Finite Group Databasebeta

← C3×#13100332 #72259825

Subgroups — C3×#13100332 #72259825

Order 4800.

656 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 656 conjugacy classes of subgroups — too many to draw. Only its 100 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
1C3×#11717448C211yesmaximal, characteristic
2C3×#11717451C211yesmaximal, characteristic
3C3×C42×C5⋊D5C211yesmaximal, characteristic
4#13100332C311yesmaximal, characteristic, direct product factor, Hall
5C2×C3×C4×C5⋊D5—21no—
6C2×C3×C4×C5⋊D5C2211yescharacteristic
7C3×C22⋊#1044—21no—
8C3×#87867—21no—
9C3×C4⋊#1044C412yescomplemented
10C3×C4×#1044—21no—
11C3×#87863C412yescomplemented
12C3×#87862—21no—
13C3×C42×C52C411yescharacteristic, complemented
14C3×C4×#1039—21no—
15C3×C4×#1039C411yescharacteristic, complemented
16C3×#64862—52nomaximal
17#11717451C2×C311yescharacteristic
18#11717448C2×C311yescharacteristic
19C42×C5⋊D5C2×C311yescharacteristic
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—
32C3×#4130—52no—
33C3×D5⋊M16—52no—
34C3×C42×D5—52no—
35C3×C42×D5—102no—
36C2×C4×C5⋊D5—21no—
37C2×C4×C5⋊D5C22×C311yescharacteristic
38C22⋊#1044—21no—
39#87867—21no—
40C4⋊#1044C3×C412yescomplemented
41C4×#1044—21no—
42#87863C3×C412yescomplemented
43#87862—21no—
44C42×C52C3×C411yescharacteristic, complemented
45C4×#1039—21no—
46C4×#1039C3×C411yescharacteristic, complemented
47#64862—52no—
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—
63C3×C22⋊AGL(1,𝔽5)—102no—
64C3×#704—102no—
65C3×C4⋊AGL(1,𝔽5)—54no—
66C3×C4×AGL(1,𝔽5)—102no—
67C3×#700—54no—
68C3×D5⋊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⋊D5C3×D411yescharacteristic
76C2×#1044—22no—
77C2×C4×C52—21no—
78C2×C4×C52C2×C3×C411yescharacteristic
79C2×#1039—21no—
80C2×#1039C3×D411yescharacteristic
81C4×C5⋊D5—24no—
82C4×C5⋊D5C2×C3×C411yescharacteristic
83C4×C5⋊D5C2×C3×C411yescharacteristic
84C4×C5⋊D5C3×D411yescharacteristic
85C4×C5⋊D5C3×Q811yescharacteristic
86#6034—22no—
87C3×#339—251noHall
88#4130—52no—
89D5⋊M16—52no—
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×C3×AGL(1,𝔽5)—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×C52C3×C22⋊C411yescharacteristic
119C2×C5⋊D5—22no—
120C2×C5⋊D5C3×C22⋊C411yescharacteristic
121C2×C5⋊D5C3×C4⋊C411yescharacteristic
122#1044—42no—
123C4×C52—22no—
124C4×C52C3×C4211yescharacteristic
125C4×C52C3×C4⋊C411yescharacteristic
126#1039—22no—
127#1039C3×C22⋊C411yescharacteristic
128#1039C3×C4⋊C411yescharacteristic
129C2×C3×M16—251no—
130C3×#117—251no—
131C2×C3×C42—251no—
132C2×C4×D5—52no—
133C2×C4×D5—102no—
134C2×C4×D5—102no—
135C2×C4×D5—104no—
136C22⋊AGL(1,𝔽5)—102no—
137#704—102no—
138C4⋊AGL(1,𝔽5)—54no—
139C4×AGL(1,𝔽5)—102no—
140#700—54no—
141D5⋊C8—102no—
142C42×C5—22no—
143C42×C5C3×AGL(1,𝔽5)12yescomplemented
144C4×Dic5—52no—
145C4×Dic5—102no—
146C4×Dic5—102no—
147C4×Dic5—104no—
148C3×C52#33911yescharacteristic, complemented, Hall
149#339—251noSylow
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—
158C3×AGL(1,𝔽5)—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⋊D5C3×#10412yes—
175C22×C3×C4—251no—
176C22×C3×C4—501no—
177C3×M16—252no—
178C3×M16—501no—
179C2×C3×C8—501no—
180C3×C4⋊C4—252no—
181C3×C22⋊C4—501no—
182C3×C42—251no—
183C3×C42—501no—
184C3×C42—501no—
185C3×C42#104411yescharacteristic, complemented
186C22×D5—52no—
187C22×D5—102no—
188C2×AGL(1,𝔽5)—104no—
189C2×C4×C5—24no—
190C2×C4×C5—22no—
191C2×C4×C5—22no—
192C2×C4×C5C2×C3×AGL(1,𝔽5)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—104no—
208C2×M16—251no—
209#117—251no—
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—
219C52C3×#33911yescharacteristic, 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×C4—502no—
230C2×C3×C4C2×#104411yescharacteristic
231C3×C8—502no—
232C22×C5—22no—
233C22×C5C3×C22⋊AGL(1,𝔽5)12yes—
234C2×D5—52no—
235C2×D5—52no—
236C2×D5—102no—
237C2×D5—102no—
238C2×D5—104no—
239C2×D5—108no—
240AGL(1,𝔽5)—204no—
241C4×C5—28no—
242C4×C5—24no—
243C4×C5—22no—
244C4×C5—22no—
245C4×C5C3×C4×AGL(1,𝔽5)12yes—
246C4×C5C3×C4⋊AGL(1,𝔽5)12yes—
247Dic5—52no—
248Dic5—52no—
249Dic5—102no—
250Dic5—102no—
251Dic5—108no—
252Dic5—104no—
253C22×C4—251no—
254C22×C4—501no—
255M16—252no—
256M16—501no—
257C2×C8—501no—
258C4⋊C4—252no—
259C22⋊C4—501no—
260C42—251no—
261C42—501no—
262C42—501no—
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×C4—1002no—
275C3×C4C4×#104411yescharacteristic
276C3×C4C4⋊#104411yescharacteristic
277C2×C5—24no—
278C2×C5—22no—
279C2×C5—22no—
280C2×C5C3×(C2×AGL(1,𝔽5))⋊C412yes—
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—502no—
295C2×C4C2×C3×#104411yescharacteristic
296C8—502no—
297C2×C3—21no—
298C2×C3—252no—
299C2×C3—501no—
300C2×C3#1171689911yescharacteristic
301C5—22no—
302C5C3×#6486212yescomplemented
303C22—251no—
304C22—251no—
305C22—502no—
306C22C3×C22⋊#104411yescharacteristic
307C4—22no—
308C4—251no—
309C4—251no—
310C4—502no—
311C4—1002no—
312C4C3×C4×#104411yescharacteristic
313C4C3×C4⋊#104411yescharacteristic
314C3#1310033211yescharacteristic, direct product factor, Sylow
315C2—21no—
316C2—252no—
317C2—501no—
318C2C3×#1171689911yescharacteristic
subgroupG/H
centerC3×C4C4⋊#1044
commutatorC4×C52C3×C42
FrattiniC2×C4C2×C3×#1044
FittingC3×C42×C52C4
socleC2×C3×C52#95
prime radicalC22×C3×C5⋊D5D4
cosocleC2×C4×C5⋊D5C22×C3
hypercenterC3×C42#1044
hypocenterC52C3×#339
LCSG ▷ C4×C52 ▷ C2×C52 ▷ C52
UCSC1 ◁ C3×C4 ◁ C2×C3×C4 ◁ C3×C42

Complements

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

NH ≅ G/Ntype
4. #13100332C3direct (N × H)
9. C3×C4⋊#1044C4semidirect
11. C3×#87863C4semidirect
13. C3×C42×C52C4semidirect
15. C3×C4×#1039C4semidirect
40. C4⋊#1044C3×C4semidirect
42. #87863C3×C4semidirect
44. C42×C52C3×C4semidirect
46. C4×#1039C3×C4semidirect
70. C3×C42×C5AGL(1,𝔽5)semidirect
143. C42×C5C3×AGL(1,𝔽5)semidirect
148. C3×C52#339semidirect
185. C3×C42#1044semidirect
219. C52C3×#339semidirect
263. C42C3×#1044semidirect
265. C3×C5#64862semidirect
302. C5C3×#64862semidirect
314. C3#13100332direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#117174482
C3×#117174512
C3×C42×C5⋊D52
#131003323
C3×#648625