Finite Group Databasebeta

← #13285384

Subgroups — #13285384

Order 1792.

3494 conjugacy classes of subgroups.

The subgroup lattice has 634 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C7×C22⋊C4C211yesmaximal, characteristic
2C2×#90019C212yesmaximal
3C2×#89982C212yesmaximal, complemented
4C2×#89984C211yesmaximal, characteristic, complemented
5C2×#89721C212yesmaximal, complemented
6#11729881C211yesmaximal, characteristic, complemented
7#11729301C214yesmaximal, complemented
8#11729300C212yesmaximal, complemented
9C26×C7C2211yescharacteristic
10C24×C4×C7C2211yescharacteristic
11C24×C4×C7C2211yescharacteristic
12C22×C7×C22⋊C4—24no—
13C22×C7×C22⋊C4—28no—
14C22×C7×C22⋊C4C2212yes—
15C22×C7×C22⋊C4C2212yes—
16C22×C22⋊Dic7C2212yes—
17C22×C22⋊Dic7C2214yes—
18C22×C22⋊Dic7C2214yes—
19C22×C22⋊Dic7C2212yes—
20C22×C4⋊Dic7C2211yescharacteristic, complemented
21C22×C4⋊Dic7C2211yescharacteristic, complemented
22C22×Dic7⋊C4C2214yescomplemented
23#90019—28no—
24#89984—24no—
25#89982—28no—
26C2×C7⋊#95C2212yescomplemented
27C2×C7⋊#95C2214yescomplemented
28C2×C7⋊#95C2212yescomplemented
29C2×C7⋊#95C2212yescomplemented
30#89721—28no—
31#14734—71nomaximal, Sylow
32C25×C7—24no—
33C25×C7—22no—
34C25×C7—24no—
35C25×C7—22no—
36C25×C7—24no—
37C25×C7—24no—
38C25×C7—42no—
39C25×C7—42no—
40C25×C7Q814yes—
41C25×C7C2312yes—
42C25×C7C2311yescharacteristic
43C23×C4×C7—22no—
44C23×C4×C7—24no—
45C23×C4×C7—24no—
46C23×C4×C7—24no—
47C23×C4×C7—24no—
48C23×C4×C7—42no—
49C23×C4×C7—42no—
50C23×C4×C7D414yes—
51C23×C4×C7C2312yes—
52C23×C4×C7C2311yescharacteristic
53C23×C4×C7C2311yescharacteristic
54C23×Dic7C2312yes—
55C23×Dic7C2312yes—
56C23×Dic7C2314yes—
57C2×C7×C22⋊C4—232no—
58C2×C7×C22⋊C4—24no—
59C2×C7×C22⋊C4—24no—
60C2×C7×C22⋊C4—24no—
61C2×C7×C22⋊C4—24no—
62C2×C7×C22⋊C4—24no—
63C2×C7×C22⋊C4D418yes—
64C2×C22⋊Dic7—24no—
65C2×C22⋊Dic7—24no—
66C2×C22⋊Dic7—28no—
67C2×C22⋊Dic7—24no—
68C2×C22⋊Dic7—24no—
69C2×C22⋊Dic7—24no—
70C2×C22⋊Dic7—28no—
71C2×C22⋊Dic7—28no—
72C2×C22⋊Dic7—28no—
73C2×C22⋊Dic7—44no—
74C2×C22⋊Dic7—44no—
75C2×C22⋊Dic7D418yes—
76C2×C4⋊Dic7—24no—
77C2×C4⋊Dic7—22no—
78C2×C4⋊Dic7—22no—
79C2×C4⋊Dic7—42no—
80C2×Dic7⋊C4—28no—
81C2×Dic7⋊C4—28no—
82C2×Dic7⋊C4—44no—
83C7⋊#95—24no—
84C7⋊#95—24no—
85C7⋊#95—28no—
86C7⋊#95—42no—
87C23×C22⋊C4D711yescharacteristic
88C2×#399—72no—
89C2×#385—72no—
90C2×#380—71no—
91C2×#379—72no—
92#1439—72no—
93#1439—74no—
94C22⋊#95—71no—
95C24×C7—22no—
96C24×C7—22no—
97C24×C7—22no—
98C24×C7—22no—
99C24×C7—28no—
100C24×C7—28no—
101C24×C7—216no—
102C24×C7—216no—
103C24×C7—28no—
104C24×C7—24no—
105C24×C7—22no—
106C24×C7—22no—
107C24×C7—22no—
108C24×C7—22no—
109C24×C7—24no—
110C24×C7—24no—
111C24×C7—24no—
112C24×C7—216no—
113C24×C7—48no—
114C24×C7—416no—
115C24×C7—416no—
116C24×C7—416no—
117C24×C7—416no—
118C24×C7—44no—
119C24×C7—44no—
120C24×C7—48no—
121C24×C7—48no—
122C24×C7—44no—
123C24×C7—44no—
124C24×C7C2×D414yes—
125C24×C7C2×D414yes—
126C24×C7C2×D412yes—
127C24×C7C2×Q814yes—
128C24×C7C2×Q812yes—
129C24×C7#4114yes—
130C24×C7#4112yes—
131C24×C7#4114yes—
132C24×C7C2411yescharacteristic
133C22×C4×C7—22no—
134C22×C4×C7—22no—
135C22×C4×C7—28no—
136C22×C4×C7—28no—
137C22×C4×C7—24no—
138C22×C4×C7—22no—
139C22×C4×C7—22no—
140C22×C4×C7—22no—
141C22×C4×C7—22no—
142C22×C4×C7—22no—
143C22×C4×C7—24no—
144C22×C4×C7—28no—
145C22×C4×C7—44no—
146C22×C4×C7—416no—
147C22×C4×C7—416no—
148C22×C4×C7—44no—
149C22×C4×C7—44no—
150C22×C4×C7C2×D412yes—
151C22×C4×C7C2×D414yes—
152C22×C4×C7C2×D414yes—
153C22×C4×C7C2×D412yes—
154C22×Dic7—24no—
155C22×Dic7—24no—
156C22×Dic7—24no—
157C22×Dic7—24no—
158C22×Dic7—24no—
159C22×Dic7—24no—
160C22×Dic7—22no—
161C22×Dic7—22no—
162C22×Dic7—22no—
163C22×Dic7—22no—
164C22×Dic7—24no—
165C22×Dic7—22no—
166C22×Dic7—22no—
167C22×Dic7—24no—
168C22×Dic7—24no—
169C22×Dic7—24no—
170C22×Dic7C2×D418yes—
171C7×C22⋊C4—216no—
172C7×C22⋊C4—216no—
173C22⋊Dic7—216no—
174C22⋊Dic7—416no—
175C22⋊Dic7—416no—
176C22⋊Dic7—48no—
177C4⋊Dic7—44no—
178C4⋊Dic7—44no—
179Dic7⋊C4—416no—
180C26C2×D711yescharacteristic
181C24×C4C2×D711yescharacteristic
182C24×C4C2×D711yescharacteristic
183C22×C4⋊C4—71no—
184C22×C4⋊C4—74no—
185C22×C4⋊C4—71no—
186C22×C22⋊C4—24no—
187C22×C22⋊C4—28no—
188C22×C22⋊C4—72no—
189C22×C22⋊C4—72no—
190C22×C22⋊C4—74no—
191C22×C22⋊C4—74no—
192C22×C22⋊C4C2×D712yes—
193C22×C22⋊C4C2×D712yes—
194#399—148no—
195#385—148no—
196#380—144no—
197#379—148no—
198C2×#95—72no—
199C2×#95—72no—
200C2×#95—74no—
201C2×#95—72no—
202C23×C7—22no—
203C23×C7—24no—
204C23×C7—22no—
205C23×C7—24no—
206C23×C7—22no—
207C23×C7—22no—
208C23×C7—24no—
209C23×C7—24no—
210C23×C7—28no—
211C23×C7—28no—
212C23×C7—28no—
213C23×C7—28no—
214C23×C7—232no—
215C23×C7—24no—
216C23×C7—28no—
217C23×C7—24no—
218C23×C7—24no—
219C23×C7—24no—
220C23×C7—24no—
221C23×C7—24no—
222C23×C7—28no—
223C23×C7—28no—
224C23×C7—28no—
225C23×C7—28no—
226C23×C7—28no—
227C23×C7—464no—
228C23×C7—416no—
229C23×C7—416no—
230C23×C7—464no—
231C23×C7—416no—
232C23×C7—416no—
233C23×C7—416no—
234C23×C7—44no—
235C23×C7—42no—
236C23×C7—42no—
237C23×C7—42no—
238C23×C7—42no—
239C23×C7—44no—
240C23×C7—44no—
241C23×C7—44no—
242C23×C7—44no—
243C23×C7—44no—
244C23×C7—416no—
245C23×C7C22⋊Q818yes—
246C23×C7C22⋊Q818yes—
247C23×C7C22⋊Q814yes—
248C23×C7#12318yes—
249C23×C7#12314yes—
250C23×C7#12714yes—
251C23×C7C22×D412yes—
252C23×C7C22×D411yescharacteristic
253C23×C7C22×D412yes—
254C23×C7C22×Q811yescharacteristic
255C23×C7C2×#4111yescharacteristic
256C23×C7C2×#4112yes—
257C23×C7C2×#4112yes—
258C23×C7H(5,𝔽2)12yes—
259C23×C7H(5,𝔽2)12yes—
260C2×C4×C7—22no—
261C2×C4×C7—22no—
262C2×C4×C7—28no—
263C2×C4×C7—22no—
264C2×C4×C7—24no—
265C2×C4×C7—22no—
266C2×C4×C7—22no—
267C2×C4×C7—28no—
268C2×C4×C7—416no—
269C2×C4×C7—416no—
270C2×C4×C7—42no—
271C2×C4×C7—42no—
272C2×C4×C7—44no—
273C2×C4×C7—44no—
274C2×C4×C7#12014yes—
275C2×Dic7—28no—
276C2×Dic7—28no—
277C2×Dic7—28no—
278C2×Dic7—44no—
279C2×Dic7—44no—
280C2×Dic7—44no—
281C2×Dic7—44no—
282C2×Dic7—42no—
283C2×Dic7—42no—
284C2×Dic7—42no—
285C2×Dic7—42no—
286C2×Dic7—42no—
287C2×Dic7—42no—
288C2×Dic7—44no—
289C2×Dic7—44no—
290C2×Dic7—44no—
291C2×Dic7—44no—
292C25—24no—
293C25—22no—
294C25—24no—
295C25—22no—
296C25—24no—
297C25—24no—
298C25—42no—
299C25—42no—
300C25Dic1414yes—
301C25C22×D712yes—
302C25C22×D711yescharacteristic
303C23×C4—24no—
304C23×C4—22no—
305C23×C4—24no—
306C23×C4—24no—
307C23×C4—24no—
308C23×C4—42no—
309C23×C4—42no—
310C23×C4—74no—
311C23×C4—72no—
312C23×C4—72no—
313C23×C4C22⋊D714yes—
314C23×C4C22×D712yes—
315C23×C4C22×D711yescharacteristic
316C23×C4C22×D711yescharacteristic
317C2×C4⋊C4—148no—
318C2×C4⋊C4—148no—
319C2×C4⋊C4—142no—
320C2×C4⋊C4—142no—
321C2×C4⋊C4—144no—
322C2×C4⋊C4—284no—
323C2×C4⋊C4—282no—
324C2×C22⋊C4—232no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—78no—
331C2×C22⋊C4—144no—
332C2×C22⋊C4—144no—
333C2×C22⋊C4—148no—
334C2×C22⋊C4—148no—
335C2×C22⋊C4—148no—
336C2×C22⋊C4—144no—
337C2×C22⋊C4—144no—
338C2×C22⋊C4—144no—
339C2×C22⋊C4—148no—
340C2×C22⋊C4—284no—
341C2×C22⋊C4—284no—
342C2×C22⋊C4C22⋊D718yes—
343#95—148no—
344#95—144no—
345#95—144no—
346#95—282no—
347C22×C7—24no—
348C22×C7—24no—
349C22×C7—24no—
350C22×C7—216no—
351C22×C7—24no—
352C22×C7—216no—
353C22×C7—24no—
354C22×C7—24no—
355C22×C7—28no—
356C22×C7—216no—
357C22×C7—464no—
358C22×C7—44no—
359C22×C7—44no—
360C22×C7—44no—
361C22×C7—48no—
362C22×C7—48no—
363C22×C7—44no—
364C22×C7—48no—
365C22×C7—44no—
366C22×C7—44no—
367C22×C7#39918yescomplemented
368C22×C7C2×#12011yescharacteristic
369C22×C7C2×C22⋊Q812yes—
370C22×C7C2×C22⋊Q811yescharacteristic
371C22×C7C2×C22⋊Q812yes—
372C22×C7C2×#12312yes—
373C22×C7C2×#12311yescharacteristic
374C22×C7#52512yes—
375C22×C7#52511yescharacteristic
376C22×C7C2×#12711yescharacteristic
377C22×C7#53411yescharacteristic
378C22×C7#53511yescharacteristic
379C22×C7#53812yes—
380C22×C7#54312yes—
381C22×C7D4214yes—
382C22×C7#54614yes—
383C22×C7#54614yes—
384C22×C7#55114yes—
385C4×C7—24no—
386C4×C7—42no—
387C4×C7—44no—
388Dic7—48no—
389Dic7—82no—
390Dic7—82no—
391C24—22no—
392C24—22no—
393C24—22no—
394C24—22no—
395C24—28no—
396C24—216no—
397C24—28no—
398C24—216no—
399C24—28no—
400C24—22no—
401C24—22no—
402C24—24no—
403C24—24no—
404C24—24no—
405C24—24no—
406C24—22no—
407C24—22no—
408C24—216no—
409C24—48no—
410C24—416no—
411C24—416no—
412C24—416no—
413C24—416no—
414C24—48no—
415C24—44no—
416C24—44no—
417C24—48no—
418C24—44no—
419C24—44no—
420C24C2×Dic1414yes—
421C24C2×Dic1412yes—
422C24#115714yes—
423C24#115712yes—
424C24D4×D714yes—
425C24#115914yes—
426C24C2×C22⋊D714yes—
427C24C2×C22⋊D712yes—
428C24C23×D711yescharacteristic
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—28no—
432C22×C4—28no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—28no—
441C22×C4—44no—
442C22×C4—416no—
443C22×C4—416no—
444C22×C4—44no—
445C22×C4—44no—
446C22×C4—78no—
447C22×C4—144no—
448C22×C4—144no—
449C22×C4—144no—
450C22×C4—144no—
451C22×C4—144no—
452C22×C4—144no—
453C22×C4—144no—
454C22×C4—142no—
455C22×C4—142no—
456C22×C4—142no—
457C22×C4—142no—
458C22×C4—144no—
459C22×C4—144no—
460C22×C4—142no—
461C22×C4—142no—
462C22×C4—144no—
463C22×C4C2×C22⋊D712yes—
464C22×C4C2×C22⋊D714yes—
465C22×C4C2×C22⋊D714yes—
466C22×C4C2×C22⋊D712yes—
467C4⋊C4—2816no—
468C4⋊C4—284no—
469C4⋊C4—284no—
470C22⋊C4—216no—
471C22⋊C4—216no—
472C22⋊C4—1416no—
473C22⋊C4—2816no—
474C22⋊C4—2816no—
475C22⋊C4—288no—
476C2×C7—28no—
477C2×C7—48no—
478C2×C7C2×#39911yescharacteristic
479C2×C7#240511yescharacteristic
480C2×C7#241814yes—
481C2×C7#243311yescharacteristic
482C2×C7#243714yes—
483C2×C7#248211yescharacteristic
484C2×C7#253911yescharacteristic
485C2×C7#262611yescharacteristic
486C2×C7#284811yescharacteristic
487C23—24no—
488C23—28no—
489C23—24no—
490C23—28no—
491C23—24no—
492C23—24no—
493C23—22no—
494C23—28no—
495C23—22no—
496C23—22no—
497C23—28no—
498C23—22no—
499C23—232no—
500C23—28no—
501C23—28no—
502C23—28no—
503C23—28no—
504C23—24no—
505C23—24no—
506C23—24no—
507C23—24no—
508C23—24no—
509C23—28no—
510C23—28no—
511C23—24no—
512C23—464no—
513C23—416no—
514C23—416no—
515C23—416no—
516C23—416no—
517C23—416no—
518C23—464no—
519C23—44no—
520C23—44no—
521C23—44no—
522C23—44no—
523C23—42no—
524C23—416no—
525C23—44no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—44no—
530C23#644118yes—
531C23#644618yes—
532C23#648718yes—
533C23#648714yes—
534C23#649814yes—
535C23#650314yes—
536C23C22×Dic1411yescharacteristic
537C23C2×#115711yescharacteristic
538C23C2×#115712yes—
539C23C2×D4×D712yes—
540C23C2×#115912yes—
541C23#654812yes—
542C23#654812yes—
543C23C22×C22⋊D711yescharacteristic
544C23C22×C22⋊D712yes—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—28no—
548C2×C4—22no—
549C2×C4—28no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—24no—
553C2×C4—416no—
554C2×C4—416no—
555C2×C4—44no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—44no—
559C2×C4—148no—
560C2×C4—148no—
561C2×C4—148no—
562C2×C4—284no—
563C2×C4—284no—
564C2×C4—284no—
565C2×C4—284no—
566C2×C4—284no—
567C2×C4—284no—
568C2×C4—282no—
569C2×C4—284no—
570C2×C4—284no—
571C2×C4—282no—
572C2×C4—282no—
573C2×C4—282no—
574C2×C4—282no—
575C2×C4—282no—
576C2×C4#651614yes—
577C7#1473411yescharacteristic, complemented, Sylow
578C22—24no—
579C22—24no—
580C22—24no—
581C22—216no—
582C22—24no—
583C22—24no—
584C22—24no—
585C22—216no—
586C22—28no—
587C22—216no—
588C22—464no—
589C22—44no—
590C22—44no—
591C22—44no—
592C22—48no—
593C22—44no—
594C22—44no—
595C22—44no—
596C22—48no—
597C22—48no—
598C22#8972118yescomplemented
599C22C2×#644112yes—
600C22#9017412yes—
601C22C2×#644612yes—
602C22#9018114yes—
603C22#9018512yes—
604C22#9018614yes—
605C22C2×#648711yescharacteristic
606C22C2×#648712yes—
607C22#9048211yescharacteristic
608C22#9048212yes—
609C22C2×#649811yescharacteristic
610C22D4×C22⋊D714yes—
611C22C2×#650311yescharacteristic
612C22#9049511yescharacteristic
613C22#9049611yescharacteristic
614C22#9049714yes—
615C22C2×#651611yescharacteristic
616C4—24no—
617C4—42no—
618C4—44no—
619C4—288no—
620C4—562no—
621C4—562no—
622C2—28no—
623C2—48no—
624C2C2×#8972111yescharacteristic
625C2#1173955514yes—
626C2#1173955811yescharacteristic
627C2#1173956011yescharacteristic
628C2#1173956211yescharacteristic
629C2#1173957314yes—
630C2#1174147611yescharacteristic
631C2#1174149211yescharacteristic
632C2#1174149411yescharacteristic
subgroupG/H
center, FrattiniC24C23×D7
commutator, socle, cosocleC24×C7C24
FittingC23×C7×C22⋊C4C2
prime radicalC26×C7C22
hypercenterC23×C22⋊C4D7
hypocenterC7#14734
LCSG ▷ C24×C7 ▷ C7
UCSC1 ◁ C24 ◁ C23×C22⋊C4

Complements

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

NH ≅ G/Ntype
3. C2×#89982C2semidirect
4. C2×#89984C2semidirect
5. C2×#89721C2semidirect
6. #11729881C2semidirect
7. #11729301C2semidirect
8. #11729300C2semidirect
20. C22×C4⋊Dic7C22semidirect
21. C22×C4⋊Dic7C22semidirect
22. C22×Dic7⋊C4C22semidirect
26. C2×C7⋊#95C22semidirect
27. C2×C7⋊#95C22semidirect
28. C2×C7⋊#95C22semidirect
29. C2×C7⋊#95C22semidirect
367. C22×C7#399semidirect
577. C7#14734semidirect
598. C22#89721semidirect

Maximal subgroups

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

subgroupindex
C23×C7×C22⋊C42
C2×#900192
C2×#899822
C2×#899842
C2×#897212
#117298812
#117293012
#117293002
#147347