Finite Group Databasebeta

← C3×#10604880 #14280963

Subgroups — C3×#10604880 #14280963

Order 1920.

880 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 880 conjugacy classes of subgroups — too many to draw. Only its 140 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
1C22×C3×#685C211yesmaximal, characteristic
2C3×#65798C212yesmaximal, complemented
3C3×#64915C214yesmaximal, complemented
4#10604880C311yesmaximal, characteristic, direct product factor, Hall
5C24×C3×D5C411yescharacteristic
6C23×C3×C4×C5C411yescharacteristic
7C23×C3×Dic5C2211yescharacteristic
8C2×C3×#704—22no—
9C2×C3×#685—22no—
10C2×C3×#685—22no—
11C2×C3×#685C412yes—
12C2×C3×#685C2212yes—
13C3×C22⋊#170—22no—
14C3×C22⋊#170C2214yescomplemented
15C3×#1462—51nomaximal, Hall
16C22×#685C2×C311yescharacteristic
17#65798C2×C312yescomplemented
18#64915C2×C314yescomplemented
19C24×C3×C5C2×C411yescharacteristic
20C23×C3×D5—21no—
21C23×C3×D5—21no—
22C23×C3×D5—24no—
23C23×C3×D5—42no—
24C23×C3×D5—42no—
25C23×C3×D5C2×C412yes—
26C22×C3×C4×C5—21no—
27C22×C3×C4×C5—21no—
28C22×C3×C4×C5—42no—
29C22×C3×C4×C5C2×C412yes—
30C22×C3×Dic5—42no—
31C22×C3×Dic5C2×C411yescharacteristic
32C22×C3×Dic5D412yes—
33C22×C3×Dic5D412yes—
34C22×C3×Dic5C2311yescharacteristic
35C3×#704—44no—
36C2×C3×#170—24no—
37C3×#685—24no—
38C3×#685—24no—
39C22×C3×C22⋊C4—51no—
40C3×#407—52no—
41C3×#323—54no—
42C24×D5C3×C411yescharacteristic
43C23×C4×C5C3×C411yescharacteristic
44C23×Dic5C22×C311yescharacteristic
45C2×#704—22no—
46C2×#685—22no—
47C2×#685—22no—
48C2×#685C3×C412yes—
49C2×#685C22×C312yes—
50C22⋊#170—22no—
51C22⋊#170C22×C314yescomplemented
52#1462—51noSylow
53C23×C3×C5—22no—
54C23×C3×C5—22no—
55C23×C3×C5—22no—
56C23×C3×C5C22⋊C411yescharacteristic
57C23×C3×C5C22⋊C411yescharacteristic
58C23×C3×C5C22×C411yescharacteristic
59C22×C3×D5—24no—
60C22×C3×D5—24no—
61C22×C3×D5—22no—
62C22×C3×D5—22no—
63C22×C3×D5—416no—
64C22×C3×D5—42no—
65C22×C3×D5—42no—
66C22×C3×D5—42no—
67C22×C3×D5—42no—
68C22×C3×D5—42no—
69C22×C3×D5—42no—
70C22×C3×D5M1614yescomplemented
71C2×C3×C4×C5—22no—
72C2×C3×C4×C5—42no—
73C2×C3×C4×C5—42no—
74C2×C3×C4×C5—42no—
75C2×C3×Dic5—24no—
76C2×C3×Dic5—41no—
77C2×C3×Dic5—42no—
78C2×C3×Dic5—41no—
79C2×C3×Dic5C22⋊C412yes—
80C2×C3×Dic5C2×D412yes—
81C3×#170—44no—
82C25×C3—51no—
83C23×C3×C4—51no—
84C23×C3×C4AGL(1,𝔽5)11yescharacteristic
85C2×C3×M16—102no—
86C2×C3×C22⋊C4—52no—
87C2×C3×C22⋊C4—52no—
88C2×C3×C22⋊C4—102no—
89C2×C3×C22⋊C4—102no—
90C3×C22⋊C8—54no—
91C3×C22⋊C8—102no—
92C24×C5C2×C3×C411yescharacteristic
93C23×D5—21no—
94C23×D5—21no—
95C23×D5—24no—
96C23×D5—42no—
97C23×D5—42no—
98C23×D5C2×C3×C412yes—
99C22×C4×C5—21no—
100C22×C4×C5—21no—
101C22×C4×C5—42no—
102C22×C4×C5C2×C3×C412yes—
103C22×Dic5—42no—
104C22×Dic5C2×C3×C411yescharacteristic
105C22×Dic5C3×D412yes—
106C22×Dic5C3×D412yes—
107C22×Dic5C23×C311yescharacteristic
108#704—44no—
109C2×#170—24no—
110#685—24no—
111#685—24no—
112C22×C22⋊C4—51no—
113#407—52no—
114#323—54no—
115C22×C3×C5—28no—
116C22×C3×C5—21no—
117C22×C3×C5—21no—
118C22×C3×C5—21no—
119C22×C3×C5—21no—
120C22×C3×C5—21no—
121C22×C3×C5—21no—
122C22×C3×C5#9912yes—
123C22×C3×C5#10012yes—
124C22×C3×C5C2×C22⋊C411yescharacteristic
125C22×C3×C5C2×M1612yes—
126C2×C3×D5—24no—
127C2×C3×D5—48no—
128C2×C3×D5—48no—
129C2×C3×D5—44no—
130C2×C3×D5—42no—
131C2×C3×D5—42no—
132C2×C3×D5—44no—
133C3×C4×C5—42no—
134C3×Dic5—22no—
135C3×Dic5—22no—
136C24×C3—52no—
137C24×C3—104no—
138C24×C3—101no—
139C24×C3—101no—
140C24×C3—202no—
141C24×C3—202no—
142C24×C3C2×AGL(1,𝔽5)11yescharacteristic
143C22×C3×C4—21no—
144C22×C3×C4—21no—
145C22×C3×C4—42no—
146C22×C3×C4—52no—
147C22×C3×C4—52no—
148C22×C3×C4—51no—
149C22×C3×C4—51no—
150C22×C3×C4—202no—
151C22×C3×C4C2×AGL(1,𝔽5)12yes—
152C3×M16—204no—
153C2×C3×C8—104no—
154C3×C22⋊C4—104no—
155C3×C22⋊C4—104no—
156C23×C5—22no—
157C23×C5—22no—
158C23×C5—22no—
159C23×C5C3×C22⋊C411yescharacteristic
160C23×C5C3×C22⋊C411yescharacteristic
161C23×C5C22×C3×C411yescharacteristic
162C22×D5—24no—
163C22×D5—24no—
164C22×D5—22no—
165C22×D5—22no—
166C22×D5—416no—
167C22×D5—42no—
168C22×D5—42no—
169C22×D5—42no—
170C22×D5—42no—
171C22×D5—42no—
172C22×D5—42no—
173C22×D5C3×M1614yescomplemented
174C2×C4×C5—22no—
175C2×C4×C5—42no—
176C2×C4×C5—42no—
177C2×C4×C5—42no—
178C2×Dic5—24no—
179C2×Dic5—41no—
180C2×Dic5—42no—
181C2×Dic5—41no—
182C2×Dic5C3×C22⋊C412yes—
183C2×Dic5C2×C3×D412yes—
184#170—44no—
185C25—51no—
186C23×C4—51no—
187C23×C4C3×AGL(1,𝔽5)11yescharacteristic
188C2×M16—102no—
189C2×C22⋊C4—52no—
190C2×C22⋊C4—52no—
191C2×C22⋊C4—102no—
192C2×C22⋊C4—102no—
193C22⋊C8—54no—
194C22⋊C8—102no—
195C2×C3×C5—22no—
196C2×C3×C5—22no—
197C2×C3×C5—22no—
198C2×C3×C5#40711yescharacteristic
199C2×C3×C5C2×#9911yescharacteristic
200C2×C3×C5C2×#10011yescharacteristic
201C3×D5—44no—
202C23×C3—22no—
203C23×C3—22no—
204C23×C3—22no—
205C23×C3—54no—
206C23×C3—104no—
207C23×C3—104no—
208C23×C3—102no—
209C23×C3—102no—
210C23×C3—202no—
211C23×C3—202no—
212C23×C3—2016no—
213C23×C3—202no—
214C23×C3—202no—
215C23×C3—202no—
216C23×C3—202no—
217C23×C3C22⋊AGL(1,𝔽5)11yescharacteristic
218C23×C3C22⋊AGL(1,𝔽5)11yescharacteristic
219C23×C3C22×AGL(1,𝔽5)11yescharacteristic
220C2×C3×C4—22no—
221C2×C3×C4—42no—
222C2×C3×C4—42no—
223C2×C3×C4—42no—
224C2×C3×C4—52no—
225C2×C3×C4—52no—
226C2×C3×C4—104no—
227C2×C3×C4—201no—
228C2×C3×C4—202no—
229C2×C3×C4—201no—
230C3×C8—204no—
231C22×C5—28no—
232C22×C5—21no—
233C22×C5—21no—
234C22×C5—21no—
235C22×C5—21no—
236C22×C5—21no—
237C22×C5—21no—
238C22×C5C3×#9912yes—
239C22×C5C3×#10012yes—
240C22×C5C2×C3×C22⋊C411yescharacteristic
241C22×C5C2×C3×M1612yes—
242C2×D5—24no—
243C2×D5—48no—
244C2×D5—48no—
245C2×D5—44no—
246C2×D5—42no—
247C2×D5—42no—
248C2×D5—44no—
249C4×C5—42no—
250Dic5—22no—
251Dic5—22no—
252C24—52no—
253C24—104no—
254C24—101no—
255C24—101no—
256C24—202no—
257C24—202no—
258C24C2×C3×AGL(1,𝔽5)11yescharacteristic
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—42no—
262C22×C4—52no—
263C22×C4—52no—
264C22×C4—51no—
265C22×C4—51no—
266C22×C4—202no—
267C22×C4C2×C3×AGL(1,𝔽5)12yes—
268M16—204no—
269C2×C8—104no—
270C22⋊C4—104no—
271C22⋊C4—104no—
272C3×C5#146211yescharacteristic, complemented, Hall
273C22×C3—28no—
274C22×C3—21no—
275C22×C3—21no—
276C22×C3—21no—
277C22×C3—21no—
278C22×C3—21no—
279C22×C3—21no—
280C22×C3—104no—
281C22×C3—204no—
282C22×C3—204no—
283C22×C3—208no—
284C22×C3—208no—
285C22×C3—202no—
286C22×C3—202no—
287C22×C3#399912yes—
288C22×C3#401312yes—
289C22×C3D5⋊M1612yes—
290C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
291C3×C4—42no—
292C3×C4—102no—
293C3×C4—102no—
294C2×C5—22no—
295C2×C5—22no—
296C2×C5—22no—
297C2×C5C3×#40711yescharacteristic
298C2×C5C2×C3×#9911yescharacteristic
299C2×C5C2×C3×#10011yescharacteristic
300D5—44no—
301C23—22no—
302C23—22no—
303C23—22no—
304C23—54no—
305C23—104no—
306C23—104no—
307C23—102no—
308C23—102no—
309C23—202no—
310C23—202no—
311C23—2016no—
312C23—202no—
313C23—202no—
314C23—202no—
315C23—202no—
316C23C3×C22⋊AGL(1,𝔽5)11yescharacteristic
317C23C3×C22⋊AGL(1,𝔽5)11yescharacteristic
318C23C22×C3×AGL(1,𝔽5)11yescharacteristic
319C2×C4—22no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—52no—
324C2×C4—52no—
325C2×C4—104no—
326C2×C4—201no—
327C2×C4—202no—
328C2×C4—201no—
329C8—204no—
330C2×C3—22no—
331C2×C3—22no—
332C2×C3—22no—
333C2×C3—204no—
334C2×C3C2×#399911yescharacteristic
335C2×C3#6575511yescharacteristic
336C2×C3C2×#401311yescharacteristic
337C5C3×#146211yescharacteristic, complemented, Sylow
338C22—28no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—104no—
346C22—204no—
347C22—208no—
348C22—204no—
349C22—208no—
350C22—202no—
351C22—202no—
352C22C3×#399912yes—
353C22C3×#401312yes—
354C22C3×D5⋊M1612yes—
355C22C2×C3×C22⋊AGL(1,𝔽5)11yescharacteristic
356C4—42no—
357C4—102no—
358C4—102no—
359C3#1060488011yescharacteristic, direct product factor, Sylow
360C2—22no—
361C2—22no—
362C2—22no—
363C2—204no—
364C2C2×C3×#399911yescharacteristic
365C2C3×#6575511yescharacteristic
366C2C2×C3×#401311yescharacteristic
subgroupG/H
centerC22×C3C2×C22⋊AGL(1,𝔽5)
commutatorC23×C5C22×C3×C4
FrattiniC23C22×C3×AGL(1,𝔽5)
FittingC23×C3×C4×C5C4
socleC22×C3×C5C2×C22⋊C4
prime radicalC24×C3×D5C4
cosocleC22×Dic5C23×C3
hypercenterC23×C3×C4AGL(1,𝔽5)
hypocenterC5C3×#1462
LCSG ▷ C23×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22×C3 ◁ C24×C3 ◁ C23×C3×C4

Complements

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

NH ≅ G/Ntype
2. C3×#65798C2semidirect
3. C3×#64915C2semidirect
4. #10604880C3direct (N × H)
14. C3×C22⋊#170C22semidirect
17. #65798C2×C3semidirect
18. #64915C2×C3semidirect
51. C22⋊#170C22×C3semidirect
70. C22×C3×D5M16semidirect
173. C22×D5C3×M16semidirect
272. C3×C5#1462semidirect
337. C5C3×#1462semidirect
359. C3#10604880direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#6852
C3×#657982
C3×#649152
#106048803
C3×#14625