Finite Group Databasebeta

← C3×#10599764 #14308984

Subgroups — C3×#10599764 #14308984

Order 1920.

932 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 932 conjugacy classes of subgroups — too many to draw. Only its 134 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×#65817C211yesmaximal, characteristic, complemented
2C3×#65244C211yesmaximal, characteristic
3C3×#65244C212yesmaximal
4C3×C5×#342C211yesmaximal, characteristic, complemented
5C3×#64747C212yesmaximal, complemented
6#10599764C311yesmaximal, characteristic, direct product factor, Hall
7C24×C3×D5C2211yescharacteristic
8C2×C3×C5×C22⋊C4C2212yes—
9C2×C3×C5×C22⋊C4C2211yescharacteristic
10C2×C3×C22⋊Dic5C2211yescharacteristic
11C2×C3×C22⋊Dic5C2212yes—
12C2×C3×#685—22no—
13C2×C3×#685—24no—
14C3×#4070—22no—
15C3×#4024—24no—
16C3×#2029—51nomaximal, Hall
17#65817C2×C311yescharacteristic, complemented
18#65244C2×C311yescharacteristic
19#65244C2×C312yes—
20C5×#342C2×C311yescharacteristic, complemented
21#64747C2×C312yescomplemented
22C24×C3×C5C2311yescharacteristic
23C23×C3×D5—21no—
24C23×C3×D5—22no—
25C23×C3×D5—22no—
26C23×C3×D5—21no—
27C23×C3×D5—44no—
28C23×C3×D5Q812yescomplemented
29C22×C3×C4×C5—22no—
30C22×C3×C4×C5D412yes—
31C22×C3×Dic5—21no—
32C22×C3×Dic5D414yes—
33C2×C3×Dic10—22no—
34C3×C5×C22⋊C4—22no—
35C3×C5×C22⋊C4—22no—
36C3×C5×C22⋊C4—22no—
37C3×C22⋊Dic5—22no—
38C3×C22⋊Dic5—22no—
39C3×C22⋊Dic5—22no—
40C3×#685—48no—
41C3×#685—44no—
42C3×C4⋊Dic5—22no—
43C3×#683—24no—
44C3×#543—51no—
45C3×#379—52no—
46C3×#379—51no—
47C3×#342—52no—
48C3×#342D511yescharacteristic, complemented
49C24×D5C22×C311yescharacteristic
50C2×C5×C22⋊C4C22×C312yes—
51C2×C5×C22⋊C4C22×C311yescharacteristic
52C2×C22⋊Dic5C22×C311yescharacteristic
53C2×C22⋊Dic5C22×C312yes—
54C2×#685—22no—
55C2×#685—24no—
56#4070—22no—
57#4024—24no—
58#2029—51noSylow
59C23×C3×C5—42no—
60C23×C3×C5C2×D411yescharacteristic
61C23×C3×C5C2×D412yes—
62C23×C3×C5C2×Q811yescharacteristic
63C23×C3×C5#4112yes—
64C23×C3×C5#4111yescharacteristic
65C22×C3×D5—24no—
66C22×C3×D5—28no—
67C22×C3×D5—24no—
68C22×C3×D5—22no—
69C22×C3×D5—44no—
70C22×C3×D5—48no—
71C22×C3×D5—44no—
72C22×C3×D5—42no—
73C22×C3×D5—44no—
74C22×C3×D5—41no—
75C22×C3×D5—41no—
76C22×C3×D5—42no—
77C2×C3×C4×C5—22no—
78C2×C3×C4×C5—42no—
79C2×C3×C4×C5—44no—
80C2×C3×C4×C5—42no—
81C2×C3×Dic5—24no—
82C2×C3×Dic5—44no—
83C2×C3×Dic5—42no—
84C2×C3×Dic5—41no—
85C3×Dic10—82no—
86C25×C3—51no—
87C3×C22⋊Q8—104no—
88C3×C22⋊Q8—102no—
89C2×C3×C22⋊C4—52no—
90C2×C3×C22⋊C4—51no—
91C2×C3×C22⋊C4—104no—
92C2×C3×C22⋊C4—102no—
93C2×C3×C22⋊C4C2×D512yes—
94C2×C3×C22⋊C4C2×D511yescharacteristic
95C24×C5C23×C311yescharacteristic
96C23×D5—21no—
97C23×D5—22no—
98C23×D5—22no—
99C23×D5—21no—
100C23×D5—44no—
101C23×D5C3×Q812yescomplemented
102C22×C4×C5—22no—
103C22×C4×C5C3×D412yes—
104C22×Dic5—21no—
105C22×Dic5C3×D414yes—
106C2×Dic10—22no—
107C5×C22⋊C4—22no—
108C5×C22⋊C4—22no—
109C5×C22⋊C4—22no—
110C22⋊Dic5—22no—
111C22⋊Dic5—22no—
112C22⋊Dic5—22no—
113#685—48no—
114#685—44no—
115C4⋊Dic5—22no—
116#683—24no—
117#543—51no—
118#379—52no—
119#379—51no—
120#342—52no—
121#342C3×D511yescharacteristic, complemented
122C22×C3×C5—22no—
123C22×C3×C5—24no—
124C22×C3×C5—41no—
125C22×C3×C5—42no—
126C22×C3×C5—41no—
127C22×C3×C5#12011yescharacteristic
128C22×C3×C5C22⋊Q812yes—
129C22×C3×C5C22⋊Q811yescharacteristic
130C22×C3×C5#12411yescharacteristic
131C22×C3×C5#12412yes—
132C2×C3×D5—28no—
133C2×C3×D5—24no—
134C2×C3×D5—44no—
135C2×C3×D5—48no—
136C2×C3×D5—44no—
137C2×C3×D5—42no—
138C2×C3×D5—42no—
139C2×C3×D5—44no—
140C3×C4×C5—42no—
141C3×C4×C5—82no—
142C3×Dic5—44no—
143C3×Dic5—81no—
144C24×C3—52no—
145C24×C3—101no—
146C24×C3—102no—
147C24×C3—102no—
148C24×C3—101no—
149C24×C3—204no—
150C24×C3C22×D511yescharacteristic
151C2×C3×Q8—102no—
152C22×C3×C4—22no—
153C22×C3×C4—54no—
154C22×C3×C4—101no—
155C22×C3×C4D2012yes—
156C3×C4⋊C4—104no—
157C3×C4⋊C4—102no—
158C3×C22⋊C4—22no—
159C3×C22⋊C4—22no—
160C3×C22⋊C4—22no—
161C3×C22⋊C4—102no—
162C3×C22⋊C4—102no—
163C3×C22⋊C4—102no—
164C3×C22⋊C4—204no—
165C3×C22⋊C4—208no—
166C23×C5—42no—
167C23×C5C2×C3×D411yescharacteristic
168C23×C5C2×C3×D412yes—
169C23×C5C2×C3×Q811yescharacteristic
170C23×C5C3×#4112yes—
171C23×C5C3×#4111yescharacteristic
172C22×D5—24no—
173C22×D5—28no—
174C22×D5—24no—
175C22×D5—22no—
176C22×D5—44no—
177C22×D5—48no—
178C22×D5—44no—
179C22×D5—42no—
180C22×D5—44no—
181C22×D5—41no—
182C22×D5—41no—
183C22×D5—42no—
184C2×C4×C5—22no—
185C2×C4×C5—42no—
186C2×C4×C5—44no—
187C2×C4×C5—42no—
188C2×Dic5—24no—
189C2×Dic5—44no—
190C2×Dic5—42no—
191C2×Dic5—41no—
192Dic10—82no—
193C25—51no—
194C22⋊Q8—104no—
195C22⋊Q8—102no—
196C2×C22⋊C4—52no—
197C2×C22⋊C4—51no—
198C2×C22⋊C4—104no—
199C2×C22⋊C4—102no—
200C2×C22⋊C4C2×C3×D512yes—
201C2×C22⋊C4C2×C3×D511yescharacteristic
202C2×C3×C5—21no—
203C2×C3×C5—22no—
204C2×C3×C5—22no—
205C2×C3×C5—21no—
206C2×C3×C5#39311yescharacteristic
207C2×C3×C5B(4,𝔽2)12yes—
208C3×D5—44no—
209C23×C3—42no—
210C23×C3—104no—
211C23×C3—108no—
212C23×C3—104no—
213C23×C3—102no—
214C23×C3—202no—
215C23×C3—204no—
216C23×C3—204no—
217C23×C3—208no—
218C23×C3—204no—
219C23×C3—201no—
220C23×C3—201no—
221C23×C3—202no—
222C23×C3C2×D2011yescharacteristic
223C23×C3#70912yes—
224C23×C3D4×D512yes—
225C23×C3#71111yescharacteristic
226C23×C3Q8×D511yescharacteristic
227C3×Q8—402no—
228C2×C3×C4—22no—
229C2×C3×C4—42no—
230C2×C3×C4—44no—
231C2×C3×C4—42no—
232C2×C3×C4—104no—
233C2×C3×C4—204no—
234C2×C3×C4—202no—
235C2×C3×C4—201no—
236C22×C5—22no—
237C22×C5—24no—
238C22×C5—41no—
239C22×C5—42no—
240C22×C5—41no—
241C22×C5C3×#12011yescharacteristic
242C22×C5C3×C22⋊Q812yes—
243C22×C5C3×C22⋊Q811yescharacteristic
244C22×C5C3×#12411yescharacteristic
245C22×C5C3×#12412yes—
246C2×D5—28no—
247C2×D5—24no—
248C2×D5—44no—
249C2×D5—48no—
250C2×D5—44no—
251C2×D5—42no—
252C2×D5—42no—
253C2×D5—44no—
254C4×C5—42no—
255C4×C5—82no—
256Dic5—44no—
257Dic5—81no—
258C24—52no—
259C24—101no—
260C24—102no—
261C24—102no—
262C24—101no—
263C24—204no—
264C24C22×C3×D511yescharacteristic
265C2×Q8—102no—
266C22×C4—22no—
267C22×C4—54no—
268C22×C4—101no—
269C22×C4C3×D2012yes—
270C4⋊C4—104no—
271C4⋊C4—102no—
272C22⋊C4—22no—
273C22⋊C4—22no—
274C22⋊C4—22no—
275C22⋊C4—102no—
276C22⋊C4—102no—
277C22⋊C4—102no—
278C22⋊C4—204no—
279C22⋊C4—208no—
280C3×C5#202911yescharacteristic, complemented, Hall
281C22×C3—22no—
282C22×C3—24no—
283C22×C3—42no—
284C22×C3—41no—
285C22×C3—41no—
286C22×C3—108no—
287C22×C3—104no—
288C22×C3—208no—
289C22×C3—204no—
290C22×C3—204no—
291C22×C3—202no—
292C22×C3—204no—
293C22×C3—202no—
294C22×C3#402111yescharacteristic
295C22×C3#402811yescharacteristic
296C22×C3#403112yes—
297C22×C3#404212yes—
298C22×C3#404311yescharacteristic
299C3×C4—42no—
300C3×C4—82no—
301C3×C4—204no—
302C3×C4—401no—
303C2×C5—21no—
304C2×C5—22no—
305C2×C5—22no—
306C2×C5—21no—
307C2×C5C3×#39311yescharacteristic
308C2×C5C3×B(4,𝔽2)12yes—
309D5—44no—
310C23—42no—
311C23—104no—
312C23—108no—
313C23—104no—
314C23—102no—
315C23—202no—
316C23—204no—
317C23—204no—
318C23—208no—
319C23—204no—
320C23—201no—
321C23—201no—
322C23—202no—
323C23C2×C3×D2011yescharacteristic
324C23C3×#70912yes—
325C23C3×D4×D512yes—
326C23C3×#71111yescharacteristic
327C23C3×Q8×D511yescharacteristic
328Q8—402no—
329C2×C4—22no—
330C2×C4—42no—
331C2×C4—44no—
332C2×C4—42no—
333C2×C4—104no—
334C2×C4—204no—
335C2×C4—202no—
336C2×C4—201no—
337C2×C3—22no—
338C2×C3—21no—
339C2×C3—22no—
340C2×C3—21no—
341C2×C3—204no—
342C2×C3#6496211yescharacteristic
343C2×C3#6503012yes—
344C5C3×#202911yescharacteristic, complemented, Sylow
345C22—22no—
346C22—24no—
347C22—42no—
348C22—41no—
349C22—41no—
350C22—108no—
351C22—104no—
352C22—208no—
353C22—204no—
354C22—204no—
355C22—202no—
356C22—204no—
357C22—202no—
358C22C3×#402111yescharacteristic
359C22C3×#402811yescharacteristic
360C22C3×#403112yes—
361C22C3×#404212yes—
362C22C3×#404311yescharacteristic
363C4—42no—
364C4—82no—
365C4—204no—
366C4—401no—
367C3#1059976411yescharacteristic, direct product factor, Sylow
368C2—22no—
369C2—21no—
370C2—22no—
371C2—21no—
372C2—204no—
373C2C3×#6496211yescharacteristic
374C2C3×#6503012yes—
subgroupG/H
centerC22×C3#4028
commutator, cosocleC24×C5C23×C3
FrattiniC24C22×C3×D5
FittingC3×C5×#342C2
socleC22×C3×C5#120
prime radicalC24×C3×D5C22
hypercenterC3×#342D5
hypocenterC5C3×#2029
LCSG ▷ C24×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22×C3 ◁ C24×C3 ◁ C3×#342

Complements

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

NH ≅ G/Ntype
1. C3×#65817C2semidirect
4. C3×C5×#342C2semidirect
5. C3×#64747C2semidirect
6. #10599764C3direct (N × H)
17. #65817C2×C3semidirect
20. C5×#342C2×C3semidirect
21. #64747C2×C3semidirect
28. C23×C3×D5Q8semidirect
48. C3×#342D5semidirect
101. C23×D5C3×Q8semidirect
121. #342C3×D5semidirect
280. C3×C5#2029semidirect
344. C5C3×#2029semidirect
367. C3#10599764direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#658172
C3×#652442
C3×C5×#3422
C3×#647472
#105997643
C3×#20295