Finite Group Databasebeta

← #13281034

Subgroups — #13281034

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×(C2×D7)⋊C4C211yesmaximal, characteristic
2C7×#1434C211yesmaximal, characteristic, complemented
3#11729288C211yesmaximal, characteristic, complemented
4#11727183C212yesmaximal
5#11727182C212yesmaximal
6(C2×D7)⋊#95C211yesmaximal, characteristic
7(C2×D7)⋊#95C211yesmaximal, characteristic
8#11727171C212yesmaximal, complemented
9#11727167C212yesmaximal, complemented
10Dic7⋊#95C211yesmaximal, characteristic, complemented
11#11727157C211yesmaximal, characteristic, complemented
12C23×C4×D7C2211yescharacteristic
13C22×C42×C7C2211yescharacteristic
14C22×(C2×D7)⋊C4C2211yescharacteristic
15C22×(C2×D7)⋊C4C2211yescharacteristic
16C22×(C2×D7)⋊C4C2212yes—
17C22×(C2×D7)⋊C4C2212yes—
18C22×Dic7⋊C4C2212yes—
19C22×C4×Dic7C2211yescharacteristic
20C2×C7×#95C2211yescharacteristic
21C2×C7×#95C2211yescharacteristic
22C2×C7×#95C2212yes—
23C2×C7×#95C2212yes—
24C2×C7⋊#95C2212yes—
25C2×C7⋊#95C2211yescharacteristic
26C2×C7⋊#95C2211yescharacteristic
27C2×C7⋊#95C2212yes—
28C2×C7⋊#95C2212yes—
29C2×C7⋊#95C2212yes—
30C2×C7⋊#95C2212yes—
31C2×C7⋊#95C2212yes—
32C2×C7⋊#95C2212yes—
33C2×C7⋊#95C2211yescharacteristic
34C2×C7⋊#95C2211yescharacteristic
35C4×(C2×D7)⋊C4C418yescomplemented
36#14351—71nomaximal, Sylow
37C24×D7C2311yescharacteristic
38C23×C4×C7C2312yes—
39C23×C4×C7C2312yes—
40C23×C4×C7C2311yescharacteristic
41C23×C4×C7C2311yescharacteristic
42C23×C4×C7C2311yescharacteristic
43C23×Dic7C2312yes—
44C23×Dic7C2312yes—
45C23×Dic7C2311yescharacteristic
46C23×Dic7C2311yescharacteristic
47C23×Dic7C2311yescharacteristic
48C22×C4×D7—22no—
49C22×C4×D7—22no—
50C22×C4×D7—42no—
51C22×C4×D7—42no—
52C22×C4×D7C2×C414yes—
53C2×C42×C7—42no—
54C2×C42×C7C2×C414yes—
55C2×(C2×D7)⋊C4—24no—
56C2×(C2×D7)⋊C4—24no—
57C2×(C2×D7)⋊C4—24no—
58C2×(C2×D7)⋊C4—24no—
59C2×(C2×D7)⋊C4—42no—
60C2×(C2×D7)⋊C4—42no—
61C2×(C2×D7)⋊C4—42no—
62C2×(C2×D7)⋊C4—42no—
63C2×(C2×D7)⋊C4C2×C414yes—
64C2×(C2×D7)⋊C4C2×C414yes—
65C2×Dic7⋊C4—24no—
66C2×Dic7⋊C4—42no—
67C2×Dic7⋊C4—42no—
68C2×C4×Dic7—42no—
69C2×C4×Dic7C2×C414yes—
70C7×#95—22no—
71C7×#95—22no—
72C7×#95—42no—
73C7×#95—42no—
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⋊#95C2×C414yes—
84C7⋊#95C2×C414yes—
85C2×C4×C22⋊C4—71no—
86#1443—72no—
87#1442—72no—
88#1440—72no—
89#1439—72no—
90C4⋊#95—71no—
91#1434—71no—
92#1434—71no—
93#1434D711yescharacteristic, complemented
94C22⋊#95—71no—
95C22⋊#95—71no—
96C24×C7C2411yescharacteristic
97C23×D7—22no—
98C23×D7—21no—
99C23×D7—21no—
100C23×D7—21no—
101C23×D7—21no—
102C23×D7—22no—
103C23×D7—22no—
104C23×D7—21no—
105C23×D7—21no—
106C23×D7C22×C412yes—
107C23×D7#4114yes—
108C22×C4×C7—22no—
109C22×C4×C7—22no—
110C22×C4×C7—22no—
111C22×C4×C7—22no—
112C22×C4×C7—21no—
113C22×C4×C7—22no—
114C22×C4×C7—21no—
115C22×C4×C7—22no—
116C22×C4×C7—21no—
117C22×C4×C7—21no—
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—22no—
126C22×C4×C7—22no—
127C22×C4×C7—22no—
128C22×C4×C7—22no—
129C22×C4×C7—22no—
130C22×C4×C7—22no—
131C22×C4×C7—22no—
132C22×C4×C7C22×C412yes—
133C22×C4×C7C22×C412yes—
134C22×C4×C7C22×C412yes—
135C22×C4×C7#4112yes—
136C22×C4×C7#4112yes—
137C22×Dic7—22no—
138C22×Dic7—21no—
139C22×Dic7—21no—
140C22×Dic7—22no—
141C22×Dic7—21no—
142C22×Dic7—21no—
143C22×Dic7—21no—
144C22×Dic7—22no—
145C22×Dic7—22no—
146C22×Dic7—22no—
147C22×Dic7—21no—
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×Dic7C22×C412yes—
163C22×Dic7C22×C412yes—
164C22×Dic7C22×C412yes—
165C22×Dic7#4114yes—
166C2×C4×D7—28no—
167C2×C4×D7—44no—
168C2×C4×D7—44no—
169C2×C4×D7—48no—
170C2×C4×D7—48no—
171C42×C7—44no—
172(C2×D7)⋊C4—28no—
173(C2×D7)⋊C4—28no—
174(C2×D7)⋊C4—48no—
175(C2×D7)⋊C4—48no—
176Dic7⋊C4—48no—
177C4×Dic7—44no—
178C24×C4—71no—
179C22×C4⋊C4—72no—
180C22×C22⋊C4—71no—
181C22×C22⋊C4—71no—
182C22×C22⋊C4—72no—
183C22×C22⋊C4—72no—
184C22×C42—71no—
185C22×C42C2×D711yescharacteristic
186C4×C22⋊C4—78no—
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×#95C2×D711yescharacteristic
199C2×#95C2×D711yescharacteristic
200C2×#95C2×D712yes—
201C2×#95C2×D712yes—
202C23×C7C23×C411yescharacteristic
203C23×C7C2×#4111yescharacteristic
204C23×C7C2×#4111yescharacteristic
205C23×C7C2×#4112yes—
206C23×C7C2×#4112yes—
207C23×C7H(5,𝔽2)12yes—
208C23×C7H(5,𝔽2)11yescharacteristic
209C23×C7H(5,𝔽2)12yes—
210C23×C7H(5,𝔽2)11yescharacteristic
211C23×C72−1+411yescharacteristic
212C23×C72−1+411yescharacteristic
213C22×D7—22no—
214C22×D7—22no—
215C22×D7—22no—
216C22×D7—22no—
217C22×D7—22no—
218C22×D7—22no—
219C22×D7—24no—
220C22×D7—24no—
221C22×D7—24no—
222C22×D7—24no—
223C22×D7—24no—
224C22×D7—24no—
225C22×D7—42no—
226C22×D7—42no—
227C22×D7—42no—
228C22×D7—42no—
229C22×D7—41no—
230C22×D7—42no—
231C22×D7—41no—
232C22×D7—41no—
233C22×D7—41no—
234C22×D7—42no—
235C22×D7#11714yes—
236C2×C4×C7—22no—
237C2×C4×C7—22no—
238C2×C4×C7—24no—
239C2×C4×C7—24no—
240C2×C4×C7—22no—
241C2×C4×C7—22no—
242C2×C4×C7—22no—
243C2×C4×C7—22no—
244C2×C4×C7—22no—
245C2×C4×C7—22no—
246C2×C4×C7—42no—
247C2×C4×C7—42no—
248C2×C4×C7—42no—
249C2×C4×C7—42no—
250C2×C4×C7—42no—
251C2×C4×C7—42no—
252C2×C4×C7—42no—
253C2×C4×C7—42no—
254C2×C4×C7—42no—
255C2×C4×C7—42no—
256C2×C4×C7—42no—
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#11714yes—
265C2×Dic7—22no—
266C2×Dic7—22no—
267C2×Dic7—24no—
268C2×Dic7—24no—
269C2×Dic7—22no—
270C2×Dic7—22no—
271C2×Dic7—22no—
272C2×Dic7—22no—
273C2×Dic7—22no—
274C2×Dic7—22no—
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#11714yes—
294C4×D7—416no—
295C25—71no—
296C23×C4—71no—
297C23×C4—71no—
298C23×C4—71no—
299C23×C4—74no—
300C23×C4—72no—
301C23×C4—72no—
302C23×C4—142no—
303C23×C4—142no—
304C23×C4—282no—
305C23×C4—282no—
306C23×C4C22×D712yes—
307C23×C4C22×D712yes—
308C23×C4C22×D711yescharacteristic
309C23×C4C22×D711yescharacteristic
310C23×C4C22×D711yescharacteristic
311C2×C4⋊C4—144no—
312C2×C4⋊C4—282no—
313C2×C4⋊C4—282no—
314C2×C22⋊C4—74no—
315C2×C22⋊C4—74no—
316C2×C22⋊C4—144no—
317C2×C22⋊C4—144no—
318C2×C22⋊C4—144no—
319C2×C22⋊C4—144no—
320C2×C22⋊C4—282no—
321C2×C22⋊C4—282no—
322C2×C22⋊C4—282no—
323C2×C22⋊C4—282no—
324C2×C42—42no—
325C2×C42—74no—
326C2×C42—282no—
327C2×C42C4×D714yes—
328#95—22no—
329#95—22no—
330#95—42no—
331#95—42no—
332#95—74no—
333#95—74no—
334#95—144no—
335#95—142no—
336#95—142no—
337#95—282no—
338#95—282no—
339#95—282no—
340#95—282no—
341#95—282no—
342#95—282no—
343C22×C7C2×#11711yescharacteristic
344C22×C7C2×#11711yescharacteristic
345C22×C7C2×#11711yescharacteristic
346C22×C7#51811yescharacteristic
347C22×C7#51811yescharacteristic
348C22×C7#52011yescharacteristic
349C22×C7#52011yescharacteristic
350C22×C7#52912yes—
351C22×C7#52912yes—
352C22×C7#53812yes—
353C22×C7#53812yes—
354C22×C7#53812yes—
355C22×C7#53812yes—
356C22×C7#53812yes—
357C22×C7#53912yes—
358C22×C7#53912yes—
359C22×C7#53912yes—
360C22×C7#54011yescharacteristic
361C22×C7#54012yes—
362C22×C7#54111yescharacteristic
363C22×C7#54211yescharacteristic
364C22×C7#54211yescharacteristic
365C22×C7#54212yes—
366C2×D7—24no—
367C2×D7—24no—
368C2×D7—24no—
369C2×D7—42no—
370C2×D7—42no—
371C2×D7—42no—
372C2×D7—42no—
373C2×D7—44no—
374C2×D7—44no—
375C2×D7—44no—
376C2×D7—44no—
377C4×C7—24no—
378C4×C7—42no—
379C4×C7—42no—
380C4×C7—82no—
381C4×C7—82no—
382Dic7—24no—
383Dic7—42no—
384Dic7—42no—
385Dic7—82no—
386Dic7—82no—
387C24—72no—
388C24—74no—
389C24—141no—
390C24—141no—
391C24—141no—
392C24—142no—
393C24—141no—
394C24—142no—
395C24—141no—
396C24—142no—
397C24—141no—
398C24C23×D711yescharacteristic
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—21no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—21no—
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—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—72no—
424C22×C4—74no—
425C22×C4—72no—
426C22×C4—72no—
427C22×C4—142no—
428C22×C4—141no—
429C22×C4—141no—
430C22×C4—141no—
431C22×C4—141no—
432C22×C4—141no—
433C22×C4—142no—
434C22×C4—141no—
435C22×C4—142no—
436C22×C4—142no—
437C22×C4—142no—
438C22×C4—148no—
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—284no—
454C22×C4—284no—
455C22×C4—288no—
456C22×C4—288no—
457C22×C4C2×C4×D712yes—
458C22×C4C2×C4×D712yes—
459C22×C4C2×C4×D712yes—
460C22×C4#115712yes—
461C22×C4#115712yes—
462C4⋊C4—288no—
463C22⋊C4—148no—
464C22⋊C4—148no—
465C22⋊C4—288no—
466C22⋊C4—288no—
467C42—44no—
468C42—284no—
469C2×C7#228311yescharacteristic
470C2×C7#231211yescharacteristic
471C2×C7#231411yescharacteristic
472C2×C7#231811yescharacteristic
473C2×C7#231811yescharacteristic
474C2×C7#231811yescharacteristic
475C2×C7#231911yescharacteristic
476C2×C7#264812yes—
477C2×C7#264912yes—
478C2×C7#265212yes—
479C2×C7#265312yes—
480D7—44no—
481C23—74no—
482C23—144no—
483C23—144no—
484C23—142no—
485C23—142no—
486C23—142no—
487C23—142no—
488C23—142no—
489C23—142no—
490C23—144no—
491C23—144no—
492C23—144no—
493C23—144no—
494C23—282no—
495C23—282no—
496C23—282no—
497C23—282no—
498C23—281no—
499C23—281no—
500C23—281no—
501C23—281no—
502C23—282no—
503C23—282no—
504C23C22×C4×D711yescharacteristic
505C23C2×#115711yescharacteristic
506C23C2×#115711yescharacteristic
507C23#654811yescharacteristic
508C23#654812yes—
509C23#655111yescharacteristic
510C23D7×#4112yes—
511C23D7×#4112yes—
512C23#655312yes—
513C23#655311yescharacteristic
514C23#655411yescharacteristic
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—74no—
544C2×C4—144no—
545C2×C4—144no—
546C2×C4—142no—
547C2×C4—142no—
548C2×C4—142no—
549C2×C4—142no—
550C2×C4—142no—
551C2×C4—142no—
552C2×C4—142no—
553C2×C4—142no—
554C2×C4—2816no—
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—282no—
573C2×C4#643514yes—
574C7#1435111yescharacteristic, complemented, Sylow
575C22—144no—
576C22—144no—
577C22—144no—
578C22—282no—
579C22—282no—
580C22—282no—
581C22—284no—
582C22—284no—
583C22—284no—
584C22—284no—
585C22—282no—
586C22C2×#643511yescharacteristic
587C22#9017711yescharacteristic
588C22#9019511yescharacteristic
589C22D7×#11711yescharacteristic
590C22D7×#11711yescharacteristic
591C22#9021211yescharacteristic
592C22#9021411yescharacteristic
593C22#9021912yes—
594C22#9021912yes—
595C22#9025112yes—
596C22#9025112yes—
597C22#9025211yescharacteristic
598C22#9025311yescharacteristic
599C22#9027311yescharacteristic
600C22#9027411yescharacteristic
601C22#9030612yes—
602C22#9033112yes—
603C22#9034812yes—
604C22#9035112yes—
605C22#9036912yes—
606C22#9038312yes—
607C22#9039512yes—
608C22#9039812yes—
609C4—24no—
610C4—42no—
611C4—42no—
612C4—82no—
613C4—82no—
614C4—144no—
615C4—282no—
616C4—282no—
617C4—562no—
618C4—562no—
619C2—284no—
620C2#1173085811yescharacteristic
621C2#1173101211yescharacteristic
622C2#1173120511yescharacteristic
623C2#1173131511yescharacteristic
624C2#1173156511yescharacteristic
625C2#1173167111yescharacteristic
626C2#1173174011yescharacteristic
627C2#1173283012yes—
628C2#1173283112yes—
629C2#1173334812yes—
630C2#1173335012yes—
subgroupG/H
center, FrattiniC24C23×D7
commutatorC23×C7C23×C4
FittingC7×#1434C2
socle, cosocleC24×C7C24
prime radicalC24×D7C23
hypercenter#1434D7
hypocenterC7#14351
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C24 ◁ #1434

Complements

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

NH ≅ G/Ntype
2. C7×#1434C2semidirect
3. #11729288C2semidirect
8. #11727171C2semidirect
9. #11727167C2semidirect
10. Dic7⋊#95C2semidirect
11. #11727157C2semidirect
35. C4×(C2×D7)⋊C4C4semidirect
93. #1434D7semidirect
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×(C2×D7)⋊C42
C7×#14342
#117292882
#117271832
#117271822
(C2×D7)⋊#952
#117271712
#117271672
Dic7⋊#952
#117271572
#143517