Finite Group Databasebeta

← #13105147

Subgroups — #13105147

Order 1600.

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

normal subgroups only

This group has 618 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
1C22×#6039C211yesmaximal, characteristic, complemented
2C2×#87879C211yesmaximal, characteristic
3#11717206C211yesmaximal, characteristic, complemented
4C5×#4032C211yesmaximal, characteristic, complemented
5C5×#4023C211yesmaximal, characteristic, complemented
6#11717084C211yesmaximal, characteristic, complemented
7#11717080C211yesmaximal, characteristic, complemented
8#11717036C212yesmaximal, complemented
9#11717009C212yesmaximal, complemented
10#11717007C212yesmaximal, complemented
11#11716970C212yesmaximal, complemented
12C2×C22⋊(C5⋊D5)C2211yescharacteristic
13C22×#1039C2211yescharacteristic
14C2×C5×C22⋊D5C2211yescharacteristic
15C22×C5×Dic5C2211yescharacteristic
16C22×C5×Dic5C2211yescharacteristic
17C2×#6039—24no—
18C2×#6039C2212yes—
19C2×#6039C2211yescharacteristic
20C2×#6039C2212yes—
21C2×#6038C2211yescharacteristic
22C2×#6036C2211yescharacteristic
23#87879—24no—
24C2×D5×Dic5C2211yescharacteristic
25C52×C22⋊C4C2211yescharacteristic
26#87808C2212yes—
27C4⋊#1039C2212yes—
28C5×C22⋊Dic5C2211yescharacteristic
29C5×#685C2212yes—
30C5×C4⋊Dic5C2212yes—
31C5×#683C2212yes—
32C4×C5×Dic5C2212yes—
33#87780C2212yes—
34#87780C2212yes—
35#87779C2212yes—
36#87778C2212yes—
37#65982—51nomaximal
38#65860—51nomaximal
39C23×C52C2311yescharacteristic
40C22×C5⋊D5C2311yescharacteristic
41C22×C5×D5C2311yescharacteristic
42C2×C4×C52C2312yes—
43C22⋊(C5⋊D5)—22no—
44C2×#1039—22no—
45C2×#1039C2312yes—
46C2×#1039C2311yescharacteristic
47C5×C22⋊D5—22no—
48C2×C5×Dic5—22no—
49C2×C5×Dic5D414yes—
50C2×C5×Dic5C2311yescharacteristic
51C2×C5×Dic5C2312yes—
52C2×C5×Dic5C2312yes—
53C2×C5×Dic5C2311yescharacteristic
54C2×C5×Dic5C2311yescharacteristic
55#6039—22no—
56#6039—24no—
57#6039—24no—
58#6038—22no—
59#6036—22no—
60D5×Dic5—22no—
61C2×#711—51no—
62C2×#709—51no—
63C22×Dic10—51no—
64C22×Dic10—51no—
65C5×#123—51no—
66C5×#117—51no—
67#4081—51no—
68#4075—51no—
69#4043—52no—
70#4043—52no—
71#4042—52no—
72#4034—52no—
73#4032—51no—
74#4032—51no—
75#4032—51no—
76#4032—51no—
77#4032—102no—
78#4032D511yescharacteristic, complemented
79#4031—52no—
80#4024—52no—
81#4023D511yescharacteristic, complemented
82#4021—52no—
83#4015—52no—
84C22×C52—22no—
85C22×C52C2×D412yes—
86C22×C52C2411yescharacteristic
87C2×C5⋊D5—21no—
88C2×C5⋊D5—22no—
89C2×C5×D5—21no—
90C2×C5×D5—22no—
91C4×C52—22no—
92#1039—21no—
93#1039—22no—
94C5×Dic5—21no—
95C5×Dic5—22no—
96C5×Dic5—22no—
97C5×Dic5C2×D412yes—
98C5×Dic5C2×D412yes—
99C2×C5×D4—51no—
100C22×C4×C5—51no—
101C22×C4×C5—51no—
102C2×C22⋊D5—51no—
103C2×C22⋊D5—51no—
104C2×C22⋊D5—51no—
105C2×C22⋊D5—102no—
106C2×C22⋊D5C2×D511yescharacteristic
107C22×Dic5—51no—
108C22×Dic5—51no—
109C22×Dic5—51no—
110C22×Dic5—102no—
111C22×Dic5C2×D511yescharacteristic
112C22×Dic5C2×D511yescharacteristic
113#711—104no—
114#709—104no—
115C2×D20—51no—
116C2×C4×D5—51no—
117C2×C4×D5—51no—
118C2×C4×D5—51no—
119C2×Dic10—52no—
120C2×Dic10—52no—
121C2×Dic10—52no—
122C2×Dic10—52no—
123C2×Dic10—51no—
124C2×Dic10—51no—
125C2×Dic10—104no—
126C2×Dic10—104no—
127C5×C4⋊C4—52no—
128C5×C4⋊C4—52no—
129C5×C22⋊C4—22no—
130C5×C22⋊C4—51no—
131C5×C22⋊C4—52no—
132C5×C22⋊C4C2×D511yescharacteristic
133C5×C22⋊C4C2×D511yescharacteristic
134C42×C5—52no—
135C22⋊Dic5—52no—
136C22⋊Dic5C2×D511yescharacteristic
137#685—52no—
138#685—52no—
139#685—52no—
140#685—52no—
141#685—52no—
142#685—104no—
143#685C2×D512yes—
144C4⋊Dic5—52no—
145C4⋊Dic5—52no—
146C4⋊Dic5—52no—
147C4⋊Dic5—52no—
148C4⋊Dic5—104no—
149C4⋊Dic5C2×D512yes—
150#683—52no—
151#683—52no—
152#683C2×D512yes—
153C4×Dic5C2×D512yes—
154#536—251noSylow
155C2×C52—22no—
156C2×C52C22×D411yescharacteristic
157C2×C522−1+412yes—
158C5⋊D5—41no—
159C5×D5—41no—
160C23×C5—22no—
161C23×C5—51no—
162C23×C5C22×D511yescharacteristic
163C23×C5C22×D511yescharacteristic
164C22×D5—51no—
165C22×D5—51no—
166C22×D5—102no—
167C22×D5C22×D511yescharacteristic
168C5×D4—102no—
169C2×C4×C5—24no—
170C2×C4×C5—54no—
171C2×C4×C5—51no—
172C2×C4×C5—51no—
173C2×C4×C5—51no—
174C2×C4×C5—52no—
175C2×C4×C5—52no—
176C2×C4×C5—102no—
177C2×C4×C5C22×D512yes—
178C2×C4×C5C22×D512yes—
179C22⋊D5—22no—
180C22⋊D5—102no—
181C22⋊D5—102no—
182C22⋊D5—102no—
183C22⋊D5—204no—
184C2×Dic5—22no—
185C2×Dic5—51no—
186C2×Dic5—52no—
187C2×Dic5—51no—
188C2×Dic5—52no—
189C2×Dic5—102no—
190C2×Dic5—104no—
191C2×Dic5—102no—
192C2×Dic5—102no—
193C2×Dic5—102no—
194C2×Dic5—204no—
195C2×Dic5D2014yes—
196C2×Dic5C22×D512yes—
197C2×Dic5C22×D511yescharacteristic
198C2×Dic5C22×D512yes—
199C2×Dic5C22×D511yescharacteristic
200C2×Dic5C22×D511yescharacteristic
201D20—102no—
202C4×D5—102no—
203C4×D5—102no—
204C4×D5—102no—
205Dic10—104no—
206Dic10—104no—
207Dic10—102no—
208Dic10—104no—
209Dic10—104no—
210Dic10—102no—
211C2×#41—251no—
212C22×Q8—251no—
213C4⋊Q8—252no—
214#124—252no—
215#123—51no—
216#123—251no—
217#123—251no—
218#123—251no—
219C22⋊Q8—252no—
220C22⋊Q8—252no—
221#117—51no—
222C52#53611yescharacteristic, complemented, Sylow
223C22×C5—22no—
224C22×C5—22no—
225C22×C5—24no—
226C22×C5—22no—
227C22×C5—44no—
228C22×C5—101no—
229C22×C5—102no—
230C22×C5C2×D2012yes—
231C22×C5D4×D512yes—
232C22×C5C23×D511yescharacteristic
233C22×C5C23×D511yescharacteristic
234C2×D5—21no—
235C2×D5—22no—
236C2×D5—101no—
237C2×D5—101no—
238C2×D5—102no—
239C2×D5—102no—
240C2×D5—202no—
241C2×D5—204no—
242C4×C5—22no—
243C4×C5—22no—
244C4×C5—44no—
245C4×C5—52no—
246C4×C5—52no—
247C4×C5—102no—
248C4×C5—101no—
249C4×C5—102no—
250Dic5—22no—
251Dic5—21no—
252Dic5—22no—
253Dic5—102no—
254Dic5—101no—
255Dic5—102no—
256Dic5—101no—
257Dic5—204no—
258Dic5—202no—
259Dic5C2×D2012yes—
260Dic5C2×D2012yes—
261#41—504no—
262C2×Q8—252no—
263C2×Q8—252no—
264C2×Q8—251no—
265C2×Q8—504no—
266C2×D4—51no—
267C2×D4—251no—
268C2×D4—251no—
269C22×C4—51no—
270C22×C4—51no—
271C22×C4—251no—
272C22×C4—251no—
273C22×C4—251no—
274C4⋊C4—52no—
275C4⋊C4—52no—
276C4⋊C4—252no—
277C4⋊C4—252no—
278C4⋊C4—252no—
279C22⋊C4—51no—
280C22⋊C4—52no—
281C22⋊C4—252no—
282C22⋊C4—252no—
283C22⋊C4—252no—
284C22⋊C4D5211yescharacteristic
285C42—52no—
286C2×C5—22no—
287C2×C5—22no—
288C2×C5—22no—
289C2×C5—24no—
290C2×C5—44no—
291C2×C5—201no—
292C2×C5C22×D2011yescharacteristic
293C2×C5C2×D4×D511yescharacteristic
294C2×C5#415012yes—
295C2×C5#415012yes—
296D5—41no—
297D5—201no—
298D5—201no—
299D5—402no—
300C23—51no—
301C23—251no—
302C23C2×D5211yescharacteristic
303Q8—504no—
304Q8—504no—
305Q8—502no—
306D4—102no—
307D4—502no—
308D4—502no—
309C2×C4—54no—
310C2×C4—51no—
311C2×C4—51no—
312C2×C4—52no—
313C2×C4—52no—
314C2×C4—51no—
315C2×C4—102no—
316C2×C4—252no—
317C2×C4—251no—
318C2×C4—502no—
319C2×C4—502no—
320C2×C4—502no—
321C2×C4C2×D5212yes—
322C5—22no—
323C5#6586011yescharacteristic, complemented
324C5#6598211yescharacteristic, complemented
325C22—22no—
326C22—101no—
327C22—102no—
328C22—501no—
329C22—502no—
330C22D5×D2012yes—
331C22C22×D5211yescharacteristic
332C4—22no—
333C4—52no—
334C4—52no—
335C4—102no—
336C4—101no—
337C4—102no—
338C4—501no—
339C4—502no—
340C2—22no—
341C2—201no—
342C2—1001no—
343C2C2×D5×D2011yescharacteristic
344C2#1171747412yes—
subgroupG/H
center, FrattiniC22C22×D52
commutator, socle, cosocleC22×C52C24
FittingC52×C22⋊C4C22
prime radicalC2×#87879C2
hypercenterC22⋊C4D52
hypocenterC52#536
LCSG ▷ C22×C52 ▷ C52
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C22×#6039C2semidirect
3. #11717206C2semidirect
4. C5×#4032C2semidirect
5. C5×#4023C2semidirect
6. #11717084C2semidirect
7. #11717080C2semidirect
8. #11717036C2semidirect
9. #11717009C2semidirect
10. #11717007C2semidirect
11. #11716970C2semidirect
78. #4032D5semidirect
81. #4023D5semidirect
222. C52#536semidirect
323. C5#65860semidirect
324. C5#65982semidirect

Maximal subgroups

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

subgroupindex
C22×#60392
C2×#878792
#117172062
C5×#40322
C5×#40232
#117170842
#117170802
#117170362
#117170092
#117170072
#117169702
#659825
#658605