Finite Group Databasebeta

← C3×#10598569 #14279079

Subgroups — C3×#10598569 #14279079

Order 1920.

332 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 76 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×#65775C211yesmaximal, characteristic
2C3×#65771C211yesmaximal, characteristic
3C3×#65029C211yesmaximal, characteristic
4#10598569C311yesmaximal, characteristic, direct product factor, Hall
5C2×C3×#711C2211yescharacteristic
6C2×C3×#704C411yescharacteristic, complemented
7C22×C3×#170C411yescharacteristic, complemented
8C3×#4130—21no—
9C3×#4023—21no—
10C3×C22⋊#170—21no—
11C3×#4011—21no—
12C3×#4011C412yescomplemented
13C3×#4003—21no—
14C3×C5×#99C411yescharacteristic, complemented
15C3×#3966C411yescharacteristic, complemented
16C3×#3938—21no—
17C3×#1391—51nomaximal, Hall
18#65775C2×C311yescharacteristic
19#65771C2×C311yescharacteristic
20#65029C2×C311yescharacteristic
21C2×C3×C5×D4C2×C411yescharacteristic
22C2×C3×C22⋊D5C2×C411yescharacteristic
23C2×C3×C22⋊D5D411yescharacteristic
24C22×C3×Dic5C2×C411yescharacteristic
25C22×C3×Dic5D411yescharacteristic
26C3×#711—41no—
27C3×#711—41no—
28C2×C3×C4×D5D411yescharacteristic
29C2×C3×Dic10Q811yescharacteristic
30C3×C22⋊AGL(1,𝔽5)—22no—
31C3×#704—41no—
32C2×C3×#170—21no—
33C2×C3×#170—21no—
34C2×C3×#170—41no—
35C3×C4⋊AGL(1,𝔽5)—41no—
36C3×C4×AGL(1,𝔽5)—41no—
37C3×C5×C22⋊C4—21no—
38C3×C22⋊Dic5—21no—
39C3×#683—41no—
40C3×C4×Dic5—41no—
41C3×#410—51no—
42C3×#410—51no—
43C3×#408—51no—
44C2×#711C22×C311yescharacteristic
45C2×#704C3×C411yescharacteristic, complemented
46C22×#170C3×C411yescharacteristic, complemented
47#4130—21no—
48#4023—21no—
49C22⋊#170—21no—
50#4011—21no—
51#4011C3×C412yescomplemented
52#4003—21no—
53C5×#99C3×C411yescharacteristic, complemented
54#3966C3×C411yescharacteristic, complemented
55#3938—21no—
56#1391—51noSylow
57C23×C3×C5C4211yescharacteristic
58C23×C3×C5C22⋊C411yescharacteristic
59C22×C3×D5C22⋊C411yescharacteristic
60C2×C3×AGL(1,𝔽5)—42no—
61C3×C5×D4—41no—
62C2×C3×C4×C5—41no—
63C2×C3×C4×C5C4⋊C411yescharacteristic
64C3×C22⋊D5—41no—
65C3×C22⋊D5—41no—
66C2×C3×Dic5—21no—
67C2×C3×Dic5—21no—
68C2×C3×Dic5—21no—
69C2×C3×Dic5—21no—
70C2×C3×Dic5—41no—
71C2×C3×Dic5C22⋊C411yescharacteristic
72C2×C3×Dic5C4⋊C411yescharacteristic
73C2×C3×Dic5C4⋊C411yescharacteristic
74C3×C4×D5—21no—
75C3×C4×D5—21no—
76C3×Dic10—41no—
77C3×#170—41no—
78C3×#170—41no—
79C2×C3×#41—51no—
80C2×C3×M16—51no—
81C22×C3×C8—51no—
82C3×#117—101no—
83C3×#117—101no—
84C3×#99—52no—
85C3×#99—51no—
86C3×#99—101no—
87C3×#99—101no—
88C3×#99AGL(1,𝔽5)11yescharacteristic, complemented
89C3×C22⋊C8—101no—
90C3×C22⋊C8—101no—
91C2×C5×D4C2×C3×C411yescharacteristic
92C2×C22⋊D5C2×C3×C411yescharacteristic
93C2×C22⋊D5C3×D411yescharacteristic
94C22×Dic5C2×C3×C411yescharacteristic
95C22×Dic5C3×D411yescharacteristic
96#711—41no—
97#711—41no—
98C2×C4×D5C3×D411yescharacteristic
99C2×Dic10C3×Q811yescharacteristic
100C22⋊AGL(1,𝔽5)—22no—
101#704—41no—
102C2×#170—21no—
103C2×#170—21no—
104C2×#170—41no—
105C4⋊AGL(1,𝔽5)—41no—
106C4×AGL(1,𝔽5)—41no—
107C5×C22⋊C4—21no—
108C22⋊Dic5—21no—
109#683—41no—
110C4×Dic5—41no—
111#410—51no—
112#410—51no—
113#408—51no—
114C22×C3×C5—21no—
115C22×C3×C5—21no—
116C22×C3×C5—41no—
117C22×C3×C5—41no—
118C22×C3×C5#9511yescharacteristic
119C2×C3×D5—21no—
120C2×C3×D5—41no—
121C3×AGL(1,𝔽5)—82no—
122C3×C4×C5—21no—
123C3×C4×C5—81no—
124C3×Dic5—21no—
125C3×Dic5—21no—
126C3×Dic5—81no—
127C3×Dic5#9911yescharacteristic
128C3×Dic5#9911yescharacteristic
129C3×#41—201no—
130C3×#41—201no—
131C2×C3×Q8—51no—
132C2×C3×D4—51no—
133C2×C3×D4—51no—
134C2×C3×D4C2×AGL(1,𝔽5)11yescharacteristic
135C22×C3×C4—51no—
136C22×C3×C4—51no—
137C22×C3×C4—51no—
138C3×M16—201no—
139C2×C3×C8—101no—
140C2×C3×C8—101no—
141C2×C3×C8—201no—
142C3×C4⋊C4—201no—
143C3×C4⋊C4—201no—
144C3×C22⋊C4—21no—
145C3×C22⋊C4—102no—
146C3×C22⋊C4—101no—
147C3×C42—201no—
148C3×C42—201no—
149C23×C5C3×C4211yescharacteristic
150C23×C5C3×C22⋊C411yescharacteristic
151C22×D5C3×C22⋊C411yescharacteristic
152C2×AGL(1,𝔽5)—42no—
153C5×D4—41no—
154C2×C4×C5—41no—
155C2×C4×C5C3×C4⋊C411yescharacteristic
156C22⋊D5—41no—
157C22⋊D5—41no—
158C2×Dic5—21no—
159C2×Dic5—21no—
160C2×Dic5—21no—
161C2×Dic5—21no—
162C2×Dic5—41no—
163C2×Dic5C3×C22⋊C411yescharacteristic
164C2×Dic5C3×C4⋊C411yescharacteristic
165C2×Dic5C3×C4⋊C411yescharacteristic
166C4×D5—21no—
167C4×D5—21no—
168Dic10—41no—
169#170—41no—
170#170—41no—
171C2×#41—51no—
172C2×M16—51no—
173C22×C8—51no—
174#117—101no—
175#117—101no—
176#99—52no—
177#99—51no—
178#99—101no—
179#99—101no—
180#99C3×AGL(1,𝔽5)11yescharacteristic, complemented
181C22⋊C8—101no—
182C22⋊C8—101no—
183C2×C3×C5—21no—
184C2×C3×C5—41no—
185C2×C3×C5—41no—
186C2×C3×C5#34211yescharacteristic
187C3×D5—41no—
188C23×C3—51no—
189C23×C3C4×AGL(1,𝔽5)11yescharacteristic
190C23×C3C22⋊AGL(1,𝔽5)11yescharacteristic
191C3×Q8—201no—
192C3×D4—41no—
193C3×D4—201no—
194C3×D4—201no—
195C2×C3×C4—41no—
196C2×C3×C4—51no—
197C2×C3×C4—51no—
198C2×C3×C4—51no—
199C2×C3×C4—101no—
200C2×C3×C4—101no—
201C2×C3×C4—101no—
202C2×C3×C4—101no—
203C2×C3×C4—101no—
204C2×C3×C4—101no—
205C2×C3×C4—201no—
206C2×C3×C4—202no—
207C2×C3×C4C4⋊AGL(1,𝔽5)11yescharacteristic
208C3×C8—201no—
209C3×C8—201no—
210C22×C5—21no—
211C22×C5—21no—
212C22×C5—41no—
213C22×C5—41no—
214C22×C5C3×#9511yescharacteristic
215C2×D5—21no—
216C2×D5—41no—
217AGL(1,𝔽5)—82no—
218C4×C5—21no—
219C4×C5—81no—
220Dic5—21no—
221Dic5—21no—
222Dic5—81no—
223Dic5C3×#9911yescharacteristic
224Dic5C3×#9911yescharacteristic
225#41—201no—
226#41—201no—
227C2×Q8—51no—
228C2×D4—51no—
229C2×D4—51no—
230C2×D4C2×C3×AGL(1,𝔽5)11yescharacteristic
231C22×C4—51no—
232C22×C4—51no—
233C22×C4—51no—
234M16—201no—
235C2×C8—101no—
236C2×C8—101no—
237C2×C8—201no—
238C4⋊C4—201no—
239C4⋊C4—201no—
240C22⋊C4—21no—
241C22⋊C4—102no—
242C22⋊C4—101no—
243C42—201no—
244C42—201no—
245C3×C5#139111yescharacteristic, complemented, Hall
246C22×C3—21no—
247C22×C3—21no—
248C22×C3—41no—
249C22×C3—41no—
250C22×C3—101no—
251C22×C3—201no—
252C22×C3(C2×AGL(1,𝔽5))⋊C411yescharacteristic
253C3×C4—21no—
254C3×C4—51no—
255C3×C4—51no—
256C3×C4—81no—
257C3×C4—101no—
258C3×C4—101no—
259C3×C4—401no—
260C3×C4—402no—
261C2×C5—21no—
262C2×C5—41no—
263C2×C5—41no—
264C2×C5C3×#34211yescharacteristic
265D5—41no—
266C23—51no—
267C23C3×C4×AGL(1,𝔽5)11yescharacteristic
268C23C3×C22⋊AGL(1,𝔽5)11yescharacteristic
269Q8—201no—
270D4—41no—
271D4—201no—
272D4—201no—
273C2×C4—41no—
274C2×C4—51no—
275C2×C4—51no—
276C2×C4—51no—
277C2×C4—101no—
278C2×C4—101no—
279C2×C4—101no—
280C2×C4—101no—
281C2×C4—101no—
282C2×C4—101no—
283C2×C4—201no—
284C2×C4—202no—
285C2×C4C3×C4⋊AGL(1,𝔽5)11yescharacteristic
286C8—201no—
287C8—201no—
288C2×C3—21no—
289C2×C3—41no—
290C2×C3—41no—
291C2×C3—201no—
292C2×C3#6486611yescharacteristic
293C5C3×#139111yescharacteristic, complemented, Sylow
294C22—21no—
295C22—21no—
296C22—41no—
297C22—41no—
298C22—101no—
299C22—201no—
300C22C3×(C2×AGL(1,𝔽5))⋊C411yescharacteristic
301C4—21no—
302C4—51no—
303C4—51no—
304C4—81no—
305C4—101no—
306C4—101no—
307C4—401no—
308C4—402no—
309C3#1059856911yescharacteristic, direct product factor, Sylow
310C2—21no—
311C2—41no—
312C2—41no—
313C2—201no—
314C2C3×#6486611yescharacteristic
subgroupG/H
centerC2×C3#64866
commutatorC23×C5C3×C42
FrattiniC2×D4C2×C3×AGL(1,𝔽5)
FittingC3×C5×#99C4
socleC2×C3×C5#342
prime radicalC2×C3×#711C22
cosocleC2×#711C22×C3
hypercenterC3×#99AGL(1,𝔽5)
hypocenterC5C3×#1391
LCSG ▷ C23×C5 ▷ C22×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C3 ◁ C22×C3 ◁ C2×C3×D4 ◁ C3×#99

Complements

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

NH ≅ G/Ntype
4. #10598569C3direct (N × H)
6. C2×C3×#704C4semidirect
7. C22×C3×#170C4semidirect
12. C3×#4011C4semidirect
14. C3×C5×#99C4semidirect
15. C3×#3966C4semidirect
45. C2×#704C3×C4semidirect
46. C22×#170C3×C4semidirect
51. #4011C3×C4semidirect
53. C5×#99C3×C4semidirect
54. #3966C3×C4semidirect
88. C3×#99AGL(1,𝔽5)semidirect
180. #99C3×AGL(1,𝔽5)semidirect
245. C3×C5#1391semidirect
293. C5C3×#1391semidirect
309. C3#10598569direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#657752
C3×#657712
C3×#650292
#105985693
C3×#13915