Finite Group Databasebeta

← #13560607

Subgroups — #13560607

Order 1792.

2326 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6509C211yesmaximal, characteristic
2C2×#90314C211yesmaximal, characteristic, complemented
3C7×#2434C211yesmaximal, characteristic, complemented
4#11741510C211yesmaximal, characteristic, complemented
5#11740711C214yesmaximal, complemented
6#11740264C211yesmaximal, characteristic
7#11734706C214yesmaximal, complemented
8#11734659C214yesmaximal, complemented
9#11734008C214yesmaximal, complemented
10#11732100C212yesmaximal
11#11732059C212yesmaximal
12#11731947C212yesmaximal, complemented
13#11731901C212yesmaximal, complemented
14#11731050C211yesmaximal, characteristic
15#11730890C211yesmaximal, characteristic, complemented
16C23×C7×Q8C2211yescharacteristic
17C23×C4×D7C2211yescharacteristic
18C2×C7×C4⋊Q8C2212yescomplemented
19C2×C7×C22⋊Q8C2211yescharacteristic
20C2×#6509—28no—
21C2×#6509C2214yes—
22C2×#6509C2214yes—
23C2×#6507C2212yescomplemented
24C2×#6507C2212yescomplemented
25C22×(C2×D7)⋊C4C2211yescharacteristic
26C22×(C2×D7)⋊C4C2211yescharacteristic
27C22×C4⋊Dic7C2211yescharacteristic
28C2×#6487C2211yescharacteristic
29C22×Dic7⋊C4C2211yescharacteristic
30C22×Dic7⋊C4C2211yescharacteristic
31#90314—28no—
32C2×#6459C2214yes—
33C2×#6452C2212yescomplemented
34C2×#6452C2212yescomplemented
35C2×#6451C2212yescomplemented
36C2×#6441C2212yes—
37C2×#6440C2212yes—
38C7×#395C2214yescomplemented
39C7×#391C2212yescomplemented
40C7×#385C2212yes—
41C7×#385C2212yes—
42C4×C7×C22⋊C4C2211yescharacteristic
43#90003C2212yes—
44#90002C2212yescomplemented
45#90001C2214yescomplemented
46#89982C2212yes—
47#89982C2212yes—
48C4×C22⋊Dic7C2211yescharacteristic
49#89765C2212yes—
50#89764C2214yes—
51#89762C2214yes—
52#89762C2214yes—
53#89761C2214yes—
54#89756C2214yescomplemented
55#89753C2214yescomplemented
56#89752C2214yescomplemented
57#89751C2214yescomplemented
58#89727C2212yes—
59#89727C2212yes—
60#89715C2212yes—
61#89715C2212yes—
62#89715C2212yes—
63#89714C2212yes—
64#89712C2214yes—
65#89705C2214yescomplemented
66#89447C2214yes—
67#89446C2212yes—
68#89445C2212yes—
69#89441C2214yes—
70#89440C2214yes—
71#89439C2214yes—
72#89432C2214yescomplemented
73#89430C2214yescomplemented
74#89429C2214yescomplemented
75#89427C2214yescomplemented
76#89424C2214yescomplemented
77#36708—71nomaximal, Sylow
78C24×D7C2311yescharacteristic
79C22×C7×Q8—24no—
80C22×C7×Q8—44no—
81C22×C7×Q8C2312yes—
82C22×C7×Q8C2312yes—
83C22×C7×Q8C2311yescharacteristic
84C23×C4×C7C2311yescharacteristic
85C23×C4×C7C2311yescharacteristic
86C23×C4×C7C2311yescharacteristic
87C23×Dic7C2311yescharacteristic
88C23×Dic7C2311yescharacteristic
89C23×Dic7C2311yescharacteristic
90C22×C4×D7—24no—
91C22×C4×D7—44no—
92C22×C4×D7C2314yes—
93C22×Dic14C2312yes—
94C22×Dic14C2311yescharacteristic
95C7×C4⋊Q8—44no—
96C7×C22⋊Q8—24no—
97C2×C7×C4⋊C4C2312yes—
98C2×C7×C4⋊C4C2311yescharacteristic
99C2×C7×C4⋊C4C2314yes—
100C2×C7×C4⋊C4C2314yes—
101C2×C7×C22⋊C4C2312yes—
102C2×C7×C22⋊C4C2312yes—
103C2×C42×C7C2312yes—
104C2×C22⋊Dic7C2312yes—
105C2×C22⋊Dic7C2312yes—
106#6509—216no—
107#6509—216no—
108#6507—28no—
109#6507—44no—
110C2×(C2×D7)⋊C4—24no—
111C2×(C2×D7)⋊C4—24no—
112C2×(C2×D7)⋊C4C2314yes—
113C2×(C2×D7)⋊C4C2314yes—
114C2×(C2×D7)⋊C4C2314yes—
115C2×(C2×D7)⋊C4C2312yes—
116C2×(C2×D7)⋊C4C2312yes—
117C2×(C2×D7)⋊C4C2314yes—
118C2×(C2×D7)⋊C4C2314yes—
119C2×C4⋊Dic7—24no—
120C2×C4⋊Dic7C2312yes—
121C2×C4⋊Dic7C2311yescharacteristic
122C2×C4⋊Dic7C2314yes—
123#6487—24no—
124C2×Dic7⋊C4—24no—
125C2×Dic7⋊C4D418yes—
126C2×Dic7⋊C4C2312yes—
127C2×Dic7⋊C4C2312yes—
128C2×Dic7⋊C4C2312yes—
129C2×Dic7⋊C4C2312yes—
130C2×Dic7⋊C4C2312yes—
131C2×Dic7⋊C4C2314yes—
132C2×Dic7⋊C4C2314yes—
133C2×Dic7⋊C4C2314yes—
134C2×Dic7⋊C4C2314yes—
135C2×Dic7⋊C4C2312yes—
136C2×Dic7⋊C4C2312yes—
137C2×C4×Dic7C2312yes—
138C2×C4×Dic7C2312yes—
139C2×C4×Dic7C2312yes—
140#6459—48no—
141#6452—28no—
142#6452—28no—
143#6451—28no—
144#6441—28no—
145#6440—28no—
146C7×#95C2312yes—
147C7×#95C2314yes—
148C7×#95C2314yes—
149C7⋊#95C2314yes—
150C7⋊#95C2314yes—
151C7⋊#95C2314yes—
152C7⋊#95C2314yes—
153C7⋊#95C2314yes—
154C7⋊#95C2314yes—
155C7⋊#95C2314yes—
156C7⋊#95C2314yes—
157C7⋊#95C2312yes—
158C7⋊#95C2314yes—
159C2×#544—71no—
160C22×C22⋊Q8—71no—
161#2848—72no—
162#2690—74no—
163#2690—74no—
164#2677—74no—
165#2677—74no—
166#2627—72no—
167#2561—71no—
168#2561—71no—
169#2484—72no—
170#2482—72no—
171#2434—71no—
172#2434D711yescharacteristic, complemented
173#2313—71no—
174C24×C7C2411yescharacteristic
175C23×D7—22no—
176C23×D7—24no—
177C23×D7—24no—
178C23×D7—42no—
179C23×D7C2412yes—
180C2×C7×Q8—24no—
181C2×C7×Q8—24no—
182C2×C7×Q8—22no—
183C2×C7×Q8—24no—
184C2×C7×Q8—24no—
185C2×C7×Q8—42no—
186C2×C7×Q8—42no—
187C2×C7×Q8—42no—
188C2×C7×Q8—416no—
189C22×C4×C7—22no—
190C22×C4×C7—24no—
191C22×C4×C7—22no—
192C22×C4×C7—42no—
193C22×C4×C7—42no—
194C22×C4×C7C2×D414yes—
195C22×C4×C7C2414yes—
196C22×C4×C7C2412yes—
197C22×C4×C7C2412yes—
198C22×C4×C7C2412yes—
199C22×C4×C7C2411yescharacteristic
200C22×C4×C7C2411yescharacteristic
201C22×C4×C7C2412yes—
202C22×Dic7—24no—
203C22×Dic7—22no—
204C22×Dic7—24no—
205C22×Dic7—42no—
206C22×Dic7C2×D414yes—
207C22×Dic7C2×D414yes—
208C22×Dic7C2414yes—
209C22×Dic7C2412yes—
210C22×Dic7C2412yes—
211C22×Dic7C2412yes—
212C22×Dic7C2412yes—
213C22×Dic7C2412yes—
214C2×C4×D7—28no—
215C2×C4×D7—28no—
216C2×C4×D7—44no—
217C2×C4×D7—416no—
218C2×Dic14—22no—
219C2×Dic14—24no—
220C2×Dic14—44no—
221C2×Dic14—42no—
222C7×C4⋊C4—24no—
223C7×C4⋊C4—22no—
224C7×C4⋊C4—28no—
225C7×C4⋊C4—44no—
226C7×C22⋊C4—24no—
227C7×C22⋊C4—24no—
228C42×C7—42no—
229C22⋊Dic7—24no—
230C22⋊Dic7—24no—
231(C2×D7)⋊C4—28no—
232(C2×D7)⋊C4—28no—
233(C2×D7)⋊C4—24no—
234(C2×D7)⋊C4—24no—
235(C2×D7)⋊C4—28no—
236(C2×D7)⋊C4—44no—
237(C2×D7)⋊C4—44no—
238C4⋊Dic7—22no—
239C4⋊Dic7—24no—
240C4⋊Dic7—28no—
241Dic7⋊C4—24no—
242Dic7⋊C4—28no—
243Dic7⋊C4—24no—
244Dic7⋊C4—24no—
245Dic7⋊C4—24no—
246Dic7⋊C4—24no—
247Dic7⋊C4—28no—
248Dic7⋊C4—44no—
249Dic7⋊C4—44no—
250Dic7⋊C4C2×D418yes—
251Dic7⋊C4C2×D418yes—
252C4×Dic7—24no—
253C4×Dic7—24no—
254C4×Dic7—42no—
255C23×Q8C2×D711yescharacteristic
256C24×C4—71no—
257#544—148no—
258C2×C4⋊Q8—72no—
259C2×C4⋊Q8—72no—
260C2×C4⋊Q8—72no—
261C2×C4⋊Q8C2×D712yescomplemented
262C2×#125—72no—
263C2×#125—72no—
264C2×C22⋊Q8—74no—
265C2×C22⋊Q8—74no—
266C2×C22⋊Q8—71no—
267C2×C22⋊Q8—74no—
268C2×C22⋊Q8—72no—
269C2×C22⋊Q8—148no—
270C2×C22⋊Q8C2×D711yescharacteristic
271C2×#117—72no—
272C22×C4⋊C4—71no—
273C22×C4⋊C4—71no—
274C22×C4⋊C4—71no—
275C22×C22⋊C4—71no—
276C22×C22⋊C4—71no—
277#400—74no—
278#400—74no—
279#399—74no—
280#398—74no—
281#398—74no—
282#398—74no—
283#397—72no—
284#397—72no—
285#397—74no—
286#395—74no—
287#395—74no—
288#395C2×D714yescomplemented
289#393—72no—
290#393—72no—
291#391—72no—
292#391C2×D712yescomplemented
293#388—74no—
294#388—74no—
295#388—74no—
296#388—74no—
297#387—74no—
298#387—74no—
299#387—74no—
300#387—74no—
301#385—72no—
302#385—72no—
303#385—74no—
304#385—74no—
305#385C2×D712yes—
306#385C2×D712yes—
307#381—72no—
308#381—72no—
309#381—72no—
310#380—72no—
311#380—74no—
312#379—72no—
313#379—72no—
314C4×C22⋊C4—71no—
315C4×C22⋊C4C2×D711yescharacteristic
316C23×C7—22no—
317C23×C7—22no—
318C23×C7C22×D412yes—
319C23×C72−1+414yes—
320C23×C7C2511yescharacteristic
321C22×D7—28no—
322C22×D7—22no—
323C22×D7—24no—
324C22×D7—24no—
325C22×D7—22no—
326C22×D7—416no—
327C22×D7—44no—
328C22×D7—44no—
329C22×D72−1+414yes—
330C7×Q8—48no—
331C7×Q8—48no—
332C7×Q8—82no—
333C2×C4×C7—28no—
334C2×C4×C7—22no—
335C2×C4×C7—22no—
336C2×C4×C7—22no—
337C2×C4×C7—22no—
338C2×C4×C7—22no—
339C2×C4×C7—22no—
340C2×C4×C7—22no—
341C2×C4×C7—22no—
342C2×C4×C7—24no—
343C2×C4×C7—24no—
344C2×C4×C7—22no—
345C2×C4×C7—24no—
346C2×C4×C7—24no—
347C2×C4×C7—24no—
348C2×C4×C7—44no—
349C2×C4×C7—44no—
350C2×C4×C7C22×D412yes—
351C2×C4×C7C22×D412yes—
352C2×C4×C7H(5,𝔽2)14yes—
353C2×Dic7—28no—
354C2×Dic7—24no—
355C2×Dic7—24no—
356C2×Dic7—22no—
357C2×Dic7—22no—
358C2×Dic7—22no—
359C2×Dic7—24no—
360C2×Dic7—24no—
361C2×Dic7—24no—
362C2×Dic7—24no—
363C2×Dic7—24no—
364C2×Dic7—24no—
365C2×Dic7—44no—
366C2×Dic7—44no—
367C2×Dic7C22×D414yes—
368C2×Dic7C22×D414yes—
369C4×D7—416no—
370Dic14—84no—
371Dic14—82no—
372C25—71no—
373C22×Q8—24no—
374C22×Q8—44no—
375C22×Q8—72no—
376C22×Q8—71no—
377C22×Q8C22×D712yes—
378C22×Q8C22×D712yes—
379C22×Q8C22×D711yescharacteristic
380C23×C4—71no—
381C23×C4—71no—
382C23×C4—71no—
383C23×C4—74no—
384C23×C4—144no—
385C23×C4—284no—
386C23×C4C22×D711yescharacteristic
387C23×C4C22×D711yescharacteristic
388C23×C4C22×D711yescharacteristic
389C4⋊Q8—44no—
390C4⋊Q8—148no—
391C4⋊Q8—148no—
392C4⋊Q8—284no—
393#125—148no—
394#125—148no—
395C22⋊Q8—24no—
396C22⋊Q8—148no—
397C22⋊Q8—1416no—
398C22⋊Q8—1416no—
399C22⋊Q8—144no—
400C22⋊Q8—288no—
401#117—148no—
402C2×C4⋊C4—78no—
403C2×C4⋊C4—72no—
404C2×C4⋊C4—72no—
405C2×C4⋊C4—74no—
406C2×C4⋊C4—74no—
407C2×C4⋊C4—72no—
408C2×C4⋊C4—72no—
409C2×C4⋊C4—72no—
410C2×C4⋊C4—71no—
411C2×C4⋊C4—72no—
412C2×C4⋊C4—72no—
413C2×C4⋊C4—72no—
414C2×C4⋊C4—74no—
415C2×C4⋊C4—74no—
416C2×C4⋊C4—74no—
417C2×C4⋊C4—144no—
418C2×C4⋊C4—144no—
419C2×C4⋊C4C22×D714yes—
420C2×C4⋊C4C22×D711yescharacteristic
421C2×C4⋊C4C22×D712yes—
422C2×C4⋊C4C22×D714yes—
423C2×C22⋊C4—72no—
424C2×C22⋊C4—72no—
425C2×C22⋊C4—72no—
426C2×C22⋊C4—72no—
427C2×C22⋊C4—74no—
428C2×C22⋊C4—74no—
429C2×C22⋊C4—74no—
430C2×C22⋊C4—74no—
431C2×C22⋊C4—74no—
432C2×C22⋊C4—144no—
433C2×C22⋊C4—144no—
434C2×C22⋊C4C22×D712yes—
435C2×C22⋊C4C22×D712yes—
436C2×C42—72no—
437C2×C42—72no—
438C2×C42—72no—
439C2×C42C22×D712yes—
440#95—72no—
441#95—74no—
442#95—74no—
443#95—74no—
444#95—74no—
445#95—74no—
446#95—74no—
447#95—74no—
448#95—74no—
449#95—74no—
450#95C22×D712yes—
451#95C22×D714yes—
452#95C22×D714yes—
453C22×C7—22no—
454C22×C7—22no—
455C22×C7—28no—
456C22×C7#53614yes—
457C22×C7C23×D411yescharacteristic
458C22×C7C2×H(5,𝔽2)12yes—
459C22×C7C2×2−1+412yes—
460C22×C7C2×2−1+412yes—
461C2×D7—24no—
462C2×D7—28no—
463C2×D7—416no—
464C2×D7—42no—
465C2×D7—42no—
466C2×D7—44no—
467C4×C7—22no—
468C4×C7—24no—
469C4×C7—22no—
470C4×C7—42no—
471C4×C7—42no—
472C4×C7—42no—
473C4×C7—44no—
474Dic7—24no—
475Dic7—24no—
476Dic7—42no—
477Dic7—42no—
478Dic7—42no—
479Dic7—44no—
480C24—72no—
481C24—144no—
482C24—142no—
483C24—144no—
484C24—282no—
485C24C23×D711yescharacteristic
486C2×Q8—24no—
487C2×Q8—24no—
488C2×Q8—22no—
489C2×Q8—24no—
490C2×Q8—24no—
491C2×Q8—42no—
492C2×Q8—42no—
493C2×Q8—42no—
494C2×Q8—416no—
495C2×Q8—144no—
496C2×Q8—142no—
497C2×Q8—284no—
498C2×Q8—282no—
499C22×C4—22no—
500C22×C4—24no—
501C22×C4—22no—
502C22×C4—42no—
503C22×C4—42no—
504C22×C4—74no—
505C22×C4—72no—
506C22×C4—72no—
507C22×C4—74no—
508C22×C4—74no—
509C22×C4—72no—
510C22×C4—72no—
511C22×C4—72no—
512C22×C4—148no—
513C22×C4—144no—
514C22×C4—142no—
515C22×C4—144no—
516C22×C4—148no—
517C22×C4—284no—
518C22×C4—2816no—
519C22×C4—282no—
520C22×C4D4×D714yes—
521C22×C4C23×D714yes—
522C22×C4C23×D712yes—
523C22×C4C23×D712yes—
524C22×C4C23×D712yes—
525C22×C4C23×D711yescharacteristic
526C22×C4C23×D711yescharacteristic
527C22×C4C23×D712yes—
528C4⋊C4—22no—
529C4⋊C4—24no—
530C4⋊C4—28no—
531C4⋊C4—44no—
532C4⋊C4—78no—
533C4⋊C4—78no—
534C4⋊C4—148no—
535C4⋊C4—144no—
536C4⋊C4—144no—
537C4⋊C4—144no—
538C4⋊C4—144no—
539C4⋊C4—148no—
540C4⋊C4—144no—
541C4⋊C4—144no—
542C4⋊C4—142no—
543C4⋊C4—148no—
544C4⋊C4—284no—
545C4⋊C4—284no—
546C22⋊C4—24no—
547C22⋊C4—24no—
548C22⋊C4—148no—
549C22⋊C4—148no—
550C22⋊C4—148no—
551C22⋊C4—144no—
552C22⋊C4—144no—
553C22⋊C4—144no—
554C22⋊C4—144no—
555C22⋊C4—284no—
556C22⋊C4—284no—
557C42—42no—
558C42—144no—
559C42—144no—
560C42—282no—
561C2×C7—24no—
562C2×C7C2×#53611yescharacteristic
563C2×C7#348411yescharacteristic
564C2×C7#348611yescharacteristic
565C2×C7#353814yes—
566D7—44no—
567C23—22no—
568C23—22no—
569C23—74no—
570C23—148no—
571C23—142no—
572C23—144no—
573C23—144no—
574C23—142no—
575C23—2816no—
576C23—284no—
577C23—284no—
578C23C2×D4×D712yes—
579C23#655114yes—
580C23C24×D711yescharacteristic
581Q8—48no—
582Q8—48no—
583Q8—82no—
584Q8—562no—
585Q8—564no—
586C2×C4—28no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—74no—
604C2×C4—74no—
605C2×C4—148no—
606C2×C4—144no—
607C2×C4—144no—
608C2×C4—142no—
609C2×C4—142no—
610C2×C4—142no—
611C2×C4—144no—
612C2×C4—144no—
613C2×C4—144no—
614C2×C4—144no—
615C2×C4—144no—
616C2×C4—144no—
617C2×C4—2816no—
618C2×C4—284no—
619C2×C4—284no—
620C2×C4C2×D4×D712yes—
621C2×C4C2×D4×D712yes—
622C2×C4#654814yes—
623C7#3670811yescharacteristic, complemented, Sylow
624C22—22no—
625C22—28no—
626C22—22no—
627C22—148no—
628C22—144no—
629C22—2816no—
630C22—282no—
631C22—282no—
632C22—284no—
633C22#9041514yes—
634C22C22×D4×D711yescharacteristic
635C22C2×#654812yes—
636C22C2×#655112yes—
637C22D7×2−1+412yes—
638C4—24no—
639C4—22no—
640C4—22no—
641C4—42no—
642C4—42no—
643C4—44no—
644C4—42no—
645C4—144no—
646C4—144no—
647C4—282no—
648C4—282no—
649C4—284no—
650C4—282no—
651C2—24no—
652C2—284no—
653C2#1174298711yescharacteristic
654C2C2×#9041511yescharacteristic
655C2#1174342311yescharacteristic
656C2#1174383014yes—
subgroupG/H
center, FrattiniC23C24×D7
commutator, socle, cosocleC23×C7C25
FittingC7×#2434C2
prime radicalC24×D7C23
hypercenter#2434D7
hypocenterC7#36708
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #2434

Complements

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

NH ≅ G/Ntype
2. C2×#90314C2semidirect
3. C7×#2434C2semidirect
4. #11741510C2semidirect
5. #11740711C2semidirect
7. #11734706C2semidirect
8. #11734659C2semidirect
9. #11734008C2semidirect
12. #11731947C2semidirect
13. #11731901C2semidirect
15. #11730890C2semidirect
18. C2×C7×C4⋊Q8C22semidirect
23. C2×#6507C22semidirect
24. C2×#6507C22semidirect
33. C2×#6452C22semidirect
34. C2×#6452C22semidirect
35. C2×#6451C22semidirect
38. C7×#395C22semidirect
39. C7×#391C22semidirect
44. #90002C22semidirect
45. #90001C22semidirect
54. #89756C22semidirect
55. #89753C22semidirect
56. #89752C22semidirect
57. #89751C22semidirect
65. #89705C22semidirect
72. #89432C22semidirect
73. #89430C22semidirect
74. #89429C22semidirect
75. #89427C22semidirect
76. #89424C22semidirect
172. #2434D7semidirect
261. C2×C4⋊Q8C2×D7semidirect
288. #395C2×D7semidirect
292. #391C2×D7semidirect
623. C7#36708semidirect

Maximal subgroups

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

subgroupindex
C22×#65092
C2×#903142
C7×#24342
#117415102
#117407112
#117402642
#117347062
#117346592
#117340082
#117321002
#117320592
#117319472
#117319012
#117310502
#117308902
#367087