Finite Group Databasebeta

← C3×#10604676 #14301387

Subgroups — C3×#10604676 #14301387

Order 1920.

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

normal subgroups only

This group has 664 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, complemented
2C3×#66059C211yesmaximal, characteristic
3C3×#65785C211yesmaximal, characteristic, complemented
4C3×#65784C212yesmaximal, complemented
5C3×#64862C212yesmaximal, complemented
6#10604676C311yesmaximal, characteristic, direct product factor, Hall
7C2×C3×#713C2211yescharacteristic
8C2×C3×#713C2211yescharacteristic
9C3×#4133—24no—
10C3×#4130C2212yes—
11C3×D5⋊M16—22no—
12C3×D5⋊M16C2212yes—
13C3×C5×C4⋊Q8C411yescharacteristic, complemented
14C3×#4093C411yescharacteristic, complemented
15C3×#4093C411yescharacteristic, complemented
16C3×#4041—22no—
17C3×#4039—22no—
18C3×#4020C411yescharacteristic, complemented
19C3×C42×D5C2211yescharacteristic
20C3×#4010—24no—
21C3×#3999—22no—
22C3×#1884—51nomaximal, Hall
23#66260C2×C311yescharacteristic, complemented
24#66059C2×C311yescharacteristic
25#65785C2×C311yescharacteristic, complemented
26#65784C2×C312yescomplemented
27#64862C2×C312yescomplemented
28C2×C3×C5×Q8C2×C411yescharacteristic
29C2×C3×C5×Q8C2×C411yescharacteristic
30C3×#713—22no—
31C3×#713—22no—
32C3×#713—22no—
33C3×#713—22no—
34C2×C3×D20—22no—
35C2×C3×D20—21no—
36C2×C3×D20C2×C411yescharacteristic
37C2×C3×D20C2×C411yescharacteristic
38C2×C3×D20D412yes—
39C2×C3×C4×D5—21no—
40C2×C3×C4×D5—21no—
41C2×C3×C4×D5D412yes—
42C2×C3×C4×D5C2311yescharacteristic
43C3×C22⋊AGL(1,𝔽5)—22no—
44C3×#704—22no—
45C3×C4⋊AGL(1,𝔽5)—22no—
46C3×C4×AGL(1,𝔽5)—22no—
47C3×#700—24no—
48C3×#700D414yescomplemented
49C3×D5⋊C8—24no—
50C3×D5⋊C8—22no—
51C3×C5×C4⋊C4—22no—
52C3×C42×C5C2×C411yescharacteristic
53C3×#685—22no—
54C3×#685—22no—
55C3×C4×Dic5—21no—
56C3×C4×Dic5C2×C411yescharacteristic
57C3×#568—51no—
58C3×#532—51no—
59C3×#421—52no—
60C3×#410—51no—
61C3×#339—52no—
62C2×#713C22×C311yescharacteristic
63C2×#713C22×C311yescharacteristic
64#4133—24no—
65#4130C22×C312yes—
66D5⋊M16—22no—
67D5⋊M16C22×C312yes—
68C5×C4⋊Q8C3×C411yescharacteristic, complemented
69#4093C3×C411yescharacteristic, complemented
70#4093C3×C411yescharacteristic, complemented
71#4041—22no—
72#4039—22no—
73#4020C3×C411yescharacteristic, complemented
74C42×D5C22×C311yescharacteristic
75#4010—24no—
76#3999—22no—
77#1884—51noSylow
78C22×C3×D5—21no—
79C22×C3×D5C22⋊C412yes—
80C22×C3×D5C2×D411yescharacteristic
81C2×C3×AGL(1,𝔽5)—42no—
82C3×C5×Q8—22no—
83C3×C5×Q8—22no—
84C2×C3×C4×C5—21no—
85C2×C3×C4×C5—21no—
86C2×C3×C4×C5C22⋊C412yes—
87C2×C3×C4×C5C22×C411yescharacteristic
88C2×C3×Dic5—21no—
89C2×C3×Dic5C2×D411yescharacteristic
90C3×D20—24no—
91C3×D20—22no—
92C3×D20—22no—
93C3×D20—44no—
94C3×D20—42no—
95C3×C4×D5—22no—
96C3×C4×D5—22no—
97C3×C4×D5—42no—
98C3×C4×D5—41no—
99C3×C4×D5—41no—
100C3×C4×D5C2×D412yes—
101C3×C4×D5#4112yes—
102C3×#170—24no—
103C2×C3×#41—51no—
104C2×C3×#41—51no—
105C3×#131—104no—
106C2×C3×M16—52no—
107C2×C3×M16—102no—
108C3×C4⋊Q8AGL(1,𝔽5)11yescharacteristic, complemented
109C3×#127—51no—
110C3×#124—51no—
111C3×#124—51no—
112C3×#121—102no—
113C3×C4×D4—102no—
114C3×#117—52no—
115C2×C3×C42—51no—
116C3×#104—104no—
117C3×#99—102no—
118C2×C5×Q8C2×C3×C411yescharacteristic
119C2×C5×Q8C2×C3×C411yescharacteristic
120#713—22no—
121#713—22no—
122#713—22no—
123#713—22no—
124C2×D20—22no—
125C2×D20—21no—
126C2×D20C2×C3×C411yescharacteristic
127C2×D20C2×C3×C411yescharacteristic
128C2×D20C3×D412yes—
129C2×C4×D5—21no—
130C2×C4×D5—21no—
131C2×C4×D5C3×D412yes—
132C2×C4×D5C23×C311yescharacteristic
133C22⋊AGL(1,𝔽5)—22no—
134#704—22no—
135C4⋊AGL(1,𝔽5)—22no—
136C4×AGL(1,𝔽5)—22no—
137#700—24no—
138#700C3×D414yescomplemented
139D5⋊C8—24no—
140D5⋊C8—22no—
141C5×C4⋊C4—22no—
142C42×C5C2×C3×C411yescharacteristic
143#685—22no—
144#685—22no—
145C4×Dic5—21no—
146C4×Dic5C2×C3×C411yescharacteristic
147#568—51no—
148#532—51no—
149#421—52no—
150#410—51no—
151#339—52no—
152C22×C3×C5C2×C22⋊C411yescharacteristic
153C2×C3×D5—22no—
154C2×C3×D5—41no—
155C2×C3×D5—42no—
156C2×C3×D5—42no—
157C2×C3×D5—41no—
158C2×C3×D5#12112yes—
159C3×AGL(1,𝔽5)—82no—
160C3×C4×C5—22no—
161C3×C4×C5—22no—
162C3×C4×C5—41no—
163C3×C4×C5C4×D412yes—
164C3×Dic5—41no—
165C3×Dic5#12011yescharacteristic
166C3×Dic5#12311yescharacteristic
167C3×#41—102no—
168C3×#41—102no—
169C3×#41—102no—
170C3×#41—102no—
171C2×C3×Q8C2×AGL(1,𝔽5)11yescharacteristic
172C2×C3×Q8C2×AGL(1,𝔽5)11yescharacteristic
173C2×C3×D4—52no—
174C2×C3×D4—51no—
175C2×C3×D4—51no—
176C2×C3×D4—102no—
177C2×C3×D4—101no—
178C22×C3×C4—51no—
179C22×C3×C4—52no—
180C22×C3×C4—101no—
181C22×C3×C4—101no—
182C3×M16—54no—
183C3×M16—102no—
184C3×M16—104no—
185C2×C3×C8—102no—
186C2×C3×C8—104no—
187C3×C4⋊C4—22no—
188C3×C4⋊C4—102no—
189C3×C22⋊C4—102no—
190C3×C22⋊C4—102no—
191C3×C22⋊C4—102no—
192C3×C42—51no—
193C3×C42—101no—
194C3×C42—102no—
195C3×C42C2×AGL(1,𝔽5)11yescharacteristic
196C22×D5—21no—
197C22×D5C3×C22⋊C412yes—
198C22×D5C2×C3×D411yescharacteristic
199C2×AGL(1,𝔽5)—42no—
200C5×Q8—22no—
201C5×Q8—22no—
202C2×C4×C5—21no—
203C2×C4×C5—21no—
204C2×C4×C5C3×C22⋊C412yes—
205C2×C4×C5C22×C3×C411yescharacteristic
206C2×Dic5—21no—
207C2×Dic5C2×C3×D411yescharacteristic
208D20—24no—
209D20—22no—
210D20—22no—
211D20—44no—
212D20—42no—
213C4×D5—22no—
214C4×D5—22no—
215C4×D5—42no—
216C4×D5—41no—
217C4×D5—41no—
218C4×D5C2×C3×D412yes—
219C4×D5C3×#4112yes—
220#170—24no—
221C2×#41—51no—
222C2×#41—51no—
223#131—104no—
224C2×M16—52no—
225C2×M16—102no—
226C4⋊Q8C3×AGL(1,𝔽5)11yescharacteristic, complemented
227#127—51no—
228#124—51no—
229#124—51no—
230#121—102no—
231C4×D4—102no—
232#117—52no—
233C2×C42—51no—
234#104—104no—
235#99—102no—
236C2×C3×C5—21no—
237C2×C3×C5#38611yescharacteristic
238C3×D5—22no—
239C3×D5—42no—
240C3×D5—81no—
241C23×C3—51no—
242C23×C3—52no—
243C23×C3—101no—
244C3×Q8—22no—
245C3×Q8—22no—
246C3×D4—104no—
247C3×D4—102no—
248C3×D4—102no—
249C3×D4—204no—
250C3×D4—202no—
251C2×C3×C4—21no—
252C2×C3×C4—21no—
253C2×C3×C4—51no—
254C2×C3×C4—52no—
255C2×C3×C4—52no—
256C2×C3×C4—102no—
257C2×C3×C4—102no—
258C2×C3×C4—101no—
259C2×C3×C4—202no—
260C2×C3×C4—201no—
261C2×C3×C4—201no—
262C2×C3×C4—202no—
263C2×C3×C4C22⋊AGL(1,𝔽5)12yes—
264C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
265C3×C8—104no—
266C22×C5C2×C3×C22⋊C411yescharacteristic
267C2×D5—22no—
268C2×D5—41no—
269C2×D5—42no—
270C2×D5—42no—
271C2×D5—41no—
272C2×D5C3×#12112yes—
273AGL(1,𝔽5)—82no—
274C4×C5—22no—
275C4×C5—22no—
276C4×C5—41no—
277C4×C5C3×C4×D412yes—
278Dic5—41no—
279Dic5C3×#12011yescharacteristic
280Dic5C3×#12311yescharacteristic
281#41—102no—
282#41—102no—
283#41—102no—
284#41—102no—
285C2×Q8C2×C3×AGL(1,𝔽5)11yescharacteristic
286C2×Q8C2×C3×AGL(1,𝔽5)11yescharacteristic
287C2×D4—52no—
288C2×D4—51no—
289C2×D4—51no—
290C2×D4—102no—
291C2×D4—101no—
292C22×C4—51no—
293C22×C4—52no—
294C22×C4—101no—
295C22×C4—101no—
296M16—54no—
297M16—102no—
298M16—104no—
299C2×C8—102no—
300C2×C8—104no—
301C4⋊C4—22no—
302C4⋊C4—102no—
303C22⋊C4—102no—
304C22⋊C4—102no—
305C22⋊C4—102no—
306C42—51no—
307C42—101no—
308C42—102no—
309C42C2×C3×AGL(1,𝔽5)11yescharacteristic
310C3×C5#188411yescharacteristic, complemented, Hall
311C22×C3—52no—
312C22×C3—102no—
313C22×C3—201no—
314C22×C3—202no—
315C22×C3—202no—
316C22×C3—201no—
317C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
318C3×C4—22no—
319C3×C4—22no—
320C3×C4—41no—
321C3×C4—51no—
322C3×C4—51no—
323C3×C4—201no—
324C3×C4—402no—
325C3×C4D4×AGL(1,𝔽5)12yes—
326C2×C5—21no—
327C2×C5C3×#38611yescharacteristic
328D5—22no—
329D5—42no—
330D5—81no—
331C23—51no—
332C23—52no—
333C23—101no—
334Q8—22no—
335Q8—22no—
336D4—104no—
337D4—102no—
338D4—102no—
339D4—204no—
340D4—202no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—51no—
344C2×C4—52no—
345C2×C4—52no—
346C2×C4—102no—
347C2×C4—102no—
348C2×C4—101no—
349C2×C4—202no—
350C2×C4—201no—
351C2×C4—201no—
352C2×C4—202no—
353C2×C4C3×C22⋊AGL(1,𝔽5)12yes—
354C2×C4C22×C3×AGL(1,𝔽5)11yescharacteristic
355C8—104no—
356C2×C3—21no—
357C2×C3—102no—
358C2×C3—202no—
359C2×C3—401no—
360C2×C3#6577911yescharacteristic
361C5C3×#188411yescharacteristic, complemented, Sylow
362C22—52no—
363C22—102no—
364C22—201no—
365C22—202no—
366C22—202no—
367C22—201no—
368C22C2×C3×C22⋊AGL(1,𝔽5)11yescharacteristic
369C4—22no—
370C4—22no—
371C4—41no—
372C4—51no—
373C4—51no—
374C4—201no—
375C4—402no—
376C4C3×D4×AGL(1,𝔽5)12yes—
377C3#1060467611yescharacteristic, direct product factor, Sylow
378C2—21no—
379C2—102no—
380C2—202no—
381C2—401no—
382C2C3×#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 radicalC3×#66059C2
cosocleC2×C4×D5C23×C3
hypercenterC3×C4⋊Q8AGL(1,𝔽5)
hypocenterC5C3×#1884
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
1. C3×#66260C2semidirect
3. C3×#65785C2semidirect
4. C3×#65784C2semidirect
5. C3×#64862C2semidirect
6. #10604676C3direct (N × H)
13. C3×C5×C4⋊Q8C4semidirect
14. C3×#4093C4semidirect
15. C3×#4093C4semidirect
18. C3×#4020C4semidirect
23. #66260C2×C3semidirect
25. #65785C2×C3semidirect
26. #65784C2×C3semidirect
27. #64862C2×C3semidirect
48. C3×#700D4semidirect
68. C5×C4⋊Q8C3×C4semidirect
69. #4093C3×C4semidirect
70. #4093C3×C4semidirect
73. #4020C3×C4semidirect
108. C3×C4⋊Q8AGL(1,𝔽5)semidirect
138. #700C3×D4semidirect
226. C4⋊Q8C3×AGL(1,𝔽5)semidirect
310. C3×C5#1884semidirect
361. C5C3×#1884semidirect
377. C3#10604676direct (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×#660592
C3×#657852
C3×#657842
C3×#648622
#106046763
C3×#18845