Finite Group Databasebeta

← C113×#12687 #43280707

Subgroups — C113×#12687 #43280707

Order 28928.

2508 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 2508 conjugacy classes of subgroups — too many to draw. Only its 158 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
1C22×#120×C113C211yesmaximal, characteristic
2C2×AΣL(1,𝔽16)×C113C214yesmaximal, complemented
3C113×#1783C211yesmaximal, characteristic, complemented
4C113×#1781C211yesmaximal, characteristic, complemented
5C26×C113C411yescharacteristic, complemented
6C23×D4×C113—21no—
7C23×D4×C113C2211yescharacteristic
8C2×#120×C113—21no—
9C2×#120×C113—21no—
10C2×#120×C113—28no—
11C2×#120×C113C412yescomplemented
12C2×#120×C113C2212yes—
13C22×C22⋊C4×C113—21no—
14C22×C22⋊C4×C113C411yescharacteristic, complemented
15C2×#100×C113—21no—
16C2×#100×C113C2212yescomplemented
17C2×#99×C113—21no—
18C2×#99×C113C2212yescomplemented
19#407×C113—21no—
20#379×C113—21no—
21AΣL(1,𝔽16)×C113—28no—
22C25×C113—21no—
23C25×C113—21no—
24C25×C113—24no—
25C25×C113—48no—
26C25×C113—42no—
27C25×C113—42no—
28C25×C113C2×C411yescharacteristic
29C25×C113C2×C412yes—
30C25×C113D411yescharacteristic
31C22×D4×C113—28no—
32C22×D4×C113—22no—
33C22×D4×C113—22no—
34C22×D4×C113—24no—
35C22×D4×C113—42no—
36C22×D4×C113—42no—
37C22×D4×C113—48no—
38C22×D4×C113C2×C411yescharacteristic
39C22×D4×C113D411yescharacteristic
40C22×D4×C113D411yescharacteristic
41C22×D4×C113C2311yescharacteristic
42C23×C4×C113—21no—
43C23×C4×C113D411yescharacteristic
44C2×M16×C113—22no—
45#120×C113—28no—
46#120×C113—28no—
47#120×C113—24no—
48#120×C113—24no—
49#120×C113—24no—
50#120×C113D418yes—
51C2×C22⋊C4×C113—28no—
52C2×C22⋊C4×C113—21no—
53C2×C22⋊C4×C113—22no—
54C2×C22⋊C4×C113—22no—
55C2×C22⋊C4×C113—24no—
56C2×C22⋊C4×C113—22no—
57C2×C22⋊C4×C113—42no—
58C2×C22⋊C4×C113C2×C412yes—
59#100×C113—24no—
60#100×C113—42no—
61#99×C113—24no—
62#99×C113—42no—
63C22⋊C8×C113—42no—
64C24×C113—24no—
65C24×C113—28no—
66C24×C113—22no—
67C24×C113—22no—
68C24×C113—22no—
69C24×C113—22no—
70C24×C113—24no—
71C24×C113—22no—
72C24×C113—22no—
73C24×C113—22no—
74C24×C113—41no—
75C24×C113—44no—
76C24×C113—48no—
77C24×C113—42no—
78C24×C113—42no—
79C24×C113—44no—
80C24×C113—42no—
81C24×C113—48no—
82C24×C113—432no—
83C24×C113—416no—
84C24×C113—42no—
85C24×C113—44no—
86C24×C113—48no—
87C24×C113—416no—
88C24×C113—432no—
89C24×C113—48no—
90C24×C113—42no—
91C24×C113—42no—
92C24×C113C22⋊C411yescharacteristic
93C24×C113C22⋊C411yescharacteristic
94C24×C113C22⋊C411yescharacteristic
95C24×C113C22⋊C414yescomplemented
96C24×C113C22×C411yescharacteristic
97C24×C113C2×D411yescharacteristic
98C2×D4×C113—28no—
99C2×D4×C113—22no—
100C2×D4×C113—22no—
101C2×D4×C113—22no—
102C2×D4×C113—22no—
103C2×D4×C113—28no—
104C2×D4×C113—22no—
105C2×D4×C113—41no—
106C2×D4×C113—41no—
107C2×D4×C113—41no—
108C2×D4×C113—41no—
109C2×D4×C113—41no—
110C2×D4×C113—41no—
111C2×D4×C113—416no—
112C2×D4×C113—48no—
113C2×D4×C113—48no—
114C2×D4×C113—432no—
115C2×D4×C113C2×D414yes—
116C22×C4×C113—24no—
117C22×C4×C113—22no—
118C22×C4×C113—22no—
119C22×C4×C113—22no—
120C22×C4×C113—41no—
121C22×C4×C113—44no—
122C22×C4×C113—42no—
123C22×C4×C113C22⋊C411yescharacteristic
124C22×C4×C113C2×D411yescharacteristic
125M16×C113—42no—
126M16×C113—42no—
127C2×C8×C113—42no—
128C22⋊C4×C113—28no—
129C22⋊C4×C113—24no—
130C22⋊C4×C113—22no—
131C22⋊C4×C113—28no—
132C22⋊C4×C113—42no—
133C22⋊C4×C113—42no—
134C22⋊C4×C113—84no—
135C22⋊C4×C113C22⋊C414yescomplemented
136C23×C113—24no—
137C23×C113—21no—
138C23×C113—21no—
139C23×C113—216no—
140C23×C113—28no—
141C23×C113—28no—
142C23×C113—21no—
143C23×C113—24no—
144C23×C113—21no—
145C23×C113—21no—
146C23×C113—21no—
147C23×C113—21no—
148C23×C113—21no—
149C23×C113—21no—
150C23×C113—21no—
151C23×C113—21no—
152C23×C113—44no—
153C23×C113—42no—
154C23×C113—48no—
155C23×C113—44no—
156C23×C113—42no—
157C23×C113—48no—
158C23×C113—48no—
159C23×C113—48no—
160C23×C113—416no—
161C23×C113—44no—
162C23×C113—48no—
163C23×C113—44no—
164C23×C113—48no—
165C23×C113—48no—
166C23×C113—416no—
167C23×C113—48no—
168C23×C113—44no—
169C23×C113—42no—
170C23×C113—42no—
171C23×C113—48no—
172C23×C113—48no—
173C23×C113—416no—
174C23×C113—416no—
175C23×C113—464no—
176C23×C113—464no—
177C23×C113—48no—
178C23×C113—416no—
179C23×C113—48no—
180C23×C113—44no—
181C23×C113—42no—
182C23×C113—42no—
183C23×C113—44no—
184C23×C113—82no—
185C23×C113—84no—
186C23×C113#9912yes—
187C23×C113#9912yes—
188C23×C113C2×C22⋊C411yescharacteristic
189C23×C113C2×C22⋊C412yes—
190C23×C113#12012yes—
191D4×C113—44no—
192D4×C113—44no—
193D4×C113—44no—
194D4×C113—416no—
195D4×C113—416no—
196D4×C113—44no—
197C2×C4×C113—21no—
198C2×C4×C113—21no—
199C2×C4×C113—21no—
200C2×C4×C113—24no—
201C2×C4×C113—21no—
202C2×C4×C113—21no—
203C2×C4×C113—44no—
204C2×C4×C113—48no—
205C2×C4×C113—44no—
206C2×C4×C113—82no—
207C2×C4×C113—84no—
208C2×C4×C113#12012yes—
209C8×C113—82no—
210C22×C113—24no—
211C22×C113—24no—
212C22×C113—22no—
213C22×C113—22no—
214C22×C113—22no—
215C22×C113—28no—
216C22×C113—22no—
217C22×C113—22no—
218C22×C113—22no—
219C22×C113—42no—
220C22×C113—42no—
221C22×C113—42no—
222C22×C113—42no—
223C22×C113—44no—
224C22×C113—432no—
225C22×C113—416no—
226C22×C113—48no—
227C22×C113—42no—
228C22×C113—42no—
229C22×C113—48no—
230C22×C113—48no—
231C22×C113—416no—
232C22×C113—432no—
233C22×C113—42no—
234C22×C113—48no—
235C22×C113—42no—
236C22×C113—42no—
237C22×C113—44no—
238C22×C113—42no—
239C22×C113—41no—
240C22×C113—81no—
241C22×C113—88no—
242C22×C113—81no—
243C22×C113AΣL(1,𝔽16)14yescomplemented
244C22×C113#37911yescharacteristic
245C22×C113C2×#9911yescharacteristic
246C22×C113C2×#9911yescharacteristic
247C4×C113—22no—
248C4×C113—44no—
249C4×C113—41no—
250C4×C113—162no—
251#12687C11311yesmaximal, characteristic, direct product factor, Sylow
252C2×C113—21no—
253C2×C113—24no—
254C2×C113—21no—
255C2×C113—42no—
256C2×C113—48no—
257C2×C113—42no—
258C2×C113—42no—
259C2×C113—81no—
260C2×C113#178111yescharacteristic
261C2×C113C2×AΣL(1,𝔽16)12yes—
262C22×#120C2×C11311yescharacteristic
263C2×AΣL(1,𝔽16)C2×C11314yescomplemented
264#1783C2×C11311yescharacteristic, complemented
265#1781C2×C11311yescharacteristic, complemented
266C113#1268711yescharacteristic, direct product factor, Sylow
267C26C4×C11311yescharacteristic, complemented
268C23×D4—21no—
269C23×D4C22×C11311yescharacteristic
270C2×#120—21no—
271C2×#120—21no—
272C2×#120—28no—
273C2×#120C4×C11312yescomplemented
274C2×#120C22×C11312yes—
275C22×C22⋊C4—21no—
276C22×C22⋊C4C4×C11311yescharacteristic, complemented
277C2×#100—21no—
278C2×#100C22×C11312yescomplemented
279C2×#99—21no—
280C2×#99C22×C11312yescomplemented
281#407—21no—
282#379—21no—
283AΣL(1,𝔽16)—28no—
284C25—21no—
285C25—21no—
286C25—24no—
287C25—48no—
288C25—42no—
289C25—42no—
290C25C2×C4×C11311yescharacteristic
291C25C2×C4×C11312yes—
292C25D4×C11311yescharacteristic
293C22×D4—28no—
294C22×D4—22no—
295C22×D4—22no—
296C22×D4—24no—
297C22×D4—42no—
298C22×D4—42no—
299C22×D4—48no—
300C22×D4C2×C4×C11311yescharacteristic
301C22×D4D4×C11311yescharacteristic
302C22×D4D4×C11311yescharacteristic
303C22×D4C23×C11311yescharacteristic
304C23×C4—21no—
305C23×C4D4×C11311yescharacteristic
306C2×M16—22no—
307#120—28no—
308#120—28no—
309#120—24no—
310#120—24no—
311#120—24no—
312#120D4×C11318yes—
313C2×C22⋊C4—28no—
314C2×C22⋊C4—21no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—24no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—42no—
320C2×C22⋊C4C2×C4×C11312yes—
321#100—24no—
322#100—42no—
323#99—24no—
324#99—42no—
325C22⋊C8—42no—
326C24—24no—
327C24—28no—
328C24—22no—
329C24—22no—
330C24—22no—
331C24—22no—
332C24—24no—
333C24—22no—
334C24—22no—
335C24—22no—
336C24—41no—
337C24—44no—
338C24—48no—
339C24—42no—
340C24—42no—
341C24—44no—
342C24—42no—
343C24—48no—
344C24—432no—
345C24—416no—
346C24—42no—
347C24—44no—
348C24—48no—
349C24—416no—
350C24—432no—
351C24—48no—
352C24—42no—
353C24—42no—
354C24C22⋊C4×C11311yescharacteristic
355C24C22⋊C4×C11311yescharacteristic
356C24C22⋊C4×C11311yescharacteristic
357C24C22⋊C4×C11314yescomplemented
358C24C22×C4×C11311yescharacteristic
359C24C2×D4×C11311yescharacteristic
360C2×D4—28no—
361C2×D4—22no—
362C2×D4—22no—
363C2×D4—22no—
364C2×D4—22no—
365C2×D4—28no—
366C2×D4—22no—
367C2×D4—41no—
368C2×D4—41no—
369C2×D4—41no—
370C2×D4—41no—
371C2×D4—41no—
372C2×D4—41no—
373C2×D4—416no—
374C2×D4—48no—
375C2×D4—48no—
376C2×D4—432no—
377C2×D4C2×D4×C11314yes—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—41no—
383C22×C4—44no—
384C22×C4—42no—
385C22×C4C22⋊C4×C11311yescharacteristic
386C22×C4C2×D4×C11311yescharacteristic
387M16—42no—
388M16—42no—
389C2×C8—42no—
390C22⋊C4—28no—
391C22⋊C4—24no—
392C22⋊C4—22no—
393C22⋊C4—28no—
394C22⋊C4—42no—
395C22⋊C4—42no—
396C22⋊C4—84no—
397C22⋊C4C22⋊C4×C11314yescomplemented
398C23—24no—
399C23—21no—
400C23—21no—
401C23—216no—
402C23—28no—
403C23—28no—
404C23—21no—
405C23—24no—
406C23—21no—
407C23—21no—
408C23—21no—
409C23—21no—
410C23—21no—
411C23—21no—
412C23—21no—
413C23—21no—
414C23—44no—
415C23—42no—
416C23—48no—
417C23—44no—
418C23—42no—
419C23—48no—
420C23—48no—
421C23—48no—
422C23—416no—
423C23—48no—
424C23—44no—
425C23—48no—
426C23—44no—
427C23—48no—
428C23—416no—
429C23—48no—
430C23—44no—
431C23—42no—
432C23—42no—
433C23—48no—
434C23—48no—
435C23—416no—
436C23—416no—
437C23—464no—
438C23—464no—
439C23—48no—
440C23—416no—
441C23—48no—
442C23—44no—
443C23—42no—
444C23—42no—
445C23—44no—
446C23—82no—
447C23—84no—
448C23#99×C11312yes—
449C23#99×C11312yes—
450C23C2×C22⋊C4×C11311yescharacteristic
451C23C2×C22⋊C4×C11312yes—
452C23#120×C11312yes—
453D4—44no—
454D4—44no—
455D4—44no—
456D4—416no—
457D4—416no—
458D4—44no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—24no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—44no—
466C2×C4—48no—
467C2×C4—44no—
468C2×C4—82no—
469C2×C4—84no—
470C2×C4#120×C11312yes—
471C8—82no—
472C22—24no—
473C22—24no—
474C22—22no—
475C22—22no—
476C22—22no—
477C22—28no—
478C22—22no—
479C22—22no—
480C22—22no—
481C22—42no—
482C22—42no—
483C22—42no—
484C22—42no—
485C22—44no—
486C22—432no—
487C22—416no—
488C22—48no—
489C22—42no—
490C22—42no—
491C22—48no—
492C22—48no—
493C22—416no—
494C22—432no—
495C22—44no—
496C22—42no—
497C22—48no—
498C22—42no—
499C22—42no—
500C22—41no—
501C22—42no—
502C22—81no—
503C22—88no—
504C22—81no—
505C22AΣL(1,𝔽16)×C11314yescomplemented
506C22#379×C11311yescharacteristic
507C22C2×#99×C11311yescharacteristic
508C22C2×#99×C11311yescharacteristic
509C4—22no—
510C4—44no—
511C4—41no—
512C4—162no—
513C2—21no—
514C2—24no—
515C2—21no—
516C2—42no—
517C2—48no—
518C2—42no—
519C2—42no—
520C2—81no—
521C2C113×#178111yescharacteristic
522C2C2×AΣL(1,𝔽16)×C11312yes—
subgroupG/H
center, socleC22×C113C2×#99
commutatorC24C22×C4×C113
Frattini, cosocleC22×D4C23×C113
prime radicalC22×#120×C113C2
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C113 ◁ C24×C113 ◁ C23×D4×C113 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×AΣL(1,𝔽16)×C113C2semidirect
3. C113×#1783C2semidirect
4. C113×#1781C2semidirect
5. C26×C113C4semidirect
11. C2×#120×C113C4semidirect
14. C22×C22⋊C4×C113C4semidirect
16. C2×#100×C113C22semidirect
18. C2×#99×C113C22semidirect
95. C24×C113C22⋊C4semidirect
135. C22⋊C4×C113C22⋊C4semidirect
243. C22×C113AΣL(1,𝔽16)semidirect
251. #12687C113direct (N × H)
263. C2×AΣL(1,𝔽16)C2×C113semidirect
264. #1783C2×C113semidirect
265. #1781C2×C113semidirect
266. C113#12687direct (N × H)
267. C26C4×C113semidirect
273. C2×#120C4×C113semidirect
276. C22×C22⋊C4C4×C113semidirect
278. C2×#100C22×C113semidirect
280. C2×#99C22×C113semidirect
357. C24C22⋊C4×C113semidirect
397. C22⋊C4C22⋊C4×C113semidirect
505. C22AΣL(1,𝔽16)×C113semidirect

Maximal subgroups

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

subgroupindex
C22×#120×C1132
C2×AΣL(1,𝔽16)×C1132
C113×#17832
C113×#17812
#12687113