Finite Group Databasebeta

← #13360626

Subgroups — #13360626

Order 1792.

938 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11741240C211yesmaximal, characteristic, complemented
2C2×#89916C211yesmaximal, characteristic, complemented
3#11736284C211yesmaximal, characteristic
4#11735999C211yesmaximal, characteristic, complemented
5C2×#89479C211yesmaximal, characteristic, complemented
6C7×#1765C211yesmaximal, characteristic, complemented
7#11728136C211yesmaximal, characteristic, complemented
8#11728128C211yesmaximal, characteristic, complemented
9#11728041C211yesmaximal, characteristic, complemented
10#11727643C211yesmaximal, characteristic, complemented
11#11727593C211yesmaximal, characteristic, complemented
12#11726694C211yesmaximal, characteristic, complemented
13#11726672C211yesmaximal, characteristic, complemented
14#11726510C211yesmaximal, characteristic, complemented
15#11726502C211yesmaximal, characteristic, complemented
16C2×C7×#131C2211yescharacteristic
17#90519C2211yescharacteristic
18C2×#6513C2211yescharacteristic
19C2×#6464C2211yescharacteristic
20C22×#1131C2211yescharacteristic
21C22×#1131C2211yescharacteristic
22#90215C2211yescharacteristic
23C2×C42×D7C2211yescharacteristic
24C2×C7×#104C2211yescharacteristic
25C7×#416C2211yescharacteristic
26C7×#405C2211yescharacteristic
27C2×C7×#97C2211yescharacteristic
28C2×#6411C2211yescharacteristic
29#90006C2211yescharacteristic
30#89916—24no—
31C2×#6399C2211yescharacteristic
32#89901C2211yescharacteristic
33#89900—22no—
34#89898C2211yescharacteristic
35#89898C2211yescharacteristic
36C2×#6396C2211yescharacteristic
37C2×#6396C2211yescharacteristic
38#89891C2211yescharacteristic
39#89885C2211yescharacteristic
40#89883—22no—
41#89873C2211yescharacteristic
42C2×C8×Dic7C2211yescharacteristic
43C2×C8×Dic7C2211yescharacteristic
44#89771C2211yescharacteristic
45#89770C2211yescharacteristic
46C2×#6379C2211yescharacteristic
47C2×#6377C2211yescharacteristic
48#89588C412yescomplemented
49#89584C412yescomplemented
50#89554C412yescomplemented
51#89552C412yescomplemented
52#89479—24no—
53#89478—22no—
54#89460—22no—
55C7×#341C2211yescharacteristic
56C7×#339C2211yescharacteristic
57#89350C2211yescharacteristic
58#89347C2211yescharacteristic
59#89327C2211yescharacteristic
60#89318C2211yescharacteristic
61#21549—71nomaximal, Sylow
62C2×C7×#41C2311yescharacteristic
63C2×#1157C2311yescharacteristic
64C22×C4×D7—21no—
65C22×C4×D7C2311yescharacteristic
66C7×#131—22no—
67C7×#131—22no—
68C2×C7×M16—21no—
69C2×C7×M16C2311yescharacteristic
70C2×C7×M16C2311yescharacteristic
71C22×C7×C8—21no—
72C22×C7×C8C2311yescharacteristic
73C22×C7×C8C2311yescharacteristic
74C7×#117C2311yescharacteristic
75C2×C42×C7C2311yescharacteristic
76#6513—22no—
77#6513—22no—
78#6509C2×C412yes—
79C2×#1144C2×C411yescharacteristic, complemented
80C2×#1143C2×C411yescharacteristic, complemented
81#6501C2×C412yes—
82#6500—21no—
83#6498—21no—
84C2×#1142C2×C411yescharacteristic, complemented
85C2×#1141C2×C411yescharacteristic, complemented
86C4×C22⋊D7—22no—
87#6489C2311yescharacteristic
88C2×C4×Dic7—21no—
89C2×C4×Dic7C2311yescharacteristic
90C2×#1136C2311yescharacteristic
91C2×#1136C2311yescharacteristic
92C22×C7⋊C8C2311yescharacteristic
93C22×C7⋊C8C2311yescharacteristic
94#6464—22no—
95#6464—22no—
96C2×#1131—22no—
97C2×#1131—22no—
98C2×#1131—22no—
99C2×#1131—22no—
100C2×#1131—21no—
101C2×#1131C2×C412yes—
102C2×#1131C2×C412yes—
103C2×#1131D412yes—
104C2×#1131D412yes—
105C2×C8×D7—21no—
106#6460C2×C412yes—
107#6458C2×C412yes—
108#6449—21no—
109#6445—21no—
110C4×D28—22no—
111C42×D7—22no—
112C42×D7—22no—
113C7×#104—22no—
114C7×#103C2×C412yescomplemented
115C7×#102C2×C412yescomplemented
116C7×#97—21no—
117C7×#97—22no—
118C4×C7×C8—21no—
119#6411—22no—
120C22⋊(C7⋊C8)—22no—
121#6399—22no—
122#6396—22no—
123#6396—22no—
124#6395C2×C412yescomplemented
125(C2×D7)⋊C8—22no—
126(C2×D7)⋊C8—22no—
127(C2×D7)⋊C8—22no—
128#6390C2×C412yescomplemented
129#6389—21no—
130Dic7⋊C8—22no—
131C8×Dic7—21no—
132C8×Dic7—22no—
133C8×Dic7D414yescomplemented
134#6382C2×C412yescomplemented
135#6381C2×C412yescomplemented
136#6379—22no—
137C4⋊(C7⋊C8)—22no—
138#6377—22no—
139#3054—71no—
140C2×#443—71no—
141#2930—71no—
142#2930—71no—
143C2×C4×M16—71no—
144#1975—71no—
145#1946—71no—
146#1937—71no—
147#1937—71no—
148#1896—71no—
149#1896—71no—
150#1811—71no—
151#1772—71no—
152#1765—71no—
153#1765D711yescharacteristic, complemented
154C23×D7#4111yescharacteristic
155C7×#41—21no—
156C7×#41—21no—
157C7×#41—21no—
158C7×#41—21no—
159C2×C7×Q8C22×C411yescharacteristic
160C2×C7×D4—21no—
161C2×C7×D4C22×C411yescharacteristic
162C22×C4×C7—21no—
163C22×C4×C7—21no—
164C22×C4×C7C2411yescharacteristic
165C2×C22⋊D7—21no—
166C2×C22⋊D7—22no—
167C22×Dic7—21no—
168C22×Dic7#4111yescharacteristic
169#1157—21no—
170#1157—21no—
171#1157—21no—
172#1157—21no—
173C2×D28—22no—
174C2×D28C22×C411yescharacteristic
175C2×C4×D7—21no—
176C2×C4×D7—21no—
177C2×C4×D7—21no—
178C2×C4×D7—21no—
179C2×C4×D7—24no—
180C2×C4×D7—21no—
181C2×C4×D7—44no—
182C2×C4×D7C22×C412yes—
183C2×C4×D7C2×D412yes—
184C2×Dic14C22×C411yescharacteristic
185C7×M16—21no—
186C7×M16—21no—
187C7×M16—21no—
188C7×M16—21no—
189C7×M16—42no—
190C2×C7×C8—21no—
191C2×C7×C8—21no—
192C2×C7×C8—21no—
193C2×C7×C8—21no—
194C2×C7×C8—42no—
195C2×C7×C8C22×C412yes—
196C2×C7×C8C2×D411yescharacteristic
197C2×C7×C8C2×D411yescharacteristic
198C2×C7×C8C2×D412yes—
199C2×C7×C8#4112yes—
200C7×C4⋊C4C22×C412yes—
201C7×C22⋊C4—21no—
202C42×C7—21no—
203C42×C7—21no—
204C42×C7#4112yes—
205C22⋊Dic7—21no—
206C22⋊Dic7—22no—
207#1144—22no—
208#1143—22no—
209#1142—22no—
210#1141—22no—
211(C2×D7)⋊C4—22no—
212(C2×D7)⋊C4—22no—
213(C2×D7)⋊C4—22no—
214C4⋊Dic7—22no—
215C4⋊Dic7C22×C412yes—
216Dic7⋊C4—22no—
217C4×Dic7—24no—
218C4×Dic7—21no—
219C4×Dic7—21no—
220C4×Dic7C2×D412yes—
221#1136—21no—
222#1136—21no—
223#1136—21no—
224#1136—21no—
225C2×C7⋊C8—21no—
226C2×C7⋊C8—21no—
227C2×C7⋊C8—21no—
228C2×C7⋊C8—21no—
229C2×C7⋊C8C22×C411yescharacteristic
230C2×C7⋊C8C22×C411yescharacteristic
231C2×C7⋊C8C2×D412yes—
232C2×C7⋊C8C2×D412yes—
233C2×C7⋊C8#4112yes—
234#1131—24no—
235#1131—24no—
236#1131—24no—
237#1131—24no—
238#1131—42no—
239C8×D7—42no—
240C2×#135—71no—
241C2×#131—71no—
242C2×#131C2×D711yescharacteristic
243C22×M16—71no—
244C22×M16—71no—
245#525—71no—
246#525—71no—
247C22×C42—71no—
248#470—72no—
249#469—72no—
250#468—72no—
251#468—72no—
252#443—144no—
253#436—142no—
254#436—142no—
255#435—142no—
256#435—142no—
257C2×#108—71no—
258C2×#108—71no—
259#427—71no—
260#424—71no—
261#424—71no—
262C2×#104—71no—
263C2×#104—71no—
264C2×#104—71no—
265C2×#104C2×D711yescharacteristic
266#416—71no—
267#416C2×D711yescharacteristic
268#408—71no—
269#408—71no—
270#407—71no—
271#407—71no—
272#405—71no—
273#405C2×D711yescharacteristic
274C4×M16—144no—
275C2×#97—71no—
276C2×#97C2×D711yescharacteristic
277C2×C4×C8—71no—
278C2×C4×C8—71no—
279#341—71no—
280#341C2×D711yescharacteristic
281#339—71no—
282#339—71no—
283#339—71no—
284#339C2×D711yescharacteristic
285C23×C7—21no—
286C23×C7C2×#4111yescharacteristic
287C22×D7—21no—
288C22×D7—21no—
289C22×D7—21no—
290C22×D7—42no—
291C22×D7C4×D412yes—
292C7×Q8—21no—
293C7×Q8—21no—
294C7×D4—21no—
295C7×D4—21no—
296C7×D4—41no—
297C7×D4—41no—
298C2×C4×C7—21no—
299C2×C4×C7—22no—
300C2×C4×C7—22no—
301C2×C4×C7—42no—
302C2×C4×C7—41no—
303C2×C4×C7—41no—
304C2×C4×C7C23×C411yescharacteristic
305C2×C4×C7C22×D411yescharacteristic
306C2×C4×C7C22×D411yescharacteristic
307C2×C4×C7C2×#4111yescharacteristic
308C2×C4×C7C2×#4111yescharacteristic
309C2×C4×C7C2×#4111yescharacteristic
310C22⋊D7—41no—
311C22⋊D7—41no—
312C22⋊D7—82no—
313C2×Dic7—24no—
314C2×Dic7—21no—
315C2×Dic7—22no—
316C2×Dic7—21no—
317C2×Dic7—22no—
318C2×Dic7—21no—
319C2×Dic7C4×D412yes—
320D28—21no—
321D28—21no—
322D28—82no—
323C4×D7—22no—
324C4×D7—22no—
325C4×D7—22no—
326C4×D7—22no—
327C4×D7—48no—
328C4×D7—41no—
329C4×D7—41no—
330Dic14—21no—
331Dic14—21no—
332C7×C8—22no—
333C7×C8—41no—
334C7×C8—41no—
335C7×C8#12112yes—
336C7×C8#12112yes—
337C7⋊C8—22no—
338C7⋊C8—41no—
339C7⋊C8—41no—
340C7⋊C8C4×D412yes—
341C7⋊C8C4×D412yes—
342C2×#41—71no—
343C2×#41C22×D711yescharacteristic
344C23×C4—71no—
345C23×C4—141no—
346#135—142no—
347#135—142no—
348C2×Q16—71no—
349C2×SD16—71no—
350C2×SD16—71no—
351C2×D8—71no—
352#131—22no—
353#131—22no—
354#131—142no—
355#131—142no—
356C2×M16—21no—
357C2×M16—72no—
358C2×M16—72no—
359C2×M16—72no—
360C2×M16—72no—
361C2×M16—71no—
362C2×M16—71no—
363C2×M16—142no—
364C2×M16—142no—
365C2×M16—142no—
366C2×M16—142no—
367C2×M16—141no—
368C2×M16C22×D711yescharacteristic
369C2×M16C22×D711yescharacteristic
370C22×C8—21no—
371C22×C8—71no—
372C22×C8—71no—
373C22×C8—141no—
374C22×C8C22×D711yescharacteristic
375C22×C8C22×D711yescharacteristic
376#123—141no—
377#123—141no—
378C22⋊Q8—72no—
379C22⋊Q8—72no—
380#121—72no—
381#121—72no—
382#120—141no—
383#120—141no—
384C4×D4—142no—
385C4×D4—142no—
386#117—71no—
387#117C22×D711yescharacteristic
388C2×C42—71no—
389C2×C42—142no—
390C2×C42—141no—
391C2×C42—142no—
392C2×C42C22×D711yescharacteristic
393#108—142no—
394#108—142no—
395#107—72no—
396#106—72no—
397C4⋊C8—142no—
398C4⋊C8—142no—
399#104—22no—
400#104—142no—
401#104—142no—
402#104—142no—
403#103—72no—
404#103C4×D712yescomplemented
405#102—72no—
406#102C4×D712yescomplemented
407C22⋊C8—142no—
408C22⋊C8—142no—
409C22⋊C8—142no—
410C22⋊C8—142no—
411#97—22no—
412#97—21no—
413#97—142no—
414#97—141no—
415C4×C8—21no—
416C4×C8—74no—
417C4×C8—142no—
418C4×C8—141no—
419C22×C7—22no—
420C22×C7—42no—
421C22×C7—41no—
422C22×C7C2×C4×D411yescharacteristic
423C22×C7#52511yescharacteristic
424C22×C7#52911yescharacteristic
425C2×D7—22no—
426C2×D7—22no—
427C2×D7—22no—
428C2×D7—42no—
429C2×D7—42no—
430C2×D7—42no—
431C2×D7—41no—
432C4×C7—42no—
433C4×C7—42no—
434C4×C7—41no—
435C4×C7C2×C4×D411yescharacteristic
436C4×C7C4×#4111yescharacteristic
437C4×C7C2×#12111yescharacteristic
438C4×C7#53211yescharacteristic
439Dic7—24no—
440Dic7—22no—
441Dic7—42no—
442Dic7—41no—
443C24—71no—
444#41—21no—
445#41—21no—
446#41—21no—
447#41—21no—
448#41—141no—
449#41—141no—
450#41—141no—
451#41—141no—
452C2×Q8—71no—
453C2×Q8C2×C4×D711yescharacteristic
454C2×D4—21no—
455C2×D4—71no—
456C2×D4—142no—
457C2×D4—141no—
458C2×D4—142no—
459C2×D4C2×C4×D711yescharacteristic
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—71no—
463C22×C4—72no—
464C22×C4—72no—
465C22×C4—144no—
466C22×C4—141no—
467C22×C4—141no—
468C22×C4—141no—
469C22×C4—141no—
470C22×C4—141no—
471C22×C4—141no—
472C22×C4—284no—
473C22×C4C23×D711yescharacteristic
474Q16—142no—
475SD16—142no—
476SD16—142no—
477D8—142no—
478M16—21no—
479M16—21no—
480M16—21no—
481M16—21no—
482M16—42no—
483M16—144no—
484M16—144no—
485M16—144no—
486M16—144no—
487M16—141no—
488M16—141no—
489M16—141no—
490M16—141no—
491M16—282no—
492C2×C8—21no—
493C2×C8—21no—
494C2×C8—21no—
495C2×C8—21no—
496C2×C8—42no—
497C2×C8—71no—
498C2×C8—71no—
499C2×C8—72no—
500C2×C8—72no—
501C2×C8—72no—
502C2×C8—141no—
503C2×C8—141no—
504C2×C8—141no—
505C2×C8—141no—
506C2×C8—282no—
507C2×C8C2×C4×D712yes—
508C2×C8#115712yes—
509C2×C8D4×D712yes—
510C2×C8D4×D711yescharacteristic
511C2×C8D4×D711yescharacteristic
512C4⋊C4—72no—
513C4⋊C4—142no—
514C4⋊C4—142no—
515C4⋊C4C2×C4×D712yes—
516C22⋊C4—21no—
517C22⋊C4—142no—
518C22⋊C4—142no—
519C22⋊C4—142no—
520C22⋊C4—142no—
521C22⋊C4—141no—
522C42—21no—
523C42—21no—
524C42—72no—
525C42—144no—
526C42—141no—
527C42—141no—
528C42#115712yes—
529C2×C7—21no—
530C2×C7—21no—
531C2×C7—81no—
532C2×C7C4×#12111yescharacteristic
533C2×C7#295612yes—
534D7—42no—
535D7—81no—
536C23—21no—
537C23—72no—
538C23—141no—
539C23—141no—
540C23—141no—
541C23—282no—
542C23D7×#4111yescharacteristic
543Q8—21no—
544Q8—21no—
545Q8—141no—
546Q8—141no—
547D4—21no—
548D4—21no—
549D4—41no—
550D4—41no—
551D4—141no—
552D4—141no—
553D4—281no—
554D4—281no—
555D4—562no—
556D4—562no—
557C2×C4—22no—
558C2×C4—21no—
559C2×C4—22no—
560C2×C4—42no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—72no—
564C2×C4—144no—
565C2×C4—141no—
566C2×C4—142no—
567C2×C4—141no—
568C2×C4—142no—
569C2×C4—142no—
570C2×C4—142no—
571C2×C4—142no—
572C2×C4—142no—
573C2×C4—141no—
574C2×C4—288no—
575C2×C4—281no—
576C2×C4—281no—
577C2×C4C22×C4×D711yescharacteristic
578C2×C4C2×#115711yescharacteristic
579C2×C4C2×#115711yescharacteristic
580C2×C4C2×D4×D711yescharacteristic
581C2×C4C2×D4×D711yescharacteristic
582C2×C4D7×#4111yescharacteristic
583C8—22no—
584C8—41no—
585C8—41no—
586C8—72no—
587C8—72no—
588C8—142no—
589C8—281no—
590C8—281no—
591C8#644712yes—
592C8#644712yes—
593C7#2154911yescharacteristic, complemented, Sylow
594C22—22no—
595C22—41no—
596C22—42no—
597C22—142no—
598C22—142no—
599C22—142no—
600C22—282no—
601C22—282no—
602C22—281no—
603C22—282no—
604C22#9021911yescharacteristic
605C22C4×D4×D711yescharacteristic
606C22#9023811yescharacteristic
607C4—42no—
608C4—41no—
609C4—42no—
610C4—144no—
611C4—142no—
612C4—281no—
613C4—282no—
614C4C4×#115711yescharacteristic
615C4C2×#644711yescharacteristic
616C4C4×D4×D711yescharacteristic
617C4#9023911yescharacteristic
618C2—21no—
619C2—21no—
620C2—81no—
621C2—282no—
622C2—561no—
623C2C4×#644711yescharacteristic
624C2#1173734312yes—
subgroupG/H
centerC2×C4C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1765C2
socleC22×C7C2×C4×D4
prime radical#11736284C2
cosocleC22×C4×C7C24
hypercenter#1765D7
hypocenterC7#21549
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ #1765

Complements

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

NH ≅ G/Ntype
1. #11741240C2semidirect
2. C2×#89916C2semidirect
4. #11735999C2semidirect
5. C2×#89479C2semidirect
6. C7×#1765C2semidirect
7. #11728136C2semidirect
8. #11728128C2semidirect
9. #11728041C2semidirect
10. #11727643C2semidirect
11. #11727593C2semidirect
12. #11726694C2semidirect
13. #11726672C2semidirect
14. #11726510C2semidirect
15. #11726502C2semidirect
48. #89588C4semidirect
49. #89584C4semidirect
50. #89554C4semidirect
51. #89552C4semidirect
79. C2×#1144C2×C4semidirect
80. C2×#1143C2×C4semidirect
84. C2×#1142C2×C4semidirect
85. C2×#1141C2×C4semidirect
114. C7×#103C2×C4semidirect
115. C7×#102C2×C4semidirect
124. #6395C2×C4semidirect
128. #6390C2×C4semidirect
133. C8×Dic7D4semidirect
134. #6382C2×C4semidirect
135. #6381C2×C4semidirect
153. #1765D7semidirect
404. #103C4×D7semidirect
406. #102C4×D7semidirect
593. C7#21549semidirect

Maximal subgroups

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

subgroupindex
#117412402
C2×#899162
#117362842
#117359992
C2×#894792
C7×#17652
#117281362
#117281282
#117280412
#117276432
#117275932
#117266942
#117266722
#117265102
#117265022
#215497