Finite Group Databasebeta

← #72231022

Subgroups — #72231022

Order 2400.

596 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 596 conjugacy classes of subgroups — too many to draw. Only its 88 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×D3×D52C211yesmaximal, characteristic
2C2×D5⋊ΓL(1,𝔽16)C211yesmaximal, characteristic, complemented
3C2×C3×D5⋊AGL(1,𝔽5)C211yesmaximal, characteristic, complemented
4#11925360C211yesmaximal, characteristic, complemented
5#11925355C211yesmaximal, characteristic, complemented
6(C2×D3)⋊#1044C211yesmaximal, characteristic, complemented
7(C2×D3)⋊#1043C211yesmaximal, characteristic, complemented
8D5⋊(C22⋊AGL(1,𝔽5))—31nomaximal, Hall
9C2×D15⋊D5C2211yescharacteristic
10C2×D5×D15C412yescomplemented
11C2×D3×C5⋊D5C2211yescharacteristic
12C2×C5×D3×D5C412yescomplemented
13C2×C3×D52C2211yescharacteristic
14C2×C3⋊#1044C2211yescharacteristic, complemented
15C2×C3⋊#1043C2211yescharacteristic, complemented
16C2×C3×#1044C2211yescharacteristic, complemented
17C2×C3×#1043C2211yescharacteristic, complemented
18D3×D52—22no—
19D3×D52—22no—
20D5⋊ΓL(1,𝔽16)—22no—
21C3×D5⋊AGL(1,𝔽5)—22no—
22C2×(C2×D3)⋊AGL(1,𝔽5)—52nomaximal
23C22×D52—31no—
24C2×D5⋊AGL(1,𝔽5)—31no—
25C2×D5⋊AGL(1,𝔽5)D311yescharacteristic, complemented
26C22⋊#1044—31no—
27C22⋊#1043—31no—
28#87831—31no—
29#87831—31no—
30C2×C3⋊(C5⋊D5)C2×C411yescharacteristic, complemented
31C2×C5×D15C2×C412yescomplemented
32C2×C52×D3C2×C411yescharacteristic, complemented
33C2×C3×C5⋊D5C2311yescharacteristic
34C2×C3×C5×D5C2×C412yes—
35D15⋊D5—21no—
36D15⋊D5—21no—
37D5×D15—24no—
38D3×C5⋊D5—21no—
39D3×C5⋊D5—21no—
40C5×D3×D5—24no—
41C3×D52D412yes—
42C3×D52D412yes—
43C3⋊#1044—21no—
44C3⋊#1043—21no—
45C3×#1044—21no—
46C3×#1043—21no—
47C22×D3×D5—52no—
48C22×ΓL(1,𝔽16)—52no—
49C22×C3×AGL(1,𝔽5)—52no—
50(C2×D3)⋊AGL(1,𝔽5)—52no—
51(C2×D3)⋊AGL(1,𝔽5)—52no—
52(C2×D3)⋊AGL(1,𝔽5)—52no—
53(C2×D3)⋊AGL(1,𝔽5)—52no—
54(C2×D3)⋊AGL(1,𝔽5)—102no—
55C22×C5⋊D5—31no—
56C22×C5×D5—32no—
57C2×D52—31no—
58C2×D52—32no—
59C2×D52—62no—
60C2×D52—62no—
61C2×D52C2×D311yescharacteristic
62C2×#1044—31no—
63C2×#1044C2×D311yescharacteristic, complemented
64C2×#1043—31no—
65C2×#1043C2×D311yescharacteristic, complemented
66D5⋊AGL(1,𝔽5)—22no—
67D5⋊AGL(1,𝔽5)—62no—
68C2×C22⋊AGL(1,𝔽5)—152no—
69C2×C3×C52C22×C411yescharacteristic
70C3⋊(C5⋊D5)—21no—
71C5×D15—22no—
72C52×D3—21no—
73C3×C5⋊D5C2×D411yescharacteristic
74C3×C5⋊D5C2×D411yescharacteristic
75C3×C5×D5C22⋊C414yescomplemented
76C22×D15—52no—
77C22×C5×D3—52no—
78C22×C3×D5—52no—
79C2×D3×D5—52no—
80C2×D3×D5—52no—
81C2×D3×D5—52no—
82C2×D3×D5—104no—
83C2×D3×D5—104no—
84C2×D3×D5—102no—
85C2×D3×D5AGL(1,𝔽5)12yescomplemented
86C2×ΓL(1,𝔽16)—52no—
87C2×ΓL(1,𝔽16)—52no—
88C2×ΓL(1,𝔽16)—104no—
89C2×ΓL(1,𝔽16)—102no—
90C2×C3×AGL(1,𝔽5)—52no—
91C2×C3×AGL(1,𝔽5)—52no—
92C2×C3×AGL(1,𝔽5)—104no—
93C2×C3×AGL(1,𝔽5)—102no—
94C22×C52—31no—
95C2×C5⋊D5—31no—
96C2×C5⋊D5—61no—
97C2×C5⋊D5—61no—
98C2×C5⋊D5C22×D311yescharacteristic
99C2×C5×D5—32no—
100C2×C5×D5—64no—
101C2×C5×D5C4×D312yes—
102D52—64no—
103D52—61no—
104D52—61no—
105D52D1212yes—
106D52#6712yes—
107#1044—21no—
108#1044—61no—
109#1043—21no—
110#1043—61no—
111C2×(C2×D3)⋊C4—251noHall
112C23×D5—152no—
113C22×AGL(1,𝔽5)—52no—
114C22×AGL(1,𝔽5)—152no—
115C22⋊AGL(1,𝔽5)—152no—
116C22⋊AGL(1,𝔽5)—152no—
117C22⋊AGL(1,𝔽5)—152no—
118C22⋊AGL(1,𝔽5)—152no—
119C22⋊AGL(1,𝔽5)—302no—
120C3×C52C2×C22⋊C411yescharacteristic, complemented, Hall
121C22×C3×C5—52no—
122C2×D15—52no—
123C2×D15—102no—
124C2×D15—104no—
125C2×D15C2×AGL(1,𝔽5)12yescomplemented
126C2×C5×D3—22no—
127C2×C5×D3—52no—
128C2×C5×D3—104no—
129C2×C5×D3C2×AGL(1,𝔽5)12yescomplemented
130C2×C3×D5—52no—
131C2×C3×D5—54no—
132C2×C3×D5—54no—
133C2×C3×D5—102no—
134C2×C3×D5C2×AGL(1,𝔽5)12yes—
135D3×D5—24no—
136D3×D5—102no—
137D3×D5—102no—
138D3×D5—102no—
139D3×D5—102no—
140D3×D5—104no—
141D3×D5—202no—
142D3×D5—202no—
143ΓL(1,𝔽16)—102no—
144ΓL(1,𝔽16)—102no—
145ΓL(1,𝔽16)—202no—
146C3×AGL(1,𝔽5)—102no—
147C3×AGL(1,𝔽5)—102no—
148C3×AGL(1,𝔽5)—202no—
149C2×C52—61no—
150C2×C52C2×C4×D311yescharacteristic
151C5⋊D5—61no—
152C5⋊D5C2×D1211yescharacteristic
153C5⋊D5C2×#6711yescharacteristic
154C5×D5—62no—
155C5×D5(C2×D3)⋊C414yescomplemented
156C23×D3—251no—
157C22×C3×C4—251no—
158C22×Dic3—251no—
159(C2×D3)⋊C4—251no—
160(C2×D3)⋊C4—251no—
161(C2×D3)⋊C4—251no—
162(C2×D3)⋊C4—251no—
163C23×C5—152no—
164C22×D5—32no—
165C22×D5—52no—
166C22×D5—152no—
167C22×D5—152no—
168C22×D5—152no—
169C22×D5—152no—
170C22×D5—302no—
171C22×D5—304no—
172C22×D5—304no—
173C2×AGL(1,𝔽5)—52no—
174C2×AGL(1,𝔽5)—52no—
175C2×AGL(1,𝔽5)—102no—
176C2×AGL(1,𝔽5)—104no—
177C2×AGL(1,𝔽5)—152no—
178C2×AGL(1,𝔽5)—152no—
179C2×AGL(1,𝔽5)—302no—
180C2×AGL(1,𝔽5)—304no—
181C2×C22⋊C4—751noSylow
182C2×C3×C5—22no—
183C2×C3×C5—54no—
184C2×C3×C5C22×AGL(1,𝔽5)12yes—
185D15—22no—
186D15—102no—
187D15—202no—
188C3×D5—52no—
189C3×D5—52no—
190C3×D5—102no—
191C3×D5—102no—
192C3×D5C22⋊AGL(1,𝔽5)14yescomplemented
193C5×D3—22no—
194C5×D3—42no—
195C5×D3—102no—
196C52C2×(C2×D3)⋊C411yescharacteristic, complemented, Sylow
197C23×C3—251no—
198C22×D3—52no—
199C22×D3—251no—
200C22×D3—252no—
201C22×D3—251no—
202C22×D3—502no—
203C22×D3—502no—
204C2×C3×C4—251no—
205C2×C3×C4—251no—
206C2×C3×C4—502no—
207C2×Dic3—251no—
208C2×Dic3—251no—
209C2×Dic3—502no—
210C22×C5—32no—
211C22×C5—52no—
212C22×C5—62no—
213C22×C5—152no—
214C22×C5—304no—
215C2×D5—32no—
216C2×D5—52no—
217C2×D5—54no—
218C2×D5—54no—
219C2×D5—64no—
220C2×D5—102no—
221C2×D5—152no—
222C2×D5—302no—
223C2×D5—302no—
224C2×D5—302no—
225C2×D5—302no—
226C2×D5—302no—
227C2×D5—304no—
228C2×D5—304no—
229C2×D5—602no—
230C2×D5—602no—
231C2×D5D3×AGL(1,𝔽5)12yes—
232AGL(1,𝔽5)—102no—
233AGL(1,𝔽5)—102no—
234AGL(1,𝔽5)—202no—
235AGL(1,𝔽5)—302no—
236AGL(1,𝔽5)—302no—
237AGL(1,𝔽5)—602no—
238C24—751no—
239C22×C4—251no—
240C22×C4—751no—
241C22⋊C4—751no—
242C22⋊C4—751no—
243C22⋊C4—751no—
244C22⋊C4—751no—
245C3×C5—22no—
246C3×C5C2×C22⋊AGL(1,𝔽5)12yescomplemented
247C22×C3—52no—
248C22×C3—252no—
249C22×C3—252no—
250C22×C3—251no—
251C2×D3—52no—
252C2×D3—104no—
253C2×D3—251no—
254C2×D3—501no—
255C2×D3—501no—
256C2×D3—501no—
257C2×D3—501no—
258C2×D3—504no—
259C2×D3D5⋊AGL(1,𝔽5)11yescharacteristic, complemented
260C3×C4—501no—
261C3×C4—501no—
262Dic3—501no—
263Dic3—501no—
264C2×C5—22no—
265C2×C5—54no—
266C2×C5—62no—
267C2×C5—122no—
268C2×C5—302no—
269C2×C5C2×D3×AGL(1,𝔽5)12yes—
270D5—52no—
271D5—52no—
272D5—62no—
273D5—102no—
274D5—102no—
275D5—302no—
276D5—602no—
277D5(C2×D3)⋊AGL(1,𝔽5)14yescomplemented
278C23—152no—
279C23—251no—
280C23—751no—
281C23—752no—
282C23—751no—
283C23—1502no—
284C23—1502no—
285C2×C4—251no—
286C2×C4—251no—
287C2×C4—502no—
288C2×C4—751no—
289C2×C4—751no—
290C2×C4—1502no—
291C2×C3—54no—
292C2×C3—251no—
293C2×C3—251no—
294C2×C3C2×D5⋊AGL(1,𝔽5)11yescharacteristic
295D3—21no—
296D3—102no—
297D3—501no—
298C5—22no—
299C5C2×(C2×D3)⋊AGL(1,𝔽5)12yescomplemented
300C22—31no—
301C22—52no—
302C22—152no—
303C22—252no—
304C22—252no—
305C22—251no—
306C22—304no—
307C22—751no—
308C22—1504no—
309C22—1501no—
310C22—1501no—
311C22—1501no—
312C22—1501no—
313C4—501no—
314C4—501no—
315C4—1501no—
316C4—1501no—
317C3D5⋊(C22⋊AGL(1,𝔽5))11yescharacteristic, complemented, Sylow
318C2—54no—
319C2—61no—
320C2—251no—
321C2—251no—
322C2—302no—
323C2—1501no—
324C2D3×D5⋊AGL(1,𝔽5)11yescharacteristic
subgroupG/H
center, Frattini, hypercenterC2D3×D5⋊AGL(1,𝔽5)
commutator, Fitting, socleC2×C3×C52C22×C4
prime radicalC2×D3×D52C2
cosocleC2×C3×C5⋊D5C23
hypocenterC3×C52C2×C22⋊C4
LCSG ▷ C2×C3×C52 ▷ C3×C52
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
2. C2×D5⋊ΓL(1,𝔽16)C2semidirect
3. C2×C3×D5⋊AGL(1,𝔽5)C2semidirect
4. #11925360C2semidirect
5. #11925355C2semidirect
6. (C2×D3)⋊#1044C2semidirect
7. (C2×D3)⋊#1043C2semidirect
10. C2×D5×D15C4semidirect
12. C2×C5×D3×D5C4semidirect
14. C2×C3⋊#1044C22semidirect
15. C2×C3⋊#1043C22semidirect
16. C2×C3×#1044C22semidirect
17. C2×C3×#1043C22semidirect
25. C2×D5⋊AGL(1,𝔽5)D3semidirect
30. C2×C3⋊(C5⋊D5)C2×C4semidirect
31. C2×C5×D15C2×C4semidirect
32. C2×C52×D3C2×C4semidirect
63. C2×#1044C2×D3semidirect
65. C2×#1043C2×D3semidirect
75. C3×C5×D5C22⋊C4semidirect
85. C2×D3×D5AGL(1,𝔽5)semidirect
120. C3×C52C2×C22⋊C4semidirect
125. C2×D15C2×AGL(1,𝔽5)semidirect
129. C2×C5×D3C2×AGL(1,𝔽5)semidirect
155. C5×D5(C2×D3)⋊C4semidirect
192. C3×D5C22⋊AGL(1,𝔽5)semidirect
196. C52C2×(C2×D3)⋊C4semidirect
246. C3×C5C2×C22⋊AGL(1,𝔽5)semidirect
259. C2×D3D5⋊AGL(1,𝔽5)semidirect
277. D5(C2×D3)⋊AGL(1,𝔽5)semidirect
299. C5C2×(C2×D3)⋊AGL(1,𝔽5)semidirect
317. C3D5⋊(C22⋊AGL(1,𝔽5))semidirect

Maximal subgroups

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

subgroupindex
C2×D3×D522
C2×D5⋊ΓL(1,𝔽16)2
C2×C3×D5⋊AGL(1,𝔽5)2
#119253602
#119253552
(C2×D3)⋊#10442
(C2×D3)⋊#10432
D5⋊(C22⋊AGL(1,𝔽5))3
C2×(C2×D3)⋊AGL(1,𝔽5)5