Finite Group Databasebeta

← C3×#10610988 #14418018

Subgroups — C3×#10610988 #14418018

Order 1920.

912 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#66082C211yesmaximal, characteristic
2C2×C3×#4053C212yesmaximal, complemented
3C3×C5×#430C211yesmaximal, characteristic, complemented
4C3×#65428C212yesmaximal, complemented
5C3×#65411C211yesmaximal, characteristic, complemented
6C3×#65184C214yesmaximal, complemented
7C3×#65183C214yesmaximal, complemented
8#10610988C311yesmaximal, characteristic, direct product factor, Hall
9C2×C3×#709C2212yes—
10C2×C3×C5×M16C2212yes—
11C2×C3×C5×M16C2211yescharacteristic
12C3×#4072C2211yescharacteristic
13C22×C3×#168C2211yescharacteristic
14C3×#4055C414yescomplemented
15C3×#4054C414yescomplemented
16C3×#4053—22no—
17C3×#4053—22no—
18C3×#4053C2214yescomplemented
19C3×#4053C2214yescomplemented
20C3×#4050C2214yes—
21C2×C3×#676—22no—
22C2×C3×C8×D5—22no—
23C3×C5×#108C2214yescomplemented
24C3×#3957C2218yescomplemented
25C3×#3947C2212yescomplemented
26C3×C8×Dic5C2212yescomplemented
27C3×#2955—51nomaximal, Hall
28#66082C2×C311yescharacteristic
29C2×#4053C2×C312yescomplemented
30C5×#430C2×C311yescharacteristic, complemented
31#65428C2×C312yescomplemented
32#65411C2×C311yescharacteristic, complemented
33#65184C2×C314yescomplemented
34#65183C2×C314yescomplemented
35C22×C3×C4×C5C2311yescharacteristic
36C2×C3×C22⋊D5—22no—
37C3×#709—22no—
38C3×#709—22no—
39C3×#709C2×C414yes—
40C3×#709C2314yes—
41C2×C3×D20C2×C412yes—
42C2×C3×C4×D5—22no—
43C2×C3×Dic10C2×C412yes—
44C3×C5×M16—22no—
45C3×C5×M16C2×C414yes—
46C3×C5×M16C2314yes—
47C2×C3×C5×C8—22no—
48C2×C3×C5×C8C2312yes—
49C3×C22⋊Dic5—21no—
50C3×C4⋊Dic5—21no—
51C3×C4×Dic5C2312yes—
52C2×C3×#168D412yes—
53C2×C3×#168D412yes—
54C2×C3×#168C2311yescharacteristic
55C2×C3×#168C2311yescharacteristic
56C3×Dic20C2×C414yes—
57C3×D40C2×C414yes—
58C3×#677C2×C418yes—
59C3×#676—24no—
60C3×#676—24no—
61C3×C8×D5—24no—
62C3×C8×D5—24no—
63C3×#575—51no—
64C2×C3×#131—52no—
65C3×#444—54no—
66C3×#443—54no—
67C3×#430D511yescharacteristic, complemented
68C3×#421—52no—
69C3×#405—51no—
70C2×#709C22×C312yes—
71C2×C5×M16C22×C312yes—
72C2×C5×M16C22×C311yescharacteristic
73#4072C22×C311yescharacteristic
74C22×#168C22×C311yescharacteristic
75#4055C3×C414yescomplemented
76#4054C3×C414yescomplemented
77#4053—22no—
78#4053—22no—
79#4053C22×C314yescomplemented
80#4053C22×C314yescomplemented
81#4050C22×C314yes—
82C2×#676—22no—
83C2×C8×D5—22no—
84C5×#108C22×C314yescomplemented
85#3957C22×C318yescomplemented
86#3947C22×C312yescomplemented
87C8×Dic5C22×C312yescomplemented
88#2955—51noSylow
89C23×C3×C5#4111yescharacteristic
90C22×C3×D5—22no—
91C2×C3×C4×C5C22×C412yes—
92C2×C3×C4×C5C2×D412yes—
93C2×C3×C4×C5#4111yescharacteristic
94C2×C3×C4×C5C2411yescharacteristic
95C3×C22⋊D5—24no—
96C3×C22⋊D5—24no—
97C2×C3×Dic5—22no—
98C2×C3×Dic5—22no—
99C3×D20—22no—
100C3×D20C22×C414yes—
101C3×C4×D5—24no—
102C3×C4×D5—24no—
103C3×Dic10—22no—
104C3×Dic10C22×C414yes—
105C3×C5×C8—24no—
106C3×C5×C8C22×C414yes—
107C3×#168C2×D411yescharacteristic
108C3×#168C2×D412yes—
109C3×#168C2×D411yescharacteristic
110C2×C3×#41—52no—
111C3×#137—54no—
112C3×AGL(1,ℤ/8)—54no—
113C3×#135—54no—
114C3×#131—54no—
115C3×#131—54no—
116C3×#131—102no—
117C3×#131—102no—
118C2×C3×M16—102no—
119C2×C3×M16C2×D512yes—
120C2×C3×M16C2×D511yescharacteristic
121C22×C3×C8—51no—
122C22×C3×C8—102no—
123C3×#117—51no—
124C3×#108C2×D514yescomplemented
125C3×#104—58no—
126C3×#97—52no—
127C3×C4×C8—52no—
128C22×C4×C5C23×C311yescharacteristic
129C2×C22⋊D5—22no—
130#709—22no—
131#709—22no—
132#709C2×C3×C414yes—
133#709C23×C314yes—
134C2×D20C2×C3×C412yes—
135C2×C4×D5—22no—
136C2×Dic10C2×C3×C412yes—
137C5×M16—22no—
138C5×M16C2×C3×C414yes—
139C5×M16C23×C314yes—
140C2×C5×C8—22no—
141C2×C5×C8C23×C312yes—
142C22⋊Dic5—21no—
143C4⋊Dic5—21no—
144C4×Dic5C23×C312yes—
145C2×#168C3×D412yes—
146C2×#168C3×D412yes—
147C2×#168C23×C311yescharacteristic
148C2×#168C23×C311yescharacteristic
149Dic20C2×C3×C414yes—
150D40C2×C3×C414yes—
151#677C2×C3×C418yes—
152#676—24no—
153#676—24no—
154C8×D5—24no—
155C8×D5—24no—
156#575—51no—
157C2×#131—52no—
158#444—54no—
159#443—54no—
160#430C3×D511yescharacteristic, complemented
161#421—52no—
162#405—51no—
163C22×C3×C5—41no—
164C22×C3×C5C4×D412yes—
165C22×C3×C5C2×#4111yescharacteristic
166C2×C3×D5—24no—
167C2×C3×D5—44no—
168C3×C4×C5C4×D412yes—
169C3×C4×C5C23×C411yescharacteristic
170C3×C4×C5C22×D411yescharacteristic
171C3×Dic5—24no—
172C3×Dic5—42no—
173C3×#41—54no—
174C3×#41—54no—
175C3×#41—102no—
176C3×#41—102no—
177C2×C3×Q8—52no—
178C2×C3×D4—52no—
179C2×C3×D4—102no—
180C22×C3×C4—102no—
181C22×C3×C4C22×D511yescharacteristic
182C3×Q16—54no—
183C3×SD16—58no—
184C3×D8—54no—
185C3×M16—22no—
186C3×M16—104no—
187C3×M16—104no—
188C3×M16C4×D514yes—
189C3×M16C22×D514yes—
190C2×C3×C8—22no—
191C2×C3×C8—52no—
192C2×C3×C8—52no—
193C2×C3×C8—51no—
194C2×C3×C8—51no—
195C2×C3×C8—104no—
196C2×C3×C8—104no—
197C2×C3×C8C22×D512yes—
198C3×C4⋊C4—101no—
199C3×C22⋊C4—101no—
200C3×C42—52no—
201C23×C5C3×#4111yescharacteristic
202C22×D5—22no—
203C2×C4×C5C22×C3×C412yes—
204C2×C4×C5C2×C3×D412yes—
205C2×C4×C5C3×#4111yescharacteristic
206C2×C4×C5C24×C311yescharacteristic
207C22⋊D5—24no—
208C22⋊D5—24no—
209C2×Dic5—22no—
210C2×Dic5—22no—
211D20—22no—
212D20C22×C3×C414yes—
213C4×D5—24no—
214C4×D5—24no—
215Dic10—22no—
216Dic10C22×C3×C414yes—
217C5×C8—24no—
218C5×C8C22×C3×C414yes—
219#168C2×C3×D411yescharacteristic
220#168C2×C3×D412yes—
221#168C2×C3×D411yescharacteristic
222C2×#41—52no—
223#137—54no—
224AGL(1,ℤ/8)—54no—
225#135—54no—
226#131—54no—
227#131—54no—
228#131—102no—
229#131—102no—
230C2×M16—102no—
231C2×M16C2×C3×D512yes—
232C2×M16C2×C3×D511yescharacteristic
233C22×C8—51no—
234C22×C8—102no—
235#117—51no—
236#108C2×C3×D514yescomplemented
237#104—58no—
238#97—52no—
239C4×C8—52no—
240C2×C3×C5—21no—
241C2×C3×C5—22no—
242C2×C3×C5C2×C4×D411yescharacteristic
243C3×D5—44no—
244C23×C3—102no—
245C23×C3#71311yescharacteristic
246C3×Q8—54no—
247C3×Q8—102no—
248C3×D4—54no—
249C3×D4—104no—
250C3×D4—104no—
251C3×D4—102no—
252C2×C3×C4—102no—
253C2×C3×C4—104no—
254C2×C3×C4—104no—
255C2×C3×C4—102no—
256C2×C3×C4C2×C4×D512yes—
257C2×C3×C4D4×D512yes—
258C2×C3×C4#71311yescharacteristic
259C2×C3×C4C23×D511yescharacteristic
260C3×C8—24no—
261C3×C8—51no—
262C3×C8—52no—
263C3×C8—51no—
264C3×C8C2×C4×D514yes—
265C22×C5—41no—
266C22×C5C3×C4×D412yes—
267C22×C5C2×C3×#4111yescharacteristic
268C2×D5—24no—
269C2×D5—44no—
270C4×C5C3×C4×D412yes—
271C4×C5C23×C3×C411yescharacteristic
272C4×C5C22×C3×D411yescharacteristic
273Dic5—24no—
274Dic5—42no—
275#41—54no—
276#41—54no—
277#41—102no—
278#41—102no—
279C2×Q8—52no—
280C2×D4—52no—
281C2×D4—102no—
282C22×C4—102no—
283C22×C4C22×C3×D511yescharacteristic
284Q16—54no—
285SD16—58no—
286D8—54no—
287M16—22no—
288M16—104no—
289M16—104no—
290M16C3×C4×D514yes—
291M16C22×C3×D514yes—
292C2×C8—22no—
293C2×C8—52no—
294C2×C8—52no—
295C2×C8—51no—
296C2×C8—51no—
297C2×C8—104no—
298C2×C8—104no—
299C2×C8C22×C3×D512yes—
300C4⋊C4—101no—
301C22⋊C4—101no—
302C42—52no—
303C3×C5#295511yescharacteristic, complemented, Hall
304C22×C3—41no—
305C22×C3—104no—
306C22×C3—204no—
307C22×C3#403912yes—
308C22×C3C2×#71311yescharacteristic
309C3×C4—104no—
310C3×C4—202no—
311C3×C4#403912yes—
312C3×C4C22×C4×D511yescharacteristic
313C3×C4C2×D4×D511yescharacteristic
314C2×C5—21no—
315C2×C5—22no—
316C2×C5C2×C3×C4×D411yescharacteristic
317D5—44no—
318C23—102no—
319C23C3×#71311yescharacteristic
320Q8—54no—
321Q8—102no—
322D4—54no—
323D4—104no—
324D4—104no—
325D4—102no—
326C2×C4—102no—
327C2×C4—104no—
328C2×C4—104no—
329C2×C4—102no—
330C2×C4C2×C3×C4×D512yes—
331C2×C4C3×D4×D512yes—
332C2×C4C3×#71311yescharacteristic
333C2×C4C23×C3×D511yescharacteristic
334C8—24no—
335C8—51no—
336C8—52no—
337C8—51no—
338C8C2×C3×C4×D514yes—
339C2×C3—21no—
340C2×C3—22no—
341C2×C3—204no—
342C2×C3C2×#403911yescharacteristic
343C5C3×#295511yescharacteristic, complemented, Sylow
344C22—41no—
345C22—104no—
346C22—204no—
347C22C3×#403912yes—
348C22C2×C3×#71311yescharacteristic
349C4—104no—
350C4—202no—
351C4C3×#403912yes—
352C4C22×C3×C4×D511yescharacteristic
353C4C2×C3×D4×D511yescharacteristic
354C3#1061098811yescharacteristic, direct product factor, Sylow
355C2—21no—
356C2—22no—
357C2—204no—
358C2C2×C3×#403911yescharacteristic
subgroupG/H
centerC3×C4C2×D4×D5
commutatorC4×C5C23×C3×C4
FrattiniC2×C4C23×C3×D5
FittingC3×C5×#430C2
socleC2×C3×C5C2×C4×D4
prime radicalC3×#66082C2
cosocleC2×C4×C5C24×C3
hypercenterC3×#430D5
hypocenterC5C3×#2955
LCSG ▷ C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C3×C4 ◁ C22×C3×C4 ◁ C3×#430

Complements

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

NH ≅ G/Ntype
2. C2×C3×#4053C2semidirect
3. C3×C5×#430C2semidirect
4. C3×#65428C2semidirect
5. C3×#65411C2semidirect
6. C3×#65184C2semidirect
7. C3×#65183C2semidirect
8. #10610988C3direct (N × H)
14. C3×#4055C4semidirect
15. C3×#4054C4semidirect
18. C3×#4053C22semidirect
19. C3×#4053C22semidirect
23. C3×C5×#108C22semidirect
24. C3×#3957C22semidirect
25. C3×#3947C22semidirect
26. C3×C8×Dic5C22semidirect
29. C2×#4053C2×C3semidirect
30. C5×#430C2×C3semidirect
31. #65428C2×C3semidirect
32. #65411C2×C3semidirect
33. #65184C2×C3semidirect
34. #65183C2×C3semidirect
67. C3×#430D5semidirect
75. #4055C3×C4semidirect
76. #4054C3×C4semidirect
79. #4053C22×C3semidirect
80. #4053C22×C3semidirect
84. C5×#108C22×C3semidirect
85. #3957C22×C3semidirect
86. #3947C22×C3semidirect
87. C8×Dic5C22×C3semidirect
124. C3×#108C2×D5semidirect
160. #430C3×D5semidirect
236. #108C2×C3×D5semidirect
303. C3×C5#2955semidirect
343. C5C3×#2955semidirect
354. C3#10610988direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#660822
C2×C3×#40532
C3×C5×#4302
C3×#654282
C3×#654112
C3×#651842
C3×#651832
#106109883
C3×#29555