Finite Group Databasebeta

← #10608418

Subgroups — #10608418

Order 640.

760 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 760 conjugacy classes of subgroups — too many to draw. Only its 161 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×#4085C212yesmaximal, complemented
2C2×D4×Dic5C211yesmaximal, characteristic, complemented
3C2×#4031C211yesmaximal, characteristic, complemented
4C2×#4024C212yesmaximal, complemented
5C5×#392C211yesmaximal, characteristic, complemented
6#65512C211yesmaximal, characteristic, complemented
7#65510C212yesmaximal, complemented
8#65246C212yesmaximal, complemented
9#65238C211yesmaximal, characteristic, complemented
10#64966C211yesmaximal, characteristic, complemented
11#64938C211yesmaximal, characteristic, complemented
12C22×C5×D4C2211yescharacteristic
13C22×C5×D4C2212yes—
14C22×C22⋊D5C2211yescharacteristic
15C22×C22⋊D5C2212yes—
16C23×Dic5C2212yes—
17C22×Dic10C2211yescharacteristic
18C2×C5×C22⋊C4C2212yes—
19C2×C5×C22⋊C4C2211yescharacteristic
20C2×C22⋊Dic5C2212yes—
21C2×C22⋊Dic5C2211yescharacteristic
22C2×C22⋊Dic5C2212yes—
23C2×C22⋊Dic5C2212yes—
24#4085—28no—
25D4×Dic5—24no—
26C2×#685C2211yescharacteristic
27C2×#685C2211yescharacteristic
28C2×#685C2212yes—
29C2×C4⋊Dic5C2212yes—
30C2×C4⋊Dic5C2211yescharacteristic
31C2×#683C2212yes—
32C2×#683C2212yes—
33C2×C4×Dic5C2212yes—
34C2×C4×Dic5C2211yescharacteristic
35#4031—24no—
36#4024—28no—
37C5×#95C2211yescharacteristic
38#3963C2211yescharacteristic
39#2540—51nomaximal, Sylow
40C24×C5C2311yescharacteristic
41C24×C5C2312yes—
42C23×D5C2311yescharacteristic
43C2×C5×D4—24no—
44C2×C5×D4—24no—
45C2×C5×D4—22no—
46C2×C5×D4—24no—
47C2×C5×D4—42no—
48C22×C4×C5C2311yescharacteristic
49C22×C4×C5C2311yescharacteristic
50C22×C4×C5C2312yes—
51C2×C22⋊D5—24no—
52C2×C22⋊D5—22no—
53C2×C22⋊D5—24no—
54C2×C22⋊D5—24no—
55C2×C22⋊D5—42no—
56C22×Dic5—24no—
57C22×Dic5—24no—
58C22×Dic5—24no—
59C22×Dic5C2311yescharacteristic
60C22×Dic5C2312yes—
61C22×Dic5C2311yescharacteristic
62C22×Dic5C2312yes—
63C22×Dic5C2311yescharacteristic
64C2×Dic10—24no—
65C2×Dic10—22no—
66C5×C22⋊C4—24no—
67C5×C22⋊C4—22no—
68C22⋊Dic5—24no—
69C22⋊Dic5—22no—
70C22⋊Dic5—24no—
71C22⋊Dic5—24no—
72#685—22no—
73#685—24no—
74#685—22no—
75C4⋊Dic5—24no—
76C4⋊Dic5—22no—
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#399—51no—
87#392—51no—
88#392D511yescharacteristic, complemented
89#390—52no—
90#390—52no—
91C4×C4⋊C4—51no—
92C23×C5—22no—
93C23×C5—22no—
94C23×C5—22no—
95C23×C5—22no—
96C23×C5—22no—
97C23×C5—22no—
98C23×C5—21no—
99C23×C5—22no—
100C23×C5—22no—
101C23×C5—21no—
102C23×C5—21no—
103C23×C5#4114yes—
104C23×C5C2411yescharacteristic
105C22×D5—21no—
106C22×D5—22no—
107C22×D5—22no—
108C22×D5—21no—
109C22×D5—21no—
110C5×D4—28no—
111C5×D4—48no—
112C2×C4×C5—22no—
113C2×C4×C5—22no—
114C2×C4×C5—22no—
115C2×C4×C5—22no—
116C2×C4×C5—21no—
117C2×C4×C5C2×D412yes—
118C2×C4×C5C2×D414yes—
119C22⋊D5—48no—
120C22⋊D5—44no—
121C2×Dic5—22no—
122C2×Dic5—21no—
123C2×Dic5—22no—
124C2×Dic5—21no—
125C2×Dic5—22no—
126C2×Dic5—22no—
127C2×Dic5—22no—
128C2×Dic5—21no—
129C2×Dic5—22no—
130C2×Dic5—48no—
131C2×Dic5C2×D412yes—
132C2×Dic5C2×D414yes—
133C2×Dic5C2×D412yes—
134C2×Dic5C2×D414yes—
135Dic10—44no—
136C22×Q8—51no—
137C22×D4—51no—
138C22×D4—52no—
139C22×D4C2×D511yescharacteristic
140C22×D4C2×D512yes—
141C23×C4—52no—
142#124—104no—
143C22⋊Q8—108no—
144#121—108no—
145C4×D4—104no—
146C2×C4⋊C4—52no—
147C2×C4⋊C4—51no—
148C2×C4⋊C4—52no—
149C2×C4⋊C4—52no—
150C2×C22⋊C4—52no—
151C2×C22⋊C4—52no—
152C2×C22⋊C4—52no—
153C2×C22⋊C4—51no—
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—24no—
164C22×C5—24no—
165C22×C5—24no—
166C22×C5—42no—
167C22×C5—42no—
168C22×C5—42no—
169C22×C5—42no—
170C22×C5—41no—
171C22×C5—41no—
172C22×C5—41no—
173C22×C5—42no—
174C22×C5—42no—
175C22×C5C22×D411yescharacteristic
176C22×C5C22×D412yes—
177C22×C5C2×#4112yes—
178C22×C5H(5,𝔽2)11yescharacteristic
179C22×C52−1+411yescharacteristic
180C2×D5—41no—
181C2×D5—41no—
182C2×D5—41no—
183C2×D5—42no—
184C2×D5—42no—
185C4×C5—22no—
186C4×C5—24no—
187C4×C5—41no—
188Dic5—22no—
189Dic5—24no—
190Dic5—41no—
191Dic5—42no—
192Dic5#12714yes—
193C24—51no—
194C24C22×D511yescharacteristic
195C24C22×D512yes—
196C2×Q8—104no—
197C2×Q8—102no—
198C2×D4—24no—
199C2×D4—24no—
200C2×D4—22no—
201C2×D4—24no—
202C2×D4—42no—
203C2×D4—104no—
204C2×D4—104no—
205C2×D4—102no—
206C2×D4—104no—
207C2×D4—202no—
208C22×C4—51no—
209C22×C4—52no—
210C22×C4—51no—
211C22×C4—51no—
212C22×C4—52no—
213C22×C4—104no—
214C22×C4—104no—
215C22×C4—104no—
216C22×C4C22×D511yescharacteristic
217C22×C4C22×D511yescharacteristic
218C22×C4C22×D512yes—
219C4⋊C4—58no—
220C4⋊C4—104no—
221C4⋊C4—102no—
222C4⋊C4—104no—
223C22⋊C4—24no—
224C22⋊C4—22no—
225C22⋊C4—102no—
226C22⋊C4—104no—
227C22⋊C4—104no—
228C22⋊C4—104no—
229C22⋊C4—102no—
230C22⋊C4—104no—
231C22⋊C4—102no—
232C42—54no—
233C42—202no—
234C2×C5—44no—
235C2×C5—81no—
236C2×C5#52511yescharacteristic
237C2×C5C2×#12711yescharacteristic
238C2×C5#53711yescharacteristic
239C2×C5#54612yes—
240C2×C5#54812yes—
241D5—81no—
242C23—22no—
243C23—22no—
244C23—22no—
245C23—22no—
246C23—22no—
247C23—22no—
248C23—21no—
249C23—22no—
250C23—22no—
251C23—21no—
252C23—21no—
253C23—101no—
254C23—102no—
255C23—102no—
256C23—101no—
257C23—101no—
258C23#71114yes—
259C23C23×D511yescharacteristic
260Q8—204no—
261D4—28no—
262D4—48no—
263D4—204no—
264D4—208no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—21no—
269C2×C4—22no—
270C2×C4—52no—
271C2×C4—54no—
272C2×C4—52no—
273C2×C4—54no—
274C2×C4—102no—
275C2×C4—101no—
276C2×C4—102no—
277C2×C4—101no—
278C2×C4—102no—
279C2×C4—101no—
280C2×C4—102no—
281C2×C4—102no—
282C2×C4—102no—
283C2×C4—208no—
284C2×C4D4×D512yes—
285C2×C4D4×D514yes—
286C5#254011yescharacteristic, complemented, Sylow
287C22—24no—
288C22—24no—
289C22—24no—
290C22—42no—
291C22—42no—
292C22—42no—
293C22—42no—
294C22—41no—
295C22—41no—
296C22—42no—
297C22—42no—
298C22—41no—
299C22—201no—
300C22—201no—
301C22—202no—
302C22—202no—
303C22—201no—
304C22C2×D4×D511yescharacteristic
305C22C2×D4×D512yes—
306C22C2×#71112yes—
307C22#414411yescharacteristic
308C22#415011yescharacteristic
309C4—22no—
310C4—24no—
311C4—41no—
312C4—54no—
313C4—104no—
314C4—102no—
315C4—202no—
316C4—201no—
317C2—44no—
318C2—81no—
319C2—401no—
320C2#6592011yescharacteristic
321C2#6592512yes—
322C2#6593212yes—
323C2#6594211yescharacteristic
324C2D5×#12711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#392C2
hypercenter#392D5
hypocenterC5#2540
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C2×#4085C2semidirect
2. C2×D4×Dic5C2semidirect
3. C2×#4031C2semidirect
4. C2×#4024C2semidirect
5. C5×#392C2semidirect
6. #65512C2semidirect
7. #65510C2semidirect
8. #65246C2semidirect
9. #65238C2semidirect
10. #64966C2semidirect
11. #64938C2semidirect
78. #683D4semidirect
80. C4×Dic5D4semidirect
88. #392D5semidirect
286. C5#2540semidirect

Maximal subgroups

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

subgroupindex
C2×#40852
C2×D4×Dic52
C2×#40312
C2×#40242
C5×#3922
#655122
#655102
#652462
#652382
#649662
#649382
#25405