Finite Group Databasebeta

← #13362379

Subgroups — #13362379

Order 1792.

1062 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11740672C211yesmaximal, characteristic, complemented
2C2×#89858C211yesmaximal, characteristic
3C2×#89561C211yesmaximal, characteristic, complemented
4C2×#89589C211yesmaximal, characteristic
5C2×#89562C211yesmaximal, characteristic, complemented
6C2×#89560C211yesmaximal, characteristic
7#11731766C211yesmaximal, characteristic
8C7×#1956C211yesmaximal, characteristic, complemented
9#11728014C211yesmaximal, characteristic
10#11728013C211yesmaximal, characteristic, complemented
11#11728012C211yesmaximal, characteristic, complemented
12#11728011C211yesmaximal, characteristic
13#11727527C211yesmaximal, characteristic
14#11727503C211yesmaximal, characteristic, complemented
15#11727332C211yesmaximal, characteristic, complemented
16C2×C7×C4⋊Q8C2211yescharacteristic
17C2×Q8×Dic7C2211yescharacteristic
18C2×Q8×Dic7C2211yescharacteristic
19C22×#1144C2211yescharacteristic
20C22×#1144C2211yescharacteristic
21C22×#1143C2211yescharacteristic
22C22×#1143C2211yescharacteristic
23C2×#6456C2211yescharacteristic
24C2×#6450C2211yescharacteristic
25C2×#6438C2211yescharacteristic
26C2×C7×#103C2211yescharacteristic
27C2×C7×#103C2211yescharacteristic
28C2×C7×#103C2211yescharacteristic
29C2×C7×#103C2211yescharacteristic
30C2×C4×C7×C8C2211yescharacteristic
31C7×#384C2211yescharacteristic
32#90002C2211yescharacteristic
33#90002C2211yescharacteristic
34C2×#6395C2211yescharacteristic
35C2×#6395C2211yescharacteristic
36C2×#6390C2211yescharacteristic
37C2×#6390C2211yescharacteristic
38C2×#6389C2211yescharacteristic
39C2×#6389C2211yescharacteristic
40C2×C8×Dic7C2211yescharacteristic
41C2×C8×Dic7C2211yescharacteristic
42#89858C418yescomplemented
43#89762—22no—
44#89761—22no—
45#89760C2211yescharacteristic
46#89753—22no—
47#89753—22no—
48#89746C2211yescharacteristic
49C2×#6382C2211yescharacteristic
50C2×#6382C2211yescharacteristic
51C2×#6381C2211yescharacteristic
52C2×#6381C2211yescharacteristic
53#89708C2211yescharacteristic
54#89704C2211yescharacteristic
55C2×#6377C2211yescharacteristic
56#89589—24no—
57#89562—24no—
58#89561—24no—
59#89560—24no—
60#21703—71nomaximal, Sylow
61C22×C7×Q8C2311yescharacteristic
62C22×C7×Q8C2311yescharacteristic
63C22×D28C2311yescharacteristic
64C22×C4×D7—21no—
65C22×Dic14C2311yescharacteristic
66C22×C7×C8C2311yescharacteristic
67C22×C7×C8C2311yescharacteristic
68C7×C4⋊Q8—22no—
69C7×C4⋊Q8C2×C414yescomplemented
70C2×C7×C4⋊C4—22no—
71C2×C7×C4⋊C4—22no—
72C2×C7×C4⋊C4C2311yescharacteristic
73C2×C7×C4⋊C4C2311yescharacteristic
74C2×C42×C7C2311yescharacteristic
75Q8×Dic7—22no—
76Q8×Dic7—22no—
77Q8×Dic7—22no—
78Q8×Dic7—22no—
79C2×#1144—22no—
80C2×#1144—22no—
81C2×#1144—22no—
82C2×#1144—22no—
83C2×#1144C2×C414yescomplemented
84C2×#1144C2×C414yescomplemented
85C2×#1143—22no—
86C2×#1143—22no—
87C2×#1143—22no—
88C2×#1143—22no—
89C2×#1143C2×C414yes—
90C2×#1143C2×C414yes—
91C2×(C2×D7)⋊C4—22no—
92C2×(C2×D7)⋊C4—21no—
93C2×(C2×D7)⋊C4—22no—
94C2×C4⋊Dic7—21no—
95C2×C4⋊Dic7—21no—
96C2×C4⋊Dic7C2311yescharacteristic
97C2×C4⋊Dic7C2311yescharacteristic
98C2×Dic7⋊C4—21no—
99C2×C4×Dic7—21no—
100C2×C4×Dic7—21no—
101C2×C4×Dic7—21no—
102C2×C4×Dic7C2311yescharacteristic
103C2×C4×Dic7C2311yescharacteristic
104C22×C7⋊C8C2311yescharacteristic
105C22×C7⋊C8C2311yescharacteristic
106#6456—22no—
107#6456—22no—
108#6450—22no—
109#6450—22no—
110#6438—22no—
111#6438C2×C414yes—
112C7×#103—22no—
113C7×#103—22no—
114C7×#103—22no—
115C7×#103—22no—
116C4×C7×C8—22no—
117C7⋊#95—22no—
118C7⋊#95—22no—
119C7⋊#95—22no—
120C7⋊#95—22no—
121C7⋊#95D412yes—
122C7⋊#95D412yes—
123#6395—22no—
124#6395—22no—
125#6390—22no—
126#6390—22no—
127#6389—22no—
128#6389D414yes—
129C8×Dic7—22no—
130C8×Dic7D414yes—
131#6382—22no—
132#6382—22no—
133#6381—22no—
134#6381—22no—
135#6377C2×C414yescomplemented
136C2×#497—71no—
137C2×#441—71no—
138C2×#440—71no—
139C2×C4×Q16—71no—
140C2×C4×SD16—71no—
141#2327—71no—
142#2326—71no—
143#1973—71no—
144#1971—71no—
145#1969—71no—
146#1968—71no—
147#1958—71no—
148#1956D711yescharacteristic, complemented
149#1847—71no—
150#1726—71no—
151C23×D7—21no—
152C2×C7×Q8—21no—
153C2×C7×Q8—21no—
154C2×C7×Q8—21no—
155C2×C7×Q8—21no—
156C2×C7×Q8—21no—
157C2×C7×Q8—21no—
158C2×C7×Q8—21no—
159C2×C7×Q8—21no—
160C2×C7×Q8—21no—
161C2×C7×Q8—21no—
162C2×C7×Q8C22×C412yes—
163C2×C7×Q8C22×C412yes—
164C22×C4×C7—21no—
165C22×C4×C7—21no—
166C22×C4×C7—21no—
167C22×C4×C7—21no—
168C22×C4×C7C2×D412yes—
169C22×C4×C7C2411yescharacteristic
170C22×Dic7—21no—
171C22×Dic7—21no—
172C22×Dic7—21no—
173C22×Dic7C2×D412yes—
174C22×Dic7C2×D411yescharacteristic
175C22×Dic7C2×D411yescharacteristic
176C2×D28—21no—
177C2×D28—21no—
178C2×D28—21no—
179C2×D28—21no—
180C2×D28—21no—
181C2×D28C22×C412yes—
182C2×C4×D7—42no—
183C2×C4×D7—82no—
184C2×Dic14—21no—
185C2×Dic14—21no—
186C2×Dic14—21no—
187C2×Dic14—21no—
188C2×Dic14—21no—
189C2×Dic14C22×C412yes—
190C2×C7×C8—21no—
191C2×C7×C8—21no—
192C2×C7×C8—21no—
193C2×C7×C8—21no—
194C2×C7×C8C2×D412yes—
195C2×C7×C8C2×D412yes—
196C7×C4⋊C4—28no—
197C7×C4⋊C4—21no—
198C7×C4⋊C4—21no—
199C7×C4⋊C4—21no—
200C7×C4⋊C4—21no—
201C7×C4⋊C4—82no—
202C42×C7C22×C412yes—
203C42×C7#4112yes—
204#1144—28no—
205#1144—28no—
206#1143—28no—
207#1143—28no—
208(C2×D7)⋊C4—28no—
209(C2×D7)⋊C4—42no—
210(C2×D7)⋊C4—82no—
211C4⋊Dic7—21no—
212C4⋊Dic7—21no—
213C4⋊Dic7—21no—
214C4⋊Dic7—21no—
215C4⋊Dic7—42no—
216C4⋊Dic7—42no—
217Dic7⋊C4—42no—
218C4×Dic7—21no—
219C4×Dic7—21no—
220C4×Dic7—42no—
221C4×Dic7—42no—
222C4×Dic7—42no—
223C4×Dic7C2×D412yes—
224C4×Dic7C2×D412yes—
225C2×C7⋊C8C22×C412yes—
226C2×C7⋊C8C22×C412yes—
227C2×C7⋊C8C2×D412yes—
228C2×C7⋊C8C2×D412yes—
229C2×C7⋊C8#4112yes—
230C2×C7⋊C8#4112yes—
231C22×Q16—71no—
232C22×Q16—71no—
233C22×SD16—71no—
234C22×SD16—71no—
235C2×C4⋊Q8C2×D711yescharacteristic
236C2×#124—71no—
237C2×C4×Q8—71no—
238C2×C4×Q8—71no—
239C2×C4×Q8—71no—
240C2×C4×D4—71no—
241#497—78no—
242#441—144no—
243#440—144no—
244C4×Q16—144no—
245C4×SD16—144no—
246C2×#107—71no—
247C2×#107—71no—
248C2×#106—71no—
249C2×#106—71no—
250C2×#103—71no—
251C2×#103—71no—
252C2×#103C2×D711yescharacteristic
253C2×#103C2×D711yescharacteristic
254C2×#103C2×D711yescharacteristic
255C2×#103C2×D711yescharacteristic
256C2×#102—71no—
257C2×#102—71no—
258C2×#97—71no—
259C2×#97—71no—
260C2×#97—71no—
261C2×C4×C8—71no—
262C2×C4×C8—71no—
263C2×C4×C8C2×D711yescharacteristic
264#391—71no—
265#391—71no—
266#391—71no—
267#390—71no—
268#388—142no—
269#387—142no—
270#387—142no—
271#385—142no—
272#384C2×D711yescharacteristic
273#382—71no—
274#376—71no—
275C23×C7C22×D411yescharacteristic
276C22×D7—22no—
277C22×D7—41no—
278C22×D7—41no—
279C22×D7—42no—
280C22×D7—42no—
281C7×Q8—22no—
282C7×Q8—22no—
283C7×Q8—22no—
284C7×Q8—22no—
285C7×Q8—41no—
286C7×Q8—41no—
287C7×Q8—41no—
288C7×Q8—41no—
289C2×C4×C7—22no—
290C2×C4×C7—22no—
291C2×C4×C7—22no—
292C2×C4×C7—22no—
293C2×C4×C7—41no—
294C2×C4×C7—41no—
295C2×C4×C7—41no—
296C2×C4×C7—41no—
297C2×C4×C7—41no—
298C2×C4×C7—41no—
299C2×C4×C7—42no—
300C2×C4×C7—42no—
301C2×C4×C7C4×D414yes—
302C2×C4×C7C23×C411yescharacteristic
303C2×C4×C7C22×D411yescharacteristic
304C2×C4×C7C22×D411yescharacteristic
305C2×C4×C7C2×#4111yescharacteristic
306C2×C4×C7C2×#4111yescharacteristic
307C2×C4×C7C2×#4111yescharacteristic
308C2×Dic7—22no—
309C2×Dic7—22no—
310C2×Dic7—22no—
311C2×Dic7—22no—
312C2×Dic7—22no—
313C2×Dic7—22no—
314C2×Dic7—41no—
315C2×Dic7—41no—
316C2×Dic7—41no—
317C2×Dic7—41no—
318C2×Dic7—42no—
319C2×Dic7—42no—
320C2×Dic7#12712yes—
321C2×Dic7#12712yes—
322D28—22no—
323D28—22no—
324D28—41no—
325D28—41no—
326C4×D7—84no—
327Dic14—22no—
328Dic14—22no—
329Dic14—41no—
330Dic14—41no—
331C7×C8—22no—
332C7×C8—22no—
333C7⋊C8C4×D414yes—
334C7⋊C8C4×D414yes—
335C22×Q8—71no—
336C22×Q8C22×D711yescharacteristic
337C22×Q8C22×D711yescharacteristic
338C22×D4—71no—
339C23×C4—141no—
340C2×Q16—74no—
341C2×Q16—74no—
342C2×Q16—142no—
343C2×Q16—142no—
344C2×Q16—142no—
345C2×Q16—142no—
346C2×SD16—74no—
347C2×SD16—74no—
348C2×SD16—142no—
349C2×SD16—142no—
350C2×SD16—142no—
351C2×SD16—142no—
352C22×C8—71no—
353C22×C8—71no—
354C22×C8C22×D711yescharacteristic
355C22×C8C22×D711yescharacteristic
356C4⋊Q8—22no—
357C4⋊Q8C4×D714yescomplemented
358#124—74no—
359#124—142no—
360C4×Q8—142no—
361C4×Q8—142no—
362C4×Q8—142no—
363C4×Q8—142no—
364C4×Q8—142no—
365C4×Q8—142no—
366C4×D4—142no—
367C4×D4—142no—
368C2×C4⋊C4—22no—
369C2×C4⋊C4—22no—
370C2×C4⋊C4—71no—
371C2×C4⋊C4—71no—
372C2×C4⋊C4—141no—
373C2×C4⋊C4—141no—
374C2×C4⋊C4—141no—
375C2×C4⋊C4C22×D711yescharacteristic
376C2×C4⋊C4C22×D711yescharacteristic
377C2×C22⋊C4—142no—
378C2×C22⋊C4—141no—
379C2×C22⋊C4—142no—
380C2×C42—71no—
381C2×C42—71no—
382C2×C42—141no—
383C2×C42—141no—
384C2×C42—141no—
385C2×C42C22×D711yescharacteristic
386#107—142no—
387#107—142no—
388#106—142no—
389#106—142no—
390#103—22no—
391#103—22no—
392#103—22no—
393#103—22no—
394#103—142no—
395#103—142no—
396#102—142no—
397#102—142no—
398#97—74no—
399#97—74no—
400#97—142no—
401C4×C8—22no—
402C4×C8—74no—
403C4×C8—142no—
404#95—72no—
405#95—72no—
406#95—142no—
407#95—142no—
408#95—142no—
409#95—142no—
410C22×C7C2×C4×D411yescharacteristic
411C22×C7C2×#12711yescharacteristic
412C22×C7#53211yescharacteristic
413C22×C7#57712yes—
414C22×C7#57812yes—
415C2×D7—42no—
416C2×D7—44no—
417C2×D7—81no—
418C2×D7—81no—
419C2×D7—82no—
420C4×C7—24no—
421C4×C7—42no—
422C4×C7—42no—
423C4×C7—81no—
424C4×C7—81no—
425C4×C7C2×C4×D411yescharacteristic
426C4×C7C2×C4×D411yescharacteristic
427C4×C7#53211yescharacteristic
428C4×C7#53211yescharacteristic
429Dic7—22no—
430Dic7—22no—
431Dic7—42no—
432Dic7—42no—
433Dic7—81no—
434Dic7—81no—
435C24—141no—
436C2×Q8—21no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8—21no—
443C2×Q8—21no—
444C2×Q8—21no—
445C2×Q8—21no—
446C2×Q8—72no—
447C2×Q8—141no—
448C2×Q8—141no—
449C2×Q8—141no—
450C2×Q8—141no—
451C2×Q8—141no—
452C2×Q8C2×C4×D712yes—
453C2×Q8C2×C4×D712yes—
454C2×D4—72no—
455C2×D4—141no—
456C2×D4—141no—
457C2×D4—141no—
458C2×D4—141no—
459C2×D4—141no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—71no—
465C22×C4—71no—
466C22×C4—72no—
467C22×C4—141no—
468C22×C4—141no—
469C22×C4—141no—
470C22×C4—282no—
471C22×C4—562no—
472C22×C4D4×D712yes—
473C22×C4C23×D711yescharacteristic
474Q16—148no—
475Q16—148no—
476SD16—148no—
477SD16—148no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—72no—
483C2×C8—72no—
484C2×C8—72no—
485C2×C8—72no—
486C2×C8—72no—
487C2×C8—72no—
488C2×C8D4×D712yes—
489C2×C8D4×D712yes—
490C4⋊C4—28no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—82no—
496C4⋊C4—141no—
497C4⋊C4—141no—
498C4⋊C4—141no—
499C4⋊C4—141no—
500C4⋊C4—282no—
501C4⋊C4—282no—
502C4⋊C4—282no—
503C22⋊C4—148no—
504C22⋊C4—282no—
505C22⋊C4—562no—
506C42—72no—
507C42—72no—
508C42—141no—
509C42—141no—
510C42—282no—
511C42—282no—
512C42—282no—
513C42C2×C4×D712yes—
514C42#115912yes—
515C2×C7C4×#12711yescharacteristic
516C2×C7#294712yes—
517C2×C7#315411yescharacteristic
518C2×C7#315611yescharacteristic
519C2×C7#317812yes—
520D7—82no—
521C23—142no—
522C23—281no—
523C23—282no—
524C23—282no—
525C23—281no—
526C23C2×D4×D711yescharacteristic
527Q8—22no—
528Q8—22no—
529Q8—22no—
530Q8—22no—
531Q8—41no—
532Q8—41no—
533Q8—41no—
534Q8—41no—
535Q8—142no—
536Q8—142no—
537Q8—281no—
538Q8—281no—
539D4—142no—
540D4—142no—
541D4—281no—
542D4—281no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—72no—
556C2×C4—72no—
557C2×C4—142no—
558C2×C4—142no—
559C2×C4—142no—
560C2×C4—142no—
561C2×C4—142no—
562C2×C4—142no—
563C2×C4—281no—
564C2×C4—281no—
565C2×C4—281no—
566C2×C4—281no—
567C2×C4—282no—
568C2×C4—282no—
569C2×C4—564no—
570C2×C4(C22⋊D7)⋊C414yes—
571C2×C4C22×C4×D711yescharacteristic
572C2×C4C2×D4×D711yescharacteristic
573C2×C4C2×D4×D711yescharacteristic
574C2×C4C2×#115911yescharacteristic
575C2×C4D7×#4111yescharacteristic
576C2×C4D7×#4111yescharacteristic
577C8—22no—
578C8—22no—
579C8—74no—
580C8—74no—
581C7#2170311yescharacteristic, complemented, Sylow
582C22—284no—
583C22—282no—
584C22—561no—
585C22—561no—
586C22—562no—
587C22C2×(C22⋊D7)⋊C411yescharacteristic
588C22#9035911yescharacteristic
589C22D7×#12711yescharacteristic
590C22#9045312yes—
591C22#9045712yes—
592C4—24no—
593C4—42no—
594C4—42no—
595C4—81no—
596C4—81no—
597C4—142no—
598C4—142no—
599C4—282no—
600C4—282no—
601C4—561no—
602C4—561no—
603C4C4×D4×D711yescharacteristic
604C4C4×D4×D711yescharacteristic
605C4#9028711yescharacteristic
606C4#9028711yescharacteristic
607C2—562no—
608C2#1173373811yescharacteristic
609C2#1173621912yes—
610C2#1173903912yes—
611C2#1173920811yescharacteristic
612C2#1173924611yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1956C2
socleC23×C7C22×D4
prime radicalC22×D28C23
cosocleC22×C4×C7C24
hypercenter#1956D7
hypocenterC7#21703
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1956

Complements

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

NH ≅ G/Ntype
1. #11740672C2semidirect
3. C2×#89561C2semidirect
5. C2×#89562C2semidirect
8. C7×#1956C2semidirect
10. #11728013C2semidirect
11. #11728012C2semidirect
14. #11727503C2semidirect
15. #11727332C2semidirect
42. #89858C4semidirect
69. C7×C4⋊Q8C2×C4semidirect
83. C2×#1144C2×C4semidirect
84. C2×#1144C2×C4semidirect
135. #6377C2×C4semidirect
148. #1956D7semidirect
357. C4⋊Q8C4×D7semidirect
581. C7#21703semidirect

Maximal subgroups

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

subgroupindex
#117406722
C2×#898582
C2×#895612
C2×#895892
C2×#895622
C2×#895602
#117317662
C7×#19562
#117280142
#117280132
#117280122
#117280112
#117275272
#117275032
#117273322
#217037