Finite Group Databasebeta

← #13323513

Subgroups — #13323513

Order 1792.

660 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#6475C211yesmaximal, characteristic
2#11736968C211yesmaximal, characteristic, complemented
3#11736850C211yesmaximal, characteristic, complemented
4C7×#1586C211yesmaximal, characteristic, complemented
5#11728634C211yesmaximal, characteristic, complemented
6#11728618C211yesmaximal, characteristic, complemented
7#11728596C211yesmaximal, characteristic, complemented
8D7×#334C211yesmaximal, characteristic, complemented
9#11726276C211yesmaximal, characteristic, complemented
10#11725920C211yesmaximal, characteristic, complemented
11#11725920C211yesmaximal, characteristic, complemented
12#11725909C211yesmaximal, characteristic, complemented
13#11725908C211yesmaximal, characteristic, complemented
14#11725841C211yesmaximal, characteristic, complemented
15#11725825C211yesmaximal, characteristic, complemented
16C2×#6475C411yescharacteristic
17C4×#1159C2211yescharacteristic
18C4×#1159C2211yescharacteristic
19C4×C7×D8C2211yescharacteristic
20C7×#436C2211yescharacteristic
21C7×C8×D4C2211yescharacteristic
22D8×Dic7C411yescharacteristic
23#89899—21no—
24#89890—21no—
25C8×C22⋊D7—21no—
26C2×Dic7⋊C8—21no—
27C4×#1142C2211yescharacteristic
28C4×#1142C2211yescharacteristic
29#89784C2211yescharacteristic
30D4×C7⋊C8C2211yescharacteristic
31#89652C411yescharacteristic
32#89645C411yescharacteristic
33D7×C4⋊C8C2211yescharacteristic
34D7×C4⋊C8C2211yescharacteristic
35#89603C2211yescharacteristic
36#89602C2211yescharacteristic
37#89544C411yescharacteristic
38#89544C411yescharacteristic
39#89530C411yescharacteristic
40#89530C411yescharacteristic
41#89500—21no—
42(C22⋊D7)⋊C8—21no—
43#89491—21no—
44#89490—21no—
45C4×Dic28C2211yescharacteristic
46#89457C2211yescharacteristic
47#89457C2211yescharacteristic
48C4×C8×D7C2211yescharacteristic
49#89451C2211yescharacteristic
50C8×Dic14C2211yescharacteristic
51C7×#334C2211yescharacteristic
52C7×D4⋊C8C2211yescharacteristic
53C7×D4⋊C8C2211yescharacteristic
54C7×#322C2211yescharacteristic
55D4⋊(C7⋊C8)C2211yescharacteristic
56D4⋊(C7⋊C8)C2211yescharacteristic
57#89279C2211yescharacteristic
58#89279C2211yescharacteristic
59#89275C2211yescharacteristic
60#89275C2211yescharacteristic
61#89252C2211yescharacteristic
62#89251C2211yescharacteristic
63#89251C2211yescharacteristic
64#89250C2211yescharacteristic
65#89249C2211yescharacteristic
66#89249C2211yescharacteristic
67#17882—71nomaximal, Sylow
68C2×#1159C2×C411yescharacteristic
69C2×#1159C2×C411yescharacteristic
70C2×C7×D8C2×C411yescharacteristic
71C2×C7×M16—21no—
72C22×C7×C8—21no—
73C4×C7×D4C2311yescharacteristic
74C4×C7×D4C2311yescharacteristic
75D4×Dic7C2×C411yescharacteristic
76D4×Dic7C2×C411yescharacteristic
77C2×#1142C2×C411yescharacteristic
78C2×#1142C2×C411yescharacteristic
79C4×C22⋊D7—21no—
80C4×C22⋊D7—21no—
81C2×C4×Dic7—21no—
82C2×C4×Dic7—21no—
83C2×#1136—21no—
84C22×C7⋊C8—21no—
85#6475—24no—
86C2×Dic28C2×C411yescharacteristic
87C2×C8×D7—21no—
88C2×C8×D7—21no—
89C2×C8×D7C2×C411yescharacteristic
90C2×C8×D7C2×C411yescharacteristic
91#6455C2×C411yescharacteristic
92#6455C2×C411yescharacteristic
93#6450C2×C411yescharacteristic
94#6450C2×C411yescharacteristic
95(C22⋊D7)⋊C4—21no—
96(C22⋊D7)⋊C4—21no—
97#6440—21no—
98#6440—21no—
99C42×D7C2311yescharacteristic
100C4×Dic14C2311yescharacteristic
101C4×Dic14C2311yescharacteristic
102C7×#107C2×C411yescharacteristic
103C7×C4⋊C8C2311yescharacteristic
104C7×C4⋊C8C2311yescharacteristic
105C7×#102C2×C411yescharacteristic
106C7×#102C2×C411yescharacteristic
107C7×C22⋊C8—21no—
108C7×C22⋊C8—21no—
109C4×C7×C8C2311yescharacteristic
110C4×C7×C8C2311yescharacteristic
111C4×C7×C8C2311yescharacteristic
112#6406C2×C411yescharacteristic
113#6406C2×C411yescharacteristic
114C22⋊(C7⋊C8)—21no—
115C22⋊(C7⋊C8)—21no—
116(C2×D7)⋊C8—21no—
117(C2×D7)⋊C8D411yescharacteristic
118(C2×D7)⋊C8D411yescharacteristic
119#6392C2×C411yescharacteristic
120#6390C2×C411yescharacteristic
121#6390C2×C411yescharacteristic
122#6389—21no—
123#6389—21no—
124Dic7⋊C8—21no—
125Dic7⋊C8—21no—
126Dic7⋊C8—21no—
127Dic7⋊C8—21no—
128Dic7⋊C8—21no—
129Dic7⋊C8—21no—
130Dic7⋊C8—21no—
131Dic7⋊C8D411yescharacteristic
132Dic7⋊C8D411yescharacteristic
133C8×Dic7—21no—
134C8×Dic7—21no—
135C8×Dic7C2×C411yescharacteristic
136C8×Dic7C2×C411yescharacteristic
137#6384C2×C411yescharacteristic
138#6384C2×C411yescharacteristic
139#6382C2×C411yescharacteristic
140#6382C2×C411yescharacteristic
141C4⋊(C7⋊C8)C2311yescharacteristic
142C4⋊(C7⋊C8)C2311yescharacteristic
143C4×C7⋊C8C2311yescharacteristic
144C4×C7⋊C8C2311yescharacteristic
145C4×C7⋊C8C2311yescharacteristic
146#2989—71no—
147#2988—71no—
148C4×#135—71no—
149#1587—71no—
150#1587—71no—
151#1586—71no—
152#1586D711yescharacteristic, complemented
153#1583—71no—
154#1583—71no—
155#1582—71no—
156#1582—71no—
157C2×#334—71no—
158#1477—71no—
159#1477—71no—
160#1452—71no—
161C2×C7×D4C22×C411yescharacteristic
162C2×C7×D4C22×C411yescharacteristic
163C22×C4×C7—21no—
164C22×C4×C7—21no—
165C2×C22⋊D7—21no—
166C2×C22⋊D7—21no—
167C22×Dic7—21no—
168C22×Dic7—21no—
169C22×Dic7—21no—
170C22×Dic7—21no—
171#1159—22no—
172#1159—22no—
173#1159—22no—
174#1159M1614yescomplemented
175C2×C4×D7C22×C411yescharacteristic
176C2×C4×D7C2×D411yescharacteristic
177C2×C4×D7#4111yescharacteristic
178C2×Dic14C22×C411yescharacteristic
179C2×Dic14C22×C411yescharacteristic
180C7×D8—22no—
181C7×M16—81no—
182C2×C7×C8—21no—
183C2×C7×C8—21no—
184C2×C7×C8—21no—
185C2×C7×C8—81no—
186C2×C7×C8C22×C411yescharacteristic
187C2×C7×C8C22×C411yescharacteristic
188C2×C7×C8C2×D411yescharacteristic
189C2×C7×C8C2×D411yescharacteristic
190C7×C4⋊C4C22×C411yescharacteristic
191C7×C4⋊C4C22×C411yescharacteristic
192C7×C22⋊C4—21no—
193C7×C22⋊C4—21no—
194C42×C7C2411yescharacteristic
195C22⋊Dic7—21no—
196C22⋊Dic7—21no—
197#1142—22no—
198#1142—22no—
199(C2×D7)⋊C4—21no—
200(C2×D7)⋊C4—21no—
201C4⋊Dic7C22×C411yescharacteristic
202C4⋊Dic7C22×C411yescharacteristic
203Dic7⋊C4—21no—
204Dic7⋊C4—21no—
205Dic7⋊C4—21no—
206Dic7⋊C4—21no—
207C4×Dic7—21no—
208C4×Dic7—21no—
209C4×Dic7—21no—
210C4×Dic7—21no—
211C4×Dic7—21no—
212C4×Dic7—21no—
213C4×Dic7C22×C411yescharacteristic
214C4×Dic7C2×D411yescharacteristic
215C4×Dic7#4111yescharacteristic
216#1136—81no—
217C2×C7⋊C8—21no—
218C2×C7⋊C8—21no—
219C2×C7⋊C8—21no—
220C2×C7⋊C8—81no—
221C2×C7⋊C8C22×C411yescharacteristic
222C2×C7⋊C8C22×C411yescharacteristic
223C2×C7⋊C8C2×D411yescharacteristic
224C2×C7⋊C8C2×D411yescharacteristic
225Dic28—22no—
226C8×D7—22no—
227C8×D7—22no—
228C8×D7—81no—
229C8×D7—81no—
230C2×#135—71no—
231C4×#41—71no—
232C4×#41—71no—
233#446—71no—
234#446—71no—
235C8×Q8—71no—
236C8×Q8—71no—
237C4×Q16—71no—
238C4×Q16—71no—
239C4×SD16—71no—
240C4×SD16—71no—
241C4×SD16—71no—
242C4×SD16—71no—
243C4×D8—71no—
244C4×D8C2×D711yescharacteristic
245#436—71no—
246#436C2×D711yescharacteristic
247#435—141no—
248#435—141no—
249C8×D4—71no—
250C8×D4—141no—
251C8×D4—141no—
252C8×D4C2×D711yescharacteristic
253#432—141no—
254#431—71no—
255#431—71no—
256#431—141no—
257#427—71no—
258#423—141no—
259C2×C4⋊C8—71no—
260C2×C4⋊C8—71no—
261C2×C4⋊C8—141no—
262#416—71no—
263#416—71no—
264C2×C4×C8—71no—
265#334—71no—
266#334—71no—
267#334—71no—
268#334C2×D711yescharacteristic
269Q8⋊C8—71no—
270Q8⋊C8—71no—
271Q8⋊C8—71no—
272Q8⋊C8—71no—
273D4⋊C8—71no—
274D4⋊C8—71no—
275D4⋊C8C2×D711yescharacteristic
276D4⋊C8C2×D711yescharacteristic
277#322—71no—
278#322—71no—
279#322—71no—
280#322C2×D711yescharacteristic
281C23×C7—21no—
282C23×C7—21no—
283C22×D7C4×D411yescharacteristic
284C7×D4—21no—
285C7×D4—21no—
286C7×D4—21no—
287C7×D4C2×M1612yes—
288C2×C4×C7—21no—
289C2×C4×C7—21no—
290C2×C4×C7—41no—
291C2×C4×C7—41no—
292C2×C4×C7—41no—
293C2×C4×C7—41no—
294C2×C4×C7C23×C411yescharacteristic
295C2×C4×C7C22×D411yescharacteristic
296C2×C4×C7C2×#4111yescharacteristic
297C22⋊D7—24no—
298C22⋊D7—42no—
299C2×Dic7—21no—
300C2×Dic7—24no—
301C2×Dic7—21no—
302C2×Dic7—21no—
303C2×Dic7—21no—
304C2×Dic7—42no—
305C2×Dic7—81no—
306C2×Dic7—81no—
307C2×Dic7C4×D411yescharacteristic
308C2×Dic7C4×D411yescharacteristic
309C2×Dic7C4×D411yescharacteristic
310C4×D7—21no—
311C4×D7—21no—
312C4×D7—41no—
313C4×D7C2×M1612yes—
314C4×D7#13112yes—
315Dic14—21no—
316Dic14—21no—
317Dic14—21no—
318Dic14C2×M1612yes—
319C7×C8—21no—
320C7×C8—21no—
321C7×C8—21no—
322C7×C8—21no—
323C7×C8—41no—
324C7×C8—41no—
325C7×C8—41no—
326C7⋊C8—21no—
327C7⋊C8—21no—
328C7⋊C8—21no—
329C7⋊C8—21no—
330C7⋊C8—41no—
331C7⋊C8—41no—
332C7⋊C8—41no—
333C2×#41—71no—
334C2×#41—71no—
335#135—144no—
336C2×Q16—71no—
337C2×SD16—71no—
338C2×SD16—71no—
339C2×D8C4×D711yescharacteristic
340C2×M16—21no—
341C2×M16—141no—
342C22×C8—21no—
343C22×C8—71no—
344C22×C8—71no—
345C22×C8—141no—
346C22×C8—141no—
347C22×C8—141no—
348C4×Q8—71no—
349C4×Q8—71no—
350C4×Q8—71no—
351C4×Q8—71no—
352C4×D4—71no—
353C4×D4—71no—
354C4×D4—141no—
355C4×D4—141no—
356C4×D4—141no—
357C4×D4—141no—
358C4×D4C22×D711yescharacteristic
359C4×D4C22×D711yescharacteristic
360#117—71no—
361#117—71no—
362#117—141no—
363#117—141no—
364C2×C42—71no—
365C2×C42—141no—
366C2×C42—141no—
367#107—71no—
368#107C4×D711yescharacteristic
369#106—71no—
370#106—71no—
371C4⋊C8—71no—
372C4⋊C8—71no—
373C4⋊C8—71no—
374C4⋊C8—71no—
375C4⋊C8—141no—
376C4⋊C8—141no—
377C4⋊C8—141no—
378C4⋊C8—141no—
379C4⋊C8—141no—
380C4⋊C8—141no—
381C4⋊C8—141no—
382C4⋊C8C22×D711yescharacteristic
383C4⋊C8C22×D711yescharacteristic
384#103—71no—
385#103—71no—
386#103—71no—
387#103—71no—
388#102—71no—
389#102—71no—
390#102C4×D711yescharacteristic
391#102C4×D711yescharacteristic
392C22⋊C8—21no—
393C22⋊C8—21no—
394C22⋊C8—71no—
395C22⋊C8—71no—
396C22⋊C8—141no—
397C22⋊C8—141no—
398C22⋊C8—141no—
399#97—141no—
400#97—141no—
401C4×C8—71no—
402C4×C8—71no—
403C4×C8—71no—
404C4×C8—71no—
405C4×C8—71no—
406C4×C8—141no—
407C4×C8—141no—
408C4×C8C22×D711yescharacteristic
409C4×C8C22×D711yescharacteristic
410C4×C8C22×D711yescharacteristic
411C22×C7—22no—
412C22×C7—41no—
413C22×C7—41no—
414C22×C7—41no—
415C22×C7—41no—
416C22×C7—41no—
417C22×C7C2×C4×D411yescharacteristic
418C2×D7—21no—
419C2×D7—21no—
420C2×D7#43512yes—
421C4×C7—21no—
422C4×C7—41no—
423C4×C7—41no—
424C4×C7C22×M1611yescharacteristic
425C4×C7C2×#13111yescharacteristic
426C4×C7#57711yescharacteristic
427C4×C7#57711yescharacteristic
428Dic7—22no—
429Dic7—21no—
430Dic7—21no—
431Dic7—41no—
432Dic7—41no—
433Dic7—41no—
434Dic7#43512yes—
435#41—74no—
436#41—142no—
437#41—142no—
438#41—142no—
439C2×Q8—71no—
440C2×Q8—71no—
441C2×D4—141no—
442C2×D4—141no—
443C2×D4C2×C4×D711yescharacteristic
444C2×D4C2×C4×D711yescharacteristic
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—71no—
448C22×C4—71no—
449C22×C4—71no—
450C22×C4—141no—
451C22×C4—141no—
452C22×C4—141no—
453C22×C4—141no—
454Q16—142no—
455SD16—142no—
456SD16—142no—
457D8—22no—
458M16—81no—
459M16—561no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—71no—
464C2×C8—71no—
465C2×C8—71no—
466C2×C8—71no—
467C2×C8—81no—
468C2×C8—142no—
469C2×C8—142no—
470C2×C8—141no—
471C2×C8—141no—
472C2×C8—141no—
473C2×C8—561no—
474C2×C8—561no—
475C2×C8—561no—
476C2×C8C2×C4×D711yescharacteristic
477C2×C8C2×C4×D711yescharacteristic
478C2×C8D4×D711yescharacteristic
479C2×C8D4×D711yescharacteristic
480C4⋊C4—71no—
481C4⋊C4—71no—
482C4⋊C4—141no—
483C4⋊C4—141no—
484C4⋊C4—141no—
485C4⋊C4—141no—
486C4⋊C4C2×C4×D711yescharacteristic
487C4⋊C4C2×C4×D711yescharacteristic
488C22⋊C4—21no—
489C22⋊C4—21no—
490C22⋊C4—141no—
491C22⋊C4—141no—
492C22⋊C4—141no—
493C22⋊C4—141no—
494C42—71no—
495C42—71no—
496C42—71no—
497C42—141no—
498C42—141no—
499C42—141no—
500C42—141no—
501C42—141no—
502C42—141no—
503C42C23×D711yescharacteristic
504C2×C7—42no—
505C2×C7—81no—
506C2×C7C2×#43511yescharacteristic
507C2×C7#295211yescharacteristic
508C2×C7#295711yescharacteristic
509D7—22no—
510C23—21no—
511C23—21no—
512C23—71no—
513Q8—72no—
514Q8—141no—
515Q8—141no—
516Q8—141no—
517D4—21no—
518D4—21no—
519D4—21no—
520D4—144no—
521D4—282no—
522D4D7×M1612yes—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—71no—
530C2×C4—72no—
531C2×C4—72no—
532C2×C4—71no—
533C2×C4—71no—
534C2×C4—144no—
535C2×C4—141no—
536C2×C4—141no—
537C2×C4—141no—
538C2×C4—141no—
539C2×C4—141no—
540C2×C4—141no—
541C2×C4—281no—
542C2×C4—282no—
543C2×C4—561no—
544C2×C4—561no—
545C2×C4C22×C4×D711yescharacteristic
546C2×C4C2×D4×D711yescharacteristic
547C2×C4D7×#4111yescharacteristic
548C8—21no—
549C8—21no—
550C8—21no—
551C8—21no—
552C8—41no—
553C8—41no—
554C8—41no—
555C8—141no—
556C8—141no—
557C8—141no—
558C8—141no—
559C8—281no—
560C8—281no—
561C8—281no—
562C7#1788211yescharacteristic, complemented, Sylow
563C22—22no—
564C22—41no—
565C22—41no—
566C22—41no—
567C22—41no—
568C22—41no—
569C22—72no—
570C22—141no—
571C22—141no—
572C22C4×D4×D711yescharacteristic
573C4—21no—
574C4—41no—
575C4—41no—
576C4—72no—
577C4—142no—
578C4—141no—
579C4—141no—
580C4—281no—
581C4—281no—
582C4—281no—
583C4C2×D7×M1611yescharacteristic
584C4D7×#13111yescharacteristic
585C4#9046611yescharacteristic
586C4#9046611yescharacteristic
587C2—42no—
588C2—81no—
589C2—142no—
590C2D7×#43511yescharacteristic
591C2#1173731311yescharacteristic
592C2#1173733511yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC42C23×D7
FittingC7×#1586C2
socleC22×C7C2×C4×D4
prime radicalC2×#6475C4
cosocleC42×C7C24
hypercenter#1586D7
hypocenterC7#17882
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1586

Complements

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

NH ≅ G/Ntype
2. #11736968C2semidirect
3. #11736850C2semidirect
4. C7×#1586C2semidirect
5. #11728634C2semidirect
6. #11728618C2semidirect
7. #11728596C2semidirect
8. D7×#334C2semidirect
9. #11726276C2semidirect
10. #11725920C2semidirect
11. #11725920C2semidirect
12. #11725909C2semidirect
13. #11725908C2semidirect
14. #11725841C2semidirect
15. #11725825C2semidirect
152. #1586D7semidirect
174. #1159M16semidirect
562. C7#17882semidirect

Maximal subgroups

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

subgroupindex
C4×#64752
#117369682
#117368502
C7×#15862
#117286342
#117286182
#117285962
D7×#3342
#117262762
#117259202
#117259092
#117259082
#117258412
#117258252
#178827