Finite Group Databasebeta

← #13356471

Subgroups — #13356471

Order 1792.

1074 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89574C211yesmaximal, characteristic
2C2×D7×#103C211yesmaximal, characteristic
3C2×#89480C211yesmaximal, characteristic
4#11734055C211yesmaximal, characteristic
5#11733413C211yesmaximal, characteristic
6C7×#1935C211yesmaximal, characteristic, complemented
7#11728010C211yesmaximal, characteristic, complemented
8#11728007C211yesmaximal, characteristic, complemented
9#11727571C211yesmaximal, characteristic, complemented
10#11727428C211yesmaximal, characteristic, complemented
11#11727368C211yesmaximal, characteristic, complemented
12#11726621C211yesmaximal, characteristic
13#11726620C211yesmaximal, characteristic
14#11726609C211yesmaximal, characteristic, complemented
15#11726604C211yesmaximal, characteristic, complemented
16C22×Q8×D7C2211yescharacteristic
17C22×#1131C2211yescharacteristic
18C22×C8×D7C2211yescharacteristic
19C2×#6455C2211yescharacteristic
20C2×#6455C2211yescharacteristic
21C2×D7×C4⋊C4C2211yescharacteristic
22C2×#6435C2211yescharacteristic
23C2×C7×#103C2211yescharacteristic
24C2×C7×#103C2211yescharacteristic
25C2×C7×#97C2211yescharacteristic
26C7×#391C2211yescharacteristic
27C7×#389C2211yescharacteristic
28#90002C2211yescharacteristic
29C2×#6409C2211yescharacteristic
30C2×#6409C2211yescharacteristic
31C2×#6390C2211yescharacteristic
32C2×#6390C2211yescharacteristic
33C2×#6389C2211yescharacteristic
34C2×C8×Dic7C2211yescharacteristic
35#89761—21no—
36#89757C2211yescharacteristic
37#89748—21no—
38#89746C2211yescharacteristic
39#89745C2211yescharacteristic
40C2×#6384C2211yescharacteristic
41C2×#6384C2211yescharacteristic
42#89701C2211yescharacteristic
43#89700C2211yescharacteristic
44C2×C4×C7⋊C8C2211yescharacteristic
45#89574C418yescomplemented
46D7×#103—24no—
47#89480—24no—
48#89440—21no—
49#89436—21no—
50#89436—21no—
51#89422—21no—
52#89418—21no—
53#89418—21no—
54C7×#340C2211yescharacteristic
55C7×#340C2211yescharacteristic
56#89346C2211yescharacteristic
57#89346C2211yescharacteristic
58#89324C2211yescharacteristic
59#89324C2211yescharacteristic
60#89324C2211yescharacteristic
61#89324C2211yescharacteristic
62#21090—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×D7C2311yescharacteristic
73C22×Dic14—21no—
74C22×Dic14C2311yescharacteristic
75C22×C7×C8C2311yescharacteristic
76C22×C7×C8C2311yescharacteristic
77C2×C7×C4⋊C4—21no—
78C2×C7×C4⋊C4C2311yescharacteristic
79C2×C7×C4⋊C4C2311yescharacteristic
80C2×C7×C4⋊C4C2311yescharacteristic
81C2×C42×C7C2311yescharacteristic
82C2×(C2×D7)⋊C4—21no—
83C2×(C2×D7)⋊C4—21no—
84C2×(C2×D7)⋊C4D411yescharacteristic
85C2×(C2×D7)⋊C4D411yescharacteristic
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×Dic7⋊C4D411yescharacteristic
94C2×Dic7⋊C4D411yescharacteristic
95C2×C4×Dic7—21no—
96C2×C4×Dic7—21no—
97C2×C4×Dic7C2311yescharacteristic
98C22×C7⋊C8C2311yescharacteristic
99C22×C7⋊C8C2311yescharacteristic
100C2×#1131—22no—
101C2×#1131—22no—
102C2×#1131C2×C414yes—
103C2×C8×D7—22no—
104C2×C8×D7—22no—
105C2×C8×D7D414yes—
106#6455—22no—
107#6455—22no—
108#6455—22no—
109#6455C2×C414yes—
110D7×C4⋊C4—22no—
111D7×C4⋊C4—22no—
112#6435—22no—
113#6435—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
133#6389—22no—
134C8×Dic7—22no—
135#6384—22no—
136#6384C2×C414yescomplemented
137C4×C7⋊C8—22no—
138C2×#419—71no—
139C22×#103—71no—
140C2×#405—71no—
141#2317—71no—
142#2310—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#1735—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—42no—
180C2×C4×D7—42no—
181C2×C4×D7—42no—
182C2×C4×D7—81no—
183C2×C4×D7—81no—
184C2×C4×D7C22×C412yes—
185C2×C4×D7C2×D412yes—
186C2×C4×D7#4112yes—
187C2×C4×D7#4112yes—
188C2×C4×D7#4112yes—
189C2×C4×D7#4112yes—
190C2×Dic14—21no—
191C2×Dic14—21no—
192C2×Dic14—21no—
193C2×Dic14—21no—
194C2×Dic14—21no—
195C2×Dic14—24no—
196C2×Dic14—42no—
197C2×Dic14—42no—
198C2×Dic14C22×C412yes—
199C2×C7×C8—21no—
200C2×C7×C8—21no—
201C2×C7×C8—21no—
202C2×C7×C8—21no—
203C2×C7×C8C22×C412yes—
204C2×C7×C8C2×D412yes—
205C7×C4⋊C4—21no—
206C7×C4⋊C4—21no—
207C7×C4⋊C4—21no—
208C7×C4⋊C4—21no—
209C7×C4⋊C4—21no—
210C7×C4⋊C4—81no—
211C7×C4⋊C4C22×C412yes—
212C42×C7—21no—
213C42×C7—21no—
214(C2×D7)⋊C4—24no—
215(C2×D7)⋊C4—22no—
216(C2×D7)⋊C4—22no—
217(C2×D7)⋊C4—42no—
218C4⋊Dic7—21no—
219C4⋊Dic7—21no—
220C4⋊Dic7—21no—
221C4⋊Dic7—21no—
222C4⋊Dic7—21no—
223C4⋊Dic7—81no—
224C4⋊Dic7C22×C412yes—
225Dic7⋊C4—22no—
226Dic7⋊C4—22no—
227Dic7⋊C4—42no—
228Dic7⋊C4—81no—
229Dic7⋊C4—81no—
230C4×Dic7—21no—
231C4×Dic7—21no—
232C4×Dic7—24no—
233C4×Dic7—42no—
234C2×C7⋊C8—21no—
235C2×C7⋊C8—21no—
236C2×C7⋊C8—21no—
237C2×C7⋊C8—21no—
238C2×C7⋊C8C22×C412yes—
239C2×C7⋊C8C2×D412yes—
240#1131—28no—
241C8×D7—28no—
242C23×Q8—71no—
243C22×M16—71no—
244C23×C8—71no—
245C2×#117—71no—
246C2×#117—71no—
247C2×#117—71no—
248C22×C4⋊C4—71no—
249#419—78no—
250C2×#103—71no—
251C2×#103—71no—
252C2×#103—71no—
253C2×#103—71no—
254C2×#103—71no—
255C2×#103—71no—
256C2×#103—144no—
257C2×#103C2×D711yescharacteristic
258C2×#103C2×D711yescharacteristic
259#405—144no—
260C2×#97—71no—
261C2×#97C2×D711yescharacteristic
262C2×C4×C8—71no—
263C2×C4×C8—71no—
264#391—71no—
265#391—71no—
266#391—71no—
267#391—141no—
268#391—141no—
269#391C2×D711yescharacteristic
270#389—71no—
271#389—71no—
272#389—71no—
273#389C2×D711yescharacteristic
274#385—141no—
275#385—141no—
276#384—141no—
277#382—141no—
278#381—141no—
279#381—141no—
280#340—71no—
281#340—71no—
282#340—71no—
283#340—71no—
284#340—71no—
285#340—71no—
286#340C2×D711yescharacteristic
287#340C2×D711yescharacteristic
288C23×C7C22×D411yescharacteristic
289C22×D7—21no—
290C22×D7—21no—
291C22×D7—21no—
292C22×D7—21no—
293C22×D7C4×D412yes—
294C22×D7#12112yes—
295C22×D7#12412yes—
296C7×Q8—22no—
297C7×Q8—22no—
298C7×Q8—41no—
299C7×Q8—41no—
300C2×C4×C7—22no—
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—41no—
309C2×C4×C7—41no—
310C2×C4×C7—41no—
311C2×C4×C7—41no—
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×C7C23×C411yescharacteristic
319C2×C4×C7C22×D411yescharacteristic
320C2×C4×C7C2×#4111yescharacteristic
321C2×C4×C7C2×#4111yescharacteristic
322C2×C4×C7C2×#4111yescharacteristic
323C2×C4×C7C2×#4111yescharacteristic
324C2×Dic7—22no—
325C2×Dic7—22no—
326C2×Dic7—21no—
327C2×Dic7—21no—
328C2×Dic7—21no—
329C2×Dic7—21no—
330C2×Dic7—21no—
331C2×Dic7—21no—
332C2×Dic7—41no—
333C2×Dic7—41no—
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×Dic7C4×D412yes—
343C2×Dic7#12112yes—
344C2×Dic7#12412yes—
345C4×D7—48no—
346C4×D7—84no—
347C4×D7#11714yes—
348C4×D7C4×D414yes—
349C4×D7#12314yes—
350C4×D7#12614yes—
351Dic14—22no—
352Dic14—22no—
353Dic14—41no—
354Dic14—41no—
355Dic14—48no—
356C7×C8—22no—
357C7×C8—22no—
358C7⋊C8—22no—
359C7⋊C8—22no—
360C22×Q8—74no—
361C22×Q8—71no—
362C22×Q8—141no—
363C22×Q8—142no—
364C22×Q8—142no—
365C22×Q8—142no—
366C22×Q8—142no—
367C22×Q8—142no—
368C22×Q8C22×D711yescharacteristic
369C23×C4—71no—
370C23×C4—141no—
371C23×C4—141no—
372C2×M16—74no—
373C2×M16—142no—
374C2×M16—142no—
375C22×C8—71no—
376C22×C8—71no—
377C22×C8—74no—
378C22×C8—142no—
379C22×C8—142no—
380C22×C8C22×D711yescharacteristic
381C22×C8C22×D711yescharacteristic
382#117—74no—
383#117—142no—
384#117—142no—
385#117—142no—
386#117—142no—
387#117—142no—
388C2×C4⋊C4—21no—
389C2×C4⋊C4—71no—
390C2×C4⋊C4—71no—
391C2×C4⋊C4—71no—
392C2×C4⋊C4—71no—
393C2×C4⋊C4—71no—
394C2×C4⋊C4—142no—
395C2×C4⋊C4—142no—
396C2×C4⋊C4—141no—
397C2×C4⋊C4—141no—
398C2×C4⋊C4—141no—
399C2×C4⋊C4—141no—
400C2×C4⋊C4C22×D711yescharacteristic
401C2×C4⋊C4C22×D711yescharacteristic
402C2×C4⋊C4C22×D711yescharacteristic
403C2×C22⋊C4—71no—
404C2×C22⋊C4—71no—
405C2×C22⋊C4—141no—
406C2×C22⋊C4—141no—
407C2×C42—71no—
408C2×C42—141no—
409C2×C42—141no—
410C2×C42C22×D711yescharacteristic
411#103—22no—
412#103—74no—
413#103—74no—
414#103—74no—
415#103—142no—
416#103—142no—
417#103—142no—
418#103C4×D714yescomplemented
419#97—22no—
420#97—142no—
421C4×C8—142no—
422C4×C8—142no—
423#95—21no—
424#95—21no—
425#95—21no—
426#95—141no—
427#95—141no—
428#95—141no—
429#95—141no—
430#95—141no—
431#95—141no—
432#95—141no—
433#95—141no—
434#95—141no—
435C22×C7C2×C4×D411yescharacteristic
436C22×C7C2×#12111yescharacteristic
437C22×C7C2×#12411yescharacteristic
438C22×C7#57711yescharacteristic
439C22×C7#57711yescharacteristic
440C22×C7#57811yescharacteristic
441C22×C7#57811yescharacteristic
442C2×D7—22no—
443C2×D7—22no—
444C2×D7—22no—
445C2×D7—22no—
446C2×D7—22no—
447C2×D7—22no—
448C2×D7#38814yes—
449C4×C7—42no—
450C4×C7—42no—
451C4×C7—41no—
452C4×C7—41no—
453C4×C7—81no—
454C4×C7—81no—
455C4×C7C2×#11711yescharacteristic
456C4×C7C2×C4×D411yescharacteristic
457C4×C7C2×#12311yescharacteristic
458C4×C7C2×#12611yescharacteristic
459Dic7—42no—
460Dic7—42no—
461Dic7—41no—
462Dic7—41no—
463Dic7—81no—
464Dic7—81no—
465Dic7#38814yes—
466C24—71no—
467C2×Q8—21no—
468C2×Q8—21no—
469C2×Q8—21no—
470C2×Q8—21no—
471C2×Q8—21no—
472C2×Q8—72no—
473C2×Q8—144no—
474C2×Q8—148no—
475C2×Q8—148no—
476C2×Q8—141no—
477C2×Q8—141no—
478C2×Q8—141no—
479C2×Q8—141no—
480C2×Q8—141no—
481C2×Q8—282no—
482C2×Q8—282no—
483C2×Q8—284no—
484C2×Q8—284no—
485C2×Q8C2×C4×D712yes—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—71no—
491C22×C4—72no—
492C22×C4—72no—
493C22×C4—72no—
494C22×C4—72no—
495C22×C4—72no—
496C22×C4—72no—
497C22×C4—71no—
498C22×C4—71no—
499C22×C4—144no—
500C22×C4—141no—
501C22×C4—141no—
502C22×C4—141no—
503C22×C4—141no—
504C22×C4—282no—
505C22×C4—282no—
506C22×C4—282no—
507C22×C4—561no—
508C22×C4—561no—
509C22×C4D4×D711yescharacteristic
510C22×C4D4×D711yescharacteristic
511C22×C4C23×D711yescharacteristic
512M16—148no—
513C2×C8—21no—
514C2×C8—21no—
515C2×C8—21no—
516C2×C8—21no—
517C2×C8—72no—
518C2×C8—72no—
519C2×C8—148no—
520C2×C8—141no—
521C2×C8—141no—
522C2×C8—141no—
523C2×C8—141no—
524C2×C8C2×C4×D712yes—
525C2×C8D4×D712yes—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—21no—
530C4⋊C4—21no—
531C4⋊C4—72no—
532C4⋊C4—81no—
533C4⋊C4—142no—
534C4⋊C4—142no—
535C4⋊C4—141no—
536C4⋊C4—141no—
537C4⋊C4—141no—
538C4⋊C4—141no—
539C4⋊C4—141no—
540C4⋊C4—282no—
541C4⋊C4—561no—
542C4⋊C4—561no—
543C4⋊C4—561no—
544C4⋊C4C2×C4×D712yes—
545C22⋊C4—142no—
546C22⋊C4—142no—
547C22⋊C4—144no—
548C22⋊C4—282no—
549C42—21no—
550C42—21no—
551C42—144no—
552C42—141no—
553C42—141no—
554C42—282no—
555C2×C7C2×#38811yescharacteristic
556C2×C7#294711yescharacteristic
557C2×C7#294711yescharacteristic
558C2×C7#305711yescharacteristic
559C2×C7#305911yescharacteristic
560C2×C7#313711yescharacteristic
561C2×C7#313711yescharacteristic
562D7—24no—
563C23—72no—
564C23—72no—
565C23—72no—
566C23—141no—
567C23—141no—
568C23—141no—
569C23—141no—
570C23C2×D4×D711yescharacteristic
571Q8—22no—
572Q8—22no—
573Q8—41no—
574Q8—41no—
575Q8—142no—
576Q8—142no—
577Q8—288no—
578Q8—281no—
579Q8—281no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—72no—
599C2×C4—72no—
600C2×C4—72no—
601C2×C4—74no—
602C2×C4—74no—
603C2×C4—74no—
604C2×C4—74no—
605C2×C4—141no—
606C2×C4—141no—
607C2×C4—141no—
608C2×C4—141no—
609C2×C4—142no—
610C2×C4—142no—
611C2×C4—141no—
612C2×C4—141no—
613C2×C4—288no—
614C2×C4—281no—
615C2×C4—281no—
616C2×C4—281no—
617C2×C4—281no—
618C2×C4—281no—
619C2×C4—281no—
620C2×C4—281no—
621C2×C4—281no—
622C2×C4—281no—
623C2×C4—281no—
624C2×C4—564no—
625C2×C4C22×C4×D711yescharacteristic
626C2×C4C2×D4×D711yescharacteristic
627C2×C4D7×#4111yescharacteristic
628C2×C4D7×#4111yescharacteristic
629C2×C4D7×#4111yescharacteristic
630C2×C4D7×#4111yescharacteristic
631C8—22no—
632C8—22no—
633C8—142no—
634C8—142no—
635C7#2109011yescharacteristic, complemented, Sylow
636C22—74no—
637C22—142no—
638C22—142no—
639C22—142no—
640C22—142no—
641C22—142no—
642C22—142no—
643C22C4×D4×D711yescharacteristic
644C22D7×#12111yescharacteristic
645C22D7×#12411yescharacteristic
646C22#9046611yescharacteristic
647C22#9046911yescharacteristic
648C22#9047011yescharacteristic
649C22#9047011yescharacteristic
650C4—42no—
651C4—42no—
652C4—41no—
653C4—41no—
654C4—74no—
655C4—81no—
656C4—81no—
657C4—282no—
658C4—281no—
659C4—282no—
660C4—281no—
661C4—561no—
662C4—561no—
663C4D7×#11711yescharacteristic
664C4C4×D4×D711yescharacteristic
665C4D7×#12311yescharacteristic
666C4D7×#12611yescharacteristic
667C2—144no—
668C2D7×#38811yescharacteristic
669C2#1173722111yescharacteristic
670C2#1173727511yescharacteristic
671C2#1173825211yescharacteristic
672C2#1173835011yescharacteristic
673C2#1173912611yescharacteristic
674C2#1173918311yescharacteristic
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#21090
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. #11728010C2semidirect
8. #11728007C2semidirect
9. #11727571C2semidirect
10. #11727428C2semidirect
11. #11727368C2semidirect
14. #11726609C2semidirect
15. #11726604C2semidirect
45. #89574C4semidirect
115. C7×#103C2×C4semidirect
121. #6409C2×C4semidirect
132. #6390C2×C4semidirect
136. #6384C2×C4semidirect
146. #1935D7semidirect
418. #103C4×D7semidirect
635. C7#21090semidirect

Maximal subgroups

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

subgroupindex
C2×#895742
C2×D7×#1032
C2×#894802
#117340552
#117334132
C7×#19352
#117280102
#117280072
#117275712
#117274282
#117273682
#117266212
#117266202
#117266092
#117266042
#210907