Finite Group Databasebeta

← C3×#10600290 #14308981

Subgroups — C3×#10600290 #14308981

Order 1920.

932 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 932 conjugacy classes of subgroups — too many to draw. Only its 138 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×C5×#379C211yesmaximal, characteristic
3C3×#65536C212yesmaximal
4C3×#64747C212yesmaximal, complemented
5C3×#64746C211yesmaximal, characteristic, complemented
6#10600290C311yesmaximal, characteristic, direct product factor, Hall
7C25×C3×C5C2211yescharacteristic
8C2×C3×C5×C22⋊C4—22no—
9C2×C3×C5×C22⋊C4C2211yescharacteristic
10C2×C3×C5×C22⋊C4C2211yescharacteristic
11C2×C3×C22⋊Dic5—24no—
12C2×C3×C22⋊Dic5C2212yes—
13C2×C3×C22⋊Dic5C2212yes—
14C3×#4070—22no—
15C3×#4024—24no—
16C3×#2029—51nomaximal, Hall
17#65817C2×C311yescharacteristic, complemented
18C5×#379C2×C311yescharacteristic
19#65536C2×C312yes—
20#64747C2×C312yescomplemented
21#64746C2×C311yescharacteristic, complemented
22C24×C3×C5—22no—
23C24×C3×C5—21no—
24C24×C3×C5—21no—
25C24×C3×C5—22no—
26C24×C3×C5—44no—
27C24×C3×C5Q812yescomplemented
28C24×C3×C5C2311yescharacteristic
29C22×C3×C4×C5—21no—
30C22×C3×C4×C5D412yes—
31C22×C3×Dic5—22no—
32C22×C3×Dic5D414yes—
33C2×C3×Dic10—22no—
34C3×C5×C22⋊C4—22no—
35C3×C5×C22⋊C4—22no—
36C3×C5×C22⋊C4—44no—
37C3×C22⋊Dic5—22no—
38C3×C22⋊Dic5—22no—
39C3×C22⋊Dic5—22no—
40C3×C22⋊Dic5—22no—
41C3×C22⋊Dic5—48no—
42C3×C4⋊Dic5—22no—
43C3×#683—24no—
44C3×#543—51no—
45C3×#379—52no—
46C3×#379D511yescharacteristic
47C3×#342—51no—
48C3×#342—52no—
49C25×C5C22×C311yescharacteristic
50C2×C5×C22⋊C4—22no—
51C2×C5×C22⋊C4C22×C311yescharacteristic
52C2×C5×C22⋊C4C22×C311yescharacteristic
53C2×C22⋊Dic5—24no—
54C2×C22⋊Dic5C22×C312yes—
55C2×C22⋊Dic5C22×C312yes—
56#4070—22no—
57#4024—24no—
58#2029—51noSylow
59C23×C3×C5—28no—
60C23×C3×C5—24no—
61C23×C3×C5—24no—
62C23×C3×C5—22no—
63C23×C3×C5—44no—
64C23×C3×C5—42no—
65C23×C3×C5—44no—
66C23×C3×C5—42no—
67C23×C3×C5—48no—
68C23×C3×C5—44no—
69C23×C3×C5—41no—
70C23×C3×C5—42no—
71C23×C3×C5—41no—
72C23×C3×C5C2×D412yes—
73C23×C3×C5C2×D411yescharacteristic
74C23×C3×C5C2×Q811yescharacteristic
75C23×C3×C5#4111yescharacteristic
76C23×C3×C5#4112yes—
77C2×C3×C4×C5—22no—
78C2×C3×C4×C5—42no—
79C2×C3×C4×C5—42no—
80C2×C3×C4×C5—41no—
81C2×C3×Dic5—24no—
82C2×C3×Dic5—44no—
83C2×C3×Dic5—44no—
84C2×C3×Dic5—42no—
85C3×Dic10—82no—
86C25×C3C2×D511yescharacteristic
87C3×C22⋊Q8—104no—
88C3×C22⋊Q8—102no—
89C2×C3×C22⋊C4—22no—
90C2×C3×C22⋊C4—52no—
91C2×C3×C22⋊C4—52no—
92C2×C3×C22⋊C4—104no—
93C2×C3×C22⋊C4C2×D511yescharacteristic
94C2×C3×C22⋊C4C2×D511yescharacteristic
95C24×C5—22no—
96C24×C5—21no—
97C24×C5—21no—
98C24×C5—22no—
99C24×C5—44no—
100C24×C5C3×Q812yescomplemented
101C24×C5C23×C311yescharacteristic
102C22×C4×C5—21no—
103C22×C4×C5C3×D412yes—
104C22×Dic5—22no—
105C22×Dic5C3×D414yes—
106C2×Dic10—22no—
107C5×C22⋊C4—22no—
108C5×C22⋊C4—22no—
109C5×C22⋊C4—44no—
110C22⋊Dic5—22no—
111C22⋊Dic5—22no—
112C22⋊Dic5—22no—
113C22⋊Dic5—22no—
114C22⋊Dic5—48no—
115C4⋊Dic5—22no—
116#683—24no—
117#543—51no—
118#379—52no—
119#379C3×D511yescharacteristic
120#342—51no—
121#342—52no—
122C22×C3×C5—24no—
123C22×C3×C5—22no—
124C22×C3×C5—28no—
125C22×C3×C5—24no—
126C22×C3×C5—41no—
127C22×C3×C5—41no—
128C22×C3×C5—42no—
129C22×C3×C5—48no—
130C22×C3×C5—44no—
131C22×C3×C5—44no—
132C22×C3×C5—42no—
133C22×C3×C5—42no—
134C22×C3×C5—44no—
135C22×C3×C5#12011yescharacteristic
136C22×C3×C5C22⋊Q812yes—
137C22×C3×C5C22⋊Q811yescharacteristic
138C22×C3×C5#12412yes—
139C22×C3×C5#12411yescharacteristic
140C3×C4×C5—42no—
141C3×C4×C5—81no—
142C3×Dic5—44no—
143C3×Dic5—82no—
144C24×C3—22no—
145C24×C3—21no—
146C24×C3—21no—
147C24×C3—22no—
148C24×C3—44no—
149C24×C3Dic1012yescomplemented
150C24×C3C22×D511yescharacteristic
151C2×C3×Q8—102no—
152C22×C3×C4—21no—
153C22×C3×C4—54no—
154C22×C3×C4—102no—
155C22×C3×C4C22⋊D512yes—
156C3×C4⋊C4—104no—
157C3×C4⋊C4—102no—
158C3×C22⋊C4—22no—
159C3×C22⋊C4—22no—
160C3×C22⋊C4—44no—
161C3×C22⋊C4—102no—
162C3×C22⋊C4—102no—
163C3×C22⋊C4—102no—
164C3×C22⋊C4—102no—
165C3×C22⋊C4—208no—
166C23×C5—28no—
167C23×C5—24no—
168C23×C5—24no—
169C23×C5—22no—
170C23×C5—44no—
171C23×C5—42no—
172C23×C5—44no—
173C23×C5—42no—
174C23×C5—44no—
175C23×C5—48no—
176C23×C5—41no—
177C23×C5—41no—
178C23×C5—42no—
179C23×C5C2×C3×D412yes—
180C23×C5C2×C3×D411yescharacteristic
181C23×C5C2×C3×Q811yescharacteristic
182C23×C5C3×#4111yescharacteristic
183C23×C5C3×#4112yes—
184C2×C4×C5—22no—
185C2×C4×C5—42no—
186C2×C4×C5—42no—
187C2×C4×C5—41no—
188C2×Dic5—24no—
189C2×Dic5—44no—
190C2×Dic5—44no—
191C2×Dic5—42no—
192Dic10—82no—
193C25C2×C3×D511yescharacteristic
194C22⋊Q8—104no—
195C22⋊Q8—102no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—52no—
198C2×C22⋊C4—52no—
199C2×C22⋊C4—104no—
200C2×C22⋊C4C2×C3×D511yescharacteristic
201C2×C22⋊C4C2×C3×D511yescharacteristic
202C2×C3×C5—22no—
203C2×C3×C5—21no—
204C2×C3×C5—22no—
205C2×C3×C5—21no—
206C2×C3×C5—44no—
207C2×C3×C5#39311yescharacteristic
208C2×C3×C5B(4,𝔽2)12yes—
209C23×C3—24no—
210C23×C3—28no—
211C23×C3—24no—
212C23×C3—22no—
213C23×C3—44no—
214C23×C3—42no—
215C23×C3—42no—
216C23×C3—44no—
217C23×C3—44no—
218C23×C3—48no—
219C23×C3—42no—
220C23×C3—41no—
221C23×C3—41no—
222C23×C3C2×Dic1011yescharacteristic
223C23×C3#70911yescharacteristic
224C23×C3D4×D512yes—
225C23×C3#71112yes—
226C23×C3C2×C22⋊D511yescharacteristic
227C3×Q8—402no—
228C2×C3×C4—22no—
229C2×C3×C4—42no—
230C2×C3×C4—42no—
231C2×C3×C4—41no—
232C2×C3×C4—104no—
233C2×C3×C4—204no—
234C2×C3×C4—204no—
235C2×C3×C4—202no—
236C22×C5—24no—
237C22×C5—22no—
238C22×C5—28no—
239C22×C5—24no—
240C22×C5—41no—
241C22×C5—41no—
242C22×C5—42no—
243C22×C5—48no—
244C22×C5—44no—
245C22×C5—44no—
246C22×C5—42no—
247C22×C5—42no—
248C22×C5—44no—
249C22×C5C3×#12011yescharacteristic
250C22×C5C3×C22⋊Q812yes—
251C22×C5C3×C22⋊Q811yescharacteristic
252C22×C5C3×#12411yescharacteristic
253C22×C5C3×#12412yes—
254C4×C5—42no—
255C4×C5—81no—
256Dic5—44no—
257Dic5—82no—
258C24—22no—
259C24—21no—
260C24—21no—
261C24—22no—
262C24—44no—
263C24C3×Dic1012yescomplemented
264C24C22×C3×D511yescharacteristic
265C2×Q8—102no—
266C22×C4—21no—
267C22×C4—54no—
268C22×C4—102no—
269C22×C4C3×C22⋊D512yes—
270C4⋊C4—104no—
271C4⋊C4—102no—
272C22⋊C4—22no—
273C22⋊C4—22no—
274C22⋊C4—44no—
275C22⋊C4—102no—
276C22⋊C4—102no—
277C22⋊C4—102no—
278C22⋊C4—102no—
279C22⋊C4—208no—
280C3×C5#202911yescharacteristic, complemented, Hall
281C22×C3—22no—
282C22×C3—24no—
283C22×C3—28no—
284C22×C3—24no—
285C22×C3—41no—
286C22×C3—41no—
287C22×C3—42no—
288C22×C3—44no—
289C22×C3—48no—
290C22×C3—44no—
291C22×C3—42no—
292C22×C3—44no—
293C22×C3—42no—
294C22×C3#402412yes—
295C22×C3#403112yes—
296C22×C3#407011yescharacteristic
297C22×C3#408211yescharacteristic
298C22×C3#408311yescharacteristic
299C3×C4—42no—
300C3×C4—81no—
301C3×C4—204no—
302C3×C4—402no—
303C2×C5—22no—
304C2×C5—21no—
305C2×C5—22no—
306C2×C5—21no—
307C2×C5—44no—
308C2×C5C3×#39311yescharacteristic
309C2×C5C3×B(4,𝔽2)12yes—
310C23—24no—
311C23—28no—
312C23—24no—
313C23—22no—
314C23—44no—
315C23—42no—
316C23—42no—
317C23—44no—
318C23—44no—
319C23—48no—
320C23—41no—
321C23—41no—
322C23—42no—
323C23C2×C3×Dic1011yescharacteristic
324C23C3×#70911yescharacteristic
325C23C3×D4×D512yes—
326C23C3×#71112yes—
327C23C2×C3×C22⋊D511yescharacteristic
328Q8—402no—
329C2×C4—22no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—41no—
333C2×C4—104no—
334C2×C4—204no—
335C2×C4—204no—
336C2×C4—202no—
337C2×C3—21no—
338C2×C3—22no—
339C2×C3—22no—
340C2×C3—21no—
341C2×C3—44no—
342C2×C3#6523611yescharacteristic
343C2×C3#6532112yes—
344C5C3×#202911yescharacteristic, complemented, Sylow
345C22—22no—
346C22—24no—
347C22—28no—
348C22—24no—
349C22—41no—
350C22—41no—
351C22—42no—
352C22—44no—
353C22—48no—
354C22—44no—
355C22—42no—
356C22—44no—
357C22—42no—
358C22C3×#402412yes—
359C22C3×#403112yes—
360C22C3×#407011yescharacteristic
361C22C3×#408211yescharacteristic
362C22C3×#408311yescharacteristic
363C4—42no—
364C4—81no—
365C4—204no—
366C4—402no—
367C3#1060029011yescharacteristic, direct product factor, Sylow
368C2—21no—
369C2—22no—
370C2—22no—
371C2—21no—
372C2—44no—
373C2C3×#6523611yescharacteristic
374C2C3×#6532112yes—
subgroupG/H
centerC22×C3#4083
commutator, cosocleC24×C5C23×C3
FrattiniC24C22×C3×D5
FittingC3×C5×#379C2
socleC22×C3×C5#120
prime radicalC25×C3×C5C22
hypercenterC3×#379D5
hypocenterC5C3×#2029
LCSG ▷ C24×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22×C3 ◁ C24×C3 ◁ C3×#379

Complements

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

NH ≅ G/Ntype
1. C3×#65817C2semidirect
4. C3×#64747C2semidirect
5. C3×#64746C2semidirect
6. #10600290C3direct (N × H)
17. #65817C2×C3semidirect
20. #64747C2×C3semidirect
21. #64746C2×C3semidirect
27. C24×C3×C5Q8semidirect
100. C24×C5C3×Q8semidirect
149. C24×C3Dic10semidirect
263. C24C3×Dic10semidirect
280. C3×C5#2029semidirect
344. C5C3×#2029semidirect
367. C3#10600290direct (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×C5×#3792
C3×#655362
C3×#647472
C3×#647462
#106002903
C3×#20295