Finite Group Databasebeta

← C2×A4×ΓL(1,𝔽25) #11923299

Subgroups — C2×A4×ΓL(1,𝔽25) #11923299

Order 1152.

822 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 822 conjugacy classes of subgroups — too many to draw. Only its 123 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×D3×A4C211yesmaximal, characteristic
2C2×C3×C8×A4C211yesmaximal, characteristic, complemented
3C2×A4×C3⋊C8C211yesmaximal, characteristic, complemented
4A4×ΓL(1,𝔽25)C214yesmaximal, direct product factor
5C23×ΓL(1,𝔽25)C311yesmaximal, characteristic, complemented
6C2×A4×M16—31nomaximal
7C22×D3×A4C411yescharacteristic
8C2×C3×C4×A4C2211yescharacteristic
9C2×Dic3×A4C411yescharacteristic
10C4×D3×A4C412yes—
11C4×D3×A4C2212yes—
12C2×C3×ΓL(1,𝔽25)—41nomaximal
13C3×C8×A4C2212yescomplemented
14A4×C3⋊C8C2212yescomplemented
15C23×C4×D3C2×C311yescharacteristic
16C22×C4×A4—31no—
17C23×C3×C8C2×C311yescharacteristic, complemented
18C23×C3⋊C8C2×C311yescharacteristic, complemented
19C22×ΓL(1,𝔽25)—31no—
20C22×ΓL(1,𝔽25)—31no—
21C22×ΓL(1,𝔽25)—31no—
22C22×ΓL(1,𝔽25)—31no—
23C22×ΓL(1,𝔽25)—34no—
24C22×ΓL(1,𝔽25)C2×C314yescomplemented
25A4×M16—34no—
26C2×C8×A4—21no—
27C2×C8×A4—31no—
28C2×C8×A4D311yescharacteristic, complemented
29C22×C3×A4C2×C411yescharacteristic
30C2×D3×A4—22no—
31C2×D3×A4C2×C412yes—
32C2×C3×C4×D3—41no—
33C3×C4×A4C2×C411yescharacteristic
34C3×C4×A4C2311yescharacteristic
35Dic3×A4C2×C412yes—
36C2×C32×C8—41no—
37C2×C3×C3⋊C8—41no—
38C3×ΓL(1,𝔽25)—44no—
39C23×M16—31noSylow
40C24×D3C3×C411yescharacteristic
41C23×A4—31no—
42C23×C3×C4C22×C311yescharacteristic
43C23×Dic3C3×C411yescharacteristic
44C22×C4×D3—31no—
45C22×C4×D3—31no—
46C22×C4×D3—31no—
47C22×C4×D3—32no—
48C22×C4×D3—31no—
49C22×C4×D3—32no—
50C22×C4×D3C3×C412yes—
51C22×C4×D3C22×C312yes—
52C2×C4×A4—21no—
53C2×C4×A4—31no—
54C2×C4×A4—32no—
55C2×C4×A4—32no—
56C2×C4×A4C2×D311yescharacteristic
57C2×C3×M16—121no—
58C22×C3×C8—31no—
59C22×C3×C8—31no—
60C22×C3×C8—32no—
61C22×C3×C8C22×C312yescomplemented
62C22×C3⋊C8—31no—
63C22×C3⋊C8—31no—
64C22×C3⋊C8—32no—
65C22×C3⋊C8C22×C312yescomplemented
66C2×ΓL(1,𝔽25)—32no—
67C2×ΓL(1,𝔽25)—34no—
68C2×ΓL(1,𝔽25)—34no—
69C2×ΓL(1,𝔽25)—34no—
70C2×ΓL(1,𝔽25)—31no—
71C2×ΓL(1,𝔽25)—31no—
72C2×ΓL(1,𝔽25)—31no—
73C2×ΓL(1,𝔽25)—34no—
74C2×ΓL(1,𝔽25)—34no—
75C2×ΓL(1,𝔽25)—34no—
76C2×ΓL(1,𝔽25)—34no—
77C2×ΓL(1,𝔽25)—34no—
78C2×ΓL(1,𝔽25)A411yescharacteristic, direct product factor
79C8×A4—22no—
80C8×A4—32no—
81C8×A4C2×D312yescomplemented
82C22×C3×D3—41no—
83C2×C3×A4M1612yescomplemented
84C2×C3×A4C22×C411yescharacteristic
85D3×A4—22no—
86C2×C32×C4—41no—
87C2×C3×Dic3—41no—
88C3×C4×D3—42no—
89C3×C4×D3—42no—
90C32×C8—42no—
91C3×C3⋊C8—42no—
92C24×C4—31no—
93C22×M16—34no—
94C22×M16—91no—
95C22×M16—91no—
96C22×M16—91no—
97C22×M16—94no—
98C22×M16—91no—
99C23×C8—31no—
100C23×C8C3×D311yescharacteristic, complemented
101C24×C3C2×C3×C411yescharacteristic
102C23×D3—22no—
103C23×D3—31no—
104C23×D3—31no—
105C23×D3—32no—
106C23×D3—62no—
107C23×D3C2×C3×C412yes—
108C22×A4—21no—
109C22×A4—32no—
110C22×A4—62no—
111C22×A4C4×D311yescharacteristic
112C22×C3×C4—31no—
113C22×C3×C4—31no—
114C22×C3×C4—31no—
115C22×C3×C4—31no—
116C22×C3×C4—121no—
117C22×C3×C4C2×C3×C411yescharacteristic
118C22×C3×C4C23×C311yescharacteristic
119C22×Dic3—31no—
120C22×Dic3—31no—
121C22×Dic3—32no—
122C22×Dic3C2×C3×C412yes—
123C2×C4×D3—32no—
124C2×C4×D3—32no—
125C2×C4×D3—34no—
126C2×C4×D3—32no—
127C2×C4×D3—34no—
128C2×C4×D3—31no—
129C2×C4×D3—31no—
130C2×C4×D3—31no—
131C2×C4×D3—32no—
132C2×C4×D3—32no—
133C2×C4×D3—32no—
134C2×C4×D3—32no—
135C2×C4×D3—32no—
136C2×C4×D3—32no—
137C2×C4×D3—32no—
138C2×C4×D3—32no—
139C2×C4×D3—32no—
140C2×C4×D3—32no—
141C2×C4×D3C2×A411yescharacteristic
142C4×A4—21no—
143C4×A4—21no—
144C4×A4—32no—
145C4×A4C4×D311yescharacteristic
146C4×A4C22×D311yescharacteristic
147C3×M16—124no—
148C2×C3×C8—32no—
149C2×C3×C8—31no—
150C2×C3×C8—32no—
151C2×C3×C8—32no—
152C2×C3×C8—32no—
153C2×C3×C8—41no—
154C2×C3×C8—81no—
155C2×C3×C8—121no—
156C2×C3×C8C2×A411yescharacteristic, complemented
157C2×C3⋊C8—32no—
158C2×C3⋊C8—31no—
159C2×C3⋊C8—32no—
160C2×C3⋊C8—32no—
161C2×C3⋊C8—32no—
162C2×C3⋊C8C2×A411yescharacteristic, complemented
163ΓL(1,𝔽25)—38no—
164ΓL(1,𝔽25)—34no—
165ΓL(1,𝔽25)—34no—
166ΓL(1,𝔽25)—34no—
167ΓL(1,𝔽25)C2×A414yesdirect product factor
168C22×C32—41no—
169C2×C3×D3—42no—
170C2×C3×D3—82no—
171C3×A4C2×M1611yescharacteristic, complemented
172C32×C4—41no—
173C32×C4—41no—
174C3×Dic3—42no—
175C25—31no—
176C23×C4—31no—
177C23×C4—32no—
178C23×C4—32no—
179C23×C4—91no—
180C23×C4—91no—
181C23×C4—92no—
182C23×C4—92no—
183C23×C4—91no—
184C23×C4—91no—
185C23×C4C2×C3×D311yescharacteristic
186C2×M16—31no—
187C2×M16—91no—
188C2×M16—94no—
189C2×M16—94no—
190C2×M16—91no—
191C2×M16—94no—
192C2×M16—94no—
193C2×M16—92no—
194C2×M16—94no—
195C2×M16—94no—
196C2×M16—94no—
197C2×M16—94no—
198C2×M16—91no—
199C22×C8—31no—
200C22×C8—31no—
201C22×C8—32no—
202C22×C8—32no—
203C22×C8—91no—
204C22×C8—92no—
205C22×C8—91no—
206C22×C8C2×C3×D312yescomplemented
207C23×C3—31no—
208C23×C3—32no—
209C23×C3—31no—
210C23×C3—121no—
211C23×C3C3×M1612yescomplemented
212C23×C3C22×C3×C411yescharacteristic
213C22×D3—22no—
214C22×D3—32no—
215C22×D3—31no—
216C22×D3—32no—
217C22×D3—32no—
218C22×D3—32no—
219C22×D3—62no—
220C22×D3—62no—
221C22×D3—64no—
222C22×D3—64no—
223C22×D3—62no—
224C22×D3—62no—
225C22×D3—62no—
226C22×D3C4×A411yescharacteristic
227C2×A4—22no—
228C2×A4—21no—
229C2×A4—62no—
230C2×A4ΓL(1,𝔽25)12yesdirect product factor
231C2×A4C2×C4×D311yescharacteristic
232C2×C3×C4—32no—
233C2×C3×C4—31no—
234C2×C3×C4—31no—
235C2×C3×C4—31no—
236C2×C3×C4—31no—
237C2×C3×C4—31no—
238C2×C3×C4—31no—
239C2×C3×C4—31no—
240C2×C3×C4—41no—
241C2×C3×C4—81no—
242C2×C3×C4—122no—
243C2×C3×C4—122no—
244C2×C3×C4—121no—
245C2×C3×C4C22×A411yescharacteristic
246C2×Dic3—32no—
247C2×Dic3—31no—
248C2×Dic3—32no—
249C2×Dic3—32no—
250C2×Dic3—32no—
251C2×Dic3C4×A411yescharacteristic
252C4×D3—34no—
253C4×D3—34no—
254C4×D3—32no—
255C4×D3—32no—
256C4×D3—32no—
257C4×D3—32no—
258C4×D3—32no—
259C4×D3—32no—
260C4×D3C4×A412yes—
261C4×D3C22×A412yes—
262C3×C8—32no—
263C3×C8—42no—
264C3×C8—82no—
265C3×C8—122no—
266C3×C8C22×A412yescomplemented
267C3⋊C8—32no—
268C3⋊C8C22×A412yescomplemented
269C2×C32—41no—
270C2×C32—42no—
271C3×D3—82no—
272C24—32no—
273C24—62no—
274C24—91no—
275C24—92no—
276C24—91no—
277C24—182no—
278C24C3×C4×D311yescharacteristic
279C22×C4—31no—
280C22×C4—31no—
281C22×C4—31no—
282C22×C4—31no—
283C22×C4—31no—
284C22×C4—32no—
285C22×C4—92no—
286C22×C4—92no—
287C22×C4—92no—
288C22×C4—91no—
289C22×C4—92no—
290C22×C4—91no—
291C22×C4—91no—
292C22×C4—92no—
293C22×C4—94no—
294C22×C4—94no—
295C22×C4—92no—
296C22×C4—92no—
297C22×C4—92no—
298C22×C4—92no—
299C22×C4—92no—
300C22×C4—92no—
301C22×C4—92no—
302C22×C4—92no—
303C22×C4—91no—
304C22×C4—92no—
305C22×C4—91no—
306C22×C4C3×C4×D311yescharacteristic
307C22×C4C22×C3×D311yescharacteristic
308M16—34no—
309M16—94no—
310M16—94no—
311M16—98no—
312M16—94no—
313C2×C8—32no—
314C2×C8—31no—
315C2×C8—31no—
316C2×C8—32no—
317C2×C8—32no—
318C2×C8—32no—
319C2×C8—92no—
320C2×C8—92no—
321C2×C8—92no—
322C2×C8—91no—
323C2×C8—92no—
324C2×C8D3×A411yescharacteristic, complemented
325C22×C3—32no—
326C22×C3—31no—
327C22×C3—32no—
328C22×C3—31no—
329C22×C3—31no—
330C22×C3—32no—
331C22×C3—32no—
332C22×C3—41no—
333C22×C3—81no—
334C22×C3—122no—
335C22×C3—242no—
336C22×C3C2×C3×M1611yescharacteristic, complemented
337C22×C3C2×C4×A411yescharacteristic
338C2×D3—22no—
339C2×D3—32no—
340C2×D3—62no—
341C2×D3—64no—
342C2×D3—62no—
343C2×D3—64no—
344C2×D3—62no—
345C2×D3—62no—
346C2×D3—62no—
347C2×D3C2×C4×A412yes—
348A4—21no—
349A4C2×ΓL(1,𝔽25)11yescharacteristic, direct product factor
350C3×C4—31no—
351C3×C4—31no—
352C3×C4—41no—
353C3×C4—41no—
354C3×C4—81no—
355C3×C4—81no—
356C3×C4—122no—
357C3×C4C2×C4×A411yescharacteristic
358C3×C4C23×A411yescharacteristic
359Dic3—32no—
360Dic3C2×C4×A412yes—
361C32—41noSylow
362C23—31no—
363C23—32no—
364C23—31no—
365C23—31no—
366C23—62no—
367C23—92no—
368C23—92no—
369C23—92no—
370C23—92no—
371C23—91no—
372C23—182no—
373C23—182no—
374C23—182no—
375C23—184no—
376C23—182no—
377C23—182no—
378C23—184no—
379C23C3×ΓL(1,𝔽25)12yescomplemented
380C23C2×C3×C4×D311yescharacteristic
381C2×C4—32no—
382C2×C4—31no—
383C2×C4—31no—
384C2×C4—31no—
385C2×C4—31no—
386C2×C4—32no—
387C2×C4—31no—
388C2×C4—31no—
389C2×C4—32no—
390C2×C4—31no—
391C2×C4—31no—
392C2×C4—92no—
393C2×C4—92no—
394C2×C4—92no—
395C2×C4—92no—
396C2×C4—92no—
397C2×C4—92no—
398C2×C4—94no—
399C2×C4—94no—
400C2×C4—92no—
401C2×C4—91no—
402C2×C4—92no—
403C2×C4—92no—
404C2×C4—92no—
405C2×C4C2×D3×A411yescharacteristic
406C8—32no—
407C8—32no—
408C8—92no—
409C8C2×D3×A412yescomplemented
410C2×C3—32no—
411C2×C3—31no—
412C2×C3—31no—
413C2×C3—41no—
414C2×C3—42no—
415C2×C3—81no—
416C2×C3—82no—
417C2×C3—242no—
418C2×C3A4×M1612yescomplemented
419C2×C3C22×C4×A411yescharacteristic
420D3—22no—
421D3—62no—
422C22—32no—
423C22—31no—
424C22—32no—
425C22—31no—
426C22—31no—
427C22—32no—
428C22—32no—
429C22—32no—
430C22—62no—
431C22—92no—
432C22—182no—
433C22—182no—
434C22—184no—
435C22—182no—
436C22—182no—
437C22—184no—
438C22—182no—
439C22C2×C3×ΓL(1,𝔽25)11yescharacteristic, complemented
440C22C4×D3×A411yescharacteristic
441C4—31no—
442C4—31no—
443C4—32no—
444C4—92no—
445C4C4×D3×A411yescharacteristic
446C4C22×D3×A411yescharacteristic
447C3—41no—
448C3—81no—
449C3C2×A4×M1611yescharacteristic, complemented
450C2—31no—
451C2—32no—
452C2—31no—
453C2—62no—
454C2—182no—
455C2A4×ΓL(1,𝔽25)12yesdirect product factor
456C2C2×C4×D3×A411yescharacteristic
subgroupG/H
centerC2×C4C2×D3×A4
commutatorC23×C3C22×C3×C4
FrattiniC4C22×D3×A4
FittingC23×C3×C8C2×C3
socleC24×C3C2×C3×C4
prime radicalC22×D3×A4C4
cosocleC22×C3×C4C23×C3
hypercenterC2×C8D3×A4
hypocenterC22×C3C2×C3×M16
LCSG ▷ C23×C3 ▷ C22×C3
UCSC1 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
2. C2×C3×C8×A4C2semidirect
3. C2×A4×C3⋊C8C2semidirect
4. A4×ΓL(1,𝔽25)C2direct (N × H)
5. C23×ΓL(1,𝔽25)C3semidirect
13. C3×C8×A4C22semidirect
14. A4×C3⋊C8C22semidirect
17. C23×C3×C8C2×C3semidirect
18. C23×C3⋊C8C2×C3semidirect
24. C22×ΓL(1,𝔽25)C2×C3semidirect
28. C2×C8×A4D3semidirect
61. C22×C3×C8C22×C3semidirect
65. C22×C3⋊C8C22×C3semidirect
78. C2×ΓL(1,𝔽25)A4direct (N × H)
81. C8×A4C2×D3semidirect
83. C2×C3×A4M16semidirect
100. C23×C8C3×D3semidirect
156. C2×C3×C8C2×A4semidirect
162. C2×C3⋊C8C2×A4semidirect
167. ΓL(1,𝔽25)C2×A4direct (N × H)
171. C3×A4C2×M16semidirect
206. C22×C8C2×C3×D3semidirect
211. C23×C3C3×M16semidirect
230. C2×A4ΓL(1,𝔽25)direct (N × H)
266. C3×C8C22×A4semidirect
268. C3⋊C8C22×A4semidirect
324. C2×C8D3×A4semidirect
336. C22×C3C2×C3×M16semidirect
349. A4C2×ΓL(1,𝔽25)direct (N × H)
379. C23C3×ΓL(1,𝔽25)semidirect
409. C8C2×D3×A4semidirect
418. C2×C3A4×M16semidirect
439. C22C2×C3×ΓL(1,𝔽25)semidirect
449. C3C2×A4×M16semidirect
455. C2A4×ΓL(1,𝔽25)direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C4×D3×A42
C2×C3×C8×A42
C2×A4×C3⋊C82
A4×ΓL(1,𝔽25)2
C23×ΓL(1,𝔽25)3
C2×A4×M163
C2×C3×ΓL(1,𝔽25)4