Finite Group Databasebeta

← C2×C11×#2122 #14610538

Subgroups — C2×C11×#2122 #14610538

Order 2816.

1406 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C11×#535C211yesmaximal, characteristic
2C2×C11×#413C211yesmaximal, characteristic, complemented
3C2×C11×#410C211yesmaximal, characteristic, complemented
4C11×#2122C218yesmaximal, direct product factor
5C2×C11×AΣL(1,𝔽16)C214yesmaximal, complemented
6C22×C11×#41C2211yescharacteristic
7C23×D4×C11C411yescharacteristic, complemented
8C22×C11×M16—21no—
9C11×#535—22no—
10C11×#535—22no—
11C11×#535C414yescomplemented
12C11×#535C2214yes—
13C2×C11×#127—21no—
14C2×C11×#124—21no—
15C2×C11×#121—21no—
16C2×C11×#121—21no—
17C2×C11×#120C412yescomplemented
18C2×C11×#120C2212yes—
19C2×C11×#117—21no—
20C2×C11×#117C411yescharacteristic, complemented
21C11×#413—22no—
22C11×#413C2214yescomplemented
23C2×C11×#101—21no—
24C2×C11×#100C2212yes—
25C11×#410—22no—
26C11×#410C2214yescomplemented
27C2×C11×#99—21no—
28C2×C11×#99C2212yes—
29C11×AΣL(1,𝔽16)C22116yescomplemented
30C25×C11C2×C412yes—
31C2×C11×#41—41no—
32C2×C11×#41—41no—
33C2×C11×#41—42no—
34C2×C11×#41C2×C412yes—
35C2×C11×#41D412yes—
36C2×C11×#41D412yes—
37C2×C11×#41C2312yes—
38C22×Q8×C11D411yescharacteristic
39C22×D4×C11—22no—
40C22×D4×C11—22no—
41C22×D4×C11—21no—
42C22×D4×C11—22no—
43C22×D4×C11—21no—
44C22×D4×C11—42no—
45C22×D4×C11—41no—
46C22×D4×C11—41no—
47C22×D4×C11C2×C411yescharacteristic
48C22×D4×C11C2×C414yescomplemented
49C22×D4×C11D411yescharacteristic
50C22×D4×C11C2311yescharacteristic
51C23×C4×C11C2×C411yescharacteristic
52C23×C4×C11D411yescharacteristic
53C23×C4×C11D411yescharacteristic
54C2×C11×M16—22no—
55C2×C11×M16—24no—
56C2×C11×M16—41no—
57C2×C11×M16—42no—
58C22×C8×C11—41no—
59C11×#127—24no—
60C11×#127—24no—
61C11×#124—24no—
62C11×#124—24no—
63C11×#121—24no—
64C11×#121—24no—
65C11×#121—24no—
66C11×#121—24no—
67C11×#120—24no—
68C11×#120—22no—
69C11×#120—22no—
70C11×#120C2×C414yes—
71C11×#120C2×C418yescomplemented
72C11×#120C2314yes—
73C11×#117—22no—
74C11×#117—24no—
75C11×#117—42no—
76C11×#117C2×C414yescomplemented
77C2×C11×C4⋊C4—21no—
78C2×C11×C4⋊C4—41no—
79C2×C11×C22⋊C4—22no—
80C2×C11×C22⋊C4—22no—
81C2×C11×C22⋊C4—22no—
82C2×C11×C22⋊C4C2×C412yes—
83C2×C42×C11—21no—
84C2×C42×C11—41no—
85C11×#101—24no—
86C11×#100—22no—
87C11×#100C2314yes—
88C11×#99—24no—
89C11×#99—22no—
90C11×#99C2314yes—
91C2×#2122C1111yesmaximal, characteristic, direct product factor, Sylow
92C24×C11—22no—
93C24×C11—22no—
94C24×C11—42no—
95C24×C11—42no—
96C24×C11—84no—
97C24×C11C22⋊C411yescharacteristic
98C24×C11C22×C411yescharacteristic
99C24×C11C22×C414yes—
100C24×C11C2×D411yescharacteristic
101C11×#41—42no—
102C11×#41—42no—
103C11×#41—42no—
104C11×#41—42no—
105C11×#41—42no—
106C11×#41—42no—
107C11×#41—42no—
108C11×#41—42no—
109C2×Q8×C11—41no—
110C2×Q8×C11—41no—
111C2×Q8×C11C22⋊C412yes—
112C2×Q8×C11C2×D412yes—
113C2×D4×C11—24no—
114C2×D4×C11—28no—
115C2×D4×C11—24no—
116C2×D4×C11—24no—
117C2×D4×C11—24no—
118C2×D4×C11—41no—
119C2×D4×C11—41no—
120C2×D4×C11—41no—
121C2×D4×C11—41no—
122C2×D4×C11—41no—
123C2×D4×C11—41no—
124C2×D4×C11—42no—
125C2×D4×C11—42no—
126C2×D4×C11—44no—
127C2×D4×C11—44no—
128C2×D4×C11—44no—
129C2×D4×C11—44no—
130C2×D4×C11—42no—
131C2×D4×C11—41no—
132C2×D4×C11—41no—
133C2×D4×C11—84no—
134C2×D4×C11—84no—
135C2×D4×C11C22⋊C412yes—
136C2×D4×C11C22⋊C412yes—
137C2×D4×C11C22×C411yescharacteristic
138C2×D4×C11C22×C412yes—
139C2×D4×C11C2×D411yescharacteristic
140C2×D4×C11C2×D412yes—
141C2×D4×C11C2×D411yescharacteristic
142C2×D4×C11C2411yescharacteristic
143C22×C4×C11—21no—
144C22×C4×C11—21no—
145C22×C4×C11—22no—
146C22×C4×C11—21no—
147C22×C4×C11—22no—
148C22×C4×C11—21no—
149C22×C4×C11—22no—
150C22×C4×C11—21no—
151C22×C4×C11—21no—
152C22×C4×C11—22no—
153C22×C4×C11—42no—
154C22×C4×C11C22⋊C411yescharacteristic
155C22×C4×C11C22⋊C411yescharacteristic
156C22×C4×C11C22⋊C412yes—
157C22×C4×C11C22⋊C412yes—
158C22×C4×C11C22⋊C412yes—
159C22×C4×C11C22⋊C411yescharacteristic
160C22×C4×C11C22×C412yes—
161C22×C4×C11C2×D412yes—
162C22×C4×C11C2×D411yescharacteristic
163C22×C4×C11C2×D412yes—
164C11×M16—24no—
165C11×M16—42no—
166C11×M16—44no—
167C2×C8×C11—42no—
168C2×C8×C11—44no—
169C11×C4⋊C4—24no—
170C11×C4⋊C4—44no—
171C11×C22⋊C4—22no—
172C11×C22⋊C4—24no—
173C11×C22⋊C4—28no—
174C11×C22⋊C4—24no—
175C11×C22⋊C4—24no—
176C11×C22⋊C4—42no—
177C11×C22⋊C4C22×C414yes—
178C42×C11—24no—
179C42×C11—44no—
180C2×#535C2×C1111yescharacteristic
181C2×#413C2×C1111yescharacteristic, complemented
182C2×#410C2×C1111yescharacteristic, complemented
183#2122C2×C1118yesdirect product factor
184C2×AΣL(1,𝔽16)C2×C1114yescomplemented
185C23×C11—21no—
186C23×C11—21no—
187C23×C11—21no—
188C23×C11—21no—
189C23×C11—21no—
190C23×C11—21no—
191C23×C11—24no—
192C23×C11—42no—
193C23×C11—42no—
194C23×C11—42no—
195C23×C11—44no—
196C23×C11—44no—
197C23×C11—44no—
198C23×C11—44no—
199C23×C11—42no—
200C23×C11—42no—
201C23×C11—84no—
202C23×C11—84no—
203C23×C11—88no—
204C23×C11—88no—
205C23×C11C2×C22⋊C411yescharacteristic
206C23×C11C2×C22⋊C411yescharacteristic
207C23×C11C2×C22⋊C411yescharacteristic
208C23×C11C2×C22⋊C412yes—
209C23×C11C23×C411yescharacteristic
210C23×C11C22×D411yescharacteristic
211Q8×C11—42no—
212Q8×C11—42no—
213D4×C11—42no—
214D4×C11—42no—
215D4×C11—42no—
216D4×C11—41no—
217D4×C11—42no—
218D4×C11—41no—
219D4×C11—41no—
220D4×C11—41no—
221D4×C11—48no—
222D4×C11—44no—
223D4×C11—44no—
224D4×C11—84no—
225D4×C11—88no—
226D4×C11—84no—
227C2×C4×C11—21no—
228C2×C4×C11—21no—
229C2×C4×C11—21no—
230C2×C4×C11—21no—
231C2×C4×C11—21no—
232C2×C4×C11—21no—
233C2×C4×C11—22no—
234C2×C4×C11—22no—
235C2×C4×C11—22no—
236C2×C4×C11—22no—
237C2×C4×C11—22no—
238C2×C4×C11—22no—
239C2×C4×C11—22no—
240C2×C4×C11—22no—
241C2×C4×C11—22no—
242C2×C4×C11—22no—
243C2×C4×C11—22no—
244C2×C4×C11—22no—
245C2×C4×C11—24no—
246C2×C4×C11—44no—
247C2×C4×C11—44no—
248C2×C4×C11—84no—
249C2×C4×C11#9912yes—
250C2×C4×C11#9911yescharacteristic
251C2×C4×C11#9911yescharacteristic
252C2×C4×C11C2×C22⋊C412yes—
253C2×C4×C11C2×C22⋊C412yes—
254C2×C4×C11C2×C22⋊C411yescharacteristic
255C2×C4×C11C2×C22⋊C412yes—
256C2×C4×C11C22×D411yescharacteristic
257C8×C11—44no—
258C22×#41C22×C1111yescharacteristic
259C23×D4C4×C1111yescharacteristic, complemented
260C22×M16—21no—
261#535—22no—
262#535—22no—
263#535C4×C1114yescomplemented
264#535C22×C1114yes—
265C2×#127—21no—
266C2×#124—21no—
267C2×#121—21no—
268C2×#121—21no—
269C2×#120C4×C1112yescomplemented
270C2×#120C22×C1112yes—
271C2×#117—21no—
272C2×#117C4×C1111yescharacteristic, complemented
273#413—22no—
274#413C22×C1114yescomplemented
275C2×#101—21no—
276C2×#100C22×C1112yes—
277#410—22no—
278#410C22×C1114yescomplemented
279C2×#99—21no—
280C2×#99C22×C1112yes—
281AΣL(1,𝔽16)C22×C11116yescomplemented
282C22×C11—22no—
283C22×C11—21no—
284C22×C11—21no—
285C22×C11—21no—
286C22×C11—22no—
287C22×C11—21no—
288C22×C11—42no—
289C22×C11—41no—
290C22×C11—41no—
291C22×C11—41no—
292C22×C11—41no—
293C22×C11—42no—
294C22×C11—44no—
295C22×C11—44no—
296C22×C11—42no—
297C22×C11—42no—
298C22×C11—42no—
299C22×C11—44no—
300C22×C11—81no—
301C22×C11—81no—
302C22×C11—84no—
303C22×C11—88no—
304C22×C11—88no—
305C22×C11—84no—
306C22×C11C2×#9911yescharacteristic
307C22×C11C2×#9911yescharacteristic
308C22×C11C22×C22⋊C411yescharacteristic
309C4×C11—22no—
310C4×C11—21no—
311C4×C11—22no—
312C4×C11—21no—
313C4×C11—44no—
314C4×C11—84no—
315C4×C11C2×#9912yes—
316C4×C11C2×#9912yes—
317C25C2×C4×C1112yes—
318C2×#41—41no—
319C2×#41—41no—
320C2×#41—42no—
321C2×#41C2×C4×C1112yes—
322C2×#41D4×C1112yes—
323C2×#41D4×C1112yes—
324C2×#41C23×C1112yes—
325C22×Q8D4×C1111yescharacteristic
326C22×D4—22no—
327C22×D4—22no—
328C22×D4—21no—
329C22×D4—21no—
330C22×D4—22no—
331C22×D4—41no—
332C22×D4—41no—
333C22×D4—42no—
334C22×D4C2×C4×C1111yescharacteristic
335C22×D4C2×C4×C1114yescomplemented
336C22×D4D4×C1111yescharacteristic
337C22×D4C23×C1111yescharacteristic
338C23×C4C2×C4×C1111yescharacteristic
339C23×C4D4×C1111yescharacteristic
340C23×C4D4×C1111yescharacteristic
341C2×M16—22no—
342C2×M16—24no—
343C2×M16—41no—
344C2×M16—42no—
345C22×C8—41no—
346#127—24no—
347#127—24no—
348#124—24no—
349#124—24no—
350#121—24no—
351#121—24no—
352#121—24no—
353#121—24no—
354#120—24no—
355#120—22no—
356#120—22no—
357#120C2×C4×C1114yes—
358#120C2×C4×C1118yescomplemented
359#120C23×C1114yes—
360#117—22no—
361#117—24no—
362#117—42no—
363#117C2×C4×C1114yescomplemented
364C2×C4⋊C4—21no—
365C2×C4⋊C4—41no—
366C2×C22⋊C4—22no—
367C2×C22⋊C4—22no—
368C2×C22⋊C4—22no—
369C2×C22⋊C4C2×C4×C1112yes—
370C2×C42—21no—
371C2×C42—41no—
372#101—24no—
373#100—22no—
374#100C23×C1114yes—
375#99—24no—
376#99—22no—
377#99C23×C1114yes—
378C2×C11—21no—
379C2×C11—21no—
380C2×C11—41no—
381C2×C11—41no—
382C2×C11—41no—
383C2×C11—41no—
384C2×C11—42no—
385C2×C11—84no—
386C2×C11#212212yesdirect product factor
387C2×C11C22×#9911yescharacteristic
388C24—22no—
389C24—22no—
390C24—42no—
391C24—42no—
392C24—84no—
393C24C11×C22⋊C411yescharacteristic
394C24C22×C4×C1111yescharacteristic
395C24C22×C4×C1114yes—
396C24C2×D4×C1111yescharacteristic
397#41—42no—
398#41—42no—
399#41—42no—
400#41—42no—
401#41—42no—
402#41—42no—
403#41—42no—
404#41—42no—
405C2×Q8—41no—
406C2×Q8—41no—
407C2×Q8C11×C22⋊C412yes—
408C2×Q8C2×D4×C1112yes—
409C2×D4—24no—
410C2×D4—28no—
411C2×D4—24no—
412C2×D4—24no—
413C2×D4—24no—
414C2×D4—41no—
415C2×D4—41no—
416C2×D4—41no—
417C2×D4—41no—
418C2×D4—41no—
419C2×D4—41no—
420C2×D4—42no—
421C2×D4—42no—
422C2×D4—44no—
423C2×D4—44no—
424C2×D4—44no—
425C2×D4—44no—
426C2×D4—42no—
427C2×D4—41no—
428C2×D4—41no—
429C2×D4—84no—
430C2×D4—84no—
431C2×D4C11×C22⋊C412yes—
432C2×D4C11×C22⋊C412yes—
433C2×D4C22×C4×C1111yescharacteristic
434C2×D4C22×C4×C1112yes—
435C2×D4C2×D4×C1111yescharacteristic
436C2×D4C2×D4×C1112yes—
437C2×D4C2×D4×C1111yescharacteristic
438C2×D4C24×C1111yescharacteristic
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—22no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—22no—
449C22×C4—42no—
450C22×C4C11×C22⋊C411yescharacteristic
451C22×C4C11×C22⋊C411yescharacteristic
452C22×C4C11×C22⋊C412yes—
453C22×C4C11×C22⋊C412yes—
454C22×C4C11×C22⋊C412yes—
455C22×C4C11×C22⋊C411yescharacteristic
456C22×C4C22×C4×C1112yes—
457C22×C4C2×D4×C1112yes—
458C22×C4C2×D4×C1111yescharacteristic
459C22×C4C2×D4×C1112yes—
460M16—24no—
461M16—42no—
462M16—44no—
463C2×C8—42no—
464C2×C8—44no—
465C4⋊C4—24no—
466C4⋊C4—44no—
467C22⋊C4—22no—
468C22⋊C4—24no—
469C22⋊C4—28no—
470C22⋊C4—24no—
471C22⋊C4—24no—
472C22⋊C4—42no—
473C22⋊C4C22×C4×C1114yes—
474C42—24no—
475C42—44no—
476C11C2×#212211yescharacteristic, direct product factor, Sylow
477C23—21no—
478C23—21no—
479C23—21no—
480C23—21no—
481C23—21no—
482C23—21no—
483C23—24no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—44no—
488C23—44no—
489C23—44no—
490C23—44no—
491C23—42no—
492C23—42no—
493C23—84no—
494C23—84no—
495C23—88no—
496C23—88no—
497C23C2×C11×C22⋊C411yescharacteristic
498C23C2×C11×C22⋊C411yescharacteristic
499C23C2×C11×C22⋊C411yescharacteristic
500C23C2×C11×C22⋊C412yes—
501C23C23×C4×C1111yescharacteristic
502C23C22×D4×C1111yescharacteristic
503Q8—42no—
504Q8—42no—
505D4—42no—
506D4—42no—
507D4—42no—
508D4—41no—
509D4—41no—
510D4—42no—
511D4—41no—
512D4—41no—
513D4—48no—
514D4—44no—
515D4—44no—
516D4—84no—
517D4—88no—
518D4—84no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—24no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—84no—
541C2×C4C11×#9912yes—
542C2×C4C11×#9911yescharacteristic
543C2×C4C11×#9911yescharacteristic
544C2×C4C2×C11×C22⋊C412yes—
545C2×C4C2×C11×C22⋊C412yes—
546C2×C4C2×C11×C22⋊C411yescharacteristic
547C2×C4C2×C11×C22⋊C412yes—
548C2×C4C22×D4×C1111yescharacteristic
549C8—44no—
550C22—22no—
551C22—21no—
552C22—21no—
553C22—21no—
554C22—22no—
555C22—21no—
556C22—42no—
557C22—41no—
558C22—41no—
559C22—41no—
560C22—41no—
561C22—42no—
562C22—44no—
563C22—44no—
564C22—42no—
565C22—42no—
566C22—42no—
567C22—44no—
568C22—81no—
569C22—81no—
570C22—84no—
571C22—88no—
572C22—88no—
573C22—84no—
574C22C2×C11×#9911yescharacteristic
575C22C2×C11×#9911yescharacteristic
576C22C22×C11×C22⋊C411yescharacteristic
577C4—21no—
578C4—22no—
579C4—21no—
580C4—22no—
581C4—44no—
582C4—84no—
583C4C2×C11×#9912yes—
584C4C2×C11×#9912yes—
585C2—21no—
586C2—21no—
587C2—41no—
588C2—41no—
589C2—41no—
590C2—41no—
591C2—42no—
592C2—84no—
593C2C11×#212212yesdirect product factor
594C2C22×C11×#9911yescharacteristic
subgroupG/H
center, socleC22×C11C2×#99
commutatorC23C23×C4×C11
Frattini, cosocleC2×D4C24×C11
prime radicalC2×C11×#535C2
LCSG ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C11 ◁ C22×C4×C11 ◁ C22×C11×#41 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C11×#413C2semidirect
3. C2×C11×#410C2semidirect
4. C11×#2122C2direct (N × H)
5. C2×C11×AΣL(1,𝔽16)C2semidirect
7. C23×D4×C11C4semidirect
11. C11×#535C4semidirect
17. C2×C11×#120C4semidirect
20. C2×C11×#117C4semidirect
22. C11×#413C22semidirect
26. C11×#410C22semidirect
29. C11×AΣL(1,𝔽16)C22semidirect
48. C22×D4×C11C2×C4semidirect
71. C11×#120C2×C4semidirect
76. C11×#117C2×C4semidirect
91. C2×#2122C11direct (N × H)
181. C2×#413C2×C11semidirect
182. C2×#410C2×C11semidirect
183. #2122C2×C11direct (N × H)
184. C2×AΣL(1,𝔽16)C2×C11semidirect
259. C23×D4C4×C11semidirect
263. #535C4×C11semidirect
269. C2×#120C4×C11semidirect
272. C2×#117C4×C11semidirect
274. #413C22×C11semidirect
278. #410C22×C11semidirect
281. AΣL(1,𝔽16)C22×C11semidirect
335. C22×D4C2×C4×C11semidirect
358. #120C2×C4×C11semidirect
363. #117C2×C4×C11semidirect
386. C2×C11#2122direct (N × H)
476. C11C2×#2122direct (N × H)
593. C2C11×#2122direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C11×#5352
C2×C11×#4132
C2×C11×#4102
C11×#21222
C2×C11×AΣL(1,𝔽16)2
C2×#212211