Finite Group Databasebeta

← C3×#10604677 #14302790

Subgroups — C3×#10604677 #14302790

Order 1920.

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

normal subgroups only

This group has 624 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×#66260C211yesmaximal, characteristic
2C3×#66058C211yesmaximal, characteristic
3C3×#65785C211yesmaximal, characteristic
4C3×#65782C212yesmaximal, complemented
5C3×#64863C212yesmaximal, complemented
6#10604677C311yesmaximal, characteristic, direct product factor, Hall
7C2×C3×#713C2211yescharacteristic
8C2×C3×Q8×D5C2211yescharacteristic
9C3×#4133—24no—
10C3×#4130C2212yes—
11C3×D5⋊M16—22no—
12C3×D5⋊M16C2212yes—
13C3×C5×C4⋊Q8C411yescharacteristic, complemented
14C3×#4093C411yescharacteristic, complemented
15C3×#4090C411yescharacteristic, complemented
16C3×#4042—22no—
17C3×#4040—22no—
18C3×#4021C411yescharacteristic, complemented
19C3×#4018C2211yescharacteristic
20C3×#4009—24no—
21C3×#3999—22no—
22C3×#1909—51nomaximal, Hall
23#66260C2×C311yescharacteristic
24#66058C2×C311yescharacteristic
25#65785C2×C311yescharacteristic
26#65782C2×C312yescomplemented
27#64863C2×C312yescomplemented
28C2×C3×C5×Q8C2×C411yescharacteristic
29C2×C3×C5×Q8C2×C411yescharacteristic
30C3×#713—22no—
31C3×#713—22no—
32C3×Q8×D5—22no—
33C3×Q8×D5—22no—
34C2×C3×D20C2×C411yescharacteristic
35C2×C3×D20D412yes—
36C2×C3×C4×D5—21no—
37C2×C3×C4×D5D412yes—
38C2×C3×C4×D5C2311yescharacteristic
39C2×C3×Dic10—21no—
40C2×C3×Dic10C2×C411yescharacteristic
41C3×C22⋊AGL(1,𝔽5)—22no—
42C3×#704—22no—
43C3×C4⋊AGL(1,𝔽5)—22no—
44C3×C4×AGL(1,𝔽5)—22no—
45C3×#700—24no—
46C3×#700—22no—
47C3×D5⋊C8—24no—
48C3×D5⋊C8D414yes—
49C3×C5×C4⋊C4—22no—
50C3×C42×C5C2×C411yescharacteristic
51C3×#685—22no—
52C3×#685—22no—
53C3×#685—21no—
54C3×#683—22no—
55C3×#683—21no—
56C3×C4×Dic5C2×C411yescharacteristic
57C3×#568—51no—
58C3×#536—51no—
59C3×#419—52no—
60C3×#410—51no—
61C3×H(3,ℤ/4)—52no—
62C2×#713C22×C311yescharacteristic
63C2×Q8×D5C22×C311yescharacteristic
64#4133—24no—
65#4130C22×C312yes—
66D5⋊M16—22no—
67D5⋊M16C22×C312yes—
68C5×C4⋊Q8C3×C411yescharacteristic, complemented
69#4093C3×C411yescharacteristic, complemented
70#4090C3×C411yescharacteristic, complemented
71#4042—22no—
72#4040—22no—
73#4021C3×C411yescharacteristic, complemented
74#4018C22×C311yescharacteristic
75#4009—24no—
76#3999—22no—
77#1909—51noSylow
78C22×C3×D5C22⋊C412yes—
79C22×C3×D5C2×D411yescharacteristic
80C2×C3×AGL(1,𝔽5)—42no—
81C3×C5×Q8—22no—
82C3×C5×Q8—22no—
83C2×C3×C4×C5—21no—
84C2×C3×C4×C5—21no—
85C2×C3×C4×C5C22⋊C412yes—
86C2×C3×C4×C5C22×C411yescharacteristic
87C2×C3×Dic5—21no—
88C2×C3×Dic5—21no—
89C2×C3×Dic5C2×D411yescharacteristic
90C3×D20—24no—
91C3×D20—22no—
92C3×C4×D5—22no—
93C3×C4×D5—22no—
94C3×C4×D5—42no—
95C3×C4×D5C2×D412yes—
96C3×C4×D5#4112yes—
97C3×Dic10—22no—
98C3×Dic10—41no—
99C3×Dic10—41no—
100C3×#170—24no—
101C2×C3×#41—51no—
102C22×C3×Q8—51no—
103C3×#131—104no—
104C2×C3×M16—52no—
105C2×C3×M16—102no—
106C3×C4⋊Q8—51no—
107C3×C4⋊Q8AGL(1,𝔽5)11yescharacteristic, complemented
108C3×#124—51no—
109C3×#124—51no—
110C3×#123—102no—
111C3×C22⋊Q8—102no—
112C3×#117—51no—
113C3×#117—52no—
114C3×#103—104no—
115C3×#99—102no—
116C2×C5×Q8C2×C3×C411yescharacteristic
117C2×C5×Q8C2×C3×C411yescharacteristic
118#713—22no—
119#713—22no—
120Q8×D5—22no—
121Q8×D5—22no—
122C2×D20C2×C3×C411yescharacteristic
123C2×D20C3×D412yes—
124C2×C4×D5—21no—
125C2×C4×D5C3×D412yes—
126C2×C4×D5C23×C311yescharacteristic
127C2×Dic10—21no—
128C2×Dic10C2×C3×C411yescharacteristic
129C22⋊AGL(1,𝔽5)—22no—
130#704—22no—
131C4⋊AGL(1,𝔽5)—22no—
132C4×AGL(1,𝔽5)—22no—
133#700—24no—
134#700—22no—
135D5⋊C8—24no—
136D5⋊C8C3×D414yes—
137C5×C4⋊C4—22no—
138C42×C5C2×C3×C411yescharacteristic
139#685—22no—
140#685—22no—
141#685—21no—
142#683—22no—
143#683—21no—
144C4×Dic5C2×C3×C411yescharacteristic
145#568—51no—
146#536—51no—
147#419—52no—
148#410—51no—
149H(3,ℤ/4)—52no—
150C22×C3×C5C2×C22⋊C411yescharacteristic
151C2×C3×D5—22no—
152C2×C3×D5—41no—
153C2×C3×D5—42no—
154C2×C3×D5#12112yes—
155C3×AGL(1,𝔽5)—82no—
156C3×C4×C5—22no—
157C3×C4×C5—41no—
158C3×C4×C5—41no—
159C3×C4×C5C4×D412yes—
160C3×Dic5—41no—
161C3×Dic5—41no—
162C3×Dic5#12011yescharacteristic
163C3×Dic5#12311yescharacteristic
164C3×#41—102no—
165C3×#41—102no—
166C2×C3×Q8—51no—
167C2×C3×Q8—101no—
168C2×C3×Q8—102no—
169C2×C3×Q8—102no—
170C2×C3×Q8C2×AGL(1,𝔽5)11yescharacteristic
171C2×C3×Q8C2×AGL(1,𝔽5)11yescharacteristic
172C2×C3×D4—52no—
173C2×C3×D4—51no—
174C22×C3×C4—51no—
175C22×C3×C4—52no—
176C22×C3×C4—101no—
177C3×M16—102no—
178C3×M16—102no—
179C3×M16—104no—
180C2×C3×C8—54no—
181C2×C3×C8—104no—
182C3×C4⋊C4—22no—
183C3×C4⋊C4—101no—
184C3×C4⋊C4—102no—
185C3×C4⋊C4—102no—
186C3×C22⋊C4—101no—
187C3×C22⋊C4—102no—
188C3×C22⋊C4—102no—
189C3×C22⋊C4—102no—
190C3×C42—51no—
191C3×C42—102no—
192C3×C42C2×AGL(1,𝔽5)11yescharacteristic
193C22×D5C3×C22⋊C412yes—
194C22×D5C2×C3×D411yescharacteristic
195C2×AGL(1,𝔽5)—42no—
196C5×Q8—22no—
197C5×Q8—22no—
198C2×C4×C5—21no—
199C2×C4×C5—21no—
200C2×C4×C5C3×C22⋊C412yes—
201C2×C4×C5C22×C3×C411yescharacteristic
202C2×Dic5—21no—
203C2×Dic5—21no—
204C2×Dic5C2×C3×D411yescharacteristic
205D20—24no—
206D20—22no—
207C4×D5—22no—
208C4×D5—22no—
209C4×D5—42no—
210C4×D5C2×C3×D412yes—
211C4×D5C3×#4112yes—
212Dic10—22no—
213Dic10—41no—
214Dic10—41no—
215#170—24no—
216C2×#41—51no—
217C22×Q8—51no—
218#131—104no—
219C2×M16—52no—
220C2×M16—102no—
221C4⋊Q8—51no—
222C4⋊Q8C3×AGL(1,𝔽5)11yescharacteristic, complemented
223#124—51no—
224#124—51no—
225#123—102no—
226C22⋊Q8—102no—
227#117—51no—
228#117—52no—
229#103—104no—
230#99—102no—
231C2×C3×C5—21no—
232C2×C3×C5#38611yescharacteristic
233C3×D5—22no—
234C3×D5—42no—
235C23×C3—51no—
236C23×C3—52no—
237C3×Q8—22no—
238C3×Q8—22no—
239C3×Q8—102no—
240C3×Q8—201no—
241C3×Q8—201no—
242C3×D4—104no—
243C3×D4—102no—
244C2×C3×C4—21no—
245C2×C3×C4—21no—
246C2×C3×C4—51no—
247C2×C3×C4—52no—
248C2×C3×C4—52no—
249C2×C3×C4—102no—
250C2×C3×C4—102no—
251C2×C3×C4—101no—
252C2×C3×C4—101no—
253C2×C3×C4—202no—
254C2×C3×C4—202no—
255C2×C3×C4C22⋊AGL(1,𝔽5)12yes—
256C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
257C3×C8—104no—
258C22×C5C2×C3×C22⋊C411yescharacteristic
259C2×D5—22no—
260C2×D5—41no—
261C2×D5—42no—
262C2×D5C3×#12112yes—
263AGL(1,𝔽5)—82no—
264C4×C5—22no—
265C4×C5—41no—
266C4×C5—41no—
267C4×C5C3×C4×D412yes—
268Dic5—41no—
269Dic5—41no—
270Dic5C3×#12011yescharacteristic
271Dic5C3×#12311yescharacteristic
272#41—102no—
273#41—102no—
274C2×Q8—51no—
275C2×Q8—101no—
276C2×Q8—102no—
277C2×Q8—102no—
278C2×Q8C2×C3×AGL(1,𝔽5)11yescharacteristic
279C2×Q8C2×C3×AGL(1,𝔽5)11yescharacteristic
280C2×D4—52no—
281C2×D4—51no—
282C22×C4—51no—
283C22×C4—52no—
284C22×C4—101no—
285M16—102no—
286M16—102no—
287M16—104no—
288C2×C8—54no—
289C2×C8—104no—
290C4⋊C4—22no—
291C4⋊C4—101no—
292C4⋊C4—102no—
293C4⋊C4—102no—
294C22⋊C4—101no—
295C22⋊C4—102no—
296C22⋊C4—102no—
297C22⋊C4—102no—
298C42—51no—
299C42—102no—
300C42C2×C3×AGL(1,𝔽5)11yescharacteristic
301C3×C5#190911yescharacteristic, complemented, Hall
302C22×C3—52no—
303C22×C3—102no—
304C22×C3—201no—
305C22×C3—202no—
306C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
307C3×C4—22no—
308C3×C4—41no—
309C3×C4—41no—
310C3×C4—51no—
311C3×C4—51no—
312C3×C4—201no—
313C3×C4—201no—
314C3×C4—402no—
315C3×C4D4×AGL(1,𝔽5)12yes—
316C2×C5—21no—
317C2×C5C3×#38611yescharacteristic
318D5—22no—
319D5—42no—
320C23—51no—
321C23—52no—
322Q8—22no—
323Q8—22no—
324Q8—102no—
325Q8—201no—
326Q8—201no—
327D4—104no—
328D4—102no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—51no—
332C2×C4—52no—
333C2×C4—52no—
334C2×C4—102no—
335C2×C4—102no—
336C2×C4—101no—
337C2×C4—101no—
338C2×C4—202no—
339C2×C4—202no—
340C2×C4C3×C22⋊AGL(1,𝔽5)12yes—
341C2×C4C22×C3×AGL(1,𝔽5)11yescharacteristic
342C8—104no—
343C2×C3—21no—
344C2×C3—102no—
345C2×C3—202no—
346C2×C3#6577911yescharacteristic
347C5C3×#190911yescharacteristic, complemented, Sylow
348C22—52no—
349C22—102no—
350C22—201no—
351C22—202no—
352C22C2×C3×C22⋊AGL(1,𝔽5)11yescharacteristic
353C4—22no—
354C4—41no—
355C4—41no—
356C4—51no—
357C4—51no—
358C4—201no—
359C4—201no—
360C4—402no—
361C4C3×D4×AGL(1,𝔽5)12yes—
362C3#1060467711yescharacteristic, direct product factor, Sylow
363C2—21no—
364C2—102no—
365C2—202no—
366C2C3×#6577911yescharacteristic
subgroupG/H
centerC2×C3#65779
commutatorC2×C4×C5C22×C3×C4
FrattiniC2×C4C22×C3×AGL(1,𝔽5)
FittingC3×C5×C4⋊Q8C4
socleC2×C3×C5#386
prime radicalC2×C3×#713C22
cosocleC2×C4×D5C23×C3
hypercenterC3×C4⋊Q8AGL(1,𝔽5)
hypocenterC5C3×#1909
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C3×C4⋊Q8

Complements

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

NH ≅ G/Ntype
4. C3×#65782C2semidirect
5. C3×#64863C2semidirect
6. #10604677C3direct (N × H)
13. C3×C5×C4⋊Q8C4semidirect
14. C3×#4093C4semidirect
15. C3×#4090C4semidirect
18. C3×#4021C4semidirect
26. #65782C2×C3semidirect
27. #64863C2×C3semidirect
68. C5×C4⋊Q8C3×C4semidirect
69. #4093C3×C4semidirect
70. #4090C3×C4semidirect
73. #4021C3×C4semidirect
107. C3×C4⋊Q8AGL(1,𝔽5)semidirect
222. C4⋊Q8C3×AGL(1,𝔽5)semidirect
301. C3×C5#1909semidirect
347. C5C3×#1909semidirect
362. C3#10604677direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#662602
C3×#660582
C3×#657852
C3×#657822
C3×#648632
#106046773
C3×#19095