Finite Group Databasebeta

← #10608417

Subgroups — #10608417

Order 640.

732 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 732 conjugacy classes of subgroups — too many to draw. Only its 151 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×#4027C211yesmaximal, characteristic, complemented
4C5×#392C211yesmaximal, characteristic, complemented
5#65512C211yesmaximal, characteristic, complemented
6#65509C212yesmaximal, complemented
7#65245C212yesmaximal, complemented
8#65242C211yesmaximal, characteristic, complemented
9#65231C212yesmaximal, complemented
10#64963C211yesmaximal, characteristic, complemented
11#64939C211yesmaximal, characteristic, complemented
12C22×C5×D4C2211yescharacteristic
13C22×C5×D4C2212yes—
14C22×C22⋊D5C2212yes—
15C22×C22⋊D5C2211yescharacteristic
16C23×Dic5C2212yes—
17C23×Dic5C2211yescharacteristic
18C22×C4×D5C2211yescharacteristic
19C2×C5×C22⋊C4C2211yescharacteristic
20C2×C5×C22⋊C4C2212yes—
21C2×C22⋊Dic5C2212yes—
22C2×C22⋊Dic5C2212yes—
23C2×C22⋊Dic5C2212yes—
24#4085—28no—
25D4×Dic5—24no—
26C2×#685C2211yescharacteristic
27C2×#685C2212yes—
28C2×C4⋊Dic5C2211yescharacteristic
29C2×#683C2211yescharacteristic
30C2×#683C2212yes—
31C2×C4×Dic5C2211yescharacteristic
32#4027—24no—
33C5×#95C2211yescharacteristic
34#3963C2211yescharacteristic
35#3963C2212yes—
36#3963C2212yes—
37#3963C2212yes—
38#2535—51nomaximal, Sylow
39C24×C5C2311yescharacteristic
40C24×C5C2312yes—
41C23×D5C2311yescharacteristic
42C2×C5×D4—24no—
43C2×C5×D4—24no—
44C2×C5×D4—24no—
45C2×C5×D4—22no—
46C2×C5×D4—42no—
47C22×C4×C5C2311yescharacteristic
48C22×C4×C5C2312yes—
49C22×C4×C5C2311yescharacteristic
50C2×C22⋊D5—24no—
51C2×C22⋊D5—24no—
52C2×C22⋊D5—22no—
53C2×C22⋊D5—22no—
54C2×C22⋊D5—42no—
55C2×C22⋊D5D414yescomplemented
56C22×Dic5—22no—
57C22×Dic5—24no—
58C22×Dic5—24no—
59C22×Dic5—24no—
60C22×Dic5—42no—
61C22×Dic5C2311yescharacteristic
62C22×Dic5C2312yes—
63C22×Dic5C2312yes—
64C22×Dic5C2311yescharacteristic
65C22×Dic5C2311yescharacteristic
66C2×C4×D5—22no—
67C2×C4×D5—42no—
68C5×C22⋊C4—22no—
69C5×C22⋊C4—42no—
70C22⋊Dic5—24no—
71C22⋊Dic5—24no—
72C22⋊Dic5—42no—
73#685—24no—
74#685—22no—
75C4⋊Dic5—22no—
76#683—22no—
77#683—24no—
78C4×Dic5D414yescomplemented
79C2×#121—52no—
80C2×C4×D4—51no—
81C2×C4×D4—51no—
82#397—51no—
83#397—51no—
84#392—51no—
85#392D511yescharacteristic, complemented
86#386—52no—
87#386—52no—
88#382—51no—
89#380—52no—
90C23×C5—22no—
91C23×C5—21no—
92C23×C5—22no—
93C23×C5—22no—
94C23×C5—21no—
95C23×C5—22no—
96C23×C5—22no—
97C23×C5—22no—
98C23×C5—22no—
99C23×C5—22no—
100C23×C5C2×D412yes—
101C23×C5#4114yes—
102C23×C5C2411yescharacteristic
103C22×D5—21no—
104C22×D5—22no—
105C22×D5—21no—
106C22×D5—22no—
107C22×D5C2×D412yes—
108C5×D4—28no—
109C5×D4—48no—
110C2×C4×C5—22no—
111C2×C4×C5—22no—
112C2×C4×C5—21no—
113C2×C4×C5—22no—
114C2×C4×C5—22no—
115C2×C4×C5—22no—
116C2×C4×C5C2×D412yes—
117C22⋊D5—28no—
118C22⋊D5—48no—
119C2×Dic5—22no—
120C2×Dic5—21no—
121C2×Dic5—22no—
122C2×Dic5—22no—
123C2×Dic5—22no—
124C2×Dic5—22no—
125C2×Dic5—22no—
126C2×Dic5—22no—
127C2×Dic5—22no—
128C2×Dic5—44no—
129C2×Dic5—48no—
130C2×Dic5C2×D412yes—
131C2×Dic5C2×D412yes—
132C2×Dic5C2×D412yes—
133C2×Dic5#4112yes—
134C4×D5—44no—
135C22×D4—51no—
136C22×D4—52no—
137C22×D4C2×D511yescharacteristic
138C22×D4C2×D512yes—
139C23×C4—52no—
140C23×C4—51no—
141C23×C4—51no—
142#121—108no—
143C4×D4—104no—
144C4×D4—104no—
145C2×C4⋊C4—51no—
146C2×C4⋊C4—51no—
147C2×C4⋊C4—52no—
148C2×C22⋊C4—52no—
149C2×C22⋊C4—52no—
150C2×C22⋊C4—51no—
151C2×C22⋊C4—52no—
152C2×C22⋊C4—52no—
153C2×C22⋊C4C2×D511yescharacteristic
154C2×C22⋊C4C2×D512yes—
155C2×C42—51no—
156#95—52no—
157#95—52no—
158#95—52no—
159#95—51no—
160#95C2×D511yescharacteristic
161C22×C5—22no—
162C22×C5—24no—
163C22×C5—24no—
164C22×C5—24no—
165C22×C5—24no—
166C22×C5—42no—
167C22×C5—41no—
168C22×C5—41no—
169C22×C5—42no—
170C22×C5—42no—
171C22×C5—42no—
172C22×C5—42no—
173C22×C5C22×D411yescharacteristic
174C22×C5C22×D411yescharacteristic
175C22×C5C2×#4112yes—
176C22×C5C2×#4111yescharacteristic
177C22×C5H(5,𝔽2)12yes—
178C2×D5—22no—
179C2×D5—24no—
180C2×D5—41no—
181C2×D5—41no—
182C2×D5—42no—
183C4×C5—22no—
184C4×C5—41no—
185C4×C5—42no—
186Dic5—22no—
187Dic5—42no—
188Dic5—41no—
189Dic5—42no—
190Dic5#12114yes—
191C24—51no—
192C24C22×D511yescharacteristic
193C24C22×D512yes—
194C2×D4—24no—
195C2×D4—24no—
196C2×D4—24no—
197C2×D4—22no—
198C2×D4—42no—
199C2×D4—54no—
200C2×D4—102no—
201C2×D4—102no—
202C2×D4—104no—
203C2×D4—104no—
204C2×D4—202no—
205C22×C4—51no—
206C22×C4—51no—
207C22×C4—52no—
208C22×C4—52no—
209C22×C4—51no—
210C22×C4—102no—
211C22×C4—104no—
212C22×C4—102no—
213C22×C4—104no—
214C22×C4—104no—
215C22×C4—202no—
216C22×C4—202no—
217C22×C4C22×D512yes—
218C22×C4C22×D511yescharacteristic
219C22×C4C22×D511yescharacteristic
220C4⋊C4—102no—
221C4⋊C4—104no—
222C4⋊C4—102no—
223C22⋊C4—22no—
224C22⋊C4—42no—
225C22⋊C4—104no—
226C22⋊C4—104no—
227C22⋊C4—104no—
228C22⋊C4—102no—
229C22⋊C4—202no—
230C42—54no—
231C2×C5—42no—
232C2×C5—44no—
233C2×C5C2×#12111yescharacteristic
234C2×C5#52511yescharacteristic
235C2×C5#53511yescharacteristic
236C2×C5#54612yes—
237C2×C5#55312yes—
238D5—42no—
239C23—21no—
240C23—22no—
241C23—22no—
242C23—22no—
243C23—22no—
244C23—22no—
245C23—22no—
246C23—22no—
247C23—22no—
248C23—21no—
249C23—52no—
250C23—101no—
251C23—102no—
252C23—102no—
253C23—101no—
254C23D4×D512yes—
255C23#71114yes—
256C23C23×D511yescharacteristic
257D4—28no—
258D4—48no—
259D4—108no—
260D4—208no—
261C2×C4—22no—
262C2×C4—21no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—52no—
268C2×C4—52no—
269C2×C4—52no—
270C2×C4—52no—
271C2×C4—102no—
272C2×C4—102no—
273C2×C4—101no—
274C2×C4—102no—
275C2×C4—102no—
276C2×C4—102no—
277C2×C4—102no—
278C2×C4—102no—
279C2×C4—102no—
280C2×C4—204no—
281C2×C4—208no—
282C2×C4—204no—
283C2×C4D4×D512yes—
284C5#253511yescharacteristic, complemented, Sylow
285C22—24no—
286C22—24no—
287C22—24no—
288C22—24no—
289C22—22no—
290C22—41no—
291C22—42no—
292C22—41no—
293C22—42no—
294C22—42no—
295C22—42no—
296C22—42no—
297C22—104no—
298C22—102no—
299C22—201no—
300C22—201no—
301C22—202no—
302C22C2×D4×D511yescharacteristic
303C22C2×D4×D511yescharacteristic
304C22C2×#71112yes—
305C22#414412yes—
306C22D5×#4111yescharacteristic
307C4—22no—
308C4—41no—
309C4—42no—
310C4—54no—
311C4—102no—
312C4—201no—
313C4—202no—
314C4—202no—
315C2—44no—
316C2—42no—
317C2—202no—
318C2#6592011yescharacteristic
319C2#6592512yes—
320C2#6593512yes—
321C2D5×#12111yescharacteristic
322C2#6604911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#392C2
hypercenter#392D5
hypocenterC5#2535
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×#4027C2semidirect
4. C5×#392C2semidirect
5. #65512C2semidirect
6. #65509C2semidirect
7. #65245C2semidirect
8. #65242C2semidirect
9. #65231C2semidirect
10. #64963C2semidirect
11. #64939C2semidirect
55. C2×C22⋊D5D4semidirect
78. C4×Dic5D4semidirect
85. #392D5semidirect
284. C5#2535semidirect

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×#40272
C5×#3922
#655122
#655092
#652452
#652422
#652312
#649632
#649392
#25355