Finite Group Databasebeta

← #10614692

Subgroups — #10614692

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×C5×#124C211yesmaximal, characteristic, complemented
3C2×#4084C211yesmaximal, characteristic
4C2×#4082C211yesmaximal, characteristic, complemented
5#65519C211yesmaximal, characteristic, complemented
6#65245C214yesmaximal, complemented
7#65236C214yesmaximal, complemented
8C4×#685C211yesmaximal, characteristic, complemented
9#65218C211yesmaximal, characteristic, complemented
10C22×C5×Q8C2211yescharacteristic
11C22×C5×D4C2211yescharacteristic
12C22×C22⋊D5C2212yes—
13C2×Q8×D5—24no—
14C2×Q8×D5—24no—
15C2×Q8×D5—24no—
16C22×C4×D5C2212yes—
17C22×C4×D5C2211yescharacteristic
18C22×Dic10C2212yes—
19C22×Dic10C2211yescharacteristic
20C5×#124—24no—
21C2×C5×C22⋊C4C2214yes—
22C2×C42×C5C2211yescharacteristic
23C2×C22⋊Dic5C2214yes—
24C2×C22⋊Dic5C2212yes—
25#4084—24no—
26#4082—24no—
27C2×#685C2212yes—
28C2×C4⋊Dic5C2211yescharacteristic
29C2×C4×Dic5C2211yescharacteristic
30#3963C2214yes—
31#3963C2212yes—
32#3963C2214yes—
33#2436—51nomaximal, Sylow
34C24×C5C2312yes—
35C23×D5C2311yescharacteristic
36C2×C5×Q8—22no—
37C2×C5×Q8—22no—
38C2×C5×Q8D414yescomplemented
39C2×C5×D4—22no—
40C2×C5×D4—22no—
41C2×C5×D4—22no—
42C22×C4×C5C2312yes—
43C22×C4×C5C2312yes—
44C22×C4×C5C2311yescharacteristic
45C2×C22⋊D5—24no—
46C2×C22⋊D5—28no—
47C22×Dic5C2312yes—
48C22×Dic5C2312yes—
49C22×Dic5C2311yescharacteristic
50C22×Dic5C2312yes—
51Q8×D5—232no—
52C2×C4×D5—24no—
53C2×C4×D5—22no—
54C2×C4×D5—24no—
55C2×C4×D5—22no—
56C2×C4×D5—24no—
57C2×C4×D5D414yes—
58C2×Dic10—22no—
59C2×Dic10—24no—
60C2×Dic10—22no—
61C2×Dic10—24no—
62C2×Dic10—24no—
63C2×Dic10D414yescomplemented
64C5×C22⋊C4—28no—
65C42×C5—22no—
66C22⋊Dic5—28no—
67C22⋊Dic5—24no—
68#685—24no—
69C4⋊Dic5—22no—
70C4×Dic5—22no—
71C23×Q8—51no—
72C2×#124—51no—
73C2×#124D511yescharacteristic, complemented
74C2×#121—51no—
75#393—54no—
76#391—51no—
77#391—51no—
78#386—54no—
79C4×C22⋊C4—51no—
80C23×C5—22no—
81C23×C5—22no—
82C23×C5—22no—
83C23×C5—24no—
84C23×C5—24no—
85C23×C5C2411yescharacteristic
86C22×D5—21no—
87C22×D5—21no—
88C22×D5—22no—
89C22×D5C2×D412yes—
90C22×D5#4114yes—
91C5×Q8—28no—
92C5×D4—44no—
93C2×C4×C5—22no—
94C2×C4×C5—22no—
95C2×C4×C5—22no—
96C2×C4×C5—22no—
97C2×C4×C5—22no—
98C2×C4×C5C2×D412yes—
99C2×C4×C5C2×D412yes—
100C2×C4×C5C2×D414yes—
101C2×C4×C5C2×D412yes—
102C22⋊D5—48no—
103C2×Dic5—22no—
104C2×Dic5—21no—
105C2×Dic5—21no—
106C2×Dic5—22no—
107C2×Dic5—22no—
108C2×Dic5—22no—
109C2×Dic5—22no—
110C2×Dic5—24no—
111C2×Dic5—22no—
112C2×Dic5C2×D412yes—
113C2×Dic5C2×D414yes—
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—52no—
125C22×D4C2×D511yescharacteristic
126C23×C4—51no—
127C23×C4—52no—
128#124—24no—
129#124—104no—
130#121—104no—
131C2×C4⋊C4—51no—
132C2×C22⋊C4—52no—
133C2×C22⋊C4—54no—
134C2×C22⋊C4—52no—
135C2×C22⋊C4C2×D514yes—
136C2×C42—51no—
137C2×C42C2×D511yescharacteristic
138#95—52no—
139#95—54no—
140#95—54no—
141C22×C5—42no—
142C22×C5—42no—
143C22×C5—42no—
144C22×C5—44no—
145C22×C5—44no—
146C22×C5C22×D411yescharacteristic
147C22×C5C22×D411yescharacteristic
148C22×C5C22×D411yescharacteristic
149C22×C5C2×#4112yes—
150C22×C52−1+412yes—
151C2×D5—22no—
152C2×D5—22no—
153C2×D5—24no—
154C2×D5—24no—
155C2×D5#12414yescomplemented
156C4×C5—24no—
157C4×C5—42no—
158C4×C5#12014yes—
159Dic5—22no—
160Dic5—24no—
161Dic5—42no—
162Dic5—42no—
163C24—51no—
164C24C22×D512yes—
165C2×Q8—22no—
166C2×Q8—22no—
167C2×Q8—54no—
168C2×Q8—104no—
169C2×Q8—104no—
170C2×Q8—1032no—
171C2×Q8—102no—
172C2×Q8—104no—
173C2×Q8—102no—
174C2×Q8C22⋊D514yescomplemented
175C2×D4—22no—
176C2×D4—22no—
177C2×D4—22no—
178C2×D4—104no—
179C2×D4—108no—
180C22×C4—51no—
181C22×C4—52no—
182C22×C4—54no—
183C22×C4—52no—
184C22×C4—52no—
185C22×C4—104no—
186C22×C4—104no—
187C22×C4—102no—
188C22×C4—104no—
189C22×C4—102no—
190C22×C4C22×D512yes—
191C22×C4C22×D512yes—
192C22×C4C22×D511yescharacteristic
193C4⋊C4—102no—
194C22⋊C4—28no—
195C22⋊C4—104no—
196C22⋊C4—104no—
197C22⋊C4—108no—
198C42—22no—
199C42—102no—
200C2×C5—82no—
201C2×C5C2×#12011yescharacteristic
202C2×C5C2×#12411yescharacteristic
203C2×C5#53611yescharacteristic
204C2×C5#54812yes—
205C2×C5#54812yes—
206D5—24no—
207C23—22no—
208C23—22no—
209C23—24no—
210C23—24no—
211C23—22no—
212C23—52no—
213C23—54no—
214C23—101no—
215C23—102no—
216C23—101no—
217C23C23×D511yescharacteristic
218Q8—28no—
219Q8—1016no—
220Q8—108no—
221D4—44no—
222D4—208no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—52no—
229C2×C4—54no—
230C2×C4—1016no—
231C2×C4—102no—
232C2×C4—101no—
233C2×C4—101no—
234C2×C4—102no—
235C2×C4—108no—
236C2×C4—102no—
237C2×C4—102no—
238C2×C4—102no—
239C2×C4—104no—
240C2×C4—102no—
241C2×C4D4×D512yes—
242C2×C4D4×D512yes—
243C2×C4C2×C22⋊D514yes—
244C2×C4C2×C22⋊D512yes—
245C5#243611yescharacteristic, complemented, Sylow
246C22—42no—
247C22—44no—
248C22—42no—
249C22—42no—
250C22—44no—
251C22—54no—
252C22—102no—
253C22—104no—
254C22—102no—
255C22—104no—
256C22C2×D4×D511yescharacteristic
257C22C2×D4×D511yescharacteristic
258C22D5×#4112yes—
259C22#415012yes—
260C22C22×C22⋊D511yescharacteristic
261C4—24no—
262C4—42no—
263C4—102no—
264C4—104no—
265C4—202no—
266C4—202no—
267C4#408314yes—
268C2—82no—
269C2—104no—
270C2D5×#12411yescharacteristic
271C2#6600911yescharacteristic
272C2#6601112yes—
273C2C2×#408311yescharacteristic
274C2#6616512yes—
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC2×C5×#124C2
prime radicalC2×#4084C2
hypercenterC2×#124D5
hypocenterC5#2436
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
1. C22×Q8×D5C2semidirect
2. C2×C5×#124C2semidirect
4. C2×#4082C2semidirect
5. #65519C2semidirect
6. #65245C2semidirect
7. #65236C2semidirect
8. C4×#685C2semidirect
9. #65218C2semidirect
38. C2×C5×Q8D4semidirect
63. C2×Dic10D4semidirect
73. C2×#124D5semidirect
155. C2×D5#124semidirect
174. C2×Q8C22⋊D5semidirect
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×C5×#1242
C2×#40842
C2×#40822
#655192
#652452
#652362
C4×#6852
#652182
#24365