Finite Group Databasebeta

← C109×#13348 #42131342

Subgroups — C109×#13348 #42131342

Order 27904.

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)×C109C211yesmaximal, characteristic
2C2×#455×C109C211yesmaximal, characteristic, complemented
3C109×#2684C211yesmaximal, characteristic, complemented
4C2×#354×C109C211yesmaximal, characteristic, complemented
5C2×AΣL(1,𝔽16)×C109C211yesmaximal, characteristic, complemented
6C109×#1790C211yesmaximal, characteristic, complemented
7C109×#1390C211yesmaximal, characteristic, complemented
8C2×H(5,𝔽2)×C109C2211yescharacteristic
9C2×#137×C109—21no—
10C2×#124×C109C2211yescharacteristic
11C2×#120×C109—21no—
12C2×#120×C109C2211yescharacteristic
13B(4,𝔽2)×C109—22no—
14B(4,𝔽2)×C109—22no—
15#455×C109—22no—
16#455×C109—22no—
17C2×#104×C109—21no—
18C2×#100×C109C2211yescharacteristic
19C2×#99×C109—21no—
20C2×#99×C109—21no—
21C2×#99×C109C2211yescharacteristic
22C2×#99×C109C2211yescharacteristic
23#397×C109—21no—
24#394×C109—21no—
25#393×C109—21no—
26#388×C109—21no—
27#386×C109—21no—
28#386×C109—21no—
29#386×C109C2211yescharacteristic
30#379×C109—21no—
31#354×C109—22no—
32AΣL(1,𝔽16)×C109—22no—
33C25×C109D411yescharacteristic
34H(5,𝔽2)×C109—21no—
35H(5,𝔽2)×C109—21no—
36H(5,𝔽2)×C109—21no—
37H(5,𝔽2)×C109—42no—
38H(5,𝔽2)×C109D412yes—
39C2×#41×C109—21no—
40C2×#41×C109—41no—
41C22×Q8×C109D411yescharacteristic, complemented
42C22×D4×C109—21no—
43C22×D4×C109—21no—
44C22×D4×C109—21no—
45C22×D4×C109—41no—
46C22×D4×C109—41no—
47C22×D4×C109C2311yescharacteristic
48C23×C4×C109D411yescharacteristic, complemented
49#137×C109—24no—
50#137×C109—42no—
51C2×Q16×C109—41no—
52C2×SD16×C109—41no—
53C2×M16×C109—21no—
54#124×C109—21no—
55#124×C109—21no—
56#124×C109—21no—
57#124×C109—21no—
58#120×C109—24no—
59#120×C109—21no—
60#120×C109—21no—
61#120×C109—21no—
62#120×C109—21no—
63#120×C109—42no—
64C2×C4⋊C4×C109—21no—
65C2×C22⋊C4×C109—21no—
66C2×C22⋊C4×C109—21no—
67C2×C22⋊C4×C109—21no—
68C2×C22⋊C4×C109—21no—
69C2×C22⋊C4×C109—21no—
70C2×C22⋊C4×C109—21no—
71C2×C22⋊C4×C109—21no—
72C2×C22⋊C4×C109—21no—
73C2×C22⋊C4×C109—21no—
74C2×C22⋊C4×C109—21no—
75C2×C22⋊C4×C109—41no—
76C2×C22⋊C4×C109—41no—
77C2×C22⋊C4×C109D411yescharacteristic
78C2×C22⋊C4×C109D411yescharacteristic, complemented
79C2×C42×C109D411yescharacteristic, complemented
80#104×C109—24no—
81#100×C109—21no—
82#100×C109D412yescomplemented
83#99×C109—24no—
84#99×C109—21no—
85#99×C109—21no—
86#99×C109—21no—
87#99×C109—42no—
88#99×C109D412yes—
89#95×C109—21no—
90#95×C109—21no—
91#95×C109—21no—
92#95×C109—21no—
93#95×C109—41no—
94C24×C109—21no—
95C24×C109—21no—
96C24×C109—21no—
97C24×C109—21no—
98C24×C109—21no—
99C24×C109—41no—
100C24×C109—41no—
101C24×C109—42no—
102C24×C109—81no—
103C24×C109C2×D411yescharacteristic
104C24×C109C2×D411yescharacteristic
105#41×C109—22no—
106#41×C109—22no—
107#41×C109—42no—
108#41×C109—41no—
109#41×C109—42no—
110#41×C109—42no—
111#41×C109—42no—
112#41×C109—41no—
113C2×Q8×C109—21no—
114C2×Q8×C109—21no—
115C2×Q8×C109—21no—
116C2×Q8×C109—41no—
117C2×Q8×C109—41no—
118C2×D4×C109—21no—
119C2×D4×C109—22no—
120C2×D4×C109—22no—
121C2×D4×C109—22no—
122C2×D4×C109—22no—
123C2×D4×C109—41no—
124C2×D4×C109—41no—
125C2×D4×C109—42no—
126C2×D4×C109—42no—
127C2×D4×C109—41no—
128C2×D4×C109—41no—
129C2×D4×C109—41no—
130C2×D4×C109—41no—
131C2×D4×C109—42no—
132C2×D4×C109—42no—
133C2×D4×C109—41no—
134C2×D4×C109—41no—
135C2×D4×C109—41no—
136C2×D4×C109—44no—
137C2×D4×C109—42no—
138C2×D4×C109—41no—
139C2×D4×C109—41no—
140C2×D4×C109—81no—
141C2×D4×C109—82no—
142C2×D4×C109—82no—
143C2×D4×C109C2×D411yescharacteristic
144C2×D4×C109C2×D411yescharacteristic
145C2×D4×C109C2×D411yescharacteristic
146C2×D4×C109#4111yescharacteristic
147C22×C4×C109—21no—
148C22×C4×C109—21no—
149C22×C4×C109—21no—
150C22×C4×C109—21no—
151C22×C4×C109—21no—
152C22×C4×C109—21no—
153C22×C4×C109—21no—
154C22×C4×C109—41no—
155C22×C4×C109—41no—
156C22×C4×C109—41no—
157C22×C4×C109—41no—
158C22×C4×C109—81no—
159C22×C4×C109C2×D411yescharacteristic
160Q16×C109—44no—
161SD16×C109—44no—
162M16×C109—22no—
163M16×C109—41no—
164C2×C8×C109—41no—
165C4⋊C4×C109—81no—
166C22⋊C4×C109—21no—
167C22⋊C4×C109—22no—
168C22⋊C4×C109—21no—
169C22⋊C4×C109—44no—
170C22⋊C4×C109—42no—
171C22⋊C4×C109—41no—
172C22⋊C4×C109—41no—
173C22⋊C4×C109—41no—
174C22⋊C4×C109—41no—
175C22⋊C4×C109—41no—
176C22⋊C4×C109—41no—
177C22⋊C4×C109—41no—
178C22⋊C4×C109—41no—
179C22⋊C4×C109—41no—
180C22⋊C4×C109—41no—
181C22⋊C4×C109—81no—
182C22⋊C4×C109—81no—
183C22⋊C4×C109—81no—
184C22⋊C4×C109—81no—
185C22⋊C4×C109—82no—
186C42×C109—21no—
187C42×C109—21no—
188C23×C109—21no—
189C23×C109—21no—
190C23×C109—21no—
191C23×C109—21no—
192C23×C109—22no—
193C23×C109—22no—
194C23×C109—22no—
195C23×C109—41no—
196C23×C109—41no—
197C23×C109—41no—
198C23×C109—41no—
199C23×C109—42no—
200C23×C109—41no—
201C23×C109—41no—
202C23×C109—41no—
203C23×C109—41no—
204C23×C109—41no—
205C23×C109—42no—
206C23×C109—42no—
207C23×C109—41no—
208C23×C109—41no—
209C23×C109—41no—
210C23×C109—41no—
211C23×C109—81no—
212C23×C109—82no—
213C23×C109—81no—
214C23×C109—82no—
215C23×C109—84no—
216C23×C109—81no—
217C23×C109—81no—
218C23×C109—81no—
219C23×C109—81no—
220C23×C109—81no—
221C23×C109#12011yescharacteristic
222C23×C109#12011yescharacteristic
223C23×C109#12111yescharacteristic
224C23×C109#12111yescharacteristic
225C23×C109#12711yescharacteristic
226Q8×C109—22no—
227Q8×C109—41no—
228Q8×C109—42no—
229Q8×C109—81no—
230D4×C109—22no—
231D4×C109—41no—
232D4×C109—41no—
233D4×C109—41no—
234D4×C109—41no—
235D4×C109—41no—
236D4×C109—42no—
237D4×C109—42no—
238D4×C109—42no—
239D4×C109—42no—
240D4×C109—42no—
241D4×C109—42no—
242D4×C109—82no—
243D4×C109—82no—
244D4×C109—88no—
245C2×C4×C109—21no—
246C2×C4×C109—21no—
247C2×C4×C109—22no—
248C2×C4×C109—22no—
249C2×C4×C109—42no—
250C2×C4×C109—42no—
251C2×C4×C109—41no—
252C2×C4×C109—41no—
253C2×C4×C109—41no—
254C2×C4×C109—42no—
255C2×C4×C109—41no—
256C2×C4×C109—41no—
257C2×C4×C109—41no—
258C2×C4×C109—41no—
259C2×C4×C109—41no—
260C2×C4×C109—41no—
261C2×C4×C109—41no—
262C2×C4×C109—41no—
263C2×C4×C109—42no—
264C2×C4×C109—41no—
265C2×C4×C109—41no—
266C2×C4×C109—41no—
267C2×C4×C109—41no—
268C2×C4×C109—41no—
269C2×C4×C109—41no—
270C2×C4×C109—41no—
271C2×C4×C109—81no—
272C2×C4×C109—81no—
273C2×C4×C109—81no—
274C2×C4×C109—81no—
275C2×C4×C109—81no—
276C2×C4×C109—81no—
277C2×C4×C109#12011yescharacteristic
278C2×C4×C109#12111yescharacteristic
279C8×C109—42no—
280C22×C109—21no—
281C22×C109—21no—
282C22×C109—21no—
283C22×C109—21no—
284C22×C109—21no—
285C22×C109—21no—
286C22×C109—41no—
287C22×C109—41no—
288C22×C109—41no—
289C22×C109—41no—
290C22×C109—42no—
291C22×C109—42no—
292C22×C109—41no—
293C22×C109—41no—
294C22×C109—41no—
295C22×C109—41no—
296C22×C109—44no—
297C22×C109—42no—
298C22×C109—42no—
299C22×C109—81no—
300C22×C109—81no—
301C22×C109—81no—
302C22×C109—81no—
303C22×C109—82no—
304C22×C109—82no—
305C22×C109—82no—
306C22×C109—82no—
307C22×C109—82no—
308C22×C109—81no—
309C22×C109—81no—
310C22×C109—81no—
311C22×C109—81no—
312C22×C109—161no—
313C22×C109—161no—
314C22×C109#39211yescharacteristic
315C22×C109B(4,𝔽2)11yescharacteristic
316C22×C109B(4,𝔽2)11yescharacteristic
317C4×C109—21no—
318C4×C109—21no—
319C4×C109—42no—
320C4×C109—81no—
321C4×C109—81no—
322C4×C109—81no—
323C4×C109—82no—
324C4×C109—81no—
325C4×C109—81no—
326C4×C109—161no—
327#13348C10911yesmaximal, characteristic, direct product factor, Sylow
328C2×C109—21no—
329C2×C109—21no—
330C2×C109—41no—
331C2×C109—41no—
332C2×C109—41no—
333C2×C109—41no—
334C2×C109—82no—
335C2×C109—82no—
336C2×C109#202111yescharacteristic
337C2×C109#219711yescharacteristic
338C2×C109#219911yescharacteristic
339C2×B(4,𝔽2)C2×C10911yescharacteristic
340C2×#455C2×C10911yescharacteristic, complemented
341#2684C2×C10911yescharacteristic, complemented
342C2×#354C2×C10911yescharacteristic, complemented
343C2×AΣL(1,𝔽16)C2×C10911yescharacteristic, complemented
344#1790C2×C10911yescharacteristic, complemented
345#1390C2×C10911yescharacteristic, complemented
346C109#1334811yescharacteristic, direct product factor, Sylow
347C2×H(5,𝔽2)C22×C10911yescharacteristic
348C2×#137—21no—
349C2×#124C22×C10911yescharacteristic
350C2×#120—21no—
351C2×#120C22×C10911yescharacteristic
352B(4,𝔽2)—22no—
353B(4,𝔽2)—22no—
354#455—22no—
355#455—22no—
356C2×#104—21no—
357C2×#100C22×C10911yescharacteristic
358C2×#99—21no—
359C2×#99—21no—
360C2×#99C22×C10911yescharacteristic
361C2×#99C22×C10911yescharacteristic
362#397—21no—
363#394—21no—
364#393—21no—
365#388—21no—
366#386—21no—
367#386—21no—
368#386C22×C10911yescharacteristic
369#379—21no—
370#354—22no—
371AΣL(1,𝔽16)—22no—
372C25D4×C10911yescharacteristic
373H(5,𝔽2)—21no—
374H(5,𝔽2)—21no—
375H(5,𝔽2)—21no—
376H(5,𝔽2)—42no—
377H(5,𝔽2)D4×C10912yes—
378C2×#41—21no—
379C2×#41—41no—
380C22×Q8D4×C10911yescharacteristic, complemented
381C22×D4—21no—
382C22×D4—21no—
383C22×D4—21no—
384C22×D4—41no—
385C22×D4—41no—
386C22×D4C23×C10911yescharacteristic
387C23×C4D4×C10911yescharacteristic, 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×C10911yescharacteristic
417C2×C22⋊C4D4×C10911yescharacteristic, complemented
418C2×C42D4×C10911yescharacteristic, complemented
419#104—24no—
420#100—21no—
421#100D4×C10912yescomplemented
422#99—24no—
423#99—21no—
424#99—21no—
425#99—21no—
426#99—42no—
427#99D4×C10912yes—
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×C10911yescharacteristic
443C24C2×D4×C10911yescharacteristic
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×C10911yescharacteristic
483C2×D4C2×D4×C10911yescharacteristic
484C2×D4C2×D4×C10911yescharacteristic
485C2×D4#41×C10911yescharacteristic
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×C10911yescharacteristic
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—42no—
510C22⋊C4—41no—
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×C10911yescharacteristic
561C23#120×C10911yescharacteristic
562C23#121×C10911yescharacteristic
563C23#121×C10911yescharacteristic
564C23#127×C10911yescharacteristic
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×C10911yescharacteristic
617C2×C4#121×C10911yescharacteristic
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×C10911yescharacteristic
654C22B(4,𝔽2)×C10911yescharacteristic
655C22B(4,𝔽2)×C10911yescharacteristic
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—
674C2C109×#202111yescharacteristic
675C2C109×#219711yescharacteristic
676C2C109×#219911yescharacteristic
subgroupG/H
center, socleC22×C109B(4,𝔽2)
commutator, Frattini, cosocleC22×D4C23×C109
prime radicalC2×B(4,𝔽2)×C109C2
LCSG ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C109 ◁ C23×C109 ◁ C22×D4×C109 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×#455×C109C2semidirect
3. C109×#2684C2semidirect
4. C2×#354×C109C2semidirect
5. C2×AΣL(1,𝔽16)×C109C2semidirect
6. C109×#1790C2semidirect
7. C109×#1390C2semidirect
41. C22×Q8×C109D4semidirect
48. C23×C4×C109D4semidirect
78. C2×C22⋊C4×C109D4semidirect
79. C2×C42×C109D4semidirect
82. #100×C109D4semidirect
327. #13348C109direct (N × H)
340. C2×#455C2×C109semidirect
341. #2684C2×C109semidirect
342. C2×#354C2×C109semidirect
343. C2×AΣL(1,𝔽16)C2×C109semidirect
344. #1790C2×C109semidirect
345. #1390C2×C109semidirect
346. C109#13348direct (N × H)
380. C22×Q8D4×C109semidirect
387. C23×C4D4×C109semidirect
417. C2×C22⋊C4D4×C109semidirect
418. C2×C42D4×C109semidirect
421. #100D4×C109semidirect

Maximal subgroups

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

subgroupindex
C2×B(4,𝔽2)×C1092
C2×#455×C1092
C109×#26842
C2×#354×C1092
C2×AΣL(1,𝔽16)×C1092
C109×#17902
C109×#13902
#13348109