Finite Group Databasebeta

← #13408129

Subgroups — #13408129

Order 1792.

1114 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#529C211yesmaximal, characteristic
2C2×#89837C211yesmaximal, characteristic, complemented
3C2×#89827C211yesmaximal, characteristic, complemented
4#11739521C211yesmaximal, characteristic, complemented
5#11729495C211yesmaximal, characteristic, complemented
6#11729480C211yesmaximal, characteristic, complemented
7#11729449C212yesmaximal, complemented
8#11729434C212yesmaximal
9#11729383C212yesmaximal, complemented
10#11729360C212yesmaximal, complemented
11#11729283C211yesmaximal, characteristic, complemented
12C7×#529—24no—
13C7×#529—24no—
14C2×C7×#126—21no—
15C2×C7×#125C2211yescharacteristic
16C2×C7×#124C2211yescharacteristic
17C2×C7×#123—21no—
18C2×C7×C22⋊Q8C2211yescharacteristic
19C2×C7×#121C2211yescharacteristic
20C2×C4×C7×Q8C2211yescharacteristic
21C2×C4×C7×D4—21no—
22C2×C4×C7×D4C2211yescharacteristic
23C2×C7×#117—21no—
24C22×C42×C7C2211yescharacteristic
25C2×#6487—21no—
26C2×#6432C2211yescharacteristic
27C2×#6409C2212yes—
28C2×#6409C2212yes—
29C2×#6406C2212yes—
30C2×#6406C2212yes—
31#89984—21no—
32#89984C2212yes—
33C2×C22⋊(C7⋊C8)C2212yes—
34#89837—24no—
35#89827—24no—
36C2×#6382C2212yes—
37C2×#6381C2212yes—
38#89707C2211yescharacteristic
39#89701—21no—
40#89700—21no—
41#89700C2212yes—
42C2×#6377C2211yescharacteristic
43C2×#6377C2211yescharacteristic
44#89324C2212yes—
45#89324C2212yes—
46#89319C2212yes—
47#25678—71nomaximal, Sylow
48C22×C7×Q8C2311yescharacteristic
49C22×C7×D4—21no—
50C22×C7×D4C2311yescharacteristic
51C23×C4×C7—21no—
52C23×C4×C7D411yescharacteristic
53C23×C4×C7D411yescharacteristic
54C23×C4×C7C2311yescharacteristic
55C22×Dic14—21no—
56C7×#126—28no—
57C7×#125—22no—
58C7×#125D414yes—
59C7×#124—22no—
60C7×#124D414yes—
61C7×#123—28no—
62C7×C22⋊Q8—22no—
63C7×C22⋊Q8—22no—
64C7×#121—22no—
65C7×#121—22no—
66C4×C7×Q8—22no—
67C4×C7×Q8—22no—
68C4×C7×D4—28no—
69C4×C7×D4—22no—
70C4×C7×D4—22no—
71C7×#117—28no—
72C2×C7×C4⋊C4—21no—
73C2×C7×C4⋊C4—21no—
74C2×C7×C4⋊C4—21no—
75C2×C7×C4⋊C4—21no—
76C2×C7×C4⋊C4C2311yescharacteristic
77C2×C7×C4⋊C4C2311yescharacteristic
78C2×C7×C22⋊C4—21no—
79C2×C7×C22⋊C4—21no—
80C2×C7×C22⋊C4—21no—
81C2×C7×C22⋊C4—21no—
82C2×C7×C22⋊C4—21no—
83C2×C42×C7—21no—
84C2×C42×C7—22no—
85C2×C42×C7—22no—
86C2×C42×C7D411yescharacteristic
87C2×C42×C7D411yescharacteristic
88C2×C42×C7C2311yescharacteristic
89C2×C42×C7C2311yescharacteristic
90C2×C22⋊Dic7—22no—
91C2×C4⋊Dic7—21no—
92C2×C4⋊Dic7—21no—
93C2×C4⋊Dic7—21no—
94C2×C4⋊Dic7C2312yes—
95C2×C4⋊Dic7C2312yes—
96#6487—44no—
97C2×Dic7⋊C4—22no—
98C22×C7⋊C8C2312yes—
99C22×C7⋊C8C2312yes—
100#6432—22no—
101#6432—22no—
102#6409—24no—
103#6409—24no—
104#6406—24no—
105#6406—24no—
106C7⋊#95—22no—
107C7⋊#95—22no—
108C22⋊(C7⋊C8)—24no—
109#6382—24no—
110#6381—24no—
111#6377—22no—
112#6377—22no—
113C2×#529D711yescharacteristic
114C2×#501—71no—
115C2×#489—71no—
116#2857—71no—
117#2036—72no—
118#2034—72no—
119#1992—71no—
120#1959—71no—
121#1865—72no—
122#1862—72no—
123#1749—71no—
124C24×C7—21no—
125C24×C7C2×D411yescharacteristic
126C2×C7×Q8—21no—
127C2×C7×Q8—22no—
128C2×C7×Q8—22no—
129C2×C7×Q8C2×D412yes—
130C2×C7×D4—24no—
131C2×C7×D4—21no—
132C2×C7×D4—22no—
133C2×C7×D4—22no—
134C2×C7×D4—42no—
135C2×C7×D4—42no—
136C2×C7×D4C2×D412yes—
137C22×C4×C7—24no—
138C22×C4×C7—22no—
139C22×C4×C7—21no—
140C22×C4×C7—21no—
141C22×C4×C7—21no—
142C22×C4×C7—22no—
143C22×C4×C7—22no—
144C22×C4×C7—22no—
145C22×C4×C7—21no—
146C22×C4×C7—21no—
147C22×C4×C7—22no—
148C22×C4×C7—22no—
149C22×C4×C7—41no—
150C22×C4×C7—41no—
151C22×C4×C7—42no—
152C22×C4×C7—42no—
153C22×C4×C7C2×D411yescharacteristic
154C22×C4×C7C2×D411yescharacteristic
155C22×C4×C7C2×D411yescharacteristic
156C22×C4×C7C2×D411yescharacteristic
157C22×C4×C7C2×D411yescharacteristic
158C22×C4×C7C2411yescharacteristic
159C22×Dic7—22no—
160C22×Dic7—22no—
161C2×Dic14—42no—
162C2×Dic14—42no—
163C2×Dic14—42no—
164C7×C4⋊C4—24no—
165C7×C4⋊C4—24no—
166C7×C4⋊C4—24no—
167C7×C4⋊C4—24no—
168C7×C4⋊C4—21no—
169C7×C4⋊C4—21no—
170C7×C4⋊C4C2×D412yes—
171C7×C4⋊C4C2×D412yes—
172C7×C22⋊C4—24no—
173C7×C22⋊C4—24no—
174C7×C22⋊C4—24no—
175C7×C22⋊C4—24no—
176C7×C22⋊C4—24no—
177C42×C7—24no—
178C42×C7—22no—
179C42×C7—22no—
180C42×C7—21no—
181C42×C7—21no—
182C42×C7C2×D412yes—
183C42×C7C2×D412yes—
184C22⋊Dic7—44no—
185C4⋊Dic7—22no—
186C4⋊Dic7—22no—
187C4⋊Dic7—22no—
188C4⋊Dic7—22no—
189C4⋊Dic7—42no—
190C4⋊Dic7—42no—
191C4⋊Dic7—42no—
192Dic7⋊C4—44no—
193C2×C7⋊C8—22no—
194C2×C7⋊C8—22no—
195C2×C7⋊C8—22no—
196C2×C7⋊C8—22no—
197C2×C7⋊C8—22no—
198C2×C7⋊C8#4114yes—
199C2×C4⋊Q8—71no—
200#529—24no—
201#529—24no—
202C2×#126—21no—
203C2×#125C2×D711yescharacteristic
204C2×#124C2×D711yescharacteristic
205C2×#123—21no—
206C2×C22⋊Q8—141no—
207C2×C22⋊Q8C2×D711yescharacteristic
208C2×#121C2×D711yescharacteristic
209C2×C4×Q8C2×D711yescharacteristic
210C2×C4×D4—21no—
211C2×C4×D4C2×D711yescharacteristic
212C2×#117—21no—
213C22×C42C2×D711yescharacteristic
214#501—144no—
215#489—144no—
216C2×#107—72no—
217C2×#106—72no—
218C2×#103—72no—
219C2×#103—72no—
220C2×#102—72no—
221C2×#102—72no—
222C2×C22⋊C8—72no—
223C2×#97—71no—
224C2×#97—71no—
225#391—141no—
226#389—72no—
227#389—141no—
228#384—71no—
229#380—72no—
230#380—141no—
231#340—72no—
232#340—72no—
233#336—72no—
234C23×C7—21no—
235C23×C7—21no—
236C23×C7—21no—
237C23×C7—21no—
238C23×C7—21no—
239C23×C7—21no—
240C23×C7—22no—
241C23×C7—41no—
242C23×C7—41no—
243C23×C7—41no—
244C23×C7—41no—
245C23×C7—41no—
246C23×C7—41no—
247C23×C7#13712yes—
248C23×C7C22×D411yescharacteristic
249C7×Q8—24no—
250C7×Q8—42no—
251C7×D4—24no—
252C7×D4—48no—
253C7×D4—42no—
254C2×C4×C7—21no—
255C2×C4×C7—21no—
256C2×C4×C7—21no—
257C2×C4×C7—21no—
258C2×C4×C7—21no—
259C2×C4×C7—21no—
260C2×C4×C7—21no—
261C2×C4×C7—21no—
262C2×C4×C7—22no—
263C2×C4×C7—22no—
264C2×C4×C7—22no—
265C2×C4×C7—21no—
266C2×C4×C7—21no—
267C2×C4×C7—21no—
268C2×C4×C7—21no—
269C2×C4×C7—44no—
270C2×C4×C7—44no—
271C2×C4×C7—48no—
272C2×C4×C7—41no—
273C2×C4×C7—41no—
274C2×C4×C7—41no—
275C2×C4×C7—41no—
276C2×C4×C7—41no—
277C2×C4×C7—41no—
278C2×C4×C7—42no—
279C2×C4×C7—42no—
280C2×C4×C7#12012yes—
281C2×C4×C7#12012yes—
282C2×C4×C7AGL(1,ℤ/8)12yes—
283C2×C4×C7C22×D411yescharacteristic
284C2×C4×C7C22×D411yescharacteristic
285C2×C4×C7C2×#4112yes—
286C2×C4×C72−1+412yes—
287C2×Dic7—42no—
288C2×Dic7—42no—
289C2×Dic7—42no—
290C2×Dic7—42no—
291C2×Dic7—42no—
292C2×Dic7—42no—
293Dic14—84no—
294C7⋊C8—24no—
295C7⋊C8—42no—
296C22×Q8—141no—
297C22×Q8C22×D711yescharacteristic
298C22×D4—21no—
299C22×D4C22×D711yescharacteristic
300C23×C4—21no—
301C23×C4C22⋊D711yescharacteristic
302C23×C4C22⋊D711yescharacteristic
303C23×C4C22×D711yescharacteristic
304C22×C8—72no—
305C22×C8—72no—
306C4⋊Q8—142no—
307C4⋊Q8—142no—
308#126—28no—
309#125—22no—
310#125C22⋊D714yes—
311#124—22no—
312#124C22⋊D714yes—
313#123—28no—
314C22⋊Q8—22no—
315C22⋊Q8—22no—
316C22⋊Q8—284no—
317#121—22no—
318#121—22no—
319C4×Q8—22no—
320C4×Q8—22no—
321C4×D4—28no—
322C4×D4—22no—
323C4×D4—22no—
324#117—28no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—72no—
330C2×C4⋊C4—72no—
331C2×C4⋊C4—142no—
332C2×C4⋊C4—141no—
333C2×C4⋊C4—141no—
334C2×C4⋊C4—141no—
335C2×C4⋊C4C22×D711yescharacteristic
336C2×C4⋊C4C22×D711yescharacteristic
337C2×C22⋊C4—21no—
338C2×C22⋊C4—21no—
339C2×C22⋊C4—21no—
340C2×C22⋊C4—21no—
341C2×C22⋊C4—21no—
342C2×C22⋊C4—142no—
343C2×C42—21no—
344C2×C42—22no—
345C2×C42—22no—
346C2×C42C22⋊D711yescharacteristic
347C2×C42C22⋊D711yescharacteristic
348C2×C42C22×D711yescharacteristic
349C2×C42C22×D711yescharacteristic
350#107—144no—
351#106—144no—
352#103—144no—
353#103—144no—
354#102—144no—
355#102—144no—
356C22⋊C8—144no—
357#97—142no—
358#97—142no—
359#95—142no—
360#95—142no—
361C22×C7—22no—
362C22×C7—22no—
363C22×C7—22no—
364C22×C7—41no—
365C22×C7—41no—
366C22×C7—41no—
367C22×C7—41no—
368C22×C7—42no—
369C22×C7—42no—
370C22×C7—42no—
371C22×C7—81no—
372C22×C7—81no—
373C22×C7—81no—
374C22×C7—81no—
375C22×C7C2×#12011yescharacteristic
376C22×C7#53211yescharacteristic
377C22×C7#53611yescharacteristic
378C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
379C22×C7C2×#13711yescharacteristic
380C22×C7#57511yescharacteristic
381C22×C7#57511yescharacteristic
382C4×C7—22no—
383C4×C7—22no—
384C4×C7—22no—
385C4×C7—41no—
386C4×C7—42no—
387C4×C7—42no—
388C4×C7—41no—
389C4×C7—42no—
390C4×C7#54712yes—
391C4×C7#54812yes—
392Dic7—82no—
393Dic7—82no—
394C24—21no—
395C24C2×C22⋊D711yescharacteristic
396C2×Q8—21no—
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—282no—
400C2×Q8—282no—
401C2×Q8—282no—
402C2×Q8C2×C22⋊D712yes—
403C2×D4—24no—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4—21no—
407C2×D4—42no—
408C2×D4—42no—
409C2×D4C2×C22⋊D712yes—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—41no—
423C22×C4—41no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—142no—
427C22×C4—142no—
428C22×C4C2×C22⋊D711yescharacteristic
429C22×C4C2×C22⋊D711yescharacteristic
430C22×C4C2×C22⋊D711yescharacteristic
431C22×C4C2×C22⋊D711yescharacteristic
432C22×C4C2×C22⋊D711yescharacteristic
433C22×C4C23×D711yescharacteristic
434C2×C8—74no—
435C2×C8—142no—
436C2×C8—142no—
437C2×C8—142no—
438C2×C8—142no—
439C2×C8—142no—
440C4⋊C4—24no—
441C4⋊C4—24no—
442C4⋊C4—24no—
443C4⋊C4—24no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—142no—
447C4⋊C4—142no—
448C4⋊C4—142no—
449C4⋊C4—142no—
450C4⋊C4—282no—
451C4⋊C4—282no—
452C4⋊C4—284no—
453C4⋊C4—282no—
454C4⋊C4C2×C22⋊D712yes—
455C4⋊C4C2×C22⋊D712yes—
456C22⋊C4—24no—
457C22⋊C4—24no—
458C22⋊C4—24no—
459C22⋊C4—24no—
460C22⋊C4—24no—
461C22⋊C4—284no—
462C42—24no—
463C42—22no—
464C42—22no—
465C42—21no—
466C42—21no—
467C42C2×C22⋊D712yes—
468C42C2×C22⋊D712yes—
469C2×C7—42no—
470C2×C7—82no—
471C2×C7#243211yescharacteristic
472C2×C7#300611yescharacteristic
473C2×C7#300711yescharacteristic
474C2×C7#317111yescharacteristic
475C2×C7#318111yescharacteristic
476C2×C7#323911yescharacteristic
477C2×C7#324011yescharacteristic
478C23—21no—
479C23—21no—
480C23—21no—
481C23—21no—
482C23—21no—
483C23—21no—
484C23—22no—
485C23—41no—
486C23—41no—
487C23—41no—
488C23—41no—
489C23—41no—
490C23—41no—
491C23#651412yes—
492C23C22×C22⋊D711yescharacteristic
493Q8—24no—
494Q8—42no—
495Q8—564no—
496D4—24no—
497D4—48no—
498D4—42no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—48no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—282no—
526C2×C4—282no—
527C2×C4—282no—
528C2×C4—282no—
529C2×C4—282no—
530C2×C4—282no—
531C2×C4#651212yes—
532C2×C4#651612yes—
533C2×C4#651612yes—
534C2×C4D7×#4112yes—
535C2×C4#655412yes—
536C2×C4C22×C22⋊D711yescharacteristic
537C2×C4C22×C22⋊D711yescharacteristic
538C8—144no—
539C8—282no—
540C7#2567811yescharacteristic, complemented, Sylow
541C22—22no—
542C22—22no—
543C22—22no—
544C22—41no—
545C22—41no—
546C22—41no—
547C22—41no—
548C22—42no—
549C22—42no—
550C22—42no—
551C22—81no—
552C22—81no—
553C22—81no—
554C22—81no—
555C22C2×#651211yescharacteristic
556C22#9051311yescharacteristic
557C22#9051311yescharacteristic
558C22C2×#651411yescharacteristic
559C22#9051611yescharacteristic
560C22#9052311yescharacteristic
561C22C2×#651611yescharacteristic
562C4—22no—
563C4—22no—
564C4—22no—
565C4—42no—
566C4—41no—
567C4—42no—
568C4—41no—
569C4—42no—
570C4—562no—
571C4—562no—
572C4#9051512yes—
573C4#9052212yes—
574C2—42no—
575C2—82no—
576C2#1174056911yescharacteristic
577C2#1174057011yescharacteristic
578C2#1174058011yescharacteristic
579C2#1174058311yescharacteristic
580C2#1174152711yescharacteristic
581C2#1174152911yescharacteristic
582C2#1174153811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC2×C7×#529C2
socleC23×C7C22×D4
prime radicalC2×C7×#121C22
hypercenterC2×#529D7
hypocenterC7#25678
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#529

Complements

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

NH ≅ G/Ntype
2. C2×#89837C2semidirect
3. C2×#89827C2semidirect
4. #11739521C2semidirect
5. #11729495C2semidirect
6. #11729480C2semidirect
7. #11729449C2semidirect
9. #11729383C2semidirect
10. #11729360C2semidirect
11. #11729283C2semidirect
540. C7#25678semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#5292
C2×#898372
C2×#898272
#117395212
#117294952
#117294802
#117294492
#117294342
#117293832
#117293602
#117292832
#256787