Finite Group Databasebeta

← #13362393

Subgroups — #13362393

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
1C7×#2326C211yesmaximal, characteristic, complemented
2C2×#89830C211yesmaximal, characteristic
3C2×#89798C211yesmaximal, characteristic, complemented
4C2×C4×#1143C211yesmaximal, characteristic, complemented
5C2×#89786C211yesmaximal, characteristic, complemented
6C2×C4×#1141C211yesmaximal, characteristic, complemented
7#11730834C211yesmaximal, characteristic, complemented
8#11729479C211yesmaximal, characteristic, complemented
9#11727433C211yesmaximal, characteristic, complemented
10#11727430C211yesmaximal, characteristic, complemented
11#11727402C211yesmaximal, characteristic, complemented
12#11727401C211yesmaximal, characteristic, complemented
13#11727334C211yesmaximal, characteristic, complemented
14#11727332C211yesmaximal, characteristic, complemented
15#11727188C211yesmaximal, characteristic, complemented
16C2×C7×#124C2211yescharacteristic
17C2×C4×C7×Q8C2211yescharacteristic
18C2×C4×C7×D4C2211yescharacteristic
19C22×#1143C2211yescharacteristic
20C22×#1143C2211yescharacteristic
21C22×#1141C2211yescharacteristic
22C22×#1141C2211yescharacteristic
23C2×#6437C2211yescharacteristic
24C2×C4×D28C2211yescharacteristic
25C2×C4×D28C2211yescharacteristic
26C7×#391C2211yescharacteristic
27C7×#390C2211yescharacteristic
28C7×#388—22no—
29C7×#385—22no—
30C7×#384C2211yescharacteristic
31C7×#376C2211yescharacteristic
32C2×#6409C2211yescharacteristic
33C2×#6409C2211yescharacteristic
34C2×#6406C2211yescharacteristic
35C2×#6406C2211yescharacteristic
36#89830C418yescomplemented
37#89798—24no—
38C4×#1143—24no—
39#89786—24no—
40C4×#1141—24no—
41C2×#6383C2211yescharacteristic
42C2×#6383C2211yescharacteristic
43C2×#6383C2211yescharacteristic
44C2×#6383C2211yescharacteristic
45C2×#6382C2211yescharacteristic
46C2×#6382C2211yescharacteristic
47C2×#6381C2211yescharacteristic
48C2×#6381C2211yescharacteristic
49#89711C2211yescharacteristic
50#89711C2211yescharacteristic
51#89708C2211yescharacteristic
52C2×#6377C2211yescharacteristic
53C2×#6377C2211yescharacteristic
54C2×#6377C2211yescharacteristic
55C2×C4×C7⋊C8C2211yescharacteristic
56C2×C4×C7⋊C8C2211yescharacteristic
57C2×C4×C7⋊C8C2211yescharacteristic
58#89437—22no—
59#89437—22no—
60#21704—71nomaximal, Sylow
61C22×C7×Q8C2311yescharacteristic
62C22×C7×D4C2311yescharacteristic
63C23×C4×C7—21no—
64C22×D28—22no—
65C22×D28C2311yescharacteristic
66C22×D28C2311yescharacteristic
67C22×C4×D7—21no—
68C22×C4×D7—21no—
69C7×#124—22no—
70C7×#124C2×C414yescomplemented
71C4×C7×Q8—22no—
72C4×C7×Q8—22no—
73C4×C7×D4—22no—
74C4×C7×D4—22no—
75C2×C7×C4⋊C4—21no—
76C2×C7×C4⋊C4—22no—
77C2×C7×C4⋊C4C2311yescharacteristic
78C2×C7×C4⋊C4C2311yescharacteristic
79C2×C7×C22⋊C4—22no—
80C2×C7×C22⋊C4—21no—
81C2×C7×C22⋊C4—22no—
82C2×C42×C7—21no—
83C2×C42×C7C2311yescharacteristic
84C2×C42×C7C2311yescharacteristic
85C2×C42×C7C2311yescharacteristic
86C2×#1143—22no—
87C2×#1143—22no—
88C2×#1143—22no—
89C2×#1143—22no—
90C2×#1143C2×C414yescomplemented
91C2×#1143C2×C414yescomplemented
92C2×#1141—22no—
93C2×#1141—22no—
94C2×#1141—22no—
95C2×#1141—22no—
96C2×#1141C2×C414yescomplemented
97C2×#1141C2×C414yescomplemented
98C2×(C2×D7)⋊C4—21no—
99C2×(C2×D7)⋊C4—21no—
100C2×(C2×D7)⋊C4—22no—
101C2×(C2×D7)⋊C4—22no—
102C2×C4⋊Dic7C2311yescharacteristic
103C2×C4⋊Dic7C2311yescharacteristic
104C22×C7⋊C8C2311yescharacteristic
105C22×C7⋊C8C2311yescharacteristic
106C22×C7⋊C8C2311yescharacteristic
107C22×C7⋊C8C2311yescharacteristic
108#6437—22no—
109#6437C2×C414yescomplemented
110C4×D28—22no—
111C4×D28—22no—
112C4×D28—22no—
113C4×D28—22no—
114C7×#95—22no—
115C7×#95D412yes—
116C7×#95D412yes—
117#6409—22no—
118#6409—22no—
119#6406—22no—
120#6406—22no—
121C7⋊#95—22no—
122#6383—22no—
123#6383—22no—
124#6383—22no—
125#6383—22no—
126#6382—22no—
127#6382—22no—
128#6381—22no—
129#6381—22no—
130#6377—22no—
131#6377C2×C414yescomplemented
132#6377D414yescomplemented
133C4×C7⋊C8—22no—
134C4×C7⋊C8—22no—
135C4×C7⋊C8D414yescomplemented
136C2×#496—71no—
137C2×#442—71no—
138C2×#440—71no—
139C2×C4×SD16—71no—
140C2×C4×D8—71no—
141#2328—71no—
142#2326D711yescharacteristic, complemented
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×D412yes—
172C22×C4×C7C2×D412yes—
173C22×C4×C7C2×D411yescharacteristic
174C22×C4×C7C2×D411yescharacteristic
175C22×C4×C7C2411yescharacteristic
176C22×Dic7—21no—
177C22×Dic7—21no—
178C2×D28—21no—
179C2×D28—21no—
180C2×D28—21no—
181C2×D28—21no—
182C2×D28—21no—
183C2×D28—21no—
184C2×D28—21no—
185C2×D28—21no—
186C2×D28—21no—
187C2×D28—21no—
188C2×D28—28no—
189C2×D28—82no—
190C2×D28—82no—
191C2×D28C22×C412yes—
192C2×D28C22×C412yes—
193C2×C4×D7—42no—
194C2×C4×D7—42no—
195C2×C4×D7—82no—
196C2×C4×D7—82no—
197C7×C4⋊C4—21no—
198C7×C4⋊C4—21no—
199C7×C4⋊C4—21no—
200C7×C4⋊C4—21no—
201C7×C4⋊C4—42no—
202C7×C4⋊C4—82no—
203C7×C22⋊C4—28no—
204C7×C22⋊C4—42no—
205C7×C22⋊C4—82no—
206C42×C7—21no—
207C42×C7—21no—
208C42×C7—42no—
209C42×C7C22×C412yes—
210C42×C7C2×D412yes—
211C42×C7C2×D412yes—
212C42×C7#4112yes—
213#1143—28no—
214#1143—28no—
215#1141—28no—
216#1141—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—
225C2×C7⋊C8—21no—
226C2×C7⋊C8—21no—
227C2×C7⋊C8—21no—
228C2×C7⋊C8—21no—
229C2×C7⋊C8C22×C412yes—
230C2×C7⋊C8C22×C412yes—
231C2×C7⋊C8C2×D412yes—
232C2×C7⋊C8C2×D412yes—
233C2×C7⋊C8C2×D412yes—
234C2×C7⋊C8C2×D412yes—
235C2×C7⋊C8#4112yes—
236C2×C7⋊C8#4112yes—
237C22×SD16—71no—
238C22×SD16—71no—
239C22×D8—71no—
240C22×D8—71no—
241C2×#127—71no—
242C2×#124C2×D711yescharacteristic
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—144no—
251C4×D8—144no—
252C2×#107—71no—
253C2×#107—71no—
254C2×#106—71no—
255C2×#106—71no—
256C2×#103—71no—
257C2×#103—71no—
258C2×#102—71no—
259C2×#102—71no—
260C2×#102—71no—
261C2×#102—71no—
262C2×#102—71no—
263C2×#102—71no—
264C2×#97—71no—
265C2×#97—71no—
266C2×#97—71no—
267C2×C4×C8—71no—
268C2×C4×C8—71no—
269C2×C4×C8—71no—
270#391C2×D711yescharacteristic
271#390—71no—
272#390—71no—
273#390C2×D711yescharacteristic
274#388—22no—
275#386—142no—
276#386—142no—
277#385—22no—
278#384C2×D711yescharacteristic
279#382—71no—
280#376C2×D711yescharacteristic
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—22no—
309C2×C4×C7—22no—
310C2×C4×C7—22no—
311C2×C4×C7—22no—
312C2×C4×C7—22no—
313C2×C4×C7—41no—
314C2×C4×C7—41no—
315C2×C4×C7—41no—
316C2×C4×C7—41no—
317C2×C4×C7—42no—
318C2×C4×C7—42no—
319C2×C4×C7—84no—
320C2×C4×C7C4×D414yes—
321C2×C4×C7#12712yes—
322C2×C4×C7#12712yes—
323C2×C4×C7C23×C411yescharacteristic
324C2×C4×C7C22×D411yescharacteristic
325C2×C4×C7C22×D411yescharacteristic
326C2×C4×C7C2×#4111yescharacteristic
327C2×C4×C7C2×#4111yescharacteristic
328C2×C4×C7C2×#4111yescharacteristic
329C2×Dic7—41no—
330C2×Dic7—41no—
331C2×Dic7—42no—
332C2×Dic7—42no—
333D28—22no—
334D28—22no—
335D28—22no—
336D28—22no—
337D28—41no—
338D28—41no—
339D28—41no—
340D28—41no—
341D28—88no—
342C4×D7—84no—
343C4×D7—84no—
344C7⋊C8—22no—
345C7⋊C8—22no—
346C7⋊C8C4×D414yes—
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—74no—
357C2×SD16—74no—
358C2×SD16—142no—
359C2×SD16—142no—
360C2×SD16—142no—
361C2×SD16—142no—
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×C8—71no—
371C22×C8—71no—
372#127—74no—
373#127—142no—
374#124—22no—
375#124C4×D714yescomplemented
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—22no—
386C2×C4⋊C4—71no—
387C2×C4⋊C4—71no—
388C2×C4⋊C4C22×D711yescharacteristic
389C2×C4⋊C4C22×D711yescharacteristic
390C2×C22⋊C4—22no—
391C2×C22⋊C4—21no—
392C2×C22⋊C4—22no—
393C2×C22⋊C4—141no—
394C2×C22⋊C4—141no—
395C2×C22⋊C4—142no—
396C2×C22⋊C4—142no—
397C2×C42—21no—
398C2×C42C22×D711yescharacteristic
399C2×C42C22×D711yescharacteristic
400C2×C42C22×D711yescharacteristic
401#107—142no—
402#107—142no—
403#106—142no—
404#106—142no—
405#103—142no—
406#103—142no—
407#102—142no—
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—74no—
417C4×C8—142no—
418C4×C8—142no—
419#95—22no—
420#95—142no—
421#95C22⋊D712yes—
422#95C22⋊D712yes—
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—24no—
446C4×C7—42no—
447C4×C7—42no—
448C4×C7—81no—
449C4×C7—81no—
450C4×C7C2×C4×D411yescharacteristic
451C4×C7C2×C4×D411yescharacteristic
452C4×C7#53211yescharacteristic
453C4×C7#53211yescharacteristic
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—82no—
491C22×C4—141no—
492C22×C4—141no—
493C22×C4—282no—
494C22×C4—282no—
495C22×C4—562no—
496C22×C4—562no—
497C22×C4C2×C22⋊D712yes—
498C22×C4C2×C22⋊D712yes—
499C22×C4C2×C22⋊D711yescharacteristic
500C22×C4C2×C22⋊D711yescharacteristic
501C22×C4C23×D711yescharacteristic
502SD16—148no—
503SD16—148no—
504D8—148no—
505D8—148no—
506C2×C8—72no—
507C2×C8—72no—
508C2×C8—72no—
509C2×C8—72no—
510C2×C8—72no—
511C2×C8—72no—
512C2×C8—72no—
513C2×C8—72no—
514C2×C8—141no—
515C2×C8—141no—
516C2×C8—141no—
517C2×C8—141no—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—42no—
523C4⋊C4—82no—
524C4⋊C4—141no—
525C4⋊C4—141no—
526C4⋊C4—141no—
527C4⋊C4—141no—
528C22⋊C4—28no—
529C22⋊C4—42no—
530C22⋊C4—82no—
531C22⋊C4—282no—
532C22⋊C4—282no—
533C22⋊C4—562no—
534C22⋊C4—562no—
535C42—21no—
536C42—21no—
537C42—42no—
538C42C2×C4×D712yes—
539C42#115712yes—
540C42D4×D712yes—
541C42D4×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—41no—
552C23—41no—
553C23—42no—
554C23—42no—
555C23—142no—
556C23—142no—
557C23—281no—
558C23—281no—
559C23—281no—
560C23—281no—
561C23—282no—
562C23—282no—
563C23—282no—
564C23—282no—
565C23C22×C22⋊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—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—84no—
598C2×C4—281no—
599C2×C4—281no—
600C2×C4—282no—
601C2×C4—282no—
602C2×C4—564no—
603C2×C4—564no—
604C2×C4C4×C22⋊D714yes—
605C2×C4#650312yes—
606C2×C4#650312yes—
607C2×C4C22×C4×D711yescharacteristic
608C2×C4C2×#115711yescharacteristic
609C2×C4C2×D4×D711yescharacteristic
610C2×C4C2×D4×D711yescharacteristic
611C2×C4D7×#4111yescharacteristic
612C2×C4D7×#4111yescharacteristic
613C8—74no—
614C8—74no—
615C8—142no—
616C8—142no—
617C7#2170411yescharacteristic, complemented, Sylow
618C22—44no—
619C22—42no—
620C22—81no—
621C22—81no—
622C22—82no—
623C22—284no—
624C22—284no—
625C22—282no—
626C22—282no—
627C22—561no—
628C22—561no—
629C22—561no—
630C22—561no—
631C22—562no—
632C22—562no—
633C22C2×C4×C22⋊D711yescharacteristic
634C22C2×#650311yescharacteristic
635C22#9052311yescharacteristic
636C22#9052612yes—
637C22#9052812yes—
638C4—24no—
639C4—22no—
640C4—22no—
641C4—42no—
642C4—42no—
643C4—81no—
644C4—81no—
645C4—561no—
646C4—561no—
647C4C4×D4×D711yescharacteristic
648C4C4×D4×D711yescharacteristic
649C4#9023911yescharacteristic
650C4#9023911yescharacteristic
651C2—82no—
652C2—562no—
653C2—562no—
654C2C4×#650311yescharacteristic
655C2#1174000912yes—
656C2#1174082711yescharacteristic
657C2#1174084711yescharacteristic
658C2#1174090412yes—
subgroupG/H
centerC23C22×C22⋊D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#2326C2
socleC23×C7C22×D4
prime radicalC2×#89830C2
cosocleC22×C4×C7C24
hypercenter#2326D7
hypocenterC7#21704
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ #376 ◁ #2326

Complements

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

NH ≅ G/Ntype
1. C7×#2326C2semidirect
3. C2×#89798C2semidirect
4. C2×C4×#1143C2semidirect
5. C2×#89786C2semidirect
6. C2×C4×#1141C2semidirect
7. #11730834C2semidirect
8. #11729479C2semidirect
9. #11727433C2semidirect
10. #11727430C2semidirect
11. #11727402C2semidirect
12. #11727401C2semidirect
13. #11727334C2semidirect
14. #11727332C2semidirect
15. #11727188C2semidirect
36. #89830C4semidirect
70. C7×#124C2×C4semidirect
90. C2×#1143C2×C4semidirect
91. C2×#1143C2×C4semidirect
96. C2×#1141C2×C4semidirect
97. C2×#1141C2×C4semidirect
109. #6437C2×C4semidirect
131. #6377C2×C4semidirect
132. #6377D4semidirect
135. C4×C7⋊C8D4semidirect
142. #2326D7semidirect
375. #124C4×D7semidirect
617. C7#21704semidirect

Maximal subgroups

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

subgroupindex
C7×#23262
C2×#898302
C2×#897982
C2×C4×#11432
C2×#897862
C2×C4×#11412
#117308342
#117294792
#117274332
#117274302
#117274022
#117274012
#117273342
#117273322
#117271882
#217047