Finite Group Databasebeta

← #10614233

Subgroups — #10614233

Order 640.

492 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 145 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×C4×C5×Q8C211yesmaximal, characteristic, complemented
2C2×#4018C211yesmaximal, characteristic
3#65520C211yesmaximal, characteristic
4#65518C211yesmaximal, characteristic, complemented
5#65282C211yesmaximal, characteristic
6#65278C212yesmaximal
7#65269C211yesmaximal, characteristic, complemented
8#65267C212yesmaximal, complemented
9#65229C212yesmaximal
10#65226C211yesmaximal, characteristic, complemented
11#65222C212yesmaximal, complemented
12C22×C5×Q8C2211yescharacteristic
13C22×C4×D5C2211yescharacteristic
14C4×C5×Q8—24no—
15C2×C5×C4⋊C4C2211yescharacteristic
16C2×C5×C4⋊C4C2212yes—
17C2×C42×C5C2212yes—
18C2×C42×C5C2211yescharacteristic
19C2×#685C2211yescharacteristic
20C2×#685C2212yes—
21C2×#685C2211yescharacteristic
22C2×#685C2212yes—
23C2×#683C2211yescharacteristic
24C2×#683C2211yescharacteristic
25C2×#683C2212yes—
26C2×#683C2212yes—
27C2×C4×Dic5C2211yescharacteristic
28#4018—24no—
29#3963C2212yes—
30#3963C2211yescharacteristic
31#3963C2212yes—
32#3963C2212yes—
33#3963C2212yes—
34#3963C2211yescharacteristic
35#3963C2212yes—
36#3963C2212yes—
37#2599—51nomaximal, Sylow
38C23×D5C2311yescharacteristic
39C2×C5×Q8—22no—
40C2×C5×Q8—24no—
41C22×C4×C5C2312yes—
42C22×C4×C5C2311yescharacteristic
43C22×C4×C5C2311yescharacteristic
44C22×C4×C5C2311yescharacteristic
45C22×C4×C5C2312yes—
46C22×Dic5C2312yes—
47C22×Dic5C2312yes—
48C22×Dic5C2311yescharacteristic
49C22×Dic5C2311yescharacteristic
50C22×Dic5C2311yescharacteristic
51C2×C4×D5—22no—
52C2×C4×D5—22no—
53C2×C4×D5—22no—
54C5×C4⋊C4—24no—
55C5×C4⋊C4—22no—
56C42×C5—24no—
57C42×C5D414yes—
58#685—24no—
59#685—22no—
60#685—22no—
61#685—24no—
62#683—22no—
63#683—22no—
64#683—24no—
65#683—24no—
66C4×Dic5—22no—
67C2×C4×Q8D511yescharacteristic, complemented
68C2×#117—51no—
69#398—51no—
70#397—51no—
71#395—51no—
72#393—51no—
73#388—52no—
74#387—52no—
75#387—52no—
76#385—52no—
77#383—51no—
78C23×C5C2411yescharacteristic
79C22×D5—21no—
80C22×D5—22no—
81C22×D5—21no—
82C22×D5—22no—
83C22×D5#4112yes—
84C5×Q8—28no—
85C2×C4×C5—22no—
86C2×C4×C5—21no—
87C2×C4×C5—21no—
88C2×C4×C5—22no—
89C2×C4×C5—22no—
90C2×C4×C5—22no—
91C2×C4×C5—22no—
92C2×C4×C5—22no—
93C2×C4×C5—22no—
94C2×C4×C5C2×D412yes—
95C2×C4×C5C2×D412yes—
96C2×C4×C5C2×D412yes—
97C2×C4×C5#4114yes—
98C2×Dic5—21no—
99C2×Dic5—21no—
100C2×Dic5—21no—
101C2×Dic5—21no—
102C2×Dic5—22no—
103C2×Dic5—22no—
104C2×Dic5—22no—
105C2×Dic5—22no—
106C2×Dic5—22no—
107C2×Dic5—22no—
108C2×Dic5—22no—
109C2×Dic5—22no—
110C2×Dic5#4112yes—
111C4×D5—28no—
112C22×Q8C2×D511yescharacteristic
113C23×C4—51no—
114C4×Q8—24no—
115#117—104no—
116C2×C4⋊C4—51no—
117C2×C4⋊C4—51no—
118C2×C4⋊C4—52no—
119C2×C4⋊C4—52no—
120C2×C4⋊C4C2×D512yes—
121C2×C4⋊C4C2×D511yescharacteristic
122C2×C22⋊C4—51no—
123C2×C22⋊C4—51no—
124C2×C22⋊C4—52no—
125C2×C22⋊C4—52no—
126C2×C42—51no—
127C2×C42C2×D512yes—
128C2×C42C2×D511yescharacteristic
129#95—52no—
130#95—52no—
131#95—51no—
132#95—51no—
133#95—52no—
134#95—52no—
135#95—52no—
136#95—52no—
137C22×C5C22×D411yescharacteristic
138C22×C5C2×#4112yes—
139C22×C5C2×#4111yescharacteristic
140C22×C5C2×#4111yescharacteristic
141C22×C52−1+412yes—
142C2×D5—22no—
143C2×D5—24no—
144C2×D5—41no—
145C2×D5—41no—
146C2×D5—42no—
147C4×C5—22no—
148C4×C5—22no—
149C4×C5—24no—
150C4×C5—22no—
151C4×C5—42no—
152Dic5—22no—
153Dic5—41no—
154Dic5—41no—
155Dic5—42no—
156Dic5—42no—
157C24—51no—
158C2×Q8—24no—
159C2×Q8—22no—
160C22×C4—51no—
161C22×C4—52no—
162C22×C4—52no—
163C22×C4—51no—
164C22×C4—51no—
165C22×C4—102no—
166C22×C4—102no—
167C22×C4—102no—
168C22×C4C22×D512yes—
169C22×C4C22×D512yes—
170C22×C4C22×D511yescharacteristic
171C22×C4C22×D511yescharacteristic
172C22×C4C22×D511yescharacteristic
173C4⋊C4—24no—
174C4⋊C4—22no—
175C4⋊C4—104no—
176C4⋊C4—104no—
177C4⋊C4—102no—
178C4⋊C4—102no—
179C22⋊C4—104no—
180C22⋊C4—104no—
181C22⋊C4—102no—
182C22⋊C4—102no—
183C42—24no—
184C42—102no—
185C42C22⋊D514yes—
186C2×C5#52511yescharacteristic
187C2×C5#52511yescharacteristic
188C2×C5#53611yescharacteristic
189C2×C5#55212yes—
190C2×C5#55612yes—
191D5—42no—
192C23—52no—
193C23—102no—
194C23—102no—
195C23—101no—
196C23—101no—
197C23C23×D511yescharacteristic
198Q8—28no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—52no—
209C2×C4—108no—
210C2×C4—101no—
211C2×C4—101no—
212C2×C4—102no—
213C2×C4—102no—
214C2×C4—102no—
215C2×C4—102no—
216C2×C4—102no—
217C2×C4—102no—
218C2×C4—102no—
219C2×C4—102no—
220C2×C4—101no—
221C2×C4—101no—
222C2×C4#70914yes—
223C2×C4C2×C22⋊D512yes—
224C2×C4C2×C22⋊D512yes—
225C2×C4C2×C22⋊D512yes—
226C5#259911yescharacteristic, complemented, Sylow
227C22—102no—
228C22—104no—
229C22—201no—
230C22—201no—
231C22—202no—
232C22C2×#70912yes—
233C22#414712yes—
234C22D5×#4111yescharacteristic
235C22D5×#4111yescharacteristic
236C22C22×C22⋊D511yescharacteristic
237C4—24no—
238C4—22no—
239C4—22no—
240C4—22no—
241C4—42no—
242C4—102no—
243C4—201no—
244C4—201no—
245C4—202no—
246C4—202no—
247C2—202no—
248C2#6590212yes—
249C2#6591512yes—
250C2#6612511yescharacteristic
251C2#6615011yescharacteristic
252C2#6616211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC2×C4×C5×Q8C2
prime radicalC23×D5C23
hypercenterC2×C4×Q8D5
hypocenterC5#2599
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
1. C2×C4×C5×Q8C2semidirect
4. #65518C2semidirect
7. #65269C2semidirect
8. #65267C2semidirect
10. #65226C2semidirect
11. #65222C2semidirect
67. C2×C4×Q8D5semidirect
226. C5#2599semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C5×Q82
C2×#40182
#655202
#655182
#652822
#652782
#652692
#652672
#652292
#652262
#652222
#25995