Finite Group Databasebeta

← #13106488

Subgroups — #13106488

Order 1600.

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

normal subgroups only

This group has 502 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
1#11717266C211yesmaximal, characteristic, complemented
2C5×#4092C211yesmaximal, characteristic, complemented
3C5×Q8×Dic5C211yesmaximal, characteristic, complemented
4C4×#6038C211yesmaximal, characteristic
5#11717018C211yesmaximal, characteristic, complemented
6#11716988C211yesmaximal, characteristic, complemented
7#11716978C212yesmaximal, complemented
8#11716972C212yesmaximal
9#11716965C212yesmaximal, complemented
10#11716959C211yesmaximal, characteristic, complemented
11#11716954C212yesmaximal, complemented
12C2×C52×Q8C2211yescharacteristic
13C2×C4×C5⋊D5C2211yescharacteristic
14C2×C4×C5×D5C2211yescharacteristic
15C2×#6038C2211yescharacteristic
16C2×#6036C2212yes—
17#87808C2212yes—
18C4⋊#1039C2211yescharacteristic
19#87806C2212yes—
20C5×#685C2212yes—
21C5×C4⋊Dic5C2211yescharacteristic
22C5×C4⋊Dic5C2212yes—
23C5×C4⋊Dic5C2211yescharacteristic
24C5×#683C2212yes—
25C4×C5×Dic5C2212yes—
26C4×C5×Dic5C2211yescharacteristic
27#87781C2212yes—
28#87781C2212yes—
29#87780C2212yes—
30#87780C2211yescharacteristic
31#87779C2211yescharacteristic
32#87778C2212yes—
33#87778C2211yescharacteristic
34Dic52C2212yes—
35#65973—51nomaximal
36#65915—51nomaximal
37C22×C5⋊D5C2311yescharacteristic
38C22×C5×D5C2311yescharacteristic
39C52×Q8—22no—
40C2×C4×C52C2312yes—
41C2×C4×C52C2311yescharacteristic
42C2×#1039C2312yes—
43C2×#1039C2311yescharacteristic
44C4×C5⋊D5—22no—
45C2×C5×Dic5C2312yes—
46C2×C5×Dic5C2312yes—
47C2×C5×Dic5C2311yescharacteristic
48C2×C5×Dic5C2311yescharacteristic
49C2×C5×Dic5C2311yescharacteristic
50C4×C5×D5—22no—
51#6038—22no—
52#6036—24no—
53C5×C22⋊Q8—51no—
54C4×C5×Q8—51no—
55#4092—51no—
56#4092—51no—
57#4092—102no—
58#4092D511yescharacteristic, complemented
59Q8×Dic5D511yescharacteristic, complemented
60C4×C22⋊D5—51no—
61#4072—51no—
62#4040—52no—
63#4038—51no—
64#4038—52no—
65D5×C4⋊C4—51no—
66D5×C4⋊C4—51no—
67#4036—51no—
68#4035—52no—
69#4035—52no—
70#4029—52no—
71#4025—52no—
72#4022—52no—
73C4×D20—51no—
74#4018—52no—
75#4016—51no—
76C22×C52C2411yescharacteristic
77C2×C5⋊D5—21no—
78C2×C5⋊D5—21no—
79C2×C5⋊D5#4112yes—
80C2×C5×D5—21no—
81C2×C5×D5—21no—
82C2×C5×D5—21no—
83C4×C52—22no—
84C4×C52#4112yes—
85#1039—21no—
86#1039—22no—
87C5×Dic5—22no—
88C5×Dic5—22no—
89C5×Dic5—21no—
90C5×Dic5—21no—
91C5×Dic5—21no—
92C2×C5×Q8—22no—
93C2×C5×Q8C2×D511yescharacteristic
94C2×C5×Q8C2×D511yescharacteristic
95C22×C4×C5—51no—
96C2×C22⋊D5—51no—
97C2×D20—51no—
98C2×C4×D5—51no—
99C2×C4×D5—51no—
100C2×C4×D5—52no—
101C2×C4×D5—52no—
102C2×C4×D5—102no—
103C2×C4×D5C2×D511yescharacteristic
104C5×C4⋊C4—52no—
105C5×C4⋊C4—51no—
106C5×C4⋊C4—52no—
107C5×C4⋊C4—51no—
108C5×C22⋊C4—52no—
109C42×C5—51no—
110C42×C5—52no—
111C22⋊Dic5—51no—
112C22⋊Dic5—52no—
113#685—51no—
114#685—52no—
115#685—52no—
116#685—52no—
117#685—51no—
118#685—51no—
119#685—104no—
120#685C2×D512yes—
121C4⋊Dic5—51no—
122C4⋊Dic5—51no—
123C4⋊Dic5—51no—
124C4⋊Dic5—52no—
125C4⋊Dic5—102no—
126C4⋊Dic5C2×D511yescharacteristic
127C4⋊Dic5C2×D512yes—
128C4⋊Dic5C2×D511yescharacteristic
129#683—52no—
130#683—52no—
131#683—52no—
132#683—52no—
133#683—52no—
134#683—52no—
135#683—52no—
136#683—51no—
137#683—104no—
138#683C2×D512yes—
139C4×Dic5—52no—
140C4×Dic5—52no—
141C4×Dic5C2×D512yes—
142C4×Dic5C2×D511yescharacteristic
143#552—251noSylow
144C2×C52C2×#4111yescharacteristic
145C2×C52C2×#4111yescharacteristic
146C2×C522−1+411yescharacteristic
147C5⋊D5—22no—
148C5×D5—41no—
149C23×C5—51no—
150C22×D5—51no—
151C22×D5—51no—
152C22×D5—102no—
153C22×D5C22×D511yescharacteristic
154C5×Q8—22no—
155C5×Q8—22no—
156C5×Q8—44no—
157C2×C4×C5—22no—
158C2×C4×C5—24no—
159C2×C4×C5—51no—
160C2×C4×C5—52no—
161C2×C4×C5—51no—
162C2×C4×C5—52no—
163C2×C4×C5—51no—
164C2×C4×C5—102no—
165C2×C4×C5C22×D512yes—
166C2×C4×C5C22×D512yes—
167C2×C4×C5C22×D511yescharacteristic
168C2×C4×C5C22×D511yescharacteristic
169C22⋊D5—102no—
170C2×Dic5—52no—
171C2×Dic5—52no—
172C2×Dic5—51no—
173C2×Dic5—51no—
174C2×Dic5—102no—
175C2×Dic5—104no—
176C2×Dic5C22×D511yescharacteristic
177C2×Dic5C22×D512yes—
178C2×Dic5C22×D511yescharacteristic
179C2×Dic5C22×D511yescharacteristic
180C2×Dic5C22×D512yes—
181D20—102no—
182C4×D5—22no—
183C4×D5—104no—
184C4×D5—104no—
185C4×D5—102no—
186C4×D5—102no—
187C4×D5—204no—
188#126—252no—
189#125—251no—
190#125—252no—
191#123—252no—
192C22⋊Q8—51no—
193C22⋊Q8—251no—
194C4×Q8—51no—
195C4×D4—251no—
196#117—252no—
197#117—251no—
198C2×C4⋊C4—251no—
199C52#55211yescharacteristic, complemented, Sylow
200C22×C5—22no—
201C22×C5—101no—
202C22×C5—101no—
203C22×C5—101no—
204C22×C5C23×D511yescharacteristic
205C22×C5C23×D511yescharacteristic
206C2×D5—21no—
207C2×D5—21no—
208C2×D5—21no—
209C2×D5—52no—
210C2×D5—52no—
211C2×D5—104no—
212C2×D5—101no—
213C2×D5—101no—
214C2×D5—101no—
215C2×D5—101no—
216C2×D5—202no—
217C2×D5—202no—
218C4×C5—22no—
219C4×C5—22no—
220C4×C5—24no—
221C4×C5—44no—
222C4×C5—101no—
223C4×C5—101no—
224C4×C5—102no—
225C4×C5—102no—
226C4×C5—101no—
227C4×C5#70912yes—
228C4×C5#71112yes—
229Dic5—21no—
230Dic5—21no—
231Dic5—22no—
232Dic5—22no—
233Dic5—21no—
234Dic5—101no—
235Dic5—101no—
236Dic5—102no—
237Dic5—102no—
238Dic5—202no—
239Dic5—204no—
240C2×Q8D5211yescharacteristic
241C2×D4—251no—
242C22×C4—51no—
243C22×C4—252no—
244C22×C4—251no—
245C4⋊C4—51no—
246C4⋊C4—51no—
247C4⋊C4—52no—
248C4⋊C4—52no—
249C4⋊C4—252no—
250C4⋊C4—252no—
251C4⋊C4—251no—
252C4⋊C4—252no—
253C4⋊C4—251no—
254C4⋊C4—252no—
255C22⋊C4—52no—
256C22⋊C4—251no—
257C22⋊C4—252no—
258C22⋊C4—251no—
259C22⋊C4—252no—
260C42—52no—
261C42—51no—
262C42—252no—
263C2×C5—22no—
264C2×C5—22no—
265C2×C5—22no—
266C2×C5—201no—
267C2×C5C2×#70911yescharacteristic
268C2×C5C2×#71111yescharacteristic
269C2×C5#414711yescharacteristic
270C2×C5#414711yescharacteristic
271C2×C5D5×#4111yescharacteristic
272C2×C5D5×#4111yescharacteristic
273D5—41no—
274D5—102no—
275D5—102no—
276D5—204no—
277C23—51no—
278C23—251no—
279Q8—22no—
280D4—502no—
281C2×C4—51no—
282C2×C4—52no—
283C2×C4—51no—
284C2×C4—51no—
285C2×C4—52no—
286C2×C4—102no—
287C2×C4—252no—
288C2×C4—251no—
289C2×C4—504no—
290C2×C4—502no—
291C2×C4C2×D5212yes—
292C2×C4C2×D5211yescharacteristic
293C5—22no—
294C5#6591511yescharacteristic, complemented
295C5#6597311yescharacteristic, complemented
296C22—101no—
297C22—101no—
298C22—101no—
299C22—252no—
300C22—501no—
301C22—501no—
302C22C22×D5211yescharacteristic
303C4—22no—
304C4—101no—
305C4—102no—
306C4—102no—
307C4—101no—
308C4—101no—
309C4—502no—
310C4—501no—
311C4#8787912yes—
312C2—201no—
313C2—502no—
314C2#1171748111yescharacteristic
315C2#1171748411yescharacteristic
316C2C2×#8787911yescharacteristic
subgroupG/H
center, FrattiniC22C22×D52
commutator, socle, cosocleC22×C52C24
FittingC2×C52×Q8C22
prime radicalC2×#6038C22
hypercenterC2×Q8D52
hypocenterC52#552
LCSG ▷ C22×C52 ▷ C52
UCSC1 ◁ C22 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. #11717266C2semidirect
2. C5×#4092C2semidirect
3. C5×Q8×Dic5C2semidirect
5. #11717018C2semidirect
6. #11716988C2semidirect
7. #11716978C2semidirect
9. #11716965C2semidirect
10. #11716959C2semidirect
11. #11716954C2semidirect
58. #4092D5semidirect
59. Q8×Dic5D5semidirect
199. C52#552semidirect
294. C5#65915semidirect
295. C5#65973semidirect

Maximal subgroups

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

subgroupindex
#117172662
C5×#40922
C5×Q8×Dic52
C4×#60382
#117170182
#117169882
#117169782
#117169722
#117169652
#117169592
#117169542
#659735
#659155