Finite Group Databasebeta

← #13098957

Subgroups — #13098957

Order 1600.

501 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 501 conjugacy classes of subgroups — too many to draw. Only its 131 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×#87748C211yesmaximal, characteristic, complemented
2C22×D100C211yesmaximal, characteristic, complemented
3C25×C22⋊Q8C211yesmaximal, characteristic, complemented
4#11716509C211yesmaximal, characteristic, complemented
5C4×C22⋊D25C211yesmaximal, characteristic, complemented
6#11716457C211yesmaximal, characteristic, complemented
7#11716457C212yesmaximal, complemented
8#11716456C212yesmaximal, complemented
9#11716455C211yesmaximal, characteristic, complemented
10#11716444C212yesmaximal, complemented
11C22⋊C4×D25C212yesmaximal, complemented
12C23×D25C2212yes—
13C2×Q8×C25C2211yescharacteristic, complemented
14C22×C4×C25C2211yescharacteristic
15C2×C22⋊D25C2211yescharacteristic
16#87748—24no—
17C2×D100—24no—
18C2×D100C2212yes—
19C2×D100C2211yescharacteristic
20C2×D100C2211yescharacteristic
21C2×D100C2212yescomplemented
22C2×D100C2212yescomplemented
23C2×C4×D25C2212yescomplemented
24C2×C4×D25C2211yescharacteristic, complemented
25C2×C4×D25C2212yescomplemented
26C4⋊C4×C25C2211yescharacteristic, complemented
27C4⋊C4×C25C2212yescomplemented
28C22⋊C4×C25C2212yes—
29C22⋊Dic25C2211yescharacteristic
30#87720C2212yescomplemented
31#87720C2212yescomplemented
32#87720C2212yescomplemented
33#87720C2212yescomplemented
34#87720C2211yescharacteristic, complemented
35#87718C2211yescharacteristic, complemented
36C4×Dic25C2211yescharacteristic, complemented
37#65964—51nomaximal
38C23×C25C2311yescharacteristic
39C22×D25—24no—
40C22×D25—24no—
41C22×D25—42no—
42C22×D25C2312yes—
43C22×D25C2311yescharacteristic
44C22×D25C2312yes—
45Q8×C25—22no—
46C2×C4×C25—22no—
47C2×C4×C25C2311yescharacteristic
48C2×C4×C25C2312yes—
49C2×C4×C25C2312yes—
50C2×C4×C25C2311yescharacteristic
51C22⋊D25—22no—
52C2×Dic25C2311yescharacteristic
53C2×Dic25C2311yescharacteristic
54C2×Dic25C2311yescharacteristic
55D100—24no—
56D100—24no—
57D100—22no—
58D100—42no—
59D100D414yescomplemented
60C4×D25—24no—
61C4×D25—22no—
62C4×D25—42no—
63C2×#713—51no—
64C22×D20—51no—
65C5×C22⋊Q8D511yescharacteristic
66#4093—51no—
67C4×C22⋊D5—51no—
68#4041—51no—
69#4041—52no—
70#4040—52no—
71#4039—51no—
72#4028—52no—
73D5×C22⋊C4—52no—
74C22×C25—21no—
75C22×C25—21no—
76C22×C25#4112yescomplemented
77C22×C25C2411yescharacteristic
78C2×D25—22no—
79C2×D25—22no—
80C2×D25—21no—
81C2×D25—22no—
82C2×D25—21no—
83C2×D25—22no—
84C2×D25—21no—
85C2×D25—22no—
86C2×D25—48no—
87C2×D25C2×D414yes—
88C4×C25—22no—
89C4×C25—21no—
90C4×C25—22no—
91C4×C25C2×D412yes—
92Dic25—21no—
93Dic25—21no—
94Dic25—21no—
95C23×D5—52no—
96C2×C5×Q8C2×D511yescharacteristic
97C22×C4×C5C2×D511yescharacteristic
98C2×C22⋊D5—51no—
99#713—104no—
100C2×D20—51no—
101C2×D20—52no—
102C2×D20—51no—
103C2×D20—52no—
104C2×D20—52no—
105C2×D20—104no—
106C2×C4×D5—52no—
107C2×C4×D5—51no—
108C2×C4×D5—52no—
109C5×C4⋊C4C2×D511yescharacteristic
110C5×C4⋊C4C2×D512yes—
111C5×C22⋊C4C2×D512yes—
112C22⋊Dic5—51no—
113#685—52no—
114#685—52no—
115#685—52no—
116#685—52no—
117#685—51no—
118#683—51no—
119C4×Dic5—51no—
120#546—251noSylow
121C2×C25—22no—
122C2×C25C22×D411yescharacteristic
123C2×C25C2×#4111yescharacteristic
124C2×C25H(5,𝔽2)11yescharacteristic
125D25—24no—
126D25—41no—
127D25—42no—
128C23×C5C22×D511yescharacteristic
129C22×D5—52no—
130C22×D5—52no—
131C22×D5—51no—
132C22×D5—104no—
133C22×D5—104no—
134C22×D5—202no—
135C5×Q8—22no—
136C2×C4×C5—22no—
137C2×C4×C5C22×D511yescharacteristic
138C2×C4×C5C22×D512yes—
139C2×C4×C5C22×D512yes—
140C2×C4×C5C22×D511yescharacteristic
141C22⋊D5—102no—
142C2×Dic5—51no—
143C2×Dic5—51no—
144C2×Dic5—51no—
145D20—54no—
146D20—104no—
147D20—104no—
148D20—102no—
149D20—202no—
150C4×D5—104no—
151C4×D5—102no—
152C4×D5—202no—
153C2×#41—251no—
154C22×D4—251no—
155#124—251no—
156#123—252no—
157C22⋊Q8D2511yescharacteristic, complemented
158#121—251no—
159#121—252no—
160#120—252no—
161C4×D4—251no—
162C4×D4—251no—
163C2×C22⋊C4—252no—
164C25#54611yescharacteristic, complemented, Sylow
165C22×C5—21no—
166C22×C5—21no—
167C22×C5#71312yes—
168C22×C5C23×D511yescharacteristic
169C2×D5—54no—
170C2×D5—102no—
171C2×D5—102no—
172C2×D5—101no—
173C2×D5—102no—
174C2×D5—101no—
175C2×D5—102no—
176C2×D5—101no—
177C2×D5—102no—
178C2×D5—208no—
179C4×C5—22no—
180C4×C5—21no—
181C4×C5—22no—
182C4×C5D4×D512yes—
183Dic5—101no—
184Dic5—101no—
185Dic5—101no—
186C24—252no—
187#41—504no—
188C2×Q8C2×D2511yescharacteristic, complemented
189C2×D4—251no—
190C2×D4—251no—
191C2×D4—252no—
192C2×D4—252no—
193C2×D4—251no—
194C2×D4—252no—
195C2×D4—504no—
196C22×C4—252no—
197C22×C4—251no—
198C22×C4—252no—
199C22×C4C2×D2511yescharacteristic
200C4⋊C4—251no—
201C4⋊C4C2×D2511yescharacteristic, complemented
202C4⋊C4C2×D2512yescomplemented
203C22⋊C4—251no—
204C22⋊C4—252no—
205C22⋊C4—252no—
206C22⋊C4—252no—
207C22⋊C4—251no—
208C22⋊C4—252no—
209C22⋊C4C2×D2512yes—
210C42—251no—
211C2×C5—22no—
212C2×C5C2×D4×D511yescharacteristic
213C2×C5C2×#71311yescharacteristic
214C2×C5#414911yescharacteristic
215D5—104no—
216D5—202no—
217D5—201no—
218C23—252no—
219C23—252no—
220C23—251no—
221C23—504no—
222C23—504no—
223C23—1002no—
224C23C22×D2511yescharacteristic
225Q8—22no—
226D4—254no—
227D4—504no—
228D4—504no—
229D4—502no—
230D4—502no—
231D4—1002no—
232C2×C4—22no—
233C2×C4—251no—
234C2×C4—251no—
235C2×C4—251no—
236C2×C4—504no—
237C2×C4—502no—
238C2×C4—1002no—
239C2×C4C22×D2512yes—
240C2×C4C22×D2512yes—
241C2×C4C22×D2511yescharacteristic
242C2×C4C22×D2511yescharacteristic
243C5#6596411yescharacteristic
244C22—21no—
245C22—21no—
246C22—254no—
247C22—502no—
248C22—502no—
249C22—501no—
250C22—502no—
251C22—501no—
252C22—502no—
253C22—501no—
254C22—502no—
255C22—1008no—
256C22#8774812yescomplemented
257C22C23×D2511yescharacteristic
258C4—21no—
259C4—22no—
260C4—22no—
261C4—501no—
262C4—501no—
263C4—501no—
264C4D4×D2512yes—
265C2—22no—
266C2—504no—
267C2—1002no—
268C2—1001no—
269C2C2×D4×D2511yescharacteristic
270C2C2×#8774811yescharacteristic
271C2#1171656811yescharacteristic
subgroupG/H
centerC22C23×D25
commutator, cosocleC22×C25C24
Frattini, socleC22×C5C23×D5
FittingC25×C22⋊Q8C2
hypercenterC22⋊Q8D25
hypocenterC25#546
LCSG ▷ C22×C25 ▷ C25
UCSC1 ◁ C22 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×#87748C2semidirect
2. C22×D100C2semidirect
3. C25×C22⋊Q8C2semidirect
4. #11716509C2semidirect
5. C4×C22⋊D25C2semidirect
6. #11716457C2semidirect
7. #11716457C2semidirect
8. #11716456C2semidirect
9. #11716455C2semidirect
10. #11716444C2semidirect
11. C22⋊C4×D25C2semidirect
13. C2×Q8×C25C22semidirect
21. C2×D100C22semidirect
22. C2×D100C22semidirect
23. C2×C4×D25C22semidirect
24. C2×C4×D25C22semidirect
25. C2×C4×D25C22semidirect
26. C4⋊C4×C25C22semidirect
27. C4⋊C4×C25C22semidirect
30. #87720C22semidirect
31. #87720C22semidirect
32. #87720C22semidirect
33. #87720C22semidirect
34. #87720C22semidirect
35. #87718C22semidirect
36. C4×Dic25C22semidirect
59. D100D4semidirect
76. C22×C25#41semidirect
157. C22⋊Q8D25semidirect
164. C25#546semidirect
188. C2×Q8C2×D25semidirect
201. C4⋊C4C2×D25semidirect
202. C4⋊C4C2×D25semidirect
256. C22#87748semidirect

Maximal subgroups

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

subgroupindex
C2×#877482
C22×D1002
C25×C22⋊Q82
#117165092
C4×C22⋊D252
#117164572
#117164562
#117164552
#117164442
C22⋊C4×D252
#659645