Finite Group Databasebeta

← #13422288

Subgroups — #13422288

Order 1792.

762 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×#2490C211yesmaximal, characteristic
2#11730913C211yesmaximal, characteristic, complemented
3#11729498C211yesmaximal, characteristic, complemented
4#11729492C211yesmaximal, characteristic, complemented
5#11729439C211yesmaximal, characteristic
6#11729432C211yesmaximal, characteristic
7#11727414C211yesmaximal, characteristic, complemented
8#11727405C211yesmaximal, characteristic, complemented
9#11727399C211yesmaximal, characteristic, complemented
10#11727391C211yesmaximal, characteristic, complemented
11#11727280C211yesmaximal, characteristic, complemented
12#11727257C211yesmaximal, characteristic, complemented
13#11727229C211yesmaximal, characteristic, complemented
14#11727214C211yesmaximal, characteristic, complemented
15#11727206C211yesmaximal, characteristic, complemented
16C2×C7×#125C2211yescharacteristic
17C2×C7×#121C2211yescharacteristic
18C22×#1142C2211yescharacteristic
19C2×#6489C2211yescharacteristic
20C2×#6487C2211yescharacteristic
21C2×C4×Dic14C2211yescharacteristic
22C7×#397—21no—
23C7×#394—21no—
24C7×#390—21no—
25C7×#390C2211yescharacteristic
26C7×#390C2211yescharacteristic
27C7×#388—21no—
28C7×#380C2211yescharacteristic
29C4×C7×C4⋊C4C2211yescharacteristic
30C4×C7×C22⋊C4C2211yescharacteristic
31C2×#6406C2211yescharacteristic
32C2×#6406C2211yescharacteristic
33C2×#6406C2211yescharacteristic
34C2×C22⋊(C7⋊C8)C2211yescharacteristic
35C2×C22⋊(C7⋊C8)C2211yescharacteristic
36C2×#6384C2211yescharacteristic
37C2×#6384C2211yescharacteristic
38C2×#6384C2211yescharacteristic
39C2×#6382C2211yescharacteristic
40#89717—21no—
41#89716—21no—
42C4×C4⋊Dic7C2211yescharacteristic
43#89701C2211yescharacteristic
44#89700C2211yescharacteristic
45C2×C4⋊(C7⋊C8)C2211yescharacteristic
46C2×#6377C2211yescharacteristic
47#89424—21no—
48#89421—21no—
49#89324C2211yescharacteristic
50#89324C2211yescharacteristic
51#89324C2211yescharacteristic
52#89324C2211yescharacteristic
53#89324C2211yescharacteristic
54#89324C2211yescharacteristic
55#89324C2211yescharacteristic
56#89324C2211yescharacteristic
57#89319C2211yescharacteristic
58#89319C2211yescharacteristic
59#26751—71nomaximal, Sylow
60C22×C7×D4—21no—
61C22×C7×D4C2311yescharacteristic
62C23×C4×C7C2311yescharacteristic
63C22×Dic14C2311yescharacteristic
64C7×#125—22no—
65C7×#125—41no—
66C7×#121—22no—
67C7×#121—41no—
68C2×C7×C4⋊C4—21no—
69C2×C7×C4⋊C4—21no—
70C2×C7×C4⋊C4C2311yescharacteristic
71C2×C7×C4⋊C4C2311yescharacteristic
72C2×C7×C4⋊C4C2311yescharacteristic
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×C7×C22⋊C4—21no—
79C2×C7×C22⋊C4D411yescharacteristic
80C2×C7×C22⋊C4D411yescharacteristic
81C2×C42×C7—21no—
82C2×C42×C7C2311yescharacteristic
83C2×C42×C7C2311yescharacteristic
84C2×C22⋊Dic7—21no—
85C2×C22⋊Dic7—21no—
86C2×#1142—22no—
87C2×#1142—41no—
88C2×#1142—41no—
89#6489—22no—
90#6489—41no—
91C2×C4⋊Dic7C2311yescharacteristic
92C2×C4⋊Dic7C2311yescharacteristic
93C2×C4⋊Dic7C2311yescharacteristic
94#6487—22no—
95#6487—22no—
96C2×Dic7⋊C4—21no—
97C2×Dic7⋊C4—21no—
98C2×Dic7⋊C4—21no—
99C2×C4×Dic7—21no—
100C2×C4×Dic7—21no—
101C2×C4×Dic7—21no—
102C22×C7⋊C8C2311yescharacteristic
103C22×C7⋊C8C2311yescharacteristic
104C22×C7⋊C8C2311yescharacteristic
105C22×C7⋊C8C2311yescharacteristic
106C4×Dic14—22no—
107C4×Dic14—41no—
108C7×#95—21no—
109C7×#95—21no—
110C7×#95D411yescharacteristic
111C7×#95D411yescharacteristic
112#6406—22no—
113#6406—22no—
114#6406—41no—
115C7⋊#95—21no—
116C7⋊#95—21no—
117C7⋊#95—21no—
118C7⋊#95—21no—
119C22⋊(C7⋊C8)—41no—
120C22⋊(C7⋊C8)—41no—
121#6384—22no—
122#6384—22no—
123#6384—41no—
124#6382—22no—
125C4⋊(C7⋊C8)—41no—
126#6377—41no—
127#2490D711yescharacteristic
128#2362—71no—
129#2055—71no—
130#2035—71no—
131#2026—71no—
132#2000—71no—
133#1989—71no—
134#1936—71no—
135#1926—71no—
136#1924—71no—
137#1920—71no—
138#1880—71no—
139#1875—71no—
140#1825—71no—
141#1763—71no—
142C24×C7—21no—
143C24×C7C2×D411yescharacteristic
144C2×C7×D4—21no—
145C2×C7×D4—21no—
146C2×C7×D4—21no—
147C2×C7×D4—21no—
148C2×C7×D4—21no—
149C2×C7×D4—21no—
150C2×C7×D4—42no—
151C2×C7×D4—42no—
152C2×C7×D4—81no—
153C22×C4×C7—21no—
154C22×C4×C7—21no—
155C22×C4×C7—21no—
156C22×C4×C7—21no—
157C22×C4×C7—21no—
158C22×C4×C7—21no—
159C22×C4×C7—21no—
160C22×C4×C7—21no—
161C22×C4×C7—21no—
162C22×C4×C7C2×D411yescharacteristic
163C22×C4×C7C2×D411yescharacteristic
164C22×C4×C7C2×D411yescharacteristic
165C22×C4×C7C2×D411yescharacteristic
166C22×C4×C7C2×D411yescharacteristic
167C22×C4×C7C2411yescharacteristic
168C22×Dic7—21no—
169C22×Dic7—21no—
170C22×Dic7—21no—
171C22×Dic7—21no—
172C2×Dic14—21no—
173C2×Dic14—21no—
174C2×Dic14—21no—
175C2×Dic14—21no—
176C2×Dic14—21no—
177C2×Dic14#4112yes—
178C7×C4⋊C4—21no—
179C7×C4⋊C4—21no—
180C7×C4⋊C4—21no—
181C7×C4⋊C4—21no—
182C7×C4⋊C4—21no—
183C7×C4⋊C4—42no—
184C7×C4⋊C4—42no—
185C7×C4⋊C4#4112yes—
186C7×C22⋊C4—41no—
187C7×C22⋊C4—41no—
188C7×C22⋊C4—42no—
189C7×C22⋊C4—42no—
190C7×C22⋊C4—81no—
191C7×C22⋊C4—81no—
192C7×C22⋊C4—81no—
193C7×C22⋊C4—81no—
194C42×C7—21no—
195C42×C7—21no—
196C42×C7—21no—
197C42×C7—21no—
198C42×C7—81no—
199C22⋊Dic7—24no—
200C22⋊Dic7—42no—
201#1142—44no—
202C4⋊Dic7—21no—
203C4⋊Dic7—21no—
204C4⋊Dic7—21no—
205C4⋊Dic7—21no—
206C4⋊Dic7#4112yes—
207C4⋊Dic7#4112yes—
208Dic7⋊C4—24no—
209Dic7⋊C4—42no—
210Dic7⋊C4—42no—
211C4×Dic7—42no—
212C4×Dic7—42no—
213C4×Dic7—42no—
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—21no—
225C2×C7⋊C8—21no—
226C22×SD16—71no—
227C2×#125C2×D711yescharacteristic
228C2×C22⋊Q8—71no—
229C2×#121C2×D711yescharacteristic
230C2×C4×Q8—71no—
231C2×#117—71no—
232C2×#106—71no—
233C2×C4⋊C8—71no—
234C2×#103—71no—
235C2×#103—71no—
236C2×#103—71no—
237C2×#102—71no—
238C2×#102—71no—
239C2×#102—71no—
240C2×C22⋊C8—71no—
241C2×C22⋊C8—71no—
242C2×#97—71no—
243#397—21no—
244#394—21no—
245#391—71no—
246#390—21no—
247#390C2×D711yescharacteristic
248#390C2×D711yescharacteristic
249#389—71no—
250#388—21no—
251#388—141no—
252#388—141no—
253#387—141no—
254#380C2×D711yescharacteristic
255C4×C4⋊C4—71no—
256C4×C4⋊C4—141no—
257C4×C4⋊C4C2×D711yescharacteristic
258C4×C22⋊C4C2×D711yescharacteristic
259#340—71no—
260#340—71no—
261#340—71no—
262#340—71no—
263#340—71no—
264#340—71no—
265#340—71no—
266#340—71no—
267#336—71no—
268#336—71no—
269C23×C7—21no—
270C23×C7—21no—
271C23×C7—21no—
272C23×C7—21no—
273C23×C7—21no—
274C23×C7—21no—
275C23×C7—21no—
276C23×C7—41no—
277C23×C7—41no—
278C23×C7—41no—
279C23×C7—41no—
280C23×C7—41no—
281C23×C7—41no—
282C23×C7—41no—
283C23×C7C22×D411yescharacteristic
284C7×D4—41no—
285C7×D4—41no—
286C7×D4—41no—
287C7×D4—41no—
288C7×D4—84no—
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—22no—
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—21no—
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×C7C2×#4111yescharacteristic
318C2×C4×C7C2×#4111yescharacteristic
319C2×C4×C7C2×#4111yescharacteristic
320C2×C4×C7C2×#4111yescharacteristic
321C2×C4×C7H(5,𝔽2)11yescharacteristic
322C2×C4×C7H(5,𝔽2)11yescharacteristic
323C2×Dic7—22no—
324C2×Dic7—22no—
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—
335Dic14—22no—
336Dic14—22no—
337Dic14—41no—
338Dic14—41no—
339C7⋊C8—41no—
340C7⋊C8—41no—
341C7⋊C8—41no—
342C7⋊C8—41no—
343C22×Q8—71no—
344C22×D4—21no—
345C22×D4C22×D711yescharacteristic
346C23×C4C22×D711yescharacteristic
347C2×SD16—142no—
348C2×SD16—281no—
349C2×SD16—281no—
350C22×C8—71no—
351C22×C8—71no—
352C22×C8—71no—
353C22×C8—71no—
354#125—22no—
355#125—41no—
356C22⋊Q8—142no—
357C22⋊Q8—142no—
358#121—22no—
359#121—41no—
360C4×Q8—142no—
361C4×Q8—281no—
362#117—142no—
363#117—281no—
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⋊C4—141no—
371C2×C4⋊C4—141no—
372C2×C4⋊C4C22×D711yescharacteristic
373C2×C4⋊C4C22×D711yescharacteristic
374C2×C4⋊C4C22×D711yescharacteristic
375C2×C22⋊C4—21no—
376C2×C22⋊C4—21no—
377C2×C22⋊C4—21no—
378C2×C22⋊C4—21no—
379C2×C22⋊C4—21no—
380C2×C22⋊C4—21no—
381C2×C22⋊C4—141no—
382C2×C22⋊C4—141no—
383C2×C22⋊C4C22⋊D711yescharacteristic
384C2×C22⋊C4C22⋊D711yescharacteristic
385C2×C42—21no—
386C2×C42—141no—
387C2×C42—141no—
388C2×C42—141no—
389C2×C42C22×D711yescharacteristic
390C2×C42C22×D711yescharacteristic
391#106—142no—
392C4⋊C8—281no—
393#103—142no—
394#103—142no—
395#103—281no—
396#102—142no—
397#102—142no—
398#102—281no—
399C22⋊C8—281no—
400C22⋊C8—281no—
401#97—281no—
402#95—21no—
403#95—21no—
404#95—141no—
405#95—141no—
406#95—141no—
407#95—141no—
408#95C22⋊D711yescharacteristic
409#95C22⋊D711yescharacteristic
410C22×C7—41no—
411C22×C7—41no—
412C22×C7—41no—
413C22×C7—41no—
414C22×C7—41no—
415C22×C7—41no—
416C22×C7—41no—
417C22×C7—81no—
418C22×C7—81no—
419C22×C7—81no—
420C22×C7—81no—
421C22×C7—81no—
422C22×C7—81no—
423C22×C7—81no—
424C22×C7#52511yescharacteristic
425C22×C7#54611yescharacteristic
426C22×C7#54611yescharacteristic
427C22×C7#57511yescharacteristic
428C22×C7#57511yescharacteristic
429C22×C7#57711yescharacteristic
430C22×C7#57711yescharacteristic
431C4×C7—41no—
432C4×C7—41no—
433C4×C7—41no—
434C4×C7—42no—
435C4×C7—41no—
436C4×C7—41no—
437C4×C7—81no—
438C4×C7—81no—
439C4×C7#52911yescharacteristic
440C4×C7#53811yescharacteristic
441C4×C7#55511yescharacteristic
442C4×C7#55911yescharacteristic
443Dic7—42no—
444Dic7—42no—
445Dic7—42no—
446Dic7—81no—
447C24—21no—
448C24C2×C22⋊D711yescharacteristic
449C2×Q8—72no—
450C2×Q8—141no—
451C2×Q8—141no—
452C2×Q8—141no—
453C2×Q8—141no—
454C2×Q8—141no—
455C2×D4—21no—
456C2×D4—21no—
457C2×D4—21no—
458C2×D4—21no—
459C2×D4—21no—
460C2×D4—21no—
461C2×D4—42no—
462C2×D4—42no—
463C2×D4—81no—
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—21no—
473C22×C4—141no—
474C22×C4—141no—
475C22×C4—141no—
476C22×C4—141no—
477C22×C4C2×C22⋊D711yescharacteristic
478C22×C4C2×C22⋊D711yescharacteristic
479C22×C4C2×C22⋊D711yescharacteristic
480C22×C4C2×C22⋊D711yescharacteristic
481C22×C4C2×C22⋊D711yescharacteristic
482C22×C4C23×D711yescharacteristic
483SD16—284no—
484C2×C8—141no—
485C2×C8—141no—
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—
495C2×C8—141no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—42no—
502C4⋊C4—42no—
503C4⋊C4—72no—
504C4⋊C4—72no—
505C4⋊C4—144no—
506C4⋊C4—141no—
507C4⋊C4—141no—
508C4⋊C4—141no—
509C4⋊C4—141no—
510C4⋊C4—282no—
511C4⋊C4—282no—
512C4⋊C4#115912yes—
513C22⋊C4—41no—
514C22⋊C4—41no—
515C22⋊C4—42no—
516C22⋊C4—42no—
517C22⋊C4—81no—
518C22⋊C4—81no—
519C22⋊C4—81no—
520C22⋊C4—81no—
521C22⋊C4—144no—
522C22⋊C4—282no—
523C42—21no—
524C42—21no—
525C42—21no—
526C42—21no—
527C42—81no—
528C42—282no—
529C42—282no—
530C42—282no—
531C2×C7—81no—
532C2×C7—161no—
533C2×C7#245911yescharacteristic
534C2×C7#311111yescharacteristic
535C2×C7#312611yescharacteristic
536C2×C7#330711yescharacteristic
537C2×C7#331111yescharacteristic
538C2×C7#331711yescharacteristic
539C2×C7#332611yescharacteristic
540C23—21no—
541C23—21no—
542C23—21no—
543C23—21no—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—41no—
548C23—41no—
549C23—41no—
550C23—41no—
551C23—41no—
552C23—41no—
553C23—41no—
554C23C22×C22⋊D711yescharacteristic
555Q8—142no—
556Q8—142no—
557Q8—281no—
558Q8—281no—
559D4—41no—
560D4—41no—
561D4—41no—
562D4—41no—
563D4—84no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—22no—
568C2×C4—21no—
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—21no—
578C2×C4—21no—
579C2×C4—21no—
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—142no—
594C2×C4—142no—
595C2×C4—281no—
596C2×C4—281no—
597C2×C4—281no—
598C2×C4—281no—
599C2×C4—281no—
600C2×C4—281no—
601C2×C4—281no—
602C2×C4—281no—
603C2×C4—281no—
604C2×C4C2×#115911yescharacteristic
605C2×C4#654811yescharacteristic
606C2×C4D7×#4111yescharacteristic
607C2×C4D7×#4111yescharacteristic
608C2×C4D7×#4111yescharacteristic
609C2×C4#655311yescharacteristic
610C8—281no—
611C8—281no—
612C8—281no—
613C8—281no—
614C7#2675111yescharacteristic, complemented, Sylow
615C22—41no—
616C22—41no—
617C22—41no—
618C22—41no—
619C22—41no—
620C22—41no—
621C22—41no—
622C22—81no—
623C22—81no—
624C22—81no—
625C22—81no—
626C22—81no—
627C22—81no—
628C22—81no—
629C22#9049711yescharacteristic
630C22#9051311yescharacteristic
631C22#9051311yescharacteristic
632C22#9051911yescharacteristic
633C22#9052011yescharacteristic
634C22#9052711yescharacteristic
635C22#9052711yescharacteristic
636C4—41no—
637C4—42no—
638C4—41no—
639C4—41no—
640C4—41no—
641C4—41no—
642C4—81no—
643C4—81no—
644C4—282no—
645C4—282no—
646C4—282no—
647C4—561no—
648C4#9036111yescharacteristic
649C4#9036911yescharacteristic
650C4#9037111yescharacteristic
651C4#9037211yescharacteristic
652C2—81no—
653C2—161no—
654C2#1174044211yescharacteristic
655C2#1174058711yescharacteristic
656C2#1174088411yescharacteristic
657C2#1174094211yescharacteristic
658C2#1174101811yescharacteristic
659C2#1174106011yescharacteristic
660C2#1174106611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#2490C2
socleC23×C7C22×D4
prime radicalC2×C7×#121C22
hypercenter#2490D7
hypocenterC7#26751
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2490

Complements

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

NH ≅ G/Ntype
2. #11730913C2semidirect
3. #11729498C2semidirect
4. #11729492C2semidirect
7. #11727414C2semidirect
8. #11727405C2semidirect
9. #11727399C2semidirect
10. #11727391C2semidirect
11. #11727280C2semidirect
12. #11727257C2semidirect
13. #11727229C2semidirect
14. #11727214C2semidirect
15. #11727206C2semidirect
614. C7#26751semidirect

Maximal subgroups

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

subgroupindex
C7×#24902
#117309132
#117294982
#117294922
#117294392
#117294322
#117274142
#117274052
#117273992
#117273912
#117272802
#117272572
#117272292
#117272142
#117272062
#267517