Finite Group Databasebeta

← #14219422

Subgroups — #14219422

Order 1792.

1760 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11742870C211yesmaximal, characteristic
2C7×#2982C211yesmaximal, characteristic, complemented
3#11739882C211yesmaximal, characteristic, complemented
4#11737354C213yesmaximal, complemented
5#11737353C213yesmaximal, complemented
6#11737352C211yesmaximal, characteristic, complemented
7#11737351C211yesmaximal, characteristic, complemented
8#11737350C211yesmaximal, characteristic, complemented
9#11737349C213yesmaximal, complemented
10#11737348C213yesmaximal, complemented
11C8×#1161C211yesmaximal, characteristic, complemented
12#11736735C213yesmaximal, complemented
13#11736734C213yesmaximal, complemented
14#11736730C213yesmaximal, complemented
15#11736728C213yesmaximal, complemented
16C2×C4×C7×Q8C2211yescharacteristic
17#90508C411yescharacteristic
18#90508C411yescharacteristic
19#90276C413yes—
20#90275C413yes—
21#90271C2213yes—
22#90271C2213yes—
23#90268C411yescharacteristic
24C4×#1161C2211yescharacteristic
25C4×#1161C2211yescharacteristic
26#90261C411yescharacteristic
27#90205C413yes—
28#90205C413yes—
29#90168C2213yes—
30C4×#1157C2213yes—
31C7×C8×Q8C2211yescharacteristic
32C7×C8×Q8C2213yes—
33C7×C8×Q8C2213yes—
34C7×C8×Q8C2211yescharacteristic
35C7×#431C2213yes—
36C7×#431C2213yes—
37#89796C2213yes—
38#89796C2213yes—
39Q8×C7⋊C8C2211yescharacteristic
40Q8×C7⋊C8C2211yescharacteristic
41#89698C2213yes—
42#89697C2213yes—
43#89612C2213yes—
44#89612C2213yes—
45#89611C2213yes—
46#89611C2213yes—
47#89610C2213yes—
48#89610C2213yes—
49#89609C2213yes—
50#89608C2216yes—
51#89608C2216yes—
52#89607C2213yes—
53#89606C2213yes—
54#89606C2213yes—
55#89605C2213yes—
56D7×C4⋊C8C2213yes—
57#89499C2213yes—
58#89499C2213yes—
59#89499C2216yes—
60#89492C2211yescharacteristic
61#89492C2213yes—
62#89462C2216yes—
63#89461C2216yes—
64#89460C2213yes—
65#89460C2216yes—
66#89460C2213yes—
67C4×#1131C2213yes—
68C8×D28C2213yes—
69C8×D28C2213yes—
70C8×D28C2216yes—
71#89457C2213yes—
72#89457C2213yes—
73C4×C8×D7C2213yes—
74#60301—71nomaximal, Sylow
75C22×C7×Q8C2×C411yescharacteristic
76C2×#1161C2×C411yescharacteristic
77C2×#1161C2×C411yescharacteristic
78C2×#1157C2313yes—
79C4×C7×Q8C2×C411yescharacteristic
80C4×C7×Q8C2×C413yes—
81C4×C7×Q8C2311yescharacteristic
82C4×C7×Q8C2313yes—
83C2×C7×C4⋊C4C2×C413yes—
84C2×C42×C7C2313yes—
85#6510C2×C413yes—
86#6510C2×C413yes—
87#6509C2×C413yes—
88#6509C2×C413yes—
89Q8×Dic7C2×C411yescharacteristic
90Q8×Dic7C2×C411yescharacteristic
91#6493C2×C416yes—
92#6493C2×C413yes—
93#6493C2×C413yes—
94C4×C22⋊D7C2×C413yes—
95C4×C22⋊D7C2×C413yes—
96C4×C22⋊D7C2×C413yes—
97C4×C22⋊D7C2×C413yes—
98#6489C2311yescharacteristic
99C2×#1131C2313yes—
100C2×#1131C2316yes—
101C2×#1131C2313yes—
102C2×C8×D7C2316yes—
103C2×C8×D7C2313yes—
104C2×C8×D7C2313yes—
105#6461C2×C416yes—
106#6461C2×C416yes—
107#6458C2×C413yes—
108#6458C2×C413yes—
109#6457C2×C416yes—
110#6457C2×C416yes—
111#6456C2×C413yes—
112#6456C2×C413yes—
113#6455C2×C413yes—
114#6455C2×C413yes—
115#6453C2×C413yes—
116#6453C2×C413yes—
117#6438C2×C416yes—
118#6437C2×C413yes—
119C4×D28C2×C413yes—
120C4×D28C2×C413yes—
121C4×D28C2313yes—
122C4×D28C2313yes—
123C4×D28C2316yes—
124#6435C2313yes—
125#6435C2313yes—
126C42×D7C2313yes—
127C42×D7C2313yes—
128#6432C2×C413yes—
129C4×Dic14C2×C413yes—
130C4×Dic14C2×C413yes—
131C7×C4⋊C8C2313yes—
132C7×C4⋊C8C2313yes—
133C7×C4⋊C8C2313yes—
134C7×C4⋊C8C2313yes—
135C7×C22⋊C8C2311yescharacteristic
136C7×C22⋊C8C2313yes—
137C4×C7×C8C2313yes—
138C4×C7×C8C2313yes—
139C4×C7×C8C2316yes—
140C22⋊(C7⋊C8)C2311yescharacteristic
141C22⋊(C7⋊C8)C2313yes—
142(C2×D7)⋊C8C2316yes—
143(C2×D7)⋊C8C2313yes—
144(C2×D7)⋊C8C2316yes—
145(C2×D7)⋊C8C2313yes—
146(C2×D7)⋊C8C2313yes—
147(C2×D7)⋊C8C2313yes—
148#6389C2313yes—
149#6389C2311yescharacteristic
150Dic7⋊C8C2313yes—
151Dic7⋊C8C2313yes—
152Dic7⋊C8C2311yescharacteristic
153Dic7⋊C8C2311yescharacteristic
154C8×Dic7C2313yes—
155C8×Dic7C2311yescharacteristic
156C4⋊(C7⋊C8)C2313yes—
157C4⋊(C7⋊C8)C2313yes—
158C4⋊(C7⋊C8)C2313yes—
159C4⋊(C7⋊C8)C2313yes—
160#6377C2316yes—
161C4×C7⋊C8C2313yes—
162C4×C7⋊C8C2313yes—
163#3480—71no—
164#2994—73no—
165#2993—73no—
166#2991—71no—
167#2988—71no—
168#2984—71no—
169#2982D711yescharacteristic, complemented
170#2981—73no—
171#2979—73no—
172#2971—71no—
173#2969—73no—
174#2966—73no—
175C8×#41—71no—
176#2925—73no—
177#2922—73no—
178C2×C7×Q8—24no—
179C2×C7×Q8C22×C413yes—
180C2×C7×Q8C22×C411yescharacteristic
181C22×C4×C7C22×C413yes—
182C22×C4×C7C22×C413yes—
183C22×C4×C7C2411yescharacteristic
184C2×C22⋊D7C22×C413yes—
185C2×C22⋊D7C22×C413yes—
186#1161—24no—
187#1161—24no—
188#1157—43no—
189#1157—43no—
190C2×D28C22×C413yes—
191C2×D28C22×C413yes—
192C2×D28C22×C413yes—
193C2×D28C22×C416yes—
194C2×C4×D7C22×C413yes—
195C2×C4×D7C22×C413yes—
196C2×C4×D7C22×C413yes—
197C2×C4×D7C22×C413yes—
198C2×C4×D7C2413yes—
199C2×C4×D7C2413yes—
200C2×Dic14C22×C413yes—
201C2×C7×C8C2411yescharacteristic
202C2×C7×C8C2413yes—
203C2×C7×C8C2411yescharacteristic
204C2×C7×C8C2413yes—
205C7×C4⋊C4C22×C413yes—
206C7×C4⋊C4C22×C413yes—
207C7×C4⋊C4C22×C416yes—
208C42×C7C22×C413yes—
209C42×C7C22×C413yes—
210C42×C7C2413yes—
211C42×C7C2413yes—
212C22⋊Dic7C22×C411yescharacteristic
213C22⋊Dic7C22×C411yescharacteristic
214(C2×D7)⋊C4C22×C413yes—
215(C2×D7)⋊C4C22×C413yes—
216(C2×D7)⋊C4C22×C413yes—
217(C2×D7)⋊C4C22×C413yes—
218(C2×D7)⋊C4C22×C416yes—
219(C2×D7)⋊C4C22×C416yes—
220(C2×D7)⋊C4C22×C416yes—
221(C2×D7)⋊C4C22×C416yes—
222C4⋊Dic7C22×C411yescharacteristic
223C4⋊Dic7C22×C411yescharacteristic
224C4⋊Dic7C22×C416yes—
225C4⋊Dic7C22×C413yes—
226C4⋊Dic7C22×C413yes—
227Dic7⋊C4C22×C413yes—
228Dic7⋊C4C22×C413yes—
229Dic7⋊C4C22×C413yes—
230Dic7⋊C4C22×C413yes—
231C4×Dic7C22×C413yes—
232C4×Dic7C22×C413yes—
233C4×Dic7C22×C413yes—
234C4×Dic7C22×C413yes—
235C4×Dic7C2411yescharacteristic
236C4×Dic7C2411yescharacteristic
237C2×C7⋊C8C2411yescharacteristic
238C2×C7⋊C8C2413yes—
239C2×C7⋊C8C2413yes—
240C2×C7⋊C8C2411yescharacteristic
241#1131—43no—
242#1131—46no—
243#1131—43no—
244C8×D7—43no—
245C8×D7—46no—
246C8×D7—43no—
247#559—73no—
248#557—71no—
249#556—73no—
250#553—73no—
251#553—73no—
252#550—71no—
253#548—71no—
254#548—71no—
255#532—73no—
256#529—73no—
257#529—73no—
258C4×#41—71no—
259C4×#41—71no—
260C4×#41—73no—
261C2×C4×Q8C2×D711yescharacteristic
262#446—73no—
263#446—73no—
264C8×Q8—71no—
265C8×Q8—71no—
266C8×Q8C2×D711yescharacteristic
267C8×Q8C2×D713yes—
268C8×Q8C2×D713yes—
269C8×Q8C2×D711yescharacteristic
270#436—73no—
271#436—73no—
272#436—76no—
273#436—73no—
274#436—73no—
275#435—76no—
276#435—76no—
277C8×D4—73no—
278C8×D4—73no—
279C8×D4—73no—
280C8×D4—76no—
281C8×D4—73no—
282#433—73no—
283#433—73no—
284#433—71no—
285#433—73no—
286#433—76no—
287#432—73no—
288#432—73no—
289#431—73no—
290#431—73no—
291#431—73no—
292#431—73no—
293#431C2×D713yes—
294#431C2×D713yes—
295#424—73no—
296#424—73no—
297#423—73no—
298C2×C4⋊C8—73no—
299#408—73no—
300#408—73no—
301#408—76no—
302#405—76no—
303C4×M16—73no—
304C2×C4×C8—73no—
305C23×C7C23×C411yescharacteristic
306C22×D7C23×C413yes—
307C22×D7C23×C413yes—
308C7×Q8—26no—
309C7×Q8#13114yes—
310C2×C4×C7—26no—
311C2×C4×C7—21no—
312C2×C4×C7—21no—
313C2×C4×C7—43no—
314C2×C4×C7C23×C413yes—
315C2×C4×C7C23×C413yes—
316C2×C4×C7C23×C413yes—
317C2×C4×C7C23×C411yescharacteristic
318C2×C4×C7C23×C413yes—
319C2×C4×C7C2511yescharacteristic
320C22⋊D7—43no—
321C22⋊D7—43no—
322C2×Dic7C23×C411yescharacteristic
323C2×Dic7C23×C411yescharacteristic
324C2×Dic7C23×C411yescharacteristic
325C2×Dic7C23×C411yescharacteristic
326C2×Dic7C23×C413yes—
327C2×Dic7C23×C413yes—
328D28—26no—
329D28—26no—
330D28—46no—
331D28—43no—
332C4×D7—26no—
333C4×D7—26no—
334C4×D7—23no—
335C4×D7—23no—
336C4×D7—23no—
337C4×D7—23no—
338C4×D7—43no—
339C4×D7—43no—
340Dic14—43no—
341C7×C8—23no—
342C7×C8—21no—
343C7×C8—21no—
344C7×C8—23no—
345C7⋊C8—21no—
346C7⋊C8—23no—
347C7⋊C8—23no—
348C7⋊C8—21no—
349C2×#41—71no—
350C2×#41—71no—
351C2×#41—73no—
352C22×Q8C4×D711yescharacteristic
353C2×M16—73no—
354C2×M16—73no—
355C2×M16—76no—
356C22×C8—73no—
357C22×C8—73no—
358C22×C8—76no—
359C4⋊Q8—73no—
360#127—73no—
361#126—76no—
362#126—76no—
363#125—73no—
364#125—73no—
365#124—73no—
366#124—73no—
367#124—76no—
368#123—76no—
369#123—76no—
370C22⋊Q8—73no—
371C22⋊Q8—73no—
372#121—73no—
373#121—73no—
374#121—76no—
375#121—73no—
376#121—73no—
377C4×Q8—71no—
378C4×Q8—71no—
379C4×Q8—73no—
380C4×Q8—73no—
381C4×Q8C4×D711yescharacteristic
382C4×Q8C4×D713yes—
383C4×Q8C22×D711yescharacteristic
384C4×Q8C22×D713yes—
385C4×D4—73no—
386C4×D4—73no—
387C4×D4—73no—
388C4×D4—73no—
389C4×D4—73no—
390C4×D4—73no—
391C4×D4—73no—
392C4×D4—73no—
393C4×D4—73no—
394C4×D4—73no—
395C4×D4—76no—
396#117—71no—
397#117—73no—
398#117—73no—
399#117—73no—
400#117—73no—
401C2×C4⋊C4C4×D713yes—
402C2×C42—73no—
403C2×C42—73no—
404C2×C42C22×D713yes—
405C4⋊C8—73no—
406C4⋊C8—73no—
407C4⋊C8—71no—
408C4⋊C8—71no—
409C4⋊C8—73no—
410C4⋊C8—73no—
411C4⋊C8—73no—
412C4⋊C8—73no—
413C4⋊C8C22×D713yes—
414C4⋊C8C22×D713yes—
415C4⋊C8C22×D713yes—
416C4⋊C8C22×D713yes—
417C22⋊C8—71no—
418C22⋊C8—73no—
419C22⋊C8—73no—
420C22⋊C8—76no—
421C22⋊C8—73no—
422C22⋊C8—73no—
423C22⋊C8—73no—
424C22⋊C8—76no—
425C22⋊C8C22×D713yes—
426C22⋊C8C22×D711yescharacteristic
427#97—71no—
428#97—73no—
429#97—76no—
430C4×C8—71no—
431C4×C8—73no—
432C4×C8—73no—
433C4×C8—73no—
434C4×C8C22×D713yes—
435C4×C8C22×D713yes—
436C4×C8C22×D716yes—
437C22×C7—21no—
438C22×C7—21no—
439C22×C7—21no—
440C22×C7C24×C411yescharacteristic
441C2×D7—23no—
442C2×D7—23no—
443C2×D7—23no—
444C2×D7—23no—
445C2×D7—23no—
446C2×D7—23no—
447C4×C7—21no—
448C4×C7—23no—
449C4×C7—23no—
450C4×C7—23no—
451C4×C7C2×#13116yes—
452C4×C7#58511yescharacteristic
453C4×C7#58511yescharacteristic
454Dic7—23no—
455Dic7—23no—
456Dic7—21no—
457Dic7—21no—
458Dic7—21no—
459Dic7—21no—
460#41—144no—
461#41—144no—
462#41—283no—
463#41—283no—
464C2×Q8—24no—
465C2×Q8—73no—
466C2×Q8C2×C4×D713yes—
467C2×Q8C2×C4×D711yescharacteristic
468C2×D4—73no—
469C2×D4—73no—
470C2×D4—73no—
471C2×D4—73no—
472C2×D4—73no—
473C2×D4—76no—
474C22×C4—73no—
475C22×C4—73no—
476C22×C4—73no—
477C22×C4—73no—
478C22×C4—73no—
479C22×C4—73no—
480C22×C4C2×C4×D713yes—
481C22×C4C2×C4×D713yes—
482C22×C4C23×D711yescharacteristic
483M16—283no—
484M16—286no—
485M16—283no—
486C2×C8—71no—
487C2×C8—71no—
488C2×C8—73no—
489C2×C8—73no—
490C2×C8—283no—
491C2×C8—286no—
492C2×C8—283no—
493C2×C8C23×D711yescharacteristic
494C2×C8C23×D713yes—
495C2×C8C23×D711yescharacteristic
496C2×C8C23×D713yes—
497C4⋊C4—73no—
498C4⋊C4—73no—
499C4⋊C4—73no—
500C4⋊C4—73no—
501C4⋊C4—71no—
502C4⋊C4—71no—
503C4⋊C4—73no—
504C4⋊C4—73no—
505C4⋊C4—76no—
506C4⋊C4C2×C4×D713yes—
507C4⋊C4C2×C4×D713yes—
508C4⋊C4C2×C4×D716yes—
509C22⋊C4—71no—
510C22⋊C4—71no—
511C22⋊C4—73no—
512C22⋊C4—73no—
513C22⋊C4—76no—
514C22⋊C4—76no—
515C22⋊C4—73no—
516C22⋊C4—73no—
517C22⋊C4—76no—
518C22⋊C4—76no—
519C42—73no—
520C42—73no—
521C42—73no—
522C42—73no—
523C42—71no—
524C42—71no—
525C42C2×C4×D713yes—
526C42C2×C4×D713yes—
527C42C23×D713yes—
528C42C23×D713yes—
529C2×C7—41no—
530C2×C7#342811yescharacteristic
531C2×C7C22×#13111yescharacteristic
532C2×C7#357311yescharacteristic
533D7—43no—
534D7—43no—
535C23—73no—
536C23—73no—
537C23C22×C4×D711yescharacteristic
538Q8—26no—
539Q8—283no—
540Q8#646414yes—
541D4—146no—
542D4—146no—
543D4—286no—
544D4—283no—
545D4—283no—
546D4—283no—
547C2×C4—26no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—43no—
551C2×C4—71no—
552C2×C4—71no—
553C2×C4—71no—
554C2×C4—71no—
555C2×C4—73no—
556C2×C4—73no—
557C2×C4—146no—
558C2×C4—146no—
559C2×C4—143no—
560C2×C4—143no—
561C2×C4—143no—
562C2×C4—143no—
563C2×C4—283no—
564C2×C4—283no—
565C2×C4C22×C4×D713yes—
566C2×C4C22×C4×D711yescharacteristic
567C2×C4C22×C4×D713yes—
568C2×C4C22×C4×D713yes—
569C2×C4C22×C4×D713yes—
570C2×C4C24×D711yescharacteristic
571C8—23no—
572C8—21no—
573C8—21no—
574C8—23no—
575C8—141no—
576C8—143no—
577C8—143no—
578C8—141no—
579C7#6030111yescharacteristic, complemented, Sylow
580C22—21no—
581C22—21no—
582C22—21no—
583C22—143no—
584C22—143no—
585C22—143no—
586C22—143no—
587C22—143no—
588C22—143no—
589C22C23×C4×D711yescharacteristic
590C4—23no—
591C4—21no—
592C4—23no—
593C4—23no—
594C4—141no—
595C4—141no—
596C4—143no—
597C4—143no—
598C4—141no—
599C4—141no—
600C4C2×#646416yes—
601C4#9062011yescharacteristic
602C4#9062011yescharacteristic
603C2—41no—
604C2—283no—
605C2—283no—
606C2#1174294011yescharacteristic
607C2C22×#646411yescharacteristic
608C2#1174423411yescharacteristic
subgroupG/H
center, FrattiniC2×C4C24×D7
commutator, socleC22×C7C24×C4
FittingC7×#2982C2
prime radical#11742870C2
cosocleC2×C4×C7C25
hypercenter#2982D7
hypocenterC7#60301
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C2×C4 ◁ #2982

Complements

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

NH ≅ G/Ntype
2. C7×#2982C2semidirect
3. #11739882C2semidirect
4. #11737354C2semidirect
5. #11737353C2semidirect
6. #11737352C2semidirect
7. #11737351C2semidirect
8. #11737350C2semidirect
9. #11737349C2semidirect
10. #11737348C2semidirect
11. C8×#1161C2semidirect
12. #11736735C2semidirect
13. #11736734C2semidirect
14. #11736730C2semidirect
15. #11736728C2semidirect
169. #2982D7semidirect
579. C7#60301semidirect

Maximal subgroups

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

subgroupindex
#117428702
C7×#29822
#117398822
#117373542
#117373532
#117373522
#117373512
#117373502
#117373492
#117373482
C8×#11612
#117367352
#117367342
#117367302
#117367282
#603017