Finite Group Databasebeta

← C3×#10602671 #14287823

Subgroups — C3×#10602671 #14287823

Order 1920.

620 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 620 conjugacy classes of subgroups — too many to draw. Only its 114 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×C5×#537C211yesmaximal, characteristic, complemented
2C3×#65325C212yesmaximal, complemented
3C3×#65280C211yesmaximal, characteristic, complemented
4C3×#64757C212yesmaximal, complemented
5C3×#64748C211yesmaximal, characteristic, complemented
6#10602671C311yesmaximal, characteristic, direct product factor, Hall
7C2×C3×C5×#41—21no—
8C22×C3×D20C2211yescharacteristic
9C3×C5×C22⋊Q8—22no—
10C3×C5×#121—22no—
11C3×C5×#121—21no—
12C3×C5×#121C2212yes—
13C2×C3×C5×C4⋊C4C2211yescharacteristic
14C2×C3×#686—22no—
15C2×C3×#685—21no—
16C2×C3×#685—21no—
17C3×#3963C2211yescharacteristic
18C3×C22⋊#168C2212yes—
19C3×#3941—22no—
20C3×#1598—51nomaximal, Hall
21C5×#537C2×C311yescharacteristic, complemented
22#65325C2×C312yescomplemented
23#65280C2×C311yescharacteristic, complemented
24#64757C2×C312yescomplemented
25#64748C2×C311yescharacteristic, complemented
26C23×C3×D5—21no—
27C3×C5×#41—42no—
28C3×C5×#41—42no—
29C2×C3×C5×Q8—21no—
30C2×C3×C5×D4—21no—
31C2×C3×C5×D4—22no—
32C2×C3×C5×D4—21no—
33C2×C3×C5×D4D412yes—
34C22×C3×C4×C5—22no—
35C22×C3×C4×C5D411yescharacteristic
36C22×C3×C4×C5D411yescharacteristic
37C22×C3×C4×C5C2311yescharacteristic
38C22×C3×Dic5—21no—
39C2×C3×D20—22no—
40C2×C3×D20—42no—
41C3×C5×C4⋊C4—21no—
42C3×C5×C4⋊C4—22no—
43C3×C5×C4⋊C4D412yes—
44C3×C5×C22⋊C4—22no—
45C3×C5×C22⋊C4—22no—
46C3×#686—44no—
47C3×#685—42no—
48C3×#685—42no—
49C2×C3×#168—22no—
50C3×#537D511yescharacteristic, complemented
51C3×#447—52no—
52C3×#394—51no—
53C3×#327—52no—
54C3×#324—51no—
55C2×C5×#41—21no—
56C22×D20C22×C311yescharacteristic
57C5×C22⋊Q8—22no—
58C5×#121—22no—
59C5×#121—21no—
60C5×#121C22×C312yes—
61C2×C5×C4⋊C4C22×C311yescharacteristic
62C2×#686—22no—
63C2×#685—21no—
64C2×#685—21no—
65#3963C22×C311yescharacteristic
66C22⋊#168C22×C312yes—
67#3941—22no—
68#1598—51noSylow
69C23×C3×C5—22no—
70C23×C3×C5C2×D411yescharacteristic
71C22×C3×D5—22no—
72C22×C3×D5—41no—
73C22×C3×D5—41no—
74C22×C3×D5—41no—
75C22×C3×D5—41no—
76C22×C3×D5—41no—
77C22×C3×D5—41no—
78C3×C5×Q8—42no—
79C3×C5×D4—42no—
80C3×C5×D4—42no—
81C3×C5×D4—42no—
82C3×C5×D4—42no—
83C3×C5×D4—81no—
84C2×C3×C4×C5—21no—
85C2×C3×C4×C5—21no—
86C2×C3×C4×C5—21no—
87C2×C3×C4×C5—22no—
88C2×C3×C4×C5—41no—
89C2×C3×C4×C5—44no—
90C2×C3×C4×C5—41no—
91C2×C3×C4×C5D812yescomplemented
92C2×C3×C4×C5C2×D412yes—
93C2×C3×Dic5—41no—
94C2×C3×Dic5—41no—
95C2×C3×Dic5—41no—
96C3×D20—24no—
97C3×D20—42no—
98C3×#168—42no—
99C2×C3×#41—21no—
100C22×C3×D4—51no—
101C2×C3×D8—102no—
102C3×C22⋊Q8—22no—
103C3×#121—22no—
104C3×#121—21no—
105C3×#121C2×D512yes—
106C2×C3×C4⋊C4C2×D511yescharacteristic
107C2×C3×C22⋊C4—101no—
108C2×C3×C22⋊C4—101no—
109C3×#102—102no—
110C3×C22⋊C8—52no—
111C3×#95—51no—
112C23×D5—21no—
113C5×#41—42no—
114C5×#41—42no—
115C2×C5×Q8—21no—
116C2×C5×D4—21no—
117C2×C5×D4—22no—
118C2×C5×D4—21no—
119C2×C5×D4C3×D412yes—
120C22×C4×C5—22no—
121C22×C4×C5C3×D411yescharacteristic
122C22×C4×C5C3×D411yescharacteristic
123C22×C4×C5C23×C311yescharacteristic
124C22×Dic5—21no—
125C2×D20—22no—
126C2×D20—42no—
127C5×C4⋊C4—21no—
128C5×C4⋊C4—22no—
129C5×C4⋊C4C3×D412yes—
130C5×C22⋊C4—22no—
131C5×C22⋊C4—22no—
132#686—44no—
133#685—42no—
134#685—42no—
135C2×#168—22no—
136#537C3×D511yescharacteristic, complemented
137#447—52no—
138#394—51no—
139#327—52no—
140#324—51no—
141C22×C3×C5—21no—
142C22×C3×C5—21no—
143C22×C3×C5—42no—
144C22×C3×C5—42no—
145C22×C3×C5—42no—
146C22×C3×C5#12011yescharacteristic
147C22×C3×C5C2×D811yescharacteristic
148C22×C3×C5AGL(1,ℤ/8)11yescharacteristic
149C2×C3×D5—42no—
150C2×C3×D5—42no—
151C2×C3×D5—42no—
152C2×C3×D5—81no—
153C2×C3×D5—81no—
154C2×C3×D5—81no—
155C2×C3×D5—81no—
156C3×C4×C5—22no—
157C3×C4×C5—22no—
158C3×C4×C5—42no—
159C3×C4×C5—41no—
160C3×Dic5—81no—
161C24×C3—101no—
162C3×#41—42no—
163C3×#41—42no—
164C2×C3×Q8—21no—
165C2×C3×D4—21no—
166C2×C3×D4—22no—
167C2×C3×D4—21no—
168C2×C3×D4—102no—
169C2×C3×D4—202no—
170C2×C3×D4C22⋊D512yes—
171C22×C3×C4—22no—
172C22×C3×C4—101no—
173C22×C3×C4C22⋊D511yescharacteristic
174C22×C3×C4C22⋊D511yescharacteristic
175C22×C3×C4C22×D511yescharacteristic
176C3×D8—204no—
177C2×C3×C8—102no—
178C3×C4⋊C4—21no—
179C3×C4⋊C4—22no—
180C3×C4⋊C4C22⋊D512yes—
181C3×C22⋊C4—22no—
182C3×C22⋊C4—22no—
183C3×C22⋊C4—202no—
184C3×C22⋊C4—202no—
185C23×C5—22no—
186C23×C5C2×C3×D411yescharacteristic
187C22×D5—22no—
188C22×D5—41no—
189C22×D5—41no—
190C22×D5—41no—
191C22×D5—41no—
192C22×D5—41no—
193C22×D5—41no—
194C5×Q8—42no—
195C5×D4—42no—
196C5×D4—42no—
197C5×D4—42no—
198C5×D4—42no—
199C5×D4—81no—
200C2×C4×C5—21no—
201C2×C4×C5—21no—
202C2×C4×C5—21no—
203C2×C4×C5—22no—
204C2×C4×C5—41no—
205C2×C4×C5—44no—
206C2×C4×C5—41no—
207C2×C4×C5C3×D812yescomplemented
208C2×C4×C5C2×C3×D412yes—
209C2×Dic5—41no—
210C2×Dic5—41no—
211C2×Dic5—41no—
212D20—24no—
213D20—42no—
214#168—42no—
215C2×#41—21no—
216C22×D4—51no—
217C2×D8—102no—
218C22⋊Q8—22no—
219#121—22no—
220#121—21no—
221#121C2×C3×D512yes—
222C2×C4⋊C4C2×C3×D511yescharacteristic
223C2×C22⋊C4—101no—
224C2×C22⋊C4—101no—
225#102—102no—
226C22⋊C8—52no—
227#95—51no—
228C2×C3×C5—21no—
229C2×C3×C5—21no—
230C2×C3×C5—82no—
231C2×C3×C5#44711yescharacteristic
232C2×C3×C5#45411yescharacteristic
233C2×C3×C5B(4,𝔽2)11yescharacteristic
234C3×D5—82no—
235C23×C3—22no—
236C23×C3—102no—
237C23×C3—201no—
238C23×C3—201no—
239C23×C3—201no—
240C23×C3—201no—
241C23×C3—201no—
242C23×C3—201no—
243C23×C3C2×C22⋊D511yescharacteristic
244C3×Q8—42no—
245C3×D4—42no—
246C3×D4—42no—
247C3×D4—42no—
248C3×D4—42no—
249C3×D4—81no—
250C3×D4—104no—
251C3×D4—202no—
252C2×C3×C4—21no—
253C2×C3×C4—21no—
254C2×C3×C4—21no—
255C2×C3×C4—22no—
256C2×C3×C4—41no—
257C2×C3×C4—44no—
258C2×C3×C4—41no—
259C2×C3×C4—201no—
260C2×C3×C4—201no—
261C2×C3×C4—201no—
262C2×C3×C4#68612yescomplemented
263C2×C3×C4C2×C22⋊D512yes—
264C3×C8—202no—
265C22×C5—21no—
266C22×C5—21no—
267C22×C5—42no—
268C22×C5—42no—
269C22×C5—42no—
270C22×C5C3×#12011yescharacteristic
271C22×C5C2×C3×D811yescharacteristic
272C22×C5C3×AGL(1,ℤ/8)11yescharacteristic
273C2×D5—42no—
274C2×D5—42no—
275C2×D5—42no—
276C2×D5—81no—
277C2×D5—81no—
278C2×D5—81no—
279C2×D5—81no—
280C4×C5—22no—
281C4×C5—22no—
282C4×C5—42no—
283C4×C5—41no—
284Dic5—81no—
285C24—101no—
286#41—42no—
287#41—42no—
288C2×Q8—21no—
289C2×D4—21no—
290C2×D4—22no—
291C2×D4—21no—
292C2×D4—102no—
293C2×D4—202no—
294C2×D4C3×C22⋊D512yes—
295C22×C4—22no—
296C22×C4—101no—
297C22×C4C3×C22⋊D511yescharacteristic
298C22×C4C3×C22⋊D511yescharacteristic
299C22×C4C22×C3×D511yescharacteristic
300D8—204no—
301C2×C8—102no—
302C4⋊C4—21no—
303C4⋊C4—22no—
304C4⋊C4C3×C22⋊D512yes—
305C22⋊C4—22no—
306C22⋊C4—22no—
307C22⋊C4—202no—
308C22⋊C4—202no—
309C3×C5#159811yescharacteristic, complemented, Hall
310C22×C3—21no—
311C22×C3—21no—
312C22×C3—42no—
313C22×C3—42no—
314C22×C3—42no—
315C22×C3—202no—
316C22×C3—202no—
317C22×C3—202no—
318C22×C3—401no—
319C22×C3—401no—
320C22×C3—401no—
321C22×C3—401no—
322C22×C3C2×#68611yescharacteristic
323C22×C3#407811yescharacteristic
324C22×C3#409911yescharacteristic
325C3×C4—22no—
326C3×C4—22no—
327C3×C4—42no—
328C3×C4—41no—
329C3×C4—401no—
330C2×C5—21no—
331C2×C5—21no—
332C2×C5—82no—
333C2×C5C3×#44711yescharacteristic
334C2×C5C3×#45411yescharacteristic
335C2×C5C3×B(4,𝔽2)11yescharacteristic
336D5—82no—
337C23—22no—
338C23—102no—
339C23—201no—
340C23—201no—
341C23—201no—
342C23—201no—
343C23—201no—
344C23—201no—
345C23C2×C3×C22⋊D511yescharacteristic
346Q8—42no—
347D4—42no—
348D4—42no—
349D4—42no—
350D4—42no—
351D4—81no—
352D4—104no—
353D4—202no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—22no—
358C2×C4—41no—
359C2×C4—44no—
360C2×C4—41no—
361C2×C4—201no—
362C2×C4—201no—
363C2×C4—201no—
364C2×C4C3×#68612yescomplemented
365C2×C4C2×C3×C22⋊D512yes—
366C8—202no—
367C2×C3—21no—
368C2×C3—21no—
369C2×C3—82no—
370C2×C3—402no—
371C2×C3#6550611yescharacteristic
372C2×C3#6553311yescharacteristic
373C2×C3#6553411yescharacteristic
374C5C3×#159811yescharacteristic, complemented, Sylow
375C22—21no—
376C22—21no—
377C22—42no—
378C22—42no—
379C22—42no—
380C22—202no—
381C22—202no—
382C22—202no—
383C22—401no—
384C22—401no—
385C22—401no—
386C22—401no—
387C22C2×C3×#68611yescharacteristic
388C22C3×#407811yescharacteristic
389C22C3×#409911yescharacteristic
390C4—22no—
391C4—22no—
392C4—42no—
393C4—41no—
394C4—401no—
395C3#1060267111yescharacteristic, direct product factor, Sylow
396C2—21no—
397C2—21no—
398C2—82no—
399C2—402no—
400C2C3×#6550611yescharacteristic
401C2C3×#6553311yescharacteristic
402C2C3×#6553411yescharacteristic
subgroupG/H
centerC22×C3#4099
commutator, cosocleC22×C4×C5C23×C3
FrattiniC22×C4C22×C3×D5
FittingC3×C5×#537C2
socleC22×C3×C5#120
hypercenterC3×#537D5
hypocenterC5C3×#1598
LCSG ▷ C22×C4×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C3×#537

Complements

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

NH ≅ G/Ntype
1. C3×C5×#537C2semidirect
2. C3×#65325C2semidirect
3. C3×#65280C2semidirect
4. C3×#64757C2semidirect
5. C3×#64748C2semidirect
6. #10602671C3direct (N × H)
21. C5×#537C2×C3semidirect
22. #65325C2×C3semidirect
23. #65280C2×C3semidirect
24. #64757C2×C3semidirect
25. #64748C2×C3semidirect
50. C3×#537D5semidirect
91. C2×C3×C4×C5D8semidirect
136. #537C3×D5semidirect
207. C2×C4×C5C3×D8semidirect
262. C2×C3×C4#686semidirect
309. C3×C5#1598semidirect
364. C2×C4C3×#686semidirect
374. C5C3×#1598semidirect
395. C3#10602671direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C5×#5372
C3×#653252
C3×#652802
C3×#647572
C3×#647482
#106026713
C3×#15985