Finite Group Databasebeta

← #13390172

Subgroups — #13390172

Order 1792.

1146 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#554C211yesmaximal, characteristic, complemented
2C2×#89856C211yesmaximal, characteristic, complemented
3#11739945C211yesmaximal, characteristic, complemented
4C2×C4×#1141C211yesmaximal, characteristic
5#11729541C211yesmaximal, characteristic, complemented
6#11729497C212yesmaximal, complemented
7#11729493C212yesmaximal, complemented
8#11729388C211yesmaximal, characteristic, complemented
9#11729367C212yesmaximal
10#11729361C212yesmaximal, complemented
11#11729358C211yesmaximal, characteristic, complemented
12C7×#554—24no—
13C7×#554—24no—
14C2×C7×C4⋊Q8C2211yescharacteristic
15C2×C7×#125C2212yes—
16C2×C7×C22⋊Q8—21no—
17C2×C7×C22⋊Q8—22no—
18C2×C4×C7×Q8C2211yescharacteristic
19C2×C4×C7×D4C2212yes—
20C2×C4×C7×D4C2211yescharacteristic
21C22×C7×C4⋊C4—21no—
22C2×#6509—21no—
23C22×#1141C2211yescharacteristic
24C2×C4×D28C2211yescharacteristic
25C2×#6406C2211yescharacteristic
26C2×#6406C2212yes—
27#89856—24no—
28C4×#1141—24no—
29#89766—22no—
30#89759—21no—
31C2×#6383C2211yescharacteristic
32C2×#6383C2212yes—
33C2×#6382C2212yes—
34C2×#6381C2211yescharacteristic
35#89711C2212yes—
36#89707C2211yescharacteristic
37#89700C2212yes—
38C2×C4⋊(C7⋊C8)C2211yescharacteristic
39C2×C4×C7⋊C8C2211yescharacteristic
40#89324C2212yes—
41#89324C2212yes—
42#89324C2212yes—
43#89319C2212yes—
44#89319C2212yes—
45#24327—71nomaximal, Sylow
46C22×C7×Q8—21no—
47C22×C7×Q8D411yescharacteristic
48C22×C7×D4C2311yescharacteristic
49C23×C4×C7—21no—
50C23×C4×C7—22no—
51C22×D28C2311yescharacteristic
52C22×C4×D7—21no—
53C7×C4⋊Q8—22no—
54C7×C4⋊Q8D414yes—
55C7×#125—24no—
56C7×#125—24no—
57C7×C22⋊Q8—28no—
58C7×C22⋊Q8—216no—
59C4×C7×Q8—22no—
60C4×C7×Q8—22no—
61C4×C7×D4—24no—
62C4×C7×D4—24no—
63C4×C7×D4—22no—
64C4×C7×D4D414yes—
65C2×C7×C4⋊C4—28no—
66C2×C7×C4⋊C4—22no—
67C2×C7×C4⋊C4—21no—
68C2×C7×C4⋊C4—22no—
69C2×C7×C4⋊C4D411yescharacteristic
70C2×C7×C4⋊C4D412yes—
71C2×C7×C4⋊C4C2311yescharacteristic
72C2×C7×C4⋊C4C2312yes—
73C2×C7×C22⋊C4—22no—
74C2×C7×C22⋊C4—21no—
75C2×C42×C7C2312yes—
76C2×C42×C7C2311yescharacteristic
77#6509—44no—
78C2×#1141—22no—
79C2×#1141—22no—
80C2×#1141Q814yescomplemented
81C2×(C2×D7)⋊C4—21no—
82C2×(C2×D7)⋊C4—22no—
83C2×(C2×D7)⋊C4—22no—
84C2×C4⋊Dic7—21no—
85C2×C4⋊Dic7—21no—
86C2×C4⋊Dic7C2311yescharacteristic
87C2×C4⋊Dic7C2312yes—
88C2×Dic7⋊C4—22no—
89C22×C7⋊C8C2311yescharacteristic
90C22×C7⋊C8C2312yes—
91C22×C7⋊C8C2311yescharacteristic
92C4×D28—22no—
93C4×D28—22no—
94#6406—22no—
95#6406—24no—
96C7⋊#95—22no—
97#6383—24no—
98#6383—22no—
99#6382—24no—
100#6381—22no—
101C4⋊(C7⋊C8)—22no—
102C4×C7⋊C8—22no—
103C2×#554D711yescharacteristic, complemented
104C2×#476—71no—
105C2×C4×D8—71no—
106#2538—71no—
107#2059—72no—
108#2025—72no—
109#1990—71no—
110#1984—71no—
111#1879—72no—
112#1864—72no—
113#1802—71no—
114C24×C7—21no—
115C23×D7—21no—
116C2×C7×Q8—22no—
117C2×C7×Q8—24no—
118C2×C7×Q8—22no—
119C2×C7×Q8—22no—
120C2×C7×Q8—81no—
121C2×C7×Q8—81no—
122C2×C7×D4—21no—
123C2×C7×D4—21no—
124C2×C7×D4—21no—
125C2×C7×D4C2×D412yes—
126C2×C7×D4C2×Q812yes—
127C2×C7×D4#4112yes—
128C22×C4×C7—21no—
129C22×C4×C7—22no—
130C22×C4×C7—28no—
131C22×C4×C7—24no—
132C22×C4×C7—42no—
133C22×C4×C7—42no—
134C22×C4×C7—82no—
135C22×C4×C7—82no—
136C22×C4×C7C2×D412yes—
137C22×C4×C7C2×D412yes—
138C22×C4×C7C2×D411yescharacteristic
139C22×C4×C7C2×D411yescharacteristic
140C22×C4×C7C2411yescharacteristic
141C22×Dic7—21no—
142C22×Dic7—22no—
143C2×D28—21no—
144C2×D28—21no—
145C2×D28—21no—
146C2×D28—21no—
147C2×D28—21no—
148C2×D28C2×Q812yes—
149C2×C4×D7—42no—
150C2×C4×D7—42no—
151C2×C4×D7—42no—
152C7×C4⋊C4—28no—
153C7×C4⋊C4—28no—
154C7×C4⋊C4—22no—
155C7×C4⋊C4—24no—
156C7×C4⋊C4—24no—
157C7×C4⋊C4—21no—
158C7×C4⋊C4—22no—
159C7×C4⋊C4C2×D414yes—
160C7×C4⋊C4C2×D412yes—
161C7×C22⋊C4—28no—
162C7×C22⋊C4—24no—
163C42×C7—22no—
164C42×C7—22no—
165C42×C7C2×D412yes—
166C42×C7C2×D412yes—
167#1141—28no—
168(C2×D7)⋊C4—44no—
169(C2×D7)⋊C4—42no—
170(C2×D7)⋊C4—82no—
171C4⋊Dic7—21no—
172C4⋊Dic7—21no—
173C4⋊Dic7—22no—
174C4⋊Dic7—22no—
175C4⋊Dic7—42no—
176C4⋊Dic7—81no—
177Dic7⋊C4—44no—
178C2×C7⋊C8—21no—
179C2×C7⋊C8—21no—
180C2×C7⋊C8—21no—
181C2×C7⋊C8—22no—
182C2×C7⋊C8—22no—
183C2×C7⋊C8—21no—
184C2×C7⋊C8—21no—
185C2×C7⋊C8—22no—
186C2×C7⋊C8C2×Q812yes—
187C22×D8—71no—
188#554—24no—
189#554—24no—
190C2×C4⋊Q8C2×D711yescharacteristic
191C2×#125C2×D712yes—
192C2×C22⋊Q8—21no—
193C2×C22⋊Q8—22no—
194C2×C22⋊Q8—141no—
195C2×C4×Q8C2×D711yescharacteristic
196C2×C4×D4—71no—
197C2×C4×D4C2×D712yes—
198C2×C4×D4C2×D711yescharacteristic
199C22×C4⋊C4—21no—
200#476—144no—
201C4×D8—144no—
202C2×#107—71no—
203C2×#106—72no—
204C2×C4⋊C8—71no—
205C2×#102—72no—
206C2×#102—71no—
207C2×#102—72no—
208C2×#102—71no—
209C2×C4×C8—71no—
210#396—142no—
211#390—72no—
212#389—72no—
213#384—71no—
214#380—141no—
215#340—72no—
216#340—72no—
217#340—72no—
218#336—72no—
219#336—72no—
220C23×C7—22no—
221C23×C7—22no—
222C23×C7—22no—
223C23×C7—41no—
224C23×C7—41no—
225C23×C7—41no—
226C23×C7—41no—
227C23×C7C22×D411yescharacteristic
228C22×D7—22no—
229C22×D7—41no—
230C22×D7—41no—
231C22×D7—41no—
232C22×D7—41no—
233C22×D7—41no—
234C22×D7—41no—
235C7×Q8—44no—
236C7×Q8—84no—
237C7×D4—22no—
238C7×D4—22no—
239C7×D4—22no—
240C7×D4C22⋊Q814yescomplemented
241C2×C4×C7—21no—
242C2×C4×C7—21no—
243C2×C4×C7—22no—
244C2×C4×C7—22no—
245C2×C4×C7—22no—
246C2×C4×C7—24no—
247C2×C4×C7—22no—
248C2×C4×C7—22no—
249C2×C4×C7—22no—
250C2×C4×C7—22no—
251C2×C4×C7—41no—
252C2×C4×C7—41no—
253C2×C4×C7—42no—
254C2×C4×C7—42no—
255C2×C4×C7—48no—
256C2×C4×C7—88no—
257C2×C4×C7#12012yes—
258C2×C4×C7#12012yes—
259C2×C4×C7#13512yes—
260C2×C4×C7AGL(1,ℤ/8)12yes—
261C2×C4×C7C22×D411yescharacteristic
262C2×C4×C7C22×D411yescharacteristic
263C2×C4×C7C22×Q811yescharacteristic
264C2×C4×C7C2×#4111yescharacteristic
265C2×C4×C72−1+411yescharacteristic
266C2×C4×C72−1+411yescharacteristic
267C2×Dic7—41no—
268C2×Dic7—42no—
269C2×Dic7—41no—
270C2×Dic7—41no—
271C2×Dic7—42no—
272C2×Dic7—42no—
273D28—22no—
274D28—22no—
275D28—41no—
276D28—41no—
277C4×D7—84no—
278C7⋊C8—22no—
279C7⋊C8—41no—
280C7⋊C8—42no—
281C22×Q8—21no—
282C22×Q8C22⋊D711yescharacteristic
283C22×D4—71no—
284C22×D4C22×D711yescharacteristic
285C23×C4—21no—
286C23×C4—22no—
287C23×C4—141no—
288C2×D8—74no—
289C2×D8—142no—
290C2×D8—142no—
291C22×C8—71no—
292C22×C8—71no—
293C22×C8—72no—
294C4⋊Q8—22no—
295C4⋊Q8C22⋊D714yes—
296#125—24no—
297#125—24no—
298C22⋊Q8—28no—
299C22⋊Q8—216no—
300C22⋊Q8—284no—
301C4×Q8—22no—
302C4×Q8—22no—
303C4×D4—24no—
304C4×D4—24no—
305C4×D4—22no—
306C4×D4—142no—
307C4×D4—142no—
308C4×D4C22⋊D714yes—
309C2×C4⋊C4—28no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—71no—
314C2×C4⋊C4—72no—
315C2×C4⋊C4—142no—
316C2×C4⋊C4—141no—
317C2×C4⋊C4—141no—
318C2×C4⋊C4C22⋊D711yescharacteristic
319C2×C4⋊C4C22⋊D712yes—
320C2×C4⋊C4C22×D711yescharacteristic
321C2×C4⋊C4C22×D712yes—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—21no—
324C2×C22⋊C4—141no—
325C2×C22⋊C4—142no—
326C2×C22⋊C4—142no—
327C2×C42C22×D712yes—
328C2×C42C22×D711yescharacteristic
329#107—142no—
330#106—144no—
331C4⋊C8—142no—
332#102—144no—
333#102—142no—
334#102—142no—
335#102—144no—
336C4×C8—142no—
337#95—142no—
338C22×C7—24no—
339C22×C7—42no—
340C22×C7—42no—
341C22×C7—42no—
342C22×C7—42no—
343C22×C7—42no—
344C22×C7—42no—
345C22×C7C2×#12011yescharacteristic
346C22×C7#54811yescharacteristic
347C22×C7D4×Q811yescharacteristic
348C22×C7C2×#13511yescharacteristic
349C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
350C22×C7#57611yescharacteristic
351C22×C7#57711yescharacteristic
352C2×D7—42no—
353C2×D7—42no—
354C2×D7—42no—
355C2×D7—81no—
356C2×D7—81no—
357C2×D7—81no—
358C2×D7—81no—
359C4×C7—22no—
360C4×C7—42no—
361C4×C7—42no—
362C4×C7—42no—
363C4×C7—44no—
364C4×C7#44914yes—
365C4×C7C2×C22⋊Q811yescharacteristic
366C4×C7#53611yescharacteristic
367C4×C7#54811yescharacteristic
368C4×C7D4×Q811yescharacteristic
369Dic7—82no—
370Dic7—81no—
371C24—21no—
372C24—141no—
373C2×Q8—22no—
374C2×Q8—22no—
375C2×Q8—22no—
376C2×Q8—24no—
377C2×Q8—81no—
378C2×Q8—81no—
379C2×D4—21no—
380C2×D4—21no—
381C2×D4—21no—
382C2×D4—72no—
383C2×D4—141no—
384C2×D4—141no—
385C2×D4—141no—
386C2×D4—141no—
387C2×D4—141no—
388C2×D4Q8×D712yes—
389C2×D4#116112yes—
390C2×D4C2×C22⋊D712yes—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—28no—
394C22×C4—24no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—82no—
398C22×C4—82no—
399C22×C4—141no—
400C22×C4—142no—
401C22×C4—282no—
402C22×C4—282no—
403C22×C4—282no—
404C22×C4C2×C22⋊D712yes—
405C22×C4C2×C22⋊D712yes—
406C22×C4C2×C22⋊D711yescharacteristic
407C22×C4C2×C22⋊D711yescharacteristic
408C22×C4C23×D711yescharacteristic
409D8—148no—
410C2×C8—72no—
411C2×C8—141no—
412C2×C8—141no—
413C2×C8—142no—
414C2×C8—141no—
415C2×C8—141no—
416C2×C8—142no—
417C2×C8—141no—
418C2×C8—142no—
419C4⋊C4—28no—
420C4⋊C4—28no—
421C4⋊C4—22no—
422C4⋊C4—24no—
423C4⋊C4—24no—
424C4⋊C4—21no—
425C4⋊C4—22no—
426C4⋊C4—142no—
427C4⋊C4—142no—
428C4⋊C4—141no—
429C4⋊C4—141no—
430C4⋊C4—284no—
431C4⋊C4—282no—
432C4⋊C4—561no—
433C4⋊C4C2×C22⋊D714yes—
434C4⋊C4C2×C22⋊D712yes—
435C22⋊C4—28no—
436C22⋊C4—24no—
437C22⋊C4—284no—
438C22⋊C4—282no—
439C22⋊C4—562no—
440C42—22no—
441C42—22no—
442C42C2×C22⋊D712yes—
443C42C2×C22⋊D712yes—
444C2×C7—44no—
445C2×C7#240911yescharacteristic
446C2×C7C2×#44911yescharacteristic
447C2×C7#301211yescharacteristic
448C2×C7#336011yescharacteristic
449C2×C7#336611yescharacteristic
450C2×C7#338011yescharacteristic
451C2×C7#338411yescharacteristic
452D7—82no—
453C23—22no—
454C23—22no—
455C23—22no—
456C23—41no—
457C23—41no—
458C23—41no—
459C23—41no—
460C23—142no—
461C23—281no—
462C23—281no—
463C23—281no—
464C23—281no—
465C23—281no—
466C23—281no—
467C23C22×C22⋊D711yescharacteristic
468Q8—44no—
469Q8—84no—
470D4—22no—
471D4—22no—
472D4—22no—
473D4—142no—
474D4—142no—
475D4—281no—
476D4—281no—
477D4#650914yescomplemented
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—24no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—48no—
493C2×C4—88no—
494C2×C4—282no—
495C2×C4—282no—
496C2×C4—281no—
497C2×C4—281no—
498C2×C4—282no—
499C2×C4—281no—
500C2×C4—564no—
501C2×C4#651212yes—
502C2×C4#651312yes—
503C2×C4#651612yes—
504C2×C4#651612yes—
505C2×C4C2×Q8×D711yescharacteristic
506C2×C4C2×#116111yescharacteristic
507C2×C4#655111yescharacteristic
508C2×C4#655111yescharacteristic
509C2×C4C22×C22⋊D711yescharacteristic
510C2×C4C22×C22⋊D711yescharacteristic
511C8—142no—
512C8—281no—
513C8—282no—
514C7#2432711yescharacteristic, complemented, Sylow
515C22—24no—
516C22—42no—
517C22—42no—
518C22—42no—
519C22—42no—
520C22—42no—
521C22—42no—
522C22—282no—
523C22—282no—
524C22—282no—
525C22—561no—
526C22—561no—
527C22—561no—
528C22—561no—
529C22Q8×C22⋊D711yescharacteristic
530C22#9050811yescharacteristic
531C22C2×#651211yescharacteristic
532C22C2×#651311yescharacteristic
533C22#9052611yescharacteristic
534C22#9052711yescharacteristic
535C22C2×#651611yescharacteristic
536C4—22no—
537C4—42no—
538C4—42no—
539C4—42no—
540C4—44no—
541C4—561no—
542C4—562no—
543C4#9001014yes—
544C4C2×#650911yescharacteristic
545C4Q8×C22⋊D711yescharacteristic
546C4#9050711yescharacteristic
547C4#9050811yescharacteristic
548C2—44no—
549C2—562no—
550C2#1174107311yescharacteristic
551C2#1174107511yescharacteristic
552C2#1174108211yescharacteristic
553C2#1174108311yescharacteristic
554C2#1174151311yescharacteristic
555C2C2×#9001011yescharacteristic
556C2#1174154711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC2×C7×#554C2
socleC23×C7C22×D4
prime radicalC22×#1141C22
hypercenterC2×#554D7
hypocenterC7#24327
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×#554

Complements

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

NH ≅ G/Ntype
1. C2×C7×#554C2semidirect
2. C2×#89856C2semidirect
3. #11739945C2semidirect
5. #11729541C2semidirect
6. #11729497C2semidirect
7. #11729493C2semidirect
8. #11729388C2semidirect
10. #11729361C2semidirect
11. #11729358C2semidirect
80. C2×#1141Q8semidirect
103. C2×#554D7semidirect
240. C7×D4C22⋊Q8semidirect
477. D4#6509semidirect
514. C7#24327semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#5542
C2×#898562
#117399452
C2×C4×#11412
#117295412
#117294972
#117294932
#117293882
#117293672
#117293612
#117293582
#243277