Finite Group Databasebeta

← #13560948

Subgroups — #13560948

Order 1792.

2214 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2656C211yesmaximal, characteristic, complemented
2#11740168C212yesmaximal, complemented
3#11740151C214yesmaximal, complemented
4#11740100C214yesmaximal, complemented
5#11740079C212yesmaximal, complemented
6#11735568C214yesmaximal, complemented
7#11735520C214yesmaximal, complemented
8#11734168C214yesmaximal, complemented
9#11734155C214yesmaximal, complemented
10C2×#89439C211yesmaximal, characteristic, complemented
11#11730805C211yesmaximal, characteristic, complemented
12C23×C4×D7C2211yescharacteristic
13C2×C7×#121C2212yes—
14C2×C7×#121C2214yes—
15C2×#6502C2214yes—
16C2×#6501C2212yes—
17C2×#6501C2212yes—
18C2×#6500C2214yes—
19C2×#6498C2214yescomplemented
20C2×D4×Dic7C2214yes—
21C22×(C2×D7)⋊C4C2212yes—
22C22×Dic7⋊C4C2212yes—
23C7×#399C2214yescomplemented
24C7×#392C2214yescomplemented
25C2×C7×#95C2211yescharacteristic
26#89995C2214yes—
27#89994C2214yescomplemented
28#89992C2214yes—
29#89992C2214yes—
30#89992C2214yescomplemented
31#89984C2212yes—
32#89982C2214yes—
33C2×C7⋊#95C2211yescharacteristic
34#89766C2214yes—
35#89762C2214yes—
36#89759C2212yes—
37#89759C2212yes—
38#89755C2214yescomplemented
39#89753C2214yescomplemented
40#89750C2214yescomplemented
41#89733C2214yes—
42#89731C2214yes—
43#89730C2214yes—
44#89729C2214yes—
45#89728C2214yes—
46#89728C2214yes—
47#89724C2214yescomplemented
48#89723C2214yescomplemented
49#89722C2214yescomplemented
50#89721C2214yescomplemented
51#89718C2214yescomplemented
52#89716C2214yescomplemented
53#89439C2214yes—
54#89439C2214yes—
55#89432C2214yescomplemented
56#89429C2214yescomplemented
57#36727—71nomaximal, Sylow
58C24×D7C2311yescharacteristic
59C22×C7×D4C2314yes—
60C22×C7×D4C2312yes—
61C22×C7×D4C2314yes—
62C23×C4×C7C2312yes—
63C23×C4×C7C2311yescharacteristic
64C22×C22⋊D7C2314yes—
65C22×C22⋊D7C2314yes—
66C23×Dic7C2311yescharacteristic
67C23×Dic7C2312yes—
68C23×Dic7C2314yes—
69C22×C4×D7—24no—
70C22×C4×D7—44no—
71C22×C4×D7C2312yes—
72C22×C4×D7C2312yes—
73C7×#121—28no—
74C7×#121—48no—
75C2×C7×C4⋊C4C2312yes—
76C2×C7×C4⋊C4C2314yes—
77C2×C7×C22⋊C4C2314yes—
78C2×C7×C22⋊C4C2314yes—
79C2×C7×C22⋊C4C2314yes—
80C2×C22⋊Dic7C2314yes—
81C2×C22⋊Dic7C2314yes—
82C2×C22⋊Dic7C2314yes—
83C2×C22⋊Dic7C2314yes—
84C2×C22⋊Dic7C2314yes—
85C2×C22⋊Dic7C2314yes—
86#6502—48no—
87#6501—44no—
88#6501—44no—
89#6500—48no—
90#6498—48no—
91D4×Dic7—48no—
92C2×(C2×D7)⋊C4—44no—
93C2×(C2×D7)⋊C4C2314yes—
94C2×(C2×D7)⋊C4C2314yes—
95C2×(C2×D7)⋊C4C2314yes—
96C2×(C2×D7)⋊C4C2314yes—
97C2×C4⋊Dic7C2312yes—
98C2×C4⋊Dic7C2312yes—
99C2×C4⋊Dic7C2314yes—
100C2×Dic7⋊C4—44no—
101C2×Dic7⋊C4C2314yes—
102C2×Dic7⋊C4C2314yes—
103C2×Dic7⋊C4C2314yes—
104C2×Dic7⋊C4C2314yes—
105C2×C4×Dic7C2314yes—
106C7×#95C2314yes—
107C7⋊#95C2314yes—
108C7⋊#95C2314yes—
109C7⋊#95C2314yes—
110C7⋊#95C2314yes—
111C7⋊#95C2314yes—
112C7⋊#95C2314yes—
113C7⋊#95C2314yes—
114C7⋊#95C2314yes—
115#2847—72no—
116#2748—74no—
117#2741—74no—
118#2703—74no—
119#2671—74no—
120#2656D711yescharacteristic, complemented
121#2646—71no—
122#2443—74no—
123#2424—74no—
124#2325—72no—
125C2×#380—71no—
126C24×C7C2414yes—
127C24×C7C2411yescharacteristic
128C23×D7—22no—
129C23×D7—22no—
130C23×D7—22no—
131C23×D7—24no—
132C23×D7—42no—
133C23×D7C2411yescharacteristic
134C23×D7C2411yescharacteristic
135C2×C7×D4—28no—
136C2×C7×D4—28no—
137C2×C7×D4—28no—
138C2×C7×D4—24no—
139C2×C7×D4—44no—
140C2×C7×D4—42no—
141C2×C7×D4—42no—
142C2×C7×D4—44no—
143C2×C7×D4—44no—
144C22×C4×C7—24no—
145C22×C4×C7—42no—
146C22×C4×C7—42no—
147C22×C4×C7—42no—
148C22×C4×C7#4114yes—
149C22×C4×C7C2414yes—
150C22×C4×C7C2414yes—
151C22×C4×C7C2412yes—
152C2×C22⋊D7—28no—
153C2×C22⋊D7—28no—
154C2×C22⋊D7—44no—
155C2×C22⋊D7—44no—
156C2×C22⋊D7—44no—
157C2×C22⋊D7—44no—
158C22×Dic7—22no—
159C22×Dic7—24no—
160C22×Dic7—24no—
161C22×Dic7—42no—
162C22×Dic7—42no—
163C22×Dic7—44no—
164C22×Dic7—44no—
165C22×Dic7—44no—
166C22×Dic7C2414yes—
167C22×Dic7C2414yes—
168C22×Dic7C2414yes—
169C22×Dic7C2412yes—
170C2×C4×D7—24no—
171C2×C4×D7—24no—
172C2×C4×D7—24no—
173C2×C4×D7—24no—
174C2×C4×D7—42no—
175C2×C4×D7—42no—
176C2×C4×D7—416no—
177C7×C4⋊C4—24no—
178C7×C4⋊C4—44no—
179C7×C22⋊C4—28no—
180C7×C22⋊C4—44no—
181C7×C22⋊C4—44no—
182C22⋊Dic7—28no—
183C22⋊Dic7—28no—
184C22⋊Dic7—28no—
185C22⋊Dic7—44no—
186C22⋊Dic7—44no—
187C22⋊Dic7—44no—
188(C2×D7)⋊C4—28no—
189(C2×D7)⋊C4—44no—
190(C2×D7)⋊C4—44no—
191(C2×D7)⋊C4—44no—
192C4⋊Dic7—44no—
193C4⋊Dic7—42no—
194C4⋊Dic7—42no—
195Dic7⋊C4—28no—
196Dic7⋊C4—44no—
197Dic7⋊C4—44no—
198Dic7⋊C4—44no—
199C4×Dic7—44no—
200C24×C4—71no—
201C2×#123—74no—
202C2×#121—72no—
203C2×#121—72no—
204C2×#121—74no—
205C2×#121C2×D712yes—
206C2×#121C2×D714yes—
207C2×#120—74no—
208C2×C4×D4—74no—
209C22×C4⋊C4—72no—
210C22×C22⋊C4—72no—
211#400—74no—
212#399—74no—
213#399C2×D714yescomplemented
214#398—74no—
215#398—74no—
216#397—74no—
217#396—74no—
218#394—74no—
219#394—74no—
220#394—74no—
221#392—74no—
222#392C2×D714yescomplemented
223#390—74no—
224#388—74no—
225#388—74no—
226#388—74no—
227#387—74no—
228#387—74no—
229#386—74no—
230#386—74no—
231#386—74no—
232#386—74no—
233#386—74no—
234#386—74no—
235#385—74no—
236#385—74no—
237#385—74no—
238#380—72no—
239#380—72no—
240#380—72no—
241#380—74no—
242#380—74no—
243C2×#95—71no—
244C2×#95C2×D711yescharacteristic
245C23×C7—22no—
246C23×C7—21no—
247C23×C7—21no—
248C23×C7—24no—
249C23×C7—24no—
250C23×C7—24no—
251C23×C7—24no—
252C23×C7—24no—
253C23×C7—24no—
254C23×C7—24no—
255C23×C7C2×#4112yes—
256C23×C7H(5,𝔽2)14yes—
257C23×C7C2511yescharacteristic
258C22×D7—28no—
259C22×D7—21no—
260C22×D7—21no—
261C22×D7—21no—
262C22×D7—22no—
263C22×D7—22no—
264C22×D7—22no—
265C22×D7—22no—
266C22×D7—21no—
267C22×D7—48no—
268C22×D7—48no—
269C22×D7—44no—
270C22×D7—44no—
271C22×D7H(5,𝔽2)12yes—
272C22×D72−1+412yes—
273C7×D4—416no—
274C7×D4—84no—
275C7×D4—88no—
276C2×C4×C7—22no—
277C2×C4×C7—22no—
278C2×C4×C7—24no—
279C2×C4×C7—24no—
280C2×C4×C7—24no—
281C2×C4×C7—24no—
282C2×C4×C7—24no—
283C2×C4×C7—24no—
284C2×C4×C7—44no—
285C2×C4×C7—84no—
286C2×C4×C7C2×#4114yes—
287C22⋊D7—88no—
288C22⋊D7—88no—
289C2×Dic7—28no—
290C2×Dic7—22no—
291C2×Dic7—22no—
292C2×Dic7—24no—
293C2×Dic7—24no—
294C2×Dic7—24no—
295C2×Dic7—24no—
296C2×Dic7—24no—
297C2×Dic7—24no—
298C2×Dic7—24no—
299C2×Dic7—24no—
300C2×Dic7—24no—
301C2×Dic7—84no—
302C2×Dic7—88no—
303C2×Dic7H(5,𝔽2)14yes—
304C4×D7—48no—
305C4×D7—48no—
306C25—71no—
307C22×D4—74no—
308C22×D4—74no—
309C22×D4C22×D714yes—
310C22×D4C22×D712yes—
311C22×D4C22×D714yes—
312C23×C4—71no—
313C23×C4—72no—
314C23×C4—74no—
315C23×C4—72no—
316C23×C4—72no—
317C23×C4—144no—
318C23×C4—284no—
319C23×C4C22×D712yes—
320C23×C4C22×D711yescharacteristic
321#123—288no—
322#121—28no—
323#121—48no—
324#121—284no—
325#121—284no—
326#121—288no—
327#120—288no—
328C4×D4—288no—
329C2×C4⋊C4—74no—
330C2×C4⋊C4—74no—
331C2×C4⋊C4—74no—
332C2×C4⋊C4—74no—
333C2×C4⋊C4—72no—
334C2×C4⋊C4—72no—
335C2×C4⋊C4—74no—
336C2×C4⋊C4—284no—
337C2×C4⋊C4C22×D712yes—
338C2×C4⋊C4C22×D714yes—
339C2×C22⋊C4—74no—
340C2×C22⋊C4—74no—
341C2×C22⋊C4—74no—
342C2×C22⋊C4—74no—
343C2×C22⋊C4—74no—
344C2×C22⋊C4—74no—
345C2×C22⋊C4—74no—
346C2×C22⋊C4—74no—
347C2×C22⋊C4—74no—
348C2×C22⋊C4—74no—
349C2×C22⋊C4—284no—
350C2×C22⋊C4C22×D714yes—
351C2×C22⋊C4C22×D714yes—
352C2×C22⋊C4C22×D714yes—
353C2×C42—74no—
354#95—74no—
355#95—74no—
356#95—74no—
357#95—74no—
358#95—74no—
359#95—74no—
360#95—74no—
361#95—74no—
362#95C22×D714yes—
363C22×C7—22no—
364C22×C7—22no—
365C22×C7—24no—
366C22×C7—24no—
367C22×C7—44no—
368C22×C7—44no—
369C22×C7—44no—
370C22×C7—44no—
371C22×C7—44no—
372C22×C7—44no—
373C22×C7—44no—
374C22×C7#54014yes—
375C22×C7C22×#4111yescharacteristic
376C22×C7C2×H(5,𝔽2)12yes—
377C22×C7C2×H(5,𝔽2)12yes—
378C22×C7C2×H(5,𝔽2)11yescharacteristic
379C22×C7C2×2−1+411yescharacteristic
380C2×D7—22no—
381C2×D7—22no—
382C2×D7—24no—
383C2×D7—24no—
384C2×D7—416no—
385C2×D7—41no—
386C2×D7—41no—
387C2×D7—41no—
388C2×D7—41no—
389C2×D7—42no—
390C2×D7—42no—
391C4×C7—24no—
392C4×C7—44no—
393C4×C7—44no—
394Dic7—24no—
395Dic7—44no—
396Dic7—44no—
397Dic7—44no—
398C24—71no—
399C24—71no—
400C24—142no—
401C24—142no—
402C24—142no—
403C24—144no—
404C24—282no—
405C24C23×D714yes—
406C24C23×D711yescharacteristic
407C2×D4—28no—
408C2×D4—28no—
409C2×D4—28no—
410C2×D4—24no—
411C2×D4—44no—
412C2×D4—42no—
413C2×D4—42no—
414C2×D4—44no—
415C2×D4—44no—
416C2×D4—148no—
417C2×D4—148no—
418C2×D4—284no—
419C2×D4—284no—
420C2×D4—284no—
421C2×D4—284no—
422C22×C4—24no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—72no—
427C22×C4—74no—
428C22×C4—74no—
429C22×C4—74no—
430C22×C4—144no—
431C22×C4—144no—
432C22×C4—144no—
433C22×C4—142no—
434C22×C4—144no—
435C22×C4—144no—
436C22×C4—144no—
437C22×C4—282no—
438C22×C4—282no—
439C22×C4—2816no—
440C22×C4—282no—
441C22×C4—282no—
442C22×C4—284no—
443C22×C4—284no—
444C22×C4—284no—
445C22×C4#115914yes—
446C22×C4C23×D714yes—
447C22×C4C23×D714yes—
448C22×C4C23×D712yes—
449C4⋊C4—24no—
450C4⋊C4—44no—
451C4⋊C4—148no—
452C4⋊C4—284no—
453C4⋊C4—284no—
454C4⋊C4—284no—
455C4⋊C4—284no—
456C4⋊C4—282no—
457C4⋊C4—282no—
458C22⋊C4—28no—
459C22⋊C4—44no—
460C22⋊C4—44no—
461C22⋊C4—148no—
462C22⋊C4—148no—
463C22⋊C4—148no—
464C22⋊C4—148no—
465C22⋊C4—284no—
466C22⋊C4—284no—
467C22⋊C4—284no—
468C22⋊C4—284no—
469C22⋊C4—284no—
470C22⋊C4—284no—
471C42—284no—
472C2×C7—24no—
473C2×C7—84no—
474C2×C7C2×#54011yescharacteristic
475C2×C7#349011yescharacteristic
476C2×C7#349211yescharacteristic
477C2×C7#353612yes—
478C2×C7#353712yes—
479D7—42no—
480D7—42no—
481C23—22no—
482C23—21no—
483C23—21no—
484C23—24no—
485C23—24no—
486C23—24no—
487C23—24no—
488C23—24no—
489C23—24no—
490C23—24no—
491C23—72no—
492C23—72no—
493C23—148no—
494C23—141no—
495C23—141no—
496C23—142no—
497C23—142no—
498C23—141no—
499C23—142no—
500C23—142no—
501C23—141no—
502C23—288no—
503C23—288no—
504C23—284no—
505C23—284no—
506C23C2×#115912yes—
507C23#654814yes—
508C23C24×D711yescharacteristic
509D4—416no—
510D4—84no—
511D4—88no—
512D4—568no—
513D4—568no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—44no—
523C2×C4—74no—
524C2×C4—84no—
525C2×C4—148no—
526C2×C4—142no—
527C2×C4—142no—
528C2×C4—144no—
529C2×C4—144no—
530C2×C4—144no—
531C2×C4—144no—
532C2×C4—144no—
533C2×C4—144no—
534C2×C4—144no—
535C2×C4—144no—
536C2×C4—144no—
537C2×C4—288no—
538C2×C4—288no—
539C2×C4—564no—
540C2×C4—568no—
541C2×C4C2×#115914yes—
542C7#3672711yescharacteristic, complemented, Sylow
543C22—24no—
544C22—22no—
545C22—24no—
546C22—22no—
547C22—44no—
548C22—44no—
549C22—44no—
550C22—44no—
551C22—44no—
552C22—44no—
553C22—44no—
554C22—144no—
555C22—144no—
556C22—142no—
557C22—142no—
558C22—2816no—
559C22—281no—
560C22—281no—
561C22—282no—
562C22—281no—
563C22—281no—
564C22—282no—
565C22#9041014yes—
566C22C22×#115911yescharacteristic
567C22C2×#654812yes—
568C22D7×H(5,𝔽2)11yescharacteristic
569C22D7×H(5,𝔽2)12yes—
570C22D7×2−1+411yescharacteristic
571C4—24no—
572C4—44no—
573C4—44no—
574C4—144no—
575C4—284no—
576C4—284no—
577C4—284no—
578C2—24no—
579C2—84no—
580C2—282no—
581C2—282no—
582C2C2×#9041011yescharacteristic
583C2#1174336912yes—
584C2#1174337711yescharacteristic
585C2#1174343812yes—
586C2#1174344911yescharacteristic
subgroupG/H
center, FrattiniC23C24×D7
commutator, socle, cosocleC23×C7C25
FittingC7×#2656C2
hypercenter#2656D7
hypocenterC7#36727
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #2656

Complements

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

NH ≅ G/Ntype
1. C7×#2656C2semidirect
2. #11740168C2semidirect
3. #11740151C2semidirect
4. #11740100C2semidirect
5. #11740079C2semidirect
6. #11735568C2semidirect
7. #11735520C2semidirect
8. #11734168C2semidirect
9. #11734155C2semidirect
10. C2×#89439C2semidirect
11. #11730805C2semidirect
19. C2×#6498C22semidirect
23. C7×#399C22semidirect
24. C7×#392C22semidirect
27. #89994C22semidirect
30. #89992C22semidirect
38. #89755C22semidirect
39. #89753C22semidirect
40. #89750C22semidirect
47. #89724C22semidirect
48. #89723C22semidirect
49. #89722C22semidirect
50. #89721C22semidirect
51. #89718C22semidirect
52. #89716C22semidirect
55. #89432C22semidirect
56. #89429C22semidirect
120. #2656D7semidirect
213. #399C2×D7semidirect
222. #392C2×D7semidirect
542. C7#36727semidirect

Maximal subgroups

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

subgroupindex
C7×#26562
#117401682
#117401512
#117401002
#117400792
#117355682
#117355202
#117341682
#117341552
C2×#894392
#117308052
#367277