Finite Group Databasebeta

← C101×#9922 #38792073

Subgroups — C101×#9922 #38792073

Order 25856.

1278 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#450×C101C211yesmaximal, characteristic, complemented
2C101×#2626C211yesmaximal, characteristic, complemented
3C2×#392×C101C211yesmaximal, characteristic
4C101×#1786C211yesmaximal, characteristic, complemented
5C2×#328×C101C211yesmaximal, characteristic, complemented
6C2×#323×C101C211yesmaximal, characteristic, complemented
7C101×#1294C211yesmaximal, characteristic, complemented
8C23×D4×C101—21no—
9C23×D4×C101C2211yescharacteristic
10C22×SD16×C101—21no—
11C2×C22⋊Q8×C101C2211yescharacteristic
12C22×C22⋊C4×C101—21no—
13C22×C22⋊C4×C101C2211yescharacteristic
14#450×C101—22no—
15#450×C101—22no—
16C2×#102×C101—21no—
17C2×#102×C101—22no—
18C2×C22⋊C8×C101C2211yescharacteristic
19C2×C22⋊C8×C101C2211yescharacteristic
20#399×C101—21no—
21#396×C101—22no—
22#393×C101—21no—
23#392×C101—22no—
24#392×C101—22no—
25#385×C101—21no—
26#380×C101C2211yescharacteristic
27#379×C101—21no—
28C2×#95×C101C2211yescharacteristic
29#328×C101—22no—
30#323×C101—22no—
31C25×C101—21no—
32C25×C101D411yescharacteristic
33C22×Q8×C101D411yescharacteristic, complemented
34C22×D4×C101—24no—
35C22×D4×C101—21no—
36C22×D4×C101—21no—
37C22×D4×C101—21no—
38C22×D4×C101—21no—
39C22×D4×C101—21no—
40C22×D4×C101—44no—
41C22×D4×C101—44no—
42C22×D4×C101—44no—
43C22×D4×C101—42no—
44C22×D4×C101—42no—
45C22×D4×C101D412yes—
46C23×C4×C101—21no—
47C23×C4×C101D411yescharacteristic
48C23×C4×C101C2311yescharacteristic
49C2×SD16×C101—24no—
50C2×SD16×C101—44no—
51C22×C8×C101—21no—
52C22×C8×C101—21no—
53C22⋊Q8×C101—21no—
54C22⋊Q8×C101—21no—
55C22⋊Q8×C101—21no—
56C22⋊Q8×C101—21no—
57C2×C4⋊C4×C101—21no—
58C2×C4⋊C4×C101—22no—
59C2×C4⋊C4×C101D411yescharacteristic, complemented
60C2×C4⋊C4×C101D412yescomplemented
61C2×C22⋊C4×C101—24no—
62C2×C22⋊C4×C101—21no—
63C2×C22⋊C4×C101—21no—
64C2×C22⋊C4×C101—21no—
65C2×C22⋊C4×C101—21no—
66C2×C22⋊C4×C101—21no—
67C2×C22⋊C4×C101—21no—
68C2×C22⋊C4×C101—21no—
69C2×C22⋊C4×C101—21no—
70C2×C22⋊C4×C101—22no—
71C2×C22⋊C4×C101—22no—
72C2×C22⋊C4×C101—44no—
73#102×C101—24no—
74#102×C101—44no—
75C22⋊C8×C101—21no—
76C22⋊C8×C101—21no—
77C22⋊C8×C101—21no—
78C22⋊C8×C101D412yescomplemented
79#95×C101—22no—
80#95×C101—21no—
81#95×C101—21no—
82#95×C101D412yes—
83C24×C101—24no—
84C24×C101—21no—
85C24×C101—22no—
86C24×C101—21no—
87C24×C101—21no—
88C24×C101—21no—
89C24×C101—21no—
90C24×C101—21no—
91C24×C101—42no—
92C24×C101—42no—
93C24×C101—42no—
94C24×C101—42no—
95C24×C101—42no—
96C24×C101—41no—
97C24×C101—41no—
98C24×C101—41no—
99C24×C101—41no—
100C24×C101—41no—
101C24×C101—41no—
102C24×C101SD1612yescomplemented
103C24×C101C2×D411yescharacteristic
104C2×Q8×C101—21no—
105C2×Q8×C101—21no—
106C2×Q8×C101—42no—
107C2×D4×C101—24no—
108C2×D4×C101—24no—
109C2×D4×C101—22no—
110C2×D4×C101—22no—
111C2×D4×C101—42no—
112C2×D4×C101—42no—
113C2×D4×C101—48no—
114C2×D4×C101—44no—
115C2×D4×C101—44no—
116C2×D4×C101—44no—
117C2×D4×C101—44no—
118C2×D4×C101—48no—
119C2×D4×C101—48no—
120C2×D4×C101—416no—
121C2×D4×C101—416no—
122C2×D4×C101—41no—
123C2×D4×C101—41no—
124C22×C4×C101—24no—
125C22×C4×C101—22no—
126C22×C4×C101—21no—
127C22×C4×C101—22no—
128C22×C4×C101—21no—
129C22×C4×C101—21no—
130C22×C4×C101—21no—
131C22×C4×C101—21no—
132C22×C4×C101—42no—
133C22×C4×C101—42no—
134C22×C4×C101—41no—
135C22×C4×C101—41no—
136C22×C4×C101—42no—
137C22×C4×C101C2×D411yescharacteristic
138C22×C4×C101C2×D411yescharacteristic
139C22×C4×C101C2×D411yescharacteristic
140C22×C4×C101C2×D411yescharacteristic
141C22×C4×C101C2×D411yescharacteristic
142C22×C4×C101#4111yescharacteristic
143SD16×C101—48no—
144C2×C8×C101—22no—
145C2×C8×C101—42no—
146C2×C8×C101—42no—
147C2×C8×C101—41no—
148C4⋊C4×C101—21no—
149C4⋊C4×C101—21no—
150C4⋊C4×C101—42no—
151C4⋊C4×C101—41no—
152C4⋊C4×C101—41no—
153C4⋊C4×C101—82no—
154C22⋊C4×C101—48no—
155C22⋊C4×C101—41no—
156C22⋊C4×C101—41no—
157C22⋊C4×C101—41no—
158C22⋊C4×C101—41no—
159C22⋊C4×C101—41no—
160C22⋊C4×C101—41no—
161C22⋊C4×C101—41no—
162C22⋊C4×C101—41no—
163C22⋊C4×C101—82no—
164C22⋊C4×C101—82no—
165C23×C101—22no—
166C23×C101—22no—
167C23×C101—24no—
168C23×C101—22no—
169C23×C101—22no—
170C23×C101—22no—
171C23×C101—44no—
172C23×C101—42no—
173C23×C101—44no—
174C23×C101—44no—
175C23×C101—44no—
176C23×C101—42no—
177C23×C101—42no—
178C23×C101—42no—
179C23×C101—42no—
180C23×C101—44no—
181C23×C101—44no—
182C23×C101—44no—
183C23×C101—41no—
184C23×C101—42no—
185C23×C101—41no—
186C23×C101—42no—
187C23×C101—41no—
188C23×C101—42no—
189C23×C101—42no—
190C23×C101—42no—
191C23×C101—44no—
192C23×C101—41no—
193C23×C101—82no—
194C23×C101—82no—
195C23×C101—81no—
196C23×C101—81no—
197C23×C101—82no—
198C23×C101—82no—
199C23×C101—82no—
200C23×C101—82no—
201C23×C101—82no—
202C23×C101—81no—
203C23×C101—81no—
204C23×C101—82no—
205C23×C101—81no—
206C23×C101—81no—
207C23×C101#12011yescharacteristic
208C23×C101#12011yescharacteristic
209C23×C101#12111yescharacteristic
210C23×C101C2×SD1611yescharacteristic
211C23×C101#13511yescharacteristic
212C23×C101AGL(1,ℤ/8)11yescharacteristic
213C23×C101#13711yescharacteristic
214Q8×C101—42no—
215Q8×C101—81no—
216D4×C101—24no—
217D4×C101—24no—
218D4×C101—42no—
219D4×C101—44no—
220D4×C101—42no—
221D4×C101—44no—
222D4×C101—432no—
223C2×C4×C101—22no—
224C2×C4×C101—22no—
225C2×C4×C101—24no—
226C2×C4×C101—41no—
227C2×C4×C101—41no—
228C2×C4×C101—41no—
229C2×C4×C101—41no—
230C2×C4×C101—44no—
231C2×C4×C101—44no—
232C2×C4×C101—42no—
233C2×C4×C101—42no—
234C2×C4×C101—42no—
235C2×C4×C101—41no—
236C2×C4×C101—41no—
237C2×C4×C101—41no—
238C2×C4×C101—42no—
239C2×C4×C101—42no—
240C2×C4×C101—42no—
241C2×C4×C101—42no—
242C2×C4×C101—42no—
243C2×C4×C101—42no—
244C2×C4×C101—81no—
245C2×C4×C101#12011yescharacteristic
246C2×C4×C101#12111yescharacteristic
247C2×C4×C101#12111yescharacteristic
248C2×C4×C101#12711yescharacteristic
249C8×C101—42no—
250C8×C101—81no—
251C22×C101—22no—
252C22×C101—22no—
253C22×C101—22no—
254C22×C101—22no—
255C22×C101—21no—
256C22×C101—21no—
257C22×C101—21no—
258C22×C101—21no—
259C22×C101—44no—
260C22×C101—44no—
261C22×C101—44no—
262C22×C101—44no—
263C22×C101—42no—
264C22×C101—44no—
265C22×C101—82no—
266C22×C101—82no—
267C22×C101—82no—
268C22×C101—82no—
269C22×C101—84no—
270C22×C101—82no—
271C22×C101—84no—
272C22×C101—84no—
273C22×C101—82no—
274C22×C101—84no—
275C22×C101—82no—
276C22×C101—81no—
277C22×C101—82no—
278C22×C101—81no—
279C22×C101#39211yescharacteristic
280C22×C101#44811yescharacteristic
281C22×C101#44911yescharacteristic
282C22×C101#45011yescharacteristic
283C22×C101#45211yescharacteristic
284C22×C101#45312yes—
285C22×C101B(4,𝔽2)12yes—
286C22×C101#46511yescharacteristic
287C22×C101#46811yescharacteristic
288C4×C101—21no—
289C4×C101—21no—
290C4×C101—21no—
291C4×C101—21no—
292C4×C101—44no—
293C4×C101—82no—
294C4×C101—81no—
295C4×C101—81no—
296C4×C101—81no—
297#9922C10111yesmaximal, characteristic, direct product factor, Sylow
298C2×C101—21no—
299C2×C101—21no—
300C2×C101—21no—
301C2×C101—21no—
302C2×C101—82no—
303C2×C101—84no—
304C2×C101#159612yes—
305C2×C101#160012yes—
306C2×C101#200011yescharacteristic
307C2×C101#200211yescharacteristic
308C2×C101#202111yescharacteristic
309C2×#450C2×C10111yescharacteristic, complemented
310#2626C2×C10111yescharacteristic, complemented
311C2×#392C2×C10111yescharacteristic
312#1786C2×C10111yescharacteristic, complemented
313C2×#328C2×C10111yescharacteristic, complemented
314C2×#323C2×C10111yescharacteristic, complemented
315#1294C2×C10111yescharacteristic, complemented
316C101#992211yescharacteristic, direct product factor, Sylow
317C23×D4—21no—
318C23×D4C22×C10111yescharacteristic
319C22×SD16—21no—
320C2×C22⋊Q8C22×C10111yescharacteristic
321C22×C22⋊C4—21no—
322C22×C22⋊C4C22×C10111yescharacteristic
323#450—22no—
324#450—22no—
325C2×#102—21no—
326C2×#102—22no—
327C2×C22⋊C8C22×C10111yescharacteristic
328C2×C22⋊C8C22×C10111yescharacteristic
329#399—21no—
330#396—22no—
331#393—21no—
332#392—22no—
333#392—22no—
334#385—21no—
335#380C22×C10111yescharacteristic
336#379—21no—
337C2×#95C22×C10111yescharacteristic
338#328—22no—
339#323—22no—
340C25—21no—
341C25D4×C10111yescharacteristic
342C22×Q8D4×C10111yescharacteristic, complemented
343C22×D4—24no—
344C22×D4—21no—
345C22×D4—21no—
346C22×D4—21no—
347C22×D4—21no—
348C22×D4—21no—
349C22×D4—44no—
350C22×D4—44no—
351C22×D4—44no—
352C22×D4—42no—
353C22×D4—42no—
354C22×D4D4×C10112yes—
355C23×C4—21no—
356C23×C4D4×C10111yescharacteristic
357C23×C4C23×C10111yescharacteristic
358C2×SD16—24no—
359C2×SD16—44no—
360C22×C8—21no—
361C22×C8—21no—
362C22⋊Q8—21no—
363C22⋊Q8—21no—
364C22⋊Q8—21no—
365C22⋊Q8—21no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—22no—
368C2×C4⋊C4D4×C10111yescharacteristic, complemented
369C2×C4⋊C4D4×C10112yescomplemented
370C2×C22⋊C4—24no—
371C2×C22⋊C4—21no—
372C2×C22⋊C4—21no—
373C2×C22⋊C4—21no—
374C2×C22⋊C4—21no—
375C2×C22⋊C4—21no—
376C2×C22⋊C4—21no—
377C2×C22⋊C4—21no—
378C2×C22⋊C4—21no—
379C2×C22⋊C4—22no—
380C2×C22⋊C4—22no—
381C2×C22⋊C4—44no—
382#102—24no—
383#102—44no—
384C22⋊C8—21no—
385C22⋊C8—21no—
386C22⋊C8—21no—
387C22⋊C8D4×C10112yescomplemented
388#95—22no—
389#95—21no—
390#95—21no—
391#95D4×C10112yes—
392C24—24no—
393C24—21no—
394C24—21no—
395C24—22no—
396C24—21no—
397C24—21no—
398C24—21no—
399C24—21no—
400C24—42no—
401C24—42no—
402C24—42no—
403C24—42no—
404C24—42no—
405C24—41no—
406C24—41no—
407C24—41no—
408C24—41no—
409C24—41no—
410C24—41no—
411C24SD16×C10112yescomplemented
412C24C2×D4×C10111yescharacteristic
413C2×Q8—21no—
414C2×Q8—21no—
415C2×Q8—42no—
416C2×D4—24no—
417C2×D4—24no—
418C2×D4—22no—
419C2×D4—22no—
420C2×D4—42no—
421C2×D4—42no—
422C2×D4—48no—
423C2×D4—44no—
424C2×D4—44no—
425C2×D4—44no—
426C2×D4—44no—
427C2×D4—48no—
428C2×D4—48no—
429C2×D4—416no—
430C2×D4—416no—
431C2×D4—41no—
432C2×D4—41no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—41no—
444C22×C4—41no—
445C22×C4—42no—
446C22×C4C2×D4×C10111yescharacteristic
447C22×C4C2×D4×C10111yescharacteristic
448C22×C4C2×D4×C10111yescharacteristic
449C22×C4C2×D4×C10111yescharacteristic
450C22×C4C2×D4×C10111yescharacteristic
451C22×C4#41×C10111yescharacteristic
452SD16—48no—
453C2×C8—22no—
454C2×C8—42no—
455C2×C8—42no—
456C2×C8—41no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—42no—
460C4⋊C4—41no—
461C4⋊C4—41no—
462C4⋊C4—82no—
463C22⋊C4—48no—
464C22⋊C4—41no—
465C22⋊C4—41no—
466C22⋊C4—41no—
467C22⋊C4—41no—
468C22⋊C4—41no—
469C22⋊C4—41no—
470C22⋊C4—41no—
471C22⋊C4—41no—
472C22⋊C4—82no—
473C22⋊C4—82no—
474C23—22no—
475C23—22no—
476C23—24no—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—44no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—44no—
487C23—44no—
488C23—44no—
489C23—44no—
490C23—44no—
491C23—44no—
492C23—41no—
493C23—42no—
494C23—41no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—44no—
499C23—41no—
500C23—41no—
501C23—42no—
502C23—82no—
503C23—81no—
504C23—81no—
505C23—82no—
506C23—82no—
507C23—82no—
508C23—82no—
509C23—82no—
510C23—82no—
511C23—81no—
512C23—81no—
513C23—82no—
514C23—81no—
515C23—81no—
516C23#120×C10111yescharacteristic
517C23#120×C10111yescharacteristic
518C23#121×C10111yescharacteristic
519C23C2×SD16×C10111yescharacteristic
520C23#135×C10111yescharacteristic
521C23AGL(1,ℤ/8)×C10111yescharacteristic
522C23#137×C10111yescharacteristic
523Q8—42no—
524Q8—81no—
525D4—24no—
526D4—24no—
527D4—42no—
528D4—44no—
529D4—42no—
530D4—44no—
531D4—432no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—24no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—42no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—81no—
554C2×C4#120×C10111yescharacteristic
555C2×C4#121×C10111yescharacteristic
556C2×C4#121×C10111yescharacteristic
557C2×C4#127×C10111yescharacteristic
558C8—42no—
559C8—81no—
560C22—21no—
561C22—21no—
562C22—21no—
563C22—21no—
564C22—22no—
565C22—22no—
566C22—22no—
567C22—22no—
568C22—44no—
569C22—44no—
570C22—44no—
571C22—42no—
572C22—44no—
573C22—44no—
574C22—82no—
575C22—82no—
576C22—82no—
577C22—82no—
578C22—84no—
579C22—82no—
580C22—84no—
581C22—82no—
582C22—81no—
583C22—82no—
584C22—81no—
585C22—84no—
586C22—82no—
587C22—84no—
588C22#392×C10111yescharacteristic
589C22#448×C10111yescharacteristic
590C22#449×C10111yescharacteristic
591C22#450×C10111yescharacteristic
592C22#452×C10111yescharacteristic
593C22#453×C10112yes—
594C22B(4,𝔽2)×C10112yes—
595C22#465×C10111yescharacteristic
596C22#468×C10111yescharacteristic
597C4—21no—
598C4—21no—
599C4—21no—
600C4—21no—
601C4—44no—
602C4—81no—
603C4—81no—
604C4—81no—
605C4—82no—
606C2—21no—
607C2—21no—
608C2—21no—
609C2—21no—
610C2—82no—
611C2—84no—
612C2C101×#159612yes—
613C2C101×#160012yes—
614C2C101×#200011yescharacteristic
615C2C101×#200211yescharacteristic
616C2C101×#202111yescharacteristic
subgroupG/H
center, socleC23×C101#120
commutator, Frattini, cosocleC23×C4C23×C101
prime radicalC2×#392×C101C2
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C101 ◁ C23×C4×C101 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×#450×C101C2semidirect
2. C101×#2626C2semidirect
4. C101×#1786C2semidirect
5. C2×#328×C101C2semidirect
6. C2×#323×C101C2semidirect
7. C101×#1294C2semidirect
33. C22×Q8×C101D4semidirect
59. C2×C4⋊C4×C101D4semidirect
60. C2×C4⋊C4×C101D4semidirect
78. C22⋊C8×C101D4semidirect
102. C24×C101SD16semidirect
297. #9922C101direct (N × H)
309. C2×#450C2×C101semidirect
310. #2626C2×C101semidirect
312. #1786C2×C101semidirect
313. C2×#328C2×C101semidirect
314. C2×#323C2×C101semidirect
315. #1294C2×C101semidirect
316. C101#9922direct (N × H)
342. C22×Q8D4×C101semidirect
368. C2×C4⋊C4D4×C101semidirect
369. C2×C4⋊C4D4×C101semidirect
387. C22⋊C8D4×C101semidirect
411. C24SD16×C101semidirect

Maximal subgroups

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

subgroupindex
C2×#450×C1012
C101×#26262
C2×#392×C1012
C101×#17862
C2×#328×C1012
C2×#323×C1012
C101×#12942
#9922101