Finite Group Databasebeta

← C3×#10604912 #14283268

Subgroups — C3×#10604912 #14283268

Order 1920.

652 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 652 conjugacy classes of subgroups — too many to draw. Only its 120 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
1C2×C3×#4084C211yesmaximal, characteristic
2C3×#65764C211yesmaximal, characteristic, complemented
3C3×#65755C211yesmaximal, characteristic, complemented
4C3×#64920C214yesmaximal, complemented
5#10604912C311yesmaximal, characteristic, direct product factor, Hall
6C22×C3×C5×D4C411yescharacteristic, complemented
7C22×C3×C22⋊D5—21no—
8C22×C3×C4×D5C2211yescharacteristic
9C2×C3×C22⋊AGL(1,𝔽5)—21no—
10C2×C3×#704—21no—
11C2×C3×C22⋊Dic5—21no—
12C3×#4084—21no—
13C3×#4084—21no—
14C3×#4084C412yescomplemented
15C3×#4084C2212yes—
16C2×C3×C4⋊Dic5C411yescharacteristic, complemented
17C3×(C2×AGL(1,𝔽5))⋊C4—21no—
18C3×(C2×AGL(1,𝔽5))⋊C4C2212yescomplemented
19C3×#4003—21no—
20C3×#4003C2212yescomplemented
21C3×#1510—51nomaximal, Hall
22C2×#4084C2×C311yescharacteristic
23#65764C2×C311yescharacteristic, complemented
24#65755C2×C311yescharacteristic, complemented
25#64920C2×C314yescomplemented
26C24×C3×C5—21no—
27C23×C3×D5D411yescharacteristic
28C22×C3×AGL(1,𝔽5)—22no—
29C2×C3×C5×D4—21no—
30C2×C3×C5×D4—42no—
31C2×C3×C5×D4C2×C412yes—
32C22×C3×C4×C5C2×C411yescharacteristic
33C2×C3×C22⋊D5—22no—
34C2×C3×C22⋊D5—22no—
35C2×C3×C22⋊D5—44no—
36C22×C3×Dic5—21no—
37C22×C3×Dic5D411yescharacteristic
38C2×C3×C4×D5—42no—
39C2×C3×C4×D5C2×C411yescharacteristic
40C2×C3×C4×D5D411yescharacteristic
41C2×C3×C4×D5D411yescharacteristic
42C2×C3×C4×D5C2311yescharacteristic
43C3×C22⋊AGL(1,𝔽5)—42no—
44C3×#704—42no—
45C2×C3×#170—22no—
46C3×C22⋊Dic5—22no—
47C3×C22⋊Dic5—22no—
48C3×C4⋊Dic5—21no—
49C3×C4⋊Dic5C2×C412yes—
50C2×C3×#121—51no—
51C3×#407—51no—
52C3×#381—51no—
53C3×#327—54no—
54C22×C5×D4C3×C411yescharacteristic, complemented
55C22×C22⋊D5—21no—
56C22×C4×D5C22×C311yescharacteristic
57C2×C22⋊AGL(1,𝔽5)—21no—
58C2×#704—21no—
59C2×C22⋊Dic5—21no—
60#4084—21no—
61#4084—21no—
62#4084C3×C412yescomplemented
63#4084C22×C312yes—
64C2×C4⋊Dic5C3×C411yescharacteristic, complemented
65(C2×AGL(1,𝔽5))⋊C4—21no—
66(C2×AGL(1,𝔽5))⋊C4C22×C312yescomplemented
67#4003—21no—
68#4003C22×C312yescomplemented
69#1510—51noSylow
70C23×C3×C5—22no—
71C23×C3×C5—42no—
72C23×C3×C5—42no—
73C23×C3×C5—42no—
74C23×C3×C5C22⋊C411yescharacteristic
75C22×C3×D5—22no—
76C22×C3×D5—22no—
77C22×C3×D5—41no—
78C22×C3×D5C22⋊C411yescharacteristic
79C22×C3×D5C2×D411yescharacteristic
80C2×C3×AGL(1,𝔽5)—42no—
81C2×C3×AGL(1,𝔽5)—42no—
82C2×C3×AGL(1,𝔽5)—42no—
83C3×C5×D4—42no—
84C3×C5×D4—42no—
85C2×C3×C4×C5—22no—
86C2×C3×C4×C5C22⋊C411yescharacteristic
87C2×C3×C4×C5C22×C411yescharacteristic
88C3×C22⋊D5—44no—
89C3×C22⋊D5—44no—
90C2×C3×Dic5—22no—
91C2×C3×Dic5—22no—
92C2×C3×Dic5—42no—
93C2×C3×Dic5C22⋊C411yescharacteristic
94C2×C3×Dic5C2×D411yescharacteristic
95C3×C4×D5—41no—
96C3×C4×D5—41no—
97C3×C4×D5—41no—
98C3×C4×D5—41no—
99C3×#170—42no—
100C22×C3×D4—101no—
101C22×C3×D4AGL(1,𝔽5)11yescharacteristic, complemented
102C23×C3×C4—51no—
103C2×C3×M16—101no—
104C3×#121—52no—
105C3×#121—52no—
106C3×#121—101no—
107C3×#121—101no—
108C2×C3×C4⋊C4—51no—
109C2×C3×C22⋊C4—101no—
110C2×C3×C22⋊C4—101no—
111C3×C22⋊C8—52no—
112C3×C22⋊C8—101no—
113C3×#95—52no—
114C3×#95—101no—
115C24×C5—21no—
116C23×D5C3×D411yescharacteristic
117C22×AGL(1,𝔽5)—22no—
118C2×C5×D4—21no—
119C2×C5×D4—42no—
120C2×C5×D4C2×C3×C412yes—
121C22×C4×C5C2×C3×C411yescharacteristic
122C2×C22⋊D5—22no—
123C2×C22⋊D5—22no—
124C2×C22⋊D5—44no—
125C22×Dic5—21no—
126C22×Dic5C3×D411yescharacteristic
127C2×C4×D5—42no—
128C2×C4×D5C2×C3×C411yescharacteristic
129C2×C4×D5C3×D411yescharacteristic
130C2×C4×D5C3×D411yescharacteristic
131C2×C4×D5C23×C311yescharacteristic
132C22⋊AGL(1,𝔽5)—42no—
133#704—42no—
134C2×#170—22no—
135C22⋊Dic5—22no—
136C22⋊Dic5—22no—
137C4⋊Dic5—21no—
138C4⋊Dic5C2×C3×C412yes—
139C2×#121—51no—
140#407—51no—
141#381—51no—
142#327—54no—
143C22×C3×C5—21no—
144C22×C3×C5—21no—
145C22×C3×C5—42no—
146C22×C3×C5—42no—
147C22×C3×C5—42no—
148C22×C3×C5—84no—
149C22×C3×C5#9912yes—
150C22×C3×C5C2×C22⋊C411yescharacteristic
151C2×C3×D5—21no—
152C2×C3×D5—21no—
153C2×C3×D5—21no—
154C2×C3×D5—21no—
155C2×C3×D5—21no—
156C2×C3×D5—44no—
157C2×C3×D5AGL(1,ℤ/8)12yes—
158C3×AGL(1,𝔽5)—82no—
159C3×C4×C5—21no—
160C3×C4×C5—21no—
161C3×Dic5—21no—
162C3×Dic5—21no—
163C3×Dic5—42no—
164C24×C3—21no—
165C24×C3—51no—
166C2×C3×D4—21no—
167C2×C3×D4—42no—
168C2×C3×D4—102no—
169C2×C3×D4—102no—
170C2×C3×D4—204no—
171C2×C3×D4C2×AGL(1,𝔽5)12yes—
172C22×C3×C4—51no—
173C22×C3×C4—51no—
174C22×C3×C4—51no—
175C22×C3×C4—51no—
176C22×C3×C4—51no—
177C22×C3×C4—101no—
178C22×C3×C4—102no—
179C22×C3×C4—202no—
180C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
181C3×M16—202no—
182C2×C3×C8—102no—
183C3×C4⋊C4—52no—
184C3×C4⋊C4—101no—
185C3×C22⋊C4—102no—
186C3×C22⋊C4—102no—
187C3×C22⋊C4—202no—
188C23×C5—22no—
189C23×C5—42no—
190C23×C5—42no—
191C23×C5—42no—
192C23×C5C3×C22⋊C411yescharacteristic
193C22×D5—22no—
194C22×D5—22no—
195C22×D5—41no—
196C22×D5C3×C22⋊C411yescharacteristic
197C22×D5C2×C3×D411yescharacteristic
198C2×AGL(1,𝔽5)—42no—
199C2×AGL(1,𝔽5)—42no—
200C2×AGL(1,𝔽5)—42no—
201C5×D4—42no—
202C5×D4—42no—
203C2×C4×C5—22no—
204C2×C4×C5C3×C22⋊C411yescharacteristic
205C2×C4×C5C22×C3×C411yescharacteristic
206C22⋊D5—44no—
207C22⋊D5—44no—
208C2×Dic5—22no—
209C2×Dic5—22no—
210C2×Dic5—42no—
211C2×Dic5C3×C22⋊C411yescharacteristic
212C2×Dic5C2×C3×D411yescharacteristic
213C4×D5—41no—
214C4×D5—41no—
215C4×D5—41no—
216C4×D5—41no—
217#170—42no—
218C22×D4—101no—
219C22×D4C3×AGL(1,𝔽5)11yescharacteristic, complemented
220C23×C4—51no—
221C2×M16—101no—
222#121—52no—
223#121—52no—
224#121—101no—
225#121—101no—
226C2×C4⋊C4—51no—
227C2×C22⋊C4—101no—
228C2×C22⋊C4—101no—
229C22⋊C8—52no—
230C22⋊C8—101no—
231#95—52no—
232#95—101no—
233C2×C3×C5—22no—
234C2×C3×C5—82no—
235C2×C3×C5C2×#9911yescharacteristic
236C2×C3×C5#41811yescharacteristic
237C2×C3×C5#42111yescharacteristic
238C3×D5—22no—
239C3×D5—41no—
240C23×C3—22no—
241C23×C3—42no—
242C23×C3—42no—
243C23×C3—42no—
244C23×C3—51no—
245C23×C3—51no—
246C23×C3—102no—
247C23×C3—102no—
248C23×C3—201no—
249C23×C3C22⋊AGL(1,𝔽5)11yescharacteristic
250C3×D4—42no—
251C3×D4—42no—
252C3×D4—204no—
253C3×D4—204no—
254C2×C3×C4—22no—
255C2×C3×C4—51no—
256C2×C3×C4—51no—
257C2×C3×C4—102no—
258C2×C3×C4—102no—
259C2×C3×C4—201no—
260C2×C3×C4—201no—
261C2×C3×C4—201no—
262C2×C3×C4—201no—
263C2×C3×C4—202no—
264C2×C3×C4—202no—
265C2×C3×C4—202no—
266C2×C3×C4—202no—
267C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
268C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
269C3×C8—202no—
270C22×C5—21no—
271C22×C5—21no—
272C22×C5—42no—
273C22×C5—42no—
274C22×C5—42no—
275C22×C5—84no—
276C22×C5C3×#9912yes—
277C22×C5C2×C3×C22⋊C411yescharacteristic
278C2×D5—21no—
279C2×D5—21no—
280C2×D5—21no—
281C2×D5—21no—
282C2×D5—21no—
283C2×D5—44no—
284C2×D5C3×AGL(1,ℤ/8)12yes—
285AGL(1,𝔽5)—82no—
286C4×C5—21no—
287C4×C5—21no—
288Dic5—21no—
289Dic5—21no—
290Dic5—42no—
291C24—21no—
292C24—51no—
293C2×D4—21no—
294C2×D4—42no—
295C2×D4—102no—
296C2×D4—102no—
297C2×D4—204no—
298C2×D4C2×C3×AGL(1,𝔽5)12yes—
299C22×C4—51no—
300C22×C4—51no—
301C22×C4—51no—
302C22×C4—51no—
303C22×C4—51no—
304C22×C4—101no—
305C22×C4—102no—
306C22×C4—202no—
307C22×C4C2×C3×AGL(1,𝔽5)11yescharacteristic
308M16—202no—
309C2×C8—102no—
310C4⋊C4—52no—
311C4⋊C4—101no—
312C22⋊C4—102no—
313C22⋊C4—102no—
314C22⋊C4—202no—
315C3×C5#151011yescharacteristic, complemented, Hall
316C22×C3—21no—
317C22×C3—21no—
318C22×C3—42no—
319C22×C3—42no—
320C22×C3—42no—
321C22×C3—52no—
322C22×C3—84no—
323C22×C3—101no—
324C22×C3—101no—
325C22×C3—101no—
326C22×C3—101no—
327C22×C3—101no—
328C22×C3—204no—
329C22×C3#401112yes—
330C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
331C3×C4—21no—
332C3×C4—21no—
333C3×C4—101no—
334C3×C4—101no—
335C3×C4—202no—
336C3×C4—402no—
337C2×C5—22no—
338C2×C5—82no—
339C2×C5C2×C3×#9911yescharacteristic
340C2×C5C3×#41811yescharacteristic
341C2×C5C3×#42111yescharacteristic
342D5—22no—
343D5—41no—
344C23—22no—
345C23—42no—
346C23—42no—
347C23—42no—
348C23—51no—
349C23—51no—
350C23—102no—
351C23—102no—
352C23—201no—
353C23C3×C22⋊AGL(1,𝔽5)11yescharacteristic
354D4—42no—
355D4—42no—
356D4—204no—
357D4—204no—
358C2×C4—22no—
359C2×C4—51no—
360C2×C4—51no—
361C2×C4—102no—
362C2×C4—102no—
363C2×C4—201no—
364C2×C4—201no—
365C2×C4—201no—
366C2×C4—201no—
367C2×C4—202no—
368C2×C4—202no—
369C2×C4—202no—
370C2×C4—202no—
371C2×C4C3×C22⋊AGL(1,𝔽5)11yescharacteristic
372C2×C4C22×C3×AGL(1,𝔽5)11yescharacteristic
373C8—202no—
374C2×C3—22no—
375C2×C3—82no—
376C2×C3—102no—
377C2×C3—201no—
378C2×C3#6576711yescharacteristic
379C2×C3#6576911yescharacteristic
380C2×C3C2×#401111yescharacteristic
381C5C3×#151011yescharacteristic, complemented, Sylow
382C22—21no—
383C22—21no—
384C22—42no—
385C22—42no—
386C22—42no—
387C22—52no—
388C22—84no—
389C22—101no—
390C22—101no—
391C22—101no—
392C22—101no—
393C22—101no—
394C22—204no—
395C22C3×#401112yes—
396C22C2×C3×C22⋊AGL(1,𝔽5)11yescharacteristic
397C4—21no—
398C4—21no—
399C4—101no—
400C4—101no—
401C4—202no—
402C4—402no—
403C3#1060491211yescharacteristic, direct product factor, Sylow
404C2—22no—
405C2—82no—
406C2—102no—
407C2—201no—
408C2C3×#6576711yescharacteristic
409C2C3×#6576911yescharacteristic
410C2C2×C3×#401111yescharacteristic
subgroupG/H
centerC22×C3C2×C22⋊AGL(1,𝔽5)
commutatorC2×C4×C5C22×C3×C4
FrattiniC2×C4C22×C3×AGL(1,𝔽5)
FittingC22×C3×C5×D4C4
socleC22×C3×C5C2×C22⋊C4
prime radicalC2×C3×#4084C2
cosocleC2×C4×D5C23×C3
hypercenterC22×C3×D4AGL(1,𝔽5)
hypocenterC5C3×#1510
LCSG ▷ C2×C4×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C22×C3×D4

Complements

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

NH ≅ G/Ntype
2. C3×#65764C2semidirect
3. C3×#65755C2semidirect
4. C3×#64920C2semidirect
5. #10604912C3direct (N × H)
6. C22×C3×C5×D4C4semidirect
14. C3×#4084C4semidirect
16. C2×C3×C4⋊Dic5C4semidirect
18. C3×(C2×AGL(1,𝔽5))⋊C4C22semidirect
20. C3×#4003C22semidirect
23. #65764C2×C3semidirect
24. #65755C2×C3semidirect
25. #64920C2×C3semidirect
54. C22×C5×D4C3×C4semidirect
62. #4084C3×C4semidirect
64. C2×C4⋊Dic5C3×C4semidirect
66. (C2×AGL(1,𝔽5))⋊C4C22×C3semidirect
68. #4003C22×C3semidirect
101. C22×C3×D4AGL(1,𝔽5)semidirect
219. C22×D4C3×AGL(1,𝔽5)semidirect
315. C3×C5#1510semidirect
381. C5C3×#1510semidirect
403. C3#10604912direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#40842
C3×#657642
C3×#657552
C3×#649202
#106049123
C3×#15105