Finite Group Databasebeta

← #13355059

Subgroups — #13355059

Order 1792.

1970 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#90521C211yesmaximal, characteristic
2C2×#90518C211yesmaximal, characteristic, complemented
3C2×C7×#408C211yesmaximal, characteristic, complemented
4C2×#89903C211yesmaximal, characteristic, complemented
5C22×#6395C211yesmaximal, characteristic, complemented
6#11729739C211yesmaximal, characteristic, complemented
7#11729723C211yesmaximal, characteristic, complemented
8#11727687C214yesmaximal, complemented
9#11727686C214yesmaximal, complemented
10C22×C7×#41C2211yescharacteristic
11C23×D28C2211yescharacteristic
12C22×C7×M16C2211yescharacteristic
13C23×C7×C8C2211yescharacteristic
14#90521—24no—
15#90521C418yescomplemented
16#90518—24no—
17#90518—24no—
18C2×#6510—21no—
19C2×Q8×Dic7—21no—
20C2×#6503—21no—
21C2×#6500—22no—
22C2×D4×Dic7—21no—
23C2×D4×Dic7—22no—
24C2×#6493C2214yes—
25C2×#6489C2211yescharacteristic
26C2×#6489C2211yescharacteristic
27C2×#6489C2212yescomplemented
28C22×C4⋊Dic7C2211yescharacteristic
29C22×C4⋊Dic7C2212yescomplemented
30C7×#408—24no—
31C2×C7×C22⋊C8C2214yescomplemented
32#89903—24no—
33C2×#6395—24no—
34C2×#6395C2212yes—
35C2×#6395C2212yes—
36C2×#6395C2214yes—
37#89346C2212yescomplemented
38#89346C2212yescomplemented
39#89346C2214yescomplemented
40#20914—71nomaximal, Sylow
41C24×D7—21no—
42C2×C7×#41—22no—
43C2×C7×#41—22no—
44C2×C7×#41—22no—
45C2×C7×#41—22no—
46C2×C7×#41C2×C414yes—
47C2×C7×#41D414yes—
48C22×C7×Q8D411yescharacteristic
49C22×C7×D4D412yes—
50C22×C7×D4D411yescharacteristic
51C23×C4×C7C2312yes—
52C23×C4×C7C2311yescharacteristic
53C22×C22⋊D7—24no—
54C23×Dic7—21no—
55C23×Dic7—22no—
56C22×D28—22no—
57C22×D28—22no—
58C22×D28—22no—
59C22×D28—22no—
60C22×D28—21no—
61C22×D28—22no—
62C22×D28C2×C414yescomplemented
63C22×D28C2312yes—
64C2×C7×M16—22no—
65C2×C7×M16—22no—
66C2×C7×M16—22no—
67C22×C7×C8—22no—
68C22×C7×C8—22no—
69C22×C7×C8—22no—
70C22×C7×C8C2312yes—
71C22×C7×C8C2312yes—
72C2×C22⋊Dic7—21no—
73C2×C22⋊Dic7—21no—
74C2×C22⋊Dic7—22no—
75#6510—28no—
76Q8×Dic7—28no—
77#6503—28no—
78#6500—216no—
79D4×Dic7—216no—
80D4×Dic7—28no—
81#6493—28no—
82#6493C2×C4116yescomplemented
83C2×(C2×D7)⋊C4—24no—
84#6489—22no—
85#6489—22no—
86#6489—24no—
87#6489—24no—
88#6489C2×C414yescomplemented
89#6489D414yescomplemented
90C2×C4⋊Dic7—21no—
91C2×C4⋊Dic7—22no—
92C2×C4⋊Dic7—22no—
93C2×C4⋊Dic7—24no—
94C2×C4⋊Dic7—24no—
95C2×C4⋊Dic7D414yescomplemented
96C2×C4⋊Dic7C2312yes—
97C2×C4⋊Dic7C2312yes—
98C2×C4⋊Dic7C2311yescharacteristic
99C2×C4⋊Dic7C2311yescharacteristic
100C2×C4×Dic7—21no—
101C2×C4×Dic7—21no—
102C2×C4×Dic7—22no—
103C7×C22⋊C8—28no—
104#6395—28no—
105#6395—24no—
106#6395—24no—
107C2×#535—71no—
108C2×#520—71no—
109C2×#418—71no—
110C22×#102—71no—
111C2×#408D711yescharacteristic, complemented
112#1875—74no—
113#1874—74no—
114#1736—71no—
115#1735—71no—
116C24×C7C2×D412yes—
117C24×C7C2×D411yescharacteristic
118C23×D7—22no—
119C23×D7—24no—
120C23×D7—42no—
121C23×D7—42no—
122C23×D7—82no—
123C23×D7—82no—
124C7×#41—28no—
125C7×#41—28no—
126C7×#41—28no—
127C7×#41—28no—
128C2×C7×Q8—22no—
129C2×C7×Q8—22no—
130C2×C7×Q8C22⋊C414yes—
131C2×C7×D4—24no—
132C2×C7×D4—24no—
133C2×C7×D4—22no—
134C2×C7×D4—22no—
135C2×C7×D4C22⋊C418yes—
136C2×C7×D4C22⋊C414yes—
137C22×C4×C7—22no—
138C22×C4×C7—22no—
139C22×C4×C7—22no—
140C22×C4×C7—22no—
141C22×C4×C7C22×C414yes—
142C22×C4×C7C22×C412yes—
143C22×C4×C7C2×D412yes—
144C22×C4×C7C2×D414yes—
145C22×C4×C7C2×D411yescharacteristic
146C22×C4×C7C2×D412yes—
147C22×C4×C7C2×D412yes—
148C22×C4×C7C2×D412yes—
149C22×C4×C7#4112yes—
150C22×C4×C7#4112yes—
151C22×C4×C7C2411yescharacteristic
152C2×C22⋊D7—216no—
153C2×C22⋊D7—84no—
154C2×C22⋊D7—84no—
155C22×Dic7—21no—
156C22×Dic7—21no—
157C22×Dic7—22no—
158C22×Dic7—22no—
159C22×Dic7—28no—
160C22×Dic7—24no—
161C22×Dic7—82no—
162C22×Dic7—84no—
163C2×D28—28no—
164C2×D28—28no—
165C2×D28—22no—
166C2×D28—22no—
167C2×D28—22no—
168C2×D28—22no—
169C2×D28—24no—
170C2×D28—22no—
171C2×D28—42no—
172C2×D28—44no—
173C2×D28—44no—
174C2×D28—42no—
175C2×D28C22×C414yes—
176C7×M16—44no—
177C2×C7×C8—22no—
178C2×C7×C8—22no—
179C2×C7×C8—22no—
180C2×C7×C8—22no—
181C2×C7×C8—22no—
182C2×C7×C8—22no—
183C2×C7×C8—44no—
184C22⋊Dic7—28no—
185C22⋊Dic7—24no—
186C22⋊Dic7—24no—
187(C2×D7)⋊C4—216no—
188C4⋊Dic7—28no—
189C4⋊Dic7—24no—
190C4⋊Dic7—22no—
191C4⋊Dic7—21no—
192C4⋊Dic7—21no—
193C4⋊Dic7—22no—
194C4⋊Dic7C22×C414yes—
195C4⋊Dic7C2×D412yes—
196C4⋊Dic7C2×D414yes—
197C4⋊Dic7C2×D412yes—
198C4×Dic7—28no—
199C4×Dic7—24no—
200C4×Dic7—24no—
201C22×#41C2×D711yescharacteristic
202C23×D4—71no—
203C22×M16C2×D711yescharacteristic
204C23×C8C2×D711yescharacteristic
205#535—78no—
206#535—144no—
207C2×#127—141no—
208C2×#124—141no—
209C2×#121—74no—
210C2×#120—142no—
211#520—144no—
212#520—144no—
213C2×C4×Q8—141no—
214C2×C4×D4—142no—
215C2×C4×D4—141no—
216C2×#117—71no—
217C2×#117—71no—
218C2×#117—72no—
219C22×C4⋊C4—71no—
220C22×C4⋊C4—72no—
221#418—144no—
222C2×#102—74no—
223C2×#102—72no—
224C2×#102—72no—
225C2×#102—144no—
226#408—24no—
227C2×C22⋊C8C2×D714yescomplemented
228#340—72no—
229#340—74no—
230#340—72no—
231C23×C7—22no—
232C23×C7—22no—
233C23×C7—22no—
234C23×C7—22no—
235C23×C7—24no—
236C23×C7—24no—
237C23×C7C2×C22⋊C414yes—
238C23×C7C2×C22⋊C412yes—
239C23×C7#12012yes—
240C23×C7#12312yes—
241C23×C7C22×D411yescharacteristic
242C22×D7—24no—
243C22×D7—42no—
244C22×D7—42no—
245C22×D7—44no—
246C22×D7—44no—
247C22×D7—48no—
248C22×D7—88no—
249C22×D7—88no—
250C22×D7—84no—
251C22×D7—84no—
252C7×Q8—28no—
253C7×D4—216no—
254C7×D4—28no—
255C2×C4×C7—22no—
256C2×C4×C7—22no—
257C2×C4×C7—24no—
258C2×C4×C7—24no—
259C2×C4×C7—24no—
260C2×C4×C7—24no—
261C2×C4×C7C2×C22⋊C414yes—
262C2×C4×C7C2×C22⋊C412yes—
263C2×C4×C7C4×D414yes—
264C2×C4×C7C4×D414yes—
265C2×C4×C7#12012yes—
266C2×C4×C7#12114yes—
267C2×C4×C7#12114yes—
268C2×C4×C7#12312yes—
269C2×C4×C7C23×C411yescharacteristic
270C2×C4×C7C22×D411yescharacteristic
271C2×C4×C7C22×D411yescharacteristic
272C2×C4×C7C22×D411yescharacteristic
273C2×C4×C7C2×#4111yescharacteristic
274C2×C4×C7C2×#4111yescharacteristic
275C22⋊D7—816no—
276C2×Dic7—24no—
277C2×Dic7—22no—
278C2×Dic7—24no—
279C2×Dic7—22no—
280C2×Dic7—44no—
281C2×Dic7—42no—
282C2×Dic7—42no—
283C2×Dic7—42no—
284C2×Dic7—42no—
285C2×Dic7—84no—
286C2×Dic7—88no—
287D28—24no—
288D28—24no—
289D28—42no—
290D28—42no—
291D28—48no—
292C7×C8—42no—
293C7×C8—42no—
294C25—141no—
295C2×#41—22no—
296C2×#41—22no—
297C2×#41—22no—
298C2×#41—22no—
299C2×#41C4×D714yes—
300C2×#41C22⋊D714yes—
301C22×Q8D2811yescharacteristic
302C22×D4—74no—
303C22×D4—72no—
304C22×D4—144no—
305C22×D4—141no—
306C22×D4—142no—
307C22×D4—142no—
308C22×D4—142no—
309C22×D4—142no—
310C22×D4—142no—
311C22×D4D2812yes—
312C22×D4D2811yescharacteristic
313C23×C4—142no—
314C23×C4—141no—
315C23×C4C22×D712yes—
316C23×C4C22×D711yescharacteristic
317C2×M16—22no—
318C2×M16—22no—
319C2×M16—22no—
320C22×C8—22no—
321C22×C8—22no—
322C22×C8—22no—
323C22×C8C22×D712yes—
324C22×C8C22×D712yes—
325#127—148no—
326#124—148no—
327#121—716no—
328#121—148no—
329#120—1416no—
330C4×Q8—148no—
331C4×D4—1416no—
332C4×D4—148no—
333#117—74no—
334#117—74no—
335#117—144no—
336#117—144no—
337#117—142no—
338#117—142no—
339C2×C4⋊C4—74no—
340C2×C4⋊C4—72no—
341C2×C4⋊C4—71no—
342C2×C4⋊C4—71no—
343C2×C4⋊C4—72no—
344C2×C4⋊C4—144no—
345C2×C4⋊C4—144no—
346C2×C4⋊C4—142no—
347C2×C4⋊C4—141no—
348C2×C4⋊C4—142no—
349C2×C22⋊C4—142no—
350C2×C22⋊C4—141no—
351C2×C22⋊C4—141no—
352C2×C22⋊C4—144no—
353C2×C42—142no—
354C2×C42—141no—
355C2×C42—141no—
356#102—148no—
357#102—144no—
358#102—144no—
359C22⋊C8—28no—
360C22×C7—24no—
361C22×C7—24no—
362C22×C7—22no—
363C22×C7—22no—
364C22×C7—28no—
365C22×C7—24no—
366C22×C7—44no—
367C22×C7—42no—
368C22×C7—42no—
369C22×C7#38614yes—
370C22×C7C22×C22⋊C411yescharacteristic
371C22×C7C2×#12011yescharacteristic
372C22×C7C2×#12311yescharacteristic
373C22×C7#57612yes—
374C22×C7#57712yes—
375C2×D7—48no—
376C2×D7—44no—
377C2×D7—816no—
378C2×D7—84no—
379C2×D7—82no—
380C2×D7—82no—
381C4×C7—24no—
382C4×C7#38614yes—
383C4×C7C2×C4×D411yescharacteristic
384C4×C7C2×C4×D411yescharacteristic
385C4×C7C2×#12111yescharacteristic
386C4×C7C2×#12111yescharacteristic
387Dic7—44no—
388Dic7—44no—
389Dic7—42no—
390Dic7—42no—
391C24—142no—
392C24—144no—
393C24—282no—
394C24—282no—
395C24—562no—
396C24—562no—
397C24C2×D2812yes—
398C24C2×D2811yescharacteristic
399#41—28no—
400#41—28no—
401#41—28no—
402#41—28no—
403C2×Q8—22no—
404C2×Q8—22no—
405C2×Q8(C2×D7)⋊C414yes—
406C2×D4—24no—
407C2×D4—24no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—74no—
411C2×D4—1416no—
412C2×D4—148no—
413C2×D4—148no—
414C2×D4—144no—
415C2×D4—142no—
416C2×D4—142no—
417C2×D4—142no—
418C2×D4—142no—
419C2×D4—142no—
420C2×D4—282no—
421C2×D4—282no—
422C2×D4—284no—
423C2×D4—284no—
424C2×D4—564no—
425C2×D4—564no—
426C2×D4(C2×D7)⋊C418yes—
427C2×D4(C2×D7)⋊C414yes—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—141no—
433C22×C4—142no—
434C22×C4—142no—
435C22×C4—141no—
436C22×C4—148no—
437C22×C4—144no—
438C22×C4—562no—
439C22×C4—564no—
440C22×C4C2×C4×D714yes—
441C22×C4C2×C4×D712yes—
442C22×C4C2×D2812yes—
443C22×C4C2×D2811yescharacteristic
444C22×C4D4×D712yes—
445C22×C4D4×D712yes—
446C22×C4#115912yes—
447C22×C4#115912yes—
448C22×C4C2×C22⋊D714yes—
449C22×C4C2×C22⋊D712yes—
450C22×C4C23×D711yescharacteristic
451M16—44no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—44no—
459C4⋊C4—74no—
460C4⋊C4—74no—
461C4⋊C4—72no—
462C4⋊C4—72no—
463C4⋊C4—148no—
464C4⋊C4—144no—
465C4⋊C4—141no—
466C4⋊C4—142no—
467C4⋊C4—141no—
468C4⋊C4—142no—
469C22⋊C4—148no—
470C22⋊C4—144no—
471C22⋊C4—144no—
472C22⋊C4—1416no—
473C42—148no—
474C42—144no—
475C42—144no—
476C2×C7—24no—
477C2×C7—44no—
478C2×C7C2×#38611yescharacteristic
479C2×C7#289712yes—
480C2×C7#300811yescharacteristic
481C2×C7#301011yescharacteristic
482C2×C7#309612yes—
483D7—84no—
484C23—22no—
485C23—22no—
486C23—22no—
487C23—22no—
488C23—24no—
489C23—24no—
490C23—144no—
491C23—282no—
492C23—282no—
493C23—284no—
494C23—284no—
495C23—288no—
496C23—568no—
497C23—568no—
498C23—564no—
499C23—564no—
500C23#644512yes—
501C23#644912yes—
502C23C2×(C2×D7)⋊C414yes—
503C23C2×(C2×D7)⋊C412yes—
504C23C22×D2811yescharacteristic
505Q8—28no—
506D4—216no—
507D4—28no—
508D4—144no—
509D4—144no—
510D4—282no—
511D4—282no—
512D4—288no—
513D4—5616no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—142no—
521C2×C4—144no—
522C2×C4—142no—
523C2×C4—144no—
524C2×C4—282no—
525C2×C4—282no—
526C2×C4—284no—
527C2×C4—282no—
528C2×C4—282no—
529C2×C4—564no—
530C2×C4—568no—
531C2×C4(C22⋊D7)⋊C414yes—
532C2×C4(C22⋊D7)⋊C414yes—
533C2×C4#644512yes—
534C2×C4#644912yes—
535C2×C4C2×(C2×D7)⋊C414yes—
536C2×C4C2×(C2×D7)⋊C412yes—
537C2×C4#650214yes—
538C2×C4#650214yes—
539C2×C4C22×C4×D711yescharacteristic
540C2×C4C2×D4×D711yescharacteristic
541C2×C4C2×D4×D711yescharacteristic
542C2×C4C2×#115911yescharacteristic
543C2×C4C2×#115911yescharacteristic
544C2×C4C22×C22⋊D711yescharacteristic
545C8—42no—
546C8—42no—
547C7#2091411yescharacteristic, complemented, Sylow
548C22—24no—
549C22—22no—
550C22—24no—
551C22—22no—
552C22—28no—
553C22—24no—
554C22—44no—
555C22—42no—
556C22—42no—
557C22—288no—
558C22—284no—
559C22—5616no—
560C22—564no—
561C22—562no—
562C22—562no—
563C22#8973014yes—
564C22C2×#644511yescharacteristic
565C22C2×#644911yescharacteristic
566C22#9044212yes—
567C22#9044312yes—
568C22C22×(C2×D7)⋊C411yescharacteristic
569C4—24no—
570C4—284no—
571C4—282no—
572C4—284no—
573C4—282no—
574C4#8973014yes—
575C4C2×(C22⋊D7)⋊C411yescharacteristic
576C4C2×(C22⋊D7)⋊C411yescharacteristic
577C4C2×#650211yescharacteristic
578C4C2×#650211yescharacteristic
579C2—24no—
580C2—44no—
581C2—564no—
582C2#1173609612yes—
583C2#1173611511yescharacteristic
584C2#1173612311yescharacteristic
585C2C2×#8973011yescharacteristic
586C2#1174124212yes—
subgroupG/H
centerC23C22×D28
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC2×C7×#408C2
socleC23×C7C22×D4
prime radicalC2×#90521C2
cosocleC22×C4×C7C24
hypercenterC2×#408D7
hypocenterC7#20914
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C22×#41 ◁ C2×#408

Complements

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

NH ≅ G/Ntype
2. C2×#90518C2semidirect
3. C2×C7×#408C2semidirect
4. C2×#89903C2semidirect
5. C22×#6395C2semidirect
6. #11729739C2semidirect
7. #11729723C2semidirect
8. #11727687C2semidirect
9. #11727686C2semidirect
15. #90521C4semidirect
27. C2×#6489C22semidirect
29. C22×C4⋊Dic7C22semidirect
31. C2×C7×C22⋊C8C22semidirect
37. #89346C22semidirect
38. #89346C22semidirect
39. #89346C22semidirect
62. C22×D28C2×C4semidirect
82. #6493C2×C4semidirect
88. #6489C2×C4semidirect
89. #6489D4semidirect
95. C2×C4⋊Dic7D4semidirect
111. C2×#408D7semidirect
227. C2×C22⋊C8C2×D7semidirect
547. C7#20914semidirect

Maximal subgroups

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

subgroupindex
C2×#905212
C2×#905182
C2×C7×#4082
C2×#899032
C22×#63952
#117297392
#117297232
#117276872
#117276862
#209147