Finite Group Databasebeta

← #13281033

Subgroups — #13281033

Order 1792.

1422 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C7×C22⋊C4C211yesmaximal, characteristic
2#11729881C211yesmaximal, characteristic
3#11729881C211yesmaximal, characteristic
4#11729302C212yesmaximal
5#11729300C212yesmaximal
6#11729289C211yesmaximal, characteristic, complemented
7#11729288C211yesmaximal, characteristic, complemented
8#11729288C211yesmaximal, characteristic, complemented
9#11729288C211yesmaximal, characteristic, complemented
10#11727170C212yesmaximal, complemented
11#11727166C212yesmaximal, complemented
12C24×C4×C7C2211yescharacteristic
13C22×C7×C22⋊C4C2211yescharacteristic
14C22×C7×C22⋊C4C2211yescharacteristic
15C22×C42×C7C2211yescharacteristic
16C22×C42×C7C2211yescharacteristic
17C22×C22⋊Dic7C2212yes—
18C22×C22⋊Dic7C2212yes—
19C22×C4⋊Dic7C2212yes—
20C4×C7×C22⋊C4C418yescomplemented
21C2×C7×#95C2211yescharacteristic
22C2×C7×#95C2211yescharacteristic
23C2×C7⋊#95C2212yes—
24C2×C7⋊#95C2211yescharacteristic
25C2×C7⋊#95C2211yescharacteristic
26C2×C7⋊#95C2211yescharacteristic
27C2×C7⋊#95C2211yescharacteristic
28C2×C7⋊#95C2212yes—
29C2×C7⋊#95C2212yes—
30C2×C7⋊#95C2212yes—
31C2×C7⋊#95C2212yes—
32C2×C7⋊#95C2212yes—
33C2×C7⋊#95C2212yes—
34C2×C7⋊#95C2212yes—
35C2×C7⋊#95C2212yes—
36#14351—71nomaximal, Sylow
37C25×C7C2311yescharacteristic
38C23×C4×C7—22no—
39C23×C4×C7—22no—
40C23×C4×C7—42no—
41C23×C4×C7—42no—
42C23×C4×C7C2×C414yes—
43C23×C4×C7C2311yescharacteristic
44C23×C4×C7C2311yescharacteristic
45C23×C4×C7C2311yescharacteristic
46C23×C4×C7C2311yescharacteristic
47C23×C4×C7C2311yescharacteristic
48C23×C4×C7C2311yescharacteristic
49C23×Dic7C2312yes—
50C23×Dic7C2312yes—
51C23×Dic7C2312yes—
52C23×Dic7C2312yes—
53C2×C7×C22⋊C4—24no—
54C2×C7×C22⋊C4—24no—
55C2×C7×C22⋊C4C2×C414yes—
56C2×C7×C22⋊C4C2×C414yes—
57C2×C42×C7—42no—
58C2×C42×C7—42no—
59C2×C42×C7C2×C414yes—
60C2×C42×C7C2×C414yes—
61C2×C22⋊Dic7—24no—
62C2×C22⋊Dic7—24no—
63C2×C22⋊Dic7—42no—
64C2×C22⋊Dic7—42no—
65C2×C22⋊Dic7—42no—
66C2×C22⋊Dic7—42no—
67C2×C4⋊Dic7—24no—
68C2×C4⋊Dic7—42no—
69C2×C4⋊Dic7—42no—
70C7×#95C2×C414yes—
71C7×#95C2×C414yes—
72C7⋊#95—22no—
73C7⋊#95—22no—
74C7⋊#95—22no—
75C7⋊#95—22no—
76C7⋊#95—24no—
77C7⋊#95—42no—
78C7⋊#95—42no—
79C7⋊#95—42no—
80C7⋊#95—42no—
81C7⋊#95—42no—
82C7⋊#95—42no—
83C7⋊#95—42no—
84C7⋊#95—42no—
85C2×C4×C22⋊C4D711yescharacteristic
86#1443—72no—
87#1442—72no—
88#1440—72no—
89#1439—72no—
90C4⋊#95—71no—
91#1434—71no—
92#1434—71no—
93#1434—71no—
94C22⋊#95—71no—
95C22⋊#95—71no—
96C24×C7—22no—
97C24×C7—21no—
98C24×C7—21no—
99C24×C7—21no—
100C24×C7—21no—
101C24×C7—22no—
102C24×C7—22no—
103C24×C7—21no—
104C24×C7—21no—
105C24×C7C22×C412yes—
106C24×C7#4114yes—
107C24×C7C2411yescharacteristic
108C22×C4×C7—21no—
109C22×C4×C7—22no—
110C22×C4×C7—22no—
111C22×C4×C7—22no—
112C22×C4×C7—21no—
113C22×C4×C7—22no—
114C22×C4×C7—22no—
115C22×C4×C7—22no—
116C22×C4×C7—21no—
117C22×C4×C7—21no—
118C22×C4×C7—21no—
119C22×C4×C7—21no—
120C22×C4×C7—21no—
121C22×C4×C7—22no—
122C22×C4×C7—22no—
123C22×C4×C7—22no—
124C22×C4×C7—22no—
125C22×C4×C7—22no—
126C22×C4×C7—21no—
127C22×C4×C7—21no—
128C22×C4×C7—21no—
129C22×C4×C7—28no—
130C22×C4×C7—44no—
131C22×C4×C7—44no—
132C22×C4×C7—48no—
133C22×C4×C7—48no—
134C22×C4×C7C22×C412yes—
135C22×C4×C7C22×C412yes—
136C22×C4×C7C22×C412yes—
137C22×C4×C7C22×C412yes—
138C22×C4×C7C22×C412yes—
139C22×C4×C7C22×C412yes—
140C22×C4×C7#4112yes—
141C22×C4×C7#4112yes—
142C22×C4×C7#4114yes—
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—22no—
164C22×Dic7—22no—
165C22×Dic7—22no—
166C22×Dic7—22no—
167C22×Dic7—22no—
168C22×Dic7—22no—
169C22×Dic7—22no—
170C22×Dic7—22no—
171C7×C22⋊C4—28no—
172C7×C22⋊C4—28no—
173C42×C7—44no—
174C42×C7—44no—
175C22⋊Dic7—48no—
176C22⋊Dic7—48no—
177C4⋊Dic7—48no—
178C24×C4C2×D711yescharacteristic
179C22×C4⋊C4—72no—
180C22×C22⋊C4—72no—
181C22×C22⋊C4—72no—
182C22×C22⋊C4C2×D711yescharacteristic
183C22×C22⋊C4C2×D711yescharacteristic
184C22×C42C2×D711yescharacteristic
185C22×C42C2×D711yescharacteristic
186C4×C22⋊C4Dic718yescomplemented
187C2×#95—71no—
188C2×#95—71no—
189C2×#95—71no—
190C2×#95—71no—
191C2×#95—72no—
192C2×#95—72no—
193C2×#95—72no—
194C2×#95—72no—
195C2×#95—72no—
196C2×#95—72no—
197C2×#95—72no—
198C2×#95—72no—
199C2×#95—72no—
200C2×#95C2×D711yescharacteristic
201C2×#95C2×D711yescharacteristic
202C23×C7—22no—
203C23×C7—22no—
204C23×C7—22no—
205C23×C7—22no—
206C23×C7—22no—
207C23×C7—22no—
208C23×C7—24no—
209C23×C7—24no—
210C23×C7—24no—
211C23×C7—24no—
212C23×C7—24no—
213C23×C7—24no—
214C23×C7—42no—
215C23×C7—42no—
216C23×C7—42no—
217C23×C7—42no—
218C23×C7—41no—
219C23×C7—42no—
220C23×C7—41no—
221C23×C7—41no—
222C23×C7—41no—
223C23×C7—42no—
224C23×C7#11714yes—
225C23×C7C23×C411yescharacteristic
226C23×C7C2×#4112yes—
227C23×C7C2×#4111yescharacteristic
228C23×C7C2×#4111yescharacteristic
229C23×C7C2×#4112yes—
230C23×C7H(5,𝔽2)12yes—
231C23×C7H(5,𝔽2)12yes—
232C23×C7H(5,𝔽2)11yescharacteristic
233C23×C7H(5,𝔽2)11yescharacteristic
234C23×C72−1+411yescharacteristic
235C23×C72−1+411yescharacteristic
236C2×C4×C7—22no—
237C2×C4×C7—22no—
238C2×C4×C7—22no—
239C2×C4×C7—22no—
240C2×C4×C7—24no—
241C2×C4×C7—24no—
242C2×C4×C7—24no—
243C2×C4×C7—24no—
244C2×C4×C7—22no—
245C2×C4×C7—22no—
246C2×C4×C7—22no—
247C2×C4×C7—22no—
248C2×C4×C7—22no—
249C2×C4×C7—22no—
250C2×C4×C7—22no—
251C2×C4×C7—22no—
252C2×C4×C7—22no—
253C2×C4×C7—22no—
254C2×C4×C7—22no—
255C2×C4×C7—22no—
256C2×C4×C7—416no—
257C2×C4×C7—42no—
258C2×C4×C7—42no—
259C2×C4×C7—42no—
260C2×C4×C7—42no—
261C2×C4×C7—42no—
262C2×C4×C7—42no—
263C2×C4×C7—42no—
264C2×C4×C7—42no—
265C2×C4×C7#11714yes—
266C2×C4×C7#11714yes—
267C2×Dic7—42no—
268C2×Dic7—42no—
269C2×Dic7—42no—
270C2×Dic7—42no—
271C2×Dic7—42no—
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—
291C2×Dic7—42no—
292C2×Dic7—42no—
293C2×Dic7—42no—
294C2×Dic7—42no—
295C25C22×D711yescharacteristic
296C23×C4—22no—
297C23×C4—22no—
298C23×C4—42no—
299C23×C4—42no—
300C23×C4—72no—
301C23×C4—72no—
302C23×C4—72no—
303C23×C4—72no—
304C23×C4C2×Dic714yes—
305C23×C4C22×D711yescharacteristic
306C23×C4C22×D711yescharacteristic
307C23×C4C22×D711yescharacteristic
308C23×C4C22×D711yescharacteristic
309C23×C4C22×D711yescharacteristic
310C23×C4C22×D711yescharacteristic
311C2×C4⋊C4—144no—
312C2×C4⋊C4—282no—
313C2×C4⋊C4—282no—
314C2×C22⋊C4—24no—
315C2×C22⋊C4—24no—
316C2×C22⋊C4—144no—
317C2×C22⋊C4—144no—
318C2×C22⋊C4—282no—
319C2×C22⋊C4—282no—
320C2×C22⋊C4—282no—
321C2×C22⋊C4—282no—
322C2×C22⋊C4C2×Dic714yes—
323C2×C22⋊C4C2×Dic714yes—
324C2×C42—42no—
325C2×C42—42no—
326C2×C42C2×Dic714yes—
327C2×C42C2×Dic714yes—
328#95—142no—
329#95—142no—
330#95—142no—
331#95—142no—
332#95—144no—
333#95—282no—
334#95—282no—
335#95—282no—
336#95—282no—
337#95—282no—
338#95—282no—
339#95—282no—
340#95—282no—
341#95C2×Dic714yes—
342#95C2×Dic714yes—
343C22×C7—24no—
344C22×C7—24no—
345C22×C7—24no—
346C22×C7—42no—
347C22×C7—42no—
348C22×C7—42no—
349C22×C7—42no—
350C22×C7—44no—
351C22×C7—44no—
352C22×C7—44no—
353C22×C7—44no—
354C22×C7C2×#11711yescharacteristic
355C22×C7C2×#11711yescharacteristic
356C22×C7C2×#11711yescharacteristic
357C22×C7#51811yescharacteristic
358C22×C7#51811yescharacteristic
359C22×C7#52011yescharacteristic
360C22×C7#52011yescharacteristic
361C22×C7#52912yes—
362C22×C7#52912yes—
363C22×C7#53812yes—
364C22×C7#53812yes—
365C22×C7#53812yes—
366C22×C7#53812yes—
367C22×C7#53812yes—
368C22×C7#53912yes—
369C22×C7#53912yes—
370C22×C7#53912yes—
371C22×C7#54011yescharacteristic
372C22×C7#54012yes—
373C22×C7#54111yescharacteristic
374C22×C7#54211yescharacteristic
375C22×C7#54211yescharacteristic
376C22×C7#54212yes—
377C4×C7—24no—
378C4×C7—24no—
379C4×C7—42no—
380C4×C7—42no—
381C4×C7—42no—
382C4×C7—42no—
383Dic7—82no—
384Dic7—82no—
385Dic7—82no—
386Dic7—82no—
387C24—22no—
388C24—21no—
389C24—21no—
390C24—22no—
391C24—22no—
392C24—21no—
393C24—21no—
394C24—21no—
395C24—21no—
396C24#115714yes—
397C24C22×Dic712yes—
398C24C23×D711yescharacteristic
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—28no—
421C22×C4—44no—
422C22×C4—44no—
423C22×C4—48no—
424C22×C4—48no—
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—142no—
450C22×C4—142no—
451C22×C4—142no—
452C22×C4—142no—
453C22×C4#115712yes—
454C22×C4#115712yes—
455C22×C4#115714yes—
456C22×C4C22×Dic712yes—
457C22×C4C22×Dic712yes—
458C22×C4C22×Dic712yes—
459C22×C4C22×Dic712yes—
460C22×C4C22×Dic712yes—
461C22×C4C22×Dic712yes—
462C4⋊C4—288no—
463C22⋊C4—28no—
464C22⋊C4—28no—
465C22⋊C4—288no—
466C22⋊C4—288no—
467C42—44no—
468C42—44no—
469C2×C7—44no—
470C2×C7#228311yescharacteristic
471C2×C7#231211yescharacteristic
472C2×C7#231411yescharacteristic
473C2×C7#231811yescharacteristic
474C2×C7#231811yescharacteristic
475C2×C7#231811yescharacteristic
476C2×C7#231911yescharacteristic
477C2×C7#264812yes—
478C2×C7#264912yes—
479C2×C7#265212yes—
480C2×C7#265312yes—
481C23—24no—
482C23—24no—
483C23—22no—
484C23—22no—
485C23—22no—
486C23—22no—
487C23—24no—
488C23—24no—
489C23—24no—
490C23—24no—
491C23—22no—
492C23—22no—
493C23—42no—
494C23—42no—
495C23—41no—
496C23—41no—
497C23—41no—
498C23—41no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23#648914yes—
504C23C2×#115711yescharacteristic
505C23C2×#115711yescharacteristic
506C23C2×#115712yes—
507C23C2×#115712yes—
508C23#654812yes—
509C23#654812yes—
510C23#655311yescharacteristic
511C23#655311yescharacteristic
512C23#655411yescharacteristic
513C23#655411yescharacteristic
514C23C23×Dic711yescharacteristic
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—416no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—282no—
545C2×C4—282no—
546C2×C4—282no—
547C2×C4—282no—
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×C4—282no—
568C2×C4—282no—
569C2×C4—282no—
570C2×C4—282no—
571C2×C4—282no—
572C2×C4#648914yes—
573C2×C4#648914yes—
574C7#1435111yescharacteristic, complemented, Sylow
575C22—24no—
576C22—24no—
577C22—24no—
578C22—42no—
579C22—44no—
580C22—44no—
581C22—44no—
582C22—44no—
583C22—42no—
584C22—42no—
585C22—42no—
586C22#9017012yes—
587C22#9017012yes—
588C22#9018512yes—
589C22#9018512yes—
590C22#9020911yescharacteristic
591C22#9022511yescharacteristic
592C22#9022511yescharacteristic
593C22#9022611yescharacteristic
594C22#9025112yes—
595C22#9025112yes—
596C22#9025112yes—
597C22#9025212yes—
598C22#9025312yes—
599C22#9025512yes—
600C22#9025512yes—
601C22#9025512yes—
602C22C2×#648911yescharacteristic
603C22C2×#648911yescharacteristic
604C22C2×#648911yescharacteristic
605C22#9048911yescharacteristic
606C22#9048911yescharacteristic
607C22#9051811yescharacteristic
608C22#9051811yescharacteristic
609C4—24no—
610C4—24no—
611C4—42no—
612C4—42no—
613C4—42no—
614C4—42no—
615C4—562no—
616C4—562no—
617C4—562no—
618C4—562no—
619C2—44no—
620C2#1173100012yes—
621C2#1173100112yes—
622C2#1173105412yes—
623C2#1173105512yes—
624C2#1173950511yescharacteristic
625C2#1173953811yescharacteristic
626C2#1173967611yescharacteristic
627C2#1173976911yescharacteristic
628C2#1173976911yescharacteristic
629C2#1173976911yescharacteristic
630C2#1173977011yescharacteristic
subgroupG/H
center, FrattiniC24C23×D7
commutatorC23×C7C23×C4
FittingC2×C4×C7×C22⋊C4C2
socle, cosocleC24×C7C24
prime radicalC25×C7C23
hypercenterC2×C4×C22⋊C4D7
hypocenterC7#14351
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
6. #11729289C2semidirect
7. #11729288C2semidirect
8. #11729288C2semidirect
9. #11729288C2semidirect
10. #11727170C2semidirect
11. #11727166C2semidirect
20. C4×C7×C22⋊C4C4semidirect
186. C4×C22⋊C4Dic7semidirect
574. C7#14351semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C7×C22⋊C42
#117298812
#117293022
#117293002
#117292892
#117292882
#117271702
#117271662
#143517