Finite Group Databasebeta

← #72260013

Subgroups — #72260013

Order 4800.

656 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 656 conjugacy classes of subgroups — too many to draw. Only its 89 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#72229453C211yesmaximal, characteristic
2#72229444C211yesmaximal, characteristic
3C2×C3×C4×#1043C211yesmaximal, characteristic
4#13099900—31nomaximal, Hall
5C22×C3×#1043—21no—
6C2×C3×C4×C5⋊D5C2211yescharacteristic
7#11925540—21no—
8C4⋊(C3⋊#1044)C412yescomplemented
9C4×C3⋊#1044—21no—
10C3×C4×#1043—21no—
11C3×C4×#1043C412yescomplemented
12C5×#6710C412yescomplemented
13#11925152—21no—
14#11925150—21no—
15#11746527—52nomaximal
16#11717451—31no—
17C2×C4×#1043D311yescharacteristic
18#11716936—31no—
19C22×C3×C5⋊D5D411yescharacteristic
20C2×C3⋊#1044—22no—
21C2×C3×#1043—24no—
22C2×C3×#1043—22no—
23C2×C3×C4×C52C2×C411yescharacteristic
24C2×C3×#1039Q811yescharacteristic
25C3×C4×C5⋊D5C2×C411yescharacteristic
26C3×C4×C5⋊D5C2×C411yescharacteristic
27C3×C4×C5⋊D5D411yescharacteristic
28C3×C4×C5⋊D5D411yescharacteristic
29C5×C3⋊#168—22no—
30#91989—52no—
31C2×C3×C4×AGL(1,𝔽5)—52no—
32C2×C3×C4×AGL(1,𝔽5)—102no—
33D5×#208—52no—
34C22×#1043—21no—
35C2×C4×C5⋊D5C2×D311yescharacteristic
36C22⋊#1044—61no—
37C4⋊#1044—32no—
38C4×#1044—61no—
39C4×#1043—21no—
40C4×#1043Dic312yescomplemented
41C5×#681—32no—
42#87769—61no—
43#87767—61no—
44#64900—152no—
45C22×C3×C52C4⋊C411yescharacteristic
46C2×C3×C5⋊D5—22no—
47C2×C3×C5⋊D5C22⋊C411yescharacteristic
48C2×C3×C5⋊D5C22⋊C411yescharacteristic
49C3⋊#1044—42no—
50C3×#1043—24no—
51C3×C4×C52C4211yescharacteristic
52C3×C4×C52C22⋊C411yescharacteristic
53C3×#1039C4⋊C411yescharacteristic
54C3×#1039C4⋊C411yescharacteristic
55C22×C3×AGL(1,𝔽5)—102no—
56C22×C3×AGL(1,𝔽5)—104no—
57C2×C3×C4×D5—52no—
58C2×C3×C4×D5—102no—
59#6763—102no—
60C4⋊ΓL(1,𝔽16)—54no—
61C4×ΓL(1,𝔽16)—102no—
62C3×C4×AGL(1,𝔽5)—54no—
63C3×C4×AGL(1,𝔽5)—102no—
64C3×C4×AGL(1,𝔽5)—104no—
65C3×C4×AGL(1,𝔽5)—104no—
66#6710AGL(1,𝔽5)12yescomplemented
67C5×#208—52no—
68#6649—102no—
69D5×C3⋊C8—102no—
70C22×C5⋊D5#6711yescharacteristic
71C2×#1044—62no—
72C2×#1043—24no—
73C2×#1043—22no—
74C2×C4×C52C4×D311yescharacteristic
75C2×#1039Dic611yescharacteristic
76C4×C5⋊D5C4×D311yescharacteristic
77C4×C5⋊D5D1211yescharacteristic
78C4×C5⋊D5C2×Dic311yescharacteristic
79C4×C5⋊D5#6711yescharacteristic
80C5×#168—62no—
81#4523—251noHall
82#4130—152no—
83C2×C4×AGL(1,𝔽5)—52no—
84C2×C4×AGL(1,𝔽5)—102no—
85D5×M16—152no—
86C2×C3×C52—21no—
87C2×C3×C52#9511yescharacteristic
88C3×C5⋊D5—21no—
89C3×C5⋊D5#10412yes—
90C22×C3×D5—52no—
91C22×C3×D5—102no—
92C2×ΓL(1,𝔽16)—104no—
93C2×C3×AGL(1,𝔽5)—104no—
94C2×C3×AGL(1,𝔽5)—108no—
95C2×C3×AGL(1,𝔽5)—108no—
96C2×C3×AGL(1,𝔽5)—1016no—
97C2×C3×C4×C5—22no—
98C2×C3×C4×C5C2×AGL(1,𝔽5)12yes—
99C2×C3×Dic5—52no—
100C2×C3×Dic5—102no—
101C3×C4×D5—52no—
102C3×C4×D5—52no—
103C3×C4×D5—52no—
104C3×C4×D5—52no—
105C3×C4×D5—102no—
106C3×C4×D5—102no—
107C3×C4×D5—102no—
108C3×C4×D5—102no—
109C3⋊#168—22no—
110C5×C3⋊C8—102no—
111C22×C52Dic3⋊C411yescharacteristic
112C2×C5⋊D5—22no—
113C2×C5⋊D5(C2×D3)⋊C411yescharacteristic
114C2×C5⋊D5#21711yescharacteristic
115#1044—122no—
116#1043—24no—
117C4×C52C4×Dic311yescharacteristic
118C4×C52(C2×D3)⋊C411yescharacteristic
119#1039Dic3⋊C411yescharacteristic
120#1039C4⋊Dic311yescharacteristic
121C2×C3×C42—251no—
122#926—251no—
123C2×#208—251no—
124C22×AGL(1,𝔽5)—102no—
125C22×AGL(1,𝔽5)—104no—
126C2×C4×D5—52no—
127C2×C4×D5—102no—
128C22⋊AGL(1,𝔽5)—302no—
129C4⋊AGL(1,𝔽5)—154no—
130C4×AGL(1,𝔽5)—54no—
131C4×AGL(1,𝔽5)—104no—
132C4×AGL(1,𝔽5)—102no—
133C4×AGL(1,𝔽5)—104no—
134C4×AGL(1,𝔽5)—302no—
135C5×M16—152no—
136#681—32no—
137#676—302no—
138C8×D5—302no—
139C3×C52#33911yescharacteristic, complemented, Hall
140#339—751noSylow
141C22×C3×C5—22no—
142C22×C3×C5C4⋊AGL(1,𝔽5)12yes—
143C2×C3×D5—52no—
144C2×C3×D5—52no—
145C2×C3×D5—102no—
146C2×C3×D5—102no—
147C2×C3×D5—104no—
148C2×C3×D5—108no—
149ΓL(1,𝔽16)—204no—
150C3×AGL(1,𝔽5)—1016no—
151C3×AGL(1,𝔽5)—108no—
152C3×C4×C5—22no—
153C3×C4×C5—22no—
154C3×C4×C5C4×AGL(1,𝔽5)12yes—
155C3×C4×C5C22⋊AGL(1,𝔽5)12yes—
156C3×Dic5—52no—
157C3×Dic5—52no—
158C3×Dic5—102no—
159C3×Dic5—102no—
160C2×C52—21no—
161C2×C52C3⋊#9511yescharacteristic
162C5⋊D5—21no—
163C5⋊D5#80512yes—
164C22×C3×C4—251no—
165C22×C3×C4—501no—
166C3×C42—252no—
167C3×C42—501no—
168#217—501no—
169C4⋊Dic3—252no—
170C4×Dic3—501no—
171#208—52no—
172#208—501no—
173C2×C3⋊C8—501no—
174C22×D5—52no—
175C22×D5—102no—
176C2×AGL(1,𝔽5)—104no—
177C2×AGL(1,𝔽5)—108no—
178C2×AGL(1,𝔽5)—108no—
179C2×AGL(1,𝔽5)—1016no—
180C2×AGL(1,𝔽5)—304no—
181C2×C4×C5—22no—
182C2×C4×C5D3×AGL(1,𝔽5)12yes—
183C2×Dic5—52no—
184C2×Dic5—102no—
185C4×D5—52no—
186C4×D5—52no—
187C4×D5—52no—
188C4×D5—52no—
189C4×D5—102no—
190C4×D5—102no—
191C4×D5—102no—
192C4×D5—102no—
193C5×C8—302no—
194#168—62no—
195C2×M16—751no—
196#117—751no—
197C2×C42—251no—
198C2×C3×C5—24no—
199C2×C3×C5—22no—
200C2×C3×C5—22no—
201C2×C3×C5(C2×AGL(1,𝔽5))⋊C412yes—
202C3×D5—54no—
203C3×D5—104no—
204C3×D5—104no—
205C3×D5—102no—
206C52#452311yescharacteristic, complemented, Sylow
207C23×C3—251no—
208C2×C3×C4—251no—
209C2×C3×C4—251no—
210C2×C3×C4—251no—
211C2×C3×C4—251no—
212C2×C3×C4—251no—
213C2×C3×C4—502no—
214C2×C3×C4—504no—
215C2×C3×C4D5⋊AGL(1,𝔽5)11yescharacteristic
216C2×Dic3—502no—
217C3⋊C8—102no—
218C22×C5—22no—
219C22×C5Dic3⋊AGL(1,𝔽5)12yes—
220C2×D5—52no—
221C2×D5—52no—
222C2×D5—102no—
223C2×D5—102no—
224C2×D5—104no—
225C2×D5—108no—
226AGL(1,𝔽5)—1016no—
227AGL(1,𝔽5)—108no—
228AGL(1,𝔽5)—604no—
229C4×C5—22no—
230C4×C5—22no—
231C4×C5Dic3×AGL(1,𝔽5)12yes—
232C4×C5(C2×D3)⋊AGL(1,𝔽5)12yes—
233Dic5—52no—
234Dic5—52no—
235Dic5—102no—
236Dic5—102no—
237C22×C4—251no—
238C22×C4—501no—
239M16—152no—
240M16—1501no—
241C2×C8—1501no—
242C4⋊C4—752no—
243C22⋊C4—1501no—
244C42—252no—
245C42—501no—
246C42—1501no—
247C3×C5—22no—
248C3×C5#6490012yescomplemented
249C22×C3—251no—
250C22×C3—251no—
251C22×C3—502no—
252C22×C3#8783211yescharacteristic
253C3×C4—251no—
254C3×C4—251no—
255C3×C4—504no—
256C3×C4#8783011yescharacteristic
257C3×C4#8783111yescharacteristic
258Dic3—1002no—
259C2×C5—24no—
260C2×C5—22no—
261C2×C5—22no—
262C2×C5(C2×AGL(1,𝔽5))⋊Dic312yes—
263D5—54no—
264D5—104no—
265D5—104no—
266D5—102no—
267C23—251no—
268C2×C4—251no—
269C2×C4—251no—
270C2×C4—251no—
271C2×C4—251no—
272C2×C4—251no—
273C2×C4—502no—
274C2×C4—504no—
275C2×C4—1502no—
276C2×C4#1059718211yescharacteristic
277C8—302no—
278C2×C3—21no—
279C2×C3—252no—
280C2×C3—501no—
281C2×C3#1171677611yescharacteristic
282C5—22no—
283C5#1174652712yescomplemented
284C22—251no—
285C22—251no—
286C22—502no—
287C22#1192535611yescharacteristic
288C4—251no—
289C4—251no—
290C4—3002no—
291C4—504no—
292C4#1192535411yescharacteristic
293C4#1192535511yescharacteristic
294C3#1309990011yescharacteristic, complemented, Sylow
295C2—21no—
296C2—252no—
297C2—501no—
298C2#7226001011yescharacteristic
subgroupG/H
centerC4#11925355
commutatorC3×C4×C52C42
Frattini, hypercenterC2×C4#10597182
FittingC2×C3×C4×C52C2×C4
socleC2×C3×C52#95
prime radicalC22×C3×C5⋊D5D4
cosocleC2×C3×C4×C5⋊D5C22
hypocenterC3×C52#339
LCSG ▷ C3×C4×C52 ▷ C2×C3×C52 ▷ C3×C52
UCSC1 ◁ C4 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
8. C4⋊(C3⋊#1044)C4semidirect
11. C3×C4×#1043C4semidirect
12. C5×#6710C4semidirect
40. C4×#1043Dic3semidirect
66. #6710AGL(1,𝔽5)semidirect
139. C3×C52#339semidirect
206. C52#4523semidirect
248. C3×C5#64900semidirect
283. C5#11746527semidirect
294. C3#13099900semidirect

Maximal subgroups

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

subgroupindex
#722294532
#722294442
C2×C3×C4×#10432
#130999003
#117465275