Finite Group Databasebeta

← #10606512

Subgroups — #10606512

Order 640.

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

normal subgroups only

This group has 632 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
2C2×#4081C211yesmaximal, characteristic, complemented
3C22×#683C211yesmaximal, characteristic, complemented
4C2×#4032C211yesmaximal, characteristic, complemented
5C5×#386C211yesmaximal, characteristic, complemented
6#65502C211yesmaximal, characteristic, complemented
7#65248C212yesmaximal, complemented
8#65237C212yesmaximal, complemented
9#65231C211yesmaximal, characteristic, complemented
10#64965C212yesmaximal, complemented
11#64946C212yesmaximal, 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⋊Dic5C2212yes—
22C2×C22⋊Dic5C2211yescharacteristic
23#4085—24no—
24#4081—24no—
25C2×#685C2212yes—
26C2×#685C2211yescharacteristic
27C2×#685C2212yes—
28C2×C4⋊Dic5C2212yes—
29C2×#683—24no—
30C2×#683C2212yes—
31C2×#683C2212yes—
32C2×#683C2212yes—
33C2×#683C2211yescharacteristic
34C2×#683C2212yes—
35#4032—24no—
36C5×#95C2212yes—
37#3963C2212yes—
38#3963C2212yes—
39#2618—51nomaximal, Sylow
40C24×C5C2311yescharacteristic
41C24×C5C2311yescharacteristic
42C23×D5C2311yescharacteristic
43C2×C5×D4—22no—
44C2×C5×D4—22no—
45C2×C5×D4—22no—
46C22×C4×C5—22no—
47C22×C4×C5—42no—
48C22×C4×C5C2312yes—
49C22×C4×C5C2312yes—
50C22×C4×C5C2311yescharacteristic
51C2×C22⋊D5—22no—
52C2×C22⋊D5—22no—
53C2×C22⋊D5—22no—
54C2×C22⋊D5—22no—
55C2×C22⋊D5—42no—
56C22×Dic5—22no—
57C22×Dic5—22no—
58C22×Dic5—42no—
59C22×Dic5D414yes—
60C22×Dic5D414yescomplemented
61C22×Dic5C2312yes—
62C22×Dic5C2311yescharacteristic
63C22×Dic5C2311yescharacteristic
64C22×Dic5C2312yes—
65C22×Dic5C2311yescharacteristic
66C5×C22⋊C4—22no—
67C5×C22⋊C4—42no—
68C22⋊Dic5—22no—
69C22⋊Dic5—24no—
70C22⋊Dic5—22no—
71#685—24no—
72#685—22no—
73#685—42no—
74C4⋊Dic5—24no—
75#683—24no—
76#683—24no—
77#683—24no—
78#683—22no—
79#683—42no—
80C2×#123—51no—
81C2×#123—51no—
82C2×#121—51no—
83C22×C4⋊C4—51no—
84#398—52no—
85#396—52no—
86#396—52no—
87#394—52no—
88#386—51no—
89#386D511yescharacteristic, complemented
90#380—51no—
91C23×C5—22no—
92C23×C5—22no—
93C23×C5—21no—
94C23×C5—21no—
95C23×C5—21no—
96C23×C5—22no—
97C23×C5—22no—
98C23×C5C2×D412yes—
99C23×C5C2×D412yes—
100C23×C5#4112yes—
101C23×C5C2411yescharacteristic
102C22×D5—21no—
103C22×D5—21no—
104C22×D5—21no—
105C22×D5—22no—
106C22×D5—22no—
107C5×D4—44no—
108C2×C4×C5—22no—
109C2×C4×C5—21no—
110C2×C4×C5—21no—
111C2×C4×C5—22no—
112C2×C4×C5—21no—
113C2×C4×C5—22no—
114C2×C4×C5—22no—
115C2×C4×C5—22no—
116C2×C4×C5—22no—
117C2×C4×C5—44no—
118C22⋊D5—44no—
119C22⋊D5—44no—
120C2×Dic5—28no—
121C2×Dic5—22no—
122C2×Dic5—22no—
123C2×Dic5—22no—
124C2×Dic5—22no—
125C2×Dic5—22no—
126C2×Dic5—21no—
127C2×Dic5—21no—
128C2×Dic5—21no—
129C2×Dic5—22no—
130C2×Dic5—22no—
131C2×Dic5—44no—
132C2×Dic5C2×D412yes—
133C2×Dic5C2×D412yes—
134C2×Dic5C2×D414yes—
135C22×D4—51no—
136C22×D4—51no—
137C22×D4C2×D511yescharacteristic
138C23×C4—51no—
139C23×C4—51no—
140C23×C4C2×D511yescharacteristic
141#123—104no—
142#123—104no—
143#121—104no—
144C2×C4⋊C4—52no—
145C2×C4⋊C4—52no—
146C2×C4⋊C4—51no—
147C2×C4⋊C4—52no—
148C2×C4⋊C4—52no—
149C2×C4⋊C4—52no—
150C2×C4⋊C4—104no—
151C2×C22⋊C4—51no—
152C2×C22⋊C4—51no—
153C2×C22⋊C4—52no—
154C2×C22⋊C4—51no—
155C2×C22⋊C4—52no—
156C2×C22⋊C4—52no—
157C2×C22⋊C4C2×D511yescharacteristic
158C2×C22⋊C4C2×D512yes—
159#95—52no—
160#95—52no—
161#95C2×D512yes—
162C22×C5—22no—
163C22×C5—22no—
164C22×C5—24no—
165C22×C5—24no—
166C22×C5—41no—
167C22×C5—41no—
168C22×C5—41no—
169C22×C5—42no—
170C22×C5—42no—
171C22×C5#12114yes—
172C22×C5C22×D411yescharacteristic
173C22×C5C22×D411yescharacteristic
174C22×C5C2×#4111yescharacteristic
175C22×C5H(5,𝔽2)12yes—
176C22×C52−1+412yes—
177C2×D5—41no—
178C2×D5—41no—
179C2×D5—41no—
180C2×D5—42no—
181C2×D5—42no—
182C4×C5—41no—
183C4×C5—42no—
184C4×C5—42no—
185Dic5—24no—
186Dic5—22no—
187Dic5—22no—
188Dic5—41no—
189Dic5—42no—
190C24—51no—
191C24C22×D511yescharacteristic
192C24C22×D511yescharacteristic
193C2×D4—22no—
194C2×D4—22no—
195C2×D4—22no—
196C2×D4—102no—
197C2×D4—102no—
198C2×D4—102no—
199C2×D4—102no—
200C2×D4—202no—
201C22×C4—22no—
202C22×C4—42no—
203C22×C4—54no—
204C22×C4—51no—
205C22×C4—52no—
206C22×C4—51no—
207C22×C4—54no—
208C22×C4—52no—
209C22×C4—51no—
210C22×C4—102no—
211C22×C4—102no—
212C22×C4—202no—
213C22×C4C22×D512yes—
214C22×C4C22×D512yes—
215C22×C4C22×D511yescharacteristic
216C4⋊C4—104no—
217C4⋊C4—104no—
218C4⋊C4—102no—
219C4⋊C4—104no—
220C4⋊C4—104no—
221C4⋊C4—202no—
222C22⋊C4—22no—
223C22⋊C4—42no—
224C22⋊C4—102no—
225C22⋊C4—104no—
226C22⋊C4—102no—
227C22⋊C4—104no—
228C22⋊C4—102no—
229C22⋊C4—202no—
230C2×C5—24no—
231C2×C5—81no—
232C2×C5C2×#12111yescharacteristic
233C2×C5#53411yescharacteristic
234C2×C5#53611yescharacteristic
235C2×C5#53712yes—
236C2×C5#53912yes—
237D5—81no—
238C23—22no—
239C23—22no—
240C23—21no—
241C23—21no—
242C23—21no—
243C23—22no—
244C23—22no—
245C23—101no—
246C23—101no—
247C23—101no—
248C23—102no—
249C23—102no—
250C23#70912yes—
251C23D4×D512yes—
252C23D4×D512yes—
253C23C23×D511yescharacteristic
254D4—44no—
255D4—204no—
256D4—204no—
257C2×C4—21no—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—22no—
261C2×C4—21no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—44no—
267C2×C4—54no—
268C2×C4—52no—
269C2×C4—52no—
270C2×C4—102no—
271C2×C4—102no—
272C2×C4—102no—
273C2×C4—102no—
274C2×C4—108no—
275C2×C4—101no—
276C2×C4—102no—
277C2×C4—101no—
278C2×C4—101no—
279C2×C4—102no—
280C2×C4—102no—
281C2×C4—204no—
282C5#261811yescharacteristic, complemented, Sylow
283C22—24no—
284C22—22no—
285C22—24no—
286C22—22no—
287C22—41no—
288C22—42no—
289C22—41no—
290C22—42no—
291C22—41no—
292C22—201no—
293C22—202no—
294C22—201no—
295C22—202no—
296C22—201no—
297C22#403014yes—
298C22C2×#70911yescharacteristic
299C22C2×D4×D511yescharacteristic
300C22C2×D4×D511yescharacteristic
301C22#414412yes—
302C22#415012yes—
303C4—41no—
304C4—42no—
305C4—42no—
306C4—104no—
307C4—102no—
308C4—102no—
309C4—201no—
310C4—202no—
311C2—24no—
312C2—81no—
313C2—401no—
314C2C2×#403011yescharacteristic
315C2#6589812yes—
316C2#6592311yescharacteristic
317C2#6593112yes—
318C2#6598211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#386C2
prime radicalC2×#4085C2
hypercenter#386D5
hypocenterC5#2618
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
2. C2×#4081C2semidirect
3. C22×#683C2semidirect
4. C2×#4032C2semidirect
5. C5×#386C2semidirect
6. #65502C2semidirect
7. #65248C2semidirect
8. #65237C2semidirect
9. #65231C2semidirect
10. #64965C2semidirect
11. #64946C2semidirect
60. C22×Dic5D4semidirect
89. #386D5semidirect
282. C5#2618semidirect

Maximal subgroups

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

subgroupindex
C2×#40852
C2×#40812
C22×#6832
C2×#40322
C5×#3862
#655022
#652482
#652372
#652312
#649652
#649462
#26185