Finite Group Databasebeta

← C107×#13348 #41039312

Subgroups — C107×#13348 #41039312

Order 27392.

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

normal subgroups only

This group has 900 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×B(4,𝔽2)×C107C211yesmaximal, characteristic
2C2×#455×C107C211yesmaximal, characteristic, complemented
3C107×#2684C211yesmaximal, characteristic, complemented
4C2×#354×C107C211yesmaximal, characteristic, complemented
5C2×AΣL(1,𝔽16)×C107C211yesmaximal, characteristic, complemented
6C107×#1790C211yesmaximal, characteristic, complemented
7C107×#1390C211yesmaximal, characteristic, complemented
8C2×H(5,𝔽2)×C107C2211yescharacteristic
9C2×#137×C107—21no—
10C2×#124×C107C2211yescharacteristic
11C2×#120×C107—21no—
12C2×#120×C107C2211yescharacteristic
13B(4,𝔽2)×C107—22no—
14B(4,𝔽2)×C107—22no—
15#455×C107—22no—
16#455×C107—22no—
17C2×#104×C107—21no—
18C2×#100×C107C2211yescharacteristic
19C2×#99×C107—21no—
20C2×#99×C107—21no—
21C2×#99×C107C2211yescharacteristic
22C2×#99×C107C2211yescharacteristic
23#397×C107—21no—
24#394×C107—21no—
25#393×C107—21no—
26#388×C107—21no—
27#386×C107—21no—
28#386×C107—21no—
29#386×C107C2211yescharacteristic
30#379×C107—21no—
31#354×C107—22no—
32AΣL(1,𝔽16)×C107—22no—
33C25×C107D411yescharacteristic
34H(5,𝔽2)×C107—21no—
35H(5,𝔽2)×C107—21no—
36H(5,𝔽2)×C107—21no—
37H(5,𝔽2)×C107—42no—
38H(5,𝔽2)×C107D412yes—
39C2×#41×C107—21no—
40C2×#41×C107—41no—
41C22×Q8×C107D411yescharacteristic, complemented
42C22×D4×C107—21no—
43C22×D4×C107—21no—
44C22×D4×C107—21no—
45C22×D4×C107—41no—
46C22×D4×C107—41no—
47C22×D4×C107C2311yescharacteristic
48C23×C4×C107D411yescharacteristic, complemented
49#137×C107—24no—
50#137×C107—42no—
51C2×Q16×C107—41no—
52C2×SD16×C107—41no—
53C2×M16×C107—21no—
54#124×C107—21no—
55#124×C107—21no—
56#124×C107—21no—
57#124×C107—21no—
58#120×C107—24no—
59#120×C107—21no—
60#120×C107—21no—
61#120×C107—21no—
62#120×C107—21no—
63#120×C107—42no—
64C2×C4⋊C4×C107—21no—
65C2×C22⋊C4×C107—21no—
66C2×C22⋊C4×C107—21no—
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, complemented
79C2×C42×C107D411yescharacteristic, complemented
80#104×C107—24no—
81#100×C107—21no—
82#100×C107D412yescomplemented
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×C107—21no—
98C24×C107—21no—
99C24×C107—41no—
100C24×C107—41no—
101C24×C107—42no—
102C24×C107—81no—
103C24×C107C2×D411yescharacteristic
104C24×C107C2×D411yescharacteristic
105#41×C107—22no—
106#41×C107—22no—
107#41×C107—42no—
108#41×C107—41no—
109#41×C107—42no—
110#41×C107—42no—
111#41×C107—42no—
112#41×C107—41no—
113C2×Q8×C107—21no—
114C2×Q8×C107—21no—
115C2×Q8×C107—21no—
116C2×Q8×C107—41no—
117C2×Q8×C107—41no—
118C2×D4×C107—21no—
119C2×D4×C107—22no—
120C2×D4×C107—22no—
121C2×D4×C107—22no—
122C2×D4×C107—22no—
123C2×D4×C107—41no—
124C2×D4×C107—41no—
125C2×D4×C107—42no—
126C2×D4×C107—42no—
127C2×D4×C107—41no—
128C2×D4×C107—41no—
129C2×D4×C107—41no—
130C2×D4×C107—41no—
131C2×D4×C107—42no—
132C2×D4×C107—42no—
133C2×D4×C107—41no—
134C2×D4×C107—41no—
135C2×D4×C107—41no—
136C2×D4×C107—44no—
137C2×D4×C107—42no—
138C2×D4×C107—41no—
139C2×D4×C107—41no—
140C2×D4×C107—81no—
141C2×D4×C107—82no—
142C2×D4×C107—82no—
143C2×D4×C107C2×D411yescharacteristic
144C2×D4×C107C2×D411yescharacteristic
145C2×D4×C107C2×D411yescharacteristic
146C2×D4×C107#4111yescharacteristic
147C22×C4×C107—21no—
148C22×C4×C107—21no—
149C22×C4×C107—21no—
150C22×C4×C107—21no—
151C22×C4×C107—21no—
152C22×C4×C107—21no—
153C22×C4×C107—21no—
154C22×C4×C107—41no—
155C22×C4×C107—41no—
156C22×C4×C107—41no—
157C22×C4×C107—41no—
158C22×C4×C107—81no—
159C22×C4×C107C2×D411yescharacteristic
160Q16×C107—44no—
161SD16×C107—44no—
162M16×C107—22no—
163M16×C107—41no—
164C2×C8×C107—41no—
165C4⋊C4×C107—81no—
166C22⋊C4×C107—21no—
167C22⋊C4×C107—22no—
168C22⋊C4×C107—21no—
169C22⋊C4×C107—44no—
170C22⋊C4×C107—42no—
171C22⋊C4×C107—41no—
172C22⋊C4×C107—41no—
173C22⋊C4×C107—41no—
174C22⋊C4×C107—41no—
175C22⋊C4×C107—41no—
176C22⋊C4×C107—41no—
177C22⋊C4×C107—41no—
178C22⋊C4×C107—41no—
179C22⋊C4×C107—41no—
180C22⋊C4×C107—41no—
181C22⋊C4×C107—81no—
182C22⋊C4×C107—81no—
183C22⋊C4×C107—81no—
184C22⋊C4×C107—81no—
185C22⋊C4×C107—82no—
186C42×C107—21no—
187C42×C107—21no—
188C23×C107—21no—
189C23×C107—21no—
190C23×C107—21no—
191C23×C107—21no—
192C23×C107—22no—
193C23×C107—22no—
194C23×C107—22no—
195C23×C107—41no—
196C23×C107—41no—
197C23×C107—41no—
198C23×C107—41no—
199C23×C107—42no—
200C23×C107—41no—
201C23×C107—41no—
202C23×C107—41no—
203C23×C107—41no—
204C23×C107—41no—
205C23×C107—42no—
206C23×C107—42no—
207C23×C107—41no—
208C23×C107—41no—
209C23×C107—41no—
210C23×C107—41no—
211C23×C107—81no—
212C23×C107—82no—
213C23×C107—81no—
214C23×C107—82no—
215C23×C107—84no—
216C23×C107—81no—
217C23×C107—81no—
218C23×C107—81no—
219C23×C107—81no—
220C23×C107—81no—
221C23×C107#12011yescharacteristic
222C23×C107#12011yescharacteristic
223C23×C107#12111yescharacteristic
224C23×C107#12111yescharacteristic
225C23×C107#12711yescharacteristic
226Q8×C107—22no—
227Q8×C107—41no—
228Q8×C107—42no—
229Q8×C107—81no—
230D4×C107—22no—
231D4×C107—41no—
232D4×C107—41no—
233D4×C107—41no—
234D4×C107—41no—
235D4×C107—41no—
236D4×C107—42no—
237D4×C107—42no—
238D4×C107—42no—
239D4×C107—42no—
240D4×C107—42no—
241D4×C107—42no—
242D4×C107—82no—
243D4×C107—82no—
244D4×C107—88no—
245C2×C4×C107—21no—
246C2×C4×C107—21no—
247C2×C4×C107—22no—
248C2×C4×C107—22no—
249C2×C4×C107—42no—
250C2×C4×C107—42no—
251C2×C4×C107—41no—
252C2×C4×C107—41no—
253C2×C4×C107—41no—
254C2×C4×C107—42no—
255C2×C4×C107—41no—
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—41no—
263C2×C4×C107—42no—
264C2×C4×C107—41no—
265C2×C4×C107—41no—
266C2×C4×C107—41no—
267C2×C4×C107—41no—
268C2×C4×C107—41no—
269C2×C4×C107—41no—
270C2×C4×C107—41no—
271C2×C4×C107—81no—
272C2×C4×C107—81no—
273C2×C4×C107—81no—
274C2×C4×C107—81no—
275C2×C4×C107—81no—
276C2×C4×C107—81no—
277C2×C4×C107#12011yescharacteristic
278C2×C4×C107#12111yescharacteristic
279C8×C107—42no—
280C22×C107—21no—
281C22×C107—21no—
282C22×C107—21no—
283C22×C107—21no—
284C22×C107—21no—
285C22×C107—21no—
286C22×C107—41no—
287C22×C107—41no—
288C22×C107—41no—
289C22×C107—41no—
290C22×C107—42no—
291C22×C107—42no—
292C22×C107—41no—
293C22×C107—41no—
294C22×C107—41no—
295C22×C107—41no—
296C22×C107—44no—
297C22×C107—42no—
298C22×C107—42no—
299C22×C107—81no—
300C22×C107—81no—
301C22×C107—81no—
302C22×C107—81no—
303C22×C107—82no—
304C22×C107—82no—
305C22×C107—82no—
306C22×C107—82no—
307C22×C107—82no—
308C22×C107—81no—
309C22×C107—81no—
310C22×C107—81no—
311C22×C107—81no—
312C22×C107—161no—
313C22×C107—161no—
314C22×C107#39211yescharacteristic
315C22×C107B(4,𝔽2)11yescharacteristic
316C22×C107B(4,𝔽2)11yescharacteristic
317C4×C107—21no—
318C4×C107—21no—
319C4×C107—42no—
320C4×C107—81no—
321C4×C107—81no—
322C4×C107—81no—
323C4×C107—82no—
324C4×C107—81no—
325C4×C107—81no—
326C4×C107—161no—
327#13348C10711yesmaximal, characteristic, direct product factor, Sylow
328C2×C107—21no—
329C2×C107—21no—
330C2×C107—41no—
331C2×C107—41no—
332C2×C107—41no—
333C2×C107—41no—
334C2×C107—82no—
335C2×C107—82no—
336C2×C107#202111yescharacteristic
337C2×C107#219711yescharacteristic
338C2×C107#219911yescharacteristic
339C2×B(4,𝔽2)C2×C10711yescharacteristic
340C2×#455C2×C10711yescharacteristic, complemented
341#2684C2×C10711yescharacteristic, complemented
342C2×#354C2×C10711yescharacteristic, complemented
343C2×AΣL(1,𝔽16)C2×C10711yescharacteristic, complemented
344#1790C2×C10711yescharacteristic, complemented
345#1390C2×C10711yescharacteristic, complemented
346C107#1334811yescharacteristic, direct product factor, Sylow
347C2×H(5,𝔽2)C22×C10711yescharacteristic
348C2×#137—21no—
349C2×#124C22×C10711yescharacteristic
350C2×#120—21no—
351C2×#120C22×C10711yescharacteristic
352B(4,𝔽2)—22no—
353B(4,𝔽2)—22no—
354#455—22no—
355#455—22no—
356C2×#104—21no—
357C2×#100C22×C10711yescharacteristic
358C2×#99—21no—
359C2×#99—21no—
360C2×#99C22×C10711yescharacteristic
361C2×#99C22×C10711yescharacteristic
362#397—21no—
363#394—21no—
364#393—21no—
365#388—21no—
366#386—21no—
367#386—21no—
368#386C22×C10711yescharacteristic
369#379—21no—
370#354—22no—
371AΣL(1,𝔽16)—22no—
372C25D4×C10711yescharacteristic
373H(5,𝔽2)—21no—
374H(5,𝔽2)—21no—
375H(5,𝔽2)—21no—
376H(5,𝔽2)—42no—
377H(5,𝔽2)D4×C10712yes—
378C2×#41—21no—
379C2×#41—41no—
380C22×Q8D4×C10711yescharacteristic, complemented
381C22×D4—21no—
382C22×D4—21no—
383C22×D4—21no—
384C22×D4—41no—
385C22×D4—41no—
386C22×D4C23×C10711yescharacteristic
387C23×C4D4×C10711yescharacteristic, complemented
388#137—24no—
389#137—42no—
390C2×Q16—41no—
391C2×SD16—41no—
392C2×M16—21no—
393#124—21no—
394#124—21no—
395#124—21no—
396#124—21no—
397#120—24no—
398#120—21no—
399#120—21no—
400#120—21no—
401#120—21no—
402#120—42no—
403C2×C4⋊C4—21no—
404C2×C22⋊C4—21no—
405C2×C22⋊C4—21no—
406C2×C22⋊C4—21no—
407C2×C22⋊C4—21no—
408C2×C22⋊C4—21no—
409C2×C22⋊C4—21no—
410C2×C22⋊C4—21no—
411C2×C22⋊C4—21no—
412C2×C22⋊C4—21no—
413C2×C22⋊C4—21no—
414C2×C22⋊C4—41no—
415C2×C22⋊C4—41no—
416C2×C22⋊C4D4×C10711yescharacteristic
417C2×C22⋊C4D4×C10711yescharacteristic, complemented
418C2×C42D4×C10711yescharacteristic, complemented
419#104—24no—
420#100—21no—
421#100D4×C10712yescomplemented
422#99—24no—
423#99—21no—
424#99—21no—
425#99—21no—
426#99—42no—
427#99D4×C10712yes—
428#95—21no—
429#95—21no—
430#95—21no—
431#95—21no—
432#95—41no—
433C24—21no—
434C24—21no—
435C24—21no—
436C24—21no—
437C24—21no—
438C24—41no—
439C24—41no—
440C24—42no—
441C24—81no—
442C24C2×D4×C10711yescharacteristic
443C24C2×D4×C10711yescharacteristic
444#41—22no—
445#41—22no—
446#41—42no—
447#41—41no—
448#41—42no—
449#41—42no—
450#41—42no—
451#41—41no—
452C2×Q8—21no—
453C2×Q8—21no—
454C2×Q8—21no—
455C2×Q8—41no—
456C2×Q8—41no—
457C2×D4—21no—
458C2×D4—22no—
459C2×D4—22no—
460C2×D4—22no—
461C2×D4—22no—
462C2×D4—41no—
463C2×D4—41no—
464C2×D4—42no—
465C2×D4—42no—
466C2×D4—41no—
467C2×D4—41no—
468C2×D4—41no—
469C2×D4—41no—
470C2×D4—42no—
471C2×D4—42no—
472C2×D4—41no—
473C2×D4—41no—
474C2×D4—41no—
475C2×D4—44no—
476C2×D4—42no—
477C2×D4—41no—
478C2×D4—41no—
479C2×D4—81no—
480C2×D4—82no—
481C2×D4—82no—
482C2×D4C2×D4×C10711yescharacteristic
483C2×D4C2×D4×C10711yescharacteristic
484C2×D4C2×D4×C10711yescharacteristic
485C2×D4#41×C10711yescharacteristic
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—21no—
493C22×C4—41no—
494C22×C4—41no—
495C22×C4—41no—
496C22×C4—41no—
497C22×C4—81no—
498C22×C4C2×D4×C10711yescharacteristic
499Q16—44no—
500SD16—44no—
501M16—22no—
502M16—41no—
503C2×C8—41no—
504C4⋊C4—81no—
505C22⋊C4—21no—
506C22⋊C4—22no—
507C22⋊C4—21no—
508C22⋊C4—44no—
509C22⋊C4—41no—
510C22⋊C4—42no—
511C22⋊C4—41no—
512C22⋊C4—41no—
513C22⋊C4—41no—
514C22⋊C4—41no—
515C22⋊C4—41no—
516C22⋊C4—41no—
517C22⋊C4—41no—
518C22⋊C4—41no—
519C22⋊C4—41no—
520C22⋊C4—81no—
521C22⋊C4—81no—
522C22⋊C4—81no—
523C22⋊C4—81no—
524C22⋊C4—82no—
525C42—21no—
526C42—21no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—21no—
531C23—22no—
532C23—22no—
533C23—22no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23—42no—
539C23—41no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23—42no—
545C23—42no—
546C23—41no—
547C23—41no—
548C23—41no—
549C23—41no—
550C23—81no—
551C23—82no—
552C23—81no—
553C23—82no—
554C23—84no—
555C23—81no—
556C23—81no—
557C23—81no—
558C23—81no—
559C23—81no—
560C23#120×C10711yescharacteristic
561C23#120×C10711yescharacteristic
562C23#121×C10711yescharacteristic
563C23#121×C10711yescharacteristic
564C23#127×C10711yescharacteristic
565Q8—22no—
566Q8—41no—
567Q8—42no—
568Q8—81no—
569D4—22no—
570D4—41no—
571D4—41no—
572D4—41no—
573D4—41no—
574D4—41no—
575D4—42no—
576D4—42no—
577D4—42no—
578D4—42no—
579D4—42no—
580D4—42no—
581D4—82no—
582D4—82no—
583D4—88no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—42no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—81no—
611C2×C4—81no—
612C2×C4—81no—
613C2×C4—81no—
614C2×C4—81no—
615C2×C4—81no—
616C2×C4#120×C10711yescharacteristic
617C2×C4#121×C10711yescharacteristic
618C8—42no—
619C22—21no—
620C22—21no—
621C22—21no—
622C22—21no—
623C22—21no—
624C22—21no—
625C22—41no—
626C22—41no—
627C22—41no—
628C22—41no—
629C22—42no—
630C22—42no—
631C22—41no—
632C22—41no—
633C22—41no—
634C22—41no—
635C22—44no—
636C22—42no—
637C22—42no—
638C22—81no—
639C22—81no—
640C22—81no—
641C22—81no—
642C22—82no—
643C22—82no—
644C22—82no—
645C22—82no—
646C22—82no—
647C22—81no—
648C22—81no—
649C22—81no—
650C22—81no—
651C22—161no—
652C22—161no—
653C22#392×C10711yescharacteristic
654C22B(4,𝔽2)×C10711yescharacteristic
655C22B(4,𝔽2)×C10711yescharacteristic
656C4—21no—
657C4—21no—
658C4—42no—
659C4—81no—
660C4—81no—
661C4—81no—
662C4—82no—
663C4—81no—
664C4—81no—
665C4—161no—
666C2—21no—
667C2—21no—
668C2—41no—
669C2—41no—
670C2—41no—
671C2—41no—
672C2—82no—
673C2—82no—
674C2C107×#202111yescharacteristic
675C2C107×#219711yescharacteristic
676C2C107×#219911yescharacteristic
subgroupG/H
center, socleC22×C107B(4,𝔽2)
commutator, Frattini, cosocleC22×D4C23×C107
prime radicalC2×B(4,𝔽2)×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
2. C2×#455×C107C2semidirect
3. C107×#2684C2semidirect
4. C2×#354×C107C2semidirect
5. C2×AΣL(1,𝔽16)×C107C2semidirect
6. C107×#1790C2semidirect
7. C107×#1390C2semidirect
41. C22×Q8×C107D4semidirect
48. C23×C4×C107D4semidirect
78. C2×C22⋊C4×C107D4semidirect
79. C2×C42×C107D4semidirect
82. #100×C107D4semidirect
327. #13348C107direct (N × H)
340. C2×#455C2×C107semidirect
341. #2684C2×C107semidirect
342. C2×#354C2×C107semidirect
343. C2×AΣL(1,𝔽16)C2×C107semidirect
344. #1790C2×C107semidirect
345. #1390C2×C107semidirect
346. C107#13348direct (N × H)
380. C22×Q8D4×C107semidirect
387. C23×C4D4×C107semidirect
417. C2×C22⋊C4D4×C107semidirect
418. C2×C42D4×C107semidirect
421. #100D4×C107semidirect

Maximal subgroups

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

subgroupindex
C2×B(4,𝔽2)×C1072
C2×#455×C1072
C107×#26842
C2×#354×C1072
C2×AΣL(1,𝔽16)×C1072
C107×#17902
C107×#13902
#13348107