Finite Group Databasebeta

← #13560751

Subgroups — #13560751

Order 1792.

2406 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C7×C22⋊Q8C211yesmaximal, characteristic, complemented
2C2×#90185C211yesmaximal, characteristic
3#11741513C212yesmaximal, complemented
4#11741478C212yesmaximal, complemented
5#11740273C214yesmaximal, complemented
6#11740245C214yesmaximal, complemented
7#11740244C214yesmaximal, complemented
8#11740237C214yesmaximal, complemented
9#11739643C212yesmaximal, complemented
10#11739627C211yesmaximal, characteristic, complemented
11#11739626C211yesmaximal, characteristic, complemented
12#11739625C211yesmaximal, characteristic, complemented
13#11739623C211yesmaximal, characteristic, complemented
14#11739576C212yesmaximal, complemented
15#11739538C211yesmaximal, characteristic, complemented
16C23×C7×Q8C2211yescharacteristic
17C24×C4×C7C2211yescharacteristic
18C2×C7×C22⋊Q8—28no—
19C2×C7×C22⋊Q8C2214yes—
20C2×C7×C22⋊Q8C2214yes—
21C22×C7×C4⋊C4C2211yescharacteristic
22C22×C7×C4⋊C4C2211yescharacteristic
23C22×C7×C4⋊C4C2211yescharacteristic
24C22×C7×C22⋊C4C2211yescharacteristic
25C22×C7×C22⋊C4C2211yescharacteristic
26C2×#6516C2212yes—
27C2×#6509C2214yescomplemented
28C2×#6493C2211yescharacteristic
29C2×#6492C2214yes—
30C2×C4×C22⋊D7C2212yes—
31C2×#6487C2211yescharacteristic
32#90185—28no—
33C2×#6448C2212yes—
34C2×#6447C2212yes—
35C2×#6446C2212yes—
36C2×#6442C2212yes—
37#90019C2212yes—
38#90003C2214yescomplemented
39#90002C2214yescomplemented
40#90001C2214yescomplemented
41#89985C2212yes—
42#89985C2214yes—
43#89985C2212yes—
44#89985C2212yes—
45#89985C2212yes—
46#89984C2212yes—
47#89983C2214yes—
48#89982C2212yes—
49#89982C2214yes—
50#89982C2212yes—
51#89982C2212yes—
52#89982C2212yes—
53C4×C22⋊Dic7C2211yescharacteristic
54C4×C22⋊Dic7C2211yescharacteristic
55#89766C2214yescomplemented
56#89765C2214yescomplemented
57#89763C2212yescomplemented
58#89762C2214yescomplemented
59#89761C2214yescomplemented
60#89760C2212yescomplemented
61#89758C2212yescomplemented
62#89756C2214yescomplemented
63#89755C2212yescomplemented
64#89752C2214yescomplemented
65#89751C2212yescomplemented
66#89750C2214yescomplemented
67#89749C2212yescomplemented
68#89748C2212yescomplemented
69#89746C2212yescomplemented
70#89744C2214yescomplemented
71#89731C2214yes—
72#89731C2214yes—
73#89724C2214yes—
74#89724C2214yes—
75#89723C2214yes—
76#89723C2214yes—
77#36716—71nomaximal, Sylow
78C25×C7C2311yescharacteristic
79C22×C7×Q8—24no—
80C22×C7×Q8—44no—
81C22×C7×Q8C2314yes—
82C23×C4×C7—24no—
83C23×C4×C7—44no—
84C23×C4×C7C2311yescharacteristic
85C23×C4×C7C2314yes—
86C23×C4×C7C2311yescharacteristic
87C23×C4×C7C2311yescharacteristic
88C23×C4×C7C2311yescharacteristic
89C23×C4×C7C2311yescharacteristic
90C23×C4×C7C2311yescharacteristic
91C22×C22⋊D7C2312yes—
92C22×C22⋊D7C2312yes—
93C22×C22⋊D7C2312yes—
94C22×D28C2311yescharacteristic
95C22×C4×D7C2312yes—
96C22×Dic14C2311yescharacteristic
97C7×C22⋊Q8—216no—
98C7×C22⋊Q8—216no—
99C2×C7×C4⋊C4—24no—
100C2×C7×C4⋊C4—24no—
101C2×C7×C4⋊C4—24no—
102C2×C7×C4⋊C4C2312yes—
103C2×C7×C4⋊C4C2312yes—
104C2×C7×C4⋊C4C2312yes—
105C2×C7×C4⋊C4C2314yes—
106C2×C7×C4⋊C4C2312yes—
107C2×C7×C22⋊C4—24no—
108C2×C7×C22⋊C4D418yes—
109C2×C7×C22⋊C4C2314yes—
110C2×C7×C22⋊C4C2312yes—
111C2×C7×C22⋊C4C2312yes—
112#6516—28no—
113C2×C22⋊Dic7C2312yes—
114C2×C22⋊Dic7C2312yes—
115C2×C22⋊Dic7C2312yes—
116C2×C22⋊Dic7C2312yes—
117C2×C22⋊Dic7C2312yes—
118C2×C22⋊Dic7C2314yes—
119C2×C22⋊Dic7C2314yes—
120#6509—48no—
121#6493—24no—
122#6492—48no—
123C4×C22⋊D7—28no—
124C2×(C2×D7)⋊C4C2312yes—
125C2×(C2×D7)⋊C4C2312yes—
126C2×(C2×D7)⋊C4C2314yes—
127C2×(C2×D7)⋊C4C2314yes—
128C2×(C2×D7)⋊C4C2314yes—
129C2×(C2×D7)⋊C4C2314yes—
130C2×C4⋊Dic7C2311yescharacteristic
131C2×C4⋊Dic7C2311yescharacteristic
132C2×C4⋊Dic7C2314yes—
133#6487—24no—
134C2×Dic7⋊C4C2312yes—
135C2×Dic7⋊C4C2312yes—
136C2×Dic7⋊C4C2314yes—
137C2×Dic7⋊C4C2314yes—
138C2×Dic7⋊C4C2314yes—
139C2×Dic7⋊C4C2314yes—
140C2×C4×Dic7C2312yes—
141C2×C4×Dic7C2312yes—
142C2×C4×Dic7C2312yes—
143#6448—28no—
144#6447—28no—
145#6446—28no—
146#6442—28no—
147C7⋊#95C2312yes—
148C7⋊#95C2312yes—
149C7⋊#95C2314yes—
150C7⋊#95C2314yes—
151C7⋊#95C2314yes—
152C7⋊#95C2314yes—
153C7⋊#95C2314yes—
154C7⋊#95C2314yes—
155C7⋊#95C2314yes—
156C7⋊#95C2314yes—
157C7⋊#95C2314yes—
158C2×#538—71no—
159C22×C22⋊Q8D711yescharacteristic, complemented
160#2849—72no—
161#2700—74no—
162#2698—74no—
163#2681—74no—
164#2680—74no—
165#2622—72no—
166#2591—71no—
167#2583—71no—
168#2579—72no—
169#2504—71no—
170#2502—71no—
171#2409—72no—
172#2312—71no—
173C24×C7—24no—
174C24×C7—42no—
175C24×C7—42no—
176C24×C7C2×D414yes—
177C24×C7C2412yes—
178C24×C7C2411yescharacteristic
179C23×D7C2412yes—
180C2×C7×Q8—28no—
181C2×C7×Q8—28no—
182C2×C7×Q8—44no—
183C2×C7×Q8—416no—
184C22×C4×C7—22no—
185C22×C4×C7—24no—
186C22×C4×C7—22no—
187C22×C4×C7—22no—
188C22×C4×C7—22no—
189C22×C4×C7—28no—
190C22×C4×C7—28no—
191C22×C4×C7—44no—
192C22×C4×C7—42no—
193C22×C4×C7—42no—
194C22×C4×C7—42no—
195C22×C4×C7—42no—
196C22×C4×C7—42no—
197C22×C4×C7—416no—
198C22×C4×C7C2×D414yes—
199C22×C4×C7C2×D414yes—
200C22×C4×C7C2412yes—
201C22×C4×C7C2412yes—
202C22×C4×C7C2412yes—
203C22×C4×C7C2411yescharacteristic
204C22×C4×C7C2411yescharacteristic
205C22×C4×C7C2412yes—
206C22×C4×C7C2412yes—
207C2×C22⋊D7—24no—
208C2×C22⋊D7—28no—
209C2×C22⋊D7—24no—
210C2×C22⋊D7—24no—
211C2×C22⋊D7—44no—
212C2×C22⋊D7—44no—
213C22×Dic7C2412yes—
214C22×Dic7C2412yes—
215C22×Dic7C2414yes—
216C22×Dic7C2414yes—
217C22×Dic7C2412yes—
218C2×D28—22no—
219C2×D28—42no—
220C2×C4×D7—24no—
221C2×C4×D7—44no—
222C2×Dic14—22no—
223C2×Dic14—42no—
224C7×C4⋊C4—24no—
225C7×C4⋊C4—24no—
226C7×C4⋊C4—28no—
227C7×C4⋊C4—24no—
228C7×C4⋊C4—24no—
229C7×C22⋊C4—28no—
230C7×C22⋊C4C2×D418yes—
231C7×C22⋊C4C2×D418yes—
232C22⋊Dic7—24no—
233C22⋊Dic7—24no—
234C22⋊Dic7—24no—
235C22⋊Dic7—24no—
236C22⋊Dic7—24no—
237C22⋊Dic7—44no—
238C22⋊Dic7—44no—
239(C2×D7)⋊C4—24no—
240(C2×D7)⋊C4—24no—
241(C2×D7)⋊C4—28no—
242(C2×D7)⋊C4—44no—
243(C2×D7)⋊C4—44no—
244(C2×D7)⋊C4—44no—
245C4⋊Dic7—22no—
246C4⋊Dic7—22no—
247C4⋊Dic7—44no—
248Dic7⋊C4—24no—
249Dic7⋊C4—24no—
250Dic7⋊C4—28no—
251Dic7⋊C4—44no—
252Dic7⋊C4—44no—
253Dic7⋊C4—44no—
254C4×Dic7—24no—
255C4×Dic7—42no—
256C4×Dic7—42no—
257C23×Q8C2×D711yescharacteristic
258C24×C4C2×D711yescharacteristic
259#538—148no—
260C2×#126—72no—
261C2×#124—72no—
262C2×#123—74no—
263C2×#123—72no—
264C2×C22⋊Q8—28no—
265C2×C22⋊Q8—71no—
266C2×C22⋊Q8—74no—
267C2×C22⋊Q8C2×D714yes—
268C2×C22⋊Q8C2×D714yes—
269C2×#121—71no—
270C2×#121—72no—
271C2×#120—72no—
272C2×C4×D4—72no—
273C22×C4⋊C4C2×D711yescharacteristic
274C22×C4⋊C4C2×D711yescharacteristic
275C22×C4⋊C4C2×D711yescharacteristic
276C22×C22⋊C4C2×D711yescharacteristic
277C22×C22⋊C4C2×D711yescharacteristic
278#400—72no—
279#398—74no—
280#398—72no—
281#398—74no—
282#397—74no—
283#397—74no—
284#397—74no—
285#396—74no—
286#395—74no—
287#395—72no—
288#394—72no—
289#394—74no—
290#394—74no—
291#393—74no—
292#391—74no—
293#391—72no—
294#390—72no—
295#389—74no—
296#388—74no—
297#388—74no—
298#388—74no—
299#387—74no—
300#386—72no—
301#386—72no—
302#386—72no—
303#386—74no—
304#386—72no—
305#385—72no—
306#385—72no—
307#385—72no—
308#385—74no—
309#385—72no—
310#385—74no—
311#384—72no—
312#382—72no—
313#381—74no—
314#380—72no—
315#379—72no—
316C4×C22⋊C4—71no—
317C4×C22⋊C4—71no—
318C23×C7—22no—
319C23×C7—22no—
320C23×C7—28no—
321C23×C7—24no—
322C23×C7—24no—
323C23×C7—24no—
324C23×C7—44no—
325C23×C7—44no—
326C23×C7—416no—
327C23×C7C22×D414yes—
328C23×C7C22×D412yes—
329C23×C72−1+414yes—
330C23×C7C2511yescharacteristic
331C22×D7—22no—
332C22×D7—24no—
333C22×D7—24no—
334C22×D7—24no—
335C7×Q8—416no—
336C2×C4×C7—28no—
337C2×C4×C7—22no—
338C2×C4×C7—24no—
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—44no—
349C2×C4×C7—44no—
350C2×C4×C7—416no—
351C2×C4×C7—44no—
352C2×C4×C7—44no—
353C2×C4×C7—44no—
354C2×C4×C7C22×D414yes—
355C2×C4×C7C22×D412yes—
356C2×C4×C7C22×D412yes—
357C2×C4×C7H(5,𝔽2)14yes—
358C22⋊D7—48no—
359C22⋊D7—84no—
360C22⋊D7—84no—
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—24no—
371C2×Dic7—24no—
372C2×Dic7—24no—
373D28—82no—
374C4×D7—84no—
375Dic14—82no—
376C25C22×D711yescharacteristic
377C22×Q8—24no—
378C22×Q8—44no—
379C22×Q8—71no—
380C22×Q8C22×D714yes—
381C22×D4—72no—
382C22×D4—72no—
383C22×D4—72no—
384C22×D4—71no—
385C23×C4—24no—
386C23×C4—44no—
387C23×C4—72no—
388C23×C4C22×D711yescharacteristic
389C23×C4C22×D714yes—
390C23×C4C22×D711yescharacteristic
391C23×C4C22×D711yescharacteristic
392C23×C4C22×D711yescharacteristic
393C23×C4C22×D711yescharacteristic
394C23×C4C22×D711yescharacteristic
395#126—148no—
396#124—148no—
397#123—148no—
398#123—288no—
399C22⋊Q8—216no—
400C22⋊Q8—216no—
401C22⋊Q8—144no—
402C22⋊Q8—288no—
403#121—144no—
404#121—148no—
405#120—148no—
406C4×D4—148no—
407C2×C4⋊C4—24no—
408C2×C4⋊C4—24no—
409C2×C4⋊C4—24no—
410C2×C4⋊C4—72no—
411C2×C4⋊C4—72no—
412C2×C4⋊C4—71no—
413C2×C4⋊C4—71no—
414C2×C4⋊C4—74no—
415C2×C4⋊C4—74no—
416C2×C4⋊C4—74no—
417C2×C4⋊C4—74no—
418C2×C4⋊C4—74no—
419C2×C4⋊C4C22×D712yes—
420C2×C4⋊C4C22×D712yes—
421C2×C4⋊C4C22×D714yes—
422C2×C4⋊C4C22×D712yes—
423C2×C4⋊C4C22×D712yes—
424C2×C22⋊C4—24no—
425C2×C22⋊C4—72no—
426C2×C22⋊C4—72no—
427C2×C22⋊C4—72no—
428C2×C22⋊C4—72no—
429C2×C22⋊C4—72no—
430C2×C22⋊C4—72no—
431C2×C22⋊C4—72no—
432C2×C22⋊C4—74no—
433C2×C22⋊C4—74no—
434C2×C22⋊C4—74no—
435C2×C22⋊C4—74no—
436C2×C22⋊C4—74no—
437C2×C22⋊C4—74no—
438C2×C22⋊C4C22⋊D718yes—
439C2×C22⋊C4C22×D714yes—
440C2×C22⋊C4C22×D712yes—
441C2×C22⋊C4C22×D712yes—
442C2×C42—72no—
443C2×C42—72no—
444C2×C42—72no—
445#95—72no—
446#95—72no—
447#95—74no—
448#95—74no—
449#95—74no—
450#95—74no—
451#95—74no—
452#95—74no—
453#95—74no—
454#95—74no—
455#95—74no—
456C22×C7—22no—
457C22×C7—22no—
458C22×C7—28no—
459C22×C7—24no—
460C22×C7—24no—
461C22×C7—24no—
462C22×C7—416no—
463C22×C7—48no—
464C22×C7#53614yes—
465C22×C7C23×D411yescharacteristic
466C22×C7C2×H(5,𝔽2)12yes—
467C22×C7C2×2−1+412yes—
468C22×C7#58512yes—
469C2×D7—42no—
470C2×D7—42no—
471C2×D7—42no—
472C2×D7—48no—
473C4×C7—24no—
474C4×C7—24no—
475C4×C7—22no—
476C4×C7—22no—
477C4×C7—42no—
478C4×C7—42no—
479C4×C7—42no—
480Dic7—42no—
481Dic7—42no—
482Dic7—42no—
483Dic7—44no—
484Dic7—44no—
485C24—24no—
486C24—42no—
487C24—42no—
488C24—72no—
489C24C2×C22⋊D714yes—
490C24C23×D712yes—
491C24C23×D711yescharacteristic
492C2×Q8—28no—
493C2×Q8—28no—
494C2×Q8—44no—
495C2×Q8—416no—
496C2×Q8—142no—
497C2×Q8—282no—
498C2×D4—148no—
499C2×D4—144no—
500C2×D4—144no—
501C2×D4—144no—
502C2×D4—142no—
503C2×D4—282no—
504C2×D4—284no—
505C2×D4—284no—
506C22×C4—22no—
507C22×C4—24no—
508C22×C4—22no—
509C22×C4—22no—
510C22×C4—22no—
511C22×C4—28no—
512C22×C4—28no—
513C22×C4—44no—
514C22×C4—42no—
515C22×C4—42no—
516C22×C4—42no—
517C22×C4—42no—
518C22×C4—42no—
519C22×C4—416no—
520C22×C4—72no—
521C22×C4—74no—
522C22×C4—74no—
523C22×C4—72no—
524C22×C4—72no—
525C22×C4—144no—
526C22×C4—284no—
527C22×C4C2×C22⋊D714yes—
528C22×C4C2×C22⋊D714yes—
529C22×C4C23×D712yes—
530C22×C4C23×D712yes—
531C22×C4C23×D712yes—
532C22×C4C23×D712yes—
533C22×C4C23×D711yescharacteristic
534C22×C4C23×D711yescharacteristic
535C22×C4C23×D712yes—
536C4⋊C4—24no—
537C4⋊C4—24no—
538C4⋊C4—28no—
539C4⋊C4—24no—
540C4⋊C4—24no—
541C4⋊C4—148no—
542C4⋊C4—144no—
543C4⋊C4—144no—
544C4⋊C4—142no—
545C4⋊C4—142no—
546C4⋊C4—284no—
547C4⋊C4—284no—
548C4⋊C4—284no—
549C4⋊C4—284no—
550C22⋊C4—28no—
551C22⋊C4—144no—
552C22⋊C4—144no—
553C22⋊C4—144no—
554C22⋊C4—144no—
555C22⋊C4—144no—
556C22⋊C4—148no—
557C22⋊C4—144no—
558C22⋊C4—144no—
559C22⋊C4—284no—
560C22⋊C4—284no—
561C22⋊C4—284no—
562C22⋊C4—284no—
563C22⋊C4—284no—
564C22⋊C4C2×C22⋊D718yes—
565C22⋊C4C2×C22⋊D718yes—
566C42—144no—
567C42—282no—
568C42—282no—
569C2×C7—24no—
570C2×C7—44no—
571C2×C7C2×#53611yescharacteristic
572C2×C7#348411yescharacteristic
573C2×C7#348511yescharacteristic
574C2×C7#354114yes—
575D7—82no—
576C23—22no—
577C23—22no—
578C23—28no—
579C23—24no—
580C23—24no—
581C23—24no—
582C23—44no—
583C23—44no—
584C23—416no—
585C23—144no—
586C23—144no—
587C23—144no—
588C23—142no—
589C23#655114yes—
590C23C22×C22⋊D714yes—
591C23C22×C22⋊D712yes—
592C23C24×D711yescharacteristic
593Q8—416no—
594Q8—562no—
595D4—288no—
596D4—562no—
597D4—564no—
598D4—564no—
599C2×C4—28no—
600C2×C4—24no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—24no—
606C2×C4—24no—
607C2×C4—24no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—416no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—144no—
618C2×C4—144no—
619C2×C4—144no—
620C2×C4—144no—
621C2×C4—144no—
622C2×C4—144no—
623C2×C4—144no—
624C2×C4—144no—
625C2×C4—144no—
626C2×C4—142no—
627C2×C4—142no—
628C2×C4—142no—
629C2×C4—564no—
630C2×C4#654814yes—
631C2×C4C22×C22⋊D714yes—
632C2×C4C22×C22⋊D712yes—
633C2×C4C22×C22⋊D712yes—
634C7#3671611yescharacteristic, complemented, Sylow
635C22—22no—
636C22—28no—
637C22—22no—
638C22—24no—
639C22—24no—
640C22—24no—
641C22—416no—
642C22—48no—
643C22—282no—
644C22—282no—
645C22—288no—
646C22—282no—
647C22#9050714yes—
648C22C2×#654812yes—
649C22C2×#655112yes—
650C22#9061612yes—
651C22C23×C22⋊D711yescharacteristic
652C4—24no—
653C4—24no—
654C4—22no—
655C4—22no—
656C4—42no—
657C4—42no—
658C4—42no—
659C4—282no—
660C4—282no—
661C4—284no—
662C4—284no—
663C4—282no—
664C2—24no—
665C2—44no—
666C2—562no—
667C2#1174305914yes—
668C2#1174430711yescharacteristic
669C2C2×#9050711yescharacteristic
670C2#1174434111yescharacteristic
subgroupG/H
center, FrattiniC23C24×D7
commutator, socle, cosocleC23×C7C25
FittingC22×C7×C22⋊Q8C2
prime radicalC2×#90185C2
hypercenterC22×C22⋊Q8D7
hypocenterC7#36716
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ C22×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C22×C7×C22⋊Q8C2semidirect
3. #11741513C2semidirect
4. #11741478C2semidirect
5. #11740273C2semidirect
6. #11740245C2semidirect
7. #11740244C2semidirect
8. #11740237C2semidirect
9. #11739643C2semidirect
10. #11739627C2semidirect
11. #11739626C2semidirect
12. #11739625C2semidirect
13. #11739623C2semidirect
14. #11739576C2semidirect
15. #11739538C2semidirect
27. C2×#6509C22semidirect
38. #90003C22semidirect
39. #90002C22semidirect
40. #90001C22semidirect
55. #89766C22semidirect
56. #89765C22semidirect
57. #89763C22semidirect
58. #89762C22semidirect
59. #89761C22semidirect
60. #89760C22semidirect
61. #89758C22semidirect
62. #89756C22semidirect
63. #89755C22semidirect
64. #89752C22semidirect
65. #89751C22semidirect
66. #89750C22semidirect
67. #89749C22semidirect
68. #89748C22semidirect
69. #89746C22semidirect
70. #89744C22semidirect
159. C22×C22⋊Q8D7semidirect
634. C7#36716semidirect

Maximal subgroups

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

subgroupindex
C22×C7×C22⋊Q82
C2×#901852
#117415132
#117414782
#117402732
#117402452
#117402442
#117402372
#117396432
#117396272
#117396262
#117396252
#117396232
#117395762
#117395382
#367167