Finite Group Databasebeta

← #13106419

Subgroups — #13106419

Order 1600.

736 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 736 conjugacy classes of subgroups — too many to draw. Only its 138 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×Q8×C5⋊D5C211yesmaximal, characteristic, complemented
2C2×#87870C211yesmaximal, characteristic, complemented
3C5×#4093C211yesmaximal, characteristic, complemented
4C5×#4092C211yesmaximal, characteristic, complemented
5#11717034C211yesmaximal, characteristic, complemented
6#11717031C212yesmaximal, complemented
7#11717030C212yesmaximal, complemented
8C4×#6038C211yesmaximal, characteristic, complemented
9#11717015C212yesmaximal, complemented
10#11716999C212yesmaximal, complemented
11#11716962C211yesmaximal, characteristic, complemented
12C2×C52×Q8C2211yescharacteristic
13Q8×C5⋊D5—24no—
14C2×C4×C5⋊D5C2212yes—
15C2×C4×C5⋊D5C2211yescharacteristic
16C2×#6045C2212yes—
17C2×#6045C2211yescharacteristic
18C2×C5×D20C2211yescharacteristic
19C2×C4×C5×D5C2211yescharacteristic
20C2×#6038C2212yes—
21C2×#6038C2211yescharacteristic
22C2×#6037C2212yes—
23C2×D5×Dic5C2212yes—
24#87870—24no—
25C5×#685C2212yes—
26C5×#685C2212yes—
27C5×#685C2212yes—
28C5×C4⋊Dic5C2211yescharacteristic
29C5×#683C2212yes—
30C4×C5×Dic5C2211yescharacteristic
31#87781C2212yes—
32#87780C2211yescharacteristic
33#87779C2211yescharacteristic
34#87778C2212yes—
35#87778C2211yescharacteristic
36Dic52C2212yes—
37#66011—51nomaximal
38#65967—51nomaximal
39C22×C5⋊D5C2311yescharacteristic
40C22×C5×D5C2312yes—
41C22×C5×D5C2311yescharacteristic
42C52×Q8—22no—
43C2×C4×C52C2312yes—
44C2×C4×C52C2311yescharacteristic
45C2×#1039C2311yescharacteristic
46C2×#1039C2312yes—
47C4×C5⋊D5—22no—
48C4×C5⋊D5—24no—
49#6045—24no—
50#6045D414yescomplemented
51C2×C5×Dic5C2311yescharacteristic
52C2×C5×Dic5C2312yes—
53C2×C5×Dic5C2311yescharacteristic
54C2×C5×Dic5C2311yescharacteristic
55C5×D20—22no—
56C4×C5×D5—22no—
57#6038—24no—
58#6038—22no—
59#6037—24no—
60D5×Dic5—24no—
61C2×Q8×D5—51no—
62C2×Q8×D5—51no—
63C2×Q8×D5—102no—
64C2×#711—51no—
65C2×#709—51no—
66C5×#124—51no—
67C5×C22⋊Q8—51no—
68#4093D511yescharacteristic, complemented
69#4092D511yescharacteristic, complemented
70#4084—51no—
71C4×C22⋊D5—51no—
72#4042—52no—
73#4041—51no—
74#4039—52no—
75#4033—51no—
76#4031—52no—
77#4031—52no—
78#4030—52no—
79#4030—52no—
80#4027—52no—
81#4024—52no—
82C4×D20—51no—
83C4×Dic10—51no—
84C22×C52C2411yescharacteristic
85C2×C5⋊D5—21no—
86C2×C5⋊D5—21no—
87C2×C5⋊D5#4112yescomplemented
88C2×C5×D5—22no—
89C2×C5×D5—21no—
90C2×C5×D5—22no—
91C2×C5×D5—21no—
92C2×C5×D5—21no—
93C2×C5×D5—22no—
94C4×C52—22no—
95C4×C52C2×D412yes—
96#1039—21no—
97#1039C2×D414yes—
98C5×Dic5—21no—
99C5×Dic5—22no—
100C5×Dic5—21no—
101C5×Dic5—21no—
102C2×C5×Q8—22no—
103C2×C5×Q8C2×D511yescharacteristic
104C2×C5×Q8C2×D511yescharacteristic
105C2×C5×D4—51no—
106C22×C4×C5—51no—
107C2×C22⋊D5—52no—
108C2×C22⋊D5—52no—
109C2×C22⋊D5—52no—
110C2×C22⋊D5—51no—
111C22×Dic5—52no—
112Q8×D5—104no—
113Q8×D5—104no—
114Q8×D5—1016no—
115#711—104no—
116#709—104no—
117C2×D20—51no—
118C2×D20—52no—
119C2×D20C2×D511yescharacteristic
120C2×C4×D5—52no—
121C2×C4×D5—51no—
122C2×C4×D5—52no—
123C2×C4×D5—52no—
124C2×C4×D5—51no—
125C2×C4×D5—104no—
126C2×C4×D5—102no—
127C2×C4×D5C2×D511yescharacteristic
128C2×Dic10—51no—
129C2×Dic10—51no—
130C2×Dic10—52no—
131C2×Dic10—52no—
132C2×Dic10—102no—
133C2×Dic10—104no—
134C5×C4⋊C4—51no—
135C5×C4⋊C4—52no—
136C5×C22⋊C4—52no—
137C5×C22⋊C4—52no—
138C5×C22⋊C4—52no—
139C42×C5—51no—
140C22⋊Dic5—52no—
141C22⋊Dic5—51no—
142#685—51no—
143#685—52no—
144#685—51no—
145#685—51no—
146#685C2×D512yes—
147#685C2×D512yes—
148#685C2×D512yes—
149C4⋊Dic5—51no—
150C4⋊Dic5C2×D511yescharacteristic
151#683—52no—
152#683—52no—
153#683—51no—
154#683C2×D512yes—
155C4×Dic5—52no—
156C4×Dic5—52no—
157C4×Dic5C2×D511yescharacteristic
158#548—251noSylow
159C2×C52C22×D411yescharacteristic
160C2×C52C2×#4111yescharacteristic
161C2×C522−1+411yescharacteristic
162C5⋊D5—22no—
163C5×D5—41no—
164C5×D5—42no—
165C23×C5—52no—
166C23×C5—51no—
167C22×D5—51no—
168C22×D5—51no—
169C22×D5—102no—
170C22×D5C22×D512yes—
171C22×D5C22×D511yescharacteristic
172C5×Q8—28no—
173C5×Q8—22no—
174C5×Q8—22no—
175C5×D4—102no—
176C2×C4×C5—24no—
177C2×C4×C5—22no—
178C2×C4×C5—51no—
179C2×C4×C5—52no—
180C2×C4×C5—51no—
181C2×C4×C5—51no—
182C2×C4×C5—102no—
183C2×C4×C5C22×D512yes—
184C2×C4×C5C22×D512yes—
185C2×C4×C5C22×D511yescharacteristic
186C2×C4×C5C22×D511yescharacteristic
187C22⋊D5—102no—
188C22⋊D5—104no—
189C22⋊D5—104no—
190C22⋊D5—104no—
191C2×Dic5—52no—
192C2×Dic5—52no—
193C2×Dic5—51no—
194C2×Dic5—51no—
195C2×Dic5—104no—
196C2×Dic5—102no—
197C2×Dic5—104no—
198C2×Dic5C22×D511yescharacteristic
199C2×Dic5C22×D512yes—
200C2×Dic5C22×D511yescharacteristic
201C2×Dic5C22×D511yescharacteristic
202D20—22no—
203D20—104no—
204D20—102no—
205C4×D5—22no—
206C4×D5—104no—
207C4×D5—104no—
208C4×D5—104no—
209C4×D5—1016no—
210C4×D5—108no—
211C4×D5—102no—
212C4×D5—102no—
213Dic10—54no—
214Dic10—54no—
215Dic10—108no—
216Dic10—104no—
217Dic10—104no—
218Dic10—1016no—
219C2×#41—251no—
220C22×Q8—251no—
221#124—51no—
222#124—252no—
223C22⋊Q8—51no—
224C22⋊Q8—252no—
225#121—252no—
226#121—251no—
227C4×Q8—251no—
228C4×D4—251no—
229C4×D4—252no—
230C52#54811yescharacteristic, complemented, Sylow
231C22×C5—22no—
232C22×C5—102no—
233C22×C5—101no—
234C22×C5—102no—
235C22×C5—102no—
236C22×C5—101no—
237C22×C5—101no—
238C22×C5C23×D511yescharacteristic
239C22×C5C23×D511yescharacteristic
240C2×D5—21no—
241C2×D5—22no—
242C2×D5—21no—
243C2×D5—21no—
244C2×D5—22no—
245C2×D5—22no—
246C2×D5—52no—
247C2×D5—52no—
248C2×D5—104no—
249C2×D5—104no—
250C2×D5—104no—
251C2×D5—101no—
252C2×D5—101no—
253C2×D5—101no—
254C2×D5—101no—
255C4×C5—28no—
256C4×C5—22no—
257C4×C5—22no—
258C4×C5—24no—
259C4×C5—101no—
260C4×C5—101no—
261C4×C5—101no—
262C4×C5—102no—
263C4×C5D4×D512yes—
264C4×C5D4×D512yes—
265Dic5—21no—
266Dic5—22no—
267Dic5—21no—
268Dic5—21no—
269Dic5—54no—
270Dic5—54no—
271Dic5—108no—
272Dic5—104no—
273Dic5—101no—
274Dic5—101no—
275#41—504no—
276C2×Q8—251no—
277C2×Q8—252no—
278C2×Q8—504no—
279C2×Q8D5211yescharacteristic
280C2×D4—51no—
281C2×D4—251no—
282C2×D4—252no—
283C2×D4—252no—
284C22×C4—51no—
285C22×C4—252no—
286C22×C4—252no—
287C22×C4—251no—
288C4⋊C4—52no—
289C4⋊C4—51no—
290C4⋊C4—251no—
291C4⋊C4—252no—
292C22⋊C4—52no—
293C22⋊C4—52no—
294C22⋊C4—52no—
295C22⋊C4—251no—
296C22⋊C4—251no—
297C22⋊C4—252no—
298C42—51no—
299C42—252no—
300C2×C5—22no—
301C2×C5—22no—
302C2×C5—22no—
303C2×C5—202no—
304C2×C5—201no—
305C2×C5C2×D4×D511yescharacteristic
306C2×C5C2×D4×D511yescharacteristic
307C2×C5#414711yescharacteristic
308C2×C5D5×#4111yescharacteristic
309C2×C5D5×#4111yescharacteristic
310C2×C5#415011yescharacteristic
311D5—41no—
312D5—42no—
313D5—108no—
314D5—102no—
315D5—102no—
316C23—51no—
317C23—52no—
318C23—251no—
319Q8—22no—
320Q8—254no—
321Q8—504no—
322D4—102no—
323D4—502no—
324D4—504no—
325D4—504no—
326C2×C4—51no—
327C2×C4—51no—
328C2×C4—52no—
329C2×C4—51no—
330C2×C4—102no—
331C2×C4—252no—
332C2×C4—251no—
333C2×C4—504no—
334C2×C4—504no—
335C2×C4—502no—
336C2×C4C2×D5212yes—
337C2×C4C2×D5211yescharacteristic
338C5—22no—
339C5#6596711yescharacteristic, complemented
340C5#6601111yescharacteristic, complemented
341C22—101no—
342C22—102no—
343C22—101no—
344C22—102no—
345C22—101no—
346C22—102no—
347C22—252no—
348C22—501no—
349C22—501no—
350C22C22×D5211yescharacteristic
351C4—22no—
352C4—101no—
353C4—101no—
354C4—102no—
355C4—101no—
356C4—254no—
357C4—501no—
358C4#8788612yes—
359C2—201no—
360C2—202no—
361C2—502no—
362C2#1171748011yescharacteristic
363C2#1171748411yescharacteristic
364C2C2×#8788611yescharacteristic
subgroupG/H
center, FrattiniC22C22×D52
commutator, socle, cosocleC22×C52C24
FittingC2×C52×Q8C22
hypercenterC2×Q8D52
hypocenterC52#548
LCSG ▷ C22×C52 ▷ C52
UCSC1 ◁ C22 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. C2×Q8×C5⋊D5C2semidirect
2. C2×#87870C2semidirect
3. C5×#4093C2semidirect
4. C5×#4092C2semidirect
5. #11717034C2semidirect
6. #11717031C2semidirect
7. #11717030C2semidirect
8. C4×#6038C2semidirect
9. #11717015C2semidirect
10. #11716999C2semidirect
11. #11716962C2semidirect
50. #6045D4semidirect
68. #4093D5semidirect
69. #4092D5semidirect
87. C2×C5⋊D5#41semidirect
230. C52#548semidirect
339. C5#65967semidirect
340. C5#66011semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×C5⋊D52
C2×#878702
C5×#40932
C5×#40922
#117170342
#117170312
#117170302
C4×#60382
#117170152
#117169992
#117169622
#660115
#659675