Finite Group Databasebeta

← #13098954

Subgroups — #13098954

Order 1600.

495 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 495 conjugacy classes of subgroups — too many to draw. Only its 133 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×C4×D25C211yesmaximal, characteristic, complemented
3C25×C22⋊Q8C211yesmaximal, characteristic, complemented
4#11716508C211yesmaximal, characteristic, complemented
5#11716492C211yesmaximal, characteristic, complemented
6#11716457C211yesmaximal, characteristic, complemented
7#11716456C212yesmaximal, complemented
8#11716455C212yesmaximal, complemented
9#11716454C211yesmaximal, characteristic, complemented
10#11716444C212yesmaximal, complemented
11#11716443C212yesmaximal, complemented
12C23×D25C2211yescharacteristic
13C2×Q8×C25C2211yescharacteristic, complemented
14C22×C4×C25C2211yescharacteristic
15C2×C22⋊D25C2212yes—
16C22×Dic25C2211yescharacteristic
17#87748—24no—
18C2×D100C2212yes—
19C2×D100C2212yes—
20C2×D100C2211yescharacteristic, complemented
21C2×C4×D25—24no—
22C2×C4×D25C2211yescharacteristic
23C2×C4×D25C2211yescharacteristic
24C2×C4×D25C2211yescharacteristic
25C2×C4×D25C2211yescharacteristic
26C2×C4×D25C2212yescomplemented
27C4⋊C4×C25C2211yescharacteristic, complemented
28C4⋊C4×C25C2212yescomplemented
29C22⋊C4×C25C2212yes—
30#87720C2212yes—
31#87720C2212yes—
32#87720C2212yescomplemented
33#87720C2212yescomplemented
34C4⋊Dic25C2211yescharacteristic, complemented
35#87718C2212yescomplemented
36C4×Dic25C2212yescomplemented
37#65961—51nomaximal
38C23×C25C2311yescharacteristic
39C22×D25—22no—
40C22×D25—22no—
41C22×D25—22no—
42C22×D25C2312yes—
43C22×D25C2311yescharacteristic
44C22×D25C2311yescharacteristic
45Q8×C25—22no—
46C2×C4×C25—22no—
47C2×C4×C25C2312yes—
48C2×C4×C25C2312yes—
49C2×C4×C25C2311yescharacteristic
50C2×C4×C25C2311yescharacteristic
51C22⋊D25—24no—
52C2×Dic25—22no—
53C2×Dic25C2312yes—
54C2×Dic25C2311yescharacteristic
55C2×Dic25C2311yescharacteristic
56D100—24no—
57D100—24no—
58D100—22no—
59C4×D25—22no—
60C4×D25—22no—
61C4×D25—22no—
62C4×D25—24no—
63C4×D25D414yescomplemented
64C2×#713—51no—
65C22×C4×D5—51no—
66C5×C22⋊Q8D511yescharacteristic
67#4092—51no—
68#4076—51no—
69#4041—51no—
70#4040—52no—
71#4039—52no—
72#4038—51no—
73#4028—52no—
74#4027—52no—
75C22×C25—21no—
76C22×C25—21no—
77C22×C25#4112yescomplemented
78C22×C25C2411yescharacteristic
79C2×D25—28no—
80C2×D25—21no—
81C2×D25—21no—
82C2×D25—21no—
83C2×D25—21no—
84C2×D25—22no—
85C2×D25—22no—
86C2×D25—22no—
87C2×D25C2×D412yes—
88C2×D25#4112yescomplemented
89C4×C25—21no—
90C4×C25—22no—
91C4×C25—22no—
92C4×C25C2×D412yes—
93Dic25—21no—
94Dic25—22no—
95Dic25C2×D412yes—
96C23×D5—51no—
97C2×C5×Q8C2×D511yescharacteristic
98C22×C4×C5C2×D511yescharacteristic
99C2×C22⋊D5—52no—
100C22×Dic5—51no—
101#713—104no—
102C2×D20—52no—
103C2×D20—51no—
104C2×D20—52no—
105C2×C4×D5—51no—
106C2×C4×D5—51no—
107C2×C4×D5—51no—
108C2×C4×D5—51no—
109C2×C4×D5—52no—
110C2×C4×D5—104no—
111C5×C4⋊C4C2×D511yescharacteristic
112C5×C4⋊C4C2×D512yes—
113C5×C22⋊C4C2×D512yes—
114#685—52no—
115#685—52no—
116#685—52no—
117#685—52no—
118C4⋊Dic5—51no—
119#683—52no—
120C4×Dic5—52no—
121#525—251noSylow
122C2×C25—22no—
123C2×C25C22×D411yescharacteristic
124C2×C25C2×#4111yescharacteristic
125C2×C25C2×#4111yescharacteristic
126D25—22no—
127D25—22no—
128D25—42no—
129C23×C5C22×D511yescharacteristic
130C22×D5—51no—
131C22×D5—51no—
132C22×D5—52no—
133C22×D5—102no—
134C22×D5—102no—
135C22×D5—102no—
136C5×Q8—22no—
137C2×C4×C5—22no—
138C2×C4×C5C22×D512yes—
139C2×C4×C5C22×D512yes—
140C2×C4×C5C22×D511yescharacteristic
141C2×C4×C5C22×D511yescharacteristic
142C22⋊D5—104no—
143C2×Dic5—51no—
144C2×Dic5—52no—
145C2×Dic5—51no—
146C2×Dic5—102no—
147D20—104no—
148D20—104no—
149D20—102no—
150C4×D5—54no—
151C4×D5—102no—
152C4×D5—104no—
153C4×D5—102no—
154C4×D5—102no—
155C2×#41—251no—
156C23×C4—251no—
157#123—252no—
158C22⋊Q8—251no—
159C22⋊Q8D2511yescharacteristic, complemented
160#121—251no—
161#121—251no—
162#120—252no—
163C4×D4—252no—
164C4×D4—252no—
165#117—251no—
166C25#52511yescharacteristic, complemented, Sylow
167C22×C5—21no—
168C22×C5—21no—
169C22×C5#71312yes—
170C22×C5C23×D511yescharacteristic
171C2×D5—52no—
172C2×D5—52no—
173C2×D5—108no—
174C2×D5—101no—
175C2×D5—101no—
176C2×D5—101no—
177C2×D5—101no—
178C2×D5—102no—
179C2×D5—102no—
180C2×D5—102no—
181C4×C5—21no—
182C4×C5—22no—
183C4×C5—22no—
184C4×C5D4×D512yes—
185Dic5—52no—
186Dic5—101no—
187Dic5—102no—
188C24—251no—
189#41—504no—
190C2×Q8C2×D2511yescharacteristic, complemented
191C2×D4—252no—
192C2×D4—252no—
193C2×D4—252no—
194C2×D4—251no—
195C22×C4—251no—
196C22×C4—251no—
197C22×C4—251no—
198C22×C4—252no—
199C22×C4—251no—
200C22×C4—251no—
201C22×C4—504no—
202C22×C4C2×D2511yescharacteristic
203C4⋊C4—251no—
204C4⋊C4—252no—
205C4⋊C4C2×D2511yescharacteristic, complemented
206C4⋊C4C2×D2512yescomplemented
207C22⋊C4—252no—
208C22⋊C4—252no—
209C22⋊C4—252no—
210C22⋊C4—252no—
211C22⋊C4C2×D2512yes—
212C42—252no—
213C2×C5—22no—
214C2×C5C2×D4×D511yescharacteristic
215C2×C5C2×#71311yescharacteristic
216C2×C5D5×#4111yescharacteristic
217D5—102no—
218D5—102no—
219D5—202no—
220C23—251no—
221C23—251no—
222C23—252no—
223C23—502no—
224C23—502no—
225C23—502no—
226C23C22×D2511yescharacteristic
227Q8—22no—
228D4—504no—
229D4—504no—
230D4—504no—
231D4—502no—
232C2×C4—22no—
233C2×C4—251no—
234C2×C4—254no—
235C2×C4—252no—
236C2×C4—251no—
237C2×C4—502no—
238C2×C4—504no—
239C2×C4—502no—
240C2×C4—502no—
241C2×C4—502no—
242C2×C4C22×D2512yes—
243C2×C4C22×D2511yescharacteristic
244C2×C4C22×D2512yes—
245C2×C4C22×D2511yescharacteristic
246C5#6596111yescharacteristic
247C22—21no—
248C22—21no—
249C22—252no—
250C22—252no—
251C22—508no—
252C22—501no—
253C22—501no—
254C22—501no—
255C22—501no—
256C22—502no—
257C22—502no—
258C22—502no—
259C22#8774812yescomplemented
260C22C23×D2511yescharacteristic
261C4—21no—
262C4—22no—
263C4—22no—
264C4—252no—
265C4—501no—
266C4—502no—
267C4D4×D2512yes—
268C2—22no—
269C2—502no—
270C2—502no—
271C2—1002no—
272C2C2×D4×D2511yescharacteristic
273C2C2×#8774811yescharacteristic
274C2#41×D2511yescharacteristic
subgroupG/H
centerC22C23×D25
commutator, cosocleC22×C25C24
Frattini, socleC22×C5C23×D5
FittingC25×C22⋊Q8C2
hypercenterC22⋊Q8D25
hypocenterC25#525
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×C4×D25C2semidirect
3. C25×C22⋊Q8C2semidirect
4. #11716508C2semidirect
5. #11716492C2semidirect
6. #11716457C2semidirect
7. #11716456C2semidirect
8. #11716455C2semidirect
9. #11716454C2semidirect
10. #11716444C2semidirect
11. #11716443C2semidirect
13. C2×Q8×C25C22semidirect
20. C2×D100C22semidirect
26. C2×C4×D25C22semidirect
27. C4⋊C4×C25C22semidirect
28. C4⋊C4×C25C22semidirect
32. #87720C22semidirect
33. #87720C22semidirect
34. C4⋊Dic25C22semidirect
35. #87718C22semidirect
36. C4×Dic25C22semidirect
63. C4×D25D4semidirect
77. C22×C25#41semidirect
88. C2×D25#41semidirect
159. C22⋊Q8D25semidirect
166. C25#525semidirect
190. C2×Q8C2×D25semidirect
205. C4⋊C4C2×D25semidirect
206. C4⋊C4C2×D25semidirect
259. 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×C4×D252
C25×C22⋊Q82
#117165082
#117164922
#117164572
#117164562
#117164552
#117164542
#117164442
#117164432
#659615