Finite Group Databasebeta

← #13354245

Subgroups — #13354245

Order 1792.

1070 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89653C211yesmaximal, characteristic
2C2×D7×#107C211yesmaximal, characteristic
3C2×#89633C211yesmaximal, characteristic
4C2×#89632C211yesmaximal, characteristic
5C2×#89480C211yesmaximal, characteristic
6#11731804C211yesmaximal, characteristic
7#11731670C211yesmaximal, characteristic
8C7×#1847C211yesmaximal, characteristic, complemented
9#11728362C211yesmaximal, characteristic, complemented
10#11728360C211yesmaximal, characteristic, complemented
11#11728310C211yesmaximal, characteristic, complemented
12#11728309C211yesmaximal, characteristic, complemented
13#11727570C211yesmaximal, characteristic, complemented
14#11727331C211yesmaximal, characteristic, complemented
15#11727303C211yesmaximal, characteristic, complemented
16C22×#1131C2211yescharacteristic
17C22×C8×D7C2211yescharacteristic
18C2×#6455C2211yescharacteristic
19C2×#6455C2211yescharacteristic
20C2×D7×C4⋊C4C2211yescharacteristic
21C2×D7×C4⋊C4C2211yescharacteristic
22C2×#6435C2211yescharacteristic
23C2×C7×#107C2211yescharacteristic
24C2×C7×#107C2211yescharacteristic
25C2×C7×#106C2211yescharacteristic
26C2×C7×#106C2211yescharacteristic
27C2×C7×#97C2211yescharacteristic
28C7×#384C2211yescharacteristic
29C7×#382C2211yescharacteristic
30C2×#6392C2211yescharacteristic
31C2×#6392C2211yescharacteristic
32C2×#6391C2211yescharacteristic
33C2×#6391C2211yescharacteristic
34C2×#6389C2211yescharacteristic
35C2×C8×Dic7C2211yescharacteristic
36#89761—22no—
37#89749C2211yescharacteristic
38#89749C2211yescharacteristic
39#89748C2211yescharacteristic
40#89748C2211yescharacteristic
41#89744—22no—
42C2×#6382C2211yescharacteristic
43C2×#6382C2211yescharacteristic
44C2×#6381C2211yescharacteristic
45C2×#6381C2211yescharacteristic
46C2×#6381C2211yescharacteristic
47C2×#6381C2211yescharacteristic
48C2×#6381C2211yescharacteristic
49C2×#6381C2211yescharacteristic
50#89708C2211yescharacteristic
51#89707C2211yescharacteristic
52C2×C4×C7⋊C8C2211yescharacteristic
53#89653—24no—
54D7×#107—24no—
55#89633—24no—
56#89632—24no—
57#89480C418yescomplemented
58#89440—22no—
59#89419—22no—
60#20803—71nomaximal, Sylow
61C22×C4×D7—21no—
62C22×C4×D7—21no—
63C22×C4×D7C2311yescharacteristic
64C22×C7×C8C2311yescharacteristic
65C22×C7×C8C2311yescharacteristic
66C2×C7×C4⋊C4—22no—
67C2×C7×C4⋊C4C2311yescharacteristic
68C2×C7×C4⋊C4C2311yescharacteristic
69C2×C7×C4⋊C4C2311yescharacteristic
70C2×C7×C4⋊C4C2311yescharacteristic
71C2×C42×C7C2311yescharacteristic
72C2×(C2×D7)⋊C4—21no—
73C2×(C2×D7)⋊C4—21no—
74C2×(C2×D7)⋊C4D412yes—
75C2×C4⋊Dic7—22no—
76C2×C4⋊Dic7C2311yescharacteristic
77C2×C4⋊Dic7C2311yescharacteristic
78C2×C4⋊Dic7C2311yescharacteristic
79C2×C4⋊Dic7C2311yescharacteristic
80C2×Dic7⋊C4—21no—
81C2×Dic7⋊C4—21no—
82C2×Dic7⋊C4—22no—
83C2×Dic7⋊C4—22no—
84C2×Dic7⋊C4D412yes—
85C2×C4×Dic7—21no—
86C2×C4×Dic7—21no—
87C2×C4×Dic7C2311yescharacteristic
88C22×C7⋊C8C2311yescharacteristic
89C22×C7⋊C8C2311yescharacteristic
90C2×#1131C2×C414yes—
91C2×#1131D414yes—
92C2×#1131Q814yes—
93C2×C8×D7C2×C414yes—
94C2×C8×D7D414yes—
95C2×C8×D7Q814yes—
96#6455—22no—
97#6455—22no—
98#6455—22no—
99#6455—22no—
100D7×C4⋊C4—22no—
101D7×C4⋊C4—22no—
102D7×C4⋊C4—22no—
103D7×C4⋊C4—22no—
104#6435C2×C414yes—
105#6435Q814yes—
106C7×#107—22no—
107C7×#107—22no—
108C7×#106—22no—
109C7×#106—22no—
110C7×#97C2×C414yescomplemented
111C7×#95—22no—
112C7⋊#95—22no—
113C7⋊#95—22no—
114C7⋊#95—22no—
115#6392—22no—
116#6392—22no—
117#6391—22no—
118#6391—22no—
119#6389C2×C414yescomplemented
120C8×Dic7C2×C414yescomplemented
121#6382—22no—
122#6382—22no—
123#6381—22no—
124#6381—22no—
125#6381—22no—
126#6381—22no—
127#6381—22no—
128#6381—22no—
129C4×C7⋊C8C2×C414yescomplemented
130C2×#428—71no—
131C2×#428—71no—
132C2×#427—71no—
133C22×#107—71no—
134C2×#405—71no—
135#2317—71no—
136#2317—71no—
137#1847—71no—
138#1847—71no—
139#1847—71no—
140#1847D711yescharacteristic, complemented
141#1846—71no—
142#1846—71no—
143#1845—71no—
144#1845—71no—
145C23×D7C2×D411yescharacteristic
146C22×C4×C7—21no—
147C22×C4×C7—21no—
148C22×C4×C7—21no—
149C22×C4×C7—21no—
150C22×C4×C7C2×D412yes—
151C22×C4×C7C2411yescharacteristic
152C22×Dic7—21no—
153C22×Dic7—21no—
154C22×Dic7—21no—
155C22×Dic7—21no—
156C22×Dic7C2×D412yes—
157C22×Dic7C2×D411yescharacteristic
158C2×C4×D7—42no—
159C2×C4×D7—42no—
160C2×C4×D7—82no—
161C2×C4×D7—82no—
162C2×C4×D7C22×C412yes—
163C2×C4×D7C2×D412yes—
164C2×C4×D7C2×D412yes—
165C2×C4×D7C2×Q812yes—
166C2×C4×D7C2×Q812yes—
167C2×C4×D7C2×Q812yes—
168C2×C7×C8C22×C412yes—
169C2×C7×C8C22×C412yes—
170C2×C7×C8C2×D412yes—
171C2×C7×C8C2×D412yes—
172C2×C7×C8C2×Q812yes—
173C2×C7×C8C2×Q812yes—
174C7×C4⋊C4—21no—
175C7×C4⋊C4—21no—
176C7×C4⋊C4—21no—
177C7×C4⋊C4—21no—
178C7×C4⋊C4—21no—
179C7×C4⋊C4—21no—
180C7×C4⋊C4—21no—
181C7×C4⋊C4—21no—
182C7×C4⋊C4—82no—
183C42×C7C22×C412yes—
184C42×C7C2×Q812yes—
185(C2×D7)⋊C4—42no—
186(C2×D7)⋊C4—42no—
187(C2×D7)⋊C4C4⋊C418yes—
188C4⋊Dic7—21no—
189C4⋊Dic7—21no—
190C4⋊Dic7—21no—
191C4⋊Dic7—21no—
192C4⋊Dic7—21no—
193C4⋊Dic7—21no—
194C4⋊Dic7—21no—
195C4⋊Dic7—21no—
196C4⋊Dic7—82no—
197Dic7⋊C4—42no—
198Dic7⋊C4—42no—
199Dic7⋊C4—82no—
200Dic7⋊C4—82no—
201Dic7⋊C4C4⋊C418yes—
202C4×Dic7—42no—
203C4×Dic7—42no—
204C4×Dic7C22×C412yes—
205C4×Dic7C2×Q812yes—
206C2×C7⋊C8C22×C412yes—
207C2×C7⋊C8C22×C412yes—
208C2×C7⋊C8C2×D412yes—
209C2×C7⋊C8C2×D412yes—
210C2×C7⋊C8C2×Q812yes—
211C2×C7⋊C8C2×Q812yes—
212#1131C4⋊C4116yescomplemented
213C8×D7C4⋊C4116yescomplemented
214C22×M16—71no—
215C23×C8—71no—
216C2×#117—71no—
217C2×#117—71no—
218C2×#117—71no—
219C22×C4⋊C4—71no—
220C22×C4⋊C4—71no—
221#428—144no—
222#428—144no—
223#427—144no—
224C2×#107—71no—
225C2×#107—71no—
226C2×#107—71no—
227C2×#107—71no—
228C2×#107—71no—
229C2×#107—71no—
230C2×#107—71no—
231C2×#107—71no—
232C2×#107—144no—
233C2×#107C2×D711yescharacteristic
234C2×#107C2×D711yescharacteristic
235C2×#106—71no—
236C2×#106—71no—
237C2×#106—71no—
238C2×#106—71no—
239C2×#106C2×D711yescharacteristic
240C2×#106C2×D711yescharacteristic
241#405—78no—
242C2×#97—71no—
243C2×#97C2×D711yescharacteristic
244C2×C4×C8—71no—
245C2×C4×C8—71no—
246#389—142no—
247#389—142no—
248#385—142no—
249#385—142no—
250#384—71no—
251#384—71no—
252#384—71no—
253#384C2×D711yescharacteristic
254#382—71no—
255#382—71no—
256#382—71no—
257#382C2×D711yescharacteristic
258C23×C7C22×D411yescharacteristic
259C22×D7—22no—
260C22×D7—22no—
261C22×D7C2×C4⋊C412yes—
262C22×D7#12712yes—
263C22×D7C4⋊Q812yes—
264C2×C4×C7—22no—
265C2×C4×C7—22no—
266C2×C4×C7—41no—
267C2×C4×C7—41no—
268C2×C4×C7—41no—
269C2×C4×C7—41no—
270C2×C4×C7—42no—
271C2×C4×C7—42no—
272C2×C4×C7—42no—
273C2×C4×C7—42no—
274C2×C4×C7C2×C4⋊C414yes—
275C2×C4×C7C23×C411yescharacteristic
276C2×C4×C7C22×D411yescharacteristic
277C2×C4×C7C22×D411yescharacteristic
278C2×C4×C7C22×Q811yescharacteristic
279C2×C4×C7C22×Q811yescharacteristic
280C2×C4×C7C22×Q811yescharacteristic
281C2×Dic7—22no—
282C2×Dic7—22no—
283C2×Dic7—41no—
284C2×Dic7—41no—
285C2×Dic7—41no—
286C2×Dic7—41no—
287C2×Dic7—42no—
288C2×Dic7—42no—
289C2×Dic7—42no—
290C2×Dic7—42no—
291C2×Dic7C2×C4⋊C414yes—
292C2×Dic7C2×C4⋊C412yes—
293C2×Dic7#12712yes—
294C2×Dic7C4⋊Q812yes—
295C4×D7—84no—
296C4×D7—84no—
297C4×D7C2×C4⋊C414yes—
298C4×D7C2×C4⋊C414yes—
299C4×D7C4⋊Q814yes—
300C4×D7C4⋊Q814yes—
301C7×C8C2×C4⋊C414yes—
302C7×C8C2×C4⋊C414yes—
303C7⋊C8C2×C4⋊C414yes—
304C7⋊C8C2×C4⋊C414yes—
305C23×C4—71no—
306C23×C4—141no—
307C23×C4—141no—
308C2×M16—74no—
309C2×M16—74no—
310C2×M16—74no—
311C22×C8—71no—
312C22×C8—71no—
313C22×C8—74no—
314C22×C8—74no—
315C22×C8—74no—
316C22×C8C22×D711yescharacteristic
317C22×C8C22×D711yescharacteristic
318#117—74no—
319#117—74no—
320#117—142no—
321#117—142no—
322#117—142no—
323#117—142no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—72no—
326C2×C4⋊C4—71no—
327C2×C4⋊C4—71no—
328C2×C4⋊C4—71no—
329C2×C4⋊C4—71no—
330C2×C4⋊C4—142no—
331C2×C4⋊C4—142no—
332C2×C4⋊C4—142no—
333C2×C4⋊C4—142no—
334C2×C4⋊C4—141no—
335C2×C4⋊C4—141no—
336C2×C4⋊C4—142no—
337C2×C4⋊C4—142no—
338C2×C4⋊C4—142no—
339C2×C4⋊C4C22×D711yescharacteristic
340C2×C4⋊C4C22×D711yescharacteristic
341C2×C4⋊C4C22×D711yescharacteristic
342C2×C4⋊C4C22×D711yescharacteristic
343C2×C22⋊C4—72no—
344C2×C22⋊C4—141no—
345C2×C22⋊C4—141no—
346C2×C42—71no—
347C2×C42—141no—
348C2×C42—141no—
349C2×C42C22×D711yescharacteristic
350#107—22no—
351#107—22no—
352#107—142no—
353#107—142no—
354#107—142no—
355#107—142no—
356#107—142no—
357#107—142no—
358#107—142no—
359#107—142no—
360#106—22no—
361#106—22no—
362#106—142no—
363#106—142no—
364#106—142no—
365#106—142no—
366#97—74no—
367#97C4×D714yescomplemented
368C4×C8—74no—
369C4×C8—74no—
370#95—22no—
371#95—142no—
372#95—142no—
373#95—142no—
374C22×C7C22×C4⋊C411yescharacteristic
375C22×C7C2×#12711yescharacteristic
376C22×C7C2×C4⋊Q811yescharacteristic
377C22×C7#57612yes—
378C22×C7#57812yes—
379C2×D7—22no—
380C2×D7—22no—
381C2×D7—24no—
382C2×D7—24no—
383C2×D7#38414yes—
384C4×C7—24no—
385C4×C7—81no—
386C4×C7—81no—
387C4×C7—81no—
388C4×C7—81no—
389C4×C7C22×C4⋊C411yescharacteristic
390C4×C7C22×C4⋊C411yescharacteristic
391C4×C7C2×C4⋊Q811yescharacteristic
392C4×C7C2×C4⋊Q811yescharacteristic
393Dic7—24no—
394Dic7—81no—
395Dic7—81no—
396Dic7—81no—
397Dic7—81no—
398Dic7#38414yes—
399C24—71no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—71no—
405C22×C4—72no—
406C22×C4—72no—
407C22×C4—72no—
408C22×C4—72no—
409C22×C4—72no—
410C22×C4—72no—
411C22×C4—72no—
412C22×C4—141no—
413C22×C4—141no—
414C22×C4—141no—
415C22×C4—141no—
416C22×C4—282no—
417C22×C4—282no—
418C22×C4—562no—
419C22×C4—562no—
420C22×C4D4×D712yes—
421C22×C4C23×D711yescharacteristic
422M16—716no—
423C2×C8—72no—
424C2×C8—72no—
425C2×C8—72no—
426C2×C8—72no—
427C2×C8—72no—
428C2×C8—72no—
429C2×C8—716no—
430C2×C8C2×C4×D712yes—
431C2×C8C2×C4×D712yes—
432C2×C8D4×D712yes—
433C2×C8D4×D712yes—
434C2×C8Q8×D712yes—
435C2×C8Q8×D712yes—
436C4⋊C4—21no—
437C4⋊C4—21no—
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—78no—
445C4⋊C4—82no—
446C4⋊C4—141no—
447C4⋊C4—141no—
448C4⋊C4—141no—
449C4⋊C4—141no—
450C4⋊C4—141no—
451C4⋊C4—141no—
452C4⋊C4—141no—
453C4⋊C4—141no—
454C4⋊C4—282no—
455C4⋊C4—282no—
456C4⋊C4—562no—
457C4⋊C4—562no—
458C4⋊C4—562no—
459C22⋊C4—78no—
460C22⋊C4—282no—
461C22⋊C4—282no—
462C42—72no—
463C42—72no—
464C42—282no—
465C42—282no—
466C42C2×C4×D712yes—
467C42Q8×D712yes—
468C2×C7C2×#38411yescharacteristic
469C2×C7#291312yes—
470C2×C7#315511yescharacteristic
471C2×C7#315611yescharacteristic
472C2×C7#316412yes—
473D7—24no—
474C23—72no—
475C23—72no—
476C23—72no—
477C23—142no—
478C23—142no—
479C23C2×D4×D711yescharacteristic
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—72no—
491C2×C4—72no—
492C2×C4—72no—
493C2×C4—74no—
494C2×C4—74no—
495C2×C4—74no—
496C2×C4—74no—
497C2×C4—74no—
498C2×C4—142no—
499C2×C4—142no—
500C2×C4—281no—
501C2×C4—281no—
502C2×C4—281no—
503C2×C4—281no—
504C2×C4—282no—
505C2×C4—282no—
506C2×C4—282no—
507C2×C4—282no—
508C2×C4—564no—
509C2×C4—564no—
510C2×C4D7×C4⋊C414yes—
511C2×C4C22×C4×D711yescharacteristic
512C2×C4C2×D4×D711yescharacteristic
513C2×C4C2×D4×D711yescharacteristic
514C2×C4C2×Q8×D711yescharacteristic
515C2×C4C2×Q8×D711yescharacteristic
516C2×C4C2×Q8×D711yescharacteristic
517C8—74no—
518C8—74no—
519C8D7×C4⋊C414yes—
520C8D7×C4⋊C414yes—
521C7#2080311yescharacteristic, complemented, Sylow
522C22—74no—
523C22—144no—
524C22—144no—
525C22—142no—
526C22—142no—
527C22C2×D7×C4⋊C411yescharacteristic
528C22D7×#12711yescharacteristic
529C22D7×C4⋊Q811yescharacteristic
530C22#9046512yes—
531C22#9047012yes—
532C4—24no—
533C4—74no—
534C4—81no—
535C4—81no—
536C4—81no—
537C4—81no—
538C4—144no—
539C4—561no—
540C4—561no—
541C4—561no—
542C4—561no—
543C4C2×D7×C4⋊C411yescharacteristic
544C4C2×D7×C4⋊C411yescharacteristic
545C4D7×C4⋊Q811yescharacteristic
546C4D7×C4⋊Q811yescharacteristic
547C2—144no—
548C2D7×#38411yescharacteristic
549C2#1173665512yes—
550C2#1173929311yescharacteristic
551C2#1173931811yescharacteristic
552C2#1173942012yes—
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1847C2
socleC23×C7C22×D4
prime radicalC23×D7C2×D4
cosocleC22×C4×C7C24
hypercenter#1847D7
hypocenterC7#20803
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1847

Complements

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

NH ≅ G/Ntype
8. C7×#1847C2semidirect
9. #11728362C2semidirect
10. #11728360C2semidirect
11. #11728310C2semidirect
12. #11728309C2semidirect
13. #11727570C2semidirect
14. #11727331C2semidirect
15. #11727303C2semidirect
57. #89480C4semidirect
110. C7×#97C2×C4semidirect
119. #6389C2×C4semidirect
120. C8×Dic7C2×C4semidirect
129. C4×C7⋊C8C2×C4semidirect
140. #1847D7semidirect
212. #1131C4⋊C4semidirect
213. C8×D7C4⋊C4semidirect
367. #97C4×D7semidirect
521. C7#20803semidirect

Maximal subgroups

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

subgroupindex
C2×#896532
C2×D7×#1072
C2×#896332
C2×#896322
C2×#894802
#117318042
#117316702
C7×#18472
#117283622
#117283602
#117283102
#117283092
#117275702
#117273312
#117273032
#208037