Finite Group Databasebeta

← #10606602

Subgroups — #10606602

Order 640.

582 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 582 conjugacy classes of subgroups — too many to draw. Only its 153 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×#4081C211yesmaximal, characteristic
2C22×C4×Dic5C211yesmaximal, characteristic, complemented
3C5×#386C211yesmaximal, characteristic
4#65509C211yesmaximal, characteristic
5#65499C211yesmaximal, characteristic, complemented
6#65240C212yesmaximal, complemented
7#65239C211yesmaximal, characteristic, complemented
8#65234C212yesmaximal, complemented
9#65232C211yesmaximal, characteristic, complemented
10#64943C212yesmaximal, complemented
11#64941C212yesmaximal, complemented
12C22×C5×D4C2211yescharacteristic
13C23×C4×C5C2211yescharacteristic
14C23×Dic5C2211yescharacteristic
15C23×Dic5C2211yescharacteristic
16C2×C5×C22⋊C4C2211yescharacteristic
17C2×C5×C22⋊C4C2212yes—
18C2×C22⋊Dic5C2211yescharacteristic
19C2×C22⋊Dic5C2211yescharacteristic
20C2×C22⋊Dic5C2211yescharacteristic
21C2×C22⋊Dic5C2211yescharacteristic
22C2×C22⋊Dic5C2212yes—
23#4081—24no—
24C2×C4⋊Dic5C2212yes—
25C2×#683C2211yescharacteristic
26C2×#683C2211yescharacteristic
27C2×#683C2212yes—
28C2×C4×Dic5—24no—
29C2×C4×Dic5C2212yes—
30C2×C4×Dic5C2212yes—
31C5×#95C2212yes—
32#3963C2212yes—
33#3963C2212yes—
34#3963C2212yes—
35#3963C2212yes—
36#3963C2212yes—
37#2853—51nomaximal, Sylow
38C24×C5C2311yescharacteristic
39C24×C5C2311yescharacteristic
40C2×C5×D4—22no—
41C2×C5×D4—22no—
42C2×C5×D4—22no—
43C22×C4×C5—22no—
44C22×C4×C5—24no—
45C22×C4×C5C2312yes—
46C22×C4×C5C2311yescharacteristic
47C22×C4×C5C2312yes—
48C22×Dic5—22no—
49C22×Dic5—22no—
50C22×Dic5—22no—
51C22×Dic5—24no—
52C22×Dic5D414yes—
53C22×Dic5C2312yes—
54C22×Dic5C2311yescharacteristic
55C22×Dic5C2311yescharacteristic
56C22×Dic5C2311yescharacteristic
57C22×Dic5C2311yescharacteristic
58C22×Dic5C2312yes—
59C5×C22⋊C4—22no—
60C5×C22⋊C4—24no—
61C22⋊Dic5—22no—
62C22⋊Dic5—22no—
63C22⋊Dic5—22no—
64C22⋊Dic5—22no—
65C22⋊Dic5—24no—
66C4⋊Dic5—24no—
67#683—24no—
68#683—22no—
69#683—22no—
70C4×Dic5—24no—
71C4×Dic5—24no—
72C2×#123—51no—
73C22×C42—51no—
74#388—52no—
75#388—52no—
76#387—52no—
77#387—52no—
78#386—51no—
79#386D511yescharacteristic
80#385—51no—
81#385—51no—
82#381—51no—
83C23×C5—22no—
84C23×C5—22no—
85C23×C5—21no—
86C23×C5—22no—
87C23×C5—22no—
88C23×C5—21no—
89C23×C5—21no—
90C23×C5C2×D412yes—
91C23×C5#4112yes—
92C23×C5#4112yes—
93C23×C5C2411yescharacteristic
94C5×D4—44no—
95C2×C4×C5—28no—
96C2×C4×C5—22no—
97C2×C4×C5—22no—
98C2×C4×C5—21no—
99C2×C4×C5—22no—
100C2×C4×C5—22no—
101C2×C4×C5—22no—
102C2×C4×C5—21no—
103C2×C4×C5—21no—
104C2×C4×C5#4114yes—
105C2×Dic5—28no—
106C2×Dic5—28no—
107C2×Dic5—22no—
108C2×Dic5—21no—
109C2×Dic5—21no—
110C2×Dic5—21no—
111C2×Dic5—21no—
112C2×Dic5—22no—
113C2×Dic5—21no—
114C2×Dic5—21no—
115C2×Dic5—22no—
116C2×Dic5—22no—
117C2×Dic5—22no—
118C2×Dic5—22no—
119C2×Dic5—22no—
120C2×Dic5C2×D412yes—
121C2×Dic5C2×D412yes—
122C2×Dic5#4114yes—
123C22×D4C2×D511yescharacteristic
124C23×C4—51no—
125C23×C4—51no—
126C23×C4C2×D511yescharacteristic
127#123—104no—
128C2×C4⋊C4—51no—
129C2×C4⋊C4—51no—
130C2×C4⋊C4—52no—
131C2×C4⋊C4—52no—
132C2×C22⋊C4—51no—
133C2×C22⋊C4—51no—
134C2×C22⋊C4—51no—
135C2×C22⋊C4—51no—
136C2×C22⋊C4—52no—
137C2×C22⋊C4C2×D511yescharacteristic
138C2×C22⋊C4C2×D512yes—
139C2×C42—52no—
140C2×C42—52no—
141C2×C42—104no—
142#95—52no—
143#95—52no—
144#95—52no—
145#95—52no—
146#95—52no—
147#95C2×D512yes—
148C22×C5—22no—
149C22×C5—22no—
150C22×C5—24no—
151C22×C5—24no—
152C22×C5—41no—
153C22×C5—41no—
154C22×C5—41no—
155C22×C5—42no—
156C22×C5—42no—
157C22×C5#12314yes—
158C22×C5C22×D411yescharacteristic
159C22×C5C2×#4111yescharacteristic
160C22×C5C2×#4111yescharacteristic
161C22×C5C2×#4112yes—
162C22×C5C2×#4112yes—
163C4×C5—24no—
164C4×C5—41no—
165C4×C5—42no—
166Dic5—22no—
167Dic5—22no—
168Dic5—24no—
169Dic5—41no—
170Dic5—41no—
171Dic5—42no—
172C24C22×D511yescharacteristic
173C24C22×D511yescharacteristic
174C2×D4—22no—
175C2×D4—22no—
176C2×D4—22no—
177C22×C4—22no—
178C22×C4—24no—
179C22×C4—54no—
180C22×C4—51no—
181C22×C4—51no—
182C22×C4—52no—
183C22×C4—52no—
184C22×C4—51no—
185C22×C4—51no—
186C22×C4—104no—
187C22×C4—102no—
188C22×C4—102no—
189C22×C4—102no—
190C22×C4C22×D512yes—
191C22×C4C22×D511yescharacteristic
192C22×C4C22×D512yes—
193C4⋊C4—102no—
194C4⋊C4—102no—
195C4⋊C4—104no—
196C4⋊C4—104no—
197C22⋊C4—22no—
198C22⋊C4—24no—
199C22⋊C4—102no—
200C22⋊C4—102no—
201C22⋊C4—104no—
202C22⋊C4—102no—
203C22⋊C4—102no—
204C42—104no—
205C42—104no—
206C2×C5—24no—
207C2×C5—81no—
208C2×C5C2×#12311yescharacteristic
209C2×C5#52511yescharacteristic
210C2×C5#52511yescharacteristic
211C2×C5#52912yes—
212C2×C5#52912yes—
213C23—22no—
214C23—22no—
215C23—21no—
216C23—21no—
217C23—22no—
218C23—21no—
219C23—22no—
220C23#70912yes—
221C23D4×D512yes—
222C23#71112yes—
223C23C23×D511yescharacteristic
224D4—44no—
225C2×C4—28no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—21no—
233C2×C4—22no—
234C2×C4—52no—
235C2×C4—52no—
236C2×C4—54no—
237C2×C4—108no—
238C2×C4—101no—
239C2×C4—101no—
240C2×C4—102no—
241C2×C4—101no—
242C2×C4—101no—
243C2×C4—102no—
244C2×C4—108no—
245C2×C4—102no—
246C2×C4—101no—
247C2×C4—101no—
248C2×C4—102no—
249C2×C4—102no—
250C2×C4—102no—
251C2×C4—102no—
252C2×C4#71114yes—
253C5#285311yescharacteristic, complemented, Sylow
254C22—24no—
255C22—22no—
256C22—24no—
257C22—22no—
258C22—41no—
259C22—42no—
260C22—41no—
261C22—42no—
262C22—41no—
263C22#402914yes—
264C22C2×#70911yescharacteristic
265C22C2×D4×D511yescharacteristic
266C22C2×#71111yescharacteristic
267C22C2×#71112yes—
268C22D5×#4112yes—
269C4—24no—
270C4—41no—
271C4—42no—
272C4—102no—
273C4—102no—
274C4—104no—
275C4—202no—
276C4—201no—
277C4—201no—
278C2—24no—
279C2—81no—
280C2C2×#402911yescharacteristic
281C2#6587212yes—
282C2#6592011yescharacteristic
283C2#6593412yes—
284C2#6599011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#386C2
prime radicalC22×C5×D4C22
hypercenter#386D5
hypocenterC5#2853
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
2. C22×C4×Dic5C2semidirect
5. #65499C2semidirect
6. #65240C2semidirect
7. #65239C2semidirect
8. #65234C2semidirect
9. #65232C2semidirect
10. #64943C2semidirect
11. #64941C2semidirect
253. C5#2853semidirect

Maximal subgroups

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

subgroupindex
C2×#40812
C22×C4×Dic52
C5×#3862
#655092
#654992
#652402
#652392
#652342
#652322
#649432
#649412
#28535