Finite Group Databasebeta

← #13526987

Subgroups — #13526987

Order 1792.

2042 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2851C211yesmaximal, characteristic, complemented
2#11740473C212yesmaximal, complemented
3#11739519C211yesmaximal, characteristic, complemented
4C2×C4×(C2×D7)⋊C4C211yesmaximal, characteristic
5#11733448C212yesmaximal, complemented
6#11733447C212yesmaximal, complemented
7#11732807C214yesmaximal, complemented
8#11732806C214yesmaximal, complemented
9#11732036C214yesmaximal
10#11731949C214yesmaximal, complemented
11#11731608C212yesmaximal, complemented
12D7×#380C212yesmaximal
13#11731443C212yesmaximal, complemented
14C23×C4×D7C2211yescharacteristic
15C2×C7×#125C2211yescharacteristic, complemented
16C22×C42×C7C2211yescharacteristic
17C22×(C2×D7)⋊C4C2212yes—
18C22×C4×Dic7C2211yescharacteristic
19C2×D7×C4⋊C4C2212yes—
20C2×D7×C4⋊C4C2212yes—
21C2×#6453C2212yescomplemented
22C2×D7×C22⋊C4C2214yes—
23C2×#6433C2211yescharacteristic, complemented
24C7×#389C2212yescomplemented
25C7×#387C2214yescomplemented
26C7×#385C2214yes—
27C7×#382C2211yescharacteristic, complemented
28C7×#380C2212yes—
29#89984C2212yes—
30#89982C2214yes—
31C2×C7⋊#95C2212yes—
32#89761C2214yes—
33#89761C2214yes—
34#89759C2214yes—
35#89759C2212yes—
36#89759C2214yes—
37#89759C2212yes—
38#89757C2212yescomplemented
39#89756C2214yescomplemented
40#89755C2214yescomplemented
41#89754C2214yescomplemented
42#89752C2214yescomplemented
43#89749C2212yescomplemented
44#89745C2212yescomplemented
45#89722C2214yes—
46#89718C2214yes—
47#89714C2214yes—
48C4×(C2×D7)⋊C4C2212yes—
49C4×(C2×D7)⋊C4C2212yes—
50C4×(C2×D7)⋊C4C2212yes—
51C4×(C2×D7)⋊C4C2212yes—
52#89708C2211yescharacteristic, complemented
53#89705C2214yescomplemented
54#89700C2212yescomplemented
55#89440C2214yes—
56#89440C2214yes—
57#89439C2214yes—
58#89439C2214yes—
59#89439C2214yes—
60D7×#95C2214yes—
61#89431C2214yescomplemented
62#89430C2214yescomplemented
63#89430C2214yescomplemented
64#89429C2214yescomplemented
65#89427C2214yescomplemented
66#89426C2214yescomplemented
67#35020—71nomaximal, Sylow
68C24×D7C2311yescharacteristic
69C23×C4×C7C2311yescharacteristic
70C23×C4×C7C2312yes—
71C23×Dic7C2312yes—
72C23×Dic7C2311yescharacteristic
73C22×C4×D7—42no—
74C22×C4×D7—42no—
75C22×C4×D7Q818yes—
76C22×C4×D7C2312yes—
77C22×C4×D7C2314yes—
78C22×C4×D7C2314yes—
79C22×C4×D7C2312yes—
80C7×#125—42no—
81C2×C7×C4⋊C4C2312yes—
82C2×C7×C4⋊C4C2312yes—
83C2×C7×C4⋊C4C2314yes—
84C2×C7×C4⋊C4C2314yes—
85C2×C7×C22⋊C4C2314yes—
86C2×C42×C7—42no—
87C2×C42×C7C2311yescharacteristic
88C2×C42×C7C2311yescharacteristic
89C2×C42×C7C2312yes—
90C2×C22⋊Dic7C2314yes—
91C2×(C2×D7)⋊C4—28no—
92C2×(C2×D7)⋊C4C2314yes—
93C2×(C2×D7)⋊C4C2314yes—
94C2×(C2×D7)⋊C4C2312yes—
95C2×(C2×D7)⋊C4C2312yes—
96C2×(C2×D7)⋊C4C2312yes—
97C2×(C2×D7)⋊C4C2312yes—
98C2×C4⋊Dic7C2312yes—
99C2×C4⋊Dic7C2312yes—
100C2×C4⋊Dic7C2314yes—
101C2×C4⋊Dic7C2314yes—
102C2×Dic7⋊C4C2314yes—
103C2×Dic7⋊C4C2314yes—
104C2×Dic7⋊C4C2314yes—
105C2×Dic7⋊C4C2314yes—
106C2×Dic7⋊C4C2314yes—
107C2×Dic7⋊C4C2314yes—
108C2×C4×Dic7—42no—
109C2×C4×Dic7C2312yes—
110C2×C4×Dic7C2311yescharacteristic
111C2×C4×Dic7C2311yescharacteristic
112D7×C4⋊C4—44no—
113D7×C4⋊C4—44no—
114#6453—44no—
115D7×C22⋊C4—216no—
116#6433—42no—
117C7×#95C2314yes—
118C7×#95C2314yes—
119C7×#95C2314yes—
120C7⋊#95C2314yes—
121C7⋊#95C2314yes—
122C7⋊#95C2314yes—
123C7⋊#95C2314yes—
124C7⋊#95C2314yes—
125C7⋊#95C2312yes—
126C7⋊#95C2314yes—
127C7⋊#95C2312yes—
128C7⋊#95C2312yes—
129C7⋊#95C2314yes—
130C7⋊#95C2312yes—
131C7⋊#95C2314yes—
132C7⋊#95C2314yes—
133#2851—71no—
134#2851—72no—
135#2851D711yescharacteristic, complemented
136#2646—74no—
137#2569—74no—
138#2569—74no—
139#2442—72no—
140#2442—72no—
141#2442—72no—
142#2442—72no—
143#2315—74no—
144C2×#380—72no—
145C2×C4×C22⋊C4—71no—
146C24×C7C2411yescharacteristic
147C23×D7—41no—
148C23×D7—41no—
149C23×D7—42no—
150C23×D7C2×Q814yes—
151C23×D7#4118yes—
152C23×D7C2412yes—
153C22×C4×C7—22no—
154C22×C4×C7—22no—
155C22×C4×C7—24no—
156C22×C4×C7—42no—
157C22×C4×C7—42no—
158C22×C4×C7C2×Q814yes—
159C22×C4×C7C2412yes—
160C22×C4×C7C2412yes—
161C22×C4×C7C2414yes—
162C22×C4×C7C2414yes—
163C22×C4×C7C2411yescharacteristic
164C22×C4×C7C2411yescharacteristic
165C22×Dic7—24no—
166C22×Dic7—42no—
167C22×Dic7—41no—
168C22×Dic7—41no—
169C22×Dic7—42no—
170C22×Dic7C2×Q814yes—
171C22×Dic7C2412yes—
172C22×Dic7C2412yes—
173C22×Dic7C2414yes—
174C22×Dic7C2414yes—
175C22×Dic7C2412yes—
176C2×C4×D7—24no—
177C2×C4×D7—24no—
178C2×C4×D7—28no—
179C2×C4×D7—28no—
180C2×C4×D7—416no—
181C2×C4×D7—42no—
182C2×C4×D7—42no—
183C2×C4×D7—44no—
184C2×C4×D7—44no—
185C2×C4×D7—44no—
186C2×C4×D7—44no—
187C2×C4×D7C2×Q818yes—
188C2×C4×D7C2×Q818yes—
189C7×C4⋊C4—24no—
190C7×C4⋊C4—24no—
191C7×C4⋊C4—44no—
192C7×C4⋊C4—44no—
193C7×C22⋊C4—28no—
194C42×C7—22no—
195C42×C7—22no—
196C42×C7—42no—
197C22⋊Dic7—28no—
198(C2×D7)⋊C4—28no—
199(C2×D7)⋊C4—28no—
200(C2×D7)⋊C4—24no—
201(C2×D7)⋊C4—24no—
202(C2×D7)⋊C4—24no—
203(C2×D7)⋊C4—24no—
204C4⋊Dic7—24no—
205C4⋊Dic7—24no—
206C4⋊Dic7—44no—
207C4⋊Dic7—44no—
208Dic7⋊C4—44no—
209Dic7⋊C4—44no—
210Dic7⋊C4—44no—
211Dic7⋊C4—44no—
212Dic7⋊C4—44no—
213Dic7⋊C4—44no—
214C4×Dic7—24no—
215C4×Dic7—41no—
216C4×Dic7—41no—
217C24×C4—71no—
218C2×#125—72no—
219C2×#125—71no—
220C2×#125C2×D711yescharacteristic, complemented
221C22×C4⋊C4—72no—
222C22×C4⋊C4—72no—
223C22×C22⋊C4—72no—
224C22×C22⋊C4—74no—
225C22×C42—71no—
226C22×C42C2×D711yescharacteristic
227#400—74no—
228#400—74no—
229#400—74no—
230#400—74no—
231#398—74no—
232#398—74no—
233#398—74no—
234#398—74no—
235#389—72no—
236#389—72no—
237#389—72no—
238#389C2×D712yescomplemented
239#387—74no—
240#387—74no—
241#387—74no—
242#387C2×D714yescomplemented
243#385—74no—
244#385—74no—
245#385—74no—
246#385—74no—
247#385—74no—
248#385—74no—
249#385—74no—
250#385C2×D714yes—
251#382—72no—
252#382—71no—
253#382C2×D711yescharacteristic, complemented
254#380—72no—
255#380—74no—
256#380—74no—
257#380—72no—
258#380—72no—
259#380—74no—
260#380—74no—
261#380—74no—
262#380—74no—
263#380C2×D712yes—
264C4×C22⋊C4—72no—
265C4×C22⋊C4—72no—
266C4×C22⋊C4—72no—
267C4×C22⋊C4—72no—
268C2×#95—72no—
269C2×#95—74no—
270C23×C7—21no—
271C23×C7—21no—
272C23×C7—22no—
273C23×C7C22×Q812yes—
274C23×C7C2×#4114yes—
275C23×C7C2511yescharacteristic
276C22×D7—22no—
277C22×D7—22no—
278C22×D7—22no—
279C22×D7—22no—
280C22×D7—44no—
281C22×D7—44no—
282C22×D7—48no—
283C22×D7—48no—
284C22×D7#125116yescomplemented
285C22×D7C22×Q814yes—
286C22×D7C2×#4114yes—
287C22×D7C2×#4114yes—
288C2×C4×C7—21no—
289C2×C4×C7—21no—
290C2×C4×C7—28no—
291C2×C4×C7—22no—
292C2×C4×C7—22no—
293C2×C4×C7—24no—
294C2×C4×C7—24no—
295C2×C4×C7—24no—
296C2×C4×C7—22no—
297C2×C4×C7—24no—
298C2×C4×C7—22no—
299C2×C4×C7—22no—
300C2×C4×C7—24no—
301C2×C4×C7—24no—
302C2×C4×C7—22no—
303C2×C4×C7—48no—
304C2×C4×C7C22×Q812yes—
305C2×C4×C7C22×Q812yes—
306C2×C4×C7H(5,𝔽2)12yes—
307C2×C4×C7H(5,𝔽2)12yes—
308C2×Dic7—22no—
309C2×Dic7—22no—
310C2×Dic7—22no—
311C2×Dic7—22no—
312C2×Dic7—24no—
313C2×Dic7—24no—
314C2×Dic7—22no—
315C2×Dic7—22no—
316C2×Dic7—24no—
317C2×Dic7—24no—
318C2×Dic7—24no—
319C2×Dic7—22no—
320C2×Dic7—24no—
321C2×Dic7—22no—
322C2×Dic7—44no—
323C2×Dic7—48no—
324C2×Dic7C22×Q814yes—
325C4×D7—216no—
326C4×D7—48no—
327C4×D7—88no—
328C4×D7—88no—
329C25—71no—
330C23×C4—71no—
331C23×C4—72no—
332C23×C4—78no—
333C23×C4—72no—
334C23×C4—72no—
335C23×C4—74no—
336C23×C4—74no—
337C23×C4—282no—
338C23×C4—282no—
339C23×C4C22×D711yescharacteristic
340C23×C4C22×D712yes—
341#125—42no—
342#125—282no—
343#125—284no—
344C2×C4⋊C4—72no—
345C2×C4⋊C4—72no—
346C2×C4⋊C4—74no—
347C2×C4⋊C4—74no—
348C2×C4⋊C4—74no—
349C2×C4⋊C4—74no—
350C2×C4⋊C4—74no—
351C2×C4⋊C4—74no—
352C2×C4⋊C4—74no—
353C2×C4⋊C4—74no—
354C2×C4⋊C4—284no—
355C2×C4⋊C4—284no—
356C2×C4⋊C4C22×D712yes—
357C2×C4⋊C4C22×D712yes—
358C2×C4⋊C4C22×D714yes—
359C2×C4⋊C4C22×D714yes—
360C2×C22⋊C4—74no—
361C2×C22⋊C4—72no—
362C2×C22⋊C4—72no—
363C2×C22⋊C4—72no—
364C2×C22⋊C4—72no—
365C2×C22⋊C4—74no—
366C2×C22⋊C4—74no—
367C2×C22⋊C4—1416no—
368C2×C22⋊C4—148no—
369C2×C22⋊C4C22×D714yes—
370C2×C42—42no—
371C2×C42—72no—
372C2×C42—71no—
373C2×C42—71no—
374C2×C42—282no—
375C2×C42C22×D711yescharacteristic
376C2×C42C22×D711yescharacteristic
377C2×C42C22×D712yes—
378#95—72no—
379#95—72no—
380#95—72no—
381#95—72no—
382#95—74no—
383#95—74no—
384#95—74no—
385#95—74no—
386#95—74no—
387#95—74no—
388#95—74no—
389#95—74no—
390#95—74no—
391#95C22×D714yes—
392#95C22×D714yes—
393#95C22×D714yes—
394C22×C7—22no—
395C22×C7—22no—
396C22×C7—24no—
397C22×C7—24no—
398C22×C7C2×#12514yes—
399C22×C7C23×Q811yescharacteristic
400C22×C7C22×#4112yes—
401C22×C7C2×H(5,𝔽2)11yescharacteristic
402C22×C7C2×H(5,𝔽2)11yescharacteristic
403C22×C7#58512yes—
404C2×D7—24no—
405C2×D7—24no—
406C2×D7—24no—
407C2×D7—24no—
408C2×D7—24no—
409C2×D7—24no—
410C2×D7—416no—
411C2×D7C2×#12518yes—
412C4×C7—22no—
413C4×C7—44no—
414C4×C7—44no—
415C4×C7—42no—
416C4×C7—42no—
417C4×C7#54314yes—
418Dic7—24no—
419Dic7—44no—
420Dic7—44no—
421Dic7—42no—
422Dic7—42no—
423C24—72no—
424C24—74no—
425C24—78no—
426C24—281no—
427C24—281no—
428C24—282no—
429C24C23×D711yescharacteristic
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—72no—
436C22×C4—74no—
437C22×C4—78no—
438C22×C4—78no—
439C22×C4—72no—
440C22×C4—72no—
441C22×C4—74no—
442C22×C4—74no—
443C22×C4—148no—
444C22×C4—148no—
445C22×C4—144no—
446C22×C4—144no—
447C22×C4—144no—
448C22×C4—2816no—
449C22×C4—282no—
450C22×C4—282no—
451C22×C4—284no—
452C22×C4—284no—
453C22×C4—284no—
454C22×C4—284no—
455C22×C4—282no—
456C22×C4—281no—
457C22×C4—281no—
458C22×C4—282no—
459C22×C4Q8×D714yes—
460C22×C4C23×D712yes—
461C22×C4C23×D712yes—
462C22×C4C23×D714yes—
463C22×C4C23×D714yes—
464C22×C4C23×D711yescharacteristic
465C22×C4C23×D711yescharacteristic
466C4⋊C4—24no—
467C4⋊C4—24no—
468C4⋊C4—44no—
469C4⋊C4—44no—
470C4⋊C4—144no—
471C4⋊C4—144no—
472C4⋊C4—284no—
473C4⋊C4—284no—
474C4⋊C4—284no—
475C4⋊C4—284no—
476C4⋊C4—284no—
477C4⋊C4—284no—
478C4⋊C4—284no—
479C4⋊C4—284no—
480C22⋊C4—28no—
481C22⋊C4—148no—
482C22⋊C4—148no—
483C22⋊C4—144no—
484C22⋊C4—144no—
485C22⋊C4—144no—
486C22⋊C4—144no—
487C22⋊C4—148no—
488C42—22no—
489C42—22no—
490C42—42no—
491C42—144no—
492C42—281no—
493C42—281no—
494C2×C7—24no—
495C2×C7C22×#12511yescharacteristic
496C2×C7C2×#54311yescharacteristic
497C2×C7#345811yescharacteristic
498C2×C7#350512yes—
499C2×C7#350512yes—
500D7—28no—
501C23—21no—
502C23—21no—
503C23—22no—
504C23—74no—
505C23—74no—
506C23—74no—
507C23—716no—
508C23—142no—
509C23—142no—
510C23—142no—
511C23—142no—
512C23—284no—
513C23—284no—
514C23—288no—
515C23—288no—
516C23C2×Q8×D712yes—
517C23D7×#4114yes—
518C23C24×D711yescharacteristic
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—28no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—24no—
530C2×C4—22no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—22no—
534C2×C4—48no—
535C2×C4—74no—
536C2×C4—142no—
537C2×C4—142no—
538C2×C4—1416no—
539C2×C4—144no—
540C2×C4—144no—
541C2×C4—142no—
542C2×C4—142no—
543C2×C4—144no—
544C2×C4—142no—
545C2×C4—142no—
546C2×C4—144no—
547C2×C4—144no—
548C2×C4—142no—
549C2×C4—142no—
550C2×C4—144no—
551C2×C4—288no—
552C2×C4—284no—
553C2×C4—288no—
554C2×C4—568no—
555C2×C4—568no—
556C2×C4#654812yes—
557C2×C4C2×Q8×D712yes—
558C2×C4C2×Q8×D712yes—
559C2×C4#655312yes—
560C7#3502011yescharacteristic, complemented, Sylow
561C22—22no—
562C22—22no—
563C22—24no—
564C22—24no—
565C22—78no—
566C22—144no—
567C22—144no—
568C22—144no—
569C22—144no—
570C22—144no—
571C22—144no—
572C22—2816no—
573C22D7×#12514yes—
574C22C2×#654811yescharacteristic
575C22C22×Q8×D711yescharacteristic
576C22C2×D7×#4112yes—
577C22C2×#655311yescharacteristic
578C22#9061612yes—
579C4—22no—
580C4—42no—
581C4—42no—
582C4—44no—
583C4—44no—
584C4—144no—
585C4—284no—
586C4—282no—
587C4—282no—
588C4—284no—
589C4#9032514yes—
590C2—24no—
591C2—148no—
592C2C2×#9032511yescharacteristic
593C2#1174311212yes—
594C2C2×D7×#12511yescharacteristic
595C2#1174322612yes—
596C2#1174334211yescharacteristic
subgroupG/H
center, FrattiniC23C24×D7
commutator, socle, cosocleC23×C7C25
FittingC7×#2851C2
prime radicalC24×D7C23
hypercenter#2851D7
hypocenterC7#35020
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #2851

Complements

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

NH ≅ G/Ntype
1. C7×#2851C2semidirect
2. #11740473C2semidirect
3. #11739519C2semidirect
5. #11733448C2semidirect
6. #11733447C2semidirect
7. #11732807C2semidirect
8. #11732806C2semidirect
10. #11731949C2semidirect
11. #11731608C2semidirect
13. #11731443C2semidirect
15. C2×C7×#125C22semidirect
21. C2×#6453C22semidirect
23. C2×#6433C22semidirect
24. C7×#389C22semidirect
25. C7×#387C22semidirect
27. C7×#382C22semidirect
38. #89757C22semidirect
39. #89756C22semidirect
40. #89755C22semidirect
41. #89754C22semidirect
42. #89752C22semidirect
43. #89749C22semidirect
44. #89745C22semidirect
52. #89708C22semidirect
53. #89705C22semidirect
54. #89700C22semidirect
61. #89431C22semidirect
62. #89430C22semidirect
63. #89430C22semidirect
64. #89429C22semidirect
65. #89427C22semidirect
66. #89426C22semidirect
135. #2851D7semidirect
220. C2×#125C2×D7semidirect
238. #389C2×D7semidirect
242. #387C2×D7semidirect
253. #382C2×D7semidirect
284. C22×D7#125semidirect
560. C7#35020semidirect

Maximal subgroups

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

subgroupindex
C7×#28512
#117404732
#117395192
C2×C4×(C2×D7)⋊C42
#117334482
#117334472
#117328072
#117328062
#117320362
#117319492
#117316082
D7×#3802
#117314432
#350207