristic
19C2×C5×C4⋊C4C2212yes—
20C2×C5×C22⋊C4C2212yes—
21C2×C5×C22⋊C4C2211yescharacteristic
22C2×C22⋊Dic5C2211yescharacteristic
23C2×#685C2212yes—
24C2×#685C2211yescharacteristic
25C2×#685C2211yescharacteristic
26C2×#685C2212yes—
27C2×#685C2212yes—
28C2×#685C2212yes—
29C2×#683C2212yes—
30C2×#683C2212yes—
31C2×C4×Dic5C2212yes—
32C2×C4×Dic5C2211yescharacteristic
33#4042—28no—
34#4039—24no—
35#4031—24no—
36#4030—28no—
37C5×#95C2211yescharacteristic
38#3963C2211yescharacteristic
39#2540—51nomaximal, Sylow
40C24×C5C2311yescharacteristic
41C23×D5C2311yescharacteristic
42C23×D5C2312yes—
43C22×C4×C5C2311yescharacteristic
44C22×C4×C5C2312yes—
45C22×C4×C5C2311yescharacteristic
46C22×C4×C5C2312yes—
47C2×C22⋊D5—24no—
48C2×C22⋊D5—24no—
49C2×C22⋊D5—24no—
50C2×C22⋊D5—22no—
51C2×C22⋊D5—42no—
52C22×Dic5C2311yescharacteristic
53C22×Dic5C2311yescharacteristic
54C22×Dic5C2312yes—
55C22×Dic5C2311yescharacteristic
56C2×D20—24no—
57C2×D20—22no—
58C2×D20—24no—
59C2×D20—24no—
60C2×D20—42no—
61C2×C4×D5—24no—
62C2×C4×D5—24no—
63C2×C4×D5—24no—
64C2×Dic10—22no—
65C2×Dic10—24no—
66C5×C4⋊C4—22no—
67C5×C4⋊C4—24no—
68C5×C22⋊C4—24no—
69C5×C22⋊C4—22no—
70C22⋊Dic5—22no—
71#685—22no—
72#685—24no—
73#685—24no—
74#685—22no—
75#685—24no—
76#685—24no—
77#683—24no—
78#683D418yescomplemented
79C4×Dic5—42no—
80C4×Dic5D414yescomplemented
81C2×#124—51no—
82C2×C22⋊Q8—52no—
83C2×#121—52no—
84C2×C4×D4—51no—
85#399—51no—
86#399D511yescharacteristic, complemented
87#392—51no—
88#392—51no—
89#390—52no—
90#390—52no—
91C4×C4⋊C4—51no—
92C23×C5—21no—
93C23×C5—22no—
94C23×C5—22no—
95C23×C5—21no—
96C23×C5—21no—
97C23×C5C2411yescharacteristic
98C22×D5—22no—
99C22×D5—22no—
100C22×D5—22no—
101C22×D5—22no—
102C22×D5—22no—
103C22×D5—22no—
104C22×D5—21no—
105C22×D5—22no—
106C22×D5—22no—
107C22×D5—21no—
108C22×D5—21no—
109C22×D5#4114yes—
110C2×C4×C5—22no—
111C2×C4×C5—22no—
112C2×C4×C5—22no—
113C2×C4×C5—22no—
114C2×C4×C5—21no—
115C2×C4×C5—22no—
116C2×C4×C5—22no—
117C2×C4×C5—22no—
118C2×C4×C5C2×D412yes—
119C2×C4×C5C2×D414yes—
120C22⋊D5—48no—
121C22⋊D5—44no—
122C2×Dic5—22no—
123C2×Dic5—21no—
124C2×Dic5—22no—
125C2×Dic5—21no—
126C2×Dic5—22no—
127C2×Dic5—21no—
128C2×Dic5C2×D412yes—
129C2×Dic5C2×D414yes—
130C2×Dic5C2×D412yes—
131C2×Dic5C2×D414yes—
132D20—28no—
133D20—48no—
134C4×D5—48no—
135Dic10—44no—
136C22×Q8—51no—
137C22×D4—51no—
138C22×D4—52no—
139C22×D4—51no—
140C22×D4—52no—
141C23×C4—52no—
142#124—104no—
143C22⋊Q8—108no—
144#121—108no—
145C4×D4—104no—
146C2×C4⋊C4—52no—
147C2×C4⋊C4—52no—
148C2×C4⋊C4C2×D511yescharacteristic
149C2×C4⋊C4C2×D512yes—
150C2×C22⋊C4—51no—
151C2×C22⋊C4—52no—
152C2×C22⋊C4—52no—
153C2×C22⋊C4—52no—
154C2×C22⋊C4—52no—
155C2×C22⋊C4—51no—
156C2×C22⋊C4—51no—
157C2×C22⋊C4C2×D512yes—
158C2×C22⋊C4C2×D511yescharacteristic
159C2×C42—51no—
160C2×C42—52no—
161#95—51no—
162#95C2×D511yescharacteristic
163C22×C5—41no—
164C22×C5—41no—
165C22×C5—41no—
166C22×C5—42no—
167C22×C5—42no—
168C22×C5C22×D411yescharacteristic
169C22×C5C22×D412yes—
170C22×C5C2×#4112yes—
171C22×C5H(5,𝔽2)11yescharacteristic
172C22×C52−1+411yescharacteristic
173C2×D5—24no—
174C2×D5—24no—
175C2×D5—24no—
176C2×D5—42no—
177C2×D5—42no—
178C2×D5—42no—
179C2×D5—42no—
180C2×D5—41no—
181C2×D5—41no—
182C2×D5—41no—
183C2×D5—42no—
184C2×D5—42no—
185C4×C5—22no—
186C4×C5—24no—
187C4×C5—41no—
188C4×C5—42no—
189Dic5—22no—
190Dic5—24no—
191Dic5—41no—
192Dic5#12714yes—
193C24—52no—
194C24—51no—
195C24C22×D511yescharacteristic
196C2×Q8—104no—
197C2×Q8—102no—
198C2×D4—104no—
199C2×D4—104no—
200C2×D4—104no—
201C2×D4—104no—
202C2×D4—102no—
203C2×D4—104no—
204C2×D4—102no—
205C2×D4—104no—
206C2×D4—202no—
207C2×D4—202no—
208C22×C4—51no—
209C22×C4—52no—
210C22×C4—51no—
211C22×C4—51no—
212C22×C4—104no—
213C22×C4—104no—
214C22×C4—104no—
215C22×C4C22×D511yescharacteristic
216C22×C4C22×D512yes—
217C22×C4C22×D511yescharacteristic
218C22×C4C22×D512yes—
219C4⋊C4—22no—
220C4⋊C4—24no—
221C4⋊C4—58no—
222C4⋊C4—104no—
223C22⋊C4—24no—
224C22⋊C4—22no—
225C22⋊C4—104no—
226C22⋊C4—104no—
227C22⋊C4—104no—
228C22⋊C4—102no—
229C22⋊C4—102no—
230C22⋊C4—104no—
231C22⋊C4—102no—
232C42—54no—
233C42—202no—
234C2×C5—81no—
235C2×C5#52511yescharacteristic
236C2×C5C2×#12711yescharacteristic
237C2×C5#53711yescharacteristic
238C2×C5#54612yes—
239C2×C5#54812yes—
240D5—44no—
241D5—81no—
242C23—21no—
243C23—22no—
244C23—21no—
245C23—22no—
246C23—21no—
247C23—54no—
248C23—102no—
249C23—102no—
250C23—102no—
251C23—102no—
252C23—102no—
253C23—102no—
254C23—101no—
255C23—102no—
256C23—101no—
257C23—102no—
258C23—101no—
259C23C23×D511yescharacteristic
260Q8—204no—
261D4—108no—
262D4—208no—
263D4—204no—
264D4—208no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—21no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—52no—
274C2×C4—54no—
275C2×C4—52no—
276C2×C4—54no—
277C2×C4—101no—
278C2×C4—102no—
279C2×C4—101no—
280C2×C4—102no—
281C2×C4—101no—
282C2×C4—102no—
283C2×C4—208no—
284C2×C4D4×D512yes—
285C2×C4D4×D514yes—
286C5#254011yescharacteristic, complemented, Sylow
287C22—41no—
288C22—41no—
289C22—42no—
290C22—42no—
291C22—41no—
292C22—104no—
293C22—104no—
294C22—104no—
295C22—202no—
296C22—202no—
297C22—202no—
298C22—202no—
299C22—201no—
300C22—201no—
301C22—202no—
302C22—202no—
303C22—201no—
304C22C2×D4×D511yescharacteristic
305C22C2×D4×D512yes—
306C22#414711yescharacteristic
307C22D5×#4112yes—
308C22#414911yescharacteristic
309C4—24no—
310C4—22no—
311C4—42no—
312C4—41no—
313C4—54no—
314C4—102no—
315C4—104no—
316C4—201no—
317C2—81no—
318C2—204no—
319C2—401no—
320C2#6596311yescharacteristic
321C2#6596712yes—
322C2#6598812yes—
323C2#6599011yescharacteristic
324C2D5×#12711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#399C2
hypercenter#399D5
hypocenterC5#2540
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
1. C2×#4042C2semidirect
2. C2×#4039C2semidirect
3. C2×#4031C2semidirect
4. C2×#4030C2semidirect
5. C5×#399C2semidirect
6. #65281C2semidirect
7. #65277C2semidirect
8. #65249C2semidirect
9. #65246C2semidirect
10. #64960C2semidirect
11. #64938C2semidirect
78. #683D4semidirect
80. C4×Dic5D4semidirect
86. #399D5semidirect
286. C5#2540semidirect

Maximal subgroups

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

subgroupindex
C2×#40422
C2×#40392
C2×#40312
C2×#40302
C5×#3992
#652812
#652772
#652492
#652462
#649602
#649382
#25405