Finite Group Databasebeta

← #13371026

Subgroups — #13371026

Order 1792.

1010 conjugacy classes of subgroups.

The subgroup lattice has 696 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2345C211yesmaximal, characteristic, complemented
2C2×#89738C211yesmaximal, characteristic
3#11731018C211yesmaximal, characteristic
4#11729307C211yesmaximal, characteristic, complemented
5#11727380C211yesmaximal, characteristic, complemented
6#11727379C211yesmaximal, characteristic, complemented
7#11727370C211yesmaximal, characteristic, complemented
8#11727368C211yesmaximal, characteristic, complemented
9#11727276C211yesmaximal, characteristic
10#11727267C211yesmaximal, characteristic, complemented
11#11727228C211yesmaximal, characteristic, complemented
12C4×#6383C211yesmaximal, characteristic, complemented
13#11727219C211yesmaximal, characteristic, complemented
14#11727218C211yesmaximal, characteristic, complemented
15#11727202C211yesmaximal, characteristic, complemented
16C22×#1157C2211yescharacteristic
17C22×C7×C4⋊C4C2211yescharacteristic
18C22×#1136C2211yescharacteristic
19C2×C4×D28C2211yescharacteristic
20C2×C4×D28C2211yescharacteristic
21C7×#389C2211yescharacteristic
22C7×#389C2211yescharacteristic
23C7×#387—21no—
24C7×#387—21no—
25C7×#380—21no—
26C7×#380C2211yescharacteristic
27C4×C7×C4⋊C4C2211yescharacteristic
28C4×C7×C4⋊C4C2211yescharacteristic
29C4×C7×C22⋊C4—21no—
30C4×C7×C22⋊C4C2211yescharacteristic
31#89984C2211yescharacteristic
32C2×C22⋊(C7⋊C8)C2211yescharacteristic
33C2×#6384C2211yescharacteristic
34C2×#6384C2211yescharacteristic
35C2×#6384C2211yescharacteristic
36C2×#6384C2211yescharacteristic
37#89738C418yescomplemented
38C2×#6383C2211yescharacteristic
39C2×#6383C2211yescharacteristic
40C2×#6383C2211yescharacteristic
41C2×#6383C2211yescharacteristic
42C2×#6382C2211yescharacteristic
43C2×#6381C2211yescharacteristic
44#89728—21no—
45#89728—21no—
46#89701C2211yescharacteristic
47#89701C2211yescharacteristic
48C2×#6377C2211yescharacteristic
49C2×C4×C7⋊C8C2211yescharacteristic
50#89436—21no—
51D7×#95—21no—
52#89324C2211yescharacteristic
53#89324C2211yescharacteristic
54#89324C2211yescharacteristic
55#89324C2211yescharacteristic
56#89324C2211yescharacteristic
57#89324C2211yescharacteristic
58#89319C2211yescharacteristic
59#89319C2211yescharacteristic
60#22578—71nomaximal, Sylow
61C23×C4×C7—21no—
62C23×C4×C7C2311yescharacteristic
63C22×C22⋊D7—21no—
64C2×#1157—22no—
65C2×#1157—22no—
66C2×#1157—22no—
67C2×#1157—41no—
68C2×#1157—41no—
69C2×#1157C2×C414yes—
70C22×D28C2311yescharacteristic
71C22×C4×D7—21no—
72C22×C4×D7—21no—
73C22×C4×D7—21no—
74C22×Dic14C2311yescharacteristic
75C2×C7×C4⋊C4—21no—
76C2×C7×C4⋊C4—22no—
77C2×C7×C4⋊C4—21no—
78C2×C7×C4⋊C4—21no—
79C2×C7×C4⋊C4C2×C414yescomplemented
80C2×C7×C4⋊C4C2311yescharacteristic
81C2×C7×C4⋊C4C2311yescharacteristic
82C2×C7×C4⋊C4C2311yescharacteristic
83C2×C7×C4⋊C4C2311yescharacteristic
84C2×C7×C22⋊C4—21no—
85C2×C7×C22⋊C4D411yescharacteristic
86C2×C7×C22⋊C4D411yescharacteristic
87C2×C42×C7—21no—
88C2×C42×C7—21no—
89C2×C42×C7C2311yescharacteristic
90C2×C42×C7C2311yescharacteristic
91C2×C22⋊Dic7—21no—
92C2×(C2×D7)⋊C4—21no—
93C2×(C2×D7)⋊C4—21no—
94C2×C4⋊Dic7C2311yescharacteristic
95C2×C4⋊Dic7C2311yescharacteristic
96C2×#1136—41no—
97C2×#1136—41no—
98C2×#1136C2×C414yescomplemented
99C22×C7⋊C8C2311yescharacteristic
100C22×C7⋊C8C2311yescharacteristic
101C22×C7⋊C8C2311yescharacteristic
102C22×C7⋊C8C2311yescharacteristic
103C4×D28—22no—
104C4×D28—22no—
105C4×D28—41no—
106C4×D28—41no—
107C7×#95—21no—
108C7×#95—21no—
109C7×#95—21no—
110C7×#95—21no—
111C7×#95—21no—
112C7×#95D411yescharacteristic
113C7×#95D411yescharacteristic
114C7⋊#95—21no—
115C7⋊#95—21no—
116C7⋊#95—21no—
117C7⋊#95—21no—
118C7⋊#95—21no—
119C22⋊(C7⋊C8)—22no—
120#6384—41no—
121#6384—41no—
122#6384C2×C414yescomplemented
123#6384C2×C414yescomplemented
124#6383—22no—
125#6383—22no—
126#6383C2×C414yes—
127#6383C2×C414yes—
128#6382—41no—
129#6381—41no—
130#6377—41no—
131C4×C7⋊C8—41no—
132C2×#417—71no—
133#2345D711yescharacteristic, complemented
134#2339—71no—
135#1935—71no—
136#1933—71no—
137#1928—71no—
138#1927—71no—
139#1864—71no—
140#1862—71no—
141#1823—71no—
142#1809—71no—
143#1788—71no—
144#1762—71no—
145C4×#102—71no—
146#1753—71no—
147C24×C7C2×D411yescharacteristic
148C23×D7—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—24no—
156C22×C4×C7—21no—
157C22×C4×C7—21no—
158C22×C4×C7—81no—
159C22×C4×C7—81no—
160C22×C4×C7C22×C412yes—
161C22×C4×C7C2×D411yescharacteristic
162C22×C4×C7C2×D411yescharacteristic
163C22×C4×C7C2×D411yescharacteristic
164C22×C4×C7C2×D411yescharacteristic
165C22×C4×C7C2×D411yescharacteristic
166C22×C4×C7#4112yes—
167C22×C4×C7C2411yescharacteristic
168C2×C22⋊D7—24no—
169C2×C22⋊D7—42no—
170C2×C22⋊D7—42no—
171C22×Dic7—21no—
172C22×Dic7—21no—
173C22×Dic7—21no—
174#1157—28no—
175#1157—44no—
176#1157—44no—
177#1157—44no—
178C2×D28—21no—
179C2×D28—21no—
180C2×D28—21no—
181C2×D28C22×C412yes—
182C2×D28#4112yes—
183C2×D28#4112yes—
184C2×C4×D7—24no—
185C2×C4×D7—42no—
186C2×C4×D7—42no—
187C2×C4×D7—42no—
188C2×C4×D7—42no—
189C2×C4×D7—42no—
190C2×C4×D7—42no—
191C2×C4×D7—81no—
192C2×C4×D7—81no—
193C2×Dic14—21no—
194C2×Dic14—21no—
195C2×Dic14—21no—
196C2×Dic14—21no—
197C2×Dic14—21no—
198C2×Dic14C22×C412yes—
199C7×C4⋊C4—24no—
200C7×C4⋊C4—21no—
201C7×C4⋊C4—21no—
202C7×C4⋊C4—21no—
203C7×C4⋊C4—21no—
204C7×C4⋊C4—21no—
205C7×C4⋊C4—21no—
206C7×C4⋊C4—81no—
207C7×C4⋊C4—81no—
208C7×C4⋊C4C22×C412yes—
209C7×C4⋊C4C22×C412yes—
210C7×C22⋊C4—22no—
211C7×C22⋊C4—41no—
212C7×C22⋊C4—42no—
213C42×C7—21no—
214C42×C7—21no—
215C42×C7—21no—
216C42×C7—21no—
217C42×C7—81no—
218C42×C7—81no—
219C22⋊Dic7—81no—
220(C2×D7)⋊C4—42no—
221(C2×D7)⋊C4—42no—
222C4⋊Dic7—21no—
223C4⋊Dic7—21no—
224C4⋊Dic7—21no—
225C4⋊Dic7—21no—
226#1136—44no—
227C2×C7⋊C8—21no—
228C2×C7⋊C8—21no—
229C2×C7⋊C8—21no—
230C2×C7⋊C8—21no—
231C2×C7⋊C8—21no—
232C2×C7⋊C8—21no—
233C2×C7⋊C8—21no—
234C2×C7⋊C8—21no—
235C2×C7⋊C8—21no—
236C2×C7⋊C8—21no—
237C2×C7⋊C8C22×C412yes—
238C2×C7⋊C8C22×C412yes—
239C22×#41—71no—
240C22×M16—71no—
241C2×C4×D4—71no—
242C2×C4×D4—71no—
243C22×C4⋊C4C2×D711yescharacteristic
244C2×#107—71no—
245C2×#106—71no—
246#417—78no—
247C2×#103—71no—
248C2×#103—71no—
249C2×#103—71no—
250C2×#103—71no—
251C2×#102—71no—
252C2×#102—71no—
253C2×#102—71no—
254C2×#102—71no—
255C2×C22⋊C8—71no—
256C2×#97—71no—
257C2×C4×C8—71no—
258#391—71no—
259#391—71no—
260#389C2×D711yescharacteristic
261#389C2×D711yescharacteristic
262#387—21no—
263#387—21no—
264#386—141no—
265#386—141no—
266#381—141no—
267#380—21no—
268#380—71no—
269#380C2×D711yescharacteristic
270C4×C4⋊C4C2×D711yescharacteristic
271C4×C4⋊C4C2×D711yescharacteristic
272C4×C22⋊C4—21no—
273C4×C22⋊C4C2×D711yescharacteristic
274C2×#95—141no—
275#340—71no—
276#340—71no—
277#340—71no—
278#340—71no—
279#340—71no—
280#340—71no—
281#336—71no—
282#336—71no—
283C23×C7—21no—
284C23×C7—21no—
285C23×C7—21no—
286C23×C7—21no—
287C23×C7—21no—
288C23×C7—21no—
289C23×C7C4×D412yes—
290C23×C7C22×D411yescharacteristic
291C22×D7—22no—
292C22×D7—22no—
293C22×D7—22no—
294C22×D7—41no—
295C22×D7—41no—
296C22×D7—41no—
297C22×D7—41no—
298C2×C4×C7—21no—
299C2×C4×C7—21no—
300C2×C4×C7—21no—
301C2×C4×C7—21no—
302C2×C4×C7—22no—
303C2×C4×C7—22no—
304C2×C4×C7—22no—
305C2×C4×C7—22no—
306C2×C4×C7—22no—
307C2×C4×C7—22no—
308C2×C4×C7—21no—
309C2×C4×C7—21no—
310C2×C4×C7—21no—
311C2×C4×C7—21no—
312C2×C4×C7—21no—
313C2×C4×C7—21no—
314C2×C4×C7—21no—
315C2×C4×C7—21no—
316C2×C4×C7—21no—
317C2×C4×C7—41no—
318C2×C4×C7—41no—
319C2×C4×C7—41no—
320C2×C4×C7—41no—
321C2×C4×C7—41no—
322C2×C4×C7—41no—
323C2×C4×C7—41no—
324C2×C4×C7—41no—
325C2×C4×C7—41no—
326C2×C4×C7—41no—
327C2×C4×C7—84no—
328C2×C4×C7C4×D412yes—
329C2×C4×C7#13712yes—
330C2×C4×C7C23×C411yescharacteristic
331C2×C4×C7C2×#4111yescharacteristic
332C2×C4×C7C2×#4111yescharacteristic
333C2×C4×C7C2×#4111yescharacteristic
334C2×C4×C7H(5,𝔽2)11yescharacteristic
335C2×C4×C7H(5,𝔽2)11yescharacteristic
336C22⋊D7—48no—
337C2×Dic7—22no—
338C2×Dic7—41no—
339C2×Dic7—41no—
340C2×Dic7—41no—
341C2×Dic7—41no—
342C2×Dic7—41no—
343C2×Dic7—41no—
344C2×Dic7—41no—
345C2×Dic7—41no—
346D28—22no—
347D28—22no—
348D28—22no—
349D28#11714yes—
350C4×D7—48no—
351C4×D7—84no—
352C4×D7—84no—
353Dic14—22no—
354Dic14—22no—
355Dic14—41no—
356Dic14—41no—
357C7⋊C8—22no—
358C7⋊C8—22no—
359C7⋊C8—41no—
360C7⋊C8—41no—
361C2×#41—74no—
362C2×#41—142no—
363C2×#41—142no—
364C2×#41—142no—
365C2×#41—281no—
366C2×#41—281no—
367C22×Q8—71no—
368C22×D4—71no—
369C22×D4—141no—
370C23×C4—21no—
371C23×C4—141no—
372C23×C4—141no—
373C23×C4—141no—
374C23×C4C22×D711yescharacteristic
375C2×M16—74no—
376C2×M16—281no—
377C2×M16—281no—
378C22×C8—71no—
379C22×C8—71no—
380C22×C8—71no—
381C22×C8—71no—
382C4×D4—142no—
383C4×D4—142no—
384C4×D4—281no—
385C4×D4—281no—
386C2×C4⋊C4—22no—
387C2×C4⋊C4—21no—
388C2×C4⋊C4—21no—
389C2×C4⋊C4—21no—
390C2×C4⋊C4—71no—
391C2×C4⋊C4—71no—
392C2×C4⋊C4C4×D714yescomplemented
393C2×C4⋊C4C22×D711yescharacteristic
394C2×C4⋊C4C22×D711yescharacteristic
395C2×C4⋊C4C22×D711yescharacteristic
396C2×C4⋊C4C22×D711yescharacteristic
397C2×C22⋊C4—21no—
398C2×C22⋊C4—141no—
399C2×C22⋊C4—141no—
400C2×C22⋊C4—141no—
401C2×C22⋊C4C22⋊D711yescharacteristic
402C2×C22⋊C4C22⋊D711yescharacteristic
403C2×C42—21no—
404C2×C42—21no—
405C2×C42C22×D711yescharacteristic
406C2×C42C22×D711yescharacteristic
407#107—281no—
408#106—281no—
409#103—74no—
410#103—74no—
411#103—281no—
412#103—281no—
413#102—74no—
414#102—74no—
415#102—142no—
416#102—142no—
417C22⋊C8—142no—
418#97—281no—
419C4×C8—281no—
420#95—21no—
421#95—21no—
422#95—21no—
423#95—21no—
424#95—21no—
425#95—141no—
426#95—141no—
427#95—141no—
428#95—141no—
429#95—141no—
430#95C22⋊D711yescharacteristic
431#95C22⋊D711yescharacteristic
432C22×C7—22no—
433C22×C7—22no—
434C22×C7—22no—
435C22×C7—41no—
436C22×C7—41no—
437C22×C7—41no—
438C22×C7—41no—
439C22×C7C2×C4×D411yescharacteristic
440C22×C7#54611yescharacteristic
441C22×C7#54611yescharacteristic
442C22×C7C2×#13711yescharacteristic
443C22×C7#57511yescharacteristic
444C22×C7#57611yescharacteristic
445C22×C7#57711yescharacteristic
446C2×D7—24no—
447C2×D7—42no—
448C2×D7—42no—
449C2×D7—42no—
450C2×D7—42no—
451C2×D7—42no—
452C2×D7—42no—
453C4×C7—22no—
454C4×C7—22no—
455C4×C7—41no—
456C4×C7—42no—
457C4×C7—42no—
458C4×C7—41no—
459C4×C7—41no—
460C4×C7—81no—
461C4×C7—81no—
462C4×C7C2×#11711yescharacteristic
463C4×C7#51811yescharacteristic
464C4×C7#55111yescharacteristic
465C4×C7#55111yescharacteristic
466Dic7—42no—
467Dic7—81no—
468Dic7—81no—
469C24—141no—
470C24C2×C22⋊D711yescharacteristic
471#41—148no—
472#41—284no—
473#41—284no—
474#41—284no—
475C2×Q8—72no—
476C2×Q8—141no—
477C2×Q8—141no—
478C2×Q8—141no—
479C2×Q8—141no—
480C2×Q8—141no—
481C2×D4—72no—
482C2×D4—72no—
483C2×D4—72no—
484C2×D4—144no—
485C2×D4—141no—
486C2×D4—141no—
487C2×D4—141no—
488C2×D4—282no—
489C2×D4—282no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—21no—
496C22×C4—24no—
497C22×C4—21no—
498C22×C4—21no—
499C22×C4—81no—
500C22×C4—81no—
501C22×C4—141no—
502C22×C4—144no—
503C22×C4—141no—
504C22×C4—141no—
505C22×C4—282no—
506C22×C4—282no—
507C22×C4—282no—
508C22×C4—282no—
509C22×C4—282no—
510C22×C4—282no—
511C22×C4—561no—
512C22×C4—561no—
513C22×C4C2×C4×D712yes—
514C22×C4#115712yes—
515C22×C4C2×C22⋊D711yescharacteristic
516C22×C4C2×C22⋊D711yescharacteristic
517C22×C4C2×C22⋊D711yescharacteristic
518C22×C4C2×C22⋊D711yescharacteristic
519C22×C4C2×C22⋊D711yescharacteristic
520C22×C4C23×D711yescharacteristic
521M16—284no—
522C2×C8—72no—
523C2×C8—72no—
524C2×C8—141no—
525C2×C8—141no—
526C2×C8—141no—
527C2×C8—141no—
528C2×C8—141no—
529C2×C8—141no—
530C2×C8—141no—
531C2×C8—141no—
532C2×C8—141no—
533C2×C8—141no—
534C4⋊C4—24no—
535C4⋊C4—21no—
536C4⋊C4—21no—
537C4⋊C4—21no—
538C4⋊C4—21no—
539C4⋊C4—21no—
540C4⋊C4—21no—
541C4⋊C4—81no—
542C4⋊C4—81no—
543C4⋊C4—141no—
544C4⋊C4—141no—
545C4⋊C4—141no—
546C4⋊C4—141no—
547C4⋊C4C2×C4×D712yes—
548C4⋊C4C2×C4×D712yes—
549C22⋊C4—22no—
550C22⋊C4—41no—
551C22⋊C4—42no—
552C22⋊C4—282no—
553C22⋊C4—282no—
554C22⋊C4—561no—
555C42—21no—
556C42—21no—
557C42—21no—
558C42—21no—
559C42—81no—
560C42—81no—
561C2×C7—42no—
562C2×C7#234211yescharacteristic
563C2×C7C4×#13711yescharacteristic
564C2×C7#294611yescharacteristic
565C2×C7#330711yescharacteristic
566C2×C7#331411yescharacteristic
567C2×C7#332211yescharacteristic
568C2×C7#333111yescharacteristic
569D7—44no—
570C23—21no—
571C23—21no—
572C23—21no—
573C23—21no—
574C23—21no—
575C23—21no—
576C23—142no—
577C23—142no—
578C23—142no—
579C23—281no—
580C23—281no—
581C23—281no—
582C23—281no—
583C23C4×C22⋊D712yes—
584C23C22×C22⋊D711yescharacteristic
585Q8—142no—
586Q8—142no—
587Q8—281no—
588Q8—281no—
589D4—74no—
590D4—142no—
591D4—142no—
592D4—142no—
593D4—288no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—84no—
624C2×C4—142no—
625C2×C4—288no—
626C2×C4—281no—
627C2×C4—281no—
628C2×C4—281no—
629C2×C4—281no—
630C2×C4—281no—
631C2×C4—281no—
632C2×C4—281no—
633C2×C4—281no—
634C2×C4—564no—
635C2×C4—564no—
636C2×C4C4×C22⋊D712yes—
637C2×C4#650512yes—
638C2×C4C22×C4×D711yescharacteristic
639C2×C4C2×#115711yescharacteristic
640C2×C4D7×#4111yescharacteristic
641C2×C4D7×#4111yescharacteristic
642C2×C4#655311yescharacteristic
643C2×C4#655311yescharacteristic
644C8—142no—
645C8—142no—
646C8—281no—
647C8—281no—
648C7#2257811yescharacteristic, complemented, Sylow
649C22—22no—
650C22—22no—
651C22—22no—
652C22—41no—
653C22—41no—
654C22—41no—
655C22—41no—
656C22—144no—
657C22—282no—
658C22—282no—
659C22—282no—
660C22—282no—
661C22—282no—
662C22—282no—
663C22C2×C4×C22⋊D711yescharacteristic
664C22C2×#650511yescharacteristic
665C22#9051311yescharacteristic
666C22#9052011yescharacteristic
667C22#9052011yescharacteristic
668C22#9052611yescharacteristic
669C22#9052711yescharacteristic
670C4—22no—
671C4—22no—
672C4—42no—
673C4—42no—
674C4—41no—
675C4—41no—
676C4—41no—
677C4—81no—
678C4—81no—
679C4—282no—
680C4—561no—
681C4—561no—
682C4D7×#11711yescharacteristic
683C4#9021211yescharacteristic
684C4#9021811yescharacteristic
685C4#9021811yescharacteristic
686C2—42no—
687C2—284no—
688C2#1173971611yescharacteristic
689C2C4×#650511yescharacteristic
690C2#1173998211yescharacteristic
691C2#1174106011yescharacteristic
692C2#1174106211yescharacteristic
693C2#1174108011yescharacteristic
694C2#1174108111yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#2345C2
socleC23×C7C22×D4
prime radicalC22×#1157C22
cosocleC22×C4×C7C24
hypercenter#2345D7
hypocenterC7#22578
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2345

Complements

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

NH ≅ G/Ntype
1. C7×#2345C2semidirect
4. #11729307C2semidirect
5. #11727380C2semidirect
6. #11727379C2semidirect
7. #11727370C2semidirect
8. #11727368C2semidirect
10. #11727267C2semidirect
11. #11727228C2semidirect
12. C4×#6383C2semidirect
13. #11727219C2semidirect
14. #11727218C2semidirect
15. #11727202C2semidirect
37. #89738C4semidirect
79. C2×C7×C4⋊C4C2×C4semidirect
98. C2×#1136C2×C4semidirect
122. #6384C2×C4semidirect
123. #6384C2×C4semidirect
133. #2345D7semidirect
392. C2×C4⋊C4C4×D7semidirect
648. C7#22578semidirect

Maximal subgroups

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

subgroupindex
C7×#23452
C2×#897382
#117310182
#117293072
#117273802
#117273792
#117273702
#117273682
#117272762
#117272672
#117272282
C4×#63832
#117272192
#117272182
#117272022
#225787