Finite Group Databasebeta

← C3×#10604345 #14280270

Subgroups — C3×#10604345 #14280270

Order 1920.

1040 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×D5×C4⋊C4C211yesmaximal, characteristic
2C2×C3×(C2×AGL(1,𝔽5))⋊C4C214yesmaximal
3C2×C3×(C2×AGL(1,𝔽5))⋊C4C211yesmaximal, characteristic
4C2×C3×(C2×AGL(1,𝔽5))⋊C4C211yesmaximal, characteristic
5#10604345C311yesmaximal, characteristic, direct product factor, Hall
6C23×C3×AGL(1,𝔽5)C2212yes—
7C23×C3×AGL(1,𝔽5)C2212yes—
8C22×C3×C4×D5C2212yes—
9C22×C3×C4×D5C2211yescharacteristic
10C2×C3×C5×C4⋊C4C411yescharacteristic, complemented
11C2×C3×C4⋊Dic5C411yescharacteristic, complemented
12C2×C3×#683C412yescomplemented
13C3×D5×C4⋊C4—22no—
14C3×D5×C4⋊C4—22no—
15C3×(C2×AGL(1,𝔽5))⋊C4—28no—
16C3×(C2×AGL(1,𝔽5))⋊C4—22no—
17C3×(C2×AGL(1,𝔽5))⋊C4—22no—
18C3×#1445—51nomaximal, Hall
19C2×D5×C4⋊C4C2×C311yescharacteristic
20C2×(C2×AGL(1,𝔽5))⋊C4C2×C314yes—
21C2×(C2×AGL(1,𝔽5))⋊C4C2×C311yescharacteristic
22C2×(C2×AGL(1,𝔽5))⋊C4C2×C311yescharacteristic
23C23×C3×D5C2311yescharacteristic
24C22×C3×AGL(1,𝔽5)—22no—
25C22×C3×AGL(1,𝔽5)—24no—
26C22×C3×AGL(1,𝔽5)—22no—
27C22×C3×AGL(1,𝔽5)—22no—
28C22×C3×AGL(1,𝔽5)—24no—
29C22×C3×AGL(1,𝔽5)—22no—
30C22×C3×AGL(1,𝔽5)—24no—
31C22×C3×AGL(1,𝔽5)—24no—
32C22×C3×AGL(1,𝔽5)D414yes—
33C22×C3×AGL(1,𝔽5)Q814yes—
34C22×C3×C4×C5C2×C412yes—
35C22×C3×C4×C5C2×C411yescharacteristic
36C22×C3×Dic5C2×C412yes—
37C22×C3×Dic5C2×C411yescharacteristic
38C2×C3×C4×D5—24no—
39C2×C3×C4×D5—24no—
40C2×C3×C4×D5—22no—
41C2×C3×C4×D5—22no—
42C2×C3×C4×D5—24no—
43C2×C3×C4×D5D412yes—
44C2×C3×C4×D5D412yes—
45C3×C5×C4⋊C4—22no—
46C3×C4⋊Dic5—22no—
47C3×#683—24no—
48C22×C3×C4⋊C4—51no—
49C2×C3×#95—51no—
50C2×C3×#95—51no—
51C2×C3×#95—54no—
52C23×AGL(1,𝔽5)C22×C312yes—
53C23×AGL(1,𝔽5)C22×C312yes—
54C22×C4×D5C22×C312yes—
55C22×C4×D5C22×C311yescharacteristic
56C2×C5×C4⋊C4C3×C411yescharacteristic, complemented
57C2×C4⋊Dic5C3×C411yescharacteristic, complemented
58C2×#683C3×C412yescomplemented
59D5×C4⋊C4—22no—
60D5×C4⋊C4—22no—
61(C2×AGL(1,𝔽5))⋊C4—28no—
62(C2×AGL(1,𝔽5))⋊C4—22no—
63(C2×AGL(1,𝔽5))⋊C4—22no—
64#1445—51noSylow
65C23×C3×C5C22×C411yescharacteristic
66C22×C3×D5C2×D411yescharacteristic
67C22×C3×D5C2×D412yes—
68C22×C3×D5C2×D411yescharacteristic
69C22×C3×D5C2×Q812yes—
70C22×C3×D5#4112yes—
71C22×C3×D5#4114yes—
72C22×C3×D5#4112yes—
73C2×C3×AGL(1,𝔽5)—24no—
74C2×C3×AGL(1,𝔽5)—28no—
75C2×C3×AGL(1,𝔽5)—24no—
76C2×C3×AGL(1,𝔽5)—28no—
77C2×C3×AGL(1,𝔽5)—42no—
78C2×C3×AGL(1,𝔽5)—44no—
79C2×C3×AGL(1,𝔽5)—42no—
80C2×C3×AGL(1,𝔽5)—42no—
81C2×C3×AGL(1,𝔽5)—44no—
82C2×C3×AGL(1,𝔽5)—42no—
83C2×C3×C4×C5—22no—
84C2×C3×C4×C5—22no—
85C2×C3×C4×C5—24no—
86C2×C3×C4×C5C22⋊C412yes—
87C2×C3×Dic5—22no—
88C2×C3×Dic5—22no—
89C2×C3×Dic5—24no—
90C2×C3×Dic5C22⋊C412yes—
91C3×C4×D5—24no—
92C3×C4×D5—24no—
93C3×C4×D5—44no—
94C3×C4×D5—42no—
95C3×C4×D5—42no—
96C23×C3×C4—51no—
97C23×C3×C4—52no—
98C23×C3×C4—52no—
99C23×C3×C4—52no—
100C2×C3×C4⋊C4—52no—
101C2×C3×C4⋊C4—51no—
102C2×C3×C4⋊C4—102no—
103C2×C3×C4⋊C4—102no—
104C2×C3×C4⋊C4AGL(1,𝔽5)11yescharacteristic, complemented
105C3×#95—102no—
106C3×#95—102no—
107C3×#95—108no—
108C23×D5C23×C311yescharacteristic
109C22×AGL(1,𝔽5)—22no—
110C22×AGL(1,𝔽5)—24no—
111C22×AGL(1,𝔽5)—22no—
112C22×AGL(1,𝔽5)—22no—
113C22×AGL(1,𝔽5)—24no—
114C22×AGL(1,𝔽5)—22no—
115C22×AGL(1,𝔽5)—24no—
116C22×AGL(1,𝔽5)—24no—
117C22×AGL(1,𝔽5)C3×D414yes—
118C22×AGL(1,𝔽5)C3×Q814yes—
119C22×C4×C5C2×C3×C412yes—
120C22×C4×C5C2×C3×C411yescharacteristic
121C22×Dic5C2×C3×C412yes—
122C22×Dic5C2×C3×C411yescharacteristic
123C2×C4×D5—24no—
124C2×C4×D5—24no—
125C2×C4×D5—22no—
126C2×C4×D5—22no—
127C2×C4×D5—24no—
128C2×C4×D5C3×D412yes—
129C2×C4×D5C3×D412yes—
130C5×C4⋊C4—22no—
131C4⋊Dic5—22no—
132#683—24no—
133C22×C4⋊C4—51no—
134C2×#95—51no—
135C2×#95—51no—
136C2×#95—54no—
137C22×C3×C5C2×C22⋊C411yescharacteristic
138C22×C3×C5#11712yes—
139C22×C3×C5C4×D412yes—
140C22×C3×C5C4×Q812yes—
141C2×C3×D5#12112yes—
142C2×C3×D5C22⋊Q814yes—
143C2×C3×D5C22⋊Q812yes—
144C2×C3×D5#12312yes—
145C2×C3×D5#12314yes—
146C2×C3×D5#12312yes—
147C2×C3×D5#12411yescharacteristic
148C2×C3×D5#12411yescharacteristic
149C2×C3×D5#12514yes—
150C2×C3×D5#12614yes—
151C2×C3×D5#12711yescharacteristic
152C2×C3×D5C4⋊Q811yescharacteristic
153C3×AGL(1,𝔽5)—44no—
154C3×AGL(1,𝔽5)—44no—
155C3×C4×C5—22no—
156C3×C4×C5—42no—
157C3×Dic5—22no—
158C3×Dic5—42no—
159C24×C3—51no—
160C22×C3×C4—51no—
161C22×C3×C4—52no—
162C22×C3×C4—52no—
163C22×C3×C4—54no—
164C22×C3×C4—54no—
165C22×C3×C4—52no—
166C22×C3×C4—104no—
167C22×C3×C4—104no—
168C22×C3×C4—102no—
169C22×C3×C4—102no—
170C22×C3×C4—104no—
171C22×C3×C4—104no—
172C22×C3×C4—102no—
173C22×C3×C4—102no—
174C22×C3×C4—104no—
175C22×C3×C4—104no—
176C22×C3×C4—102no—
177C22×C3×C4—104no—
178C22×C3×C4—102no—
179C22×C3×C4C2×AGL(1,𝔽5)12yes—
180C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
181C3×C4⋊C4—22no—
182C3×C4⋊C4—104no—
183C3×C4⋊C4—102no—
184C23×C5C22×C3×C411yescharacteristic
185C22×D5C2×C3×D411yescharacteristic
186C22×D5C2×C3×D412yes—
187C22×D5C2×C3×D411yescharacteristic
188C22×D5C2×C3×Q812yes—
189C22×D5C3×#4112yes—
190C22×D5C3×#4114yes—
191C22×D5C3×#4112yes—
192C2×AGL(1,𝔽5)—24no—
193C2×AGL(1,𝔽5)—28no—
194C2×AGL(1,𝔽5)—24no—
195C2×AGL(1,𝔽5)—28no—
196C2×AGL(1,𝔽5)—42no—
197C2×AGL(1,𝔽5)—44no—
198C2×AGL(1,𝔽5)—42no—
199C2×AGL(1,𝔽5)—42no—
200C2×AGL(1,𝔽5)—44no—
201C2×AGL(1,𝔽5)—42no—
202C2×C4×C5—22no—
203C2×C4×C5—22no—
204C2×C4×C5—24no—
205C2×C4×C5C3×C22⋊C412yes—
206C2×Dic5—22no—
207C2×Dic5—22no—
208C2×Dic5—24no—
209C2×Dic5C3×C22⋊C412yes—
210C4×D5—24no—
211C4×D5—24no—
212C4×D5—44no—
213C4×D5—42no—
214C4×D5—42no—
215C23×C4—51no—
216C23×C4—52no—
217C23×C4—52no—
218C23×C4—52no—
219C2×C4⋊C4—52no—
220C2×C4⋊C4—51no—
221C2×C4⋊C4—102no—
222C2×C4⋊C4—102no—
223C2×C4⋊C4C3×AGL(1,𝔽5)11yescharacteristic, complemented
224#95—102no—
225#95—102no—
226#95—108no—
227C2×C3×C5#38111yescharacteristic
228C2×C3×C5#38714yes—
229C2×C3×C5#39011yescharacteristic
230C2×C3×C5#39111yescharacteristic
231C3×D5#39712yes—
232C3×D5#39812yes—
233C3×D5#39912yes—
234C3×D5#40012yes—
235C23×C3—54no—
236C23×C3—52no—
237C23×C3—52no—
238C23×C3—52no—
239C23×C3—52no—
240C23×C3—51no—
241C23×C3—51no—
242C23×C3C22×AGL(1,𝔽5)11yescharacteristic
243C2×C3×C4—22no—
244C2×C3×C4—24no—
245C2×C3×C4—22no—
246C2×C3×C4—52no—
247C2×C3×C4—104no—
248C2×C3×C4—104no—
249C2×C3×C4—102no—
250C2×C3×C4—108no—
251C2×C3×C4—108no—
252C2×C3×C4—104no—
253C2×C3×C4—104no—
254C2×C3×C4—104no—
255C2×C3×C4—102no—
256C2×C3×C4—204no—
257C2×C3×C4—202no—
258C2×C3×C4—202no—
259C2×C3×C4—204no—
260C2×C3×C4—204no—
261C2×C3×C4—202no—
262C2×C3×C4—202no—
263C2×C3×C4—202no—
264C2×C3×C4—202no—
265C2×C3×C4C22⋊AGL(1,𝔽5)12yes—
266C22×C5C2×C3×C22⋊C411yescharacteristic
267C22×C5C3×#11712yes—
268C22×C5C3×C4×D412yes—
269C22×C5C3×C4×Q812yes—
270C2×D5C3×#12112yes—
271C2×D5C3×C22⋊Q814yes—
272C2×D5C3×C22⋊Q812yes—
273C2×D5C3×#12312yes—
274C2×D5C3×#12314yes—
275C2×D5C3×#12312yes—
276C2×D5C3×#12411yescharacteristic
277C2×D5C3×#12411yescharacteristic
278C2×D5C3×#12514yes—
279C2×D5C3×#12614yes—
280C2×D5C3×#12711yescharacteristic
281C2×D5C3×C4⋊Q811yescharacteristic
282AGL(1,𝔽5)—44no—
283AGL(1,𝔽5)—44no—
284C4×C5—22no—
285C4×C5—42no—
286Dic5—22no—
287Dic5—42no—
288C24—51no—
289C22×C4—51no—
290C22×C4—52no—
291C22×C4—52no—
292C22×C4—54no—
293C22×C4—54no—
294C22×C4—52no—
295C22×C4—104no—
296C22×C4—104no—
297C22×C4—102no—
298C22×C4—102no—
299C22×C4—104no—
300C22×C4—104no—
301C22×C4—102no—
302C22×C4—102no—
303C22×C4—104no—
304C22×C4—104no—
305C22×C4—102no—
306C22×C4—104no—
307C22×C4—102no—
308C22×C4C2×C3×AGL(1,𝔽5)12yes—
309C22×C4C2×C3×AGL(1,𝔽5)11yescharacteristic
310C4⋊C4—22no—
311C4⋊C4—104no—
312C4⋊C4—102no—
313C3×C5#144511yescharacteristic, complemented, Hall
314C22×C3—54no—
315C22×C3—54no—
316C22×C3—54no—
317C22×C3—54no—
318C22×C3—52no—
319C22×C3—52no—
320C22×C3—52no—
321C22×C3—51no—
322C22×C3—51no—
323C22×C3—52no—
324C22×C3—51no—
325C22×C3—51no—
326C22×C3#413012yes—
327C22×C3D4×AGL(1,𝔽5)12yes—
328C22×C3Q8×AGL(1,𝔽5)12yes—
329C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
330C3×C4—22no—
331C3×C4—42no—
332C3×C4—102no—
333C3×C4—204no—
334C3×C4—204no—
335C3×C4—202no—
336C2×C5C3×#38111yescharacteristic
337C2×C5C3×#38714yes—
338C2×C5C3×#39011yescharacteristic
339C2×C5C3×#39111yescharacteristic
340D5C3×#39712yes—
341D5C3×#39812yes—
342D5C3×#39912yes—
343D5C3×#40012yes—
344C23—54no—
345C23—52no—
346C23—52no—
347C23—52no—
348C23—52no—
349C23—51no—
350C23—51no—
351C23C22×C3×AGL(1,𝔽5)11yescharacteristic
352C2×C4—22no—
353C2×C4—24no—
354C2×C4—22no—
355C2×C4—52no—
356C2×C4—104no—
357C2×C4—104no—
358C2×C4—102no—
359C2×C4—108no—
360C2×C4—108no—
361C2×C4—104no—
362C2×C4—104no—
363C2×C4—104no—
364C2×C4—102no—
365C2×C4—204no—
366C2×C4—202no—
367C2×C4—202no—
368C2×C4—204no—
369C2×C4—204no—
370C2×C4—202no—
371C2×C4—202no—
372C2×C4—202no—
373C2×C4—202no—
374C2×C4C3×C22⋊AGL(1,𝔽5)12yes—
375C2×C3—52no—
376C2×C3—52no—
377C2×C3—52no—
378C2×C3—52no—
379C2×C3#6571114yes—
380C2×C3#6576411yescharacteristic
381C2×C3#6578011yescharacteristic
382C2×C3(C2×AGL(1,𝔽5))⋊Q811yescharacteristic
383C5C3×#144511yescharacteristic, complemented, Sylow
384C22—54no—
385C22—54no—
386C22—54no—
387C22—54no—
388C22—52no—
389C22—52no—
390C22—52no—
391C22—51no—
392C22—51no—
393C22—52no—
394C22—51no—
395C22—51no—
396C22C3×#413012yes—
397C22C3×D4×AGL(1,𝔽5)12yes—
398C22C3×Q8×AGL(1,𝔽5)12yes—
399C22C2×C3×C22⋊AGL(1,𝔽5)11yescharacteristic
400C4—22no—
401C4—42no—
402C4—102no—
403C4—204no—
404C4—204no—
405C4—202no—
406C3#1060434511yescharacteristic, direct product factor, Sylow
407C2—52no—
408C2—52no—
409C2—52no—
410C2—52no—
411C2C3×#6571114yes—
412C2C3×#6576411yescharacteristic
413C2C3×#6578011yescharacteristic
414C2C3×(C2×AGL(1,𝔽5))⋊Q811yescharacteristic
subgroupG/H
centerC23×C3C22×AGL(1,𝔽5)
commutatorC23×C5C22×C3×C4
FrattiniC23C22×C3×AGL(1,𝔽5)
FittingC2×C3×C5×C4⋊C4C4
socleC23×C3×C5C22×C4
prime radicalC23×C3×D5C23
cosocleC23×D5C23×C3
hypercenterC2×C3×C4⋊C4AGL(1,𝔽5)
hypocenterC5C3×#1445
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23×C3 ◁ C2×C3×C4⋊C4

Complements

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

NH ≅ G/Ntype
5. #10604345C3direct (N × H)
10. C2×C3×C5×C4⋊C4C4semidirect
11. C2×C3×C4⋊Dic5C4semidirect
12. C2×C3×#683C4semidirect
56. C2×C5×C4⋊C4C3×C4semidirect
57. C2×C4⋊Dic5C3×C4semidirect
58. C2×#683C3×C4semidirect
104. C2×C3×C4⋊C4AGL(1,𝔽5)semidirect
223. C2×C4⋊C4C3×AGL(1,𝔽5)semidirect
313. C3×C5#1445semidirect
383. C5C3×#1445semidirect
406. C3#10604345direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×D5×C4⋊C42
C2×C3×(C2×AGL(1,𝔽5))⋊C42
#106043453
C3×#14455