Finite Group Databasebeta

← C3×#10604615 #14302890

Subgroups — C3×#10604615 #14302890

Order 1920.

724 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 724 conjugacy classes of subgroups — too many to draw. Only its 132 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×#66258C211yesmaximal, characteristic, complemented
2C3×#66049C211yesmaximal, characteristic
3C3×D5⋊#100C211yesmaximal, characteristic, complemented
4C3×#65769C212yesmaximal, complemented
5C3×#64863C212yesmaximal, complemented
6#10604615C311yesmaximal, characteristic, direct product factor, Hall
7C2×C3×#711C2211yescharacteristic
8C2×C3×D4×D5C2211yescharacteristic
9C2×C3×C22⋊AGL(1,𝔽5)—22no—
10C3×D4×AGL(1,𝔽5)—24no—
11C3×#4130C2212yes—
12C3×D5⋊M16C2212yes—
13C3×C5×#127C411yescharacteristic, complemented
14C3×#4086C411yescharacteristic, complemented
15C3×#4085—22no—
16C3×#4083—22no—
17C3×#4082C411yescharacteristic, complemented
18C3×#4021C411yescharacteristic, complemented
19C3×#4018C2211yescharacteristic
20C3×#4013—22no—
21C3×#4008—24no—
22C3×#1911—51nomaximal, Hall
23#66258C2×C311yescharacteristic, complemented
24#66049C2×C311yescharacteristic
25D5⋊#100C2×C311yescharacteristic, complemented
26#65769C2×C312yescomplemented
27#64863C2×C312yescomplemented
28C23×C3×D5D412yes—
29C22×C3×AGL(1,𝔽5)—24no—
30C2×C3×C5×D4—22no—
31C2×C3×C5×D4C2×C411yescharacteristic
32C2×C3×C5×D4C2×C411yescharacteristic, complemented
33C2×C3×C22⋊D5—21no—
34C2×C3×C22⋊D5—22no—
35C2×C3×C22⋊D5D412yes—
36C22×C3×Dic5—21no—
37C3×#711—22no—
38C3×#711—22no—
39C3×D4×D5—22no—
40C3×D4×D5—22no—
41C2×C3×D20C2×C411yescharacteristic
42C2×C3×C4×D5C2311yescharacteristic
43C2×C3×Dic10C2×C411yescharacteristic, complemented
44C3×C22⋊AGL(1,𝔽5)—24no—
45C3×C22⋊AGL(1,𝔽5)—22no—
46C3×#704—22no—
47C3×C4⋊AGL(1,𝔽5)—22no—
48C3×C4×AGL(1,𝔽5)D414yescomplemented
49C3×#700—22no—
50C3×D5⋊C8—22no—
51C3×C42×C5C2×C411yescharacteristic, complemented
52C3×C22⋊Dic5—22no—
53C3×#685—22no—
54C3×#685—21no—
55C3×#683—21no—
56C3×C4×Dic5C2×C411yescharacteristic, complemented
57C3×#535—51no—
58C3×#518—51no—
59C3×#421—52no—
60C2×C3×#100—51no—
61C3×H(3,ℤ/4)—52no—
62C2×#711C22×C311yescharacteristic
63C2×D4×D5C22×C311yescharacteristic
64C2×C22⋊AGL(1,𝔽5)—22no—
65D4×AGL(1,𝔽5)—24no—
66#4130C22×C312yes—
67D5⋊M16C22×C312yes—
68C5×#127C3×C411yescharacteristic, complemented
69#4086C3×C411yescharacteristic, complemented
70#4085—22no—
71#4083—22no—
72#4082C3×C411yescharacteristic, complemented
73#4021C3×C411yescharacteristic, complemented
74#4018C22×C311yescharacteristic
75#4013—22no—
76#4008—24no—
77#1911—51noSylow
78C23×C3×C5—21no—
79C23×C3×C5C22⋊C412yes—
80C22×C3×D5—22no—
81C22×C3×D5—22no—
82C22×C3×D5—82no—
83C22×C3×D5C22⋊C412yes—
84C22×C3×D5C2×D411yescharacteristic
85C2×C3×AGL(1,𝔽5)—24no—
86C2×C3×AGL(1,𝔽5)—84no—
87C3×C5×D4—22no—
88C3×C5×D4—22no—
89C3×C5×D4—82no—
90C2×C3×C4×C5—21no—
91C2×C3×C4×C5C22×C411yescharacteristic
92C3×C22⋊D5—24no—
93C3×C22⋊D5—41no—
94C3×C22⋊D5—41no—
95C3×C22⋊D5—82no—
96C2×C3×Dic5—21no—
97C2×C3×Dic5—21no—
98C2×C3×Dic5—42no—
99C2×C3×Dic5C2×D411yescharacteristic
100C3×D20—22no—
101C3×C4×D5C2×D412yes—
102C3×C4×D5#4112yes—
103C3×Dic10—22no—
104C3×#170—42no—
105C2×C3×#41—51no—
106C22×C3×D4—51no—
107C2×C3×M16—52no—
108C3×#127—51no—
109C3×#127AGL(1,𝔽5)11yescharacteristic, complemented
110C3×#124—51no—
111C3×#124—51no—
112C3×#121—102no—
113C3×#120—102no—
114C3×C4×D4—104no—
115C3×#117—52no—
116C3×#117—51no—
117C2×C3×C22⋊C4—102no—
118C3×#104—104no—
119C3×#100—102no—
120C23×D5C3×D412yes—
121C22×AGL(1,𝔽5)—24no—
122C2×C5×D4—22no—
123C2×C5×D4C2×C3×C411yescharacteristic
124C2×C5×D4C2×C3×C411yescharacteristic, complemented
125C2×C22⋊D5—21no—
126C2×C22⋊D5—22no—
127C2×C22⋊D5C3×D412yes—
128C22×Dic5—21no—
129#711—22no—
130#711—22no—
131D4×D5—22no—
132D4×D5—22no—
133C2×D20C2×C3×C411yescharacteristic
134C2×C4×D5C23×C311yescharacteristic
135C2×Dic10C2×C3×C411yescharacteristic, complemented
136C22⋊AGL(1,𝔽5)—24no—
137C22⋊AGL(1,𝔽5)—22no—
138#704—22no—
139C4⋊AGL(1,𝔽5)—22no—
140C4×AGL(1,𝔽5)C3×D414yescomplemented
141#700—22no—
142D5⋊C8—22no—
143C42×C5C2×C3×C411yescharacteristic, complemented
144C22⋊Dic5—22no—
145#685—22no—
146#685—21no—
147#683—21no—
148C4×Dic5C2×C3×C411yescharacteristic, complemented
149#535—51no—
150#518—51no—
151#421—52no—
152C2×#100—51no—
153H(3,ℤ/4)—52no—
154C22×C3×C5—22no—
155C22×C3×C5—42no—
156C22×C3×C5—42no—
157C22×C3×C5—41no—
158C22×C3×C5C2×C22⋊C411yescharacteristic
159C2×C3×D5—22no—
160C2×C3×D5—41no—
161C2×C3×D5—44no—
162C2×C3×D5—42no—
163C2×C3×D5—82no—
164C2×C3×D5#12011yescharacteristic
165C2×C3×D5#12311yescharacteristic
166C3×AGL(1,𝔽5)—44no—
167C3×C4×C5—41no—
168C3×C4×C5C4×D412yes—
169C3×Dic5—41no—
170C3×Dic5—41no—
171C3×Dic5#12112yes—
172C24×C3—52no—
173C3×#41—102no—
174C3×#41—102no—
175C2×C3×Q8—51no—
176C2×C3×D4—22no—
177C2×C3×D4—52no—
178C2×C3×D4—51no—
179C2×C3×D4—102no—
180C2×C3×D4—102no—
181C2×C3×D4—101no—
182C2×C3×D4—102no—
183C2×C3×D4C2×AGL(1,𝔽5)11yescharacteristic
184C2×C3×D4C2×AGL(1,𝔽5)11yescharacteristic, complemented
185C22×C3×C4—51no—
186C22×C3×C4—101no—
187C22×C3×C4—104no—
188C3×M16—102no—
189C3×M16—102no—
190C2×C3×C8—102no—
191C3×C4⋊C4—101no—
192C3×C4⋊C4—102no—
193C3×C22⋊C4—101no—
194C3×C22⋊C4—102no—
195C3×C22⋊C4—102no—
196C3×C22⋊C4—104no—
197C3×C22⋊C4—102no—
198C3×C42—54no—
199C3×C42—51no—
200C3×C42C2×AGL(1,𝔽5)11yescharacteristic, complemented
201C23×C5—21no—
202C23×C5C3×C22⋊C412yes—
203C22×D5—22no—
204C22×D5—22no—
205C22×D5—82no—
206C22×D5C3×C22⋊C412yes—
207C22×D5C2×C3×D411yescharacteristic
208C2×AGL(1,𝔽5)—24no—
209C2×AGL(1,𝔽5)—84no—
210C5×D4—22no—
211C5×D4—22no—
212C5×D4—82no—
213C2×C4×C5—21no—
214C2×C4×C5C22×C3×C411yescharacteristic
215C22⋊D5—24no—
216C22⋊D5—41no—
217C22⋊D5—41no—
218C22⋊D5—82no—
219C2×Dic5—21no—
220C2×Dic5—21no—
221C2×Dic5—42no—
222C2×Dic5C2×C3×D411yescharacteristic
223D20—22no—
224C4×D5C2×C3×D412yes—
225C4×D5C3×#4112yes—
226Dic10—22no—
227#170—42no—
228C2×#41—51no—
229C22×D4—51no—
230C2×M16—52no—
231#127—51no—
232#127C3×AGL(1,𝔽5)11yescharacteristic, complemented
233#124—51no—
234#124—51no—
235#121—102no—
236#120—102no—
237C4×D4—104no—
238#117—52no—
239#117—51no—
240C2×C22⋊C4—102no—
241#104—104no—
242#100—102no—
243C2×C3×C5—21no—
244C2×C3×C5—42no—
245C2×C3×C5—81no—
246C2×C3×C5#38611yescharacteristic
247C3×D5—21no—
248C3×D5—21no—
249C3×D5—42no—
250C23×C3—21no—
251C23×C3—51no—
252C23×C3—52no—
253C23×C3—102no—
254C23×C3—102no—
255C23×C3—402no—
256C23×C3C22⋊AGL(1,𝔽5)12yes—
257C3×Q8—102no—
258C3×D4—22no—
259C3×D4—22no—
260C3×D4—82no—
261C3×D4—104no—
262C3×D4—102no—
263C3×D4—201no—
264C3×D4—201no—
265C3×D4—402no—
266C2×C3×C4—21no—
267C2×C3×C4—51no—
268C2×C3×C4—52no—
269C2×C3×C4—52no—
270C2×C3×C4—104no—
271C2×C3×C4—101no—
272C2×C3×C4—101no—
273C2×C3×C4—202no—
274C2×C3×C4—404no—
275C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
276C3×C8—202no—
277C22×C5—22no—
278C22×C5—42no—
279C22×C5—42no—
280C22×C5—41no—
281C22×C5C2×C3×C22⋊C411yescharacteristic
282C2×D5—22no—
283C2×D5—41no—
284C2×D5—44no—
285C2×D5—42no—
286C2×D5—82no—
287C2×D5C3×#12011yescharacteristic
288C2×D5C3×#12311yescharacteristic
289AGL(1,𝔽5)—44no—
290C4×C5—41no—
291C4×C5C3×C4×D412yes—
292Dic5—41no—
293Dic5—41no—
294Dic5C3×#12112yes—
295C24—52no—
296#41—102no—
297#41—102no—
298C2×Q8—51no—
299C2×D4—22no—
300C2×D4—52no—
301C2×D4—51no—
302C2×D4—102no—
303C2×D4—102no—
304C2×D4—101no—
305C2×D4—102no—
306C2×D4C2×C3×AGL(1,𝔽5)11yescharacteristic
307C2×D4C2×C3×AGL(1,𝔽5)11yescharacteristic, complemented
308C22×C4—51no—
309C22×C4—101no—
310C22×C4—104no—
311M16—102no—
312M16—102no—
313C2×C8—102no—
314C4⋊C4—101no—
315C4⋊C4—102no—
316C22⋊C4—101no—
317C22⋊C4—102no—
318C22⋊C4—102no—
319C22⋊C4—104no—
320C22⋊C4—102no—
321C42—54no—
322C42—51no—
323C42C2×C3×AGL(1,𝔽5)11yescharacteristic, complemented
324C3×C5#191111yescharacteristic, complemented, Hall
325C22×C3—22no—
326C22×C3—42no—
327C22×C3—42no—
328C22×C3—41no—
329C22×C3—51no—
330C22×C3—51no—
331C22×C3—102no—
332C22×C3—201no—
333C22×C3—202no—
334C22×C3—204no—
335C22×C3—402no—
336C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
337C3×C4—41no—
338C3×C4—52no—
339C3×C4—201no—
340C3×C4—204no—
341C3×C4—201no—
342C3×C4D4×AGL(1,𝔽5)12yes—
343C2×C5—21no—
344C2×C5—42no—
345C2×C5—81no—
346C2×C5C3×#38611yescharacteristic
347D5—21no—
348D5—21no—
349D5—42no—
350C23—21no—
351C23—51no—
352C23—52no—
353C23—102no—
354C23—102no—
355C23—402no—
356C23C3×C22⋊AGL(1,𝔽5)12yes—
357Q8—102no—
358D4—22no—
359D4—22no—
360D4—82no—
361D4—104no—
362D4—102no—
363D4—201no—
364D4—201no—
365D4—402no—
366C2×C4—21no—
367C2×C4—51no—
368C2×C4—52no—
369C2×C4—52no—
370C2×C4—104no—
371C2×C4—101no—
372C2×C4—101no—
373C2×C4—202no—
374C2×C4—404no—
375C2×C4C22×C3×AGL(1,𝔽5)11yescharacteristic
376C8—202no—
377C2×C3—21no—
378C2×C3—42no—
379C2×C3—81no—
380C2×C3—101no—
381C2×C3—101no—
382C2×C3—202no—
383C2×C3#6577911yescharacteristic
384C5C3×#191111yescharacteristic, complemented, Sylow
385C22—22no—
386C22—42no—
387C22—42no—
388C22—41no—
389C22—51no—
390C22—51no—
391C22—102no—
392C22—201no—
393C22—202no—
394C22—204no—
395C22—402no—
396C22C2×C3×C22⋊AGL(1,𝔽5)11yescharacteristic
397C4—41no—
398C4—52no—
399C4—201no—
400C4—204no—
401C4—201no—
402C4C3×D4×AGL(1,𝔽5)12yes—
403C3#1060461511yescharacteristic, direct product factor, Sylow
404C2—21no—
405C2—42no—
406C2—81no—
407C2—101no—
408C2—101no—
409C2—202no—
410C2C3×#6577911yescharacteristic
subgroupG/H
centerC2×C3#65779
commutatorC2×C4×C5C22×C3×C4
FrattiniC2×C4C22×C3×AGL(1,𝔽5)
FittingC3×C5×#127C4
socleC2×C3×C5#386
prime radicalC3×#66049C2
cosocleC2×C4×D5C23×C3
hypercenterC3×#127AGL(1,𝔽5)
hypocenterC5C3×#1911
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C3×#127

Complements

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

NH ≅ G/Ntype
1. C3×#66258C2semidirect
3. C3×D5⋊#100C2semidirect
4. C3×#65769C2semidirect
5. C3×#64863C2semidirect
6. #10604615C3direct (N × H)
13. C3×C5×#127C4semidirect
14. C3×#4086C4semidirect
17. C3×#4082C4semidirect
18. C3×#4021C4semidirect
23. #66258C2×C3semidirect
25. D5⋊#100C2×C3semidirect
26. #65769C2×C3semidirect
27. #64863C2×C3semidirect
32. C2×C3×C5×D4C2×C4semidirect
43. C2×C3×Dic10C2×C4semidirect
48. C3×C4×AGL(1,𝔽5)D4semidirect
51. C3×C42×C5C2×C4semidirect
56. C3×C4×Dic5C2×C4semidirect
68. C5×#127C3×C4semidirect
69. #4086C3×C4semidirect
72. #4082C3×C4semidirect
73. #4021C3×C4semidirect
109. C3×#127AGL(1,𝔽5)semidirect
124. C2×C5×D4C2×C3×C4semidirect
135. C2×Dic10C2×C3×C4semidirect
140. C4×AGL(1,𝔽5)C3×D4semidirect
143. C42×C5C2×C3×C4semidirect
148. C4×Dic5C2×C3×C4semidirect
184. C2×C3×D4C2×AGL(1,𝔽5)semidirect
200. C3×C42C2×AGL(1,𝔽5)semidirect
232. #127C3×AGL(1,𝔽5)semidirect
307. C2×D4C2×C3×AGL(1,𝔽5)semidirect
323. C42C2×C3×AGL(1,𝔽5)semidirect
324. C3×C5#1911semidirect
384. C5C3×#1911semidirect
403. C3#10604615direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#662582
C3×#660492
C3×D5⋊#1002
C3×#657692
C3×#648632
#106046153
C3×#19115