Finite Group Databasebeta

← #13410982

Subgroups — #13410982

Order 1792.

966 conjugacy classes of subgroups.

The subgroup lattice has 702 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#416C211yesmaximal, characteristic
2#11741534C211yesmaximal, characteristic
3#11739693C211yesmaximal, characteristic, complemented
4C2×#89566C211yesmaximal, characteristic, complemented
5#11729889C211yesmaximal, characteristic
6#11729728C211yesmaximal, characteristic, complemented
7#11729725C211yesmaximal, characteristic, complemented
8#11729330C211yesmaximal, characteristic, complemented
9#11727991C211yesmaximal, characteristic, complemented
10#11727990C211yesmaximal, characteristic, complemented
11#11727976C211yesmaximal, characteristic, complemented
12#11727948C211yesmaximal, characteristic, complemented
13#11727935C211yesmaximal, characteristic, complemented
14#11727934C211yesmaximal, characteristic, complemented
15#11727917C211yesmaximal, characteristic, complemented
16C22×C7×#41C2211yescharacteristic
17C23×C7×C8C2211yescharacteristic
18C2×C7×#117C2211yescharacteristic
19C2×Q8×Dic7C2211yescharacteristic
20C2×Q8×Dic7C2211yescharacteristic
21C2×#6451C2211yescharacteristic
22C7×#416—24no—
23C2×C7×#103C2211yescharacteristic
24C2×C7×#103C2211yescharacteristic
25C2×C7×#102C2211yescharacteristic
26C2×C7×#102C2211yescharacteristic
27C2×#6409C2211yescharacteristic
28C2×#6409C2211yescharacteristic
29#89993—21no—
30#89993—21no—
31#89993C2211yescharacteristic
32#89993C2211yescharacteristic
33C2×#6406C2211yescharacteristic
34C2×#6406C2211yescharacteristic
35#89984—21no—
36#89984C2211yescharacteristic
37#89984C2211yescharacteristic
38C4×C22⋊Dic7—21no—
39C4×C22⋊Dic7C2211yescharacteristic
40C2×C22⋊(C7⋊C8)C2211yescharacteristic
41C2×#6392C2211yescharacteristic
42C2×#6392C2211yescharacteristic
43C2×Dic7⋊C8C2211yescharacteristic
44C2×Dic7⋊C8C2211yescharacteristic
45#89757C2211yescharacteristic
46#89749C2211yescharacteristic
47#89745C2211yescharacteristic
48C2×#6382C2211yescharacteristic
49C2×#6381C2211yescharacteristic
50#89720—21no—
51#89719—21no—
52#89705—21no—
53#89701—21no—
54#89566—24no—
55#89346C2211yescharacteristic
56#89346C2211yescharacteristic
57#89345C2211yescharacteristic
58#89345C2211yescharacteristic
59#89324C2211yescharacteristic
60#89324C2211yescharacteristic
61#25959—71nomaximal, Sylow
62C2×C7×#41—22no—
63C2×C7×#41—22no—
64C2×C7×#41—22no—
65C2×C7×#41—41no—
66C2×C7×#41—41no—
67C2×C7×#41Q814yescomplemented
68C22×C7×Q8C2311yescharacteristic
69C22×C7×D4—21no—
70C22×C7×D4C2311yescharacteristic
71C23×C4×C7—21no—
72C23×C4×C7C2311yescharacteristic
73C22×Dic14—21no—
74C22×C7×C8—22no—
75C22×C7×C8—22no—
76C22×C7×C8—41no—
77C22×C7×C8C2311yescharacteristic
78C22×C7×C8C2311yescharacteristic
79C7×#117—22no—
80C7×#117—41no—
81C2×C7×C4⋊C4C2311yescharacteristic
82C2×C7×C4⋊C4C2311yescharacteristic
83C2×C7×C22⋊C4—21no—
84C2×C42×C7—21no—
85C2×C22⋊Dic7—21no—
86C2×C22⋊Dic7—21no—
87C2×C22⋊Dic7—21no—
88C2×C22⋊Dic7—21no—
89C2×C22⋊Dic7—21no—
90C2×C22⋊Dic7—21no—
91C2×C22⋊Dic7D411yescharacteristic
92C2×C22⋊Dic7D411yescharacteristic
93Q8×Dic7—22no—
94Q8×Dic7—22no—
95Q8×Dic7—41no—
96Q8×Dic7—41no—
97C2×C4⋊Dic7—21no—
98C2×C4⋊Dic7—21no—
99C2×C4⋊Dic7—21no—
100C2×C4⋊Dic7C2311yescharacteristic
101C2×C4⋊Dic7C2311yescharacteristic
102C2×C4⋊Dic7C2311yescharacteristic
103C2×C4⋊Dic7C2311yescharacteristic
104C2×Dic7⋊C4—21no—
105C2×Dic7⋊C4—21no—
106C2×C4×Dic7—21no—
107C2×C4×Dic7—21no—
108C2×C4×Dic7C2311yescharacteristic
109C2×C4×Dic7C2311yescharacteristic
110C22×C7⋊C8C2311yescharacteristic
111C22×C7⋊C8C2311yescharacteristic
112#6451—22no—
113#6451—41no—
114C7×#103—22no—
115C7×#103D414yes—
116C7×#102—22no—
117C7×#102—22no—
118#6409—22no—
119#6409—22no—
120#6406—41no—
121#6406—41no—
122C7⋊#95—21no—
123C7⋊#95—21no—
124C7⋊#95—21no—
125C7⋊#95—21no—
126C7⋊#95—21no—
127C7⋊#95—21no—
128C7⋊#95D411yescharacteristic
129C7⋊#95D411yescharacteristic
130C22⋊(C7⋊C8)—22no—
131#6392—22no—
132#6392—41no—
133Dic7⋊C8—22no—
134Dic7⋊C8—41no—
135#6382—41no—
136#6381—22no—
137C2×#477—71no—
138C2×#416D711yescharacteristic
139#2439—71no—
140#2341—71no—
141#2092—71no—
142#2067—71no—
143#2067—71no—
144#2023—71no—
145#1865—71no—
146#1863—71no—
147#1854—71no—
148#1850—71no—
149#1808—71no—
150#1805—71no—
151#1788—71no—
152C24×C7—21no—
153C24×C7C2×D411yescharacteristic
154C7×#41—28no—
155C7×#41—44no—
156C7×#41—44no—
157C7×#41—44no—
158C2×C7×Q8—21no—
159C2×C7×Q8—21no—
160C2×C7×Q8—21no—
161C2×C7×Q8C2×D412yes—
162C2×C7×Q8C2×Q812yes—
163C2×C7×Q8#4112yes—
164C2×C7×D4—24no—
165C2×C7×D4—21no—
166C2×C7×D4—21no—
167C2×C7×D4—21no—
168C2×C7×D4—21no—
169C2×C7×D4—21no—
170C2×C7×D4—42no—
171C2×C7×D4—42no—
172C2×C7×D4C2×Q812yes—
173C22×C4×C7—21no—
174C22×C4×C7—21no—
175C22×C4×C7—21no—
176C22×C4×C7—21no—
177C22×C4×C7—21no—
178C22×C4×C7—21no—
179C22×C4×C7—24no—
180C22×C4×C7—21no—
181C22×C4×C7—42no—
182C22×C4×C7—42no—
183C22×C4×C7C2×D411yescharacteristic
184C22×C4×C7C2×Q812yes—
185C22×C4×C7#4112yes—
186C22×C4×C7C2411yescharacteristic
187C22×Dic7—21no—
188C22×Dic7—21no—
189C22×Dic7—21no—
190C22×Dic7—21no—
191C22×Dic7C2×D411yescharacteristic
192C22×Dic7C2×D411yescharacteristic
193C22×Dic7C2×D411yescharacteristic
194C22×Dic7C2×D411yescharacteristic
195C2×Dic14—42no—
196C2×Dic14—42no—
197C2×Dic14—81no—
198C2×C7×C8—21no—
199C2×C7×C8—21no—
200C2×C7×C8—21no—
201C2×C7×C8—21no—
202C2×C7×C8—21no—
203C2×C7×C8—44no—
204C2×C7×C8C2×D412yes—
205C7×C4⋊C4—21no—
206C7×C4⋊C4—21no—
207C7×C4⋊C4—21no—
208C7×C4⋊C4C2×D412yes—
209C7×C22⋊C4—42no—
210C42×C7—42no—
211C22⋊Dic7—22no—
212C22⋊Dic7—41no—
213C22⋊Dic7—42no—
214C22⋊Dic7—42no—
215C22⋊Dic7—81no—
216C22⋊Dic7—81no—
217C22⋊Dic7—81no—
218C22⋊Dic7—81no—
219C4⋊Dic7—21no—
220C4⋊Dic7—21no—
221C4⋊Dic7—21no—
222C4⋊Dic7—21no—
223C4⋊Dic7—21no—
224C4⋊Dic7—21no—
225C4⋊Dic7—21no—
226C4⋊Dic7—21no—
227C4⋊Dic7—42no—
228C4⋊Dic7—42no—
229C4⋊Dic7—81no—
230Dic7⋊C4—42no—
231Dic7⋊C4—42no—
232C4×Dic7—21no—
233C4×Dic7—21no—
234C4×Dic7—21no—
235C4×Dic7—21no—
236C4×Dic7—42no—
237C4×Dic7—42no—
238C2×C7⋊C8—21no—
239C2×C7⋊C8—21no—
240C2×C7⋊C8—21no—
241C2×C7⋊C8—21no—
242C2×C7⋊C8—21no—
243C2×C7⋊C8—21no—
244C22×#41C2×D711yescharacteristic
245C23×C8C2×D711yescharacteristic
246C2×C4⋊Q8—71no—
247C2×C4×Q8—71no—
248C2×C4×Q8—71no—
249C2×#117C2×D711yescharacteristic
250#477—144no—
251C2×#107—71no—
252C2×#107—71no—
253C2×#107—71no—
254C2×#106—71no—
255C2×C4⋊C8—71no—
256C2×C4⋊C8—71no—
257#416—24no—
258C2×#103—71no—
259C2×#103—71no—
260C2×#103C2×D711yescharacteristic
261C2×#103C2×D711yescharacteristic
262C2×#102—71no—
263C2×#102—71no—
264C2×#102C2×D711yescharacteristic
265C2×#102C2×D711yescharacteristic
266C2×C22⋊C8—71no—
267#396—141no—
268#393—141no—
269#391—141no—
270#390—71no—
271#390—71no—
272#390—141no—
273#390—141no—
274#389—71no—
275#389—71no—
276#387—141no—
277#382—71no—
278#380—71no—
279#380—71no—
280#380—141no—
281C4×C22⋊C4—71no—
282C4×C22⋊C4—141no—
283#340—71no—
284#340—71no—
285#340—71no—
286#340—71no—
287#336—71no—
288#336—71no—
289C23×C7—21no—
290C23×C7—21no—
291C23×C7—21no—
292C23×C7—21no—
293C23×C7—21no—
294C23×C7—21no—
295C23×C7—22no—
296C23×C7—41no—
297C23×C7—41no—
298C23×C7—41no—
299C23×C7—41no—
300C23×C7—41no—
301C23×C7—41no—
302C23×C7C22⋊Q812yes—
303C23×C7C22×D411yescharacteristic
304C7×Q8—22no—
305C7×Q8—22no—
306C7×Q8—22no—
307C7×Q8C22⋊Q814yes—
308C7×D4—22no—
309C7×D4—22no—
310C7×D4—48no—
311C7×D4—41no—
312C7×D4—41no—
313C2×C4×C7—22no—
314C2×C4×C7—22no—
315C2×C4×C7—21no—
316C2×C4×C7—22no—
317C2×C4×C7—21no—
318C2×C4×C7—22no—
319C2×C4×C7—22no—
320C2×C4×C7—22no—
321C2×C4×C7—22no—
322C2×C4×C7—22no—
323C2×C4×C7—48no—
324C2×C4×C7—41no—
325C2×C4×C7—41no—
326C2×C4×C7—41no—
327C2×C4×C7—41no—
328C2×C4×C7—41no—
329C2×C4×C7C22⋊Q812yes—
330C2×C4×C7C22×D411yescharacteristic
331C2×C4×C7C22×Q811yescharacteristic
332C2×C4×C7C2×#4111yescharacteristic
333C2×C4×C7C2×#4111yescharacteristic
334C2×C4×C7H(5,𝔽2)11yescharacteristic
335C2×C4×C7H(5,𝔽2)11yescharacteristic
336C2×Dic7—21no—
337C2×Dic7—21no—
338C2×Dic7—21no—
339C2×Dic7—21no—
340C2×Dic7—21no—
341C2×Dic7—21no—
342C2×Dic7—21no—
343C2×Dic7—21no—
344C2×Dic7—21no—
345C2×Dic7—21no—
346C2×Dic7—21no—
347C2×Dic7—41no—
348C2×Dic7—41no—
349C2×Dic7—41no—
350C2×Dic7—41no—
351C2×Dic7—41no—
352C2×Dic7—41no—
353C2×Dic7—41no—
354C2×Dic7—41no—
355C2×Dic7—41no—
356C2×Dic7—41no—
357C2×Dic7—41no—
358C2×Dic7—41no—
359C2×Dic7#13512yes—
360Dic14—84no—
361C7×C8—22no—
362C7×C8—41no—
363C7⋊C8—41no—
364C7⋊C8—41no—
365C2×#41—22no—
366C2×#41—22no—
367C2×#41—22no—
368C2×#41—41no—
369C2×#41—41no—
370C2×#41Dic1414yescomplemented
371C22×Q8—141no—
372C22×Q8C22×D711yescharacteristic
373C22×D4—21no—
374C22×D4C22×D711yescharacteristic
375C23×C4—21no—
376C23×C4C22×D711yescharacteristic
377C22×C8—22no—
378C22×C8—22no—
379C22×C8—41no—
380C22×C8—71no—
381C22×C8—71no—
382C22×C8C22×D711yescharacteristic
383C22×C8C22×D711yescharacteristic
384C4⋊Q8—142no—
385C4⋊Q8—281no—
386C4×Q8—142no—
387C4×Q8—142no—
388C4×Q8—281no—
389C4×Q8—281no—
390#117—22no—
391#117—41no—
392C2×C4⋊C4—71no—
393C2×C4⋊C4—71no—
394C2×C4⋊C4—71no—
395C2×C4⋊C4—71no—
396C2×C4⋊C4—141no—
397C2×C4⋊C4—141no—
398C2×C4⋊C4—141no—
399C2×C4⋊C4—141no—
400C2×C4⋊C4—141no—
401C2×C4⋊C4C22×D711yescharacteristic
402C2×C4⋊C4C22×D711yescharacteristic
403C2×C22⋊C4—21no—
404C2×C22⋊C4—71no—
405C2×C22⋊C4—71no—
406C2×C22⋊C4—141no—
407C2×C22⋊C4—141no—
408C2×C22⋊C4—141no—
409C2×C22⋊C4—141no—
410C2×C22⋊C4—141no—
411C2×C22⋊C4—141no—
412C2×C42—21no—
413C2×C42—71no—
414C2×C42—71no—
415C2×C42—141no—
416C2×C42—141no—
417#107—142no—
418#107—142no—
419#107—281no—
420#106—281no—
421C4⋊C8—142no—
422C4⋊C8—281no—
423#103—22no—
424#103—142no—
425#103—142no—
426#103C22⋊D714yes—
427#102—22no—
428#102—22no—
429#102—281no—
430#102—281no—
431C22⋊C8—142no—
432#95—71no—
433#95—71no—
434#95—141no—
435#95—141no—
436#95—141no—
437#95—141no—
438#95—141no—
439#95—141no—
440C22×C7—22no—
441C22×C7—22no—
442C22×C7—22no—
443C22×C7—41no—
444C22×C7—41no—
445C22×C7—41no—
446C22×C7—41no—
447C22×C7—42no—
448C22×C7—42no—
449C22×C7—42no—
450C22×C7—81no—
451C22×C7—81no—
452C22×C7—81no—
453C22×C7—81no—
454C22×C7C2×C22⋊Q811yescharacteristic
455C22×C7D4211yescharacteristic
456C22×C7#54611yescharacteristic
457C22×C7C2×#13511yescharacteristic
458C22×C7#57511yescharacteristic
459C22×C7#57611yescharacteristic
460C22×C7#57811yescharacteristic
461C4×C7—22no—
462C4×C7—24no—
463C4×C7—42no—
464C4×C7—81no—
465C4×C7C2×C22⋊Q811yescharacteristic
466C4×C7#54311yescharacteristic
467C4×C7#54611yescharacteristic
468C4×C7#55111yescharacteristic
469Dic7—22no—
470Dic7—41no—
471Dic7—41no—
472Dic7—41no—
473Dic7—81no—
474Dic7—81no—
475Dic7—81no—
476Dic7—81no—
477C24—21no—
478C24D4×D711yescharacteristic
479#41—28no—
480#41—44no—
481#41—44no—
482#41—44no—
483C2×Q8—21no—
484C2×Q8—21no—
485C2×Q8—21no—
486C2×Q8—282no—
487C2×Q8—282no—
488C2×Q8—561no—
489C2×Q8C2×Dic1412yes—
490C2×Q8#115712yes—
491C2×Q8C2×C22⋊D712yes—
492C2×D4—24no—
493C2×D4—21no—
494C2×D4—21no—
495C2×D4—21no—
496C2×D4—21no—
497C2×D4—21no—
498C2×D4—42no—
499C2×D4—42no—
500C2×D4C2×Dic1412yes—
501C22×C4—21no—
502C22×C4—21no—
503C22×C4—21no—
504C22×C4—21no—
505C22×C4—21no—
506C22×C4—21no—
507C22×C4—24no—
508C22×C4—21no—
509C22×C4—42no—
510C22×C4—42no—
511C22×C4—71no—
512C22×C4—71no—
513C22×C4—71no—
514C22×C4—71no—
515C22×C4—141no—
516C22×C4—141no—
517C22×C4—141no—
518C22×C4—141no—
519C22×C4C2×Dic1412yes—
520C22×C4D4×D711yescharacteristic
521C22×C4#115912yes—
522C22×C4C23×D711yescharacteristic
523C2×C8—21no—
524C2×C8—21no—
525C2×C8—21no—
526C2×C8—21no—
527C2×C8—21no—
528C2×C8—44no—
529C2×C8—141no—
530C2×C8—141no—
531C2×C8—141no—
532C2×C8—141no—
533C2×C8—141no—
534C2×C8—141no—
535C2×C8C2×C22⋊D712yes—
536C4⋊C4—21no—
537C4⋊C4—21no—
538C4⋊C4—21no—
539C4⋊C4—141no—
540C4⋊C4—141no—
541C4⋊C4—141no—
542C4⋊C4—141no—
543C4⋊C4—141no—
544C4⋊C4—141no—
545C4⋊C4—141no—
546C4⋊C4—141no—
547C4⋊C4—282no—
548C4⋊C4—282no—
549C4⋊C4—282no—
550C4⋊C4—282no—
551C4⋊C4—561no—
552C4⋊C4C2×C22⋊D712yes—
553C22⋊C4—42no—
554C22⋊C4—142no—
555C22⋊C4—282no—
556C22⋊C4—281no—
557C22⋊C4—282no—
558C22⋊C4—561no—
559C22⋊C4—561no—
560C22⋊C4—561no—
561C22⋊C4—561no—
562C42—42no—
563C42—141no—
564C42—141no—
565C42—141no—
566C42—141no—
567C42—282no—
568C42—282no—
569C2×C7—42no—
570C2×C7—82no—
571C2×C7#243711yescharacteristic
572C2×C7#307611yescharacteristic
573C2×C7#308311yescharacteristic
574C2×C7#326811yescharacteristic
575C2×C7#328511yescharacteristic
576C2×C7#330511yescharacteristic
577C2×C7#331311yescharacteristic
578C23—21no—
579C23—21no—
580C23—21no—
581C23—21no—
582C23—21no—
583C23—21no—
584C23—22no—
585C23—41no—
586C23—41no—
587C23—41no—
588C23—41no—
589C23—41no—
590C23—41no—
591C23#644112yes—
592C23C2×D4×D711yescharacteristic
593Q8—22no—
594Q8—22no—
595Q8—22no—
596Q8—564no—
597Q8#648714yes—
598D4—22no—
599D4—22no—
600D4—48no—
601D4—41no—
602D4—41no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—48no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—72no—
620C2×C4—141no—
621C2×C4—141no—
622C2×C4—141no—
623C2×C4—141no—
624C2×C4—141no—
625C2×C4—141no—
626C2×C4—141no—
627C2×C4—141no—
628C2×C4—141no—
629C2×C4—141no—
630C2×C4—141no—
631C2×C4—281no—
632C2×C4—281no—
633C2×C4—281no—
634C2×C4—281no—
635C2×C4—281no—
636C2×C4—281no—
637C2×C4—281no—
638C2×C4—281no—
639C2×C4—281no—
640C2×C4—281no—
641C2×C4—281no—
642C2×C4—281no—
643C2×C4#644112yes—
644C2×C4C22×Dic1411yescharacteristic
645C2×C4C2×#115711yescharacteristic
646C2×C4C2×#115911yescharacteristic
647C2×C4#654811yescharacteristic
648C2×C4#654811yescharacteristic
649C2×C4C22×C22⋊D711yescharacteristic
650C8—22no—
651C8—41no—
652C8—281no—
653C8—281no—
654C7#2595911yescharacteristic, complemented, Sylow
655C22—22no—
656C22—22no—
657C22—22no—
658C22—41no—
659C22—41no—
660C22—41no—
661C22—41no—
662C22—42no—
663C22—42no—
664C22—42no—
665C22—81no—
666C22—81no—
667C22—81no—
668C22—81no—
669C22C2×#644111yescharacteristic
670C22#9018111yescharacteristic
671C22D7×#13511yescharacteristic
672C22#9045911yescharacteristic
673C22#9046011yescharacteristic
674C22#9046211yescharacteristic
675C22D4×C22⋊D711yescharacteristic
676C4—22no—
677C4—24no—
678C4—42no—
679C4—81no—
680C4—142no—
681C4—281no—
682C4—281no—
683C4—281no—
684C4—561no—
685C4—561no—
686C4—561no—
687C4—561no—
688C4#9017411yescharacteristic
689C4#9018611yescharacteristic
690C4C2×#648711yescharacteristic
691C4#9049711yescharacteristic
692C2—42no—
693C2—82no—
694C2#1173626711yescharacteristic
695C2#1173626811yescharacteristic
696C2#1173629711yescharacteristic
697C2#1173630111yescharacteristic
698C2#1173955511yescharacteristic
699C2#1174136511yescharacteristic
700C2#1174136711yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC2×C7×#416C2
socleC23×C7C22×D4
prime radicalC22×C7×#41C22
hypercenterC2×#416D7
hypocenterC7#25959
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#416

Complements

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

NH ≅ G/Ntype
3. #11739693C2semidirect
4. C2×#89566C2semidirect
6. #11729728C2semidirect
7. #11729725C2semidirect
8. #11729330C2semidirect
9. #11727991C2semidirect
10. #11727990C2semidirect
11. #11727976C2semidirect
12. #11727948C2semidirect
13. #11727935C2semidirect
14. #11727934C2semidirect
15. #11727917C2semidirect
67. C2×C7×#41Q8semidirect
370. C2×#41Dic14semidirect
654. C7#25959semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#4162
#117415342
#117396932
C2×#895662
#117298892
#117297282
#117297252
#117293302
#117279912
#117279902
#117279762
#117279482
#117279352
#117279342
#117279172
#259597