Finite Group Databasebeta

← C3×#10604996 #14313414

Subgroups — C3×#10604996 #14313414

Order 1920.

516 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 108 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×#66166C211yesmaximal, characteristic
2C3×#65745C212yesmaximal, complemented
3C3×#64928C212yesmaximal, complemented
4C3×#64922C212yesmaximal, complemented
5#10604996C311yesmaximal, characteristic, direct product factor, Hall
6C2×C3×C5×#41C411yescharacteristic, complemented
7C2×C3×#709C2211yescharacteristic
8C3×#4130—22no—
9C3×#4093C2211yescharacteristic
10C3×#4090C411yescharacteristic, complemented
11C3×#4086C2211yescharacteristic
12C3×#4085—22no—
13C3×#4082C411yescharacteristic, complemented
14C3×C4×C22⋊D5—22no—
15C2×C3×C4×Dic5C411yescharacteristic, complemented
16C3×#4011—22no—
17C3×#3999C2214yes—
18C3×#2126—51nomaximal, Hall
19#66166C2×C311yescharacteristic
20#65745C2×C312yescomplemented
21#64928C2×C312yescomplemented
22#64922C2×C312yescomplemented
23C3×C5×#41—41no—
24C3×C5×#41—41no—
25C2×C3×C5×Q8C2×C411yescharacteristic
26C2×C3×C5×D4—21no—
27C2×C3×C5×D4C2×C411yescharacteristic
28C22×C3×C4×C5—21no—
29C22×C3×C4×C5C2×C411yescharacteristic
30C2×C3×C22⋊D5—22no—
31C2×C3×C22⋊D5D412yes—
32C22×C3×Dic5—21no—
33C3×#709—41no—
34C3×#709—41no—
35C2×C3×D20C2311yescharacteristic
36C2×C3×C4×D5D412yes—
37C2×C3×Dic10C2×C411yescharacteristic
38C3×C22⋊AGL(1,𝔽5)—24no—
39C3×C4⋊AGL(1,𝔽5)—42no—
40C3×C4×AGL(1,𝔽5)—42no—
41C3×C22⋊Dic5—22no—
42C3×#685—22no—
43C3×#683—22no—
44C3×C4×Dic5—21no—
45C3×C4×Dic5C2×C411yescharacteristic
46C3×C4×Dic5C2×C411yescharacteristic
47C3×#532—51no—
48C3×#410—52no—
49C3×#354—52no—
50C3×#353—52no—
51C2×C5×#41C3×C411yescharacteristic, complemented
52C2×#709C22×C311yescharacteristic
53#4130—22no—
54#4093C22×C311yescharacteristic
55#4090C3×C411yescharacteristic, complemented
56#4086C22×C311yescharacteristic
57#4085—22no—
58#4082C3×C411yescharacteristic, complemented
59C4×C22⋊D5—22no—
60C2×C4×Dic5C3×C411yescharacteristic, complemented
61#4011—22no—
62#3999C22×C314yes—
63#2126—51noSylow
64C23×C3×C5—21no—
65C23×C3×C5C22⋊C411yescharacteristic
66C22×C3×D5C2×D412yes—
67C2×C3×AGL(1,𝔽5)—44no—
68C3×C5×Q8—41no—
69C3×C5×D4—41no—
70C3×C5×D4—42no—
71C2×C3×C4×C5—21no—
72C2×C3×C4×C5—21no—
73C2×C3×C4×C5—21no—
74C2×C3×C4×C5—41no—
75C2×C3×C4×C5—41no—
76C2×C3×C4×C5C22⋊C411yescharacteristic
77C2×C3×C4×C5C22×C411yescharacteristic
78C3×C22⋊D5—42no—
79C3×C22⋊D5—44no—
80C2×C3×Dic5—21no—
81C2×C3×Dic5—21no—
82C2×C3×Dic5—42no—
83C2×C3×Dic5C22⋊C412yes—
84C3×D20—41no—
85C3×C4×D5—22no—
86C3×C4×D5—22no—
87C3×Dic10—41no—
88C2×C3×#41—51no—
89C2×C3×#41AGL(1,𝔽5)11yescharacteristic, complemented
90C3×C4⋊Q8—51no—
91C3×#127—51no—
92C3×#124—51no—
93C3×#124—51no—
94C3×#121—102no—
95C3×C4×D4—102no—
96C3×#117—102no—
97C2×C3×C42—51no—
98C3×#99—54no—
99C3×#99—102no—
100C5×#41—41no—
101C5×#41—41no—
102C2×C5×Q8C2×C3×C411yescharacteristic
103C2×C5×D4—21no—
104C2×C5×D4C2×C3×C411yescharacteristic
105C22×C4×C5—21no—
106C22×C4×C5C2×C3×C411yescharacteristic
107C2×C22⋊D5—22no—
108C2×C22⋊D5C3×D412yes—
109C22×Dic5—21no—
110#709—41no—
111#709—41no—
112C2×D20C23×C311yescharacteristic
113C2×C4×D5C3×D412yes—
114C2×Dic10C2×C3×C411yescharacteristic
115C22⋊AGL(1,𝔽5)—24no—
116C4⋊AGL(1,𝔽5)—42no—
117C4×AGL(1,𝔽5)—42no—
118C22⋊Dic5—22no—
119#685—22no—
120#683—22no—
121C4×Dic5—21no—
122C4×Dic5C2×C3×C411yescharacteristic
123C4×Dic5C2×C3×C411yescharacteristic
124#532—51no—
125#410—52no—
126#354—52no—
127#353—52no—
128C22×C3×C5—21no—
129C22×C3×C5—41no—
130C22×C3×C5—42no—
131C22×C3×C5—41no—
132C22×C3×C5C2×C22⋊C411yescharacteristic
133C2×C3×D5—22no—
134C2×C3×D5—42no—
135C3×AGL(1,𝔽5)—84no—
136C3×C4×C5—21no—
137C3×C4×C5—41no—
138C3×C4×C5#9911yescharacteristic
139C3×C4×C5#9911yescharacteristic
140C3×Dic5—22no—
141C3×Dic5—22no—
142C3×Dic5—41no—
143C3×#41—41no—
144C3×#41—41no—
145C3×#41—201no—
146C3×#41—201no—
147C2×C3×Q8—51no—
148C2×C3×Q8C2×AGL(1,𝔽5)11yescharacteristic
149C2×C3×D4—21no—
150C2×C3×D4—52no—
151C2×C3×D4—51no—
152C2×C3×D4—102no—
153C2×C3×D4C2×AGL(1,𝔽5)11yescharacteristic
154C22×C3×C4—21no—
155C22×C3×C4—52no—
156C22×C3×C4—101no—
157C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
158C3×C4⋊C4—102no—
159C3×C4⋊C4—202no—
160C3×C22⋊C4—102no—
161C3×C22⋊C4—102no—
162C3×C22⋊C4—104no—
163C3×C42—51no—
164C3×C42—51no—
165C3×C42—101no—
166C3×C42—202no—
167C23×C5—21no—
168C23×C5C3×C22⋊C411yescharacteristic
169C22×D5C2×C3×D412yes—
170C2×AGL(1,𝔽5)—44no—
171C5×Q8—41no—
172C5×D4—41no—
173C5×D4—42no—
174C2×C4×C5—21no—
175C2×C4×C5—21no—
176C2×C4×C5—21no—
177C2×C4×C5—41no—
178C2×C4×C5—41no—
179C2×C4×C5C3×C22⋊C411yescharacteristic
180C2×C4×C5C22×C3×C411yescharacteristic
181C22⋊D5—42no—
182C22⋊D5—44no—
183C2×Dic5—21no—
184C2×Dic5—21no—
185C2×Dic5—42no—
186C2×Dic5C3×C22⋊C412yes—
187D20—41no—
188C4×D5—22no—
189C4×D5—22no—
190Dic10—41no—
191C2×#41—51no—
192C2×#41C3×AGL(1,𝔽5)11yescharacteristic, complemented
193C4⋊Q8—51no—
194#127—51no—
195#124—51no—
196#124—51no—
197#121—102no—
198C4×D4—102no—
199#117—102no—
200C2×C42—51no—
201#99—54no—
202#99—102no—
203C2×C3×C5—21no—
204C2×C3×C5—41no—
205C2×C3×C5—81no—
206C2×C3×C5C2×#9911yescharacteristic
207C3×D5—42no—
208C23×C3—21no—
209C23×C3—52no—
210C23×C3C22⋊AGL(1,𝔽5)11yescharacteristic
211C3×Q8—41no—
212C3×Q8—201no—
213C3×D4—41no—
214C3×D4—42no—
215C3×D4—201no—
216C3×D4—202no—
217C3×D4—204no—
218C2×C3×C4—21no—
219C2×C3×C4—21no—
220C2×C3×C4—21no—
221C2×C3×C4—41no—
222C2×C3×C4—41no—
223C2×C3×C4—52no—
224C2×C3×C4—101no—
225C2×C3×C4—102no—
226C2×C3×C4—102no—
227C2×C3×C4—101no—
228C2×C3×C4—202no—
229C2×C3×C4—204no—
230C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
231C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
232C22×C5—21no—
233C22×C5—41no—
234C22×C5—42no—
235C22×C5—41no—
236C22×C5C2×C3×C22⋊C411yescharacteristic
237C2×D5—22no—
238C2×D5—42no—
239AGL(1,𝔽5)—84no—
240C4×C5—21no—
241C4×C5—41no—
242C4×C5C3×#9911yescharacteristic
243C4×C5C3×#9911yescharacteristic
244Dic5—22no—
245Dic5—22no—
246Dic5—41no—
247#41—41no—
248#41—41no—
249#41—201no—
250#41—201no—
251C2×Q8—51no—
252C2×Q8C2×C3×AGL(1,𝔽5)11yescharacteristic
253C2×D4—21no—
254C2×D4—52no—
255C2×D4—51no—
256C2×D4—102no—
257C2×D4C2×C3×AGL(1,𝔽5)11yescharacteristic
258C22×C4—21no—
259C22×C4—52no—
260C22×C4—101no—
261C22×C4C2×C3×AGL(1,𝔽5)11yescharacteristic
262C4⋊C4—102no—
263C4⋊C4—202no—
264C22⋊C4—102no—
265C22⋊C4—102no—
266C22⋊C4—104no—
267C42—51no—
268C42—51no—
269C42—101no—
270C42—202no—
271C3×C5#212611yescharacteristic, complemented, Hall
272C22×C3—21no—
273C22×C3—41no—
274C22×C3—42no—
275C22×C3—41no—
276C22×C3—102no—
277C22×C3—202no—
278C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
279C3×C4—21no—
280C3×C4—41no—
281C3×C4—102no—
282C3×C4—102no—
283C3×C4—201no—
284C3×C4—404no—
285C3×C4#401111yescharacteristic
286C3×C4#401111yescharacteristic
287C2×C5—21no—
288C2×C5—41no—
289C2×C5—81no—
290C2×C5C2×C3×#9911yescharacteristic
291D5—42no—
292C23—21no—
293C23—52no—
294C23C3×C22⋊AGL(1,𝔽5)11yescharacteristic
295Q8—41no—
296Q8—201no—
297D4—41no—
298D4—42no—
299D4—201no—
300D4—202no—
301D4—204no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—52no—
308C2×C4—101no—
309C2×C4—102no—
310C2×C4—102no—
311C2×C4—101no—
312C2×C4—202no—
313C2×C4—204no—
314C2×C4C3×C22⋊AGL(1,𝔽5)11yescharacteristic
315C2×C4C22×C3×AGL(1,𝔽5)11yescharacteristic
316C2×C3—21no—
317C2×C3—41no—
318C2×C3—81no—
319C2×C3—202no—
320C2×C3C2×#401111yescharacteristic
321C5C3×#212611yescharacteristic, complemented, Sylow
322C22—21no—
323C22—41no—
324C22—42no—
325C22—41no—
326C22—102no—
327C22—202no—
328C22C2×C3×C22⋊AGL(1,𝔽5)11yescharacteristic
329C4—21no—
330C4—41no—
331C4—102no—
332C4—102no—
333C4—201no—
334C4—404no—
335C4C3×#401111yescharacteristic
336C4C3×#401111yescharacteristic
337C3#1060499611yescharacteristic, direct product factor, Sylow
338C2—21no—
339C2—41no—
340C2—81no—
341C2—202no—
342C2C2×C3×#401111yescharacteristic
subgroupG/H
centerC3×C4#4011
commutatorC2×C4×C5C22×C3×C4
FrattiniC2×C4C22×C3×AGL(1,𝔽5)
FittingC2×C3×C5×#41C4
socleC2×C3×C5C2×#99
prime radicalC3×#66166C2
cosocleC2×D20C23×C3
hypercenterC2×C3×#41AGL(1,𝔽5)
hypocenterC5C3×#2126
LCSG ▷ C2×C4×C5 ▷ C22×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C3×C4 ◁ C2×C3×C4 ◁ C22×C3×C4 ◁ C2×C3×#41

Complements

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

NH ≅ G/Ntype
2. C3×#65745C2semidirect
3. C3×#64928C2semidirect
4. C3×#64922C2semidirect
5. #10604996C3direct (N × H)
6. C2×C3×C5×#41C4semidirect
10. C3×#4090C4semidirect
13. C3×#4082C4semidirect
15. C2×C3×C4×Dic5C4semidirect
20. #65745C2×C3semidirect
21. #64928C2×C3semidirect
22. #64922C2×C3semidirect
51. C2×C5×#41C3×C4semidirect
55. #4090C3×C4semidirect
58. #4082C3×C4semidirect
60. C2×C4×Dic5C3×C4semidirect
89. C2×C3×#41AGL(1,𝔽5)semidirect
192. C2×#41C3×AGL(1,𝔽5)semidirect
271. C3×C5#2126semidirect
321. C5C3×#2126semidirect
337. C3#10604996direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#661662
C3×#657452
C3×#649282
C3×#649222
#106049963
C3×#21265