Finite Group Databasebeta

← #13354522

Subgroups — #13354522

Order 1792.

1590 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6455C211yesmaximal, characteristic
2C2×#89632C212yesmaximal, complemented
3C2×D7×C22⋊C8C211yesmaximal, characteristic
4#11731307C211yesmaximal, characteristic
5C7×#1853C211yesmaximal, characteristic, complemented
6#11729309C211yesmaximal, characteristic, complemented
7#11728334C212yesmaximal, complemented
8#11727682C211yesmaximal, characteristic, complemented
9#11727258C211yesmaximal, characteristic, complemented
10#11726630C212yesmaximal, complemented
11#11726620C212yesmaximal, complemented
12C23×C4×D7C2211yescharacteristic
13C22×C7×C4⋊C4C2211yescharacteristic
14C22×(C2×D7)⋊C4—21no—
15C22×C4⋊Dic7C2211yescharacteristic
16C22×C4×Dic7—21no—
17C22×C8×D7C2212yes—
18C2×#6455—21no—
19C2×#6455—21no—
20C2×#6455—28no—
21C2×#6455C2211yescharacteristic
22C2×#6455C2212yes—
23C2×#6455C2211yescharacteristic
24C2×#6455C2212yes—
25C2×C7×#106C2212yescomplemented
26C2×C7×C22⋊C8C2211yescharacteristic
27C7×#380C2211yescharacteristic
28#89984C2211yescharacteristic
29C2×C22⋊(C7⋊C8)C2211yescharacteristic
30C2×(C2×D7)⋊C8C2212yes—
31C2×#6391C2212yescomplemented
32#89759C2211yescharacteristic
33#89759C2211yescharacteristic
34C2×#6381C2212yescomplemented
35C2×#6381C2212yescomplemented
36#89727—21no—
37C22⋊C4×Dic7—21no—
38#89632—28no—
39D7×C22⋊C8C418yescomplemented
40#89438—21no—
41#89436—21no—
42C7×#340C2212yescomplemented
43#89346C2212yescomplemented
44#89324C2212yescomplemented
45#89324C2212yescomplemented
46#20843—71nomaximal, Sylow
47C24×D7D411yescharacteristic
48C23×C4×C7—21no—
49C23×C4×C7C2311yescharacteristic
50C23×Dic7—21no—
51C23×Dic7D411yescharacteristic
52C22×C4×D7—24no—
53C22×C4×D7—24no—
54C22×C4×D7C2×C414yes—
55C22×C4×D7D411yescharacteristic
56C22×C4×D7D411yescharacteristic
57C22×C4×D7Q814yes—
58C22×C4×D7C2311yescharacteristic
59C22×C4×D7C2311yescharacteristic
60C22×C7×C8C2312yes—
61C2×C7×C4⋊C4—21no—
62C2×C7×C4⋊C4—24no—
63C2×C7×C4⋊C4C2311yescharacteristic
64C2×C7×C4⋊C4C2311yescharacteristic
65C2×C7×C4⋊C4C2312yes—
66C2×C7×C22⋊C4—21no—
67C2×C22⋊Dic7—21no—
68C2×(C2×D7)⋊C4—28no—
69C2×(C2×D7)⋊C4—22no—
70C2×(C2×D7)⋊C4—22no—
71C2×(C2×D7)⋊C4—21no—
72C2×(C2×D7)⋊C4—21no—
73C2×(C2×D7)⋊C4—21no—
74C2×(C2×D7)⋊C4—21no—
75C2×C4⋊Dic7—21no—
76C2×C4⋊Dic7—24no—
77C2×C4⋊Dic7C2311yescharacteristic
78C2×C4⋊Dic7C2311yescharacteristic
79C2×C4⋊Dic7C2312yes—
80C2×C4×Dic7—28no—
81C2×C4×Dic7—22no—
82C2×C4×Dic7—22no—
83C2×C4×Dic7—21no—
84C2×C4×Dic7—21no—
85C22×C7⋊C8C2312yes—
86C2×C8×D7—24no—
87C2×C8×D7—24no—
88C2×C8×D7C2×C418yescomplemented
89#6455—28no—
90#6455—28no—
91#6455—24no—
92#6455—24no—
93#6455—24no—
94#6455—24no—
95#6455—24no—
96#6455D418yes—
97C7×#106—24no—
98C7×C22⋊C8C2×C414yes—
99C7×#95—21no—
100C7⋊#95—21no—
101C7⋊#95—21no—
102C7⋊#95—21no—
103C22⋊(C7⋊C8)C2×C414yes—
104(C2×D7)⋊C8C2×C418yescomplemented
105#6391—24no—
106#6381—24no—
107#6381—24no—
108C22×#117—71no—
109C2×#427—72no—
110C22×C22⋊C8—71no—
111#2280—71no—
112#1854—72no—
113#1854—72no—
114#1853—71no—
115#1853D711yescharacteristic, complemented
116#1852—71no—
117#1852—71no—
118#1735—72no—
119C24×C7C2×D411yescharacteristic
120C23×D7—22no—
121C23×D7—22no—
122C23×D7—22no—
123C23×D7—22no—
124C23×D7—41no—
125C23×D7—41no—
126C23×D7C4⋊C414yes—
127C23×D7C2×D411yescharacteristic
128C23×D7C2×D411yescharacteristic
129C22×C4×C7—22no—
130C22×C4×C7—22no—
131C22×C4×C7—24no—
132C22×C4×C7—22no—
133C22×C4×C7—21no—
134C22×C4×C7—21no—
135C22×C4×C7—42no—
136C22×C4×C7—42no—
137C22×C4×C7C22×C412yes—
138C22×C4×C7C2×D411yescharacteristic
139C22×C4×C7C2×Q812yes—
140C22×C4×C7C2411yescharacteristic
141C22×Dic7—22no—
142C22×Dic7—22no—
143C22×Dic7—24no—
144C22×Dic7—22no—
145C22×Dic7—21no—
146C22×Dic7—21no—
147C22×Dic7—42no—
148C22×Dic7—42no—
149C22×Dic7C4⋊C414yes—
150C22×Dic7C2×D411yescharacteristic
151C22×Dic7C2×D411yescharacteristic
152C2×C4×D7—22no—
153C2×C4×D7—22no—
154C2×C4×D7—22no—
155C2×C4×D7—22no—
156C2×C4×D7—22no—
157C2×C4×D7—22no—
158C2×C4×D7—216no—
159C2×C4×D7—216no—
160C2×C4×D7C4⋊C414yes—
161C2×C4×D7C4⋊C414yes—
162C2×C4×D7C22×C412yes—
163C2×C4×D7C22×C412yes—
164C2×C4×D7C2×D414yes—
165C2×C4×D7C2×Q812yes—
166C2×C4×D7C2×Q812yes—
167C2×C4×D7#4114yes—
168C2×C7×C8—22no—
169C2×C7×C8—22no—
170C2×C7×C8C22×C414yes—
171C7×C4⋊C4—24no—
172C7×C4⋊C4—22no—
173C7×C4⋊C4—22no—
174C7×C4⋊C4—22no—
175C7×C4⋊C4C2×D414yes—
176C7×C22⋊C4—42no—
177C22⋊Dic7—42no—
178(C2×D7)⋊C4—28no—
179(C2×D7)⋊C4—28no—
180(C2×D7)⋊C4—42no—
181(C2×D7)⋊C4—42no—
182(C2×D7)⋊C4—42no—
183(C2×D7)⋊C4—42no—
184C4⋊Dic7—24no—
185C4⋊Dic7—22no—
186C4⋊Dic7—22no—
187C4⋊Dic7—22no—
188C4⋊Dic7C2×D414yes—
189C4×Dic7—28no—
190C4×Dic7—28no—
191C4×Dic7—42no—
192C4×Dic7—42no—
193C2×C7⋊C8—22no—
194C2×C7⋊C8—22no—
195C2×C7⋊C8C22×C414yes—
196C8×D7—216no—
197C24×C4—71no—
198C23×C8—72no—
199C2×#117—72no—
200C2×#117—72no—
201C2×#117—71no—
202C2×#117—71no—
203C2×#117—148no—
204C2×#117—141no—
205C2×#117—141no—
206C22×C4⋊C4—71no—
207C22×C4⋊C4C2×D711yescharacteristic
208C22×C22⋊C4—141no—
209C22×C42—141no—
210#427—148no—
211C2×#107—72no—
212C2×#107—72no—
213C2×#106—72no—
214C2×#106C2×D712yescomplemented
215C2×C22⋊C8—71no—
216C2×C22⋊C8—72no—
217C2×C22⋊C8—78no—
218C2×C22⋊C8C2×D711yescharacteristic
219#381—141no—
220#380—71no—
221#380—71no—
222#380—71no—
223#380C2×D711yescharacteristic
224#379—141no—
225C4×C22⋊C4—141no—
226C4×C22⋊C4—141no—
227#340—72no—
228#340—72no—
229#340—72no—
230#340C2×D712yescomplemented
231C23×C7—21no—
232C23×C7—21no—
233C23×C7—21no—
234C23×C7—21no—
235C23×C7C2×C4⋊C412yes—
236C23×C7#13512yes—
237C23×C7#13512yes—
238C23×C7C22×D411yescharacteristic
239C22×D7—21no—
240C22×D7—21no—
241C22×D7—21no—
242C22×D7—21no—
243C22×D7—21no—
244C22×D7—21no—
245C22×D7—21no—
246C22×D7—21no—
247C22×D7—21no—
248C22×D7—21no—
249C22×D7—28no—
250C22×D7—28no—
251C22×D7—44no—
252C22×D7—44no—
253C22×D7—44no—
254C22×D7—44no—
255C22×D7—44no—
256C22×D7C2×C4⋊C412yes—
257C22×D7C2×C4⋊C412yes—
258C22×D7#12012yes—
259C22×D7#12312yes—
260C2×C4×C7—21no—
261C2×C4×C7—21no—
262C2×C4×C7—24no—
263C2×C4×C7—24no—
264C2×C4×C7—24no—
265C2×C4×C7—48no—
266C2×C4×C7—42no—
267C2×C4×C7—42no—
268C2×C4×C7—41no—
269C2×C4×C7—41no—
270C2×C4×C7—41no—
271C2×C4×C7—41no—
272C2×C4×C7—41no—
273C2×C4×C7—41no—
274C2×C4×C7C2×C4⋊C412yes—
275C2×C4×C7C23×C411yescharacteristic
276C2×C4×C7C22×D412yes—
277C2×C4×C7C22×Q811yescharacteristic
278C2×C4×C7C2×#4112yes—
279C2×Dic7—21no—
280C2×Dic7—21no—
281C2×Dic7—28no—
282C2×Dic7—24no—
283C2×Dic7—48no—
284C2×Dic7—42no—
285C2×Dic7—42no—
286C2×Dic7—41no—
287C2×Dic7—41no—
288C2×Dic7—41no—
289C2×Dic7—41no—
290C2×Dic7—41no—
291C2×Dic7—41no—
292C2×Dic7C2×C4⋊C412yes—
293C2×Dic7C2×C4⋊C412yes—
294C2×Dic7#12012yes—
295C2×Dic7#12312yes—
296C4×D7—28no—
297C4×D7—24no—
298C4×D7—24no—
299C4×D7—28no—
300C4×D7C4×D418yes—
301C4×D7C22⋊Q818yes—
302C7×C8—24no—
303C7⋊C8—24no—
304C25—71no—
305C23×C4—21no—
306C23×C4—71no—
307C23×C4—71no—
308C23×C4—74no—
309C23×C4—71no—
310C23×C4—71no—
311C23×C4—74no—
312C23×C4—71no—
313C23×C4—144no—
314C23×C4—144no—
315C23×C4—141no—
316C23×C4C22×D711yescharacteristic
317C22×C8—72no—
318C22×C8—78no—
319C22×C8—144no—
320C22×C8—144no—
321C22×C8C22×D712yes—
322#117—78no—
323#117—144no—
324#117—144no—
325#117—148no—
326#117—148no—
327#117—144no—
328#117—144no—
329#117—144no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—24no—
332C2×C4⋊C4—72no—
333C2×C4⋊C4—71no—
334C2×C4⋊C4—71no—
335C2×C4⋊C4—144no—
336C2×C4⋊C4—141no—
337C2×C4⋊C4C22×D711yescharacteristic
338C2×C4⋊C4C22×D711yescharacteristic
339C2×C4⋊C4C22×D712yes—
340C2×C22⋊C4—21no—
341C2×C22⋊C4—142no—
342C2×C22⋊C4—142no—
343C2×C22⋊C4—148no—
344C2×C22⋊C4—141no—
345C2×C22⋊C4—141no—
346C2×C22⋊C4—141no—
347C2×C22⋊C4—141no—
348C2×C22⋊C4—141no—
349C2×C42—142no—
350C2×C42—142no—
351C2×C42—148no—
352C2×C42—141no—
353C2×C42—141no—
354#107—144no—
355#107—144no—
356#106—24no—
357#106—144no—
358C22⋊C8—74no—
359C22⋊C8—78no—
360C22⋊C8C4×D714yes—
361#95—21no—
362#95—141no—
363#95—141no—
364#95—141no—
365C22×C7—22no—
366C22×C7—22no—
367C22×C7—22no—
368C22×C7—22no—
369C22×C7—22no—
370C22×C7—22no—
371C22×C7#42714yescomplemented
372C22×C7C22×C4⋊C411yescharacteristic
373C22×C7C2×#12011yescharacteristic
374C22×C7C2×#12311yescharacteristic
375C22×C7C2×#13511yescharacteristic
376C22×C7C2×#13511yescharacteristic
377C22×C7#57511yescharacteristic
378C22×C7#57511yescharacteristic
379C2×D7—22no—
380C2×D7—22no—
381C2×D7—22no—
382C2×D7—22no—
383C2×D7—22no—
384C2×D7—22no—
385C2×D7—22no—
386C2×D7—22no—
387C2×D7—22no—
388C2×D7—416no—
389C2×D7—41no—
390C2×D7—41no—
391C2×D7—41no—
392C2×D7—41no—
393C2×D7#38514yes—
394C4×C7—22no—
395C4×C7—44no—
396C4×C7—81no—
397C4×C7—81no—
398C4×C7C2×C4×D412yes—
399C4×C7C2×C22⋊Q812yes—
400Dic7—22no—
401Dic7—44no—
402Dic7—81no—
403Dic7—81no—
404Dic7#38514yes—
405C24—71no—
406C24—71no—
407C24—74no—
408C24—142no—
409C24—142no—
410C24—142no—
411C24—142no—
412C24—281no—
413C24—281no—
414C24D4×D711yescharacteristic
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—71no—
424C22×C4—72no—
425C22×C4—74no—
426C22×C4—72no—
427C22×C4—74no—
428C22×C4—74no—
429C22×C4—71no—
430C22×C4—74no—
431C22×C4—74no—
432C22×C4—72no—
433C22×C4—72no—
434C22×C4—142no—
435C22×C4—142no—
436C22×C4—1416no—
437C22×C4—1416no—
438C22×C4—142no—
439C22×C4—142no—
440C22×C4—142no—
441C22×C4—142no—
442C22×C4—142no—
443C22×C4—142no—
444C22×C4—144no—
445C22×C4—142no—
446C22×C4—141no—
447C22×C4—141no—
448C22×C4—282no—
449C22×C4—282no—
450C22×C4C2×C4×D712yes—
451C22×C4D4×D711yescharacteristic
452C22×C4Q8×D712yes—
453C22×C4C23×D711yescharacteristic
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—74no—
457C2×C8—1416no—
458C2×C8—142no—
459C2×C8—142no—
460C2×C8C2×C4×D714yes—
461C4⋊C4—24no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—74no—
466C4⋊C4—144no—
467C4⋊C4—142no—
468C4⋊C4—142no—
469C4⋊C4—142no—
470C4⋊C4D4×D714yes—
471C22⋊C4—42no—
472C22⋊C4—148no—
473C22⋊C4—148no—
474C22⋊C4—282no—
475C22⋊C4—282no—
476C22⋊C4—282no—
477C22⋊C4—282no—
478C22⋊C4—282no—
479C42—148no—
480C42—148no—
481C42—282no—
482C42—282no—
483C2×C7—24no—
484C2×C7C2×#38511yescharacteristic
485C2×C7C2×#42711yescharacteristic
486C2×C7#291111yescharacteristic
487C2×C7#300211yescharacteristic
488C2×C7#300211yescharacteristic
489C2×C7#309111yescharacteristic
490C2×C7#309111yescharacteristic
491D7—24no—
492D7—42no—
493C23—21no—
494C23—21no—
495C23—21no—
496C23—21no—
497C23—72no—
498C23—72no—
499C23—72no—
500C23—72no—
501C23—148no—
502C23—141no—
503C23—141no—
504C23—141no—
505C23—141no—
506C23—148no—
507C23—141no—
508C23—141no—
509C23—141no—
510C23—141no—
511C23—141no—
512C23—141no—
513C23—284no—
514C23—284no—
515C23—284no—
516C23—284no—
517C23—284no—
518C23D7×C4⋊C412yes—
519C23#647912yes—
520C23#647912yes—
521C23C2×D4×D711yescharacteristic
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—48no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—72no—
537C2×C4—72no—
538C2×C4—72no—
539C2×C4—78no—
540C2×C4—78no—
541C2×C4—72no—
542C2×C4—148no—
543C2×C4—148no—
544C2×C4—148no—
545C2×C4—141no—
546C2×C4—141no—
547C2×C4—144no—
548C2×C4—144no—
549C2×C4—144no—
550C2×C4—288no—
551C2×C4—282no—
552C2×C4—282no—
553C2×C4—281no—
554C2×C4—281no—
555C2×C4—281no—
556C2×C4—281no—
557C2×C4—281no—
558C2×C4—281no—
559C2×C4D7×C4⋊C412yes—
560C2×C4C22×C4×D711yescharacteristic
561C2×C4C2×D4×D712yes—
562C2×C4C2×Q8×D711yescharacteristic
563C2×C4D7×#4112yes—
564C8—24no—
565C8—144no—
566C7#2084311yescharacteristic, complemented, Sylow
567C22—22no—
568C22—22no—
569C22—22no—
570C22—22no—
571C22—22no—
572C22—22no—
573C22—74no—
574C22—142no—
575C22—142no—
576C22—142no—
577C22—142no—
578C22—142no—
579C22—142no—
580C22—142no—
581C22—142no—
582C22—142no—
583C22—2816no—
584C22—281no—
585C22—281no—
586C22—281no—
587C22—281no—
588C22#8963214yescomplemented
589C22C2×D7×C4⋊C411yescharacteristic
590C22D7×#12011yescharacteristic
591C22D7×#12311yescharacteristic
592C22C2×#647911yescharacteristic
593C22C2×#647911yescharacteristic
594C22#9046411yescharacteristic
595C22#9046811yescharacteristic
596C4—22no—
597C4—44no—
598C4—74no—
599C4—81no—
600C4—81no—
601C4—142no—
602C4—284no—
603C4—561no—
604C4—561no—
605C4C4×D4×D712yes—
606C4D7×C22⋊Q812yes—
607C2—24no—
608C2—144no—
609C2—282no—
610C2D7×#38511yescharacteristic
611C2C2×#8963211yescharacteristic
612C2#1173665311yescharacteristic
613C2#1173748311yescharacteristic
614C2#1173759011yescharacteristic
615C2#1173876911yescharacteristic
616C2#1173887111yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1853C2
socleC23×C7C22×D4
prime radicalC24×D7D4
cosocleC22×C4×C7C24
hypercenter#1853D7
hypocenterC7#20843
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1853

Complements

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

NH ≅ G/Ntype
2. C2×#89632C2semidirect
5. C7×#1853C2semidirect
6. #11729309C2semidirect
7. #11728334C2semidirect
8. #11727682C2semidirect
9. #11727258C2semidirect
10. #11726630C2semidirect
11. #11726620C2semidirect
25. C2×C7×#106C22semidirect
31. C2×#6391C22semidirect
34. C2×#6381C22semidirect
35. C2×#6381C22semidirect
39. D7×C22⋊C8C4semidirect
42. C7×#340C22semidirect
43. #89346C22semidirect
44. #89324C22semidirect
45. #89324C22semidirect
88. C2×C8×D7C2×C4semidirect
104. (C2×D7)⋊C8C2×C4semidirect
115. #1853D7semidirect
214. C2×#106C2×D7semidirect
230. #340C2×D7semidirect
371. C22×C7#427semidirect
566. C7#20843semidirect
588. C22#89632semidirect

Maximal subgroups

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

subgroupindex
C22×#64552
C2×#896322
C2×D7×C22⋊C82
#117313072
C7×#18532
#117293092
#117283342
#117276822
#117272582
#117266302
#117266202
#208437