Finite Group Databasebeta

← #72259372

Subgroups — #72259372

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#72230829C212yesmaximal, complemented
2#72230769C212yesmaximal, complemented
3#72229453C211yesmaximal, characteristic
4C3×#11717451C211yesmaximal, characteristic
5C2×C4⋊(C3⋊#1043)C211yesmaximal, characteristic
6#13103962—31nomaximal, Hall
7C22×C3⋊#1043—21no—
8C2×C3×C4×C5⋊D5C2211yescharacteristic
9C4⋊(C3⋊#1043)—21no—
10C4⋊(C3⋊#1043)C2212yes—
11C3×C22⋊#1044—21no—
12C3×C4⋊#1044C2211yescharacteristic
13C3×C4⋊#1044C2211yescharacteristic
14C3×C4×#1044—21no—
15C5×#6710C412yescomplemented
16#11925152C412yescomplemented
17#11925150C2212yes—
18#11751728—52nomaximal
19#11717451D311yescharacteristic
20C2×C4⋊#1043—31no—
21#11716936—31no—
22#11716772—32no—
23#11716771—32no—
24C22×C3×C5⋊D5D411yescharacteristic
25C2×C3⋊#1043—22no—
26C2×C3⋊#1043—42no—
27C2×C3×#1044—21no—
28C2×C3×#1044—21no—
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
36C2×C4⋊ΓL(1,𝔽16)—52no—
37C2×C4⋊ΓL(1,𝔽16)—102no—
38C3×#4130—52no—
39D5×#208—52no—
40#91160—54no—
41#91159—54no—
42C22×#1043—61no—
43C2×C4×C5⋊D5C2×D311yescharacteristic
44C22⋊#1044—21no—
45C4⋊#1044C2×D311yescharacteristic
46C4⋊#1044C2×D311yescharacteristic
47C4×#1044—21no—
48C4⋊#1043—32no—
49C4⋊#1043—61no—
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⋊#1043—42no—
60C3×#1044—41no—
61C3×#1044—41no—
62C3×C4×C52C4⋊C411yescharacteristic
63C3×C4×C52C22×C411yescharacteristic
64C3×#1039C4⋊C411yescharacteristic
65C3×#1039C2×Q811yescharacteristic
66C22×ΓL(1,𝔽16)—102no—
67C22×ΓL(1,𝔽16)—104no—
68C2×C3×C4×D5—52no—
69C2×C3×C4×D5—102no—
70C4⋊ΓL(1,𝔽16)—54no—
71C4⋊ΓL(1,𝔽16)—102no—
72C4⋊ΓL(1,𝔽16)—104no—
73C4⋊ΓL(1,𝔽16)—104no—
74C3×C22⋊AGL(1,𝔽5)—102no—
75C3×C4⋊AGL(1,𝔽5)—52no—
76C3×C4⋊AGL(1,𝔽5)—52no—
77C3×C4×AGL(1,𝔽5)—102no—
78#6710AGL(1,𝔽5)12yescomplemented
79C5×#208—52no—
80#6649—54no—
81D5×C3⋊C8—54no—
82C22×C5⋊D5#6711yescharacteristic
83C2×#1044—21no—
84C2×#1044—21no—
85C2×#1043—62no—
86C2×#1043—122no—
87C2×C4×C52C4×D311yescharacteristic
88C2×#1039Dic611yescharacteristic
89C4×C5⋊D5Dic611yescharacteristic
90C4×C5⋊D5C4×D311yescharacteristic
91C4×C5⋊D5#6711yescharacteristic
92C4×C5⋊D5C22×D311yescharacteristic
93C5×#168—34no—
94#4995—251noHall
95#4130—52no—
96C2×C4⋊AGL(1,𝔽5)—152no—
97C2×C4⋊AGL(1,𝔽5)—302no—
98D5×M16—152no—
99#3994—154no—
100#3993—154no—
101C2×C3×C52—21no—
102C2×C3×C52C2×C4⋊C411yescharacteristic
103C3×C5⋊D5—21no—
104C3×C5⋊D5AGL(1,ℤ/8)11yescharacteristic
105C3×C5⋊D5#13711yescharacteristic
106C22×C3×D5—52no—
107C22×C3×D5—102no—
108C2×ΓL(1,𝔽16)—104no—
109C2×ΓL(1,𝔽16)—108no—
110C2×ΓL(1,𝔽16)—204no—
111C2×ΓL(1,𝔽16)—208no—
112C2×C3×AGL(1,𝔽5)—102no—
113C2×C3×AGL(1,𝔽5)—102no—
114C2×C3×C4×C5—22no—
115C2×C3×C4×C5C2×AGL(1,𝔽5)12yes—
116C2×C3×Dic5—52no—
117C2×C3×Dic5—102no—
118C3×C4×D5—52no—
119C3×C4×D5—52no—
120C3×C4×D5—52no—
121C3×C4×D5—52no—
122C3×C4×D5—102no—
123C3×C4×D5—102no—
124C3×C4×D5—102no—
125C3×C4×D5—102no—
126C3⋊#168C2×AGL(1,𝔽5)14yescomplemented
127C5×C3⋊C8—54no—
128C22×C52Dic3⋊C411yescharacteristic
129C2×C5⋊D5—21no—
130C2×C5⋊D5—21no—
131C2×C5⋊D5Dic3⋊C411yescharacteristic
132C2×C5⋊D5C2×#6711yescharacteristic
133#1044—41no—
134#1044—41no—
135#1043—122no—
136C4×C52Dic3⋊C411yescharacteristic
137C4×C52C2×C4×D311yescharacteristic
138#1039Dic3⋊C411yescharacteristic
139#1039C2×Dic611yescharacteristic
140C3×#117—251no—
141C2×C4⋊Dic3—251no—
142C2×#208—251no—
143#808—252no—
144#807—252no—
145C22×AGL(1,𝔽5)—302no—
146C22×AGL(1,𝔽5)—304no—
147C2×C4×D5—52no—
148C2×C4×D5—102no—
149C22⋊AGL(1,𝔽5)—102no—
150C4⋊AGL(1,𝔽5)—52no—
151C4⋊AGL(1,𝔽5)—52no—
152C4⋊AGL(1,𝔽5)—154no—
153C4⋊AGL(1,𝔽5)—304no—
154C4⋊AGL(1,𝔽5)—302no—
155C4⋊AGL(1,𝔽5)—304no—
156C4×AGL(1,𝔽5)—102no—
157C5×M16—152no—
158#681—32no—
159#676—154no—
160C8×D5—154no—
161C3×C52#42811yescharacteristic, complemented, Hall
162#428—751noSylow
163C22×C3×C5—22no—
164C22×C3×C5C4⋊AGL(1,𝔽5)12yes—
165C2×C3×D5—52no—
166C2×C3×D5—52no—
167C2×C3×D5—102no—
168C2×C3×D5—102no—
169C2×C3×D5—102no—
170C2×C3×D5—102no—
171C2×C3×D5—104no—
172C2×C3×D5—104no—
173ΓL(1,𝔽16)—204no—
174ΓL(1,𝔽16)—208no—
175C3×AGL(1,𝔽5)—202no—
176C3×AGL(1,𝔽5)—202no—
177C3×C4×C5—22no—
178C3×C4×C5—22no—
179C3×C4×C5C4⋊AGL(1,𝔽5)12yes—
180C3×C4×C5C22×AGL(1,𝔽5)12yes—
181C3×Dic5—52no—
182C3×Dic5—52no—
183C3×Dic5—102no—
184C3×Dic5—102no—
185C2×C52—21no—
186C2×C52C2×Dic3⋊C411yescharacteristic
187C5⋊D5—21no—
188C5⋊D5#94911yescharacteristic
189C5⋊D5#95111yescharacteristic
190C22×C3×C4—251no—
191C22×Dic3—501no—
192C3×C4⋊C4—251no—
193C3×C4⋊C4—251no—
194C3×C22⋊C4—501no—
195C3×C42—501no—
196C4⋊Dic3—252no—
197C4⋊Dic3—501no—
198#208—52no—
199#208—252no—
200C2×C3⋊C8—252no—
201C22×D5—52no—
202C22×D5—102no—
203C2×AGL(1,𝔽5)—102no—
204C2×AGL(1,𝔽5)—102no—
205C2×AGL(1,𝔽5)—304no—
206C2×AGL(1,𝔽5)—308no—
207C2×AGL(1,𝔽5)—604no—
208C2×AGL(1,𝔽5)—608no—
209C2×C4×C5—22no—
210C2×C4×C5D3×AGL(1,𝔽5)12yes—
211C2×Dic5—52no—
212C2×Dic5—102no—
213C4×D5—52no—
214C4×D5—52no—
215C4×D5—52no—
216C4×D5—52no—
217C4×D5—102no—
218C4×D5—102no—
219C4×D5—102no—
220C4×D5—102no—
221C5×C8—154no—
222#168—34no—
223C2×M16—751no—
224#117—251no—
225C2×C4⋊C4—751no—
226#107—752no—
227#106—752no—
228C2×C3×C5—24no—
229C2×C3×C5—22no—
230C2×C3×C5—22no—
231C2×C3×C5C2×C4⋊AGL(1,𝔽5)12yes—
232C3×D5—52no—
233C3×D5—52no—
234C3×D5—102no—
235C3×D5—102no—
236C3×D5—104no—
237C3×D5—102no—
238C52#499511yescharacteristic, complemented, Sylow
239C23×C3—251no—
240C2×C3×C4—251no—
241C2×C3×C4—251no—
242C2×C3×C4—251no—
243C2×C3×C4—251no—
244C2×C3×C4—251no—
245C2×C3×C4—501no—
246C2×C3×C4—501no—
247C2×C3×C4D5⋊AGL(1,𝔽5)11yescharacteristic
248C2×Dic3—502no—
249C2×Dic3—1002no—
250C3⋊C8—54no—
251C22×C5—22no—
252C22×C5Dic3⋊AGL(1,𝔽5)12yes—
253C2×D5—52no—
254C2×D5—52no—
255C2×D5—102no—
256C2×D5—102no—
257C2×D5—102no—
258C2×D5—102no—
259C2×D5—104no—
260C2×D5—104no—
261AGL(1,𝔽5)—202no—
262AGL(1,𝔽5)—202no—
263AGL(1,𝔽5)—604no—
264AGL(1,𝔽5)—608no—
265C4×C5—22no—
266C4×C5—22no—
267C4×C5Dic3⋊AGL(1,𝔽5)12yes—
268C4×C5C2×D3×AGL(1,𝔽5)12yes—
269Dic5—52no—
270Dic5—52no—
271Dic5—102no—
272Dic5—102no—
273C22×C4—251no—
274C22×C4—1501no—
275M16—152no—
276M16—752no—
277C2×C8—752no—
278C4⋊C4—251no—
279C4⋊C4—251no—
280C4⋊C4—752no—
281C4⋊C4—1501no—
282C22⋊C4—501no—
283C42—501no—
284C3×C5—22no—
285C3×C5#6572712yescomplemented
286C22×C3—251no—
287C22×C3—251no—
288C22×C3—501no—
289C22×C3—501no—
290C22×C3#8783211yescharacteristic
291C3×C4—251no—
292C3×C4—251no—
293C3×C4—1001no—
294C3×C4—1001no—
295C3×C4#8783211yescharacteristic
296C3×C4C2×D5⋊AGL(1,𝔽5)11yescharacteristic
297Dic3—1002no—
298C2×C5—24no—
299C2×C5—22no—
300C2×C5—22no—
301C2×C5C2×Dic3⋊AGL(1,𝔽5)12yes—
302D5—52no—
303D5—52no—
304D5—102no—
305D5—102no—
306D5—104no—
307D5—102no—
308C23—251no—
309C2×C4—251no—
310C2×C4—251no—
311C2×C4—251no—
312C2×C4—251no—
313C2×C4—251no—
314C2×C4—3002no—
315C2×C4—501no—
316C2×C4—501no—
317C2×C4—1502no—
318C2×C4#1059718311yescharacteristic
319C8—154no—
320C2×C3—21no—
321C2×C3—251no—
322C2×C3—251no—
323C2×C3—501no—
324C2×C3C2×#8783211yescharacteristic
325C5—22no—
326C5#1175172812yescomplemented
327C22—251no—
328C22—251no—
329C22—501no—
330C22—501no—
331C22#1192536111yescharacteristic
332C4—251no—
333C4—251no—
334C4—3002no—
335C4—1001no—
336C4—1001no—
337C4#1192536111yescharacteristic
338C4C2×#1059718311yescharacteristic
339C3#1310396211yescharacteristic, complemented, Sylow
340C2—21no—
341C2—251no—
342C2—251no—
343C2—501no—
344C2C2×#1192536111yescharacteristic
subgroupG/H
centerC2C2×#11925361
commutatorC3×C4×C52C22×C4
FrattiniC4C2×#10597183
FittingC2×C3×C4×C52C2×C4
socleC2×C3×C52C2×C4⋊C4
prime radicalC22×C3×C5⋊D5D4
cosocleC3×C4×C5⋊D5C23
hypercenterC2×C4#10597183
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. #72230829C2semidirect
2. #72230769C2semidirect
15. C5×#6710C4semidirect
16. #11925152C4semidirect
35. C5×C3⋊#168C2×C4semidirect
78. #6710AGL(1,𝔽5)semidirect
126. C3⋊#168C2×AGL(1,𝔽5)semidirect
161. C3×C52#428semidirect
238. C52#4995semidirect
285. C3×C5#65727semidirect
326. C5#11751728semidirect
339. C3#13103962semidirect

Maximal subgroups

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

subgroupindex
#722308292
#722307692
#722294532
C3×#117174512
C2×C4⋊(C3⋊#1043)2
#131039623
#117517285