Finite Group Databasebeta

← #13322066

Subgroups — #13322066

Order 1792.

900 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11742917C211yesmaximal, characteristic
2C7×#2971C211yesmaximal, characteristic, complemented
3#11739963C211yesmaximal, characteristic, complemented
4#11728569C211yesmaximal, characteristic, complemented
5#11728564C212yesmaximal, complemented
6#11728561C212yesmaximal, complemented
7#11728558C211yesmaximal, characteristic, complemented
8#11727616C211yesmaximal, characteristic, complemented
9#11727615C212yesmaximal, complemented
10#11727614C211yesmaximal, characteristic, complemented
11#11727612C212yesmaximal, complemented
12C4×C7×#41C2211yescharacteristic
13#90521C411yescharacteristic
14#90516C411yescharacteristic
15#90259—21no—
16#90259—21no—
17#90250—21no—
18#90250—21no—
19#90247—22no—
20#90247—22no—
21#90225C412yes—
22#90225C412yes—
23#90212C411yescharacteristic
24#90207C411yescharacteristic
25#90170C2212yes—
26#90170C2212yes—
27C2×#6438C2211yescharacteristic
28#90161C2211yescharacteristic
29C7×C8×Q8—21no—
30C7×#436—21no—
31C7×#436—22no—
32C7×#432C2211yescharacteristic
33C7×#432C2212yescomplemented
34C4×C7×M16C2211yescharacteristic
35C4×C7×M16C2212yescomplemented
36#89796—21no—
37#89783—22no—
38D4×C7⋊C8—21no—
39#89698C2211yescharacteristic
40#89698C2212yescomplemented
41C4×#1136C2211yescharacteristic
42C2×#6377C2212yescomplemented
43#89260C2211yescharacteristic, complemented
44#89260C2211yescharacteristic, complemented
45#89260C2211yescharacteristic, complemented
46#89260C2211yescharacteristic, complemented
47#89260C2212yescomplemented
48#89260C2212yescomplemented
49#89259C2212yes—
50#89259C2212yes—
51#89259C2212yes—
52#89259C2212yes—
53#17691—71nomaximal, Sylow
54C2×C7×#41C2×C411yescharacteristic
55C22×D28C2×C411yescharacteristic
56C22×Dic14C2×C411yescharacteristic
57C2×C7×M16—21no—
58C2×C7×M16—22no—
59C4×C7×Q8D411yescharacteristic
60C4×C7×Q8D411yescharacteristic
61C4×C7×D4D412yes—
62C4×C7×D4D411yescharacteristic
63C4×C7×D4D412yes—
64C4×C7×D4D411yescharacteristic
65C7×#117C2×C411yescharacteristic
66C7×#117C2×C412yes—
67C2×C42×C7C2311yescharacteristic
68C2×C42×C7C2312yes—
69#6510—21no—
70#6507—21no—
71#6503—21no—
72#6500—22no—
73#6499—21no—
74#6498—22no—
75#6493C2×C412yes—
76#6493C2×C412yes—
77#6492—22no—
78#6492—22no—
79C4×C22⋊D7—22no—
80C4×C22⋊D7—22no—
81C2×(C2×D7)⋊C4—21no—
82C2×(C2×D7)⋊C4—21no—
83#6489C2×C411yescharacteristic
84#6489C2×C411yescharacteristic
85#6487—21no—
86#6487—21no—
87#6487C2×C412yes—
88#6487C2×C412yes—
89C2×#1136—21no—
90C22×C7⋊C8—22no—
91#6461—22no—
92#6461—22no—
93#6459—22no—
94#6459—22no—
95#6456—21no—
96#6456—21no—
97#6450—21no—
98#6450—21no—
99#6448—22no—
100#6448—22no—
101#6446—22no—
102#6446—22no—
103D7×C22⋊C4—22no—
104#6440—22no—
105#6439—22no—
106#6439—22no—
107#6438—21no—
108#6438—21no—
109#6438C2×C412yes—
110#6438C2312yes—
111C4×D28C2×C412yes—
112C4×D28C2×C412yes—
113#6435—22no—
114#6435—22no—
115#6433C2311yescharacteristic
116#6433C2311yescharacteristic
117#6432C2×C411yescharacteristic
118#6432C2×C411yescharacteristic
119C4×Dic14—21no—
120C4×Dic14—21no—
121C4×Dic14C2×C412yes—
122C4×Dic14C2×C412yes—
123C7×C4⋊C8—21no—
124C7×C4⋊C8—22no—
125C7×C22⋊C8—21no—
126C7×C22⋊C8—22no—
127C7×#97C2311yescharacteristic
128C7×#97C2311yescharacteristic
129C7×#97C2312yes—
130C4×C7×C8—21no—
131C4×C7×C8—22no—
132C22⋊(C7⋊C8)—21no—
133C22⋊(C7⋊C8)—22no—
134C4⋊(C7⋊C8)—21no—
135C4⋊(C7⋊C8)—22no—
136#6377—22no—
137#6377C2311yescharacteristic
138#6377C2312yes—
139#6377C2311yescharacteristic
140C4×C7⋊C8—21no—
141#3510—71no—
142#2971D711yescharacteristic, complemented
143#2970—71no—
144#1537—71no—
145#1535—72no—
146#1532—72no—
147#1529—72no—
148#1527—71no—
149#1527—71no—
150#1526—71no—
151#1524—72no—
152C23×D7—21no—
153C7×#41—42no—
154C2×C7×Q8C22⋊C411yescharacteristic
155C2×C7×D4C22⋊C411yescharacteristic
156C2×C7×D4C22⋊C412yes—
157C22×C4×C7C22×C411yescharacteristic
158C22×C4×C7C22×C412yes—
159C22×C4×C7C2×D412yes—
160C22×C4×C7C2×D411yescharacteristic
161C22×C4×C7C2×D411yescharacteristic
162C22×C4×C7C2×D412yes—
163C2×C22⋊D7—22no—
164C2×C22⋊D7—22no—
165C22×Dic7—21no—
166C2×D28—21no—
167C2×D28—42no—
168C2×D28C22×C412yes—
169C2×C4×D7—22no—
170C2×C4×D7—22no—
171C2×Dic14—21no—
172C2×Dic14—21no—
173C2×Dic14—21no—
174C2×Dic14—42no—
175C2×Dic14C22×C412yes—
176C7×M16—42no—
177C7×M16—44no—
178C2×C7×C8—21no—
179C2×C7×C8—21no—
180C2×C7×C8—22no—
181C7×C4⋊C4C22⋊C411yescharacteristic
182C7×C4⋊C4C22⋊C411yescharacteristic
183C7×C4⋊C4C22⋊C412yes—
184C7×C4⋊C4C22⋊C412yes—
185C7×C22⋊C4C22⋊C412yes—
186C7×C22⋊C4C22⋊C412yes—
187C7×C22⋊C4C22⋊C412yes—
188C42×C7C22×C412yes—
189C42×C7C22×C411yescharacteristic
190C42×C7C2×D411yescharacteristic
191C42×C7C2×D412yes—
192C42×C7C2×D411yescharacteristic
193C42×C7C2×D412yes—
194C42×C7C2411yescharacteristic
195C22⋊Dic7—21no—
196C22⋊Dic7—21no—
197C22⋊Dic7—22no—
198C22⋊Dic7—22no—
199(C2×D7)⋊C4—22no—
200(C2×D7)⋊C4—22no—
201(C2×D7)⋊C4—22no—
202(C2×D7)⋊C4—22no—
203(C2×D7)⋊C4—22no—
204(C2×D7)⋊C4—22no—
205(C2×D7)⋊C4—22no—
206(C2×D7)⋊C4—22no—
207C4⋊Dic7—21no—
208C4⋊Dic7—21no—
209C4⋊Dic7C22×C411yescharacteristic
210C4⋊Dic7C22×C411yescharacteristic
211C4⋊Dic7C22×C411yescharacteristic
212C4⋊Dic7C22×C411yescharacteristic
213Dic7⋊C4—22no—
214Dic7⋊C4—22no—
215Dic7⋊C4—22no—
216Dic7⋊C4—22no—
217Dic7⋊C4—21no—
218Dic7⋊C4—21no—
219Dic7⋊C4—21no—
220Dic7⋊C4—21no—
221C4×Dic7—21no—
222C4×Dic7—21no—
223C4×Dic7—22no—
224C4×Dic7—22no—
225#1136—42no—
226C2×C7⋊C8—21no—
227C2×C7⋊C8—22no—
228C2×C7⋊C8—21no—
229C2×C7⋊C8—44no—
230#559—141no—
231#559—141no—
232#556—141no—
233#556—141no—
234#551—142no—
235#551—142no—
236#542—72no—
237#542—72no—
238#536—71no—
239#535—71no—
240#533—71no—
241#529—72no—
242#529—72no—
243C2×#124—71no—
244#519—71no—
245#518—71no—
246C4×#41C2×D711yescharacteristic
247#446—141no—
248C8×Q8—21no—
249#436—21no—
250#436—22no—
251#435—142no—
252C8×D4—141no—
253#432—71no—
254#432—72no—
255#432C2×D711yescharacteristic
256#432C2×D712yescomplemented
257C4×M16—71no—
258C4×M16C2×D711yescharacteristic
259C4×M16C2×D712yescomplemented
260C2×#97—72no—
261#330—72no—
262#330—72no—
263#330—71no—
264#330—71no—
265#330—71no—
266#330—71no—
267#329—72no—
268#329—72no—
269#329—72no—
270#329—72no—
271C23×C7C2×C22⋊C411yescharacteristic
272C23×C7C2×C22⋊C412yes—
273C22×D7—22no—
274C22×D7—42no—
275C22×D7—81no—
276C22×D7—81no—
277C7×Q8—22no—
278C7×D4—22no—
279C7×D4—42no—
280C2×C4×C7—22no—
281C2×C4×C7—42no—
282C2×C4×C7—41no—
283C2×C4×C7—41no—
284C2×C4×C7—42no—
285C2×C4×C7—42no—
286C2×C4×C7C2×C22⋊C412yes—
287C2×C4×C7C2×C22⋊C412yes—
288C2×C4×C7C2×C22⋊C412yes—
289C2×C4×C7C2×C22⋊C411yescharacteristic
290C2×C4×C7C2×C22⋊C411yescharacteristic
291C2×C4×C7C2×C22⋊C411yescharacteristic
292C2×C4×C7C23×C411yescharacteristic
293C2×C4×C7C22×D411yescharacteristic
294C2×C4×C7C22×D411yescharacteristic
295C22⋊D7—82no—
296C22⋊D7—82no—
297C2×Dic7—21no—
298C2×Dic7—21no—
299C2×Dic7—22no—
300C2×Dic7—21no—
301C2×Dic7—21no—
302C2×Dic7—42no—
303D28—42no—
304D28—42no—
305C4×D7—82no—
306C4×D7—82no—
307Dic14—42no—
308Dic14—42no—
309Dic14—81no—
310Dic14—81no—
311C7×C8—22no—
312C7×C8—41no—
313C7×C8—42no—
314C7⋊C8—22no—
315C7⋊C8—42no—
316C7⋊C8—41no—
317C2×#41C4×D711yescharacteristic
318C22×Q8—71no—
319C22×D4—71no—
320C2×M16—21no—
321C2×M16—22no—
322C2×M16—141no—
323C22×C8—142no—
324C4⋊Q8—71no—
325C4⋊Q8—71no—
326C4⋊Q8—141no—
327#127—141no—
328#126—142no—
329#126—142no—
330#126—142no—
331#126—142no—
332#125—71no—
333#125—71no—
334#124—72no—
335#124—72no—
336#124—141no—
337#124—141no—
338#124—142no—
339#124—142no—
340#124—141no—
341#124—141no—
342#123—142no—
343#123—142no—
344#123—142no—
345#123—142no—
346#123—142no—
347C22⋊Q8—72no—
348C22⋊Q8—72no—
349C22⋊Q8—141no—
350C22⋊Q8—141no—
351C22⋊Q8—142no—
352C22⋊Q8—142no—
353#121—72no—
354#121—72no—
355#120—142no—
356C4×Q8—72no—
357C4×Q8—72no—
358C4×Q8—141no—
359C4×Q8—141no—
360C4×Q8—141no—
361C4×Q8—141no—
362C4×Q8D2811yescharacteristic
363C4×Q8C22⋊D711yescharacteristic
364C4×D4—72no—
365C4×D4—72no—
366C4×D4—142no—
367C4×D4—142no—
368C4×D4—141no—
369C4×D4—141no—
370C4×D4D2812yes—
371C4×D4D2811yescharacteristic
372C4×D4C22⋊D712yes—
373C4×D4C22⋊D711yescharacteristic
374#117—71no—
375#117—71no—
376#117—142no—
377#117—142no—
378#117—142no—
379#117C4×D711yescharacteristic
380#117C4×D712yes—
381C2×C22⋊C4—141no—
382C2×C22⋊C4—141no—
383C2×C22⋊C4—142no—
384C2×C42C22×D711yescharacteristic
385C2×C42C22×D712yes—
386C4⋊C8—21no—
387C4⋊C8—22no—
388C4⋊C8—141no—
389C4⋊C8—142no—
390C22⋊C8—21no—
391C22⋊C8—22no—
392C22⋊C8—142no—
393C22⋊C8—141no—
394#97—71no—
395#97—72no—
396#97—71no—
397#97—142no—
398#97C22×D711yescharacteristic
399#97C22×D711yescharacteristic
400#97C22×D712yes—
401C4×C8—21no—
402C4×C8—22no—
403C4×C8—141no—
404C22×C7—21no—
405C22×C7—22no—
406C22×C7—22no—
407C22×C7—22no—
408C22×C7—21no—
409C22×C7—21no—
410C22×C7C22×C22⋊C411yescharacteristic
411C2×D7—42no—
412C2×D7—42no—
413C2×D7—42no—
414C2×D7—162no—
415C4×C7—21no—
416C4×C7—22no—
417C4×C7—21no—
418C4×C7—21no—
419C4×C7—21no—
420C4×C7—21no—
421C4×C7—22no—
422C4×C7—22no—
423C4×C7#41312yes—
424Dic7—42no—
425Dic7—41no—
426Dic7—41no—
427Dic7—41no—
428Dic7—41no—
429C24—141no—
430#41—42no—
431C2×Q8—72no—
432C2×Q8—141no—
433C2×Q8—141no—
434C2×Q8—141no—
435C2×Q8—282no—
436C2×Q8(C2×D7)⋊C411yescharacteristic
437C2×D4—72no—
438C2×D4—142no—
439C2×D4—142no—
440C2×D4—141no—
441C2×D4—282no—
442C2×D4(C2×D7)⋊C411yescharacteristic
443C2×D4(C2×D7)⋊C412yes—
444C22×C4—141no—
445C22×C4—142no—
446C22×C4—142no—
447C22×C4C2×C4×D711yescharacteristic
448C22×C4C2×C4×D712yes—
449C22×C4C2×D2811yescharacteristic
450C22×C4C2×D2812yes—
451C22×C4C2×C22⋊D711yescharacteristic
452C22×C4C2×C22⋊D712yes—
453M16—42no—
454M16—44no—
455M16—282no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—22no—
459C2×C8—141no—
460C2×C8—141no—
461C2×C8—142no—
462C2×C8—284no—
463C4⋊C4—71no—
464C4⋊C4—71no—
465C4⋊C4—71no—
466C4⋊C4—71no—
467C4⋊C4—142no—
468C4⋊C4—142no—
469C4⋊C4—142no—
470C4⋊C4—142no—
471C4⋊C4—141no—
472C4⋊C4—141no—
473C4⋊C4—141no—
474C4⋊C4—141no—
475C4⋊C4—141no—
476C4⋊C4—141no—
477C4⋊C4(C2×D7)⋊C411yescharacteristic
478C4⋊C4(C2×D7)⋊C411yescharacteristic
479C4⋊C4(C2×D7)⋊C412yes—
480C4⋊C4(C2×D7)⋊C412yes—
481C22⋊C4—141no—
482C22⋊C4—141no—
483C22⋊C4—142no—
484C22⋊C4—142no—
485C22⋊C4—142no—
486C22⋊C4—142no—
487C22⋊C4—142no—
488C22⋊C4—142no—
489C22⋊C4—142no—
490C22⋊C4—142no—
491C22⋊C4—142no—
492C22⋊C4—142no—
493C22⋊C4(C2×D7)⋊C412yes—
494C22⋊C4(C2×D7)⋊C412yes—
495C22⋊C4(C2×D7)⋊C412yes—
496C42—141no—
497C42—141no—
498C42—142no—
499C42—142no—
500C42C2×C4×D712yes—
501C42C2×C4×D711yescharacteristic
502C42C2×D2811yescharacteristic
503C42C2×D2812yes—
504C42C2×C22⋊D712yes—
505C42C2×C22⋊D711yescharacteristic
506C42C23×D711yescharacteristic
507C2×C7—41no—
508C2×C7—42no—
509C2×C7C2×#41311yescharacteristic
510C2×C7#290111yescharacteristic
511C2×C7#290111yescharacteristic
512D7—82no—
513C23—142no—
514C23—282no—
515C23—561no—
516C23—561no—
517C23C2×(C2×D7)⋊C411yescharacteristic
518C23C2×(C2×D7)⋊C412yes—
519Q8—22no—
520Q8—282no—
521Q8—282no—
522Q8—561no—
523Q8—561no—
524D4—22no—
525D4—42no—
526D4—282no—
527D4—282no—
528D4—562no—
529D4—562no—
530C2×C4—22no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—42no—
536C2×C4—141no—
537C2×C4—141no—
538C2×C4—142no—
539C2×C4—141no—
540C2×C4—141no—
541C2×C4—282no—
542C2×C4—562no—
543C2×C4—562no—
544C2×C4C2×(C2×D7)⋊C412yes—
545C2×C4C2×(C2×D7)⋊C412yes—
546C2×C4C2×(C2×D7)⋊C411yescharacteristic
547C2×C4C2×(C2×D7)⋊C411yescharacteristic
548C2×C4C2×(C2×D7)⋊C412yes—
549C2×C4C2×(C2×D7)⋊C411yescharacteristic
550C2×C4C22×C4×D711yescharacteristic
551C2×C4C22×D2811yescharacteristic
552C2×C4C22×C22⋊D711yescharacteristic
553C8—22no—
554C8—41no—
555C8—42no—
556C8—142no—
557C8—282no—
558C8—281no—
559C7#1769111yescharacteristic, complemented, Sylow
560C22—21no—
561C22—22no—
562C22—22no—
563C22—21no—
564C22—22no—
565C22—21no—
566C22—282no—
567C22—282no—
568C22—282no—
569C22—1122no—
570C22C22×(C2×D7)⋊C411yescharacteristic
571C4—21no—
572C4—22no—
573C4—21no—
574C4—22no—
575C4—21no—
576C4—21no—
577C4—22no—
578C4—21no—
579C4—282no—
580C4—281no—
581C4—281no—
582C4—281no—
583C4—281no—
584C4#8990412yes—
585C2—41no—
586C2—42no—
587C2—562no—
588C2#1173998111yescharacteristic
589C2C2×#8990411yescharacteristic
590C2#1174126211yescharacteristic
subgroupG/H
centerC22C22×(C2×D7)⋊C4
commutatorC2×C4×C7C23×C4
FrattiniC42C23×D7
FittingC7×#2971C2
socleC22×C7C22×C22⋊C4
prime radical#90521C4
cosocleC42×C7C24
hypercenter#2971D7
hypocenterC7#17691
LCSG ▷ C2×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C4×#41 ◁ #2971

Complements

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

NH ≅ G/Ntype
2. C7×#2971C2semidirect
3. #11739963C2semidirect
4. #11728569C2semidirect
5. #11728564C2semidirect
6. #11728561C2semidirect
7. #11728558C2semidirect
8. #11727616C2semidirect
9. #11727615C2semidirect
10. #11727614C2semidirect
11. #11727612C2semidirect
33. C7×#432C22semidirect
35. C4×C7×M16C22semidirect
40. #89698C22semidirect
42. C2×#6377C22semidirect
43. #89260C22semidirect
44. #89260C22semidirect
45. #89260C22semidirect
46. #89260C22semidirect
47. #89260C22semidirect
48. #89260C22semidirect
142. #2971D7semidirect
256. #432C2×D7semidirect
259. C4×M16C2×D7semidirect
559. C7#17691semidirect

Maximal subgroups

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

subgroupindex
#117429172
C7×#29712
#117399632
#117285692
#117285642
#117285612
#117285582
#117276162
#117276152
#117276142
#117276122
#176917