Finite Group Databasebeta

← #13276411

Subgroups — #13276411

Order 1792.

1486 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
1C2×C4×(C2×D7)⋊C4C211yesmaximal, characteristic
2C7×#1434C211yesmaximal, characteristic, complemented
3#11729288C211yesmaximal, characteristic, complemented
4#11727183C212yesmaximal
5(C2×D7)⋊#95C211yesmaximal, characteristic
6(C2×D7)⋊#95C211yesmaximal, characteristic
7#11727171C212yesmaximal, complemented
8#11727162C212yesmaximal
9#11727156C212yesmaximal, complemented
10#11727155C211yesmaximal, characteristic, complemented
11Dic7×#95C211yesmaximal, characteristic, complemented
12C23×C4×D7C2211yescharacteristic
13C22×C42×C7C2211yescharacteristic
14C22×(C2×D7)⋊C4C2212yes—
15C22×(C2×D7)⋊C4C2212yes—
16C22×(C2×D7)⋊C4C2211yescharacteristic
17C22×(C2×D7)⋊C4C2211yescharacteristic
18C22×C4×Dic7C2212yes—
19C22×C4×Dic7C2211yescharacteristic
20C2×C7×#95C2211yescharacteristic
21C2×C7×#95C2211yescharacteristic
22C2×C7×#95C2212yes—
23C2×C7×#95C2212yes—
24C2×C7⋊#95C2212yes—
25C2×C7⋊#95C2211yescharacteristic
26C2×C7⋊#95C2212yes—
27C2×C7⋊#95C2211yescharacteristic
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#13947—71nomaximal, Sylow
37C24×D7C2311yescharacteristic
38C23×C4×C7C2312yes—
39C23×C4×C7C2311yescharacteristic
40C23×C4×C7C2311yescharacteristic
41C23×C4×C7C2312yes—
42C23×C4×C7C2311yescharacteristic
43C23×Dic7C2312yes—
44C23×Dic7C2311yescharacteristic
45C23×Dic7C2311yescharacteristic
46C23×Dic7C2312yes—
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—22no—
56C2×(C2×D7)⋊C4—22no—
57C2×(C2×D7)⋊C4—22no—
58C2×(C2×D7)⋊C4—22no—
59C2×(C2×D7)⋊C4—24no—
60C2×(C2×D7)⋊C4—24no—
61C2×(C2×D7)⋊C4—24no—
62C2×(C2×D7)⋊C4—24no—
63C2×(C2×D7)⋊C4—42no—
64C2×(C2×D7)⋊C4—42no—
65C2×(C2×D7)⋊C4C2×C414yes—
66C2×(C2×D7)⋊C4C2×C414yes—
67C2×C4×Dic7—24no—
68C2×C4×Dic7—24no—
69C2×C4×Dic7—42no—
70C2×C4×Dic7—42no—
71C2×C4×Dic7C2×C414yes—
72C7×#95—24no—
73C7×#95—22no—
74C7×#95—22no—
75C7×#95—42no—
76C7⋊#95—22no—
77C7⋊#95—22no—
78C7⋊#95—24no—
79C7⋊#95—24no—
80C7⋊#95—24no—
81C7⋊#95—42no—
82C7⋊#95—42no—
83C7⋊#95—42no—
84C7⋊#95—42no—
85C7⋊#95C2×C414yes—
86C7⋊#95C2×C414yes—
87C2×C4×C22⋊C4—71no—
88#1443—72no—
89#1440—72no—
90#1437—72no—
91#1434—71no—
92#1434—71no—
93#1434—72no—
94#1434D711yescharacteristic, complemented
95C22⋊#95—71no—
96C22⋊#95—71no—
97C4×#95—71no—
98C24×C7C2411yescharacteristic
99C23×D7—21no—
100C23×D7—21no—
101C23×D7—22no—
102C23×D7—22no—
103C23×D7—21no—
104C23×D7—21no—
105C23×D7—22no—
106C23×D7—22no—
107C23×D7C22×C412yes—
108C23×D7#4112yes—
109C23×D7#4112yes—
110C22×C4×C7—22no—
111C22×C4×C7—22no—
112C22×C4×C7—22no—
113C22×C4×C7—22no—
114C22×C4×C7—22no—
115C22×C4×C7—22no—
116C22×C4×C7—22no—
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—22no—
127C22×C4×C7—22no—
128C22×C4×C7—22no—
129C22×C4×C7—22no—
130C22×C4×C7—22no—
131C22×C4×C7—22no—
132C22×C4×C7—22no—
133C22×C4×C7C22×C412yes—
134C22×C4×C7C22×C412yes—
135C22×C4×C7C22×C412yes—
136C22×C4×C7#4114yes—
137C22×C4×C7#4112yes—
138C22×C4×C7#4112yes—
139C22×Dic7—22no—
140C22×Dic7—22no—
141C22×Dic7—22no—
142C22×Dic7—22no—
143C22×Dic7—22no—
144C22×Dic7—22no—
145C22×Dic7—21no—
146C22×Dic7—21no—
147C22×Dic7—22no—
148C22×Dic7—21no—
149C22×Dic7—21no—
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—
166C22×Dic7#4112yes—
167C22×Dic7#4112yes—
168C2×C4×D7—28no—
169C2×C4×D7—44no—
170C2×C4×D7—44no—
171C2×C4×D7—48no—
172C2×C4×D7—48no—
173C42×C7—44no—
174(C2×D7)⋊C4—28no—
175(C2×D7)⋊C4—28no—
176(C2×D7)⋊C4—48no—
177(C2×D7)⋊C4—48no—
178C4×Dic7—44no—
179C4×Dic7—48no—
180C24×C4—71no—
181C22×C22⋊C4—71no—
182C22×C22⋊C4—71no—
183C22×C22⋊C4—72no—
184C22×C22⋊C4—72no—
185C22×C42—71no—
186C22×C42—72no—
187C22×C42C2×D711yescharacteristic
188C4×C22⋊C4—78no—
189C2×#95—71no—
190C2×#95—71no—
191C2×#95—71no—
192C2×#95—72no—
193C2×#95—71no—
194C2×#95—72no—
195C2×#95—72no—
196C2×#95—72no—
197C2×#95—72no—
198C2×#95—72no—
199C2×#95—72no—
200C2×#95C2×D711yescharacteristic
201C2×#95C2×D711yescharacteristic
202C2×#95C2×D712yes—
203C2×#95C2×D712yes—
204C23×C7C23×C411yescharacteristic
205C23×C7C2×#4112yes—
206C23×C7C2×#4111yescharacteristic
207C23×C7C2×#4111yescharacteristic
208C23×C7C2×#4111yescharacteristic
209C23×C7C2×#4111yescharacteristic
210C23×C7C2×#4111yescharacteristic
211C23×C7C2×#4111yescharacteristic
212C23×C7C2×#4112yes—
213C23×C7H(5,𝔽2)12yes—
214C23×C7H(5,𝔽2)12yes—
215C22×D7—22no—
216C22×D7—22no—
217C22×D7—22no—
218C22×D7—22no—
219C22×D7—22no—
220C22×D7—22no—
221C22×D7—22no—
222C22×D7—22no—
223C22×D7—22no—
224C22×D7—22no—
225C22×D7—24no—
226C22×D7—24no—
227C22×D7—24no—
228C22×D7—24no—
229C22×D7—42no—
230C22×D7—42no—
231C22×D7—42no—
232C22×D7—42no—
233C22×D7—42no—
234C22×D7—42no—
235C22×D7—42no—
236C22×D7—42no—
237C22×D7#11714yes—
238C2×C4×C7—22no—
239C2×C4×C7—22no—
240C2×C4×C7—24no—
241C2×C4×C7—24no—
242C2×C4×C7—22no—
243C2×C4×C7—22no—
244C2×C4×C7—22no—
245C2×C4×C7—22no—
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—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—42no—
265C2×C4×C7—42no—
266C2×C4×C7#11714yes—
267C2×Dic7—22no—
268C2×Dic7—22no—
269C2×Dic7—24no—
270C2×Dic7—24no—
271C2×Dic7—22no—
272C2×Dic7—22no—
273C2×Dic7—22no—
274C2×Dic7—22no—
275C2×Dic7—22no—
276C2×Dic7—22no—
277C2×Dic7—24no—
278C2×Dic7—24no—
279C2×Dic7—24no—
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—
295C2×Dic7#11714yes—
296C4×D7—416no—
297C25—71no—
298C23×C4—71no—
299C23×C4—71no—
300C23×C4—72no—
301C23×C4—71no—
302C23×C4—74no—
303C23×C4—72no—
304C23×C4—142no—
305C23×C4—142no—
306C23×C4—282no—
307C23×C4—282no—
308C23×C4C22×D712yes—
309C23×C4C22×D711yescharacteristic
310C23×C4C22×D711yescharacteristic
311C23×C4C22×D712yes—
312C23×C4C22×D711yescharacteristic
313C2×C22⋊C4—74no—
314C2×C22⋊C4—74no—
315C2×C22⋊C4—142no—
316C2×C22⋊C4—142no—
317C2×C22⋊C4—142no—
318C2×C22⋊C4—142no—
319C2×C22⋊C4—144no—
320C2×C22⋊C4—144no—
321C2×C22⋊C4—144no—
322C2×C22⋊C4—144no—
323C2×C22⋊C4—282no—
324C2×C22⋊C4—282no—
325C2×C42—42no—
326C2×C42—74no—
327C2×C42—144no—
328C2×C42—144no—
329C2×C42—282no—
330C2×C42—282no—
331C2×C42C4×D714yes—
332#95—24no—
333#95—22no—
334#95—22no—
335#95—42no—
336#95—74no—
337#95—74no—
338#95—144no—
339#95—144no—
340#95—144no—
341#95—142no—
342#95—142no—
343#95—282no—
344#95—282no—
345#95—282no—
346#95—282no—
347C22×C7C2×#11711yescharacteristic
348C22×C7C2×#11711yescharacteristic
349C22×C7C2×#11711yescharacteristic
350C22×C7C4×#4111yescharacteristic
351C22×C7C4×#4111yescharacteristic
352C22×C7#51811yescharacteristic
353C22×C7#51811yescharacteristic
354C22×C7#52911yescharacteristic
355C22×C7#52911yescharacteristic
356C22×C7#52911yescharacteristic
357C22×C7#52911yescharacteristic
358C22×C7#52912yes—
359C22×C7#52912yes—
360C22×C7#53812yes—
361C22×C7#53812yes—
362C22×C7#55112yes—
363C22×C7#55112yes—
364C22×C7#55112yes—
365C22×C7#55312yes—
366C22×C7#55312yes—
367C22×C7#55312yes—
368C22×C7#55312yes—
369C22×C7#55912yes—
370C2×D7—24no—
371C2×D7—24no—
372C2×D7—24no—
373C2×D7—42no—
374C2×D7—44no—
375C2×D7—44no—
376C2×D7—44no—
377C2×D7—44no—
378C2×D7—42no—
379C2×D7—42no—
380C2×D7—42no—
381C4×C7—24no—
382C4×C7—44no—
383C4×C7—42no—
384C4×C7—42no—
385C4×C7—82no—
386Dic7—24no—
387Dic7—42no—
388Dic7—44no—
389Dic7—42no—
390Dic7—82no—
391C24—72no—
392C24—72no—
393C24—72no—
394C24—142no—
395C24—141no—
396C24—142no—
397C24—141no—
398C24—141no—
399C24—141no—
400C24—142no—
401C24—142no—
402C24C23×D711yescharacteristic
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—22no—
412C22×C4—22no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—72no—
427C22×C4—72no—
428C22×C4—72no—
429C22×C4—72no—
430C22×C4—72no—
431C22×C4—74no—
432C22×C4—142no—
433C22×C4—142no—
434C22×C4—142no—
435C22×C4—141no—
436C22×C4—142no—
437C22×C4—142no—
438C22×C4—142no—
439C22×C4—141no—
440C22×C4—142no—
441C22×C4—142no—
442C22×C4—142no—
443C22×C4—141no—
444C22×C4—141no—
445C22×C4—142no—
446C22×C4—142no—
447C22×C4—142no—
448C22×C4—148no—
449C22×C4—142no—
450C22×C4—142no—
451C22×C4—142no—
452C22×C4—142no—
453C22×C4—142no—
454C22×C4—142no—
455C22×C4—142no—
456C22×C4—284no—
457C22×C4—284no—
458C22×C4—288no—
459C22×C4—288no—
460C22×C4C2×C4×D712yes—
461C22×C4C2×C4×D712yes—
462C22×C4C2×C4×D712yes—
463C22×C4#115714yes—
464C22×C4#115912yes—
465C22×C4#116112yes—
466C22⋊C4—148no—
467C22⋊C4—148no—
468C22⋊C4—288no—
469C22⋊C4—288no—
470C42—44no—
471C42—284no—
472C42—288no—
473C2×C7#228311yescharacteristic
474C2×C7#229811yescharacteristic
475C2×C7#231211yescharacteristic
476C2×C7#235311yescharacteristic
477C2×C7#235311yescharacteristic
478C2×C7#235311yescharacteristic
479C2×C7#235711yescharacteristic
480C2×C7#257612yes—
481C2×C7#257812yes—
482C2×C7#259512yes—
483C2×C7#259712yes—
484D7—44no—
485C23—74no—
486C23—142no—
487C23—142no—
488C23—142no—
489C23—142no—
490C23—144no—
491C23—144no—
492C23—142no—
493C23—142no—
494C23—142no—
495C23—142no—
496C23—142no—
497C23—142no—
498C23—144no—
499C23—144no—
500C23—282no—
501C23—282no—
502C23—282no—
503C23—282no—
504C23—282no—
505C23—282no—
506C23—282no—
507C23—282no—
508C23C22×C4×D711yescharacteristic
509C23C2×#115712yes—
510C23C2×#115911yescharacteristic
511C23#654812yes—
512C23C2×#116111yescharacteristic
513C23D7×#4111yescharacteristic
514C23D7×#4111yescharacteristic
515C23D7×#4111yescharacteristic
516C23D7×#4111yescharacteristic
517C23D7×#4112yes—
518C23#655312yes—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
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—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—74no—
548C2×C4—142no—
549C2×C4—142no—
550C2×C4—144no—
551C2×C4—144no—
552C2×C4—144no—
553C2×C4—142no—
554C2×C4—142no—
555C2×C4—142no—
556C2×C4—142no—
557C2×C4—142no—
558C2×C4—142no—
559C2×C4—144no—
560C2×C4—144no—
561C2×C4—2816no—
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—282no—
574C2×C4—282no—
575C2×C4—282no—
576C2×C4—282no—
577C2×C4#645514yes—
578C7#1394711yescharacteristic, complemented, Sylow
579C22—144no—
580C22—144no—
581C22—144no—
582C22—282no—
583C22—284no—
584C22—284no—
585C22—284no—
586C22—284no—
587C22—282no—
588C22—282no—
589C22—282no—
590C22C4×#115711yescharacteristic
591C22#9017711yescharacteristic
592C22C2×#645511yescharacteristic
593C22D7×#11711yescharacteristic
594C22D7×#11711yescharacteristic
595C22#9021211yescharacteristic
596C22#9021311yescharacteristic
597C22#9021812yes—
598C22#9022112yes—
599C22#9022912yes—
600C22#9024712yes—
601C22#9024912yes—
602C22#9027112yes—
603C22#9030812yes—
604C22#9033512yes—
605C22#9035212yes—
606C22#9035512yes—
607C22#9037111yescharacteristic
608C22#9037111yescharacteristic
609C22#9038211yescharacteristic
610C22#9038211yescharacteristic
611C22#9039612yes—
612C22#9040012yes—
613C4—24no—
614C4—42no—
615C4—44no—
616C4—42no—
617C4—82no—
618C4—144no—
619C4—282no—
620C4—284no—
621C4—282no—
622C4—562no—
623C2—284no—
624C2#1173086211yescharacteristic
625C2#1173102111yescharacteristic
626C2#1173106511yescharacteristic
627C2#1173144911yescharacteristic
628C2#1173150611yescharacteristic
629C2#1173159211yescharacteristic
630C2#1173174211yescharacteristic
631C2#1173253412yes—
632C2#1173256912yes—
633C2#1173305612yes—
634C2#1173307712yes—
subgroupG/H
center, FrattiniC24C23×D7
commutatorC23×C7C23×C4
FittingC7×#1434C2
socle, cosocleC24×C7C24
prime radicalC24×D7C23
hypercenter#1434D7
hypocenterC7#13947
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
7. #11727171C2semidirect
9. #11727156C2semidirect
10. #11727155C2semidirect
11. Dic7×#95C2semidirect
35. C4×(C2×D7)⋊C4C4semidirect
94. #1434D7semidirect
578. C7#13947semidirect

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
(C2×D7)⋊#952
#117271712
#117271622
#117271562
#117271552
Dic7×#952
#139477