Finite Group Databasebeta

← #13523767

Subgroups — #13523767

Order 1792.

1994 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
1C7×#2280C211yesmaximal, characteristic
2#11741472C211yesmaximal, characteristic
3C2×C4×C22⋊Dic7C211yesmaximal, characteristic
4#11739672C211yesmaximal, characteristic, complemented
5#11739671C211yesmaximal, characteristic, complemented
6#11739669C211yesmaximal, characteristic, complemented
7Dic7×#117C211yesmaximal, characteristic, complemented
8#11731523C214yesmaximal, complemented
9#11731370C212yesmaximal, complemented
10#11731368C212yesmaximal, complemented
11#11731366C212yesmaximal, complemented
12#11731364C212yesmaximal, complemented
13#11731275C214yesmaximal
14#11730927C214yesmaximal, complemented
15C4×#6442C214yesmaximal, complemented
16C24×C4×C7C2211yescharacteristic
17C2×C7×#117C2211yescharacteristic
18C2×C7×#117C2211yescharacteristic
19C22×C22⋊Dic7C2212yes—
20C2×#6489C2211yescharacteristic
21C2×#6489C2211yescharacteristic
22C22×C4×Dic7C2211yescharacteristic
23C22×C4×Dic7C2211yescharacteristic
24C2×#6442C2214yes—
25C7×#381C2212yes—
26C7×#380C2212yes—
27C7×#380C2212yes—
28C7×#379C2212yes—
29C4×C7×C22⋊C4C2214yes—
30#90019C2212yes—
31#89984C2212yes—
32#89984C2212yes—
33#89983C2212yes—
34C4×C22⋊Dic7C2212yes—
35C4×C22⋊Dic7C2212yes—
36C4×C22⋊Dic7C2212yes—
37C4×C22⋊Dic7C2212yes—
38C4×C22⋊Dic7C2214yes—
39C2×C7⋊#95C2212yes—
40#89757C2212yescomplemented
41#89757C2212yescomplemented
42#89749C2211yescharacteristic, complemented
43#89749C2211yescharacteristic, complemented
44#89748C2211yescharacteristic, complemented
45#89748C2211yescharacteristic, complemented
46C4⋊C4×Dic7C2212yescomplemented
47C4⋊C4×Dic7C2212yescomplemented
48#89745C2212yescomplemented
49#89745C2212yescomplemented
50#89725C2214yes—
51#89724C2214yes—
52#89724C2214yes—
53#89723C2214yes—
54#89722C2214yes—
55#89720C2214yes—
56#89718C2214yes—
57#89717C2214yes—
58#89716C2214yes—
59#89714C2214yes—
60C22⋊C4×Dic7C2214yes—
61C4×C4⋊Dic7C2214yescomplemented
62#89704C2212yescomplemented
63C4×Dic7⋊C4C2214yescomplemented
64C4×Dic7⋊C4C2212yescomplemented
65C4×Dic7⋊C4C2212yescomplemented
66C42×Dic7C2212yescomplemented
67C7⋊#401C2214yescomplemented
68#89432C2214yescomplemented
69#89431C2214yescomplemented
70#89430C2214yescomplemented
71#89427C2214yescomplemented
72#89425C2214yescomplemented
73#89424C2214yescomplemented
74#89423C2214yescomplemented
75#34814—71nomaximal, Sylow
76C25×C7C2311yescharacteristic
77C23×C4×C7—24no—
78C23×C4×C7—24no—
79C23×C4×C7—24no—
80C23×C4×C7C2311yescharacteristic
81C23×C4×C7C2312yes—
82C23×C4×C7C2312yes—
83C23×C4×C7C2311yescharacteristic
84C23×Dic7C2312yes—
85C23×Dic7C2312yes—
86C7×#117—24no—
87C7×#117—24no—
88C2×C7×C4⋊C4C2312yes—
89C2×C7×C4⋊C4C2311yescharacteristic
90C2×C7×C4⋊C4C2311yescharacteristic
91C2×C7×C22⋊C4C2314yes—
92C2×C7×C22⋊C4C2312yes—
93C2×C7×C22⋊C4C2312yes—
94C2×C7×C22⋊C4C2314yes—
95C2×C42×C7C2312yes—
96C2×C42×C7C2312yes—
97C2×C22⋊Dic7—28no—
98C2×C22⋊Dic7C2314yes—
99C2×C22⋊Dic7C2312yes—
100C2×C22⋊Dic7C2312yes—
101C2×C22⋊Dic7C2312yes—
102C2×C22⋊Dic7C2312yes—
103C2×C22⋊Dic7C2312yes—
104C2×C22⋊Dic7C2314yes—
105C2×C22⋊Dic7C2312yes—
106#6489—24no—
107#6489—24no—
108C2×C4⋊Dic7C2314yes—
109C2×C4⋊Dic7C2311yescharacteristic
110C2×C4⋊Dic7C2311yescharacteristic
111C2×C4⋊Dic7C2312yes—
112C2×C4⋊Dic7C2314yes—
113C2×Dic7⋊C4C2314yes—
114C2×Dic7⋊C4C2314yes—
115C2×Dic7⋊C4C2314yes—
116C2×Dic7⋊C4C2314yes—
117C2×C4×Dic7—24no—
118C2×C4×Dic7—24no—
119C2×C4×Dic7C2314yes—
120C2×C4×Dic7C2311yescharacteristic
121C2×C4×Dic7C2312yes—
122C2×C4×Dic7C2312yes—
123C2×C4×Dic7C2311yescharacteristic
124C2×C4×Dic7C2312yes—
125C2×C4×Dic7C2312yes—
126C2×C4×Dic7C2311yescharacteristic
127C2×C4×Dic7C2311yescharacteristic
128C2×C4×Dic7C2314yes—
129#6442—216no—
130C7×#95C2314yes—
131C7×#95C2314yes—
132C7⋊#95C2312yes—
133C7⋊#95C2314yes—
134C7⋊#95C2314yes—
135C7⋊#95C2314yes—
136C7⋊#95C2314yes—
137C7⋊#95C2312yes—
138C7⋊#95C2314yes—
139C7⋊#95C2314yes—
140C7⋊#95C2314yes—
141C7⋊#95C2312yes—
142C7⋊#95C2314yes—
143C7⋊#95C2312yes—
144C7⋊#95C2314yes—
145C7⋊#95C2314yes—
146#2497—72no—
147#2496—72no—
148#2495—72no—
149#2494—72no—
150#2493—74no—
151#2480—74no—
152#2401—74no—
153C4×#126—74no—
154#2296—71no—
155#2296—71no—
156#2285—71no—
157#2280—71no—
158#2280D711yescharacteristic
159C4×#117—71no—
160C2×C4×C22⋊C4—71no—
161C24×C7—24no—
162C24×C7—22no—
163C24×C7—22no—
164C24×C7—22no—
165C24×C7—42no—
166C24×C7C2412yes—
167C24×C7C2411yescharacteristic
168C22×C4×C7—22no—
169C22×C4×C7—22no—
170C22×C4×C7—22no—
171C22×C4×C7—24no—
172C22×C4×C7—24no—
173C22×C4×C7—24no—
174C22×C4×C7—24no—
175C22×C4×C7—24no—
176C22×C4×C7—24no—
177C22×C4×C7—416no—
178C22×C4×C7#4114yes—
179C22×C4×C7#4114yes—
180C22×C4×C7#4114yes—
181C22×C4×C7C2412yes—
182C22×C4×C7C2412yes—
183C22×C4×C7C2412yes—
184C22×C4×C7C2414yes—
185C22×C4×C7C2411yescharacteristic
186C22×C4×C7C2411yescharacteristic
187C22×Dic7—24no—
188C22×Dic7—24no—
189C22×Dic7—24no—
190C22×Dic7—24no—
191C22×Dic7—42no—
192C22×Dic7—42no—
193C22×Dic7C2412yes—
194C22×Dic7C2412yes—
195C22×Dic7C2414yes—
196C22×Dic7C2412yes—
197C22×Dic7C2412yes—
198C22×Dic7C2412yes—
199C22×Dic7C2412yes—
200C7×C4⋊C4—24no—
201C7×C4⋊C4—22no—
202C7×C4⋊C4—22no—
203C7×C22⋊C4—28no—
204C7×C22⋊C4—24no—
205C7×C22⋊C4—24no—
206C7×C22⋊C4—44no—
207C42×C7—24no—
208C42×C7—24no—
209C22⋊Dic7—28no—
210C22⋊Dic7—24no—
211C22⋊Dic7—24no—
212C22⋊Dic7—24no—
213C22⋊Dic7—24no—
214C22⋊Dic7—24no—
215C22⋊Dic7—44no—
216C22⋊Dic7#4118yes—
217C4⋊Dic7—24no—
218C4⋊Dic7—22no—
219C4⋊Dic7—22no—
220C4⋊Dic7—28no—
221C4⋊Dic7—44no—
222Dic7⋊C4—28no—
223Dic7⋊C4—28no—
224Dic7⋊C4—44no—
225Dic7⋊C4—44no—
226C4×Dic7—22no—
227C4×Dic7—24no—
228C4×Dic7—22no—
229C4×Dic7—22no—
230C4×Dic7—24no—
231C4×Dic7—22no—
232C4×Dic7—24no—
233C4×Dic7—24no—
234C4×Dic7—28no—
235C4×Dic7—44no—
236C24×C4C2×D711yescharacteristic
237C2×#126—74no—
238C2×#117—71no—
239C2×#117—71no—
240C2×#117C2×D711yescharacteristic
241C2×#117C2×D711yescharacteristic
242C22×C22⋊C4—72no—
243C22×C42—71no—
244C22×C42—71no—
245#401—74no—
246#400—74no—
247#400—74no—
248#400—74no—
249#397—74no—
250#397—74no—
251#397—74no—
252#396—74no—
253#389—72no—
254#389—72no—
255#389—72no—
256#389—72no—
257#388—74no—
258#388—74no—
259#388—74no—
260#387—74no—
261#387—74no—
262#387—74no—
263#385—74no—
264#385—74no—
265#384—71no—
266#384—71no—
267#383—74no—
268#382—71no—
269#382—71no—
270#381—72no—
271#381C2×D712yes—
272#380—72no—
273#380—72no—
274#380—74no—
275#380C2×D712yes—
276#380C2×D712yes—
277#379—72no—
278#379C2×D712yes—
279C4×C4⋊C4—72no—
280C4×C4⋊C4—72no—
281C4×C4⋊C4—72no—
282C4×C4⋊C4—74no—
283C4×C4⋊C4—74no—
284C4×C4⋊C4—72no—
285C4×C22⋊C4—72no—
286C4×C22⋊C4—74no—
287C4×C22⋊C4—72no—
288C4×C22⋊C4—72no—
289C4×C22⋊C4—72no—
290C4×C22⋊C4—74no—
291C4×C22⋊C4C2×D714yes—
292#376—72no—
293C2×#95—72no—
294C43—72no—
295C23×C7—22no—
296C23×C7—22no—
297C23×C7—28no—
298C23×C7—22no—
299C23×C7—22no—
300C23×C7—22no—
301C23×C7—22no—
302C23×C7—22no—
303C23×C7—22no—
304C23×C7—48no—
305C23×C7—48no—
306C23×C7—44no—
307C23×C7—44no—
308C23×C7C2×#4112yes—
309C23×C7C2×#4112yes—
310C23×C7C2×#4112yes—
311C23×C7C2×#4114yes—
312C23×C7C2511yescharacteristic
313C2×C4×C7—24no—
314C2×C4×C7—24no—
315C2×C4×C7—22no—
316C2×C4×C7—22no—
317C2×C4×C7—24no—
318C2×C4×C7—22no—
319C2×C4×C7—22no—
320C2×C4×C7—22no—
321C2×C4×C7—24no—
322C2×C4×C7—24no—
323C2×C4×C7—44no—
324C2×C4×C7—48no—
325C2×C4×C7—48no—
326C2×C4×C7#126116yescomplemented
327C2×C4×C7C2×#4112yes—
328C2×C4×C7C2×#4112yes—
329C2×C4×C7C2×#4112yes—
330C2×C4×C7C2×#4112yes—
331C2×C4×C7C2×#4112yes—
332C2×C4×C7C2×#4112yes—
333C2×Dic7—22no—
334C2×Dic7—22no—
335C2×Dic7—22no—
336C2×Dic7—22no—
337C2×Dic7—22no—
338C2×Dic7—22no—
339C2×Dic7—24no—
340C2×Dic7—22no—
341C2×Dic7—22no—
342C2×Dic7—24no—
343C2×Dic7—24no—
344C2×Dic7—22no—
345C2×Dic7—22no—
346C2×Dic7—24no—
347C2×Dic7—24no—
348C2×Dic7—22no—
349C2×Dic7—22no—
350C2×Dic7—48no—
351C2×Dic7—48no—
352C2×Dic7C2×#4114yes—
353C2×Dic7C2×#4114yes—
354C25C22×D711yescharacteristic
355C23×C4—24no—
356C23×C4—24no—
357C23×C4—24no—
358C23×C4—72no—
359C23×C4—72no—
360C23×C4C22×D711yescharacteristic
361C23×C4C22×D712yes—
362C23×C4C22×D712yes—
363C23×C4C22×D711yescharacteristic
364#126—1416no—
365#117—24no—
366#117—24no—
367#117—144no—
368#117—144no—
369C2×C4⋊C4—72no—
370C2×C4⋊C4—71no—
371C2×C4⋊C4—71no—
372C2×C4⋊C4—74no—
373C2×C4⋊C4—74no—
374C2×C4⋊C4—74no—
375C2×C4⋊C4—74no—
376C2×C4⋊C4—74no—
377C2×C4⋊C4—74no—
378C2×C4⋊C4C22×D711yescharacteristic
379C2×C4⋊C4C22×D711yescharacteristic
380C2×C4⋊C4C22×D712yes—
381C2×C22⋊C4—72no—
382C2×C22⋊C4—72no—
383C2×C22⋊C4—72no—
384C2×C22⋊C4—72no—
385C2×C22⋊C4—72no—
386C2×C22⋊C4—72no—
387C2×C22⋊C4—74no—
388C2×C22⋊C4—74no—
389C2×C22⋊C4—148no—
390C2×C22⋊C4C22×D712yes—
391C2×C22⋊C4C22×D712yes—
392C2×C22⋊C4C22×D714yes—
393C2×C22⋊C4C22×D714yes—
394C2×C42—74no—
395C2×C42—72no—
396C2×C42—71no—
397C2×C42—71no—
398C2×C42—72no—
399C2×C42—72no—
400C2×C42—71no—
401C2×C42—71no—
402C2×C42—72no—
403C2×C42—74no—
404C2×C42—144no—
405C2×C42—144no—
406C2×C42C22×D712yes—
407C2×C42C22×D712yes—
408#95—72no—
409#95—72no—
410#95—72no—
411#95—72no—
412#95—74no—
413#95—74no—
414#95—74no—
415#95—74no—
416#95—74no—
417#95—74no—
418#95—74no—
419#95—74no—
420#95—74no—
421#95—74no—
422#95C22×D714yes—
423#95C22×D714yes—
424C22×C7—22no—
425C22×C7—22no—
426C22×C7—24no—
427C22×C7—24no—
428C22×C7—24no—
429C22×C7—24no—
430C22×C7—24no—
431C22×C7—416no—
432C22×C7—42no—
433C22×C7—42no—
434C22×C7—42no—
435C22×C7—42no—
436C22×C7C2×#12614yes—
437C22×C7C22×#4111yescharacteristic
438C22×C7C22×#4111yescharacteristic
439C22×C7C22×#4111yescharacteristic
440C22×C7C22×#4112yes—
441C22×C7#58512yes—
442C4×C7—42no—
443C4×C7—42no—
444C4×C7—42no—
445C4×C7—44no—
446C4×C7C2×#12614yes—
447C4×C7C2×#12614yes—
448Dic7—24no—
449Dic7—24no—
450Dic7—42no—
451Dic7—42no—
452Dic7—44no—
453Dic7—42no—
454Dic7—42no—
455C24—22no—
456C24—24no—
457C24—22no—
458C24—22no—
459C24—42no—
460C24C23×D712yes—
461C24C23×D711yescharacteristic
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—24no—
468C22×C4—24no—
469C22×C4—24no—
470C22×C4—24no—
471C22×C4—416no—
472C22×C4—72no—
473C22×C4—72no—
474C22×C4—72no—
475C22×C4—72no—
476C22×C4—74no—
477C22×C4—72no—
478C22×C4—72no—
479C22×C4—144no—
480C22×C4—144no—
481C22×C4—144no—
482C22×C4—144no—
483C22×C4—282no—
484C22×C4—282no—
485C22×C4#115714yes—
486C22×C4#115914yes—
487C22×C4#115914yes—
488C22×C4C23×D712yes—
489C22×C4C23×D712yes—
490C22×C4C23×D712yes—
491C22×C4C23×D714yes—
492C22×C4C23×D711yescharacteristic
493C22×C4C23×D711yescharacteristic
494C4⋊C4—24no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—148no—
498C4⋊C4—148no—
499C4⋊C4—148no—
500C4⋊C4—144no—
501C4⋊C4—142no—
502C4⋊C4—142no—
503C4⋊C4—284no—
504C4⋊C4—284no—
505C4⋊C4—284no—
506C22⋊C4—24no—
507C22⋊C4—24no—
508C22⋊C4—28no—
509C22⋊C4—44no—
510C22⋊C4—78no—
511C22⋊C4—144no—
512C22⋊C4—144no—
513C22⋊C4—144no—
514C22⋊C4—144no—
515C22⋊C4—144no—
516C22⋊C4—148no—
517C22⋊C4—284no—
518C42—24no—
519C42—24no—
520C42—148no—
521C42—142no—
522C42—144no—
523C42—142no—
524C42—144no—
525C42—142no—
526C42—144no—
527C42—142no—
528C42—144no—
529C42—284no—
530C2×C7—24no—
531C2×C7—44no—
532C2×C7C22×#12611yescharacteristic
533C2×C7#344411yescharacteristic
534C2×C7#345511yescharacteristic
535C2×C7#347512yes—
536C2×C7#347512yes—
537C23—22no—
538C23—22no—
539C23—28no—
540C23—22no—
541C23—22no—
542C23—22no—
543C23—22no—
544C23—22no—
545C23—22no—
546C23—48no—
547C23—48no—
548C23—44no—
549C23—44no—
550C23C2×#115712yes—
551C23C2×#115912yes—
552C23C2×#115912yes—
553C23D7×#4114yes—
554C23C24×D711yescharacteristic
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—24no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—44no—
566C2×C4—48no—
567C2×C4—48no—
568C2×C4—74no—
569C2×C4—74no—
570C2×C4—142no—
571C2×C4—142no—
572C2×C4—142no—
573C2×C4—142no—
574C2×C4—142no—
575C2×C4—144no—
576C2×C4—142no—
577C2×C4—142no—
578C2×C4—142no—
579C2×C4—144no—
580C2×C4—142no—
581C2×C4—142no—
582C2×C4—144no—
583C2×C4—144no—
584C2×C4—144no—
585C2×C4—142no—
586C2×C4—142no—
587C2×C4—288no—
588C2×C4—288no—
589C2×C4#6442116yescomplemented
590C2×C4C2×#115712yes—
591C2×C4C2×#115712yes—
592C2×C4C2×#115912yes—
593C2×C4C2×#115912yes—
594C2×C4C2×#115912yes—
595C2×C4C2×#115912yes—
596C7#3481411yescharacteristic, complemented, Sylow
597C22—24no—
598C22—22no—
599C22—22no—
600C22—24no—
601C22—24no—
602C22—24no—
603C22—24no—
604C22—416no—
605C22—42no—
606C22—42no—
607C22—42no—
608C22—42no—
609C22C2×#644214yes—
610C22C22×#115711yescharacteristic
611C22C22×#115911yescharacteristic
612C22C22×#115911yescharacteristic
613C22C2×D7×#4112yes—
614C22#9061612yes—
615C4—42no—
616C4—42no—
617C4—42no—
618C4—44no—
619C4—144no—
620C4—144no—
621C4—282no—
622C4—284no—
623C4—282no—
624C4—282no—
625C4—282no—
626C4C2×#644214yes—
627C4C2×#644214yes—
628C2—24no—
629C2—44no—
630C2C22×#644211yescharacteristic
631C2#1174281212yes—
632C2#1174308611yescharacteristic
633C2#1174308711yescharacteristic
634C2#1174308912yes—
subgroupG/H
centerC22×C4C23×D7
commutator, socle, cosocleC23×C7C25
FrattiniC23C24×D7
FittingC7×#2280C2
prime radicalC25×C7C23
hypercenter#2280D7
hypocenterC7#34814
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C22×C4 ◁ #2280

Complements

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

NH ≅ G/Ntype
4. #11739672C2semidirect
5. #11739671C2semidirect
6. #11739669C2semidirect
7. Dic7×#117C2semidirect
8. #11731523C2semidirect
9. #11731370C2semidirect
10. #11731368C2semidirect
11. #11731366C2semidirect
12. #11731364C2semidirect
14. #11730927C2semidirect
15. C4×#6442C2semidirect
40. #89757C22semidirect
41. #89757C22semidirect
42. #89749C22semidirect
43. #89749C22semidirect
44. #89748C22semidirect
45. #89748C22semidirect
46. C4⋊C4×Dic7C22semidirect
47. C4⋊C4×Dic7C22semidirect
48. #89745C22semidirect
49. #89745C22semidirect
61. C4×C4⋊Dic7C22semidirect
62. #89704C22semidirect
63. C4×Dic7⋊C4C22semidirect
64. C4×Dic7⋊C4C22semidirect
65. C4×Dic7⋊C4C22semidirect
66. C42×Dic7C22semidirect
67. C7⋊#401C22semidirect
68. #89432C22semidirect
69. #89431C22semidirect
70. #89430C22semidirect
71. #89427C22semidirect
72. #89425C22semidirect
73. #89424C22semidirect
74. #89423C22semidirect
326. C2×C4×C7#126semidirect
589. C2×C4#6442semidirect
596. C7#34814semidirect

Maximal subgroups

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

subgroupindex
C7×#22802
#117414722
C2×C4×C22⋊Dic72
#117396722
#117396712
#117396692
Dic7×#1172
#117315232
#117313702
#117313682
#117313662
#117313642
#117312752
#117309272
C4×#64422
#348147