Finite Group Databasebeta

← #13105148

Subgroups — #13105148

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×#87880C211yesmaximal, characteristic, complemented
2C22×D5×Dic5C211yesmaximal, characteristic, complemented
3#11717197C211yesmaximal, characteristic, complemented
4C5×#4032C211yesmaximal, characteristic, complemented
5C5×#4028C211yesmaximal, characteristic, complemented
6#11717094C211yesmaximal, characteristic, complemented
7#11717079C211yesmaximal, characteristic, complemented
8#11717020C212yesmaximal, complemented
9#11717017C212yesmaximal, complemented
10#11717014C212yesmaximal, complemented
11#11717004C212yesmaximal, complemented
12C23×C5×D5C2211yescharacteristic
13C22×#1039C2211yescharacteristic
14C2×C5×C22⋊D5C2211yescharacteristic
15C2×C5×C22⋊D5C2211yescharacteristic
16C22×C5×Dic5C2211yescharacteristic
17C2×C5×D20C2212yes—
18C2×#6039C2211yescharacteristic, complemented
19C2×#6037C2212yes—
20C2×#6037C2211yescharacteristic, complemented
21#87880—24no—
22C2×D5×Dic5—24no—
23C2×D5×Dic5C2212yes—
24C2×D5×Dic5C2212yes—
25C2×D5×Dic5C2211yescharacteristic, complemented
26C2×D5×Dic5C2211yescharacteristic, complemented
27C52×C22⋊C4C2211yescharacteristic
28C22⋊#1039C2211yescharacteristic
29#87806C2212yescomplemented
30C4×#1039C2212yescomplemented
31C5×#685C2212yes—
32C5×#685C2212yescomplemented
33C5×C4⋊Dic5C2212yescomplemented
34#87781C2212yescomplemented
35#87778C2212yes—
36#87778C2212yescomplemented
37#65990—51nomaximal
38#65920—51nomaximal
39C23×C52C2311yescharacteristic
40C22×C5×D5—22no—
41C22×C5×D5—24no—
42C22×C5×D5C2311yescharacteristic
43C22×C5×D5C2311yescharacteristic
44C22×C5×D5C2312yes—
45C2×C4×C52C2312yes—
46C2×#1039D414yescomplemented
47C2×#1039C2312yes—
48C2×#1039C2311yescharacteristic
49C2×#1039C2311yescharacteristic
50C5×C22⋊D5—22no—
51C5×C22⋊D5—22no—
52C2×C5×Dic5—22no—
53C2×C5×Dic5C2312yes—
54C2×C5×Dic5C2311yescharacteristic
55C2×C5×Dic5C2311yescharacteristic
56C5×D20—24no—
57#6039—22no—
58#6037—24no—
59#6037—22no—
60D5×Dic5—24no—
61D5×Dic5—24no—
62D5×Dic5—22no—
63D5×Dic5—22no—
64C23×Dic5—51no—
65C2×#711—51no—
66C2×#711—51no—
67C22×C4×D5—51no—
68C5×#123—51no—
69C5×#120—51no—
70#4099—51no—
71#4085—52no—
72#4083—51no—
73#4081—51no—
74D4×Dic5—52no—
75#4075—51no—
76#4042—52no—
77#4039—52no—
78#4032D511yescharacteristic, complemented
79#4030—52no—
80#4028D511yescharacteristic, complemented
81#4027—52no—
82#4027—52no—
83#4024—52no—
84#4023—51no—
85#4023—51no—
86#4023—102no—
87C22×C52—22no—
88C22×C52C2×D412yes—
89C22×C52C2411yescharacteristic
90C2×C5×D5—28no—
91C2×C5×D5—22no—
92C2×C5×D5—21no—
93C2×C5×D5—21no—
94C2×C5×D5—22no—
95C2×C5×D5—22no—
96C2×C5×D5—22no—
97C2×C5×D5#4114yescomplemented
98C4×C52—22no—
99#1039—22no—
100#1039C2×D412yes—
101#1039C2×D412yes—
102C5×Dic5—22no—
103C5×Dic5—21no—
104C5×Dic5—21no—
105C24×C5—51no—
106C23×D5C2×D511yescharacteristic
107C2×C5×D4—51no—
108C2×C5×D4—51no—
109C2×C5×D4—52no—
110C22×C4×C5—51no—
111C2×C22⋊D5—52no—
112C2×C22⋊D5—51no—
113C2×C22⋊D5—51no—
114C2×C22⋊D5—52no—
115C2×C22⋊D5C2×D511yescharacteristic
116C2×C22⋊D5C2×D511yescharacteristic
117C22×Dic5—51no—
118C22×Dic5—51no—
119C22×Dic5—51no—
120C22×Dic5—51no—
121C22×Dic5—52no—
122C22×Dic5—52no—
123C22×Dic5—102no—
124C22×Dic5—104no—
125C22×Dic5C2×D511yescharacteristic
126#711—104no—
127#711—104no—
128C2×D20C2×D512yes—
129C2×C4×D5—52no—
130C2×C4×D5—52no—
131C2×C4×D5—51no—
132C2×C4×D5—51no—
133C2×C4×D5—104no—
134C2×Dic10—51no—
135C2×Dic10—51no—
136C5×C4⋊C4—52no—
137C5×C22⋊C4—22no—
138C5×C22⋊C4—52no—
139C5×C22⋊C4—52no—
140C5×C22⋊C4C2×D511yescharacteristic
141C5×C22⋊C4C2×D511yescharacteristic
142C22⋊Dic5—51no—
143C22⋊Dic5—51no—
144C22⋊Dic5—52no—
145C22⋊Dic5—52no—
146C22⋊Dic5—102no—
147#685—52no—
148#685—52no—
149#685C2×D512yes—
150#685C2×D512yescomplemented
151C4⋊Dic5C2×D512yescomplemented
152#683—52no—
153#683—52no—
154#683—52no—
155#683—52no—
156#683—104no—
157C4×Dic5—52no—
158C4×Dic5—52no—
159C4×Dic5—104no—
160#525—251noSylow
161C2×C52—22no—
162C2×C52C22×D411yescharacteristic
163C2×C52C2×#4112yes—
164C5×D5—24no—
165C5×D5—41no—
166C5×D5—41no—
167C23×C5—22no—
168C23×C5—52no—
169C23×C5—51no—
170C23×C5—51no—
171C23×C5—104no—
172C23×C5—102no—
173C23×C5C22×D511yescharacteristic
174C23×C5C22×D511yescharacteristic
175C22×D5—22no—
176C22×D5—24no—
177C22×D5C22×D511yescharacteristic
178C22×D5C22×D511yescharacteristic
179C22×D5C22×D512yes—
180C5×D4—104no—
181C5×D4—102no—
182C5×D4—102no—
183C2×C4×C5—24no—
184C2×C4×C5—51no—
185C2×C4×C5—52no—
186C2×C4×C5—51no—
187C2×C4×C5—102no—
188C2×C4×C5C22×D512yes—
189C2×C4×C5C22×D512yes—
190C22⋊D5—22no—
191C22⋊D5—22no—
192C22⋊D5—102no—
193C22⋊D5—102no—
194C22⋊D5—104no—
195C22⋊D5—104no—
196C2×Dic5—22no—
197C2×Dic5—54no—
198C2×Dic5—54no—
199C2×Dic5—51no—
200C2×Dic5—51no—
201C2×Dic5—51no—
202C2×Dic5—51no—
203C2×Dic5—52no—
204C2×Dic5—52no—
205C2×Dic5—108no—
206C2×Dic5—102no—
207C2×Dic5—102no—
208C2×Dic5—104no—
209C2×Dic5—102no—
210C2×Dic5—102no—
211C2×Dic5—104no—
212C2×Dic5—104no—
213C2×Dic5C22×D511yescharacteristic
214C2×Dic5C22×D512yes—
215C2×Dic5C22×D511yescharacteristic
216D20—24no—
217C4×D5—104no—
218C4×D5—104no—
219C4×D5—102no—
220C4×D5—102no—
221Dic10—102no—
222Dic10—102no—
223C2×#41—251no—
224C23×C4—251no—
225#123—51no—
226#123—251no—
227C22⋊Q8—252no—
228#121—252no—
229#120—51no—
230#120—251no—
231C4×D4—252no—
232C4×D4—252no—
233#117—251no—
234C52#52511yescharacteristic, complemented, Sylow
235C22×C5—22no—
236C22×C5—22no—
237C22×C5—24no—
238C22×C5—22no—
239C22×C5—44no—
240C22×C5—54no—
241C22×C5—108no—
242C22×C5—102no—
243C22×C5—102no—
244C22×C5—101no—
245C22×C5—101no—
246C22×C5—102no—
247C22×C5—102no—
248C22×C5D4×D512yes—
249C22×C5D4×D512yes—
250C22×C5C23×D511yescharacteristic
251C22×C5C23×D511yescharacteristic
252C2×D5—28no—
253C2×D5—22no—
254C2×D5—22no—
255C2×D5—21no—
256C2×D5—21no—
257C2×D5—22no—
258C2×D5—22no—
259C2×D5#71114yescomplemented
260C4×C5—22no—
261C4×C5—22no—
262C4×C5—44no—
263C4×C5—101no—
264C4×C5—101no—
265C4×C5—102no—
266Dic5—21no—
267Dic5—21no—
268Dic5—22no—
269Dic5—52no—
270Dic5—52no—
271Dic5—52no—
272Dic5—52no—
273Dic5—104no—
274Dic5—104no—
275Dic5—102no—
276Dic5—102no—
277Dic5—204no—
278C24—51no—
279#41—504no—
280C2×Q8—251no—
281C2×D4—51no—
282C2×D4—51no—
283C2×D4—52no—
284C2×D4—252no—
285C2×D4—251no—
286C22×C4—51no—
287C22×C4—252no—
288C22×C4—251no—
289C22×C4—251no—
290C22×C4—252no—
291C22×C4—251no—
292C22×C4—504no—
293C4⋊C4—52no—
294C4⋊C4—252no—
295C4⋊C4—252no—
296C22⋊C4—52no—
297C22⋊C4—52no—
298C22⋊C4—252no—
299C22⋊C4—252no—
300C22⋊C4—251no—
301C22⋊C4D5211yescharacteristic
302C42—252no—
303C2×C5—22no—
304C2×C5—22no—
305C2×C5—22no—
306C2×C5—24no—
307C2×C5—44no—
308C2×C5—104no—
309C2×C5—201no—
310C2×C5—201no—
311C2×C5C2×D4×D511yescharacteristic
312C2×C5C2×D4×D511yescharacteristic
313C2×C5C2×#71112yes—
314C2×C5D5×#4112yes—
315D5—24no—
316D5—41no—
317D5—41no—
318C23—52no—
319C23—51no—
320C23—51no—
321C23—104no—
322C23—102no—
323C23C2×D5211yescharacteristic
324Q8—502no—
325D4—104no—
326D4—102no—
327D4—102no—
328D4—502no—
329D4—504no—
330C2×C4—51no—
331C2×C4—52no—
332C2×C4—51no—
333C2×C4—102no—
334C2×C4—254no—
335C2×C4—251no—
336C2×C4—251no—
337C2×C4—252no—
338C2×C4—504no—
339C2×C4—504no—
340C2×C4—502no—
341C2×C4—502no—
342C2×C4C2×D5212yes—
343C5—22no—
344C5#6592011yescharacteristic, complemented
345C5#6599011yescharacteristic, complemented
346C22—22no—
347C22—54no—
348C22—108no—
349C22—102no—
350C22—102no—
351C22—101no—
352C22—101no—
353C22—102no—
354C22—102no—
355C22#8787712yes—
356C22C22×D5211yescharacteristic
357C4—22no—
358C4—102no—
359C4—101no—
360C4—101no—
361C4—252no—
362C4—252no—
363C4—502no—
364C2—22no—
365C2—104no—
366C2—201no—
367C2—201no—
368C2C2×#8787711yescharacteristic
369C2D5×#71112yes—
subgroupG/H
center, FrattiniC22C22×D52
commutator, socle, cosocleC22×C52C24
FittingC52×C22⋊C4C22
hypercenterC22⋊C4D52
hypocenterC52#525
LCSG ▷ C22×C52 ▷ C52
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#87880C2semidirect
2. C22×D5×Dic5C2semidirect
3. #11717197C2semidirect
4. C5×#4032C2semidirect
5. C5×#4028C2semidirect
6. #11717094C2semidirect
7. #11717079C2semidirect
8. #11717020C2semidirect
9. #11717017C2semidirect
10. #11717014C2semidirect
11. #11717004C2semidirect
18. C2×#6039C22semidirect
20. C2×#6037C22semidirect
25. C2×D5×Dic5C22semidirect
26. C2×D5×Dic5C22semidirect
29. #87806C22semidirect
30. C4×#1039C22semidirect
32. C5×#685C22semidirect
33. C5×C4⋊Dic5C22semidirect
34. #87781C22semidirect
36. #87778C22semidirect
46. C2×#1039D4semidirect
78. #4032D5semidirect
80. #4028D5semidirect
97. C2×C5×D5#41semidirect
150. #685C2×D5semidirect
151. C4⋊Dic5C2×D5semidirect
234. C52#525semidirect
259. C2×D5#711semidirect
344. C5#65920semidirect
345. C5#65990semidirect

Maximal subgroups

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

subgroupindex
C2×#878802
C22×D5×Dic52
#117171972
C5×#40322
C5×#40282
#117170942
#117170792
#117170202
#117170172
#117170142
#117170042
#659905
#659205