Finite Group Databasebeta

← #13407133

Subgroups — #13407133

Order 1792.

1190 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#553C211yesmaximal, characteristic
2C2×#89841C211yesmaximal, characteristic, complemented
3#11739824C211yesmaximal, characteristic, complemented
4C2×#89791C211yesmaximal, characteristic, complemented
5#11729492C211yesmaximal, characteristic, complemented
6#11729439C212yesmaximal, complemented
7#11729439C212yesmaximal, complemented
8#11729369C211yesmaximal, characteristic, complemented
9#11729363C211yesmaximal, characteristic, complemented
10#11729339C212yesmaximal, complemented
11#11729313C212yesmaximal, complemented
12C7×#553—24no—
13C7×#553—24no—
14C2×C7×#126—21no—
15C2×C7×#125C2211yescharacteristic
16C2×C7×#123—21no—
17C2×C7×#121—21no—
18C2×C7×#121C2211yescharacteristic
19C2×C7×#121C2211yescharacteristic
20C2×C4×C7×D4—21no—
21C2×C4×C7×D4C2211yescharacteristic
22C2×C4×C7×D4C2211yescharacteristic
23C2×C7×#117C2211yescharacteristic
24C22×C7×C4⋊C4C2211yescharacteristic
25C22×#1142C2212yes—
26C2×#6487C2212yes—
27C2×#6487C2212yes—
28C2×#6432C2211yescharacteristic
29#89993—21no—
30C2×#6406C2212yes—
31C2×C22⋊(C7⋊C8)C2212yes—
32C2×C22⋊(C7⋊C8)C2212yes—
33#89841—24no—
34#89791—24no—
35#89748—21no—
36C2×#6384C2212yes—
37C2×#6384C2212yes—
38C2×#6384C2212yes—
39#89721—21no—
40#89721—21no—
41C4×C4⋊Dic7C2211yescharacteristic
42C2×C4⋊(C7⋊C8)C2211yescharacteristic
43C2×#6377C2211yescharacteristic
44#89324C2212yes—
45#89324C2212yes—
46#89324C2212yes—
47#25582—71nomaximal, Sylow
48C22×C7×D4—21no—
49C22×C7×D4—21no—
50C22×C7×D4D411yescharacteristic
51C22×C7×D4C2311yescharacteristic
52C23×C4×C7—21no—
53C23×C4×C7—21no—
54C23×C4×C7C2311yescharacteristic
55C23×C4×C7C2311yescharacteristic
56C22×Dic14C2312yes—
57C7×#126—28no—
58C7×#125—22no—
59C7×#125D414yes—
60C7×#123—28no—
61C7×#121—28no—
62C7×#121—22no—
63C7×#121—22no—
64C7×#121—22no—
65C7×#121—22no—
66C4×C7×D4—28no—
67C4×C7×D4—22no—
68C4×C7×D4—22no—
69C4×C7×D4—22no—
70C4×C7×D4D414yes—
71C7×#117—22no—
72C7×#117—22no—
73C2×C7×C4⋊C4—21no—
74C2×C7×C4⋊C4—21no—
75C2×C7×C4⋊C4—21no—
76C2×C7×C4⋊C4—22no—
77C2×C7×C4⋊C4—22no—
78C2×C7×C4⋊C4D411yescharacteristic
79C2×C7×C4⋊C4C2311yescharacteristic
80C2×C7×C4⋊C4C2311yescharacteristic
81C2×C7×C4⋊C4C2311yescharacteristic
82C2×C7×C22⋊C4—21no—
83C2×C7×C22⋊C4—21no—
84C2×C7×C22⋊C4—21no—
85C2×C7×C22⋊C4—21no—
86C2×C7×C22⋊C4D411yescharacteristic
87C2×C7×C22⋊C4D411yescharacteristic
88C2×C42×C7—21no—
89C2×C42×C7C2311yescharacteristic
90C2×C22⋊Dic7—22no—
91C2×C22⋊Dic7—22no—
92C2×#1142—24no—
93C2×#1142—42no—
94C2×#1142—42no—
95C2×C4⋊Dic7—21no—
96C2×C4⋊Dic7—21no—
97C2×C4⋊Dic7C2312yes—
98#6487—24no—
99#6487—24no—
100#6487—42no—
101#6487—42no—
102C2×Dic7⋊C4—22no—
103C2×Dic7⋊C4—22no—
104C2×C4×Dic7—21no—
105C22×C7⋊C8C2312yes—
106C22×C7⋊C8C2312yes—
107#6432—22no—
108#6432—22no—
109#6406—24no—
110C7⋊#95—21no—
111C22⋊(C7⋊C8)—42no—
112C22⋊(C7⋊C8)—42no—
113#6384—24no—
114#6384—24no—
115#6384—24no—
116C4⋊(C7⋊C8)—22no—
117#6377—22no—
118C2×#553D711yescharacteristic
119C2×#491—71no—
120C2×#461—71no—
121#2542—71no—
122#2000—72no—
123#2000—72no—
124#1989—71no—
125#1955—71no—
126#1873—72no—
127#1872—72no—
128#1762—71no—
129C24×C7—21no—
130C24×C7C2×D411yescharacteristic
131C24×C7C2×D411yescharacteristic
132C2×C7×D4—24no—
133C2×C7×D4—22no—
134C2×C7×D4—24no—
135C2×C7×D4—22no—
136C2×C7×D4—21no—
137C2×C7×D4—22no—
138C2×C7×D4—22no—
139C2×C7×D4—41no—
140C2×C7×D4—42no—
141C2×C7×D4—42no—
142C2×C7×D4—82no—
143C2×C7×D4C2×D412yes—
144C22×C4×C7—21no—
145C22×C4×C7—21no—
146C22×C4×C7—21no—
147C22×C4×C7—24no—
148C22×C4×C7—21no—
149C22×C4×C7—21no—
150C22×C4×C7—21no—
151C22×C4×C7—21no—
152C22×C4×C7—22no—
153C22×C4×C7—22no—
154C22×C4×C7—22no—
155C22×C4×C7—22no—
156C22×C4×C7—24no—
157C22×C4×C7—82no—
158C22×C4×C7—82no—
159C22×C4×C7C2×D411yescharacteristic
160C22×C4×C7C2×D411yescharacteristic
161C22×C4×C7C2×D411yescharacteristic
162C22×C4×C7C2×D411yescharacteristic
163C22×C4×C7C2411yescharacteristic
164C22×Dic7—22no—
165C22×Dic7—22no—
166C2×Dic14—22no—
167C2×Dic14—22no—
168C2×Dic14—22no—
169C2×Dic14—22no—
170C2×Dic14—22no—
171C2×Dic14—22no—
172C7×C4⋊C4—24no—
173C7×C4⋊C4—24no—
174C7×C4⋊C4—24no—
175C7×C4⋊C4—22no—
176C7×C4⋊C4—21no—
177C7×C4⋊C4—21no—
178C7×C4⋊C4—21no—
179C7×C4⋊C4C2×D412yes—
180C7×C4⋊C4C2×D412yes—
181C7×C4⋊C4C2×D412yes—
182C7×C22⋊C4—24no—
183C7×C22⋊C4—24no—
184C7×C22⋊C4—24no—
185C7×C22⋊C4—24no—
186C7×C22⋊C4—22no—
187C7×C22⋊C4—22no—
188C42×C7—24no—
189C42×C7C2×D412yes—
190C42×C7C2×D412yes—
191C22⋊Dic7—44no—
192C22⋊Dic7—44no—
193#1142—48no—
194C4⋊Dic7—22no—
195C4⋊Dic7—42no—
196C4⋊Dic7—42no—
197C4⋊Dic7#4114yes—
198Dic7⋊C4—44no—
199Dic7⋊C4—44no—
200C4×Dic7—42no—
201C2×C7⋊C8—22no—
202C2×C7⋊C8—22no—
203C2×C7⋊C8—22no—
204C2×C7⋊C8—22no—
205C2×C7⋊C8—22no—
206C2×C7⋊C8—22no—
207C22×SD16—72no—
208#553—24no—
209#553—24no—
210C2×C4⋊Q8—71no—
211C2×#126—21no—
212C2×#125C2×D711yescharacteristic
213C2×#123—21no—
214C2×C22⋊Q8—72no—
215C2×C22⋊Q8—72no—
216C2×#121—21no—
217C2×#121C2×D711yescharacteristic
218C2×#121C2×D711yescharacteristic
219C2×C4×D4—21no—
220C2×C4×D4C2×D711yescharacteristic
221C2×C4×D4C2×D711yescharacteristic
222C2×#117C2×D711yescharacteristic
223C22×C4⋊C4C2×D711yescharacteristic
224#491—144no—
225#461—144no—
226C2×C4⋊C8—71no—
227C2×#103—72no—
228C2×#103—72no—
229C2×#103—72no—
230C2×#102—72no—
231C2×C22⋊C8—72no—
232C2×C22⋊C8—72no—
233C2×#97—71no—
234#399—141no—
235#399—141no—
236#390—141no—
237#384—141no—
238C4×C4⋊C4—71no—
239#340—72no—
240#340—72no—
241#340—72no—
242C23×C7—21no—
243C23×C7—22no—
244C23×C7—21no—
245C23×C7—22no—
246C23×C7—21no—
247C23×C7—22no—
248C23×C7—22no—
249C23×C7—21no—
250C23×C7—21no—
251C23×C7—22no—
252C23×C7—42no—
253C23×C7—41no—
254C23×C7—41no—
255C23×C7—42no—
256C23×C7#13712yes—
257C23×C7C22×D411yescharacteristic
258C7×D4—22no—
259C7×D4—22no—
260C7×D4—44no—
261C7×D4—48no—
262C7×D4—42no—
263C7×D4—84no—
264C2×C4×C7—21no—
265C2×C4×C7—21no—
266C2×C4×C7—22no—
267C2×C4×C7—21no—
268C2×C4×C7—22no—
269C2×C4×C7—22no—
270C2×C4×C7—21no—
271C2×C4×C7—22no—
272C2×C4×C7—22no—
273C2×C4×C7—22no—
274C2×C4×C7—41no—
275C2×C4×C7—41no—
276C2×C4×C7—42no—
277C2×C4×C7—42no—
278C2×C4×C7—41no—
279C2×C4×C7—41no—
280C2×C4×C7—41no—
281C2×C4×C7—41no—
282C2×C4×C7—42no—
283C2×C4×C7—42no—
284C2×C4×C7—84no—
285C2×C4×C7—84no—
286C2×C4×C7#12012yes—
287C2×C4×C7#12012yes—
288C2×C4×C7C22×D411yescharacteristic
289C2×C4×C7C22×D411yescharacteristic
290C2×C4×C7C2×#4112yes—
291C2×C4×C7H(5,𝔽2)12yes—
292C2×Dic7—24no—
293C2×Dic7—42no—
294C2×Dic7—42no—
295C2×Dic7—42no—
296C2×Dic7—42no—
297C2×Dic7—42no—
298Dic14—42no—
299Dic14—42no—
300Dic14—42no—
301Dic14—42no—
302C7⋊C8—42no—
303C7⋊C8—42no—
304C22×Q8—72no—
305C22×D4—21no—
306C22×D4—21no—
307C22×D4C22⋊D711yescharacteristic
308C22×D4C22×D711yescharacteristic
309C23×C4—21no—
310C23×C4—21no—
311C23×C4C22×D711yescharacteristic
312C23×C4C22×D711yescharacteristic
313C2×SD16—144no—
314C2×SD16—282no—
315C2×SD16—282no—
316C22×C8—72no—
317C22×C8—72no—
318C4⋊Q8—142no—
319C4⋊Q8—142no—
320#126—28no—
321#125—22no—
322#125C22⋊D714yes—
323#123—28no—
324C22⋊Q8—144no—
325C22⋊Q8—144no—
326C22⋊Q8—282no—
327C22⋊Q8—282no—
328#121—28no—
329#121—22no—
330#121—22no—
331#121—22no—
332#121—22no—
333C4×D4—28no—
334C4×D4—22no—
335C4×D4—22no—
336C4×D4—22no—
337C4×D4C22⋊D714yes—
338#117—22no—
339#117—22no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—72no—
346C2×C4⋊C4—142no—
347C2×C4⋊C4—142no—
348C2×C4⋊C4—141no—
349C2×C4⋊C4—141no—
350C2×C4⋊C4C22⋊D711yescharacteristic
351C2×C4⋊C4C22×D711yescharacteristic
352C2×C4⋊C4C22×D711yescharacteristic
353C2×C4⋊C4C22×D711yescharacteristic
354C2×C22⋊C4—21no—
355C2×C22⋊C4—21no—
356C2×C22⋊C4—21no—
357C2×C22⋊C4—21no—
358C2×C22⋊C4—142no—
359C2×C22⋊C4—142no—
360C2×C22⋊C4C22⋊D711yescharacteristic
361C2×C22⋊C4C22⋊D711yescharacteristic
362C2×C42—21no—
363C2×C42—141no—
364C2×C42C22×D711yescharacteristic
365C4⋊C8—142no—
366#103—144no—
367#103—144no—
368#103—144no—
369#102—144no—
370C22⋊C8—282no—
371C22⋊C8—282no—
372#97—142no—
373#95—141no—
374C22×C7—22no—
375C22×C7—24no—
376C22×C7—41no—
377C22×C7—41no—
378C22×C7—42no—
379C22×C7—41no—
380C22×C7—42no—
381C22×C7—41no—
382C22×C7—42no—
383C22×C7—42no—
384C22×C7—41no—
385C22×C7—42no—
386C22×C7—42no—
387C22×C7—82no—
388C22×C7—82no—
389C22×C7C2×#12011yescharacteristic
390C22×C7#54612yes—
391C22×C7C2×#13711yescharacteristic
392C22×C7#57511yescharacteristic
393C22×C7#57712yes—
394C4×C7—22no—
395C4×C7—22no—
396C4×C7—41no—
397C4×C7—41no—
398C4×C7—42no—
399C4×C7—42no—
400C4×C7—42no—
401C4×C7#53211yescharacteristic
402C4×C7#53511yescharacteristic
403C4×C7#55012yes—
404Dic7—44no—
405Dic7—82no—
406C24—21no—
407C24C2×C22⋊D711yescharacteristic
408C24C2×C22⋊D711yescharacteristic
409C2×Q8—142no—
410C2×Q8—142no—
411C2×Q8—142no—
412C2×Q8—142no—
413C2×Q8—142no—
414C2×Q8—142no—
415C2×D4—24no—
416C2×D4—22no—
417C2×D4—24no—
418C2×D4—22no—
419C2×D4—22no—
420C2×D4—22no—
421C2×D4—21no—
422C2×D4—41no—
423C2×D4—42no—
424C2×D4—42no—
425C2×D4—82no—
426C2×D4C2×C22⋊D712yes—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—24no—
440C22×C4—82no—
441C22×C4—82no—
442C22×C4—142no—
443C22×C4—142no—
444C22×C4C2×C22⋊D711yescharacteristic
445C22×C4C2×C22⋊D711yescharacteristic
446C22×C4C2×C22⋊D711yescharacteristic
447C22×C4C2×C22⋊D711yescharacteristic
448C22×C4C23×D711yescharacteristic
449SD16—288no—
450C2×C8—142no—
451C2×C8—142no—
452C2×C8—142no—
453C2×C8—142no—
454C2×C8—142no—
455C2×C8—142no—
456C4⋊C4—24no—
457C4⋊C4—24no—
458C4⋊C4—24no—
459C4⋊C4—22no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—74no—
464C4⋊C4—142no—
465C4⋊C4—284no—
466C4⋊C4—284no—
467C4⋊C4—282no—
468C4⋊C4—282no—
469C4⋊C4C2×C22⋊D712yes—
470C4⋊C4C2×C22⋊D712yes—
471C4⋊C4C2×C22⋊D712yes—
472C22⋊C4—24no—
473C22⋊C4—24no—
474C22⋊C4—24no—
475C22⋊C4—24no—
476C22⋊C4—22no—
477C22⋊C4—22no—
478C22⋊C4—284no—
479C22⋊C4—284no—
480C42—24no—
481C42—282no—
482C42C2×C22⋊D712yes—
483C42C2×C22⋊D712yes—
484C2×C7—42no—
485C2×C7—81no—
486C2×C7—82no—
487C2×C7#243111yescharacteristic
488C2×C7#300711yescharacteristic
489C2×C7#300911yescharacteristic
490C2×C7#330712yes—
491C2×C7#332012yes—
492C23—21no—
493C23—22no—
494C23—22no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—22no—
499C23—22no—
500C23—21no—
501C23—22no—
502C23—41no—
503C23—41no—
504C23—42no—
505C23—42no—
506C23#651412yes—
507C23C22×C22⋊D711yescharacteristic
508Q8—282no—
509Q8—282no—
510Q8—282no—
511Q8—282no—
512D4—22no—
513D4—22no—
514D4—44no—
515D4—48no—
516D4—42no—
517D4—84no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—21no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—21no—
527C2×C4—22no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—84no—
539C2×C4—84no—
540C2×C4—144no—
541C2×C4—282no—
542C2×C4—282no—
543C2×C4—282no—
544C2×C4—282no—
545C2×C4—282no—
546C2×C4#651612yes—
547C2×C4#651612yes—
548C2×C4D7×#4112yes—
549C2×C4#655312yes—
550C2×C4C22×C22⋊D711yescharacteristic
551C2×C4C22×C22⋊D711yescharacteristic
552C8—282no—
553C8—282no—
554C7#2558211yescharacteristic, complemented, Sylow
555C22—22no—
556C22—24no—
557C22—41no—
558C22—41no—
559C22—42no—
560C22—41no—
561C22—42no—
562C22—41no—
563C22—42no—
564C22—42no—
565C22—41no—
566C22—42no—
567C22—42no—
568C22—82no—
569C22—82no—
570C22#9051311yescharacteristic
571C22C2×#651411yescharacteristic
572C22#9052012yes—
573C22#9052712yes—
574C22C2×#651611yescharacteristic
575C4—22no—
576C4—22no—
577C4—42no—
578C4—41no—
579C4—41no—
580C4—42no—
581C4—42no—
582C4—284no—
583C4—562no—
584C4#9052111yescharacteristic
585C4#9052311yescharacteristic
586C4#9052512yes—
587C2—42no—
588C2—81no—
589C2—82no—
590C2#1174106012yes—
591C2#1174106512yes—
592C2#1174152911yescharacteristic
593C2#1174153711yescharacteristic
594C2#1174155011yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
Fitting, prime radicalC2×C7×#553C2
socleC23×C7C22×D4
hypercenterC2×#553D7
hypocenterC7#25582
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#553

Complements

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

NH ≅ G/Ntype
2. C2×#89841C2semidirect
3. #11739824C2semidirect
4. C2×#89791C2semidirect
5. #11729492C2semidirect
6. #11729439C2semidirect
7. #11729439C2semidirect
8. #11729369C2semidirect
9. #11729363C2semidirect
10. #11729339C2semidirect
11. #11729313C2semidirect
554. C7#25582semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#5532
C2×#898412
#117398242
C2×#897912
#117294922
#117294392
#117293692
#117293632
#117293392
#117293132
#255827