Finite Group Databasebeta

← #10608622

Subgroups — #10608622

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×D4×Dic5C211yesmaximal, characteristic, complemented
2C2×#4038C211yesmaximal, characteristic, complemented
3C2×#4027C212yesmaximal, complemented
4C5×#394C211yesmaximal, characteristic, complemented
5#65512C211yesmaximal, characteristic
6#65272C211yesmaximal, characteristic, complemented
7#65248C212yesmaximal, complemented
8#65237C212yesmaximal, complemented
9C22⋊C4×Dic5C212yesmaximal, complemented
10#64963C211yesmaximal, characteristic, complemented
11#64938C211yesmaximal, characteristic, complemented
12C22×C5×D4C2211yescharacteristic
13C22×C22⋊D5C2212yes—
14C23×Dic5C2212yes—
15C22×C4×D5C2211yescharacteristic
16C2×C5×C4⋊C4C2211yescharacteristic
17C2×C5×C22⋊C4C2212yes—
18C2×C5×C22⋊C4C2212yes—
19C2×C22⋊Dic5C2212yes—
20C2×C22⋊Dic5C2212yes—
21D4×Dic5—24no—
22C2×#685C2212yes—
23C2×#685C2211yescharacteristic
24C2×#685C2211yescharacteristic
25C2×C4⋊Dic5C2211yescharacteristic
26C2×C4⋊Dic5C2211yescharacteristic
27C2×#683C2212yes—
28C2×#683C2212yes—
29C2×C4×Dic5C2212yes—
30C2×C4×Dic5C2211yescharacteristic
31C2×C4×Dic5C2211yescharacteristic
32#4038—24no—
33#4027—28no—
34C5×#95C2211yescharacteristic
35#3963C2211yescharacteristic
36#3963C2212yes—
37#3963C2212yes—
38#2392—51nomaximal, Sylow
39C24×C5C2312yes—
40C23×D5C2311yescharacteristic
41C2×C5×D4—24no—
42C2×C5×D4—22no—
43C22×C4×C5C2311yescharacteristic
44C22×C4×C5C2312yes—
45C22×C4×C5C2311yescharacteristic
46C22×C4×C5C2311yescharacteristic
47C2×C22⋊D5—24no—
48C2×C22⋊D5—24no—
49C2×C22⋊D5—24no—
50C22×Dic5—24no—
51C22×Dic5—24no—
52C22×Dic5—24no—
53C22×Dic5C2311yescharacteristic
54C22×Dic5C2312yes—
55C22×Dic5C2311yescharacteristic
56C22×Dic5C2312yes—
57C22×Dic5C2311yescharacteristic
58C2×C4×D5—22no—
59C2×C4×D5—22no—
60C2×C4×D5D414yescomplemented
61C5×C4⋊C4—22no—
62C5×C22⋊C4—24no—
63C5×C22⋊C4—24no—
64C22⋊Dic5—24no—
65C22⋊Dic5—24no—
66#685—24no—
67#685—22no—
68#685—22no—
69C4⋊Dic5—22no—
70C4⋊Dic5—22no—
71#683—24no—
72#683—24no—
73C4×Dic5—24no—
74C4×Dic5—22no—
75C4×Dic5D414yescomplemented
76C2×C4×D4—51no—
77C2×C4×D4—52no—
78C2×#117—51no—
79#398—51no—
80#397—51no—
81#396—52no—
82#394—52no—
83#394D511yescharacteristic, complemented
84#392—51no—
85C4×C4⋊C4—51no—
86C4×C22⋊C4—52no—
87C23×C5—22no—
88C23×C5—22no—
89C23×C5—22no—
90C23×C5—22no—
91C23×C5—22no—
92C23×C5—22no—
93C23×C5#4114yes—
94C23×C5C2411yescharacteristic
95C22×D5—21no—
96C22×D5—21no—
97C22×D5—22no—
98C22×D5—22no—
99C22×D5C2×D412yes—
100C5×D4—28no—
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×C5C2×D412yes—
110C2×C4×C5C2×D412yes—
111C22⋊D5—216no—
112C2×Dic5—22no—
113C2×Dic5—22no—
114C2×Dic5—22no—
115C2×Dic5—21no—
116C2×Dic5—22no—
117C2×Dic5—22no—
118C2×Dic5—22no—
119C2×Dic5—22no—
120C2×Dic5—48no—
121C2×Dic5C2×D412yes—
122C2×Dic5C2×D412yes—
123C2×Dic5C2×D412yes—
124C2×Dic5#4114yes—
125C2×Dic5#4112yes—
126C4×D5—28no—
127C22×D4—52no—
128C22×D4C2×D511yescharacteristic
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⋊C4—51no—
137C2×C4⋊C4—52no—
138C2×C4⋊C4C2×D511yescharacteristic
139C2×C22⋊C4—52no—
140C2×C22⋊C4—52no—
141C2×C22⋊C4—51no—
142C2×C22⋊C4—51no—
143C2×C22⋊C4—52no—
144C2×C22⋊C4C2×D512yes—
145C2×C22⋊C4C2×D512yes—
146C2×C42—51no—
147C2×C42—51no—
148C2×C42—52no—
149#95—52no—
150#95—52no—
151#95—51no—
152#95C2×D511yescharacteristic
153C22×C5—24no—
154C22×C5—24no—
155C22×C5—24no—
156C22×C5—42no—
157C22×C5—42no—
158C22×C5—42no—
159C22×C5—42no—
160C22×C5C22×D411yescharacteristic
161C22×C5C22×D411yescharacteristic
162C22×C5C2×#4112yes—
163C22×C5C2×#4111yescharacteristic
164C22×C5C2×#4112yes—
165C2×D5—22no—
166C2×D5—24no—
167C2×D5—41no—
168C2×D5—42no—
169C2×D5—41no—
170C4×C5—22no—
171C4×C5—22no—
172C4×C5—41no—
173C4×C5—42no—
174Dic5—24no—
175Dic5—22no—
176Dic5—42no—
177Dic5—41no—
178Dic5#12114yes—
179C24—51no—
180C24C22×D512yes—
181C2×D4—24no—
182C2×D4—22no—
183C2×D4—104no—
184C2×D4—104no—
185C2×D4—104no—
186C22×C4—51no—
187C22×C4—52no—
188C22×C4—51no—
189C22×C4—54no—
190C22×C4—51no—
191C22×C4—52no—
192C22×C4—102no—
193C22×C4—102no—
194C22×C4—104no—
195C22×C4—104no—
196C22×C4—104no—
197C22×C4C22×D511yescharacteristic
198C22×C4C22×D512yes—
199C22×C4C22×D511yescharacteristic
200C22×C4C22×D511yescharacteristic
201C4⋊C4—22no—
202C4⋊C4—102no—
203C4⋊C4—104no—
204C4⋊C4—104no—
205C4⋊C4—102no—
206C22⋊C4—24no—
207C22⋊C4—24no—
208C22⋊C4—102no—
209C22⋊C4—104no—
210C22⋊C4—104no—
211C22⋊C4—104no—
212C22⋊C4—102no—
213C42—54no—
214C42—102no—
215C42—104no—
216C2×C5—44no—
217C2×C5C2×#12111yescharacteristic
218C2×C5#52511yescharacteristic
219C2×C5#52512yes—
220C2×C5#52912yes—
221C2×C5#53211yescharacteristic
222D5—42no—
223C23—22no—
224C23—22no—
225C23—22no—
226C23—22no—
227C23—22no—
228C23—22no—
229C23—52no—
230C23—101no—
231C23—102no—
232C23—101no—
233C23—102no—
234C23#71114yes—
235C23C23×D511yescharacteristic
236D4—28no—
237D4—1016no—
238C2×C4—21no—
239C2×C4—22no—
240C2×C4—21no—
241C2×C4—22no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—52no—
247C2×C4—52no—
248C2×C4—52no—
249C2×C4—52no—
250C2×C4—54no—
251C2×C4—108no—
252C2×C4—102no—
253C2×C4—102no—
254C2×C4—102no—
255C2×C4—101no—
256C2×C4—102no—
257C2×C4—102no—
258C2×C4—102no—
259C2×C4—102no—
260C2×C4—208no—
261C2×C4D4×D512yes—
262C2×C4D4×D512yes—
263C5#239211yescharacteristic, complemented, Sylow
264C22—24no—
265C22—24no—
266C22—24no—
267C22—42no—
268C22—42no—
269C22—42no—
270C22—42no—
271C22—102no—
272C22—104no—
273C22—201no—
274C22—202no—
275C22—201no—
276C22C2×D4×D511yescharacteristic
277C22C2×D4×D511yescharacteristic
278C22C2×#71112yes—
279C22D5×#4111yescharacteristic
280C22D5×#4112yes—
281C4—22no—
282C4—22no—
283C4—41no—
284C4—42no—
285C4—54no—
286C4—104no—
287C4—102no—
288C4—202no—
289C4—201no—
290C2—44no—
291C2—202no—
292C2#6592011yescharacteristic
293C2D5×#12111yescharacteristic
294C2#6594012yes—
295C2#6598312yes—
296C2#6600211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#394C2
prime radical#65512C2
hypercenter#394D5
hypocenterC5#2392
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. C2×D4×Dic5C2semidirect
2. C2×#4038C2semidirect
3. C2×#4027C2semidirect
4. C5×#394C2semidirect
6. #65272C2semidirect
7. #65248C2semidirect
8. #65237C2semidirect
9. C22⋊C4×Dic5C2semidirect
10. #64963C2semidirect
11. #64938C2semidirect
60. C2×C4×D5D4semidirect
75. C4×Dic5D4semidirect
83. #394D5semidirect
263. C5#2392semidirect

Maximal subgroups

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

subgroupindex
C2×D4×Dic52
C2×#40382
C2×#40272
C5×#3942
#655122
#652722
#652482
#652372
C22⋊C4×Dic52
#649632
#649382
#23925