Finite Group Databasebeta

← #13106394

Subgroups — #13106394

Order 1600.

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

normal subgroups only

This group has 776 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
1C2×C5×#711C211yesmaximal, characteristic, complemented
2C2×D5×D20C211yesmaximal, characteristic, complemented
3#11717260C211yesmaximal, characteristic, complemented
4C5×#4086C211yesmaximal, characteristic, complemented
5#11717101C214yesmaximal, complemented
6#11717096C214yesmaximal, complemented
7#11716982C211yesmaximal, characteristic, complemented
8#11716981C211yesmaximal, characteristic, complemented
9#11716973C211yesmaximal, characteristic, complemented
10C22×D52C2212yes—
11C2×C52×D4C2211yescharacteristic, complemented
12C2×C22⋊(C5⋊D5)C2214yescomplemented
13C2×C4⋊(C5⋊D5)C2211yescharacteristic
14C2×C5×C22⋊D5C2212yes—
15C2×C5×C22⋊D5C2214yescomplemented
16C22×C5×Dic5C2212yescomplemented
17C5×#711—24no—
18C2×C5×D20C2211yescharacteristic
19C2×C4×C5×D5C2211yescharacteristic
20C2×C5×Dic10C2211yescharacteristic, complemented
21C2×#6038C2212yes—
22D5×D20—24no—
23C4×#1039C2211yescharacteristic, complemented
24C4×C5×Dic5C2211yescharacteristic, complemented
25#87780C2212yescomplemented
26#87779C2214yescomplemented
27#87778C2212yes—
28#87778C2214yescomplemented
29#66164—51nomaximal
30#66049—51nomaximal
31C23×C52C2312yes—
32C22×C5⋊D5C2312yes—
33C22×C5×D5C2312yes—
34C22×C5×D5C2311yescharacteristic
35C2×D52—24no—
36C2×D52—44no—
37C52×D4—22no—
38C2×C4×C52C2311yescharacteristic
39C22⋊(C5⋊D5)—44no—
40C2×#1039C2312yes—
41C4⋊(C5⋊D5)—22no—
42C5×C22⋊D5—24no—
43C5×C22⋊D5—44no—
44C2×C5×Dic5—24no—
45C2×C5×Dic5C2312yes—
46C2×C5×Dic5C2312yes—
47C2×C5×Dic5C2311yescharacteristic
48C5×D20—22no—
49C4×C5×D5D414yescomplemented
50C5×Dic10—22no—
51#6038—24no—
52C2×C5×#41—51no—
53C2×#711D511yescharacteristic, complemented
54C2×D4×D5—51no—
55C22×D20—51no—
56C5×#127—51no—
57#4093—51no—
58#4093—51no—
59#4086—51no—
60#4086—51no—
61#4086—102no—
62#4086D511yescharacteristic, complemented
63#4085—54no—
64#4083—54no—
65#4083—54no—
66#4082—51no—
67#4076—54no—
68#4072—51no—
69#4021—51no—
70#4018—51no—
71C22×C52—22no—
72C22×C52—24no—
73C22×C52C2411yescharacteristic
74C2×C5⋊D5—22no—
75C2×C5⋊D5—24no—
76C2×C5×D5—22no—
77C2×C5×D5—22no—
78C2×C5×D5—24no—
79C2×C5×D5C2×D412yes—
80D52—48no—
81C4×C52C2×D412yes—
82#1039—22no—
83C5×Dic5—22no—
84C5×Dic5—22no—
85C5×Dic5C2×D412yes—
86C23×D5—52no—
87C23×D5—52no—
88C5×#41—104no—
89C2×C5×Q8—51no—
90C2×C5×D4—22no—
91C2×C5×D4—52no—
92C2×C5×D4—54no—
93C2×C5×D4—51no—
94C2×C5×D4C2×D511yescharacteristic, complemented
95C2×C5×D4C2×D511yescharacteristic, complemented
96C22×C4×C5—51no—
97C22×C4×C5—52no—
98C2×C22⋊D5—54no—
99C2×C22⋊D5—54no—
100C2×C22⋊D5—52no—
101C2×C22⋊D5—108no—
102C2×C22⋊D5C2×D512yes—
103C2×C22⋊D5C2×D514yescomplemented
104C22×Dic5C2×D512yescomplemented
105#711—24no—
106D4×D5—104no—
107C2×D20—52no—
108C2×D20—51no—
109C2×D20—51no—
110C2×D20—102no—
111C2×D20—104no—
112C2×D20C2×D511yescharacteristic
113C2×C4×D5C2×D511yescharacteristic
114C2×Dic10C2×D511yescharacteristic, complemented
115C42×C5—51no—
116C22⋊Dic5—52no—
117C22⋊Dic5—54no—
118#685—52no—
119#685—54no—
120#685—54no—
121#685—54no—
122C4⋊Dic5—52no—
123#683—52no—
124C4×Dic5—51no—
125C4×Dic5—51no—
126C4×Dic5—102no—
127C4×Dic5C2×D511yescharacteristic, complemented
128#535—251noSylow
129C2×C52—42no—
130C2×C52C22×D411yescharacteristic
131C2×C52H(5,𝔽2)12yes—
132C5⋊D5—42no—
133C5×D5—22no—
134C5×D5—42no—
135C23×C5—24no—
136C23×C5—51no—
137C23×C5—52no—
138C23×C5C22×D512yes—
139C23×C5C22×D512yes—
140C22×D5—52no—
141C22×D5—52no—
142C22×D5—104no—
143C22×D5—104no—
144C22×D5—104no—
145C22×D5—204no—
146C22×D5—204no—
147C22×D5C22×D512yes—
148C22×D5C22×D511yescharacteristic
149C5×Q8—102no—
150C5×D4—22no—
151C5×D4—22no—
152C5×D4—44no—
153C5×D4—104no—
154C5×D4—102no—
155C5×D4—204no—
156C2×C4×C5—22no—
157C2×C4×C5—51no—
158C2×C4×C5—54no—
159C2×C4×C5—52no—
160C2×C4×C5—52no—
161C2×C4×C5—104no—
162C2×C4×C5C22×D511yescharacteristic
163C2×C4×C5C22×D511yescharacteristic
164C22⋊D5—24no—
165C22⋊D5—44no—
166C22⋊D5—104no—
167C22⋊D5—204no—
168C22⋊D5—204no—
169C22⋊D5—2016no—
170C2×Dic5—24no—
171C2×Dic5—52no—
172C2×Dic5—52no—
173C2×Dic5—104no—
174C2×Dic5C22×D512yes—
175C2×Dic5C22×D512yes—
176C2×Dic5C22×D511yescharacteristic
177D20—22no—
178D20—104no—
179D20—102no—
180D20—102no—
181D20—204no—
182C4×D5C22⋊D514yescomplemented
183Dic10—22no—
184C2×#41—51no—
185C22×D4—251no—
186#127—51no—
187#127—251no—
188#124—251no—
189#124—251no—
190#121—254no—
191#120—254no—
192#117—251no—
193C52#53511yescharacteristic, complemented, Sylow
194C22×C5—22no—
195C22×C5—24no—
196C22×C5—24no—
197C22×C5—22no—
198C22×C5—22no—
199C22×C5—48no—
200C22×C5—44no—
201C22×C5—52no—
202C22×C5—102no—
203C22×C5—104no—
204C22×C5—102no—
205C22×C5C23×D511yescharacteristic
206C22×C5C23×D511yescharacteristic
207C2×D5—22no—
208C2×D5—22no—
209C2×D5—24no—
210C2×D5—102no—
211C2×D5—104no—
212C2×D5—104no—
213C2×D5—102no—
214C2×D5—204no—
215C2×D5—208no—
216C2×D5—208no—
217C2×D5—208no—
218C2×D5C2×C22⋊D512yes—
219C4×C5—24no—
220C4×C5—52no—
221C4×C5—102no—
222C4×C5—102no—
223C4×C5D4×D512yes—
224C4×C5C2×C22⋊D512yes—
225Dic5—22no—
226Dic5—22no—
227Dic5—108no—
228Dic5—102no—
229Dic5—102no—
230Dic5C2×C22⋊D512yes—
231C24—252no—
232#41—104no—
233C2×Q8—51no—
234C2×D4—52no—
235C2×D4—54no—
236C2×D4—51no—
237C2×D4—252no—
238C2×D4—254no—
239C2×D4—251no—
240C2×D4—504no—
241C2×D4D5211yescharacteristic, complemented
242C22×C4—51no—
243C22×C4—52no—
244C4⋊C4—252no—
245C22⋊C4—254no—
246C22⋊C4—252no—
247C22⋊C4—254no—
248C42—51no—
249C42—251no—
250C2×C5—24no—
251C2×C5—22no—
252C2×C5—42no—
253C2×C5—42no—
254C2×C5—84no—
255C2×C5—102no—
256C2×C5—202no—
257C2×C5C2×D4×D511yescharacteristic
258C2×C5#414412yes—
259C2×C5#414912yes—
260C2×C5C22×C22⋊D511yescharacteristic
261D5—22no—
262D5—42no—
263D5—202no—
264D5—202no—
265D5—404no—
266C23—51no—
267C23—52no—
268C23—252no—
269C23—504no—
270C23—1004no—
271C23C2×D5212yes—
272Q8—102no—
273D4—22no—
274D4—104no—
275D4—102no—
276D4—204no—
277D4—504no—
278D4—502no—
279D4—1004no—
280C2×C4—51no—
281C2×C4—54no—
282C2×C4—52no—
283C2×C4—52no—
284C2×C4—104no—
285C2×C4—252no—
286C2×C4C2×D5211yescharacteristic
287C5—22no—
288C5#6604911yescharacteristic, complemented
289C5#6616411yescharacteristic, complemented
290C22—24no—
291C22—22no—
292C22—52no—
293C22—102no—
294C22—104no—
295C22—102no—
296C22—504no—
297C22—502no—
298C22—1008no—
299C22C22×D5211yescharacteristic
300C4—52no—
301C4—102no—
302C4—102no—
303C4—502no—
304C4D5×C22⋊D512yes—
305C2—42no—
306C2—102no—
307C2—202no—
308C2—1002no—
309C2#1171747912yes—
310C2C2×D5×C22⋊D511yescharacteristic
subgroupG/H
center, FrattiniC22C22×D52
commutator, socle, cosocleC22×C52C24
FittingC2×C52×D4C22
hypercenterC2×D4D52
hypocenterC52#535
LCSG ▷ C22×C52 ▷ C52
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×C5×#711C2semidirect
2. C2×D5×D20C2semidirect
3. #11717260C2semidirect
4. C5×#4086C2semidirect
5. #11717101C2semidirect
6. #11717096C2semidirect
7. #11716982C2semidirect
8. #11716981C2semidirect
9. #11716973C2semidirect
11. C2×C52×D4C22semidirect
12. C2×C22⋊(C5⋊D5)C22semidirect
15. C2×C5×C22⋊D5C22semidirect
16. C22×C5×Dic5C22semidirect
20. C2×C5×Dic10C22semidirect
23. C4×#1039C22semidirect
24. C4×C5×Dic5C22semidirect
25. #87780C22semidirect
26. #87779C22semidirect
28. #87778C22semidirect
49. C4×C5×D5D4semidirect
53. C2×#711D5semidirect
62. #4086D5semidirect
94. C2×C5×D4C2×D5semidirect
95. C2×C5×D4C2×D5semidirect
103. C2×C22⋊D5C2×D5semidirect
104. C22×Dic5C2×D5semidirect
114. C2×Dic10C2×D5semidirect
127. C4×Dic5C2×D5semidirect
182. C4×D5C22⋊D5semidirect
193. C52#535semidirect
241. C2×D4D52semidirect
288. C5#66049semidirect
289. C5#66164semidirect

Maximal subgroups

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

subgroupindex
C2×C5×#7112
C2×D5×D202
#117172602
C5×#40862
#117171012
#117170962
#117169822
#117169812
#117169732
#661645
#660495