Finite Group Databasebeta

← #13106033

Subgroups — #13106033

Order 1600.

558 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 558 conjugacy classes of subgroups — too many to draw. Only its 144 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
1#11717244C211yesmaximal, characteristic, complemented
2C2×C5×#685C211yesmaximal, characteristic
3C5×#4070C211yesmaximal, characteristic, complemented
4D5×C22⋊Dic5C212yesmaximal
5#11717082C212yesmaximal, complemented
6#11717015C212yesmaximal, complemented
7#11717014C212yesmaximal, complemented
8#11717007C212yesmaximal, complemented
9#11717006C212yesmaximal, complemented
10C23×C5×D5C2211yescharacteristic
11C22×C4×C52C2211yescharacteristic
12C2×#6045C2211yescharacteristic, complemented
13C22×C5×Dic5C2211yescharacteristic
14C2×C5×Dic10C2211yescharacteristic, complemented
15C2×#6039C2212yescomplemented
16C2×D5×Dic5C2212yes—
17C2×D5×Dic5C2212yes—
18C22⋊#1039C2212yes—
19C4⋊#1039C2211yescharacteristic, complemented
20#87806C2212yescomplemented
21C5×C22⋊Dic5C2212yes—
22C5×#685C2212yes—
23C5×#685C2212yes—
24C5×C4⋊Dic5C2211yescharacteristic, complemented
25C5×#683C2212yescomplemented
26#87781C2212yescomplemented
27#87780C2212yescomplemented
28#87780C2212yescomplemented
29#87778C2212yes—
30#87778C2212yes—
31#65968—51nomaximal
32#65817—51nomaximal
33C23×C52C2311yescharacteristic
34C22×C5×D5—41no—
35C22×C5×D5—41no—
36C22×C5×D5Q814yescomplemented
37C22×C5×D5C2312yes—
38C2×C4×C52—41no—
39C2×C4×C52C2311yescharacteristic
40C2×C4×C52C2311yescharacteristic
41C2×#1039C2312yes—
42C2×#1039C2312yes—
43#6045—41no—
44C2×C5×Dic5—41no—
45C2×C5×Dic5C2312yes—
46C2×C5×Dic5C2312yes—
47C2×C5×Dic5C2312yes—
48C5×Dic10—41no—
49#6039—42no—
50D5×Dic5—42no—
51D5×Dic5—42no—
52C5×C22⋊Q8—51no—
53C2×C5×C22⋊C4—51no—
54C2×C22⋊Dic5—52no—
55#4092—52no—
56C2×#685D511yescharacteristic
57#4070—51no—
58#4070—51no—
59#4070—52no—
60#4070—102no—
61#4070D511yescharacteristic, complemented
62#4043—52no—
63#4042—52no—
64#4042—52no—
65D5×C22⋊C4—52no—
66#4024—52no—
67#4024—52no—
68#4024—52no—
69#4024—52no—
70#4024—52no—
71C22×C52—21no—
72C22×C52—21no—
73C22×C52C2×Q812yes—
74C22×C52C2411yescharacteristic
75C2×C5×D5—22no—
76C2×C5×D5—22no—
77C2×C5×D5—44no—
78C2×C5×D5C2×Q814yes—
79C4×C52—21no—
80C4×C52—21no—
81#1039—22no—
82#1039—22no—
83C5×Dic5—22no—
84C5×Dic5—22no—
85C5×Dic5—22no—
86C24×C5—51no—
87C23×D5C2×D511yescharacteristic
88C2×C5×Q8—51no—
89C22×C4×C5—22no—
90C22×C4×C5—51no—
91C22×C4×C5C2×D511yescharacteristic
92C22×C4×C5C2×D511yescharacteristic
93C22×Dic5—52no—
94C22×Dic5—52no—
95C22×Dic5C2×D511yescharacteristic
96C2×C4×D5—52no—
97C2×C4×D5—52no—
98C2×Dic10—51no—
99C2×Dic10—51no—
100C2×Dic10—52no—
101C2×Dic10—52no—
102C2×Dic10—102no—
103C2×Dic10C2×D511yescharacteristic, complemented
104C5×C4⋊C4—51no—
105C5×C4⋊C4—52no—
106C5×C22⋊C4—52no—
107C5×C22⋊C4—52no—
108C5×C22⋊C4—52no—
109C22⋊Dic5—52no—
110C22⋊Dic5—52no—
111C22⋊Dic5—52no—
112C22⋊Dic5—52no—
113C22⋊Dic5—104no—
114C22⋊Dic5C2×D512yes—
115#685—52no—
116#685—52no—
117#685C2×D512yes—
118#685C2×D512yes—
119C4⋊Dic5—51no—
120C4⋊Dic5—51no—
121C4⋊Dic5—52no—
122C4⋊Dic5—52no—
123C4⋊Dic5—102no—
124C4⋊Dic5C2×D511yescharacteristic, complemented
125#683—52no—
126#683—52no—
127#683—52no—
128#683—52no—
129#683—52no—
130#683—52no—
131#683—104no—
132#683C2×D512yescomplemented
133#543—251noSylow
134C2×C52—22no—
135C2×C52C22×Q811yescharacteristic
136C2×C52H(5,𝔽2)11yescharacteristic
137C2×C52H(5,𝔽2)11yescharacteristic
138C5×D5—24no—
139C23×C5—22no—
140C23×C5—52no—
141C23×C5—54no—
142C23×C5—201no—
143C23×C5—201no—
144C23×C5C22×D511yescharacteristic
145C23×C5C22×D511yescharacteristic
146C22×D5—41no—
147C22×D5—41no—
148C22×D5Dic1014yescomplemented
149C22×D5C22×D512yes—
150C5×Q8—201no—
151C2×C4×C5—22no—
152C2×C4×C5—22no—
153C2×C4×C5—41no—
154C2×C4×C5—41no—
155C2×C4×C5—44no—
156C2×C4×C5—52no—
157C2×C4×C5—52no—
158C2×C4×C5—52no—
159C2×C4×C5—201no—
160C2×C4×C5C22×D511yescharacteristic
161C2×C4×C5C22×D511yescharacteristic
162C2×C4×C5C22×D511yescharacteristic
163C2×C4×C5C22×D511yescharacteristic
164C2×Dic5—41no—
165C2×Dic5—52no—
166C2×Dic5—52no—
167C2×Dic5—52no—
168C2×Dic5—52no—
169C2×Dic5—104no—
170C2×Dic5—104no—
171C2×Dic5—202no—
172C2×Dic5—202no—
173C2×Dic5C22×D512yes—
174C2×Dic5C22×D512yes—
175C2×Dic5C22×D512yes—
176C4×D5—202no—
177C4×D5—202no—
178Dic10—41no—
179Dic10—202no—
180Dic10—202no—
181Dic10—201no—
182Dic10—201no—
183Dic10—204no—
184C22⋊Q8—51no—
185C22⋊Q8—252no—
186C22⋊Q8—252no—
187C22⋊Q8—252no—
188C22⋊Q8—252no—
189C22⋊Q8—252no—
190C22⋊Q8—251no—
191C2×C22⋊C4—51no—
192C2×C22⋊C4—252no—
193C52#54311yescharacteristic, complemented, Sylow
194C22×C5—21no—
195C22×C5—21no—
196C22×C5—21no—
197C22×C5—21no—
198C22×C5—24no—
199C22×C5—22no—
200C22×C5—42no—
201C22×C5—42no—
202C22×C5—54no—
203C22×C5—102no—
204C22×C5—102no—
205C22×C5—204no—
206C22×C5C2×Dic1012yes—
207C22×C5Q8×D512yes—
208C22×C5C23×D511yescharacteristic
209C22×C5C23×D511yescharacteristic
210C2×D5—22no—
211C2×D5—22no—
212C2×D5—44no—
213C2×D5C2×Dic1014yes—
214C4×C5—24no—
215C4×C5—21no—
216C4×C5—21no—
217C4×C5—21no—
218C4×C5—21no—
219C4×C5—42no—
220C4×C5—102no—
221C4×C5—102no—
222C4×C5—102no—
223Dic5—22no—
224Dic5—22no—
225Dic5—22no—
226Dic5—102no—
227Dic5—102no—
228Dic5—102no—
229Dic5—102no—
230Dic5—204no—
231Dic5—204no—
232C24—51no—
233C2×Q8—51no—
234C2×Q8—252no—
235C2×Q8—251no—
236C22×C4—51no—
237C22×C4—252no—
238C22×C4—252no—
239C22×C4D5211yescharacteristic
240C4⋊C4—52no—
241C4⋊C4—51no—
242C4⋊C4—252no—
243C4⋊C4—252no—
244C4⋊C4—252no—
245C4⋊C4—252no—
246C4⋊C4—251no—
247C22⋊C4—52no—
248C22⋊C4—52no—
249C22⋊C4—52no—
250C22⋊C4—252no—
251C22⋊C4—252no—
252C22⋊C4—252no—
253C2×C5—22no—
254C2×C5—22no—
255C2×C5—22no—
256C2×C5—22no—
257C2×C5—22no—
258C2×C5—44no—
259C2×C5—104no—
260C2×C5C22×Dic1011yescharacteristic
261C2×C5#414411yescharacteristic
262C2×C5#414411yescharacteristic
263C2×C5#414411yescharacteristic
264C2×C5C2×Q8×D511yescharacteristic
265C2×C5#414911yescharacteristic
266D5—24no—
267C23—52no—
268C23—54no—
269C23—201no—
270C23—201no—
271C23C2×D5211yescharacteristic
272Q8—201no—
273Q8—1001no—
274Q8—1002no—
275C2×C4—41no—
276C2×C4—52no—
277C2×C4—52no—
278C2×C4—52no—
279C2×C4—201no—
280C2×C4—252no—
281C2×C4—252no—
282C2×C4—1002no—
283C2×C4—1002no—
284C2×C4C2×D5211yescharacteristic
285C2×C4C2×D5211yescharacteristic
286C5—22no—
287C5#6581711yescharacteristic, complemented
288C5#6596811yescharacteristic, complemented
289C22—21no—
290C22—21no—
291C22—54no—
292C22—102no—
293C22—102no—
294C22—204no—
295C22D5×Dic1012yes—
296C22C22×D5211yescharacteristic
297C4—21no—
298C4—21no—
299C4—102no—
300C4—102no—
301C4—102no—
302C4—502no—
303C4—502no—
304C2—22no—
305C2—104no—
306C2C2×D5×Dic1011yescharacteristic
307C2#1171747211yescharacteristic
308C2#1171749511yescharacteristic
subgroupG/H
center, FrattiniC22C22×D52
commutator, socle, cosocleC22×C52C24
FittingC22×C4×C52C22
prime radicalC23×C5×D5C22
hypercenterC22×C4D52
hypocenterC52#543
LCSG ▷ C22×C52 ▷ C52
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. #11717244C2semidirect
3. C5×#4070C2semidirect
5. #11717082C2semidirect
6. #11717015C2semidirect
7. #11717014C2semidirect
8. #11717007C2semidirect
9. #11717006C2semidirect
12. C2×#6045C22semidirect
14. C2×C5×Dic10C22semidirect
15. C2×#6039C22semidirect
19. C4⋊#1039C22semidirect
20. #87806C22semidirect
24. C5×C4⋊Dic5C22semidirect
25. C5×#683C22semidirect
26. #87781C22semidirect
27. #87780C22semidirect
28. #87780C22semidirect
36. C22×C5×D5Q8semidirect
61. #4070D5semidirect
103. C2×Dic10C2×D5semidirect
124. C4⋊Dic5C2×D5semidirect
132. #683C2×D5semidirect
148. C22×D5Dic10semidirect
193. C52#543semidirect
287. C5#65817semidirect
288. C5#65968semidirect

Maximal subgroups

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

subgroupindex
#117172442
C2×C5×#6852
C5×#40702
D5×C22⋊Dic52
#117170822
#117170152
#117170142
#117170072
#117170062
#659685
#658175