Finite Group Databasebeta

← C101×#13043 #38795194

Subgroups — C101×#13043 #38795194

Order 25856.

844 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×B(4,𝔽2)×C101C211yesmaximal, characteristic
2C101×#2375C211yesmaximal, characteristic, complemented
3C2×AΣL(1,𝔽16)×C101C211yesmaximal, characteristic, complemented
4C101×#1792C211yesmaximal, characteristic, complemented
5C101×#1790C211yesmaximal, characteristic, complemented
6C101×#1393C211yesmaximal, characteristic, complemented
7C101×#1390C211yesmaximal, characteristic, complemented
8C2×H(5,𝔽2)×C101C2211yescharacteristic
9C2×#120×C101—21no—
10C2×#120×C101C2211yescharacteristic
11B(4,𝔽2)×C101—22no—
12B(4,𝔽2)×C101C414yescomplemented
13C2×#104×C101—21no—
14#417×C101—21no—
15C2×#100×C101C2211yescharacteristic
16C2×#99×C101—21no—
17C2×#99×C101—21no—
18C2×#99×C101C2211yescharacteristic
19C2×#99×C101C2211yescharacteristic
20#390×C101—21no—
21#390×C101C2211yescharacteristic
22#388×C101—21no—
23#388×C101—21no—
24#388×C101—21no—
25#386×C101—21no—
26#386×C101—21no—
27#386×C101C2211yescharacteristic
28#379×C101—21no—
29AΣL(1,𝔽16)×C101—22no—
30C25×C101D411yescharacteristic
31H(5,𝔽2)×C101—21no—
32H(5,𝔽2)×C101—21no—
33H(5,𝔽2)×C101—21no—
34H(5,𝔽2)×C101—42no—
35H(5,𝔽2)×C101C2×C412yes—
36C2×#41×C101—21no—
37C2×#41×C101—41no—
38C22×D4×C101—21no—
39C22×D4×C101—21no—
40C22×D4×C101—21no—
41C22×D4×C101—41no—
42C22×D4×C101—41no—
43C22×D4×C101C2311yescharacteristic
44C23×C4×C101D411yescharacteristic, complemented
45C2×M16×C101—21no—
46#120×C101—21no—
47#120×C101—21no—
48#120×C101—24no—
49#120×C101—42no—
50#120×C101C2×C412yes—
51#120×C101D412yes—
52C2×C4⋊C4×C101—21no—
53C2×C4⋊C4×C101D411yescharacteristic, complemented
54C2×C22⋊C4×C101—21no—
55C2×C22⋊C4×C101—21no—
56C2×C22⋊C4×C101—21no—
57C2×C22⋊C4×C101—21no—
58C2×C22⋊C4×C101—21no—
59C2×C22⋊C4×C101—21no—
60C2×C22⋊C4×C101—21no—
61C2×C22⋊C4×C101—21no—
62C2×C22⋊C4×C101—21no—
63C2×C22⋊C4×C101—21no—
64C2×C22⋊C4×C101—41no—
65C2×C22⋊C4×C101—41no—
66C2×C22⋊C4×C101D411yescharacteristic
67C2×C22⋊C4×C101D411yescharacteristic, complemented
68C2×C42×C101—21no—
69C2×C42×C101D411yescharacteristic, complemented
70#104×C101—42no—
71#103×C101—41no—
72#102×C101—41no—
73#100×C101—21no—
74#100×C101—21no—
75#99×C101—21no—
76#99×C101—24no—
77#99×C101—21no—
78#99×C101—21no—
79#99×C101—42no—
80#99×C101C2×C412yes—
81#95×C101—21no—
82#95×C101—21no—
83#95×C101—21no—
84#95×C101—41no—
85C24×C101—21no—
86C24×C101—21no—
87C24×C101—21no—
88C24×C101—21no—
89C24×C101—41no—
90C24×C101—41no—
91C24×C101—81no—
92C24×C101—81no—
93C24×C101C22⋊C412yescomplemented
94C24×C101C2×D411yescharacteristic
95C24×C101C2×D411yescharacteristic
96#41×C101—22no—
97#41×C101—22no—
98#41×C101—42no—
99#41×C101—41no—
100#41×C101—42no—
101#41×C101—42no—
102#41×C101—42no—
103#41×C101—41no—
104C2×Q8×C101—21no—
105C2×D4×C101—21no—
106C2×D4×C101—22no—
107C2×D4×C101—22no—
108C2×D4×C101—22no—
109C2×D4×C101—22no—
110C2×D4×C101—22no—
111C2×D4×C101—22no—
112C2×D4×C101—41no—
113C2×D4×C101—41no—
114C2×D4×C101—42no—
115C2×D4×C101—42no—
116C2×D4×C101—41no—
117C2×D4×C101—41no—
118C2×D4×C101—41no—
119C2×D4×C101—41no—
120C2×D4×C101—42no—
121C2×D4×C101—42no—
122C2×D4×C101—41no—
123C2×D4×C101—41no—
124C2×D4×C101—41no—
125C2×D4×C101—44no—
126C2×D4×C101—81no—
127C2×D4×C101—81no—
128C2×D4×C101—82no—
129C2×D4×C101—82no—
130C2×D4×C101C22×C411yescharacteristic
131C2×D4×C101C2×D411yescharacteristic
132C2×D4×C101#4111yescharacteristic
133C2×D4×C101#4111yescharacteristic
134C22×C4×C101—21no—
135C22×C4×C101—21no—
136C22×C4×C101—21no—
137C22×C4×C101—21no—
138C22×C4×C101—21no—
139C22×C4×C101—21no—
140C22×C4×C101—21no—
141C22×C4×C101—41no—
142C22×C4×C101—41no—
143C22×C4×C101—41no—
144C22×C4×C101—41no—
145C22×C4×C101—81no—
146C22×C4×C101C2×D411yescharacteristic
147M16×C101—41no—
148M16×C101—41no—
149C2×C8×C101—41no—
150C4⋊C4×C101—41no—
151C4⋊C4×C101—81no—
152C22⋊C4×C101—21no—
153C22⋊C4×C101—22no—
154C22⋊C4×C101—22no—
155C22⋊C4×C101—22no—
156C22⋊C4×C101—44no—
157C22⋊C4×C101—41no—
158C22⋊C4×C101—41no—
159C22⋊C4×C101—41no—
160C22⋊C4×C101—41no—
161C22⋊C4×C101—81no—
162C22⋊C4×C101—81no—
163C22⋊C4×C101—81no—
164C22⋊C4×C101—81no—
165C22⋊C4×C101—81no—
166C22⋊C4×C101—81no—
167C22⋊C4×C101—81no—
168C22⋊C4×C101—82no—
169C22⋊C4×C101C22⋊C412yescomplemented
170C42×C101—41no—
171C42×C101—81no—
172C23×C101—21no—
173C23×C101—21no—
174C23×C101—21no—
175C23×C101—21no—
176C23×C101—22no—
177C23×C101—22no—
178C23×C101—22no—
179C23×C101—42no—
180C23×C101—41no—
181C23×C101—41no—
182C23×C101—41no—
183C23×C101—41no—
184C23×C101—41no—
185C23×C101—41no—
186C23×C101—41no—
187C23×C101—41no—
188C23×C101—41no—
189C23×C101—41no—
190C23×C101—42no—
191C23×C101—42no—
192C23×C101—41no—
193C23×C101—41no—
194C23×C101—41no—
195C23×C101—81no—
196C23×C101—82no—
197C23×C101—82no—
198C23×C101—81no—
199C23×C101—81no—
200C23×C101—84no—
201C23×C101—81no—
202C23×C101—81no—
203C23×C101—81no—
204C23×C101—81no—
205C23×C101C2×C22⋊C411yescharacteristic
206C23×C101C4×D411yescharacteristic
207C23×C101#12011yescharacteristic
208C23×C101#12111yescharacteristic
209C23×C101#12311yescharacteristic
210Q8×C101—22no—
211Q8×C101—41no—
212D4×C101—22no—
213D4×C101—41no—
214D4×C101—41no—
215D4×C101—41no—
216D4×C101—41no—
217D4×C101—41no—
218D4×C101—42no—
219D4×C101—42no—
220D4×C101—42no—
221D4×C101—42no—
222D4×C101—42no—
223D4×C101—42no—
224D4×C101—82no—
225D4×C101—82no—
226D4×C101—88no—
227C2×C4×C101—21no—
228C2×C4×C101—21no—
229C2×C4×C101—22no—
230C2×C4×C101—22no—
231C2×C4×C101—22no—
232C2×C4×C101—42no—
233C2×C4×C101—42no—
234C2×C4×C101—41no—
235C2×C4×C101—41no—
236C2×C4×C101—41no—
237C2×C4×C101—41no—
238C2×C4×C101—41no—
239C2×C4×C101—41no—
240C2×C4×C101—41no—
241C2×C4×C101—42no—
242C2×C4×C101—41no—
243C2×C4×C101—41no—
244C2×C4×C101—41no—
245C2×C4×C101—41no—
246C2×C4×C101—41no—
247C2×C4×C101—41no—
248C2×C4×C101—42no—
249C2×C4×C101—41no—
250C2×C4×C101—41no—
251C2×C4×C101—41no—
252C2×C4×C101—41no—
253C2×C4×C101—81no—
254C2×C4×C101—81no—
255C2×C4×C101—81no—
256C2×C4×C101—81no—
257C2×C4×C101—81no—
258C2×C4×C101—81no—
259C2×C4×C101C4×D411yescharacteristic
260C2×C4×C101#12111yescharacteristic
261C8×C101—81no—
262C22×C101—21no—
263C22×C101—21no—
264C22×C101—21no—
265C22×C101—21no—
266C22×C101—21no—
267C22×C101—21no—
268C22×C101—41no—
269C22×C101—41no—
270C22×C101—41no—
271C22×C101—41no—
272C22×C101—42no—
273C22×C101—42no—
274C22×C101—41no—
275C22×C101—41no—
276C22×C101—41no—
277C22×C101—41no—
278C22×C101—44no—
279C22×C101—42no—
280C22×C101—42no—
281C22×C101—81no—
282C22×C101—81no—
283C22×C101—81no—
284C22×C101—81no—
285C22×C101—82no—
286C22×C101—82no—
287C22×C101—82no—
288C22×C101—82no—
289C22×C101—82no—
290C22×C101—81no—
291C22×C101—81no—
292C22×C101—81no—
293C22×C101—81no—
294C22×C101—161no—
295C22×C101—161no—
296C22×C101#38611yescharacteristic
297C22×C101B(4,𝔽2)11yescharacteristic
298C22×C101B(4,𝔽2)11yescharacteristic
299C4×C101—21no—
300C4×C101—21no—
301C4×C101—42no—
302C4×C101—42no—
303C4×C101—81no—
304C4×C101—81no—
305C4×C101—82no—
306C4×C101—81no—
307C4×C101—81no—
308C4×C101—161no—
309#13043C10111yesmaximal, characteristic, direct product factor, Sylow
310C2×C101—21no—
311C2×C101—21no—
312C2×C101—41no—
313C2×C101—41no—
314C2×C101—41no—
315C2×C101—41no—
316C2×C101—82no—
317C2×C101—82no—
318C2×C101#188911yescharacteristic
319C2×C101#219911yescharacteristic
320C2×C101#220011yescharacteristic
321C2×B(4,𝔽2)C2×C10111yescharacteristic
322#2375C2×C10111yescharacteristic, complemented
323C2×AΣL(1,𝔽16)C2×C10111yescharacteristic, complemented
324#1792C2×C10111yescharacteristic, complemented
325#1790C2×C10111yescharacteristic, complemented
326#1393C2×C10111yescharacteristic, complemented
327#1390C2×C10111yescharacteristic, complemented
328C101#1304311yescharacteristic, direct product factor, Sylow
329C2×H(5,𝔽2)C22×C10111yescharacteristic
330C2×#120—21no—
331C2×#120C22×C10111yescharacteristic
332B(4,𝔽2)—22no—
333B(4,𝔽2)C4×C10114yescomplemented
334C2×#104—21no—
335#417—21no—
336C2×#100C22×C10111yescharacteristic
337C2×#99—21no—
338C2×#99—21no—
339C2×#99C22×C10111yescharacteristic
340C2×#99C22×C10111yescharacteristic
341#390—21no—
342#390C22×C10111yescharacteristic
343#388—21no—
344#388—21no—
345#388—21no—
346#386—21no—
347#386—21no—
348#386C22×C10111yescharacteristic
349#379—21no—
350AΣL(1,𝔽16)—22no—
351C25D4×C10111yescharacteristic
352H(5,𝔽2)—21no—
353H(5,𝔽2)—21no—
354H(5,𝔽2)—21no—
355H(5,𝔽2)—42no—
356H(5,𝔽2)C2×C4×C10112yes—
357C2×#41—21no—
358C2×#41—41no—
359C22×D4—21no—
360C22×D4—21no—
361C22×D4—21no—
362C22×D4—41no—
363C22×D4—41no—
364C22×D4C23×C10111yescharacteristic
365C23×C4D4×C10111yescharacteristic, complemented
366C2×M16—21no—
367#120—21no—
368#120—21no—
369#120—24no—
370#120—42no—
371#120C2×C4×C10112yes—
372#120D4×C10112yes—
373C2×C4⋊C4—21no—
374C2×C4⋊C4D4×C10111yescharacteristic, complemented
375C2×C22⋊C4—21no—
376C2×C22⋊C4—21no—
377C2×C22⋊C4—21no—
378C2×C22⋊C4—21no—
379C2×C22⋊C4—21no—
380C2×C22⋊C4—21no—
381C2×C22⋊C4—21no—
382C2×C22⋊C4—21no—
383C2×C22⋊C4—21no—
384C2×C22⋊C4—21no—
385C2×C22⋊C4—41no—
386C2×C22⋊C4—41no—
387C2×C22⋊C4D4×C10111yescharacteristic
388C2×C22⋊C4D4×C10111yescharacteristic, complemented
389C2×C42—21no—
390C2×C42D4×C10111yescharacteristic, complemented
391#104—42no—
392#103—41no—
393#102—41no—
394#100—21no—
395#100—21no—
396#99—21no—
397#99—24no—
398#99—21no—
399#99—21no—
400#99—42no—
401#99C2×C4×C10112yes—
402#95—21no—
403#95—21no—
404#95—21no—
405#95—41no—
406C24—21no—
407C24—21no—
408C24—21no—
409C24—21no—
410C24—41no—
411C24—41no—
412C24—81no—
413C24—81no—
414C24C22⋊C4×C10112yescomplemented
415C24C2×D4×C10111yescharacteristic
416C24C2×D4×C10111yescharacteristic
417#41—22no—
418#41—22no—
419#41—42no—
420#41—41no—
421#41—42no—
422#41—42no—
423#41—42no—
424#41—41no—
425C2×Q8—21no—
426C2×D4—21no—
427C2×D4—22no—
428C2×D4—22no—
429C2×D4—22no—
430C2×D4—22no—
431C2×D4—22no—
432C2×D4—22no—
433C2×D4—41no—
434C2×D4—41no—
435C2×D4—42no—
436C2×D4—42no—
437C2×D4—41no—
438C2×D4—41no—
439C2×D4—41no—
440C2×D4—41no—
441C2×D4—42no—
442C2×D4—42no—
443C2×D4—41no—
444C2×D4—41no—
445C2×D4—41no—
446C2×D4—44no—
447C2×D4—81no—
448C2×D4—81no—
449C2×D4—82no—
450C2×D4—82no—
451C2×D4C22×C4×C10111yescharacteristic
452C2×D4C2×D4×C10111yescharacteristic
453C2×D4#41×C10111yescharacteristic
454C2×D4#41×C10111yescharacteristic
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—41no—
463C22×C4—41no—
464C22×C4—41no—
465C22×C4—41no—
466C22×C4—81no—
467C22×C4C2×D4×C10111yescharacteristic
468M16—41no—
469M16—41no—
470C2×C8—41no—
471C4⋊C4—41no—
472C4⋊C4—81no—
473C22⋊C4—21no—
474C22⋊C4—22no—
475C22⋊C4—22no—
476C22⋊C4—22no—
477C22⋊C4—44no—
478C22⋊C4—41no—
479C22⋊C4—41no—
480C22⋊C4—41no—
481C22⋊C4—41no—
482C22⋊C4—81no—
483C22⋊C4—81no—
484C22⋊C4—81no—
485C22⋊C4—81no—
486C22⋊C4—81no—
487C22⋊C4—81no—
488C22⋊C4—81no—
489C22⋊C4—82no—
490C22⋊C4C22⋊C4×C10112yescomplemented
491C42—41no—
492C42—81no—
493C23—21no—
494C23—21no—
495C23—21no—
496C23—21no—
497C23—22no—
498C23—22no—
499C23—22no—
500C23—42no—
501C23—41no—
502C23—41no—
503C23—41no—
504C23—41no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23—42no—
512C23—42no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23—81no—
517C23—82no—
518C23—82no—
519C23—81no—
520C23—81no—
521C23—84no—
522C23—81no—
523C23—81no—
524C23—81no—
525C23—81no—
526C23C2×C22⋊C4×C10111yescharacteristic
527C23C4×D4×C10111yescharacteristic
528C23#120×C10111yescharacteristic
529C23#121×C10111yescharacteristic
530C23#123×C10111yescharacteristic
531Q8—22no—
532Q8—41no—
533D4—22no—
534D4—41no—
535D4—41no—
536D4—41no—
537D4—41no—
538D4—41no—
539D4—42no—
540D4—42no—
541D4—42no—
542D4—42no—
543D4—42no—
544D4—42no—
545D4—82no—
546D4—82no—
547D4—88no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—42no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—42no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—81no—
575C2×C4—81no—
576C2×C4—81no—
577C2×C4—81no—
578C2×C4—81no—
579C2×C4—81no—
580C2×C4C4×D4×C10111yescharacteristic
581C2×C4#121×C10111yescharacteristic
582C8—81no—
583C22—21no—
584C22—21no—
585C22—21no—
586C22—21no—
587C22—21no—
588C22—21no—
589C22—41no—
590C22—41no—
591C22—41no—
592C22—41no—
593C22—42no—
594C22—42no—
595C22—41no—
596C22—41no—
597C22—41no—
598C22—41no—
599C22—44no—
600C22—42no—
601C22—42no—
602C22—81no—
603C22—81no—
604C22—81no—
605C22—81no—
606C22—82no—
607C22—82no—
608C22—82no—
609C22—82no—
610C22—82no—
611C22—81no—
612C22—81no—
613C22—81no—
614C22—81no—
615C22—161no—
616C22—161no—
617C22#386×C10111yescharacteristic
618C22B(4,𝔽2)×C10111yescharacteristic
619C22B(4,𝔽2)×C10111yescharacteristic
620C4—21no—
621C4—21no—
622C4—42no—
623C4—42no—
624C4—81no—
625C4—81no—
626C4—82no—
627C4—81no—
628C4—81no—
629C4—161no—
630C2—21no—
631C2—21no—
632C2—41no—
633C2—41no—
634C2—41no—
635C2—41no—
636C2—82no—
637C2—82no—
638C2C101×#188911yescharacteristic
639C2C101×#219911yescharacteristic
640C2C101×#220011yescharacteristic
subgroupG/H
center, socleC22×C101B(4,𝔽2)
commutatorC2×D4C22×C4×C101
Frattini, cosocleC22×D4C23×C101
prime radicalC2×B(4,𝔽2)×C101C2
LCSG ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C101 ◁ C23×C101 ◁ C22×D4×C101 ◁ G

Complements

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

NH ≅ G/Ntype
2. C101×#2375C2semidirect
3. C2×AΣL(1,𝔽16)×C101C2semidirect
4. C101×#1792C2semidirect
5. C101×#1790C2semidirect
6. C101×#1393C2semidirect
7. C101×#1390C2semidirect
12. B(4,𝔽2)×C101C4semidirect
44. C23×C4×C101D4semidirect
53. C2×C4⋊C4×C101D4semidirect
67. C2×C22⋊C4×C101D4semidirect
69. C2×C42×C101D4semidirect
93. C24×C101C22⋊C4semidirect
169. C22⋊C4×C101C22⋊C4semidirect
309. #13043C101direct (N × H)
322. #2375C2×C101semidirect
323. C2×AΣL(1,𝔽16)C2×C101semidirect
324. #1792C2×C101semidirect
325. #1790C2×C101semidirect
326. #1393C2×C101semidirect
327. #1390C2×C101semidirect
328. C101#13043direct (N × H)
333. B(4,𝔽2)C4×C101semidirect
365. C23×C4D4×C101semidirect
374. C2×C4⋊C4D4×C101semidirect
388. C2×C22⋊C4D4×C101semidirect
390. C2×C42D4×C101semidirect
414. C24C22⋊C4×C101semidirect
490. C22⋊C4C22⋊C4×C101semidirect

Maximal subgroups

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

subgroupindex
C2×B(4,𝔽2)×C1012
C101×#23752
C2×AΣL(1,𝔽16)×C1012
C101×#17922
C101×#17902
C101×#13932
C101×#13902
#13043101