Finite Group Databasebeta

← C3×#10604619 #14302839

Subgroups — C3×#10604619 #14302839

Order 1920.

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

normal subgroups only

This group has 688 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×#66256C211yesmaximal, characteristic, complemented
2C3×#66049C211yesmaximal, characteristic
3C3×#65771C211yesmaximal, characteristic, complemented
4C3×#65767C212yesmaximal, complemented
5C3×#64863C212yesmaximal, complemented
6#10604619C311yesmaximal, characteristic, direct product factor, Hall
7C2×C3×#711C2211yescharacteristic
8C2×C3×D4×D5C2211yescharacteristic
9C2×C3×#704—22no—
10C3×#4131—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×#4011—22no—
21C3×#4007—24no—
22C3×#1910—51nomaximal, Hall
23#66256C2×C311yescharacteristic, complemented
24#66049C2×C311yescharacteristic
25#65771C2×C311yescharacteristic, complemented
26#65767C2×C312yescomplemented
27#64863C2×C312yescomplemented
28C23×C3×D5—21no—
29C2×C3×C5×D4—22no—
30C2×C3×C5×D4C2×C411yescharacteristic
31C2×C3×C5×D4C2×C411yescharacteristic
32C2×C3×C22⋊D5—22no—
33C2×C3×C22⋊D5—21no—
34C2×C3×C22⋊D5D412yes—
35C22×C3×Dic5D412yes—
36C3×#711—22no—
37C3×#711—22no—
38C3×D4×D5—22no—
39C3×D4×D5—22no—
40C2×C3×D20C2×C411yescharacteristic
41C2×C3×C4×D5C2311yescharacteristic
42C2×C3×Dic10C2×C411yescharacteristic
43C3×C22⋊AGL(1,𝔽5)—22no—
44C3×#704—24no—
45C3×#704—22no—
46C2×C3×#170—24no—
47C3×C4⋊AGL(1,𝔽5)—22no—
48C3×C4×AGL(1,𝔽5)—22no—
49C3×#700—22no—
50C3×D5⋊C8D414yescomplemented
51C3×C42×C5C2×C411yescharacteristic
52C3×C22⋊Dic5—22no—
53C3×#685—21no—
54C3×#685—22no—
55C3×#683—21no—
56C3×C4×Dic5C2×C411yescharacteristic
57C3×#568—51no—
58C3×#535—51no—
59C3×#418—52no—
60C3×#410—51no—
61C3×H(3,ℤ/4)—52no—
62C2×#711C22×C311yescharacteristic
63C2×D4×D5C22×C311yescharacteristic
64C2×#704—22no—
65#4131—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#4011—22no—
76#4007—24no—
77#1910—51noSylow
78C23×C3×C5—21no—
79C23×C3×C5C22⋊C412yes—
80C22×C3×D5—21no—
81C22×C3×D5—42no—
82C22×C3×D5—82no—
83C22×C3×D5C2×D411yescharacteristic
84C2×C3×AGL(1,𝔽5)—42no—
85C3×C5×D4—22no—
86C3×C5×D4—22no—
87C3×C5×D4—82no—
88C2×C3×C4×C5—21no—
89C2×C3×C4×C5C22×C411yescharacteristic
90C3×C22⋊D5—24no—
91C3×C22⋊D5—41no—
92C3×C22⋊D5—41no—
93C3×C22⋊D5—82no—
94C2×C3×Dic5—22no—
95C2×C3×Dic5—22no—
96C2×C3×Dic5—21no—
97C2×C3×Dic5C22⋊C412yes—
98C2×C3×Dic5C2×D411yescharacteristic
99C3×D20—22no—
100C3×C4×D5C2×D412yes—
101C3×C4×D5#4112yes—
102C3×Dic10—22no—
103C3×#170—24no—
104C2×C3×#41—51no—
105C22×C3×D4—51no—
106C3×#131—104no—
107C2×C3×M16—52no—
108C2×C3×M16—102no—
109C3×#127—51no—
110C3×#127AGL(1,𝔽5)11yescharacteristic, complemented
111C3×#124—51no—
112C3×#124—51no—
113C3×#121—102no—
114C3×#120—102no—
115C3×#117—51no—
116C3×#117—52no—
117C3×#102—104no—
118C3×#99—102no—
119C23×D5—21no—
120C2×C5×D4—22no—
121C2×C5×D4C2×C3×C411yescharacteristic
122C2×C5×D4C2×C3×C411yescharacteristic
123C2×C22⋊D5—22no—
124C2×C22⋊D5—21no—
125C2×C22⋊D5C3×D412yes—
126C22×Dic5C3×D412yes—
127#711—22no—
128#711—22no—
129D4×D5—22no—
130D4×D5—22no—
131C2×D20C2×C3×C411yescharacteristic
132C2×C4×D5C23×C311yescharacteristic
133C2×Dic10C2×C3×C411yescharacteristic
134C22⋊AGL(1,𝔽5)—22no—
135#704—24no—
136#704—22no—
137C2×#170—24no—
138C4⋊AGL(1,𝔽5)—22no—
139C4×AGL(1,𝔽5)—22no—
140#700—22no—
141D5⋊C8C3×D414yescomplemented
142C42×C5C2×C3×C411yescharacteristic
143C22⋊Dic5—22no—
144#685—21no—
145#685—22no—
146#683—21no—
147C4×Dic5C2×C3×C411yescharacteristic
148#568—51no—
149#535—51no—
150#418—52no—
151#410—51no—
152H(3,ℤ/4)—52no—
153C22×C3×C5—22no—
154C22×C3×C5—42no—
155C22×C3×C5—42no—
156C22×C3×C5—41no—
157C22×C3×C5C2×C22⋊C411yescharacteristic
158C2×C3×D5—41no—
159C2×C3×D5—42no—
160C2×C3×D5—41no—
161C2×C3×D5—84no—
162C2×C3×D5#12112yes—
163C3×AGL(1,𝔽5)—82no—
164C3×C4×C5—41no—
165C3×C4×C5C4×D412yes—
166C3×Dic5—22no—
167C3×Dic5—41no—
168C3×Dic5#12011yescharacteristic
169C3×Dic5#12311yescharacteristic
170C24×C3—101no—
171C3×#41—102no—
172C3×#41—102no—
173C2×C3×Q8—51no—
174C2×C3×D4—22no—
175C2×C3×D4—52no—
176C2×C3×D4—51no—
177C2×C3×D4—102no—
178C2×C3×D4—102no—
179C2×C3×D4—101no—
180C2×C3×D4—102no—
181C2×C3×D4C2×AGL(1,𝔽5)11yescharacteristic
182C2×C3×D4C2×AGL(1,𝔽5)11yescharacteristic
183C22×C3×C4—51no—
184C22×C3×C4—52no—
185C3×M16—102no—
186C3×M16—102no—
187C3×M16—104no—
188C2×C3×C8—54no—
189C2×C3×C8—104no—
190C3×C4⋊C4—101no—
191C3×C4⋊C4—102no—
192C3×C22⋊C4—102no—
193C3×C22⋊C4—101no—
194C3×C22⋊C4—102no—
195C3×C22⋊C4—102no—
196C3×C42—51no—
197C3×C42—102no—
198C3×C42C2×AGL(1,𝔽5)11yescharacteristic
199C23×C5—21no—
200C23×C5C3×C22⋊C412yes—
201C22×D5—21no—
202C22×D5—42no—
203C22×D5—82no—
204C22×D5C2×C3×D411yescharacteristic
205C2×AGL(1,𝔽5)—42no—
206C5×D4—22no—
207C5×D4—22no—
208C5×D4—82no—
209C2×C4×C5—21no—
210C2×C4×C5C22×C3×C411yescharacteristic
211C22⋊D5—24no—
212C22⋊D5—41no—
213C22⋊D5—41no—
214C22⋊D5—82no—
215C2×Dic5—22no—
216C2×Dic5—22no—
217C2×Dic5—21no—
218C2×Dic5C3×C22⋊C412yes—
219C2×Dic5C2×C3×D411yescharacteristic
220D20—22no—
221C4×D5C2×C3×D412yes—
222C4×D5C3×#4112yes—
223Dic10—22no—
224#170—24no—
225C2×#41—51no—
226C22×D4—51no—
227#131—104no—
228C2×M16—52no—
229C2×M16—102no—
230#127—51no—
231#127C3×AGL(1,𝔽5)11yescharacteristic, complemented
232#124—51no—
233#124—51no—
234#121—102no—
235#120—102no—
236#117—51no—
237#117—52no—
238#102—104no—
239#99—102no—
240C2×C3×C5—21no—
241C2×C3×C5—42no—
242C2×C3×C5—81no—
243C2×C3×C5#38611yescharacteristic
244C3×D5—22no—
245C3×D5—81no—
246C23×C3—21no—
247C23×C3—51no—
248C23×C3—101no—
249C23×C3—202no—
250C23×C3—402no—
251C23×C3C22⋊AGL(1,𝔽5)12yes—
252C3×Q8—102no—
253C3×D4—22no—
254C3×D4—22no—
255C3×D4—82no—
256C3×D4—104no—
257C3×D4—102no—
258C3×D4—201no—
259C3×D4—201no—
260C3×D4—402no—
261C2×C3×C4—21no—
262C2×C3×C4—51no—
263C2×C3×C4—52no—
264C2×C3×C4—52no—
265C2×C3×C4—52no—
266C2×C3×C4—102no—
267C2×C3×C4—102no—
268C2×C3×C4—101no—
269C2×C3×C4—202no—
270C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
271C3×C8—104no—
272C22×C5—22no—
273C22×C5—42no—
274C22×C5—42no—
275C22×C5—41no—
276C22×C5C2×C3×C22⋊C411yescharacteristic
277C2×D5—41no—
278C2×D5—42no—
279C2×D5—41no—
280C2×D5—84no—
281C2×D5C3×#12112yes—
282AGL(1,𝔽5)—82no—
283C4×C5—41no—
284C4×C5C3×C4×D412yes—
285Dic5—22no—
286Dic5—41no—
287Dic5C3×#12011yescharacteristic
288Dic5C3×#12311yescharacteristic
289C24—101no—
290#41—102no—
291#41—102no—
292C2×Q8—51no—
293C2×D4—22no—
294C2×D4—52no—
295C2×D4—51no—
296C2×D4—102no—
297C2×D4—102no—
298C2×D4—101no—
299C2×D4—102no—
300C2×D4C2×C3×AGL(1,𝔽5)11yescharacteristic
301C2×D4C2×C3×AGL(1,𝔽5)11yescharacteristic
302C22×C4—51no—
303C22×C4—52no—
304M16—102no—
305M16—102no—
306M16—104no—
307C2×C8—54no—
308C2×C8—104no—
309C4⋊C4—101no—
310C4⋊C4—102no—
311C22⋊C4—102no—
312C22⋊C4—101no—
313C22⋊C4—102no—
314C22⋊C4—102no—
315C42—51no—
316C42—102no—
317C42C2×C3×AGL(1,𝔽5)11yescharacteristic
318C3×C5#191011yescharacteristic, complemented, Hall
319C22×C3—22no—
320C22×C3—42no—
321C22×C3—42no—
322C22×C3—41no—
323C22×C3—52no—
324C22×C3—201no—
325C22×C3—202no—
326C22×C3—201no—
327C22×C3—404no—
328C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
329C3×C4—41no—
330C3×C4—51no—
331C3×C4—51no—
332C3×C4—102no—
333C3×C4—201no—
334C3×C4—402no—
335C3×C4D4×AGL(1,𝔽5)12yes—
336C2×C5—21no—
337C2×C5—42no—
338C2×C5—81no—
339C2×C5C3×#38611yescharacteristic
340D5—22no—
341D5—81no—
342C23—21no—
343C23—51no—
344C23—101no—
345C23—202no—
346C23—402no—
347C23C3×C22⋊AGL(1,𝔽5)12yes—
348Q8—102no—
349D4—22no—
350D4—22no—
351D4—82no—
352D4—104no—
353D4—102no—
354D4—201no—
355D4—201no—
356D4—402no—
357C2×C4—21no—
358C2×C4—51no—
359C2×C4—52no—
360C2×C4—52no—
361C2×C4—52no—
362C2×C4—102no—
363C2×C4—102no—
364C2×C4—101no—
365C2×C4—202no—
366C2×C4C22×C3×AGL(1,𝔽5)11yescharacteristic
367C8—104no—
368C2×C3—21no—
369C2×C3—42no—
370C2×C3—81no—
371C2×C3—102no—
372C2×C3—401no—
373C2×C3#6577911yescharacteristic
374C5C3×#191011yescharacteristic, complemented, Sylow
375C22—22no—
376C22—42no—
377C22—42no—
378C22—41no—
379C22—52no—
380C22—201no—
381C22—202no—
382C22—201no—
383C22—404no—
384C22C2×C3×C22⋊AGL(1,𝔽5)11yescharacteristic
385C4—41no—
386C4—51no—
387C4—51no—
388C4—102no—
389C4—201no—
390C4—402no—
391C4C3×D4×AGL(1,𝔽5)12yes—
392C3#1060461911yescharacteristic, direct product factor, Sylow
393C2—21no—
394C2—42no—
395C2—81no—
396C2—102no—
397C2—401no—
398C2C3×#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×#1910
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×#66256C2semidirect
3. C3×#65771C2semidirect
4. C3×#65767C2semidirect
5. C3×#64863C2semidirect
6. #10604619C3direct (N × H)
13. C3×C5×#127C4semidirect
14. C3×#4086C4semidirect
17. C3×#4082C4semidirect
18. C3×#4021C4semidirect
23. #66256C2×C3semidirect
25. #65771C2×C3semidirect
26. #65767C2×C3semidirect
27. #64863C2×C3semidirect
50. C3×D5⋊C8D4semidirect
68. C5×#127C3×C4semidirect
69. #4086C3×C4semidirect
72. #4082C3×C4semidirect
73. #4021C3×C4semidirect
110. C3×#127AGL(1,𝔽5)semidirect
141. D5⋊C8C3×D4semidirect
231. #127C3×AGL(1,𝔽5)semidirect
318. C3×C5#1910semidirect
374. C5C3×#1910semidirect
392. C3#10604619direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#662562
C3×#660492
C3×#657712
C3×#657672
C3×#648632
#106046193
C3×#19105