Finite Group Databasebeta

← #72260007

Subgroups — #72260007

Order 4800.

656 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 656 conjugacy classes of subgroups — too many to draw. Only its 93 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
1#72229455C211yesmaximal, characteristic
2#72229444C211yesmaximal, characteristic
3C3×C42×C5⋊D5C211yesmaximal, characteristic
4#13100332—31nomaximal, Hall
5C2×C3×C4×C5⋊D5—21no—
6C2×C3×C4×C5⋊D5C2211yescharacteristic
7#11925540—21no—
8#11925539—21no—
9C4⋊(C3⋊#1044)C412yescomplemented
10C4×C3⋊#1044—21no—
11#11925535C412yescomplemented
12#11925534—21no—
13C3×C42×C52C411yescharacteristic, complemented
14C3×C4×#1039—21no—
15C3×C4×#1039C411yescharacteristic, complemented
16#11746733—52nomaximal
17#11717451—31no—
18#11717448—31no—
19C42×C5⋊D5D311yescharacteristic
20C22×C3×C5⋊D5D411yescharacteristic
21C2×C3⋊#1044—22no—
22C2×C3×C4×C52—21no—
23C2×C3×C4×C52C2×C411yescharacteristic
24C2×C3×#1039—21no—
25C2×C3×#1039D411yescharacteristic
26C3×C4×C5⋊D5—24no—
27C3×C4×C5⋊D5C2×C411yescharacteristic
28C3×C4×C5⋊D5C2×C411yescharacteristic
29C3×C4×C5⋊D5D411yescharacteristic
30C3×C4×C5⋊D5Q811yescharacteristic
31C3⋊#6034—22no—
32#91989—52no—
33D5⋊#208—52no—
34C3×C42×D5—52no—
35C3×C42×D5—102no—
36C2×C4×C5⋊D5—21no—
37C2×C4×C5⋊D5C2×D311yescharacteristic
38C22⋊#1044—61no—
39#87867—61no—
40C4⋊#1044—32no—
41C4×#1044—61no—
42#87863—32no—
43#87862—61no—
44C42×C52Dic311yescharacteristic, complemented
45C4×#1039—21no—
46C4×#1039Dic311yescharacteristic, complemented
47#64862—152no—
48C22×C3×C52C22⋊C411yescharacteristic
49C2×C3×C5⋊D5—22no—
50C2×C3×C5⋊D5C22⋊C411yescharacteristic
51C2×C3×C5⋊D5C4⋊C411yescharacteristic
52C3⋊#1044—42no—
53C3×C4×C52—22no—
54C3×C4×C52C4211yescharacteristic
55C3×C4×C52C4⋊C411yescharacteristic
56C3×#1039—22no—
57C3×#1039C22⋊C411yescharacteristic
58C3×#1039C4⋊C411yescharacteristic
59C2×C3×C4×D5—52no—
60C2×C3×C4×D5—102no—
61C2×C3×C4×D5—102no—
62C2×C3×C4×D5—104no—
63#6763—102no—
64#6762—102no—
65C4⋊ΓL(1,𝔽16)—54no—
66C4×ΓL(1,𝔽16)—102no—
67#6758—54no—
68D5⋊(C3⋊C8)—102no—
69C3×C42×C5—22no—
70C3×C42×C5AGL(1,𝔽5)12yescomplemented
71C3×C4×Dic5—52no—
72C3×C4×Dic5—102no—
73C3×C4×Dic5—102no—
74C3×C4×Dic5—104no—
75C22×C5⋊D5#6711yescharacteristic
76C2×#1044—62no—
77C2×C4×C52—21no—
78C2×C4×C52C2×Dic311yescharacteristic
79C2×#1039—21no—
80C2×#1039#6711yescharacteristic
81C4×C5⋊D5—24no—
82C4×C5⋊D5Dic611yescharacteristic
83C4×C5⋊D5C4×D311yescharacteristic
84C4×C5⋊D5C4×D311yescharacteristic
85C4×C5⋊D5D1211yescharacteristic
86#6034—62no—
87#4523—251noHall
88#4130—152no—
89D5⋊M16—152no—
90C42×D5—52no—
91C42×D5—102no—
92C2×C3×C52—21no—
93C2×C3×C52#9511yescharacteristic
94C3×C5⋊D5—21no—
95C3×C5⋊D5#10412yes—
96C22×C3×D5—52no—
97C22×C3×D5—102no—
98C2×ΓL(1,𝔽16)—104no—
99C2×C3×C4×C5—24no—
100C2×C3×C4×C5—22no—
101C2×C3×C4×C5—22no—
102C2×C3×C4×C5C2×AGL(1,𝔽5)12yes—
103C2×C3×Dic5—52no—
104C2×C3×Dic5—102no—
105C2×C3×Dic5—104no—
106C2×C3×Dic5—102no—
107C3×C4×D5—52no—
108C3×C4×D5—52no—
109C3×C4×D5—52no—
110C3×C4×D5—52no—
111C3×C4×D5—102no—
112C3×C4×D5—102no—
113C3×C4×D5—102no—
114C3×C4×D5—102no—
115C3×C4×D5—108no—
116C3×C4×D5—1016no—
117C3⋊#170—104no—
118C22×C52#21711yescharacteristic
119C2×C5⋊D5—22no—
120C2×C5⋊D5Dic3⋊C411yescharacteristic
121C2×C5⋊D5(C2×D3)⋊C411yescharacteristic
122#1044—122no—
123C4×C52—22no—
124C4×C52C4×Dic311yescharacteristic
125C4×C52C4⋊Dic311yescharacteristic
126#1039—22no—
127#1039Dic3⋊C411yescharacteristic
128#1039(C2×D3)⋊C411yescharacteristic
129C2×C3×C42—251no—
130#926—251no—
131C2×#208—251no—
132C2×C4×D5—52no—
133C2×C4×D5—102no—
134C2×C4×D5—102no—
135C2×C4×D5—104no—
136C22⋊AGL(1,𝔽5)—302no—
137#704—302no—
138C4⋊AGL(1,𝔽5)—154no—
139C4×AGL(1,𝔽5)—302no—
140#700—154no—
141D5⋊C8—302no—
142C42×C5—22no—
143C42×C5ΓL(1,𝔽16)12yescomplemented
144C4×Dic5—52no—
145C4×Dic5—102no—
146C4×Dic5—102no—
147C4×Dic5—104no—
148C3×C52#33911yescharacteristic, complemented, Hall
149#339—751noSylow
150C22×C3×C5—22no—
151C22×C3×C5C22⋊AGL(1,𝔽5)12yes—
152C2×C3×D5—52no—
153C2×C3×D5—52no—
154C2×C3×D5—102no—
155C2×C3×D5—102no—
156C2×C3×D5—104no—
157C2×C3×D5—108no—
158ΓL(1,𝔽16)—204no—
159C3×C4×C5—28no—
160C3×C4×C5—24no—
161C3×C4×C5—22no—
162C3×C4×C5—22no—
163C3×C4×C5C4×AGL(1,𝔽5)12yes—
164C3×C4×C5C4⋊AGL(1,𝔽5)12yes—
165C3×Dic5—52no—
166C3×Dic5—52no—
167C3×Dic5—102no—
168C3×Dic5—102no—
169C3×Dic5—108no—
170C3×Dic5—104no—
171C2×C52—21no—
172C2×C52C3⋊#9511yescharacteristic
173C5⋊D5—21no—
174C5⋊D5#80512yes—
175C22×C3×C4—251no—
176C22×C3×C4—501no—
177C3×C42—251no—
178C3×C42—501no—
179C3×C42#104411yescharacteristic, complemented
180#217—501no—
181C4⋊Dic3—252no—
182C4×Dic3—501no—
183#208—252no—
184#208—501no—
185C2×C3⋊C8—501no—
186C22×D5—52no—
187C22×D5—102no—
188C2×AGL(1,𝔽5)—304no—
189C2×C4×C5—24no—
190C2×C4×C5—22no—
191C2×C4×C5—22no—
192C2×C4×C5C2×ΓL(1,𝔽16)12yes—
193C2×Dic5—52no—
194C2×Dic5—102no—
195C2×Dic5—104no—
196C2×Dic5—102no—
197C4×D5—52no—
198C4×D5—52no—
199C4×D5—52no—
200C4×D5—52no—
201C4×D5—102no—
202C4×D5—102no—
203C4×D5—102no—
204C4×D5—102no—
205C4×D5—108no—
206C4×D5—1016no—
207#170—304no—
208C2×M16—751no—
209#117—751no—
210C2×C42—251no—
211C2×C3×C5—24no—
212C2×C3×C5—22no—
213C2×C3×C5—22no—
214C2×C3×C5(C2×AGL(1,𝔽5))⋊C412yes—
215C3×D5—54no—
216C3×D5—104no—
217C3×D5—104no—
218C3×D5—102no—
219C52#452311yescharacteristic, complemented, Sylow
220C23×C3—251no—
221C2×C3×C4—21no—
222C2×C3×C4—251no—
223C2×C3×C4—251no—
224C2×C3×C4—251no—
225C2×C3×C4—251no—
226C2×C3×C4—251no—
227C2×C3×C4—504no—
228C2×C3×C4—501no—
229C2×C3×C4C2×#104411yescharacteristic
230C2×Dic3—502no—
231C3⋊C8—502no—
232C22×C5—22no—
233C22×C5#676312yes—
234C2×D5—52no—
235C2×D5—52no—
236C2×D5—102no—
237C2×D5—102no—
238C2×D5—104no—
239C2×D5—108no—
240AGL(1,𝔽5)—604no—
241C4×C5—28no—
242C4×C5—24no—
243C4×C5—22no—
244C4×C5—22no—
245C4×C5C4×ΓL(1,𝔽16)12yes—
246C4×C5C4⋊ΓL(1,𝔽16)12yes—
247Dic5—52no—
248Dic5—52no—
249Dic5—102no—
250Dic5—102no—
251Dic5—108no—
252Dic5—104no—
253C22×C4—251no—
254C22×C4—501no—
255M16—752no—
256M16—1501no—
257C2×C8—1501no—
258C4⋊C4—752no—
259C22⋊C4—1501no—
260C42—251no—
261C42—501no—
262C42—1501no—
263C42C3⋊#104411yescharacteristic, complemented
264C3×C5—22no—
265C3×C5#6486212yescomplemented
266C22×C3—251no—
267C22×C3—251no—
268C22×C3—502no—
269C22×C3C22⋊#104411yescharacteristic
270C3×C4—22no—
271C3×C4—251no—
272C3×C4—251no—
273C3×C4—502no—
274C3×C4C4×#104411yescharacteristic
275C3×C4C4⋊#104411yescharacteristic
276Dic3—1002no—
277C2×C5—24no—
278C2×C5—22no—
279C2×C5—22no—
280C2×C5#9123512yes—
281D5—54no—
282D5—104no—
283D5—104no—
284D5—102no—
285C23—251no—
286C2×C4—21no—
287C2×C4—251no—
288C2×C4—251no—
289C2×C4—251no—
290C2×C4—251no—
291C2×C4—251no—
292C2×C4—504no—
293C2×C4—501no—
294C2×C4—1502no—
295C2×C4C2×C3⋊#104411yescharacteristic
296C8—1502no—
297C2×C3—21no—
298C2×C3—252no—
299C2×C3—501no—
300C2×C3#1171689911yescharacteristic
301C5—22no—
302C5#1174673312yescomplemented
303C22—251no—
304C22—251no—
305C22—502no—
306C22#1192554011yescharacteristic
307C4—22no—
308C4—251no—
309C4—251no—
310C4—3002no—
311C4—502no—
312C4C4×C3⋊#104411yescharacteristic
313C4C4⋊(C3⋊#1044)11yescharacteristic
314C3#1310033211yescharacteristic, complemented, Sylow
315C2—21no—
316C2—252no—
317C2—501no—
318C2#7226000411yescharacteristic
subgroupG/H
centerC4C4⋊(C3⋊#1044)
commutatorC3×C4×C52C42
FrattiniC2×C4C2×C3⋊#1044
FittingC3×C42×C52C4
socleC2×C3×C52#95
prime radicalC22×C3×C5⋊D5D4
cosocleC2×C3×C4×C5⋊D5C22
hypercenterC42C3⋊#1044
hypocenterC3×C52#339
LCSG ▷ C3×C4×C52 ▷ C2×C3×C52 ▷ C3×C52
UCSC1 ◁ C4 ◁ C2×C4 ◁ C42

Complements

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

NH ≅ G/Ntype
9. C4⋊(C3⋊#1044)C4semidirect
11. #11925535C4semidirect
13. C3×C42×C52C4semidirect
15. C3×C4×#1039C4semidirect
44. C42×C52Dic3semidirect
46. C4×#1039Dic3semidirect
70. C3×C42×C5AGL(1,𝔽5)semidirect
143. C42×C5ΓL(1,𝔽16)semidirect
148. C3×C52#339semidirect
179. C3×C42#1044semidirect
219. C52#4523semidirect
263. C42C3⋊#1044semidirect
265. C3×C5#64862semidirect
302. C5#11746733semidirect
314. C3#13100332semidirect

Maximal subgroups

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

subgroupindex
#722294552
#722294442
C3×C42×C5⋊D52
#131003323
#117467335