Finite Group Databasebeta

← #10608680

Subgroups — #10608680

Order 640.

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

normal subgroups only

This group has 696 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×#4085C212yesmaximal, complemented
2C2×#4031C212yesmaximal, complemented
3C5×#394C211yesmaximal, characteristic, complemented
4#65277C211yesmaximal, characteristic, complemented
5#65268C211yesmaximal, characteristic, complemented
6#65249C212yesmaximal, complemented
7#65247C212yesmaximal, complemented
8C22⋊C4×Dic5C212yesmaximal, complemented
9#64958C212yesmaximal, complemented
10C22×C5×D4C2211yescharacteristic
11C22×C22⋊D5C2212yes—
12C22×C22⋊D5C2212yes—
13C23×Dic5C2212yes—
14C22×D20C2211yescharacteristic
15C22×Dic10C2211yescharacteristic
16C2×C5×C4⋊C4C2211yescharacteristic
17C2×C5×C22⋊C4C2212yes—
18C2×C5×C22⋊C4C2212yes—
19C2×C22⋊Dic5C2212yes—
20#4085—28no—
21C2×#685C2212yes—
22C2×#685C2212yes—
23C2×#685C2212yes—
24C2×#685C2212yes—
25C2×#683C2212yes—
26C2×C4×Dic5C2212yes—
27C2×C4×Dic5C2211yescharacteristic
28#4031—28no—
29C5×#95C2211yescharacteristic
30#3963C2212yes—
31#3963C2212yes—
32#3963C2211yescharacteristic
33#2555—51nomaximal, Sylow
34C24×C5C2312yes—
35C23×D5C2312yes—
36C2×C5×D4—24no—
37C2×C5×D4—41no—
38C22×C4×C5C2311yescharacteristic
39C22×C4×C5C2312yes—
40C22×C4×C5C2311yescharacteristic
41C22×C4×C5C2311yescharacteristic
42C2×C22⋊D5—24no—
43C2×C22⋊D5—24no—
44C2×C22⋊D5—24no—
45C2×C22⋊D5—24no—
46C2×C22⋊D5—42no—
47C2×C22⋊D5—42no—
48C22×Dic5—24no—
49C22×Dic5—24no—
50C22×Dic5—42no—
51C22×Dic5C2312yes—
52C22×Dic5C2312yes—
53C22×Dic5C2312yes—
54C2×D20—22no—
55C2×D20—41no—
56C2×D20—41no—
57C2×Dic10—24no—
58C2×Dic10—41no—
59C5×C4⋊C4—41no—
60C5×C22⋊C4—24no—
61C5×C22⋊C4—24no—
62C22⋊Dic5D418yescomplemented
63#685—24no—
64#685—24no—
65#685—24no—
66#685—24no—
67#683—24no—
68C4×Dic5—24no—
69C4×Dic5—41no—
70C2×#124—52no—
71C2×#121—52no—
72#395—51no—
73#394D511yescharacteristic, complemented
74#392—52no—
75#390—51no—
76#388—52no—
77#386—52no—
78C4×C22⋊C4—52no—
79C23×C5—22no—
80C23×C5—22no—
81C23×C5—22no—
82C23×C5—22no—
83C23×C5—22no—
84C23×C5—22no—
85C23×C5C2×D414yes—
86C23×C5C2411yescharacteristic
87C22×D5—22no—
88C22×D5—22no—
89C22×D5—22no—
90C22×D5—22no—
91C22×D5—22no—
92C22×D5—22no—
93C22×D5—22no—
94C5×D4—44no—
95C2×C4×C5—21no—
96C2×C4×C5—21no—
97C2×C4×C5—22no—
98C2×C4×C5—22no—
99C2×C4×C5—22no—
100C2×C4×C5—22no—
101C2×C4×C5—22no—
102C2×C4×C5—21no—
103C2×C4×C5—21no—
104C2×C4×C5#4112yes—
105C22⋊D5—48no—
106C22⋊D5—48no—
107C2×Dic5—22no—
108C2×Dic5—22no—
109C2×Dic5—22no—
110C2×Dic5—22no—
111C2×Dic5—22no—
112C2×Dic5—22no—
113C2×Dic5—48no—
114C2×Dic5C2×D414yes—
115C2×Dic5C2×D414yes—
116C2×Dic5#4114yes—
117D20—44no—
118Dic10—44no—
119C22×Q8—51no—
120C22×D4—52no—
121C22×D4—52no—
122C22×D4—51no—
123C22×D4C2×D511yescharacteristic
124C23×C4—52no—
125#124—108no—
126#121—108no—
127C2×C4⋊C4—52no—
128C2×C4⋊C4C2×D511yescharacteristic
129C2×C22⋊C4—52no—
130C2×C22⋊C4—52no—
131C2×C22⋊C4—52no—
132C2×C22⋊C4—52no—
133C2×C22⋊C4—52no—
134C2×C22⋊C4C2×D512yes—
135C2×C22⋊C4C2×D512yes—
136C2×C42—52no—
137C2×C42—51no—
138#95—52no—
139#95—52no—
140#95—51no—
141#95C2×D511yescharacteristic
142C22×C5—24no—
143C22×C5—24no—
144C22×C5—24no—
145C22×C5—42no—
146C22×C5—42no—
147C22×C5—42no—
148C22×C5—42no—
149C22×C5C22×D412yes—
150C22×C5C2×#4111yescharacteristic
151C22×C5C2×#4112yes—
152C22×C5H(5,𝔽2)11yescharacteristic
153C22×C5H(5,𝔽2)11yescharacteristic
154C2×D5—42no—
155C2×D5—42no—
156C2×D5—42no—
157C2×D5—42no—
158C2×D5—42no—
159C2×D5—42no—
160C2×D5—42no—
161C4×C5—22no—
162C4×C5—42no—
163C4×C5—41no—
164C4×C5—41no—
165Dic5—24no—
166Dic5—24no—
167Dic5—24no—
168C24—52no—
169C24C22×D512yes—
170C2×Q8—104no—
171C2×Q8—201no—
172C2×D4—24no—
173C2×D4—41no—
174C2×D4—104no—
175C2×D4—104no—
176C2×D4—104no—
177C2×D4—104no—
178C2×D4—102no—
179C2×D4—201no—
180C2×D4—201no—
181C2×D4—202no—
182C2×D4—202no—
183C22×C4—52no—
184C22×C4—52no—
185C22×C4—52no—
186C22×C4—104no—
187C22×C4—104no—
188C22×C4—202no—
189C22×C4C22×D512yes—
190C22×C4C22×D511yescharacteristic
191C22×C4C22×D511yescharacteristic
192C22×C4C22×D511yescharacteristic
193C4⋊C4—41no—
194C4⋊C4—104no—
195C22⋊C4—24no—
196C22⋊C4—24no—
197C22⋊C4—58no—
198C22⋊C4—104no—
199C22⋊C4—104no—
200C22⋊C4—104no—
201C22⋊C4—104no—
202C42—104no—
203C42—201no—
204C2×C5—44no—
205C2×C5#53212yes—
206C2×C5#53811yescharacteristic
207C2×C5D4211yescharacteristic
208C2×C5#54612yes—
209C2×C5#55511yescharacteristic
210D5—82no—
211C23—22no—
212C23—22no—
213C23—22no—
214C23—22no—
215C23—22no—
216C23—22no—
217C23—102no—
218C23—102no—
219C23—102no—
220C23—102no—
221C23—102no—
222C23—102no—
223C23—102no—
224C23D4×D514yes—
225C23C23×D511yescharacteristic
226Q8—204no—
227D4—44no—
228D4—204no—
229D4—208no—
230D4—208no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—22no—
240C2×C4—54no—
241C2×C4—54no—
242C2×C4—54no—
243C2×C4—102no—
244C2×C4—102no—
245C2×C4—102no—
246C2×C4—102no—
247C2×C4—102no—
248C2×C4—102no—
249C2×C4—208no—
250C2×C4#71112yes—
251C5#255511yescharacteristic, complemented, Sylow
252C22—24no—
253C22—24no—
254C22—24no—
255C22—42no—
256C22—42no—
257C22—42no—
258C22—42no—
259C22—202no—
260C22—202no—
261C22—202no—
262C22—202no—
263C22—202no—
264C22—202no—
265C22—202no—
266C22C2×D4×D512yes—
267C22C2×#71111yescharacteristic
268C22#414411yescharacteristic
269C22D5×#4112yes—
270C22#414911yescharacteristic
271C4—22no—
272C4—41no—
273C4—41no—
274C4—42no—
275C4—104no—
276C4—104no—
277C4—104no—
278C2—44no—
279C2—402no—
280C2#6592411yescharacteristic
281C2#6593012yes—
282C2#6595111yescharacteristic
283C2#6598812yes—
284C2#6600411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#394C2
hypercenter#394D5
hypocenterC5#2555
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. C2×#4085C2semidirect
2. C2×#4031C2semidirect
3. C5×#394C2semidirect
4. #65277C2semidirect
5. #65268C2semidirect
6. #65249C2semidirect
7. #65247C2semidirect
8. C22⋊C4×Dic5C2semidirect
9. #64958C2semidirect
62. C22⋊Dic5D4semidirect
73. #394D5semidirect
251. C5#2555semidirect

Maximal subgroups

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

subgroupindex
C2×#40852
C2×#40312
C5×#3942
#652772
#652682
#652492
#652472
C22⋊C4×Dic52
#649582
#25555