Finite Group Databasebeta

← #13560608

Subgroups — #13560608

Order 1792.

2326 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6487C211yesmaximal, characteristic
2C2×#90314C211yesmaximal, characteristic, complemented
3C7×#2848C211yesmaximal, characteristic
4#11741476C211yesmaximal, characteristic
5#11740236C214yesmaximal, complemented
6#11740233C211yesmaximal, characteristic, complemented
7#11731959C214yesmaximal, complemented
8#11731955C214yesmaximal, complemented
9#11731901C212yesmaximal, complemented
10#11731857C212yesmaximal, complemented
11#11731405C214yesmaximal, complemented
12#11731398C211yesmaximal, characteristic, complemented
13#11731312C211yesmaximal, characteristic, complemented
14#11731280C212yesmaximal
15#11731277C212yesmaximal
16C24×C4×C7C2211yescharacteristic
17C23×Dic14C2211yescharacteristic
18C2×C7×C22⋊Q8C2211yescharacteristic
19C2×C7×C22⋊Q8C2212yes—
20C22×C22⋊Dic7C2211yescharacteristic
21C22×C22⋊Dic7C2211yescharacteristic
22C2×#6507C2212yescomplemented
23C22×C4⋊Dic7C2211yescharacteristic
24C2×#6487—28no—
25C2×#6487C2214yes—
26C2×#6487C2214yes—
27C2×#6487C2211yescharacteristic
28C2×#6487C2212yes—
29C22×Dic7⋊C4C2211yescharacteristic
30C22×Dic7⋊C4C2211yescharacteristic
31#90314—28no—
32C2×#6452C2212yescomplemented
33C2×#6452C2212yescomplemented
34C2×#6451C2212yescomplemented
35C2×#6451C2214yescomplemented
36C2×#6441C2212yes—
37C2×#6440C2212yes—
38C7×#385C2212yes—
39C7×#385C2214yes—
40C7×#380C2211yescharacteristic
41C7×#380C2212yes—
42C7×#379C2212yes—
43#90019C2212yes—
44#90001C2212yescomplemented
45#89984C2211yescharacteristic
46#89984C2212yes—
47#89982C2214yes—
48#89982C2212yes—
49#89756C2214yescomplemented
50#89756C2214yescomplemented
51#89755C2214yescomplemented
52#89754C2212yescomplemented
53#89753C2214yescomplemented
54#89751C2214yescomplemented
55#89750C2214yescomplemented
56#89724C2214yes—
57#89724C2214yes—
58#89723C2214yes—
59#89723C2214yes—
60#89722C2212yes—
61#89721C2214yes—
62#89719C2214yes—
63#89718C2212yes—
64#89717C2214yes—
65#89716C2214yes—
66#89715C2212yes—
67#89715C2212yes—
68#89715C2212yes—
69C22⋊C4×Dic7C2212yes—
70#89432C2214yescomplemented
71#89431C2212yescomplemented
72#89428C2214yescomplemented
73#89428C2212yescomplemented
74#89427C2214yescomplemented
75#89427C2214yescomplemented
76#89418C2214yescomplemented
77#36708—71nomaximal, Sylow
78C25×C7C2311yescharacteristic
79C22×C7×Q8C2311yescharacteristic
80C23×C4×C7—24no—
81C23×C4×C7—44no—
82C23×C4×C7C2311yescharacteristic
83C23×C4×C7C2312yes—
84C23×C4×C7C2312yes—
85C23×C4×C7C2311yescharacteristic
86C23×Dic7C2311yescharacteristic
87C23×Dic7C2311yescharacteristic
88C23×Dic7C2311yescharacteristic
89C23×Dic7C2311yescharacteristic
90C22×Dic14—24no—
91C22×Dic14—44no—
92C22×Dic14C2314yes—
93C22×Dic14C2312yes—
94C22×Dic14C2311yescharacteristic
95C7×C22⋊Q8—24no—
96C7×C22⋊Q8—44no—
97C2×C7×C4⋊C4C2312yes—
98C2×C7×C4⋊C4C2314yes—
99C2×C7×C4⋊C4C2311yescharacteristic
100C2×C7×C4⋊C4C2312yes—
101C2×C7×C22⋊C4C2312yes—
102C2×C7×C22⋊C4C2312yes—
103C2×C7×C22⋊C4C2314yes—
104C2×C7×C22⋊C4C2314yes—
105C2×C22⋊Dic7—24no—
106C2×C22⋊Dic7—24no—
107C2×C22⋊Dic7C2312yes—
108C2×C22⋊Dic7C2314yes—
109C2×C22⋊Dic7C2314yes—
110C2×C22⋊Dic7C2312yes—
111C2×C22⋊Dic7C2314yes—
112C2×C22⋊Dic7C2314yes—
113#6507—28no—
114C2×C4⋊Dic7—24no—
115C2×C4⋊Dic7C2312yes—
116C2×C4⋊Dic7C2311yescharacteristic
117C2×C4⋊Dic7C2314yes—
118C2×C4⋊Dic7C2314yes—
119C2×C4⋊Dic7C2312yes—
120#6487—24no—
121#6487—216no—
122#6487—216no—
123#6487—44no—
124C2×Dic7⋊C4—24no—
125C2×Dic7⋊C4D418yes—
126C2×Dic7⋊C4C2312yes—
127C2×Dic7⋊C4C2312yes—
128C2×Dic7⋊C4C2312yes—
129C2×Dic7⋊C4C2312yes—
130C2×Dic7⋊C4C2314yes—
131C2×Dic7⋊C4C2312yes—
132C2×Dic7⋊C4C2312yes—
133C2×Dic7⋊C4C2312yes—
134C2×Dic7⋊C4C2314yes—
135C2×Dic7⋊C4C2312yes—
136C2×Dic7⋊C4C2312yes—
137C2×C4×Dic7C2312yes—
138C2×C4×Dic7C2312yes—
139C2×C4×Dic7C2314yes—
140#6452—28no—
141#6452—28no—
142#6451—28no—
143#6451—48no—
144#6441—28no—
145#6440—28no—
146C7×#95C2314yes—
147C7×#95C2312yes—
148C7⋊#95C2314yes—
149C7⋊#95C2314yes—
150C7⋊#95C2314yes—
151C7⋊#95C2314yes—
152C7⋊#95C2314yes—
153C7⋊#95C2314yes—
154C7⋊#95C2314yes—
155C7⋊#95C2314yes—
156C7⋊#95C2314yes—
157C7⋊#95C2314yes—
158C7⋊#95C2312yes—
159C2×#544—71no—
160C22×C22⋊Q8—71no—
161#2848—71no—
162#2848D711yescharacteristic
163#2690—74no—
164#2690—74no—
165#2677—74no—
166#2677—74no—
167#2627—71no—
168#2627—71no—
169#2561—72no—
170#2484—72no—
171#2482—72no—
172#2434—72no—
173#2313—71no—
174C24×C7—22no—
175C24×C7—24no—
176C24×C7—22no—
177C24×C7—22no—
178C24×C7—42no—
179C24×C7C2412yes—
180C24×C7C2411yescharacteristic
181C2×C7×Q8—22no—
182C2×C7×Q8—42no—
183C22×C4×C7—22no—
184C22×C4×C7—24no—
185C22×C4×C7—24no—
186C22×C4×C7—24no—
187C22×C4×C7—24no—
188C22×C4×C7—24no—
189C22×C4×C7—42no—
190C22×C4×C7—42no—
191C22×C4×C7—42no—
192C22×C4×C7—416no—
193C22×C4×C7C2×D414yes—
194C22×C4×C7C2414yes—
195C22×C4×C7C2412yes—
196C22×C4×C7C2412yes—
197C22×C4×C7C2411yescharacteristic
198C22×C4×C7C2411yescharacteristic
199C22×C4×C7C2412yes—
200C22×Dic7—22no—
201C22×Dic7—22no—
202C22×Dic7—22no—
203C22×Dic7—22no—
204C22×Dic7—22no—
205C22×Dic7—22no—
206C22×Dic7—42no—
207C22×Dic7—42no—
208C22×Dic7C2×D414yes—
209C22×Dic7C2×D414yes—
210C22×Dic7C2414yes—
211C22×Dic7C2412yes—
212C22×Dic7C2412yes—
213C22×Dic7C2412yes—
214C22×Dic7C2412yes—
215C22×Dic7C2412yes—
216C22×Dic7C2412yes—
217C2×Dic14—28no—
218C2×Dic14—28no—
219C2×Dic14—22no—
220C2×Dic14—24no—
221C2×Dic14—44no—
222C2×Dic14—416no—
223C2×Dic14—44no—
224C2×Dic14—42no—
225C7×C4⋊C4—24no—
226C7×C4⋊C4—22no—
227C7×C4⋊C4—28no—
228C7×C4⋊C4—42no—
229C7×C22⋊C4—24no—
230C7×C22⋊C4—28no—
231C7×C22⋊C4—24no—
232C7×C22⋊C4—44no—
233C22⋊Dic7—24no—
234C22⋊Dic7—24no—
235C22⋊Dic7—28no—
236C22⋊Dic7—28no—
237C22⋊Dic7—28no—
238C22⋊Dic7—44no—
239C4⋊Dic7—22no—
240C4⋊Dic7—24no—
241C4⋊Dic7—28no—
242C4⋊Dic7—44no—
243C4⋊Dic7—42no—
244Dic7⋊C4—24no—
245Dic7⋊C4—24no—
246Dic7⋊C4—24no—
247Dic7⋊C4—28no—
248Dic7⋊C4—24no—
249Dic7⋊C4—24no—
250Dic7⋊C4—24no—
251Dic7⋊C4—24no—
252Dic7⋊C4—44no—
253Dic7⋊C4C2×D418yes—
254Dic7⋊C4C2×D418yes—
255C4×Dic7—24no—
256C4×Dic7—24no—
257C4×Dic7—44no—
258C23×Q8—71no—
259C24×C4C2×D711yescharacteristic
260#544—148no—
261C2×C4⋊Q8—72no—
262C2×C4⋊Q8—72no—
263C2×C4⋊Q8—74no—
264C2×#125—72no—
265C2×#125—72no—
266C2×C22⋊Q8—71no—
267C2×C22⋊Q8—74no—
268C2×C22⋊Q8—74no—
269C2×C22⋊Q8—72no—
270C2×C22⋊Q8—72no—
271C2×C22⋊Q8—148no—
272C2×C22⋊Q8C2×D711yescharacteristic
273C2×C22⋊Q8C2×D712yes—
274C2×#117—72no—
275C22×C4⋊C4—71no—
276C22×C4⋊C4—71no—
277C22×C4⋊C4—71no—
278C22×C22⋊C4—71no—
279C22×C22⋊C4—71no—
280#400—72no—
281#400—72no—
282#400—74no—
283#399—74no—
284#398—74no—
285#398—74no—
286#398—74no—
287#397—74no—
288#397—74no—
289#395—72no—
290#395—74no—
291#395—72no—
292#395—74no—
293#393—74no—
294#391—74no—
295#388—74no—
296#388—74no—
297#388—74no—
298#388—74no—
299#387—74no—
300#387—74no—
301#387—74no—
302#387—74no—
303#385—74no—
304#385—72no—
305#385—72no—
306#385—72no—
307#385C2×D714yes—
308#385C2×D712yes—
309#381—72no—
310#381—72no—
311#381—72no—
312#380—71no—
313#380—72no—
314#380C2×D711yescharacteristic
315#380C2×D712yes—
316#379—72no—
317#379C2×D712yes—
318C4×C22⋊C4—72no—
319C23×C7—22no—
320C23×C7—22no—
321C23×C7—28no—
322C23×C7—22no—
323C23×C7—22no—
324C23×C7—22no—
325C23×C7—22no—
326C23×C7—22no—
327C23×C7—22no—
328C23×C7—48no—
329C23×C7—48no—
330C23×C7—44no—
331C23×C7—44no—
332C23×C7C22×D412yes—
333C23×C72−1+412yes—
334C23×C72−1+414yes—
335C23×C72−1+412yes—
336C23×C7C2511yescharacteristic
337C7×Q8—82no—
338C2×C4×C7—28no—
339C2×C4×C7—22no—
340C2×C4×C7—22no—
341C2×C4×C7—22no—
342C2×C4×C7—22no—
343C2×C4×C7—22no—
344C2×C4×C7—22no—
345C2×C4×C7—24no—
346C2×C4×C7—24no—
347C2×C4×C7—24no—
348C2×C4×C7—24no—
349C2×C4×C7—24no—
350C2×C4×C7—44no—
351C2×C4×C7—48no—
352C2×C4×C7—48no—
353C2×C4×C7C22×D412yes—
354C2×C4×C7C22×D412yes—
355C2×C4×C7H(5,𝔽2)14yes—
356C2×Dic7—28no—
357C2×Dic7—24no—
358C2×Dic7—22no—
359C2×Dic7—22no—
360C2×Dic7—22no—
361C2×Dic7—22no—
362C2×Dic7—22no—
363C2×Dic7—22no—
364C2×Dic7—24no—
365C2×Dic7—24no—
366C2×Dic7—24no—
367C2×Dic7—24no—
368C2×Dic7—24no—
369C2×Dic7—24no—
370C2×Dic7—22no—
371C2×Dic7—22no—
372C2×Dic7—44no—
373C2×Dic7—44no—
374C2×Dic7—44no—
375C2×Dic7C22×D414yes—
376C2×Dic7C22×D414yes—
377Dic14—416no—
378Dic14—84no—
379Dic14—82no—
380C25C22×D711yescharacteristic
381C22×Q8—74no—
382C22×Q8—71no—
383C22×Q8—72no—
384C22×Q8—144no—
385C22×Q8—284no—
386C22×Q8C22×D711yescharacteristic
387C23×C4—24no—
388C23×C4—44no—
389C23×C4—71no—
390C23×C4—71no—
391C23×C4—71no—
392C23×C4—71no—
393C23×C4C22×D711yescharacteristic
394C23×C4C22×D712yes—
395C23×C4C22×D712yes—
396C23×C4C22×D711yescharacteristic
397C4⋊Q8—148no—
398C4⋊Q8—148no—
399C4⋊Q8—288no—
400#125—148no—
401#125—148no—
402C22⋊Q8—24no—
403C22⋊Q8—44no—
404C22⋊Q8—148no—
405C22⋊Q8—1416no—
406C22⋊Q8—1416no—
407C22⋊Q8—144no—
408C22⋊Q8—284no—
409#117—148no—
410C2×C4⋊C4—78no—
411C2×C4⋊C4—72no—
412C2×C4⋊C4—72no—
413C2×C4⋊C4—72no—
414C2×C4⋊C4—72no—
415C2×C4⋊C4—74no—
416C2×C4⋊C4—72no—
417C2×C4⋊C4—72no—
418C2×C4⋊C4—72no—
419C2×C4⋊C4—71no—
420C2×C4⋊C4—72no—
421C2×C4⋊C4—72no—
422C2×C4⋊C4—72no—
423C2×C4⋊C4—74no—
424C2×C4⋊C4—72no—
425C2×C4⋊C4—74no—
426C2×C4⋊C4—74no—
427C2×C4⋊C4—144no—
428C2×C4⋊C4—144no—
429C2×C4⋊C4C22×D711yescharacteristic
430C2×C4⋊C4C22×D712yes—
431C2×C4⋊C4C22×D714yes—
432C2×C4⋊C4C22×D712yes—
433C2×C22⋊C4—74no—
434C2×C22⋊C4—74no—
435C2×C22⋊C4—72no—
436C2×C22⋊C4—72no—
437C2×C22⋊C4—74no—
438C2×C22⋊C4—74no—
439C2×C22⋊C4—144no—
440C2×C22⋊C4—144no—
441C2×C22⋊C4C22×D712yes—
442C2×C22⋊C4C22×D712yes—
443C2×C22⋊C4C22×D714yes—
444C2×C22⋊C4C22×D714yes—
445C2×C42—72no—
446C2×C42—72no—
447C2×C42—74no—
448#95—74no—
449#95—72no—
450#95—74no—
451#95—74no—
452#95—74no—
453#95—74no—
454#95—74no—
455#95—74no—
456#95—74no—
457#95—74no—
458#95—74no—
459#95C22×D712yes—
460#95C22×D714yes—
461C22×C7—22no—
462C22×C7—22no—
463C22×C7—24no—
464C22×C7—24no—
465C22×C7—24no—
466C22×C7—24no—
467C22×C7—24no—
468C22×C7—416no—
469C22×C7—42no—
470C22×C7—42no—
471C22×C7—42no—
472C22×C7—42no—
473C22×C7#53614yes—
474C22×C7C23×D411yescharacteristic
475C22×C7C2×H(5,𝔽2)12yes—
476C22×C7C2×2−1+411yescharacteristic
477C22×C7C2×2−1+411yescharacteristic
478C22×C7C2×2−1+412yes—
479C4×C7—24no—
480C4×C7—22no—
481C4×C7—22no—
482C4×C7—42no—
483C4×C7—42no—
484C4×C7—44no—
485Dic7—24no—
486Dic7—24no—
487Dic7—42no—
488Dic7—42no—
489Dic7—42no—
490Dic7—42no—
491Dic7—44no—
492C24—22no—
493C24—24no—
494C24—22no—
495C24—22no—
496C24—42no—
497C24C23×D712yes—
498C24C23×D711yescharacteristic
499C2×Q8—22no—
500C2×Q8—42no—
501C2×Q8—148no—
502C2×Q8—144no—
503C2×Q8—142no—
504C2×Q8—148no—
505C2×Q8—284no—
506C2×Q8—282no—
507C2×Q8—284no—
508C2×Q8—2816no—
509C22×C4—22no—
510C22×C4—24no—
511C22×C4—24no—
512C22×C4—24no—
513C22×C4—24no—
514C22×C4—24no—
515C22×C4—42no—
516C22×C4—42no—
517C22×C4—42no—
518C22×C4—416no—
519C22×C4—74no—
520C22×C4—72no—
521C22×C4—72no—
522C22×C4—74no—
523C22×C4—74no—
524C22×C4—72no—
525C22×C4—72no—
526C22×C4—72no—
527C22×C4—72no—
528C22×C4—142no—
529C22×C4—142no—
530C22×C4—142no—
531C22×C4—142no—
532C22×C4—142no—
533C22×C4—142no—
534C22×C4—282no—
535C22×C4—282no—
536C22×C4D4×D714yes—
537C22×C4C23×D714yes—
538C22×C4C23×D712yes—
539C22×C4C23×D712yes—
540C22×C4C23×D711yescharacteristic
541C22×C4C23×D711yescharacteristic
542C22×C4C23×D712yes—
543C4⋊C4—22no—
544C4⋊C4—24no—
545C4⋊C4—28no—
546C4⋊C4—42no—
547C4⋊C4—78no—
548C4⋊C4—78no—
549C4⋊C4—144no—
550C4⋊C4—144no—
551C4⋊C4—144no—
552C4⋊C4—144no—
553C4⋊C4—148no—
554C4⋊C4—144no—
555C4⋊C4—144no—
556C4⋊C4—144no—
557C4⋊C4—144no—
558C4⋊C4—142no—
559C4⋊C4—148no—
560C4⋊C4—284no—
561C4⋊C4—282no—
562C4⋊C4—284no—
563C22⋊C4—24no—
564C22⋊C4—28no—
565C22⋊C4—24no—
566C22⋊C4—44no—
567C22⋊C4—148no—
568C22⋊C4—148no—
569C22⋊C4—144no—
570C22⋊C4—144no—
571C22⋊C4—148no—
572C22⋊C4—284no—
573C42—144no—
574C42—144no—
575C42—284no—
576C2×C7—24no—
577C2×C7—44no—
578C2×C7C2×#53611yescharacteristic
579C2×C7#348411yescharacteristic
580C2×C7#348611yescharacteristic
581C2×C7#353812yes—
582C2×C7#353812yes—
583C23—22no—
584C23—22no—
585C23—28no—
586C23—22no—
587C23—22no—
588C23—22no—
589C23—22no—
590C23—22no—
591C23—22no—
592C23—48no—
593C23—48no—
594C23—44no—
595C23—44no—
596C23C2×D4×D712yes—
597C23#655112yes—
598C23#655414yes—
599C23#655412yes—
600C23C24×D711yescharacteristic
601Q8—82no—
602Q8—2816no—
603Q8—562no—
604Q8—564no—
605C2×C4—28no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—24no—
613C2×C4—24no—
614C2×C4—24no—
615C2×C4—24no—
616C2×C4—24no—
617C2×C4—44no—
618C2×C4—48no—
619C2×C4—48no—
620C2×C4—74no—
621C2×C4—74no—
622C2×C4—148no—
623C2×C4—144no—
624C2×C4—142no—
625C2×C4—142no—
626C2×C4—142no—
627C2×C4—142no—
628C2×C4—142no—
629C2×C4—142no—
630C2×C4—144no—
631C2×C4—144no—
632C2×C4—144no—
633C2×C4—144no—
634C2×C4—142no—
635C2×C4—144no—
636C2×C4—144no—
637C2×C4—142no—
638C2×C4—284no—
639C2×C4—284no—
640C2×C4—284no—
641C2×C4C2×D4×D712yes—
642C2×C4C2×D4×D712yes—
643C2×C4#654814yes—
644C7#3670811yescharacteristic, complemented, Sylow
645C22—22no—
646C22—24no—
647C22—22no—
648C22—24no—
649C22—24no—
650C22—24no—
651C22—24no—
652C22—416no—
653C22—42no—
654C22—42no—
655C22—42no—
656C22—42no—
657C22#9031914yes—
658C22C22×D4×D711yescharacteristic
659C22C2×#654812yes—
660C22C2×#655111yescharacteristic
661C22C2×#655412yes—
662C22C2×#655411yescharacteristic
663C4—24no—
664C4—22no—
665C4—22no—
666C4—42no—
667C4—42no—
668C4—44no—
669C4—144no—
670C4—144no—
671C4—282no—
672C4—282no—
673C4—284no—
674C4—282no—
675C4—282no—
676C2—24no—
677C2—44no—
678C2#1174298711yescharacteristic
679C2#1174304112yes—
680C2C2×#9031911yescharacteristic
681C2#1174309011yescharacteristic
682C2#1174309912yes—
subgroupG/H
center, FrattiniC23C24×D7
commutator, socle, cosocleC23×C7C25
FittingC7×#2848C2
prime radicalC25×C7C23
hypercenter#2848D7
hypocenterC7#36708
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #2848

Complements

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

NH ≅ G/Ntype
2. C2×#90314C2semidirect
5. #11740236C2semidirect
6. #11740233C2semidirect
7. #11731959C2semidirect
8. #11731955C2semidirect
9. #11731901C2semidirect
10. #11731857C2semidirect
11. #11731405C2semidirect
12. #11731398C2semidirect
13. #11731312C2semidirect
22. C2×#6507C22semidirect
32. C2×#6452C22semidirect
33. C2×#6452C22semidirect
34. C2×#6451C22semidirect
35. C2×#6451C22semidirect
44. #90001C22semidirect
49. #89756C22semidirect
50. #89756C22semidirect
51. #89755C22semidirect
52. #89754C22semidirect
53. #89753C22semidirect
54. #89751C22semidirect
55. #89750C22semidirect
70. #89432C22semidirect
71. #89431C22semidirect
72. #89428C22semidirect
73. #89428C22semidirect
74. #89427C22semidirect
75. #89427C22semidirect
76. #89418C22semidirect
644. C7#36708semidirect

Maximal subgroups

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

subgroupindex
C22×#64872
C2×#903142
C7×#28482
#117414762
#117402362
#117402332
#117319592
#117319552
#117319012
#117318572
#117314052
#117313982
#117313122
#117312802
#117312772
#367087