Finite Group Databasebeta

← #72231001

Subgroups — #72231001

Order 4800.

656 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 656 conjugacy classes of subgroups — too many to draw. Only its 105 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#72230822C212yesmaximal, complemented
2#72229676C212yesmaximal, complemented
3#72229453C211yesmaximal, characteristic
4#72229444C211yesmaximal, characteristic
5C2×C3×C4⋊#1043C211yesmaximal, characteristic
6#13103962—31nomaximal, Hall
7C22×C3×#1043—21no—
8C2×C3×C4×C5⋊D5C2211yescharacteristic
9#11925540—21no—
10C4⋊(C3⋊#1044)C2211yescharacteristic
11C4⋊(C3⋊#1044)C2211yescharacteristic
12C4×C3⋊#1044—21no—
13C3×C4⋊#1043—21no—
14C3×C4⋊#1043C2212yes—
15C5×#6710C412yescomplemented
16#11925152C412yescomplemented
17#11925150C2212yes—
18#11751727—52nomaximal
19#11717451—31no—
20C2×C4⋊#1043D311yescharacteristic
21#11716936—31no—
22#11716772—32no—
23#11716771—32no—
24C22×C3×C5⋊D5D411yescharacteristic
25C2×C3⋊#1044—21no—
26C2×C3⋊#1044—21no—
27C2×C3×#1043—22no—
28C2×C3×#1043—42no—
29C2×C3×C4×C52C2×C411yescharacteristic
30C2×C3×#1039Q811yescharacteristic
31C3×C4×C5⋊D5C2×C411yescharacteristic
32C3×C4×C5⋊D5D411yescharacteristic
33C3×C4×C5⋊D5Q811yescharacteristic
34C3×C4×C5⋊D5C2311yescharacteristic
35C5×C3⋊#168C2×C414yescomplemented
36#91989—52no—
37C2×C3×C4⋊AGL(1,𝔽5)—52no—
38C2×C3×C4⋊AGL(1,𝔽5)—102no—
39D5×#208—52no—
40#91160—54no—
41#91159—54no—
42C22×#1043—21no—
43C2×C4×C5⋊D5C2×D311yescharacteristic
44C22⋊#1044—61no—
45C4⋊#1044—31no—
46C4⋊#1044—31no—
47C4×#1044—61no—
48C4⋊#1043—21no—
49C4⋊#1043C2×D312yes—
50C5×#681—32no—
51#87769—32no—
52#87767—32no—
53#65727—152no—
54C22×C3×C52C4⋊C411yescharacteristic
55C2×C3×C5⋊D5—21no—
56C2×C3×C5⋊D5—21no—
57C2×C3×C5⋊D5C4⋊C411yescharacteristic
58C2×C3×C5⋊D5C2×D411yescharacteristic
59C3⋊#1044—41no—
60C3⋊#1044—41no—
61C3×#1043—42no—
62C3×C4×C52C4⋊C411yescharacteristic
63C3×C4×C52C22×C411yescharacteristic
64C3×#1039C4⋊C411yescharacteristic
65C3×#1039C2×Q811yescharacteristic
66C22×C3×AGL(1,𝔽5)—102no—
67C22×C3×AGL(1,𝔽5)—102no—
68C22×C3×AGL(1,𝔽5)—102no—
69C2×C3×C4×D5—52no—
70C2×C3×C4×D5—102no—
71#6763—102no—
72C4⋊ΓL(1,𝔽16)—52no—
73C4⋊ΓL(1,𝔽16)—52no—
74C4×ΓL(1,𝔽16)—102no—
75C3×C4⋊AGL(1,𝔽5)—54no—
76C3×C4⋊AGL(1,𝔽5)—102no—
77C3×C4⋊AGL(1,𝔽5)—102no—
78C3×C4⋊AGL(1,𝔽5)—102no—
79C3×C4⋊AGL(1,𝔽5)—104no—
80#6710AGL(1,𝔽5)12yescomplemented
81C5×#208—52no—
82#6649—54no—
83D5×C3⋊C8—54no—
84C22×C5⋊D5#6711yescharacteristic
85C2×#1044—61no—
86C2×#1044—61no—
87C2×#1043—22no—
88C2×#1043—42no—
89C2×C4×C52C4×D311yescharacteristic
90C2×#1039Dic611yescharacteristic
91C4×C5⋊D5Dic611yescharacteristic
92C4×C5⋊D5C4×D311yescharacteristic
93C4×C5⋊D5#6711yescharacteristic
94C4×C5⋊D5C22×D311yescharacteristic
95C5×#168—34no—
96#4964—251noHall
97#4130—152no—
98C2×C4⋊AGL(1,𝔽5)—52no—
99C2×C4⋊AGL(1,𝔽5)—102no—
100D5×M16—152no—
101#3994—154no—
102#3993—154no—
103C2×C3×C52—21no—
104C2×C3×C52C2×C4⋊C411yescharacteristic
105C3×C5⋊D5—21no—
106C3×C5⋊D5AGL(1,ℤ/8)11yescharacteristic
107C3×C5⋊D5#13711yescharacteristic
108C22×C3×D5—52no—
109C22×C3×D5—102no—
110C2×ΓL(1,𝔽16)—102no—
111C2×ΓL(1,𝔽16)—102no—
112C2×C3×AGL(1,𝔽5)—104no—
113C2×C3×AGL(1,𝔽5)—104no—
114C2×C3×AGL(1,𝔽5)—104no—
115C2×C3×AGL(1,𝔽5)—204no—
116C2×C3×AGL(1,𝔽5)—204no—
117C2×C3×AGL(1,𝔽5)—204no—
118C2×C3×C4×C5—22no—
119C2×C3×C4×C5C2×AGL(1,𝔽5)12yes—
120C2×C3×Dic5—52no—
121C2×C3×Dic5—102no—
122C3×C4×D5—52no—
123C3×C4×D5—52no—
124C3×C4×D5—52no—
125C3×C4×D5—52no—
126C3×C4×D5—102no—
127C3×C4×D5—102no—
128C3×C4×D5—102no—
129C3×C4×D5—102no—
130C3⋊#168C2×AGL(1,𝔽5)14yescomplemented
131C5×C3⋊C8—54no—
132C22×C52Dic3⋊C411yescharacteristic
133C2×C5⋊D5—21no—
134C2×C5⋊D5—21no—
135C2×C5⋊D5Dic3⋊C411yescharacteristic
136C2×C5⋊D5C2×#6711yescharacteristic
137#1044—121no—
138#1044—121no—
139#1043—42no—
140C4×C52Dic3⋊C411yescharacteristic
141C4×C52C2×C4×D311yescharacteristic
142#1039Dic3⋊C411yescharacteristic
143#1039C2×Dic611yescharacteristic
144C2×C3×C4⋊C4—251no—
145#926—251no—
146C2×#208—251no—
147#808—252no—
148#807—252no—
149C22×AGL(1,𝔽5)—102no—
150C22×AGL(1,𝔽5)—102no—
151C22×AGL(1,𝔽5)—102no—
152C2×C4×D5—52no—
153C2×C4×D5—102no—
154C22⋊AGL(1,𝔽5)—302no—
155C4⋊AGL(1,𝔽5)—54no—
156C4⋊AGL(1,𝔽5)—104no—
157C4⋊AGL(1,𝔽5)—102no—
158C4⋊AGL(1,𝔽5)—102no—
159C4⋊AGL(1,𝔽5)—102no—
160C4⋊AGL(1,𝔽5)—152no—
161C4⋊AGL(1,𝔽5)—152no—
162C4×AGL(1,𝔽5)—302no—
163C5×M16—152no—
164#681—32no—
165#676—154no—
166C8×D5—154no—
167C3×C52#42811yescharacteristic, complemented, Hall
168#428—751noSylow
169C22×C3×C5—22no—
170C22×C3×C5C4⋊AGL(1,𝔽5)12yes—
171C2×C3×D5—52no—
172C2×C3×D5—52no—
173C2×C3×D5—102no—
174C2×C3×D5—102no—
175C2×C3×D5—102no—
176C2×C3×D5—102no—
177C2×C3×D5—102no—
178C2×C3×D5—102no—
179C2×C3×D5—102no—
180C2×C3×D5—102no—
181ΓL(1,𝔽16)—202no—
182ΓL(1,𝔽16)—202no—
183C3×AGL(1,𝔽5)—204no—
184C3×AGL(1,𝔽5)—204no—
185C3×AGL(1,𝔽5)—204no—
186C3×C4×C5—22no—
187C3×C4×C5—22no—
188C3×C4×C5C4⋊AGL(1,𝔽5)12yes—
189C3×C4×C5C22×AGL(1,𝔽5)12yes—
190C3×Dic5—52no—
191C3×Dic5—52no—
192C3×Dic5—102no—
193C3×Dic5—102no—
194C2×C52—21no—
195C2×C52C2×Dic3⋊C411yescharacteristic
196C5⋊D5—21no—
197C5⋊D5#93211yescharacteristic
198C5⋊D5#94211yescharacteristic
199C22×C3×C4—251no—
200C22×C3×C4—501no—
201C3×C4⋊C4—252no—
202C3×C4⋊C4—501no—
203#217—501no—
204C4⋊Dic3—251no—
205C4⋊Dic3—251no—
206C4×Dic3—501no—
207#208—52no—
208#208—252no—
209C2×C3⋊C8—252no—
210C22×D5—52no—
211C22×D5—102no—
212C2×AGL(1,𝔽5)—104no—
213C2×AGL(1,𝔽5)—104no—
214C2×AGL(1,𝔽5)—104no—
215C2×AGL(1,𝔽5)—204no—
216C2×AGL(1,𝔽5)—204no—
217C2×AGL(1,𝔽5)—204no—
218C2×AGL(1,𝔽5)—302no—
219C2×AGL(1,𝔽5)—302no—
220C2×C4×C5—22no—
221C2×C4×C5D3×AGL(1,𝔽5)12yes—
222C2×Dic5—52no—
223C2×Dic5—102no—
224C4×D5—52no—
225C4×D5—52no—
226C4×D5—52no—
227C4×D5—52no—
228C4×D5—102no—
229C4×D5—102no—
230C4×D5—102no—
231C4×D5—102no—
232C5×C8—154no—
233#168—34no—
234C2×M16—751no—
235#117—751no—
236C2×C4⋊C4—251no—
237#107—752no—
238#106—752no—
239C2×C3×C5—22no—
240C2×C3×C5—22no—
241C2×C3×C5—22no—
242C2×C3×C5—22no—
243C2×C3×C5C2×C4⋊AGL(1,𝔽5)12yes—
244C3×D5—52no—
245C3×D5—52no—
246C3×D5—102no—
247C3×D5—102no—
248C3×D5—102no—
249C3×D5—102no—
250C3×D5—102no—
251C52#496411yescharacteristic, complemented, Sylow
252C23×C3—251no—
253C2×C3×C4—251no—
254C2×C3×C4—251no—
255C2×C3×C4—251no—
256C2×C3×C4—251no—
257C2×C3×C4—251no—
258C2×C3×C4—502no—
259C2×C3×C4—1002no—
260C2×C3×C4D5⋊AGL(1,𝔽5)11yescharacteristic
261C2×Dic3—501no—
262C2×Dic3—501no—
263C3⋊C8—54no—
264C22×C5—22no—
265C22×C5Dic3⋊AGL(1,𝔽5)12yes—
266C2×D5—52no—
267C2×D5—52no—
268C2×D5—102no—
269C2×D5—102no—
270C2×D5—102no—
271C2×D5—102no—
272C2×D5—102no—
273C2×D5—102no—
274C2×D5—102no—
275C2×D5—102no—
276AGL(1,𝔽5)—204no—
277AGL(1,𝔽5)—204no—
278AGL(1,𝔽5)—204no—
279AGL(1,𝔽5)—602no—
280AGL(1,𝔽5)—602no—
281C4×C5—22no—
282C4×C5—22no—
283C4×C5Dic3⋊AGL(1,𝔽5)12yes—
284C4×C5C2×D3×AGL(1,𝔽5)12yes—
285Dic5—52no—
286Dic5—52no—
287Dic5—102no—
288Dic5—102no—
289C22×C4—251no—
290C22×C4—501no—
291M16—152no—
292M16—752no—
293C2×C8—752no—
294C4⋊C4—252no—
295C4⋊C4—501no—
296C4⋊C4—751no—
297C4⋊C4—751no—
298C22⋊C4—1501no—
299C42—1501no—
300C3×C5—22no—
301C3×C5#6572712yescomplemented
302C22×C3—251no—
303C22×C3—251no—
304C22×C3—501no—
305C22×C3—501no—
306C22×C3#8783211yescharacteristic
307C3×C4—251no—
308C3×C4—251no—
309C3×C4—1002no—
310C3×C4#8783211yescharacteristic
311C3×C4C2×D5⋊AGL(1,𝔽5)11yescharacteristic
312Dic3—1001no—
313Dic3—1001no—
314C2×C5—22no—
315C2×C5—22no—
316C2×C5—22no—
317C2×C5—22no—
318C2×C5C2×Dic3⋊AGL(1,𝔽5)12yes—
319D5—52no—
320D5—52no—
321D5—102no—
322D5—102no—
323D5—102no—
324D5—102no—
325D5—102no—
326C23—251no—
327C2×C4—251no—
328C2×C4—251no—
329C2×C4—251no—
330C2×C4—251no—
331C2×C4—251no—
332C2×C4—502no—
333C2×C4—1002no—
334C2×C4—1501no—
335C2×C4—1501no—
336C2×C4#1059718211yescharacteristic
337C8—154no—
338C2×C3—21no—
339C2×C3—251no—
340C2×C3—251no—
341C2×C3—501no—
342C2×C3C2×#8783211yescharacteristic
343C5—22no—
344C5#1175172712yescomplemented
345C22—251no—
346C22—251no—
347C22—501no—
348C22—501no—
349C22#1192535611yescharacteristic
350C4—251no—
351C4—251no—
352C4—3001no—
353C4—3001no—
354C4—1002no—
355C4#1192535611yescharacteristic
356C4C2×#1059718211yescharacteristic
357C3#1310396211yescharacteristic, complemented, Sylow
358C2—21no—
359C2—251no—
360C2—251no—
361C2—501no—
362C2C2×#1192535611yescharacteristic
subgroupG/H
centerC2C2×#11925356
commutatorC3×C4×C52C22×C4
FrattiniC4C2×#10597182
FittingC2×C3×C4×C52C2×C4
socleC2×C3×C52C2×C4⋊C4
prime radicalC22×C3×C5⋊D5D4
cosocleC3×C4×C5⋊D5C23
hypercenterC2×C4#10597182
hypocenterC3×C52#428
LCSG ▷ C3×C4×C52 ▷ C2×C3×C52 ▷ C3×C52
UCSC1 ◁ C2 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. #72230822C2semidirect
2. #72229676C2semidirect
15. C5×#6710C4semidirect
16. #11925152C4semidirect
35. C5×C3⋊#168C2×C4semidirect
80. #6710AGL(1,𝔽5)semidirect
130. C3⋊#168C2×AGL(1,𝔽5)semidirect
167. C3×C52#428semidirect
251. C52#4964semidirect
301. C3×C5#65727semidirect
344. C5#11751727semidirect
357. C3#13103962semidirect

Maximal subgroups

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

subgroupindex
#722308222
#722296762
#722294532
#722294442
C2×C3×C4⋊#10432
#131039623
#117517275