Finite Group Databasebeta

← #13106114

Subgroups — #13106114

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
2C4×C5×C22⋊D5C212yesmaximal, complemented
3#11717100C211yesmaximal, characteristic, complemented
4#11717099C212yesmaximal, complemented
5#11717098C211yesmaximal, characteristic, complemented
6#11717024C212yesmaximal, complemented
7#11716987C212yesmaximal, complemented
8#11716970C212yesmaximal, complemented
9#11716968C212yesmaximal, complemented
10C23×C5⋊D5C2211yescharacteristic
11C22×C4×C52C2211yescharacteristic
12C22×#1039C2211yescharacteristic
13C2×C5×C22⋊D5C2212yes—
14C2×C4×C5×D5C2212yescomplemented
15C2×#6039C2211yescharacteristic, complemented
16C2×#6038C2212yes—
17C2×#6038C2212yes—
18C2×#6037C2211yescharacteristic, complemented
19#87808C2212yes—
20#87808C2212yes—
21C5×C22⋊Dic5C2212yes—
22C5×#685C2212yescomplemented
23C5×#683C2212yescomplemented
24C4×C5×Dic5C2212yescomplemented
25#87781C2211yescharacteristic, complemented
26#87781C2211yescharacteristic, complemented
27#87780C2212yescomplemented
28#87779C2212yes—
29#87779C2212yes—
30#87778C2212yescomplemented
31#65894—52nomaximal
32C23×C52C2311yescharacteristic
33C22×C5⋊D5—22no—
34C22×C5⋊D5—41no—
35C22×C5⋊D5—41no—
36C22×C5⋊D5C2312yes—
37C22×C5×D5C2312yes—
38C2×C4×C52—41no—
39C2×C4×C52C2312yes—
40C2×#1039—41no—
41C2×#1039C2311yescharacteristic
42C2×#1039C2311yescharacteristic
43C5×C22⋊D5—24no—
44C2×C5×Dic5C2312yes—
45C2×C5×Dic5C2312yes—
46C2×C5×Dic5C2312yes—
47C4×C5×D5—42no—
48#6039—41no—
49#6038—42no—
50#6038—42no—
51#6037—41no—
52C4×C5×D4—52no—
53#4085—52no—
54#4083—52no—
55#4076—52no—
56#4075—52no—
57C4×C22⋊D5D512yescomplemented
58C2×#685—52no—
59C2×#685—102no—
60#4044—52no—
61#4040—52no—
62#4031—52no—
63#4030—52no—
64#4028—52no—
65#4024—52no—
66#4022—52no—
67#4021—52no—
68C22×C52—21no—
69C22×C52—21no—
70C22×C52#4112yes—
71C22×C52C2411yescharacteristic
72C2×C5⋊D5—22no—
73C2×C5⋊D5—22no—
74C2×C5⋊D5—22no—
75C2×C5⋊D5—44no—
76C2×C5×D5—22no—
77C2×C5×D5—22no—
78C2×C5×D5—22no—
79C4×C52—22no—
80#1039—21no—
81#1039—21no—
82C5×Dic5—22no—
83C5×Dic5—22no—
84C5×Dic5—22no—
85C23×D5—52no—
86C23×D5—102no—
87C2×C5×D4—52no—
88C22×C4×C5—22no—
89C22×C4×C5—52no—
90C22×C4×C5C2×D512yes—
91C2×C22⋊D5—52no—
92C2×C22⋊D5—52no—
93C2×C22⋊D5—52no—
94C2×C22⋊D5C2×D512yes—
95C22×Dic5—52no—
96C22×Dic5—102no—
97C2×D20—52no—
98C2×D20—52no—
99C2×C4×D5C2×D512yescomplemented
100C2×Dic10—52no—
101C5×C4⋊C4—52no—
102C5×C22⋊C4—52no—
103C5×C22⋊C4—52no—
104C42×C5—52no—
105C22⋊Dic5—52no—
106C22⋊Dic5C2×D512yes—
107#685—52no—
108#685—52no—
109#685—52no—
110#685—52no—
111#685—52no—
112#685—52no—
113#685—52no—
114#685—52no—
115#685—52no—
116#685—104no—
117#685—104no—
118#685C2×D512yescomplemented
119C4⋊Dic5—52no—
120#683—52no—
121#683—52no—
122#683—52no—
123#683C2×D512yescomplemented
124C4×Dic5C2×D512yescomplemented
125#538—251noSylow
126C2×C52—22no—
127C2×C52C2×#4111yescharacteristic
128C2×C52H(5,𝔽2)11yescharacteristic
129C2×C52H(5,𝔽2)11yescharacteristic
130C5⋊D5—42no—
131C5×D5—42no—
132C23×C5—22no—
133C23×C5—52no—
134C23×C5C22×D512yes—
135C22×D5—52no—
136C22×D5—52no—
137C22×D5—104no—
138C22×D5—108no—
139C22×D5—104no—
140C22×D5—202no—
141C22×D5—202no—
142C22×D5—204no—
143C22×D5—204no—
144C22×D5C22×D512yes—
145C5×D4—104no—
146C2×C4×C5—24no—
147C2×C4×C5—42no—
148C2×C4×C5—44no—
149C2×C4×C5—52no—
150C2×C4×C5—52no—
151C2×C4×C5—52no—
152C2×C4×C5—202no—
153C2×C4×C5C22×D512yes—
154C2×C4×C5C22×D512yes—
155C22⋊D5—24no—
156C22⋊D5—202no—
157C22⋊D5—202no—
158C22⋊D5—202no—
159C2×Dic5—52no—
160C2×Dic5—52no—
161C2×Dic5—102no—
162C2×Dic5—102no—
163C2×Dic5—202no—
164C2×Dic5—204no—
165C2×Dic5C22×D512yes—
166C2×Dic5C22×D512yes—
167C2×Dic5C22×D512yes—
168D20—202no—
169D20—202no—
170C4×D5—42no—
171Dic10—202no—
172#126—252no—
173#124—252no—
174#123—252no—
175C22⋊Q8—251no—
176#121—252no—
177#121—251no—
178#120—252no—
179C4×D4—52no—
180C2×C22⋊C4—251no—
181C52#53811yescharacteristic, complemented, Sylow
182C22×C5—24no—
183C22×C5—24no—
184C22×C5—22no—
185C22×C5—22no—
186C22×C5—24no—
187C22×C5—22no—
188C22×C5—102no—
189C22×C5—102no—
190C22×C5—102no—
191C22×C5#70914yes—
192C22×C5C23×D512yes—
193C2×D5—22no—
194C2×D5—22no—
195C2×D5—22no—
196C2×D5—108no—
197C2×D5—102no—
198C2×D5—102no—
199C2×D5—102no—
200C2×D5—102no—
201C2×D5—102no—
202C2×D5—102no—
203C2×D5—204no—
204C2×D5—204no—
205C2×D5—2016no—
206C2×D5—208no—
207C4×C5—22no—
208C4×C5—22no—
209C4×C5—44no—
210C4×C5—102no—
211C4×C5—102no—
212C4×C5—102no—
213Dic5—22no—
214Dic5—22no—
215Dic5—22no—
216Dic5—102no—
217Dic5—102no—
218Dic5—202no—
219Dic5—202no—
220C24—251no—
221C2×Q8—251no—
222C2×D4—52no—
223C2×D4—252no—
224C2×D4—252no—
225C2×D4—251no—
226C22×C4—52no—
227C22×C4—251no—
228C22×C4D5211yescharacteristic
229C4⋊C4—52no—
230C4⋊C4—252no—
231C4⋊C4—251no—
232C4⋊C4—251no—
233C22⋊C4—52no—
234C22⋊C4—52no—
235C22⋊C4—252no—
236C22⋊C4—252no—
237C22⋊C4—252no—
238C22⋊C4—252no—
239C22⋊C4—252no—
240C42—52no—
241C2×C5—28no—
242C2×C5—24no—
243C2×C5—22no—
244C2×C5—22no—
245C2×C5—22no—
246C2×C5—202no—
247C2×C5C2×#70912yes—
248C2×C5#414412yes—
249C2×C5#414912yes—
250D5—42no—
251D5—208no—
252D5—202no—
253D5—202no—
254C23—52no—
255C23—252no—
256C23—502no—
257C23—1001no—
258C23—1001no—
259C23C2×D5211yescharacteristic
260Q8—1001no—
261D4—104no—
262D4—1002no—
263D4—1002no—
264D4—1001no—
265C2×C4—41no—
266C2×C4—52no—
267C2×C4—52no—
268C2×C4—52no—
269C2×C4—202no—
270C2×C4—251no—
271C2×C4—251no—
272C2×C4—1001no—
273C2×C4C2×D5212yes—
274C5—22no—
275C5#6589412yescomplemented
276C22—21no—
277C22—21no—
278C22—102no—
279C22—102no—
280C22—102no—
281C22—502no—
282C22—502no—
283C22—502no—
284C22—1004no—
285C22#8787312yes—
286C22C22×D5211yescharacteristic
287C4—22no—
288C4—102no—
289C4—102no—
290C4—102no—
291C4—501no—
292C4—501no—
293C2—22no—
294C2—202no—
295C2—1002no—
296C2C2×#8787311yescharacteristic
297C2#1171747311yescharacteristic
298C2#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. C4×C5×C22⋊D5C2semidirect
3. #11717100C2semidirect
4. #11717099C2semidirect
5. #11717098C2semidirect
6. #11717024C2semidirect
7. #11716987C2semidirect
8. #11716970C2semidirect
9. #11716968C2semidirect
14. C2×C4×C5×D5C22semidirect
15. C2×#6039C22semidirect
18. C2×#6037C22semidirect
22. C5×#685C22semidirect
23. C5×#683C22semidirect
24. C4×C5×Dic5C22semidirect
25. #87781C22semidirect
26. #87781C22semidirect
27. #87780C22semidirect
30. #87778C22semidirect
57. C4×C22⋊D5D5semidirect
99. C2×C4×D5C2×D5semidirect
118. #685C2×D5semidirect
123. #683C2×D5semidirect
124. C4×Dic5C2×D5semidirect
181. C52#538semidirect
275. C5#65894semidirect

Maximal subgroups

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

subgroupindex
C2×#878082
C4×C5×C22⋊D52
#117171002
#117170992
#117170982
#117170242
#117169872
#117169702
#117169682
#658945