Finite Group Databasebeta

← #13360581

Subgroups — #13360581

Order 1792.

906 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11740367C211yesmaximal, characteristic
2#11740362C211yesmaximal, characteristic, complemented
3C2×#89769C211yesmaximal, characteristic, complemented
4C2×#89596C211yesmaximal, characteristic, complemented
5#11736072C211yesmaximal, characteristic, complemented
6C7×#1897C211yesmaximal, characteristic, complemented
7#11729345C211yesmaximal, characteristic, complemented
8#11728127C211yesmaximal, characteristic, complemented
9#11728126C211yesmaximal, characteristic, complemented
10#11728124C211yesmaximal, characteristic, complemented
11#11727736C211yesmaximal, characteristic, complemented
12#11726501C211yesmaximal, characteristic, complemented
13#11726500C211yesmaximal, characteristic, complemented
14#11726495C211yesmaximal, characteristic, complemented
15#11726494C211yesmaximal, characteristic, complemented
16C7×#525C2211yescharacteristic
17C2×C7×#117C2211yescharacteristic
18C2×#6513C2211yescharacteristic
19C2×#6511C2211yescharacteristic
20#90482C2211yescharacteristic
21C22×#1136C2211yescharacteristic
22C23×C7⋊C8C2211yescharacteristic
23C2×#6470C2211yescharacteristic
24C2×C7×#104C2211yescharacteristic
25C2×C7×#104C2211yescharacteristic
26C7×#407C2211yescharacteristic
27#90006C2211yescharacteristic
28#90005C2211yescharacteristic
29#89979C2211yescharacteristic
30#89901C2211yescharacteristic
31C2×#6396C2211yescharacteristic
32C2×#6396C2211yescharacteristic
33#89894C2211yescharacteristic
34#89891C2211yescharacteristic
35#89823C412yescomplemented
36#89820C412yescomplemented
37#89813C412yescomplemented
38#89810C412yescomplemented
39#89784—22no—
40#89783—22no—
41#89774C2211yescharacteristic
42#89771C2211yescharacteristic
43#89770C2211yescharacteristic
44#89769—24no—
45#89769C2211yescharacteristic
46C2×#6379C2211yescharacteristic
47C2×#6379C2211yescharacteristic
48C2×#6377C2211yescharacteristic
49C2×#6377C2211yescharacteristic
50C2×C4×C7⋊C8C2211yescharacteristic
51C2×C4×C7⋊C8C2211yescharacteristic
52#89596—24no—
53#89501—22no—
54#89500—22no—
55C7×#339C2211yescharacteristic
56C7×#339C2211yescharacteristic
57#89353C2211yescharacteristic
58#89353C2211yescharacteristic
59#89318C2211yescharacteristic
60#89318C2211yescharacteristic
61#21545—71nomaximal, Sylow
62C2×C7×#41C2311yescharacteristic
63C23×C4×C7C2311yescharacteristic
64C2×#1157C2311yescharacteristic
65C2×C7×M16C2311yescharacteristic
66C2×C7×M16C2311yescharacteristic
67C7×#123—21no—
68C7×C22⋊Q8C2×C412yes—
69C7×#121C2×C412yes—
70C7×#120—21no—
71C4×C7×D4—22no—
72C7×#117—22no—
73C7×#117—22no—
74C7×#117C2311yescharacteristic
75C2×C7×C4⋊C4—21no—
76C2×C7×C22⋊C4—21no—
77C2×C42×C7C2311yescharacteristic
78C2×C42×C7C2311yescharacteristic
79#6516—21no—
80#6513—22no—
81#6513—22no—
82#6511—22no—
83#6511—22no—
84C2×#1144C2×C411yescharacteristic
85C2×#1143C2×C411yescharacteristic
86C2×#1142C2×C411yescharacteristic
87C2×#1141C2×C411yescharacteristic
88#6493C2×C412yes—
89#6492—21no—
90C4×C22⋊D7—22no—
91#6489C2311yescharacteristic
92#6487C2×C412yes—
93C2×#1136—22no—
94C2×#1136—21no—
95C2×#1136—22no—
96C2×#1136D412yes—
97C2×#1136D412yes—
98C2×#1136C2311yescharacteristic
99C2×#1136C2311yescharacteristic
100C22×C7⋊C8—22no—
101C22×C7⋊C8—22no—
102C22×C7⋊C8—21no—
103C22×C7⋊C8C2×C412yes—
104C22×C7⋊C8C2×C412yes—
105C22×C7⋊C8C2311yescharacteristic
106C22×C7⋊C8C2311yescharacteristic
107C22×C7⋊C8C2311yescharacteristic
108C22×C7⋊C8C2311yescharacteristic
109#6470—22no—
110#6470—22no—
111C2×#1131—21no—
112C2×C8×D7—21no—
113C7×#104—22no—
114C7×#104—22no—
115C7×C22⋊C8—22no—
116#6409C2×C412yes—
117#6406C2×C412yes—
118C22⋊(C7⋊C8)—22no—
119C22⋊(C7⋊C8)—22no—
120#6396—22no—
121#6396—22no—
122(C2×D7)⋊C8—22no—
123#6389—21no—
124Dic7⋊C8—22no—
125C8×Dic7—21no—
126#6384C2×C412yes—
127#6383C2×C412yes—
128#6382C2×C412yes—
129#6381C2×C412yes—
130#6379—22no—
131#6379—22no—
132C4⋊(C7⋊C8)—22no—
133#6377—22no—
134#6377—22no—
135#6377—21no—
136C4×C7⋊C8—22no—
137C4×C7⋊C8—21no—
138C4×C7⋊C8D414yescomplemented
139#3050—71no—
140C2×#443—71no—
141#2930—71no—
142#2930—71no—
143C2×#405—71no—
144#1974—71no—
145#1945—71no—
146#1938—71no—
147#1938—71no—
148#1897—71no—
149#1897D711yescharacteristic, complemented
150#1810—71no—
151#1772—71no—
152#1765—71no—
153#1765—71no—
154C24×C7#4111yescharacteristic
155C7×#41—21no—
156C7×#41—21no—
157C7×#41—21no—
158C7×#41—21no—
159C2×C7×Q8C22×C411yescharacteristic
160C2×C7×D4—21no—
161C2×C7×D4—22no—
162C2×C7×D4C22×C411yescharacteristic
163C22×C4×C7—21no—
164C22×C4×C7—21no—
165C22×C4×C7—21no—
166C22×C4×C7—21no—
167C22×C4×C7—21no—
168C22×C4×C7—21no—
169C22×C4×C7—21no—
170C22×C4×C7C22×C412yes—
171C22×C4×C7C2×D412yes—
172C22×C4×C7#4111yescharacteristic
173C22×C4×C7C2411yescharacteristic
174C2×C22⋊D7—21no—
175C2×C22⋊D7—22no—
176#1157—21no—
177#1157—21no—
178#1157—21no—
179#1157—21no—
180C2×D28C22×C411yescharacteristic
181C2×C4×D7—21no—
182C2×Dic14C22×C411yescharacteristic
183C7×M16—21no—
184C7×M16—21no—
185C7×M16—21no—
186C7×M16—21no—
187C2×C7×C8—21no—
188C2×C7×C8—21no—
189C7×C4⋊C4—24no—
190C7×C4⋊C4—22no—
191C7×C4⋊C4C22×C412yes—
192C7×C22⋊C4—24no—
193C7×C22⋊C4—22no—
194C7×C22⋊C4—21no—
195C7×C22⋊C4—22no—
196C42×C7—21no—
197C42×C7—21no—
198C42×C7—21no—
199C42×C7C2×D412yes—
200C42×C7#4112yes—
201C22⋊Dic7—22no—
202C22⋊Dic7—21no—
203#1144—22no—
204#1143—22no—
205#1142—22no—
206#1141—22no—
207(C2×D7)⋊C4—22no—
208C4⋊Dic7C22×C412yes—
209Dic7⋊C4—22no—
210C4×Dic7—21no—
211#1136—24no—
212#1136—24no—
213#1136—21no—
214#1136—21no—
215#1136—21no—
216#1136—21no—
217#1136—42no—
218C2×C7⋊C8—21no—
219C2×C7⋊C8—21no—
220C2×C7⋊C8—21no—
221C2×C7⋊C8—21no—
222C2×C7⋊C8—24no—
223C2×C7⋊C8—24no—
224C2×C7⋊C8—21no—
225C2×C7⋊C8—21no—
226C2×C7⋊C8—42no—
227C2×C7⋊C8C22×C412yes—
228C2×C7⋊C8C22×C411yescharacteristic
229C2×C7⋊C8C22×C411yescharacteristic
230C2×C7⋊C8C2×D412yes—
231C2×C7⋊C8C2×D412yes—
232C2×C7⋊C8C2×D411yescharacteristic
233C2×C7⋊C8C2×D411yescharacteristic
234C2×C7⋊C8C2×D412yes—
235C2×C7⋊C8#4112yes—
236C2×C7⋊C8#4112yes—
237#1131—42no—
238C8×D7—42no—
239C2×#135—71no—
240C2×#131—71no—
241C2×#131—71no—
242C22×M16—71no—
243C23×C8—71no—
244#525—71no—
245#525C2×D711yescharacteristic
246C2×#117C2×D711yescharacteristic
247#467—72no—
248#466—72no—
249#465—72no—
250#465—72no—
251#443—144no—
252#436—142no—
253#436—142no—
254#435—142no—
255#435—142no—
256C2×#108—71no—
257C2×#108—71no—
258#427—71no—
259#424—71no—
260#424—71no—
261C2×#104—71no—
262C2×#104—71no—
263C2×#104C2×D711yescharacteristic
264C2×#104C2×D711yescharacteristic
265#416—71no—
266#416—71no—
267#408—71no—
268#408—71no—
269#407—71no—
270#407C2×D711yescharacteristic
271#405—71no—
272#405—71no—
273#405—144no—
274C2×#97—71no—
275C2×#97—71no—
276C2×C4×C8—71no—
277C2×C4×C8—71no—
278#341—71no—
279#341—71no—
280#339—71no—
281#339—71no—
282#339C2×D711yescharacteristic
283#339C2×D711yescharacteristic
284C23×C7—21no—
285C23×C7—21no—
286C23×C7—21no—
287C23×C7—42no—
288C23×C7C4×D412yes—
289C23×C7C2×#4111yescharacteristic
290C22×D7—21no—
291C7×Q8—21no—
292C7×Q8—21no—
293C7×D4—21no—
294C7×D4—21no—
295C7×D4—41no—
296C7×D4—41no—
297C7×D4—82no—
298C2×C4×C7—21no—
299C2×C4×C7—21no—
300C2×C4×C7—22no—
301C2×C4×C7—22no—
302C2×C4×C7—22no—
303C2×C4×C7—22no—
304C2×C4×C7—22no—
305C2×C4×C7—21no—
306C2×C4×C7—22no—
307C2×C4×C7—22no—
308C2×C4×C7—42no—
309C2×C4×C7—42no—
310C2×C4×C7—41no—
311C2×C4×C7—41no—
312C2×C4×C7C4×D412yes—
313C2×C4×C7C23×C411yescharacteristic
314C2×C4×C7C22×D411yescharacteristic
315C2×C4×C7C22×D411yescharacteristic
316C2×C4×C7C2×#4111yescharacteristic
317C2×C4×C7C2×#4111yescharacteristic
318C2×C4×C7C2×#4111yescharacteristic
319C22⋊D7—41no—
320C22⋊D7—41no—
321C22⋊D7—82no—
322C2×Dic7—22no—
323C2×Dic7—21no—
324D28—21no—
325D28—21no—
326C4×D7—41no—
327C4×D7—41no—
328Dic14—21no—
329Dic14—21no—
330C7×C8—41no—
331C7×C8—41no—
332C7⋊C8—22no—
333C7⋊C8—22no—
334C7⋊C8—41no—
335C7⋊C8—41no—
336C7⋊C8C4×D412yes—
337C7⋊C8C4×D412yes—
338C7⋊C8#12112yes—
339C7⋊C8#12112yes—
340C2×#41—71no—
341C2×#41C22×D711yescharacteristic
342C23×C4C22×D711yescharacteristic
343#135—142no—
344#135—142no—
345C2×Q16—71no—
346C2×SD16—71no—
347C2×SD16—71no—
348C2×D8—71no—
349#131—142no—
350#131—142no—
351#131—142no—
352#131—142no—
353C2×M16—72no—
354C2×M16—72no—
355C2×M16—71no—
356C2×M16—71no—
357C2×M16—141no—
358C2×M16—142no—
359C2×M16—142no—
360C2×M16—141no—
361C2×M16C22×D711yescharacteristic
362C2×M16C22×D711yescharacteristic
363C22×C8—71no—
364C22×C8—71no—
365C22×C8—72no—
366C22×C8—72no—
367C22×C8—71no—
368C22×C8—71no—
369C22×C8—141no—
370C22×C8—142no—
371C22×C8—142no—
372C22×C8—141no—
373#123—21no—
374#123—141no—
375C22⋊Q8—72no—
376C22⋊Q8C4×D712yes—
377#121—72no—
378#121C4×D712yes—
379#120—21no—
380#120—141no—
381C4×D4—22no—
382C4×D4—142no—
383#117—22no—
384#117—22no—
385#117—71no—
386#117C22×D711yescharacteristic
387C2×C4⋊C4—21no—
388C2×C22⋊C4—21no—
389C2×C42C22×D711yescharacteristic
390C2×C42C22×D711yescharacteristic
391#108—142no—
392#108—142no—
393#107—72no—
394#106—72no—
395C4⋊C8—142no—
396C4⋊C8—142no—
397#104—22no—
398#104—22no—
399#104—142no—
400#104—142no—
401#103—72no—
402#103—72no—
403#102—72no—
404#102—72no—
405C22⋊C8—22no—
406C22⋊C8—142no—
407C22⋊C8—142no—
408C22⋊C8—142no—
409#97—142no—
410#97—141no—
411#97—141no—
412#97—142no—
413C4×C8—74no—
414C4×C8—142no—
415C4×C8—141no—
416C4×C8—141no—
417C22×C7—22no—
418C22×C7—22no—
419C22×C7—22no—
420C22×C7—22no—
421C22×C7—42no—
422C22×C7—42no—
423C22×C7—42no—
424C22×C7—41no—
425C22×C7C2×C4×D411yescharacteristic
426C22×C7#52511yescharacteristic
427C22×C7#52911yescharacteristic
428C2×D7—42no—
429C2×D7—41no—
430C4×C7—22no—
431C4×C7—42no—
432C4×C7—42no—
433C4×C7—42no—
434C4×C7—41no—
435C4×C7C2×C4×D411yescharacteristic
436C4×C7C4×#4111yescharacteristic
437C4×C7C2×#12111yescharacteristic
438C4×C7#53211yescharacteristic
439Dic7—42no—
440Dic7—41no—
441C24#115911yescharacteristic
442#41—21no—
443#41—21no—
444#41—21no—
445#41—21no—
446#41—141no—
447#41—141no—
448#41—141no—
449#41—141no—
450C2×Q8—71no—
451C2×Q8C2×C4×D711yescharacteristic
452C2×D4—21no—
453C2×D4—22no—
454C2×D4—71no—
455C2×D4—141no—
456C2×D4—142no—
457C2×D4C2×C4×D711yescharacteristic
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—141no—
466C22×C4C2×C4×D712yes—
467C22×C4D4×D712yes—
468C22×C4#115911yescharacteristic
469C22×C4C23×D711yescharacteristic
470Q16—142no—
471SD16—142no—
472SD16—142no—
473D8—142no—
474M16—21no—
475M16—21no—
476M16—21no—
477M16—21no—
478M16—144no—
479M16—144no—
480M16—141no—
481M16—141no—
482M16—141no—
483M16—141no—
484M16—282no—
485M16—282no—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—71no—
489C2×C8—71no—
490C2×C8—72no—
491C2×C8—72no—
492C2×C8—71no—
493C2×C8—71no—
494C2×C8—72no—
495C2×C8—72no—
496C2×C8—72no—
497C2×C8—72no—
498C2×C8—144no—
499C2×C8—144no—
500C2×C8—141no—
501C2×C8—141no—
502C2×C8—141no—
503C2×C8—141no—
504C2×C8—141no—
505C2×C8—141no—
506C2×C8—282no—
507C2×C8—282no—
508C4⋊C4—24no—
509C4⋊C4—22no—
510C4⋊C4—72no—
511C4⋊C4—142no—
512C4⋊C4C2×C4×D712yes—
513C22⋊C4—24no—
514C22⋊C4—22no—
515C22⋊C4—21no—
516C22⋊C4—22no—
517C22⋊C4—142no—
518C22⋊C4—141no—
519C22⋊C4—142no—
520C42—21no—
521C42—21no—
522C42—21no—
523C42—141no—
524C42D4×D712yes—
525C42#115912yes—
526C2×C7—21no—
527C2×C7—21no—
528C2×C7—42no—
529C2×C7—81no—
530C2×C7C4×#12111yescharacteristic
531C2×C7#295712yes—
532D7—81no—
533C23—21no—
534C23—21no—
535C23—21no—
536C23—42no—
537C23—141no—
538C23(C22⋊D7)⋊C412yes—
539C23C2×#115911yescharacteristic
540Q8—21no—
541Q8—21no—
542Q8—141no—
543Q8—141no—
544D4—21no—
545D4—21no—
546D4—41no—
547D4—41no—
548D4—82no—
549D4—141no—
550D4—141no—
551D4—281no—
552D4—281no—
553D4—562no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—21no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—142no—
569C2×C4—141no—
570C2×C4—281no—
571C2×C4—281no—
572C2×C4(C22⋊D7)⋊C412yes—
573C2×C4C22×C4×D711yescharacteristic
574C2×C4C2×D4×D711yescharacteristic
575C2×C4C2×D4×D711yescharacteristic
576C2×C4C2×#115911yescharacteristic
577C2×C4D7×#4111yescharacteristic
578C2×C4D7×#4111yescharacteristic
579C8—41no—
580C8—41no—
581C8—72no—
582C8—72no—
583C8—72no—
584C8—72no—
585C8—142no—
586C8—142no—
587C8—281no—
588C8—281no—
589C7#2154511yescharacteristic, complemented, Sylow
590C22—22no—
591C22—22no—
592C22—22no—
593C22—22no—
594C22—42no—
595C22—42no—
596C22—42no—
597C22—41no—
598C22—282no—
599C22—281no—
600C22C2×(C22⋊D7)⋊C411yescharacteristic
601C22#9027711yescharacteristic
602C22#9034011yescharacteristic
603C4—22no—
604C4—42no—
605C4—42no—
606C4—42no—
607C4—41no—
608C4—282no—
609C4—281no—
610C4C4×#115911yescharacteristic
611C4C4×D4×D711yescharacteristic
612C4#9028711yescharacteristic
613C4D7×#12111yescharacteristic
614C2—21no—
615C2—21no—
616C2—42no—
617C2—81no—
618C2—561no—
619C2#1173210511yescharacteristic
620C2#1173643512yes—
subgroupG/H
centerC2×C4C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1897C2
socleC22×C7C2×C4×D4
prime radical#11740367C2
cosocleC22×C4×C7C24
hypercenter#1897D7
hypocenterC7#21545
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #1897

Complements

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

NH ≅ G/Ntype
2. #11740362C2semidirect
3. C2×#89769C2semidirect
4. C2×#89596C2semidirect
5. #11736072C2semidirect
6. C7×#1897C2semidirect
7. #11729345C2semidirect
8. #11728127C2semidirect
9. #11728126C2semidirect
10. #11728124C2semidirect
11. #11727736C2semidirect
12. #11726501C2semidirect
13. #11726500C2semidirect
14. #11726495C2semidirect
15. #11726494C2semidirect
35. #89823C4semidirect
36. #89820C4semidirect
37. #89813C4semidirect
38. #89810C4semidirect
138. C4×C7⋊C8D4semidirect
149. #1897D7semidirect
589. C7#21545semidirect

Maximal subgroups

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

subgroupindex
#117403672
#117403622
C2×#897692
C2×#895962
#117360722
C7×#18972
#117293452
#117281272
#117281262
#117281242
#117277362
#117265012
#117265002
#117264952
#117264942
#215457