Finite Group Databasebeta

← #13410925

Subgroups — #13410925

Order 1792.

990 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2440C211yesmaximal, characteristic
2C2×#89866C211yesmaximal, characteristic, complemented
3C22×#6384C211yesmaximal, characteristic
4C22⋊C4×Dic14C211yesmaximal, characteristic
5#11729548C211yesmaximal, characteristic, complemented
6#11729310C211yesmaximal, characteristic
7#11727388C211yesmaximal, characteristic, complemented
8#11727388C211yesmaximal, characteristic, complemented
9#11727386C211yesmaximal, characteristic, complemented
10#11727386C211yesmaximal, characteristic, complemented
11#11727345C211yesmaximal, characteristic, complemented
12#11727342C211yesmaximal, characteristic, complemented
13#11727279C211yesmaximal, characteristic
14#11727265C211yesmaximal, characteristic
15#11727203C211yesmaximal, characteristic
16C23×Dic14C2211yescharacteristic
17C2×C7×C4⋊Q8C2211yescharacteristic
18C2×C7×C22⋊Q8—21no—
19C22×C7×C4⋊C4C2211yescharacteristic
20C23×C7⋊C8C2211yescharacteristic
21C2×C4×Dic14C2211yescharacteristic
22C2×C4×Dic14C2211yescharacteristic
23C7×#398—21no—
24C7×#395—21no—
25C7×#389C2211yescharacteristic
26C7×#389C2211yescharacteristic
27C7×#385—21no—
28C7×#384C2211yescharacteristic
29C7×#380C2211yescharacteristic
30C4×C7×C22⋊C4C2211yescharacteristic
31#89984C2211yescharacteristic
32C2×C22⋊(C7⋊C8)C2211yescharacteristic
33#89866—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×#6382C2211yescharacteristic
46C2×#6382C2211yescharacteristic
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#25955—71nomaximal, Sylow
62C22×C7×Q8—21no—
63C23×C4×C7—21no—
64C23×C4×C7C2311yescharacteristic
65C23×Dic7—21no—
66C22×Dic14—22no—
67C22×Dic14—22no—
68C22×Dic14—21no—
69C22×Dic14—22no—
70C22×Dic14—41no—
71C22×Dic14—41no—
72C22×Dic14Q814yescomplemented
73C22×Dic14C2311yescharacteristic
74C22×Dic14C2311yescharacteristic
75C7×C4⋊Q8—22no—
76C7×C4⋊Q8—41no—
77C7×C22⋊Q8—44no—
78C2×C7×C4⋊C4—22no—
79C2×C7×C4⋊C4—21no—
80C2×C7×C4⋊C4—21no—
81C2×C7×C4⋊C4—21no—
82C2×C7×C4⋊C4—21no—
83C2×C7×C4⋊C4—21no—
84C2×C7×C4⋊C4—21no—
85C2×C7×C4⋊C4—41no—
86C2×C7×C4⋊C4C2311yescharacteristic
87C2×C7×C4⋊C4C2311yescharacteristic
88C2×C7×C4⋊C4C2311yescharacteristic
89C2×C7×C4⋊C4C2311yescharacteristic
90C2×C7×C22⋊C4—21no—
91C2×C7×C22⋊C4D411yescharacteristic
92C2×C7×C22⋊C4D411yescharacteristic
93C2×C42×C7C2311yescharacteristic
94C2×C42×C7C2311yescharacteristic
95C2×C22⋊Dic7—21no—
96C2×C4⋊Dic7C2311yescharacteristic
97C2×C4⋊Dic7C2311yescharacteristic
98C2×Dic7⋊C4—21no—
99C2×Dic7⋊C4—21no—
100C2×C4×Dic7—21no—
101C2×C4×Dic7—21no—
102C22×C7⋊C8—22no—
103C22×C7⋊C8—22no—
104C22×C7⋊C8—41no—
105C22×C7⋊C8C2311yescharacteristic
106C22×C7⋊C8C2311yescharacteristic
107C22×C7⋊C8C2311yescharacteristic
108C22×C7⋊C8C2311yescharacteristic
109C4×Dic14—22no—
110C4×Dic14—22no—
111C4×Dic14—41no—
112C4×Dic14—41no—
113C7×#95—21no—
114C7×#95—21no—
115C7×#95—21no—
116C7×#95D411yescharacteristic
117C7×#95D411yescharacteristic
118C7⋊#95—21no—
119C7⋊#95—21no—
120C7⋊#95—21no—
121C7⋊#95—21no—
122C7⋊#95—21no—
123C22⋊(C7⋊C8)—22no—
124#6384—22no—
125#6384—22no—
126#6384—22no—
127#6384—22no—
128#6384—22no—
129#6384—41no—
130#6384—41no—
131#6384D414yes—
132#6382—22no—
133#6382—22no—
134#6382—41no—
135#6382—41no—
136C4⋊(C7⋊C8)—22no—
137C4⋊(C7⋊C8)—41no—
138C2×#475—71no—
139C22×#103—71no—
140#2440D711yescharacteristic
141Q8×C22⋊C4—71no—
142#2096—71no—
143#2066—71no—
144#2066—71no—
145#2024—71no—
146#1865—71no—
147#1865—71no—
148#1853—71no—
149#1850—71no—
150#1807—71no—
151#1803—71no—
152#1787—71no—
153C24×C7C2×D411yescharacteristic
154C2×C7×Q8—42no—
155C2×C7×Q8—42no—
156C2×C7×Q8—81no—
157C22×C4×C7—21no—
158C22×C4×C7—21no—
159C22×C4×C7—21no—
160C22×C4×C7—21no—
161C22×C4×C7—21no—
162C22×C4×C7—21no—
163C22×C4×C7—21no—
164C22×C4×C7—21no—
165C22×C4×C7—42no—
166C22×C4×C7—42no—
167C22×C4×C7—81no—
168C22×C4×C7C2×D411yescharacteristic
169C22×C4×C7C2×D411yescharacteristic
170C22×C4×C7C2×D411yescharacteristic
171C22×C4×C7C2×D411yescharacteristic
172C22×C4×C7C2×D411yescharacteristic
173C22×C4×C7C2×Q812yes—
174C22×C4×C7#4112yes—
175C22×C4×C7C2411yescharacteristic
176C22×Dic7—21no—
177C22×Dic7—24no—
178C22×Dic7—21no—
179C22×Dic7—21no—
180C22×Dic7—21no—
181C22×Dic7—42no—
182C22×Dic7—42no—
183C2×Dic14—21no—
184C2×Dic14—21no—
185C2×Dic14—21no—
186C2×Dic14—21no—
187C2×Dic14—21no—
188C2×Dic14—21no—
189C2×Dic14—21no—
190C2×Dic14—28no—
191C2×Dic14—24no—
192C2×Dic14—21no—
193C2×Dic14—44no—
194C2×Dic14—44no—
195C2×Dic14—44no—
196C2×Dic14—42no—
197C2×Dic14—42no—
198C2×Dic14C2×D412yes—
199C2×Dic14C2×Q812yes—
200C2×Dic14C2×Q812yes—
201C2×Dic14#4112yes—
202C7×C4⋊C4—21no—
203C7×C4⋊C4—21no—
204C7×C4⋊C4—21no—
205C7×C4⋊C4—21no—
206C7×C4⋊C4—21no—
207C7×C4⋊C4—21no—
208C7×C4⋊C4—21no—
209C7×C4⋊C4—42no—
210C7×C4⋊C4—42no—
211C7×C4⋊C4—42no—
212C7×C4⋊C4—42no—
213C7×C4⋊C4—42no—
214C7×C4⋊C4—81no—
215C7×C4⋊C4C2×D412yes—
216C7×C22⋊C4—22no—
217C7×C22⋊C4—42no—
218C7×C22⋊C4SD1614yes—
219C42×C7—21no—
220C42×C7—21no—
221C42×C7—21no—
222C42×C7—21no—
223C22⋊Dic7—42no—
224C4⋊Dic7—21no—
225C4⋊Dic7—21no—
226C4⋊Dic7—21no—
227C4⋊Dic7—21no—
228Dic7⋊C4—42no—
229Dic7⋊C4—42no—
230C4×Dic7—42no—
231C4×Dic7—42no—
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—21no—
239C2×C7⋊C8—21no—
240C2×C7⋊C8—21no—
241C2×C7⋊C8—21no—
242C2×C7⋊C8—21no—
243C2×C7⋊C8—44no—
244C2×C7⋊C8C2×D412yes—
245C23×Q8—71no—
246C23×C8—71no—
247C2×C4⋊Q8C2×D711yescharacteristic
248C2×C22⋊Q8—21no—
249C2×C4×Q8—71no—
250C2×C4×Q8—71no—
251C22×C4⋊C4C2×D711yescharacteristic
252#475—144no—
253C2×#106—71no—
254C2×#106—71no—
255C2×#106—71no—
256C2×#106—71no—
257C2×C4⋊C8—71no—
258C2×C4⋊C8—71no—
259C2×#103—71no—
260C2×#103—71no—
261C2×#103—71no—
262C2×#103—71no—
263C2×#103—71no—
264C2×#103—71no—
265C2×#103—71no—
266C2×#103—71no—
267C2×#103—144no—
268C2×C22⋊C8—71no—
269#398—21no—
270#395—21no—
271#391—71no—
272#391—71no—
273#391—141no—
274#391—141no—
275#389C2×D711yescharacteristic
276#389C2×D711yescharacteristic
277#385—21no—
278#384C2×D711yescharacteristic
279#380—71no—
280#380—141no—
281#380C2×D711yescharacteristic
282C4×C22⋊C4—141no—
283C4×C22⋊C4C2×D711yescharacteristic
284#340—71no—
285#340—71no—
286#340—71no—
287#340—71no—
288#336—71no—
289#336—71no—
290C23×C7—21no—
291C23×C7—21no—
292C23×C7—21no—
293C23×C7—21no—
294C23×C7C22⋊Q812yes—
295C23×C7C2×SD1612yes—
296C23×C7#13712yes—
297C23×C7C22×D411yescharacteristic
298C7×Q8—84no—
299C2×C4×C7—21no—
300C2×C4×C7—21no—
301C2×C4×C7—21no—
302C2×C4×C7—21no—
303C2×C4×C7—21no—
304C2×C4×C7—21no—
305C2×C4×C7—22no—
306C2×C4×C7—22no—
307C2×C4×C7—22no—
308C2×C4×C7—22no—
309C2×C4×C7—21no—
310C2×C4×C7—22no—
311C2×C4×C7—21no—
312C2×C4×C7—21no—
313C2×C4×C7—21no—
314C2×C4×C7—21no—
315C2×C4×C7—21no—
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—41no—
324C2×C4×C7—41no—
325C2×C4×C7—41no—
326C2×C4×C7—41no—
327C2×C4×C7—84no—
328C2×C4×C7C22⋊Q812yes—
329C2×C4×C7C2×SD1612yes—
330C2×C4×C7C2×SD1612yes—
331C2×C4×C7C22×D411yescharacteristic
332C2×C4×C7C22×Q811yescharacteristic
333C2×C4×C7C2×#4111yescharacteristic
334C2×C4×C7C2×#4111yescharacteristic
335C2×C4×C7H(5,𝔽2)11yescharacteristic
336C2×C4×C7H(5,𝔽2)11yescharacteristic
337C2×Dic7—22no—
338C2×Dic7—22no—
339C2×Dic7—22no—
340C2×Dic7—22no—
341C2×Dic7—48no—
342C2×Dic7—41no—
343C2×Dic7—41no—
344C2×Dic7—41no—
345C2×Dic7—41no—
346C2×Dic7—41no—
347C2×Dic7—41no—
348C2×Dic7—41no—
349C2×Dic7—41no—
350Dic14—22no—
351Dic14—22no—
352Dic14—22no—
353Dic14—22no—
354Dic14—22no—
355Dic14—48no—
356Dic14—41no—
357Dic14—41no—
358Dic14C22⋊Q814yescomplemented
359C7⋊C8—22no—
360C7⋊C8—41no—
361C7⋊C8—41no—
362C7⋊C8—41no—
363C22×Q8—21no—
364C22×Q8—74no—
365C22×Q8—71no—
366C22×Q8—71no—
367C22×Q8—142no—
368C22×Q8—142no—
369C22×Q8—142no—
370C22×Q8—141no—
371C22×Q8—281no—
372C22×Q8—281no—
373C23×C4—21no—
374C23×C4—141no—
375C23×C4C22×D711yescharacteristic
376C22×C8—71no—
377C22×C8—71no—
378C22×C8—71no—
379C22×C8—71no—
380C22×C8—142no—
381C22×C8—142no—
382C22×C8—281no—
383C4⋊Q8—22no—
384C4⋊Q8—41no—
385C22⋊Q8—44no—
386C4×Q8—142no—
387C4×Q8—142no—
388C4×Q8—281no—
389C4×Q8—281no—
390C2×C4⋊C4—22no—
391C2×C4⋊C4—21no—
392C2×C4⋊C4—21no—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—21no—
396C2×C4⋊C4—21no—
397C2×C4⋊C4—41no—
398C2×C4⋊C4—71no—
399C2×C4⋊C4—71no—
400C2×C4⋊C4—141no—
401C2×C4⋊C4—141no—
402C2×C4⋊C4C22×D711yescharacteristic
403C2×C4⋊C4C22×D711yescharacteristic
404C2×C4⋊C4C22×D711yescharacteristic
405C2×C4⋊C4C22×D711yescharacteristic
406C2×C22⋊C4—21no—
407C2×C22⋊C4—141no—
408C2×C22⋊C4C22⋊D711yescharacteristic
409C2×C22⋊C4C22⋊D711yescharacteristic
410C2×C42—141no—
411C2×C42—141no—
412C2×C42C22×D711yescharacteristic
413C2×C42C22×D711yescharacteristic
414#106—142no—
415#106—142no—
416#106—281no—
417#106—281no—
418C4⋊C8—142no—
419C4⋊C8—281no—
420#103—74no—
421#103—142no—
422#103—142no—
423#103—142no—
424#103—142no—
425#103—142no—
426#103—281no—
427#103—281no—
428C22⋊C8—142no—
429#95—21no—
430#95—21no—
431#95—21no—
432#95—141no—
433#95—141no—
434#95—141no—
435#95—141no—
436#95—141no—
437#95C22⋊D711yescharacteristic
438#95C22⋊D711yescharacteristic
439C22×C7—22no—
440C22×C7—22no—
441C22×C7—22no—
442C22×C7—22no—
443C22×C7—22no—
444C22×C7—22no—
445C22×C7#47514yescomplemented
446C22×C7C2×C22⋊Q811yescharacteristic
447C22×C7D4211yescharacteristic
448C22×C7#54611yescharacteristic
449C22×C7C22×SD1611yescharacteristic
450C22×C7C2×#13711yescharacteristic
451C22×C7#57711yescharacteristic
452C22×C7#57811yescharacteristic
453C4×C7—22no—
454C4×C7—22no—
455C4×C7—22no—
456C4×C7—41no—
457C4×C7—42no—
458C4×C7—41no—
459C4×C7—81no—
460C4×C7—81no—
461C4×C7—81no—
462C4×C7C2×C22⋊Q811yescharacteristic
463C4×C7#54311yescharacteristic
464C4×C7#54611yescharacteristic
465C4×C7#55111yescharacteristic
466Dic7—24no—
467Dic7—42no—
468Dic7—81no—
469Dic7—81no—
470C24C2×C22⋊D711yescharacteristic
471C2×Q8—42no—
472C2×Q8—42no—
473C2×Q8—72no—
474C2×Q8—72no—
475C2×Q8—72no—
476C2×Q8—72no—
477C2×Q8—81no—
478C2×Q8—148no—
479C2×Q8—144no—
480C2×Q8—141no—
481C2×Q8—141no—
482C2×Q8—141no—
483C2×Q8—141no—
484C2×Q8—141no—
485C2×Q8—141no—
486C2×Q8—141no—
487C2×Q8—141no—
488C2×Q8—282no—
489C2×Q8—282no—
490C2×Q8—284no—
491C2×Q8—284no—
492C2×Q8—284no—
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—21no—
496C22×C4—21no—
497C22×C4—21no—
498C22×C4—21no—
499C22×C4—21no—
500C22×C4—21no—
501C22×C4—42no—
502C22×C4—42no—
503C22×C4—81no—
504C22×C4—141no—
505C22×C4—144no—
506C22×C4—141no—
507C22×C4—141no—
508C22×C4—141no—
509C22×C4—282no—
510C22×C4—282no—
511C22×C4Q8×D712yes—
512C22×C4#116112yes—
513C22×C4C2×C22⋊D711yescharacteristic
514C22×C4C2×C22⋊D711yescharacteristic
515C22×C4C2×C22⋊D711yescharacteristic
516C22×C4C2×C22⋊D711yescharacteristic
517C22×C4C2×C22⋊D711yescharacteristic
518C22×C4C23×D711yescharacteristic
519C2×C8—72no—
520C2×C8—141no—
521C2×C8—141no—
522C2×C8—141no—
523C2×C8—141no—
524C2×C8—141no—
525C2×C8—141no—
526C2×C8—141no—
527C2×C8—141no—
528C2×C8—141no—
529C2×C8—141no—
530C2×C8—141no—
531C2×C8—284no—
532C4⋊C4—21no—
533C4⋊C4—21no—
534C4⋊C4—21no—
535C4⋊C4—21no—
536C4⋊C4—21no—
537C4⋊C4—21no—
538C4⋊C4—21no—
539C4⋊C4—42no—
540C4⋊C4—42no—
541C4⋊C4—42no—
542C4⋊C4—42no—
543C4⋊C4—42no—
544C4⋊C4—81no—
545C4⋊C4—141no—
546C4⋊C4—141no—
547C4⋊C4—141no—
548C4⋊C4—141no—
549C4⋊C4—282no—
550C4⋊C4—282no—
551C4⋊C4D4×D712yes—
552C22⋊C4—22no—
553C22⋊C4—42no—
554C22⋊C4—282no—
555C22⋊C4#114214yes—
556C42—21no—
557C42—21no—
558C42—21no—
559C42—21no—
560C42—282no—
561C42—282no—
562C2×C7—24no—
563C2×C7#243711yescharacteristic
564C2×C7C2×#47511yescharacteristic
565C2×C7#308211yescharacteristic
566C2×C7#327011yescharacteristic
567C2×C7D4×SD1611yescharacteristic
568C2×C7#329811yescharacteristic
569C2×C7#331611yescharacteristic
570C23—21no—
571C23—21no—
572C23—21no—
573C23—21no—
574C23C2×#114212yes—
575C23#650512yes—
576C23#650912yes—
577C23C22×C22⋊D711yescharacteristic
578Q8—74no—
579Q8—84no—
580Q8—142no—
581Q8—142no—
582Q8—142no—
583Q8—142no—
584Q8—142no—
585Q8—288no—
586Q8—281no—
587Q8—281no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—21no—
599C2×C4—22no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—84no—
617C2×C4—142no—
618C2×C4—142no—
619C2×C4—142no—
620C2×C4—142no—
621C2×C4—288no—
622C2×C4—281no—
623C2×C4—281no—
624C2×C4—281no—
625C2×C4—281no—
626C2×C4—281no—
627C2×C4—281no—
628C2×C4—281no—
629C2×C4—281no—
630C2×C4C2×#114212yes—
631C2×C4C2×#114212yes—
632C2×C4#650912yes—
633C2×C4C2×D4×D711yescharacteristic
634C2×C4#654811yescharacteristic
635C2×C4C2×Q8×D711yescharacteristic
636C2×C4C2×#116111yescharacteristic
637C2×C4D7×#4111yescharacteristic
638C2×C4#655311yescharacteristic
639C8—142no—
640C8—281no—
641C8—281no—
642C8—281no—
643C7#2595511yescharacteristic, complemented, Sylow
644C22—22no—
645C22—22no—
646C22—22no—
647C22—22no—
648C22—22no—
649C22—22no—
650C22#8986614yescomplemented
651C22C22×#114211yescharacteristic
652C22D4×C22⋊D711yescharacteristic
653C22C2×#650511yescharacteristic
654C22C2×#650911yescharacteristic
655C22#9052011yescharacteristic
656C22#9052711yescharacteristic
657C22#9052911yescharacteristic
658C4—22no—
659C4—22no—
660C4—22no—
661C4—42no—
662C4—41no—
663C4—41no—
664C4—81no—
665C4—81no—
666C4—81no—
667C4—144no—
668C4—282no—
669C4—561no—
670C4—561no—
671C4D7×C22⋊Q811yescharacteristic
672C4#9032111yescharacteristic
673C4#9032511yescharacteristic
674C4#9032811yescharacteristic
675C2—24no—
676C2#1174026111yescharacteristic
677C2C2×#8986611yescharacteristic
678C2#1174092511yescharacteristic
679C2D4×#114211yescharacteristic
680C2#1174097611yescharacteristic
681C2#1174101011yescharacteristic
682C2#1174108611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#2440C2
socleC23×C7C22×D4
prime radicalC24×C7C2×D4
hypercenter#2440D7
hypocenterC7#25955
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2440

Complements

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

NH ≅ G/Ntype
2. C2×#89866C2semidirect
5. #11729548C2semidirect
7. #11727388C2semidirect
8. #11727388C2semidirect
9. #11727386C2semidirect
10. #11727386C2semidirect
11. #11727345C2semidirect
12. #11727342C2semidirect
72. C22×Dic14Q8semidirect
358. Dic14C22⋊Q8semidirect
445. C22×C7#475semidirect
643. C7#25955semidirect
650. C22#89866semidirect

Maximal subgroups

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

subgroupindex
C7×#24402
C2×#898662
C22×#63842
C22⋊C4×Dic142
#117295482
#117293102
#117273882
#117273862
#117273452
#117273422
#117272792
#117272652
#117272032
#259557