Finite Group Databasebeta

← #13106115

Subgroups — #13106115

Order 1600.

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

normal subgroups only

This group has 632 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
1C2×#87808C211yesmaximal, characteristic, complemented
2C5×#4076C211yesmaximal, characteristic, complemented
3C5×#4070C211yesmaximal, characteristic, complemented
4#11717099C212yesmaximal, complemented
5#11717091C212yesmaximal, complemented
6#11717025C212yesmaximal, complemented
7#11717011C212yesmaximal, complemented
8#11717005C212yesmaximal, complemented
9#11716966C212yesmaximal, complemented
10C23×C5⋊D5C2211yescharacteristic
11C22×C4×C52C2211yescharacteristic
12C22×#1039C2211yescharacteristic
13C2×C5×C22⋊D5C2212yes—
14C2×C5×D20C2211yescharacteristic, complemented
15C2×C5×Dic10C2211yescharacteristic, complemented
16C2×#6038C2212yes—
17C2×#6038C2212yes—
18C2×D5×Dic5C2212yescomplemented
19#87808C2212yes—
20#87808C2212yes—
21C5×C22⋊Dic5C2212yes—
22C5×#685C2212yescomplemented
23C5×C4⋊Dic5C2211yescharacteristic, complemented
24C5×C4⋊Dic5C2211yescharacteristic, complemented
25C5×#683C2212yescomplemented
26#87780C2212yescomplemented
27#87779C2212yes—
28#87779C2212yes—
29#87778C2212yescomplemented
30Dic52C2212yescomplemented
31#65978—51nomaximal
32#65951—51nomaximal
33C23×C52C2311yescharacteristic
34C22×C5⋊D5—22no—
35C22×C5⋊D5—41no—
36C22×C5⋊D5—41no—
37C22×C5⋊D5C2312yes—
38C22×C5×D5C2312yes—
39C2×C4×C52—41no—
40C2×C4×C52C2311yescharacteristic
41C2×C4×C52C2311yescharacteristic
42C2×#1039—41no—
43C2×#1039C2312yes—
44C5×C22⋊D5—24no—
45C2×C5×Dic5C2312yes—
46C2×C5×Dic5C2312yes—
47C2×C5×Dic5C2312yes—
48C5×D20—41no—
49C5×Dic10—41no—
50#6038—42no—
51#6038—42no—
52D5×Dic5—42no—
53C5×C22⋊Q8—51no—
54C5×#121—51no—
55#4093—52no—
56#4085—52no—
57#4083—52no—
58D4×Dic5—52no—
59#4076D511yescharacteristic, complemented
60C2×#685—51no—
61C2×#685—51no—
62C2×#685—102no—
63#4070D511yescharacteristic, complemented
64#4044—52no—
65#4044—52no—
66#4041—52no—
67#4040—52no—
68#4032—52no—
69#4031—52no—
70#4028—52no—
71#4027—52no—
72C22×C52—21no—
73C22×C52—21no—
74C22×C52#4112yes—
75C22×C52C2411yescharacteristic
76C2×C5⋊D5—22no—
77C2×C5⋊D5—22no—
78C2×C5⋊D5—22no—
79C2×C5⋊D5—44no—
80C2×C5×D5—22no—
81C2×C5×D5—22no—
82C2×C5×D5—22no—
83C4×C52—21no—
84C4×C52—21no—
85#1039—22no—
86C5×Dic5—22no—
87C5×Dic5—22no—
88C5×Dic5—22no—
89C23×D5—51no—
90C23×D5—51no—
91C23×D5—102no—
92C2×C5×Q8—51no—
93C2×C5×D4—52no—
94C2×C5×D4—51no—
95C22×C4×C5—22no—
96C22×C4×C5C2×D511yescharacteristic
97C22×C4×C5C2×D511yescharacteristic
98C2×C22⋊D5—52no—
99C2×C22⋊D5—52no—
100C2×C22⋊D5C2×D512yes—
101C22×Dic5—51no—
102C22×Dic5—51no—
103C22×Dic5—52no—
104C22×Dic5—102no—
105C2×D20—52no—
106C2×D20—52no—
107C2×D20C2×D511yescharacteristic, complemented
108C2×C4×D5—52no—
109C2×Dic10C2×D511yescharacteristic, complemented
110C5×C4⋊C4—51no—
111C5×C4⋊C4—51no—
112C5×C4⋊C4—52no—
113C5×C22⋊C4—52no—
114C5×C22⋊C4—52no—
115C22⋊Dic5—52no—
116C22⋊Dic5C2×D512yes—
117#685—52no—
118#685—52no—
119#685—52no—
120#685—52no—
121#685—52no—
122#685—52no—
123#685—52no—
124#685—52no—
125#685—52no—
126#685—104no—
127#685—104no—
128#685C2×D512yescomplemented
129C4⋊Dic5—52no—
130C4⋊Dic5C2×D511yescharacteristic, complemented
131C4⋊Dic5C2×D511yescharacteristic, complemented
132#683—52no—
133#683C2×D512yescomplemented
134C4×Dic5—52no—
135C4×Dic5—52no—
136#538—251noSylow
137C2×C52—22no—
138C2×C52C2×#4111yescharacteristic
139C2×C52H(5,𝔽2)11yescharacteristic
140C2×C52H(5,𝔽2)11yescharacteristic
141C5⋊D5—42no—
142C5×D5—42no—
143C23×C5—22no—
144C23×C5—52no—
145C23×C5C22×D511yescharacteristic
146C23×C5C22×D511yescharacteristic
147C22×D5—52no—
148C22×D5—52no—
149C22×D5—104no—
150C22×D5—108no—
151C22×D5—102no—
152C22×D5—102no—
153C22×D5—201no—
154C22×D5—201no—
155C22×D5—201no—
156C22×D5—201no—
157C22×D5—204no—
158C22×D5—204no—
159C22×D5C22×D512yes—
160C5×Q8—201no—
161C5×D4—104no—
162C5×D4—201no—
163C2×C4×C5—22no—
164C2×C4×C5—22no—
165C2×C4×C5—41no—
166C2×C4×C5—41no—
167C2×C4×C5—44no—
168C2×C4×C5—52no—
169C2×C4×C5—52no—
170C2×C4×C5—52no—
171C2×C4×C5C22×D511yescharacteristic
172C2×C4×C5C22×D511yescharacteristic
173C2×C4×C5C22×D511yescharacteristic
174C2×C4×C5C22×D511yescharacteristic
175C22⋊D5—24no—
176C22⋊D5—202no—
177C22⋊D5—202no—
178C2×Dic5—52no—
179C2×Dic5—52no—
180C2×Dic5—104no—
181C2×Dic5—202no—
182C2×Dic5—201no—
183C2×Dic5—201no—
184C2×Dic5—204no—
185C2×Dic5C22×D512yes—
186C2×Dic5C22×D512yes—
187C2×Dic5C22×D512yes—
188D20—41no—
189D20—202no—
190D20—202no—
191C4×D5—202no—
192Dic10—41no—
193#126—252no—
194#124—252no—
195#123—252no—
196C22⋊Q8—51no—
197#121—51no—
198#121—252no—
199#120—252no—
200C4×D4—252no—
201C2×C22⋊C4—251no—
202C52#53811yescharacteristic, complemented, Sylow
203C22×C5—24no—
204C22×C5—24no—
205C22×C5—21no—
206C22×C5—21no—
207C22×C5—21no—
208C22×C5—21no—
209C22×C5—24no—
210C22×C5—22no—
211C22×C5—102no—
212C22×C5—102no—
213C22×C5—102no—
214C22×C5#71112yes—
215C22×C5#71312yes—
216C22×C5C23×D511yescharacteristic
217C22×C5C23×D511yescharacteristic
218C2×D5—22no—
219C2×D5—22no—
220C2×D5—22no—
221C2×D5—108no—
222C2×D5—102no—
223C2×D5—102no—
224C2×D5—102no—
225C2×D5—102no—
226C2×D5—102no—
227C2×D5—102no—
228C2×D5—204no—
229C2×D5—204no—
230C2×D5—2016no—
231C2×D5—204no—
232C2×D5—204no—
233C4×C5—21no—
234C4×C5—21no—
235C4×C5—21no—
236C4×C5—21no—
237C4×C5—42no—
238C4×C5—42no—
239C4×C5—102no—
240C4×C5—102no—
241C4×C5—102no—
242Dic5—22no—
243Dic5—22no—
244Dic5—22no—
245Dic5—102no—
246Dic5—102no—
247Dic5—204no—
248C24—251no—
249C2×Q8—51no—
250C2×D4—52no—
251C2×D4—51no—
252C2×D4—252no—
253C2×D4—252no—
254C22×C4—252no—
255C22×C4—251no—
256C22×C4D5211yescharacteristic
257C4⋊C4—51no—
258C4⋊C4—52no—
259C4⋊C4—51no—
260C4⋊C4—252no—
261C22⋊C4—52no—
262C22⋊C4—52no—
263C22⋊C4—252no—
264C22⋊C4—252no—
265C22⋊C4—252no—
266C22⋊C4—252no—
267C22⋊C4—252no—
268C42—252no—
269C2×C5—28no—
270C2×C5—22no—
271C2×C5—22no—
272C2×C5—22no—
273C2×C5—22no—
274C2×C5—22no—
275C2×C5—202no—
276C2×C5C2×#71111yescharacteristic
277C2×C5#414411yescharacteristic
278C2×C5#414411yescharacteristic
279C2×C5C2×#71311yescharacteristic
280C2×C5#414911yescharacteristic
281C2×C5#414911yescharacteristic
282D5—42no—
283D5—208no—
284D5—202no—
285D5—202no—
286C23—52no—
287C23—252no—
288C23—502no—
289C23—1001no—
290C23—1001no—
291C23C2×D5211yescharacteristic
292Q8—201no—
293D4—104no—
294D4—201no—
295D4—1002no—
296D4—1002no—
297C2×C4—41no—
298C2×C4—52no—
299C2×C4—52no—
300C2×C4—52no—
301C2×C4—252no—
302C2×C4—1002no—
303C2×C4—1001no—
304C2×C4C2×D5211yescharacteristic
305C2×C4C2×D5211yescharacteristic
306C5—22no—
307C5#6595111yescharacteristic, complemented
308C5#6597811yescharacteristic, complemented
309C22—21no—
310C22—21no—
311C22—102no—
312C22—102no—
313C22—102no—
314C22—502no—
315C22—502no—
316C22—502no—
317C22—1004no—
318C22#8787112yes—
319C22C22×D5211yescharacteristic
320C4—21no—
321C4—21no—
322C4—102no—
323C4—102no—
324C4—102no—
325C4—502no—
326C2—22no—
327C2—202no—
328C2—1002no—
329C2C2×#8787111yescharacteristic
330C2#1171747311yescharacteristic
331C2#1171749511yescharacteristic
subgroupG/H
center, FrattiniC22C22×D52
commutator, socle, cosocleC22×C52C24
FittingC22×C4×C52C22
hypercenterC22×C4D52
hypocenterC52#538
LCSG ▷ C22×C52 ▷ C52
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C2×#87808C2semidirect
2. C5×#4076C2semidirect
3. C5×#4070C2semidirect
4. #11717099C2semidirect
5. #11717091C2semidirect
6. #11717025C2semidirect
7. #11717011C2semidirect
8. #11717005C2semidirect
9. #11716966C2semidirect
14. C2×C5×D20C22semidirect
15. C2×C5×Dic10C22semidirect
18. C2×D5×Dic5C22semidirect
22. C5×#685C22semidirect
23. C5×C4⋊Dic5C22semidirect
24. C5×C4⋊Dic5C22semidirect
25. C5×#683C22semidirect
26. #87780C22semidirect
29. #87778C22semidirect
30. Dic52C22semidirect
59. #4076D5semidirect
63. #4070D5semidirect
107. C2×D20C2×D5semidirect
109. C2×Dic10C2×D5semidirect
128. #685C2×D5semidirect
130. C4⋊Dic5C2×D5semidirect
131. C4⋊Dic5C2×D5semidirect
133. #683C2×D5semidirect
202. C52#538semidirect
307. C5#65951semidirect
308. C5#65978semidirect

Maximal subgroups

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

subgroupindex
C2×#878082
C5×#40762
C5×#40702
#117170992
#117170912
#117170252
#117170112
#117170052
#117169662
#659785
#659515