Finite Group Databasebeta

← #10606473

Subgroups — #10606473

Order 640.

596 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 596 conjugacy classes of subgroups — too many to draw. Only its 149 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×#4085C211yesmaximal, characteristic
2C22×#683C211yesmaximal, characteristic, complemented
3C5×#386C211yesmaximal, characteristic, complemented
4#65509C211yesmaximal, characteristic, complemented
5#65502C211yesmaximal, characteristic, complemented
6#65247C211yesmaximal, characteristic, complemented
7#65246C212yesmaximal, complemented
8#65241C212yesmaximal, complemented
9C22⋊C4×Dic5C211yesmaximal, characteristic, complemented
10#64964C212yesmaximal, complemented
11#64939C212yesmaximal, complemented
12C22×C5×D4C2211yescharacteristic
13C23×C4×C5C2211yescharacteristic
14C22×C22⋊D5C2211yescharacteristic
15C22×C22⋊D5C2211yescharacteristic
16C23×Dic5C2211yescharacteristic
17C23×Dic5C2211yescharacteristic
18C2×C5×C22⋊C4C2211yescharacteristic
19C2×C5×C22⋊C4C2212yes—
20C2×C22⋊Dic5C2211yescharacteristic
21C2×C22⋊Dic5C2211yescharacteristic
22C2×C22⋊Dic5C2212yes—
23#4085—24no—
24C2×#685C2211yescharacteristic
25C2×#685C2212yes—
26C2×#685C2212yes—
27C2×#683—24no—
28C2×#683C2212yes—
29C2×#683C2212yes—
30C2×#683C2211yescharacteristic
31C2×C4×Dic5C2212yes—
32C5×#95C2212yes—
33#3963C2212yes—
34#3963C2212yes—
35#3963C2212yes—
36#3963C2212yes—
37#2504—51nomaximal, Sylow
38C24×C5C2311yescharacteristic
39C24×C5C2311yescharacteristic
40C23×D5C2311yescharacteristic
41C2×C5×D4—22no—
42C2×C5×D4—22no—
43C2×C5×D4—22no—
44C22×C4×C5—22no—
45C22×C4×C5—42no—
46C22×C4×C5C2312yes—
47C22×C4×C5C2312yes—
48C22×C4×C5C2311yescharacteristic
49C2×C22⋊D5—22no—
50C2×C22⋊D5—22no—
51C2×C22⋊D5—22no—
52C2×C22⋊D5—22no—
53C2×C22⋊D5—24no—
54C22×Dic5—24no—
55C22×Dic5—22no—
56C22×Dic5—42no—
57C22×Dic5D414yes—
58C22×Dic5C2312yes—
59C22×Dic5C2312yes—
60C22×Dic5C2311yescharacteristic
61C22×Dic5C2311yescharacteristic
62C22×Dic5C2311yescharacteristic
63C5×C22⋊C4—22no—
64C5×C22⋊C4—42no—
65C22⋊Dic5—22no—
66C22⋊Dic5—22no—
67C22⋊Dic5—42no—
68#685—22no—
69#685—42no—
70#685—42no—
71#683—24no—
72#683—24no—
73#683—22no—
74C4×Dic5—24no—
75C2×#121—51no—
76C22×C4⋊C4—51no—
77#397—52no—
78#397—52no—
79#390—52no—
80#386—51no—
81#386—51no—
82#386—51no—
83#386D511yescharacteristic, complemented
84#382—52no—
85C4×C22⋊C4—51no—
86C23×C5—22no—
87C23×C5—22no—
88C23×C5—21no—
89C23×C5—22no—
90C23×C5—21no—
91C23×C5—21no—
92C23×C5—22no—
93C23×C5C2×D412yes—
94C23×C5#4112yes—
95C23×C5#4112yes—
96C23×C5C2411yescharacteristic
97C22×D5—21no—
98C22×D5—21no—
99C22×D5—21no—
100C22×D5—22no—
101C22×D5—22no—
102C5×D4—44no—
103C2×C4×C5—22no—
104C2×C4×C5—21no—
105C2×C4×C5—22no—
106C2×C4×C5—22no—
107C2×C4×C5—21no—
108C2×C4×C5—21no—
109C2×C4×C5—22no—
110C2×C4×C5—22no—
111C2×C4×C5—22no—
112C2×C4×C5—44no—
113C22⋊D5—44no—
114C22⋊D5—44no—
115C2×Dic5—22no—
116C2×Dic5—28no—
117C2×Dic5—22no—
118C2×Dic5—21no—
119C2×Dic5—22no—
120C2×Dic5—22no—
121C2×Dic5—21no—
122C2×Dic5—21no—
123C2×Dic5—22no—
124C2×Dic5—22no—
125C2×Dic5—22no—
126C2×Dic5—44no—
127C2×Dic5C2×D412yes—
128C2×Dic5C2×D412yes—
129C2×Dic5#4114yes—
130C22×D4—51no—
131C22×D4—51no—
132C22×D4C2×D511yescharacteristic
133C23×C4—51no—
134C23×C4—51no—
135C23×C4C2×D511yescharacteristic
136#121—104no—
137C2×C4⋊C4—52no—
138C2×C4⋊C4—52no—
139C2×C4⋊C4—51no—
140C2×C4⋊C4—104no—
141C2×C22⋊C4—51no—
142C2×C22⋊C4—51no—
143C2×C22⋊C4—51no—
144C2×C22⋊C4—52no—
145C2×C22⋊C4—52no—
146C2×C22⋊C4—52no—
147C2×C22⋊C4C2×D511yescharacteristic
148C2×C22⋊C4C2×D512yes—
149C2×C42—52no—
150#95—52no—
151#95—52no—
152#95—52no—
153#95—52no—
154#95C2×D512yes—
155C22×C5—22no—
156C22×C5—22no—
157C22×C5—24no—
158C22×C5—24no—
159C22×C5—41no—
160C22×C5—41no—
161C22×C5—41no—
162C22×C5—42no—
163C22×C5—42no—
164C22×C5#12414yes—
165C22×C5C22×D411yescharacteristic
166C22×C5C2×#4111yescharacteristic
167C22×C5C2×#4111yescharacteristic
168C22×C5C2×#4112yes—
169C22×C5H(5,𝔽2)12yes—
170C2×D5—41no—
171C2×D5—41no—
172C2×D5—41no—
173C2×D5—42no—
174C2×D5—42no—
175C4×C5—41no—
176C4×C5—42no—
177C4×C5—42no—
178Dic5—22no—
179Dic5—41no—
180Dic5—42no—
181Dic5—42no—
182Dic5#12314yes—
183C24—51no—
184C24C22×D511yescharacteristic
185C24C22×D511yescharacteristic
186C2×D4—22no—
187C2×D4—22no—
188C2×D4—22no—
189C2×D4—102no—
190C2×D4—104no—
191C2×D4—102no—
192C2×D4—102no—
193C2×D4—102no—
194C22×C4—22no—
195C22×C4—42no—
196C22×C4—51no—
197C22×C4—54no—
198C22×C4—51no—
199C22×C4—51no—
200C22×C4—52no—
201C22×C4—52no—
202C22×C4—104no—
203C22×C4—102no—
204C22×C4—202no—
205C22×C4C22×D511yescharacteristic
206C22×C4C22×D512yes—
207C22×C4C22×D512yes—
208C4⋊C4—104no—
209C4⋊C4—104no—
210C4⋊C4—102no—
211C22⋊C4—22no—
212C22⋊C4—42no—
213C22⋊C4—102no—
214C22⋊C4—102no—
215C22⋊C4—102no—
216C22⋊C4—202no—
217C22⋊C4—202no—
218C22⋊C4—202no—
219C42—104no—
220C2×C5—24no—
221C2×C5—81no—
222C2×C5C2×#12311yescharacteristic
223C2×C5C2×#12411yescharacteristic
224C2×C5#53511yescharacteristic
225C2×C5#55312yes—
226C2×C5#55312yes—
227D5—81no—
228C23—22no—
229C23—22no—
230C23—21no—
231C23—21no—
232C23—21no—
233C23—22no—
234C23—22no—
235C23—101no—
236C23—101no—
237C23—101no—
238C23—102no—
239C23—102no—
240C23#70912yes—
241C23D4×D512yes—
242C23#71112yes—
243C23C23×D511yescharacteristic
244D4—44no—
245D4—204no—
246D4—204no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—21no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—44no—
257C2×C4—52no—
258C2×C4—54no—
259C2×C4—52no—
260C2×C4—108no—
261C2×C4—102no—
262C2×C4—102no—
263C2×C4—101no—
264C2×C4—101no—
265C2×C4—102no—
266C2×C4—101no—
267C2×C4—102no—
268C2×C4—102no—
269C2×C4—102no—
270C2×C4—102no—
271C2×C4—204no—
272C5#250411yescharacteristic, complemented, Sylow
273C22—24no—
274C22—22no—
275C22—24no—
276C22—22no—
277C22—41no—
278C22—42no—
279C22—41no—
280C22—42no—
281C22—41no—
282C22—201no—
283C22—202no—
284C22—201no—
285C22—202no—
286C22—201no—
287C22#403114yes—
288C22C2×#70911yescharacteristic
289C22C2×D4×D511yescharacteristic
290C22C2×#71111yescharacteristic
291C22#414412yes—
292C22D5×#4112yes—
293C4—41no—
294C4—42no—
295C4—42no—
296C4—54no—
297C4—102no—
298C4—201no—
299C4—202no—
300C4—202no—
301C2—24no—
302C2—81no—
303C2—401no—
304C2C2×#403111yescharacteristic
305C2#6587412yes—
306C2#6592611yescharacteristic
307C2#6593512yes—
308C2D5×#12311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#386C2
prime radicalC2×#4085C2
hypercenter#386D5
hypocenterC5#2504
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
2. C22×#683C2semidirect
3. C5×#386C2semidirect
4. #65509C2semidirect
5. #65502C2semidirect
6. #65247C2semidirect
7. #65246C2semidirect
8. #65241C2semidirect
9. C22⋊C4×Dic5C2semidirect
10. #64964C2semidirect
11. #64939C2semidirect
83. #386D5semidirect
272. C5#2504semidirect

Maximal subgroups

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

subgroupindex
C2×#40852
C22×#6832
C5×#3862
#655092
#655022
#652472
#652462
#652412
C22⋊C4×Dic52
#649642
#649392
#25045