Finite Group Databasebeta

← #10605588

Subgroups — #10605588

Order 640.

816 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 816 conjugacy classes of subgroups — too many to draw. Only its 141 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×#4099C212yesmaximal, complemented
2C2×#4030C212yesmaximal, complemented
3C5×#379C211yesmaximal, characteristic, complemented
4#65250C212yesmaximal, complemented
5#65249C211yesmaximal, characteristic, complemented
6#65245C212yesmaximal, complemented
7#65245C212yesmaximal, complemented
8#65241C212yesmaximal, complemented
9#65238C211yesmaximal, characteristic, complemented
10C25×C5C2211yescharacteristic
11C22×C22⋊D5C2212yes—
12C22×C22⋊D5C2212yes—
13C22×C22⋊D5C2212yes—
14C22×D20C2211yescharacteristic, complemented
15C22×C4×D5C2212yescomplemented
16C2×C5×C22⋊C4C2212yes—
17C2×C5×C22⋊C4C2211yescharacteristic
18C2×C5×C22⋊C4C2211yescharacteristic
19C2×C5×C22⋊C4C2212yes—
20#4099—28no—
21C2×C22⋊Dic5C2212yes—
22C2×C22⋊Dic5C2212yes—
23C2×C22⋊Dic5C2212yes—
24C2×#685C2212yescomplemented
25C2×#685C2212yescomplemented
26C2×C4⋊Dic5C2211yescharacteristic, complemented
27C2×#683C2212yescomplemented
28#4030—28no—
29#3963C2211yescharacteristic, complemented
30#3963C2211yescharacteristic, complemented
31#3963C2212yescomplemented
32#3963C2212yescomplemented
33#2668—51nomaximal, Sylow
34C24×C5—24no—
35C24×C5—22no—
36C24×C5—41no—
37C24×C5—42no—
38C24×C5—41no—
39C24×C5C2312yes—
40C24×C5C2311yescharacteristic
41C23×D5C2312yes—
42C22×C4×C5C2312yes—
43C22×C4×C5C2311yescharacteristic
44C22×C4×C5C2311yescharacteristic
45C2×C22⋊D5—24no—
46C2×C22⋊D5—24no—
47C2×C22⋊D5—24no—
48C2×C22⋊D5—24no—
49C2×C22⋊D5—42no—
50C2×C22⋊D5—42no—
51C2×C22⋊D5—42no—
52C2×C22⋊D5—42no—
53C2×C22⋊D5D418yescomplemented
54C22×Dic5C2312yes—
55C22×Dic5C2312yes—
56C22×Dic5C2312yes—
57C2×D20—24no—
58C2×D20—41no—
59C2×C4×D5—24no—
60C2×C4×D5—42no—
61C2×C4×D5—42no—
62C5×C22⋊C4—24no—
63C5×C22⋊C4—41no—
64C5×C22⋊C4—42no—
65C5×C22⋊C4—41no—
66C22⋊Dic5—24no—
67C22⋊Dic5—24no—
68C22⋊Dic5—24no—
69#685—24no—
70#685—42no—
71C4⋊Dic5—41no—
72#683—24no—
73C2×#121—52no—
74C2×#120—52no—
75#399—51no—
76#397—52no—
77#394—52no—
78#392—51no—
79#386—52no—
80#386—52no—
81#379D511yescharacteristic, complemented
82C23×C5—22no—
83C23×C5—22no—
84C23×C5—22no—
85C23×C5—21no—
86C23×C5—21no—
87C23×C5—22no—
88C23×C5—22no—
89C23×C5—22no—
90C23×C5—22no—
91C23×C5—22no—
92C23×C5—44no—
93C23×C5—48no—
94C23×C5—48no—
95C23×C5—44no—
96C23×C5—44no—
97C23×C5C2×D414yes—
98C23×C5#4112yes—
99C23×C5C2411yescharacteristic
100C22×D5—22no—
101C22×D5—22no—
102C22×D5—22no—
103C22×D5—22no—
104C22×D5—22no—
105C22×D5—22no—
106C22×D5C2×D414yes—
107C2×C4×C5—22no—
108C2×C4×C5—22no—
109C2×C4×C5—21no—
110C2×C4×C5—22no—
111C2×C4×C5—22no—
112C2×C4×C5—22no—
113C2×C4×C5—21no—
114C22⋊D5—48no—
115C22⋊D5—48no—
116C22⋊D5—48no—
117C2×Dic5—22no—
118C2×Dic5—22no—
119C2×Dic5—22no—
120C2×Dic5—22no—
121C2×Dic5—22no—
122C2×Dic5—22no—
123C2×Dic5—22no—
124C2×Dic5—22no—
125C2×Dic5C2×D414yes—
126D20—44no—
127C4×D5—48no—
128C25C2×D511yescharacteristic
129C22×D4—52no—
130C22×D4—52no—
131C22×D4—52no—
132C22×D4—51no—
133C23×C4—52no—
134#121—108no—
135#120—108no—
136C2×C4⋊C4—52no—
137C2×C4⋊C4—51no—
138C2×C22⋊C4—52no—
139C2×C22⋊C4—52no—
140C2×C22⋊C4—52no—
141C2×C22⋊C4—52no—
142C2×C22⋊C4—52no—
143C2×C22⋊C4C2×D512yes—
144C2×C22⋊C4C2×D511yescharacteristic
145C2×C22⋊C4C2×D512yes—
146C2×C22⋊C4C2×D511yescharacteristic
147#95—52no—
148#95—51no—
149#95—52no—
150#95—51no—
151C22×C5—24no—
152C22×C5—22no—
153C22×C5—24no—
154C22×C5—24no—
155C22×C5—24no—
156C22×C5—416no—
157C22×C5—42no—
158C22×C5—42no—
159C22×C5—42no—
160C22×C5—41no—
161C22×C5—42no—
162C22×C5—42no—
163C22×C5—41no—
164C22×C5C22×D412yes—
165C22×C5C2×#4111yescharacteristic
166C22×C5H(5,𝔽2)11yescharacteristic
167C22×C5H(5,𝔽2)12yes—
168C22×C5H(5,𝔽2)11yescharacteristic
169C2×D5—24no—
170C2×D5—24no—
171C2×D5—24no—
172C2×D5—42no—
173C2×D5—42no—
174C2×D5—42no—
175C2×D5—42no—
176C4×C5—42no—
177C4×C5—41no—
178C4×C5—41no—
179Dic5—24no—
180Dic5—42no—
181Dic5—42no—
182C24—22no—
183C24—24no—
184C24—42no—
185C24—41no—
186C24—41no—
187C24—52no—
188C24C22×D511yescharacteristic
189C24C22×D512yes—
190C2×D4—58no—
191C2×D4—104no—
192C2×D4—104no—
193C2×D4—104no—
194C2×D4—104no—
195C2×D4—104no—
196C2×D4—201no—
197C2×D4—202no—
198C2×D4—202no—
199C2×D4—202no—
200C2×D4—202no—
201C22×C4—52no—
202C22×C4—52no—
203C22×C4—52no—
204C22×C4—104no—
205C22×C4—202no—
206C22×C4—202no—
207C22×C4C22×D512yes—
208C22×C4C22×D511yescharacteristic
209C22×C4C22×D511yescharacteristic
210C4⋊C4—104no—
211C4⋊C4—201no—
212C22⋊C4—24no—
213C22⋊C4—42no—
214C22⋊C4—41no—
215C22⋊C4—41no—
216C22⋊C4—104no—
217C22⋊C4—104no—
218C22⋊C4—104no—
219C22⋊C4—104no—
220C22⋊C4—202no—
221C2×C5—44no—
222C2×C5—42no—
223C2×C5#53412yes—
224C2×C5#53811yescharacteristic
225C2×C5D4211yescharacteristic
226C2×C5#54612yes—
227C2×C5#56011yescharacteristic
228D5—44no—
229C23—22no—
230C23—22no—
231C23—22no—
232C23—22no—
233C23—21no—
234C23—21no—
235C23—22no—
236C23—22no—
237C23—22no—
238C23—22no—
239C23—48no—
240C23—44no—
241C23—44no—
242C23—48no—
243C23—44no—
244C23—54no—
245C23—102no—
246C23—102no—
247C23—102no—
248C23—102no—
249C23—102no—
250C23—102no—
251C23#70912yes—
252C23D4×D514yes—
253C23C23×D511yescharacteristic
254D4—204no—
255D4—208no—
256D4—208no—
257D4—208no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—21no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—21no—
265C2×C4—54no—
266C2×C4—102no—
267C2×C4—102no—
268C2×C4—102no—
269C2×C4—102no—
270C2×C4—102no—
271C2×C4—102no—
272C2×C4—102no—
273C2×C4—102no—
274C2×C4—208no—
275C5#266811yescharacteristic, complemented, Sylow
276C22—24no—
277C22—24no—
278C22—22no—
279C22—24no—
280C22—24no—
281C22—416no—
282C22—42no—
283C22—42no—
284C22—42no—
285C22—42no—
286C22—41no—
287C22—41no—
288C22—42no—
289C22—104no—
290C22—104no—
291C22—104no—
292C22—202no—
293C22—202no—
294C22—202no—
295C22—202no—
296C22C2×#70911yescharacteristic
297C22C2×D4×D512yes—
298C22#414412yes—
299C22#414411yescharacteristic
300C22#414411yescharacteristic
301C4—42no—
302C4—41no—
303C4—41no—
304C4—104no—
305C4—202no—
306C4—202no—
307C2—42no—
308C2—44no—
309C2—204no—
310C2#6582412yes—
311C2#6582811yescharacteristic
312C2#6592312yes—
313C2#6592411yescharacteristic
314C2#6592911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#379C2
hypercenter#379D5
hypocenterC5#2668
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #379

Complements

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

NH ≅ G/Ntype
1. C2×#4099C2semidirect
2. C2×#4030C2semidirect
3. C5×#379C2semidirect
4. #65250C2semidirect
5. #65249C2semidirect
6. #65245C2semidirect
7. #65245C2semidirect
8. #65241C2semidirect
9. #65238C2semidirect
14. C22×D20C22semidirect
15. C22×C4×D5C22semidirect
24. C2×#685C22semidirect
25. C2×#685C22semidirect
26. C2×C4⋊Dic5C22semidirect
27. C2×#683C22semidirect
29. #3963C22semidirect
30. #3963C22semidirect
31. #3963C22semidirect
32. #3963C22semidirect
53. C2×C22⋊D5D4semidirect
81. #379D5semidirect
275. C5#2668semidirect

Maximal subgroups

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

subgroupindex
C2×#40992
C2×#40302
C5×#3792
#652502
#652492
#652452
#652412
#652382
#26685