Finite Group Databasebeta

← #10608431

Subgroups — #10608431

Order 640.

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

normal subgroups only

This group has 628 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×D4×Dic5C211yesmaximal, characteristic, complemented
2C2×#4031C211yesmaximal, characteristic, complemented
3C5×#392C211yesmaximal, characteristic, complemented
4#65512C212yesmaximal, complemented
5#65512C211yesmaximal, characteristic, complemented
6#65509C212yesmaximal, complemented
7#65247C212yesmaximal, complemented
8#65242C211yesmaximal, characteristic, complemented
9#65236C212yesmaximal, complemented
10#64963C211yesmaximal, characteristic, complemented
11#64940C211yesmaximal, characteristic, complemented
12C22×C5×D4C2211yescharacteristic
13C22×C5×D4C2212yes—
14C22×C22⋊D5C2211yescharacteristic
15C22×C22⋊D5C2212yes—
16C23×Dic5C2212yes—
17C22×Dic10C2211yescharacteristic
18C2×C5×C22⋊C4C2211yescharacteristic
19C2×C5×C22⋊C4C2212yes—
20C2×C22⋊Dic5C2211yescharacteristic
21C2×C22⋊Dic5C2212yes—
22C2×C22⋊Dic5C2212yes—
23C2×C22⋊Dic5C2212yes—
24D4×Dic5—24no—
25C2×#685C2211yescharacteristic
26C2×#685C2211yescharacteristic
27C2×#685C2212yes—
28C2×C4⋊Dic5C2211yescharacteristic
29C2×C4×Dic5C2211yescharacteristic
30#4031—24no—
31C5×#95C2211yescharacteristic
32#3963C2212yes—
33#3963C2211yescharacteristic
34#3963C2212yes—
35#3963C2212yes—
36#3963C2212yes—
37#2631—51nomaximal, Sylow
38C24×C5C2311yescharacteristic
39C24×C5C2312yes—
40C23×D5C2311yescharacteristic
41C2×C5×D4—24no—
42C2×C5×D4—24no—
43C2×C5×D4—22no—
44C2×C5×D4—42no—
45C2×C5×D4—42no—
46C22×C4×C5C2311yescharacteristic
47C22×C4×C5C2312yes—
48C22×C4×C5C2311yescharacteristic
49C2×C22⋊D5—24no—
50C2×C22⋊D5—22no—
51C2×C22⋊D5—42no—
52C2×C22⋊D5—42no—
53C2×C22⋊D5—42no—
54C22×Dic5—24no—
55C22×Dic5—42no—
56C22×Dic5—42no—
57C22×Dic5C2311yescharacteristic
58C22×Dic5C2312yes—
59C22×Dic5C2312yes—
60C22×Dic5C2311yescharacteristic
61C22×Dic5C2311yescharacteristic
62C2×Dic10—22no—
63C2×Dic10—42no—
64C5×C22⋊C4—22no—
65C5×C22⋊C4—42no—
66C22⋊Dic5—22no—
67C22⋊Dic5—24no—
68C22⋊Dic5—42no—
69C22⋊Dic5—42no—
70#685—22no—
71#685—22no—
72#685—42no—
73C4⋊Dic5—22no—
74C4×Dic5D414yescomplemented
75C2×#124—51no—
76C2×C4×D4—51no—
77#397—51no—
78#397—51no—
79#393—52no—
80#392—52no—
81#392—51no—
82#392D511yescharacteristic, complemented
83#386—52no—
84#386—52no—
85#383—51no—
86C23×C5—22no—
87C23×C5—22no—
88C23×C5—22no—
89C23×C5—22no—
90C23×C5—22no—
91C23×C5—22no—
92C23×C5—21no—
93C23×C5—22no—
94C23×C5—21no—
95C23×C5—22no—
96C23×C5—21no—
97C23×C5#4114yes—
98C23×C5C2411yescharacteristic
99C22×D5—21no—
100C22×D5—22no—
101C22×D5—21no—
102C22×D5—22no—
103C22×D5—21no—
104C5×D4—28no—
105C5×D4—48no—
106C2×C4×C5—22no—
107C2×C4×C5—21no—
108C2×C4×C5—22no—
109C2×C4×C5—22no—
110C2×C4×C5—22no—
111C2×C4×C5—22no—
112C2×C4×C5C2×D412yes—
113C22⋊D5—44no—
114C22⋊D5—48no—
115C2×Dic5—22no—
116C2×Dic5—21no—
117C2×Dic5—21no—
118C2×Dic5—22no—
119C2×Dic5—21no—
120C2×Dic5—22no—
121C2×Dic5—22no—
122C2×Dic5—22no—
123C2×Dic5—22no—
124C2×Dic5—22no—
125C2×Dic5—22no—
126C2×Dic5—48no—
127C2×Dic5C2×D412yes—
128C2×Dic5C2×D412yes—
129Dic10—44no—
130C22×Q8—51no—
131C22×D4—51no—
132C22×D4—52no—
133C22×D4C2×D511yescharacteristic
134C22×D4C2×D512yes—
135C23×C4—52no—
136#124—104no—
137C4×D4—104no—
138C2×C4⋊C4—51no—
139C2×C22⋊C4—52no—
140C2×C22⋊C4—51no—
141C2×C22⋊C4—51no—
142C2×C22⋊C4—51no—
143C2×C22⋊C4—52no—
144C2×C22⋊C4—52no—
145C2×C22⋊C4—52no—
146C2×C22⋊C4C2×D511yescharacteristic
147C2×C22⋊C4C2×D512yes—
148C2×C42—51no—
149#95—52no—
150#95—52no—
151#95—51no—
152#95—52no—
153#95—52no—
154#95C2×D511yescharacteristic
155C22×C5—24no—
156C22×C5—24no—
157C22×C5—24no—
158C22×C5—42no—
159C22×C5—42no—
160C22×C5—42no—
161C22×C5—42no—
162C22×C5—41no—
163C22×C5—41no—
164C22×C5—41no—
165C22×C5—42no—
166C22×C5—42no—
167C22×C5C22×D411yescharacteristic
168C22×C5C2×#4112yes—
169C22×C5H(5,𝔽2)12yes—
170C22×C5H(5,𝔽2)11yescharacteristic
171C22×C5H(5,𝔽2)11yescharacteristic
172C2×D5—41no—
173C2×D5—41no—
174C2×D5—41no—
175C2×D5—42no—
176C2×D5—42no—
177C4×C5—22no—
178C4×C5—41no—
179C4×C5—42no—
180Dic5—22no—
181Dic5—22no—
182Dic5—41no—
183Dic5—42no—
184Dic5—42no—
185C24—51no—
186C24C22×D511yescharacteristic
187C24C22×D512yes—
188C2×Q8—102no—
189C2×Q8—202no—
190C2×D4—24no—
191C2×D4—24no—
192C2×D4—22no—
193C2×D4—42no—
194C2×D4—42no—
195C2×D4—102no—
196C2×D4—104no—
197C2×D4—202no—
198C2×D4—202no—
199C2×D4—202no—
200C22×C4—51no—
201C22×C4—51no—
202C22×C4—52no—
203C22×C4—51no—
204C22×C4—52no—
205C22×C4—104no—
206C22×C4—202no—
207C22×C4—202no—
208C22×C4C22×D512yes—
209C22×C4C22×D511yescharacteristic
210C22×C4C22×D511yescharacteristic
211C4⋊C4—102no—
212C22⋊C4—22no—
213C22⋊C4—42no—
214C22⋊C4—102no—
215C22⋊C4—104no—
216C22⋊C4—102no—
217C22⋊C4—102no—
218C22⋊C4—202no—
219C22⋊C4—202no—
220C22⋊C4—202no—
221C42—54no—
222C2×C5—44no—
223C2×C5—81no—
224C2×C5#52511yescharacteristic
225C2×C5#53511yescharacteristic
226C2×C5#53511yescharacteristic
227C2×C5#53812yes—
228C2×C5#53812yes—
229D5—81no—
230C23—22no—
231C23—22no—
232C23—22no—
233C23—22no—
234C23—22no—
235C23—22no—
236C23—21no—
237C23—22no—
238C23—21no—
239C23—22no—
240C23—21no—
241C23—101no—
242C23—102no—
243C23—101no—
244C23—102no—
245C23—101no—
246C23#71114yes—
247C23C23×D511yescharacteristic
248Q8—204no—
249D4—28no—
250D4—48no—
251D4—204no—
252D4—208no—
253C2×C4—22no—
254C2×C4—21no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—52no—
260C2×C4—52no—
261C2×C4—101no—
262C2×C4—102no—
263C2×C4—101no—
264C2×C4—101no—
265C2×C4—102no—
266C2×C4—102no—
267C2×C4—102no—
268C2×C4—102no—
269C2×C4—102no—
270C2×C4—102no—
271C2×C4—102no—
272C2×C4—208no—
273C2×C4D4×D512yes—
274C5#263111yescharacteristic, complemented, Sylow
275C22—24no—
276C22—24no—
277C22—24no—
278C22—42no—
279C22—42no—
280C22—42no—
281C22—42no—
282C22—41no—
283C22—41no—
284C22—42no—
285C22—41no—
286C22—42no—
287C22—201no—
288C22—201no—
289C22—202no—
290C22—201no—
291C22—202no—
292C22C2×D4×D511yescharacteristic
293C22C2×#71112yes—
294C22#414412yes—
295C22#414411yescharacteristic
296C22#414911yescharacteristic
297C4—22no—
298C4—41no—
299C4—42no—
300C4—102no—
301C4—102no—
302C4—201no—
303C4—202no—
304C4—202no—
305C2—44no—
306C2—81no—
307C2—401no—
308C2#6592011yescharacteristic
309C2#6592712yes—
310C2#6594111yescharacteristic
311C2#6595112yes—
312C2#6604911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#392C2
hypercenter#392D5
hypocenterC5#2631
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C2×D4×Dic5C2semidirect
2. C2×#4031C2semidirect
3. C5×#392C2semidirect
4. #65512C2semidirect
5. #65512C2semidirect
6. #65509C2semidirect
7. #65247C2semidirect
8. #65242C2semidirect
9. #65236C2semidirect
10. #64963C2semidirect
11. #64940C2semidirect
74. C4×Dic5D4semidirect
82. #392D5semidirect
274. C5#2631semidirect

Maximal subgroups

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

subgroupindex
C2×D4×Dic52
C2×#40312
C5×#3922
#655122
#655092
#652472
#652422
#652362
#649632
#649402
#26315