Finite Group Databasebeta

← #13360410

Subgroups — #13360410

Order 1792.

1010 conjugacy classes of subgroups.

The subgroup lattice has 694 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#405C211yesmaximal, characteristic, complemented
2C2×#89886C211yesmaximal, characteristic
3#11739681C211yesmaximal, characteristic
4#11739680C211yesmaximal, characteristic, complemented
5C2×#89482C211yesmaximal, characteristic, complemented
6#11729744C211yesmaximal, characteristic
7#11729743C211yesmaximal, characteristic, complemented
8#11729725C211yesmaximal, characteristic, complemented
9#11727580C211yesmaximal, characteristic, complemented
10#11727574C211yesmaximal, characteristic, complemented
11#11727568C211yesmaximal, characteristic, complemented
12#11727522C211yesmaximal, characteristic, complemented
13#11727521C211yesmaximal, characteristic, complemented
14C4×#6390C211yesmaximal, characteristic, complemented
15C4×#6395C211yesmaximal, characteristic, complemented
16C22×C7×M16C2211yescharacteristic
17C23×C7×C8C2211yescharacteristic
18C2×C7×#117C2211yescharacteristic
19C2×#6493C2211yescharacteristic
20C2×#6487C2211yescharacteristic
21C22×#1132C2211yescharacteristic
22C2×C4×D28C2211yescharacteristic
23C2×C4×Dic14C2211yescharacteristic
24C7×#405—24no—
25C2×C7×#97C2211yescharacteristic
26C2×C7×#97C2211yescharacteristic
27C2×C4×C7×C8C2211yescharacteristic
28C2×C4×C7×C8C2211yescharacteristic
29#89984C2211yescharacteristic
30#89984C2211yescharacteristic
31#89886C418yescomplemented
32C2×#6395C2211yescharacteristic
33C2×#6395C2211yescharacteristic
34C2×#6395C2211yescharacteristic
35C2×#6392C2211yescharacteristic
36C2×#6392C2211yescharacteristic
37C2×#6391C2211yescharacteristic
38C2×#6390C2211yescharacteristic
39C2×#6390C2211yescharacteristic
40C2×#6390C2211yescharacteristic
41#89762—21no—
42#89761—21no—
43#89750—21no—
44#89744—21no—
45#89729—21no—
46#89728—21no—
47#89717—21no—
48#89716—21no—
49#89711C2211yescharacteristic
50C4×C4⋊Dic7C2211yescharacteristic
51C4×C4⋊Dic7C2211yescharacteristic
52#89701C2211yescharacteristic
53#89700C2211yescharacteristic
54#89700C2211yescharacteristic
55#89482—24no—
56#89346C2211yescharacteristic
57#89346C2211yescharacteristic
58#89346C2211yescharacteristic
59#89346C2211yescharacteristic
60#89346C2211yescharacteristic
61#89346C2211yescharacteristic
62#21531—71nomaximal, Sylow
63C23×C4×C7C2311yescharacteristic
64C22×C22⋊D7—21no—
65C22×D28C2311yescharacteristic
66C22×C4×D7—21no—
67C22×Dic14C2311yescharacteristic
68C2×C7×M16—22no—
69C2×C7×M16—22no—
70C2×C7×M16—22no—
71C22×C7×C8—22no—
72C22×C7×C8—22no—
73C22×C7×C8C2×C414yescomplemented
74C22×C7×C8C2311yescharacteristic
75C22×C7×C8C2311yescharacteristic
76C22×C7×C8C2311yescharacteristic
77C22×C7×C8C2311yescharacteristic
78C7×#117—22no—
79C7×#117—22no—
80C2×C7×C4⋊C4D411yescharacteristic
81C2×C7×C4⋊C4D411yescharacteristic
82C2×C7×C22⋊C4D411yescharacteristic
83C2×C7×C22⋊C4D411yescharacteristic
84C2×C42×C7C2311yescharacteristic
85C2×C42×C7C2311yescharacteristic
86C2×C22⋊Dic7—21no—
87C2×C22⋊Dic7—21no—
88C2×C22⋊Dic7—21no—
89#6493—22no—
90#6493C2×C414yes—
91C2×(C2×D7)⋊C4—21no—
92C2×(C2×D7)⋊C4—21no—
93C2×(C2×D7)⋊C4—21no—
94C2×(C2×D7)⋊C4—21no—
95C2×C4⋊Dic7C2311yescharacteristic
96C2×C4⋊Dic7C2311yescharacteristic
97C2×C4⋊Dic7C2311yescharacteristic
98C2×C4⋊Dic7C2311yescharacteristic
99C2×C4⋊Dic7C2311yescharacteristic
100C2×C4⋊Dic7C2311yescharacteristic
101#6487—22no—
102#6487C2×C414yescomplemented
103C2×Dic7⋊C4—21no—
104C2×Dic7⋊C4—21no—
105C2×Dic7⋊C4—21no—
106C2×Dic7⋊C4—21no—
107C2×C4×Dic7—21no—
108C2×C4×Dic7—21no—
109C2×C4×Dic7—21no—
110C2×#1132—22no—
111C2×#1132—22no—
112C2×#1132C2×C414yes—
113C4×D28—22no—
114C4×D28—22no—
115C4×Dic14—22no—
116C4×Dic14—22no—
117C7×#97—22no—
118C7×#97D414yescomplemented
119C4×C7×C8—22no—
120C4×C7×C8—22no—
121C7⋊#95—21no—
122C7⋊#95—21no—
123C7⋊#95—21no—
124C7⋊#95—21no—
125C7⋊#95—21no—
126C7⋊#95—21no—
127C7⋊#95—21no—
128C7⋊#95—21no—
129#6395—22no—
130#6395—22no—
131#6395C2×C414yes—
132#6392—22no—
133#6392—22no—
134#6391C2×C414yescomplemented
135#6390—22no—
136#6390—22no—
137#6390C2×C414yescomplemented
138C2×#465—71no—
139C2×#440—71no—
140C2×#405D711yescharacteristic, complemented
141#2324—71no—
142#2323—71no—
143#1973—71no—
144#1944—71no—
145#1934—71no—
146#1933—71no—
147#1895—71no—
148#1894—71no—
149#1808—71no—
150#1776—71no—
151C4×#103—71no—
152C4×#102—71no—
153C24×C7C2×D411yescharacteristic
154C23×D7—21no—
155C22×C4×C7—21no—
156C22×C4×C7—21no—
157C22×C4×C7—21no—
158C22×C4×C7—21no—
159C22×C4×C7C22×C412yes—
160C22×C4×C7C2×D411yescharacteristic
161C22×C4×C7C2×D411yescharacteristic
162C22×C4×C7C2×D411yescharacteristic
163C22×C4×C7C2×D411yescharacteristic
164C22×C4×C7C2×D411yescharacteristic
165C22×C4×C7#4112yes—
166C22×C4×C7C2411yescharacteristic
167C2×C22⋊D7—24no—
168C2×C22⋊D7—81no—
169C2×C22⋊D7—81no—
170C22×Dic7—21no—
171C22×Dic7—21no—
172C22×Dic7—21no—
173C22×Dic7—21no—
174C22×Dic7—21no—
175C22×Dic7—21no—
176C22×Dic7—21no—
177C2×D28—21no—
178C2×D28—21no—
179C2×D28—21no—
180C2×D28—21no—
181C2×D28—21no—
182C2×D28C22×C412yes—
183C2×C4×D7—42no—
184C2×C4×D7—81no—
185C2×C4×D7—81no—
186C2×Dic14—21no—
187C2×Dic14—21no—
188C2×Dic14—21no—
189C2×Dic14—21no—
190C2×Dic14—21no—
191C2×Dic14C22×C412yes—
192C7×M16—28no—
193C2×C7×C8—21no—
194C2×C7×C8—21no—
195C2×C7×C8—21no—
196C2×C7×C8—21no—
197C2×C7×C8—21no—
198C2×C7×C8—21no—
199C2×C7×C8—28no—
200C2×C7×C8C22×C412yes—
201C2×C7×C8C22×C412yes—
202C2×C7×C8C2×D412yes—
203C2×C7×C8C2×D412yes—
204C2×C7×C8#4112yes—
205C2×C7×C8#4112yes—
206C7×C4⋊C4—22no—
207C7×C4⋊C4—22no—
208C7×C22⋊C4—22no—
209C7×C22⋊C4—22no—
210C42×C7—21no—
211C42×C7—21no—
212C42×C7C2×D412yes—
213C42×C7#4112yes—
214C22⋊Dic7—24no—
215C22⋊Dic7—81no—
216C22⋊Dic7—81no—
217(C2×D7)⋊C4—24no—
218(C2×D7)⋊C4—42no—
219(C2×D7)⋊C4—81no—
220(C2×D7)⋊C4—81no—
221C4⋊Dic7—21no—
222C4⋊Dic7—21no—
223C4⋊Dic7—21no—
224C4⋊Dic7—21no—
225C4⋊Dic7—21no—
226C4⋊Dic7—21no—
227C4⋊Dic7—21no—
228C4⋊Dic7—21no—
229C4⋊Dic7—21no—
230C4⋊Dic7—21no—
231C4⋊Dic7C22×C412yes—
232C4⋊Dic7C22×C412yes—
233Dic7⋊C4—24no—
234Dic7⋊C4—42no—
235Dic7⋊C4—81no—
236Dic7⋊C4—81no—
237C4×Dic7—42no—
238C4×Dic7—81no—
239C4×Dic7—81no—
240#1132—28no—
241C22×SD16—71no—
242C22×M16C2×D711yescharacteristic
243C23×C8C2×D711yescharacteristic
244C2×C22⋊Q8—71no—
245C2×#121—71no—
246C2×C4×Q8—71no—
247C2×C4×D4—71no—
248C2×#117C2×D711yescharacteristic
249#465—78no—
250#440—144no—
251C2×#107—71no—
252C2×#107—71no—
253C2×#106—71no—
254C2×#103—71no—
255C2×#103—71no—
256C2×#103—71no—
257C2×#102—71no—
258C2×#102—71no—
259C2×#102—71no—
260#405—24no—
261C2×#97C2×D711yescharacteristic
262C2×#97C2×D711yescharacteristic
263C2×C4×C8C2×D711yescharacteristic
264C2×C4×C8C2×D711yescharacteristic
265#391—71no—
266#390—71no—
267#390—141no—
268#389—71no—
269#389—71no—
270#389—141no—
271#388—141no—
272#388—141no—
273#388—141no—
274#387—141no—
275#386—141no—
276#385—141no—
277#380—71no—
278#380—71no—
279C4×C4⋊C4—71no—
280C4×C4⋊C4—71no—
281#340—71no—
282#340—71no—
283#340—71no—
284#340—71no—
285#340—71no—
286#340—71no—
287C23×C7—21no—
288C23×C7—21no—
289C23×C7—21no—
290C23×C7—21no—
291C23×C7—21no—
292C23×C7—21no—
293C23×C7C4×D412yes—
294C23×C7C22×D411yescharacteristic
295C22×D7—22no—
296C22×D7—41no—
297C22×D7—41no—
298C22×D7—41no—
299C22×D7—41no—
300C22×D7—41no—
301C22×D7—41no—
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—22no—
309C2×C4×C7—22no—
310C2×C4×C7—22no—
311C2×C4×C7—22no—
312C2×C4×C7—21no—
313C2×C4×C7—21no—
314C2×C4×C7—21no—
315C2×C4×C7—21no—
316C2×C4×C7—21no—
317C2×C4×C7—21no—
318C2×C4×C7C4×D412yes—
319C2×C4×C7#12112yes—
320C2×C4×C7#12112yes—
321C2×C4×C7C23×C411yescharacteristic
322C2×C4×C7C22×D411yescharacteristic
323C2×C4×C7C2×#4111yescharacteristic
324C2×C4×C7C2×#4111yescharacteristic
325C2×C4×C7C2×#4111yescharacteristic
326C2×C4×C7C2×#4111yescharacteristic
327C22⋊D7—84no—
328C2×Dic7—22no—
329C2×Dic7—22no—
330C2×Dic7—22no—
331C2×Dic7—41no—
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×Dic7—41no—
343C2×Dic7—41no—
344C2×Dic7—41no—
345C2×Dic7—41no—
346C2×Dic7—41no—
347C2×Dic7—41no—
348C2×Dic7—41no—
349D28—22no—
350D28—22no—
351D28—41no—
352D28—41no—
353C4×D7—84no—
354Dic14—22no—
355Dic14—22no—
356Dic14—41no—
357Dic14—41no—
358C7×C8—22no—
359C7×C8—22no—
360C7×C8—22no—
361C7×C8C4×D414yes—
362C22×Q8—71no—
363C22×D4—71no—
364C22×D4—141no—
365C23×C4—141no—
366C23×C4C22×D711yescharacteristic
367C2×SD16—74no—
368C2×SD16—142no—
369C2×SD16—142no—
370C2×M16—22no—
371C2×M16—22no—
372C2×M16—22no—
373C22×C8—22no—
374C22×C8—22no—
375C22×C8C4×D714yescomplemented
376C22×C8C22×D711yescharacteristic
377C22×C8C22×D711yescharacteristic
378C22×C8C22×D711yescharacteristic
379C22×C8C22×D711yescharacteristic
380C22⋊Q8—74no—
381C22⋊Q8—142no—
382#121—74no—
383#121—142no—
384C4×Q8—142no—
385C4×Q8—142no—
386C4×D4—142no—
387C4×D4—142no—
388#117—22no—
389#117—22no—
390C2×C4⋊C4—71no—
391C2×C4⋊C4—71no—
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⋊C4D2811yescharacteristic
401C2×C4⋊C4D2811yescharacteristic
402C2×C22⋊C4—141no—
403C2×C22⋊C4—141no—
404C2×C22⋊C4—141no—
405C2×C22⋊C4—141no—
406C2×C22⋊C4—141no—
407C2×C22⋊C4—141no—
408C2×C22⋊C4—141no—
409C2×C22⋊C4D2811yescharacteristic
410C2×C22⋊C4D2811yescharacteristic
411C2×C42—141no—
412C2×C42—141no—
413C2×C42—141no—
414C2×C42C22×D711yescharacteristic
415C2×C42C22×D711yescharacteristic
416#107—142no—
417#107—142no—
418#106—74no—
419#103—74no—
420#103—142no—
421#103—142no—
422#102—74no—
423#102—142no—
424#102—142no—
425#97—22no—
426#97C22⋊D714yescomplemented
427C4×C8—22no—
428C4×C8—22no—
429#95—141no—
430#95—141no—
431#95—141no—
432#95—141no—
433#95—141no—
434#95—141no—
435#95—141no—
436#95—141no—
437C22×C7—22no—
438C22×C7—22no—
439C22×C7—22no—
440C22×C7—41no—
441C22×C7—41no—
442C22×C7—41no—
443C22×C7—41no—
444C22×C7C2×C4×D411yescharacteristic
445C22×C7C2×#12111yescharacteristic
446C22×C7#53211yescharacteristic
447C22×C7#57611yescharacteristic
448C22×C7#57711yescharacteristic
449C22×C7#57711yescharacteristic
450C22×C7#57811yescharacteristic
451C2×D7—42no—
452C2×D7—42no—
453C2×D7—42no—
454C2×D7—81no—
455C2×D7—81no—
456C2×D7—81no—
457C2×D7—81no—
458C4×C7—22no—
459C4×C7—22no—
460C4×C7—22no—
461C4×C7—41no—
462C4×C7—41no—
463C4×C7C2×C4×D411yescharacteristic
464C4×C7C4×#4111yescharacteristic
465C4×C7#52511yescharacteristic
466C4×C7#52911yescharacteristic
467Dic7—42no—
468Dic7—42no—
469Dic7—42no—
470Dic7—81no—
471Dic7—81no—
472Dic7—81no—
473Dic7—81no—
474C24—141no—
475C24C2×D2811yescharacteristic
476C2×Q8—72no—
477C2×Q8—141no—
478C2×Q8—141no—
479C2×Q8—141no—
480C2×Q8—141no—
481C2×Q8—141no—
482C2×D4—72no—
483C2×D4—144no—
484C2×D4—141no—
485C2×D4—141no—
486C2×D4—141no—
487C2×D4—141no—
488C2×D4—141no—
489C2×D4—561no—
490C2×D4—561no—
491C22×C4—21no—
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—141no—
496C22×C4—141no—
497C22×C4—141no—
498C22×C4—141no—
499C22×C4—141no—
500C22×C4—141no—
501C22×C4—141no—
502C22×C4—282no—
503C22×C4—561no—
504C22×C4—561no—
505C22×C4C2×C4×D712yes—
506C22×C4C2×D2811yescharacteristic
507C22×C4C2×D2811yescharacteristic
508C22×C4C2×D2811yescharacteristic
509C22×C4C2×D2811yescharacteristic
510C22×C4C2×D2811yescharacteristic
511C22×C4#115712yes—
512C22×C4C23×D711yescharacteristic
513SD16—148no—
514M16—28no—
515C2×C8—21no—
516C2×C8—21no—
517C2×C8—21no—
518C2×C8—21no—
519C2×C8—21no—
520C2×C8—21no—
521C2×C8—28no—
522C2×C8C2×C4×D712yes—
523C2×C8C2×C4×D712yes—
524C2×C8#115712yes—
525C2×C8#115712yes—
526C2×C8C2×C22⋊D712yes—
527C2×C8C2×C22⋊D712yes—
528C4⋊C4—22no—
529C4⋊C4—22no—
530C4⋊C4—72no—
531C4⋊C4—72no—
532C4⋊C4—144no—
533C4⋊C4—141no—
534C4⋊C4—141no—
535C4⋊C4—141no—
536C4⋊C4—141no—
537C4⋊C4—141no—
538C4⋊C4—141no—
539C4⋊C4—141no—
540C4⋊C4—141no—
541C4⋊C4—141no—
542C4⋊C4—141no—
543C4⋊C4—282no—
544C4⋊C4—561no—
545C4⋊C4—561no—
546C22⋊C4—22no—
547C22⋊C4—22no—
548C22⋊C4—144no—
549C22⋊C4—144no—
550C22⋊C4—282no—
551C22⋊C4—561no—
552C22⋊C4—561no—
553C22⋊C4—561no—
554C22⋊C4—561no—
555C42—21no—
556C42—21no—
557C42—282no—
558C42—561no—
559C42—561no—
560C42#115712yes—
561C42C2×C22⋊D712yes—
562C2×C7—42no—
563C2×C7C4×#12111yescharacteristic
564C2×C7#294911yescharacteristic
565C2×C7#295111yescharacteristic
566C2×C7#306211yescharacteristic
567C2×C7#306311yescharacteristic
568C2×C7#317811yescharacteristic
569C2×C7#317911yescharacteristic
570D7—82no—
571C23—21no—
572C23—21no—
573C23—21no—
574C23—21no—
575C23—21no—
576C23—21no—
577C23—142no—
578C23—281no—
579C23—281no—
580C23—281no—
581C23—281no—
582C23—281no—
583C23—281no—
584C23C4×D2812yes—
585C23C22×D2811yescharacteristic
586Q8—142no—
587Q8—142no—
588Q8—281no—
589Q8—281no—
590D4—142no—
591D4—142no—
592D4—281no—
593D4—281no—
594D4—564no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—142no—
612C2×C4—142no—
613C2×C4—142no—
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—281no—
625C2×C4—281no—
626C2×C4—281no—
627C2×C4—281no—
628C2×C4—281no—
629C2×C4—281no—
630C2×C4—281no—
631C2×C4—281no—
632C2×C4—564no—
633C2×C4C4×D2812yes—
634C2×C4#649312yes—
635C2×C4#649312yes—
636C2×C4C22×C4×D711yescharacteristic
637C2×C4C2×#115711yescharacteristic
638C2×C4C2×#115711yescharacteristic
639C2×C4C2×#115711yescharacteristic
640C2×C4C2×#115711yescharacteristic
641C2×C4C22×C22⋊D711yescharacteristic
642C8—22no—
643C8—22no—
644C8—22no—
645C8C4×C22⋊D714yes—
646C7#2153111yescharacteristic, complemented, Sylow
647C22—22no—
648C22—22no—
649C22—22no—
650C22—41no—
651C22—41no—
652C22—41no—
653C22—41no—
654C22—282no—
655C22—282no—
656C22—282no—
657C22—561no—
658C22—561no—
659C22—561no—
660C22—561no—
661C22C2×C4×D2811yescharacteristic
662C22#9016811yescharacteristic
663C22#9044211yescharacteristic
664C22#9044211yescharacteristic
665C22#9044311yescharacteristic
666C22#9044411yescharacteristic
667C22C2×#649311yescharacteristic
668C4—22no—
669C4—22no—
670C4—22no—
671C4—41no—
672C4—41no—
673C4—282no—
674C4—282no—
675C4—282no—
676C4—561no—
677C4—561no—
678C4—561no—
679C4—561no—
680C4C4×#115711yescharacteristic
681C4#9017011yescharacteristic
682C4C2×C4×C22⋊D711yescharacteristic
683C4#9048211yescharacteristic
684C2—42no—
685C2—562no—
686C2#1173600711yescharacteristic
687C2#1173601511yescharacteristic
688C2#1173601711yescharacteristic
689C2#1173602311yescharacteristic
690C2C4×#649311yescharacteristic
691C2#1174124711yescharacteristic
692C2#1174125311yescharacteristic
subgroupG/H
centerC23C22×D28
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC2×C7×#405C2
socleC23×C7C22×D4
prime radicalC2×#6493C22
cosocleC22×C4×C7C24
hypercenterC2×#405D7
hypocenterC7#21531
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#405

Complements

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

NH ≅ G/Ntype
1. C2×C7×#405C2semidirect
4. #11739680C2semidirect
5. C2×#89482C2semidirect
7. #11729743C2semidirect
8. #11729725C2semidirect
9. #11727580C2semidirect
10. #11727574C2semidirect
11. #11727568C2semidirect
12. #11727522C2semidirect
13. #11727521C2semidirect
14. C4×#6390C2semidirect
15. C4×#6395C2semidirect
31. #89886C4semidirect
73. C22×C7×C8C2×C4semidirect
102. #6487C2×C4semidirect
118. C7×#97D4semidirect
134. #6391C2×C4semidirect
137. #6390C2×C4semidirect
140. C2×#405D7semidirect
375. C22×C8C4×D7semidirect
426. #97C22⋊D7semidirect
646. C7#21531semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#4052
C2×#898862
#117396812
#117396802
C2×#894822
#117297442
#117297432
#117297252
#117275802
#117275742
#117275682
#117275222
#117275212
C4×#63902
C4×#63952
#215317