Finite Group Databasebeta

← #13105066

Subgroups — #13105066

Order 1600.

508 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 508 conjugacy classes of subgroups — too many to draw. Only its 128 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#11717192C211yesmaximal, characteristic, complemented
2C4×C5×D20C211yesmaximal, characteristic, complemented
3C4×C5×Dic10C211yesmaximal, characteristic, complemented
4#11717037C211yesmaximal, characteristic, complemented
5#11717034C211yesmaximal, characteristic
6#11716990C211yesmaximal, characteristic, complemented
7#11716981C211yesmaximal, characteristic, complemented
8#11716978C212yesmaximal, complemented
9#11716970C212yesmaximal, complemented
10#11716969C212yesmaximal, complemented
11#11716968C212yesmaximal, complemented
12C2×C4×C5⋊D5C2211yescharacteristic
13C2×C5×D20C2211yescharacteristic
14C2×C4×C5×D5C2212yes—
15C2×C5×Dic10C2211yescharacteristic
16C2×#6039C2212yes—
17C2×#6038C2212yes—
18C42×C52C2211yescharacteristic
19#87808C2211yescharacteristic
20#87808C2211yescharacteristic
21#87806C2211yescharacteristic
22#87806C2211yescharacteristic
23C4×#1039C2211yescharacteristic
24C5×#685C2212yes—
25C5×C4⋊Dic5C2211yescharacteristic
26C5×C4⋊Dic5C2211yescharacteristic
27C5×#683C2212yes—
28C4×C5×Dic5C2212yes—
29#87781C2212yes—
30#87780C2212yes—
31#87779C2212yes—
32#87778C2212yes—
33#87778C2212yes—
34#87778C2212yes—
35#65916—51nomaximal
36#65895—51nomaximal
37C22×C5⋊D5C2311yescharacteristic
38C22×C5×D5C2312yes—
39C2×C4×C52C2311yescharacteristic
40C2×C4×C52C2311yescharacteristic
41C2×C4×C52C2311yescharacteristic
42C2×#1039C2311yescharacteristic
43C2×#1039C2311yescharacteristic
44C2×#1039C2311yescharacteristic
45C4×C5⋊D5—22no—
46C2×C5×Dic5C2312yes—
47C2×C5×Dic5C2312yes—
48C2×C5×Dic5C2312yes—
49C5×D20—22no—
50C4×C5×D5—42no—
51C5×Dic10—22no—
52#6039—42no—
53#6038—42no—
54C4×C5×Q8—51no—
55C4×C5×D4—51no—
56#4093—51no—
57#4092—51no—
58#4084—51no—
59#4082—51no—
60#4075—52no—
61#4070—52no—
62#4043—51no—
63#4042—52no—
64#4041—51no—
65#4040—52no—
66#4034—51no—
67#4034—51no—
68#4031—52no—
69#4025—52no—
70#4022—52no—
71#4021—52no—
72C4×D20D511yescharacteristic, complemented
73#4018—51no—
74#4018—51no—
75#4018—102no—
76C4×Dic10D511yescharacteristic, complemented
77C22×C52C2411yescharacteristic
78C2×C5⋊D5—21no—
79C2×C5⋊D5—21no—
80C2×C5⋊D5—21no—
81C2×C5×D5—22no—
82C2×C5×D5—22no—
83C2×C5×D5—22no—
84C4×C52—21no—
85C4×C52—21no—
86C4×C52#4112yes—
87#1039—21no—
88#1039—21no—
89#1039—21no—
90C5×Dic5—22no—
91C5×Dic5—22no—
92C5×Dic5—22no—
93C2×C5×Q8—51no—
94C2×C5×D4—51no—
95C22×C4×C5—52no—
96C2×C22⋊D5—52no—
97C2×D20—52no—
98C2×D20C2×D511yescharacteristic
99C2×C4×D5—51no—
100C2×C4×D5—51no—
101C2×C4×D5—102no—
102C2×C4×D5C2×D512yes—
103C2×Dic10—52no—
104C2×Dic10—52no—
105C2×Dic10C2×D511yescharacteristic
106C5×C4⋊C4—51no—
107C5×C4⋊C4—51no—
108C5×C4⋊C4—52no—
109C5×C22⋊C4—52no—
110C42×C5—22no—
111C42×C5—52no—
112C42×C5C2×D511yescharacteristic
113C42×C5C2×D511yescharacteristic
114C22⋊Dic5—52no—
115C22⋊Dic5—52no—
116C22⋊Dic5—52no—
117#685—52no—
118#685—52no—
119#685—52no—
120#685—51no—
121#685—51no—
122#685—51no—
123#685—51no—
124#685—52no—
125#685—52no—
126#685—102no—
127#685—102no—
128#685C2×D512yes—
129C4⋊Dic5—52no—
130C4⋊Dic5C2×D511yescharacteristic
131C4⋊Dic5C2×D511yescharacteristic
132#683—51no—
133#683—51no—
134#683—51no—
135#683—51no—
136#683—52no—
137#683—52no—
138#683—52no—
139#683—102no—
140#683—102no—
141#683C2×D512yes—
142C4×Dic5—51no—
143C4×Dic5—51no—
144C4×Dic5—102no—
145C4×Dic5C2×D512yes—
146#542—251noSylow
147C2×C52C2×#4111yescharacteristic
148C2×C52H(5,𝔽2)11yescharacteristic
149C2×C522−1+411yescharacteristic
150C5⋊D5—41no—
151C5×D5—42no—
152C23×C5—52no—
153C22×D5—51no—
154C22×D5—51no—
155C22×D5—102no—
156C22×D5C22×D512yes—
157C5×Q8—102no—
158C5×D4—102no—
159C2×C4×C5—22no—
160C2×C4×C5—22no—
161C2×C4×C5—22no—
162C2×C4×C5—52no—
163C2×C4×C5—52no—
164C2×C4×C5—52no—
165C2×C4×C5—202no—
166C2×C4×C5C22×D511yescharacteristic
167C2×C4×C5C22×D511yescharacteristic
168C2×C4×C5C22×D511yescharacteristic
169C2×C4×C5C22×D511yescharacteristic
170C2×C4×C5C22×D511yescharacteristic
171C2×C4×C5C22×D511yescharacteristic
172C22⋊D5—202no—
173C2×Dic5—51no—
174C2×Dic5—51no—
175C2×Dic5—51no—
176C2×Dic5—51no—
177C2×Dic5—51no—
178C2×Dic5—51no—
179C2×Dic5—102no—
180C2×Dic5—102no—
181C2×Dic5—102no—
182C2×Dic5C22×D512yes—
183C2×Dic5C22×D512yes—
184C2×Dic5C22×D512yes—
185D20—22no—
186D20—202no—
187C4×D5—42no—
188C4×D5—108no—
189C4×D5—102no—
190C4×D5—102no—
191Dic10—22no—
192Dic10—202no—
193Dic10—202no—
194C4⋊Q8—251no—
195#126—252no—
196#124—252no—
197#124—251no—
198#123—252no—
199C22⋊Q8—252no—
200C22⋊Q8—251no—
201#121—251no—
202C4×Q8—51no—
203C4×D4—51no—
204#117—251no—
205C52#54211yescharacteristic, complemented, Sylow
206C22×C5—22no—
207C22×C5—102no—
208C22×C5—102no—
209C22×C5—102no—
210C22×C5C23×D511yescharacteristic
211C22×C5C23×D511yescharacteristic
212C2×D5—22no—
213C2×D5—22no—
214C2×D5—22no—
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—
224C4×C5—21no—
225C4×C5—21no—
226C4×C5—21no—
227C4×C5—21no—
228C4×C5—24no—
229C4×C5—42no—
230C4×C5—42no—
231C4×C5—102no—
232C4×C5—102no—
233C4×C5—102no—
234C4×C5#70912yes—
235C4×C5#70912yes—
236Dic5—22no—
237Dic5—22no—
238Dic5—22no—
239Dic5—104no—
240Dic5—101no—
241Dic5—101no—
242Dic5—101no—
243Dic5—101no—
244Dic5—101no—
245Dic5—101no—
246Dic5—202no—
247Dic5—202no—
248C2×Q8—51no—
249C2×Q8—252no—
250C2×D4—51no—
251C2×D4—252no—
252C22×C4—52no—
253C22×C4—251no—
254C4⋊C4—51no—
255C4⋊C4—51no—
256C4⋊C4—52no—
257C4⋊C4—252no—
258C4⋊C4—252no—
259C4⋊C4—251no—
260C4⋊C4—251no—
261C22⋊C4—52no—
262C22⋊C4—252no—
263C22⋊C4—252no—
264C22⋊C4—252no—
265C22⋊C4—252no—
266C22⋊C4—251no—
267C22⋊C4—251no—
268C42—52no—
269C42—251no—
270C42D5211yescharacteristic
271C2×C5—22no—
272C2×C5—22no—
273C2×C5—22no—
274C2×C5—202no—
275C2×C5C2×#70911yescharacteristic
276C2×C5C2×#70911yescharacteristic
277C2×C5#414411yescharacteristic
278C2×C5#414711yescharacteristic
279C2×C5#414911yescharacteristic
280C2×C5#415011yescharacteristic
281D5—42no—
282D5—204no—
283D5—201no—
284D5—201no—
285C23—52no—
286C23—251no—
287Q8—102no—
288Q8—1002no—
289D4—102no—
290D4—1002no—
291C2×C4—52no—
292C2×C4—52no—
293C2×C4—52no—
294C2×C4—202no—
295C2×C4—251no—
296C2×C4—251no—
297C2×C4—251no—
298C2×C4—502no—
299C2×C4C2×D5211yescharacteristic
300C2×C4C2×D5211yescharacteristic
301C2×C4C2×D5211yescharacteristic
302C5—22no—
303C5#6589511yescharacteristic, complemented
304C5#6591611yescharacteristic, complemented
305C22—102no—
306C22—102no—
307C22—102no—
308C22—501no—
309C22—501no—
310C22—501no—
311C22C22×D5211yescharacteristic
312C4—21no—
313C4—21no—
314C4—102no—
315C4—102no—
316C4—102no—
317C4—501no—
318C4—501no—
319C4—501no—
320C4#8787312yes—
321C2—202no—
322C2—1001no—
323C2#1171746211yescharacteristic
324C2C2×#8787311yescharacteristic
325C2#1171747211yescharacteristic
subgroupG/H
center, FrattiniC22C22×D52
commutator, socle, cosocleC22×C52C24
FittingC42×C52C22
prime radical#11717034C2
hypercenterC42D52
hypocenterC52#542
LCSG ▷ C22×C52 ▷ C52
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. #11717192C2semidirect
2. C4×C5×D20C2semidirect
3. C4×C5×Dic10C2semidirect
4. #11717037C2semidirect
6. #11716990C2semidirect
7. #11716981C2semidirect
8. #11716978C2semidirect
9. #11716970C2semidirect
10. #11716969C2semidirect
11. #11716968C2semidirect
72. C4×D20D5semidirect
76. C4×Dic10D5semidirect
205. C52#542semidirect
303. C5#65895semidirect
304. C5#65916semidirect

Maximal subgroups

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

subgroupindex
#117171922
C4×C5×D202
C4×C5×Dic102
#117170372
#117170342
#117169902
#117169812
#117169782
#117169702
#117169692
#117169682
#659165
#658955