Finite Group Databasebeta

← C107×#13347 #41039311

Subgroups — C107×#13347 #41039311

Order 27392.

830 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 830 conjugacy classes of subgroups — too many to draw. Only its 98 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
1C2×#458×C107C211yesmaximal, characteristic, complemented
2C2×#453×C107C211yesmaximal, characteristic
3C107×#2709C211yesmaximal, characteristic, complemented
4C2×#353×C107C211yesmaximal, characteristic, complemented
5C2×#352×C107C211yesmaximal, characteristic, complemented
6C107×#1790C211yesmaximal, characteristic, complemented
7C107×#1390C211yesmaximal, characteristic, complemented
8C2×H(5,𝔽2)×C107C2211yescharacteristic
9C2×AGL(1,ℤ/8)×C107—21no—
10C2×#127×C107C2211yescharacteristic
11C2×#123×C107—21no—
12C2×#123×C107C2211yescharacteristic
13#458×C107—22no—
14#458×C107—22no—
15#453×C107—22no—
16#453×C107—22no—
17C2×#104×C107—21no—
18C2×#100×C107C2211yescharacteristic
19C2×#99×C107—21no—
20C2×#99×C107—21no—
21C2×#99×C107C2211yescharacteristic
22C2×#99×C107C2211yescharacteristic
23#394×C107—21no—
24#394×C107—21no—
25#394×C107—21no—
26#387×C107—21no—
27#386×C107—21no—
28#386×C107—21no—
29#386×C107—21no—
30#386×C107C2211yescharacteristic
31#353×C107—22no—
32#352×C107—22no—
33H(5,𝔽2)×C107—21no—
34H(5,𝔽2)×C107—21no—
35H(5,𝔽2)×C107—21no—
36H(5,𝔽2)×C107—42no—
37H(5,𝔽2)×C107D412yes—
38C2×#41×C107—21no—
39C2×#41×C107—41no—
40C22×D4×C107—21no—
41C22×D4×C107—21no—
42C22×D4×C107—21no—
43C22×D4×C107—21no—
44C22×D4×C107—41no—
45C22×D4×C107D411yescharacteristic
46C22×D4×C107C2311yescharacteristic
47C23×C4×C107D411yescharacteristic
48C23×C4×C107D411yescharacteristic
49AGL(1,ℤ/8)×C107—24no—
50AGL(1,ℤ/8)×C107—42no—
51C2×SD16×C107—41no—
52C2×D8×C107—41no—
53C2×M16×C107—21no—
54#127×C107—21no—
55#127×C107—21no—
56#127×C107—21no—
57#127×C107—21no—
58#123×C107—24no—
59#123×C107—21no—
60#123×C107—21no—
61#123×C107—21no—
62#123×C107—21no—
63#123×C107—42no—
64C2×C4⋊C4×C107—21no—
65C2×C4⋊C4×C107—21no—
66C2×C4⋊C4×C107—41no—
67C2×C22⋊C4×C107—21no—
68C2×C22⋊C4×C107—21no—
69C2×C22⋊C4×C107—21no—
70C2×C22⋊C4×C107—21no—
71C2×C22⋊C4×C107—21no—
72C2×C22⋊C4×C107—21no—
73C2×C22⋊C4×C107—21no—
74C2×C22⋊C4×C107—21no—
75C2×C22⋊C4×C107—41no—
76C2×C22⋊C4×C107—41no—
77C2×C22⋊C4×C107D411yescharacteristic
78C2×C22⋊C4×C107D411yescharacteristic
79C2×C42×C107D411yescharacteristic
80#104×C107—24no—
81#100×C107—21no—
82#100×C107D412yes—
83#99×C107—24no—
84#99×C107—21no—
85#99×C107—21no—
86#99×C107—21no—
87#99×C107—42no—
88#99×C107D412yes—
89#95×C107—21no—
90#95×C107—21no—
91#95×C107—21no—
92#95×C107—21no—
93#95×C107—41no—
94C24×C107—21no—
95C24×C107—21no—
96C24×C107—21no—
97C24×C107C2×D411yescharacteristic
98C24×C107C2×D411yescharacteristic
99#41×C107—22no—
100#41×C107—22no—
101#41×C107—42no—
102#41×C107—41no—
103#41×C107—42no—
104#41×C107—42no—
105#41×C107—42no—
106#41×C107—41no—
107C2×Q8×C107—21no—
108C2×D4×C107—21no—
109C2×D4×C107—22no—
110C2×D4×C107—22no—
111C2×D4×C107—22no—
112C2×D4×C107—22no—
113C2×D4×C107—21no—
114C2×D4×C107—21no—
115C2×D4×C107—41no—
116C2×D4×C107—41no—
117C2×D4×C107—41no—
118C2×D4×C107—41no—
119C2×D4×C107—42no—
120C2×D4×C107—42no—
121C2×D4×C107—42no—
122C2×D4×C107—42no—
123C2×D4×C107—41no—
124C2×D4×C107—41no—
125C2×D4×C107—41no—
126C2×D4×C107—41no—
127C2×D4×C107—42no—
128C2×D4×C107—42no—
129C2×D4×C107—41no—
130C2×D4×C107—41no—
131C2×D4×C107—41no—
132C2×D4×C107—41no—
133C2×D4×C107—41no—
134C2×D4×C107—41no—
135C2×D4×C107—41no—
136C2×D4×C107—81no—
137C2×D4×C107—81no—
138C2×D4×C107C2×D411yescharacteristic
139C2×D4×C107C2×D411yescharacteristic
140C2×D4×C107C2×D411yescharacteristic
141C2×D4×C107#4111yescharacteristic
142C22×C4×C107—21no—
143C22×C4×C107—21no—
144C22×C4×C107—21no—
145C22×C4×C107—21no—
146C22×C4×C107—21no—
147C22×C4×C107—21no—
148C22×C4×C107—21no—
149C22×C4×C107—21no—
150C22×C4×C107—21no—
151C22×C4×C107—41no—
152C22×C4×C107—41no—
153C22×C4×C107—41no—
154C22×C4×C107—41no—
155C22×C4×C107—41no—
156C22×C4×C107—41no—
157C22×C4×C107—81no—
158C22×C4×C107C2×D411yescharacteristic
159SD16×C107—44no—
160D8×C107—44no—
161M16×C107—22no—
162M16×C107—41no—
163C2×C8×C107—41no—
164C4⋊C4×C107—42no—
165C4⋊C4×C107—44no—
166C4⋊C4×C107—41no—
167C4⋊C4×C107—41no—
168C22⋊C4×C107—21no—
169C22⋊C4×C107—22no—
170C22⋊C4×C107—21no—
171C22⋊C4×C107—41no—
172C22⋊C4×C107—44no—
173C22⋊C4×C107—41no—
174C22⋊C4×C107—41no—
175C22⋊C4×C107—41no—
176C22⋊C4×C107—41no—
177C22⋊C4×C107—41no—
178C22⋊C4×C107—81no—
179C22⋊C4×C107—81no—
180C22⋊C4×C107—81no—
181C22⋊C4×C107—81no—
182C22⋊C4×C107—81no—
183C22⋊C4×C107—82no—
184C42×C107—21no—
185C42×C107—21no—
186C23×C107—21no—
187C23×C107—21no—
188C23×C107—21no—
189C23×C107—21no—
190C23×C107—22no—
191C23×C107—22no—
192C23×C107—41no—
193C23×C107—41no—
194C23×C107—41no—
195C23×C107—41no—
196C23×C107—41no—
197C23×C107—41no—
198C23×C107—41no—
199C23×C107—41no—
200C23×C107—41no—
201C23×C107—41no—
202C23×C107—41no—
203C23×C107—41no—
204C23×C107—41no—
205C23×C107—41no—
206C23×C107—81no—
207C23×C107—81no—
208C23×C107—81no—
209C23×C107—81no—
210C23×C107#12011yescharacteristic
211C23×C107#12011yescharacteristic
212C23×C107#12111yescharacteristic
213C23×C107#12111yescharacteristic
214C23×C107#12711yescharacteristic
215Q8×C107—22no—
216Q8×C107—41no—
217D4×C107—22no—
218D4×C107—41no—
219D4×C107—41no—
220D4×C107—41no—
221D4×C107—41no—
222D4×C107—41no—
223D4×C107—42no—
224D4×C107—42no—
225D4×C107—42no—
226D4×C107—42no—
227D4×C107—42no—
228D4×C107—42no—
229D4×C107—42no—
230D4×C107—81no—
231D4×C107—82no—
232D4×C107—82no—
233D4×C107—82no—
234D4×C107—82no—
235C2×C4×C107—21no—
236C2×C4×C107—21no—
237C2×C4×C107—22no—
238C2×C4×C107—22no—
239C2×C4×C107—22no—
240C2×C4×C107—42no—
241C2×C4×C107—42no—
242C2×C4×C107—41no—
243C2×C4×C107—41no—
244C2×C4×C107—41no—
245C2×C4×C107—41no—
246C2×C4×C107—41no—
247C2×C4×C107—42no—
248C2×C4×C107—42no—
249C2×C4×C107—41no—
250C2×C4×C107—41no—
251C2×C4×C107—41no—
252C2×C4×C107—41no—
253C2×C4×C107—41no—
254C2×C4×C107—41no—
255C2×C4×C107—42no—
256C2×C4×C107—41no—
257C2×C4×C107—41no—
258C2×C4×C107—41no—
259C2×C4×C107—41no—
260C2×C4×C107—41no—
261C2×C4×C107—41no—
262C2×C4×C107—81no—
263C2×C4×C107—81no—
264C2×C4×C107—81no—
265C2×C4×C107—81no—
266C2×C4×C107—81no—
267C2×C4×C107—81no—
268C2×C4×C107—81no—
269C2×C4×C107#12011yescharacteristic
270C2×C4×C107#12111yescharacteristic
271C8×C107—42no—
272C22×C107—21no—
273C22×C107—21no—
274C22×C107—21no—
275C22×C107—21no—
276C22×C107—21no—
277C22×C107—21no—
278C22×C107—41no—
279C22×C107—41no—
280C22×C107—41no—
281C22×C107—41no—
282C22×C107—41no—
283C22×C107—41no—
284C22×C107—41no—
285C22×C107—41no—
286C22×C107—42no—
287C22×C107—81no—
288C22×C107—81no—
289C22×C107—81no—
290C22×C107—81no—
291C22×C107—81no—
292C22×C107—81no—
293C22×C107—82no—
294C22×C107—82no—
295C22×C107—81no—
296C22×C107—81no—
297C22×C107—81no—
298C22×C107—81no—
299C22×C107—81no—
300C22×C107—161no—
301C22×C107—161no—
302C22×C107#39211yescharacteristic
303C22×C107B(4,𝔽2)11yescharacteristic
304C22×C107B(4,𝔽2)11yescharacteristic
305C4×C107—21no—
306C4×C107—21no—
307C4×C107—42no—
308C4×C107—42no—
309C4×C107—81no—
310C4×C107—81no—
311C4×C107—82no—
312C4×C107—81no—
313C4×C107—81no—
314C4×C107—161no—
315#13347C10711yesmaximal, characteristic, direct product factor, Sylow
316C2×C107—21no—
317C2×C107—21no—
318C2×C107—41no—
319C2×C107—41no—
320C2×C107—41no—
321C2×C107—41no—
322C2×C107—82no—
323C2×C107—161no—
324C2×C107#202111yescharacteristic
325C2×C107#219811yescharacteristic
326C2×C107#220011yescharacteristic
327C2×#458C2×C10711yescharacteristic, complemented
328C2×#453C2×C10711yescharacteristic
329#2709C2×C10711yescharacteristic, complemented
330C2×#353C2×C10711yescharacteristic, complemented
331C2×#352C2×C10711yescharacteristic, complemented
332#1790C2×C10711yescharacteristic, complemented
333#1390C2×C10711yescharacteristic, complemented
334C107#1334711yescharacteristic, direct product factor, Sylow
335C2×H(5,𝔽2)C22×C10711yescharacteristic
336C2×AGL(1,ℤ/8)—21no—
337C2×#127C22×C10711yescharacteristic
338C2×#123—21no—
339C2×#123C22×C10711yescharacteristic
340#458—22no—
341#458—22no—
342#453—22no—
343#453—22no—
344C2×#104—21no—
345C2×#100C22×C10711yescharacteristic
346C2×#99—21no—
347C2×#99—21no—
348C2×#99C22×C10711yescharacteristic
349C2×#99C22×C10711yescharacteristic
350#394—21no—
351#394—21no—
352#394—21no—
353#387—21no—
354#386—21no—
355#386—21no—
356#386—21no—
357#386C22×C10711yescharacteristic
358#353—22no—
359#352—22no—
360H(5,𝔽2)—21no—
361H(5,𝔽2)—21no—
362H(5,𝔽2)—21no—
363H(5,𝔽2)—42no—
364H(5,𝔽2)D4×C10712yes—
365C2×#41—21no—
366C2×#41—41no—
367C22×D4—21no—
368C22×D4—21no—
369C22×D4—21no—
370C22×D4—21no—
371C22×D4—41no—
372C22×D4D4×C10711yescharacteristic
373C22×D4C23×C10711yescharacteristic
374C23×C4D4×C10711yescharacteristic
375C23×C4D4×C10711yescharacteristic
376AGL(1,ℤ/8)—24no—
377AGL(1,ℤ/8)—42no—
378C2×SD16—41no—
379C2×D8—41no—
380C2×M16—21no—
381#127—21no—
382#127—21no—
383#127—21no—
384#127—21no—
385#123—24no—
386#123—21no—
387#123—21no—
388#123—21no—
389#123—21no—
390#123—42no—
391C2×C4⋊C4—21no—
392C2×C4⋊C4—21no—
393C2×C4⋊C4—41no—
394C2×C22⋊C4—21no—
395C2×C22⋊C4—21no—
396C2×C22⋊C4—21no—
397C2×C22⋊C4—21no—
398C2×C22⋊C4—21no—
399C2×C22⋊C4—21no—
400C2×C22⋊C4—21no—
401C2×C22⋊C4—21no—
402C2×C22⋊C4—41no—
403C2×C22⋊C4—41no—
404C2×C22⋊C4D4×C10711yescharacteristic
405C2×C22⋊C4D4×C10711yescharacteristic
406C2×C42D4×C10711yescharacteristic
407#104—24no—
408#100—21no—
409#100D4×C10712yes—
410#99—24no—
411#99—21no—
412#99—21no—
413#99—21no—
414#99—42no—
415#99D4×C10712yes—
416#95—21no—
417#95—21no—
418#95—21no—
419#95—21no—
420#95—41no—
421C24—21no—
422C24—21no—
423C24—21no—
424C24C2×D4×C10711yescharacteristic
425C24C2×D4×C10711yescharacteristic
426#41—22no—
427#41—22no—
428#41—42no—
429#41—41no—
430#41—42no—
431#41—42no—
432#41—42no—
433#41—41no—
434C2×Q8—21no—
435C2×D4—21no—
436C2×D4—22no—
437C2×D4—22no—
438C2×D4—22no—
439C2×D4—22no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4—41no—
443C2×D4—41no—
444C2×D4—41no—
445C2×D4—41no—
446C2×D4—42no—
447C2×D4—42no—
448C2×D4—42no—
449C2×D4—42no—
450C2×D4—41no—
451C2×D4—41no—
452C2×D4—41no—
453C2×D4—41no—
454C2×D4—42no—
455C2×D4—42no—
456C2×D4—41no—
457C2×D4—41no—
458C2×D4—41no—
459C2×D4—41no—
460C2×D4—41no—
461C2×D4—41no—
462C2×D4—41no—
463C2×D4—81no—
464C2×D4—81no—
465C2×D4C2×D4×C10711yescharacteristic
466C2×D4C2×D4×C10711yescharacteristic
467C2×D4C2×D4×C10711yescharacteristic
468C2×D4#41×C10711yescharacteristic
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—41no—
479C22×C4—41no—
480C22×C4—41no—
481C22×C4—41no—
482C22×C4—41no—
483C22×C4—41no—
484C22×C4—81no—
485C22×C4C2×D4×C10711yescharacteristic
486SD16—44no—
487D8—44no—
488M16—22no—
489M16—41no—
490C2×C8—41no—
491C4⋊C4—42no—
492C4⋊C4—44no—
493C4⋊C4—41no—
494C4⋊C4—41no—
495C22⋊C4—21no—
496C22⋊C4—22no—
497C22⋊C4—21no—
498C22⋊C4—41no—
499C22⋊C4—44no—
500C22⋊C4—41no—
501C22⋊C4—41no—
502C22⋊C4—41no—
503C22⋊C4—41no—
504C22⋊C4—41no—
505C22⋊C4—81no—
506C22⋊C4—81no—
507C22⋊C4—81no—
508C22⋊C4—81no—
509C22⋊C4—81no—
510C22⋊C4—82no—
511C42—21no—
512C42—21no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23—21no—
517C23—22no—
518C23—22no—
519C23—41no—
520C23—41no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23—41no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23—41no—
533C23—81no—
534C23—81no—
535C23—81no—
536C23—81no—
537C23#120×C10711yescharacteristic
538C23#120×C10711yescharacteristic
539C23#121×C10711yescharacteristic
540C23#121×C10711yescharacteristic
541C23#127×C10711yescharacteristic
542Q8—22no—
543Q8—41no—
544D4—22no—
545D4—41no—
546D4—41no—
547D4—41no—
548D4—41no—
549D4—41no—
550D4—42no—
551D4—42no—
552D4—42no—
553D4—42no—
554D4—42no—
555D4—42no—
556D4—42no—
557D4—81no—
558D4—82no—
559D4—82no—
560D4—82no—
561D4—82no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—81no—
590C2×C4—81no—
591C2×C4—81no—
592C2×C4—81no—
593C2×C4—81no—
594C2×C4—81no—
595C2×C4—81no—
596C2×C4#120×C10711yescharacteristic
597C2×C4#121×C10711yescharacteristic
598C8—42no—
599C22—21no—
600C22—21no—
601C22—21no—
602C22—21no—
603C22—21no—
604C22—21no—
605C22—41no—
606C22—41no—
607C22—41no—
608C22—41no—
609C22—41no—
610C22—41no—
611C22—41no—
612C22—41no—
613C22—42no—
614C22—81no—
615C22—81no—
616C22—81no—
617C22—81no—
618C22—81no—
619C22—81no—
620C22—82no—
621C22—82no—
622C22—81no—
623C22—81no—
624C22—81no—
625C22—81no—
626C22—81no—
627C22—161no—
628C22—161no—
629C22#392×C10711yescharacteristic
630C22B(4,𝔽2)×C10711yescharacteristic
631C22B(4,𝔽2)×C10711yescharacteristic
632C4—21no—
633C4—21no—
634C4—42no—
635C4—42no—
636C4—81no—
637C4—81no—
638C4—82no—
639C4—81no—
640C4—81no—
641C4—161no—
642C2—21no—
643C2—21no—
644C2—41no—
645C2—41no—
646C2—41no—
647C2—41no—
648C2—82no—
649C2—161no—
650C2C107×#202111yescharacteristic
651C2C107×#219811yescharacteristic
652C2C107×#220011yescharacteristic
subgroupG/H
center, socleC22×C107B(4,𝔽2)
commutator, Frattini, cosocleC22×D4C23×C107
prime radicalC2×#453×C107C2
LCSG ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C107 ◁ C23×C107 ◁ C22×D4×C107 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×#458×C107C2semidirect
3. C107×#2709C2semidirect
4. C2×#353×C107C2semidirect
5. C2×#352×C107C2semidirect
6. C107×#1790C2semidirect
7. C107×#1390C2semidirect
315. #13347C107direct (N × H)
327. C2×#458C2×C107semidirect
329. #2709C2×C107semidirect
330. C2×#353C2×C107semidirect
331. C2×#352C2×C107semidirect
332. #1790C2×C107semidirect
333. #1390C2×C107semidirect
334. C107#13347direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#458×C1072
C2×#453×C1072
C107×#27092
C2×#353×C1072
C2×#352×C1072
C107×#17902
C107×#13902
#13347107