Finite Group Databasebeta

← C113×#13353 #43281373

Subgroups — C113×#13353 #43281373

Order 28928.

720 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 720 conjugacy classes of subgroups — too many to draw. Only its 94 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
1C113×#3199C211yesmaximal, characteristic, complemented
2C113×#3027C211yesmaximal, characteristic, complemented
3C113×#3021C211yesmaximal, characteristic, complemented
4C113×#2134C211yesmaximal, characteristic, complemented
5C113×#2122C211yesmaximal, characteristic, complemented
6C113×#1791C211yesmaximal, characteristic, complemented
7C113×#1395C211yesmaximal, characteristic, complemented
8#585×C113C2211yescharacteristic
9C2×AGL(1,ℤ/8)×C113—21no—
10#535×C113—21no—
11#535×C113C2211yescharacteristic
12#535×C113C2211yescharacteristic
13#482×C113—21no—
14#482×C113—21no—
15#469×C113—21no—
16#466×C113—21no—
17B(4,𝔽2)×C113—22no—
18B(4,𝔽2)×C113—22no—
19#454×C113—22no—
20#453×C113—22no—
21#447×C113—21no—
22#447×C113—21no—
23#421×C113—21no—
24#413×C113—21no—
25#413×C113C2211yescharacteristic
26#413×C113C2211yescharacteristic
27#410×C113—21no—
28#410×C113C2211yescharacteristic
29#408×C113—21no—
30#408×C113C2211yescharacteristic
31#355×C113—22no—
32AΣL(1,𝔽16)×C113—22no—
332−1+4×C113—21no—
342−1+4×C113—41no—
35H(5,𝔽2)×C113—21no—
36H(5,𝔽2)×C113—21no—
37H(5,𝔽2)×C113—41no—
38H(5,𝔽2)×C113D412yes—
39C2×#41×C113—21no—
40C2×#41×C113—21no—
41C2×#41×C113—21no—
42C2×#41×C113—42no—
43C2×#41×C113C2311yescharacteristic
44C22×D4×C113D411yescharacteristic
45C22×D4×C113D411yescharacteristic, complemented
46AGL(1,ℤ/8)×C113—24no—
47AGL(1,ℤ/8)×C113—42no—
48C2×SD16×C113—41no—
49C2×D8×C113—21no—
50C2×D8×C113—21no—
51C2×D8×C113—41no—
52C2×M16×C113—21no—
53C2×M16×C113—21no—
54C2×M16×C113D411yescharacteristic, complemented
55C22×C8×C113D411yescharacteristic, complemented
56#127×C113—21no—
57#127×C113—21no—
58#127×C113—41no—
59#124×C113—21no—
60#124×C113—21no—
61#124×C113—41no—
62#121×C113—21no—
63#121×C113—21no—
64#121×C113—21no—
65#121×C113—21no—
66#121×C113—42no—
67#120×C113—24no—
68#120×C113—21no—
69#120×C113—21no—
70#120×C113—21no—
71#120×C113—21no—
72#117×C113—21no—
73#117×C113D411yescharacteristic
74#117×C113D411yescharacteristic, complemented
75#107×C113—21no—
76#106×C113—21no—
77#104×C113—24no—
78#102×C113—21no—
79#102×C113—21no—
80#102×C113—21no—
81#102×C113—21no—
82#101×C113—21no—
83#101×C113—41no—
84#101×C113D412yescomplemented
85#100×C113—21no—
86#100×C113—21no—
87#100×C113—41no—
88#99×C113—24no—
89#99×C113—21no—
90#99×C113D412yes—
91C22⋊C8×C113—21no—
92C22⋊C8×C113—21no—
93C22⋊C8×C113—42no—
94C24×C113—21no—
95C24×C113—21no—
96#41×C113—22no—
97#41×C113—22no—
98#41×C113—41no—
99#41×C113—41no—
100#41×C113—42no—
101#41×C113—41no—
102#41×C113—41no—
103#41×C113—41no—
104#41×C113—41no—
105#41×C113—41no—
106#41×C113—42no—
107#41×C113—42no—
108#41×C113—42no—
109#41×C113—41no—
110#41×C113—41no—
111#41×C113—41no—
112C2×Q8×C113—21no—
113C2×Q8×C113—21no—
114C2×Q8×C113—21no—
115C2×Q8×C113—42no—
116C2×Q8×C113C2×D411yescharacteristic
117C2×D4×C113—21no—
118C2×D4×C113—21no—
119C2×D4×C113—21no—
120C2×D4×C113—22no—
121C2×D4×C113—22no—
122C2×D4×C113—21no—
123C2×D4×C113—21no—
124C2×D4×C113—21no—
125C2×D4×C113—21no—
126C2×D4×C113—41no—
127C2×D4×C113—41no—
128C2×D4×C113—41no—
129C2×D4×C113—41no—
130C2×D4×C113—42no—
131C2×D4×C113—42no—
132C2×D4×C113—42no—
133C2×D4×C113—41no—
134C2×D4×C113—44no—
135C2×D4×C113—41no—
136C2×D4×C113—41no—
137C2×D4×C113—41no—
138C2×D4×C113—41no—
139C2×D4×C113C2×D411yescharacteristic
140C2×D4×C113C2×D411yescharacteristic
141C2×D4×C113#4111yescharacteristic
142C22×C4×C113—22no—
143C22×C4×C113—22no—
144C22×C4×C113—41no—
145C22×C4×C113C2×D411yescharacteristic
146C22×C4×C113C2×D411yescharacteristic
147C22×C4×C113C2×D411yescharacteristic
148SD16×C113—44no—
149D8×C113—44no—
150D8×C113—81no—
151D8×C113—81no—
152M16×C113—22no—
153M16×C113—41no—
154M16×C113—41no—
155M16×C113—41no—
156M16×C113—41no—
157C2×C8×C113—21no—
158C2×C8×C113—21no—
159C2×C8×C113—41no—
160C2×C8×C113—41no—
161C2×C8×C113—41no—
162C4⋊C4×C113—21no—
163C4⋊C4×C113—21no—
164C4⋊C4×C113—41no—
165C22⋊C4×C113—21no—
166C22⋊C4×C113—22no—
167C22⋊C4×C113—21no—
168C22⋊C4×C113—44no—
169C22⋊C4×C113—41no—
170C22⋊C4×C113—41no—
171C22⋊C4×C113—41no—
172C22⋊C4×C113—41no—
173C42×C113—21no—
174C42×C113—21no—
175C42×C113—41no—
176C23×C113—22no—
177C23×C113—22no—
178C23×C113—41no—
179C23×C113—42no—
180C23×C113—41no—
181C23×C113—41no—
182C23×C113—81no—
183C23×C113—161no—
184C23×C113—161no—
185C23×C113#12011yescharacteristic
186C23×C113#12111yescharacteristic
187C23×C113#12111yescharacteristic
188Q8×C113—22no—
189Q8×C113—41no—
190Q8×C113—41no—
191Q8×C113—41no—
192Q8×C113—41no—
193D4×C113—22no—
194D4×C113—41no—
195D4×C113—41no—
196D4×C113—41no—
197D4×C113—41no—
198D4×C113—41no—
199D4×C113—42no—
200D4×C113—42no—
201D4×C113—42no—
202D4×C113—42no—
203D4×C113—42no—
204D4×C113—41no—
205D4×C113—41no—
206D4×C113—41no—
207D4×C113—41no—
208D4×C113—41no—
209D4×C113—81no—
210D4×C113—161no—
211D4×C113—161no—
212D4×C113—161no—
213D4×C113—161no—
214D4×C113—164no—
215C2×C4×C113—21no—
216C2×C4×C113—21no—
217C2×C4×C113—21no—
218C2×C4×C113—21no—
219C2×C4×C113—21no—
220C2×C4×C113—21no—
221C2×C4×C113—22no—
222C2×C4×C113—22no—
223C2×C4×C113—42no—
224C2×C4×C113—42no—
225C2×C4×C113—42no—
226C2×C4×C113—41no—
227C2×C4×C113—42no—
228C2×C4×C113—42no—
229C2×C4×C113—41no—
230C2×C4×C113—41no—
231C2×C4×C113#12011yescharacteristic
232C2×C4×C113#12011yescharacteristic
233C2×C4×C113#12111yescharacteristic
234C2×C4×C113#12711yescharacteristic
235C8×C113—42no—
236C8×C113—41no—
237C8×C113—41no—
238C8×C113—81no—
239C22×C113—21no—
240C22×C113—21no—
241C22×C113—21no—
242C22×C113—41no—
243C22×C113—41no—
244C22×C113—41no—
245C22×C113—42no—
246C22×C113—81no—
247C22×C113—81no—
248C22×C113—81no—
249C22×C113—81no—
250C22×C113—82no—
251C22×C113—82no—
252C22×C113—84no—
253C22×C113—81no—
254C22×C113—81no—
255C22×C113—161no—
256C22×C113—162no—
257C22×C113#39211yescharacteristic
258C4×C113—21no—
259C4×C113—21no—
260C4×C113—21no—
261C4×C113—42no—
262C4×C113—81no—
263C4×C113—81no—
264C4×C113—82no—
265C4×C113B(4,𝔽2)11yescharacteristic
266C4×C113B(4,𝔽2)11yescharacteristic
267#13353C11311yesmaximal, characteristic, direct product factor, Sylow
268C2×C113—21no—
269C2×C113—41no—
270C2×C113—41no—
271C2×C113—41no—
272C2×C113—82no—
273C2×C113—161no—
274C2×C113—161no—
275C2×C113#202111yescharacteristic
276#3199C2×C11311yescharacteristic, complemented
277#3027C2×C11311yescharacteristic, complemented
278#3021C2×C11311yescharacteristic, complemented
279#2134C2×C11311yescharacteristic, complemented
280#2122C2×C11311yescharacteristic, complemented
281#1791C2×C11311yescharacteristic, complemented
282#1395C2×C11311yescharacteristic, complemented
283C113#1335311yescharacteristic, direct product factor, Sylow
284#585C22×C11311yescharacteristic
285C2×AGL(1,ℤ/8)—21no—
286#535—21no—
287#535C22×C11311yescharacteristic
288#535C22×C11311yescharacteristic
289#482—21no—
290#482—21no—
291#469—21no—
292#466—21no—
293B(4,𝔽2)—22no—
294B(4,𝔽2)—22no—
295#454—22no—
296#453—22no—
297#447—21no—
298#447—21no—
299#421—21no—
300#413—21no—
301#413C22×C11311yescharacteristic
302#413C22×C11311yescharacteristic
303#410—21no—
304#410C22×C11311yescharacteristic
305#408—21no—
306#408C22×C11311yescharacteristic
307#355—22no—
308AΣL(1,𝔽16)—22no—
3092−1+4—21no—
3102−1+4—41no—
311H(5,𝔽2)—21no—
312H(5,𝔽2)—21no—
313H(5,𝔽2)—41no—
314H(5,𝔽2)D4×C11312yes—
315C2×#41—21no—
316C2×#41—21no—
317C2×#41—21no—
318C2×#41—42no—
319C2×#41C23×C11311yescharacteristic
320C22×D4D4×C11311yescharacteristic
321C22×D4D4×C11311yescharacteristic, complemented
322AGL(1,ℤ/8)—24no—
323AGL(1,ℤ/8)—42no—
324C2×SD16—41no—
325C2×D8—21no—
326C2×D8—21no—
327C2×D8—41no—
328C2×M16—21no—
329C2×M16—21no—
330C2×M16D4×C11311yescharacteristic, complemented
331C22×C8D4×C11311yescharacteristic, complemented
332#127—21no—
333#127—21no—
334#127—41no—
335#124—21no—
336#124—21no—
337#124—41no—
338#121—21no—
339#121—21no—
340#121—21no—
341#121—21no—
342#121—42no—
343#120—24no—
344#120—21no—
345#120—21no—
346#120—21no—
347#120—21no—
348#117—21no—
349#117D4×C11311yescharacteristic
350#117D4×C11311yescharacteristic, complemented
351#107—21no—
352#106—21no—
353#104—24no—
354#102—21no—
355#102—21no—
356#102—21no—
357#102—21no—
358#101—21no—
359#101—41no—
360#101D4×C11312yescomplemented
361#100—21no—
362#100—21no—
363#100—41no—
364#99—24no—
365#99—21no—
366#99D4×C11312yes—
367C22⋊C8—21no—
368C22⋊C8—21no—
369C22⋊C8—42no—
370C24—21no—
371C24—21no—
372#41—22no—
373#41—22no—
374#41—41no—
375#41—41no—
376#41—42no—
377#41—41no—
378#41—41no—
379#41—41no—
380#41—41no—
381#41—41no—
382#41—42no—
383#41—42no—
384#41—42no—
385#41—41no—
386#41—41no—
387#41—41no—
388C2×Q8—21no—
389C2×Q8—21no—
390C2×Q8—21no—
391C2×Q8—42no—
392C2×Q8C2×D4×C11311yescharacteristic
393C2×D4—21no—
394C2×D4—21no—
395C2×D4—21no—
396C2×D4—22no—
397C2×D4—22no—
398C2×D4—21no—
399C2×D4—21no—
400C2×D4—21no—
401C2×D4—21no—
402C2×D4—41no—
403C2×D4—41no—
404C2×D4—41no—
405C2×D4—41no—
406C2×D4—42no—
407C2×D4—42no—
408C2×D4—42no—
409C2×D4—41no—
410C2×D4—44no—
411C2×D4—41no—
412C2×D4—41no—
413C2×D4—41no—
414C2×D4—41no—
415C2×D4C2×D4×C11311yescharacteristic
416C2×D4C2×D4×C11311yescharacteristic
417C2×D4#41×C11311yescharacteristic
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—41no—
421C22×C4C2×D4×C11311yescharacteristic
422C22×C4C2×D4×C11311yescharacteristic
423C22×C4C2×D4×C11311yescharacteristic
424SD16—44no—
425D8—44no—
426D8—81no—
427D8—81no—
428M16—22no—
429M16—41no—
430M16—41no—
431M16—41no—
432M16—41no—
433C2×C8—21no—
434C2×C8—21no—
435C2×C8—41no—
436C2×C8—41no—
437C2×C8—41no—
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—41no—
441C22⋊C4—21no—
442C22⋊C4—22no—
443C22⋊C4—21no—
444C22⋊C4—44no—
445C22⋊C4—41no—
446C22⋊C4—41no—
447C22⋊C4—41no—
448C22⋊C4—41no—
449C42—21no—
450C42—21no—
451C42—41no—
452C23—22no—
453C23—22no—
454C23—41no—
455C23—42no—
456C23—41no—
457C23—41no—
458C23—81no—
459C23—161no—
460C23—161no—
461C23#120×C11311yescharacteristic
462C23#121×C11311yescharacteristic
463C23#121×C11311yescharacteristic
464Q8—22no—
465Q8—41no—
466Q8—41no—
467Q8—41no—
468Q8—41no—
469D4—22no—
470D4—41no—
471D4—41no—
472D4—41no—
473D4—41no—
474D4—41no—
475D4—42no—
476D4—42no—
477D4—42no—
478D4—42no—
479D4—42no—
480D4—41no—
481D4—41no—
482D4—41no—
483D4—41no—
484D4—41no—
485D4—81no—
486D4—161no—
487D4—161no—
488D4—161no—
489D4—161no—
490D4—164no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—41no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4#120×C11311yescharacteristic
508C2×C4#120×C11311yescharacteristic
509C2×C4#121×C11311yescharacteristic
510C2×C4#127×C11311yescharacteristic
511C8—42no—
512C8—41no—
513C8—41no—
514C8—81no—
515C22—21no—
516C22—21no—
517C22—21no—
518C22—41no—
519C22—41no—
520C22—41no—
521C22—42no—
522C22—81no—
523C22—81no—
524C22—81no—
525C22—81no—
526C22—82no—
527C22—82no—
528C22—84no—
529C22—81no—
530C22—81no—
531C22—161no—
532C22—162no—
533C22#392×C11311yescharacteristic
534C4—21no—
535C4—21no—
536C4—21no—
537C4—42no—
538C4—81no—
539C4—81no—
540C4—82no—
541C4B(4,𝔽2)×C11311yescharacteristic
542C4B(4,𝔽2)×C11311yescharacteristic
543C2—21no—
544C2—41no—
545C2—41no—
546C2—41no—
547C2—82no—
548C2—161no—
549C2—161no—
550C2C113×#202111yescharacteristic
subgroupG/H
center, socleC2×C113#2021
commutator, Frattini, cosocleC2×#41C23×C113
LCSG ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C113 ◁ C2×C4×C113 ◁ C2×#41×C113 ◁ G

Complements

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

NH ≅ G/Ntype
1. C113×#3199C2semidirect
2. C113×#3027C2semidirect
3. C113×#3021C2semidirect
4. C113×#2134C2semidirect
5. C113×#2122C2semidirect
6. C113×#1791C2semidirect
7. C113×#1395C2semidirect
45. C22×D4×C113D4semidirect
54. C2×M16×C113D4semidirect
55. C22×C8×C113D4semidirect
74. #117×C113D4semidirect
84. #101×C113D4semidirect
267. #13353C113direct (N × H)
276. #3199C2×C113semidirect
277. #3027C2×C113semidirect
278. #3021C2×C113semidirect
279. #2134C2×C113semidirect
280. #2122C2×C113semidirect
281. #1791C2×C113semidirect
282. #1395C2×C113semidirect
283. C113#13353direct (N × H)
321. C22×D4D4×C113semidirect
330. C2×M16D4×C113semidirect
331. C22×C8D4×C113semidirect
350. #117D4×C113semidirect
360. #101D4×C113semidirect

Maximal subgroups

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

subgroupindex
C113×#31992
C113×#30272
C113×#30212
C113×#21342
C113×#21222
C113×#17912
C113×#13952
#13353113