Finite Group Databasebeta

← C2×C4×Dic3⋊AGL(1,𝔽5) #14319473

Subgroups — C2×C4×Dic3⋊AGL(1,𝔽5) #14319473

Order 1920.

1856 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×ΓL(1,𝔽16)C211yesmaximal, characteristic
2C22×C3×C4×AGL(1,𝔽5)C211yesmaximal, characteristic
3C22×Dic3⋊AGL(1,𝔽5)C212yesmaximal
4C2×C4×D5×Dic3C211yesmaximal, characteristic
5C2×(C2×AGL(1,𝔽5))⋊Dic3C212yesmaximal
6C4×Dic3⋊AGL(1,𝔽5)C218yesmaximal, direct product factor
7C2×C4×C4⋊AGL(1,𝔽5)—31nomaximal, Hall
8C23×ΓL(1,𝔽16)C2212yes—
9C23×C3×AGL(1,𝔽5)C2212yes—
10C22×C3×C4×D5C2211yescharacteristic
11C22×D5×Dic3C2211yescharacteristic
12C22×D5×Dic3C2211yescharacteristic
13C2×C4×ΓL(1,𝔽16)—24no—
14C2×C4×ΓL(1,𝔽16)C2214yes—
15C2×C3×C4×AGL(1,𝔽5)—24no—
16C2×C3×C4×AGL(1,𝔽5)C2214yes—
17C2×Dic3⋊AGL(1,𝔽5)C4116yesdirect product factor
18C2×Dic3⋊AGL(1,𝔽5)C2218yes—
19C2×C4×Dic15C411yescharacteristic, complemented
20C2×C4×C5×Dic3C411yescharacteristic, complemented
21C2×Dic3×Dic5C411yescharacteristic, complemented
22C2×Dic3×Dic5C411yescharacteristic, complemented
23C4×D5×Dic3C414yescomplemented
24C4×D5×Dic3C2214yes—
25(C2×AGL(1,𝔽5))⋊Dic3C2218yes—
26C2×C4×Dic3⋊C4—51nomaximal, Hall
27C22×C4⋊AGL(1,𝔽5)—32no—
28C22×C4×AGL(1,𝔽5)—31no—
29C22×C4×AGL(1,𝔽5)D311yescharacteristic
30C2×C42×D5—31no—
31C2×(C2×AGL(1,𝔽5))⋊C4—32no—
32C4×C4⋊AGL(1,𝔽5)—38no—
33C23×C3×D5C2311yescharacteristic
34C22×ΓL(1,𝔽16)—24no—
35C22×ΓL(1,𝔽16)—24no—
36C22×ΓL(1,𝔽16)C2×C418yes—
37C22×ΓL(1,𝔽16)C2314yes—
38C22×C3×AGL(1,𝔽5)—24no—
39C22×C3×AGL(1,𝔽5)—24no—
40C22×C3×AGL(1,𝔽5)C2×C418yes—
41C22×C3×AGL(1,𝔽5)C2314yes—
42C22×C3×C4×C5C2×C411yescharacteristic
43C22×Dic15C2×C411yescharacteristic
44C22×Dic15C2×C411yescharacteristic
45C22×C5×Dic3C2×C411yescharacteristic
46C22×C5×Dic3C2×C411yescharacteristic
47C22×C3×Dic5C2×C411yescharacteristic
48C2×C3×C4×D5C2×C412yes—
49C2×C3×C4×D5D414yes—
50C2×C3×C4×D5Q814yes—
51C2×C3×C4×D5C2312yes—
52C2×D5×Dic3—24no—
53C2×D5×Dic3C2×C412yes—
54C2×D5×Dic3C2×C412yes—
55C2×D5×Dic3C2×C418yes—
56C2×D5×Dic3C2312yes—
57C2×D5×Dic3C2312yes—
58C4×ΓL(1,𝔽16)—28no—
59C3×C4×AGL(1,𝔽5)—28no—
60Dic3⋊AGL(1,𝔽5)C2×C4132yesdirect product factor
61C4×Dic15C2×C414yescomplemented
62C4×C5×Dic3C2×C414yescomplemented
63Dic3×Dic5C2×C414yescomplemented
64Dic3×Dic5C2×C414yescomplemented
65C22×C3×C42—51no—
66C22×Dic3⋊C4—52no—
67C22×C4×Dic3—51no—
68C22×C4×Dic3—51no—
69C2×C3⋊#95—52no—
70C4×Dic3⋊C4—58no—
71C23×AGL(1,𝔽5)—32no—
72C23×AGL(1,𝔽5)C2×D312yes—
73C22×C4×D5—31no—
74C22×C4×D5—31no—
75C22×C4×D5C2×D311yescharacteristic
76C2×C4⋊AGL(1,𝔽5)—316no—
77C2×C4⋊AGL(1,𝔽5)—38no—
78C2×C4×AGL(1,𝔽5)—24no—
79C2×C4×AGL(1,𝔽5)—34no—
80C2×C4×AGL(1,𝔽5)—64no—
81C2×C4×AGL(1,𝔽5)C2×D314yes—
82C2×C42×C5—31no—
83C2×C4×Dic5—31no—
84C2×C4×Dic5—31no—
85C2×C4×Dic5—31no—
86C42×D5—34no—
87C42×D5—34no—
88(C2×AGL(1,𝔽5))⋊C4—38no—
89C2×C4×C4⋊C4—151noSylow
90C23×C3×C5C22×C411yescharacteristic
91C22×C3×D5C22×C411yescharacteristic
92C22×C3×D5C22×C414yes—
93C22×C3×D5C2×D412yes—
94C22×C3×D5C2×Q812yes—
95C22×C3×D5#4114yes—
96C22×C3×D5C2411yescharacteristic
97C2×ΓL(1,𝔽16)—216no—
98C2×ΓL(1,𝔽16)—24no—
99C2×ΓL(1,𝔽16)—24no—
100C2×ΓL(1,𝔽16)C22×C418yes—
101C2×C3×AGL(1,𝔽5)—216no—
102C2×C3×AGL(1,𝔽5)—24no—
103C2×C3×AGL(1,𝔽5)—24no—
104C2×C3×AGL(1,𝔽5)C22×C418yes—
105C2×C3×C4×C5C4⋊C414yescomplemented
106C2×C3×C4×C5C22×C412yes—
107C2×Dic15—22no—
108C2×Dic15C4214yescomplemented
109C2×Dic15C22×C412yes—
110C2×Dic15C22×C412yes—
111C2×C5×Dic3—22no—
112C2×C5×Dic3C4214yescomplemented
113C2×C5×Dic3C22×C412yes—
114C2×C5×Dic3C22×C412yes—
115C2×C3×Dic5C4⋊C414yescomplemented
116C2×C3×Dic5C22×C412yes—
117C3×C4×D5C4⋊C418yescomplemented
118C3×C4×D5C2×D414yes—
119C3×C4×D5C2×Q814yes—
120D5×Dic3—24no—
121D5×Dic3—24no—
122D5×Dic3C4218yescomplemented
123D5×Dic3C22×C418yes—
124C23×C3×C4—51no—
125C23×C3×C4—52no—
126C23×Dic3—51no—
127C23×Dic3—52no—
128C23×Dic3—51no—
129C2×C3×C42—54no—
130C2×C3×C42—104no—
131C2×Dic3⋊C4—516no—
132C2×Dic3⋊C4—58no—
133C2×C4×Dic3—51no—
134C2×C4×Dic3—51no—
135C2×C4×Dic3—51no—
136C2×C4×Dic3—54no—
137C2×C4×Dic3—54no—
138C2×C4×Dic3—54no—
139C2×C4×Dic3—104no—
140C2×C4×Dic3AGL(1,𝔽5)11yescharacteristic, complemented
141C3⋊#95—58no—
142C23×D5C22×D311yescharacteristic
143C22×AGL(1,𝔽5)—24no—
144C22×AGL(1,𝔽5)—24no—
145C22×AGL(1,𝔽5)—38no—
146C22×AGL(1,𝔽5)—34no—
147C22×AGL(1,𝔽5)—64no—
148C22×AGL(1,𝔽5)—64no—
149C22×AGL(1,𝔽5)C4×D318yes—
150C22×AGL(1,𝔽5)C22×D314yes—
151C22×C4×C5—31no—
152C22×C4×C5—31no—
153C22×C4×C5C4×D311yescharacteristic
154C22×Dic5—31no—
155C22×Dic5—31no—
156C22×Dic5C4×D311yescharacteristic
157C2×C4×D5—38no—
158C2×C4×D5—32no—
159C2×C4×D5—32no—
160C2×C4×D5—32no—
161C2×C4×D5—32no—
162C2×C4×D5—64no—
163C2×C4×D5Dic614yes—
164C2×C4×D5C4×D312yes—
165C2×C4×D5#6714yes—
166C2×C4×D5C22×D312yes—
167C4⋊AGL(1,𝔽5)—332no—
168C4×AGL(1,𝔽5)—28no—
169C4×AGL(1,𝔽5)—68no—
170C42×C5—34no—
171C4×Dic5—34no—
172C4×Dic5—34no—
173C4×Dic5—34no—
174C22×C4⋊C4—152no—
175C22×C42—51no—
176C22×C42—151no—
177C22×C42—151no—
178C4×C4⋊C4—158no—
179C2×#95—152no—
180C22×C3×C5C2×C4212yes—
181C22×C3×C5C2×C4⋊C412yes—
182C22×C3×C5#11712yes—
183C22×C3×C5C23×C411yescharacteristic
184C2×C3×D5C2×C4212yes—
185C2×C3×D5C2×C4⋊C412yes—
186C2×C3×D5#11712yes—
187C2×C3×D5C4×D418yes—
188C2×C3×D5C4×Q818yes—
189C2×C3×D5C23×C412yes—
190C2×C3×D5C22×D411yescharacteristic
191C2×C3×D5C22×Q811yescharacteristic
192C2×C3×D5C2×#4112yes—
193ΓL(1,𝔽16)—28no—
194C3×AGL(1,𝔽5)—28no—
195C3×C4×C5C2×C4⋊C414yescomplemented
196Dic15—22no—
197Dic15C2×C4214yescomplemented
198C3×Dic5C2×C4⋊C414yescomplemented
199C5×Dic3—22no—
200C5×Dic3C2×C4214yescomplemented
201C24×C3—51no—
202C22×C3×C4—51no—
203C22×C3×C4—54no—
204C22×C3×C4—54no—
205C22×C3×C4—52no—
206C22×C3×C4—52no—
207C22×C3×C4—54no—
208C22×C3×C4—58no—
209C22×C3×C4—104no—
210C22×C3×C4—104no—
211C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
212C22×Dic3—51no—
213C22×Dic3—58no—
214C22×Dic3—52no—
215C22×Dic3—52no—
216C22×Dic3—54no—
217C22×Dic3—58no—
218C22×Dic3—51no—
219C22×Dic3—52no—
220C22×Dic3—52no—
221C22×Dic3—104no—
222C22×Dic3—104no—
223C22×Dic3—104no—
224C22×Dic3C2×AGL(1,𝔽5)11yescharacteristic
225C22×Dic3C2×AGL(1,𝔽5)11yescharacteristic
226C3×C42—108no—
227Dic3⋊C4—532no—
228C4×Dic3—54no—
229C4×Dic3—54no—
230C4×Dic3—54no—
231C4×Dic3—108no—
232C4×Dic3C2×AGL(1,𝔽5)14yescomplemented
233C23×C5C2×C4×D311yescharacteristic
234C22×D5C2×Dic612yes—
235C22×D5C2×C4×D314yes—
236C22×D5C2×C4×D311yescharacteristic
237C22×D5#23514yes—
238C22×D5C2×#6712yes—
239C22×D5C23×D311yescharacteristic
240C2×AGL(1,𝔽5)—216no—
241C2×AGL(1,𝔽5)—24no—
242C2×AGL(1,𝔽5)—24no—
243C2×AGL(1,𝔽5)—38no—
244C2×AGL(1,𝔽5)—616no—
245C2×AGL(1,𝔽5)—64no—
246C2×AGL(1,𝔽5)—64no—
247C2×AGL(1,𝔽5)C2×C4×D318yes—
248C2×C4×C5—34no—
249C2×C4×C5—32no—
250C2×C4×C5—32no—
251C2×C4×C5—62no—
252C2×C4×C5Dic3⋊C414yescomplemented
253C2×C4×C5C2×C4×D312yes—
254C2×Dic5—34no—
255C2×Dic5—32no—
256C2×Dic5—32no—
257C2×Dic5—62no—
258C2×Dic5Dic3⋊C414yescomplemented
259C2×Dic5C2×C4×D312yes—
260C4×D5—38no—
261C4×D5—38no—
262C4×D5—64no—
263C4×D5—64no—
264C4×D5Dic3⋊C418yescomplemented
265C4×D5C2×Dic614yes—
266C4×D5C2×#6714yes—
267C23×C4—51no—
268C23×C4—52no—
269C23×C4—151no—
270C23×C4—151no—
271C23×C4—152no—
272C2×C4⋊C4—158no—
273C2×C4⋊C4—1516no—
274C2×C42—31no—
275C2×C42—54no—
276C2×C42—104no—
277C2×C42—154no—
278C2×C42—154no—
279C2×C42—151no—
280C2×C42—151no—
281C2×C42—151no—
282C2×C42—154no—
283C2×C42—304no—
284#95—158no—
285C2×C3×C5C4×C4⋊C414yescomplemented
286C2×C3×C5C22×C4211yescharacteristic
287C2×C3×C5C22×C4⋊C411yescharacteristic
288C2×C3×C5C2×#11711yescharacteristic
289C3×D5C4×C4⋊C414yescomplemented
290C3×D5C2×C4×D412yes—
291C3×D5C2×C4×Q812yes—
292C23×C3—54no—
293C23×C3—52no—
294C23×C3—52no—
295C23×C3—54no—
296C23×C3—51no—
297C23×C3—51no—
298C23×C3C22×AGL(1,𝔽5)11yescharacteristic
299C2×C3×C4—54no—
300C2×C3×C4—52no—
301C2×C3×C4—58no—
302C2×C3×C4—54no—
303C2×C3×C4—54no—
304C2×C3×C4—58no—
305C2×C3×C4—104no—
306C2×C3×C4—1016no—
307C2×C3×C4—104no—
308C2×C3×C4C4⋊AGL(1,𝔽5)14yescomplemented
309C2×C3×C4C22×AGL(1,𝔽5)12yes—
310C2×Dic3—22no—
311C2×Dic3—54no—
312C2×Dic3—52no—
313C2×Dic3—58no—
314C2×Dic3—58no—
315C2×Dic3—58no—
316C2×Dic3—52no—
317C2×Dic3—104no—
318C2×Dic3—104no—
319C2×Dic3—104no—
320C2×Dic3—1016no—
321C2×Dic3—104no—
322C2×Dic3—102no—
323C2×Dic3C4×AGL(1,𝔽5)14yescomplemented
324C2×Dic3C22×AGL(1,𝔽5)12yes—
325C2×Dic3C22×AGL(1,𝔽5)12yes—
326C22×C5C42×D312yes—
327C22×C5#87212yes—
328C22×C5C2×Dic3⋊C412yes—
329C22×C5C22×C4×D311yescharacteristic
330C2×D5C4×Dic618yes—
331C2×D5C42×D312yes—
332C2×D5#87212yes—
333C2×D5C2×Dic3⋊C412yes—
334C2×D5C4×#6718yes—
335C2×D5C22×Dic611yescharacteristic
336C2×D5C22×C4×D312yes—
337C2×D5C2×#23512yes—
338C2×D5C22×#6711yescharacteristic
339AGL(1,𝔽5)—28no—
340AGL(1,𝔽5)—68no—
341C4×C5—34no—
342C4×C5—62no—
343C4×C5C2×Dic3⋊C414yescomplemented
344Dic5—34no—
345Dic5—62no—
346Dic5C2×Dic3⋊C414yescomplemented
347C24—51no—
348C22×C4—31no—
349C22×C4—31no—
350C22×C4—51no—
351C22×C4—54no—
352C22×C4—54no—
353C22×C4—52no—
354C22×C4—52no—
355C22×C4—54no—
356C22×C4—58no—
357C22×C4—104no—
358C22×C4—104no—
359C22×C4—152no—
360C22×C4—152no—
361C22×C4—158no—
362C22×C4—152no—
363C22×C4—152no—
364C22×C4—151no—
365C22×C4—154no—
366C22×C4—158no—
367C22×C4—151no—
368C22×C4—304no—
369C22×C4—304no—
370C22×C4—304no—
371C22×C4D3×AGL(1,𝔽5)11yescharacteristic
372C4⋊C4—1532no—
373C42—34no—
374C42—108no—
375C42—154no—
376C42—154no—
377C42—154no—
378C42—308no—
379C3×C5C2×C4×C4⋊C411yescharacteristic, complemented, Hall
380C22×C3—58no—
381C22×C3—58no—
382C22×C3—52no—
383C22×C3—52no—
384C22×C3—52no—
385C22×C3—51no—
386C22×C3—51no—
387C22×C3—52no—
388C22×C3—52no—
389C22×C3C2×C4×AGL(1,𝔽5)12yes—
390C22×C3C2×C4⋊AGL(1,𝔽5)12yes—
391C22×C3#413012yes—
392C22×C3C23×AGL(1,𝔽5)11yescharacteristic
393C3×C4—54no—
394C3×C4—108no—
395C3×C4C2×C4⋊AGL(1,𝔽5)14yescomplemented
396Dic3—22no—
397Dic3—54no—
398Dic3—108no—
399Dic3—102no—
400Dic3C2×C4×AGL(1,𝔽5)14yescomplemented
401C2×C5C4×Dic3⋊C414yescomplemented
402C2×C5C2×C42×D311yescharacteristic
403C2×C5C2×#87211yescharacteristic
404C2×C5C22×Dic3⋊C411yescharacteristic
405D5C4×Dic3⋊C414yescomplemented
406D5C2×C4×Dic612yes—
407D5C2×C4×#6712yes—
408C23—54no—
409C23—54no—
410C23—52no—
411C23—52no—
412C23—51no—
413C23—51no—
414C23C2×D3×AGL(1,𝔽5)11yescharacteristic
415C2×C4—34no—
416C2×C4—32no—
417C2×C4—32no—
418C2×C4—54no—
419C2×C4—52no—
420C2×C4—54no—
421C2×C4—58no—
422C2×C4—54no—
423C2×C4—58no—
424C2×C4—62no—
425C2×C4—104no—
426C2×C4—104no—
427C2×C4—1016no—
428C2×C4—158no—
429C2×C4—158no—
430C2×C4—154no—
431C2×C4—152no—
432C2×C4—158no—
433C2×C4—152no—
434C2×C4—304no—
435C2×C4—304no—
436C2×C4—304no—
437C2×C4—304no—
438C2×C4—3016no—
439C2×C4—302no—
440C2×C4Dic3⋊AGL(1,𝔽5)14yesdirect product factor
441C2×C4C2×D3×AGL(1,𝔽5)12yes—
442C2×C3—52no—
443C2×C3—52no—
444C2×C3—54no—
445C2×C3C4×C4⋊AGL(1,𝔽5)14yescomplemented
446C2×C3C22×C4×AGL(1,𝔽5)11yescharacteristic
447C2×C3C22×C4⋊AGL(1,𝔽5)11yescharacteristic
448C2×C3C2×#413011yescharacteristic
449C5C2×C4×Dic3⋊C411yescharacteristic, complemented, Sylow
450C22—52no—
451C22—52no—
452C22—52no—
453C22—52no—
454C22—51no—
455C22—52no—
456C22—51no—
457C22—58no—
458C22—58no—
459C22#9190912yes—
460C22C4×D3×AGL(1,𝔽5)12yes—
461C22C2×Dic3⋊AGL(1,𝔽5)12yes—
462C22C22×D3×AGL(1,𝔽5)11yescharacteristic
463C4—34no—
464C4—54no—
465C4—62no—
466C4—108no—
467C4—154no—
468C4—308no—
469C4—302no—
470C4C2×Dic3⋊AGL(1,𝔽5)14yesdirect product factor
471C3C2×C4×C4⋊AGL(1,𝔽5)11yescharacteristic, complemented, Sylow
472C2—52no—
473C2—54no—
474C2—52no—
475C2C4×Dic3⋊AGL(1,𝔽5)14yesdirect product factor
476C2C2×#9190911yescharacteristic
477C2C2×C4×D3×AGL(1,𝔽5)11yescharacteristic
478C2C22×Dic3⋊AGL(1,𝔽5)11yescharacteristic
subgroupG/H
center, hypercenterC22×C4D3×AGL(1,𝔽5)
commutatorC2×C3×C5C22×C42
FrattiniC22C22×D3×AGL(1,𝔽5)
FittingC22×C3×C4×C5C2×C4
socleC23×C3×C5C22×C4
prime radicalC23×C3×D5C23
cosocleC22×C3×D5C24
hypocenterC3×C5C2×C4×C4⋊C4
LCSG ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
6. C4×Dic3⋊AGL(1,𝔽5)C2direct (N × H)
17. C2×Dic3⋊AGL(1,𝔽5)C4direct (N × H)
19. C2×C4×Dic15C4semidirect
20. C2×C4×C5×Dic3C4semidirect
21. C2×Dic3×Dic5C4semidirect
22. C2×Dic3×Dic5C4semidirect
23. C4×D5×Dic3C4semidirect
60. Dic3⋊AGL(1,𝔽5)C2×C4direct (N × H)
61. C4×Dic15C2×C4semidirect
62. C4×C5×Dic3C2×C4semidirect
63. Dic3×Dic5C2×C4semidirect
64. Dic3×Dic5C2×C4semidirect
105. C2×C3×C4×C5C4⋊C4semidirect
108. C2×Dic15C42semidirect
112. C2×C5×Dic3C42semidirect
115. C2×C3×Dic5C4⋊C4semidirect
117. C3×C4×D5C4⋊C4semidirect
122. D5×Dic3C42semidirect
140. C2×C4×Dic3AGL(1,𝔽5)semidirect
195. C3×C4×C5C2×C4⋊C4semidirect
197. Dic15C2×C42semidirect
198. C3×Dic5C2×C4⋊C4semidirect
200. C5×Dic3C2×C42semidirect
232. C4×Dic3C2×AGL(1,𝔽5)semidirect
252. C2×C4×C5Dic3⋊C4semidirect
258. C2×Dic5Dic3⋊C4semidirect
264. C4×D5Dic3⋊C4semidirect
285. C2×C3×C5C4×C4⋊C4semidirect
289. C3×D5C4×C4⋊C4semidirect
308. C2×C3×C4C4⋊AGL(1,𝔽5)semidirect
323. C2×Dic3C4×AGL(1,𝔽5)semidirect
343. C4×C5C2×Dic3⋊C4semidirect
346. Dic5C2×Dic3⋊C4semidirect
379. C3×C5C2×C4×C4⋊C4semidirect
395. C3×C4C2×C4⋊AGL(1,𝔽5)semidirect
400. Dic3C2×C4×AGL(1,𝔽5)semidirect
401. C2×C5C4×Dic3⋊C4semidirect
405. D5C4×Dic3⋊C4semidirect
440. C2×C4Dic3⋊AGL(1,𝔽5)direct (N × H)
445. C2×C3C4×C4⋊AGL(1,𝔽5)semidirect
449. C5C2×C4×Dic3⋊C4semidirect
470. C4C2×Dic3⋊AGL(1,𝔽5)direct (N × H)
471. C3C2×C4×C4⋊AGL(1,𝔽5)semidirect
475. C2C4×Dic3⋊AGL(1,𝔽5)direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4×ΓL(1,𝔽16)2
C22×C3×C4×AGL(1,𝔽5)2
C22×Dic3⋊AGL(1,𝔽5)2
C2×C4×D5×Dic32
C2×(C2×AGL(1,𝔽5))⋊Dic32
C4×Dic3⋊AGL(1,𝔽5)2
C2×C4×C4⋊AGL(1,𝔽5)3
C2×C4×Dic3⋊C45