Finite Group Databasebeta

← #13381288

Subgroups — #13381288

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×#6493C211yesmaximal, characteristic
2C7×#2280C211yesmaximal, characteristic, complemented
3C2×#89979C211yesmaximal, characteristic, complemented
4C2×#89773C211yesmaximal, characteristic, complemented
5C2×#89773C211yesmaximal, characteristic, complemented
6C2×#89767C211yesmaximal, characteristic, complemented
7#11729335C211yesmaximal, characteristic, complemented
8#11727278C214yesmaximal, complemented
9#11727277C214yesmaximal, complemented
10C24×C4×C7C2211yescharacteristic
11C23×C22⋊D7—21no—
12C23×D28C2211yescharacteristic
13C2×C7×#117C2211yescharacteristic
14C2×C7×#117C2211yescharacteristic
15C2×#6493—22no—
16C2×#6493—24no—
17C2×#6493—24no—
18C2×#6493C2214yes—
19C22×(C2×D7)⋊C4—21no—
20C22×C4⋊Dic7C2211yescharacteristic
21C22×#1136C2211yescharacteristic
22C22×#1136C2211yescharacteristic
23C7×#381—21no—
24C7×#380C2212yescomplemented
25C7×#380C2212yescomplemented
26C7×#379—21no—
27C4×C7×C22⋊C4—22no—
28#89979—24no—
29C2×C22⋊(C7⋊C8)C2214yescomplemented
30#89773—24no—
31#89773—24no—
32#89767—24no—
33C2×#6383C2212yescomplemented
34C2×#6383C2214yescomplemented
35C2×#6383C2212yescomplemented
36C2×#6382C2212yescomplemented
37C2×#6381C2212yescomplemented
38#89324C2214yescomplemented
39#23480—71nomaximal, Sylow
40C25×C7D411yescharacteristic
41C24×D7—21no—
42C23×C4×C7—22no—
43C23×C4×C7—22no—
44C23×C4×C7—22no—
45C23×C4×C7—22no—
46C23×C4×C7—22no—
47C23×C4×C7—22no—
48C23×C4×C7D411yescharacteristic
49C23×C4×C7D412yes—
50C23×C4×C7C2312yes—
51C23×C4×C7C2311yescharacteristic
52C22×C22⋊D7—28no—
53C22×C22⋊D7—24no—
54C22×C22⋊D7—48no—
55C23×Dic7—21no—
56C22×D28—22no—
57C22×D28—22no—
58C22×D28—22no—
59C22×D28—22no—
60C22×D28—21no—
61C22×D28—22no—
62C22×D28D414yes—
63C22×D28C2312yes—
64C7×#117—22no—
65C7×#117—22no—
66C7×#117—22no—
67C7×#117—22no—
68C2×C7×C4⋊C4C2311yescharacteristic
69C2×C7×C4⋊C4C2311yescharacteristic
70C2×C7×C4⋊C4C2312yes—
71C2×C7×C22⋊C4—22no—
72C2×C7×C22⋊C4—22no—
73C2×C7×C22⋊C4—21no—
74C2×C7×C22⋊C4—21no—
75C2×C42×C7—21no—
76C2×C42×C7—21no—
77#6493—216no—
78#6493—28no—
79#6493—28no—
80C2×(C2×D7)⋊C4—28no—
81C2×(C2×D7)⋊C4—24no—
82C2×C4⋊Dic7—21no—
83C2×C4⋊Dic7—22no—
84C2×C4⋊Dic7D414yes—
85C2×C4⋊Dic7C2312yes—
86C2×#1136—22no—
87C2×#1136—22no—
88C2×#1136—22no—
89C2×#1136—22no—
90C2×#1136—22no—
91C2×#1136D414yescomplemented
92C22×C7⋊C8C2312yes—
93C22×C7⋊C8C2312yes—
94C7×#95—22no—
95C7×#95—22no—
96C22⋊(C7⋊C8)—28no—
97#6383—24no—
98#6383—24no—
99#6383—28no—
100#6382—24no—
101#6381—24no—
102C22×#121—71no—
103C2×#428—71no—
104C2×#418—71no—
105C2×#418—71no—
106C2×#407—71no—
107#2280D711yescharacteristic, complemented
108#2034—74no—
109#2033—74no—
110#1736—71no—
111C24×C7—22no—
112C24×C7—24no—
113C24×C7—22no—
114C24×C7—22no—
115C24×C7—42no—
116C24×C7C2×D412yes—
117C24×C7C2×D411yescharacteristic
118C23×D7—24no—
119C23×D7—22no—
120C23×D7—44no—
121C23×D7—44no—
122C23×D7—42no—
123C23×D7—42no—
124C22×C4×C7—22no—
125C22×C4×C7—22no—
126C22×C4×C7—22no—
127C22×C4×C7—24no—
128C22×C4×C7—24no—
129C22×C4×C7—24no—
130C22×C4×C7—22no—
131C22×C4×C7—22no—
132C22×C4×C7—22no—
133C22×C4×C7—22no—
134C22×C4×C7—22no—
135C22×C4×C7—22no—
136C22×C4×C7—21no—
137C22×C4×C7—21no—
138C22×C4×C7—22no—
139C22×C4×C7—44no—
140C22×C4×C7—44no—
141C22×C4×C7—44no—
142C22×C4×C7—44no—
143C22×C4×C7C2×D412yes—
144C22×C4×C7C2×D411yescharacteristic
145C22×C4×C7C2×D412yes—
146C22×C4×C7C2×D412yes—
147C22×C4×C7C2×D412yes—
148C22×C4×C7#4112yes—
149C22×C4×C7#4112yes—
150C22×C4×C7#4112yes—
151C22×C4×C7C2411yescharacteristic
152C2×C22⋊D7—216no—
153C2×C22⋊D7—432no—
154C2×C22⋊D7—48no—
155C2×C22⋊D7—48no—
156C22×Dic7—24no—
157C22×Dic7—22no—
158C22×Dic7—44no—
159C2×D28—28no—
160C2×D28—28no—
161C2×D28—22no—
162C2×D28—22no—
163C2×D28—22no—
164C2×D28—22no—
165C2×D28—24no—
166C2×D28—22no—
167C2×D28—44no—
168C2×D28—42no—
169C2×D28—42no—
170C2×D28—44no—
171C2×D28C2×D414yes—
172C7×C4⋊C4—21no—
173C7×C4⋊C4—21no—
174C7×C4⋊C4—21no—
175C7×C4⋊C4—21no—
176C7×C4⋊C4—22no—
177C7×C4⋊C4—22no—
178C7×C22⋊C4—44no—
179C7×C22⋊C4—42no—
180C7×C22⋊C4—42no—
181C7×C22⋊C4—44no—
182C42×C7—42no—
183C42×C7—42no—
184(C2×D7)⋊C4—216no—
185C4⋊Dic7—24no—
186C4⋊Dic7—22no—
187C4⋊Dic7C2×D414yes—
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—71no—
197C23×D4—141no—
198C24×C4C2×D711yescharacteristic
199C22×M16—71no—
200C22×M16—71no—
201C2×#121—74no—
202C2×#121—144no—
203C2×#121—144no—
204C2×#121—142no—
205C2×#117C2×D711yescharacteristic
206C2×#117C2×D711yescharacteristic
207C22×C4⋊C4—71no—
208C22×C22⋊C4—141no—
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—21no—
220#380C2×D712yescomplemented
221#380C2×D712yescomplemented
222#379—21no—
223C4×C22⋊C4—22no—
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—48no—
235C23×C7—48no—
236C23×C7—44no—
237C23×C7—44no—
238C23×C7#12012yes—
239C23×C7#12112yes—
240C23×C7#12312yes—
241C23×C7AGL(1,ℤ/8)14yes—
242C23×C7C22×D411yescharacteristic
243C22×D7—24no—
244C22×D7—48no—
245C22×D7—48no—
246C22×D7—42no—
247C22×D7—42no—
248C22×D7—44no—
249C22×D7—44no—
250C22×D7—48no—
251C22×D7—88no—
252C22×D7—88no—
253C2×C4×C7—22no—
254C2×C4×C7—22no—
255C2×C4×C7—22no—
256C2×C4×C7—44no—
257C2×C4×C7—42no—
258C2×C4×C7—42no—
259C2×C4×C7—42no—
260C2×C4×C7—42no—
261C2×C4×C7—48no—
262C2×C4×C7—41no—
263C2×C4×C7—41no—
264C2×C4×C7—42no—
265C2×C4×C7—42no—
266C2×C4×C7—42no—
267C2×C4×C7—42no—
268C2×C4×C7—42no—
269C2×C4×C7#12012yes—
270C2×C4×C7#12114yes—
271C2×C4×C7#12114yes—
272C2×C4×C7#12112yes—
273C2×C4×C7#12314yes—
274C2×C4×C7#12312yes—
275C2×C4×C7#12414yes—
276C2×C4×C7C22×D411yescharacteristic
277C2×C4×C7C22×D411yescharacteristic
278C2×C4×C7C22×D411yescharacteristic
279C2×C4×C7C2×#4111yescharacteristic
280C2×C4×C7C2×#4111yescharacteristic
281C2×C4×C7C2×#4111yescharacteristic
282C22⋊D7—432no—
283C2×Dic7—24no—
284C2×Dic7—48no—
285C2×Dic7—44no—
286D28—24no—
287D28—24no—
288D28—42no—
289D28—42no—
290D28—48no—
291C7⋊C8—24no—
292C7⋊C8—42no—
293C25—141no—
294C25C22⋊D711yescharacteristic
295C22×D4—74no—
296C22×D4—72no—
297C22×D4—148no—
298C22×D4—144no—
299C22×D4—142no—
300C22×D4—142no—
301C22×D4—142no—
302C22×D4—142no—
303C22×D4—142no—
304C22×D4—141no—
305C22×D4—288no—
306C23×C4—22no—
307C23×C4—22no—
308C23×C4—22no—
309C23×C4—22no—
310C23×C4—22no—
311C23×C4—22no—
312C23×C4—141no—
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—148no—
326#121—148no—
327#121—1416no—
328#117—22no—
329#117—22no—
330#117—22no—
331#117—22no—
332C2×C4⋊C4—74no—
333C2×C4⋊C4—72no—
334C2×C4⋊C4—142no—
335C2×C4⋊C4—141no—
336C2×C4⋊C4C22×D712yes—
337C2×C4⋊C4C22×D711yescharacteristic
338C2×C4⋊C4C22×D711yescharacteristic
339C2×C22⋊C4—21no—
340C2×C22⋊C4—21no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—148no—
344C2×C22⋊C4—144no—
345C2×C42—21no—
346C2×C42—21no—
347#107—144no—
348#106—144no—
349#102—148no—
350#102—144no—
351#102—144no—
352C22⋊C8—148no—
353#95—22no—
354#95—22no—
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#39414yes—
368C22×C7C2×#12011yescharacteristic
369C22×C7C2×#12111yescharacteristic
370C22×C7C2×#12311yescharacteristic
371C22×C7C2×AGL(1,ℤ/8)12yes—
372C22×C7#57512yes—
373C2×D7—44no—
374C2×D7—44no—
375C2×D7—44no—
376C2×D7—816no—
377C2×D7—84no—
378C2×D7—84no—
379C4×C7—42no—
380C4×C7—81no—
381C4×C7—81no—
382C4×C7—82no—
383C4×C7#39414yes—
384C4×C7C2×#12111yescharacteristic
385C4×C7C2×#12111yescharacteristic
386C4×C7C2×#12311yescharacteristic
387C4×C7C2×#12411yescharacteristic
388Dic7—44no—
389C24—22no—
390C24—24no—
391C24—22no—
392C24—22no—
393C24—42no—
394C24—144no—
395C24—142no—
396C24—284no—
397C24—284no—
398C24—282no—
399C24—282no—
400C24C2×C22⋊D712yes—
401C24C2×C22⋊D711yescharacteristic
402C2×D4—74no—
403C2×D4—1416no—
404C2×D4—148no—
405C2×D4—148no—
406C2×D4—144no—
407C2×D4—142no—
408C2×D4—142no—
409C2×D4—142no—
410C2×D4—142no—
411C2×D4—142no—
412C2×D4—2832no—
413C2×D4—288no—
414C2×D4—288no—
415C2×D4—284no—
416C2×D4—284no—
417C2×D4—282no—
418C2×D4—282no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—22no—
434C22×C4—44no—
435C22×C4—44no—
436C22×C4—44no—
437C22×C4—44no—
438C22×C4—144no—
439C22×C4—142no—
440C22×C4—284no—
441C22×C4#115712yes—
442C22×C4#115712yes—
443C22×C4D4×D712yes—
444C22×C4D4×D712yes—
445C22×C4D4×D712yes—
446C22×C4#115912yes—
447C22×C4C2×C22⋊D712yes—
448C22×C4C2×C22⋊D711yescharacteristic
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—21no—
459C4⋊C4—21no—
460C4⋊C4—22no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—22no—
464C4⋊C4—74no—
465C4⋊C4—144no—
466C4⋊C4—142no—
467C22⋊C4—44no—
468C22⋊C4—42no—
469C22⋊C4—42no—
470C22⋊C4—44no—
471C22⋊C4—1416no—
472C42—42no—
473C42—42no—
474C2×C7—24no—
475C2×C7—44no—
476C2×C7C2×#39411yescharacteristic
477C2×C7#300311yescharacteristic
478C2×C7#300511yescharacteristic
479C2×C7#305512yes—
480C2×C7#309412yes—
481D7—84no—
482C23—22no—
483C23—22no—
484C23—28no—
485C23—22no—
486C23—22no—
487C23—22no—
488C23—22no—
489C23—22no—
490C23—22no—
491C23—48no—
492C23—48no—
493C23—44no—
494C23—44no—
495C23—144no—
496C23—288no—
497C23—288no—
498C23—282no—
499C23—282no—
500C23—288no—
501C23—284no—
502C23—284no—
503C23—568no—
504C23—568no—
505C23#649212yes—
506C23#650012yes—
507C23#650212yes—
508C23#651214yes—
509C23C22×C22⋊D711yescharacteristic
510D4—144no—
511D4—144no—
512D4—2832no—
513D4—282no—
514D4—282no—
515D4—288no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—44no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—48no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—144no—
533C2×C4—288no—
534C2×C4—284no—
535C2×C4#644614yes—
536C2×C4#644714yes—
537C2×C4#644714yes—
538C2×C4#644814yes—
539C2×C4#649212yes—
540C2×C4#650012yes—
541C2×C4#650212yes—
542C2×C4C2×#115711yescharacteristic
543C2×C4C2×#115711yescharacteristic
544C2×C4C2×D4×D711yescharacteristic
545C2×C4C2×D4×D711yescharacteristic
546C2×C4C2×D4×D711yescharacteristic
547C2×C4C2×#115911yescharacteristic
548C8—144no—
549C8—282no—
550C7#2348011yescharacteristic, complemented, Sylow
551C22—24no—
552C22—24no—
553C22—22no—
554C22—22no—
555C22—24no—
556C22—24no—
557C22—24no—
558C22—416no—
559C22—42no—
560C22—42no—
561C22—42no—
562C22—42no—
563C22—284no—
564C22—284no—
565C22—284no—
566C22—5616no—
567C22—564no—
568C22—564no—
569C22#8973114yes—
570C22C2×#649211yescharacteristic
571C22C2×#650011yescharacteristic
572C22C2×#650211yescharacteristic
573C22C2×#651212yes—
574C22#9051312yes—
575C4—42no—
576C4—82no—
577C4—81no—
578C4—81no—
579C4—284no—
580C4#8973114yes—
581C4C2×#644611yescharacteristic
582C4C2×#644711yescharacteristic
583C4C2×#644711yescharacteristic
584C4C2×#644811yescharacteristic
585C2—24no—
586C2—44no—
587C2—564no—
588C2C2×#8973111yescharacteristic
589C2#1173966212yes—
590C2#1174036912yes—
591C2#1174037211yescharacteristic
592C2#1174037311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#2280C2
socleC23×C7C22×D4
prime radicalC22×#6493C2
hypercenter#2280D7
hypocenterC7#23480
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C24×C4 ◁ #2280

Complements

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

NH ≅ G/Ntype
2. C7×#2280C2semidirect
3. C2×#89979C2semidirect
4. C2×#89773C2semidirect
5. C2×#89773C2semidirect
6. C2×#89767C2semidirect
7. #11729335C2semidirect
8. #11727278C2semidirect
9. #11727277C2semidirect
24. C7×#380C22semidirect
25. C7×#380C22semidirect
29. C2×C22⋊(C7⋊C8)C22semidirect
33. C2×#6383C22semidirect
34. C2×#6383C22semidirect
35. C2×#6383C22semidirect
36. C2×#6382C22semidirect
37. C2×#6381C22semidirect
38. #89324C22semidirect
91. C2×#1136D4semidirect
107. #2280D7semidirect
195. C2×C7⋊C8C2×D4semidirect
220. #380C2×D7semidirect
221. #380C2×D7semidirect
550. C7#23480semidirect

Maximal subgroups

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

subgroupindex
C22×#64932
C7×#22802
C2×#899792
C2×#897732
C2×#897672
#117293352
#117272782
#117272772
#234807