Finite Group Databasebeta

← #13105437

Subgroups — #13105437

Order 1600.

650 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 650 conjugacy classes of subgroups — too many to draw. Only its 140 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×#87870C211yesmaximal, characteristic, complemented
2C2×D5×Dic10C211yesmaximal, characteristic, complemented
3#11717207C211yesmaximal, characteristic, complemented
4C5×#4043C211yesmaximal, characteristic, complemented
5C5×#4041C211yesmaximal, characteristic, complemented
6#11717033C212yesmaximal, complemented
7C4×#6037C211yesmaximal, characteristic, complemented
8#11717017C212yesmaximal, complemented
9#11717014C212yesmaximal, complemented
10#11717004C212yesmaximal, complemented
11#11716981C211yesmaximal, characteristic, complemented
12C2×#6045C2211yescharacteristic
13C2×C5×D20C2211yescharacteristic
14C2×C5×D20C2212yes—
15C2×C4×C5×D5C2211yescharacteristic
16C2×C4×C5×D5C2211yescharacteristic
17C2×C5×Dic10C2211yescharacteristic
18C2×#6039C2212yes—
19C2×#6037C2212yes—
20C2×#6037C2211yescharacteristic
21C2×D5×Dic5C2212yes—
22C2×D5×Dic5C2212yes—
23#87870—24no—
24D5×Dic10—24no—
25C52×C4⋊C4C2211yescharacteristic
26#87806C2212yes—
27C4×#1039C2211yescharacteristic
28C4×#1039C2212yes—
29C5×#685C2212yes—
30C5×#685C2212yes—
31C5×C4⋊Dic5C2212yes—
32#87781C2211yescharacteristic
33#87778C2212yes—
34#87778C2212yes—
35#87778C2211yescharacteristic
36#87778C2211yescharacteristic
37#65967—51nomaximal
38#65932—51nomaximal
39C22×C5×D5C2312yes—
40C22×C5×D5C2311yescharacteristic
41C22×C5×D5C2311yescharacteristic
42C2×C4×C52C2312yes—
43C2×C4×C52C2311yescharacteristic
44C2×#1039C2311yescharacteristic
45C2×#1039C2311yescharacteristic
46C2×#1039C2312yes—
47#6045D414yescomplemented
48C2×C5×Dic5C2311yescharacteristic
49C2×C5×Dic5C2312yes—
50C2×C5×Dic5C2311yescharacteristic
51C5×D20—24no—
52C5×D20—22no—
53C4×C5×D5—22no—
54C4×C5×D5—22no—
55C5×Dic10—22no—
56#6039—24no—
57#6037—22no—
58#6037—24no—
59D5×Dic5—24no—
60D5×Dic5—24no—
61C2×Q8×D5—51no—
62C2×#711—51no—
63C2×#709—51no—
64C22×Dic10—51no—
65C5×C22⋊Q8—51no—
66C5×#121—51no—
67#4093—51no—
68#4085—52no—
69#4082—51no—
70D4×Dic5—52no—
71C4×C22⋊D5—51no—
72C4×C22⋊D5—51no—
73#4043D511yescharacteristic, complemented
74#4042—52no—
75#4041D511yescharacteristic, complemented
76#4039—52no—
77#4033—51no—
78#4033—51no—
79#4033—102no—
80#4031—52no—
81#4031—52no—
82#4030—52no—
83#4024—52no—
84C22×C52C2411yescharacteristic
85C2×C5×D5—21no—
86C2×C5×D5—22no—
87C2×C5×D5—21no—
88C2×C5×D5—21no—
89C2×C5×D5—22no—
90C2×C5×D5—21no—
91C2×C5×D5—21no—
92C2×C5×D5—22no—
93C2×C5×D5#4112yescomplemented
94C4×C52—22no—
95C4×C52C2×D412yes—
96#1039—21no—
97#1039—21no—
98#1039C2×D414yes—
99C5×Dic5—21no—
100C5×Dic5—22no—
101C5×Dic5—21no—
102C2×C5×Q8—51no—
103C2×C5×D4—51no—
104C2×C5×D4—52no—
105C22×C4×C5—51no—
106C22×C4×C5—51no—
107C2×C22⋊D5—52no—
108C2×C22⋊D5—52no—
109C2×C22⋊D5—51no—
110C2×C22⋊D5—51no—
111C22×Dic5—52no—
112C22×Dic5—52no—
113Q8×D5—104no—
114#711—104no—
115#709—104no—
116C2×D20C2×D511yescharacteristic
117C2×D20C2×D512yes—
118C2×C4×D5—52no—
119C2×C4×D5—52no—
120C2×C4×D5C2×D511yescharacteristic
121C2×C4×D5C2×D511yescharacteristic
122C2×Dic10—51no—
123C2×Dic10—51no—
124C2×Dic10—52no—
125C2×Dic10—52no—
126C2×Dic10—102no—
127C2×Dic10—104no—
128C2×Dic10C2×D511yescharacteristic
129C5×C4⋊C4—22no—
130C5×C4⋊C4—52no—
131C5×C4⋊C4C2×D511yescharacteristic
132C5×C4⋊C4C2×D511yescharacteristic
133C5×C22⋊C4—52no—
134C5×C22⋊C4—52no—
135C22⋊Dic5—52no—
136C22⋊Dic5—51no—
137C22⋊Dic5—52no—
138C22⋊Dic5—51no—
139#685—52no—
140#685—51no—
141#685—52no—
142#685—51no—
143#685C2×D512yes—
144#685C2×D512yes—
145C4⋊Dic5C2×D512yes—
146#683—52no—
147#683—52no—
148#683—51no—
149#683—51no—
150#683—104no—
151C4×Dic5—52no—
152C4×Dic5—52no—
153C4×Dic5—51no—
154C4×Dic5—51no—
155C4×Dic5—104no—
156C4×Dic5—102no—
157#548—251noSylow
158C2×C52C22×D411yescharacteristic
159C2×C52C2×#4111yescharacteristic
160C2×C522−1+411yescharacteristic
161C5×D5—22no—
162C5×D5—42no—
163C5×D5—41no—
164C23×C5—51no—
165C23×C5—52no—
166C23×C5—51no—
167C22×D5C22×D512yes—
168C22×D5C22×D511yescharacteristic
169C22×D5C22×D511yescharacteristic
170C5×Q8—102no—
171C5×D4—104no—
172C5×D4—102no—
173C2×C4×C5—22no—
174C2×C4×C5—24no—
175C2×C4×C5—51no—
176C2×C4×C5—51no—
177C2×C4×C5—52no—
178C2×C4×C5—102no—
179C2×C4×C5—102no—
180C2×C4×C5C22×D512yes—
181C2×C4×C5C22×D512yes—
182C2×C4×C5C22×D511yescharacteristic
183C2×C4×C5C22×D511yescharacteristic
184C22⋊D5—104no—
185C22⋊D5—104no—
186C22⋊D5—102no—
187C22⋊D5—102no—
188C2×Dic5—52no—
189C2×Dic5—52no—
190C2×Dic5—51no—
191C2×Dic5—51no—
192C2×Dic5—51no—
193C2×Dic5—51no—
194C2×Dic5—102no—
195C2×Dic5—104no—
196C2×Dic5—102no—
197C2×Dic5—104no—
198C2×Dic5—104no—
199C2×Dic5C22×D511yescharacteristic
200C2×Dic5C22×D511yescharacteristic
201C2×Dic5C22×D512yes—
202D20—24no—
203D20—22no—
204C4×D5—22no—
205C4×D5—22no—
206C4×D5—104no—
207C4×D5—104no—
208Dic10—22no—
209Dic10—54no—
210Dic10—54no—
211Dic10—108no—
212Dic10—104no—
213Dic10—104no—
214C2×#41—251no—
215C22×Q8—251no—
216#124—252no—
217#124—251no—
218C22⋊Q8—51no—
219C22⋊Q8—252no—
220#121—51no—
221#121—252no—
222C4×Q8—251no—
223C4×D4—252no—
224C4×D4—251no—
225C52#54811yescharacteristic, complemented, Sylow
226C22×C5—22no—
227C22×C5—52no—
228C22×C5—101no—
229C22×C5—101no—
230C22×C5—102no—
231C22×C5—101no—
232C22×C5—101no—
233C22×C5—102no—
234C22×C5—102no—
235C22×C5—101no—
236C22×C5C23×D511yescharacteristic
237C22×C5C23×D511yescharacteristic
238C2×D5—21no—
239C2×D5—21no—
240C2×D5—21no—
241C2×D5—22no—
242C2×D5—22no—
243C2×D5—21no—
244C2×D5—21no—
245C2×D5—22no—
246C2×D5#71112yescomplemented
247C4×C5—22no—
248C4×C5—22no—
249C4×C5—24no—
250C4×C5—44no—
251C4×C5—101no—
252C4×C5—102no—
253C4×C5—101no—
254C4×C5D4×D512yes—
255C4×C5D4×D512yes—
256Dic5—22no—
257Dic5—21no—
258Dic5—21no—
259Dic5—54no—
260Dic5—54no—
261Dic5—108no—
262Dic5—104no—
263Dic5—101no—
264Dic5—101no—
265Dic5—101no—
266Dic5—101no—
267Dic5—202no—
268#41—504no—
269C2×Q8—51no—
270C2×Q8—252no—
271C2×Q8—251no—
272C2×Q8—504no—
273C2×D4—52no—
274C2×D4—51no—
275C2×D4—251no—
276C2×D4—252no—
277C22×C4—51no—
278C22×C4—51no—
279C22×C4—252no—
280C22×C4—252no—
281C4⋊C4—52no—
282C4⋊C4—251no—
283C4⋊C4—252no—
284C4⋊C4D5211yescharacteristic
285C22⋊C4—52no—
286C22⋊C4—52no—
287C22⋊C4—251no—
288C22⋊C4—251no—
289C22⋊C4—252no—
290C22⋊C4—252no—
291C42—252no—
292C42—251no—
293C2×C5—22no—
294C2×C5—22no—
295C2×C5—22no—
296C2×C5—102no—
297C2×C5—201no—
298C2×C5—202no—
299C2×C5C2×D4×D511yescharacteristic
300C2×C5C2×D4×D511yescharacteristic
301C2×C5C2×#71111yescharacteristic
302C2×C5#414711yescharacteristic
303C2×C5D5×#4111yescharacteristic
304C2×C5#415011yescharacteristic
305D5—22no—
306D5—42no—
307D5—41no—
308C23—51no—
309C23—51no—
310C23—52no—
311Q8—102no—
312Q8—254no—
313Q8—504no—
314D4—104no—
315D4—102no—
316D4—504no—
317D4—502no—
318C2×C4—51no—
319C2×C4—52no—
320C2×C4—51no—
321C2×C4—102no—
322C2×C4—102no—
323C2×C4—252no—
324C2×C4—251no—
325C2×C4—251no—
326C2×C4—504no—
327C2×C4—504no—
328C2×C4C2×D5212yes—
329C2×C4C2×D5211yescharacteristic
330C5—22no—
331C5#6593211yescharacteristic, complemented
332C5#6596711yescharacteristic, complemented
333C22—52no—
334C22—101no—
335C22—101no—
336C22—101no—
337C22—102no—
338C22—102no—
339C22—101no—
340C22—101no—
341C22—102no—
342C22C22×D5211yescharacteristic
343C4—22no—
344C4—101no—
345C4—101no—
346C4—102no—
347C4—254no—
348C4—501no—
349C4—501no—
350C4#8787712yes—
351C2—102no—
352C2—201no—
353C2—202no—
354C2C2×#8787711yescharacteristic
355C2D5×#71111yescharacteristic
356C2#1171748011yescharacteristic
subgroupG/H
center, FrattiniC22C22×D52
commutator, socle, cosocleC22×C52C24
FittingC52×C4⋊C4C22
hypercenterC4⋊C4D52
hypocenterC52#548
LCSG ▷ C22×C52 ▷ C52
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#87870C2semidirect
2. C2×D5×Dic10C2semidirect
3. #11717207C2semidirect
4. C5×#4043C2semidirect
5. C5×#4041C2semidirect
6. #11717033C2semidirect
7. C4×#6037C2semidirect
8. #11717017C2semidirect
9. #11717014C2semidirect
10. #11717004C2semidirect
11. #11716981C2semidirect
47. #6045D4semidirect
73. #4043D5semidirect
75. #4041D5semidirect
93. C2×C5×D5#41semidirect
225. C52#548semidirect
246. C2×D5#711semidirect
331. C5#65932semidirect
332. C5#65967semidirect

Maximal subgroups

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

subgroupindex
C2×#878702
C2×D5×Dic102
#117172072
C5×#40432
C5×#40412
#117170332
C4×#60372
#117170172
#117170142
#117170042
#117169812
#659675
#659325