Finite Group Databasebeta

← #13362391

Subgroups — #13362391

Order 1792.

1174 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C7×SD16C211yesmaximal, characteristic, complemented
2C2×#89948C211yesmaximal, characteristic
3#11739900C211yesmaximal, characteristic, complemented
4C2×#89798C211yesmaximal, characteristic, complemented
5#11739774C211yesmaximal, characteristic, complemented
6C2×#89786C211yesmaximal, characteristic, complemented
7C2×C4×D56C211yesmaximal, characteristic, complemented
8#11729804C211yesmaximal, characteristic, complemented
9#11728330C211yesmaximal, characteristic, complemented
10#11728310C211yesmaximal, characteristic, complemented
11#11728011C211yesmaximal, characteristic, complemented
12#11727980C211yesmaximal, characteristic, complemented
13#11727944C211yesmaximal, characteristic, complemented
14#11727911C211yesmaximal, characteristic, complemented
15#11727503C211yesmaximal, characteristic, complemented
16C22×C7×SD16C2211yescharacteristic
17C2×C4×C7×Q8C2211yescharacteristic
18C2×C4×C7×D4C2211yescharacteristic
19C2×#6510C2211yescharacteristic
20C22×#1143C2211yescharacteristic
21C2×#6503C2211yescharacteristic
22C22×#1141C2211yescharacteristic
23C22×D56C2211yescharacteristic
24C2×C4×D28C2211yescharacteristic
25C2×C4×D28C2211yescharacteristic
26C4×C7×SD16—24no—
27C2×C7×#106C2211yescharacteristic
28C2×C7×#103C2211yescharacteristic
29C2×C7×#102C2211yescharacteristic
30C2×C4×C7×C8C2211yescharacteristic
31#90002C2211yescharacteristic
32C2×#6409C2211yescharacteristic
33#89993C2211yescharacteristic
34C2×#6406C2211yescharacteristic
35#89985—22no—
36#89948C418yescomplemented
37C2×#6395C2211yescharacteristic
38C2×#6395C2211yescharacteristic
39C2×#6392C2211yescharacteristic
40C2×#6389C2211yescharacteristic
41C2×#6389C2211yescharacteristic
42C2×C8×Dic7C2211yescharacteristic
43C2×C8×Dic7C2211yescharacteristic
44#89798—24no—
45#89786—24no—
46#89761—22no—
47#89760C2211yescharacteristic
48#89760C2211yescharacteristic
49#89749C2211yescharacteristic
50#89748C2211yescharacteristic
51C2×#6383C2211yescharacteristic
52C2×#6383C2211yescharacteristic
53C2×#6382C2211yescharacteristic
54C2×#6381C2211yescharacteristic
55#89728—22no—
56#89712—22no—
57#89704C2211yescharacteristic
58C2×#6377C2211yescharacteristic
59C4×D56—24no—
60#21704—71nomaximal, Sylow
61C22×C7×Q8C2311yescharacteristic
62C22×C7×D4C2311yescharacteristic
63C23×C4×C7—21no—
64C22×C22⋊D7—22no—
65C22×D28C2311yescharacteristic
66C22×D28C2311yescharacteristic
67C22×C4×D7—21no—
68C22×C4×D7—21no—
69C2×C7×SD16—22no—
70C2×C7×SD16—22no—
71C2×C7×SD16C2×C414yescomplemented
72C22×C7×C8C2311yescharacteristic
73C22×C7×C8C2311yescharacteristic
74C4×C7×Q8—22no—
75C4×C7×Q8—22no—
76C4×C7×D4—22no—
77C4×C7×D4—22no—
78C2×C7×C4⋊C4—21no—
79C2×C7×C4⋊C4C2311yescharacteristic
80C2×C7×C4⋊C4C2311yescharacteristic
81C2×C7×C22⋊C4—21no—
82C2×C42×C7—21no—
83C2×C42×C7C2311yescharacteristic
84C2×C22⋊Dic7—22no—
85#6510—22no—
86#6510C2×C414yescomplemented
87C2×#1143—22no—
88C2×#1143—22no—
89C2×#1143C2×C414yescomplemented
90#6503—22no—
91#6503C2×C414yescomplemented
92C2×#1141—22no—
93C2×#1141—22no—
94C2×#1141C2×C414yescomplemented
95C2×(C2×D7)⋊C4—22no—
96C2×(C2×D7)⋊C4—21no—
97C2×(C2×D7)⋊C4—21no—
98C2×(C2×D7)⋊C4—22no—
99C2×(C2×D7)⋊C4—22no—
100C2×C4⋊Dic7C2311yescharacteristic
101C2×C4⋊Dic7C2311yescharacteristic
102C2×Dic7⋊C4—22no—
103C2×C4×Dic7C2311yescharacteristic
104C2×C4×Dic7C2311yescharacteristic
105C22×C7⋊C8C2311yescharacteristic
106C22×C7⋊C8C2311yescharacteristic
107C2×D56—22no—
108C2×D56—22no—
109C2×D56C2×C414yescomplemented
110C4×D28—22no—
111C4×D28—22no—
112C4×D28—22no—
113C4×D28—22no—
114C7×#106—22no—
115C7×#103—22no—
116C7×#102—22no—
117C4×C7×C8D414yescomplemented
118#6409—22no—
119#6406—22no—
120C7⋊#95—22no—
121C7⋊#95—22no—
122C7⋊#95D412yes—
123C7⋊#95D412yes—
124#6395—22no—
125#6395—22no—
126#6392—22no—
127#6389—22no—
128#6389C2×C414yescomplemented
129C8×Dic7—22no—
130C8×Dic7—22no—
131#6383—22no—
132#6383—22no—
133#6382—22no—
134#6381—22no—
135#6377D414yescomplemented
136C2×#496—71no—
137C2×#442—71no—
138C2×#440—71no—
139C2×C4×SD16D711yescharacteristic, complemented
140C2×C4×D8—71no—
141#2328—71no—
142#2326—71no—
143#1973—71no—
144#1972—71no—
145#1970—71no—
146#1968—71no—
147#1958—71no—
148#1957—71no—
149#1847—71no—
150#1726—71no—
151C24×C7—21no—
152C23×D7—21no—
153C23×D7—21no—
154C2×C7×Q8—21no—
155C2×C7×Q8—21no—
156C2×C7×Q8—21no—
157C2×C7×Q8—21no—
158C2×C7×Q8—21no—
159C2×C7×Q8C22×C412yes—
160C2×C7×D4—21no—
161C2×C7×D4—21no—
162C2×C7×D4—21no—
163C2×C7×D4—21no—
164C2×C7×D4—21no—
165C2×C7×D4C22×C412yes—
166C22×C4×C7—21no—
167C22×C4×C7—21no—
168C22×C4×C7—21no—
169C22×C4×C7—42no—
170C22×C4×C7—82no—
171C22×C4×C7C2×D411yescharacteristic
172C22×C4×C7C2×D411yescharacteristic
173C22×C4×C7C2411yescharacteristic
174C2×C22⋊D7—28no—
175C2×C22⋊D7—82no—
176C2×C22⋊D7—82no—
177C22×Dic7—21no—
178C22×Dic7—21no—
179C22×Dic7C2×D412yes—
180C22×Dic7C2×D412yes—
181C2×D28—21no—
182C2×D28—21no—
183C2×D28—21no—
184C2×D28—21no—
185C2×D28—21no—
186C2×D28—21no—
187C2×D28—21no—
188C2×D28—21no—
189C2×D28—21no—
190C2×D28—21no—
191C2×D28C22×C412yes—
192C2×D28C22×C412yes—
193C2×C4×D7—42no—
194C2×C4×D7—42no—
195C2×C4×D7—82no—
196C2×C4×D7—82no—
197C7×SD16—28no—
198C2×C7×C8—21no—
199C2×C7×C8—21no—
200C2×C7×C8C22×C412yes—
201C2×C7×C8C2×D412yes—
202C2×C7×C8C2×D412yes—
203C2×C7×C8#4112yes—
204C7×C4⋊C4—21no—
205C7×C4⋊C4—21no—
206C7×C4⋊C4—21no—
207C7×C4⋊C4—21no—
208C7×C4⋊C4—42no—
209C7×C22⋊C4—42no—
210C42×C7—42no—
211C42×C7C2×D412yes—
212C42×C7C2×D412yes—
213C22⋊Dic7—82no—
214#1143—28no—
215#1141—28no—
216(C2×D7)⋊C4—28no—
217(C2×D7)⋊C4—42no—
218(C2×D7)⋊C4—42no—
219(C2×D7)⋊C4—82no—
220(C2×D7)⋊C4—82no—
221C4⋊Dic7—21no—
222C4⋊Dic7—21no—
223C4⋊Dic7—21no—
224C4⋊Dic7—21no—
225Dic7⋊C4—82no—
226C4×Dic7—21no—
227C4×Dic7—21no—
228C4×Dic7C22×C412yes—
229C4×Dic7#4112yes—
230C2×C7⋊C8—21no—
231C2×C7⋊C8—21no—
232C2×C7⋊C8C22×C412yes—
233C2×C7⋊C8C2×D412yes—
234C2×C7⋊C8C2×D412yes—
235C2×C7⋊C8#4112yes—
236D56—28no—
237C22×SD16—71no—
238C22×SD16C2×D711yescharacteristic
239C22×D8—71no—
240C22×D8—71no—
241C2×#127—71no—
242C2×#124—71no—
243C2×C4×Q8C2×D711yescharacteristic
244C2×C4×D4—71no—
245C2×C4×D4—71no—
246C2×C4×D4C2×D711yescharacteristic
247#496—78no—
248#442—144no—
249#440—144no—
250C4×SD16—24no—
251C4×D8—144no—
252C2×#107—71no—
253C2×#107—71no—
254C2×#106—71no—
255C2×#106C2×D711yescharacteristic
256C2×#103—71no—
257C2×#103C2×D711yescharacteristic
258C2×#102—71no—
259C2×#102—71no—
260C2×#102—71no—
261C2×#102—71no—
262C2×#102—71no—
263C2×#102C2×D711yescharacteristic
264C2×#97—71no—
265C2×#97—71no—
266C2×#97—71no—
267C2×C4×C8—71no—
268C2×C4×C8—71no—
269C2×C4×C8C2×D711yescharacteristic
270#391—71no—
271#390—71no—
272#390—71no—
273#390—71no—
274#388—142no—
275#386—142no—
276#386—142no—
277#385—142no—
278#384—71no—
279#382—71no—
280#376—71no—
281C23×C7—22no—
282C23×C7—41no—
283C23×C7—41no—
284C23×C7—42no—
285C23×C7—42no—
286C23×C7C22×D411yescharacteristic
287C22×D7—22no—
288C22×D7—22no—
289C22×D7—41no—
290C22×D7—41no—
291C22×D7—41no—
292C22×D7—41no—
293C22×D7—42no—
294C22×D7—42no—
295C22×D7—42no—
296C22×D7—42no—
297C7×Q8—22no—
298C7×Q8—22no—
299C7×Q8—41no—
300C7×Q8—41no—
301C7×D4—22no—
302C7×D4—22no—
303C7×D4—41no—
304C7×D4—41no—
305C2×C4×C7—22no—
306C2×C4×C7—22no—
307C2×C4×C7—22no—
308C2×C4×C7—41no—
309C2×C4×C7—41no—
310C2×C4×C7—41no—
311C2×C4×C7—41no—
312C2×C4×C7—42no—
313C2×C4×C7—42no—
314C2×C4×C7—84no—
315C2×C4×C7#12712yes—
316C2×C4×C7#12712yes—
317C2×C4×C7C23×C411yescharacteristic
318C2×C4×C7C22×D411yescharacteristic
319C2×C4×C7C22×D411yescharacteristic
320C2×C4×C7C2×#4111yescharacteristic
321C2×C4×C7C2×#4111yescharacteristic
322C2×C4×C7C2×#4111yescharacteristic
323C22⋊D7—88no—
324C2×Dic7—22no—
325C2×Dic7—22no—
326C2×Dic7—22no—
327C2×Dic7—22no—
328C2×Dic7—22no—
329C2×Dic7—41no—
330C2×Dic7—41no—
331C2×Dic7—42no—
332C2×Dic7—42no—
333C2×Dic7C4×D414yes—
334D28—22no—
335D28—22no—
336D28—22no—
337D28—22no—
338D28—41no—
339D28—41no—
340D28—41no—
341D28—41no—
342C4×D7—84no—
343C4×D7—84no—
344C7×C8—22no—
345C7×C8C4×D414yes—
346C7⋊C8—22no—
347C7⋊C8C4×D414yes—
348C22×Q8C22×D711yescharacteristic
349C22×D4—71no—
350C22×D4—71no—
351C22×D4—142no—
352C22×D4C22×D711yescharacteristic
353C23×C4—21no—
354C23×C4—141no—
355C23×C4—141no—
356C2×SD16—22no—
357C2×SD16—22no—
358C2×SD16—74no—
359C2×SD16—142no—
360C2×SD16—142no—
361C2×SD16C4×D714yescomplemented
362C2×D8—74no—
363C2×D8—74no—
364C2×D8—142no—
365C2×D8—142no—
366C2×D8—142no—
367C2×D8—142no—
368C22×C8—71no—
369C22×C8—71no—
370C22×C8C22×D711yescharacteristic
371C22×C8C22×D711yescharacteristic
372#127—74no—
373#127—142no—
374#124—74no—
375#124—142no—
376C4×Q8—22no—
377C4×Q8—22no—
378C4×D4—22no—
379C4×D4—22no—
380C4×D4—142no—
381C4×D4—142no—
382C4×D4—142no—
383C4×D4—142no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—71no—
386C2×C4⋊C4—71no—
387C2×C4⋊C4—142no—
388C2×C4⋊C4C22×D711yescharacteristic
389C2×C4⋊C4C22×D711yescharacteristic
390C2×C22⋊C4—21no—
391C2×C22⋊C4—142no—
392C2×C22⋊C4—141no—
393C2×C22⋊C4—141no—
394C2×C22⋊C4—142no—
395C2×C22⋊C4—142no—
396C2×C22⋊C4—142no—
397C2×C42—21no—
398C2×C42—71no—
399C2×C42—71no—
400C2×C42C22×D711yescharacteristic
401#107—142no—
402#107—142no—
403#106—22no—
404#106—142no—
405#103—22no—
406#103—142no—
407#102—22no—
408#102—142no—
409#102—142no—
410#102—142no—
411#102—142no—
412#102—142no—
413#97—74no—
414#97—74no—
415#97—142no—
416C4×C8—142no—
417C4×C8—142no—
418C4×C8C22⋊D714yescomplemented
419#95—72no—
420#95—72no—
421#95—142no—
422#95—142no—
423C22×C7—42no—
424C22×C7—44no—
425C22×C7—81no—
426C22×C7—81no—
427C22×C7—82no—
428C22×C7C2×C4×D411yescharacteristic
429C22×C7C2×#12711yescharacteristic
430C22×C7#53211yescharacteristic
431C22×C7#57612yes—
432C22×C7#57712yes—
433C2×D7—42no—
434C2×D7—42no—
435C2×D7—44no—
436C2×D7—44no—
437C2×D7—81no—
438C2×D7—81no—
439C2×D7—81no—
440C2×D7—81no—
441C2×D7—82no—
442C2×D7—82no—
443C4×C7—22no—
444C4×C7—22no—
445C4×C7—42no—
446C4×C7—81no—
447C4×C7—81no—
448C4×C7C2×C4×D411yescharacteristic
449C4×C7C2×C4×D411yescharacteristic
450C4×C7#53211yescharacteristic
451C4×C7#53211yescharacteristic
452Dic7—24no—
453Dic7—42no—
454Dic7—81no—
455Dic7—81no—
456C24—21no—
457C24—141no—
458C24—141no—
459C2×Q8—21no—
460C2×Q8—21no—
461C2×Q8—21no—
462C2×Q8—21no—
463C2×Q8—21no—
464C2×Q8C2×C4×D712yes—
465C2×D4—21no—
466C2×D4—21no—
467C2×D4—21no—
468C2×D4—21no—
469C2×D4—21no—
470C2×D4—72no—
471C2×D4—72no—
472C2×D4—148no—
473C2×D4—141no—
474C2×D4—141no—
475C2×D4—141no—
476C2×D4—141no—
477C2×D4—141no—
478C2×D4—141no—
479C2×D4—141no—
480C2×D4—141no—
481C2×D4—141no—
482C2×D4—141no—
483C2×D4—562no—
484C2×D4—562no—
485C2×D4C2×C4×D712yes—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—42no—
490C22×C4—72no—
491C22×C4—72no—
492C22×C4—82no—
493C22×C4—141no—
494C22×C4—141no—
495C22×C4—282no—
496C22×C4—282no—
497C22×C4—562no—
498C22×C4—562no—
499C22×C4D4×D711yescharacteristic
500C22×C4D4×D711yescharacteristic
501C22×C4C23×D711yescharacteristic
502SD16—28no—
503SD16—148no—
504D8—148no—
505D8—148no—
506C2×C8—21no—
507C2×C8—21no—
508C2×C8—72no—
509C2×C8—72no—
510C2×C8—72no—
511C2×C8—72no—
512C2×C8—141no—
513C2×C8—141no—
514C2×C8C2×C4×D712yes—
515C2×C8#115712yes—
516C2×C8C2×C22⋊D712yes—
517C2×C8C2×C22⋊D712yes—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—42no—
523C4⋊C4—141no—
524C4⋊C4—141no—
525C4⋊C4—141no—
526C4⋊C4—141no—
527C4⋊C4—562no—
528C22⋊C4—42no—
529C22⋊C4—148no—
530C22⋊C4—282no—
531C22⋊C4—282no—
532C22⋊C4—562no—
533C22⋊C4—562no—
534C22⋊C4—562no—
535C42—42no—
536C42—72no—
537C42—72no—
538C42—141no—
539C42—141no—
540C42D4×D712yes—
541C42C2×C22⋊D712yes—
542C2×C7—82no—
543C2×C7C4×#12711yescharacteristic
544C2×C7#294612yes—
545C2×C7#315411yescharacteristic
546C2×C7#315511yescharacteristic
547C2×C7#317912yes—
548D7—82no—
549D7—82no—
550C23—22no—
551C23—42no—
552C23—42no—
553C23—41no—
554C23—41no—
555C23—142no—
556C23—142no—
557C23—282no—
558C23—282no—
559C23—282no—
560C23—282no—
561C23—281no—
562C23—281no—
563C23—281no—
564C23—281no—
565C23C2×D4×D711yescharacteristic
566Q8—22no—
567Q8—22no—
568Q8—41no—
569Q8—41no—
570D4—22no—
571D4—22no—
572D4—41no—
573D4—41no—
574D4—142no—
575D4—142no—
576D4—142no—
577D4—142no—
578D4—281no—
579D4—281no—
580D4—281no—
581D4—281no—
582D4—568no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—74no—
593C2×C4—84no—
594C2×C4—142no—
595C2×C4—142no—
596C2×C4—142no—
597C2×C4—142no—
598C2×C4—142no—
599C2×C4—282no—
600C2×C4—282no—
601C2×C4—281no—
602C2×C4—281no—
603C2×C4—564no—
604C2×C4—564no—
605C2×C4#650312yes—
606C2×C4#650312yes—
607C2×C4C22×C4×D711yescharacteristic
608C2×C4C2×#115711yescharacteristic
609C2×C4C2×D4×D711yescharacteristic
610C2×C4D7×#4111yescharacteristic
611C2×C4D7×#4111yescharacteristic
612C2×C4C22×C22⋊D711yescharacteristic
613C8—22no—
614C8—74no—
615C8—142no—
616C8C4×C22⋊D714yes—
617C7#2170411yescharacteristic, complemented, Sylow
618C22—44no—
619C22—42no—
620C22—82no—
621C22—81no—
622C22—81no—
623C22—284no—
624C22—284no—
625C22—282no—
626C22—282no—
627C22—562no—
628C22—562no—
629C22—561no—
630C22—561no—
631C22—561no—
632C22—561no—
633C22C4×D4×D711yescharacteristic
634C22#9023911yescharacteristic
635C22#9045312yes—
636C22#9046012yes—
637C22C2×#650311yescharacteristic
638C4—22no—
639C4—22no—
640C4—42no—
641C4—81no—
642C4—81no—
643C4—144no—
644C4—282no—
645C4—561no—
646C4—561no—
647C4C4×D4×D711yescharacteristic
648C4#9023911yescharacteristic
649C4C2×C4×C22⋊D711yescharacteristic
650C4#9052311yescharacteristic
651C2—82no—
652C2—562no—
653C2—562no—
654C2#1173709212yes—
655C2#1173711412yes—
656C2C4×#650311yescharacteristic
657C2#1174129911yescharacteristic
658C2#1174135111yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC2×C4×C7×SD16C2
socleC23×C7C22×D4
prime radicalC2×#89948C2
cosocleC22×C4×C7C24
hypercenterC2×C4×SD16D7
hypocenterC7#21704
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×SD16

Complements

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

NH ≅ G/Ntype
1. C2×C4×C7×SD16C2semidirect
3. #11739900C2semidirect
4. C2×#89798C2semidirect
5. #11739774C2semidirect
6. C2×#89786C2semidirect
7. C2×C4×D56C2semidirect
8. #11729804C2semidirect
9. #11728330C2semidirect
10. #11728310C2semidirect
11. #11728011C2semidirect
12. #11727980C2semidirect
13. #11727944C2semidirect
14. #11727911C2semidirect
15. #11727503C2semidirect
36. #89948C4semidirect
71. C2×C7×SD16C2×C4semidirect
86. #6510C2×C4semidirect
89. C2×#1143C2×C4semidirect
91. #6503C2×C4semidirect
94. C2×#1141C2×C4semidirect
109. C2×D56C2×C4semidirect
117. C4×C7×C8D4semidirect
128. #6389C2×C4semidirect
135. #6377D4semidirect
139. C2×C4×SD16D7semidirect
361. C2×SD16C4×D7semidirect
418. C4×C8C22⋊D7semidirect
617. C7#21704semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C7×SD162
C2×#899482
#117399002
C2×#897982
#117397742
C2×#897862
C2×C4×D562
#117298042
#117283302
#117283102
#117280112
#117279802
#117279442
#117279112
#117275032
#217047