Finite Group Databasebeta

← #10609468

Subgroups — #10609468

Order 640.

696 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 696 conjugacy classes of subgroups — too many to draw. Only its 155 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×#4039C211yesmaximal, characteristic, complemented
2C2×#4039C212yesmaximal, complemented
3C2×#4038C211yesmaximal, characteristic, complemented
4C5×#398C211yesmaximal, characteristic, complemented
5#65283C212yesmaximal, complemented
6#65282C211yesmaximal, characteristic, complemented
7#65280C211yesmaximal, characteristic, complemented
8#64961C212yesmaximal, complemented
9#64960C211yesmaximal, characteristic
10#64955C212yesmaximal, complemented
11#64938C211yesmaximal, characteristic, complemented
12C22×D20C2211yescharacteristic
13C22×D20C2212yes—
14C22×C4×D5C2212yes—
15C22×C4×D5C2211yescharacteristic
16C2×C5×C4⋊C4C2211yescharacteristic
17C2×C5×C4⋊C4C2211yescharacteristic
18C2×C5×C4⋊C4C2212yes—
19C2×#685C2212yes—
20C2×#685C2212yes—
21C2×#685C2211yescharacteristic
22C2×#685C2212yes—
23C2×#685C2212yes—
24C2×#685C2212yes—
25C2×#685C2211yescharacteristic
26C2×C4⋊Dic5C2211yescharacteristic
27C2×#683C2212yes—
28C2×C4×Dic5C2211yescharacteristic
29C2×C4×Dic5C2212yes—
30C2×C4×Dic5C2211yescharacteristic
31#4039—28no—
32#4039—24no—
33#4038—24no—
34C5×#95C2211yescharacteristic
35C5×#95C2212yes—
36#3963C2211yescharacteristic
37#3963C2212yes—
38#2392—51nomaximal, Sylow
39C23×D5C2311yescharacteristic
40C23×D5C2312yes—
41C22×C4×C5C2312yes—
42C22×C4×C5C2311yescharacteristic
43C22×C4×C5C2311yescharacteristic
44C22×C4×C5C2311yescharacteristic
45C22×C4×C5C2312yes—
46C22×Dic5C2312yes—
47C22×Dic5C2311yescharacteristic
48C22×Dic5C2311yescharacteristic
49C22×Dic5C2311yescharacteristic
50C2×D20—24no—
51C2×D20—24no—
52C2×D20—24no—
53C2×D20—22no—
54C2×D20—24no—
55C2×C4×D5—22no—
56C2×C4×D5—22no—
57C2×C4×D5—24no—
58C2×C4×D5—24no—
59C2×C4×D5—24no—
60C2×C4×D5D414yescomplemented
61C5×C4⋊C4—22no—
62C5×C4⋊C4—24no—
63C5×C4⋊C4—22no—
64#685—24no—
65#685—24no—
66#685—22no—
67#685—24no—
68#685—24no—
69#685—22no—
70#685—24no—
71C4⋊Dic5—22no—
72#683—24no—
73C4×Dic5—22no—
74C4×Dic5—24no—
75C4×Dic5D414yescomplemented
76C2×C4×D4—51no—
77C2×C4×D4—52no—
78C2×#117—51no—
79#398D511yescharacteristic, complemented
80#397—51no—
81#396—52no—
82#394—51no—
83#394—52no—
84#392—51no—
85C4×C4⋊C4—51no—
86C4×C22⋊C4—52no—
87C23×C5C2411yescharacteristic
88C22×D5—21no—
89C22×D5—22no—
90C22×D5—22no—
91C22×D5—22no—
92C22×D5—21no—
93C22×D5—22no—
94C22×D5—22no—
95C22×D5—22no—
96C22×D5—22no—
97C22×D5—22no—
98C22×D5C2×D412yes—
99C22×D5#4114yes—
100C2×C4×C5—22no—
101C2×C4×C5—21no—
102C2×C4×C5—21no—
103C2×C4×C5—22no—
104C2×C4×C5—22no—
105C2×C4×C5—21no—
106C2×C4×C5—22no—
107C2×C4×C5—22no—
108C2×C4×C5—22no—
109C2×C4×C5—22no—
110C2×C4×C5C2×D412yes—
111C2×C4×C5C2×D412yes—
112C2×C4×C5#4114yes—
113C2×Dic5—22no—
114C2×Dic5—21no—
115C2×Dic5—22no—
116C2×Dic5—22no—
117C2×Dic5—22no—
118C2×Dic5—22no—
119C2×Dic5C2×D412yes—
120C2×Dic5C2×D412yes—
121C2×Dic5C2×D412yes—
122C2×Dic5#4112yes—
123D20—28no—
124D20—216no—
125C4×D5—28no—
126C4×D5—48no—
127C22×D4—51no—
128C22×D4—52no—
129C23×C4—51no—
130C23×C4—52no—
131C4×D4—104no—
132C4×D4—108no—
133#117—104no—
134C2×C4⋊C4—51no—
135C2×C4⋊C4—52no—
136C2×C4⋊C4C2×D511yescharacteristic
137C2×C4⋊C4C2×D512yes—
138C2×C4⋊C4C2×D511yescharacteristic
139C2×C22⋊C4—51no—
140C2×C22⋊C4—51no—
141C2×C22⋊C4—52no—
142C2×C22⋊C4—52no—
143C2×C22⋊C4—52no—
144C2×C22⋊C4—52no—
145C2×C22⋊C4—52no—
146C2×C42—51no—
147C2×C42—51no—
148C2×C42—52no—
149#95—52no—
150#95—51no—
151#95C2×D512yes—
152#95C2×D511yescharacteristic
153C22×C5C22×D411yescharacteristic
154C22×C5C22×D411yescharacteristic
155C22×C5C2×#4112yes—
156C22×C5C2×#4111yescharacteristic
157C22×C5C2×#4112yes—
158C2×D5—22no—
159C2×D5—24no—
160C2×D5—24no—
161C2×D5—24no—
162C2×D5—24no—
163C2×D5—42no—
164C2×D5—42no—
165C2×D5—41no—
166C2×D5—42no—
167C2×D5—42no—
168C2×D5—41no—
169C2×D5—42no—
170C4×C5—22no—
171C4×C5—22no—
172C4×C5—24no—
173C4×C5—42no—
174C4×C5—41no—
175Dic5—22no—
176Dic5—42no—
177Dic5—41no—
178Dic5#12114yes—
179C24—51no—
180C24—52no—
181C2×D4—104no—
182C2×D4—104no—
183C2×D4—104no—
184C2×D4—102no—
185C2×D4—104no—
186C22×C4—51no—
187C22×C4—51no—
188C22×C4—54no—
189C22×C4—52no—
190C22×C4—51no—
191C22×C4—102no—
192C22×C4—102no—
193C22×C4—104no—
194C22×C4—104no—
195C22×C4—104no—
196C22×C4C22×D512yes—
197C22×C4C22×D511yescharacteristic
198C22×C4C22×D511yescharacteristic
199C22×C4C22×D511yescharacteristic
200C22×C4C22×D512yes—
201C4⋊C4—22no—
202C4⋊C4—24no—
203C4⋊C4—22no—
204C4⋊C4—102no—
205C4⋊C4—104no—
206C22⋊C4—102no—
207C22⋊C4—104no—
208C22⋊C4—104no—
209C22⋊C4—104no—
210C22⋊C4—102no—
211C22⋊C4—104no—
212C22⋊C4—104no—
213C42—54no—
214C42—102no—
215C42—104no—
216C2×C5C2×#12111yescharacteristic
217C2×C5#52512yes—
218C2×C5#52511yescharacteristic
219C2×C5#52912yes—
220C2×C5#53211yescharacteristic
221D5—42no—
222D5—44no—
223C23—52no—
224C23—54no—
225C23—101no—
226C23—102no—
227C23—102no—
228C23—102no—
229C23—102no—
230C23—102no—
231C23—101no—
232C23—102no—
233C23—102no—
234C23—102no—
235C23C23×D511yescharacteristic
236D4—1016no—
237D4—108no—
238C2×C4—22no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—21no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—52no—
249C2×C4—52no—
250C2×C4—52no—
251C2×C4—52no—
252C2×C4—108no—
253C2×C4—102no—
254C2×C4—102no—
255C2×C4—101no—
256C2×C4—102no—
257C2×C4—102no—
258C2×C4—102no—
259C2×C4—208no—
260C2×C4D4×D512yes—
261C2×C4D4×D512yes—
262C2×C4#71314yes—
263C5#239211yescharacteristic, complemented, Sylow
264C22—102no—
265C22—104no—
266C22—104no—
267C22—104no—
268C22—104no—
269C22—201no—
270C22—202no—
271C22—202no—
272C22—202no—
273C22—201no—
274C22—202no—
275C22—202no—
276C22C2×D4×D511yescharacteristic
277C22C2×D4×D511yescharacteristic
278C22C2×#71312yes—
279C22D5×#4111yescharacteristic
280C22D5×#4112yes—
281C4—22no—
282C4—24no—
283C4—22no—
284C4—42no—
285C4—41no—
286C4—54no—
287C4—102no—
288C4—201no—
289C4—202no—
290C2—202no—
291C2—204no—
292C2D5×#12111yescharacteristic
293C2#6596112yes—
294C2#6599011yescharacteristic
295C2#6602512yes—
296C2#6605911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#398C2
prime radical#64960C2
hypercenter#398D5
hypocenterC5#2392
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
1. C2×#4039C2semidirect
2. C2×#4039C2semidirect
3. C2×#4038C2semidirect
4. C5×#398C2semidirect
5. #65283C2semidirect
6. #65282C2semidirect
7. #65280C2semidirect
8. #64961C2semidirect
10. #64955C2semidirect
11. #64938C2semidirect
60. C2×C4×D5D4semidirect
75. C4×Dic5D4semidirect
79. #398D5semidirect
263. C5#2392semidirect

Maximal subgroups

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

subgroupindex
C2×#40392
C2×#40382
C5×#3982
#652832
#652822
#652802
#649612
#649602
#649552
#649382
#23925