Finite Group Databasebeta

← #13105149

Subgroups — #13105149

Order 1600.

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

normal subgroups only

This group has 678 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×#87879C212yesmaximal, complemented
2#11717201C211yesmaximal, characteristic, complemented
3C5×#4032C212yesmaximal, complemented
4#11717098C211yesmaximal, characteristic, complemented
5C2×#87781C211yesmaximal, characteristic, complemented
6#11717037C211yesmaximal, characteristic, complemented
7#11717030C211yesmaximal, characteristic, complemented
8#11717020C211yesmaximal, characteristic, complemented
9#11717018C211yesmaximal, characteristic, complemented
10#11717014C212yesmaximal, complemented
11#11716972C212yesmaximal, complemented
12C2×C22⋊(C5⋊D5)C2211yescharacteristic
13C22×#1039C2211yescharacteristic
14C2×C4×C5⋊D5C2211yescharacteristic
15C2×C5×C22⋊D5C2212yes—
16C22×C5×Dic5C2212yes—
17C2×#6039C2212yes—
18C2×#6038C2212yes—
19C2×#6037C2211yescharacteristic
20C2×#6036C2212yes—
21#87879—28no—
22C2×D5×Dic5C2212yes—
23C52×C22⋊C4C2211yescharacteristic
24#87808C2211yescharacteristic
25#87806C2211yescharacteristic
26C4×#1039C2211yescharacteristic
27C5×#685C2212yes—
28C5×#685C2212yes—
29C5×C4⋊Dic5C2212yes—
30C5×C4⋊Dic5C2212yes—
31#87781C2212yes—
32#87781C2211yescharacteristic
33#87781C2211yescharacteristic
34#87781C2211yescharacteristic
35#87778C2212yes—
36#65989—52nomaximal
37C23×C52C2311yescharacteristic
38C22×C5⋊D5C2311yescharacteristic
39C22×C5×D5C2312yes—
40C2×C4×C52C2311yescharacteristic
41C2×C4×C52C2311yescharacteristic
42C22⋊(C5⋊D5)D414yescomplemented
43C2×#1039—22no—
44C2×#1039C2311yescharacteristic
45C2×#1039C2311yescharacteristic
46C2×#1039C2311yescharacteristic
47C4×C5⋊D5—22no—
48C5×C22⋊D5—24no—
49C2×C5×Dic5—24no—
50C2×C5×Dic5C2312yes—
51C2×C5×Dic5C2312yes—
52C2×C5×Dic5C2312yes—
53#6039—24no—
54#6038—24no—
55#6037—22no—
56#6036—24no—
57D5×Dic5—24no—
58C2×#711—52no—
59C2×#709—52no—
60C5×#123—52no—
61#4085—52no—
62C2×#683—52no—
63#4042—52no—
64#4040—52no—
65D5×C4⋊C4—52no—
66#4034—52no—
67#4032D512yescomplemented
68#4030—52no—
69#4029—52no—
70#4027—52no—
71#4027—52no—
72#4027—102no—
73#4024—52no—
74C22×C52—21no—
75C22×C52—21no—
76C22×C52C2×D412yes—
77C22×C52C2411yescharacteristic
78C2×C5⋊D5—21no—
79C2×C5⋊D5—21no—
80C2×C5⋊D5C2×D412yes—
81C2×C5×D5—22no—
82C2×C5×D5—22no—
83C2×C5×D5—22no—
84C4×C52—21no—
85C4×C52—21no—
86#1039—21no—
87#1039C2×D412yes—
88#1039#4112yescomplemented
89C5×Dic5—22no—
90C5×Dic5—22no—
91C5×Dic5—22no—
92C2×C5×D4—52no—
93C22×C4×C5—52no—
94C2×C22⋊D5—52no—
95C2×C22⋊D5—52no—
96C2×C22⋊D5—52no—
97C2×C22⋊D5—102no—
98C2×C22⋊D5C2×D512yes—
99C22×Dic5—52no—
100C22×Dic5—52no—
101C22×Dic5—102no—
102C22×Dic5C2×D512yes—
103#711—108no—
104#709—108no—
105C2×D20—52no—
106C2×C4×D5—52no—
107C2×C4×D5—52no—
108C2×C4×D5—52no—
109C2×C4×D5—52no—
110C2×C4×D5—102no—
111C2×Dic10—52no—
112C2×Dic10—52no—
113C5×C4⋊C4—52no—
114C5×C4⋊C4—52no—
115C5×C22⋊C4—22no—
116C5×C22⋊C4—52no—
117C5×C22⋊C4—52no—
118C5×C22⋊C4C2×D512yes—
119C22⋊Dic5—52no—
120#685—52no—
121#685—52no—
122#685—102no—
123#685C2×D512yes—
124#685C2×D512yes—
125C4⋊Dic5C2×D512yes—
126C4⋊Dic5C2×D512yes—
127#683—52no—
128#683—52no—
129#683—52no—
130#683—52no—
131#683—52no—
132#683—52no—
133#683—102no—
134C4×Dic5—52no—
135C4×Dic5—102no—
136#547—251noSylow
137C2×C52—22no—
138C2×C52C22×D411yescharacteristic
139C2×C52C2×#4111yescharacteristic
140C2×C522−1+411yescharacteristic
141C5⋊D5—22no—
142C5×D5—42no—
143C23×C5—22no—
144C23×C5—52no—
145C23×C5C22×D512yes—
146C22×D5—52no—
147C22×D5—102no—
148C22×D5C22×D512yes—
149C5×D4—104no—
150C2×C4×C5—22no—
151C2×C4×C5—22no—
152C2×C4×C5—52no—
153C2×C4×C5—52no—
154C2×C4×C5—52no—
155C2×C4×C5—104no—
156C2×C4×C5C22×D512yes—
157C2×C4×C5C22×D512yes—
158C22⋊D5—24no—
159C22⋊D5—58no—
160C22⋊D5—108no—
161C22⋊D5—104no—
162C22⋊D5—104no—
163C2×Dic5—24no—
164C2×Dic5—52no—
165C2×Dic5—52no—
166C2×Dic5—52no—
167C2×Dic5—102no—
168C2×Dic5—102no—
169C2×Dic5—102no—
170C2×Dic5—104no—
171C2×Dic5—104no—
172C2×Dic5—204no—
173C2×Dic5C22×D512yes—
174C2×Dic5C22×D512yes—
175C2×Dic5C22×D512yes—
176D20—104no—
177C4×D5—104no—
178C4×D5—104no—
179C4×D5—104no—
180C4×D5—104no—
181C4×D5—204no—
182Dic10—104no—
183Dic10—104no—
184C2×#41—252no—
185C4⋊Q8—251no—
186#123—52no—
187#123—252no—
188C22⋊Q8—252no—
189#121—251no—
190#121—251no—
191C4×D4—251no—
192C4×D4—251no—
193C2×C4⋊C4—251no—
194C2×C4⋊C4—251no—
195C52#54711yescharacteristic, complemented, Sylow
196C22×C5—22no—
197C22×C5—22no—
198C22×C5—24no—
199C22×C5—22no—
200C22×C5—42no—
201C22×C5—42no—
202C22×C5—102no—
203C22×C5—102no—
204C22×C5—102no—
205C22×C5D4×D514yes—
206C22×C5C23×D512yes—
207C2×D5—22no—
208C2×D5—22no—
209C2×D5—22no—
210C2×D5—54no—
211C2×D5—104no—
212C2×D5—102no—
213C2×D5—102no—
214C2×D5—202no—
215C2×D5—202no—
216C4×C5—22no—
217C4×C5—22no—
218C4×C5—42no—
219C4×C5—42no—
220C4×C5—102no—
221C4×C5—102no—
222C4×C5—102no—
223Dic5—22no—
224Dic5—22no—
225Dic5—22no—
226Dic5—54no—
227Dic5—54no—
228Dic5—104no—
229Dic5—104no—
230Dic5—102no—
231Dic5—202no—
232#41—508no—
233C2×Q8—252no—
234C2×D4—52no—
235C2×D4—252no—
236C2×D4—251no—
237C2×D4—251no—
238C22×C4—52no—
239C22×C4—252no—
240C22×C4—252no—
241C22×C4—251no—
242C22×C4—251no—
243C4⋊C4—52no—
244C4⋊C4—52no—
245C4⋊C4—251no—
246C4⋊C4—251no—
247C4⋊C4—251no—
248C4⋊C4—252no—
249C4⋊C4—251no—
250C22⋊C4—52no—
251C22⋊C4—52no—
252C22⋊C4—252no—
253C22⋊C4—251no—
254C22⋊C4D5211yescharacteristic
255C42—251no—
256C2×C5—24no—
257C2×C5—22no—
258C2×C5—22no—
259C2×C5—22no—
260C2×C5—44no—
261C2×C5—202no—
262C2×C5C2×D4×D512yes—
263C2×C5D5×#4112yes—
264C2×C5#415012yes—
265D5—42no—
266D5—104no—
267D5—204no—
268C23—52no—
269C23—251no—
270C23C2×D5211yescharacteristic
271Q8—504no—
272D4—104no—
273D4—254no—
274D4—504no—
275D4—502no—
276C2×C4—52no—
277C2×C4—52no—
278C2×C4—52no—
279C2×C4—104no—
280C2×C4—251no—
281C2×C4—251no—
282C2×C4—251no—
283C2×C4—502no—
284C2×C4—504no—
285C2×C4—504no—
286C2×C4—502no—
287C2×C4C2×D5211yescharacteristic
288C2×C4C2×D5211yescharacteristic
289C5—22no—
290C5#6598912yescomplemented
291C22—21no—
292C22—21no—
293C22—102no—
294C22—102no—
295C22—102no—
296C22—252no—
297C22—501no—
298C22—501no—
299C22#8787712yes—
300C22C22×D5211yescharacteristic
301C4—21no—
302C4—21no—
303C4—102no—
304C4—102no—
305C4—102no—
306C4—252no—
307C4—252no—
308C4—501no—
309C2—22no—
310C2—202no—
311C2—502no—
312C2C2×#8787711yescharacteristic
313C2#1171747411yescharacteristic
314C2#1171747711yescharacteristic
subgroupG/H
center, FrattiniC22C22×D52
commutator, socle, cosocleC22×C52C24
FittingC52×C22⋊C4C22
hypercenterC22⋊C4D52
hypocenterC52#547
LCSG ▷ C22×C52 ▷ C52
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#87879C2semidirect
2. #11717201C2semidirect
3. C5×#4032C2semidirect
4. #11717098C2semidirect
5. C2×#87781C2semidirect
6. #11717037C2semidirect
7. #11717030C2semidirect
8. #11717020C2semidirect
9. #11717018C2semidirect
10. #11717014C2semidirect
11. #11716972C2semidirect
42. C22⋊(C5⋊D5)D4semidirect
67. #4032D5semidirect
88. #1039#41semidirect
195. C52#547semidirect
290. C5#65989semidirect

Maximal subgroups

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

subgroupindex
C2×#878792
#117172012
C5×#40322
#117170982
C2×#877812
#117170372
#117170302
#117170202
#117170182
#117170142
#117169722
#659895