Finite Group Databasebeta

← #13356510

Subgroups — #13356510

Order 1792.

1122 conjugacy classes of subgroups.

The subgroup lattice has 688 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89574C211yesmaximal, characteristic
2C2×#89574C211yesmaximal, characteristic
3C2×#89479C211yesmaximal, characteristic
4D7×#391C211yesmaximal, characteristic
5#11733408C211yesmaximal, characteristic
6C7×#1935C211yesmaximal, characteristic, complemented
7#11728008C211yesmaximal, characteristic, complemented
8#11728007C211yesmaximal, characteristic, complemented
9#11727571C211yesmaximal, characteristic, complemented
10#11727429C211yesmaximal, characteristic, complemented
11#11727370C211yesmaximal, characteristic, complemented
12#11726621C211yesmaximal, characteristic
13#11726621C211yesmaximal, characteristic
14#11726605C211yesmaximal, characteristic, complemented
15#11726604C211yesmaximal, characteristic, complemented
16C22×Q8×D7C2211yescharacteristic
17C22×#1131C2211yescharacteristic
18C22×#1131C2211yescharacteristic
19C2×#6455C2211yescharacteristic
20C2×#6455C2211yescharacteristic
21C2×D7×C4⋊C4C2211yescharacteristic
22C2×C42×D7C2211yescharacteristic
23C2×C7×#103C2211yescharacteristic
24C2×C7×#103C2211yescharacteristic
25C2×C7×#97C2211yescharacteristic
26C7×#391C2211yescharacteristic
27C7×#389C2211yescharacteristic
28#90002C2211yescharacteristic
29C2×#6409C2211yescharacteristic
30C2×#6409C2211yescharacteristic
31C2×#6390C2211yescharacteristic
32C2×#6390C2211yescharacteristic
33C2×C8×Dic7C2211yescharacteristic
34C2×C8×Dic7C2211yescharacteristic
35#89761—21no—
36#89757C2211yescharacteristic
37C4⋊C4×Dic7—21no—
38#89746C2211yescharacteristic
39#89745C2211yescharacteristic
40C2×#6384C2211yescharacteristic
41C2×#6384C2211yescharacteristic
42#89701C2211yescharacteristic
43#89700C2211yescharacteristic
44C2×#6377C2211yescharacteristic
45#89574—24no—
46#89574C418yescomplemented
47#89479—24no—
48#89440—21no—
49D7×#95—21no—
50D7×#95—21no—
51#89421—21no—
52#89418—21no—
53#89418—21no—
54C7×#340C2211yescharacteristic
55C7×#340C2211yescharacteristic
56#89346C2211yescharacteristic
57#89346C2211yescharacteristic
58#89324C2211yescharacteristic
59#89324C2211yescharacteristic
60#89324C2211yescharacteristic
61#89324C2211yescharacteristic
62#21094—71nomaximal, Sylow
63C22×C7×Q8C2311yescharacteristic
64C2×Q8×D7—22no—
65C2×Q8×D7—22no—
66C2×Q8×D7—22no—
67C2×Q8×D7—22no—
68C2×Q8×D7—22no—
69C2×Q8×D7C2×C414yes—
70C22×C4×D7—21no—
71C22×C4×D7—21no—
72C22×C4×D7D411yescharacteristic
73C22×C4×D7D411yescharacteristic
74C22×C4×D7C2311yescharacteristic
75C22×Dic14—21no—
76C22×Dic14C2311yescharacteristic
77C22×C7×C8C2311yescharacteristic
78C22×C7×C8C2311yescharacteristic
79C2×C7×C4⋊C4—21no—
80C2×C7×C4⋊C4C2311yescharacteristic
81C2×C7×C4⋊C4C2311yescharacteristic
82C2×C7×C4⋊C4C2311yescharacteristic
83C2×C42×C7C2311yescharacteristic
84C2×(C2×D7)⋊C4—21no—
85C2×(C2×D7)⋊C4—21no—
86C2×C4⋊Dic7—21no—
87C2×C4⋊Dic7C2311yescharacteristic
88C2×C4⋊Dic7C2311yescharacteristic
89C2×C4⋊Dic7C2311yescharacteristic
90C2×Dic7⋊C4—21no—
91C2×Dic7⋊C4—21no—
92C2×Dic7⋊C4—21no—
93C2×C4×Dic7—21no—
94C2×C4×Dic7—21no—
95C2×C4×Dic7D411yescharacteristic
96C2×C4×Dic7D411yescharacteristic
97C2×C4×Dic7C2311yescharacteristic
98C22×C7⋊C8C2311yescharacteristic
99C22×C7⋊C8C2311yescharacteristic
100C2×#1131—22no—
101C2×#1131—22no—
102C2×#1131—22no—
103C2×#1131—22no—
104C2×#1131C2×C414yes—
105C2×#1131D414yes—
106#6455—22no—
107#6455—22no—
108#6455—22no—
109#6455C2×C414yes—
110D7×C4⋊C4—22no—
111D7×C4⋊C4—22no—
112C42×D7—22no—
113C42×D7—22no—
114C7×#103—22no—
115C7×#103C2×C414yescomplemented
116C7×#97—22no—
117C7×#95—21no—
118C7×#95—21no—
119C7×#95—21no—
120#6409—22no—
121#6409C2×C414yescomplemented
122C7⋊#95—21no—
123C7⋊#95—21no—
124C7⋊#95—21no—
125C7⋊#95—21no—
126C7⋊#95—21no—
127C7⋊#95—21no—
128C7⋊#95—21no—
129C7⋊#95—21no—
130C7⋊#95—21no—
131#6390—22no—
132#6390C2×C414yescomplemented
133C8×Dic7—22no—
134C8×Dic7—22no—
135#6384—22no—
136#6384C2×C414yescomplemented
137#6377—22no—
138C2×#419—71no—
139C2×#419—71no—
140C2×C4×M16—71no—
141#2296—71no—
142C2×#391—71no—
143#1935—71no—
144#1935—71no—
145#1935—71no—
146#1935D711yescharacteristic, complemented
147#1933—71no—
148#1933—71no—
149#1933—71no—
150#1933—71no—
151#1736—71no—
152#1736—71no—
153C23×D7C2×D411yescharacteristic
154C2×C7×Q8—21no—
155C2×C7×Q8—21no—
156C2×C7×Q8—21no—
157C2×C7×Q8—21no—
158C2×C7×Q8—21no—
159C2×C7×Q8C22×C412yes—
160C22×C4×C7—21no—
161C22×C4×C7—21no—
162C22×C4×C7—21no—
163C22×C4×C7—21no—
164C22×C4×C7C2×D411yescharacteristic
165C22×C4×C7C2×D411yescharacteristic
166C22×C4×C7C2411yescharacteristic
167C22×Dic7—21no—
168C22×Dic7—21no—
169C22×Dic7—21no—
170C22×Dic7—21no—
171C22×Dic7C2×D411yescharacteristic
172C22×Dic7C2×D411yescharacteristic
173C22×Dic7C2×D411yescharacteristic
174Q8×D7—28no—
175Q8×D7—28no—
176Q8×D7—44no—
177Q8×D7—44no—
178C2×C4×D7—24no—
179C2×C4×D7—22no—
180C2×C4×D7—22no—
181C2×C4×D7—22no—
182C2×C4×D7—22no—
183C2×C4×D7—41no—
184C2×C4×D7—41no—
185C2×C4×D7—42no—
186C2×C4×D7—42no—
187C2×C4×D7—42no—
188C2×C4×D7—42no—
189C2×C4×D7—42no—
190C2×C4×D7C22×C412yes—
191C2×C4×D7C2×D412yes—
192C2×C4×D7#4112yes—
193C2×C4×D7#4112yes—
194C2×C4×D7#4112yes—
195C2×C4×D7#4112yes—
196C2×Dic14—21no—
197C2×Dic14—21no—
198C2×Dic14—21no—
199C2×Dic14—21no—
200C2×Dic14—21no—
201C2×Dic14—24no—
202C2×Dic14—42no—
203C2×Dic14—42no—
204C2×Dic14C22×C412yes—
205C2×C7×C8—21no—
206C2×C7×C8—21no—
207C2×C7×C8—21no—
208C2×C7×C8—21no—
209C2×C7×C8C22×C412yes—
210C2×C7×C8C2×D412yes—
211C7×C4⋊C4—21no—
212C7×C4⋊C4—21no—
213C7×C4⋊C4—21no—
214C7×C4⋊C4—21no—
215C7×C4⋊C4—21no—
216C7×C4⋊C4—42no—
217C7×C4⋊C4C22×C412yes—
218C42×C7—21no—
219C42×C7—21no—
220(C2×D7)⋊C4—24no—
221(C2×D7)⋊C4—42no—
222C4⋊Dic7—21no—
223C4⋊Dic7—21no—
224C4⋊Dic7—21no—
225C4⋊Dic7—21no—
226C4⋊Dic7—21no—
227C4⋊Dic7—42no—
228C4⋊Dic7C22×C412yes—
229Dic7⋊C4—42no—
230Dic7⋊C4—42no—
231Dic7⋊C4—42no—
232C4×Dic7—21no—
233C4×Dic7—21no—
234C4×Dic7—24no—
235C4×Dic7—22no—
236C4×Dic7—22no—
237C4×Dic7—42no—
238C2×C7⋊C8—21no—
239C2×C7⋊C8—21no—
240C2×C7⋊C8—21no—
241C2×C7⋊C8—21no—
242C2×C7⋊C8C22×C412yes—
243C2×C7⋊C8C2×D412yes—
244#1131—28no—
245#1131—28no—
246C23×Q8—71no—
247C22×M16—71no—
248C22×M16—71no—
249C2×#117—71no—
250C2×#117—71no—
251C22×C4⋊C4—71no—
252C22×C42—71no—
253#419—78no—
254#419—144no—
255C2×#103—71no—
256C2×#103—71no—
257C2×#103—71no—
258C2×#103—71no—
259C2×#103—71no—
260C2×#103—71no—
261C2×#103C2×D711yescharacteristic
262C2×#103C2×D711yescharacteristic
263C4×M16—144no—
264C2×#97—71no—
265C2×#97C2×D711yescharacteristic
266C2×C4×C8—71no—
267C2×C4×C8—71no—
268#391—71no—
269#391—71no—
270#391—71no—
271#391—141no—
272#391—141no—
273#391C2×D711yescharacteristic
274#389—71no—
275#389—71no—
276#389—71no—
277#389C2×D711yescharacteristic
278#385—141no—
279#385—141no—
280C4×C4⋊C4—141no—
281C4×C4⋊C4—141no—
282C2×#95—141no—
283C2×#95—141no—
284#340—71no—
285#340—71no—
286#340—71no—
287#340—71no—
288#340—71no—
289#340—71no—
290#340C2×D711yescharacteristic
291#340C2×D711yescharacteristic
292C23×C7C22×D411yescharacteristic
293C22×D7—21no—
294C22×D7—21no—
295C22×D7—21no—
296C22×D7—21no—
297C22×D7C4×D412yes—
298C22×D7#12112yes—
299C22×D7#12412yes—
300C7×Q8—22no—
301C7×Q8—22no—
302C7×Q8—41no—
303C7×Q8—41no—
304C2×C4×C7—22no—
305C2×C4×C7—22no—
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×C7C23×C411yescharacteristic
323C2×C4×C7C22×D411yescharacteristic
324C2×C4×C7C2×#4111yescharacteristic
325C2×C4×C7C2×#4111yescharacteristic
326C2×C4×C7C2×#4111yescharacteristic
327C2×C4×C7C2×#4111yescharacteristic
328C2×Dic7—21no—
329C2×Dic7—21no—
330C2×Dic7—21no—
331C2×Dic7—21no—
332C2×Dic7—22no—
333C2×Dic7—22no—
334C2×Dic7—41no—
335C2×Dic7—41no—
336C2×Dic7—41no—
337C2×Dic7—41no—
338C2×Dic7—41no—
339C2×Dic7—41no—
340C2×Dic7—41no—
341C2×Dic7—41no—
342C2×Dic7—41no—
343C2×Dic7—41no—
344C2×Dic7C4×D412yes—
345C2×Dic7#12112yes—
346C2×Dic7#12412yes—
347C2×Dic7#13712yes—
348C2×Dic7#13712yes—
349C4×D7—44no—
350C4×D7—44no—
351C4×D7—48no—
352C4×D7—84no—
353C4×D7#11714yes—
354C4×D7C4×D414yes—
355C4×D7#12314yes—
356C4×D7#12614yes—
357Dic14—22no—
358Dic14—22no—
359Dic14—41no—
360Dic14—41no—
361Dic14—48no—
362C7×C8—22no—
363C7×C8—22no—
364C7⋊C8—22no—
365C7⋊C8—22no—
366C22×Q8—74no—
367C22×Q8—71no—
368C22×Q8—141no—
369C22×Q8—142no—
370C22×Q8—142no—
371C22×Q8—142no—
372C22×Q8—142no—
373C22×Q8—142no—
374C22×Q8C22×D711yescharacteristic
375C23×C4—71no—
376C23×C4—71no—
377C23×C4—71no—
378C23×C4—141no—
379C23×C4—141no—
380C2×M16—74no—
381C2×M16—74no—
382C2×M16—142no—
383C2×M16—142no—
384C2×M16—142no—
385C2×M16—142no—
386C22×C8—71no—
387C22×C8—71no—
388C22×C8C22×D711yescharacteristic
389C22×C8C22×D711yescharacteristic
390#117—74no—
391#117—142no—
392#117—142no—
393#117—142no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—71no—
396C2×C4⋊C4—71no—
397C2×C4⋊C4—71no—
398C2×C4⋊C4—142no—
399C2×C4⋊C4—142no—
400C2×C4⋊C4—141no—
401C2×C4⋊C4—141no—
402C2×C4⋊C4—141no—
403C2×C4⋊C4—141no—
404C2×C4⋊C4C22×D711yescharacteristic
405C2×C4⋊C4C22×D711yescharacteristic
406C2×C4⋊C4C22×D711yescharacteristic
407C2×C22⋊C4—141no—
408C2×C22⋊C4—141no—
409C2×C42—71no—
410C2×C42—71no—
411C2×C42—71no—
412C2×C42—142no—
413C2×C42—142no—
414C2×C42—141no—
415C2×C42—141no—
416C2×C42C22×D711yescharacteristic
417#103—22no—
418#103—74no—
419#103—74no—
420#103—74no—
421#103—142no—
422#103—142no—
423#103—142no—
424#103C4×D714yescomplemented
425#97—22no—
426#97—142no—
427C4×C8—142no—
428C4×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#95—141no—
438#95—141no—
439#95—141no—
440#95—141no—
441C22×C7C2×C4×D411yescharacteristic
442C22×C7C2×#12111yescharacteristic
443C22×C7C2×#12411yescharacteristic
444C22×C7C2×#13711yescharacteristic
445C22×C7C2×#13711yescharacteristic
446C22×C7#57511yescharacteristic
447C22×C7#57511yescharacteristic
448C2×D7—22no—
449C2×D7—22no—
450C2×D7—22no—
451C2×D7—22no—
452C2×D7—22no—
453C2×D7—22no—
454C2×D7#38814yes—
455C4×C7—42no—
456C4×C7—41no—
457C4×C7—41no—
458C4×C7—42no—
459C4×C7—81no—
460C4×C7—81no—
461C4×C7C2×#11711yescharacteristic
462C4×C7C2×C4×D411yescharacteristic
463C4×C7C2×#12311yescharacteristic
464C4×C7C2×#12611yescharacteristic
465Dic7—22no—
466Dic7—22no—
467Dic7—42no—
468Dic7—42no—
469Dic7—81no—
470Dic7—81no—
471Dic7#38814yes—
472C24—71no—
473C2×Q8—21no—
474C2×Q8—21no—
475C2×Q8—21no—
476C2×Q8—21no—
477C2×Q8—21no—
478C2×Q8—72no—
479C2×Q8—144no—
480C2×Q8—148no—
481C2×Q8—148no—
482C2×Q8—141no—
483C2×Q8—141no—
484C2×Q8—141no—
485C2×Q8—141no—
486C2×Q8—141no—
487C2×Q8—282no—
488C2×Q8—282no—
489C2×Q8—284no—
490C2×Q8—284no—
491C2×Q8C2×C4×D712yes—
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—21no—
496C22×C4—71no—
497C22×C4—72no—
498C22×C4—72no—
499C22×C4—72no—
500C22×C4—72no—
501C22×C4—72no—
502C22×C4—72no—
503C22×C4—71no—
504C22×C4—71no—
505C22×C4—142no—
506C22×C4—142no—
507C22×C4—142no—
508C22×C4—142no—
509C22×C4—144no—
510C22×C4—141no—
511C22×C4—141no—
512C22×C4—141no—
513C22×C4—141no—
514C22×C4—281no—
515C22×C4—281no—
516C22×C4—282no—
517C22×C4—282no—
518C22×C4—282no—
519C22×C4—282no—
520C22×C4—282no—
521C22×C4D4×D711yescharacteristic
522C22×C4D4×D711yescharacteristic
523C22×C4C23×D711yescharacteristic
524M16—148no—
525M16—148no—
526C2×C8—21no—
527C2×C8—21no—
528C2×C8—21no—
529C2×C8—21no—
530C2×C8—72no—
531C2×C8—72no—
532C2×C8—141no—
533C2×C8—141no—
534C2×C8—141no—
535C2×C8—141no—
536C2×C8C2×C4×D712yes—
537C2×C8D4×D712yes—
538C4⋊C4—21no—
539C4⋊C4—21no—
540C4⋊C4—21no—
541C4⋊C4—21no—
542C4⋊C4—21no—
543C4⋊C4—42no—
544C4⋊C4—72no—
545C4⋊C4—141no—
546C4⋊C4—141no—
547C4⋊C4—141no—
548C4⋊C4—141no—
549C4⋊C4—141no—
550C4⋊C4—282no—
551C4⋊C4—282no—
552C4⋊C4—282no—
553C4⋊C4—282no—
554C4⋊C4C2×C4×D712yes—
555C22⋊C4—144no—
556C22⋊C4—282no—
557C42—21no—
558C42—21no—
559C42—142no—
560C42—142no—
561C42—144no—
562C42—141no—
563C42—141no—
564C42—282no—
565C2×C7C2×#38811yescharacteristic
566C2×C7C4×#13711yescharacteristic
567C2×C7C4×#13711yescharacteristic
568C2×C7#305611yescharacteristic
569C2×C7#305611yescharacteristic
570C2×C7#313911yescharacteristic
571C2×C7#314011yescharacteristic
572D7—24no—
573C23—72no—
574C23—72no—
575C23—72no—
576C23—141no—
577C23—141no—
578C23—141no—
579C23—141no—
580C23C2×D4×D711yescharacteristic
581Q8—22no—
582Q8—22no—
583Q8—41no—
584Q8—41no—
585Q8—142no—
586Q8—142no—
587Q8—288no—
588Q8—281no—
589Q8—281no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—21no—
597C2×C4—21no—
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—41no—
607C2×C4—41no—
608C2×C4—72no—
609C2×C4—72no—
610C2×C4—72no—
611C2×C4—74no—
612C2×C4—74no—
613C2×C4—74no—
614C2×C4—74no—
615C2×C4—72no—
616C2×C4—72no—
617C2×C4—141no—
618C2×C4—141no—
619C2×C4—141no—
620C2×C4—141no—
621C2×C4—142no—
622C2×C4—142no—
623C2×C4—284no—
624C2×C4—284no—
625C2×C4—288no—
626C2×C4—281no—
627C2×C4—281no—
628C2×C4—281no—
629C2×C4—281no—
630C2×C4—281no—
631C2×C4—281no—
632C2×C4—281no—
633C2×C4—281no—
634C2×C4—281no—
635C2×C4—281no—
636C2×C4—564no—
637C2×C4C22×C4×D711yescharacteristic
638C2×C4C2×D4×D711yescharacteristic
639C2×C4D7×#4111yescharacteristic
640C2×C4D7×#4111yescharacteristic
641C2×C4D7×#4111yescharacteristic
642C2×C4D7×#4111yescharacteristic
643C8—22no—
644C8—22no—
645C8—142no—
646C8—142no—
647C7#2109411yescharacteristic, complemented, Sylow
648C22—74no—
649C22—142no—
650C22—142no—
651C22—142no—
652C22—142no—
653C22—142no—
654C22—142no—
655C22C4×D4×D711yescharacteristic
656C22D7×#12111yescharacteristic
657C22D7×#12411yescharacteristic
658C22#9046411yescharacteristic
659C22D7×#13711yescharacteristic
660C22D7×#13711yescharacteristic
661C22#9046811yescharacteristic
662C4—42no—
663C4—42no—
664C4—41no—
665C4—41no—
666C4—74no—
667C4—81no—
668C4—81no—
669C4—142no—
670C4—142no—
671C4—282no—
672C4—282no—
673C4—561no—
674C4—561no—
675C4D7×#11711yescharacteristic
676C4C4×D4×D711yescharacteristic
677C4D7×#12311yescharacteristic
678C4D7×#12611yescharacteristic
679C2—144no—
680C2D7×#38811yescharacteristic
681C2#1173721911yescharacteristic
682C2#1173727411yescharacteristic
683C2#1173825111yescharacteristic
684C2#1173834911yescharacteristic
685C2#1173912711yescharacteristic
686C2#1173918411yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1935C2
socleC23×C7C22×D4
prime radicalC23×D7C2×D4
cosocleC22×C4×C7C24
hypercenter#1935D7
hypocenterC7#21094
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1935

Complements

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

NH ≅ G/Ntype
6. C7×#1935C2semidirect
7. #11728008C2semidirect
8. #11728007C2semidirect
9. #11727571C2semidirect
10. #11727429C2semidirect
11. #11727370C2semidirect
14. #11726605C2semidirect
15. #11726604C2semidirect
46. #89574C4semidirect
115. C7×#103C2×C4semidirect
121. #6409C2×C4semidirect
132. #6390C2×C4semidirect
136. #6384C2×C4semidirect
146. #1935D7semidirect
424. #103C4×D7semidirect
647. C7#21094semidirect

Maximal subgroups

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

subgroupindex
C2×#895742
C2×#894792
D7×#3912
#117334082
C7×#19352
#117280082
#117280072
#117275712
#117274292
#117273702
#117266212
#117266052
#117266042
#210947