Finite Group Databasebeta

← #13106593

Subgroups — #13106593

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×C22⋊#1039C211yesmaximal, characteristic, complemented
2C5×#4099C212yesmaximal, complemented
3#11717100C211yesmaximal, characteristic, complemented
4#11717098C211yesmaximal, characteristic, complemented
5#11717091C212yesmaximal, complemented
6#11717088C212yesmaximal, complemented
7#11717087C212yesmaximal, complemented
8#11717077C212yesmaximal, complemented
9#11717074C212yesmaximal, complemented
10C24×C52C2211yescharacteristic
11C22×#1039C2211yescharacteristic
12C22×#1039C2211yescharacteristic
13C2×C5×C22⋊D5C2212yes—
14C2×C5×C22⋊D5C2212yes—
15C2×C5×C22⋊D5C2212yes—
16C2×#6039C2211yescharacteristic, complemented
17C2×#6037C2211yescharacteristic, complemented
18C2×D5×Dic5C2212yescomplemented
19C22⋊#1039C2212yes—
20C22⋊#1039C2212yes—
21C5×C22⋊Dic5C2212yes—
22C5×C22⋊Dic5C2212yes—
23C5×C22⋊Dic5C2212yes—
24#87781C2211yescharacteristic, complemented
25#87781C2211yescharacteristic, complemented
26#87780C2212yescomplemented
27#87780C2212yescomplemented
28#87778C2212yescomplemented
29#87778C2212yescomplemented
30Dic52C2212yescomplemented
31#65927—52nomaximal
32C23×C52—22no—
33C23×C52—41no—
34C23×C52—41no—
35C23×C52C2312yes—
36C23×C52C2311yescharacteristic
37C22×C5×D5C2312yes—
38C2×#1039—41no—
39C2×#1039—41no—
40C2×#1039C2312yes—
41C2×#1039C2311yescharacteristic
42C2×#1039C2311yescharacteristic
43C5×C22⋊D5—24no—
44C5×C22⋊D5—42no—
45C5×C22⋊D5—42no—
46C2×C5×Dic5C2312yes—
47C2×C5×Dic5C2312yes—
48C2×C5×Dic5C2312yes—
49#6039—41no—
50#6037—41no—
51D5×Dic5—42no—
52C5×#120—52no—
53#4099D512yescomplemented
54C2×C22⋊Dic5—52no—
55C2×C22⋊Dic5—102no—
56#4085—52no—
57#4085—52no—
58#4084—52no—
59#4082—52no—
60#4081—52no—
61D4×Dic5—52no—
62#4031—52no—
63#4029—52no—
64#4027—52no—
65#4025—52no—
66#4025—52no—
67#4024—52no—
68C22×C52—21no—
69C22×C52—21no—
70C22×C52—22no—
71C22×C52—22no—
72C22×C52—22no—
73C22×C52—44no—
74C22×C52#4112yes—
75C22×C52C2411yescharacteristic
76C2×C5×D5—22no—
77C2×C5×D5—22no—
78C2×C5×D5—22no—
79#1039—22no—
80#1039—21no—
81#1039—21no—
82C5×Dic5—22no—
83C5×Dic5—22no—
84C5×Dic5—22no—
85C24×C5—22no—
86C24×C5C2×D512yes—
87C2×C5×D4—52no—
88C2×C5×D4—52no—
89C2×C5×D4—52no—
90C2×C22⋊D5—52no—
91C2×C22⋊D5C2×D512yes—
92C2×C22⋊D5C2×D512yes—
93C2×C22⋊D5C2×D512yes—
94C22×Dic5—52no—
95C22×Dic5—52no—
96C22×Dic5—52no—
97C22×Dic5—102no—
98C22×Dic5—102no—
99C2×C4×D5—52no—
100C2×Dic10—52no—
101C5×C22⋊C4—52no—
102C5×C22⋊C4—52no—
103C5×C22⋊C4—52no—
104C22⋊Dic5—52no—
105C22⋊Dic5—52no—
106C22⋊Dic5—52no—
107C22⋊Dic5—52no—
108C22⋊Dic5—52no—
109C22⋊Dic5—52no—
110C22⋊Dic5—104no—
111C22⋊Dic5—104no—
112C22⋊Dic5C2×D512yes—
113C22⋊Dic5C2×D512yes—
114C22⋊Dic5C2×D512yes—
115#685—52no—
116#685—52no—
117C4⋊Dic5—52no—
118C4⋊Dic5—52no—
119#683—52no—
120#683—52no—
121#683—52no—
122#683—52no—
123C4×Dic5—52no—
124C4×Dic5—52no—
125#538—251noSylow
126C2×C52—22no—
127C2×C52—42no—
128C2×C52C2×#4111yescharacteristic
129C2×C52H(5,𝔽2)11yescharacteristic
130C2×C52H(5,𝔽2)11yescharacteristic
131C5×D5—42no—
132C23×C5—28no—
133C23×C5—24no—
134C23×C5—22no—
135C23×C5—24no—
136C23×C5—42no—
137C23×C5—42no—
138C23×C5—44no—
139C23×C5—44no—
140C23×C5—52no—
141C23×C5C22×D512yes—
142C23×C5C22×D512yes—
143C23×C5C22×D512yes—
144C22×D5C22×D512yes—
145C5×D4—104no—
146C5×D4—202no—
147C5×D4—202no—
148C2×C4×C5—52no—
149C2×C4×C5—52no—
150C2×C4×C5—52no—
151C22⋊D5—24no—
152C22⋊D5—42no—
153C22⋊D5—42no—
154C22⋊D5—202no—
155C2×Dic5—52no—
156C2×Dic5—52no—
157C2×Dic5—52no—
158C2×Dic5—52no—
159C2×Dic5—104no—
160C2×Dic5—102no—
161C2×Dic5—102no—
162C2×Dic5—202no—
163C2×Dic5—202no—
164C2×Dic5—202no—
165C2×Dic5—204no—
166C2×Dic5—204no—
167C2×Dic5C22×D512yes—
168C2×Dic5C22×D512yes—
169C2×Dic5C22×D512yes—
170C4×D5—202no—
171Dic10—202no—
172#126—252no—
173#124—252no—
174#123—252no—
175C22⋊Q8—251no—
176#121—252no—
177#121—251no—
178#120—52no—
179C4×D4—252no—
180C2×C22⋊C4—251no—
181C52#53811yescharacteristic, complemented, Sylow
182C22×C5—24no—
183C22×C5—24no—
184C22×C5—22no—
185C22×C5—22no—
186C22×C5—28no—
187C22×C5—22no—
188C22×C5—22no—
189C22×C5—22no—
190C22×C5—22no—
191C22×C5—22no—
192C22×C5—22no—
193C22×C5—24no—
194C22×C5—22no—
195C22×C5—44no—
196C22×C5—44no—
197C22×C5—416no—
198C22×C5—48no—
199C22×C5—102no—
200C22×C5—102no—
201C22×C5—102no—
202C22×C5#71114yes—
203C22×C5C23×D512yes—
204C2×D5—22no—
205C2×D5—22no—
206C2×D5—22no—
207C4×C5—102no—
208C4×C5—102no—
209C4×C5—102no—
210Dic5—22no—
211Dic5—22no—
212Dic5—22no—
213Dic5—102no—
214Dic5—102no—
215Dic5—102no—
216Dic5—102no—
217Dic5—204no—
218Dic5—202no—
219Dic5—202no—
220C24D5211yescharacteristic
221C2×Q8—251no—
222C2×D4—52no—
223C2×D4—52no—
224C2×D4—52no—
225C2×D4—251no—
226C22×C4—252no—
227C22×C4—251no—
228C22×C4—251no—
229C4⋊C4—252no—
230C4⋊C4—252no—
231C4⋊C4—251no—
232C4⋊C4—251no—
233C22⋊C4—52no—
234C22⋊C4—52no—
235C22⋊C4—52no—
236C22⋊C4—252no—
237C22⋊C4—252no—
238C22⋊C4—252no—
239C22⋊C4—252no—
240C42—252no—
241C2×C5—28no—
242C2×C5—24no—
243C2×C5—22no—
244C2×C5—22no—
245C2×C5—22no—
246C2×C5—48no—
247C2×C5—42no—
248C2×C5—42no—
249C2×C5—202no—
250C2×C5C2×#71112yes—
251C2×C5#414412yes—
252C2×C5#414412yes—
253D5—42no—
254C23—22no—
255C23—41no—
256C23—41no—
257C23—52no—
258C23C2×D5212yes—
259C23C2×D5211yescharacteristic
260Q8—1001no—
261D4—104no—
262D4—202no—
263D4—202no—
264D4—1001no—
265C2×C4—52no—
266C2×C4—52no—
267C2×C4—52no—
268C2×C4—252no—
269C2×C4—251no—
270C2×C4—251no—
271C2×C4—1002no—
272C2×C4—1001no—
273C2×C4—1001no—
274C5—22no—
275C5#6592712yescomplemented
276C22—21no—
277C22—21no—
278C22—22no—
279C22—22no—
280C22—22no—
281C22—44no—
282C22—102no—
283C22—102no—
284C22—102no—
285C22#8788012yes—
286C22C22×D5211yescharacteristic
287C4—102no—
288C4—102no—
289C4—102no—
290C4—502no—
291C4—501no—
292C4—501no—
293C2—22no—
294C2—42no—
295C2—202no—
296C2C2×#8788011yescharacteristic
297C2#1171749511yescharacteristic
298C2#1171749511yescharacteristic
subgroupG/H
center, FrattiniC22C22×D52
commutator, socle, cosocleC22×C52C24
FittingC24×C52C22
hypercenterC24D52
hypocenterC52#538
LCSG ▷ C22×C52 ▷ C52
UCSC1 ◁ C22 ◁ C24

Complements

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

NH ≅ G/Ntype
1. C2×C22⋊#1039C2semidirect
2. C5×#4099C2semidirect
3. #11717100C2semidirect
4. #11717098C2semidirect
5. #11717091C2semidirect
6. #11717088C2semidirect
7. #11717087C2semidirect
8. #11717077C2semidirect
9. #11717074C2semidirect
16. C2×#6039C22semidirect
17. C2×#6037C22semidirect
18. C2×D5×Dic5C22semidirect
24. #87781C22semidirect
25. #87781C22semidirect
26. #87780C22semidirect
27. #87780C22semidirect
28. #87778C22semidirect
29. #87778C22semidirect
30. Dic52C22semidirect
53. #4099D5semidirect
181. C52#538semidirect
275. C5#65927semidirect

Maximal subgroups

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

subgroupindex
C2×C22⋊#10392
C5×#40992
#117171002
#117170982
#117170912
#117170882
#117170872
#117170772
#117170742
#659275