Finite Group Databasebeta

← C113×#22572 #43290592

Subgroups — C113×#22572 #43290592

Order 28928.

1006 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#416×C113C211yesmaximal, characteristic
2C113×#2346C211yesmaximal, characteristic, complemented
3C113×#2339C211yesmaximal, characteristic
4C113×#1933C212yesmaximal, complemented
5C113×#1927C212yesmaximal, complemented
6C113×#1864C212yesmaximal, complemented
7C113×#1825C211yesmaximal, characteristic, complemented
8C113×#1807C211yesmaximal, characteristic, complemented
9C113×#1788C211yesmaximal, characteristic
10C4×#102×C113C212yesmaximal, complemented
11C8×C22⋊C4×C113C211yesmaximal, characteristic, complemented
12C22×#41×C113C2211yescharacteristic
13C23×C8×C113C2211yescharacteristic
14C2×C4×D4×C113C2212yes—
15C2×#117×C113C2211yescharacteristic
16C2×#106×C113C2212yes—
17#416×C113C418yescomplemented
18C2×#103×C113C2212yes—
19C2×#103×C113C2211yescharacteristic
20C2×#103×C113C2211yescharacteristic
21C2×#102×C113C2212yes—
22C2×#102×C113C2211yescharacteristic
23C2×#102×C113C2211yescharacteristic
24C2×C22⋊C8×C113C2211yescharacteristic
25C2×C4×C8×C113C2212yes—
26#391×C113C2212yes—
27#389×C113C2212yes—
28#388×C113—22no—
29#386×C113—22no—
30#382×C113—21no—
31#381×C113—21no—
32#380×C113C2211yescharacteristic
33#380×C113C2211yescharacteristic
34C4×C4⋊C4×C113C2212yes—
35C4×C22⋊C4×C113C2211yescharacteristic
36#376×C113—21no—
37C2×#95×C113—21no—
38#340×C113C2212yes—
39#340×C113C2212yes—
40#340×C113C2212yes—
41#336×C113C2212yes—
42C2×#41×C113—22no—
43C2×#41×C113—24no—
44C2×#41×C113—42no—
45C2×#41×C113C2×C414yes—
46C22×Q8×C113C2311yescharacteristic
47C22×D4×C113—21no—
48C22×D4×C113C2311yescharacteristic
49C23×C4×C113—21no—
50C23×C4×C113—22no—
51C23×C4×C113C2311yescharacteristic
52C22×C8×C113—42no—
53C22×C8×C113C2×C414yescomplemented
54C22×C8×C113C2312yes—
55C22×C8×C113C2311yescharacteristic
56C22×C8×C113C2311yescharacteristic
57C4×D4×C113—24no—
58C4×D4×C113—42no—
59#117×C113—22no—
60#117×C113C2×C414yescomplemented
61C2×C4⋊C4×C113—22no—
62C2×C4⋊C4×C113C2312yes—
63C2×C4⋊C4×C113C2312yes—
64C2×C4⋊C4×C113C2311yescharacteristic
65C2×C4⋊C4×C113C2311yescharacteristic
66C2×C22⋊C4×C113—21no—
67C2×C22⋊C4×C113—22no—
68C2×C22⋊C4×C113—21no—
69C2×C22⋊C4×C113—21no—
70C2×C22⋊C4×C113D412yes—
71C2×C42×C113—21no—
72C2×C42×C113—22no—
73C2×C42×C113C2312yes—
74#106×C113—42no—
75#103×C113—42no—
76#103×C113C2×C414yescomplemented
77#103×C113C2×C414yescomplemented
78#102×C113—24no—
79#102×C113C2×C414yes—
80#102×C113C2×C414yes—
81C22⋊C8×C113—22no—
82C4×C8×C113—42no—
83#95×C113—21no—
84#95×C113—22no—
85#95×C113—21no—
86#95×C113—22no—
87#95×C113—22no—
88#95×C113—22no—
89#95×C113D411yescharacteristic
90#95×C113D411yescharacteristic
91C24×C113—21no—
92C24×C113C2×D411yescharacteristic
93#41×C113—28no—
94#41×C113—44no—
95#41×C113—48no—
96C2×Q8×C113—21no—
97C2×Q8×C113—22no—
98C2×Q8×C113—22no—
99C2×Q8×C113C22×C412yes—
100C2×D4×C113—24no—
101C2×D4×C113—21no—
102C2×D4×C113—22no—
103C2×D4×C113—44no—
104C2×D4×C113C22×C412yes—
105C2×D4×C113#4114yes—
106C22×C4×C113—22no—
107C22×C4×C113—22no—
108C22×C4×C113—21no—
109C22×C4×C113—21no—
110C22×C4×C113—21no—
111C22×C4×C113—24no—
112C22×C4×C113—22no—
113C22×C4×C113—22no—
114C22×C4×C113—44no—
115C22×C4×C113—44no—
116C22×C4×C113—44no—
117C22×C4×C113—82no—
118C22×C4×C113C22×C412yes—
119C22×C4×C113C2×D412yes—
120C22×C4×C113C2×D411yescharacteristic
121C22×C4×C113C2×D412yes—
122C22×C4×C113#4112yes—
123C22×C4×C113C2411yescharacteristic
124C2×C8×C113—22no—
125C2×C8×C113—22no—
126C2×C8×C113—22no—
127C2×C8×C113—22no—
128C2×C8×C113—22no—
129C2×C8×C113—44no—
130C2×C8×C113C22×C412yes—
131C2×C8×C113C22×C412yes—
132C4⋊C4×C113—21no—
133C4⋊C4×C113—21no—
134C4⋊C4×C113—22no—
135C4⋊C4×C113—22no—
136C4⋊C4×C113—22no—
137C4⋊C4×C113—22no—
138C4⋊C4×C113—82no—
139C4⋊C4×C113C22×C412yes—
140C4⋊C4×C113C22×C412yes—
141C22⋊C4×C113—24no—
142C22⋊C4×C113—24no—
143C22⋊C4×C113—44no—
144C22⋊C4×C113—81no—
145C22⋊C4×C113—81no—
146C42×C113—24no—
147C42×C113—22no—
148C42×C113—22no—
149C42×C113—44no—
150C23×C113—22no—
151C23×C113—24no—
152C23×C113—22no—
153C23×C113—22no—
154C23×C113—21no—
155C23×C113—21no—
156C23×C113—42no—
157C23×C113—41no—
158C23×C113—41no—
159C23×C113C4×D412yes—
160C23×C113C22×D411yescharacteristic
161Q8×C113—22no—
162Q8×C113—22no—
163Q8×C113—42no—
164D4×C113—22no—
165D4×C113—24no—
166D4×C113—48no—
167D4×C113#11714yes—
168C2×C4×C113—22no—
169C2×C4×C113—22no—
170C2×C4×C113—22no—
171C2×C4×C113—22no—
172C2×C4×C113—22no—
173C2×C4×C113—24no—
174C2×C4×C113—22no—
175C2×C4×C113—22no—
176C2×C4×C113—22no—
177C2×C4×C113—22no—
178C2×C4×C113—22no—
179C2×C4×C113—22no—
180C2×C4×C113—48no—
181C2×C4×C113—42no—
182C2×C4×C113—42no—
183C2×C4×C113—41no—
184C2×C4×C113—41no—
185C2×C4×C113—42no—
186C2×C4×C113—42no—
187C2×C4×C113—42no—
188C2×C4×C113—42no—
189C2×C4×C113—42no—
190C2×C4×C113—42no—
191C2×C4×C113—88no—
192C2×C4×C113C4×D412yes—
193C2×C4×C113#13514yes—
194C2×C4×C113C23×C411yescharacteristic
195C2×C4×C113C2×#4112yes—
196C2×C4×C113C2×#4111yescharacteristic
197C2×C4×C113H(5,𝔽2)12yes—
198C8×C113—22no—
199C8×C113—22no—
200C8×C113—42no—
201C22×C113—22no—
202C22×C113—24no—
203C22×C113—22no—
204C22×C113—22no—
205C22×C113—42no—
206C22×C113—42no—
207C22×C113—44no—
208C22×C113—42no—
209C22×C113—42no—
210C22×C113—44no—
211C22×C113C2×C4×D411yescharacteristic
212C22×C113#54612yes—
213C22×C113C2×#13512yes—
214C22×C113#57711yescharacteristic
215C22×C113#57711yescharacteristic
216C4×C113—22no—
217C4×C113—24no—
218C4×C113—42no—
219C4×C113—42no—
220C4×C113—42no—
221C4×C113—42no—
222C4×C113—82no—
223C4×C113—82no—
224C4×C113C2×#11711yescharacteristic
225C4×C113#51811yescharacteristic
226C4×C113#55112yes—
227#22572C11311yesmaximal, characteristic, direct product factor, Sylow
228C2×C113—42no—
229C2×C113—44no—
230C2×C113#234211yescharacteristic
231C2×C113C4×#13511yescharacteristic
232C2×C113#295211yescharacteristic
233C2×C113#329612yes—
234C2×C113#330312yes—
235C2×#416C2×C11311yescharacteristic
236#2346C2×C11311yescharacteristic, complemented
237#2339C2×C11311yescharacteristic
238#1933C2×C11312yescomplemented
239#1927C2×C11312yescomplemented
240#1864C2×C11312yescomplemented
241#1825C2×C11311yescharacteristic, complemented
242#1807C2×C11311yescharacteristic, complemented
243#1788C2×C11311yescharacteristic
244C4×#102C2×C11312yescomplemented
245C8×C22⋊C4C2×C11311yescharacteristic, complemented
246C113#2257211yescharacteristic, direct product factor, Sylow
247C22×#41C22×C11311yescharacteristic
248C23×C8C22×C11311yescharacteristic
249C2×C4×D4C22×C11312yes—
250C2×#117C22×C11311yescharacteristic
251C2×#106C22×C11312yes—
252#416C4×C11318yescomplemented
253C2×#103C22×C11312yes—
254C2×#103C22×C11311yescharacteristic
255C2×#103C22×C11311yescharacteristic
256C2×#102C22×C11312yes—
257C2×#102C22×C11311yescharacteristic
258C2×#102C22×C11311yescharacteristic
259C2×C22⋊C8C22×C11311yescharacteristic
260C2×C4×C8C22×C11312yes—
261#391C22×C11312yes—
262#389C22×C11312yes—
263#388—22no—
264#386—22no—
265#382—21no—
266#381—21no—
267#380C22×C11311yescharacteristic
268#380C22×C11311yescharacteristic
269C4×C4⋊C4C22×C11312yes—
270C4×C22⋊C4C22×C11311yescharacteristic
271#376—21no—
272C2×#95—21no—
273#340C22×C11312yes—
274#340C22×C11312yes—
275#340C22×C11312yes—
276#336C22×C11312yes—
277C2×#41—22no—
278C2×#41—24no—
279C2×#41—42no—
280C2×#41C2×C4×C11314yes—
281C22×Q8C23×C11311yescharacteristic
282C22×D4—21no—
283C22×D4C23×C11311yescharacteristic
284C23×C4—21no—
285C23×C4—22no—
286C23×C4C23×C11311yescharacteristic
287C22×C8—42no—
288C22×C8C2×C4×C11314yescomplemented
289C22×C8C23×C11312yes—
290C22×C8C23×C11311yescharacteristic
291C22×C8C23×C11311yescharacteristic
292C4×D4—24no—
293C4×D4—42no—
294#117—22no—
295#117C2×C4×C11314yescomplemented
296C2×C4⋊C4—22no—
297C2×C4⋊C4C23×C11312yes—
298C2×C4⋊C4C23×C11312yes—
299C2×C4⋊C4C23×C11311yescharacteristic
300C2×C4⋊C4C23×C11311yescharacteristic
301C2×C22⋊C4—21no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—21no—
304C2×C22⋊C4—21no—
305C2×C22⋊C4D4×C11312yes—
306C2×C42—21no—
307C2×C42—22no—
308C2×C42C23×C11312yes—
309#106—42no—
310#103—42no—
311#103C2×C4×C11314yescomplemented
312#103C2×C4×C11314yescomplemented
313#102—24no—
314#102C2×C4×C11314yes—
315#102C2×C4×C11314yes—
316C22⋊C8—22no—
317C4×C8—42no—
318#95—21no—
319#95—22no—
320#95—21no—
321#95—22no—
322#95—22no—
323#95—22no—
324#95D4×C11311yescharacteristic
325#95D4×C11311yescharacteristic
326C24—21no—
327C24C2×D4×C11311yescharacteristic
328#41—28no—
329#41—44no—
330#41—48no—
331C2×Q8—21no—
332C2×Q8—22no—
333C2×Q8—22no—
334C2×Q8C22×C4×C11312yes—
335C2×D4—24no—
336C2×D4—21no—
337C2×D4—22no—
338C2×D4—44no—
339C2×D4C22×C4×C11312yes—
340C2×D4#41×C11314yes—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—24no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—44no—
350C22×C4—44no—
351C22×C4—44no—
352C22×C4—82no—
353C22×C4C22×C4×C11312yes—
354C22×C4C2×D4×C11312yes—
355C22×C4C2×D4×C11311yescharacteristic
356C22×C4C2×D4×C11312yes—
357C22×C4#41×C11312yes—
358C22×C4C24×C11311yescharacteristic
359C2×C8—22no—
360C2×C8—22no—
361C2×C8—22no—
362C2×C8—22no—
363C2×C8—22no—
364C2×C8—44no—
365C2×C8C22×C4×C11312yes—
366C2×C8C22×C4×C11312yes—
367C4⋊C4—21no—
368C4⋊C4—21no—
369C4⋊C4—22no—
370C4⋊C4—22no—
371C4⋊C4—22no—
372C4⋊C4—22no—
373C4⋊C4—82no—
374C4⋊C4C22×C4×C11312yes—
375C4⋊C4C22×C4×C11312yes—
376C22⋊C4—24no—
377C22⋊C4—24no—
378C22⋊C4—44no—
379C22⋊C4—81no—
380C22⋊C4—81no—
381C42—24no—
382C42—22no—
383C42—22no—
384C42—44no—
385C23—22no—
386C23—24no—
387C23—22no—
388C23—22no—
389C23—21no—
390C23—21no—
391C23—42no—
392C23—41no—
393C23—41no—
394C23C4×D4×C11312yes—
395C23C22×D4×C11311yescharacteristic
396Q8—22no—
397Q8—22no—
398Q8—42no—
399D4—22no—
400D4—24no—
401D4—48no—
402D4#117×C11314yes—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—24no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—48no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—88no—
427C2×C4C4×D4×C11312yes—
428C2×C4#135×C11314yes—
429C2×C4C23×C4×C11311yescharacteristic
430C2×C4C2×#41×C11312yes—
431C2×C4C2×#41×C11311yescharacteristic
432C2×C4H(5,𝔽2)×C11312yes—
433C8—22no—
434C8—22no—
435C8—42no—
436C22—22no—
437C22—24no—
438C22—22no—
439C22—22no—
440C22—42no—
441C22—44no—
442C22—42no—
443C22—42no—
444C22—44no—
445C22—42no—
446C22C2×C4×D4×C11311yescharacteristic
447C22#546×C11312yes—
448C22C2×#135×C11312yes—
449C22#577×C11311yescharacteristic
450C22#577×C11311yescharacteristic
451C4—22no—
452C4—24no—
453C4—42no—
454C4—42no—
455C4—42no—
456C4—42no—
457C4—82no—
458C4—82no—
459C4C2×#117×C11311yescharacteristic
460C4#518×C11311yescharacteristic
461C4#551×C11312yes—
462C2—42no—
463C2—44no—
464C2C113×#234211yescharacteristic
465C2C4×#135×C11311yescharacteristic
466C2C113×#295211yescharacteristic
467C2C113×#329612yes—
468C2C113×#330312yes—
subgroupG/H
center, socleC23×C113C22×D4
commutatorC2×C4C23×C4×C113
Frattini, cosocleC22×C4C24×C113
prime radicalC22×#41×C113C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C113 ◁ C4×C22⋊C4×C113 ◁ G

Complements

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

NH ≅ G/Ntype
2. C113×#2346C2semidirect
4. C113×#1933C2semidirect
5. C113×#1927C2semidirect
6. C113×#1864C2semidirect
7. C113×#1825C2semidirect
8. C113×#1807C2semidirect
10. C4×#102×C113C2semidirect
11. C8×C22⋊C4×C113C2semidirect
17. #416×C113C4semidirect
53. C22×C8×C113C2×C4semidirect
60. #117×C113C2×C4semidirect
76. #103×C113C2×C4semidirect
77. #103×C113C2×C4semidirect
227. #22572C113direct (N × H)
236. #2346C2×C113semidirect
238. #1933C2×C113semidirect
239. #1927C2×C113semidirect
240. #1864C2×C113semidirect
241. #1825C2×C113semidirect
242. #1807C2×C113semidirect
244. C4×#102C2×C113semidirect
245. C8×C22⋊C4C2×C113semidirect
246. C113#22572direct (N × H)
252. #416C4×C113semidirect
288. C22×C8C2×C4×C113semidirect
295. #117C2×C4×C113semidirect
311. #103C2×C4×C113semidirect
312. #103C2×C4×C113semidirect

Maximal subgroups

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

subgroupindex
C2×#416×C1132
C113×#23462
C113×#23392
C113×#19332
C113×#19272
C113×#18642
C113×#18252
C113×#18072
C113×#17882
C4×#102×C1132
C8×C22⋊C4×C1132
#22572113