Finite Group Databasebeta

← #13105100

Subgroups — #13105100

Order 1600.

744 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 744 conjugacy classes of subgroups — too many to draw. Only its 136 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×#87877C211yesmaximal, characteristic, complemented
2C2×#87871C211yesmaximal, characteristic, complemented
3#11717192C211yesmaximal, characteristic, complemented
4C5×#4021C211yesmaximal, characteristic, complemented
5C5×#4020C211yesmaximal, characteristic, complemented
6#11717040C214yesmaximal, complemented
7#11717032C211yesmaximal, characteristic, complemented
8#11717004C214yesmaximal, complemented
9#11716981C211yesmaximal, characteristic, complemented
10C22×D52C2212yes—
11C2×C4×C5⋊D5C2211yescharacteristic
12C2×C5×D20C2211yescharacteristic
13C2×C5×D20C2211yescharacteristic
14C2×C5×D20C2211yescharacteristic, complemented
15C2×C5×D20C2214yescomplemented
16C2×C5×Dic10C2211yescharacteristic, complemented
17C2×#6038C2212yes—
18C2×#6037C2212yes—
19C2×#6037C2214yescomplemented
20C2×D5×Dic5C2212yescomplemented
21#87877—24no—
22#87871—24no—
23C42×C52C2211yescharacteristic, complemented
24#87808C2212yes—
25#87806C2212yescomplemented
26C4×#1039C2211yescharacteristic, complemented
27C5×#685C2214yescomplemented
28#87778C2214yescomplemented
29#66049—51nomaximal
30#66008—51nomaximal
31C22×C5⋊D5C2311yescharacteristic
32C22×C5×D5C2312yes—
33C22×C5×D5C2312yes—
34C22×C5×D5C2312yes—
35C2×D52—24no—
36C2×D52—44no—
37C2×C4×C52C2312yes—
38C2×C4×C52C2311yescharacteristic
39C2×#1039C2312yes—
40C2×#1039C2311yescharacteristic
41C4×C5⋊D5D414yescomplemented
42C2×C5×Dic5C2312yes—
43C5×D20—22no—
44C5×D20—22no—
45C5×D20—22no—
46C5×D20—44no—
47C5×Dic10—22no—
48#6038—24no—
49#6037—24no—
50#6037—44no—
51D5×Dic5—24no—
52C2×#713—51no—
53C2×#711—51no—
54C2×D4×D5—51no—
55C2×D4×D5—51no—
56C5×#127—51no—
57C5×#124—51no—
58#4093—51no—
59#4086—51no—
60#4086—51no—
61#4085—54no—
62#4083—54no—
63#4082—51no—
64#4030—54no—
65#4028—54no—
66#4021D511yescharacteristic, complemented
67#4020D511yescharacteristic, complemented
68#4018—51no—
69#4018—51no—
70#4018—102no—
71C22×C52C2411yescharacteristic
72C2×C5⋊D5—22no—
73C2×C5⋊D5C2×D412yes—
74C2×C5×D5—22no—
75C2×C5×D5—22no—
76C2×C5×D5—22no—
77C2×C5×D5—24no—
78C2×C5×D5—24no—
79C2×C5×D5—24no—
80D52—48no—
81C4×C52—22no—
82C4×C52C2×D412yes—
83#1039—22no—
84#1039C2×D412yes—
85C5×Dic5—22no—
86C23×D5—52no—
87C23×D5—52no—
88C2×C5×Q8—51no—
89C2×C5×D4—51no—
90C2×C5×D4—51no—
91C2×C5×D4—51no—
92C2×C5×D4—54no—
93C2×C22⋊D5—52no—
94C2×C22⋊D5—52no—
95C2×C22⋊D5—54no—
96C2×C22⋊D5—54no—
97C2×C22⋊D5—52no—
98C22×Dic5—52no—
99#713—104no—
100#711—104no—
101D4×D5—104no—
102D4×D5—104no—
103C2×D20—52no—
104C2×D20C2×D511yescharacteristic
105C2×D20C2×D511yescharacteristic
106C2×D20C2×D511yescharacteristic, complemented
107C2×D20C2×D514yescomplemented
108C2×C4×D5—51no—
109C2×C4×D5—51no—
110C2×C4×D5—52no—
111C2×C4×D5—102no—
112C2×Dic10C2×D511yescharacteristic, complemented
113C5×C22⋊C4—54no—
114C42×C5—22no—
115C42×C5C2×D511yescharacteristic, complemented
116C42×C5C2×D511yescharacteristic, complemented
117C22⋊Dic5—54no—
118#685—54no—
119#685—52no—
120#685—52no—
121#685—104no—
122#685C2×D514yescomplemented
123#683—52no—
124#683—52no—
125#683—104no—
126C4×Dic5—51no—
127C4×Dic5—51no—
128C4×Dic5—102no—
129#535—251noSylow
130C2×C52C22×D411yescharacteristic
131C2×C52H(5,𝔽2)12yes—
132C5⋊D5—22no—
133C5×D5—42no—
134C5×D5—42no—
135C5×D5—42no—
136C23×C5—52no—
137C23×C5—52no—
138C23×C5—52no—
139C22×D5—51no—
140C22×D5—51no—
141C22×D5—102no—
142C22×D5—104no—
143C22×D5—104no—
144C22×D5—204no—
145C22×D5—204no—
146C22×D5C22×D512yes—
147C22×D5C22×D512yes—
148C22×D5C22×D512yes—
149C5×Q8—102no—
150C5×D4—102no—
151C5×D4—102no—
152C5×D4—102no—
153C5×D4—204no—
154C2×C4×C5—22no—
155C2×C4×C5—24no—
156C2×C4×C5—52no—
157C2×C4×C5C22×D512yes—
158C2×C4×C5C22×D512yes—
159C2×C4×C5C22×D511yescharacteristic
160C2×C4×C5C22×D511yescharacteristic
161C22⋊D5—104no—
162C22⋊D5—104no—
163C22⋊D5—104no—
164C22⋊D5—204no—
165C22⋊D5—204no—
166C2×Dic5—51no—
167C2×Dic5—51no—
168C2×Dic5—52no—
169C2×Dic5—52no—
170C2×Dic5—102no—
171C2×Dic5—104no—
172C2×Dic5—104no—
173C2×Dic5C22×D512yes—
174D20—22no—
175D20—22no—
176D20—22no—
177D20—44no—
178D20—104no—
179C4×D5—54no—
180C4×D5—54no—
181C4×D5—108no—
182C4×D5—104no—
183Dic10—22no—
184C2×#41—251no—
185C22×D4—251no—
186#127—51no—
187#127—251no—
188#124—51no—
189#124—251no—
190#121—254no—
191#120—254no—
192#117—251no—
193C52#53511yescharacteristic, complemented, Sylow
194C22×C5—22no—
195C22×C5—102no—
196C22×C5—102no—
197C22×C5—102no—
198C22×C5—104no—
199C22×C5—104no—
200C22×C5—104no—
201C22×C5C23×D511yescharacteristic
202C22×C5C23×D511yescharacteristic
203C2×D5—22no—
204C2×D5—22no—
205C2×D5—22no—
206C2×D5—24no—
207C2×D5—24no—
208C2×D5—24no—
209C2×D5—52no—
210C2×D5—52no—
211C2×D5—104no—
212C2×D5—102no—
213C2×D5—102no—
214C2×D5—204no—
215C2×D5—208no—
216C2×D5—208no—
217C4×C5—22no—
218C4×C5—22no—
219C4×C5—24no—
220C4×C5—44no—
221C4×C5—102no—
222C4×C5D4×D512yes—
223C4×C5D4×D512yes—
224Dic5—22no—
225Dic5—52no—
226Dic5—52no—
227Dic5—104no—
228Dic5—102no—
229Dic5—102no—
230Dic5—204no—
231C24—252no—
232#41—504no—
233C2×Q8—51no—
234C2×D4—51no—
235C2×D4—51no—
236C2×D4—51no—
237C2×D4—54no—
238C2×D4—252no—
239C2×D4—252no—
240C2×D4—254no—
241C2×D4—504no—
242C22×C4—252no—
243C22×C4—251no—
244C4⋊C4—252no—
245C22⋊C4—54no—
246C22⋊C4—254no—
247C22⋊C4—252no—
248C42—251no—
249C42D5211yescharacteristic, complemented
250C2×C5—22no—
251C2×C5—24no—
252C2×C5—202no—
253C2×C5—202no—
254C2×C5—202no—
255C2×C5C2×D4×D511yescharacteristic
256C2×C5C2×D4×D511yescharacteristic
257C2×C5#414412yes—
258C2×C5#414912yes—
259D5—42no—
260D5—42no—
261D5—42no—
262D5—102no—
263D5—102no—
264D5—204no—
265C23—52no—
266C23—52no—
267C23—52no—
268C23—251no—
269C23—504no—
270C23—1004no—
271Q8—102no—
272D4—102no—
273D4—102no—
274D4—102no—
275D4—204no—
276D4—504no—
277D4—504no—
278D4—1004no—
279C2×C4—52no—
280C2×C4—251no—
281C2×C4—254no—
282C2×C4—252no—
283C2×C4—504no—
284C2×C4C2×D5212yes—
285C2×C4C2×D5211yescharacteristic
286C5—22no—
287C5#6600811yescharacteristic, complemented
288C5#6604911yescharacteristic, complemented
289C22—102no—
290C22—102no—
291C22—102no—
292C22—104no—
293C22—104no—
294C22—104no—
295C22—252no—
296C22—502no—
297C22—1008no—
298C22C22×D5211yescharacteristic
299C4—22no—
300C4—102no—
301C4—252no—
302C4—502no—
303C4#8787712yes—
304C2—202no—
305C2—202no—
306C2—202no—
307C2—502no—
308C2C2×#8787711yescharacteristic
309C2#1171747212yes—
subgroupG/H
center, FrattiniC22C22×D52
commutator, socle, cosocleC22×C52C24
FittingC42×C52C22
hypercenterC42D52
hypocenterC52#535
LCSG ▷ C22×C52 ▷ C52
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. C2×#87877C2semidirect
2. C2×#87871C2semidirect
3. #11717192C2semidirect
4. C5×#4021C2semidirect
5. C5×#4020C2semidirect
6. #11717040C2semidirect
7. #11717032C2semidirect
8. #11717004C2semidirect
9. #11716981C2semidirect
14. C2×C5×D20C22semidirect
15. C2×C5×D20C22semidirect
16. C2×C5×Dic10C22semidirect
19. C2×#6037C22semidirect
20. C2×D5×Dic5C22semidirect
23. C42×C52C22semidirect
25. #87806C22semidirect
26. C4×#1039C22semidirect
27. C5×#685C22semidirect
28. #87778C22semidirect
41. C4×C5⋊D5D4semidirect
66. #4021D5semidirect
67. #4020D5semidirect
106. C2×D20C2×D5semidirect
107. C2×D20C2×D5semidirect
112. C2×Dic10C2×D5semidirect
115. C42×C5C2×D5semidirect
116. C42×C5C2×D5semidirect
122. #685C2×D5semidirect
193. C52#535semidirect
249. C42D52semidirect
287. C5#66008semidirect
288. C5#66049semidirect

Maximal subgroups

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

subgroupindex
C2×#878772
C2×#878712
#117171922
C5×#40212
C5×#40202
#117170402
#117170322
#117170042
#117169812
#660495
#660085