Finite Group Databasebeta

← #13403886

Subgroups — #13403886

Order 1792.

824 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2367C211yesmaximal, characteristic
2C2×#89826C211yesmaximal, characteristic
3C2×#89827C211yesmaximal, characteristic
4#11731167C211yesmaximal, characteristic, complemented
5#11729386C211yesmaximal, characteristic, complemented
6#11729381C211yesmaximal, characteristic, complemented
7#11729366C211yesmaximal, characteristic
8#11729359C211yesmaximal, characteristic
9#11727435C211yesmaximal, characteristic, complemented
10#11727425C211yesmaximal, characteristic, complemented
11#11727405C211yesmaximal, characteristic
12#11727395C211yesmaximal, characteristic
13#11727243C211yesmaximal, characteristic, complemented
14#11727239C211yesmaximal, characteristic, complemented
15#11727229C211yesmaximal, characteristic, complemented
16C2×C7×#124C2211yescharacteristic
17C2×C4×C7×Q8C2211yescharacteristic
18C2×C4×C7×D4C2211yescharacteristic
19C22×#1144C2211yescharacteristic
20C22×#1142C2211yescharacteristic
21C2×#6432C2211yescharacteristic
22C2×C4×Dic14C2211yescharacteristic
23C7×#391C2211yescharacteristic
24C7×#390C2211yescharacteristic
25C7×#388—21no—
26C7×#388—21no—
27C7×#385—21no—
28C4×C7×C4⋊C4C2211yescharacteristic
29C4×C7×C4⋊C4C2211yescharacteristic
30C4×C7×C22⋊C4—21no—
31C2×#6409C2211yescharacteristic
32C2×#6409C2211yescharacteristic
33C2×#6409C2211yescharacteristic
34C2×#6406C2211yescharacteristic
35C2×#6406C2211yescharacteristic
36C2×#6406C2211yescharacteristic
37#89827—24no—
38#89826—24no—
39#89746—21no—
40#89746—21no—
41C2×#6384C2211yescharacteristic
42C2×#6384C2211yescharacteristic
43#89708C2211yescharacteristic
44#89708C2211yescharacteristic
45#89701C2211yescharacteristic
46#89701C2211yescharacteristic
47C2×C4⋊(C7⋊C8)C2211yescharacteristic
48C2×C4⋊(C7⋊C8)C2211yescharacteristic
49C2×C4⋊(C7⋊C8)C2211yescharacteristic
50C2×#6377C2211yescharacteristic
51#89431—21no—
52#89431—21no—
53#89324C2211yescharacteristic
54#89324C2211yescharacteristic
55#89324C2211yescharacteristic
56#89324C2211yescharacteristic
57#89324C2211yescharacteristic
58#89324C2211yescharacteristic
59#89319C2211yescharacteristic
60#89319C2211yescharacteristic
61#25337—71nomaximal, Sylow
62C22×C7×Q8C2311yescharacteristic
63C22×C7×D4C2311yescharacteristic
64C23×C4×C7—21no—
65C22×Dic14—21no—
66C22×Dic14C2311yescharacteristic
67C7×#124—22no—
68C7×#124D414yes—
69C4×C7×Q8—22no—
70C4×C7×Q8—41no—
71C4×C7×D4—22no—
72C4×C7×D4—41no—
73C2×C7×C4⋊C4—21no—
74C2×C7×C4⋊C4D411yescharacteristic
75C2×C7×C4⋊C4D411yescharacteristic
76C2×C7×C4⋊C4C2311yescharacteristic
77C2×C7×C4⋊C4C2311yescharacteristic
78C2×C7×C22⋊C4—21no—
79C2×C7×C22⋊C4—21no—
80C2×C7×C22⋊C4—21no—
81C2×C7×C22⋊C4—21no—
82C2×C7×C22⋊C4—21no—
83C2×C42×C7—21no—
84C2×C42×C7—21no—
85C2×C42×C7C2311yescharacteristic
86C2×C42×C7C2311yescharacteristic
87C2×C42×C7C2311yescharacteristic
88C2×#1144—22no—
89C2×#1144—22no—
90C2×#1144—41no—
91C2×#1142—22no—
92C2×#1142—22no—
93C2×#1142—41no—
94C2×C4⋊Dic7—21no—
95C2×C4⋊Dic7C2311yescharacteristic
96C2×C4⋊Dic7C2311yescharacteristic
97C2×C4⋊Dic7C2311yescharacteristic
98C2×Dic7⋊C4—21no—
99C2×C4×Dic7—21no—
100C22×C7⋊C8C2311yescharacteristic
101C22×C7⋊C8C2311yescharacteristic
102C22×C7⋊C8C2311yescharacteristic
103C22×C7⋊C8C2311yescharacteristic
104#6432—22no—
105#6432—22no—
106C4×Dic14—22no—
107C4×Dic14—22no—
108C7×#95—21no—
109C7×#95—21no—
110C7×#95—21no—
111C7×#95D411yescharacteristic
112C7×#95D411yescharacteristic
113#6409—22no—
114#6409—22no—
115#6409—22no—
116#6406—22no—
117#6406—22no—
118#6406—22no—
119C7⋊#95—21no—
120C7⋊#95—21no—
121C7⋊#95—21no—
122C7⋊#95—21no—
123C7⋊#95—21no—
124C7⋊#95—21no—
125C7⋊#95—21no—
126C7⋊#95—21no—
127#6384—22no—
128#6384—22no—
129C4⋊(C7⋊C8)—22no—
130C4⋊(C7⋊C8)—41no—
131C4⋊(C7⋊C8)—41no—
132#6377—22no—
133C2×#489—71no—
134C2×#464—71no—
135#2568—71no—
136#2367D711yescharacteristic
137#2074—71no—
138#2073—71no—
139#2056—71no—
140#2055—71no—
141#1987—71no—
142#1940—71no—
143#1878—71no—
144#1877—71no—
145#1805—71no—
146#1804—71no—
147#1763—71no—
148C24×C7—21no—
149C2×C7×Q8—21no—
150C2×C7×Q8—21no—
151C2×C7×Q8—21no—
152C2×C7×Q8—21no—
153C2×C7×Q8—21no—
154C2×C7×Q8C2×D412yes—
155C2×C7×D4—21no—
156C2×C7×D4—21no—
157C2×C7×D4—21no—
158C2×C7×D4—21no—
159C2×C7×D4—21no—
160C2×C7×D4C2×D412yes—
161C22×C4×C7—21no—
162C22×C4×C7—21no—
163C22×C4×C7—21no—
164C22×C4×C7—42no—
165C22×C4×C7—42no—
166C22×C4×C7—81no—
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—42no—
184C2×Dic14—42no—
185C2×Dic14—42no—
186C2×Dic14#4112yes—
187C7×C4⋊C4—21no—
188C7×C4⋊C4—21no—
189C7×C4⋊C4—21no—
190C7×C4⋊C4—41no—
191C7×C4⋊C4—41no—
192C7×C4⋊C4—42no—
193C7×C4⋊C4#4112yes—
194C7×C22⋊C4—24no—
195C7×C22⋊C4—24no—
196C7×C22⋊C4—42no—
197C7×C22⋊C4—42no—
198C7×C22⋊C4—81no—
199C42×C7—21no—
200C42×C7—21no—
201C42×C7—21no—
202C42×C7—21no—
203C42×C7—42no—
204C42×C7—81no—
205C42×C7C2×D412yes—
206C42×C7#4112yes—
207#1144—44no—
208#1142—44no—
209C4⋊Dic7—21no—
210C4⋊Dic7—21no—
211C4⋊Dic7—21no—
212C4⋊Dic7—21no—
213C4⋊Dic7—21no—
214C4⋊Dic7—21no—
215C4⋊Dic7—42no—
216Dic7⋊C4—42no—
217C4×Dic7—42no—
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⋊C8—21no—
227C2×C7⋊C8—21no—
228C2×C7⋊C8—21no—
229C2×C7⋊C8—21no—
230C22×Q16—71no—
231C22×SD16—71no—
232C2×C4⋊Q8—71no—
233C2×#124C2×D711yescharacteristic
234C2×C4×Q8—71no—
235C2×C4×Q8C2×D711yescharacteristic
236C2×C4×D4C2×D711yescharacteristic
237#489—144no—
238#464—144no—
239C2×C4⋊C8—71no—
240C2×C4⋊C8—71no—
241C2×C4⋊C8—71no—
242C2×#103—71no—
243C2×#103—71no—
244C2×#103—71no—
245C2×#103—71no—
246C2×#103—71no—
247C2×#102—71no—
248C2×#102—71no—
249C2×#102—71no—
250C2×#97—71no—
251#400—141no—
252#400—141no—
253#391—71no—
254#391—71no—
255#391—141no—
256#391—141no—
257#391C2×D711yescharacteristic
258#390C2×D711yescharacteristic
259#388—21no—
260#388—21no—
261#385—21no—
262#382—71no—
263#382—71no—
264C4×C4⋊C4C2×D711yescharacteristic
265C4×C4⋊C4C2×D711yescharacteristic
266C4×C22⋊C4—21no—
267#340—71no—
268#340—71no—
269#340—71no—
270#340—71no—
271#340—71no—
272#340—71no—
273#336—71no—
274#336—71no—
275C23×C7—22no—
276C23×C7—41no—
277C23×C7—41no—
278C23×C7—41no—
279C23×C7—41no—
280C23×C7—41no—
281C23×C7—41no—
282C23×C7C22×D411yescharacteristic
283C7×Q8—22no—
284C7×Q8—22no—
285C7×Q8—41no—
286C7×Q8—41no—
287C7×D4—22no—
288C7×D4—22no—
289C7×D4—41no—
290C7×D4—41no—
291C2×C4×C7—21no—
292C2×C4×C7—21no—
293C2×C4×C7—21no—
294C2×C4×C7—21no—
295C2×C4×C7—22no—
296C2×C4×C7—21no—
297C2×C4×C7—21no—
298C2×C4×C7—21no—
299C2×C4×C7—21no—
300C2×C4×C7—22no—
301C2×C4×C7—21no—
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—21no—
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—84no—
317C2×C4×C7#12112yes—
318C2×C4×C7#12112yes—
319C2×C4×C7C22×D411yescharacteristic
320C2×C4×C7C2×#4111yescharacteristic
321C2×C4×C7C2×#4111yescharacteristic
322C2×C4×C7C2×#4111yescharacteristic
323C2×C4×C7H(5,𝔽2)11yescharacteristic
324C2×C4×C72−1+411yescharacteristic
325C2×Dic7—22no—
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—
335C2×Dic7—41no—
336C2×Dic7—41no—
337Dic14—22no—
338Dic14—22no—
339Dic14—41no—
340Dic14—41no—
341Dic14—84no—
342C7⋊C8—41no—
343C7⋊C8—41no—
344C7⋊C8—41no—
345C7⋊C8—41no—
346C22×Q8—71no—
347C22×Q8—141no—
348C22×Q8C22×D711yescharacteristic
349C22×D4C22×D711yescharacteristic
350C23×C4—21no—
351C2×Q16—142no—
352C2×Q16—142no—
353C2×Q16—281no—
354C2×SD16—142no—
355C2×SD16—142no—
356C2×SD16—281no—
357C22×C8—71no—
358C22×C8—71no—
359C22×C8—71no—
360C22×C8—71no—
361C4⋊Q8—142no—
362C4⋊Q8—142no—
363#124—22no—
364#124D2814yes—
365C4×Q8—22no—
366C4×Q8—41no—
367C4×Q8—142no—
368C4×Q8—142no—
369C4×D4—22no—
370C4×D4—41no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—71no—
373C2×C4⋊C4—71no—
374C2×C4⋊C4—71no—
375C2×C4⋊C4—141no—
376C2×C4⋊C4—141no—
377C2×C4⋊C4C22⋊D711yescharacteristic
378C2×C4⋊C4C22⋊D711yescharacteristic
379C2×C4⋊C4C22×D711yescharacteristic
380C2×C4⋊C4C22×D711yescharacteristic
381C2×C22⋊C4—21no—
382C2×C22⋊C4—21no—
383C2×C22⋊C4—21no—
384C2×C22⋊C4—21no—
385C2×C22⋊C4—21no—
386C2×C42—21no—
387C2×C42—21no—
388C2×C42—141no—
389C2×C42C22×D711yescharacteristic
390C2×C42C22×D711yescharacteristic
391C2×C42C22×D711yescharacteristic
392C4⋊C8—142no—
393C4⋊C8—281no—
394C4⋊C8—281no—
395#103—142no—
396#103—142no—
397#103—142no—
398#103—142no—
399#103—142no—
400#102—142no—
401#102—142no—
402#102—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—
411#95—141no—
412#95—141no—
413#95—141no—
414#95—141no—
415#95C22⋊D711yescharacteristic
416#95C22⋊D711yescharacteristic
417C22×C7—42no—
418C22×C7—42no—
419C22×C7—42no—
420C22×C7—81no—
421C22×C7—81no—
422C22×C7—81no—
423C22×C7—81no—
424C22×C7C2×#12111yescharacteristic
425C22×C7#54611yescharacteristic
426C22×C7#54811yescharacteristic
427C22×C7#57511yescharacteristic
428C22×C7#57511yescharacteristic
429C22×C7#57711yescharacteristic
430C22×C7#57811yescharacteristic
431C4×C7—22no—
432C4×C7—22no—
433C4×C7—42no—
434C4×C7—41no—
435C4×C7—41no—
436C4×C7—42no—
437C4×C7—41no—
438C4×C7—41no—
439C4×C7—81no—
440C4×C7#52911yescharacteristic
441C4×C7#54211yescharacteristic
442C4×C7#54611yescharacteristic
443C4×C7#54711yescharacteristic
444Dic7—42no—
445Dic7—81no—
446Dic7—81no—
447Dic7—81no—
448C24—21no—
449C2×Q8—21no—
450C2×Q8—21no—
451C2×Q8—21no—
452C2×Q8—21no—
453C2×Q8—21no—
454C2×Q8—72no—
455C2×Q8—141no—
456C2×Q8—141no—
457C2×Q8—141no—
458C2×Q8—141no—
459C2×Q8—141no—
460C2×Q8—282no—
461C2×Q8—282no—
462C2×Q8—282no—
463C2×Q8C2×D2812yes—
464C2×D4—21no—
465C2×D4—21no—
466C2×D4—21no—
467C2×D4—21no—
468C2×D4—21no—
469C2×D4C2×D2812yes—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—42no—
474C22×C4—42no—
475C22×C4—81no—
476C22×C4—141no—
477C22×C4—141no—
478C22×C4—141no—
479C22×C4—141no—
480C22×C4C2×C22⋊D711yescharacteristic
481C22×C4C2×C22⋊D711yescharacteristic
482C22×C4C2×C22⋊D711yescharacteristic
483C22×C4C2×C22⋊D711yescharacteristic
484C22×C4C2×C22⋊D711yescharacteristic
485C22×C4C2×C22⋊D711yescharacteristic
486C22×C4C23×D711yescharacteristic
487Q16—284no—
488SD16—284no—
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—
499C2×C8—141no—
500C2×C8—141no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—41no—
505C4⋊C4—41no—
506C4⋊C4—42no—
507C4⋊C4—141no—
508C4⋊C4—141no—
509C4⋊C4—141no—
510C4⋊C4—141no—
511C4⋊C4—141no—
512C4⋊C4—141no—
513C4⋊C4—282no—
514C4⋊C4—282no—
515C4⋊C4#115912yes—
516C22⋊C4—24no—
517C22⋊C4—24no—
518C22⋊C4—42no—
519C22⋊C4—42no—
520C22⋊C4—81no—
521C42—21no—
522C42—21no—
523C42—21no—
524C42—21no—
525C42—42no—
526C42—81no—
527C42—282no—
528C42C2×D2812yes—
529C42#115712yes—
530C2×C7—82no—
531C2×C7#242711yescharacteristic
532C2×C7#304011yescharacteristic
533C2×C7#304611yescharacteristic
534C2×C7#331111yescharacteristic
535C2×C7#331311yescharacteristic
536C2×C7#337111yescharacteristic
537C2×C7#337711yescharacteristic
538C23—22no—
539C23—41no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23—41no—
545C23C22×C22⋊D711yescharacteristic
546Q8—22no—
547Q8—22no—
548Q8—41no—
549Q8—41no—
550Q8—142no—
551Q8—142no—
552Q8—281no—
553Q8—281no—
554Q8—564no—
555D4—22no—
556D4—22no—
557D4—41no—
558D4—41no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—22no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—84no—
585C2×C4—142no—
586C2×C4—281no—
587C2×C4—281no—
588C2×C4—281no—
589C2×C4—281no—
590C2×C4—281no—
591C2×C4—281no—
592C2×C4—281no—
593C2×C4—281no—
594C2×C4—281no—
595C2×C4—281no—
596C2×C4—281no—
597C2×C4#649312yes—
598C2×C4#649312yes—
599C2×C4C22×D2811yescharacteristic
600C2×C4C2×#115711yescharacteristic
601C2×C4C2×#115911yescharacteristic
602C2×C4#654811yescharacteristic
603C2×C4D7×#4111yescharacteristic
604C2×C4#655411yescharacteristic
605C8—281no—
606C8—281no—
607C8—281no—
608C8—281no—
609C7#2533711yescharacteristic, complemented, Sylow
610C22—42no—
611C22—42no—
612C22—42no—
613C22—81no—
614C22—81no—
615C22—81no—
616C22—81no—
617C22C2×#649311yescharacteristic
618C22#9049711yescharacteristic
619C22#9051311yescharacteristic
620C22#9051311yescharacteristic
621C22#9052211yescharacteristic
622C22#9052711yescharacteristic
623C22#9052911yescharacteristic
624C4—22no—
625C4—22no—
626C4—41no—
627C4—41no—
628C4—42no—
629C4—41no—
630C4—41no—
631C4—42no—
632C4—81no—
633C4—282no—
634C4—561no—
635C4—561no—
636C4—561no—
637C4#9024511yescharacteristic
638C4#9024611yescharacteristic
639C4#9024811yescharacteristic
640C4#9025211yescharacteristic
641C2—82no—
642C2#1173985311yescharacteristic
643C2#1173997211yescharacteristic
644C2#1174001711yescharacteristic
645C2#1174094211yescharacteristic
646C2#1174098211yescharacteristic
647C2#1174112311yescharacteristic
648C2#1174116311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#2367C2
socleC23×C7C22×D4
prime radicalC22×C7×D4C23
hypercenter#2367D7
hypocenterC7#25337
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2367

Complements

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

NH ≅ G/Ntype
4. #11731167C2semidirect
5. #11729386C2semidirect
6. #11729381C2semidirect
9. #11727435C2semidirect
10. #11727425C2semidirect
13. #11727243C2semidirect
14. #11727239C2semidirect
15. #11727229C2semidirect
609. C7#25337semidirect

Maximal subgroups

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

subgroupindex
C7×#23672
C2×#898262
C2×#898272
#117311672
#117293862
#117293812
#117293662
#117293592
#117274352
#117274252
#117274052
#117273952
#117272432
#117272392
#117272292
#253377