Finite Group Databasebeta

← C3×#10617212 #14327278

Subgroups — C3×#10617212 #14327278

Order 1920.

1104 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×Q8×AGL(1,𝔽5)C212yesmaximal
2C3×C4×Q8×D5C211yesmaximal, characteristic
3C3×#65712C216yesmaximal
4C3×#65689C213yesmaximal
5C3×C4×C4⋊AGL(1,𝔽5)C213yesmaximal
6#10617212C311yesmaximal, characteristic, direct product factor, Hall
7C2×C3×Q8×D5C2211yescharacteristic
8C3×Q8×AGL(1,𝔽5)—28no—
9C2×C3×C4⋊AGL(1,𝔽5)C2216yes—
10C2×C3×C4⋊AGL(1,𝔽5)C2216yes—
11C2×C3×C4×AGL(1,𝔽5)C2216yes—
12C3×C4×C5×Q8C411yescharacteristic, complemented
13C3×Q8×Dic5C411yescharacteristic, complemented
14C3×D5×C4⋊C4C2213yes—
15C3×#4033C413yescomplemented
16C3×C42×D5C2213yes—
17C3×C4×Dic10C413yescomplemented
18C3×(C2×AGL(1,𝔽5))⋊C4C2211yescharacteristic
19C3×(C2×AGL(1,𝔽5))⋊C4C2213yes—
20C3×(C2×AGL(1,𝔽5))⋊C4C2216yes—
21C3×#2351—51nomaximal, Hall
22C2×Q8×AGL(1,𝔽5)C2×C312yes—
23C4×Q8×D5C2×C311yescharacteristic
24#65712C2×C316yes—
25#65689C2×C313yes—
26C4×C4⋊AGL(1,𝔽5)C2×C313yes—
27C22×C3×AGL(1,𝔽5)C2316yes—
28C22×C3×AGL(1,𝔽5)C2312yes—
29C2×C3×C5×Q8C2×C411yescharacteristic
30C3×Q8×D5D414yes—
31C3×Q8×D5Q814yes—
32C2×C3×C4×D5C2313yes—
33C2×C3×C4×D5C2311yescharacteristic
34C2×C3×C4×D5C2313yes—
35C2×C3×Dic10C2×C413yes—
36C3×C4⋊AGL(1,𝔽5)—212no—
37C3×C4⋊AGL(1,𝔽5)—212no—
38C3×C4×AGL(1,𝔽5)—212no—
39C3×C5×C4⋊C4C2×C413yes—
40C3×C42×C5C2×C413yes—
41C3×C4⋊Dic5C2×C413yes—
42C3×#683C2×C416yes—
43C3×C4×Dic5C2×C413yes—
44C3×C4×Dic5C2×C413yes—
45C3×C4×Dic5C2×C413yes—
46C2×C3×C4×Q8—51no—
47C2×C3×C4×Q8—52no—
48C3×#389—56no—
49C3×#382—53no—
50C3×C4×C4⋊C4—53no—
51C2×Q8×D5C22×C311yescharacteristic
52Q8×AGL(1,𝔽5)—28no—
53C2×C4⋊AGL(1,𝔽5)C22×C316yes—
54C2×C4⋊AGL(1,𝔽5)C22×C316yes—
55C2×C4×AGL(1,𝔽5)C22×C316yes—
56C4×C5×Q8C3×C411yescharacteristic, complemented
57Q8×Dic5C3×C411yescharacteristic, complemented
58D5×C4⋊C4C22×C313yes—
59#4033C3×C413yescomplemented
60C42×D5C22×C313yes—
61C4×Dic10C3×C413yescomplemented
62(C2×AGL(1,𝔽5))⋊C4C22×C311yescharacteristic
63(C2×AGL(1,𝔽5))⋊C4C22×C313yes—
64(C2×AGL(1,𝔽5))⋊C4C22×C316yes—
65#2351—51noSylow
66C22×C3×D5C2411yescharacteristic
67C2×C3×AGL(1,𝔽5)—22no—
68C2×C3×AGL(1,𝔽5)—22no—
69C2×C3×AGL(1,𝔽5)—26no—
70C2×C3×AGL(1,𝔽5)—26no—
71C2×C3×AGL(1,𝔽5)—26no—
72C2×C3×AGL(1,𝔽5)#4114yes—
73C3×C5×Q8C4⋊C414yescomplemented
74C2×C3×C4×C5C22×C411yescharacteristic
75C2×C3×C4×C5C22×C413yes—
76C2×C3×C4×C5C22×C413yes—
77C2×C3×Dic5C22×C411yescharacteristic
78C2×C3×Dic5C22×C413yes—
79C2×C3×Dic5C22×C413yes—
80C3×C4×D5—26no—
81C3×C4×D5—22no—
82C3×C4×D5C2×D416yes—
83C3×C4×D5C2×Q816yes—
84C3×Dic10C4⋊C4112yescomplemented
85C22×C3×Q8—51no—
86C3×C4×Q8—51no—
87C3×C4×Q8—53no—
88C3×C4×Q8—53no—
89C3×C4×Q8—108no—
90C3×C4×Q8AGL(1,𝔽5)11yescharacteristic, complemented
91C2×C3×C4⋊C4—53no—
92C2×C3×C4⋊C4—56no—
93C2×C3×C4⋊C4—56no—
94C2×C3×C42—53no—
95C2×C3×C42—56no—
96C3×#95—51no—
97C3×#95—53no—
98C3×#95—56no—
99C22×AGL(1,𝔽5)C23×C316yes—
100C22×AGL(1,𝔽5)C23×C312yes—
101C2×C5×Q8C2×C3×C411yescharacteristic
102Q8×D5C3×D414yes—
103Q8×D5C3×Q814yes—
104C2×C4×D5C23×C313yes—
105C2×C4×D5C23×C311yescharacteristic
106C2×C4×D5C23×C313yes—
107C2×Dic10C2×C3×C413yes—
108C4⋊AGL(1,𝔽5)—212no—
109C4⋊AGL(1,𝔽5)—212no—
110C4×AGL(1,𝔽5)—212no—
111C5×C4⋊C4C2×C3×C413yes—
112C42×C5C2×C3×C413yes—
113C4⋊Dic5C2×C3×C413yes—
114#683C2×C3×C416yes—
115C4×Dic5C2×C3×C413yes—
116C4×Dic5C2×C3×C413yes—
117C4×Dic5C2×C3×C413yes—
118C2×C4×Q8—51no—
119C2×C4×Q8—52no—
120#389—56no—
121#382—53no—
122C4×C4⋊C4—53no—
123C22×C3×C5C23×C411yescharacteristic
124C2×C3×D5C22×D411yescharacteristic
125C2×C3×D5C22×Q811yescharacteristic
126C2×C3×D5C2×#4112yes—
127C2×C3×D52−1+412yes—
128C3×AGL(1,𝔽5)—24no—
129C3×AGL(1,𝔽5)—46no—
130C3×C4×C5—23no—
131C3×C4×C5—21no—
132C3×C4×C5C2×C4⋊C416yes—
133C3×Dic5—23no—
134C3×Dic5—21no—
135C3×Dic5C2×C4⋊C416yes—
136C2×C3×Q8—54no—
137C2×C3×Q8—54no—
138C2×C3×Q8—53no—
139C2×C3×Q8C2×AGL(1,𝔽5)11yescharacteristic
140C22×C3×C4—53no—
141C22×C3×C4—52no—
142C22×C3×C4—51no—
143C22×C3×C4—53no—
144C22×C3×C4—56no—
145C3×C4⋊C4—56no—
146C3×C4⋊C4—53no—
147C3×C4⋊C4—1012no—
148C3×C4⋊C4—1012no—
149C3×C4⋊C4C2×AGL(1,𝔽5)13yes—
150C3×C42—53no—
151C3×C42—53no—
152C3×C42—53no—
153C3×C42—1012no—
154C3×C42C2×AGL(1,𝔽5)13yes—
155C22×D5C24×C311yescharacteristic
156C2×AGL(1,𝔽5)—22no—
157C2×AGL(1,𝔽5)—22no—
158C2×AGL(1,𝔽5)—26no—
159C2×AGL(1,𝔽5)—26no—
160C2×AGL(1,𝔽5)—26no—
161C2×AGL(1,𝔽5)C3×#4114yes—
162C5×Q8C3×C4⋊C414yescomplemented
163C2×C4×C5C22×C3×C411yescharacteristic
164C2×C4×C5C22×C3×C413yes—
165C2×C4×C5C22×C3×C413yes—
166C2×Dic5C22×C3×C411yescharacteristic
167C2×Dic5C22×C3×C413yes—
168C2×Dic5C22×C3×C413yes—
169C4×D5—26no—
170C4×D5—22no—
171C4×D5C2×C3×D416yes—
172C4×D5C2×C3×Q816yes—
173Dic10C3×C4⋊C4112yescomplemented
174C22×Q8—51no—
175C4×Q8—51no—
176C4×Q8—53no—
177C4×Q8—53no—
178C4×Q8—108no—
179C4×Q8C3×AGL(1,𝔽5)11yescharacteristic, complemented
180C2×C4⋊C4—53no—
181C2×C4⋊C4—56no—
182C2×C4⋊C4—56no—
183C2×C42—53no—
184C2×C42—56no—
185#95—51no—
186#95—53no—
187#95—56no—
188C2×C3×C5C22×C4⋊C411yescharacteristic
189C2×C3×C5C4×#4111yescharacteristic
190C2×C3×C5#51911yescharacteristic
191C3×D5#54812yes—
192C3×D5#55712yes—
193C23×C3—51no—
194C3×Q8—512no—
195C3×Q8C4⋊AGL(1,𝔽5)14yescomplemented
196C2×C3×C4—53no—
197C2×C3×C4—56no—
198C2×C3×C4—56no—
199C2×C3×C4—54no—
200C2×C3×C4—51no—
201C2×C3×C4—53no—
202C2×C3×C4—106no—
203C2×C3×C4—102no—
204C2×C3×C4—102no—
205C2×C3×C4—102no—
206C2×C3×C4—106no—
207C2×C3×C4—106no—
208C2×C3×C4—106no—
209C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
210C2×C3×C4C22×AGL(1,𝔽5)13yes—
211C2×C3×C4C22×AGL(1,𝔽5)13yes—
212C22×C5C23×C3×C411yescharacteristic
213C2×D5C22×C3×D411yescharacteristic
214C2×D5C22×C3×Q811yescharacteristic
215C2×D5C2×C3×#4112yes—
216C2×D5C3×2−1+412yes—
217AGL(1,𝔽5)—24no—
218AGL(1,𝔽5)—46no—
219C4×C5—23no—
220C4×C5—21no—
221C4×C5C2×C3×C4⋊C416yes—
222Dic5—23no—
223Dic5—21no—
224Dic5C2×C3×C4⋊C416yes—
225C2×Q8—54no—
226C2×Q8—54no—
227C2×Q8—53no—
228C2×Q8C2×C3×AGL(1,𝔽5)11yescharacteristic
229C22×C4—53no—
230C22×C4—52no—
231C22×C4—51no—
232C22×C4—53no—
233C22×C4—56no—
234C4⋊C4—56no—
235C4⋊C4—53no—
236C4⋊C4—1012no—
237C4⋊C4—1012no—
238C4⋊C4C2×C3×AGL(1,𝔽5)13yes—
239C42—53no—
240C42—53no—
241C42—53no—
242C42—1012no—
243C42C2×C3×AGL(1,𝔽5)13yes—
244C3×C5#235111yescharacteristic, complemented, Hall
245C22×C3—51no—
246C22×C3—51no—
247C22×C3—52no—
248C22×C3—52no—
249C22×C3C23×AGL(1,𝔽5)11yescharacteristic
250C3×C4—21no—
251C3×C4—23no—
252C3×C4—56no—
253C3×C4—104no—
254C3×C4—101no—
255C3×C4—103no—
256C3×C4—206no—
257C3×C4C2×C4⋊AGL(1,𝔽5)16yes—
258C2×C5C22×C3×C4⋊C411yescharacteristic
259C2×C5C3×C4×#4111yescharacteristic
260C2×C5C3×#51911yescharacteristic
261D5C3×#54812yes—
262D5C3×#55712yes—
263C23—51no—
264Q8—512no—
265Q8C3×C4⋊AGL(1,𝔽5)14yescomplemented
266C2×C4—53no—
267C2×C4—56no—
268C2×C4—56no—
269C2×C4—54no—
270C2×C4—51no—
271C2×C4—53no—
272C2×C4—106no—
273C2×C4—102no—
274C2×C4—102no—
275C2×C4—102no—
276C2×C4—106no—
277C2×C4—106no—
278C2×C4—106no—
279C2×C4C22×C3×AGL(1,𝔽5)11yescharacteristic
280C2×C4C22×C3×AGL(1,𝔽5)13yes—
281C2×C4C22×C3×AGL(1,𝔽5)13yes—
282C2×C3—52no—
283C2×C3—52no—
284C2×C3C22×C4⋊AGL(1,𝔽5)11yescharacteristic
285C2×C3#6626211yescharacteristic
286C2×C3#41×AGL(1,𝔽5)11yescharacteristic
287C5C3×#235111yescharacteristic, complemented, Sylow
288C22—51no—
289C22—51no—
290C22—52no—
291C22—52no—
292C22C23×C3×AGL(1,𝔽5)11yescharacteristic
293C4—21no—
294C4—23no—
295C4—56no—
296C4—104no—
297C4—101no—
298C4—103no—
299C4—206no—
300C4C2×C3×C4⋊AGL(1,𝔽5)16yes—
301C3#1061721211yescharacteristic, direct product factor, Sylow
302C2—52no—
303C2—52no—
304C2C22×C3×C4⋊AGL(1,𝔽5)11yescharacteristic
305C2C3×#6626211yescharacteristic
306C2C3×#41×AGL(1,𝔽5)11yescharacteristic
subgroupG/H
centerC22×C3C23×AGL(1,𝔽5)
commutatorC22×C5C23×C3×C4
FrattiniC22C23×C3×AGL(1,𝔽5)
FittingC3×C4×C5×Q8C4
socleC22×C3×C5C23×C4
prime radicalC22×C3×D5C24
cosocleC22×D5C24×C3
hypercenterC3×C4×Q8AGL(1,𝔽5)
hypocenterC5C3×#2351
LCSG ▷ C22×C5 ▷ C5
UCSC1 ◁ C22×C3 ◁ C3×C4×Q8

Complements

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

NH ≅ G/Ntype
6. #10617212C3direct (N × H)
12. C3×C4×C5×Q8C4semidirect
13. C3×Q8×Dic5C4semidirect
15. C3×#4033C4semidirect
17. C3×C4×Dic10C4semidirect
56. C4×C5×Q8C3×C4semidirect
57. Q8×Dic5C3×C4semidirect
59. #4033C3×C4semidirect
61. C4×Dic10C3×C4semidirect
73. C3×C5×Q8C4⋊C4semidirect
84. C3×Dic10C4⋊C4semidirect
90. C3×C4×Q8AGL(1,𝔽5)semidirect
162. C5×Q8C3×C4⋊C4semidirect
173. Dic10C3×C4⋊C4semidirect
179. C4×Q8C3×AGL(1,𝔽5)semidirect
195. C3×Q8C4⋊AGL(1,𝔽5)semidirect
244. C3×C5#2351semidirect
265. Q8C3×C4⋊AGL(1,𝔽5)semidirect
287. C5C3×#2351semidirect
301. C3#10617212direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×Q8×AGL(1,𝔽5)2
C3×C4×Q8×D52
C3×#657122
C3×#656892
C3×C4×C4⋊AGL(1,𝔽5)2
#106172123
C3×#23515