Finite Group Databasebeta

← #10608715

Subgroups — #10608715

Order 640.

584 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 133 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×#4086C211yesmaximal, characteristic
2C2×#4035C211yesmaximal, characteristic, complemented
3C5×#394C211yesmaximal, characteristic, complemented
4#65511C211yesmaximal, characteristic, complemented
5#65280C211yesmaximal, characteristic, complemented
6#65250C212yesmaximal, complemented
7#65248C212yesmaximal, complemented
8#65246C212yesmaximal, complemented
9#65246C212yesmaximal, complemented
10#64961C211yesmaximal, characteristic, complemented
11#64937C211yesmaximal, characteristic, complemented
12C22×C5×D4C2211yescharacteristic
13C22×C22⋊D5C2212yes—
14C22×C22⋊D5C2212yes—
15C22×D20C2211yescharacteristic
16C2×C5×C4⋊C4C2211yescharacteristic
17C2×C5×C22⋊C4C2212yes—
18C2×C5×C22⋊C4C2212yes—
19C2×C22⋊Dic5C2212yes—
20C2×C22⋊Dic5C2212yes—
21#4086—24no—
22C2×#685C2212yes—
23C2×#685C2212yes—
24C2×#685C2212yes—
25C2×#685C2212yes—
26C2×C4⋊Dic5C2211yescharacteristic
27C2×#683C2211yescharacteristic
28C2×#683C2211yescharacteristic
29C2×#683C2211yescharacteristic
30C2×#683C2211yescharacteristic
31C2×C4×Dic5C2212yes—
32C2×C4×Dic5C2211yescharacteristic
33#4035—24no—
34C5×#95C2211yescharacteristic
35#3963C2211yescharacteristic
36#3963C2212yes—
37#2641—51nomaximal, Sylow
38C24×C5C2312yes—
39C23×D5C2312yes—
40C2×C5×D4—22no—
41C2×C5×D4—42no—
42C22×C4×C5C2311yescharacteristic
43C22×C4×C5C2311yescharacteristic
44C22×C4×C5C2312yes—
45C22×C4×C5C2311yescharacteristic
46C2×C22⋊D5—24no—
47C2×C22⋊D5—24no—
48C2×C22⋊D5—24no—
49C2×C22⋊D5—24no—
50C2×C22⋊D5—42no—
51C2×C22⋊D5—42no—
52C22×Dic5C2312yes—
53C22×Dic5C2311yescharacteristic
54C22×Dic5C2311yescharacteristic
55C22×Dic5C2311yescharacteristic
56C22×Dic5C2311yescharacteristic
57C2×D20—22no—
58C2×D20—42no—
59C5×C4⋊C4—22no—
60C5×C22⋊C4—42no—
61C5×C22⋊C4—42no—
62C22⋊Dic5—42no—
63C22⋊Dic5—42no—
64#685—42no—
65#685—42no—
66#685—42no—
67#685—42no—
68C4⋊Dic5—22no—
69#683—22no—
70#683—22no—
71#683—22no—
72#683—22no—
73C4×Dic5—24no—
74C4×Dic5D414yes—
75C2×#127—51no—
76C2×#125—51no—
77#394—51no—
78#394—51no—
79#394—52no—
80#394—52no—
81#394—51no—
82#394D511yescharacteristic, complemented
83#390—52no—
84#390—52no—
85#376—51no—
86C23×C5—22no—
87C23×C5—22no—
88C23×C5—22no—
89C23×C5—22no—
90C23×C5—22no—
91C23×C5—22no—
92C23×C5—22no—
93C23×C5C2411yescharacteristic
94C22×D5—22no—
95C22×D5—22no—
96C22×D5—22no—
97C22×D5—22no—
98C22×D5—22no—
99C22×D5—22no—
100C22×D5—22no—
101C5×D4—44no—
102C2×C4×C5—22no—
103C2×C4×C5—21no—
104C2×C4×C5—21no—
105C2×C4×C5—22no—
106C2×C4×C5—21no—
107C2×C4×C5—22no—
108C2×C4×C5—22no—
109C2×C4×C5—22no—
110C2×C4×C5—21no—
111C2×C4×C5C2×D412yes—
112C22⋊D5—48no—
113C22⋊D5—48no—
114C2×Dic5—21no—
115C2×Dic5—21no—
116C2×Dic5—21no—
117C2×Dic5—21no—
118C2×Dic5—22no—
119C2×Dic5—21no—
120C2×Dic5—22no—
121C2×Dic5—22no—
122C2×Dic5—22no—
123C2×Dic5—21no—
124C2×Dic5C2×D412yes—
125C2×Dic5C2×D412yes—
126C2×Dic5#4114yes—
127D20—44no—
128C22×D4—52no—
129C22×D4—52no—
130C22×D4—51no—
131C22×D4C2×D511yescharacteristic
132#127—104no—
133#125—104no—
134C2×C4⋊C4—51no—
135C2×C4⋊C4—51no—
136C2×C4⋊C4—51no—
137C2×C4⋊C4—51no—
138C2×C4⋊C4—51no—
139C2×C4⋊C4C2×D511yescharacteristic
140C2×C22⋊C4—52no—
141C2×C22⋊C4—52no—
142C2×C22⋊C4—52no—
143C2×C22⋊C4—52no—
144C2×C22⋊C4—52no—
145C2×C22⋊C4—52no—
146C2×C22⋊C4C2×D512yes—
147C2×C22⋊C4C2×D512yes—
148C2×C42—51no—
149C2×C42—52no—
150#95—52no—
151#95—51no—
152#95C2×D511yescharacteristic
153C22×C5—42no—
154C22×C5—42no—
155C22×C5—42no—
156C22×C5—42no—
157C22×C5—42no—
158C22×C5—42no—
159C22×C5—42no—
160C22×C5C22×D411yescharacteristic
161C22×C5C2×#4112yes—
162C22×C5H(5,𝔽2)12yes—
163C22×C5H(5,𝔽2)11yescharacteristic
164C22×C5H(5,𝔽2)11yescharacteristic
165C2×D5—42no—
166C2×D5—42no—
167C2×D5—42no—
168C2×D5—42no—
169C2×D5—42no—
170C2×D5—42no—
171C2×D5—42no—
172C4×C5—22no—
173C4×C5—41no—
174C4×C5—41no—
175C4×C5—42no—
176Dic5—22no—
177Dic5—41no—
178Dic5—41no—
179Dic5—42no—
180Dic5#12414yes—
181C24—52no—
182C24C22×D512yes—
183C2×D4—22no—
184C2×D4—42no—
185C2×D4—104no—
186C2×D4—104no—
187C2×D4—104no—
188C2×D4—104no—
189C2×D4—102no—
190C2×D4—202no—
191C2×D4—202no—
192C2×D4—202no—
193C22×C4—51no—
194C22×C4—51no—
195C22×C4—52no—
196C22×C4—51no—
197C22×C4—51no—
198C22×C4C22×D512yes—
199C22×C4C22×D511yescharacteristic
200C22×C4C22×D511yescharacteristic
201C22×C4C22×D511yescharacteristic
202C4⋊C4—22no—
203C4⋊C4—102no—
204C4⋊C4—102no—
205C4⋊C4—102no—
206C4⋊C4—102no—
207C4⋊C4—102no—
208C22⋊C4—42no—
209C22⋊C4—42no—
210C22⋊C4—202no—
211C22⋊C4—202no—
212C22⋊C4—202no—
213C22⋊C4—202no—
214C22⋊C4—202no—
215C22⋊C4—202no—
216C42—54no—
217C42—104no—
218C2×C5—82no—
219C2×C5C2×#12411yescharacteristic
220C2×C5#53511yescharacteristic
221C2×C5#53511yescharacteristic
222C2×C5#54012yes—
223C2×C5#54012yes—
224D5—82no—
225C23—22no—
226C23—22no—
227C23—22no—
228C23—22no—
229C23—22no—
230C23—22no—
231C23—22no—
232C23—102no—
233C23—102no—
234C23—102no—
235C23—102no—
236C23—102no—
237C23—102no—
238C23—102no—
239C23C23×D511yescharacteristic
240D4—44no—
241D4—204no—
242D4—208no—
243D4—208no—
244C2×C4—22no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—21no—
253C2×C4—52no—
254C2×C4—54no—
255C2×C4—52no—
256C2×C4—101no—
257C2×C4—101no—
258C2×C4—101no—
259C2×C4—101no—
260C2×C4—102no—
261C2×C4—101no—
262C2×C4—102no—
263C2×C4—102no—
264C2×C4—101no—
265C2×C4—102no—
266C2×C4D4×D512yes—
267C5#264111yescharacteristic, complemented, Sylow
268C22—42no—
269C22—42no—
270C22—42no—
271C22—42no—
272C22—42no—
273C22—42no—
274C22—42no—
275C22—202no—
276C22—202no—
277C22—202no—
278C22—202no—
279C22—202no—
280C22—202no—
281C22—202no—
282C22C2×D4×D511yescharacteristic
283C22#414412yes—
284C22#414411yescharacteristic
285C22D5×#4112yes—
286C22#414911yescharacteristic
287C4—22no—
288C4—41no—
289C4—41no—
290C4—42no—
291C4—54no—
292C4—102no—
293C4—201no—
294C4—201no—
295C4—202no—
296C2—82no—
297C2—402no—
298C2#6592611yescharacteristic
299C2#6594111yescharacteristic
300C2#6594912yes—
301C2#6599712yes—
302C2D5×#12411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#394C2
prime radicalC2×#4086C2
hypercenter#394D5
hypocenterC5#2641
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
2. C2×#4035C2semidirect
3. C5×#394C2semidirect
4. #65511C2semidirect
5. #65280C2semidirect
6. #65250C2semidirect
7. #65248C2semidirect
8. #65246C2semidirect
9. #65246C2semidirect
10. #64961C2semidirect
11. #64937C2semidirect
82. #394D5semidirect
267. C5#2641semidirect

Maximal subgroups

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

subgroupindex
C2×#40862
C2×#40352
C5×#3942
#655112
#652802
#652502
#652482
#652462
#649612
#649372
#26415