Finite Group Databasebeta

← #13321596

Subgroups — #13321596

Order 1792.

898 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3507C211yesmaximal, characteristic
2#11739908C211yesmaximal, characteristic, complemented
3#11739881C211yesmaximal, characteristic, complemented
4#11729591C211yesmaximal, characteristic, complemented
5#11729590C211yesmaximal, characteristic, complemented
6#11729567C211yesmaximal, characteristic
7#11729566C211yesmaximal, characteristic
8#11729511C212yesmaximal, complemented
9#11729510C212yesmaximal, complemented
10#11729509C212yesmaximal, complemented
11#11729508C212yesmaximal, complemented
12C7×#559C412yescomplemented
13C7×#557C412yescomplemented
14C7×#556—22no—
15C7×#556C412yescomplemented
16C7×#554C412yescomplemented
17C7×#552—22no—
18C7×#539—21no—
19C7×#539—21no—
20C7×#536—21no—
21C7×#533C2211yescharacteristic
22C7×#532C2211yescharacteristic
23C7×#529C2212yes—
24C7×#529C2212yes—
25C7×#520—21no—
26C2×C4×C7×Q8C2211yescharacteristic
27#90002—21no—
28#89796—22no—
29#89796—22no—
30#89753—21no—
31#89753—21no—
32C4⋊C4×Dic7—21no—
33#89709C2211yescharacteristic
34#89709C2211yescharacteristic
35#89708C2211yescharacteristic
36C4×C4⋊Dic7C2211yescharacteristic
37#89704C2211yescharacteristic
38#89704C2211yescharacteristic
39#89698C2212yes—
40#89698C2211yescharacteristic
41#89698C2211yescharacteristic
42#89697C2212yes—
43#89333C2212yes—
44#89333C2212yes—
45#89333C2212yes—
46#89333C2212yes—
47#89332C2212yes—
48#89332C2212yes—
49#89332C2212yes—
50#89332C2212yes—
51#17624—71nomaximal, Sylow
52C2×C7×#41—21no—
53C22×C7×Q8D411yescharacteristic
54C7×C4⋊Q8—22no—
55C7×C4⋊Q8C2×C411yescharacteristic
56C7×C4⋊Q8C2×C411yescharacteristic
57C7×C4⋊Q8C2×C411yescharacteristic
58C7×#127C2×C411yescharacteristic
59C7×#126—22no—
60C7×#126—22no—
61C7×#126—22no—
62C7×#126—22no—
63C7×#125—22no—
64C7×#125—22no—
65C7×#125—21no—
66C7×#125C2×C412yes—
67C7×#125C2×C412yes—
68C7×#124—22no—
69C7×#124—21no—
70C7×#124C2×C412yes—
71C7×#124C2×C412yes—
72C7×#123—22no—
73C7×#123—22no—
74C7×#123—22no—
75C7×#123—22no—
76C7×C22⋊Q8—22no—
77C7×C22⋊Q8—22no—
78C7×C22⋊Q8—21no—
79C7×C22⋊Q8—22no—
80C7×C22⋊Q8—22no—
81C7×C22⋊Q8C2312yes—
82C7×#121—21no—
83C7×#121C2312yes—
84C4×C7×Q8—22no—
85C4×C7×Q8—21no—
86C4×C7×Q8C2×C412yes—
87C4×C7×Q8C2×C412yes—
88C4×C7×Q8C2×C412yes—
89C4×C7×Q8C2×C412yes—
90C4×C7×Q8C2×C412yes—
91C4×C7×Q8D412yes—
92C4×C7×Q8D412yes—
93C4×C7×D4—22no—
94C4×C7×D4—22no—
95C4×C7×D4—22no—
96C4×C7×D4—21no—
97C4×C7×D4C2×C412yes—
98C4×C7×D4C2×C412yes—
99C4×C7×D4C2×C412yes—
100C7×#117—21no—
101C7×#117—22no—
102C7×#117—22no—
103C7×#117—22no—
104C2×C7×C4⋊C4—22no—
105C2×C7×C4⋊C4D411yescharacteristic
106C2×C7×C4⋊C4D411yescharacteristic
107C2×C7×C4⋊C4D411yescharacteristic
108C2×C42×C7C2311yescharacteristic
109C2×C42×C7C2311yescharacteristic
110C2×C42×C7C2311yescharacteristic
111C2×C4⋊Dic7—21no—
112C2×C4⋊Dic7—21no—
113C2×C4×Dic7—21no—
114C2×C4×Dic7—21no—
115C2×C4×Dic7—21no—
116C7⋊#95—21no—
117C7⋊#95—21no—
118C7⋊#95—21no—
119C7⋊#95—21no—
120C7⋊#95—21no—
121C7⋊#95—21no—
122C7⋊#95—21no—
123C7⋊#95C2311yescharacteristic
124C7⋊#95C2311yescharacteristic
125C7⋊#95C2311yescharacteristic
126C7⋊#95C2311yescharacteristic
127C22⋊(C7⋊C8)C2312yes—
128C22⋊(C7⋊C8)C2312yes—
129C4⋊(C7⋊C8)—22no—
130C4⋊(C7⋊C8)—22no—
131C4⋊(C7⋊C8)—21no—
132C4⋊(C7⋊C8)—21no—
133#6377—22no—
134#6377—21no—
135#6377—21no—
136C4×C7⋊C8—22no—
137#3507D711yescharacteristic
138#2983—71no—
139#2356—71no—
140#1521—72no—
141#1515—71no—
142#1514—71no—
143#1512—72no—
144#1509—72no—
145#1506—71no—
146#1505—71no—
147#1500—72no—
148C7×#41—42no—
149C7×#41—42no—
150C2×C7×Q8—21no—
151C2×C7×Q8—21no—
152C2×C7×Q8—41no—
153C2×C7×Q8—41no—
154C2×C7×Q8C22⋊C412yes—
155C2×C7×Q8C22⋊C412yes—
156C2×C7×Q8C22×C412yes—
157C2×C7×D4—22no—
158C2×C7×D4—21no—
159C2×C7×D4—21no—
160C2×C7×D4C22×C412yes—
161C22×C4×C7—22no—
162C22×C4×C7—22no—
163C22×C4×C7—22no—
164C22×C4×C7C2×D411yescharacteristic
165C22×C4×C7C2×D411yescharacteristic
166C22×C4×C7C2×D411yescharacteristic
167C22×C4×C7C2×D411yescharacteristic
168C22×C4×C7C2×D411yescharacteristic
169C22×C4×C7C2×D411yescharacteristic
170C22×C4×C7C2411yescharacteristic
171C22×Dic7—21no—
172C22×Dic7—21no—
173C22×Dic7—21no—
174C22×Dic7—21no—
175C7×C4⋊C4—21no—
176C7×C4⋊C4—22no—
177C7×C4⋊C4—22no—
178C7×C4⋊C4—21no—
179C7×C4⋊C4—22no—
180C7×C4⋊C4—22no—
181C7×C4⋊C4—21no—
182C7×C4⋊C4—21no—
183C7×C4⋊C4—22no—
184C7×C4⋊C4—22no—
185C7×C4⋊C4—22no—
186C7×C4⋊C4—22no—
187C7×C4⋊C4C22⋊C412yes—
188C7×C4⋊C4C22⋊C412yes—
189C7×C4⋊C4C22⋊C412yes—
190C7×C4⋊C4C22⋊C412yes—
191C7×C4⋊C4C22⋊C412yes—
192C7×C4⋊C4C22⋊C412yes—
193C7×C4⋊C4C22×C412yes—
194C7×C4⋊C4C22×C412yes—
195C7×C22⋊C4—22no—
196C7×C22⋊C4—22no—
197C7×C22⋊C4—22no—
198C7×C22⋊C4—22no—
199C7×C22⋊C4—22no—
200C7×C22⋊C4—22no—
201C7×C22⋊C4—22no—
202C7×C22⋊C4—22no—
203C7×C22⋊C4—22no—
204C42×C7—22no—
205C42×C7—22no—
206C42×C7—22no—
207C42×C7C22×C411yescharacteristic
208C42×C7C22×C412yes—
209C42×C7C22×C412yes—
210C42×C7C22×C411yescharacteristic
211C42×C7C2×D412yes—
212C42×C7C2×D412yes—
213C42×C7C2×D411yescharacteristic
214C42×C7C2×D411yescharacteristic
215C4⋊Dic7—42no—
216C4⋊Dic7—42no—
217C4×Dic7—42no—
218C4×Dic7—42no—
219C4×Dic7—42no—
220C2×C7⋊C8—22no—
221C2×C7⋊C8—22no—
222#559Dic712yescomplemented
223#557Dic712yescomplemented
224#556—22no—
225#556Dic712yescomplemented
226#554Dic712yescomplemented
227#552—22no—
228#539—21no—
229#539—21no—
230#536—21no—
231#533C2×D711yescharacteristic
232#532C2×D711yescharacteristic
233#529C2×D712yes—
234#529C2×D712yes—
235#520—21no—
236C2×C4×Q8C2×D711yescharacteristic
237#446—142no—
238#446—142no—
239#432—71no—
240#432—71no—
241#432—72no—
242#431—72no—
243#391—141no—
244#387—141no—
245#387—141no—
246#383—71no—
247#383—71no—
248#382—71no—
249C4×C4⋊C4—71no—
250C4×C4⋊C4—141no—
251#376—71no—
252#376—71no—
253#328—72no—
254#328—72no—
255#328—72no—
256#328—72no—
257#327—72no—
258#327—72no—
259#327—72no—
260#327—72no—
261C23×C7—22no—
262C23×C7C22×D411yescharacteristic
263C7×Q8—42no—
264C7×Q8—42no—
265C7×Q8—42no—
266C7×Q8—42no—
267C7×Q8—81no—
268C7×D4—42no—
269C7×D4—42no—
270C7×D4—42no—
271C7×D4—42no—
272C7×D4—81no—
273C2×C4×C7—21no—
274C2×C4×C7—21no—
275C2×C4×C7—21no—
276C2×C4×C7—21no—
277C2×C4×C7—21no—
278C2×C4×C7—21no—
279C2×C4×C7—22no—
280C2×C4×C7—22no—
281C2×C4×C7—22no—
282C2×C4×C7—41no—
283C2×C4×C7—41no—
284C2×C4×C7—41no—
285C2×C4×C7—41no—
286C2×C4×C7—44no—
287C2×C4×C7—82no—
288C2×C4×C7—82no—
289C2×C4×C7C2×C22⋊C411yescharacteristic
290C2×C4×C7C2×C22⋊C412yes—
291C2×C4×C7C2×C22⋊C412yes—
292C2×C4×C7C2×C22⋊C412yes—
293C2×C4×C7C2×C22⋊C412yes—
294C2×C4×C7C2×C22⋊C411yescharacteristic
295C2×C4×C7C2×C22⋊C411yescharacteristic
296C2×C4×C7C2×C22⋊C411yescharacteristic
297C2×C4×C7C23×C411yescharacteristic
298C2×C4×C7C22×D411yescharacteristic
299C2×Dic7—22no—
300C2×Dic7—41no—
301C2×Dic7—41no—
302C2×Dic7—41no—
303C2×Dic7—41no—
304C2×Dic7—41no—
305C2×Dic7—41no—
306C2×Dic7—41no—
307C2×Dic7—41no—
308C2×Dic7—41no—
309C2×Dic7—41no—
310C2×Dic7—41no—
311C7⋊C8—42no—
312C7⋊C8—42no—
313C2×#41—21no—
314C22×Q8C22⋊D711yescharacteristic
315C4⋊Q8—22no—
316C4⋊Q8C2×Dic711yescharacteristic
317C4⋊Q8C2×Dic711yescharacteristic
318C4⋊Q8C2×Dic711yescharacteristic
319#127C2×Dic711yescharacteristic
320#126—22no—
321#126—22no—
322#126—22no—
323#126—22no—
324#125—22no—
325#125—22no—
326#125—21no—
327#125C2×Dic712yes—
328#125C2×Dic712yes—
329#124—22no—
330#124—21no—
331#124C2×Dic712yes—
332#124C2×Dic712yes—
333#123—22no—
334#123—22no—
335#123—22no—
336#123—22no—
337C22⋊Q8—22no—
338C22⋊Q8—22no—
339C22⋊Q8—21no—
340C22⋊Q8—22no—
341C22⋊Q8—22no—
342C22⋊Q8C22×D712yes—
343#121—21no—
344#121C22×D712yes—
345C4×Q8—22no—
346C4×Q8—21no—
347C4×Q8C2×Dic712yes—
348C4×Q8C2×Dic712yes—
349C4×Q8C2×Dic712yes—
350C4×Q8C2×Dic712yes—
351C4×Q8C2×Dic712yes—
352C4×Q8C22⋊D712yes—
353C4×Q8C22⋊D712yes—
354C4×D4—22no—
355C4×D4—22no—
356C4×D4—22no—
357C4×D4—21no—
358C4×D4C2×Dic712yes—
359C4×D4C2×Dic712yes—
360C4×D4C2×Dic712yes—
361#117—21no—
362#117—22no—
363#117—22no—
364#117—22no—
365C2×C4⋊C4—22no—
366C2×C4⋊C4—141no—
367C2×C4⋊C4—141no—
368C2×C4⋊C4C22⋊D711yescharacteristic
369C2×C4⋊C4C22⋊D711yescharacteristic
370C2×C4⋊C4C22⋊D711yescharacteristic
371C2×C42—141no—
372C2×C42—141no—
373C2×C42—141no—
374C2×C42C22×D711yescharacteristic
375C2×C42C22×D711yescharacteristic
376C2×C42C22×D711yescharacteristic
377C4⋊C8—142no—
378C4⋊C8—142no—
379C4⋊C8—141no—
380C4⋊C8—141no—
381C22⋊C8—72no—
382C22⋊C8—72no—
383#97—142no—
384#97—141no—
385#97—141no—
386C4×C8—142no—
387#95—71no—
388#95—71no—
389#95—71no—
390#95—71no—
391#95—141no—
392#95—141no—
393#95—141no—
394#95—141no—
395#95—141no—
396#95—141no—
397#95—141no—
398C22×C7—21no—
399C22×C7—21no—
400C22×C7—42no—
401C22×C7—42no—
402C22×C7—42no—
403C22×C7C22×C22⋊C411yescharacteristic
404C22×C7#57511yescharacteristic
405C22×C7#57511yescharacteristic
406C4×C7—21no—
407C4×C7—21no—
408C4×C7—21no—
409C4×C7—21no—
410C4×C7—22no—
411C4×C7—22no—
412C4×C7—22no—
413C4×C7—22no—
414C4×C7—21no—
415C4×C7—21no—
416C4×C7—42no—
417C4×C7—42no—
418C4×C7—42no—
419Dic7—42no—
420Dic7—81no—
421Dic7—81no—
422Dic7—81no—
423#41—42no—
424#41—42no—
425C2×Q8—21no—
426C2×Q8—21no—
427C2×Q8—41no—
428C2×Q8—41no—
429C2×Q8C22⋊Dic712yes—
430C2×Q8C22⋊Dic712yes—
431C2×Q8C22×Dic712yes—
432C2×D4—22no—
433C2×D4—21no—
434C2×D4—21no—
435C2×D4C22×Dic712yes—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—141no—
440C22×C4—141no—
441C22×C4—141no—
442C22×C4—141no—
443C22×C4C2×C22⋊D711yescharacteristic
444C22×C4C2×C22⋊D711yescharacteristic
445C22×C4C2×C22⋊D711yescharacteristic
446C22×C4C2×C22⋊D711yescharacteristic
447C22×C4C2×C22⋊D711yescharacteristic
448C22×C4C2×C22⋊D711yescharacteristic
449C22×C4C23×D711yescharacteristic
450C2×C8—142no—
451C2×C8—142no—
452C4⋊C4—21no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—21no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—282no—
465C4⋊C4—282no—
466C4⋊C4C22⋊Dic712yes—
467C4⋊C4C22⋊Dic712yes—
468C4⋊C4C22⋊Dic712yes—
469C4⋊C4C22⋊Dic712yes—
470C4⋊C4C22⋊Dic712yes—
471C4⋊C4C22⋊Dic712yes—
472C4⋊C4C22×Dic712yes—
473C4⋊C4C22×Dic712yes—
474C22⋊C4—22no—
475C22⋊C4—22no—
476C22⋊C4—22no—
477C22⋊C4—22no—
478C22⋊C4—22no—
479C22⋊C4—22no—
480C22⋊C4—22no—
481C22⋊C4—22no—
482C22⋊C4—22no—
483C42—22no—
484C42—22no—
485C42—22no—
486C42—282no—
487C42—282no—
488C42—282no—
489C42C22×Dic711yescharacteristic
490C42C22×Dic712yes—
491C42C22×Dic712yes—
492C42C22×Dic711yescharacteristic
493C42C2×C22⋊D712yes—
494C42C2×C22⋊D712yes—
495C42C2×C22⋊D711yescharacteristic
496C42C2×C22⋊D711yescharacteristic
497C2×C7—21no—
498C2×C7—21no—
499C2×C7—82no—
500C2×C7#288411yescharacteristic
501C2×C7#289611yescharacteristic
502C2×C7#290111yescharacteristic
503C23—22no—
504C23C22×C22⋊D711yescharacteristic
505Q8—42no—
506Q8—42no—
507Q8—42no—
508Q8—42no—
509Q8—81no—
510D4—42no—
511D4—42no—
512D4—42no—
513D4—42no—
514D4—81no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—44no—
529C2×C4—82no—
530C2×C4—82no—
531C2×C4—142no—
532C2×C4—281no—
533C2×C4—281no—
534C2×C4—281no—
535C2×C4—281no—
536C2×C4—281no—
537C2×C4—281no—
538C2×C4—281no—
539C2×C4—281no—
540C2×C4—281no—
541C2×C4—281no—
542C2×C4—281no—
543C2×C4C2×C22⋊Dic711yescharacteristic
544C2×C4C2×C22⋊Dic712yes—
545C2×C4C2×C22⋊Dic712yes—
546C2×C4C2×C22⋊Dic712yes—
547C2×C4C2×C22⋊Dic712yes—
548C2×C4C2×C22⋊Dic711yescharacteristic
549C2×C4C2×C22⋊Dic711yescharacteristic
550C2×C4C2×C22⋊Dic711yescharacteristic
551C2×C4C23×Dic711yescharacteristic
552C2×C4C22×C22⋊D711yescharacteristic
553C8—282no—
554C8—282no—
555C7#1762411yescharacteristic, complemented, Sylow
556C22—21no—
557C22—21no—
558C22—42no—
559C22—42no—
560C22—42no—
561C22#9051311yescharacteristic
562C22#9051311yescharacteristic
563C22C22×C22⋊Dic711yescharacteristic
564C4—21no—
565C4—21no—
566C4—22no—
567C4—22no—
568C4—21no—
569C4—21no—
570C4—22no—
571C4—22no—
572C4—21no—
573C4—21no—
574C4—42no—
575C4—42no—
576C4—42no—
577C4—282no—
578C4—561no—
579C4—561no—
580C4—561no—
581C2—21no—
582C2—21no—
583C2—82no—
584C2#1174151711yescharacteristic
585C2#1174156411yescharacteristic
586C2#1174156511yescharacteristic
subgroupG/H
centerC22C22×C22⋊Dic7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#3507C2
socleC22×C7C22×C22⋊C4
prime radicalC7×#532C22
cosocleC22×C4×C7C24
hypercenter#3507D7
hypocenterC7#17624
LCSG ▷ C2×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C4×Q8 ◁ #3507

Complements

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

NH ≅ G/Ntype
2. #11739908C2semidirect
3. #11739881C2semidirect
4. #11729591C2semidirect
5. #11729590C2semidirect
8. #11729511C2semidirect
9. #11729510C2semidirect
10. #11729509C2semidirect
11. #11729508C2semidirect
12. C7×#559C4semidirect
13. C7×#557C4semidirect
15. C7×#556C4semidirect
16. C7×#554C4semidirect
222. #559Dic7semidirect
223. #557Dic7semidirect
225. #556Dic7semidirect
226. #554Dic7semidirect
555. C7#17624semidirect

Maximal subgroups

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

subgroupindex
C7×#35072
#117399082
#117398812
#117295912
#117295902
#117295672
#117295662
#117295112
#117295102
#117295092
#117295082
#176247