Finite Group Databasebeta

← #13105014

Subgroups — #13105014

Order 1600.

522 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 522 conjugacy classes of subgroups — too many to draw. Only its 130 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
1C4×C4⋊(C5⋊D5)C211yesmaximal, characteristic
2C5×#4016C211yesmaximal, characteristic, complemented
3C4×C5×Dic10C211yesmaximal, characteristic, complemented
4#11717018C211yesmaximal, characteristic, complemented
5#11717016C212yesmaximal
6#11716990C211yesmaximal, characteristic, complemented
7#11716990C211yesmaximal, characteristic, complemented
8#11716987C212yesmaximal, complemented
9#11716967C211yesmaximal, characteristic, complemented
10#11716963C212yesmaximal, complemented
11#11716954C212yesmaximal, complemented
12C2×C4⋊(C5⋊D5)C2211yescharacteristic
13C2×C4×C5⋊D5C2211yescharacteristic
14C2×C4×C5⋊D5C2211yescharacteristic
15C2×C5×Dic10C2211yescharacteristic
16C2×#6036C2212yes—
17C42×C52C2211yescharacteristic
18#87808C2211yescharacteristic
19#87808C2211yescharacteristic
20C4⋊#1039C2211yescharacteristic
21C5×C4⋊Dic5C2211yescharacteristic
22C5×C4⋊Dic5C2211yescharacteristic
23C5×C4⋊Dic5C2211yescharacteristic
24C5×#683C2212yes—
25C5×#683C2212yes—
26C5×#683C2212yes—
27C4×C5×Dic5C2212yes—
28#87781C2212yes—
29#87780C2212yes—
30#87780C2212yes—
31#87779C2212yes—
32#87779C2212yes—
33#87779C2212yes—
34Dic52C2212yes—
35#66028—51nomaximal
36#65915—51nomaximal
37C22×C5⋊D5C2311yescharacteristic
38C22×C5⋊D5C2311yescharacteristic
39C2×C4×C52C2311yescharacteristic
40C2×C4×C52C2311yescharacteristic
41C2×C4×C52C2311yescharacteristic
42C2×#1039C2311yescharacteristic
43C2×#1039C2311yescharacteristic
44C4⋊(C5⋊D5)—22no—
45C4×C5⋊D5—22no—
46C4×C5⋊D5—22no—
47C2×C5×Dic5C2312yes—
48C2×C5×Dic5C2312yes—
49C2×C5×Dic5C2312yes—
50C2×C5×Dic5C2312yes—
51C5×Dic10—22no—
52#6036—24no—
53C5×#125—51no—
54C4×C5×Q8—51no—
55#4092—51no—
56#4092—51no—
57#4044—52no—
58#4043—51no—
59#4043—51no—
60#4040—52no—
61#4040—52no—
62#4038—51no—
63#4038—51no—
64#4038—52no—
65D5×C4⋊C4—51no—
66D5×C4⋊C4—51no—
67#4036—52no—
68#4035—52no—
69#4022—52no—
70C4×D20—51no—
71C4×D20—51no—
72C4×D20—102no—
73#4018—52no—
74#4016D511yescharacteristic, complemented
75C4×Dic10D511yescharacteristic, complemented
76C22×C52C2411yescharacteristic
77C2×C5⋊D5—21no—
78C2×C5⋊D5—21no—
79C2×C5⋊D5—21no—
80C2×C5⋊D5—21no—
81C2×C5⋊D5—21no—
82C2×C5⋊D5#4112yes—
83C4×C52—21no—
84C4×C52—21no—
85C4×C52#4112yes—
86#1039—21no—
87#1039—21no—
88C5×Dic5—22no—
89C5×Dic5—22no—
90C5×Dic5—22no—
91C5×Dic5—22no—
92C2×C5×Q8—51no—
93C2×D20—51no—
94C2×D20—51no—
95C2×D20—102no—
96C2×C4×D5—51no—
97C2×C4×D5—51no—
98C2×C4×D5—51no—
99C2×C4×D5—51no—
100C2×C4×D5—52no—
101C2×C4×D5—52no—
102C2×C4×D5—102no—
103C2×C4×D5—102no—
104C2×Dic10C2×D511yescharacteristic
105C5×C4⋊C4—52no—
106C5×C4⋊C4—51no—
107C5×C4⋊C4—52no—
108C5×C4⋊C4—51no—
109C5×C4⋊C4—51no—
110C5×C4⋊C4—52no—
111C42×C5—22no—
112C42×C5—52no—
113C42×C5C2×D511yescharacteristic
114C42×C5C2×D511yescharacteristic
115#685—51no—
116#685—51no—
117#685—51no—
118#685—51no—
119#685—52no—
120#685—52no—
121#685—52no—
122#685—52no—
123#685—52no—
124#685—52no—
125#685—102no—
126#685—102no—
127C4⋊Dic5—51no—
128C4⋊Dic5—51no—
129C4⋊Dic5—52no—
130C4⋊Dic5—52no—
131C4⋊Dic5—102no—
132C4⋊Dic5C2×D511yescharacteristic
133C4⋊Dic5C2×D511yescharacteristic
134C4⋊Dic5C2×D511yescharacteristic
135#683—52no—
136#683—52no—
137#683—52no—
138#683—52no—
139#683C2×D512yes—
140#683C2×D512yes—
141#683C2×D512yes—
142C4×Dic5—52no—
143C4×Dic5—52no—
144C4×Dic5C2×D512yes—
145#552—251noSylow
146C2×C52C2×#4111yescharacteristic
147C2×C52C2×#4111yescharacteristic
148C2×C522−1+411yescharacteristic
149C5⋊D5—22no—
150C5⋊D5—41no—
151C22×D5—51no—
152C22×D5—51no—
153C22×D5—51no—
154C22×D5—51no—
155C22×D5—102no—
156C22×D5—102no—
157C5×Q8—102no—
158C2×C4×C5—22no—
159C2×C4×C5—22no—
160C2×C4×C5—22no—
161C2×C4×C5—52no—
162C2×C4×C5—52no—
163C2×C4×C5—52no—
164C2×C4×C5—52no—
165C2×C4×C5C22×D511yescharacteristic
166C2×C4×C5C22×D511yescharacteristic
167C2×C4×C5C22×D511yescharacteristic
168C2×C4×C5C22×D511yescharacteristic
169C2×C4×C5C22×D511yescharacteristic
170C2×C4×C5C22×D511yescharacteristic
171C2×Dic5—51no—
172C2×Dic5—51no—
173C2×Dic5—51no—
174C2×Dic5—51no—
175C2×Dic5—102no—
176C2×Dic5—102no—
177C2×Dic5C22×D512yes—
178C2×Dic5C22×D512yes—
179C2×Dic5C22×D512yes—
180C2×Dic5C22×D512yes—
181D20—102no—
182D20—102no—
183D20—108no—
184C4×D5—104no—
185C4×D5—104no—
186C4×D5—102no—
187C4×D5—102no—
188C4×D5—102no—
189C4×D5—102no—
190C4×D5—204no—
191C4×D5—204no—
192Dic10—22no—
193#126—252no—
194#125—51no—
195#125—252no—
196#123—252no—
197C22⋊Q8—251no—
198C22⋊Q8—251no—
199C4×Q8—51no—
200C4×D4—251no—
201#117—252no—
202#117—251no—
203C2×C4⋊C4—251no—
204C52#55211yescharacteristic, complemented, Sylow
205C22×C5—22no—
206C22×C5C23×D511yescharacteristic
207C22×C5C23×D511yescharacteristic
208C2×D5—52no—
209C2×D5—52no—
210C2×D5—104no—
211C2×D5—101no—
212C2×D5—101no—
213C2×D5—101no—
214C2×D5—101no—
215C2×D5—104no—
216C2×D5—101no—
217C2×D5—101no—
218C2×D5—101no—
219C2×D5—101no—
220C2×D5—101no—
221C2×D5—101no—
222C2×D5—202no—
223C2×D5—202no—
224C2×D5—202no—
225C2×D5—202no—
226C4×C5—24no—
227C4×C5—21no—
228C4×C5—21no—
229C4×C5—21no—
230C4×C5—21no—
231C4×C5—24no—
232C4×C5—42no—
233C4×C5—102no—
234C4×C5—102no—
235C4×C5—102no—
236C4×C5—102no—
237C4×C5#70912yes—
238C4×C5#71312yes—
239Dic5—22no—
240Dic5—22no—
241Dic5—22no—
242Dic5—22no—
243Dic5—101no—
244Dic5—101no—
245Dic5—101no—
246Dic5—101no—
247Dic5—202no—
248Dic5—202no—
249C2×Q8—51no—
250C2×D4—251no—
251C22×C4—252no—
252C22×C4—251no—
253C22×C4—251no—
254C4⋊C4—51no—
255C4⋊C4—51no—
256C4⋊C4—51no—
257C4⋊C4—52no—
258C4⋊C4—52no—
259C4⋊C4—52no—
260C4⋊C4—252no—
261C4⋊C4—252no—
262C4⋊C4—252no—
263C4⋊C4—251no—
264C22⋊C4—252no—
265C22⋊C4—252no—
266C22⋊C4—252no—
267C22⋊C4—251no—
268C22⋊C4—251no—
269C42—52no—
270C42—252no—
271C42D5211yescharacteristic
272C2×C5—22no—
273C2×C5—22no—
274C2×C5—22no—
275C2×C5C2×#70911yescharacteristic
276C2×C5C2×#71311yescharacteristic
277C2×C5#414711yescharacteristic
278C2×C5D5×#4111yescharacteristic
279C2×C5D5×#4111yescharacteristic
280C2×C5#415011yescharacteristic
281D5—102no—
282D5—102no—
283D5—204no—
284D5—204no—
285D5—201no—
286D5—201no—
287C23—251no—
288C23—251no—
289Q8—102no—
290D4—502no—
291C2×C4—52no—
292C2×C4—52no—
293C2×C4—52no—
294C2×C4—52no—
295C2×C4—251no—
296C2×C4—251no—
297C2×C4—504no—
298C2×C4—502no—
299C2×C4—502no—
300C2×C4C2×D5211yescharacteristic
301C2×C4C2×D5211yescharacteristic
302C2×C4C2×D5211yescharacteristic
303C5—22no—
304C5#6591511yescharacteristic, complemented
305C5#6602811yescharacteristic, complemented
306C22—252no—
307C22—501no—
308C22—501no—
309C22—501no—
310C22—501no—
311C22—501no—
312C22C22×D5211yescharacteristic
313C4—21no—
314C4—21no—
315C4—102no—
316C4—102no—
317C4—102no—
318C4—102no—
319C4—501no—
320C4—501no—
321C4#8787412yes—
322C2—502no—
323C2—1001no—
324C2#1171746211yescharacteristic
325C2C2×#8787411yescharacteristic
326C2#1171747111yescharacteristic
subgroupG/H
center, FrattiniC22C22×D52
commutator, socle, cosocleC22×C52C24
FittingC42×C52C22
prime radicalC2×C4⋊(C5⋊D5)C22
hypercenterC42D52
hypocenterC52#552
LCSG ▷ C22×C52 ▷ C52
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
2. C5×#4016C2semidirect
3. C4×C5×Dic10C2semidirect
4. #11717018C2semidirect
6. #11716990C2semidirect
7. #11716990C2semidirect
8. #11716987C2semidirect
9. #11716967C2semidirect
10. #11716963C2semidirect
11. #11716954C2semidirect
74. #4016D5semidirect
75. C4×Dic10D5semidirect
204. C52#552semidirect
304. C5#65915semidirect
305. C5#66028semidirect

Maximal subgroups

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

subgroupindex
C4×C4⋊(C5⋊D5)2
C5×#40162
C4×C5×Dic102
#117170182
#117170162
#117169902
#117169872
#117169672
#117169632
#117169542
#660285
#659155