Finite Group Databasebeta

← #13106402

Subgroups — #13106402

Order 1600.

594 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 594 conjugacy classes of subgroups — too many to draw. Only its 122 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#11717258C211yesmaximal, characteristic, complemented
2C5×#4085C212yesmaximal, complemented
3#11717101C212yesmaximal, complemented
4#11717079C212yesmaximal, complemented
5#11717077C212yesmaximal, complemented
6#11717070C212yesmaximal, complemented
7#11717039C211yesmaximal, characteristic, complemented
8#11716984C211yesmaximal, characteristic, complemented
9#11716977C212yesmaximal, complemented
10C22×D52C2211yescharacteristic
11C2×C52×D4C2211yescharacteristic, complemented
12C2×C22⋊(C5⋊D5)C2212yes—
13C2×C4×C5⋊D5C2211yescharacteristic
14C2×C5×C22⋊D5C2212yes—
15C2×C5×C22⋊D5C2212yes—
16C22×C5×Dic5C2212yescomplemented
17C2×#6039C2211yescharacteristic, complemented
18C2×#6036C2211yescharacteristic
19C22⋊#1039C2212yescomplemented
20C4⋊#1039C2211yescharacteristic, complemented
21C5×C22⋊Dic5C2212yescomplemented
22C5×#685C2212yes—
23C5×#683C2212yescomplemented
24#87781C2212yescomplemented
25#87781C2211yescharacteristic, complemented
26#87779C2212yes—
27#87778C2212yes—
28#87778C2212yes—
29#87778C2212yes—
30Dic52C2212yescomplemented
31#65947—52nomaximal
32C23×C52C2312yes—
33C22×C5⋊D5C2311yescharacteristic
34C22×C5×D5C2312yes—
35C2×D52—22no—
36C2×D52—41no—
37C2×D52—41no—
38C52×D4—41no—
39C2×C4×C52C2311yescharacteristic
40C22⋊(C5⋊D5)—24no—
41C2×#1039C2311yescharacteristic
42C2×#1039C2312yes—
43C4×C5⋊D5—41no—
44C5×C22⋊D5—42no—
45C5×C22⋊D5—42no—
46C2×C5×Dic5—42no—
47C2×C5×Dic5C2312yes—
48C2×C5×Dic5C2312yes—
49C2×C5×Dic5C2312yes—
50#6039—41no—
51#6036—41no—
52C5×#121—52no—
53#4085D512yescomplemented
54#4084—52no—
55#4084—102no—
56#4083—52no—
57#4083—52no—
58#4082—52no—
59#4081—52no—
60#4075—52no—
61C4×C22⋊D5—52no—
62#4044—52no—
63#4042—52no—
64#4031—52no—
65#4027—52no—
66D5×C22⋊C4—52no—
67#4025—52no—
68C22×C52—22no—
69C22×C52—22no—
70C22×C52—22no—
71C22×C52C2411yescharacteristic
72C2×C5⋊D5—21no—
73C2×C5⋊D5—21no—
74C2×C5⋊D5#4112yes—
75C2×C5×D5—22no—
76C2×C5×D5—22no—
77C2×C5×D5—22no—
78D52—44no—
79C4×C52—21no—
80#1039—22no—
81#1039—21no—
82C5×Dic5—22no—
83C5×Dic5—22no—
84C5×Dic5—22no—
85C23×D5—52no—
86C2×C5×D4—22no—
87C2×C5×D4—52no—
88C2×C5×D4—52no—
89C2×C5×D4C2×D512yescomplemented
90C22×C4×C5—52no—
91C2×C22⋊D5—52no—
92C2×C22⋊D5—52no—
93C2×C22⋊D5—104no—
94C2×C22⋊D5C2×D512yes—
95C2×C22⋊D5C2×D512yes—
96C22×Dic5C2×D512yescomplemented
97C2×C4×D5—52no—
98C2×C4×D5—52no—
99C2×C4×D5—102no—
100C2×Dic10—52no—
101C5×C4⋊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—104no—
110C22⋊Dic5C2×D512yescomplemented
111#685—52no—
112#685—52no—
113#685—52no—
114#685—52no—
115#685—52no—
116#685C2×D512yes—
117C4⋊Dic5—52no—
118C4⋊Dic5—102no—
119#683—52no—
120#683—52no—
121#683—52no—
122#683C2×D512yescomplemented
123C4×Dic5—52no—
124C4×Dic5—52no—
125#538—251noSylow
126C2×C52—42no—
127C2×C52C2×#4111yescharacteristic
128C2×C52H(5,𝔽2)11yescharacteristic
129C2×C52H(5,𝔽2)11yescharacteristic
130C5⋊D5—22no—
131C5×D5—42no—
132C23×C5—24no—
133C23×C5—52no—
134C23×C5C22×D512yes—
135C23×C5C22×D512yes—
136C22×D5—52no—
137C22×D5—102no—
138C22×D5—104no—
139C22×D5—202no—
140C22×D5—202no—
141C22×D5C22×D512yes—
142C5×D4—42no—
143C5×D4—44no—
144C5×D4—202no—
145C5×D4—202no—
146C2×C4×C5—22no—
147C2×C4×C5—52no—
148C2×C4×C5—52no—
149C2×C4×C5—52no—
150C2×C4×C5—202no—
151C2×C4×C5C22×D512yes—
152C22⋊D5—42no—
153C22⋊D5—42no—
154C22⋊D5—104no—
155C22⋊D5—104no—
156C22⋊D5—208no—
157C2×Dic5—42no—
158C2×Dic5—52no—
159C2×Dic5—52no—
160C2×Dic5—52no—
161C2×Dic5—104no—
162C2×Dic5—102no—
163C2×Dic5C22×D512yes—
164C2×Dic5C22×D512yes—
165C2×Dic5C22×D512yes—
166C4×D5—202no—
167C4×D5—202no—
168C4×D5—204no—
169Dic10—202no—
170#126—252no—
171#124—252no—
172#123—252no—
173C22⋊Q8—251no—
174#121—52no—
175#121—251no—
176#120—252no—
177C4×D4—252no—
178C2×C22⋊C4—251no—
179C52#53811yescharacteristic, complemented, Sylow
180C22×C5—22no—
181C22×C5—22no—
182C22×C5—22no—
183C22×C5—22no—
184C22×C5—22no—
185C22×C5—22no—
186C22×C5—22no—
187C22×C5—44no—
188C22×C5—44no—
189C22×C5—44no—
190C22×C5—102no—
191C22×C5—102no—
192C22×C5—102no—
193C22×C5C23×D512yes—
194C2×D5—22no—
195C2×D5—22no—
196C2×D5—22no—
197C2×D5—54no—
198C2×D5—104no—
199C2×D5—102no—
200C2×D5—102no—
201C2×D5—202no—
202C2×D5—202no—
203C2×D5—208no—
204C4×C5—24no—
205C4×C5—22no—
206C4×C5—102no—
207C4×C5—102no—
208C4×C5—102no—
209Dic5—22no—
210Dic5—22no—
211Dic5—22no—
212Dic5—102no—
213Dic5—102no—
214Dic5—102no—
215Dic5—202no—
216Dic5—204no—
217C24—251no—
218C2×Q8—251no—
219C2×D4—52no—
220C2×D4—52no—
221C2×D4—252no—
222C2×D4D5211yescharacteristic, complemented
223C22×C4—52no—
224C22×C4—251no—
225C22×C4—251no—
226C4⋊C4—52no—
227C4⋊C4—252no—
228C4⋊C4—251no—
229C4⋊C4—251no—
230C22⋊C4—52no—
231C22⋊C4—52no—
232C22⋊C4—252no—
233C22⋊C4—252no—
234C22⋊C4—252no—
235C22⋊C4—252no—
236C22⋊C4—252no—
237C42—252no—
238C2×C5—22no—
239C2×C5—22no—
240C2×C5—22no—
241C2×C5—42no—
242C2×C5—42no—
243C2×C5—84no—
244C2×C5—202no—
245C2×C5#414412yes—
246C2×C5#414412yes—
247C2×C5D5×#4112yes—
248D5—42no—
249D5—104no—
250D5—204no—
251C23—52no—
252C23—251no—
253C23—502no—
254C23—1001no—
255C23—1001no—
256C23C2×D5212yes—
257Q8—1001no—
258D4—41no—
259D4—202no—
260D4—202no—
261D4—504no—
262C2×C4—52no—
263C2×C4—52no—
264C2×C4—52no—
265C2×C4—202no—
266C2×C4—252no—
267C2×C4—251no—
268C2×C4—1001no—
269C2×C4—1001no—
270C2×C4C2×D5211yescharacteristic
271C5—22no—
272C5#6594712yescomplemented
273C22—22no—
274C22—22no—
275C22—22no—
276C22—102no—
277C22—102no—
278C22—102no—
279C22—252no—
280C22—501no—
281C22—501no—
282C22—1004no—
283C22C22×D5211yescharacteristic
284C4—21no—
285C4—102no—
286C4—102no—
287C4—102no—
288C4—502no—
289C4—501no—
290C2—42no—
291C2—202no—
292C2—502no—
293C2#1171747711yescharacteristic
294C2#1171747811yescharacteristic
295C2#1171749511yescharacteristic
subgroupG/H
center, FrattiniC22C22×D52
commutator, socle, cosocleC22×C52C24
FittingC2×C52×D4C22
hypercenterC2×D4D52
hypocenterC52#538
LCSG ▷ C22×C52 ▷ C52
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. #11717258C2semidirect
2. C5×#4085C2semidirect
3. #11717101C2semidirect
4. #11717079C2semidirect
5. #11717077C2semidirect
6. #11717070C2semidirect
7. #11717039C2semidirect
8. #11716984C2semidirect
9. #11716977C2semidirect
11. C2×C52×D4C22semidirect
16. C22×C5×Dic5C22semidirect
17. C2×#6039C22semidirect
19. C22⋊#1039C22semidirect
20. C4⋊#1039C22semidirect
21. C5×C22⋊Dic5C22semidirect
23. C5×#683C22semidirect
24. #87781C22semidirect
25. #87781C22semidirect
30. Dic52C22semidirect
53. #4085D5semidirect
89. C2×C5×D4C2×D5semidirect
96. C22×Dic5C2×D5semidirect
110. C22⋊Dic5C2×D5semidirect
122. #683C2×D5semidirect
179. C52#538semidirect
222. C2×D4D52semidirect
272. C5#65947semidirect

Maximal subgroups

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

subgroupindex
#117172582
C5×#40852
#117171012
#117170792
#117170772
#117170702
#117170392
#117169842
#117169772
#659475