Finite Group Databasebeta

← #13321648

Subgroups — #13321648

Order 1792.

1066 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11743291C211yesmaximal, characteristic
2#11736733C211yesmaximal, characteristic, complemented
3#11731215C211yesmaximal, characteristic, complemented
4C7×#1502C211yesmaximal, characteristic, complemented
5#11729568C211yesmaximal, characteristic, complemented
6#11728547C211yesmaximal, characteristic, complemented
7#11727225C211yesmaximal, characteristic, complemented
8#11726059C214yesmaximal, complemented
9#11726056C214yesmaximal, complemented
10C7×#532C2211yescharacteristic
11#90521—21no—
12#90518—21no—
13#90423C412yescomplemented
14#90409C412yescomplemented
15#90372—22no—
16#90308—22no—
17#90297C2214yes—
18#90256—22no—
19#90254C412yescomplemented
20#90246C412yescomplemented
21C2×C4×D28—21no—
22C2×C4×D28C2211yescharacteristic
23#90161C2211yescharacteristic
24C7×#431C2211yescharacteristic
25C4×C7×C4⋊C4C2211yescharacteristic
26#89761—22no—
27#89708C2211yescharacteristic
28#89700—21no—
29#89697C2211yescharacteristic
30#89610—22no—
31#89498C2214yes—
32#89460—22no—
33#89439C2212yes—
34D7×#95C2212yes—
35C7×#327C2212yes—
36C7×#327C2212yes—
37#89332C2214yes—
38#89262C2214yes—
39#89247C2212yes—
40#89247C2212yes—
41#17630—71nomaximal, Sylow
42C2×C7×#41—21no—
43C2×#1159C2×C414yescomplemented
44C22×D28D411yescharacteristic
45C22×C4×D7—22no—
46C22×C4×D7C2312yes—
47C7×C4⋊Q8C2×C411yescharacteristic, complemented
48C7×#127C2×C411yescharacteristic, complemented
49C7×#124—21no—
50C7×#121—21no—
51C7×#121C2312yes—
52C4×C7×D4—21no—
53C4×C7×D4C2×C412yescomplemented
54C2×C7×C4⋊C4—21no—
55C2×C42×C7—21no—
56C2×C42×C7C2311yescharacteristic
57#6510—22no—
58Q8×Dic7—22no—
59#6503—22no—
60#6502—24no—
61#6501C2×C414yescomplemented
62#6500—24no—
63D4×Dic7—24no—
64D4×Dic7—22no—
65#6493—22no—
66#6493—22no—
67C2×(C2×D7)⋊C4—22no—
68C2×(C2×D7)⋊C4D412yes—
69#6489—21no—
70#6489—22no—
71C2×C4⋊Dic7—22no—
72C2×C4⋊Dic7—21no—
73C2×C4⋊Dic7D411yescharacteristic
74#6487—21no—
75#6487C2312yes—
76C2×Dic7⋊C4—22no—
77C2×#1131—24no—
78#6461—24no—
79#6460C2×C414yescomplemented
80#6455C2×C414yescomplemented
81#6449—24no—
82#6448—24no—
83#6446—24no—
84(C22⋊D7)⋊C4—24no—
85C4×D28—44no—
86C4×D28C2×C412yes—
87C4×D28C2×C412yes—
88C4×D28D412yes—
89C4×D28D412yes—
90#6433—21no—
91#6432C2×C411yescharacteristic, complemented
92#6432C2×C411yescharacteristic, complemented
93C4×Dic14—21no—
94C4×Dic14C2×C412yescomplemented
95C7×C4⋊C8—21no—
96C7×C22⋊C8C2312yes—
97C4×C7×C8—21no—
98C7×#95C2311yescharacteristic
99C7×#95C2311yescharacteristic
100C7⋊#95—21no—
101C7⋊#95—22no—
102C7⋊#95C2312yes—
103C22⋊(C7⋊C8)C2312yes—
104(C2×D7)⋊C8—24no—
105C4⋊(C7⋊C8)—21no—
106C4×C7⋊C8—21no—
107#3506—71no—
108#2975—71no—
109#2349—71no—
110#1518—71no—
111#1517—71no—
112#1513—74no—
113#1504—74no—
114#1503—71no—
115#1502D711yescharacteristic, complemented
116C23×D7C2×D412yes—
117C7×#41—42no—
118C7×#41—42no—
119C2×C7×Q8—21no—
120C2×C7×D4—22no—
121C2×C7×D4—21no—
122C2×C7×D4—21no—
123C2×C7×D4C22×C412yes—
124C22×C4×C7—21no—
125C22×C4×C7—22no—
126C22×C4×C7—21no—
127C22×C4×C7C2×D411yescharacteristic
128C22×C4×C7C2×D411yescharacteristic
129C22×C4×C7C2411yescharacteristic
130C2×C22⋊D7—24no—
131C2×C22⋊D7—24no—
132C22×Dic7—24no—
133C22×Dic7—22no—
134C22×Dic7C2×D412yes—
135#1159—84no—
136C2×D28—22no—
137C2×D28—22no—
138C2×D28C22⋊C412yes—
139C2×D28C22⋊C412yes—
140C2×C4×D7—44no—
141C2×C4×D7—44no—
142C2×C4×D7—82no—
143C2×C4×D7—82no—
144C2×C4×D7C22×C414yes—
145C2×C4×D7C2×D414yes—
146C2×Dic14—21no—
147C2×Dic14C22×C412yes—
148C2×C7×C8—22no—
149C7×C4⋊C4—21no—
150C7×C4⋊C4—42no—
151C7×C4⋊C4C22×C412yes—
152C7×C22⋊C4—22no—
153C7×C22⋊C4—22no—
154C42×C7—42no—
155C42×C7C22×C411yescharacteristic
156C42×C7C22×C411yescharacteristic
157C42×C7C2×D411yescharacteristic
158C42×C7C2×D411yescharacteristic
159C22⋊Dic7—22no—
160C22⋊Dic7—24no—
161(C2×D7)⋊C4—24no—
162(C2×D7)⋊C4—24no—
163(C2×D7)⋊C4—44no—
164(C2×D7)⋊C4C22⋊C414yes—
165(C2×D7)⋊C4C22⋊C414yes—
166C4⋊Dic7—21no—
167C4⋊Dic7—21no—
168C4⋊Dic7—21no—
169C4⋊Dic7—24no—
170C4⋊Dic7—42no—
171C4⋊Dic7C22⋊C412yes—
172C4⋊Dic7C22⋊C412yes—
173C4⋊Dic7C22×C412yes—
174Dic7⋊C4—22no—
175Dic7⋊C4—22no—
176Dic7⋊C4—82no—
177C4×Dic7—22no—
178C4×Dic7—24no—
179C2×C7⋊C8—22no—
180#1131—84no—
181#559—142no—
182#556—72no—
183#555—72no—
184#553—142no—
185#550—72no—
186#547—72no—
187#538—142no—
188#535—141no—
189#533—71no—
190#532C2×D711yescharacteristic
191#525—74no—
192#520—141no—
193C2×C4×D4—71no—
194C2×C4×D4—141no—
195#436—142no—
196#436—142no—
197#431—71no—
198#431C2×D711yescharacteristic
199#407—74no—
200#389—141no—
201#385—142no—
202#382—71no—
203#380—72no—
204C4×C4⋊C4C2×D711yescharacteristic
205C2×#95—72no—
206#328—74no—
207#328—72no—
208#328—72no—
209#327—74no—
210#327C2×D712yes—
211#327C2×D712yes—
212C23×C7—22no—
213C23×C7C22×D411yescharacteristic
214C22×D7—22no—
215C22×D7—22no—
216C22×D7—42no—
217C22×D7—42no—
218C22×D7C2×C22⋊C414yes—
219C7×Q8—42no—
220C7×D4—42no—
221C7×D4—42no—
222C7×D4—42no—
223C7×D4—42no—
224C7×D4—81no—
225C2×C4×C7—22no—
226C2×C4×C7—22no—
227C2×C4×C7—41no—
228C2×C4×C7—41no—
229C2×C4×C7—41no—
230C2×C4×C7—41no—
231C2×C4×C7—44no—
232C2×C4×C7—41no—
233C2×C4×C7—41no—
234C2×C4×C7—41no—
235C2×C4×C7C2×C22⋊C411yescharacteristic
236C2×C4×C7C2×C22⋊C411yescharacteristic
237C2×C4×C7C2×C22⋊C411yescharacteristic
238C2×C4×C7C2×C22⋊C411yescharacteristic
239C2×C4×C7AGL(1,ℤ/8)12yes—
240C2×C4×C7#13712yes—
241C2×C4×C7C23×C411yescharacteristic
242C2×C4×C7C22×D411yescharacteristic
243C22⋊D7—84no—
244C22⋊D7—84no—
245C2×Dic7—22no—
246C2×Dic7—22no—
247C2×Dic7—42no—
248C2×Dic7—42no—
249C2×Dic7—42no—
250C2×Dic7—42no—
251C2×Dic7—84no—
252C2×Dic7C2×C22⋊C414yes—
253D28—24no—
254D28—24no—
255C4×D7—44no—
256C4×D7—44no—
257C4×D7—88no—
258Dic14—42no—
259Dic14—81no—
260C7×C8—42no—
261C7⋊C8—42no—
262C2×#41—21no—
263C2×#41—74no—
264C22×D4—71no—
265C23×C4—72no—
266C23×C4—142no—
267C2×M16—144no—
268C4⋊Q8—71no—
269C4⋊Q8—71no—
270C4⋊Q8C4×D711yescharacteristic, complemented
271#127—142no—
272#127C4×D711yescharacteristic, complemented
273#126—144no—
274#125—141no—
275#124—21no—
276#124—144no—
277#124—142no—
278#123—144no—
279#123—144no—
280C22⋊Q8—72no—
281C22⋊Q8—74no—
282C22⋊Q8—141no—
283#121—21no—
284#121—74no—
285#121—144no—
286#121—142no—
287#121—142no—
288#121C22×D712yes—
289#120—144no—
290C4×Q8—72no—
291C4×Q8—142no—
292C4×Q8—141no—
293C4×D4—21no—
294C4×D4—72no—
295C4×D4—72no—
296C4×D4—72no—
297C4×D4—72no—
298C4×D4—144no—
299C4×D4—142no—
300C4×D4—144no—
301C4×D4—284no—
302C4×D4C4×D712yescomplemented
303#117—74no—
304#117—142no—
305#117—141no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—71no—
308C2×C4⋊C4—142no—
309C2×C4⋊C4—141no—
310C2×C4⋊C4—142no—
311C2×C22⋊C4—72no—
312C2×C22⋊C4—142no—
313C2×C42—21no—
314C2×C42C22×D711yescharacteristic
315C4⋊C8—21no—
316C4⋊C8—141no—
317C22⋊C8—72no—
318C22⋊C8—144no—
319C22⋊C8C22×D712yes—
320C4×C8—21no—
321C4×C8—141no—
322#95—72no—
323#95—142no—
324#95—141no—
325#95C22×D711yescharacteristic
326#95C22×D711yescharacteristic
327C22×C7—21no—
328C22×C7—21no—
329C22×C7—42no—
330C22×C7—42no—
331C22×C7—42no—
332C22×C7C22×C22⋊C411yescharacteristic
333C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
334C22×C7C2×#13711yescharacteristic
335C2×D7—24no—
336C2×D7—24no—
337C2×D7—24no—
338C2×D7—44no—
339C2×D7—44no—
340C4×C7—21no—
341C4×C7—21no—
342C4×C7—21no—
343C4×C7—21no—
344C4×C7—42no—
345C4×C7—42no—
346C4×C7—81no—
347C4×C7#41712yes—
348C4×C7#41712yes—
349Dic7—24no—
350Dic7—42no—
351Dic7—42no—
352Dic7—82no—
353C24—72no—
354#41—42no—
355#41—42no—
356#41—564no—
357C2×Q8—21no—
358C2×Q8—72no—
359C2×Q8—141no—
360C2×D4—22no—
361C2×D4—21no—
362C2×D4—21no—
363C2×D4—72no—
364C2×D4—72no—
365C2×D4—144no—
366C2×D4—144no—
367C2×D4—142no—
368C2×D4—142no—
369C2×D4C2×C4×D712yes—
370C22×C4—21no—
371C22×C4—22no—
372C22×C4—21no—
373C22×C4—72no—
374C22×C4—74no—
375C22×C4—74no—
376C22×C4—144no—
377C22×C4—142no—
378C22×C4—284no—
379C22×C4—284no—
380C22×C4—562no—
381C22×C4—562no—
382C22×C4D4×D711yescharacteristic
383C22×C4D4×D711yescharacteristic
384C22×C4C23×D711yescharacteristic
385M16—564no—
386C2×C8—22no—
387C2×C8—142no—
388C4⋊C4—21no—
389C4⋊C4—42no—
390C4⋊C4—72no—
391C4⋊C4—72no—
392C4⋊C4—72no—
393C4⋊C4—144no—
394C4⋊C4—142no—
395C4⋊C4—141no—
396C4⋊C4—141no—
397C4⋊C4—141no—
398C4⋊C4—142no—
399C4⋊C4—282no—
400C4⋊C4—562no—
401C4⋊C4C2×C4×D712yes—
402C22⋊C4—22no—
403C22⋊C4—22no—
404C22⋊C4—74no—
405C22⋊C4—74no—
406C22⋊C4—144no—
407C22⋊C4—144no—
408C22⋊C4—144no—
409C22⋊C4—142no—
410C22⋊C4—284no—
411C42—42no—
412C42—144no—
413C42—142no—
414C42C2×C4×D711yescharacteristic
415C42C2×C4×D711yescharacteristic
416C42D4×D711yescharacteristic
417C42D4×D711yescharacteristic
418C2×C7—21no—
419C2×C7—21no—
420C2×C7—82no—
421C2×C7#288311yescharacteristic
422C2×C7C2×#41711yescharacteristic
423C2×C7#290111yescharacteristic
424D7—44no—
425C23—22no—
426C23—74no—
427C23—142no—
428C23—142no—
429C23—282no—
430C23—282no—
431C23C2×D4×D711yescharacteristic
432Q8—42no—
433Q8—282no—
434Q8—561no—
435D4—42no—
436D4—42no—
437D4—42no—
438D4—42no—
439D4—81no—
440D4—144no—
441D4—144no—
442D4—564no—
443D4—564no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—44no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—74no—
455C2×C4—142no—
456C2×C4—142no—
457C2×C4—284no—
458C2×C4—284no—
459C2×C4—282no—
460C2×C4—282no—
461C2×C4—282no—
462C2×C4—282no—
463C2×C4—568no—
464C2×C4—564no—
465C2×C4D7×C22⋊C411yescharacteristic
466C2×C4D7×C22⋊C411yescharacteristic
467C2×C4D7×C22⋊C411yescharacteristic
468C2×C4D7×C22⋊C411yescharacteristic
469C2×C4#647412yes—
470C2×C4#647812yes—
471C2×C4C22×C4×D711yescharacteristic
472C2×C4C2×D4×D711yescharacteristic
473C8—42no—
474C8—282no—
475C7#1763011yescharacteristic, complemented, Sylow
476C22—21no—
477C22—21no—
478C22—42no—
479C22—42no—
480C22—42no—
481C22—144no—
482C22—144no—
483C22—144no—
484C22—284no—
485C22—284no—
486C22C2×D7×C22⋊C411yescharacteristic
487C22C2×#647411yescharacteristic
488C22C2×#647811yescharacteristic
489C4—21no—
490C4—21no—
491C4—21no—
492C4—21no—
493C4—42no—
494C4—42no—
495C4—81no—
496C4—144no—
497C4—282no—
498C4—282no—
499C4—562no—
500C4#8954312yes—
501C4#8954312yes—
502C2—21no—
503C2—21no—
504C2—82no—
505C2—284no—
506C2#1173613911yescharacteristic
507C2C2×#8954311yescharacteristic
508C2#1173642811yescharacteristic
subgroupG/H
centerC22C2×D7×C22⋊C4
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1502C2
socleC22×C7C22×C22⋊C4
prime radical#11743291C2
cosocleC22×C4×C7C24
hypercenter#1502D7
hypocenterC7#17630
LCSG ▷ C2×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1502

Complements

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

NH ≅ G/Ntype
2. #11736733C2semidirect
3. #11731215C2semidirect
4. C7×#1502C2semidirect
5. #11729568C2semidirect
6. #11728547C2semidirect
7. #11727225C2semidirect
8. #11726059C2semidirect
9. #11726056C2semidirect
13. #90423C4semidirect
14. #90409C4semidirect
19. #90254C4semidirect
20. #90246C4semidirect
43. C2×#1159C2×C4semidirect
47. C7×C4⋊Q8C2×C4semidirect
48. C7×#127C2×C4semidirect
53. C4×C7×D4C2×C4semidirect
61. #6501C2×C4semidirect
79. #6460C2×C4semidirect
80. #6455C2×C4semidirect
91. #6432C2×C4semidirect
92. #6432C2×C4semidirect
94. C4×Dic14C2×C4semidirect
115. #1502D7semidirect
270. C4⋊Q8C4×D7semidirect
272. #127C4×D7semidirect
302. C4×D4C4×D7semidirect
475. C7#17630semidirect

Maximal subgroups

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

subgroupindex
#117432912
#117367332
#117312152
C7×#15022
#117295682
#117285472
#117272252
#117260592
#117260562
#176307