Finite Group Databasebeta

← #13410955

Subgroups — #13410955

Order 1792.

946 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2442C211yesmaximal, characteristic
2C2×#89835C211yesmaximal, characteristic, complemented
3C22×#6384C211yesmaximal, characteristic
4C22⋊C4×Dic14C211yesmaximal, characteristic
5#11729498C211yesmaximal, characteristic, complemented
6#11729309C211yesmaximal, characteristic
7#11727389C211yesmaximal, characteristic, complemented
8#11727388C211yesmaximal, characteristic, complemented
9#11727386C211yesmaximal, characteristic, complemented
10#11727385C211yesmaximal, characteristic, complemented
11#11727315C211yesmaximal, characteristic, complemented
12#11727313C211yesmaximal, characteristic, complemented
13#11727279C211yesmaximal, characteristic
14#11727265C211yesmaximal, characteristic
15#11727203C211yesmaximal, characteristic
16C23×Dic14C2211yescharacteristic
17C2×C7×#125C2211yescharacteristic
18C22×C7×C4⋊C4C2211yescharacteristic
19C23×C7⋊C8C2211yescharacteristic
20C2×C4×Dic14C2211yescharacteristic
21C2×C4×Dic14C2211yescharacteristic
22C7×#400—21no—
23C7×#398—21no—
24C7×#389C2211yescharacteristic
25C7×#389C2211yescharacteristic
26C7×#385—21no—
27C7×#382C2211yescharacteristic
28C7×#380—21no—
29C7×#380C2211yescharacteristic
30C4×C7×C22⋊C4C2211yescharacteristic
31#89984C2211yescharacteristic
32C2×C22⋊(C7⋊C8)C2211yescharacteristic
33#89835—24no—
34C2×#6384—24no—
35C2×#6384C2211yescharacteristic
36C2×#6384C2211yescharacteristic
37C2×#6384C2211yescharacteristic
38C2×#6384C2211yescharacteristic
39C2×#6384C2211yescharacteristic
40C2×#6384C2211yescharacteristic
41C2×#6384C2211yescharacteristic
42C2×#6384C2211yescharacteristic
43C2×#6382C2211yescharacteristic
44C2×#6382C2211yescharacteristic
45C2×#6381C2211yescharacteristic
46C2×#6381C2211yescharacteristic
47#89714—21no—
48C22⋊C4×Dic7—21no—
49#89701C2211yescharacteristic
50#89701C2211yescharacteristic
51C2×C4⋊(C7⋊C8)C2211yescharacteristic
52C2×C4⋊(C7⋊C8)C2211yescharacteristic
53#89418—21no—
54#89418—21no—
55#89324C2211yescharacteristic
56#89324C2211yescharacteristic
57#89324C2211yescharacteristic
58#89324C2211yescharacteristic
59#89319C2211yescharacteristic
60#89319C2211yescharacteristic
61#25957—71nomaximal, Sylow
62C23×C4×C7—21no—
63C23×C4×C7C2311yescharacteristic
64C23×Dic7—21no—
65C22×Dic14—22no—
66C22×Dic14—22no—
67C22×Dic14—21no—
68C22×Dic14—22no—
69C22×Dic14—41no—
70C22×Dic14—41no—
71C22×Dic14Q814yes—
72C22×Dic14C2311yescharacteristic
73C22×Dic14C2311yescharacteristic
74C7×#125—22no—
75C7×#125—41no—
76C2×C7×C4⋊C4—22no—
77C2×C7×C4⋊C4—21no—
78C2×C7×C4⋊C4—21no—
79C2×C7×C4⋊C4—21no—
80C2×C7×C4⋊C4—21no—
81C2×C7×C4⋊C4—21no—
82C2×C7×C4⋊C4—41no—
83C2×C7×C4⋊C4C2311yescharacteristic
84C2×C7×C4⋊C4C2311yescharacteristic
85C2×C7×C4⋊C4C2311yescharacteristic
86C2×C7×C4⋊C4C2311yescharacteristic
87C2×C7×C22⋊C4—21no—
88C2×C7×C22⋊C4D411yescharacteristic
89C2×C7×C22⋊C4D411yescharacteristic
90C2×C42×C7C2311yescharacteristic
91C2×C42×C7C2311yescharacteristic
92C2×C22⋊Dic7—21no—
93C2×C4⋊Dic7C2311yescharacteristic
94C2×C4⋊Dic7C2311yescharacteristic
95C2×Dic7⋊C4—21no—
96C2×Dic7⋊C4—21no—
97C2×C4×Dic7—21no—
98C2×C4×Dic7—21no—
99C22×C7⋊C8—22no—
100C22×C7⋊C8—22no—
101C22×C7⋊C8—41no—
102C22×C7⋊C8C2311yescharacteristic
103C22×C7⋊C8C2311yescharacteristic
104C22×C7⋊C8C2311yescharacteristic
105C22×C7⋊C8C2311yescharacteristic
106C4×Dic14—22no—
107C4×Dic14—22no—
108C4×Dic14—41no—
109C4×Dic14—41no—
110C7×#95—21no—
111C7×#95—21no—
112C7×#95—21no—
113C7×#95—21no—
114C7×#95—21no—
115C7×#95D411yescharacteristic
116C7×#95D411yescharacteristic
117C7⋊#95—21no—
118C7⋊#95—21no—
119C7⋊#95—21no—
120C7⋊#95—21no—
121C7⋊#95—21no—
122C22⋊(C7⋊C8)—22no—
123#6384—22no—
124#6384—22no—
125#6384—22no—
126#6384—22no—
127#6384—22no—
128#6384—41no—
129#6384—41no—
130#6384D414yes—
131#6382—22no—
132#6382—41no—
133#6381—22no—
134#6381—41no—
135C4⋊(C7⋊C8)—22no—
136C4⋊(C7⋊C8)—41no—
137C2×#479—71no—
138C22×#103—71no—
139#2442D711yescharacteristic
140Q8×C22⋊C4—71no—
141#2097—71no—
142#2068—71no—
143#2066—71no—
144#2026—71no—
145#1865—71no—
146#1863—71no—
147#1852—71no—
148#1850—71no—
149#1807—71no—
150#1805—71no—
151#1787—71no—
152C24×C7C2×D411yescharacteristic
153C22×C4×C7—21no—
154C22×C4×C7—21no—
155C22×C4×C7—21no—
156C22×C4×C7—21no—
157C22×C4×C7—21no—
158C22×C4×C7—21no—
159C22×C4×C7—21no—
160C22×C4×C7—21no—
161C22×C4×C7—42no—
162C22×C4×C7—42no—
163C22×C4×C7—81no—
164C22×C4×C7C2×D411yescharacteristic
165C22×C4×C7C2×D411yescharacteristic
166C22×C4×C7C2×D411yescharacteristic
167C22×C4×C7C2×D411yescharacteristic
168C22×C4×C7C2×D411yescharacteristic
169C22×C4×C7C2×Q812yes—
170C22×C4×C7#4112yes—
171C22×C4×C7C2411yescharacteristic
172C22×Dic7—21no—
173C22×Dic7—24no—
174C22×Dic7—21no—
175C22×Dic7—21no—
176C22×Dic7—21no—
177C22×Dic7—42no—
178C22×Dic7—42no—
179C2×Dic14—21no—
180C2×Dic14—21no—
181C2×Dic14—21no—
182C2×Dic14—21no—
183C2×Dic14—21no—
184C2×Dic14—21no—
185C2×Dic14—21no—
186C2×Dic14—28no—
187C2×Dic14—24no—
188C2×Dic14—21no—
189C2×Dic14—44no—
190C2×Dic14—44no—
191C2×Dic14—44no—
192C2×Dic14—42no—
193C2×Dic14—42no—
194C2×Dic14C2×D412yes—
195C2×Dic14C2×Q812yes—
196C2×Dic14C2×Q812yes—
197C2×Dic14#4112yes—
198C7×C4⋊C4—21no—
199C7×C4⋊C4—21no—
200C7×C4⋊C4—21no—
201C7×C4⋊C4—21no—
202C7×C4⋊C4—21no—
203C7×C4⋊C4—21no—
204C7×C4⋊C4—21no—
205C7×C4⋊C4—42no—
206C7×C4⋊C4—42no—
207C7×C4⋊C4—42no—
208C7×C4⋊C4—81no—
209C7×C4⋊C4—81no—
210C7×C4⋊C4C2×D412yes—
211C7×C22⋊C4—22no—
212C7×C22⋊C4—22no—
213C7×C22⋊C4—42no—
214C42×C7—21no—
215C42×C7—21no—
216C42×C7—21no—
217C42×C7—21no—
218C22⋊Dic7—42no—
219C4⋊Dic7—21no—
220C4⋊Dic7—21no—
221C4⋊Dic7—21no—
222C4⋊Dic7—21no—
223Dic7⋊C4—42no—
224Dic7⋊C4—42no—
225C4×Dic7—42no—
226C4×Dic7—42no—
227C2×C7⋊C8—21no—
228C2×C7⋊C8—21no—
229C2×C7⋊C8—21no—
230C2×C7⋊C8—21no—
231C2×C7⋊C8—21no—
232C2×C7⋊C8—21no—
233C2×C7⋊C8—21no—
234C2×C7⋊C8—21no—
235C2×C7⋊C8—21no—
236C2×C7⋊C8—21no—
237C2×C7⋊C8—21no—
238C2×C7⋊C8—44no—
239C2×C7⋊C8C2×D412yes—
240C23×Q8—71no—
241C23×C8—71no—
242C2×#125C2×D711yescharacteristic
243C2×C4×Q8—71no—
244C2×C4×Q8—71no—
245C22×C4⋊C4C2×D711yescharacteristic
246#479—144no—
247C2×#107—71no—
248C2×#107—71no—
249C2×#106—71no—
250C2×#106—71no—
251C2×C4⋊C8—71no—
252C2×C4⋊C8—71no—
253C2×#103—71no—
254C2×#103—71no—
255C2×#103—71no—
256C2×#103—71no—
257C2×#103—71no—
258C2×#103—71no—
259C2×#103—71no—
260C2×#103—71no—
261C2×#103—144no—
262C2×C22⋊C8—71no—
263#400—21no—
264#398—21no—
265#391—71no—
266#391—71no—
267#391—141no—
268#391—141no—
269#389C2×D711yescharacteristic
270#389C2×D711yescharacteristic
271#385—21no—
272#382C2×D711yescharacteristic
273#380—21no—
274#380—71no—
275#380—141no—
276#380C2×D711yescharacteristic
277C4×C22⋊C4—141no—
278C4×C22⋊C4C2×D711yescharacteristic
279#340—71no—
280#340—71no—
281#340—71no—
282#340—71no—
283#336—71no—
284#336—71no—
285C23×C7—21no—
286C23×C7—21no—
287C23×C7—21no—
288C23×C7—21no—
289C23×C7C22⋊Q812yes—
290C23×C7#13512yes—
291C23×C7#13712yes—
292C23×C7C22×D411yescharacteristic
293C2×C4×C7—21no—
294C2×C4×C7—21no—
295C2×C4×C7—22no—
296C2×C4×C7—22no—
297C2×C4×C7—22no—
298C2×C4×C7—22no—
299C2×C4×C7—21no—
300C2×C4×C7—21no—
301C2×C4×C7—22no—
302C2×C4×C7—21no—
303C2×C4×C7—21no—
304C2×C4×C7—21no—
305C2×C4×C7—21no—
306C2×C4×C7—21no—
307C2×C4×C7—21no—
308C2×C4×C7—21no—
309C2×C4×C7—21no—
310C2×C4×C7—21no—
311C2×C4×C7—21no—
312C2×C4×C7—41no—
313C2×C4×C7—41no—
314C2×C4×C7—41no—
315C2×C4×C7—41no—
316C2×C4×C7—41no—
317C2×C4×C7—41no—
318C2×C4×C7—41no—
319C2×C4×C7—41no—
320C2×C4×C7—41no—
321C2×C4×C7—41no—
322C2×C4×C7—41no—
323C2×C4×C7—84no—
324C2×C4×C7C22⋊Q812yes—
325C2×C4×C7C22×D411yescharacteristic
326C2×C4×C7C22×Q811yescharacteristic
327C2×C4×C7C2×#4111yescharacteristic
328C2×C4×C7C2×#4111yescharacteristic
329C2×C4×C7H(5,𝔽2)11yescharacteristic
330C2×C4×C7H(5,𝔽2)11yescharacteristic
331C2×Dic7—22no—
332C2×Dic7—22no—
333C2×Dic7—22no—
334C2×Dic7—22no—
335C2×Dic7—48no—
336C2×Dic7—41no—
337C2×Dic7—41no—
338C2×Dic7—41no—
339C2×Dic7—41no—
340C2×Dic7—41no—
341C2×Dic7—41no—
342C2×Dic7—41no—
343C2×Dic7—41no—
344Dic14—22no—
345Dic14—22no—
346Dic14—22no—
347Dic14—22no—
348Dic14—22no—
349Dic14—48no—
350Dic14—41no—
351Dic14—41no—
352Dic14C22⋊Q814yes—
353C7⋊C8—22no—
354C7⋊C8—41no—
355C7⋊C8—41no—
356C7⋊C8—41no—
357C22×Q8—74no—
358C22×Q8—71no—
359C22×Q8—71no—
360C22×Q8—142no—
361C22×Q8—142no—
362C22×Q8—142no—
363C22×Q8—141no—
364C22×Q8—281no—
365C22×Q8—281no—
366C23×C4—21no—
367C23×C4—141no—
368C23×C4C22×D711yescharacteristic
369C22×C8—71no—
370C22×C8—71no—
371C22×C8—71no—
372C22×C8—71no—
373C22×C8—142no—
374C22×C8—142no—
375C22×C8—281no—
376#125—22no—
377#125—41no—
378C4×Q8—142no—
379C4×Q8—142no—
380C4×Q8—281no—
381C4×Q8—281no—
382C2×C4⋊C4—22no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—21no—
388C2×C4⋊C4—41no—
389C2×C4⋊C4—71no—
390C2×C4⋊C4—71no—
391C2×C4⋊C4—141no—
392C2×C4⋊C4—141no—
393C2×C4⋊C4C22×D711yescharacteristic
394C2×C4⋊C4C22×D711yescharacteristic
395C2×C4⋊C4C22×D711yescharacteristic
396C2×C4⋊C4C22×D711yescharacteristic
397C2×C22⋊C4—21no—
398C2×C22⋊C4—141no—
399C2×C22⋊C4C22⋊D711yescharacteristic
400C2×C22⋊C4C22⋊D711yescharacteristic
401C2×C42—141no—
402C2×C42—141no—
403C2×C42C22×D711yescharacteristic
404C2×C42C22×D711yescharacteristic
405#107—142no—
406#107—281no—
407#106—142no—
408#106—281no—
409C4⋊C8—142no—
410C4⋊C8—281no—
411#103—74no—
412#103—142no—
413#103—142no—
414#103—142no—
415#103—142no—
416#103—142no—
417#103—281no—
418#103—281no—
419C22⋊C8—142no—
420#95—21no—
421#95—21no—
422#95—21no—
423#95—21no—
424#95—21no—
425#95—141no—
426#95—141no—
427#95—141no—
428#95—141no—
429#95—141no—
430#95C22⋊D711yescharacteristic
431#95C22⋊D711yescharacteristic
432C22×C7—22no—
433C22×C7—22no—
434C22×C7—22no—
435C22×C7—22no—
436C22×C7—22no—
437C22×C7—22no—
438C22×C7#47914yescomplemented
439C22×C7C2×C22⋊Q811yescharacteristic
440C22×C7D4211yescharacteristic
441C22×C7#54611yescharacteristic
442C22×C7C2×#13511yescharacteristic
443C22×C7C2×#13711yescharacteristic
444C22×C7#57711yescharacteristic
445C22×C7#57811yescharacteristic
446C4×C7—22no—
447C4×C7—41no—
448C4×C7—41no—
449C4×C7—42no—
450C4×C7—41no—
451C4×C7—41no—
452C4×C7—81no—
453C4×C7—81no—
454C4×C7—81no—
455C4×C7C2×C22⋊Q811yescharacteristic
456C4×C7#54311yescharacteristic
457C4×C7#54611yescharacteristic
458C4×C7#55111yescharacteristic
459Dic7—24no—
460Dic7—42no—
461Dic7—81no—
462Dic7—81no—
463C24C2×C22⋊D711yescharacteristic
464C2×Q8—72no—
465C2×Q8—72no—
466C2×Q8—72no—
467C2×Q8—72no—
468C2×Q8—148no—
469C2×Q8—144no—
470C2×Q8—141no—
471C2×Q8—141no—
472C2×Q8—141no—
473C2×Q8—141no—
474C2×Q8—141no—
475C2×Q8—141no—
476C2×Q8—141no—
477C2×Q8—141no—
478C2×Q8—282no—
479C2×Q8—282no—
480C2×Q8—284no—
481C2×Q8—284no—
482C2×Q8—284no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—42no—
492C22×C4—42no—
493C22×C4—81no—
494C22×C4—141no—
495C22×C4—144no—
496C22×C4—141no—
497C22×C4—141no—
498C22×C4—141no—
499C22×C4—282no—
500C22×C4—282no—
501C22×C4Q8×D712yes—
502C22×C4#116112yes—
503C22×C4C2×C22⋊D711yescharacteristic
504C22×C4C2×C22⋊D711yescharacteristic
505C22×C4C2×C22⋊D711yescharacteristic
506C22×C4C2×C22⋊D711yescharacteristic
507C22×C4C2×C22⋊D711yescharacteristic
508C22×C4C23×D711yescharacteristic
509C2×C8—72no—
510C2×C8—141no—
511C2×C8—141no—
512C2×C8—141no—
513C2×C8—141no—
514C2×C8—141no—
515C2×C8—141no—
516C2×C8—141no—
517C2×C8—141no—
518C2×C8—141no—
519C2×C8—141no—
520C2×C8—141no—
521C2×C8—284no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—42no—
530C4⋊C4—42no—
531C4⋊C4—42no—
532C4⋊C4—81no—
533C4⋊C4—81no—
534C4⋊C4—141no—
535C4⋊C4—141no—
536C4⋊C4—141no—
537C4⋊C4—141no—
538C4⋊C4—282no—
539C4⋊C4—282no—
540C4⋊C4D4×D712yes—
541C22⋊C4—22no—
542C22⋊C4—22no—
543C22⋊C4—42no—
544C22⋊C4—282no—
545C42—21no—
546C42—21no—
547C42—21no—
548C42—21no—
549C42—282no—
550C42—282no—
551C2×C7—24no—
552C2×C7#243711yescharacteristic
553C2×C7C2×#47911yescharacteristic
554C2×C7#308211yescharacteristic
555C2×C7#327011yescharacteristic
556C2×C7#328211yescharacteristic
557C2×C7#330011yescharacteristic
558C2×C7#332011yescharacteristic
559C23—21no—
560C23—21no—
561C23—21no—
562C23—21no—
563C23#650912yes—
564C23#651312yes—
565C23#651412yes—
566C23C22×C22⋊D711yescharacteristic
567Q8—74no—
568Q8—142no—
569Q8—142no—
570Q8—142no—
571Q8—142no—
572Q8—142no—
573Q8—288no—
574Q8—281no—
575Q8—281no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—22no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—84no—
607C2×C4—142no—
608C2×C4—142no—
609C2×C4—142no—
610C2×C4—142no—
611C2×C4—288no—
612C2×C4—281no—
613C2×C4—281no—
614C2×C4—281no—
615C2×C4—281no—
616C2×C4—281no—
617C2×C4—281no—
618C2×C4—281no—
619C2×C4—281no—
620C2×C4#650912yes—
621C2×C4C2×D4×D711yescharacteristic
622C2×C4#654811yescharacteristic
623C2×C4C2×Q8×D711yescharacteristic
624C2×C4C2×#116111yescharacteristic
625C2×C4D7×#4111yescharacteristic
626C2×C4#655311yescharacteristic
627C8—142no—
628C8—281no—
629C8—281no—
630C8—281no—
631C7#2595711yescharacteristic, complemented, Sylow
632C22—22no—
633C22—22no—
634C22—22no—
635C22—22no—
636C22—22no—
637C22—22no—
638C22#8983514yescomplemented
639C22D4×C22⋊D711yescharacteristic
640C22C2×#650911yescharacteristic
641C22C2×#651311yescharacteristic
642C22C2×#651411yescharacteristic
643C22#9052011yescharacteristic
644C22#9052711yescharacteristic
645C22#9052911yescharacteristic
646C4—22no—
647C4—41no—
648C4—42no—
649C4—41no—
650C4—41no—
651C4—41no—
652C4—81no—
653C4—81no—
654C4—81no—
655C4—144no—
656C4—282no—
657C4—561no—
658C4—561no—
659C4D7×C22⋊Q811yescharacteristic
660C4#9032111yescharacteristic
661C4#9032511yescharacteristic
662C4#9032811yescharacteristic
663C2—24no—
664C2#1174026111yescharacteristic
665C2C2×#8983511yescharacteristic
666C2#1174092511yescharacteristic
667C2#1174095411yescharacteristic
668C2#1174099111yescharacteristic
669C2#1174104711yescharacteristic
670C2#1174106511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#2442C2
socleC23×C7C22×D4
prime radicalC24×C7C2×D4
hypercenter#2442D7
hypocenterC7#25957
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2442

Complements

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

NH ≅ G/Ntype
2. C2×#89835C2semidirect
5. #11729498C2semidirect
7. #11727389C2semidirect
8. #11727388C2semidirect
9. #11727386C2semidirect
10. #11727385C2semidirect
11. #11727315C2semidirect
12. #11727313C2semidirect
438. C22×C7#479semidirect
631. C7#25957semidirect
638. C22#89835semidirect

Maximal subgroups

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

subgroupindex
C7×#24422
C2×#898352
C22×#63842
C22⋊C4×Dic142
#117294982
#117293092
#117273892
#117273882
#117273862
#117273852
#117273152
#117273132
#117272792
#117272652
#117272032
#259577