Finite Group Databasebeta

← #13403362

Subgroups — #13403362

Order 1792.

800 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2592C211yesmaximal, characteristic
2C2×#89826C211yesmaximal, characteristic
3#11739925C211yesmaximal, characteristic, complemented
4C2×Q8×C7⋊C8C211yesmaximal, characteristic, complemented
5#11729479C211yesmaximal, characteristic
6#11727436C211yesmaximal, characteristic, complemented
7#11727435C211yesmaximal, characteristic, complemented
8#11727431C211yesmaximal, characteristic, complemented
9#11727428C211yesmaximal, characteristic, complemented
10#11727407C211yesmaximal, characteristic
11#11727405C211yesmaximal, characteristic
12#11727400C211yesmaximal, characteristic
13#11727398C211yesmaximal, characteristic
14C4×#6384C211yesmaximal, characteristic, complemented
15#11727222C211yesmaximal, characteristic, complemented
16C2×C7×#124C2211yescharacteristic
17C2×C4×C7×Q8C2211yescharacteristic
18C22×#1144C2211yescharacteristic
19C22×#1142C2211yescharacteristic
20C2×C4×Dic14C2211yescharacteristic
21C7×#397—21no—
22C7×#393—21no—
23C7×#391C2211yescharacteristic
24C7×#391C2211yescharacteristic
25C7×#390C2211yescharacteristic
26C7×#390C2211yescharacteristic
27C7×#388—21no—
28C7×#388—21no—
29C4×C7×C4⋊C4C2211yescharacteristic
30#90001—21no—
31C2×#6409C2211yescharacteristic
32C2×#6409C2211yescharacteristic
33C2×#6409C2211yescharacteristic
34C2×#6406C2211yescharacteristic
35C2×#6406C2211yescharacteristic
36C2×#6406C2211yescharacteristic
37#89826—24no—
38Q8×C7⋊C8—24no—
39#89754—21no—
40#89750—21no—
41#89750—21no—
42C2×#6384C2211yescharacteristic
43C2×#6384C2211yescharacteristic
44C2×#6382C2211yescharacteristic
45C2×#6381C2211yescharacteristic
46#89708C2211yescharacteristic
47#89701C2211yescharacteristic
48#89701C2211yescharacteristic
49#89700C2211yescharacteristic
50#89700C2211yescharacteristic
51C2×C4⋊(C7⋊C8)C2211yescharacteristic
52C2×C4⋊(C7⋊C8)C2211yescharacteristic
53C2×C4⋊(C7⋊C8)C2211yescharacteristic
54C2×C4×C7⋊C8C2211yescharacteristic
55C2×C4×C7⋊C8C2211yescharacteristic
56C2×C4×C7⋊C8C2211yescharacteristic
57#89324C2211yescharacteristic
58#89324C2211yescharacteristic
59#89324C2211yescharacteristic
60#89324C2211yescharacteristic
61#25297—71nomaximal, Sylow
62C22×C7×Q8D411yescharacteristic
63C22×C7×Q8C2311yescharacteristic
64C22×C7×D4C2311yescharacteristic
65C22×Dic14C2311yescharacteristic
66C7×#124—22no—
67C7×#124—22no—
68C4×C7×Q8—22no—
69C4×C7×Q8—22no—
70C2×C7×C4⋊C4D411yescharacteristic
71C2×C7×C4⋊C4D411yescharacteristic
72C2×C7×C4⋊C4D411yescharacteristic
73C2×C7×C4⋊C4C2311yescharacteristic
74C2×C7×C4⋊C4C2311yescharacteristic
75C2×C7×C22⋊C4—21no—
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—21no—
82C2×C42×C7C2311yescharacteristic
83C2×C42×C7C2311yescharacteristic
84C2×C42×C7C2311yescharacteristic
85C2×#1144—22no—
86C2×#1144—22no—
87C2×#1144—41no—
88C2×#1142—22no—
89C2×#1142—22no—
90C2×#1142—41no—
91C2×C4⋊Dic7C2311yescharacteristic
92C2×C4⋊Dic7C2311yescharacteristic
93C2×C4⋊Dic7C2311yescharacteristic
94C2×Dic7⋊C4—21no—
95C2×Dic7⋊C4—21no—
96C2×Dic7⋊C4—21no—
97C2×C4×Dic7—21no—
98C22×C7⋊C8C2311yescharacteristic
99C22×C7⋊C8C2311yescharacteristic
100C22×C7⋊C8C2311yescharacteristic
101C22×C7⋊C8C2311yescharacteristic
102C4×Dic14—22no—
103C4×Dic14—22no—
104C7×#95—21no—
105C7×#95—21no—
106C7×#95—21no—
107C7×#95—21no—
108C7×#95—21no—
109#6409—22no—
110#6409—22no—
111#6409—41no—
112#6406—22no—
113#6406—22no—
114#6406—41no—
115C7⋊#95—21no—
116C7⋊#95—21no—
117C7⋊#95—21no—
118C7⋊#95—21no—
119C7⋊#95—21no—
120C7⋊#95—21no—
121C7⋊#95—21no—
122C7⋊#95—21no—
123#6384—22no—
124#6384—22no—
125#6382—41no—
126#6381—41no—
127C4⋊(C7⋊C8)—22no—
128C4⋊(C7⋊C8)—22no—
129C4⋊(C7⋊C8)D414yes—
130C4×C7⋊C8—22no—
131C4×C7⋊C8—22no—
132C4×C7⋊C8—22no—
133C2×#464—71no—
134C2×C8×Q8—71no—
135#2592D711yescharacteristic
136#2581—71no—
137#2067—71no—
138#2066—71no—
139#2056—71no—
140#2055—71no—
141#1969—71no—
142#1968—71no—
143#1958—71no—
144#1949—71no—
145#1934—71no—
146#1933—71no—
147C4×#103—71no—
148C24×C7—21no—
149C2×C7×Q8—22no—
150C2×C7×Q8—22no—
151C2×C7×Q8—22no—
152C2×C7×Q8—21no—
153C2×C7×Q8—21no—
154C2×C7×Q8—21no—
155C2×C7×Q8—21no—
156C2×C7×Q8—21no—
157C2×C7×Q8—21no—
158C2×C7×D4—21no—
159C2×C7×D4—21no—
160C2×C7×D4—21no—
161C2×C7×D4—21no—
162C2×C7×D4—21no—
163C2×C7×D4—21no—
164C22×C4×C7—21no—
165C22×C4×C7—21no—
166C22×C4×C7—21no—
167C22×C4×C7C2×D411yescharacteristic
168C22×C4×C7C2×D411yescharacteristic
169C22×C4×C7C2×D411yescharacteristic
170C22×C4×C7C2×D411yescharacteristic
171C22×C4×C7C2×D411yescharacteristic
172C22×C4×C7C2×D411yescharacteristic
173C22×C4×C7C2411yescharacteristic
174C22×Dic7—21no—
175C22×Dic7—21no—
176C22×Dic7—21no—
177C22×Dic7—21no—
178C2×Dic14—21no—
179C2×Dic14—21no—
180C2×Dic14—21no—
181C2×Dic14—21no—
182C2×Dic14—21no—
183C2×Dic14#4112yes—
184C7×C4⋊C4—22no—
185C7×C4⋊C4—22no—
186C7×C4⋊C4—22no—
187C7×C4⋊C4—21no—
188C7×C4⋊C4—21no—
189C7×C4⋊C4—21no—
190C7×C4⋊C4—21no—
191C7×C22⋊C4—42no—
192C7×C22⋊C4—42no—
193C7×C22⋊C4—42no—
194C7×C22⋊C4—42no—
195C7×C22⋊C4—81no—
196C7×C22⋊C4—81no—
197C42×C7—21no—
198C42×C7—21no—
199C42×C7—21no—
200C42×C7—21no—
201C42×C7—81no—
202C42×C7C2×D412yes—
203C42×C7#4112yes—
204#1144—44no—
205#1142—44no—
206C4⋊Dic7—21no—
207C4⋊Dic7—21no—
208C4⋊Dic7—21no—
209C4⋊Dic7—21no—
210C4⋊Dic7—21no—
211C4⋊Dic7—21no—
212Dic7⋊C4—42no—
213Dic7⋊C4—42no—
214Dic7⋊C4—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⋊C8C2×D412yes—
226C2×C7⋊C8C2×D412yes—
227C2×C7⋊C8#4112yes—
228C22×Q16—71no—
229C22×SD16—71no—
230C2×#124C2×D711yescharacteristic
231C2×C4×Q8—71no—
232C2×C4×Q8C2×D711yescharacteristic
233#464—144no—
234C8×Q8—144no—
235C2×#107—71no—
236C2×#106—71no—
237C2×C4⋊C8—71no—
238C2×C4⋊C8—71no—
239C2×C4⋊C8—71no—
240C2×#103—71no—
241C2×#103—71no—
242C2×#103—71no—
243C2×#103—71no—
244C2×#103—71no—
245C2×#102—71no—
246C2×#102—71no—
247C2×#102—71no—
248C2×C4×C8—71no—
249C2×C4×C8—71no—
250C2×C4×C8—71no—
251#400—141no—
252#397—21no—
253#395—141no—
254#393—21no—
255#391—71no—
256#391—71no—
257#391C2×D711yescharacteristic
258#391C2×D711yescharacteristic
259#390C2×D711yescharacteristic
260#390C2×D711yescharacteristic
261#389—71no—
262#389—71no—
263#388—21no—
264#388—21no—
265#387—141no—
266#387—141no—
267#382—71no—
268C4×C4⋊C4C2×D711yescharacteristic
269#340—71no—
270#340—71no—
271#340—71no—
272#340—71no—
273C23×C7—41no—
274C23×C7—41no—
275C23×C7—41no—
276C23×C7—41no—
277C23×C7—41no—
278C23×C7—41no—
279C23×C7—41no—
280C23×C7C22×D411yescharacteristic
281C7×Q8—28no—
282C7×Q8—41no—
283C7×Q8—41no—
284C7×Q8—41no—
285C7×Q8—41no—
286C7×D4—41no—
287C7×D4—41no—
288C7×D4—41no—
289C7×D4—41no—
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—21no—
301C2×C4×C7—21no—
302C2×C4×C7—21no—
303C2×C4×C7—21no—
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#12112yes—
314C2×C4×C7#12112yes—
315C2×C4×C7#13512yes—
316C2×C4×C7#13512yes—
317C2×C4×C7C22×D411yescharacteristic
318C2×C4×C7C2×#4111yescharacteristic
319C2×C4×C7C2×#4111yescharacteristic
320C2×C4×C7C2×#4111yescharacteristic
321C2×C4×C7H(5,𝔽2)11yescharacteristic
322C2×C4×C72−1+411yescharacteristic
323C2×Dic7—22no—
324C2×Dic7—41no—
325C2×Dic7—41no—
326C2×Dic7—41no—
327C2×Dic7—41no—
328C2×Dic7—41no—
329C2×Dic7—41no—
330C2×Dic7—41no—
331C2×Dic7—41no—
332C2×Dic7—41no—
333C2×Dic7—41no—
334C2×Dic7—41no—
335Dic14—22no—
336Dic14—22no—
337Dic14—41no—
338Dic14—41no—
339C7⋊C8—22no—
340C7⋊C8—22no—
341C7⋊C8—22no—
342C7⋊C8—41no—
343C22×Q8—71no—
344C22×Q8C22⋊D711yescharacteristic
345C22×Q8C22×D711yescharacteristic
346C22×D4C22×D711yescharacteristic
347C2×Q16—142no—
348C2×Q16—142no—
349C2×Q16—281no—
350C2×SD16—142no—
351C2×SD16—142no—
352C2×SD16—281no—
353C22×C8—71no—
354C22×C8—71no—
355C22×C8—71no—
356C22×C8—71no—
357#124—22no—
358#124—22no—
359C4×Q8—22no—
360C4×Q8—22no—
361C4×Q8—142no—
362C4×Q8—142no—
363C2×C4⋊C4—71no—
364C2×C4⋊C4—71no—
365C2×C4⋊C4—71no—
366C2×C4⋊C4—141no—
367C2×C4⋊C4—141no—
368C2×C4⋊C4—141no—
369C2×C4⋊C4C22⋊D711yescharacteristic
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—21no—
377C2×C22⋊C4—21no—
378C2×C22⋊C4—21no—
379C2×C22⋊C4—21no—
380C2×C42—21no—
381C2×C42—141no—
382C2×C42C22×D711yescharacteristic
383C2×C42C22×D711yescharacteristic
384C2×C42C22×D711yescharacteristic
385#107—281no—
386#106—281no—
387C4⋊C8—74no—
388C4⋊C8—142no—
389C4⋊C8—142no—
390#103—142no—
391#103—142no—
392#103—142no—
393#103—142no—
394#103—281no—
395#102—142no—
396#102—142no—
397#102—281no—
398C4×C8—142no—
399C4×C8—142no—
400C4×C8—142no—
401#95—21no—
402#95—21no—
403#95—21no—
404#95—21no—
405#95—21no—
406#95—141no—
407#95—141no—
408#95—141no—
409#95—141no—
410#95—141no—
411#95—141no—
412#95—141no—
413#95—141no—
414C22×C7—81no—
415C22×C7—81no—
416C22×C7—81no—
417C22×C7—81no—
418C22×C7—81no—
419C22×C7—81no—
420C22×C7—81no—
421C22×C7C2×#12111yescharacteristic
422C22×C7#54811yescharacteristic
423C22×C7#55011yescharacteristic
424C22×C7C2×#13511yescharacteristic
425C22×C7C2×#13511yescharacteristic
426C22×C7#57711yescharacteristic
427C22×C7#57811yescharacteristic
428C4×C7—22no—
429C4×C7—22no—
430C4×C7—22no—
431C4×C7—22no—
432C4×C7—41no—
433C4×C7—41no—
434C4×C7—81no—
435C4×C7—81no—
436C4×C7—81no—
437C4×C7#52911yescharacteristic
438C4×C7#54211yescharacteristic
439C4×C7#54611yescharacteristic
440C4×C7#54811yescharacteristic
441Dic7—42no—
442Dic7—81no—
443Dic7—81no—
444Dic7—81no—
445C24—21no—
446C2×Q8—22no—
447C2×Q8—22no—
448C2×Q8—22no—
449C2×Q8—21no—
450C2×Q8—21no—
451C2×Q8—21no—
452C2×Q8—21no—
453C2×Q8—21no—
454C2×Q8—21no—
455C2×Q8—72no—
456C2×Q8—141no—
457C2×Q8—141no—
458C2×Q8—141no—
459C2×Q8—141no—
460C2×Q8—141no—
461C2×D4—21no—
462C2×D4—21no—
463C2×D4—21no—
464C2×D4—21no—
465C2×D4—21no—
466C2×D4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—141no—
471C22×C4—141no—
472C22×C4—141no—
473C22×C4—141no—
474C22×C4C2×C22⋊D711yescharacteristic
475C22×C4C2×C22⋊D711yescharacteristic
476C22×C4C2×C22⋊D711yescharacteristic
477C22×C4C2×C22⋊D711yescharacteristic
478C22×C4C2×C22⋊D711yescharacteristic
479C22×C4C2×C22⋊D711yescharacteristic
480C22×C4C23×D711yescharacteristic
481Q16—284no—
482SD16—284no—
483C2×C8—72no—
484C2×C8—72no—
485C2×C8—72no—
486C2×C8—141no—
487C2×C8—141no—
488C2×C8—141no—
489C2×C8—141no—
490C2×C8—141no—
491C2×C8—141no—
492C2×C8—141no—
493C2×C8—141no—
494C2×C8—141no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—141no—
503C4⋊C4—141no—
504C4⋊C4—141no—
505C4⋊C4—141no—
506C4⋊C4—141no—
507C4⋊C4—141no—
508C4⋊C4—282no—
509C4⋊C4—282no—
510C4⋊C4—282no—
511C22⋊C4—42no—
512C22⋊C4—42no—
513C22⋊C4—42no—
514C22⋊C4—42no—
515C22⋊C4—81no—
516C22⋊C4—81no—
517C42—21no—
518C42—21no—
519C42—21no—
520C42—21no—
521C42—81no—
522C42—282no—
523C42D4×D712yes—
524C42#115912yes—
525C2×C7—161no—
526C2×C7#242611yescharacteristic
527C2×C7#305011yescharacteristic
528C2×C7#306311yescharacteristic
529C2×C7#336411yescharacteristic
530C2×C7#336511yescharacteristic
531C2×C7#339311yescharacteristic
532C2×C7#339411yescharacteristic
533C23—41no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23C22×C22⋊D711yescharacteristic
541Q8—28no—
542Q8—41no—
543Q8—41no—
544Q8—41no—
545Q8—41no—
546Q8—142no—
547Q8—142no—
548Q8—281no—
549Q8—281no—
550D4—41no—
551D4—41no—
552D4—41no—
553D4—41no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—142no—
578C2×C4—281no—
579C2×C4—281no—
580C2×C4—281no—
581C2×C4—281no—
582C2×C4—281no—
583C2×C4—281no—
584C2×C4—281no—
585C2×C4—281no—
586C2×C4—281no—
587C2×C4—281no—
588C2×C4—281no—
589C2×C4#650212yes—
590C2×C4#650212yes—
591C2×C4#651312yes—
592C2×C4#651312yes—
593C2×C4C2×D4×D711yescharacteristic
594C2×C4C2×#115911yescharacteristic
595C2×C4D7×#4111yescharacteristic
596C2×C4D7×#4111yescharacteristic
597C2×C4#655311yescharacteristic
598C2×C4#655411yescharacteristic
599C8—142no—
600C8—142no—
601C8—142no—
602C8—281no—
603C7#2529711yescharacteristic, complemented, Sylow
604C22—81no—
605C22—81no—
606C22—81no—
607C22—81no—
608C22—81no—
609C22—81no—
610C22—81no—
611C22C2×#650211yescharacteristic
612C22C2×#651311yescharacteristic
613C22C2×#651311yescharacteristic
614C22#9052211yescharacteristic
615C22#9052511yescharacteristic
616C22#9052711yescharacteristic
617C22#9052911yescharacteristic
618C4—22no—
619C4—22no—
620C4—22no—
621C4—22no—
622C4—41no—
623C4—41no—
624C4—81no—
625C4—81no—
626C4—81no—
627C4—282no—
628C4—561no—
629C4—561no—
630C4—561no—
631C4#9036711yescharacteristic
632C4#9036811yescharacteristic
633C4#9037111yescharacteristic
634C4#9037311yescharacteristic
635C2—161no—
636C2#1174036711yescharacteristic
637C2#1174044011yescharacteristic
638C2#1174086811yescharacteristic
639C2#1174111511yescharacteristic
640C2#1174112111yescharacteristic
641C2#1174115511yescharacteristic
642C2#1174116211yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#2592C2
socleC23×C7C22×D4
prime radicalC22×C7×D4C23
hypercenter#2592D7
hypocenterC7#25297
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #2592

Complements

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

NH ≅ G/Ntype
3. #11739925C2semidirect
4. C2×Q8×C7⋊C8C2semidirect
6. #11727436C2semidirect
7. #11727435C2semidirect
8. #11727431C2semidirect
9. #11727428C2semidirect
14. C4×#6384C2semidirect
15. #11727222C2semidirect
603. C7#25297semidirect

Maximal subgroups

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

subgroupindex
C7×#25922
C2×#898262
#117399252
C2×Q8×C7⋊C82
#117294792
#117274362
#117274352
#117274312
#117274282
#117274072
#117274052
#117274002
#117273982
C4×#63842
#117272222
#252977