Finite Group Databasebeta

← #13360613

Subgroups — #13360613

Order 1792.

914 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
1C7×#2929C211yesmaximal, characteristic, complemented
2C2×#89916C211yesmaximal, characteristic, complemented
3#11741179C211yesmaximal, characteristic, complemented
4C2×#89873C211yesmaximal, characteristic, complemented
5#11740368C211yesmaximal, characteristic
6#11729762C211yesmaximal, characteristic, complemented
7#11729466C211yesmaximal, characteristic, complemented
8#11728292C211yesmaximal, characteristic, complemented
9#11728290C211yesmaximal, characteristic, complemented
10#11727754C211yesmaximal, characteristic, complemented
11#11727752C211yesmaximal, characteristic, complemented
12#11727737C211yesmaximal, characteristic, complemented
13#11727720C211yesmaximal, characteristic, complemented
14#11727643C211yesmaximal, characteristic, complemented
15#11727641C211yesmaximal, characteristic, complemented
16C2×C7×#131C2211yescharacteristic
17C23×C7×C8C2211yescharacteristic
18C7×#525C2211yescharacteristic
19C2×#6513C2211yescharacteristic
20#90482C2211yescharacteristic
21C2×#6489C2211yescharacteristic
22C22×#1136C2211yescharacteristic
23C2×#6464C2211yescharacteristic
24C7×#435—22no—
25C7×C8×D4—22no—
26C7×#424C2211yescharacteristic
27C7×#408C2211yescharacteristic
28C7×#407C2211yescharacteristic
29C7×#405C2211yescharacteristic
30C2×#6411C2211yescharacteristic
31C2×#6411C2211yescharacteristic
32#90005C2211yescharacteristic
33#89979C2211yescharacteristic
34#89916—24no—
35C2×#6399C2211yescharacteristic
36C2×#6399C2211yescharacteristic
37C2×#6396C2211yescharacteristic
38C2×#6396C2211yescharacteristic
39#89883—22no—
40#89882C2211yescharacteristic
41C8×C22⋊D7—22no—
42#89874C2211yescharacteristic
43#89873—24no—
44#89873C2211yescharacteristic
45C2×#6389C2211yescharacteristic
46C2×#6389C2211yescharacteristic
47C2×C8×Dic7C2211yescharacteristic
48C2×C8×Dic7C2211yescharacteristic
49#89824C412yescomplemented
50#89821C412yescomplemented
51#89814C412yescomplemented
52#89811C412yescomplemented
53#89774C2211yescharacteristic
54#89770C2211yescharacteristic
55#89353C2211yescharacteristic
56#89353C2211yescharacteristic
57#89350C2211yescharacteristic
58#89350C2211yescharacteristic
59#89327C2211yescharacteristic
60#89327C2211yescharacteristic
61#21548—71nomaximal, Sylow
62C2×C7×#41C2311yescharacteristic
63C23×C4×C7C2311yescharacteristic
64C2×#1157C2311yescharacteristic
65C7×#131—22no—
66C7×#131—22no—
67C2×C7×M16—21no—
68C2×C7×M16C2311yescharacteristic
69C2×C7×M16C2311yescharacteristic
70C22×C7×C8—22no—
71C22×C7×C8—22no—
72C22×C7×C8—21no—
73C22×C7×C8D412yes—
74C22×C7×C8D412yes—
75C22×C7×C8C2311yescharacteristic
76C22×C7×C8C2311yescharacteristic
77C7×#123—21no—
78C7×C22⋊Q8C2×C412yes—
79C7×#121C2×C412yes—
80C7×#120—21no—
81C4×C7×D4—22no—
82C7×#117C2311yescharacteristic
83#6516—21no—
84C2×C22⋊Dic7—21no—
85#6513—22no—
86#6513—22no—
87C2×#1144C2×C411yescharacteristic
88C2×#1143C2×C411yescharacteristic
89C2×#1142C2×C411yescharacteristic
90C2×#1141C2×C411yescharacteristic
91#6493C2×C412yes—
92#6492—21no—
93C4×C22⋊D7—22no—
94#6489—22no—
95#6489—22no—
96#6489C2311yescharacteristic
97C2×C4⋊Dic7—21no—
98#6487C2×C412yes—
99C2×C4×Dic7C2311yescharacteristic
100C2×C4×Dic7C2311yescharacteristic
101C2×#1136—22no—
102C2×#1136—22no—
103C2×#1136C2×C412yes—
104C2×#1136C2×C412yes—
105C2×#1136C2311yescharacteristic
106C2×#1136C2311yescharacteristic
107C22×C7⋊C8C2311yescharacteristic
108C22×C7⋊C8C2311yescharacteristic
109#6464—22no—
110#6464—22no—
111C2×#1131—21no—
112C2×C8×D7—21no—
113C7×C4⋊C8—22no—
114C7×C22⋊C8—22no—
115C7×C22⋊C8—22no—
116C7×#97—21no—
117C4×C7×C8—21no—
118#6411—22no—
119#6411—22no—
120#6409C2×C412yes—
121#6406C2×C412yes—
122C22⋊(C7⋊C8)—22no—
123#6399—22no—
124#6399—22no—
125#6396—22no—
126#6396—22no—
127(C2×D7)⋊C8—22no—
128#6389—21no—
129#6389—22no—
130#6389—22no—
131Dic7⋊C8—22no—
132C8×Dic7—22no—
133C8×Dic7—21no—
134C8×Dic7D414yescomplemented
135#6384C2×C412yes—
136#6383C2×C412yes—
137#6382C2×C412yes—
138#6381C2×C412yes—
139#3054—71no—
140C2×#443—71no—
141#2929—71no—
142#2929D711yescharacteristic, complemented
143C2×#405—71no—
144#1975—71no—
145#1946—71no—
146#1938—71no—
147#1938—71no—
148#1897—71no—
149#1897—71no—
150#1811—71no—
151#1772—71no—
152#1764—71no—
153#1764—71no—
154C24×C7#4111yescharacteristic
155C7×#41—21no—
156C7×#41—21no—
157C7×#41—21no—
158C7×#41—21no—
159C2×C7×Q8C22×C411yescharacteristic
160C2×C7×D4—22no—
161C2×C7×D4—21no—
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×C7C22×C412yes—
169C22×C4×C7C2×D412yes—
170C22×C4×C7#4111yescharacteristic
171C22×C4×C7C2411yescharacteristic
172C2×C22⋊D7—22no—
173C2×C22⋊D7—21no—
174C22×Dic7—21no—
175C22×Dic7—21no—
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—
187C7×M16—42no—
188C2×C7×C8—21no—
189C2×C7×C8—21no—
190C2×C7×C8—24no—
191C2×C7×C8—24no—
192C2×C7×C8—21no—
193C2×C7×C8—21no—
194C2×C7×C8—42no—
195C2×C7×C8C2×D412yes—
196C2×C7×C8C2×D412yes—
197C2×C7×C8C2×D411yescharacteristic
198C2×C7×C8C2×D411yescharacteristic
199C2×C7×C8#4112yes—
200C7×C4⋊C4—22no—
201C7×C4⋊C4C22×C412yes—
202C7×C22⋊C4—22no—
203C7×C22⋊C4—21no—
204C7×C22⋊C4—22no—
205C42×C7—21no—
206C22⋊Dic7—24no—
207C22⋊Dic7—22no—
208C22⋊Dic7—21no—
209#1144—22no—
210#1143—22no—
211#1142—22no—
212#1141—22no—
213(C2×D7)⋊C4—22no—
214C4⋊Dic7—24no—
215C4⋊Dic7C22×C412yes—
216Dic7⋊C4—22no—
217C4×Dic7—21no—
218C4×Dic7—21no—
219C4×Dic7—21no—
220C4×Dic7C2×D412yes—
221C4×Dic7#4112yes—
222#1136—24no—
223#1136—24no—
224#1136—21no—
225#1136—21no—
226#1136—21no—
227#1136—21no—
228C2×C7⋊C8—21no—
229C2×C7⋊C8—21no—
230C2×C7⋊C8—21no—
231C2×C7⋊C8—21no—
232C2×C7⋊C8C22×C412yes—
233C2×C7⋊C8C22×C411yescharacteristic
234C2×C7⋊C8C22×C411yescharacteristic
235C2×C7⋊C8C2×D412yes—
236C2×C7⋊C8#4112yes—
237#1131—42no—
238C8×D7—42no—
239C2×#135—71no—
240C2×#131—71no—
241C2×#131C2×D711yescharacteristic
242C22×M16—71no—
243C23×C8C2×D711yescharacteristic
244#525—71no—
245#525C2×D711yescharacteristic
246C2×#117—71no—
247#470—72no—
248#469—72no—
249#468—72no—
250#468—72no—
251#443—144no—
252#435—22no—
253#435—142no—
254C8×D4—22no—
255C8×D4—142no—
256C2×#108—71no—
257C2×#108—71no—
258#427—71no—
259#424—71no—
260#424C2×D711yescharacteristic
261C2×#104—71no—
262C2×#104—71no—
263C2×#104—71no—
264C2×#104—71no—
265#416—71no—
266#416—71no—
267#408—71no—
268#408C2×D711yescharacteristic
269#407—71no—
270#407C2×D711yescharacteristic
271#405—71no—
272#405—144no—
273#405C2×D711yescharacteristic
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#339—71no—
283#339—71no—
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—44no—
296C7×D4—41no—
297C7×D4—41no—
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—41no—
307C2×C4×C7—41no—
308C2×C4×C7C4×D412yes—
309C2×C4×C7C23×C411yescharacteristic
310C2×C4×C7C22×D411yescharacteristic
311C2×C4×C7C22×D411yescharacteristic
312C2×C4×C7C2×#4111yescharacteristic
313C2×C4×C7C2×#4111yescharacteristic
314C2×C4×C7C2×#4111yescharacteristic
315C22⋊D7—44no—
316C22⋊D7—41no—
317C22⋊D7—41no—
318C2×Dic7—22no—
319C2×Dic7—21no—
320C2×Dic7—22no—
321C2×Dic7—22no—
322C2×Dic7—42no—
323C2×Dic7—42no—
324D28—21no—
325D28—21no—
326C4×D7—41no—
327C4×D7—41no—
328Dic14—21no—
329Dic14—21no—
330C7×C8—22no—
331C7×C8—41no—
332C7×C8—41no—
333C7×C8#12112yes—
334C7×C8#12112yes—
335C7⋊C8—22no—
336C7⋊C8—41no—
337C7⋊C8—41no—
338C7⋊C8C4×D412yes—
339C7⋊C8C4×D412yes—
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—22no—
350#131—22no—
351#131—142no—
352#131—142no—
353C2×M16—21no—
354C2×M16—72no—
355C2×M16—72no—
356C2×M16—71no—
357C2×M16—71no—
358C2×M16—142no—
359C2×M16—142no—
360C2×M16—141no—
361C2×M16C22×D711yescharacteristic
362C2×M16C22×D711yescharacteristic
363C22×C8—22no—
364C22×C8—22no—
365C22×C8—21no—
366C22×C8—71no—
367C22×C8—71no—
368C22×C8—141no—
369C22×C8C22⋊D712yes—
370C22×C8C22⋊D712yes—
371C22×C8C22×D711yescharacteristic
372C22×C8C22×D711yescharacteristic
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—71no—
384#117—142no—
385#117—142no—
386#117C22×D711yescharacteristic
387C2×C4⋊C4—141no—
388C2×C22⋊C4—141no—
389C2×C42—71no—
390C2×C42—71no—
391#108—142no—
392#108—142no—
393#107—72no—
394#106—72no—
395C4⋊C8—22no—
396C4⋊C8—142no—
397#104—142no—
398#104—142no—
399#104—142no—
400#104—142no—
401#103—72no—
402#103—72no—
403#102—72no—
404#102—72no—
405C22⋊C8—22no—
406C22⋊C8—22no—
407C22⋊C8—142no—
408C22⋊C8—142no—
409#97—21no—
410#97—142no—
411#97—142no—
412#97—141no—
413C4×C8—21no—
414C4×C8—74no—
415C4×C8—142no—
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—41no—
433C4×C7C2×C4×D411yescharacteristic
434C4×C7C4×#4111yescharacteristic
435C4×C7C2×#12111yescharacteristic
436C4×C7#53211yescharacteristic
437Dic7—42no—
438Dic7—42no—
439Dic7—42no—
440Dic7—41no—
441C24#115711yescharacteristic
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—22no—
453C2×D4—21no—
454C2×D4—71no—
455C2×D4—142no—
456C2×D4—141no—
457C2×D4C2×C4×D711yescharacteristic
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—141no—
464C22×C4—141no—
465C22×C4—141no—
466C22×C4C2×C4×D712yes—
467C22×C4#115711yescharacteristic
468C22×C4C2×C22⋊D712yes—
469C22×C4C23×D711yescharacteristic
470Q16—142no—
471SD16—142no—
472SD16—142no—
473D8—142no—
474M16—21no—
475M16—21no—
476M16—21no—
477M16—21no—
478M16—42no—
479M16—144no—
480M16—144no—
481M16—141no—
482M16—141no—
483M16—141no—
484M16—141no—
485M16—282no—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—24no—
489C2×C8—24no—
490C2×C8—21no—
491C2×C8—21no—
492C2×C8—42no—
493C2×C8—71no—
494C2×C8—71no—
495C2×C8—72no—
496C2×C8—72no—
497C2×C8—72no—
498C2×C8—141no—
499C2×C8—141no—
500C2×C8—141no—
501C2×C8—141no—
502C2×C8—282no—
503C2×C8D4×D712yes—
504C2×C8#115912yes—
505C2×C8C2×C22⋊D712yes—
506C2×C8C2×C22⋊D711yescharacteristic
507C2×C8C2×C22⋊D711yescharacteristic
508C4⋊C4—22no—
509C4⋊C4—72no—
510C4⋊C4—144no—
511C4⋊C4—142no—
512C4⋊C4C2×C4×D712yes—
513C22⋊C4—22no—
514C22⋊C4—21no—
515C22⋊C4—22no—
516C22⋊C4—144no—
517C22⋊C4—142no—
518C22⋊C4—141no—
519C22⋊C4—142no—
520C42—21no—
521C42—72no—
522C42—72no—
523C42—141no—
524C42—141no—
525C42—141no—
526C2×C7—21no—
527C2×C7—21no—
528C2×C7—42no—
529C2×C7—81no—
530C2×C7C4×#12111yescharacteristic
531C2×C7#295512yes—
532D7—81no—
533C23—21no—
534C23—21no—
535C23—21no—
536C23—42no—
537C23—141no—
538C23C4×C22⋊D712yes—
539C23C2×#115711yescharacteristic
540Q8—21no—
541Q8—21no—
542Q8—141no—
543Q8—141no—
544D4—21no—
545D4—21no—
546D4—44no—
547D4—41no—
548D4—41no—
549D4—141no—
550D4—141no—
551D4—284no—
552D4—281no—
553D4—281no—
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—41no—
563C2×C4—41no—
564C2×C4—142no—
565C2×C4—141no—
566C2×C4—142no—
567C2×C4—142no—
568C2×C4—282no—
569C2×C4—282no—
570C2×C4—281no—
571C2×C4—281no—
572C2×C4C4×C22⋊D712yes—
573C2×C4C22×C4×D711yescharacteristic
574C2×C4C2×D4×D711yescharacteristic
575C2×C4C2×#115911yescharacteristic
576C2×C4D7×#4111yescharacteristic
577C2×C4D7×#4111yescharacteristic
578C2×C4C22×C22⋊D711yescharacteristic
579C8—22no—
580C8—41no—
581C8—41no—
582C8—72no—
583C8—72no—
584C8—142no—
585C8—281no—
586C8—281no—
587C8#650212yes—
588C8#650212yes—
589C7#2154811yescharacteristic, complemented, Sylow
590C22—22no—
591C22—22no—
592C22—22no—
593C22—22no—
594C22—42no—
595C22—42no—
596C22—42no—
597C22—41no—
598C22—282no—
599C22—281no—
600C22#9022911yescharacteristic
601C22#9023811yescharacteristic
602C22C2×C4×C22⋊D711yescharacteristic
603C4—22no—
604C4—42no—
605C4—41no—
606C4—282no—
607C4—282no—
608C4—282no—
609C4—281no—
610C4C4×#115911yescharacteristic
611C4C4×D4×D711yescharacteristic
612C4C2×#650211yescharacteristic
613C4#9052311yescharacteristic
614C2—21no—
615C2—21no—
616C2—42no—
617C2—81no—
618C2—561no—
619C2C4×#650211yescharacteristic
620C2#1174123712yes—
subgroupG/H
centerC2×C8C2×C22⋊D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#2929C2
socleC22×C7C2×C4×D4
prime radical#11740368C2
cosocleC22×C4×C7C24
hypercenter#2929D7
hypocenterC7#21548
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2×C8 ◁ C23×C8 ◁ #2929

Complements

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

NH ≅ G/Ntype
1. C7×#2929C2semidirect
2. C2×#89916C2semidirect
3. #11741179C2semidirect
4. C2×#89873C2semidirect
6. #11729762C2semidirect
7. #11729466C2semidirect
8. #11728292C2semidirect
9. #11728290C2semidirect
10. #11727754C2semidirect
11. #11727752C2semidirect
12. #11727737C2semidirect
13. #11727720C2semidirect
14. #11727643C2semidirect
15. #11727641C2semidirect
49. #89824C4semidirect
50. #89821C4semidirect
51. #89814C4semidirect
52. #89811C4semidirect
134. C8×Dic7D4semidirect
142. #2929D7semidirect
589. C7#21548semidirect

Maximal subgroups

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

subgroupindex
C7×#29292
C2×#899162
#117411792
C2×#898732
#117403682
#117297622
#117294662
#117282922
#117282902
#117277542
#117277522
#117277372
#117277202
#117276432
#117276412
#215487