Finite Group Databasebeta

← #10608816

Subgroups — #10608816

Order 640.

528 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 125 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×#4084C211yesmaximal, characteristic
2C2×#4042C211yesmaximal, characteristic, complemented
3C5×#394C211yesmaximal, characteristic, complemented
4#65511C211yesmaximal, characteristic, complemented
5#65272C211yesmaximal, characteristic, complemented
6#65250C212yesmaximal, complemented
7#65248C212yesmaximal, complemented
8#65238C212yesmaximal, complemented
9#65236C212yesmaximal, complemented
10#64953C211yesmaximal, characteristic, complemented
11#64950C211yesmaximal, characteristic, complemented
12C22×C5×D4C2211yescharacteristic
13C22×C22⋊D5C2212yes—
14C22×C4×D5C2211yescharacteristic
15C22×Dic10C2211yescharacteristic
16C2×C5×C4⋊C4C2211yescharacteristic
17C2×C5×C22⋊C4C2212yes—
18C2×C5×C22⋊C4C2212yes—
19C2×C22⋊Dic5C2212yes—
20C2×C22⋊Dic5C2212yes—
21C2×C22⋊Dic5C2212yes—
22#4084—24no—
23C2×#685C2211yescharacteristic
24C2×#685C2211yescharacteristic
25C2×#685C2212yes—
26C2×C4⋊Dic5C2211yescharacteristic
27C2×C4⋊Dic5C2212yes—
28C2×#683C2211yescharacteristic
29C2×#683C2211yescharacteristic
30C2×#683C2212yes—
31#4042—24no—
32C5×#95C2211yescharacteristic
33#3963C2212yes—
34#3963C2212yes—
35#3963C2211yescharacteristic
36#3963C2212yes—
37#2816—51nomaximal, Sylow
38C24×C5C2312yes—
39C23×D5C2311yescharacteristic
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—42no—
48C2×C22⋊D5—42no—
49C22×Dic5C2311yescharacteristic
50C22×Dic5C2312yes—
51C22×Dic5C2311yescharacteristic
52C22×Dic5C2312yes—
53C22×Dic5C2311yescharacteristic
54C2×C4×D5—42no—
55C2×C4×D5D414yes—
56C2×Dic10—22no—
57C2×Dic10—42no—
58C5×C4⋊C4—22no—
59C5×C22⋊C4—42no—
60C5×C22⋊C4—42no—
61C22⋊Dic5—24no—
62C22⋊Dic5—42no—
63C22⋊Dic5—42no—
64#685—22no—
65#685—22no—
66#685—42no—
67C4⋊Dic5—22no—
68C4⋊Dic5—42no—
69#683—22no—
70#683—22no—
71#683—42no—
72C2×C22⋊Q8—51no—
73C2×#121—51no—
74#399—52no—
75#398—51no—
76#398—51no—
77#394—51no—
78#394—52no—
79#394—52no—
80#394D511yescharacteristic, complemented
81#393—52no—
82#381—51no—
83C23×C5—22no—
84C23×C5—22no—
85C23×C5—22no—
86C23×C5—22no—
87C23×C5—22no—
88C23×C5—22no—
89C23×C5—22no—
90C23×C5C2411yescharacteristic
91C22×D5—22no—
92C22×D5—21no—
93C22×D5—21no—
94C22×D5—22no—
95C22×D5C2×D412yes—
96C5×D4—44no—
97C2×C4×C5—22no—
98C2×C4×C5—21no—
99C2×C4×C5—21no—
100C2×C4×C5—21no—
101C2×C4×C5—22no—
102C2×C4×C5—22no—
103C2×C4×C5—21no—
104C2×C4×C5—22no—
105C2×C4×C5—22no—
106C2×C4×C5C2×D412yes—
107C22⋊D5—48no—
108C2×Dic5—22no—
109C2×Dic5—21no—
110C2×Dic5—21no—
111C2×Dic5—22no—
112C2×Dic5—22no—
113C2×Dic5—22no—
114C2×Dic5—21no—
115C2×Dic5—22no—
116C2×Dic5—22no—
117C2×Dic5—21no—
118C2×Dic5—22no—
119C2×Dic5—22no—
120C2×Dic5C2×D412yes—
121C4×D5—44no—
122Dic10—44no—
123C22×Q8—51no—
124C22×D4—52no—
125C22×D4C2×D511yescharacteristic
126C23×C4—51no—
127C22⋊Q8—104no—
128#121—104no—
129C2×C4⋊C4—51no—
130C2×C4⋊C4—51no—
131C2×C4⋊C4—51no—
132C2×C4⋊C4—52no—
133C2×C4⋊C4—52no—
134C2×C4⋊C4C2×D511yescharacteristic
135C2×C22⋊C4—52no—
136C2×C22⋊C4—51no—
137C2×C22⋊C4—51no—
138C2×C22⋊C4—52no—
139C2×C22⋊C4—52no—
140C2×C22⋊C4—52no—
141C2×C22⋊C4C2×D512yes—
142C2×C22⋊C4C2×D512yes—
143#95—52no—
144#95—51no—
145#95—52no—
146#95—52no—
147#95C2×D511yescharacteristic
148C22×C5—42no—
149C22×C5—42no—
150C22×C5—42no—
151C22×C5—42no—
152C22×C5—42no—
153C22×C5—42no—
154C22×C5—42no—
155C22×C5C22×D411yescharacteristic
156C22×C5H(5,𝔽2)12yes—
157C22×C5H(5,𝔽2)11yescharacteristic
158C22×C5H(5,𝔽2)11yescharacteristic
159C22×C52−1+412yes—
160C2×D5—22no—
161C2×D5—22no—
162C2×D5—22no—
163C2×D5—42no—
164C2×D5—42no—
165C4×C5—22no—
166C4×C5—41no—
167C4×C5—41no—
168C4×C5—42no—
169Dic5—22no—
170Dic5—41no—
171Dic5—41no—
172Dic5—42no—
173Dic5—42no—
174C24—51no—
175C24C22×D512yes—
176C2×Q8—102no—
177C2×Q8—202no—
178C2×D4—22no—
179C2×D4—42no—
180C2×D4—104no—
181C2×D4—202no—
182C2×D4—202no—
183C22×C4—51no—
184C22×C4—54no—
185C22×C4—52no—
186C22×C4—51no—
187C22×C4—52no—
188C22×C4—51no—
189C22×C4—202no—
190C22×C4C22×D512yes—
191C22×C4C22×D511yescharacteristic
192C22×C4C22×D511yescharacteristic
193C22×C4C22×D511yescharacteristic
194C4⋊C4—22no—
195C4⋊C4—102no—
196C4⋊C4—102no—
197C4⋊C4—102no—
198C4⋊C4—202no—
199C4⋊C4—202no—
200C22⋊C4—42no—
201C22⋊C4—42no—
202C22⋊C4—102no—
203C22⋊C4—102no—
204C22⋊C4—104no—
205C22⋊C4—202no—
206C22⋊C4—202no—
207C22⋊C4—202no—
208C2×C5—82no—
209C2×C5#53411yescharacteristic
210C2×C5#53511yescharacteristic
211C2×C5#53611yescharacteristic
212C2×C5#56212yes—
213C2×C5#56212yes—
214D5—42no—
215C23—22no—
216C23—22no—
217C23—22no—
218C23—22no—
219C23—22no—
220C23—22no—
221C23—22no—
222C23—52no—
223C23—102no—
224C23—101no—
225C23—101no—
226C23—102no—
227C23C23×D511yescharacteristic
228Q8—204no—
229D4—44no—
230D4—208no—
231C2×C4—22no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—21no—
239C2×C4—22no—
240C2×C4—52no—
241C2×C4—102no—
242C2×C4—101no—
243C2×C4—102no—
244C2×C4—101no—
245C2×C4—102no—
246C2×C4—102no—
247C2×C4—101no—
248C2×C4—102no—
249C2×C4—101no—
250C2×C4—102no—
251C2×C4—102no—
252C2×C4—102no—
253C2×C4—204no—
254C2×C4D4×D512yes—
255C5#281611yescharacteristic, complemented, Sylow
256C22—42no—
257C22—42no—
258C22—42no—
259C22—42no—
260C22—42no—
261C22—42no—
262C22—42no—
263C22—102no—
264C22—102no—
265C22—102no—
266C22—202no—
267C22—202no—
268C22C2×D4×D511yescharacteristic
269C22#414412yes—
270C22#414411yescharacteristic
271C22#414911yescharacteristic
272C22#415012yes—
273C4—22no—
274C4—41no—
275C4—41no—
276C4—42no—
277C4—102no—
278C4—201no—
279C4—202no—
280C4—201no—
281C4—202no—
282C2—82no—
283C2—202no—
284C2#6592311yescharacteristic
285C2#6594111yescharacteristic
286C2#6595512yes—
287C2#6600112yes—
288C2#6600911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#394C2
prime radicalC2×#4084C2
hypercenter#394D5
hypocenterC5#2816
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
2. C2×#4042C2semidirect
3. C5×#394C2semidirect
4. #65511C2semidirect
5. #65272C2semidirect
6. #65250C2semidirect
7. #65248C2semidirect
8. #65238C2semidirect
9. #65236C2semidirect
10. #64953C2semidirect
11. #64950C2semidirect
80. #394D5semidirect
255. C5#2816semidirect

Maximal subgroups

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

subgroupindex
C2×#40842
C2×#40422
C5×#3942
#655112
#652722
#652502
#652482
#652382
#652362
#649532
#649502
#28165