Finite Group Databasebeta

← #13105097

Subgroups — #13105097

Order 1600.

736 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 736 conjugacy classes of subgroups — too many to draw. Only its 144 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×C4×D52C211yesmaximal, characteristic, complemented
2C2×#87873C211yesmaximal, characteristic, complemented
3#11717192C211yesmaximal, characteristic, complemented
4C4×C5×D20C212yesmaximal, complemented
5#11717040C212yesmaximal, complemented
6C4×#6037C212yesmaximal, complemented
7#11717014C212yesmaximal, complemented
8#11717004C212yesmaximal, complemented
9#11716972C212yesmaximal, complemented
10C22×D52C2211yescharacteristic
11C2×C4×C5⋊D5C2211yescharacteristic
12C2×C5×D20C2212yes—
13C2×C4×C5×D5C2212yes—
14C2×C4×C5×D5C2212yescomplemented
15C2×#6039C2211yescharacteristic, complemented
16C2×#6038C2212yes—
17C2×#6037C2212yes—
18C2×#6037C2211yescharacteristic, complemented
19C2×#6036C2211yescharacteristic
20C2×D5×Dic5C2212yes—
21C4×D52—24no—
22#87873—24no—
23C42×C52C2211yescharacteristic, complemented
24#87808C2212yes—
25#87806C2212yescomplemented
26C4×#1039C2211yescharacteristic, complemented
27C5×#685C2212yes—
28C5×#685C2212yescomplemented
29C5×C4⋊Dic5C2212yescomplemented
30#87781C2212yescomplemented
31#87778C2212yes—
32#87778C2212yescomplemented
33#65881—52nomaximal
34C22×C5⋊D5C2311yescharacteristic
35C22×C5×D5C2312yes—
36C22×C5×D5C2312yes—
37C2×D52—22no—
38C2×D52—24no—
39C2×C4×C52C2312yes—
40C2×C4×C52C2311yescharacteristic
41C2×#1039C2312yes—
42C2×#1039C2311yescharacteristic
43C4×C5⋊D5D414yescomplemented
44C2×C5×Dic5C2312yes—
45C2×C5×Dic5C2312yes—
46C5×D20—24no—
47C4×C5×D5—24no—
48C4×C5×D5—24no—
49#6039—22no—
50#6038—24no—
51#6037—24no—
52#6037—22no—
53#6036—22no—
54D5×Dic5—24no—
55C2×#709—52no—
56C22×C4×D5—52no—
57C4×C5×D4—52no—
58#4085—52no—
59#4083—52no—
60C4×C22⋊D5—52no—
61C4×C22⋊D5—52no—
62#4042—52no—
63#4040—52no—
64#4030—52no—
65#4029—52no—
66#4028—52no—
67#4024—52no—
68C4×D20D512yescomplemented
69#4018—52no—
70#4018—102no—
71C22×C52C2411yescharacteristic
72C2×C5⋊D5—22no—
73C2×C5⋊D5C2×D412yes—
74C2×C5×D5—22no—
75C2×C5×D5—22no—
76C2×C5×D5—22no—
77C2×C5×D5—22no—
78C2×C5×D5—22no—
79C2×C5×D5#4114yescomplemented
80D52—28no—
81C4×C52—22no—
82C4×C52C2×D412yes—
83#1039—22no—
84#1039C2×D412yes—
85C5×Dic5—22no—
86C5×Dic5—22no—
87C23×D5—52no—
88C2×C5×D4—52no—
89C22×C4×C5—52no—
90C22×C4×C5—52no—
91C2×C22⋊D5—52no—
92C2×C22⋊D5—52no—
93C2×C22⋊D5—52no—
94C2×C22⋊D5—52no—
95C22×Dic5—52no—
96#709—108no—
97C2×D20—52no—
98C2×D20C2×D512yes—
99C2×C4×D5—52no—
100C2×C4×D5—52no—
101C2×C4×D5—52no—
102C2×C4×D5—102no—
103C2×C4×D5—108no—
104C2×C4×D5C2×D512yes—
105C2×C4×D5C2×D512yescomplemented
106C2×Dic10—52no—
107C5×C4⋊C4—52no—
108C5×C22⋊C4—52no—
109C5×C22⋊C4—52no—
110C42×C5—22no—
111C42×C5C2×D512yescomplemented
112C22⋊Dic5—52no—
113C22⋊Dic5—52no—
114#685—52no—
115#685—52no—
116#685—52no—
117#685—52no—
118#685—104no—
119#685C2×D512yes—
120#685C2×D512yescomplemented
121C4⋊Dic5C2×D512yescomplemented
122#683—52no—
123#683—52no—
124#683—52no—
125#683—52no—
126#683—104no—
127C4×Dic5—52no—
128C4×Dic5—102no—
129#525—251noSylow
130C2×C52C22×D411yescharacteristic
131C2×C52C2×#4112yes—
132C5⋊D5—22no—
133C5×D5—24no—
134C5×D5—42no—
135C23×C5—52no—
136C23×C5—52no—
137C22×D5—52no—
138C22×D5—102no—
139C22×D5—104no—
140C22×D5—104no—
141C22×D5—104no—
142C22×D5C22×D512yes—
143C22×D5C22×D512yes—
144C5×D4—104no—
145C2×C4×C5—24no—
146C2×C4×C5—22no—
147C2×C4×C5—52no—
148C2×C4×C5—52no—
149C2×C4×C5—104no—
150C2×C4×C5—104no—
151C2×C4×C5C22×D512yes—
152C2×C4×C5C22×D512yes—
153C2×C4×C5C22×D512yes—
154C22⋊D5—104no—
155C22⋊D5—104no—
156C22⋊D5—104no—
157C22⋊D5—104no—
158C2×Dic5—52no—
159C2×Dic5—52no—
160C2×Dic5—52no—
161C2×Dic5—102no—
162C2×Dic5—104no—
163C2×Dic5—104no—
164C2×Dic5C22×D512yes—
165C2×Dic5C22×D512yes—
166D20—24no—
167D20—104no—
168C4×D5—24no—
169C4×D5—24no—
170C4×D5—58no—
171C4×D5—108no—
172C4×D5—104no—
173C4×D5—104no—
174Dic10—104no—
175C2×#41—251no—
176C23×C4—251no—
177#123—252no—
178C22⋊Q8—252no—
179#121—252no—
180#120—252no—
181C4×D4—52no—
182C4×D4—252no—
183#117—251no—
184C52#52511yescharacteristic, complemented, Sylow
185C22×C5—22no—
186C22×C5—54no—
187C22×C5—102no—
188C22×C5—102no—
189C22×C5—102no—
190C22×C5—102no—
191C22×C5—102no—
192C22×C5C23×D512yes—
193C2×D5—22no—
194C2×D5—22no—
195C2×D5—22no—
196C2×D5—22no—
197C2×D5—22no—
198C2×D5—54no—
199C2×D5—104no—
200C2×D5—1016no—
201C2×D5—102no—
202C2×D5—102no—
203C2×D5—204no—
204C2×D5#70914yescomplemented
205C4×C5—22no—
206C4×C5—22no—
207C4×C5—24no—
208C4×C5—44no—
209C4×C5—102no—
210C4×C5—102no—
211C4×C5D4×D514yes—
212Dic5—22no—
213Dic5—22no—
214Dic5—54no—
215Dic5—104no—
216Dic5—102no—
217Dic5—102no—
218Dic5—204no—
219C24—251no—
220#41—504no—
221C2×Q8—251no—
222C2×D4—52no—
223C2×D4—252no—
224C2×D4—252no—
225C2×D4—251no—
226C22×C4—52no—
227C22×C4—52no—
228C22×C4—251no—
229C22×C4—252no—
230C22×C4—251no—
231C22×C4—504no—
232C4⋊C4—52no—
233C4⋊C4—252no—
234C4⋊C4—252no—
235C22⋊C4—52no—
236C22⋊C4—52no—
237C22⋊C4—252no—
238C22⋊C4—252no—
239C22⋊C4—252no—
240C42—251no—
241C42D5211yescharacteristic, complemented
242C2×C5—22no—
243C2×C5—24no—
244C2×C5—104no—
245C2×C5—202no—
246C2×C5C2×#70912yes—
247C2×C5C2×D4×D512yes—
248C2×C5D5×#4112yes—
249D5—24no—
250D5—42no—
251D5—104no—
252D5—204no—
253C23—52no—
254C23—52no—
255C23—251no—
256C23—504no—
257C23—502no—
258Q8—502no—
259D4—104no—
260D4—504no—
261D4—502no—
262D4—504no—
263C2×C4—52no—
264C2×C4—52no—
265C2×C4—104no—
266C2×C4—104no—
267C2×C4—251no—
268C2×C4—254no—
269C2×C4—252no—
270C2×C4—502no—
271C2×C4—504no—
272C2×C4C2×D5212yes—
273C2×C4C2×D5211yescharacteristic
274C5—22no—
275C5#6588112yescomplemented
276C22—54no—
277C22—102no—
278C22—102no—
279C22—102no—
280C22—102no—
281C22—102no—
282C22—252no—
283C22—508no—
284C22—502no—
285C22C22×D5211yescharacteristic
286C4—22no—
287C4—102no—
288C4—102no—
289C4—252no—
290C4—502no—
291C4#8787712yes—
292C2—104no—
293C2—202no—
294C2—502no—
295C2C2×#8787711yescharacteristic
296C2D5×#70912yes—
subgroupG/H
centerC2×C4C2×D52
commutator, socle, cosocleC22×C52C24
FrattiniC22C22×D52
FittingC42×C52C22
hypercenterC42D52
hypocenterC52#525
LCSG ▷ C22×C52 ▷ C52
UCSC1 ◁ C2×C4 ◁ C42

Complements

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

NH ≅ G/Ntype
1. C2×C4×D52C2semidirect
2. C2×#87873C2semidirect
3. #11717192C2semidirect
4. C4×C5×D20C2semidirect
5. #11717040C2semidirect
6. C4×#6037C2semidirect
7. #11717014C2semidirect
8. #11717004C2semidirect
9. #11716972C2semidirect
14. C2×C4×C5×D5C22semidirect
15. C2×#6039C22semidirect
18. C2×#6037C22semidirect
23. C42×C52C22semidirect
25. #87806C22semidirect
26. C4×#1039C22semidirect
28. C5×#685C22semidirect
29. C5×C4⋊Dic5C22semidirect
30. #87781C22semidirect
32. #87778C22semidirect
43. C4×C5⋊D5D4semidirect
68. C4×D20D5semidirect
79. C2×C5×D5#41semidirect
105. C2×C4×D5C2×D5semidirect
111. C42×C5C2×D5semidirect
120. #685C2×D5semidirect
121. C4⋊Dic5C2×D5semidirect
184. C52#525semidirect
204. C2×D5#709semidirect
241. C42D52semidirect
275. C5#65881semidirect

Maximal subgroups

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

subgroupindex
C2×C4×D522
C2×#878732
#117171922
C4×C5×D202
#117170402
C4×#60372
#117170142
#117170042
#117169722
#658815