Finite Group Databasebeta

← #13426989

Subgroups — #13426989

Order 1792.

762 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2512C211yesmaximal, characteristic
2#11731096C211yesmaximal, characteristic, complemented
3#11727415C211yesmaximal, characteristic
4#11727414C211yesmaximal, characteristic
5#11727414C211yesmaximal, characteristic
6#11727405C211yesmaximal, characteristic
7#11727400C211yesmaximal, characteristic
8#11727391C211yesmaximal, characteristic
9#11727387C211yesmaximal, characteristic, complemented
10#11727355C211yesmaximal, characteristic, complemented
11#11727345C211yesmaximal, characteristic, complemented
12#11727342C211yesmaximal, characteristic, complemented
13#11727236C211yesmaximal, characteristic, complemented
14C4×#6384C211yesmaximal, characteristic, complemented
15#11727218C211yesmaximal, characteristic, complemented
16C2×C7×#126—21no—
17C22×#1142C2211yescharacteristic
18C22×#1142C2211yescharacteristic
19C2×#6432C2211yescharacteristic
20C2×C4×Dic14C2211yescharacteristic
21C2×C4×Dic14C2211yescharacteristic
22C7×#397—21no—
23C7×#396—21no—
24C7×#390C2211yescharacteristic
25C7×#390C2211yescharacteristic
26C7×#390C2211yescharacteristic
27C7×#389C2211yescharacteristic
28C7×#388—21no—
29C7×#384C2211yescharacteristic
30C4×C7×C4⋊C4C2211yescharacteristic
31C4×C7×C4⋊C4C2211yescharacteristic
32C2×#6406C2211yescharacteristic
33C2×#6406C2211yescharacteristic
34C4⋊C4×Dic7—21no—
35#89745—21no—
36C2×#6384C2211yescharacteristic
37C2×#6384C2211yescharacteristic
38C2×#6384C2211yescharacteristic
39C2×#6384C2211yescharacteristic
40C2×#6384C2211yescharacteristic
41C2×#6384C2211yescharacteristic
42C2×#6382C2211yescharacteristic
43C2×#6382C2211yescharacteristic
44C4×C4⋊Dic7C2211yescharacteristic
45#89701C2211yescharacteristic
46#89701C2211yescharacteristic
47C2×C4⋊(C7⋊C8)C2211yescharacteristic
48C2×C4×C7⋊C8C2211yescharacteristic
49#89426—21no—
50#89426—21no—
51#89324C2211yescharacteristic
52#89324C2211yescharacteristic
53#89324C2211yescharacteristic
54#89324C2211yescharacteristic
55#89319C2211yescharacteristic
56#89319C2211yescharacteristic
57#89319C2211yescharacteristic
58#89319C2211yescharacteristic
59#27066—71nomaximal, Sylow
60C22×C7×D4C2311yescharacteristic
61C22×Dic14C2311yescharacteristic
62C22×Dic14C2311yescharacteristic
63C7×#126—44no—
64C2×C7×C4⋊C4—21no—
65C2×C7×C4⋊C4—21no—
66C2×C7×C4⋊C4—21no—
67C2×C7×C4⋊C4D411yescharacteristic
68C2×C7×C4⋊C4D411yescharacteristic
69C2×C7×C4⋊C4C2311yescharacteristic
70C2×C7×C4⋊C4C2311yescharacteristic
71C2×C7×C4⋊C4C2311yescharacteristic
72C2×C7×C22⋊C4—21no—
73C2×C7×C22⋊C4—21no—
74C2×C7×C22⋊C4—21no—
75C2×C7×C22⋊C4—21no—
76C2×C7×C22⋊C4—21no—
77C2×C7×C22⋊C4—21no—
78C2×C42×C7—21no—
79C2×C42×C7C2311yescharacteristic
80C2×C42×C7C2311yescharacteristic
81C2×C42×C7C2311yescharacteristic
82C2×#1142—22no—
83C2×#1142—22no—
84C2×#1142—22no—
85C2×#1142—41no—
86C2×#1142—41no—
87C2×#1142—41no—
88C2×C4⋊Dic7—21no—
89C2×C4⋊Dic7—21no—
90C2×C4⋊Dic7C2311yescharacteristic
91C2×C4⋊Dic7C2311yescharacteristic
92C2×Dic7⋊C4—21no—
93C2×Dic7⋊C4—21no—
94C2×C4×Dic7—21no—
95C2×C4×Dic7—21no—
96C2×C4×Dic7—21no—
97C22×C7⋊C8C2311yescharacteristic
98C22×C7⋊C8C2311yescharacteristic
99C22×C7⋊C8C2311yescharacteristic
100C22×C7⋊C8C2311yescharacteristic
101#6432—22no—
102#6432—22no—
103C4×Dic14—22no—
104C4×Dic14—22no—
105C4×Dic14—41no—
106C4×Dic14—41no—
107C7×#95—21no—
108C7×#95—21no—
109C7×#95D411yescharacteristic
110C7×#95D411yescharacteristic
111#6406—22no—
112#6406—41no—
113C7⋊#95—21no—
114C7⋊#95—21no—
115C7⋊#95—21no—
116C7⋊#95—21no—
117C7⋊#95—21no—
118#6384—22no—
119#6384—22no—
120#6384—22no—
121#6384—41no—
122#6384—41no—
123#6384—41no—
124#6382—22no—
125#6382—41no—
126C4⋊(C7⋊C8)—41no—
127C4×C7⋊C8—41no—
128#2512D711yescharacteristic
129#2365—71no—
130#2096—71no—
131#2055—71no—
132#2043—71no—
133#2013—71no—
134#1968—71no—
135#1927—71no—
136#1926—71no—
137#1919—71no—
138#1916—71no—
139#1880—71no—
140#1880—71no—
141#1803—71no—
142C4×#103—71no—
143C24×C7—21no—
144C2×C7×D4—21no—
145C2×C7×D4—21no—
146C2×C7×D4—21no—
147C2×C7×D4—21no—
148C2×C7×D4—21no—
149C2×C7×D4—21no—
150C22×C4×C7—21no—
151C22×C4×C7—21no—
152C22×C4×C7—21no—
153C22×C4×C7C2×D411yescharacteristic
154C22×C4×C7C2×D411yescharacteristic
155C22×C4×C7C2×D411yescharacteristic
156C22×C4×C7C2×D411yescharacteristic
157C22×C4×C7C2×D411yescharacteristic
158C22×C4×C7C2×D411yescharacteristic
159C22×C4×C7C2411yescharacteristic
160C22×Dic7—21no—
161C22×Dic7—21no—
162C22×Dic7—21no—
163C22×Dic7—21no—
164C2×Dic14—21no—
165C2×Dic14—21no—
166C2×Dic14—21no—
167C2×Dic14—21no—
168C2×Dic14—21no—
169C2×Dic14—21no—
170C2×Dic14—21no—
171C2×Dic14—21no—
172C2×Dic14—21no—
173C2×Dic14—21no—
174C2×Dic14#4112yes—
175C2×Dic14#4112yes—
176C7×C4⋊C4—21no—
177C7×C4⋊C4—21no—
178C7×C4⋊C4—21no—
179C7×C4⋊C4—21no—
180C7×C4⋊C4—21no—
181C7×C4⋊C4—21no—
182C7×C4⋊C4—41no—
183C7×C4⋊C4—42no—
184C7×C4⋊C4—42no—
185C7×C4⋊C4—81no—
186C7×C4⋊C4SD1614yescomplemented
187C7×C22⋊C4—42no—
188C7×C22⋊C4—42no—
189C7×C22⋊C4—42no—
190C7×C22⋊C4—42no—
191C7×C22⋊C4—81no—
192C7×C22⋊C4—81no—
193C42×C7—21no—
194C42×C7—21no—
195C42×C7—21no—
196C42×C7—21no—
197C42×C7—21no—
198C42×C7—21no—
199C42×C7—42no—
200#1142—44no—
201#1142—44no—
202C4⋊Dic7—24no—
203C4⋊Dic7—24no—
204C4⋊Dic7—21no—
205C4⋊Dic7—21no—
206C4⋊Dic7—21no—
207C4⋊Dic7#4112yes—
208Dic7⋊C4—42no—
209Dic7⋊C4—42no—
210C4×Dic7—42no—
211C4×Dic7—42no—
212C4×Dic7—42no—
213C2×C7⋊C8—21no—
214C2×C7⋊C8—21no—
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#4112yes—
225C22×SD16—71no—
226C22×SD16—71no—
227C2×C4⋊Q8—71no—
228C2×#126—21no—
229C2×C4×Q8—71no—
230C2×C4×Q8—71no—
231C2×#106—71no—
232C2×#106—71no—
233C2×C4⋊C8—71no—
234C2×#103—71no—
235C2×#103—71no—
236C2×#103—71no—
237C2×#103—71no—
238C2×#103—71no—
239C2×#103—71no—
240C2×#102—71no—
241C2×#102—71no—
242C2×C4×C8—71no—
243#397—21no—
244#396—21no—
245#391—71no—
246#391—71no—
247#390C2×D711yescharacteristic
248#390C2×D711yescharacteristic
249#390C2×D711yescharacteristic
250#389—141no—
251#389C2×D711yescharacteristic
252#388—21no—
253#387—141no—
254#387—141no—
255#384C2×D711yescharacteristic
256C4×C4⋊C4—71no—
257C4×C4⋊C4—141no—
258C4×C4⋊C4C2×D711yescharacteristic
259C4×C4⋊C4C2×D711yescharacteristic
260#340—71no—
261#340—71no—
262#340—71no—
263#340—71no—
264#336—71no—
265#336—71no—
266#336—71no—
267#336—71no—
268C23×C7—41no—
269C23×C7—41no—
270C23×C7—41no—
271C23×C7—41no—
272C23×C7—41no—
273C23×C7—41no—
274C23×C7—41no—
275C23×C7C22×D411yescharacteristic
276C7×D4—41no—
277C7×D4—41no—
278C7×D4—41no—
279C7×D4—41no—
280C2×C4×C7—21no—
281C2×C4×C7—21no—
282C2×C4×C7—21no—
283C2×C4×C7—21no—
284C2×C4×C7—21no—
285C2×C4×C7—21no—
286C2×C4×C7—21no—
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—41no—
296C2×C4×C7—41no—
297C2×C4×C7—41no—
298C2×C4×C7—41no—
299C2×C4×C7—41no—
300C2×C4×C7—41no—
301C2×C4×C7—41no—
302C2×C4×C7—41no—
303C2×C4×C7—41no—
304C2×C4×C7C2×SD1612yes—
305C2×C4×C7C2×SD1612yes—
306C2×C4×C7C2×SD1612yes—
307C2×C4×C7C2×#4111yescharacteristic
308C2×C4×C7C2×#4111yescharacteristic
309C2×C4×C7C2×#4111yescharacteristic
310C2×C4×C7C2×#4111yescharacteristic
311C2×C4×C7H(5,𝔽2)11yescharacteristic
312C2×C4×C7H(5,𝔽2)11yescharacteristic
313C2×Dic7—22no—
314C2×Dic7—22no—
315C2×Dic7—22no—
316C2×Dic7—41no—
317C2×Dic7—41no—
318C2×Dic7—41no—
319C2×Dic7—41no—
320C2×Dic7—41no—
321C2×Dic7—41no—
322C2×Dic7—41no—
323C2×Dic7—41no—
324C2×Dic7—41no—
325Dic14—22no—
326Dic14—22no—
327Dic14—22no—
328Dic14—22no—
329Dic14—41no—
330Dic14—41no—
331Dic14—41no—
332Dic14—41no—
333C7⋊C8—22no—
334C7⋊C8—41no—
335C7⋊C8—41no—
336C7⋊C8—41no—
337C22×Q8—71no—
338C22×Q8—71no—
339C22×D4C22×D711yescharacteristic
340C2×SD16—142no—
341C2×SD16—142no—
342C2×SD16—142no—
343C2×SD16—281no—
344C2×SD16—281no—
345C2×SD16—281no—
346C22×C8—71no—
347C22×C8—71no—
348C22×C8—71no—
349C22×C8—71no—
350C4⋊Q8—142no—
351C4⋊Q8—142no—
352#126—44no—
353C4×Q8—142no—
354C4×Q8—142no—
355C4×Q8—281no—
356C4×Q8—281no—
357C2×C4⋊C4—21no—
358C2×C4⋊C4—21no—
359C2×C4⋊C4—21no—
360C2×C4⋊C4—71no—
361C2×C4⋊C4—71no—
362C2×C4⋊C4—141no—
363C2×C4⋊C4—141no—
364C2×C4⋊C4—141no—
365C2×C4⋊C4—141no—
366C2×C4⋊C4C22⋊D711yescharacteristic
367C2×C4⋊C4C22⋊D711yescharacteristic
368C2×C4⋊C4C22×D711yescharacteristic
369C2×C4⋊C4C22×D711yescharacteristic
370C2×C4⋊C4C22×D711yescharacteristic
371C2×C22⋊C4—21no—
372C2×C22⋊C4—21no—
373C2×C22⋊C4—21no—
374C2×C22⋊C4—21no—
375C2×C22⋊C4—21no—
376C2×C22⋊C4—21no—
377C2×C42—21no—
378C2×C42—141no—
379C2×C42—141no—
380C2×C42—141no—
381C2×C42C22×D711yescharacteristic
382C2×C42C22×D711yescharacteristic
383C2×C42C22×D711yescharacteristic
384#106—142no—
385#106—281no—
386C4⋊C8—281no—
387#103—142no—
388#103—142no—
389#103—142no—
390#103—281no—
391#103—281no—
392#103—281no—
393#102—142no—
394#102—281no—
395C4×C8—281no—
396#95—21no—
397#95—21no—
398#95—141no—
399#95—141no—
400#95—141no—
401#95—141no—
402#95—141no—
403#95C22⋊D711yescharacteristic
404#95C22⋊D711yescharacteristic
405C22×C7—81no—
406C22×C7—81no—
407C22×C7—81no—
408C22×C7—81no—
409C22×C7—81no—
410C22×C7—81no—
411C22×C7—81no—
412C22×C7#52511yescharacteristic
413C22×C7#54611yescharacteristic
414C22×C7#55011yescharacteristic
415C22×C7C22×SD1611yescharacteristic
416C22×C7#57511yescharacteristic
417C22×C7#57711yescharacteristic
418C22×C7#57811yescharacteristic
419C4×C7—22no—
420C4×C7—22no—
421C4×C7—22no—
422C4×C7—41no—
423C4×C7—41no—
424C4×C7—41no—
425C4×C7—81no—
426C4×C7—81no—
427C4×C7—81no—
428C4×C7#52911yescharacteristic
429C4×C7#53811yescharacteristic
430C4×C7#55111yescharacteristic
431C4×C7#55311yescharacteristic
432Dic7—42no—
433Dic7—42no—
434Dic7—42no—
435Dic7—81no—
436C24—21no—
437C2×Q8—72no—
438C2×Q8—72no—
439C2×Q8—141no—
440C2×Q8—141no—
441C2×Q8—141no—
442C2×Q8—141no—
443C2×Q8—141no—
444C2×Q8—141no—
445C2×Q8—141no—
446C2×Q8—141no—
447C2×Q8—141no—
448C2×Q8—141no—
449C2×D4—21no—
450C2×D4—21no—
451C2×D4—21no—
452C2×D4—21no—
453C2×D4—21no—
454C2×D4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—141no—
459C22×C4—141no—
460C22×C4—141no—
461C22×C4—141no—
462C22×C4C2×C22⋊D711yescharacteristic
463C22×C4C2×C22⋊D711yescharacteristic
464C22×C4C2×C22⋊D711yescharacteristic
465C22×C4C2×C22⋊D711yescharacteristic
466C22×C4C2×C22⋊D711yescharacteristic
467C22×C4C2×C22⋊D711yescharacteristic
468C22×C4C23×D711yescharacteristic
469SD16—284no—
470SD16—284no—
471C2×C8—72no—
472C2×C8—141no—
473C2×C8—141no—
474C2×C8—141no—
475C2×C8—141no—
476C2×C8—141no—
477C2×C8—141no—
478C2×C8—141no—
479C2×C8—141no—
480C2×C8—141no—
481C2×C8—141no—
482C2×C8—141no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—41no—
490C4⋊C4—42no—
491C4⋊C4—42no—
492C4⋊C4—72no—
493C4⋊C4—81no—
494C4⋊C4—144no—
495C4⋊C4—144no—
496C4⋊C4—141no—
497C4⋊C4—141no—
498C4⋊C4—141no—
499C4⋊C4—282no—
500C4⋊C4—282no—
501C4⋊C4#114214yescomplemented
502C22⋊C4—42no—
503C22⋊C4—42no—
504C22⋊C4—42no—
505C22⋊C4—42no—
506C22⋊C4—81no—
507C22⋊C4—81no—
508C42—21no—
509C42—21no—
510C42—21no—
511C42—21no—
512C42—21no—
513C42—21no—
514C42—42no—
515C42—282no—
516C42—282no—
517C42—282no—
518C2×C7—161no—
519C2×C7#246711yescharacteristic
520C2×C7#310311yescharacteristic
521C2×C7#312111yescharacteristic
522C2×C7#329811yescharacteristic
523C2×C7#330711yescharacteristic
524C2×C7#339211yescharacteristic
525C2×C7#340211yescharacteristic
526C23—41no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23—41no—
533C23C22×C22⋊D711yescharacteristic
534Q8—142no—
535Q8—142no—
536Q8—142no—
537Q8—142no—
538Q8—281no—
539Q8—281no—
540Q8—281no—
541Q8—281no—
542D4—41no—
543D4—41no—
544D4—41no—
545D4—41no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—142no—
571C2×C4—142no—
572C2×C4—142no—
573C2×C4—281no—
574C2×C4—281no—
575C2×C4—281no—
576C2×C4—281no—
577C2×C4—281no—
578C2×C4—281no—
579C2×C4—281no—
580C2×C4—281no—
581C2×C4—281no—
582C2×C4C2×#114212yes—
583C2×C4C2×#114212yes—
584C2×C4C2×#114212yes—
585C2×C4D7×#4111yescharacteristic
586C2×C4D7×#4111yescharacteristic
587C2×C4D7×#4111yescharacteristic
588C2×C4D7×#4111yescharacteristic
589C2×C4#655311yescharacteristic
590C2×C4#655311yescharacteristic
591C8—142no—
592C8—281no—
593C8—281no—
594C8—281no—
595C7#2706611yescharacteristic, complemented, Sylow
596C22—81no—
597C22—81no—
598C22—81no—
599C22—81no—
600C22—81no—
601C22—81no—
602C22—81no—
603C22C22×#114211yescharacteristic
604C22#9051311yescharacteristic
605C22#9051911yescharacteristic
606C22#9052011yescharacteristic
607C22#9052511yescharacteristic
608C22#9052711yescharacteristic
609C22#9052911yescharacteristic
610C4—22no—
611C4—22no—
612C4—22no—
613C4—41no—
614C4—41no—
615C4—41no—
616C4—81no—
617C4—81no—
618C4—81no—
619C4—282no—
620C4—282no—
621C4—282no—
622C4—561no—
623C4#9039511yescharacteristic
624C4#9039611yescharacteristic
625C4#9039711yescharacteristic
626C4#9040011yescharacteristic
627C2—161no—
628C2#1174055111yescharacteristic
629C2#1174076011yescharacteristic
630C2#1174091211yescharacteristic
631C2#1174097611yescharacteristic
632C2#1174103311yescharacteristic
633C2#1174106011yescharacteristic
634C2#1174112611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#2512C2
socleC23×C7C22×D4
prime radicalC22×C7×D4C23
hypercenter#2512D7
hypocenterC7#27066
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2512

Complements

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

NH ≅ G/Ntype
2. #11731096C2semidirect
9. #11727387C2semidirect
10. #11727355C2semidirect
11. #11727345C2semidirect
12. #11727342C2semidirect
13. #11727236C2semidirect
14. C4×#6384C2semidirect
15. #11727218C2semidirect
186. C7×C4⋊C4SD16semidirect
501. C4⋊C4#1142semidirect
595. C7#27066semidirect

Maximal subgroups

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

subgroupindex
C7×#25122
#117310962
#117274152
#117274142
#117274052
#117274002
#117273912
#117273872
#117273552
#117273452
#117273422
#117272362
C4×#63842
#117272182
#270667