Finite Group Databasebeta

← #13354799

Subgroups — #13354799

Order 1792.

1458 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#90214C211yesmaximal, characteristic
2C7×#2316C211yesmaximal, characteristic, complemented
3C2×#89774C211yesmaximal, characteristic
4C2×#89767C211yesmaximal, characteristic, complemented
5C2×#89738C211yesmaximal, characteristic
6#11729333C211yesmaximal, characteristic, complemented
7C2×#89324C211yesmaximal, characteristic, complemented
8#11727386C212yesmaximal, complemented
9#11727385C212yesmaximal, complemented
10#11727380C212yesmaximal, complemented
11#11727379C212yesmaximal, complemented
12C22×#1157C2211yescharacteristic
13C2×C7×#117C2211yescharacteristic
14C2×C7×#117C2211yescharacteristic
15C22×C7×C4⋊C4C2211yescharacteristic
16C22×C4⋊Dic7C2211yescharacteristic
17C22×#1136C2211yescharacteristic
18C23×C7⋊C8C2211yescharacteristic
19#90214—24no—
20#90214—24no—
21C2×#6455—21no—
22C2×D7×C4⋊C4—21no—
23C2×(C22⋊D7)⋊C4—22no—
24C2×C4×D28C2212yes—
25C2×C4×Dic14C2212yes—
26C7×#389—21no—
27C7×#389—21no—
28C7×#389C2212yes—
29C7×#389C2212yes—
30C7×#380—22no—
31#89774—24no—
32#89767—24no—
33C2×#6384C2211yescharacteristic
34C2×#6384C2211yescharacteristic
35C2×#6384C2212yes—
36#89738C418yescomplemented
37C2×#6383C2211yescharacteristic
38C2×#6383C2211yescharacteristic
39C2×#6383C2212yes—
40C2×#6382C2212yes—
41C2×#6381C2212yes—
42#89324C2212yes—
43#89324C2212yes—
44#89319C2212yes—
45#89319C2212yes—
46#20878—71nomaximal, Sylow
47C23×C4×C7—21no—
48C23×C4×C7C2311yescharacteristic
49C22×C22⋊D7—21no—
50C23×Dic7—21no—
51C2×#1157—22no—
52C2×#1157—22no—
53C2×#1157—22no—
54C2×#1157C2×C414yes—
55C2×#1157D414yes—
56C2×#1157Q814yes—
57C22×D28C2311yescharacteristic
58C22×C4×D7—21no—
59C22×C4×D7—22no—
60C22×Dic14C2311yescharacteristic
61C7×#117—22no—
62C7×#117—22no—
63C7×#117—22no—
64C7×#117—22no—
65C2×C7×C4⋊C4—21no—
66C2×C7×C4⋊C4—22no—
67C2×C7×C4⋊C4—22no—
68C2×C7×C4⋊C4—22no—
69C2×C7×C4⋊C4C2×C414yescomplemented
70C2×C7×C4⋊C4D411yescharacteristic
71C2×C7×C4⋊C4D411yescharacteristic
72C2×C7×C4⋊C4C2311yescharacteristic
73C2×C7×C4⋊C4C2311yescharacteristic
74C2×C7×C4⋊C4C2312yes—
75C2×C7×C22⋊C4—21no—
76C2×C7×C22⋊C4D412yes—
77C2×C42×C7—21no—
78C2×C42×C7C2312yes—
79C2×(C2×D7)⋊C4—22no—
80C2×C4⋊Dic7—21no—
81C2×C4⋊Dic7—22no—
82C2×C4⋊Dic7D414yes—
83C2×C4⋊Dic7C2312yes—
84C2×Dic7⋊C4—22no—
85C2×C4×Dic7—22no—
86C2×#1136—22no—
87C2×#1136—22no—
88C2×#1136C2×C414yescomplemented
89C22×C7⋊C8—22no—
90C22×C7⋊C8—22no—
91C22×C7⋊C8—22no—
92C22×C7⋊C8C2311yescharacteristic
93C22×C7⋊C8C2311yescharacteristic
94C22×C7⋊C8C2312yes—
95#6455—28no—
96D7×C4⋊C4—28no—
97(C22⋊D7)⋊C4—216no—
98C4×D28—24no—
99C4×D28—24no—
100C4×Dic14—24no—
101C4×Dic14—24no—
102C7×#95—22no—
103C7×#95—22no—
104#6384—22no—
105#6384—22no—
106#6384C2×C418yescomplemented
107#6383—22no—
108#6383—22no—
109#6383C2×C418yescomplemented
110#6382—24no—
111#6381—24no—
112C2×#520—71no—
113C2×#428—71no—
114C2×#417—71no—
115C2×#416—71no—
116#2316D711yescharacteristic, complemented
117#1865—72no—
118#1864—72no—
119#1863—72no—
120#1862—72no—
121C2×#340—71no—
122#1729—71no—
123C24×C7C2×D411yescharacteristic
124C23×D7—21no—
125C22×C4×C7—22no—
126C22×C4×C7—24no—
127C22×C4×C7—21no—
128C22×C4×C7—21no—
129C22×C4×C7—81no—
130C22×C4×C7—81no—
131C22×C4×C7C22×C412yes—
132C22×C4×C7C2×D412yes—
133C22×C4×C7C2×D412yes—
134C22×C4×C7C2×D412yes—
135C22×C4×C7C2×D411yescharacteristic
136C22×C4×C7C2×D412yes—
137C22×C4×C7C2×Q812yes—
138C22×C4×C7#4112yes—
139C22×C4×C7#4112yes—
140C22×C4×C7C2411yescharacteristic
141C2×C22⋊D7—24no—
142C2×C22⋊D7—24no—
143C2×C22⋊D7—24no—
144C22×Dic7—21no—
145C22×Dic7—22no—
146C22×Dic7—24no—
147C22×Dic7—81no—
148C22×Dic7—81no—
149#1157—28no—
150#1157—28no—
151#1157—28no—
152#1157C4⋊C4116yescomplemented
153C2×D28—21no—
154C2×D28—21no—
155C2×D28—21no—
156C2×D28C22×C412yes—
157C2×D28C2×D412yes—
158C2×D28C2×Q812yes—
159C2×C4×D7—24no—
160C2×C4×D7—24no—
161C2×C4×D7—24no—
162C2×C4×D7—28no—
163C2×C4×D7—82no—
164C2×C4×D7—82no—
165C2×Dic14—21no—
166C2×Dic14—21no—
167C2×Dic14—21no—
168C2×Dic14C22×C412yes—
169C2×Dic14C2×D412yes—
170C2×Dic14C2×Q812yes—
171C7×C4⋊C4—24no—
172C7×C4⋊C4—22no—
173C7×C4⋊C4—22no—
174C7×C4⋊C4—22no—
175C7×C4⋊C4—21no—
176C7×C4⋊C4—21no—
177C7×C4⋊C4—21no—
178C7×C4⋊C4—21no—
179C7×C4⋊C4—82no—
180C7×C4⋊C4—82no—
181C7×C4⋊C4C22×C414yes—
182C7×C22⋊C4—24no—
183C7×C22⋊C4—42no—
184C42×C7—22no—
185C42×C7—22no—
186C42×C7—42no—
187(C2×D7)⋊C4—28no—
188C4⋊Dic7—24no—
189C4⋊Dic7—22no—
190C4⋊Dic7C2×D414yes—
191Dic7⋊C4—28no—
192C4×Dic7—28no—
193#1136—28no—
194C2×C7⋊C8—22no—
195C2×C7⋊C8—22no—
196C2×C7⋊C8—21no—
197C2×C7⋊C8—21no—
198C2×C7⋊C8—21no—
199C2×C7⋊C8—21no—
200C2×C7⋊C8—21no—
201C2×C7⋊C8—21no—
202C2×C7⋊C8—44no—
203C2×C7⋊C8C22×C414yes—
204C22×#41—71no—
205C22×M16—71no—
206C23×C8—71no—
207#520—144no—
208#520—144no—
209C2×C4×Q8—72no—
210C2×C4×D4—72no—
211C2×C4×D4—142no—
212C2×#117—141no—
213C2×#117C2×D711yescharacteristic
214C2×#117C2×D711yescharacteristic
215C22×C4⋊C4—71no—
216C22×C4⋊C4—141no—
217C22×C4⋊C4C2×D711yescharacteristic
218#428—144no—
219C2×#107—72no—
220C2×#106—72no—
221#417—78no—
222#416—144no—
223C2×#103—72no—
224C2×#103—71no—
225C2×#103—71no—
226C2×#102—72no—
227C2×#102—71no—
228C2×#102—71no—
229#389—21no—
230#389—21no—
231#389C2×D712yes—
232#389C2×D712yes—
233#380—22no—
234#340—72no—
235#340—72no—
236#336—72no—
237#336—72no—
238C23×C7—21no—
239C23×C7—21no—
240C23×C7—21no—
241C23×C7—21no—
242C23×C7C4×D412yes—
243C23×C7#12012yes—
244C23×C7C22⋊Q812yes—
245C23×C7C22×D411yescharacteristic
246C22×D7—22no—
247C22×D7—22no—
248C22×D7—22no—
249C22×D7—41no—
250C22×D7—41no—
251C22×D7—41no—
252C22×D7—41no—
253C2×C4×C7—24no—
254C2×C4×C7—22no—
255C2×C4×C7—22no—
256C2×C4×C7—22no—
257C2×C4×C7—22no—
258C2×C4×C7—22no—
259C2×C4×C7—22no—
260C2×C4×C7—42no—
261C2×C4×C7—42no—
262C2×C4×C7—41no—
263C2×C4×C7—41no—
264C2×C4×C7—41no—
265C2×C4×C7—41no—
266C2×C4×C7—41no—
267C2×C4×C7—41no—
268C2×C4×C7—84no—
269C2×C4×C7C2×C4⋊C414yes—
270C2×C4×C7C4×D414yes—
271C2×C4×C7C4×D412yes—
272C2×C4×C7#12012yes—
273C2×C4×C7C22⋊Q814yes—
274C2×C4×C7C22⋊Q812yes—
275C2×C4×C7#12314yes—
276C2×C4×C7C23×C411yescharacteristic
277C2×C4×C7C22×D411yescharacteristic
278C2×C4×C7C22×D411yescharacteristic
279C2×C4×C7C22×Q811yescharacteristic
280C2×C4×C7C2×#4111yescharacteristic
281C2×C4×C7C2×#4111yescharacteristic
282C22⋊D7—216no—
283C2×Dic7—22no—
284C2×Dic7—22no—
285C2×Dic7—22no—
286C2×Dic7—24no—
287C2×Dic7—42no—
288C2×Dic7—42no—
289C2×Dic7—84no—
290D28—22no—
291D28—22no—
292D28—22no—
293D28C2×C4⋊C414yes—
294C4×D7—216no—
295C4×D7—88no—
296Dic14—22no—
297Dic14—22no—
298Dic14—22no—
299Dic14C2×C4⋊C414yes—
300C7⋊C8—24no—
301C7⋊C8—41no—
302C7⋊C8—41no—
303C2×#41—74no—
304C2×#41—74no—
305C2×#41—74no—
306C2×#41—142no—
307C2×#41—142no—
308C2×#41—142no—
309C22×Q8—71no—
310C22×D4—71no—
311C22×D4—141no—
312C23×C4—21no—
313C23×C4—141no—
314C23×C4—141no—
315C23×C4—142no—
316C23×C4C22×D711yescharacteristic
317C2×M16—74no—
318C2×M16—142no—
319C2×M16—142no—
320C22×C8—72no—
321C22×C8—71no—
322C22×C8—71no—
323C22×C8—142no—
324C22×C8—142no—
325C22×C8—142no—
326C4×Q8—144no—
327C4×Q8—144no—
328C4×D4—144no—
329C4×D4—144no—
330C4×D4—1416no—
331#117—22no—
332#117—22no—
333#117—22no—
334#117—22no—
335#117—148no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—74no—
341C2×C4⋊C4—72no—
342C2×C4⋊C4—148no—
343C2×C4⋊C4—142no—
344C2×C4⋊C4—141no—
345C2×C4⋊C4—142no—
346C2×C4⋊C4C4×D714yescomplemented
347C2×C4⋊C4C22⋊D711yescharacteristic
348C2×C4⋊C4C22⋊D711yescharacteristic
349C2×C4⋊C4C22×D711yescharacteristic
350C2×C4⋊C4C22×D711yescharacteristic
351C2×C4⋊C4C22×D712yes—
352C2×C22⋊C4—21no—
353C2×C22⋊C4—142no—
354C2×C22⋊C4C22⋊D712yes—
355C2×C42—21no—
356C2×C42—142no—
357C2×C42C22×D712yes—
358#107—144no—
359#106—144no—
360#103—78no—
361#103—142no—
362#103—142no—
363#102—78no—
364#102—142no—
365#102—142no—
366#95—22no—
367#95—22no—
368C22×C7—22no—
369C22×C7—22no—
370C22×C7—22no—
371C22×C7—22no—
372C22×C7—22no—
373C22×C7—22no—
374C22×C7#38514yes—
375C22×C7C2×C4×D411yescharacteristic
376C22×C7C2×#12011yescharacteristic
377C22×C7C2×C22⋊Q811yescharacteristic
378C22×C7#57611yescharacteristic
379C22×C7#57711yescharacteristic
380C22×C7#57711yescharacteristic
381C22×C7#57811yescharacteristic
382C2×D7—24no—
383C2×D7—42no—
384C2×D7—42no—
385C2×D7—42no—
386C2×D7—42no—
387C2×D7—42no—
388C2×D7—42no—
389C4×C7—44no—
390C4×C7—42no—
391C4×C7—42no—
392C4×C7—81no—
393C4×C7—81no—
394C4×C7#38514yes—
395C4×C7C22×C4⋊C411yescharacteristic
396C4×C7C2×C4×D411yescharacteristic
397C4×C7C2×C22⋊Q811yescharacteristic
398C4×C7C2×#12311yescharacteristic
399Dic7—24no—
400Dic7—44no—
401C24—141no—
402C24C2×C22⋊D711yescharacteristic
403#41—716no—
404#41—148no—
405#41—148no—
406#41—148no—
407C2×Q8—72no—
408C2×Q8—72no—
409C2×Q8—72no—
410C2×Q8—141no—
411C2×Q8—141no—
412C2×Q8—141no—
413C2×D4—72no—
414C2×D4—72no—
415C2×D4—72no—
416C2×D4—144no—
417C2×D4—144no—
418C2×D4—144no—
419C2×D4—141no—
420C2×D4—141no—
421C2×D4—141no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—81no—
427C22×C4—81no—
428C22×C4—148no—
429C22×C4—141no—
430C22×C4—144no—
431C22×C4—144no—
432C22×C4—144no—
433C22×C4—142no—
434C22×C4—144no—
435C22×C4—562no—
436C22×C4—562no—
437C22×C4—561no—
438C22×C4—561no—
439C22×C4C2×C4×D712yes—
440C22×C4#115712yes—
441C22×C4D4×D712yes—
442C22×C4D4×D712yes—
443C22×C4Q8×D712yes—
444C22×C4#116112yes—
445C22×C4C2×C22⋊D712yes—
446C22×C4C2×C22⋊D712yes—
447C22×C4C2×C22⋊D711yescharacteristic
448C22×C4C23×D711yescharacteristic
449M16—148no—
450C2×C8—74no—
451C2×C8—142no—
452C2×C8—142no—
453C2×C8—141no—
454C2×C8—141no—
455C2×C8—141no—
456C2×C8—141no—
457C2×C8—141no—
458C2×C8—141no—
459C2×C8—284no—
460C4⋊C4—24no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—74no—
469C4⋊C4—82no—
470C4⋊C4—82no—
471C4⋊C4—148no—
472C4⋊C4—144no—
473C4⋊C4—142no—
474C4⋊C4C2×C4×D714yes—
475C22⋊C4—24no—
476C22⋊C4—42no—
477C22⋊C4—148no—
478C42—22no—
479C42—22no—
480C42—42no—
481C42—148no—
482C2×C7—24no—
483C2×C7C2×#38511yescharacteristic
484C2×C7#294611yescharacteristic
485C2×C7#294711yescharacteristic
486C2×C7#301211yescharacteristic
487C2×C7#301311yescharacteristic
488C2×C7#308311yescharacteristic
489C2×C7#308411yescharacteristic
490D7—44no—
491C23—21no—
492C23—21no—
493C23—21no—
494C23—21no—
495C23—142no—
496C23—142no—
497C23—142no—
498C23—281no—
499C23—281no—
500C23—281no—
501C23—281no—
502C23C4×C22⋊D712yes—
503C23#650012yes—
504C23#650912yes—
505C23C22×C22⋊D711yescharacteristic
506Q8—74no—
507Q8—142no—
508Q8—142no—
509Q8—142no—
510D4—74no—
511D4—1416no—
512D4—142no—
513D4—142no—
514D4—142no—
515C2×C4—24no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—84no—
531C2×C4—142no—
532C2×C4—142no—
533C2×C4—142no—
534C2×C4—1416no—
535C2×C4—144no—
536C2×C4—282no—
537C2×C4—282no—
538C2×C4—568no—
539C2×C4—564no—
540C2×C4D7×C4⋊C414yes—
541C2×C4#645614yes—
542C2×C4#645714yes—
543C2×C4#645914yes—
544C2×C4C4×C22⋊D712yes—
545C2×C4#650012yes—
546C2×C4#650912yes—
547C2×C4C22×C4×D711yescharacteristic
548C2×C4C2×#115711yescharacteristic
549C2×C4C2×D4×D711yescharacteristic
550C2×C4C2×D4×D711yescharacteristic
551C2×C4C2×Q8×D711yescharacteristic
552C2×C4C2×#116111yescharacteristic
553C8—144no—
554C8—281no—
555C8—281no—
556C7#2087811yescharacteristic, complemented, Sylow
557C22—22no—
558C22—22no—
559C22—22no—
560C22—22no—
561C22—22no—
562C22—22no—
563C22—144no—
564C22—282no—
565C22—282no—
566C22—282no—
567C22—282no—
568C22—282no—
569C22—282no—
570C22#8976114yes—
571C22C2×C4×C22⋊D711yescharacteristic
572C22C2×#650011yescharacteristic
573C22C2×#650911yescharacteristic
574C22#9052611yescharacteristic
575C22#9052711yescharacteristic
576C22#9052811yescharacteristic
577C22#9052911yescharacteristic
578C4—44no—
579C4—42no—
580C4—42no—
581C4—81no—
582C4—81no—
583C4—144no—
584C4—284no—
585C4#8976114yes—
586C4C2×D7×C4⋊C411yescharacteristic
587C4C2×#645611yescharacteristic
588C4C2×#645711yescharacteristic
589C4C2×#645911yescharacteristic
590C2—24no—
591C2—284no—
592C2C2×#8976111yescharacteristic
593C2#1173998211yescharacteristic
594C2#1173999211yescharacteristic
595C2#1174086911yescharacteristic
596C2#1174087211yescharacteristic
597C2#1174092011yescharacteristic
598C2#1174092111yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#2316C2
socleC23×C7C22×D4
prime radicalC22×#1157C22
cosocleC22×C4×C7C24
hypercenter#2316D7
hypocenterC7#20878
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2316

Complements

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

NH ≅ G/Ntype
2. C7×#2316C2semidirect
4. C2×#89767C2semidirect
6. #11729333C2semidirect
7. C2×#89324C2semidirect
8. #11727386C2semidirect
9. #11727385C2semidirect
10. #11727380C2semidirect
11. #11727379C2semidirect
36. #89738C4semidirect
69. C2×C7×C4⋊C4C2×C4semidirect
88. C2×#1136C2×C4semidirect
106. #6384C2×C4semidirect
109. #6383C2×C4semidirect
116. #2316D7semidirect
152. #1157C4⋊C4semidirect
346. C2×C4⋊C4C4×D7semidirect
556. C7#20878semidirect

Maximal subgroups

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

subgroupindex
C2×#902142
C7×#23162
C2×#897742
C2×#897672
C2×#897382
#117293332
C2×#893242
#117273862
#117273852
#117273802
#117273792
#208787