Finite Group Databasebeta

← #13560950

Subgroups — #13560950

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
1C2×C7×#380C211yesmaximal, characteristic, complemented
2#11741477C212yesmaximal, complemented
3#11739640C214yesmaximal, complemented
4#11739554C214yesmaximal, complemented
5#11731480C214yesmaximal, complemented
6#11731431C214yesmaximal, complemented
7#11731422C214yesmaximal, complemented
8#11731411C214yesmaximal, complemented
9#11730826C211yesmaximal, characteristic, complemented
10#11730805C211yesmaximal, characteristic, complemented
11#11730778C212yesmaximal, complemented
12C24×C4×C7C2211yescharacteristic
13C22×C7×C4⋊C4C2212yes—
14C22×C7×C22⋊C4C2212yes—
15C2×#6516C2214yes—
16C2×#6493C2212yes—
17C2×#6493C2212yes—
18C2×#6493C2212yes—
19C2×#6493C2214yes—
20C2×C4×C22⋊D7C2214yes—
21C2×#6457C2214yescomplemented
22C2×#6447C2214yes—
23C7×#380C2214yes—
24C7×#380C2214yes—
25C2×C7×#95C2211yescharacteristic
26C2×C7×#95C2211yescharacteristic
27#89985C2214yes—
28#89985C2214yes—
29#89985C2214yes—
30#89985C2214yes—
31#89984C2212yes—
32#89984C2212yes—
33#89984C2212yes—
34#89982C2214yes—
35#89765C2214yescomplemented
36#89764C2214yescomplemented
37#89763C2214yescomplemented
38#89762C2214yescomplemented
39#89761C2214yescomplemented
40#89756C2214yescomplemented
41#89753C2214yescomplemented
42#89732C2214yes—
43#89731C2214yes—
44#89731C2214yes—
45#89729C2214yes—
46#89728C2214yes—
47#89723C2214yes—
48#89720C2214yes—
49#89449C2214yescomplemented
50#89443C2214yescomplemented
51#89442C2214yescomplemented
52#89440C2214yescomplemented
53#89437C2214yescomplemented
54#89432C2214yescomplemented
55#89429C2214yescomplemented
56#89425C2214yescomplemented
57#36727—71nomaximal, Sylow
58C25×C7C2311yescharacteristic
59C23×C4×C7—24no—
60C23×C4×C7—44no—
61C23×C4×C7C2311yescharacteristic
62C23×C4×C7C2312yes—
63C23×C4×C7C2312yes—
64C23×C4×C7C2312yes—
65C23×C4×C7C2312yes—
66C23×C4×C7C2311yescharacteristic
67C22×C22⋊D7C2314yes—
68C22×C22⋊D7C2314yes—
69C22×C22⋊D7C2314yes—
70C22×D28C2312yes—
71C22×D28C2314yes—
72C22×C4×D7C2314yes—
73C2×C7×C4⋊C4—44no—
74C2×C7×C4⋊C4C2314yes—
75C2×C7×C4⋊C4C2314yes—
76C2×C7×C22⋊C4—44no—
77C2×C7×C22⋊C4C2314yes—
78C2×C7×C22⋊C4C2314yes—
79#6516—48no—
80C2×C22⋊Dic7C2314yes—
81C2×C22⋊Dic7C2314yes—
82C2×C22⋊Dic7C2314yes—
83#6493—28no—
84#6493—44no—
85#6493—44no—
86#6493—48no—
87C4×C22⋊D7—48no—
88C2×(C2×D7)⋊C4C2314yes—
89C2×(C2×D7)⋊C4C2314yes—
90C2×(C2×D7)⋊C4C2314yes—
91C2×(C2×D7)⋊C4C2314yes—
92C2×(C2×D7)⋊C4C2314yes—
93C2×(C2×D7)⋊C4C2314yes—
94C2×(C2×D7)⋊C4C2314yes—
95C2×(C2×D7)⋊C4C2314yes—
96C2×C4⋊Dic7C2312yes—
97C2×C4⋊Dic7C2312yes—
98C2×C4⋊Dic7C2312yes—
99C2×C4⋊Dic7C2314yes—
100C2×Dic7⋊C4C2314yes—
101C2×Dic7⋊C4C2314yes—
102C2×Dic7⋊C4C2314yes—
103C2×C4×Dic7C2314yes—
104#6457—48no—
105#6447—48no—
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#2656—71no—
121#2646—71no—
122#2443—74no—
123#2424—74no—
124#2325—72no—
125C2×#380D711yescharacteristic, complemented
126C24×C7—22no—
127C24×C7—22no—
128C24×C7—22no—
129C24×C7—24no—
130C24×C7—42no—
131C24×C7C2411yescharacteristic
132C24×C7C2411yescharacteristic
133C24×C7C2411yescharacteristic
134C23×D7C2414yes—
135C22×C4×C7—22no—
136C22×C4×C7—24no—
137C22×C4×C7—24no—
138C22×C4×C7—24no—
139C22×C4×C7—24no—
140C22×C4×C7—24no—
141C22×C4×C7—24no—
142C22×C4×C7—24no—
143C22×C4×C7—42no—
144C22×C4×C7—42no—
145C22×C4×C7—42no—
146C22×C4×C7—42no—
147C22×C4×C7—42no—
148C22×C4×C7—42no—
149C22×C4×C7—42no—
150C22×C4×C7—416no—
151C22×C4×C7#4114yes—
152C22×C4×C7C2414yes—
153C22×C4×C7C2414yes—
154C22×C4×C7C2412yes—
155C22×C4×C7C2412yes—
156C2×C22⋊D7—28no—
157C2×C22⋊D7—28no—
158C2×C22⋊D7—28no—
159C2×C22⋊D7—28no—
160C2×C22⋊D7—28no—
161C2×C22⋊D7—44no—
162C2×C22⋊D7—44no—
163C2×C22⋊D7—44no—
164C2×C22⋊D7—44no—
165C22×Dic7C2414yes—
166C22×Dic7C2414yes—
167C22×Dic7C2414yes—
168C2×D28—24no—
169C2×D28—44no—
170C2×D28—44no—
171C2×D28—42no—
172C2×D28—42no—
173C2×D28—44no—
174C2×C4×D7—44no—
175C2×C4×D7—44no—
176C2×C4×D7—44no—
177C7×C4⋊C4—28no—
178C7×C4⋊C4—44no—
179C7×C22⋊C4—28no—
180C7×C22⋊C4—44no—
181C22⋊Dic7—28no—
182C22⋊Dic7—44no—
183C22⋊Dic7—44no—
184(C2×D7)⋊C4—28no—
185(C2×D7)⋊C4—28no—
186(C2×D7)⋊C4—28no—
187(C2×D7)⋊C4—44no—
188(C2×D7)⋊C4—44no—
189(C2×D7)⋊C4—44no—
190(C2×D7)⋊C4—44no—
191(C2×D7)⋊C4—44no—
192C4⋊Dic7—24no—
193C4⋊Dic7—44no—
194C4⋊Dic7—42no—
195C4⋊Dic7—42no—
196Dic7⋊C4—44no—
197Dic7⋊C4—44no—
198Dic7⋊C4—44no—
199C4×Dic7—44no—
200C24×C4C2×D711yescharacteristic
201C2×#123—74no—
202C2×#121—72no—
203C2×#121—72no—
204C2×#121—72no—
205C2×#121—74no—
206C2×#121—74no—
207C2×#120—74no—
208C2×C4×D4—74no—
209C22×C4⋊C4C2×D712yes—
210C22×C22⋊C4C2×D712yes—
211#400—74no—
212#399—74no—
213#399—74no—
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#392—74no—
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#380C2×D714yes—
242#380C2×D714yes—
243C2×#95C2×D711yescharacteristic
244C2×#95C2×D711yescharacteristic
245C23×C7—22no—
246C23×C7—21no—
247C23×C7—21no—
248C23×C7—28no—
249C23×C7—21no—
250C23×C7—21no—
251C23×C7—22no—
252C23×C7—22no—
253C23×C7—21no—
254C23×C7—22no—
255C23×C7—22no—
256C23×C7—21no—
257C23×C7—48no—
258C23×C7—48no—
259C23×C7—44no—
260C23×C7—44no—
261C23×C7C2×#4112yes—
262C23×C7H(5,𝔽2)12yes—
263C23×C7H(5,𝔽2)14yes—
264C23×C72−1+412yes—
265C23×C7C2511yescharacteristic
266C22×D7—24no—
267C22×D7—24no—
268C22×D7—24no—
269C22×D7—24no—
270C22×D7—24no—
271C22×D7—24no—
272C22×D7—24no—
273C2×C4×C7—28no—
274C2×C4×C7—22no—
275C2×C4×C7—22no—
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—48no—
286C2×C4×C7—48no—
287C2×C4×C7—84no—
288C2×C4×C7—84no—
289C2×C4×C7C2×#4114yes—
290C2×C4×C7H(5,𝔽2)14yes—
291C22⋊D7—416no—
292C22⋊D7—88no—
293C22⋊D7—88no—
294C2×Dic7—24no—
295C2×Dic7—24no—
296C2×Dic7—24no—
297C2×Dic7—24no—
298C2×Dic7—24no—
299C2×Dic7—24no—
300C2×Dic7—24no—
301C2×Dic7—24no—
302C2×Dic7—24no—
303D28—84no—
304D28—88no—
305C4×D7—88no—
306C25C22×D711yescharacteristic
307C22×D4—74no—
308C22×D4—74no—
309C22×D4—74no—
310C22×D4—72no—
311C22×D4—74no—
312C23×C4—24no—
313C23×C4—44no—
314C23×C4—74no—
315C23×C4C22×D711yescharacteristic
316C23×C4C22×D712yes—
317C23×C4C22×D712yes—
318C23×C4C22×D712yes—
319C23×C4C22×D712yes—
320C23×C4C22×D711yescharacteristic
321#123—288no—
322#121—148no—
323#121—288no—
324#121—284no—
325#121—284no—
326#121—288no—
327#120—288no—
328C4×D4—288no—
329C2×C4⋊C4—44no—
330C2×C4⋊C4—72no—
331C2×C4⋊C4—74no—
332C2×C4⋊C4—74no—
333C2×C4⋊C4—72no—
334C2×C4⋊C4—72no—
335C2×C4⋊C4—74no—
336C2×C4⋊C4—74no—
337C2×C4⋊C4C22×D714yes—
338C2×C4⋊C4C22×D714yes—
339C2×C22⋊C4—44no—
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—74no—
350C2×C22⋊C4—74no—
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#95C22×D714yes—
362#95C22×D714yes—
363C22×C7—22no—
364C22×C7—22no—
365C22×C7—24no—
366C22×C7—24no—
367C22×C7—22no—
368C22×C7—22no—
369C22×C7—24no—
370C22×C7—24no—
371C22×C7—416no—
372C22×C7—41no—
373C22×C7—41no—
374C22×C7—41no—
375C22×C7—41no—
376C22×C7—42no—
377C22×C7—42no—
378C22×C7#54014yes—
379C22×C7C22×#4111yescharacteristic
380C22×C7C2×H(5,𝔽2)12yes—
381C22×C7C2×H(5,𝔽2)11yescharacteristic
382C22×C7C2×H(5,𝔽2)12yes—
383C22×C7C2×2−1+411yescharacteristic
384C2×D7—44no—
385C2×D7—44no—
386C2×D7—44no—
387C2×D7—44no—
388C2×D7—44no—
389C2×D7—44no—
390C2×D7—44no—
391C4×C7—24no—
392C4×C7—24no—
393C4×C7—44no—
394C4×C7—44no—
395Dic7—44no—
396Dic7—44no—
397Dic7—44no—
398C24—22no—
399C24—22no—
400C24—22no—
401C24—24no—
402C24—42no—
403C24—74no—
404C24C23×D711yescharacteristic
405C24C23×D711yescharacteristic
406C24C23×D711yescharacteristic
407C2×D4—148no—
408C2×D4—148no—
409C2×D4—148no—
410C2×D4—148no—
411C2×D4—148no—
412C2×D4—144no—
413C2×D4—282no—
414C2×D4—282no—
415C2×D4—284no—
416C2×D4—284no—
417C2×D4—284no—
418C2×D4—284no—
419C2×D4—284no—
420C2×D4—284no—
421C2×D4—284no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—416no—
438C22×C4—74no—
439C22×C4—74no—
440C22×C4—74no—
441C22×C4—284no—
442C22×C4—284no—
443C22×C4—284no—
444C22×C4#115714yes—
445C22×C4C23×D714yes—
446C22×C4C23×D714yes—
447C22×C4C23×D712yes—
448C22×C4C23×D712yes—
449C4⋊C4—28no—
450C4⋊C4—44no—
451C4⋊C4—144no—
452C4⋊C4—284no—
453C4⋊C4—284no—
454C4⋊C4—284no—
455C4⋊C4—282no—
456C4⋊C4—282no—
457C4⋊C4—284no—
458C22⋊C4—28no—
459C22⋊C4—44no—
460C22⋊C4—148no—
461C22⋊C4—148no—
462C22⋊C4—148no—
463C22⋊C4—148no—
464C22⋊C4—284no—
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—42no—
474C2×C7—42no—
475C2×C7C2×#54011yescharacteristic
476C2×C7#349011yescharacteristic
477C2×C7#349211yescharacteristic
478C2×C7#353612yes—
479C2×C7#353712yes—
480D7—84no—
481C23—22no—
482C23—21no—
483C23—21no—
484C23—28no—
485C23—21no—
486C23—21no—
487C23—21no—
488C23—22no—
489C23—22no—
490C23—21no—
491C23—22no—
492C23—22no—
493C23—48no—
494C23—48no—
495C23—44no—
496C23—44no—
497C23—144no—
498C23—144no—
499C23—144no—
500C23—144no—
501C23—144no—
502C23—144no—
503C23—144no—
504C23C2×#115712yes—
505C23#654812yes—
506C23#655112yes—
507C23#655314yes—
508C23C24×D711yescharacteristic
509D4—2816no—
510D4—564no—
511D4—568no—
512D4—568no—
513D4—568no—
514C2×C4—28no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—44no—
526C2×C4—48no—
527C2×C4—48no—
528C2×C4—84no—
529C2×C4—84no—
530C2×C4—144no—
531C2×C4—144no—
532C2×C4—144no—
533C2×C4—144no—
534C2×C4—144no—
535C2×C4—144no—
536C2×C4—144no—
537C2×C4—144no—
538C2×C4—144no—
539C2×C4—568no—
540C2×C4C2×#115714yes—
541C2×C4#654814yes—
542C7#3672711yescharacteristic, complemented, Sylow
543C22—24no—
544C22—22no—
545C22—22no—
546C22—24no—
547C22—24no—
548C22—22no—
549C22—22no—
550C22—24no—
551C22—416no—
552C22—42no—
553C22—41no—
554C22—41no—
555C22—42no—
556C22—41no—
557C22—41no—
558C22—284no—
559C22—284no—
560C22—284no—
561C22—284no—
562C22—284no—
563C22—284no—
564C22—284no—
565C22#9022614yes—
566C22C22×#115711yescharacteristic
567C22C2×#654811yescharacteristic
568C22C2×#654812yes—
569C22C2×#655111yescharacteristic
570C22C2×#655312yes—
571C4—24no—
572C4—24no—
573C4—44no—
574C4—44no—
575C4—284no—
576C4—284no—
577C4—284no—
578C2—24no—
579C2—42no—
580C2—42no—
581C2—564no—
582C2#1174276511yescharacteristic
583C2#1174278611yescharacteristic
584C2C2×#9022611yescharacteristic
585C2#1174300612yes—
586C2#1174305812yes—
subgroupG/H
center, FrattiniC23C24×D7
commutator, socle, cosocleC23×C7C25
FittingC2×C7×#380C2
hypercenterC2×#380D7
hypocenterC7#36727
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×#380

Complements

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

NH ≅ G/Ntype
1. C2×C7×#380C2semidirect
2. #11741477C2semidirect
3. #11739640C2semidirect
4. #11739554C2semidirect
5. #11731480C2semidirect
6. #11731431C2semidirect
7. #11731422C2semidirect
8. #11731411C2semidirect
9. #11730826C2semidirect
10. #11730805C2semidirect
11. #11730778C2semidirect
21. C2×#6457C22semidirect
35. #89765C22semidirect
36. #89764C22semidirect
37. #89763C22semidirect
38. #89762C22semidirect
39. #89761C22semidirect
40. #89756C22semidirect
41. #89753C22semidirect
49. #89449C22semidirect
50. #89443C22semidirect
51. #89442C22semidirect
52. #89440C22semidirect
53. #89437C22semidirect
54. #89432C22semidirect
55. #89429C22semidirect
56. #89425C22semidirect
125. C2×#380D7semidirect
542. C7#36727semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#3802
#117414772
#117396402
#117395542
#117314802
#117314312
#117314222
#117314112
#117308262
#117308052
#117307782
#367277