Finite Group Databasebeta

← #13361017

Subgroups — #13361017

Order 1792.

960 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×C7×#121C211yesmaximal, characteristic
2C2×#89807C211yesmaximal, characteristic
3C4×#6489C211yesmaximal, characteristic, complemented
4#11729432C211yesmaximal, characteristic
5#11729363C211yesmaximal, characteristic, complemented
6#11729363C211yesmaximal, characteristic, complemented
7#11729283C211yesmaximal, characteristic, complemented
8#11727391C211yesmaximal, characteristic, complemented
9#11727390C211yesmaximal, characteristic, complemented
10#11727355C211yesmaximal, characteristic, complemented
11#11727354C211yesmaximal, characteristic, complemented
12#11727260C211yesmaximal, characteristic, complemented
13#11727257C211yesmaximal, characteristic, complemented
14#11727235C211yesmaximal, characteristic, complemented
15#11727235C211yesmaximal, characteristic, complemented
16C2×C7×#121C2211yescharacteristic
17C2×C4×C7×D4—21no—
18C2×C4×C7×D4C2211yescharacteristic
19C22×C42×C7C2211yescharacteristic
20C2×#6489C2211yescharacteristic
21C2×#6489C2211yescharacteristic
22C7×#390C2211yescharacteristic
23C7×#389C2211yescharacteristic
24C7×#388—21no—
25C7×#386—21no—
26C7×#380C2211yescharacteristic
27C4×C7×C4⋊C4C2211yescharacteristic
28C4×C7×C22⋊C4—21no—
29C2×#6406C2211yescharacteristic
30C2×#6406C2211yescharacteristic
31C2×#6406C2211yescharacteristic
32C2×#6406C2211yescharacteristic
33C4×C22⋊Dic7—21no—
34C4×C22⋊Dic7—21no—
35C2×C22⋊(C7⋊C8)C2211yescharacteristic
36C2×C22⋊(C7⋊C8)C2211yescharacteristic
37#89807C418yescomplemented
38C2×#6382C2211yescharacteristic
39C2×#6382C2211yescharacteristic
40C2×#6381C2211yescharacteristic
41C2×#6381C2211yescharacteristic
42C4×C4⋊Dic7C2211yescharacteristic
43C4×C4⋊Dic7C2211yescharacteristic
44C4×C4⋊Dic7C2211yescharacteristic
45C4×C4⋊Dic7C2211yescharacteristic
46#89704—21no—
47C42×Dic7—21no—
48C2×#6377C2211yescharacteristic
49C2×#6377C2211yescharacteristic
50#89324C2211yescharacteristic
51#89324C2211yescharacteristic
52#89324C2211yescharacteristic
53#89324C2211yescharacteristic
54#89324C2211yescharacteristic
55#89324C2211yescharacteristic
56#89324C2211yescharacteristic
57#89324C2211yescharacteristic
58#89319C2211yescharacteristic
59#89319C2211yescharacteristic
60#21582—71nomaximal, Sylow
61C22×C7×D4—21no—
62C22×C7×D4C2311yescharacteristic
63C23×C4×C7—21no—
64C23×C4×C7D411yescharacteristic
65C23×C4×C7D411yescharacteristic
66C23×C4×C7C2311yescharacteristic
67C7×#121—22no—
68C7×#121C2×C414yes—
69C4×C7×D4—22no—
70C4×C7×D4—22no—
71C4×C7×D4—44no—
72C2×C7×C4⋊C4—21no—
73C2×C7×C4⋊C4C2311yescharacteristic
74C2×C7×C4⋊C4C2311yescharacteristic
75C2×C7×C4⋊C4C2311yescharacteristic
76C2×C7×C22⋊C4—21no—
77C2×C7×C22⋊C4—21no—
78C2×C7×C22⋊C4—21no—
79C2×C7×C22⋊C4—21no—
80C2×C7×C22⋊C4—21no—
81C2×C42×C7—22no—
82C2×C42×C7—22no—
83C2×C42×C7—21no—
84C2×C42×C7D411yescharacteristic
85C2×C42×C7D411yescharacteristic
86C2×C42×C7C2311yescharacteristic
87C2×C42×C7C2311yescharacteristic
88C2×C22⋊Dic7—21no—
89C2×C22⋊Dic7—21no—
90#6489—22no—
91#6489—22no—
92#6489—22no—
93#6489C2×C414yescomplemented
94C2×C4⋊Dic7C2311yescharacteristic
95C2×C4⋊Dic7C2311yescharacteristic
96C2×C4⋊Dic7C2311yescharacteristic
97C2×C4⋊Dic7C2311yescharacteristic
98C2×C4×Dic7—21no—
99C2×C4×Dic7—21no—
100C2×C4×Dic7—21no—
101C2×C4×Dic7—21no—
102C2×C4×Dic7—21no—
103C2×C4×Dic7—21no—
104C22×C7⋊C8C2311yescharacteristic
105C22×C7⋊C8C2311yescharacteristic
106C22×C7⋊C8C2311yescharacteristic
107C22×C7⋊C8C2311yescharacteristic
108C7×#95—21no—
109C7×#95—21no—
110C7×#95—21no—
111#6406—22no—
112#6406—22no—
113#6406C2×C414yescomplemented
114#6406C2×C414yescomplemented
115C7⋊#95—21no—
116C7⋊#95—21no—
117C7⋊#95—21no—
118C7⋊#95—21no—
119C22⋊(C7⋊C8)—22no—
120C22⋊(C7⋊C8)C2×C414yescomplemented
121#6382—22no—
122#6382C2×C414yescomplemented
123#6381—22no—
124#6381C2×C414yescomplemented
125#6377—22no—
126#6377—22no—
127C2×#482—71no—
128C4×#121D711yescharacteristic
129C4×#117—71no—
130#1926—71no—
131#1925—71no—
132#1919—71no—
133#1918—71no—
134#1875—71no—
135#1854—71no—
136#1825—71no—
137#1776—71no—
138#1776—71no—
139#1762—71no—
140#1762—71no—
141#1749—71no—
142C24×C7—21no—
143C24×C7C2×D411yescharacteristic
144C2×C7×D4—24no—
145C2×C7×D4—21no—
146C2×C7×D4—21no—
147C2×C7×D4—21no—
148C2×C7×D4—21no—
149C2×C7×D4—21no—
150C2×C7×D4—42no—
151C2×C7×D4—42no—
152C2×C7×D4C22×C412yes—
153C22×C4×C7—21no—
154C22×C4×C7—21no—
155C22×C4×C7—21no—
156C22×C4×C7—21no—
157C22×C4×C7—21no—
158C22×C4×C7—22no—
159C22×C4×C7—22no—
160C22×C4×C7—22no—
161C22×C4×C7—21no—
162C22×C4×C7—21no—
163C22×C4×C7—41no—
164C22×C4×C7—41no—
165C22×C4×C7—41no—
166C22×C4×C7—42no—
167C22×C4×C7—42no—
168C22×C4×C7—42no—
169C22×C4×C7C22×C412yes—
170C22×C4×C7C2×D411yescharacteristic
171C22×C4×C7C2×D411yescharacteristic
172C22×C4×C7C2×D411yescharacteristic
173C22×C4×C7C2×D411yescharacteristic
174C22×C4×C7C2×D411yescharacteristic
175C22×C4×C7#4112yes—
176C22×C4×C7C2411yescharacteristic
177C22×Dic7—21no—
178C22×Dic7—21no—
179C22×Dic7—21no—
180C22×Dic7—21no—
181C7×C4⋊C4—21no—
182C7×C4⋊C4—21no—
183C7×C4⋊C4—21no—
184C7×C4⋊C4—21no—
185C7×C4⋊C4—21no—
186C7×C4⋊C4—42no—
187C7×C4⋊C4C22×C412yes—
188C7×C22⋊C4—24no—
189C7×C22⋊C4—42no—
190C7×C22⋊C4—42no—
191C7×C22⋊C4—42no—
192C7×C22⋊C4—42no—
193C42×C7—22no—
194C42×C7—22no—
195C42×C7—21no—
196C42×C7—21no—
197C42×C7—42no—
198C42×C7#4112yes—
199C42×C7#4112yes—
200C22⋊Dic7—24no—
201C22⋊Dic7—24no—
202C4⋊Dic7—21no—
203C4⋊Dic7—21no—
204C4⋊Dic7—21no—
205C4⋊Dic7—21no—
206C4⋊Dic7C22×C412yes—
207C4⋊Dic7C22×C412yes—
208C4⋊Dic7#4112yes—
209C4⋊Dic7#4112yes—
210C4×Dic7—24no—
211C4×Dic7—24no—
212C4×Dic7—42no—
213C4×Dic7—42no—
214C4×Dic7—42no—
215C4×Dic7—42no—
216C2×C7⋊C8—21no—
217C2×C7⋊C8—21no—
218C2×C7⋊C8—21no—
219C2×C7⋊C8—21no—
220C2×C7⋊C8—21no—
221C2×C7⋊C8—21no—
222C2×C7⋊C8—21no—
223C2×C7⋊C8—21no—
224C2×C7⋊C8—21no—
225C2×C7⋊C8—21no—
226C2×C7⋊C8C22×C412yes—
227C2×C7⋊C8C22×C412yes—
228C2×#121C2×D711yescharacteristic
229C2×C4×D4—21no—
230C2×C4×D4C2×D711yescharacteristic
231C2×#117—71no—
232C2×#117—71no—
233C22×C42C2×D711yescharacteristic
234#482—78no—
235C2×#107—71no—
236C2×#107—71no—
237C2×#106—71no—
238C2×#106—71no—
239C2×#102—71no—
240C2×#102—71no—
241C2×#102—71no—
242C2×#102—71no—
243C2×C22⋊C8—71no—
244C2×C22⋊C8—71no—
245C2×#97—71no—
246C2×#97—71no—
247#390C2×D711yescharacteristic
248#389C2×D711yescharacteristic
249#388—21no—
250#386—21no—
251#380C2×D711yescharacteristic
252C4×C4⋊C4—71no—
253C4×C4⋊C4—71no—
254C4×C4⋊C4—71no—
255C4×C4⋊C4—71no—
256C4×C4⋊C4C2×D711yescharacteristic
257C4×C22⋊C4—21no—
258C4×C22⋊C4—141no—
259C4×C22⋊C4—141no—
260#376—141no—
261C43—141no—
262#340—71no—
263#340—71no—
264#340—71no—
265#340—71no—
266#340—71no—
267#340—71no—
268#340—71no—
269#340—71no—
270#336—71no—
271#336—71no—
272C23×C7—21no—
273C23×C7—21no—
274C23×C7—21no—
275C23×C7—21no—
276C23×C7—21no—
277C23×C7—21no—
278C23×C7—22no—
279C23×C7—41no—
280C23×C7—41no—
281C23×C7—41no—
282C23×C7—41no—
283C23×C7—41no—
284C23×C7—41no—
285C23×C7C4×D412yes—
286C23×C7C22×D411yescharacteristic
287C7×D4—22no—
288C7×D4—22no—
289C7×D4—48no—
290C7×D4—41no—
291C7×D4—41no—
292C2×C4×C7—21no—
293C2×C4×C7—21no—
294C2×C4×C7—21no—
295C2×C4×C7—21no—
296C2×C4×C7—21no—
297C2×C4×C7—21no—
298C2×C4×C7—21no—
299C2×C4×C7—21no—
300C2×C4×C7—22no—
301C2×C4×C7—22no—
302C2×C4×C7—22no—
303C2×C4×C7—22no—
304C2×C4×C7—22no—
305C2×C4×C7—21no—
306C2×C4×C7—21no—
307C2×C4×C7—21no—
308C2×C4×C7—44no—
309C2×C4×C7—44no—
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×C7—84no—
319C2×C4×C7C4×D412yes—
320C2×C4×C7#12312yes—
321C2×C4×C7#12312yes—
322C2×C4×C7AGL(1,ℤ/8)12yes—
323C2×C4×C7C23×C411yescharacteristic
324C2×C4×C7C2×#4111yescharacteristic
325C2×C4×C7C2×#4111yescharacteristic
326C2×C4×C7C2×#4111yescharacteristic
327C2×C4×C7C2×#4111yescharacteristic
328C2×C4×C7C2×#4111yescharacteristic
329C2×Dic7—22no—
330C2×Dic7—22no—
331C2×Dic7—22no—
332C2×Dic7—22no—
333C2×Dic7—41no—
334C2×Dic7—41no—
335C2×Dic7—41no—
336C2×Dic7—41no—
337C2×Dic7—41no—
338C2×Dic7—41no—
339C2×Dic7—41no—
340C2×Dic7—41no—
341C7⋊C8—22no—
342C7⋊C8—22no—
343C7⋊C8—41no—
344C7⋊C8—41no—
345C22×D4—21no—
346C22×D4C22×D711yescharacteristic
347C23×C4—21no—
348C23×C4C22⋊D711yescharacteristic
349C23×C4C22⋊D711yescharacteristic
350C23×C4C22×D711yescharacteristic
351C22×C8—71no—
352C22×C8—71no—
353C22×C8—71no—
354C22×C8—71no—
355#121—22no—
356#121C4×D714yes—
357C4×D4—22no—
358C4×D4—22no—
359C4×D4—44no—
360#117—74no—
361#117—142no—
362#117—142no—
363#117—142no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—71no—
366C2×C4⋊C4—71no—
367C2×C4⋊C4—71no—
368C2×C4⋊C4—71no—
369C2×C4⋊C4C22×D711yescharacteristic
370C2×C4⋊C4C22×D711yescharacteristic
371C2×C4⋊C4C22×D711yescharacteristic
372C2×C22⋊C4—21no—
373C2×C22⋊C4—21no—
374C2×C22⋊C4—21no—
375C2×C22⋊C4—21no—
376C2×C22⋊C4—21no—
377C2×C22⋊C4—141no—
378C2×C22⋊C4—141no—
379C2×C42—22no—
380C2×C42—22no—
381C2×C42—21no—
382C2×C42—141no—
383C2×C42—141no—
384C2×C42—141no—
385C2×C42—141no—
386C2×C42—141no—
387C2×C42—141no—
388C2×C42C22⋊D711yescharacteristic
389C2×C42C22⋊D711yescharacteristic
390C2×C42C22×D711yescharacteristic
391C2×C42C22×D711yescharacteristic
392#107—74no—
393#107—142no—
394#106—74no—
395#106—142no—
396#102—74no—
397#102—74no—
398#102—142no—
399#102—142no—
400C22⋊C8—74no—
401C22⋊C8—142no—
402#97—142no—
403#97—142no—
404#95—21no—
405#95—21no—
406#95—21no—
407#95—141no—
408#95—141no—
409#95—141no—
410#95—141no—
411C22×C7—22no—
412C22×C7—22no—
413C22×C7—22no—
414C22×C7—41no—
415C22×C7—41no—
416C22×C7—41no—
417C22×C7—41no—
418C22×C7—42no—
419C22×C7—42no—
420C22×C7—42no—
421C22×C7—81no—
422C22×C7—81no—
423C22×C7—81no—
424C22×C7—81no—
425C22×C7C2×C4×D411yescharacteristic
426C22×C7C2×#12311yescharacteristic
427C22×C7#52511yescharacteristic
428C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
429C22×C7#57511yescharacteristic
430C22×C7#57511yescharacteristic
431C22×C7#57511yescharacteristic
432C4×C7—22no—
433C4×C7—22no—
434C4×C7—22no—
435C4×C7—22no—
436C4×C7—41no—
437C4×C7—42no—
438C4×C7—81no—
439C4×C7—81no—
440C4×C7C4×#4111yescharacteristic
441C4×C7C4×#4111yescharacteristic
442C4×C7#52911yescharacteristic
443C4×C7#52911yescharacteristic
444Dic7—42no—
445Dic7—42no—
446Dic7—42no—
447Dic7—42no—
448C24—21no—
449C24C2×C22⋊D711yescharacteristic
450C2×D4—24no—
451C2×D4—21no—
452C2×D4—21no—
453C2×D4—21no—
454C2×D4—21no—
455C2×D4—21no—
456C2×D4—42no—
457C2×D4—42no—
458C2×D4C2×C4×D712yes—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—41no—
470C22×C4—41no—
471C22×C4—41no—
472C22×C4—42no—
473C22×C4—42no—
474C22×C4—42no—
475C22×C4—141no—
476C22×C4—141no—
477C22×C4—141no—
478C22×C4—141no—
479C22×C4C2×C4×D712yes—
480C22×C4#115712yes—
481C22×C4C2×C22⋊D711yescharacteristic
482C22×C4C2×C22⋊D711yescharacteristic
483C22×C4C2×C22⋊D711yescharacteristic
484C22×C4C2×C22⋊D711yescharacteristic
485C22×C4C2×C22⋊D711yescharacteristic
486C22×C4C23×D711yescharacteristic
487C2×C8—72no—
488C2×C8—72no—
489C2×C8—141no—
490C2×C8—141no—
491C2×C8—141no—
492C2×C8—141no—
493C2×C8—141no—
494C2×C8—141no—
495C2×C8—141no—
496C2×C8—141no—
497C2×C8—141no—
498C2×C8—141no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—42no—
505C4⋊C4—72no—
506C4⋊C4—72no—
507C4⋊C4—72no—
508C4⋊C4—72no—
509C4⋊C4—141no—
510C4⋊C4—141no—
511C4⋊C4—141no—
512C4⋊C4—141no—
513C4⋊C4C2×C4×D712yes—
514C22⋊C4—24no—
515C22⋊C4—42no—
516C22⋊C4—42no—
517C22⋊C4—42no—
518C22⋊C4—42no—
519C22⋊C4—144no—
520C22⋊C4—144no—
521C42—22no—
522C42—22no—
523C42—21no—
524C42—21no—
525C42—42no—
526C42—144no—
527C42—144no—
528C42—282no—
529C42—282no—
530C42—282no—
531C42—282no—
532C42#115912yes—
533C42#115912yes—
534C2×C7—42no—
535C2×C7—82no—
536C2×C7C4×#12311yescharacteristic
537C2×C7#294311yescharacteristic
538C2×C7C4×AGL(1,ℤ/8)11yescharacteristic
539C2×C7#309311yescharacteristic
540C2×C7#309411yescharacteristic
541C2×C7#311111yescharacteristic
542C2×C7#311511yescharacteristic
543C23—21no—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—22no—
550C23—41no—
551C23—41no—
552C23—41no—
553C23—41no—
554C23—41no—
555C23—41no—
556C23C4×C22⋊D712yes—
557C23C22×C22⋊D711yescharacteristic
558D4—22no—
559D4—22no—
560D4—48no—
561D4—41no—
562D4—41no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—84no—
590C2×C4—142no—
591C2×C4—142no—
592C2×C4—142no—
593C2×C4—142no—
594C2×C4—281no—
595C2×C4—281no—
596C2×C4—281no—
597C2×C4—281no—
598C2×C4—281no—
599C2×C4—281no—
600C2×C4—281no—
601C2×C4—281no—
602C2×C4C4×C22⋊D712yes—
603C2×C4#649512yes—
604C2×C4#649812yes—
605C2×C4#649812yes—
606C2×C4C22×C4×D711yescharacteristic
607C2×C4C2×#115711yescharacteristic
608C2×C4C2×#115911yescharacteristic
609C2×C4C2×#115911yescharacteristic
610C2×C4D7×#4111yescharacteristic
611C2×C4D7×#4111yescharacteristic
612C8—142no—
613C8—142no—
614C8—281no—
615C8—281no—
616C7#2158211yescharacteristic, complemented, Sylow
617C22—22no—
618C22—22no—
619C22—22no—
620C22—41no—
621C22—41no—
622C22—41no—
623C22—41no—
624C22—42no—
625C22—42no—
626C22—42no—
627C22—81no—
628C22—81no—
629C22—81no—
630C22—81no—
631C22C2×C4×C22⋊D711yescharacteristic
632C22C2×#649511yescharacteristic
633C22C2×#649811yescharacteristic
634C22#9051311yescharacteristic
635C22#9051311yescharacteristic
636C22#9051311yescharacteristic
637C22#9051911yescharacteristic
638C4—22no—
639C4—22no—
640C4—22no—
641C4—22no—
642C4—42no—
643C4—41no—
644C4—81no—
645C4—81no—
646C4—282no—
647C4—282no—
648C4—282no—
649C4—282no—
650C4C4×#115911yescharacteristic
651C4C4×#115911yescharacteristic
652C4#9022911yescharacteristic
653C4#9022911yescharacteristic
654C2—42no—
655C2—82no—
656C2C4×#649511yescharacteristic
657C2C4×#649811yescharacteristic
658C2#1173995111yescharacteristic
659C2#1174006111yescharacteristic
660C2#1174036011yescharacteristic
661C2#1174058711yescharacteristic
662C2#1174074611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC4×C7×#121C2
socleC23×C7C22×D4
prime radicalC2×C7×#121C22
cosocleC22×C4×C7C24
hypercenterC4×#121D7
hypocenterC7#21582
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C22×C42 ◁ C4×#121

Complements

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

NH ≅ G/Ntype
3. C4×#6489C2semidirect
5. #11729363C2semidirect
6. #11729363C2semidirect
7. #11729283C2semidirect
8. #11727391C2semidirect
9. #11727390C2semidirect
10. #11727355C2semidirect
11. #11727354C2semidirect
12. #11727260C2semidirect
13. #11727257C2semidirect
14. #11727235C2semidirect
15. #11727235C2semidirect
37. #89807C4semidirect
93. #6489C2×C4semidirect
113. #6406C2×C4semidirect
114. #6406C2×C4semidirect
120. C22⋊(C7⋊C8)C2×C4semidirect
122. #6382C2×C4semidirect
124. #6381C2×C4semidirect
616. C7#21582semidirect

Maximal subgroups

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

subgroupindex
C4×C7×#1212
C2×#898072
C4×#64892
#117294322
#117293632
#117292832
#117273912
#117273902
#117273552
#117273542
#117272602
#117272572
#117272352
#215827