Finite Group Databasebeta

← #14217988

Subgroups — #14217988

Order 1792.

2666 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#90503C211yesmaximal, characteristic, complemented
2C22×#6505C211yesmaximal, characteristic
3#11742799C211yesmaximal, characteristic
4C2×C7×#419C211yesmaximal, characteristic, complemented
5#11741213C211yesmaximal, characteristic
6#11741192C211yesmaximal, characteristic, complemented
7#11739652C211yesmaximal, characteristic, complemented
8#11736393C214yesmaximal, complemented
9#11736392C214yesmaximal, complemented
10#11736391C218yesmaximal, complemented
11#11736219C218yesmaximal, complemented
12C23×C7×Q8C2211yescharacteristic
13C22×#1157C2211yescharacteristic
14C22×C7×M16C2211yescharacteristic
15C2×C7×#117C2211yescharacteristic
16#90503—22no—
17#90503—22no—
18#90503C2214yes—
19#90503C2214yescomplemented
20C2×Q8×Dic7—22no—
21C2×Q8×Dic7C2214yescomplemented
22C22×#1144C2212yes—
23C2×#6505C4116yescomplemented
24C2×#6505C2214yes—
25C2×#6505C2214yes—
26C22×#1143C2212yes—
27C2×#6489—22no—
28C2×#6489C2211yescharacteristic
29C2×#6489C2211yescharacteristic
30C22×#1136C2211yescharacteristic
31#90214—22no—
32#90213C2214yes—
33#90212C2212yes—
34D7×#117—22no—
35#90207C2212yes—
36#90195C2214yes—
37#90177—22no—
38C4×#1157—22no—
39C7×#419C2214yes—
40C7×#419C2214yescomplemented
41C2×C7×#103C2214yescomplemented
42#89905C2214yes—
43#89903C2212yes—
44#89894C2214yes—
45#89892C2212yes—
46M16×Dic7C2214yescomplemented
47#89885C2214yes—
48#89873C2214yescomplemented
49#89770C2214yes—
50#89767C2214yescomplemented
51#89736C2214yescomplemented
52#89589C22116yescomplemented
53#89562C22116yescomplemented
54#89561C22116yescomplemented
55#89560C22116yescomplemented
56#60169—71nomaximal, Sylow
57C22×C7×Q8—21no—
58C22×C7×Q8—21no—
59C22×C7×Q8—21no—
60C22×C7×Q8—24no—
61C22×C7×Q8C2×C418yescomplemented
62C22×C7×Q8C2312yes—
63C22×C7×Q8C2312yes—
64C22×C7×Q8C2312yes—
65C23×C4×C7—21no—
66C23×C4×C7C2311yescharacteristic
67C22×C22⋊D7—21no—
68C2×#1157—21no—
69C2×#1157—21no—
70C2×#1157—24no—
71C2×#1157C2×C418yes—
72C2×#1157C2312yes—
73C2×#1157C2312yes—
74C22×D28C2311yescharacteristic
75C22×C4×D7—21no—
76C22×Dic14C2311yescharacteristic
77C2×C7×M16—24no—
78C2×C7×M16C2312yes—
79C2×C7×M16C2312yes—
80C22×C7×C8C2312yes—
81C7×#117—21no—
82C7×#117—21no—
83C7×#117C2312yes—
84C7×#117C2312yes—
85C2×C7×C4⋊C4C2312yes—
86C2×C7×C22⋊C4—21no—
87C2×C42×C7—21no—
88C2×C22⋊Dic7—21no—
89C2×C22⋊Dic7D412yes—
90Q8×Dic7—28no—
91Q8×Dic7—28no—
92Q8×Dic7—28no—
93Q8×Dic7C23116yes—
94C2×#1144C2×C4116yescomplemented
95C2×#1144C2318yes—
96#6505C2×C4132yescomplemented
97#6505C2×C4132yescomplemented
98C2×#1143C2×C4116yescomplemented
99C2×#1143C2318yes—
100C4×C22⋊D7—28no—
101#6489—21no—
102#6489—21no—
103#6489—28no—
104#6489—28no—
105#6489D412yes—
106#6489D412yes—
107#6489C2312yes—
108#6489C2312yes—
109#6489C2312yes—
110#6489C2312yes—
111C2×C4⋊Dic7—24no—
112C2×C4⋊Dic7D412yes—
113C2×C4⋊Dic7C2312yes—
114C2×C4×Dic7—21no—
115C2×C4×Dic7—24no—
116C2×C4×Dic7C2312yes—
117C2×#1136C2×C418yescomplemented
118C2×#1136C2312yes—
119C2×#1136C2312yes—
120C22×C7⋊C8C2312yes—
121#6456C2318yes—
122#6455—24no—
123D7×C4⋊C4—24no—
124#6450C2318yes—
125(C22⋊D7)⋊C4—28no—
126D7×C22⋊C4—24no—
127#6440—24no—
128C4×D28—24no—
129#6435—24no—
130C42×D7—24no—
131C4×Dic14—24no—
132C7×#103C23116yes—
133#6395C2318yes—
134#6390C2318yes—
135#6389C2318yes—
136C8×Dic7C2318yes—
137#6382C2318yes—
138#6381C2318yes—
139C22×#137—71no—
140#3428—71no—
141C2×#519—71no—
142#2950—74no—
143#2949—74no—
144#2947—78no—
145C4×#137—78no—
146#2911—71no—
147#2896—71no—
148C2×#419D711yescharacteristic, complemented
149#2873—71no—
150C24×C7C2×D411yescharacteristic
151C23×D7—21no—
152C2×C7×Q8—28no—
153C2×C7×Q8—28no—
154C2×C7×Q8—28no—
155C2×C7×Q8—22no—
156C2×C7×Q8—22no—
157C2×C7×Q8—22no—
158C2×C7×Q8—24no—
159C2×C7×Q8—24no—
160C2×C7×Q8—24no—
161C2×C7×Q8C22×C418yes—
162C2×C7×Q8C22×C418yes—
163C2×C7×Q8C22×C418yes—
164C2×C7×Q8C2414yes—
165C22×C4×C7—28no—
166C22×C4×C7—22no—
167C22×C4×C7—22no—
168C22×C4×C7—22no—
169C22×C4×C7—22no—
170C22×C4×C7C22×C414yes—
171C22×C4×C7C2×D411yescharacteristic
172C22×C4×C7C2×D411yescharacteristic
173C22×C4×C7C2×D411yescharacteristic
174C22×C4×C7#4114yes—
175C22×C4×C7C2411yescharacteristic
176C22×C4×C7C2411yescharacteristic
177C22×C4×C7C2411yescharacteristic
178C2×C22⋊D7—28no—
179C2×C22⋊D7—22no—
180C2×C22⋊D7—22no—
181C22×Dic7—21no—
182C22×Dic7—22no—
183C22×Dic7C2×D414yes—
184#1157—28no—
185#1157—28no—
186#1157—24no—
187#1157—24no—
188#1157C22×C418yes—
189#1157C22×C418yes—
190C2×D28—21no—
191C2×D28—22no—
192C2×D28C22×C414yes—
193C2×D28C2412yes—
194C2×C4×D7—28no—
195C2×C4×D7—22no—
196C2×C4×D7—22no—
197C2×C4×D7—28no—
198C2×Dic14—21no—
199C2×Dic14—22no—
200C2×Dic14C22×C414yes—
201C2×Dic14C2412yes—
202C7×M16—24no—
203C7×M16—24no—
204C2×C7×C8—24no—
205C2×C7×C8C2414yes—
206C7×C4⋊C4—22no—
207C7×C4⋊C4C2414yes—
208C7×C22⋊C4—22no—
209C7×C22⋊C4—22no—
210C42×C7—22no—
211C42×C7—22no—
212C22⋊Dic7—22no—
213C22⋊Dic7—22no—
214C22⋊Dic7C2×D414yes—
215C22⋊Dic7C2×D414yes—
216#1144C22×C4132yes—
217#1143C22×C4132yes—
218(C2×D7)⋊C4—28no—
219C4⋊Dic7—22no—
220C4⋊Dic7—216no—
221C4⋊Dic7C2×D414yes—
222C4⋊Dic7C2×D414yes—
223C4⋊Dic7C2414yes—
224Dic7⋊C4—28no—
225C4×Dic7—22no—
226C4×Dic7—22no—
227C4×Dic7—28no—
228C4×Dic7—216no—
229C4×Dic7C2×D414yes—
230C4×Dic7C2414yes—
231#1136C22×C418yes—
232#1136C22×C418yes—
233C2×C7⋊C8C22×C418yes—
234C2×C7⋊C8C2414yes—
235C22×#41—71no—
236C23×Q8C2×D711yescharacteristic
237C2×#137—716no—
238C2×#137—74no—
239C2×#137—74no—
240C22×Q16—72no—
241C22×SD16—72no—
242C22×M16—71no—
243C22×M16C2×D711yescharacteristic
244#520—74no—
245#520—142no—
246#519—74no—
247#519—74no—
248#519—72no—
249#519—142no—
250#519—142no—
251#518—72no—
252#518—142no—
253C4×#41—74no—
254C4×#41—142no—
255C2×C4×Q8—74no—
256C2×C4×Q8—142no—
257C2×#117—71no—
258C2×#117—71no—
259C2×#117—142no—
260C2×#117—142no—
261C2×#117C2×D711yescharacteristic
262#441—716no—
263#440—716no—
264C4×Q16—716no—
265C4×SD16—716no—
266#428—74no—
267#428—74no—
268#427—74no—
269#419—72no—
270#419C2×D714yes—
271#419C2×D714yescomplemented
272#418—72no—
273#417—74no—
274#416—74no—
275C2×#103C2×D714yescomplemented
276#405—74no—
277#405—74no—
278C4×M16—74no—
279C23×C7—42no—
280C23×C7C4×D414yes—
281C23×C7C22×D411yescharacteristic
282C23×C7C22×D411yescharacteristic
283C23×C7C22×D411yescharacteristic
284C22×D7—22no—
285C22×D7—24no—
286C22×D7—82no—
287C7×Q8—28no—
288C7×Q8—28no—
289C7×Q8—28no—
290C7×Q8—24no—
291C7×Q8C23×C418yes—
292C2×C4×C7—28no—
293C2×C4×C7—28no—
294C2×C4×C7—28no—
295C2×C4×C7—24no—
296C2×C4×C7—24no—
297C2×C4×C7—82no—
298C2×C4×C7C4×D414yes—
299C2×C4×C7C4×D414yes—
300C2×C4×C7C4×D414yes—
301C2×C4×C7C23×C412yes—
302C2×C4×C7C23×C412yes—
303C2×C4×C7C23×C412yes—
304C2×C4×C7C22×D411yescharacteristic
305C2×C4×C7C22×D411yescharacteristic
306C2×C4×C7C22×D411yescharacteristic
307C2×C4×C7C2×#4112yes—
308C2×C4×C7C2×#4112yes—
309C2×C4×C7C2×#4112yes—
310C2×C4×C7C2511yescharacteristic
311C22⋊D7—28no—
312C22⋊D7—28no—
313C2×Dic7—24no—
314C2×Dic7—22no—
315C2×Dic7—24no—
316C2×Dic7—44no—
317C2×Dic7—44no—
318C2×Dic7C22×D418yes—
319D28—24no—
320D28—22no—
321D28C23×C414yes—
322C4×D7—28no—
323C4×D7—28no—
324C4×D7—88no—
325Dic14—24no—
326Dic14—22no—
327Dic14C23×C414yes—
328C7×C8—24no—
329C7⋊C8C23×C418yes—
330C2×#41—78no—
331C2×#41—72no—
332C2×#41—72no—
333C2×#41—141no—
334C2×#41—141no—
335C2×#41—144no—
336C22×Q8—21no—
337C22×Q8—21no—
338C22×Q8—21no—
339C22×Q8—24no—
340C22×Q8—71no—
341C22×Q8C4×D718yescomplemented
342C22×Q8C22×D712yes—
343C22×Q8C22×D712yes—
344C22×Q8C22×D712yes—
345C22×D4—71no—
346C22×D4—141no—
347C23×C4—21no—
348C23×C4—141no—
349C23×C4C22×D711yescharacteristic
350#137—732no—
351#137—732no—
352C2×Q16—716no—
353C2×Q16—78no—
354C2×SD16—716no—
355C2×SD16—78no—
356C2×M16—24no—
357C2×M16—78no—
358C2×M16—72no—
359C2×M16—72no—
360C2×M16C22×D712yes—
361C2×M16C22×D712yes—
362C22×C8—72no—
363C22×C8C22×D712yes—
364C4×Q8—716no—
365C4×Q8—78no—
366C4×Q8—144no—
367C4×Q8—148no—
368C4×Q8—148no—
369C4×Q8—148no—
370C4×D4—78no—
371C4×D4—144no—
372C4×D4—148no—
373C4×D4—148no—
374#117—21no—
375#117—21no—
376#117—72no—
377#117—72no—
378#117—72no—
379#117—72no—
380#117—72no—
381#117—72no—
382#117—144no—
383#117—144no—
384#117—148no—
385#117—148no—
386#117—144no—
387#117—141no—
388#117—141no—
389#117C22×D712yes—
390#117C22×D712yes—
391C2×C4⋊C4—72no—
392C2×C4⋊C4—72no—
393C2×C4⋊C4—144no—
394C2×C4⋊C4—144no—
395C2×C4⋊C4C22×D712yes—
396C2×C22⋊C4—21no—
397C2×C22⋊C4—72no—
398C2×C22⋊C4—144no—
399C2×C22⋊C4—141no—
400C2×C42—21no—
401C2×C42—72no—
402C2×C42—144no—
403C2×C42—144no—
404C2×C42—141no—
405#107—78no—
406#106—78no—
407#103—78no—
408#103C22×D7116yes—
409#102—78no—
410#97—78no—
411C4×C8—78no—
412C22×C7—22no—
413C22×C7—22no—
414C22×C7—22no—
415C22×C7—44no—
416C22×C7C2×C4×D412yes—
417C22×C7C2×C4×D412yes—
418C22×C7C2×C4×D412yes—
419C22×C7C23×D411yescharacteristic
420C2×D7—24no—
421C2×D7—44no—
422C2×D7—84no—
423C4×C7—28no—
424C4×C7—44no—
425C4×C7C2×C4×D412yes—
426C4×C7C2×C4×D412yes—
427C4×C7C2×C4×D412yes—
428C4×C7C24×C411yescharacteristic
429C4×C7C22×#4111yescharacteristic
430Dic7—28no—
431Dic7—24no—
432Dic7—44no—
433C24—141no—
434C24D4×D711yescharacteristic
435#41—78no—
436#41—78no—
437#41—148no—
438#41—148no—
439#41—144no—
440#41—144no—
441C2×Q8—28no—
442C2×Q8—28no—
443C2×Q8—28no—
444C2×Q8—22no—
445C2×Q8—22no—
446C2×Q8—22no—
447C2×Q8—24no—
448C2×Q8—24no—
449C2×Q8—24no—
450C2×Q8—74no—
451C2×Q8—72no—
452C2×Q8—141no—
453C2×Q8—142no—
454C2×Q8C2×C4×D718yes—
455C2×Q8C2×C4×D718yes—
456C2×Q8C2×C4×D718yes—
457C2×Q8C23×D714yes—
458C2×D4—74no—
459C2×D4—72no—
460C2×D4—148no—
461C2×D4—142no—
462C2×D4—142no—
463C2×D4—141no—
464C2×D4—142no—
465C22×C4—28no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—74no—
471C22×C4—148no—
472C22×C4—141no—
473C22×C4—148no—
474C22×C4—142no—
475C22×C4—142no—
476C22×C4—142no—
477C22×C4C2×C4×D714yes—
478C22×C4D4×D711yescharacteristic
479C22×C4D4×D711yescharacteristic
480C22×C4D4×D711yescharacteristic
481C22×C4#115914yes—
482C22×C4C23×D711yescharacteristic
483C22×C4C23×D711yescharacteristic
484C22×C4C23×D711yescharacteristic
485Q16—732no—
486SD16—732no—
487M16—24no—
488M16—24no—
489M16—78no—
490M16—78no—
491C2×C8—24no—
492C2×C8—78no—
493C2×C8—74no—
494C2×C8C23×D714yes—
495C4⋊C4—22no—
496C4⋊C4—74no—
497C4⋊C4—74no—
498C4⋊C4—74no—
499C4⋊C4—148no—
500C4⋊C4—1416no—
501C4⋊C4—142no—
502C4⋊C4C23×D714yes—
503C22⋊C4—22no—
504C22⋊C4—22no—
505C22⋊C4—74no—
506C22⋊C4—74no—
507C22⋊C4—148no—
508C22⋊C4—142no—
509C22⋊C4—142no—
510C42—22no—
511C42—22no—
512C42—74no—
513C42—74no—
514C42—148no—
515C42—1416no—
516C42—142no—
517C42—142no—
518C2×C7—22no—
519C2×C7—22no—
520C2×C7—22no—
521C2×C7C22×C4×D411yescharacteristic
522C2×C7#358612yes—
523D7—44no—
524C23—42no—
525C23—142no—
526C23—144no—
527C23—562no—
528C23(C22⋊D7)⋊C414yes—
529C23C2×D4×D711yescharacteristic
530C23C2×D4×D711yescharacteristic
531C23C2×D4×D711yescharacteristic
532Q8—28no—
533Q8—28no—
534Q8—28no—
535Q8—24no—
536Q8—74no—
537Q8—144no—
538Q8—142no—
539Q8C22×C4×D718yes—
540D4—74no—
541D4—148no—
542D4—148no—
543D4—144no—
544D4—142no—
545C2×C4—28no—
546C2×C4—28no—
547C2×C4—28no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—78no—
551C2×C4—82no—
552C2×C4—144no—
553C2×C4—142no—
554C2×C4—148no—
555C2×C4—148no—
556C2×C4—144no—
557C2×C4—284no—
558C2×C4—284no—
559C2×C4—568no—
560C2×C4(C22⋊D7)⋊C414yes—
561C2×C4(C22⋊D7)⋊C414yes—
562C2×C4(C22⋊D7)⋊C414yes—
563C2×C4C22×C4×D712yes—
564C2×C4C22×C4×D712yes—
565C2×C4C22×C4×D712yes—
566C2×C4C2×D4×D711yescharacteristic
567C2×C4C2×D4×D711yescharacteristic
568C2×C4C2×D4×D711yescharacteristic
569C2×C4C2×#115912yes—
570C2×C4C2×#115912yes—
571C2×C4C2×#115912yes—
572C2×C4C24×D711yescharacteristic
573C8—24no—
574C8—78no—
575C7#6016911yescharacteristic, complemented, Sylow
576C22—22no—
577C22—22no—
578C22—22no—
579C22—44no—
580C22—144no—
581C22—284no—
582C22—564no—
583C22C2×(C22⋊D7)⋊C412yes—
584C22C2×(C22⋊D7)⋊C412yes—
585C22C2×(C22⋊D7)⋊C412yes—
586C22C22×D4×D711yescharacteristic
587C4—28no—
588C4—44no—
589C4—148no—
590C4—144no—
591C4—284no—
592C4C2×(C22⋊D7)⋊C412yes—
593C4C2×(C22⋊D7)⋊C412yes—
594C4C2×(C22⋊D7)⋊C412yes—
595C4C23×C4×D711yescharacteristic
596C4C22×#115911yescharacteristic
597C2—22no—
598C2—22no—
599C2—22no—
600C2—284no—
601C2C22×(C22⋊D7)⋊C411yescharacteristic
602C2#1174426512yes—
subgroupG/H
centerC22C22×D4×D7
commutatorC4×C7C24×C4
FrattiniC2×C4C24×D7
FittingC2×C7×#419C2
socleC22×C7C23×D4
prime radicalC22×#1157C22
cosocleC2×C4×C7C25
hypercenterC2×#419D7
hypocenterC7#60169
LCSG ▷ C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#419

Complements

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

NH ≅ G/Ntype
1. C2×#90503C2semidirect
4. C2×C7×#419C2semidirect
6. #11741192C2semidirect
7. #11739652C2semidirect
8. #11736393C2semidirect
9. #11736392C2semidirect
10. #11736391C2semidirect
11. #11736219C2semidirect
19. #90503C22semidirect
21. C2×Q8×Dic7C22semidirect
23. C2×#6505C4semidirect
40. C7×#419C22semidirect
41. C2×C7×#103C22semidirect
46. M16×Dic7C22semidirect
48. #89873C22semidirect
50. #89767C22semidirect
51. #89736C22semidirect
52. #89589C22semidirect
53. #89562C22semidirect
54. #89561C22semidirect
55. #89560C22semidirect
61. C22×C7×Q8C2×C4semidirect
94. C2×#1144C2×C4semidirect
96. #6505C2×C4semidirect
97. #6505C2×C4semidirect
98. C2×#1143C2×C4semidirect
117. C2×#1136C2×C4semidirect
148. C2×#419D7semidirect
271. #419C2×D7semidirect
275. C2×#103C2×D7semidirect
341. C22×Q8C4×D7semidirect
575. C7#60169semidirect

Maximal subgroups

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

subgroupindex
C2×#905032
C22×#65052
#117427992
C2×C7×#4192
#117412132
#117411922
#117396522
#117363932
#117363922
#117363912
#117362192
#601697