Finite Group Databasebeta

← #13395061

Subgroups — #13395061

Order 1792.

868 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2851C211yesmaximal, characteristic, complemented
2C2×#89838C211yesmaximal, characteristic, complemented
3C4×#6493C211yesmaximal, characteristic
4C2×#89698C211yesmaximal, characteristic, complemented
5#11729492C211yesmaximal, characteristic, complemented
6#11727375C211yesmaximal, characteristic, complemented
7#11727374C211yesmaximal, characteristic, complemented
8#11727371C211yesmaximal, characteristic, complemented
9#11727371C211yesmaximal, characteristic, complemented
10#11727310C211yesmaximal, characteristic, complemented
11#11727294C211yesmaximal, characteristic, complemented
12#11727278C211yesmaximal, characteristic
13#11727277C211yesmaximal, characteristic
14#11727272C211yesmaximal, characteristic, complemented
15#11727261C211yesmaximal, characteristic, complemented
16C2×C7×#125C2211yescharacteristic
17C22×C42×C7C2211yescharacteristic
18C2×#6493C2211yescharacteristic
19C2×C4×C22⋊D7—21no—
20C2×C4×D28C2211yescharacteristic
21C7×#389C2211yescharacteristic
22C7×#389C2211yescharacteristic
23C7×#387—21no—
24C7×#387—21no—
25C7×#385—21no—
26C7×#385—21no—
27C7×#382C2211yescharacteristic
28C7×#380C2211yescharacteristic
29C7×#380C2211yescharacteristic
30#89985—21no—
31#89984C2211yescharacteristic
32C2×C22⋊(C7⋊C8)C2211yescharacteristic
33C2×C22⋊(C7⋊C8)C2211yescharacteristic
34#89838—24no—
35C2×#6383C2211yescharacteristic
36C2×#6383C2211yescharacteristic
37C2×#6383C2211yescharacteristic
38C2×#6383C2211yescharacteristic
39C2×#6382C2211yescharacteristic
40C2×#6382C2211yescharacteristic
41C2×#6381C2211yescharacteristic
42C2×#6381C2211yescharacteristic
43#89712—21no—
44#89711C2211yescharacteristic
45C4×(C2×D7)⋊C4—21no—
46C4×C4⋊Dic7C2211yescharacteristic
47#89700C2211yescharacteristic
48#89698—24no—
49C2×C4⋊(C7⋊C8)C2211yescharacteristic
50C2×C4⋊(C7⋊C8)C2211yescharacteristic
51C2×#6377C2211yescharacteristic
52C2×#6377C2211yescharacteristic
53#89324C2211yescharacteristic
54#89324C2211yescharacteristic
55#89324C2211yescharacteristic
56#89324C2211yescharacteristic
57#89324C2211yescharacteristic
58#89324C2211yescharacteristic
59#89324C2211yescharacteristic
60#89324C2211yescharacteristic
61#24706—71nomaximal, Sylow
62C23×C4×C7D411yescharacteristic
63C23×C4×C7D411yescharacteristic
64C23×C4×C7C2311yescharacteristic
65C22×C22⋊D7—21no—
66C22×D28C2311yescharacteristic
67C22×C4×D7—21no—
68C7×#125—22no—
69C7×#125—22no—
70C2×C7×C4⋊C4—21no—
71C2×C7×C4⋊C4—21no—
72C2×C7×C4⋊C4—21no—
73C2×C7×C4⋊C4—21no—
74C2×C7×C4⋊C4C2311yescharacteristic
75C2×C7×C4⋊C4C2311yescharacteristic
76C2×C7×C4⋊C4C2311yescharacteristic
77C2×C7×C4⋊C4C2311yescharacteristic
78C2×C7×C22⋊C4—21no—
79C2×C7×C22⋊C4—21no—
80C2×C42×C7—22no—
81C2×C42×C7—22no—
82C2×C42×C7D411yescharacteristic
83C2×C42×C7D411yescharacteristic
84C2×C42×C7C2311yescharacteristic
85C2×C42×C7C2311yescharacteristic
86C2×C22⋊Dic7—21no—
87#6493—22no—
88#6493—22no—
89C4×C22⋊D7—44no—
90C2×(C2×D7)⋊C4—21no—
91C2×(C2×D7)⋊C4—21no—
92C2×(C2×D7)⋊C4—21no—
93C2×(C2×D7)⋊C4—21no—
94C2×C4⋊Dic7C2311yescharacteristic
95C2×C4⋊Dic7C2311yescharacteristic
96C2×C4⋊Dic7C2311yescharacteristic
97C2×Dic7⋊C4—21no—
98C2×C4×Dic7—21no—
99C22×C7⋊C8C2311yescharacteristic
100C22×C7⋊C8C2311yescharacteristic
101C22×C7⋊C8C2311yescharacteristic
102C22×C7⋊C8C2311yescharacteristic
103C4×D28—22no—
104C4×D28—22no—
105C7×#95—21no—
106C7×#95—21no—
107C7×#95—21no—
108C7×#95—21no—
109C7×#95—21no—
110C7×#95—21no—
111C7⋊#95—21no—
112C7⋊#95—21no—
113C7⋊#95—21no—
114C22⋊(C7⋊C8)—22no—
115C22⋊(C7⋊C8)—22no—
116#6383—22no—
117#6383—22no—
118#6383—22no—
119#6383—22no—
120#6382—22no—
121#6382—22no—
122#6381—22no—
123#6381—22no—
124C4⋊(C7⋊C8)—22no—
125C4⋊(C7⋊C8)D414yescomplemented
126#6377—22no—
127#6377—22no—
128C2×#478—71no—
129C2×#432—71no—
130#2851D711yescharacteristic, complemented
131C4×#121—71no—
132#2077—71no—
133#2075—71no—
134#2034—71no—
135#2033—71no—
136#1989—71no—
137#1949—71no—
138#1944—71no—
139#1943—71no—
140#1936—71no—
141#1936—71no—
142#1804—71no—
143C24×C7C2×D411yescharacteristic
144C23×D7—21no—
145C22×C4×C7—22no—
146C22×C4×C7—22no—
147C22×C4×C7—22no—
148C22×C4×C7—21no—
149C22×C4×C7—21no—
150C22×C4×C7—21no—
151C22×C4×C7—21no—
152C22×C4×C7—21no—
153C22×C4×C7—21no—
154C22×C4×C7—21no—
155C22×C4×C7—21no—
156C22×C4×C7—21no—
157C22×C4×C7—21no—
158C22×C4×C7—41no—
159C22×C4×C7—41no—
160C22×C4×C7—41no—
161C22×C4×C7C2×D411yescharacteristic
162C22×C4×C7C2×D411yescharacteristic
163C22×C4×C7C2×D411yescharacteristic
164C22×C4×C7C2×D411yescharacteristic
165C22×C4×C7C2×D411yescharacteristic
166C22×C4×C7C2411yescharacteristic
167C2×C22⋊D7—24no—
168C2×C22⋊D7—42no—
169C2×C22⋊D7—42no—
170C22×Dic7—21no—
171C22×Dic7—21no—
172C22×Dic7—21no—
173C2×D28—21no—
174C2×D28—21no—
175C2×D28—21no—
176C2×D28—21no—
177C2×D28—21no—
178C2×D28#4112yes—
179C2×C4×D7—42no—
180C2×C4×D7—42no—
181C2×C4×D7—42no—
182C7×C4⋊C4—21no—
183C7×C4⋊C4—21no—
184C7×C4⋊C4—21no—
185C7×C4⋊C4—21no—
186C7×C4⋊C4—21no—
187C7×C4⋊C4—21no—
188C7×C4⋊C4—21no—
189C7×C4⋊C4—21no—
190C7×C4⋊C4—42no—
191C7×C4⋊C4—42no—
192C7×C4⋊C4—42no—
193C7×C4⋊C4—42no—
194C7×C22⋊C4—42no—
195C7×C22⋊C4—42no—
196C42×C7—22no—
197C42×C7—22no—
198C42×C7—21no—
199C42×C7—21no—
200C42×C7C2×D412yes—
201C42×C7#4112yes—
202C22⋊Dic7—42no—
203(C2×D7)⋊C4—24no—
204(C2×D7)⋊C4—42no—
205(C2×D7)⋊C4—42no—
206(C2×D7)⋊C4—42no—
207C4⋊Dic7—21no—
208C4⋊Dic7—21no—
209C4⋊Dic7—21no—
210C4⋊Dic7—21no—
211C4⋊Dic7—21no—
212C4⋊Dic7#4112yes—
213Dic7⋊C4—42no—
214C4×Dic7—42no—
215C2×C7⋊C8—21no—
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⋊C8C2×D412yes—
226C2×C7⋊C8C2×D412yes—
227C2×#125C2×D711yescharacteristic
228C2×#121—71no—
229C2×C4×D4—71no—
230C2×C4×D4—141no—
231C22×C42C2×D711yescharacteristic
232#478—144no—
233#432—144no—
234C2×#107—71no—
235C2×#107—71no—
236C2×#106—71no—
237C2×#106—71no—
238C2×C4⋊C8—71no—
239C2×C4⋊C8—71no—
240C2×#102—71no—
241C2×#102—71no—
242C2×#102—71no—
243C2×#102—71no—
244C2×C22⋊C8—71no—
245C2×C22⋊C8—71no—
246C2×#97—71no—
247C2×#97—71no—
248#390—71no—
249#389—71no—
250#389C2×D711yescharacteristic
251#389C2×D711yescharacteristic
252#388—141no—
253#387—21no—
254#387—21no—
255#386—141no—
256#385—21no—
257#385—21no—
258#382C2×D711yescharacteristic
259#380—71no—
260#380C2×D711yescharacteristic
261#380C2×D711yescharacteristic
262C4×C4⋊C4—71no—
263C4×C22⋊C4—141no—
264#340—71no—
265#340—71no—
266#340—71no—
267#340—71no—
268#340—71no—
269#340—71no—
270#340—71no—
271#340—71no—
272C23×C7—21no—
273C23×C7—21no—
274C23×C7—21no—
275C23×C7—21no—
276C23×C7—21no—
277C23×C7—21no—
278C23×C7AGL(1,ℤ/8)12yes—
279C23×C7C22×D411yescharacteristic
280C22×D7—22no—
281C22×D7—41no—
282C22×D7—41no—
283C22×D7—41no—
284C22×D7—41no—
285C22×D7—41no—
286C22×D7—41no—
287C2×C4×C7—21no—
288C2×C4×C7—21no—
289C2×C4×C7—21no—
290C2×C4×C7—21no—
291C2×C4×C7—21no—
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—44no—
301C2×C4×C7—44no—
302C2×C4×C7—41no—
303C2×C4×C7—41no—
304C2×C4×C7—41no—
305C2×C4×C7—41no—
306C2×C4×C7—41no—
307C2×C4×C7—41no—
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×C7#12112yes—
319C2×C4×C7#12112yes—
320C2×C4×C7C22×D411yescharacteristic
321C2×C4×C7C2×#4111yescharacteristic
322C2×C4×C7C2×#4111yescharacteristic
323C2×C4×C7C2×#4111yescharacteristic
324C2×C4×C72−1+411yescharacteristic
325C2×C4×C72−1+411yescharacteristic
326C22⋊D7—48no—
327C2×Dic7—22no—
328C2×Dic7—41no—
329C2×Dic7—41no—
330C2×Dic7—41no—
331C2×Dic7—41no—
332C2×Dic7—41no—
333C2×Dic7—41no—
334C2×Dic7—41no—
335C2×Dic7—41no—
336D28—22no—
337D28—22no—
338D28—41no—
339D28—41no—
340C4×D7—84no—
341C7⋊C8—22no—
342C7⋊C8—22no—
343C7⋊C8—41no—
344C7⋊C8—41no—
345C22×D4—71no—
346C22×D4—141no—
347C23×C4—141no—
348C23×C4C22⋊D711yescharacteristic
349C23×C4C22⋊D711yescharacteristic
350C23×C4C22×D711yescharacteristic
351C22×C8—71no—
352C22×C8—71no—
353C22×C8—71no—
354C22×C8—71no—
355#125—22no—
356#125—22no—
357#121—142no—
358#121—142no—
359C4×D4—142no—
360C4×D4—142no—
361C4×D4—284no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—21no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—71no—
367C2×C4⋊C4—71no—
368C2×C4⋊C4—71no—
369C2×C4⋊C4—141no—
370C2×C4⋊C4C22×D711yescharacteristic
371C2×C4⋊C4C22×D711yescharacteristic
372C2×C4⋊C4C22×D711yescharacteristic
373C2×C4⋊C4C22×D711yescharacteristic
374C2×C22⋊C4—21no—
375C2×C22⋊C4—21no—
376C2×C22⋊C4—141no—
377C2×C22⋊C4—141no—
378C2×C22⋊C4—141no—
379C2×C22⋊C4—141no—
380C2×C22⋊C4—141no—
381C2×C42—22no—
382C2×C42—22no—
383C2×C42—141no—
384C2×C42C22⋊D711yescharacteristic
385C2×C42C22⋊D711yescharacteristic
386C2×C42C22×D711yescharacteristic
387C2×C42C22×D711yescharacteristic
388#107—142no—
389#107—142no—
390#106—142no—
391#106—142no—
392C4⋊C8—74no—
393C4⋊C8—142no—
394#102—142no—
395#102—142no—
396#102—142no—
397#102—142no—
398C22⋊C8—142no—
399C22⋊C8—142no—
400#97—142no—
401#97—142no—
402#95—21no—
403#95—21no—
404#95—21no—
405#95—21no—
406#95—21no—
407#95—21no—
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×C7C2×#12111yescharacteristic
419C22×C7#52511yescharacteristic
420C22×C7#53611yescharacteristic
421C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
422C22×C7#57511yescharacteristic
423C22×C7#57511yescharacteristic
424C22×C7#57511yescharacteristic
425C2×D7—42no—
426C2×D7—42no—
427C2×D7—42no—
428C2×D7—81no—
429C2×D7—81no—
430C2×D7—81no—
431C2×D7—81no—
432C4×C7—22no—
433C4×C7—22no—
434C4×C7—41no—
435C4×C7—41no—
436C4×C7—41no—
437C4×C7—81no—
438C4×C7—81no—
439C4×C7—81no—
440C4×C7—81no—
441C4×C7#54711yescharacteristic
442C4×C7#54811yescharacteristic
443C4×C7#55211yescharacteristic
444C4×C7#55611yescharacteristic
445Dic7—42no—
446Dic7—81no—
447Dic7—81no—
448C24—141no—
449C24C2×C22⋊D711yescharacteristic
450C2×D4—72no—
451C2×D4—144no—
452C2×D4—141no—
453C2×D4—141no—
454C2×D4—141no—
455C2×D4—141no—
456C2×D4—141no—
457C2×D4—282no—
458C2×D4—282no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—41no—
473C22×C4—41no—
474C22×C4—41no—
475C22×C4—141no—
476C22×C4—141no—
477C22×C4—141no—
478C22×C4—282no—
479C22×C4—282no—
480C22×C4—282no—
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—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—42no—
508C4⋊C4—42no—
509C4⋊C4—42no—
510C4⋊C4—42no—
511C4⋊C4—72no—
512C4⋊C4—141no—
513C4⋊C4—141no—
514C4⋊C4—141no—
515C4⋊C4—141no—
516C4⋊C4—141no—
517C4⋊C4—282no—
518C22⋊C4—42no—
519C22⋊C4—42no—
520C22⋊C4—144no—
521C22⋊C4—282no—
522C22⋊C4—282no—
523C22⋊C4—282no—
524C22⋊C4—282no—
525C42—22no—
526C42—22no—
527C42—21no—
528C42—21no—
529C42—282no—
530C42D4×D712yes—
531C42#115912yes—
532C2×C7—42no—
533C2×C7#241511yescharacteristic
534C2×C7#305411yescharacteristic
535C2×C7#305511yescharacteristic
536C2×C7#311111yescharacteristic
537C2×C7#311411yescharacteristic
538C2×C7#323411yescharacteristic
539C2×C7#324011yescharacteristic
540D7—82no—
541C23—21no—
542C23—21no—
543C23—21no—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—142no—
548C23—281no—
549C23—281no—
550C23—281no—
551C23—281no—
552C23—281no—
553C23—281no—
554C23#651212yes—
555C23C22×C22⋊D711yescharacteristic
556D4—142no—
557D4—142no—
558D4—288no—
559D4—281no—
560D4—281no—
561C2×C4—21no—
562C2×C4—21no—
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—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
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—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—142no—
593C2×C4—281no—
594C2×C4—281no—
595C2×C4—281no—
596C2×C4—281no—
597C2×C4—281no—
598C2×C4—281no—
599C2×C4—281no—
600C2×C4—281no—
601C2×C4—564no—
602C2×C4#650212yes—
603C2×C4#650212yes—
604C2×C4C2×D4×D711yescharacteristic
605C2×C4C2×#115911yescharacteristic
606C2×C4#655111yescharacteristic
607C2×C4#655111yescharacteristic
608C2×C4D7×#4111yescharacteristic
609C2×C4D7×#4111yescharacteristic
610C8—142no—
611C8—142no—
612C8—281no—
613C8—281no—
614C7#2470611yescharacteristic, complemented, Sylow
615C22—22no—
616C22—22no—
617C22—22no—
618C22—41no—
619C22—41no—
620C22—41no—
621C22—41no—
622C22—282no—
623C22—282no—
624C22—282no—
625C22—561no—
626C22—561no—
627C22—561no—
628C22—561no—
629C22C2×#650211yescharacteristic
630C22#9050711yescharacteristic
631C22C2×#651211yescharacteristic
632C22#9051311yescharacteristic
633C22#9051311yescharacteristic
634C22#9051311yescharacteristic
635C22#9051911yescharacteristic
636C4—22no—
637C4—22no—
638C4—41no—
639C4—41no—
640C4—41no—
641C4—81no—
642C4—81no—
643C4—81no—
644C4—81no—
645C4—282no—
646C4—561no—
647C4—561no—
648C4#9032211yescharacteristic
649C4#9032411yescharacteristic
650C4#9033011yescharacteristic
651C4#9033211yescharacteristic
652C2—42no—
653C2—562no—
654C2#1174025911yescharacteristic
655C2#1174036811yescharacteristic
656C2#1174036911yescharacteristic
657C2#1174046411yescharacteristic
658C2#1174058711yescharacteristic
659C2#1174058811yescharacteristic
660C2#1174080711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#2851C2
socleC23×C7C22×D4
prime radicalC2×#6493C22
hypercenter#2851D7
hypocenterC7#24706
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2851

Complements

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

NH ≅ G/Ntype
1. C7×#2851C2semidirect
2. C2×#89838C2semidirect
4. C2×#89698C2semidirect
5. #11729492C2semidirect
6. #11727375C2semidirect
7. #11727374C2semidirect
8. #11727371C2semidirect
9. #11727371C2semidirect
10. #11727310C2semidirect
11. #11727294C2semidirect
14. #11727272C2semidirect
15. #11727261C2semidirect
125. C4⋊(C7⋊C8)D4semidirect
130. #2851D7semidirect
614. C7#24706semidirect

Maximal subgroups

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

subgroupindex
C7×#28512
C2×#898382
C4×#64932
C2×#896982
#117294922
#117273752
#117273742
#117273712
#117273102
#117272942
#117272782
#117272772
#117272722
#117272612
#247067