Finite Group Databasebeta

← #13104665

Subgroups — #13104665

Order 1600.

599 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 599 conjugacy classes of subgroups — too many to draw. Only its 113 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
1C23×#1042C211yesmaximal, characteristic, complemented
2C2×C5×D4×D5C211yesmaximal, characteristic
3C2×C22⋊#1042C211yesmaximal, characteristic, complemented
4C2×C22⋊#1042C212yesmaximal, complemented
5#11716849C212yesmaximal, complemented
6C23×C5×D5C2211yescharacteristic
7C23×C5×D5C2211yescharacteristic
8C22×#1042—22no—
9C22×#1042—24no—
10C22×#1042C2212yes—
11C22×#1042C2212yes—
12C2×C52×D4C411yescharacteristic, complemented
13C2×C5×C22⋊D5C411yescharacteristic, complemented
14C2×C5×C22⋊D5C411yescharacteristic, complemented
15C5×D4×D5—22no—
16C5×D4×D5—22no—
17C2×C5×D20C411yescharacteristic, complemented
18C2×C4×C5×D5C2211yescharacteristic
19C22⋊#1042—22no—
20C22⋊#1042—24no—
21#65779—51nomaximal
22#65509—51nomaximal
23C23×C52C2×C411yescharacteristic
24C23×C52C2×C411yescharacteristic, complemented
25C22×C5×D5—22no—
26C22×C5×D5—22no—
27C22×C5×D5—22no—
28C22×C5×D5—21no—
29C22×C5×D5—21no—
30C22×C5×D5—22no—
31C22×C5×D5C2×C411yescharacteristic
32C22×C5×D5C2×C411yescharacteristic, complemented
33C22×C5×D5D412yes—
34C22×C5×D5D412yes—
35C22×C5×D5C2311yescharacteristic
36C2×#1042—28no—
37C2×#1042—22no—
38C2×#1042—22no—
39C2×#1042—22no—
40C2×#1042—22no—
41C2×#1042—22no—
42C2×#1042D414yescomplemented
43C52×D4—22no—
44C2×C4×C52C2×C411yescharacteristic, complemented
45C5×C22⋊D5—22no—
46C5×C22⋊D5—22no—
47C2×C5×Dic5C2×C411yescharacteristic, complemented
48C5×D20—22no—
49C4×C5×D5—21no—
50C4×C5×D5—21no—
51C23×AGL(1,𝔽5)—51no—
52C22×C5×D4—51no—
53C23×Dic5—51no—
54C2×D4×D5D511yescharacteristic
55C2×C22⋊AGL(1,𝔽5)—52no—
56C2×C22⋊AGL(1,𝔽5)—51no—
57C2×C22⋊Dic5—52no—
58C2×C22⋊Dic5—51no—
59(C2×AGL(1,𝔽5))⋊C4—52no—
60#3963—52no—
61C22×C52—22no—
62C22×C52—21no—
63C22×C52—22no—
64C22×C52C22⋊C412yescomplemented
65C22×C52C22×C411yescharacteristic
66C2×C5×D5—22no—
67C2×C5×D5—24no—
68C2×C5×D5—22no—
69C2×C5×D5—21no—
70C2×C5×D5—22no—
71C2×C5×D5—22no—
72C2×C5×D5—41no—
73C2×C5×D5—42no—
74C2×C5×D5—41no—
75C2×C5×D5C22⋊C412yescomplemented
76C2×C5×D5C2×D411yescharacteristic
77C2×C5×D5C2×D411yescharacteristic
78C2×C5×D5C2×D412yes—
79C2×C5×D5#4112yes—
80#1042—24no—
81#1042—42no—
82C4×C52—21no—
83C5×Dic5—21no—
84C24×C5—51no—
85C24×C5—51no—
86C23×D5C2×D511yescharacteristic
87C23×D5C2×D511yescharacteristic
88C22×AGL(1,𝔽5)—52no—
89C22×AGL(1,𝔽5)—52no—
90C22×AGL(1,𝔽5)—104no—
91C22×AGL(1,𝔽5)—102no—
92C2×C5×D4—41no—
93C2×C5×D4—51no—
94C2×C5×D4—51no—
95C2×C5×D4—51no—
96C2×C5×D4—102no—
97C2×C5×D4—102no—
98C2×C5×D4Dic511yescharacteristic, complemented
99C2×C5×D4AGL(1,𝔽5)11yescharacteristic, complemented
100C22×C4×C5—51no—
101C2×C22⋊D5Dic511yescharacteristic, complemented
102C2×C22⋊D5Dic511yescharacteristic, complemented
103C22×Dic5—52no—
104C22×Dic5—52no—
105C22×Dic5—104no—
106C22×Dic5—102no—
107D4×D5—22no—
108D4×D5—22no—
109C2×D20Dic511yescharacteristic, complemented
110C2×C4×D5C2×D511yescharacteristic
111C22⋊AGL(1,𝔽5)—104no—
112C22⋊AGL(1,𝔽5)—102no—
113C22⋊Dic5—104no—
114C22⋊Dic5—102no—
115#386—251noSylow
116C2×C52—22no—
117C2×C52—41no—
118C2×C52C2×C22⋊C411yescharacteristic
119C2×C52C4×D412yes—
120C5×D5—22no—
121C5×D5—41no—
122C5×D5#12011yescharacteristic
123C5×D5#12112yes—
124C5×D5#12311yescharacteristic
125C23×C5—41no—
126C23×C5—41no—
127C23×C5—51no—
128C23×C5—52no—
129C23×C5—52no—
130C23×C5—51no—
131C23×C5—51no—
132C23×C5—102no—
133C23×C5—102no—
134C23×C5—102no—
135C23×C5—101no—
136C23×C5—101no—
137C23×C5—102no—
138C23×C5C2×Dic511yescharacteristic
139C23×C5C2×Dic511yescharacteristic, complemented
140C23×C5C2×AGL(1,𝔽5)11yescharacteristic
141C23×C5C2×AGL(1,𝔽5)11yescharacteristic, complemented
142C22×D5—22no—
143C22×D5—22no—
144C22×D5—22no—
145C22×D5—22no—
146C22×D5—21no—
147C22×D5—21no—
148C22×D5C2×Dic511yescharacteristic
149C22×D5C2×Dic511yescharacteristic, complemented
150C22×D5C22⋊D512yes—
151C22×D5C22⋊D512yes—
152C22×D5C22×D511yescharacteristic
153C2×AGL(1,𝔽5)—54no—
154C2×AGL(1,𝔽5)—108no—
155C2×AGL(1,𝔽5)—102no—
156C2×AGL(1,𝔽5)—102no—
157C2×AGL(1,𝔽5)—102no—
158C2×AGL(1,𝔽5)—102no—
159C2×AGL(1,𝔽5)—102no—
160C5×D4—22no—
161C5×D4—22no—
162C5×D4—44no—
163C5×D4—102no—
164C5×D4—102no—
165C5×D4—102no—
166C2×C4×C5—41no—
167C2×C4×C5—51no—
168C2×C4×C5—101no—
169C2×C4×C5—101no—
170C2×C4×C5C2×Dic511yescharacteristic, complemented
171C2×C4×C5C2×AGL(1,𝔽5)11yescharacteristic, complemented
172C22⋊D5—22no—
173C22⋊D5—22no—
174C2×Dic5—54no—
175C2×Dic5—102no—
176C2×Dic5—102no—
177C2×Dic5—108no—
178C2×Dic5—102no—
179C2×Dic5—102no—
180C2×Dic5—102no—
181C2×Dic5C2×Dic511yescharacteristic, complemented
182D20—22no—
183C4×D5—21no—
184C4×D5—21no—
185C22×D4—51no—
186C23×C4—251no—
187C2×C22⋊C4—252no—
188C2×C22⋊C4—251no—
189#95—252no—
190C52#38611yescharacteristic, complemented, Sylow
191C22×C5—22no—
192C22×C5—22no—
193C22×C5—21no—
194C22×C5—22no—
195C22×C5—21no—
196C22×C5—22no—
197C22×C5—42no—
198C22×C5—42no—
199C22×C5—41no—
200C22×C5—52no—
201C22×C5—52no—
202C22×C5—51no—
203C22×C5—51no—
204C22×C5—52no—
205C22×C5—82no—
206C22×C5—82no—
207C22×C5—102no—
208C22×C5—104no—
209C22×C5—102no—
210C22×C5—102no—
211C22×C5—102no—
212C22×C5—101no—
213C22×C5—201no—
214C22×C5—202no—
215C22×C5—201no—
216C22×C5C22⋊Dic512yescomplemented
217C22×C5C22⋊AGL(1,𝔽5)12yescomplemented
218C22×C5C22×Dic511yescharacteristic
219C22×C5C22×AGL(1,𝔽5)11yescharacteristic
220C2×D5—22no—
221C2×D5—22no—
222C2×D5—22no—
223C2×D5—24no—
224C2×D5—21no—
225C2×D5—22no—
226C2×D5—41no—
227C2×D5—41no—
228C2×D5—42no—
229C2×D5C22⋊Dic512yescomplemented
230C2×D5D4×D512yes—
231C2×D5#71112yes—
232C2×D5C2×C22⋊D511yescharacteristic
233C2×D5C2×C22⋊D511yescharacteristic
234AGL(1,𝔽5)—104no—
235AGL(1,𝔽5)—202no—
236C4×C5—21no—
237C4×C5—21no—
238C4×C5—42no—
239C4×C5—101no—
240Dic5—21no—
241Dic5—104no—
242Dic5—202no—
243C24—51no—
244C24—51no—
245C2×D4—51no—
246C2×D4—51no—
247C2×D4—51no—
248C2×D4—102no—
249C2×D4—102no—
250C2×D4#104211yescharacteristic, complemented
251C22×C4—51no—
252C22×C4—252no—
253C22×C4—252no—
254C22×C4—504no—
255C22×C4—502no—
256C22⋊C4—504no—
257C22⋊C4—502no—
258C2×C5—22no—
259C2×C5—22no—
260C2×C5—41no—
261C2×C5—41no—
262C2×C5—42no—
263C2×C5—41no—
264C2×C5—51no—
265C2×C5—52no—
266C2×C5—51no—
267C2×C5—82no—
268C2×C5—82no—
269C2×C5—102no—
270C2×C5—201no—
271C2×C5D4×Dic512yes—
272C2×C5C2×C22⋊Dic511yescharacteristic
273C2×C5D4×AGL(1,𝔽5)12yes—
274C2×C5C2×C22⋊AGL(1,𝔽5)11yescharacteristic
275D5—22no—
276D5—41no—
277D5#408111yescharacteristic
278D5#408311yescharacteristic
279D5#408512yes—
280C23—51no—
281C23—52no—
282C23—52no—
283C23—51no—
284C23—51no—
285C23—102no—
286C23—102no—
287C23—102no—
288C23—101no—
289C23—101no—
290C23—102no—
291C23C2×#104211yescharacteristic
292C23C2×#104211yescharacteristic, complemented
293D4—22no—
294D4—102no—
295D4—102no—
296D4—102no—
297C2×C4—51no—
298C2×C4—101no—
299C2×C4—101no—
300C2×C4—254no—
301C2×C4—502no—
302C2×C4—502no—
303C2×C4—508no—
304C2×C4—502no—
305C2×C4—502no—
306C2×C4—502no—
307C2×C4C2×#104211yescharacteristic, complemented
308C5—41no—
309C5#6550911yescharacteristic, complemented
310C5#6577911yescharacteristic, complemented
311C22—22no—
312C22—22no—
313C22—21no—
314C22—52no—
315C22—52no—
316C22—51no—
317C22—51no—
318C22—52no—
319C22—102no—
320C22—104no—
321C22—102no—
322C22—102no—
323C22—102no—
324C22—101no—
325C22—201no—
326C22—202no—
327C22—201no—
328C22C22⋊#104212yescomplemented
329C22C22×#104211yescharacteristic
330C4—21no—
331C4—101no—
332C4—504no—
333C4—1002no—
334C2—22no—
335C2—41no—
336C2—51no—
337C2—52no—
338C2—51no—
339C2—102no—
340C2—201no—
341C2D4×#104212yes—
342C2C2×C22⋊#104211yescharacteristic
subgroupG/H
center, FrattiniC22C22×#1042
commutator, socleC22×C52C22×C4
FittingC2×C52×D4C4
prime radicalC2×C5×D4×D5C2
cosocleC22×C5×D5C23
hypercenterC2×D4#1042
hypocenterC52#386
LCSG ▷ C22×C52 ▷ C52
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C23×#1042C2semidirect
3. C2×C22⋊#1042C2semidirect
4. C2×C22⋊#1042C2semidirect
5. #11716849C2semidirect
12. C2×C52×D4C4semidirect
13. C2×C5×C22⋊D5C4semidirect
14. C2×C5×C22⋊D5C4semidirect
17. C2×C5×D20C4semidirect
24. C23×C52C2×C4semidirect
32. C22×C5×D5C2×C4semidirect
42. C2×#1042D4semidirect
44. C2×C4×C52C2×C4semidirect
47. C2×C5×Dic5C2×C4semidirect
64. C22×C52C22⋊C4semidirect
75. C2×C5×D5C22⋊C4semidirect
98. C2×C5×D4Dic5semidirect
99. C2×C5×D4AGL(1,𝔽5)semidirect
101. C2×C22⋊D5Dic5semidirect
102. C2×C22⋊D5Dic5semidirect
109. C2×D20Dic5semidirect
139. C23×C5C2×Dic5semidirect
141. C23×C5C2×AGL(1,𝔽5)semidirect
149. C22×D5C2×Dic5semidirect
170. C2×C4×C5C2×Dic5semidirect
171. C2×C4×C5C2×AGL(1,𝔽5)semidirect
181. C2×Dic5C2×Dic5semidirect
190. C52#386semidirect
216. C22×C5C22⋊Dic5semidirect
217. C22×C5C22⋊AGL(1,𝔽5)semidirect
229. C2×D5C22⋊Dic5semidirect
250. C2×D4#1042semidirect
292. C23C2×#1042semidirect
307. C2×C4C2×#1042semidirect
309. C5#65509semidirect
310. C5#65779semidirect
328. C22C22⋊#1042semidirect

Maximal subgroups

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

subgroupindex
C23×#10422
C2×C5×D4×D52
C2×C22⋊#10422
#117168492
#657795
#655095