Finite Group Databasebeta

← #10608231

Subgroups — #10608231

Order 640.

852 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 852 conjugacy classes of subgroups — too many to draw. Only its 173 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
1C22×Q8×D5C211yesmaximal, characteristic, complemented
2C2×#4093C211yesmaximal, characteristic, complemented
3C2×#4041C211yesmaximal, characteristic
4C2×#4021C211yesmaximal, characteristic, complemented
5C5×#391C211yesmaximal, characteristic, complemented
6#65263C211yesmaximal, characteristic, complemented
7C4×#685C211yesmaximal, characteristic, complemented
8#64962C214yesmaximal, complemented
9#64954C214yesmaximal, complemented
10C22×C5×Q8C2211yescharacteristic
11C2×Q8×D5—24no—
12C2×Q8×D5—24no—
13C2×Q8×D5—24no—
14C22×D20C2211yescharacteristic
15C22×D20C2212yes—
16C22×C4×D5C2212yes—
17C22×C4×D5C2211yescharacteristic
18C22×Dic10C2212yes—
19C22×Dic10C2211yescharacteristic
20C2×C5×C4⋊C4C2211yescharacteristic
21C2×C42×C5C2211yescharacteristic
22#4093—24no—
23C2×#685C2214yes—
24C2×#685C2212yes—
25C2×#685C2214yes—
26C2×#685C2212yes—
27C2×C4×Dic5C2211yescharacteristic
28#4041—24no—
29#4021—24no—
30C5×#95C2214yes—
31#3963C2214yes—
32#3963C2212yes—
33#2436—51nomaximal, Sylow
34C23×D5C2312yes—
35C23×D5C2311yescharacteristic
36C2×C5×Q8—22no—
37C2×C5×Q8—22no—
38C2×C5×Q8D414yescomplemented
39C22×C4×C5C2312yes—
40C22×C4×C5C2312yes—
41C22×C4×C5C2312yes—
42C22×C4×C5C2311yescharacteristic
43C22×Dic5C2312yes—
44C22×Dic5C2311yescharacteristic
45C22×Dic5C2312yes—
46Q8×D5—232no—
47C2×D20—24no—
48C2×D20—28no—
49C2×D20—22no—
50C2×D20—22no—
51C2×D20—22no—
52C2×C4×D5—24no—
53C2×C4×D5—24no—
54C2×C4×D5—22no—
55C2×C4×D5—22no—
56C2×C4×D5—24no—
57C2×C4×D5D414yes—
58C2×Dic10—24no—
59C2×Dic10—22no—
60C2×Dic10—22no—
61C2×Dic10—24no—
62C2×Dic10—24no—
63C2×Dic10D414yescomplemented
64C5×C4⋊C4—22no—
65C42×C5—22no—
66#685—24no—
67#685—28no—
68#685—28no—
69#685—24no—
70C4×Dic5—22no—
71C23×Q8—51no—
72C2×#124—51no—
73C2×#124—51no—
74C2×#121—51no—
75#393—54no—
76#391—51no—
77#391D511yescharacteristic, complemented
78#386—54no—
79C4×C22⋊C4—51no—
80C23×C5C2411yescharacteristic
81C22×D5—21no—
82C22×D5—21no—
83C22×D5—22no—
84C22×D5—22no—
85C22×D5—22no—
86C22×D5—22no—
87C22×D5—24no—
88C22×D5—24no—
89C22×D5C2×D412yes—
90C22×D5#4114yes—
91C5×Q8—28no—
92C2×C4×C5—22no—
93C2×C4×C5—22no—
94C2×C4×C5—22no—
95C2×C4×C5—22no—
96C2×C4×C5—22no—
97C2×C4×C5—24no—
98C2×C4×C5—22no—
99C2×C4×C5C2×D414yes—
100C2×C4×C5C2×D412yes—
101C2×C4×C5C2×D412yes—
102C2×C4×C5C2×D412yes—
103C2×Dic5—21no—
104C2×Dic5—22no—
105C2×Dic5—22no—
106C2×Dic5—21no—
107C2×Dic5—22no—
108C2×Dic5—22no—
109C2×Dic5—22no—
110C2×Dic5C2×D412yes—
111C2×Dic5C2×D414yes—
112D20—48no—
113D20—44no—
114C4×D5—216no—
115C4×D5—28no—
116Dic10—216no—
117Dic10—28no—
118C22×Q8—51no—
119C22×Q8—52no—
120C22×Q8—104no—
121C22×Q8—104no—
122C22×Q8—104no—
123C22×Q8C2×D511yescharacteristic
124C22×D4—51no—
125C22×D4—52no—
126C23×C4—51no—
127C23×C4—52no—
128#124—104no—
129#124—104no—
130#121—104no—
131C2×C4⋊C4C2×D511yescharacteristic
132C2×C22⋊C4—52no—
133C2×C22⋊C4—54no—
134C2×C22⋊C4—54no—
135C2×C22⋊C4—52no—
136C2×C42—51no—
137C2×C42C2×D511yescharacteristic
138#95—52no—
139#95—54no—
140#95C2×D514yes—
141C22×C5C22×D411yescharacteristic
142C22×C5C22×D411yescharacteristic
143C22×C5C22×D411yescharacteristic
144C22×C5C2×#4112yes—
145C22×C52−1+412yes—
146C2×D5—22no—
147C2×D5—22no—
148C2×D5—24no—
149C2×D5—24no—
150C2×D5—42no—
151C2×D5—42no—
152C2×D5—42no—
153C2×D5—44no—
154C2×D5—44no—
155C2×D5#12414yescomplemented
156C4×C5—24no—
157C4×C5—42no—
158C4×C5—42no—
159C4×C5#12014yes—
160Dic5—22no—
161Dic5—24no—
162Dic5—42no—
163C24—51no—
164C24—52no—
165C2×Q8—22no—
166C2×Q8—22no—
167C2×Q8—54no—
168C2×Q8—104no—
169C2×Q8—104no—
170C2×Q8—1032no—
171C2×Q8—104no—
172C2×Q8—102no—
173C2×Q8—102no—
174C2×Q8D2014yescomplemented
175C2×D4—104no—
176C2×D4—108no—
177C2×D4—102no—
178C2×D4—102no—
179C2×D4—102no—
180C22×C4—51no—
181C22×C4—52no—
182C22×C4—54no—
183C22×C4—52no—
184C22×C4—104no—
185C22×C4—104no—
186C22×C4—104no—
187C22×C4—102no—
188C22×C4—102no—
189C22×C4C22×D512yes—
190C22×C4C22×D512yes—
191C22×C4C22×D512yes—
192C22×C4C22×D511yescharacteristic
193C4⋊C4—22no—
194C22⋊C4—104no—
195C22⋊C4—108no—
196C22⋊C4—104no—
197C22⋊C4—108no—
198C42—22no—
199C42—102no—
200C2×C5C2×#12011yescharacteristic
201C2×C5C2×#12411yescharacteristic
202C2×C5#53611yescharacteristic
203C2×C5#54812yes—
204C2×C5#54812yes—
205D5—24no—
206D5—82no—
207C23—52no—
208C23—54no—
209C23—101no—
210C23—101no—
211C23—102no—
212C23—102no—
213C23—102no—
214C23—102no—
215C23—104no—
216C23—104no—
217C23C23×D511yescharacteristic
218Q8—28no—
219Q8—1016no—
220Q8—108no—
221D4—208no—
222D4—204no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—24no—
230C2×C4—52no—
231C2×C4—54no—
232C2×C4—1016no—
233C2×C4—101no—
234C2×C4—102no—
235C2×C4—102no—
236C2×C4—101no—
237C2×C4—108no—
238C2×C4—102no—
239C2×C4—102no—
240C2×C4—102no—
241C2×C4C2×D2014yes—
242C2×C4C2×D2012yes—
243C2×C4D4×D512yes—
244C2×C4D4×D512yes—
245C5#243611yescharacteristic, complemented, Sylow
246C22—54no—
247C22—102no—
248C22—102no—
249C22—104no—
250C22—104no—
251C22—202no—
252C22—202no—
253C22—204no—
254C22—204no—
255C22—202no—
256C22C22×D2011yescharacteristic
257C22C2×D4×D511yescharacteristic
258C22C2×D4×D511yescharacteristic
259C22#414712yes—
260C22D5×#4112yes—
261C4—24no—
262C4—42no—
263C4—42no—
264C4—102no—
265C4—104no—
266C4—202no—
267C4#402814yes—
268C2—104no—
269C2—402no—
270C2C2×#402811yescharacteristic
271C2#6591012yes—
272C2#6596712yes—
273C2D5×#12411yescharacteristic
274C2#6605811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#391C2
prime radicalC2×#4041C2
hypercenter#391D5
hypocenterC5#2436
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
1. C22×Q8×D5C2semidirect
2. C2×#4093C2semidirect
4. C2×#4021C2semidirect
5. C5×#391C2semidirect
6. #65263C2semidirect
7. C4×#685C2semidirect
8. #64962C2semidirect
9. #64954C2semidirect
38. C2×C5×Q8D4semidirect
63. C2×Dic10D4semidirect
77. #391D5semidirect
155. C2×D5#124semidirect
174. C2×Q8D20semidirect
245. C5#2436semidirect

Maximal subgroups

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

subgroupindex
C22×Q8×D52
C2×#40932
C2×#40412
C2×#40212
C5×#3912
#652632
C4×#6852
#649622
#649542
#24365