Finite Group Databasebeta

← #13321649

Subgroups — #13321649

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#11742856C211yesmaximal, characteristic
2C7×#2349C211yesmaximal, characteristic, complemented
3#11739749C211yesmaximal, characteristic, complemented
4#11727300C211yesmaximal, characteristic, complemented
5#11727299C211yesmaximal, characteristic, complemented
6#11727262C214yesmaximal, complemented
7#11727225C211yesmaximal, characteristic, complemented
8#11727223C211yesmaximal, characteristic, complemented
9#11727152C214yesmaximal, complemented
10C2×C4×C7×D4—21no—
11C2×C4×C7×D4C2211yescharacteristic
12#90496—21no—
13#90482C2214yes—
14#90254C412yescomplemented
15#90250—22no—
16#90249—22no—
17#90244C412yescomplemented
18#90242C412yescomplemented
19#90233C412yescomplemented
20#90195—21no—
21#90185—22no—
22#90168C2211yescharacteristic
23#90161C2211yescharacteristic
24C7×#389—21no—
25C7×#385—22no—
26C7×#382C2211yescharacteristic
27C7×#380C2212yes—
28C4×C7×C4⋊C4C2211yescharacteristic
29C2×C7×#95C2212yes—
30#89979C2214yes—
31#89784—22no—
32#89784—22no—
33#89697C2211yescharacteristic
34#89697C2211yescharacteristic
35#89247C2212yes—
36#89247C2214yes—
37#89247C2212yes—
38#89246C2212yes—
39#89246C2214yes—
40#89246C2212yes—
41#17630—71nomaximal, Sylow
42C22×C7×D4D411yescharacteristic
43C23×C4×C7—22no—
44C23×C4×C7C2312yes—
45C2×#1157—21no—
46C2×#1157C2×C414yescomplemented
47C4×C7×D4—44no—
48C4×C7×D4C2×C412yes—
49C4×C7×D4C2×C412yes—
50C4×C7×D4D412yes—
51C4×C7×D4D412yes—
52C2×C7×C4⋊C4—21no—
53C2×C7×C4⋊C4—21no—
54C2×C7×C4⋊C4—22no—
55C2×C7×C4⋊C4D411yescharacteristic
56C2×C7×C22⋊C4—22no—
57C2×C7×C22⋊C4D412yes—
58C2×C42×C7—21no—
59C2×C42×C7C2311yescharacteristic
60#6516—24no—
61#6503—22no—
62#6501—22no—
63#6501—22no—
64#6499—22no—
65#6493—21no—
66#6493C2×C414yescomplemented
67#6493C2312yes—
68#6492—24no—
69C4×C22⋊D7—24no—
70C4×C22⋊D7—24no—
71#6489—21no—
72#6489C2×C414yescomplemented
73#6487—21no—
74#6487C2×C414yescomplemented
75#6487C2312yes—
76C2×#1136—24no—
77#6456—22no—
78#6455—22no—
79D7×C4⋊C4—22no—
80#6450—22no—
81#6448—24no—
82#6447—24no—
83#6446—24no—
84#6442—24no—
85#6438—21no—
86#6437C2×C411yescharacteristic, complemented
87C4×D28—21no—
88C4×D28C2×C412yescomplemented
89#6433—21no—
90#6432C2×C411yescharacteristic, complemented
91#6432C2×C411yescharacteristic, complemented
92#6432C2×C411yescharacteristic, complemented
93C4×Dic14—21no—
94C4×Dic14C2×C412yescomplemented
95C7×#95—22no—
96C7×#95—21no—
97C7×#95C2311yescharacteristic
98C7×#95C2312yes—
99C7×#95C2311yescharacteristic
100C22⋊(C7⋊C8)—24no—
101C22⋊(C7⋊C8)C2312yes—
102C22⋊(C7⋊C8)C2312yes—
103C4⋊(C7⋊C8)—21no—
104C4⋊(C7⋊C8)—21no—
105C4×C7⋊C8—21no—
106C4×C7⋊C8—21no—
107#3506—71no—
108#2975—71no—
109#2349D711yescharacteristic, complemented
110#1518—71no—
111#1517—71no—
112#1513—74no—
113#1504—74no—
114#1503—71no—
115#1502—71no—
116C24×C7C2×D412yes—
117C2×C7×D4—22no—
118C2×C7×D4—22no—
119C2×C7×D4C22⋊C412yes—
120C2×C7×D4C22⋊C412yes—
121C22×C4×C7—21no—
122C22×C4×C7—22no—
123C22×C4×C7—21no—
124C22×C4×C7—44no—
125C22×C4×C7—44no—
126C22×C4×C7—82no—
127C22×C4×C7—82no—
128C22×C4×C7C22×C414yes—
129C22×C4×C7C2×D411yescharacteristic
130C22×C4×C7C2×D412yes—
131C22×C4×C7C2×D414yes—
132C22×C4×C7C2×D411yescharacteristic
133C22×C4×C7C2411yescharacteristic
134C2×C22⋊D7—22no—
135C2×C22⋊D7—24no—
136C2×C22⋊D7—24no—
137#1157—42no—
138#1157—42no—
139#1157—84no—
140C2×D28—21no—
141C2×D28—21no—
142C2×D28C22×C412yes—
143C2×C4×D7—22no—
144C2×C4×D7—24no—
145C2×Dic14—21no—
146C2×Dic14—21no—
147C2×Dic14C22×C412yes—
148C7×C4⋊C4—42no—
149C7×C4⋊C4—42no—
150C7×C4⋊C4—82no—
151C7×C4⋊C4C22⋊C412yes—
152C7×C4⋊C4C22⋊C412yes—
153C7×C22⋊C4—44no—
154C7×C22⋊C4C22⋊C414yes—
155C7×C22⋊C4C22⋊C414yes—
156C42×C7—42no—
157C42×C7C22×C411yescharacteristic
158C42×C7C22×C411yescharacteristic
159C42×C7C2×D411yescharacteristic
160C42×C7C2×D411yescharacteristic
161C22⋊Dic7—24no—
162C22⋊Dic7—24no—
163C22⋊Dic7—22no—
164(C2×D7)⋊C4—22no—
165(C2×D7)⋊C4—24no—
166(C2×D7)⋊C4—22no—
167C4⋊Dic7—21no—
168C4⋊Dic7—21no—
169C4⋊Dic7—21no—
170C4⋊Dic7—21no—
171C4⋊Dic7C22×C412yes—
172C4⋊Dic7C22×C412yes—
173Dic7⋊C4—22no—
174Dic7⋊C4—24no—
175Dic7⋊C4—22no—
176C4×Dic7—22no—
177C4×Dic7—24no—
178#1136—84no—
179C2×C7⋊C8—22no—
180C2×C7⋊C8—22no—
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#532—71no—
191#525—74no—
192#520—141no—
193C2×C4×D4—21no—
194C2×C4×D4C2×D711yescharacteristic
195#436—142no—
196#436—142no—
197#431—71no—
198#431—71no—
199#407—74no—
200#389—21no—
201#385—22no—
202#382C2×D711yescharacteristic
203#380C2×D712yes—
204C4×C4⋊C4C2×D711yescharacteristic
205C2×#95C2×D712yes—
206#328—72no—
207#328—72no—
208#328—74no—
209#327—72no—
210#327—72no—
211#327—74no—
212C23×C7—22no—
213C23×C7—22no—
214C23×C7—42no—
215C23×C7—42no—
216C23×C7C2×C22⋊C414yes—
217C23×C7C22×D411yescharacteristic
218C22×D7—22no—
219C7×D4—24no—
220C7×D4—24no—
221C2×C4×C7—22no—
222C2×C4×C7—44no—
223C2×C4×C7—41no—
224C2×C4×C7—42no—
225C2×C4×C7—44no—
226C2×C4×C7—41no—
227C2×C4×C7—41no—
228C2×C4×C7—41no—
229C2×C4×C7—41no—
230C2×C4×C7—41no—
231C2×C4×C7—42no—
232C2×C4×C7—42no—
233C2×C4×C7—41no—
234C2×C4×C7—42no—
235C2×C4×C7—88no—
236C2×C4×C7C2×C22⋊C414yes—
237C2×C4×C7C2×C22⋊C411yescharacteristic
238C2×C4×C7C2×C22⋊C411yescharacteristic
239C2×C4×C7C2×C22⋊C411yescharacteristic
240C2×C4×C7C2×C22⋊C411yescharacteristic
241C2×C4×C7AGL(1,ℤ/8)12yes—
242C2×C4×C7#13712yes—
243C2×C4×C7C23×C411yescharacteristic
244C2×C4×C7C22×D411yescharacteristic
245C22⋊D7—42no—
246C22⋊D7—42no—
247C22⋊D7—84no—
248C22⋊D7—84no—
249C2×Dic7—22no—
250C2×Dic7—22no—
251C2×Dic7—22no—
252D28—42no—
253D28—42no—
254D28—81no—
255C4×D7—44no—
256C4×D7—84no—
257Dic14—42no—
258Dic14—42no—
259Dic14—81no—
260C7⋊C8—42no—
261C7⋊C8—42no—
262C2×#41—74no—
263C2×#41—141no—
264C22×D4C22⋊D711yescharacteristic
265C23×C4—22no—
266C23×C4C22×D712yes—
267C2×M16—144no—
268C4⋊Q8—71no—
269C4⋊Q8—71no—
270C4⋊Q8—71no—
271#127—71no—
272#127—142no—
273#126—144no—
274#125—141no—
275#124—142no—
276#124—144no—
277#124—141no—
278#123—144no—
279#123—144no—
280C22⋊Q8—74no—
281C22⋊Q8—72no—
282C22⋊Q8—141no—
283#121—74no—
284#121—72no—
285#121—142no—
286#121—142no—
287#121—141no—
288#121—144no—
289#120—144no—
290C4×Q8—72no—
291C4×Q8—142no—
292C4×Q8—141no—
293C4×D4—44no—
294C4×D4—72no—
295C4×D4—144no—
296C4×D4—144no—
297C4×D4—142no—
298C4×D4—141no—
299C4×D4C4×D712yes—
300C4×D4C4×D712yes—
301C4×D4D2812yes—
302C4×D4D2812yes—
303#117—74no—
304#117—141no—
305#117—142no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—142no—
310C2×C4⋊C4C22⋊D711yescharacteristic
311C2×C22⋊C4—22no—
312C2×C22⋊C4C22⋊D712yes—
313C2×C42—21no—
314C2×C42C22×D711yescharacteristic
315C4⋊C8—141no—
316C4⋊C8—141no—
317C22⋊C8—72no—
318C22⋊C8—72no—
319C22⋊C8—144no—
320C4×C8—141no—
321C4×C8—141no—
322#95—22no—
323#95—21no—
324#95C22×D711yescharacteristic
325#95C22×D712yes—
326#95C22×D711yescharacteristic
327C22×C7—21no—
328C22×C7—21no—
329C22×C7—24no—
330C22×C7—24no—
331C22×C7—24no—
332C22×C7—44no—
333C22×C7—44no—
334C22×C7C22×C22⋊C411yescharacteristic
335C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
336C22×C7C2×#13711yescharacteristic
337C2×D7—42no—
338C2×D7—42no—
339C2×D7—42no—
340C4×C7—21no—
341C4×C7—21no—
342C4×C7—24no—
343C4×C7—21no—
344C4×C7—21no—
345C4×C7—42no—
346C4×C7—81no—
347C4×C7—82no—
348C4×C7#41712yes—
349C4×C7#41712yes—
350Dic7—42no—
351Dic7—42no—
352Dic7—42no—
353C24C2×C22⋊D712yes—
354#41—282no—
355#41—282no—
356#41—564no—
357C2×Q8—72no—
358C2×Q8—141no—
359C2×Q8—141no—
360C2×D4—22no—
361C2×D4—22no—
362C2×D4—72no—
363C2×D4—142no—
364C2×D4—144no—
365C2×D4—144no—
366C2×D4—141no—
367C2×D4—141no—
368C2×D4(C2×D7)⋊C412yes—
369C2×D4(C2×D7)⋊C412yes—
370C22×C4—21no—
371C22×C4—22no—
372C22×C4—21no—
373C22×C4—44no—
374C22×C4—44no—
375C22×C4—82no—
376C22×C4—82no—
377C22×C4—142no—
378C22×C4—144no—
379C22×C4C2×C4×D714yes—
380C22×C4C2×D2814yes—
381C22×C4C2×C22⋊D711yescharacteristic
382C22×C4C2×C22⋊D712yes—
383C22×C4C2×C22⋊D711yescharacteristic
384C22×C4C23×D711yescharacteristic
385M16—564no—
386C2×C8—142no—
387C2×C8—142no—
388C4⋊C4—42no—
389C4⋊C4—42no—
390C4⋊C4—72no—
391C4⋊C4—72no—
392C4⋊C4—82no—
393C4⋊C4—144no—
394C4⋊C4—142no—
395C4⋊C4—141no—
396C4⋊C4—141no—
397C4⋊C4—141no—
398C4⋊C4—141no—
399C4⋊C4—142no—
400C4⋊C4(C2×D7)⋊C412yes—
401C4⋊C4(C2×D7)⋊C412yes—
402C22⋊C4—44no—
403C22⋊C4—144no—
404C22⋊C4—144no—
405C22⋊C4—142no—
406C22⋊C4—144no—
407C22⋊C4—142no—
408C22⋊C4—142no—
409C22⋊C4(C2×D7)⋊C414yes—
410C22⋊C4(C2×D7)⋊C414yes—
411C42—42no—
412C42—144no—
413C42—142no—
414C42C2×C4×D711yescharacteristic
415C42C2×C4×D711yescharacteristic
416C42C2×D2811yescharacteristic
417C42C2×D2811yescharacteristic
418C2×C7—21no—
419C2×C7—21no—
420C2×C7—44no—
421C2×C7#288311yescharacteristic
422C2×C7C2×#41711yescharacteristic
423C2×C7#290111yescharacteristic
424D7—82no—
425C23—22no—
426C23—22no—
427C23—42no—
428C23—42no—
429C23—142no—
430C23C2×(C2×D7)⋊C414yes—
431C23C22×C22⋊D711yescharacteristic
432Q8—282no—
433Q8—282no—
434Q8—561no—
435D4—24no—
436D4—24no—
437D4—282no—
438D4—282no—
439D4—282no—
440D4—282no—
441D4—561no—
442D4—564no—
443D4—564no—
444C2×C4—22no—
445C2×C4—41no—
446C2×C4—42no—
447C2×C4—44no—
448C2×C4—41no—
449C2×C4—44no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—41no—
457C2×C4—42no—
458C2×C4—88no—
459C2×C4—142no—
460C2×C4—142no—
461C2×C4—142no—
462C2×C4—284no—
463C2×C4—564no—
464C2×C4C2×(C2×D7)⋊C414yes—
465C2×C4C2×(C2×D7)⋊C411yescharacteristic
466C2×C4C2×(C2×D7)⋊C411yescharacteristic
467C2×C4C2×(C2×D7)⋊C411yescharacteristic
468C2×C4C2×(C2×D7)⋊C411yescharacteristic
469C2×C4#649512yes—
470C2×C4#650512yes—
471C2×C4C22×C4×D711yescharacteristic
472C2×C4C22×D2811yescharacteristic
473C8—282no—
474C8—282no—
475C7#1763011yescharacteristic, complemented, Sylow
476C22—21no—
477C22—21no—
478C22—24no—
479C22—24no—
480C22—24no—
481C22—44no—
482C22—44no—
483C22—282no—
484C22—282no—
485C22—282no—
486C22C22×(C2×D7)⋊C411yescharacteristic
487C22C2×#649511yescharacteristic
488C22C2×#650511yescharacteristic
489C4—21no—
490C4—21no—
491C4—21no—
492C4—21no—
493C4—24no—
494C4—42no—
495C4—82no—
496C4—81no—
497C4—282no—
498C4—282no—
499C4—282no—
500C4#8973812yes—
501C4#8973812yes—
502C2—21no—
503C2—21no—
504C2—44no—
505C2—562no—
506C2C2×#8973811yescharacteristic
507C2#1173998111yescharacteristic
508C2#1173999111yescharacteristic
subgroupG/H
centerC22C22×(C2×D7)⋊C4
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#2349C2
socleC22×C7C22×C22⋊C4
prime radical#11742856C2
cosocleC22×C4×C7C24
hypercenter#2349D7
hypocenterC7#17630
LCSG ▷ C2×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C4×D4 ◁ #2349

Complements

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

NH ≅ G/Ntype
2. C7×#2349C2semidirect
3. #11739749C2semidirect
4. #11727300C2semidirect
5. #11727299C2semidirect
6. #11727262C2semidirect
7. #11727225C2semidirect
8. #11727223C2semidirect
9. #11727152C2semidirect
14. #90254C4semidirect
17. #90244C4semidirect
18. #90242C4semidirect
19. #90233C4semidirect
46. C2×#1157C2×C4semidirect
66. #6493C2×C4semidirect
72. #6489C2×C4semidirect
74. #6487C2×C4semidirect
86. #6437C2×C4semidirect
88. C4×D28C2×C4semidirect
90. #6432C2×C4semidirect
91. #6432C2×C4semidirect
92. #6432C2×C4semidirect
94. C4×Dic14C2×C4semidirect
109. #2349D7semidirect
475. C7#17630semidirect

Maximal subgroups

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

subgroupindex
#117428562
C7×#23492
#117397492
#117273002
#117272992
#117272622
#117272252
#117272232
#117271522
#176307