Finite Group Databasebeta

← #13283892

Subgroups — #13283892

Order 1792.

1706 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#393C211yesmaximal, characteristic
2C2×#90002C212yesmaximal, complemented
3C2×#89722C212yesmaximal
4#11729300C211yesmaximal, characteristic
5#11729299C212yesmaximal
6#11729298C211yesmaximal, characteristic
7#11727173C212yesmaximal, complemented
8#11727166C212yesmaximal, complemented
9#11727155C212yesmaximal, complemented
10C23×C7×Q8C2211yescharacteristic
11C24×Dic7C2211yescharacteristic
12C22×C7×C22⋊C4C2212yes—
13C22×C7×C22⋊C4C2211yescharacteristic
14C22×C22⋊Dic7C2212yes—
15C22×C22⋊Dic7C2211yescharacteristic
16C22×C4⋊Dic7C2212yes—
17C22×C4⋊Dic7C2212yes—
18C22×C4⋊Dic7C2211yescharacteristic
19C22×C4×Dic7C2212yes—
20C7×#393C418yescomplemented
21C2×C7×#95C2212yes—
22C2×C7×#95C2211yescharacteristic
23#90002—28no—
24C2×C7⋊#95C2212yes—
25C2×C7⋊#95C2212yes—
26C2×C7⋊#95C2212yes—
27C2×C7⋊#95C2212yes—
28C2×C7⋊#95C2211yescharacteristic
29C2×C7⋊#95C2212yes—
30C2×C7⋊#95C2212yes—
31C2×C7⋊#95C2212yes—
32C2×C7⋊#95C2212yes—
33#89722—28no—
34#14575—71nomaximal, Sylow
35C25×C7C2311yescharacteristic
36C22×C7×Q8—24no—
37C22×C7×Q8—24no—
38C22×C7×Q8—41no—
39C22×C7×Q8—41no—
40C22×C7×Q8C2×C414yescomplemented
41C23×C4×C7C2311yescharacteristic
42C23×C4×C7C2312yes—
43C23×C4×C7C2311yescharacteristic
44C23×C4×C7C2312yes—
45C23×Dic7—22no—
46C23×Dic7—24no—
47C23×Dic7—42no—
48C23×Dic7—41no—
49C23×Dic7—41no—
50C23×Dic7C2312yes—
51C23×Dic7C2312yes—
52C23×Dic7C2312yes—
53C23×Dic7C2312yes—
54C2×C7×C22⋊C4—24no—
55C2×C7×C22⋊C4—24no—
56C2×C7×C22⋊C4—41no—
57C2×C7×C22⋊C4—41no—
58C2×C7×C22⋊C4C2×C418yes—
59C2×C7×C22⋊C4C2×C414yes—
60C2×C22⋊Dic7—24no—
61C2×C22⋊Dic7—24no—
62C2×C22⋊Dic7—41no—
63C2×C22⋊Dic7—41no—
64C2×C22⋊Dic7—42no—
65C2×C22⋊Dic7Q814yescomplemented
66C2×C4⋊Dic7—24no—
67C2×C4⋊Dic7—24no—
68C2×C4⋊Dic7—24no—
69C2×C4⋊Dic7—42no—
70C2×C4⋊Dic7—42no—
71C2×C4⋊Dic7—42no—
72C2×C4⋊Dic7—41no—
73C2×C4⋊Dic7—42no—
74C2×C4×Dic7—24no—
75C2×C4×Dic7—24no—
76C2×C4×Dic7—42no—
77C7×#95C2×C414yescomplemented
78C7×#95C2×C418yescomplemented
79C7⋊#95—24no—
80C7⋊#95—24no—
81C7⋊#95—24no—
82C7⋊#95—24no—
83C7⋊#95—42no—
84C7⋊#95—42no—
85C7⋊#95—41no—
86C7⋊#95—42no—
87C7⋊#95D418yes—
88C2×#393D711yescharacteristic
89C2×#391—72no—
90C2×#385—72no—
91#1444—72no—
92#1442—72no—
93#1439—71no—
94#1437—72no—
95#1434—72no—
96C22⋊#95—71no—
97C24×C7—22no—
98C24×C7—22no—
99C24×C7—21no—
100C24×C7—21no—
101C24×C7—22no—
102C24×C7—21no—
103C24×C7—21no—
104C24×C7—22no—
105C24×C7C22×C412yes—
106C24×C7C2×Q812yes—
107C24×C7#4112yes—
108C24×C7C2411yescharacteristic
109C2×C7×Q8—216no—
110C2×C7×Q8—44no—
111C2×C7×Q8—44no—
112C2×C7×Q8—48no—
113C2×C7×Q8—44no—
114C22×C4×C7—22no—
115C22×C4×C7—22no—
116C22×C4×C7—22no—
117C22×C4×C7—22no—
118C22×C4×C7—22no—
119C22×C4×C7—22no—
120C22×C4×C7—22no—
121C22×C4×C7—22no—
122C22×C4×C7—22no—
123C22×C4×C7—22no—
124C22×C4×C7—22no—
125C22×C4×C7—21no—
126C22×C4×C7—21no—
127C22×C4×C7—21no—
128C22×C4×C7—21no—
129C22×C4×C7—22no—
130C22×C4×C7—22no—
131C22×C4×C7—22no—
132C22×C4×C7C22×C414yes—
133C22×C4×C7C22×C412yes—
134C22×C4×C7C22×C412yes—
135C22×C4×C7C22×C414yes—
136C22×C4×C7C2×D414yes—
137C22×C4×C7#4114yes—
138C22×Dic7—22no—
139C22×Dic7—22no—
140C22×Dic7—22no—
141C22×Dic7—22no—
142C22×Dic7—22no—
143C22×Dic7—22no—
144C22×Dic7—22no—
145C22×Dic7—22no—
146C22×Dic7—22no—
147C22×Dic7—22no—
148C22×Dic7—22no—
149C22×Dic7—22no—
150C22×Dic7—22no—
151C22×Dic7—22no—
152C22×Dic7—22no—
153C22×Dic7—22no—
154C22×Dic7—22no—
155C22×Dic7—22no—
156C22×Dic7—22no—
157C22×Dic7—22no—
158C22×Dic7—22no—
159C22×Dic7—22no—
160C22×Dic7—22no—
161C22×Dic7—22no—
162C22×Dic7—22no—
163C22×Dic7—44no—
164C22×Dic7—48no—
165C22×Dic7—48no—
166C22×Dic7—44no—
167C22×Dic7—44no—
168C22×Dic7C2×D414yes—
169C22×Dic7C2×D414yes—
170C22×Dic7C2×Q814yes—
171C7×C22⋊C4—216no—
172C7×C22⋊C4—44no—
173C22⋊Dic7—44no—
174C22⋊Dic7—48no—
175C4⋊Dic7—48no—
176C4⋊Dic7—48no—
177C4⋊Dic7—44no—
178C4×Dic7—48no—
179C23×Q8C2×D711yescharacteristic
180C24×C4—71no—
181C22×C4⋊C4—72no—
182C22×C4⋊C4—72no—
183C22×C4⋊C4—71no—
184C22×C22⋊C4—71no—
185C22×C22⋊C4—72no—
186C22×C22⋊C4C2×D712yes—
187C22×C22⋊C4C2×D711yescharacteristic
188C22×C42—72no—
189#393Dic718yescomplemented
190#391—148no—
191#385—148no—
192C2×#95—72no—
193C2×#95—72no—
194C2×#95—72no—
195C2×#95—72no—
196C2×#95—72no—
197C2×#95—72no—
198C2×#95—71no—
199C2×#95—72no—
200C2×#95—72no—
201C2×#95C2×D712yes—
202C2×#95C2×D711yescharacteristic
203C23×C7—24no—
204C23×C7—22no—
205C23×C7—22no—
206C23×C7—24no—
207C23×C7—22no—
208C23×C7—22no—
209C23×C7—22no—
210C23×C7—22no—
211C23×C7—24no—
212C23×C7—24no—
213C23×C7—24no—
214C23×C7—24no—
215C23×C7—42no—
216C23×C7—42no—
217C23×C7—42no—
218C23×C7—42no—
219C23×C7—41no—
220C23×C7—42no—
221C23×C7—41no—
222C23×C7—41no—
223C23×C7—42no—
224C23×C7—41no—
225C23×C7C4×Q814yes—
226C23×C7C23×C411yescharacteristic
227C23×C7C22×D412yes—
228C23×C7C22×Q811yescharacteristic
229C23×C7C2×#4112yes—
230C23×C7C2×#4111yescharacteristic
231C23×C7H(5,𝔽2)11yescharacteristic
232C23×C7H(5,𝔽2)11yescharacteristic
233C23×C7H(5,𝔽2)11yescharacteristic
234C23×C7H(5,𝔽2)12yes—
235C23×C72−1+411yescharacteristic
236C23×C72−1+412yes—
237C7×Q8—416no—
238C2×C4×C7—24no—
239C2×C4×C7—24no—
240C2×C4×C7—24no—
241C2×C4×C7—24no—
242C2×C4×C7—24no—
243C2×C4×C7—24no—
244C2×C4×C7—24no—
245C2×C4×C7—24no—
246C2×C4×C7—22no—
247C2×C4×C7—22no—
248C2×C4×C7—24no—
249C2×C4×C7—24no—
250C2×C4×C7—24no—
251C2×C4×C7—42no—
252C2×C4×C7—42no—
253C2×C4×C7—42no—
254C2×C4×C7—42no—
255C2×C4×C7—41no—
256C2×C4×C7—42no—
257C2×C4×C7—41no—
258C2×C4×C7—41no—
259C2×C4×C7—42no—
260C2×C4×C7—41no—
261C2×C4×C7C4×D418yes—
262C2×Dic7—24no—
263C2×Dic7—24no—
264C2×Dic7—24no—
265C2×Dic7—24no—
266C2×Dic7—24no—
267C2×Dic7—24no—
268C2×Dic7—24no—
269C2×Dic7—24no—
270C2×Dic7—24no—
271C2×Dic7—416no—
272C2×Dic7—42no—
273C2×Dic7—42no—
274C2×Dic7—42no—
275C2×Dic7—42no—
276C2×Dic7—42no—
277C2×Dic7—42no—
278C2×Dic7—42no—
279C2×Dic7—42no—
280C2×Dic7—42no—
281C2×Dic7—42no—
282C2×Dic7—42no—
283C2×Dic7—42no—
284C2×Dic7—42no—
285C2×Dic7—42no—
286C2×Dic7—42no—
287C2×Dic7—42no—
288C2×Dic7—42no—
289C2×Dic7—42no—
290C2×Dic7—42no—
291C25C22×D711yescharacteristic
292C22×Q8—24no—
293C22×Q8—24no—
294C22×Q8—41no—
295C22×Q8—41no—
296C22×Q8C2×Dic714yescomplemented
297C23×C4—72no—
298C23×C4—72no—
299C23×C4—72no—
300C23×C4—72no—
301C23×C4—144no—
302C23×C4—142no—
303C23×C4—281no—
304C23×C4—282no—
305C23×C4—281no—
306C23×C4C22×D712yes—
307C23×C4C22×D711yescharacteristic
308C23×C4C22×D711yescharacteristic
309C23×C4C22×D712yes—
310C2×C4⋊C4—144no—
311C2×C4⋊C4—144no—
312C2×C4⋊C4—144no—
313C2×C4⋊C4—282no—
314C2×C4⋊C4—282no—
315C2×C4⋊C4—282no—
316C2×C4⋊C4—281no—
317C2×C4⋊C4—282no—
318C2×C22⋊C4—24no—
319C2×C22⋊C4—24no—
320C2×C22⋊C4—41no—
321C2×C22⋊C4—41no—
322C2×C22⋊C4—74no—
323C2×C22⋊C4—144no—
324C2×C22⋊C4—144no—
325C2×C22⋊C4—281no—
326C2×C22⋊C4—281no—
327C2×C22⋊C4—282no—
328C2×C22⋊C4C2×Dic718yes—
329C2×C22⋊C4C2×Dic714yes—
330C2×C42—144no—
331C2×C42—144no—
332C2×C42—282no—
333#95—78no—
334#95—144no—
335#95—144no—
336#95—144no—
337#95—144no—
338#95—281no—
339#95—282no—
340#95—282no—
341#95—282no—
342#95C2×Dic714yescomplemented
343#95C2×Dic718yescomplemented
344C22×C7—24no—
345C22×C7—28no—
346C22×C7—44no—
347C22×C7—42no—
348C22×C7—42no—
349C22×C7—42no—
350C22×C7—42no—
351C22×C7—44no—
352C22×C7—44no—
353C22×C7—42no—
354C22×C7—42no—
355C22×C7C2×C4×D412yes—
356C22×C7C2×C4×Q811yescharacteristic
357C22×C7#51811yescharacteristic
358C22×C7#52011yescharacteristic
359C22×C7#52012yes—
360C22×C7#53512yes—
361C22×C7#53612yes—
362C22×C7#53811yescharacteristic
363C22×C7#53911yescharacteristic
364C22×C7#54212yes—
365C22×C7#54212yes—
366C22×C7#54311yescharacteristic
367C22×C7#54411yescharacteristic
368C22×C7D4211yescharacteristic
369C22×C7#54612yes—
370C22×C7#54612yes—
371C22×C7D4×Q812yes—
372C22×C7#55111yescharacteristic
373C22×C7#55212yes—
374C22×C7#55412yes—
375C22×C7H(3,𝔽4)11yescharacteristic
376C22×C7#56212yes—
377C22×C7#56311yescharacteristic
378C4×C7—28no—
379C4×C7—44no—
380C4×C7—42no—
381C4×C7—42no—
382Dic7—44no—
383Dic7—44no—
384Dic7—44no—
385Dic7—82no—
386C24—22no—
387C24—22no—
388C24—21no—
389C24—21no—
390C24—22no—
391C24—21no—
392C24—21no—
393C24—22no—
394C24Q8×D712yes—
395C24#116112yes—
396C24C22×Dic712yes—
397C24C23×D711yescharacteristic
398C2×Q8—216no—
399C2×Q8—44no—
400C2×Q8—44no—
401C2×Q8—44no—
402C2×Q8—48no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—74no—
422C22×C4—74no—
423C22×C4—74no—
424C22×C4—142no—
425C22×C4—142no—
426C22×C4—142no—
427C22×C4—142no—
428C22×C4—142no—
429C22×C4—142no—
430C22×C4—142no—
431C22×C4—142no—
432C22×C4—142no—
433C22×C4—142no—
434C22×C4—142no—
435C22×C4—142no—
436C22×C4—142no—
437C22×C4—142no—
438C22×C4—142no—
439C22×C4—142no—
440C22×C4—142no—
441C22×C4—142no—
442C22×C4—142no—
443C22×C4—142no—
444C22×C4—142no—
445C22×C4—142no—
446C22×C4—142no—
447C22×C4—142no—
448C22×C4—142no—
449C22×C4—284no—
450C22×C4—288no—
451C22×C4—288no—
452C22×C4—284no—
453C22×C4—284no—
454C22×C4D4×D714yes—
455C22×C4#115914yes—
456C22×C4C22×Dic714yes—
457C22×C4C22×Dic712yes—
458C22×C4C22×Dic712yes—
459C22×C4C22×Dic714yes—
460C4⋊C4—288no—
461C4⋊C4—288no—
462C4⋊C4—284no—
463C22⋊C4—216no—
464C22⋊C4—44no—
465C22⋊C4—284no—
466C22⋊C4—288no—
467C42—288no—
468C2×C7—44no—
469C2×C7#232612yes—
470C2×C7#232911yescharacteristic
471C2×C7#235811yescharacteristic
472C2×C7#236012yes—
473C2×C7#237911yescharacteristic
474C2×C7#267011yescharacteristic
475C2×C7#267711yescharacteristic
476C2×C7#268712yes—
477C2×C7#269212yes—
478C2×C7#276311yescharacteristic
479C2×C7#276411yescharacteristic
480C23—24no—
481C23—22no—
482C23—22no—
483C23—24no—
484C23—22no—
485C23—24no—
486C23—24no—
487C23—22no—
488C23—24no—
489C23—24no—
490C23—22no—
491C23—22no—
492C23—42no—
493C23—42no—
494C23—41no—
495C23—42no—
496C23—41no—
497C23—41no—
498C23—42no—
499C23—41no—
500C23—42no—
501C23—42no—
502C23Q8×Dic714yes—
503C23C2×D4×D712yes—
504C23C2×#115912yes—
505C23#654811yescharacteristic
506C23#654811yescharacteristic
507C23C2×Q8×D711yescharacteristic
508C23C2×#116111yescharacteristic
509C23#655311yescharacteristic
510C23#655312yes—
511C23#655411yescharacteristic
512C23#655412yes—
513C23C23×Dic711yescharacteristic
514Q8—416no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—41no—
531C2×C4—42no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—42no—
535C2×C4—41no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—144no—
539C2×C4—144no—
540C2×C4—144no—
541C2×C4—144no—
542C2×C4—144no—
543C2×C4—144no—
544C2×C4—144no—
545C2×C4—144no—
546C2×C4—144no—
547C2×C4—2816no—
548C2×C4—282no—
549C2×C4—282no—
550C2×C4—282no—
551C2×C4—282no—
552C2×C4—282no—
553C2×C4—282no—
554C2×C4—282no—
555C2×C4—282no—
556C2×C4—282no—
557C2×C4—282no—
558C2×C4—282no—
559C2×C4—282no—
560C2×C4—282no—
561C2×C4—282no—
562C2×C4—282no—
563C2×C4—282no—
564C2×C4—282no—
565C2×C4—282no—
566C2×C4—282no—
567C2×C4D4×Dic718yes—
568C7#1457511yescharacteristic, complemented, Sylow
569C22—24no—
570C22—28no—
571C22—42no—
572C22—42no—
573C22—42no—
574C22—44no—
575C22—42no—
576C22—44no—
577C22—42no—
578C22—42no—
579C22—44no—
580C22#9027811yescharacteristic
581C22#9028111yescharacteristic
582C22#9028212yes—
583C22#9031411yescharacteristic
584C22#9032112yes—
585C22#9032312yes—
586C22#9032511yescharacteristic
587C22#9032612yes—
588C22#9033312yes—
589C22#9033511yescharacteristic
590C22#9033611yescharacteristic
591C22#9036311yescharacteristic
592C22#9036512yes—
593C22#9036612yes—
594C22#9037312yes—
595C22#9037312yes—
596C22#9037411yescharacteristic
597C22#9037512yes—
598C22C2×D4×Dic712yes—
599C22#9048911yescharacteristic
600C22C2×Q8×Dic711yescharacteristic
601C22#9051811yescharacteristic
602C22#9051812yes—
603C4—28no—
604C4—42no—
605C4—44no—
606C4—42no—
607C4—284no—
608C4—284no—
609C4—284no—
610C4—562no—
611C2—44no—
612C2#1173426811yescharacteristic
613C2#1173427012yes—
614C2#1173428811yescharacteristic
615C2#1173429011yescharacteristic
616C2#1173431811yescharacteristic
617C2#1173432312yes—
618C2#1174002411yescharacteristic
619C2#1174019011yescharacteristic
620C2#1174019212yes—
621C2#1174039812yes—
622C2#1174040211yescharacteristic
subgroupG/H
center, FrattiniC24C23×D7
commutatorC23×C7C23×C4
FittingC2×C7×#393C2
socle, cosocleC24×C7C24
prime radicalC25×C7C23
hypercenterC2×#393D7
hypocenterC7#14575
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C24 ◁ C2×#393

Complements

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

NH ≅ G/Ntype
2. C2×#90002C2semidirect
7. #11727173C2semidirect
8. #11727166C2semidirect
9. #11727155C2semidirect
20. C7×#393C4semidirect
40. C22×C7×Q8C2×C4semidirect
65. C2×C22⋊Dic7Q8semidirect
77. C7×#95C2×C4semidirect
78. C7×#95C2×C4semidirect
189. #393Dic7semidirect
296. C22×Q8C2×Dic7semidirect
342. #95C2×Dic7semidirect
343. #95C2×Dic7semidirect
568. C7#14575semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#3932
C2×#900022
C2×#897222
#117293002
#117292992
#117292982
#117271732
#117271662
#117271552
#145757