Finite Group Databasebeta

← #13381284

Subgroups — #13381284

Order 1792.

1902 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C7×#121C211yesmaximal, characteristic
2C2×#89987C211yesmaximal, characteristic, complemented
3C2×#89987C211yesmaximal, characteristic, complemented
4#11741472C211yesmaximal, characteristic, complemented
5C2×#89979C211yesmaximal, characteristic, complemented
6C2×#89736C211yesmaximal, characteristic, complemented
7#11729434C214yesmaximal, complemented
8#11729433C214yesmaximal, complemented
9#11729312C211yesmaximal, characteristic, complemented
10C23×C7×D4—21no—
11C23×C7×D4C2211yescharacteristic
12C24×C4×C7C2211yescharacteristic
13C2×C7×#121—24no—
14C2×C7×#121—22no—
15C2×C7×#121—24no—
16C2×C7×#121C2214yes—
17C22×C7×C4⋊C4C2211yescharacteristic
18C22×C7×C22⋊C4—21no—
19C2×#6489C2211yescharacteristic
20C2×#6489C2211yescharacteristic
21C22×#1136C2211yescharacteristic
22C22×#1136C2211yescharacteristic
23#90019—21no—
24#89987—24no—
25#89987—24no—
26C2×#6406C2212yescomplemented
27C2×#6406C2214yescomplemented
28C2×#6406C2212yescomplemented
29#89984C2212yescomplemented
30#89984C2212yescomplemented
31#89983—21no—
32C4×C22⋊Dic7—22no—
33#89979—24no—
34C2×C22⋊(C7⋊C8)C2214yescomplemented
35#89736—24no—
36C2×#6382C2212yescomplemented
37C2×#6381C2212yescomplemented
38#89324C2214yescomplemented
39#23480—71nomaximal, Sylow
40C25×C7—21no—
41C25×C7D411yescharacteristic
42C22×C7×D4—28no—
43C22×C7×D4—24no—
44C22×C7×D4—22no—
45C22×C7×D4—21no—
46C22×C7×D4—22no—
47C22×C7×D4—22no—
48C22×C7×D4—22no—
49C22×C7×D4—22no—
50C22×C7×D4—48no—
51C22×C7×D4D414yes—
52C22×C7×D4C2312yes—
53C23×C4×C7—21no—
54C23×C4×C7—22no—
55C23×C4×C7—22no—
56C23×C4×C7—22no—
57C23×C4×C7—22no—
58C23×C4×C7—22no—
59C23×C4×C7—22no—
60C23×C4×C7D412yes—
61C23×C4×C7D411yescharacteristic
62C23×C4×C7C2312yes—
63C23×C4×C7C2311yescharacteristic
64C7×#121—216no—
65C7×#121—28no—
66C7×#121—28no—
67C2×C7×C4⋊C4—21no—
68C2×C7×C4⋊C4—22no—
69C2×C7×C4⋊C4D414yes—
70C2×C7×C4⋊C4C2312yes—
71C2×C7×C22⋊C4—28no—
72C2×C7×C22⋊C4—24no—
73C2×C22⋊Dic7—21no—
74C2×C22⋊Dic7—21no—
75C2×C22⋊Dic7—22no—
76C2×C22⋊Dic7—22no—
77#6489—22no—
78#6489—22no—
79#6489—22no—
80#6489—22no—
81C2×C4⋊Dic7C2311yescharacteristic
82C2×C4⋊Dic7C2311yescharacteristic
83C2×C4⋊Dic7C2312yes—
84C2×C4×Dic7—21no—
85C2×C4×Dic7—21no—
86C2×#1136—22no—
87C2×#1136—22no—
88C2×#1136—22no—
89C2×#1136—22no—
90C2×#1136—22no—
91C2×#1136D414yescomplemented
92C22×C7⋊C8C2312yes—
93C22×C7⋊C8C2312yes—
94#6406—24no—
95#6406—24no—
96#6406—28no—
97C7⋊#95—22no—
98C7⋊#95—22no—
99C22⋊(C7⋊C8)—28no—
100#6382—24no—
101#6381—24no—
102C22×#121D711yescharacteristic
103C2×#428—71no—
104C2×#418—71no—
105C2×#418—71no—
106C2×#407—71no—
107#2280—71no—
108#2034—74no—
109#2033—74no—
110#1736—71no—
111C24×C7—22no—
112C24×C7—24no—
113C24×C7—24no—
114C24×C7—22no—
115C24×C7—22no—
116C24×C7—22no—
117C24×C7—42no—
118C24×C7—44no—
119C24×C7—44no—
120C24×C7—42no—
121C24×C7—42no—
122C24×C7C2×D412yes—
123C24×C7C2×D411yescharacteristic
124C2×C7×D4—216no—
125C2×C7×D4—24no—
126C2×C7×D4—28no—
127C2×C7×D4—28no—
128C2×C7×D4—22no—
129C2×C7×D4—22no—
130C2×C7×D4—22no—
131C2×C7×D4—22no—
132C2×C7×D4—22no—
133C2×C7×D4—432no—
134C2×C7×D4—48no—
135C2×C7×D4—48no—
136C2×C7×D4—44no—
137C2×C7×D4—42no—
138C2×C7×D4—42no—
139C2×C7×D4—44no—
140C2×C7×D4C2×D414yes—
141C22×C4×C7—22no—
142C22×C4×C7—24no—
143C22×C4×C7—22no—
144C22×C4×C7—22no—
145C22×C4×C7—22no—
146C22×C4×C7—24no—
147C22×C4×C7—24no—
148C22×C4×C7—24no—
149C22×C4×C7—22no—
150C22×C4×C7—22no—
151C22×C4×C7—22no—
152C22×C4×C7—22no—
153C22×C4×C7—22no—
154C22×C4×C7—22no—
155C22×C4×C7—44no—
156C22×C4×C7—44no—
157C22×C4×C7—44no—
158C22×C4×C7—44no—
159C22×C4×C7—44no—
160C22×C4×C7C2×D412yes—
161C22×C4×C7C2×D411yescharacteristic
162C22×C4×C7C2×D412yes—
163C22×C4×C7C2×D412yes—
164C22×C4×C7C2×D412yes—
165C22×C4×C7#4112yes—
166C22×C4×C7#4112yes—
167C22×C4×C7#4112yes—
168C22×C4×C7C2411yescharacteristic
169C22×Dic7—21no—
170C22×Dic7—21no—
171C22×Dic7—22no—
172C7×C4⋊C4—24no—
173C7×C4⋊C4—22no—
174C7×C4⋊C4C2×D414yes—
175C7×C22⋊C4—216no—
176C22⋊Dic7—44no—
177C22⋊Dic7—42no—
178C22⋊Dic7—42no—
179C22⋊Dic7—44no—
180C4⋊Dic7—21no—
181C4⋊Dic7—21no—
182C4⋊Dic7—22no—
183C4⋊Dic7—21no—
184C4⋊Dic7—21no—
185C4⋊Dic7—22no—
186C4×Dic7—42no—
187C4×Dic7—42no—
188#1136—28no—
189#1136—44no—
190C2×C7⋊C8—22no—
191C2×C7⋊C8—22no—
192C2×C7⋊C8—22no—
193C2×C7⋊C8—22no—
194C2×C7⋊C8—22no—
195C2×C7⋊C8C2×D414yescomplemented
196C23×D4—21no—
197C23×D4C2×D711yescharacteristic
198C24×C4C2×D711yescharacteristic
199C22×M16—71no—
200C22×M16—71no—
201C2×#121—24no—
202C2×#121—22no—
203C2×#121—24no—
204C2×#121C2×D714yes—
205C2×#117—71no—
206C2×#117—71no—
207C22×C4⋊C4C2×D711yescharacteristic
208C22×C22⋊C4—21no—
209#428—144no—
210C2×#107—72no—
211C2×#106—72no—
212#418—144no—
213#418—144no—
214C2×#102—72no—
215C2×#102—74no—
216C2×#102—72no—
217#407—144no—
218C2×C22⋊C8—74no—
219#381—141no—
220#380—72no—
221#380—72no—
222#379—141no—
223C4×C22⋊C4—142no—
224#340—74no—
225C23×C7—22no—
226C23×C7—22no—
227C23×C7—28no—
228C23×C7—22no—
229C23×C7—22no—
230C23×C7—22no—
231C23×C7—22no—
232C23×C7—22no—
233C23×C7—22no—
234C23×C7—24no—
235C23×C7—48no—
236C23×C7—48no—
237C23×C7—44no—
238C23×C7—48no—
239C23×C7—48no—
240C23×C7—44no—
241C23×C7—42no—
242C23×C7—42no—
243C23×C7—44no—
244C23×C7—44no—
245C23×C7—48no—
246C23×C7—88no—
247C23×C7—88no—
248C23×C7#12012yes—
249C23×C7#12112yes—
250C23×C7#12312yes—
251C23×C7AGL(1,ℤ/8)14yes—
252C23×C7C22×D411yescharacteristic
253C7×D4—24no—
254C7×D4—24no—
255C7×D4—432no—
256C7×D4—42no—
257C7×D4—42no—
258C7×D4—48no—
259C2×C4×C7—22no—
260C2×C4×C7—24no—
261C2×C4×C7—22no—
262C2×C4×C7—22no—
263C2×C4×C7—48no—
264C2×C4×C7—44no—
265C2×C4×C7—44no—
266C2×C4×C7—42no—
267C2×C4×C7—42no—
268C2×C4×C7—42no—
269C2×C4×C7—42no—
270C2×C4×C7—48no—
271C2×C4×C7#12012yes—
272C2×C4×C7#12114yes—
273C2×C4×C7#12114yes—
274C2×C4×C7#12112yes—
275C2×C4×C7#12314yes—
276C2×C4×C7#12312yes—
277C2×C4×C7#12414yes—
278C2×C4×C7C22×D411yescharacteristic
279C2×C4×C7C22×D411yescharacteristic
280C2×C4×C7C22×D411yescharacteristic
281C2×C4×C7C2×#4111yescharacteristic
282C2×C4×C7C2×#4111yescharacteristic
283C2×C4×C7C2×#4111yescharacteristic
284C2×Dic7—41no—
285C2×Dic7—41no—
286C2×Dic7—42no—
287C2×Dic7—42no—
288C2×Dic7—42no—
289C2×Dic7—42no—
290C2×Dic7—42no—
291C7⋊C8—24no—
292C7⋊C8—42no—
293C25—21no—
294C25C22⋊D711yescharacteristic
295C22×D4—28no—
296C22×D4—24no—
297C22×D4—22no—
298C22×D4—22no—
299C22×D4—22no—
300C22×D4—22no—
301C22×D4—21no—
302C22×D4—22no—
303C22×D4—48no—
304C22×D4C22⋊D714yes—
305C22×D4C22×D712yes—
306C23×C4—21no—
307C23×C4—22no—
308C23×C4—22no—
309C23×C4—22no—
310C23×C4—22no—
311C23×C4—22no—
312C23×C4—22no—
313C23×C4C22⋊D711yescharacteristic
314C23×C4C22⋊D712yes—
315C23×C4C22×D712yes—
316C23×C4C22×D711yescharacteristic
317C2×M16—74no—
318C2×M16—142no—
319C2×M16—142no—
320C2×M16—142no—
321C2×M16—142no—
322C2×M16—142no—
323C22×C8—72no—
324C22×C8—72no—
325#121—216no—
326#121—28no—
327#121—28no—
328#117—142no—
329#117—142no—
330#117—142no—
331#117—142no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—72no—
335C2×C4⋊C4—71no—
336C2×C4⋊C4—71no—
337C2×C4⋊C4C22⋊D714yes—
338C2×C4⋊C4C22×D712yes—
339C2×C22⋊C4—28no—
340C2×C22⋊C4—24no—
341C2×C22⋊C4—141no—
342C2×C22⋊C4—141no—
343C2×C22⋊C4—142no—
344C2×C22⋊C4—142no—
345C2×C42—141no—
346C2×C42—141no—
347#107—144no—
348#106—144no—
349#102—148no—
350#102—144no—
351#102—144no—
352C22⋊C8—148no—
353#95—142no—
354#95—142no—
355C22×C7—24no—
356C22×C7—22no—
357C22×C7—22no—
358C22×C7—24no—
359C22×C7—24no—
360C22×C7—24no—
361C22×C7—24no—
362C22×C7—416no—
363C22×C7—42no—
364C22×C7—42no—
365C22×C7—42no—
366C22×C7—42no—
367C22×C7—44no—
368C22×C7—44no—
369C22×C7—44no—
370C22×C7—816no—
371C22×C7—84no—
372C22×C7—84no—
373C22×C7#39414yes—
374C22×C7C2×#12011yescharacteristic
375C22×C7C2×#12111yescharacteristic
376C22×C7C2×#12311yescharacteristic
377C22×C7C2×AGL(1,ℤ/8)12yes—
378C22×C7#57512yes—
379C4×C7—42no—
380C4×C7—44no—
381C4×C7#39414yes—
382C4×C7C2×#12111yescharacteristic
383C4×C7C2×#12111yescharacteristic
384C4×C7C2×#12311yescharacteristic
385C4×C7C2×#12411yescharacteristic
386Dic7—81no—
387Dic7—81no—
388Dic7—82no—
389C24—24no—
390C24—22no—
391C24—22no—
392C24—22no—
393C24—24no—
394C24—22no—
395C24—42no—
396C24—44no—
397C24—44no—
398C24—42no—
399C24—42no—
400C24C2×C22⋊D712yes—
401C24C2×C22⋊D711yescharacteristic
402C2×D4—216no—
403C2×D4—28no—
404C2×D4—28no—
405C2×D4—24no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—432no—
412C2×D4—48no—
413C2×D4—48no—
414C2×D4—42no—
415C2×D4—44no—
416C2×D4—44no—
417C2×D4—42no—
418C2×D4C2×C22⋊D714yes—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—44no—
434C22×C4—44no—
435C22×C4—44no—
436C22×C4—44no—
437C22×C4—44no—
438C22×C4—141no—
439C22×C4—142no—
440C22×C4—141no—
441C22×C4D4×D712yes—
442C22×C4#115912yes—
443C22×C4#115912yes—
444C22×C4#115912yes—
445C22×C4C2×C22⋊D712yes—
446C22×C4C2×C22⋊D711yescharacteristic
447C22×C4C2×C22⋊D712yes—
448C22×C4C2×C22⋊D712yes—
449C22×C4C23×D711yescharacteristic
450M16—148no—
451M16—284no—
452C2×C8—74no—
453C2×C8—142no—
454C2×C8—142no—
455C2×C8—142no—
456C2×C8—142no—
457C2×C8—142no—
458C4⋊C4—24no—
459C4⋊C4—22no—
460C4⋊C4—142no—
461C4⋊C4—142no—
462C4⋊C4—141no—
463C4⋊C4—141no—
464C4⋊C4—141no—
465C4⋊C4—141no—
466C4⋊C4C2×C22⋊D714yes—
467C22⋊C4—216no—
468C22⋊C4—284no—
469C22⋊C4—284no—
470C22⋊C4—282no—
471C22⋊C4—282no—
472C42—282no—
473C42—282no—
474C2×C7—24no—
475C2×C7—44no—
476C2×C7—84no—
477C2×C7C2×#39411yescharacteristic
478C2×C7#300311yescharacteristic
479C2×C7#300511yescharacteristic
480C2×C7#305512yes—
481C2×C7#309412yes—
482C23—22no—
483C23—22no—
484C23—28no—
485C23—22no—
486C23—22no—
487C23—22no—
488C23—22no—
489C23—22no—
490C23—22no—
491C23—24no—
492C23—48no—
493C23—48no—
494C23—44no—
495C23—48no—
496C23—48no—
497C23—44no—
498C23—44no—
499C23—42no—
500C23—42no—
501C23—48no—
502C23—44no—
503C23—88no—
504C23—88no—
505C23#649514yes—
506C23#649812yes—
507C23#650212yes—
508C23#651612yes—
509C23C22×C22⋊D711yescharacteristic
510D4—24no—
511D4—24no—
512D4—432no—
513D4—42no—
514D4—42no—
515D4—48no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—24no—
520C2×C4—48no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—48no—
528C2×C4—282no—
529C2×C4—282no—
530C2×C4—282no—
531C2×C4—282no—
532C2×C4—282no—
533C2×C4—281no—
534C2×C4—281no—
535C2×C4#649814yes—
536C2×C4#649812yes—
537C2×C4#649914yes—
538C2×C4#650114yes—
539C2×C4#650214yes—
540C2×C4#650212yes—
541C2×C4#651612yes—
542C2×C4C2×D4×D711yescharacteristic
543C2×C4C2×#115911yescharacteristic
544C2×C4C2×#115911yescharacteristic
545C2×C4C2×#115911yescharacteristic
546C2×C4C22×C22⋊D711yescharacteristic
547C2×C4C22×C22⋊D711yescharacteristic
548C8—144no—
549C8—282no—
550C7#2348011yescharacteristic, complemented, Sylow
551C22—22no—
552C22—22no—
553C22—24no—
554C22—24no—
555C22—24no—
556C22—24no—
557C22—24no—
558C22—416no—
559C22—42no—
560C22—42no—
561C22—42no—
562C22—42no—
563C22—44no—
564C22—44no—
565C22—44no—
566C22—816no—
567C22—84no—
568C22—84no—
569C22#8999414yes—
570C22C2×#649512yes—
571C22C2×#649811yescharacteristic
572C22C2×#650211yescharacteristic
573C22#9051312yes—
574C22C2×#651611yescharacteristic
575C4—44no—
576C4—42no—
577C4—561no—
578C4—561no—
579C4—562no—
580C4#8999414yes—
581C4C2×#649811yescharacteristic
582C4C2×#649911yescharacteristic
583C4C2×#650111yescharacteristic
584C4C2×#650211yescharacteristic
585C2—24no—
586C2—44no—
587C2—84no—
588C2#1174006112yes—
589C2#1174006612yes—
590C2#1174148311yescharacteristic
591C2C2×#8999411yescharacteristic
592C2#1174152611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
Fitting, prime radicalC22×C7×#121C2
socleC23×C7C22×D4
hypercenterC22×#121D7
hypocenterC7#23480
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C24×C4 ◁ C22×#121

Complements

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

NH ≅ G/Ntype
2. C2×#89987C2semidirect
3. C2×#89987C2semidirect
4. #11741472C2semidirect
5. C2×#89979C2semidirect
6. C2×#89736C2semidirect
7. #11729434C2semidirect
8. #11729433C2semidirect
9. #11729312C2semidirect
26. C2×#6406C22semidirect
27. C2×#6406C22semidirect
28. C2×#6406C22semidirect
29. #89984C22semidirect
30. #89984C22semidirect
34. C2×C22⋊(C7⋊C8)C22semidirect
36. C2×#6382C22semidirect
37. C2×#6381C22semidirect
38. #89324C22semidirect
91. C2×#1136D4semidirect
195. C2×C7⋊C8C2×D4semidirect
550. C7#23480semidirect

Maximal subgroups

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

subgroupindex
C22×C7×#1212
C2×#899872
#117414722
C2×#899792
C2×#897362
#117294342
#117294332
#117293122
#234807