Finite Group Databasebeta

← C3×#10601994 #14292094

Subgroups — C3×#10601994 #14292094

Order 1920.

364 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C5×#486C211yesmaximal, characteristic, complemented
2C3×#65372C211yesmaximal, characteristic, complemented
3C3×#65361C211yesmaximal, characteristic
4C3×C8×D20C211yesmaximal, characteristic, complemented
5C3×#64767C211yesmaximal, characteristic, complemented
6C3×#64764C211yesmaximal, characteristic, complemented
7C3×#64678C211yesmaximal, characteristic, complemented
8#10601994C311yesmaximal, characteristic, direct product factor, Hall
9C3×C5×C4⋊Q8C2211yescharacteristic
10C3×C5×#127C2211yescharacteristic
11C2×C3×#688—21no—
12C2×C3×#686—21no—
13C2×C3×C8×D5—21no—
14C3×C4×D20C2211yescharacteristic
15C3×C5×#102—21no—
16C3×C5×#102—21no—
17C3×C4×C5×C8C2211yescharacteristic
18C3×#3967—21no—
19C3×#3964—21no—
20C3×#3952—21no—
21C3×C4⋊#168C2211yescharacteristic
22C3×C4⋊#168C2211yescharacteristic
23C3×C4⋊#168C2211yescharacteristic
24C3×#1681—51nomaximal, Hall
25C5×#486C2×C311yescharacteristic, complemented
26#65372C2×C311yescharacteristic, complemented
27#65361C2×C311yescharacteristic
28C8×D20C2×C311yescharacteristic, complemented
29#64767C2×C311yescharacteristic, complemented
30#64764C2×C311yescharacteristic, complemented
31#64678C2×C311yescharacteristic, complemented
32C2×C3×C5×Q8—21no—
33C2×C3×C5×D4—21no—
34C2×C3×C5×D4—21no—
35C2×C3×C5×D4—21no—
36C2×C3×D20D411yescharacteristic
37C2×C3×C4×D5—21no—
38C2×C3×C5×C8D411yescharacteristic
39C2×C3×C5×C8D411yescharacteristic
40C3×C5×C4⋊C4—21no—
41C3×C5×C4⋊C4—21no—
42C3×C42×C5C2311yescharacteristic
43C3×#688—42no—
44C3×#686—42no—
45C3×#685—21no—
46C3×C4⋊Dic5D411yescharacteristic
47C2×C3×#168—21no—
48C2×C3×#168—21no—
49C2×C3×#168—21no—
50C3×C8×D5—42no—
51C3×#486D511yescharacteristic, complemented
52C3×#461—51no—
53C3×#459—51no—
54C3×C8×D4—51no—
55C3×#335—51no—
56C3×#332—51no—
57C3×#331—51no—
58C5×C4⋊Q8C22×C311yescharacteristic
59C5×#127C22×C311yescharacteristic
60C2×#688—21no—
61C2×#686—21no—
62C2×C8×D5—21no—
63C4×D20C22×C311yescharacteristic
64C5×#102—21no—
65C5×#102—21no—
66C4×C5×C8C22×C311yescharacteristic
67#3967—21no—
68#3964—21no—
69#3952—21no—
70C4⋊#168C22×C311yescharacteristic
71C4⋊#168C22×C311yescharacteristic
72C4⋊#168C22×C311yescharacteristic
73#1681—51noSylow
74C23×C3×C5—41no—
75C22×C3×D5—21no—
76C3×C5×Q8—41no—
77C3×C5×Q8—41no—
78C3×C5×D4—41no—
79C3×C5×D4—41no—
80C3×C5×D4—41no—
81C3×C5×D4—41no—
82C3×C5×D4—81no—
83C2×C3×C4×C5—41no—
84C2×C3×C4×C5C2×D411yescharacteristic
85C2×C3×C4×C5C2×D411yescharacteristic
86C2×C3×C4×C5#4111yescharacteristic
87C2×C3×Dic5—21no—
88C3×D20—21no—
89C3×D20D812yes—
90C3×C4×D5—41no—
91C3×C4×D5—41no—
92C3×C5×C8—21no—
93C3×C5×C8—21no—
94C3×#168—41no—
95C3×#168—41no—
96C3×#168—41no—
97C2×C3×SD16—101no—
98C2×C3×D8—101no—
99C22×C3×C8—101no—
100C3×C4⋊Q8C2×D511yescharacteristic
101C3×#127C2×D511yescharacteristic
102C3×C4×D4—51no—
103C3×C4⋊C8—51no—
104C3×C4⋊C8—51no—
105C3×C4⋊C8—51no—
106C3×#103—101no—
107C3×#102—21no—
108C3×#102—21no—
109C3×#102—101no—
110C3×C22⋊C8—101no—
111C3×C4×C8C2×D511yescharacteristic
112C2×C5×Q8—21no—
113C2×C5×D4—21no—
114C2×C5×D4—21no—
115C2×C5×D4—21no—
116C2×D20C3×D411yescharacteristic
117C2×C4×D5—21no—
118C2×C5×C8C3×D411yescharacteristic
119C2×C5×C8C3×D411yescharacteristic
120C5×C4⋊C4—21no—
121C5×C4⋊C4—21no—
122C42×C5C23×C311yescharacteristic
123#688—42no—
124#686—42no—
125#685—21no—
126C4⋊Dic5C3×D411yescharacteristic
127C2×#168—21no—
128C2×#168—21no—
129C2×#168—21no—
130C8×D5—42no—
131#486C3×D511yescharacteristic, complemented
132#461—51no—
133#459—51no—
134C8×D4—51no—
135#335—51no—
136#332—51no—
137#331—51no—
138C22×C3×C5—81no—
139C22×C3×C5—81no—
140C22×C3×C5—81no—
141C22×C3×C5#12111yescharacteristic
142C2×C3×D5—22no—
143C2×C3×D5—41no—
144C2×C3×D5—41no—
145C3×C4×C5—21no—
146C3×C4×C5—81no—
147C3×C4×C5C2×D811yescharacteristic
148C3×C4×C5#13511yescharacteristic
149C3×C4×C5#13511yescharacteristic
150C3×C4×C5AGL(1,ℤ/8)11yescharacteristic
151C3×Dic5—41no—
152C2×C3×Q8—21no—
153C2×C3×D4—21no—
154C2×C3×D4—21no—
155C2×C3×D4—21no—
156C2×C3×D4—51no—
157C22×C3×C4—101no—
158C3×SD16—202no—
159C3×D8—202no—
160C2×C3×C8—101no—
161C2×C3×C8—101no—
162C2×C3×C8—101no—
163C2×C3×C8—202no—
164C2×C3×C8C22⋊D511yescharacteristic
165C2×C3×C8C22⋊D511yescharacteristic
166C3×C4⋊C4—21no—
167C3×C4⋊C4—21no—
168C3×C4⋊C4—51no—
169C3×C22⋊C4—101no—
170C3×C42C22×D511yescharacteristic
171C23×C5—41no—
172C22×D5—21no—
173C5×Q8—41no—
174C5×Q8—41no—
175C5×D4—41no—
176C5×D4—41no—
177C5×D4—41no—
178C5×D4—41no—
179C5×D4—81no—
180C2×C4×C5—41no—
181C2×C4×C5C2×C3×D411yescharacteristic
182C2×C4×C5C2×C3×D411yescharacteristic
183C2×C4×C5C3×#4111yescharacteristic
184C2×Dic5—21no—
185D20—21no—
186D20C3×D812yes—
187C4×D5—41no—
188C4×D5—41no—
189C5×C8—21no—
190C5×C8—21no—
191#168—41no—
192#168—41no—
193#168—41no—
194C2×SD16—101no—
195C2×D8—101no—
196C22×C8—101no—
197C4⋊Q8C2×C3×D511yescharacteristic
198#127C2×C3×D511yescharacteristic
199C4×D4—51no—
200C4⋊C8—51no—
201C4⋊C8—51no—
202C4⋊C8—51no—
203#103—101no—
204#102—21no—
205#102—21no—
206#102—101no—
207C22⋊C8—101no—
208C4×C8C2×C3×D511yescharacteristic
209C2×C3×C5—161no—
210C2×C3×C5#46311yescharacteristic
211C2×C3×C5#46611yescharacteristic
212C2×C3×C5#47111yescharacteristic
213C3×D5—42no—
214C23×C3—41no—
215C23×C3—101no—
216C3×Q8—41no—
217C3×Q8—41no—
218C3×D4—41no—
219C3×D4—41no—
220C3×D4—41no—
221C3×D4—41no—
222C3×D4—52no—
223C3×D4—81no—
224C3×D4—101no—
225C2×C3×C4—41no—
226C2×C3×C4—101no—
227C2×C3×C4—201no—
228C2×C3×C4—201no—
229C2×C3×C4D4×D511yescharacteristic
230C2×C3×C4#71111yescharacteristic
231C2×C3×C4C2×C22⋊D511yescharacteristic
232C3×C8—21no—
233C3×C8—21no—
234C3×C8—201no—
235C3×C8—201no—
236C3×C8—201no—
237C22×C5—81no—
238C22×C5—81no—
239C22×C5—81no—
240C22×C5C3×#12111yescharacteristic
241C2×D5—22no—
242C2×D5—41no—
243C2×D5—41no—
244C4×C5—21no—
245C4×C5—81no—
246C4×C5C2×C3×D811yescharacteristic
247C4×C5C3×#13511yescharacteristic
248C4×C5C3×#13511yescharacteristic
249C4×C5C3×AGL(1,ℤ/8)11yescharacteristic
250Dic5—41no—
251C2×Q8—21no—
252C2×D4—21no—
253C2×D4—21no—
254C2×D4—21no—
255C2×D4—51no—
256C22×C4—101no—
257SD16—202no—
258D8—202no—
259C2×C8—101no—
260C2×C8—101no—
261C2×C8—101no—
262C2×C8—202no—
263C2×C8C3×C22⋊D511yescharacteristic
264C2×C8C3×C22⋊D511yescharacteristic
265C4⋊C4—21no—
266C4⋊C4—21no—
267C4⋊C4—51no—
268C22⋊C4—101no—
269C42C22×C3×D511yescharacteristic
270C3×C5#168111yescharacteristic, complemented, Hall
271C22×C3—81no—
272C22×C3—81no—
273C22×C3—81no—
274C22×C3—102no—
275C22×C3—201no—
276C22×C3—201no—
277C22×C3#408411yescharacteristic
278C3×C4—21no—
279C3×C4—81no—
280C3×C4—201no—
281C3×C4D5×D811yescharacteristic
282C3×C4#405811yescharacteristic
283C3×C4#409511yescharacteristic
284C3×C4#409611yescharacteristic
285C2×C5—161no—
286C2×C5C3×#46311yescharacteristic
287C2×C5C3×#46611yescharacteristic
288C2×C5C3×#47111yescharacteristic
289D5—42no—
290C23—41no—
291C23—101no—
292Q8—41no—
293Q8—41no—
294D4—41no—
295D4—41no—
296D4—41no—
297D4—41no—
298D4—52no—
299D4—81no—
300D4—101no—
301C2×C4—41no—
302C2×C4—101no—
303C2×C4—201no—
304C2×C4—201no—
305C2×C4C3×D4×D511yescharacteristic
306C2×C4C3×#71111yescharacteristic
307C2×C4C2×C3×C22⋊D511yescharacteristic
308C8—21no—
309C8—21no—
310C8—201no—
311C8—201no—
312C8—201no—
313C2×C3—161no—
314C2×C3—202no—
315C2×C3#6534611yescharacteristic
316C2×C3#6544611yescharacteristic
317C2×C3#6546611yescharacteristic
318C5C3×#168111yescharacteristic, complemented, Sylow
319C22—81no—
320C22—81no—
321C22—81no—
322C22—102no—
323C22—201no—
324C22—201no—
325C22C3×#408411yescharacteristic
326C4—21no—
327C4—81no—
328C4—201no—
329C4C3×D5×D811yescharacteristic
330C4C3×#405811yescharacteristic
331C4C3×#409511yescharacteristic
332C4C3×#409611yescharacteristic
333C3#1060199411yescharacteristic, direct product factor, Sylow
334C2—161no—
335C2—202no—
336C2C3×#6534611yescharacteristic
337C2C3×#6544611yescharacteristic
338C2C3×#6546611yescharacteristic
subgroupG/H
centerC22×C3#4084
commutator, cosocleC42×C5C23×C3
FrattiniC42C22×C3×D5
FittingC3×C5×#486C2
socleC22×C3×C5#121
prime radicalC3×#65361C2
hypercenterC3×#486D5
hypocenterC5C3×#1681
LCSG ▷ C42×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C22×C3 ◁ C3×C42 ◁ C3×#486

Complements

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

NH ≅ G/Ntype
1. C3×C5×#486C2semidirect
2. C3×#65372C2semidirect
4. C3×C8×D20C2semidirect
5. C3×#64767C2semidirect
6. C3×#64764C2semidirect
7. C3×#64678C2semidirect
8. #10601994C3direct (N × H)
25. C5×#486C2×C3semidirect
26. #65372C2×C3semidirect
28. C8×D20C2×C3semidirect
29. #64767C2×C3semidirect
30. #64764C2×C3semidirect
31. #64678C2×C3semidirect
51. C3×#486D5semidirect
131. #486C3×D5semidirect
270. C3×C5#1681semidirect
318. C5C3×#1681semidirect
333. C3#10601994direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C5×#4862
C3×#653722
C3×#653612
C3×C8×D202
C3×#647672
C3×#647642
C3×#646782
#106019943
C3×#16815