Finite Group Databasebeta

← S4×#681 #14512980

Subgroups — S4×#681 #14512980

Order 1920.

594 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 594 conjugacy classes of subgroups — too many to draw. Only its 79 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
1C2×C4×C5×S4C211yesmaximal, characteristic
2A4×#681C211yesmaximal, characteristic, complemented
3#11752128C211yesmaximal, characteristic, complemented
4#11751571C212yesmaximal, complemented
5S4×#168C212yesmaximal, complemented
6D4×#681—31nomaximal, Hall
7C22×C5×S4C411yescharacteristic
8C2×C4×C5×A4C2211yescharacteristic
9C2×C5×SL(2,ℤ/4)C411yescharacteristic
10C4×C5×S4C411yescharacteristic
11C4×C5×S4C411yescharacteristic
12C4×C5×S4C2211yescharacteristic
13C4×C5×S4C2211yescharacteristic
14D3×#681—41nomaximal
15A4×#168C2212yescomplemented
16A4⋊#168C2212yescomplemented
17M16×S4—51nomaximal, Hall
18C2×C4×C5×D4—31no—
19C22×#681—31no—
20C22×#681D311yescharacteristic, complemented
21#65496—31no—
22#65496—31no—
23#65301—32no—
24#65300—32no—
25#65300—32no—
26D4×#168—32no—
27#65213—31no—
28C4×#681—31no—
29C22×C5×A4C2×C411yescharacteristic
30C2×C5×S4—22no—
31C2×C5×S4C2×C411yescharacteristic
32C2×C5×S4C2×C411yescharacteristic
33C2×C4×C5×D3—41no—
34C4×C5×A4C2×C411yescharacteristic
35C4×C5×A4C2311yescharacteristic
36C5×SL(2,ℤ/4)C2×C411yescharacteristic
37C5×SL(2,ℤ/4)C2×C411yescharacteristic
38#6710—41no—
39C3×#681—41no—
40#6650—42no—
41D3×#168—42no—
42C2×C4×S4D511yescharacteristic
43A4×M16—51no—
44#5409—51no—
45#5400—52no—
46C8×S4—52no—
47C22×C5×D4—31no—
48C23×C4×C5—31no—
49C23×C4×C5C2×D311yescharacteristic
50C4×C5×D4—31no—
51C4×C5×D4—31no—
52C4×C5×D4—31no—
53C4×C5×D4—31no—
54C4×C5×D4—31no—
55C4×C5×D4—31no—
56C4×C5×D4—31no—
57C4×C5×D4—31no—
58C2×C5×C4⋊C4—31no—
59C2×C5×C22⋊C4—31no—
60C2×C5×C22⋊C4—31no—
61C2×C42×C5—31no—
62C2×#681—31no—
63C2×#681—32no—
64C2×#681—31no—
65C2×#681—32no—
66C2×#681—31no—
67C2×#681—31no—
68C2×#681—61no—
69C2×#681—62no—
70C2×#681—61no—
71C22×#168—32no—
72C22×#168C2×D312yescomplemented
73C22⋊#168—32no—
74C22⋊#168—32no—
75C22⋊#168—32no—
76C22⋊#168—32no—
77C4⋊#168—32no—
78C4⋊#168—32no—
79#3935—32no—
80C4×#168—32no—
81D4×M16—151noSylow
82C22×C5×D3—41no—
83C2×C5×A4—21no—
84C2×C5×A4C22×C411yescharacteristic
85C5×S4—21no—
86C5×S4M1612yescomplemented
87C2×C3×C4×C5—41no—
88C2×C5×Dic3—41no—
89C4×C5×D3—41no—
90C4×C5×D3—41no—
91C4×C5×D3—41no—
92C4×C5×D3—41no—
93C3⋊#168—42no—
94C3×#168—42no—
95C22×S4Dic511yescharacteristic
96C2×C4×A4C2×D511yescharacteristic
97C2×SL(2,ℤ/4)Dic511yescharacteristic
98C4×S4Dic511yescharacteristic
99C4×S4Dic511yescharacteristic
100C4×S4C2×D511yescharacteristic
101C4×S4C2×D511yescharacteristic
102D3×M16—201no—
103C8×A4—52no—
104A4⋊C8—52no—
105C24×C5—31no—
106C24×C5C4×D311yescharacteristic
107C2×C5×D4—31no—
108C2×C5×D4—31no—
109C2×C5×D4—31no—
110C2×C5×D4—31no—
111C2×C5×D4—62no—
112C2×C5×D4—62no—
113C22×C4×C5—31no—
114C22×C4×C5—31no—
115C22×C4×C5—31no—
116C22×C4×C5—31no—
117C22×C4×C5—31no—
118C22×C4×C5—31no—
119C22×C4×C5—31no—
120C22×C4×C5—31no—
121C22×C4×C5—31no—
122C22×C4×C5—31no—
123C22×C4×C5—31no—
124C22×C4×C5—61no—
125C22×C4×C5—61no—
126C22×C4×C5—61no—
127C22×C4×C5—61no—
128C22×C4×C5C4×D311yescharacteristic
129C22×C4×C5C22×D311yescharacteristic
130C5×C4⋊C4—31no—
131C5×C4⋊C4—31no—
132C5×C4⋊C4—31no—
133C5×C4⋊C4—31no—
134C5×C22⋊C4—31no—
135C5×C22⋊C4—31no—
136C5×C22⋊C4—31no—
137C5×C22⋊C4—31no—
138C5×C22⋊C4—31no—
139C5×C22⋊C4—31no—
140C5×C22⋊C4—31no—
141C5×C22⋊C4—31no—
142C42×C5—31no—
143C42×C5—31no—
144C42×C5—31no—
145C42×C5—31no—
146#681—31no—
147#681—32no—
148#681—61no—
149#681—62no—
150#681—61no—
151#681—61no—
152#681—61no—
153#681—61no—
154#681—62no—
155#681S411yescharacteristic, direct product factor
156C2×#168—32no—
157C2×#168—32no—
158C2×#168—32no—
159C2×#168—32no—
160C2×#168—62no—
161C2×#168—62no—
162C22×M16—51no—
163C22×M16—151no—
164C2×C4×D4—31no—
165#436—152no—
166#435—152no—
167#435—152no—
168C8×D4—152no—
169#423—151no—
170#407—151no—
171#407—151no—
172C4×M16—151no—
173C22×C3×C5—41no—
174C2×C5×D3—41no—
175C2×C5×D3—41no—
176C2×C5×D3—82no—
177C5×A4C2×M1611yescharacteristic, complemented
178C3×C4×C5—41no—
179C3×C4×C5—41no—
180C5×Dic3—41no—
181C5×Dic3—41no—
182C22×A4C2×Dic511yescharacteristic
183C2×S4—22no—
184C2×S4C2×Dic511yescharacteristic
185C2×S4C2×Dic511yescharacteristic
186C2×C4×D3—41no—
187C4×A4C2×Dic511yescharacteristic
188C4×A4C22×D511yescharacteristic
189SL(2,ℤ/4)C2×Dic511yescharacteristic
190SL(2,ℤ/4)C2×Dic511yescharacteristic
191C3×M16—201no—
192#208—201no—
193ΓL(1,𝔽25)—202no—
194D3×C8—202no—
195C23×C5—21no—
196C23×C5—31no—
197C23×C5—31no—
198C23×C5—31no—
199C23×C5—31no—
200C23×C5—61no—
201C23×C5—61no—
202C23×C5—62no—
203C23×C5—61no—
204C23×C5—121no—
205C23×C5C2×C4×D311yescharacteristic
206C5×D4—32no—
207C5×D4—32no—
208C5×D4—62no—
209C5×D4—61no—
210C5×D4—62no—
211C5×D4—61no—
212C2×C4×C5—31no—
213C2×C4×C5—31no—
214C2×C4×C5—31no—
215C2×C4×C5—31no—
216C2×C4×C5—31no—
217C2×C4×C5—31no—
218C2×C4×C5—31no—
219C2×C4×C5—31no—
220C2×C4×C5—31no—
221C2×C4×C5—31no—
222C2×C4×C5—31no—
223C2×C4×C5—31no—
224C2×C4×C5—31no—
225C2×C4×C5—61no—
226C2×C4×C5—61no—
227C2×C4×C5—61no—
228C2×C4×C5—61no—
229C2×C4×C5—61no—
230C2×C4×C5—62no—
231C2×C4×C5—61no—
232C2×C4×C5—61no—
233C2×C4×C5—61no—
234C2×C4×C5—61no—
235C2×C4×C5—61no—
236C2×C4×C5—61no—
237C2×C4×C5—121no—
238C2×C4×C5C2×S411yescharacteristic
239#168—32no—
240#168—62no—
241#168—62no—
242#168C2×S412yescomplemented
243C22×D4—31no—
244C23×C4—31no—
245C23×C4D3×D511yescharacteristic
246C2×M16—151no—
247C2×M16—151no—
248C2×M16—152no—
249C2×M16—151no—
250C2×M16—152no—
251C2×M16—151no—
252C2×M16—301no—
253C2×M16—302no—
254C2×M16—301no—
255C22×C8—52no—
256C22×C8—152no—
257C4×D4—31no—
258C4×D4—31no—
259C4×D4—31no—
260C4×D4—31no—
261C4×D4—31no—
262C4×D4—31no—
263C4×D4—31no—
264C4×D4—31no—
265C2×C4⋊C4—31no—
266C2×C22⋊C4—31no—
267C2×C22⋊C4—31no—
268C2×C42—31no—
269C4⋊C8—152no—
270C4⋊C8—152no—
271C22⋊C8—152no—
272C22⋊C8—152no—
273C22⋊C8—152no—
274C22⋊C8—152no—
275#97—152no—
276C4×C8—152no—
277C2×C3×C5—41no—
278C2×C3×C5—81no—
279C5×D3—42no—
280C5×D3—81no—
281C22×D3—41no—
282C2×A4—21no—
283C2×A4C22×Dic511yescharacteristic
284S4—21no—
285S4#68112yesdirect product factor
286C2×C3×C4—41no—
287C2×Dic3—41no—
288C4×D3—41no—
289C4×D3—41no—
290C4×D3—41no—
291C4×D3—41no—
292C3×C8—202no—
293C3⋊C8—202no—
294C22×C5—31no—
295C22×C5—31no—
296C22×C5—32no—
297C22×C5—31no—
298C22×C5—61no—
299C22×C5—61no—
300C22×C5—61no—
301C22×C5—61no—
302C22×C5—61no—
303C22×C5—61no—
304C22×C5—61no—
305C22×C5—61no—
306C22×C5—61no—
307C22×C5—122no—
308C22×C5—122no—
309C22×C5D3×M1611yescharacteristic, complemented
310C22×C5C4×S411yescharacteristic
311C4×C5—32no—
312C4×C5—31no—
313C4×C5—31no—
314C4×C5—61no—
315C4×C5—61no—
316C4×C5—61no—
317C4×C5—61no—
318C4×C5—61no—
319C4×C5C4×S411yescharacteristic
320C4×C5C22×S411yescharacteristic
321C24—31no—
322C24D3×Dic511yescharacteristic
323C2×D4—31no—
324C2×D4—31no—
325C2×D4—31no—
326C2×D4—31no—
327C2×D4—62no—
328C2×D4—62no—
329C22×C4—31no—
330C22×C4—31no—
331C22×C4—31no—
332C22×C4—31no—
333C22×C4—31no—
334C22×C4—31no—
335C22×C4—31no—
336C22×C4—31no—
337C22×C4—31no—
338C22×C4—31no—
339C22×C4—31no—
340C22×C4—61no—
341C22×C4—61no—
342C22×C4—61no—
343C22×C4—61no—
344C22×C4D3×Dic511yescharacteristic
345C22×C4C2×D3×D511yescharacteristic
346M16—51no—
347M16—152no—
348M16—151no—
349M16—302no—
350M16—301no—
351M16—302no—
352M16—301no—
353M16—301no—
354M16—301no—
355M16—301no—
356C2×C8—152no—
357C2×C8—152no—
358C2×C8—152no—
359C2×C8—152no—
360C2×C8—302no—
361C2×C8—302no—
362C4⋊C4—31no—
363C4⋊C4—31no—
364C4⋊C4—31no—
365C4⋊C4—31no—
366C22⋊C4—31no—
367C22⋊C4—31no—
368C22⋊C4—31no—
369C22⋊C4—31no—
370C22⋊C4—31no—
371C22⋊C4—31no—
372C22⋊C4—31no—
373C22⋊C4—31no—
374C42—31no—
375C42—31no—
376C42—31no—
377C42—31no—
378C3×C5—41noHall
379C22×C3—41no—
380C2×D3—41no—
381C2×D3—41no—
382C2×D3—82no—
383A4C2×#68111yescharacteristic, complemented
384C3×C4—41no—
385C3×C4—41no—
386Dic3—41no—
387Dic3—41no—
388C2×C5—21no—
389C2×C5—31no—
390C2×C5—31no—
391C2×C5—61no—
392C2×C5—62no—
393C2×C5—121no—
394C2×C5C2×C4×S411yescharacteristic
395C23—21no—
396C23—31no—
397C23—31no—
398C23—31no—
399C23—31no—
400C23—61no—
401C23—61no—
402C23—62no—
403C23—61no—
404C23—121no—
405C23C2×D3×Dic511yescharacteristic
406D4—32no—
407D4—32no—
408D4—62no—
409D4—61no—
410D4—62no—
411D4—61no—
412C2×C4—31no—
413C2×C4—31no—
414C2×C4—31no—
415C2×C4—31no—
416C2×C4—31no—
417C2×C4—31no—
418C2×C4—31no—
419C2×C4—31no—
420C2×C4—31no—
421C2×C4—31no—
422C2×C4—31no—
423C2×C4—31no—
424C2×C4—31no—
425C2×C4—61no—
426C2×C4—61no—
427C2×C4—61no—
428C2×C4—61no—
429C2×C4—61no—
430C2×C4—62no—
431C2×C4—61no—
432C2×C4—61no—
433C2×C4—61no—
434C2×C4—61no—
435C2×C4—61no—
436C2×C4—61no—
437C2×C4—121no—
438C2×C4D5×S411yescharacteristic
439C8—52no—
440C8—152no—
441C8—302no—
442C8—302no—
443C2×C3—41no—
444C2×C3—81no—
445D3—42no—
446D3—81no—
447C5M16×S411yescharacteristic, complemented, Sylow
448C22—31no—
449C22—31no—
450C22—32no—
451C22—31no—
452C22—61no—
453C22—61no—
454C22—61no—
455C22—61no—
456C22—61no—
457C22—61no—
458C22—61no—
459C22—61no—
460C22—61no—
461C22—122no—
462C22—122no—
463C22D3×#68111yescharacteristic, complemented
464C22Dic5×S411yescharacteristic
465C4—32no—
466C4—31no—
467C4—31no—
468C4—61no—
469C4—61no—
470C4—61no—
471C4—61no—
472C4—61no—
473C4Dic5×S411yescharacteristic
474C4C2×D5×S411yescharacteristic
475C3—41noSylow
476C2—21no—
477C2—31no—
478C2—31no—
479C2—61no—
480C2—62no—
481C2—121no—
482C2C2×Dic5×S411yescharacteristic
subgroupG/H
center, FrattiniC4C2×D5×S4
commutatorC2×C5×A4C22×C4
FittingC23×C4×C5C2×D3
socleC23×C5C2×C4×D3
prime radicalC22×C5×S4C4
cosocleC4×C5×A4C23
hypercenterC2×C4D5×S4
hypocenterC5×A4C2×M16
LCSG ▷ C2×C5×A4 ▷ C5×A4
UCSC1 ◁ C4 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. A4×#681C2semidirect
3. #11752128C2semidirect
4. #11751571C2semidirect
5. S4×#168C2semidirect
15. A4×#168C22semidirect
16. A4⋊#168C22semidirect
20. C22×#681D3semidirect
72. C22×#168C2×D3semidirect
86. C5×S4M16semidirect
155. #681S4direct (N × H)
177. C5×A4C2×M16semidirect
242. #168C2×S4semidirect
285. S4#681direct (N × H)
309. C22×C5D3×M16semidirect
383. A4C2×#681semidirect
447. C5M16×S4semidirect
463. C22D3×#681semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C5×S42
A4×#6812
#117521282
#117515712
S4×#1682
D4×#6813
D3×#6814
M16×S45