Finite Group Databasebeta

← #72231060

Subgroups — #72231060

Order 2400.

484 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 70 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
1C22×C3⋊#1044C211yesmaximal, characteristic, complemented
2C2×D3×#1044C211yesmaximal, characteristic, complemented
3#67×C5⋊D5C211yesmaximal, characteristic
4C3×C22⋊#1044C211yesmaximal, characteristic, complemented
5Dic3⋊#1044C211yesmaximal, characteristic, complemented
6(C2×D3)⋊#1044C211yesmaximal, characteristic, complemented
7Dic3×#1044C211yesmaximal, characteristic, complemented
8D4×#1044—31nomaximal, Hall
9C22×C3×C5⋊D5C2211yescharacteristic
10C2×D3×C5⋊D5C2211yescharacteristic
11C2×C3⋊#1044—22no—
12C2×C3⋊#1044C2211yescharacteristic
13C2×C3⋊#1044C2211yescharacteristic
14C2×C3×#1044C2211yescharacteristic
15C2×C3×#1044C2211yescharacteristic
16D3×#1044—22no—
17#10597159C411yescharacteristic, complemented
18C52×#67C411yescharacteristic, complemented
19#10597115C411yescharacteristic, complemented
20#10597114C411yescharacteristic, complemented
21Dic3×C5⋊D5C2211yescharacteristic
22#67⋊AGL(1,𝔽5)—52nomaximal
23C22×#1044—31no—
24C22×#1044—31no—
25D4×C5⋊D5—31no—
26C22⋊#1044—31no—
27C22⋊#1044D311yescharacteristic, complemented
28C4⋊#1044—31no—
29C4×#1044—31no—
30C22×C3×C52C2×C411yescharacteristic, complemented
31C2×C3⋊(C5⋊D5)C2×C411yescharacteristic, complemented
32C2×C52×D3C2×C411yescharacteristic, complemented
33C2×C3×C5⋊D5—21no—
34C2×C3×C5⋊D5—21no—
35C2×C3×C5⋊D5C2×C411yescharacteristic, complemented
36C2×C3×C5⋊D5C2311yescharacteristic
37D3×C5⋊D5—21no—
38D3×C5⋊D5—21no—
39C3⋊#1044—21no—
40C3⋊#1044D412yescomplemented
41C3×#1044—21no—
42C3×#1044—21no—
43C3⋊#1039C2×C411yescharacteristic, complemented
44C52×Dic3C2×C411yescharacteristic, complemented
45C22×ΓL(1,𝔽16)—52no—
46C2×D3×AGL(1,𝔽5)—52no—
47D5×#67—52no—
48D5×#67—102no—
49C3×C22⋊AGL(1,𝔽5)—52no—
50Dic3⋊AGL(1,𝔽5)—52no—
51(C2×D3)⋊AGL(1,𝔽5)—52no—
52Dic3×AGL(1,𝔽5)—52no—
53C22×C5⋊D5—31no—
54C22×C5⋊D5C2×D311yescharacteristic
55C2×#1044—31no—
56C2×#1044—31no—
57C2×#1044—62no—
58C2×#1044—62no—
59C2×#1044C2×D311yescharacteristic
60C2×#1044C2×D311yescharacteristic
61C52×D4—31no—
62C22⋊(C5⋊D5)—31no—
63C22⋊(C5⋊D5)—31no—
64C4⋊(C5⋊D5)—31no—
65C4×C5⋊D5—31no—
66D4×AGL(1,𝔽5)—152no—
67C2×C3×C52—21no—
68C2×C3×C52C22×C411yescharacteristic
69C3⋊(C5⋊D5)—21no—
70C52×D3—21no—
71C3×C5⋊D5—21no—
72C3×C5⋊D5C2×D411yescharacteristic
73C3×C5⋊D5#4111yescharacteristic
74C22×C3×D5—52no—
75C22×C3×D5—102no—
76C2×D3×D5—52no—
77C2×D3×D5—102no—
78C2×ΓL(1,𝔽16)—52no—
79C2×ΓL(1,𝔽16)—52no—
80C2×ΓL(1,𝔽16)—104no—
81C2×C3×AGL(1,𝔽5)—52no—
82C2×C3×AGL(1,𝔽5)—52no—
83D3×AGL(1,𝔽5)—104no—
84#1220—52no—
85#1220—102no—
86C5×#67—22no—
87C5×#67AGL(1,𝔽5)12yescomplemented
88#1202—52no—
89#1202—102no—
90#1201—52no—
91#1201—102no—
92D5×Dic3—52no—
93D5×Dic3—102no—
94C22×C52—31no—
95C22×C52C4×D311yescharacteristic, complemented
96C2×C5⋊D5—21no—
97C2×C5⋊D5—21no—
98C2×C5⋊D5—31no—
99C2×C5⋊D5—61no—
100C2×C5⋊D5—61no—
101C2×C5⋊D5C4×D311yescharacteristic, complemented
102C2×C5⋊D5C22×D311yescharacteristic
103#1044—21no—
104#1044—21no—
105#1044—32no—
106#1044—61no—
107C4×C52—31no—
108#1039—31no—
109#67⋊C4—251noHall
110C22×AGL(1,𝔽5)—152no—
111C22×AGL(1,𝔽5)—152no—
112D4×D5—152no—
113D4×D5—302no—
114C22⋊AGL(1,𝔽5)—52no—
115C22⋊AGL(1,𝔽5)—152no—
116C4⋊AGL(1,𝔽5)—152no—
117C4×AGL(1,𝔽5)—152no—
118C3×C52C4×D411yescharacteristic, complemented, Hall
119C22×C3×C5—22no—
120C22×C3×C5C2×AGL(1,𝔽5)12yescomplemented
121C2×D15—52no—
122C2×D15—102no—
123C2×C5×D3—22no—
124C2×C5×D3C2×AGL(1,𝔽5)12yescomplemented
125C2×C3×D5—52no—
126C2×C3×D5—52no—
127C2×C3×D5—102no—
128C2×C3×D5—102no—
129C2×C3×D5—102no—
130C2×C3×D5—102no—
131C2×C3×D5—202no—
132C2×C3×D5—202no—
133D3×D5—102no—
134D3×D5—102no—
135D3×D5—202no—
136D3×D5—202no—
137ΓL(1,𝔽16)—54no—
138ΓL(1,𝔽16)—102no—
139C3×AGL(1,𝔽5)—102no—
140C3×AGL(1,𝔽5)—102no—
141Dic15—52no—
142Dic15—102no—
143C5×Dic3—22no—
144C5×Dic3C2×AGL(1,𝔽5)12yescomplemented
145C2×C52—21no—
146C2×C52—61no—
147C2×C52C2×C4×D311yescharacteristic
148C5⋊D5—21no—
149C5⋊D5—61no—
150C5⋊D5D3×D411yescharacteristic
151C5⋊D5#23711yescharacteristic
152C2×#67—251no—
153C22×Dic3—251no—
154C2×C4×D3—251no—
155C3×C22⋊C4—251no—
156(C2×D3)⋊C4—251no—
157Dic3⋊C4—251no—
158C4×Dic3—251no—
159C22×D5—52no—
160C22×D5—102no—
161C22×D5—152no—
162C22×D5—302no—
163C2×AGL(1,𝔽5)—52no—
164C2×AGL(1,𝔽5)—52no—
165C2×AGL(1,𝔽5)—152no—
166C2×AGL(1,𝔽5)—152no—
167C2×AGL(1,𝔽5)—304no—
168C2×AGL(1,𝔽5)—304no—
169C5×D4—32no—
170C5×D4—62no—
171C22⋊D5—152no—
172C22⋊D5—152no—
173C22⋊D5—302no—
174C22⋊D5—302no—
175D20—152no—
176D20—302no—
177C4×D5—152no—
178C4×D5—302no—
179C4×D4—751noSylow
180C2×C3×C5—22no—
181C2×C3×C5—22no—
182C2×C3×C5—42no—
183C2×C3×C5C22×AGL(1,𝔽5)12yes—
184D15—102no—
185D15—202no—
186C3×D5—52no—
187C3×D5—52no—
188C3×D5—102no—
189C3×D5—102no—
190C3×D5—102no—
191C3×D5—202no—
192C5×D3—22no—
193C5×D3—42no—
194C52#67⋊C411yescharacteristic, complemented, Sylow
195C23×C3—251no—
196C22×D3—251no—
197C2×C3×C4—251no—
198C2×C3×C4—251no—
199#67—251no—
200#67—251no—
201#67—251no—
202#67#104411yescharacteristic, complemented
203C2×Dic3—251no—
204C2×Dic3—251no—
205C2×Dic3—251no—
206C2×Dic3—502no—
207C4×D3—502no—
208C22×C5—22no—
209C22×C5—32no—
210C22×C5—62no—
211C22×C5D3×AGL(1,𝔽5)12yescomplemented
212C2×D5—52no—
213C2×D5—52no—
214C2×D5—102no—
215C2×D5—102no—
216C2×D5—102no—
217C2×D5—102no—
218C2×D5—152no—
219C2×D5—202no—
220C2×D5—202no—
221C2×D5—302no—
222C2×D5—302no—
223C2×D5—302no—
224C2×D5—602no—
225C2×D5—602no—
226AGL(1,𝔽5)—102no—
227AGL(1,𝔽5)—102no—
228AGL(1,𝔽5)—154no—
229AGL(1,𝔽5)—302no—
230C4×C5—32no—
231C4×C5—62no—
232Dic5—152no—
233Dic5—302no—
234C2×D4—751no—
235C22×C4—751no—
236C22×C4—751no—
237C4⋊C4—751no—
238C22⋊C4—251no—
239C22⋊C4—751no—
240C42—751no—
241C3×C5—22no—
242C3×C5D4×AGL(1,𝔽5)12yescomplemented
243C22×C3—251no—
244C22×C3—251no—
245C22×C3—501no—
246C22×C3—501no—
247C22×C3C2×#104411yescharacteristic, complemented
248C2×D3—251no—
249C2×D3—501no—
250C2×D3—501no—
251C2×D3C2×#104411yescharacteristic, complemented
252C3×C4—501no—
253C3×C4—501no—
254Dic3—252no—
255Dic3—251no—
256Dic3—501no—
257Dic3C2×#104411yescharacteristic, complemented
258C2×C5—22no—
259C2×C5—22no—
260C2×C5—42no—
261C2×C5—62no—
262C2×C5—122no—
263C2×C5C2×D3×AGL(1,𝔽5)12yes—
264D5—52no—
265D5—52no—
266D5—102no—
267D5—102no—
268D5—102no—
269D5—202no—
270D5—302no—
271D5—602no—
272C23—251no—
273C23—751no—
274D4—31no—
275D4—751no—
276D4—751no—
277D4—751no—
278C2×C4—251no—
279C2×C4—251no—
280C2×C4—751no—
281C2×C4—751no—
282C2×C4—751no—
283C2×C4—1502no—
284C2×C4—1502no—
285C2×C3—21no—
286C2×C3—251no—
287C2×C3—251no—
288C2×C3—501no—
289C2×C3C22×#104411yescharacteristic
290D3—21no—
291D3—501no—
292C5—22no—
293C5#67⋊AGL(1,𝔽5)12yescomplemented
294C22—31no—
295C22—251no—
296C22—251no—
297C22—501no—
298C22—501no—
299C22—751no—
300C22—1501no—
301C22—1501no—
302C22D3×#104411yescharacteristic, complemented
303C4—31no—
304C4—501no—
305C4—501no—
306C4—752no—
307C4—751no—
308C4—1501no—
309C3D4×#104411yescharacteristic, complemented, Sylow
310C2—21no—
311C2—61no—
312C2—251no—
313C2—251no—
314C2—501no—
315C2—1501no—
316C2C2×D3×#104411yescharacteristic
subgroupG/H
center, FrattiniC2C2×D3×#1044
commutator, socleC2×C3×C52C22×C4
FittingC22×C3×C52C2×C4
prime radical#67×C5⋊D5C2
cosocleC2×C3×C5⋊D5C23
hypercenterC22D3×#1044
hypocenterC3×C52C4×D4
LCSG ▷ C2×C3×C52 ▷ C3×C52
UCSC1 ◁ C2 ◁ C22

Complements

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

NH ≅ G/Ntype
1. C22×C3⋊#1044C2semidirect
2. C2×D3×#1044C2semidirect
4. C3×C22⋊#1044C2semidirect
5. Dic3⋊#1044C2semidirect
6. (C2×D3)⋊#1044C2semidirect
7. Dic3×#1044C2semidirect
17. #10597159C4semidirect
18. C52×#67C4semidirect
19. #10597115C4semidirect
20. #10597114C4semidirect
27. C22⋊#1044D3semidirect
30. C22×C3×C52C2×C4semidirect
31. C2×C3⋊(C5⋊D5)C2×C4semidirect
32. C2×C52×D3C2×C4semidirect
35. C2×C3×C5⋊D5C2×C4semidirect
40. C3⋊#1044D4semidirect
43. C3⋊#1039C2×C4semidirect
44. C52×Dic3C2×C4semidirect
87. C5×#67AGL(1,𝔽5)semidirect
95. C22×C52C4×D3semidirect
101. C2×C5⋊D5C4×D3semidirect
118. C3×C52C4×D4semidirect
120. C22×C3×C5C2×AGL(1,𝔽5)semidirect
124. C2×C5×D3C2×AGL(1,𝔽5)semidirect
144. C5×Dic3C2×AGL(1,𝔽5)semidirect
194. C52#67⋊C4semidirect
202. #67#1044semidirect
211. C22×C5D3×AGL(1,𝔽5)semidirect
242. C3×C5D4×AGL(1,𝔽5)semidirect
247. C22×C3C2×#1044semidirect
251. C2×D3C2×#1044semidirect
257. Dic3C2×#1044semidirect
293. C5#67⋊AGL(1,𝔽5)semidirect
302. C22D3×#1044semidirect
309. C3D4×#1044semidirect

Maximal subgroups

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

subgroupindex
C22×C3⋊#10442
C2×D3×#10442
#67×C5⋊D52
C3×C22⋊#10442
Dic3⋊#10442
(C2×D3)⋊#10442
Dic3×#10442
D4×#10443
#67⋊AGL(1,𝔽5)5