Finite Group Databasebeta

← #13355490

Subgroups — #13355490

Order 1792.

1914 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#1131C211yesmaximal, characteristic, complemented
2C2×#90215C211yesmaximal, characteristic
3C2×#89885C211yesmaximal, characteristic, complemented
4#11739712C211yesmaximal, characteristic, complemented
5C2×#89774C211yesmaximal, characteristic, complemented
6C7×#1735C211yesmaximal, characteristic, complemented
7#11729334C211yesmaximal, characteristic, complemented
8#11726639C214yesmaximal, complemented
9#11726638C214yesmaximal, complemented
10C22×#1157C2211yescharacteristic
11C23×C4×D7C2211yescharacteristic
12C23×C7×C8C2211yescharacteristic
13C2×C7×#117C2211yescharacteristic
14C2×C7×#117C2211yescharacteristic
15C2×#6489C2211yescharacteristic
16C23×C7⋊C8C2211yescharacteristic
17C22×#1131—24no—
18C22×#1131—24no—
19C22×#1131—24no—
20C22×#1131C2214yes—
21#90215—24no—
22#90215C418yescomplemented
23C2×#6460C2212yes—
24C2×#6458C2212yes—
25C2×#6449—21no—
26C2×#6445—21no—
27C2×C4×D28—22no—
28#89885—24no—
29C2×#6395C2212yescomplemented
30C2×#6390C2212yescomplemented
31#89774—24no—
32#89759C2214yes—
33C2×#6384C2212yescomplemented
34C2×#6383C2212yescomplemented
35#89727—21no—
36#89715—21no—
37C4×(C2×D7)⋊C4—22no—
38C7×#340C2214yescomplemented
39#89324C2214yescomplemented
40#20970—71nomaximal, Sylow
41C24×D7D411yescharacteristic
42C23×C4×C7C2311yescharacteristic
43C22×C22⋊D7—21no—
44C23×Dic7D411yescharacteristic
45C2×#1157—22no—
46C2×#1157—22no—
47C2×#1157—22no—
48C2×#1157—22no—
49C2×#1157—22no—
50C2×#1157C2×C414yes—
51C22×D28—22no—
52C22×D28C2311yescharacteristic
53C22×C4×D7—22no—
54C22×C4×D7—22no—
55C22×C4×D7—22no—
56C22×C4×D7—22no—
57C22×C4×D7—21no—
58C22×C4×D7C2×C414yes—
59C22×C4×D7D412yes—
60C22×C4×D7D414yes—
61C22×C4×D7C2312yes—
62C22×Dic14C2311yescharacteristic
63C22×C7×C8—22no—
64C22×C7×C8—22no—
65C22×C7×C8D414yescomplemented
66C22×C7×C8C2312yes—
67C7×#117—22no—
68C7×#117—22no—
69C7×#117—22no—
70C7×#117C2×C414yes—
71C2×C7×C4⋊C4C2312yes—
72C2×C7×C4⋊C4C2312yes—
73C2×C7×C22⋊C4—21no—
74C2×C7×C22⋊C4—21no—
75C2×C42×C7—21no—
76C2×C42×C7—21no—
77C2×C22⋊Dic7—21no—
78C2×(C2×D7)⋊C4—22no—
79C2×(C2×D7)⋊C4—22no—
80C2×(C2×D7)⋊C4—24no—
81#6489—22no—
82#6489—22no—
83C2×C4⋊Dic7—22no—
84C2×C4⋊Dic7C2312yes—
85C2×C4×Dic7—21no—
86C22×C7⋊C8—22no—
87C22×C7⋊C8—22no—
88C22×C7⋊C8D414yescomplemented
89C22×C7⋊C8C2312yes—
90C2×#1131—232no—
91C2×#1131—28no—
92C2×#1131—28no—
93C2×#1131—28no—
94#6460—24no—
95#6460C2×C418yes—
96#6458—24no—
97#6458C2×C418yes—
98#6449—28no—
99#6445—28no—
100C4×D28—216no—
101C7⋊#95—24no—
102#6395—24no—
103#6390—24no—
104#6384—24no—
105#6383—24no—
106C23×M16—71no—
107C2×#525—71no—
108C2×#416—71no—
109C2×#416—71no—
110#2280—71no—
111#1895—74no—
112#1894—74no—
113#1735—71no—
114#1735D711yescharacteristic, complemented
115C24×C7C2×D411yescharacteristic
116C23×D7—22no—
117C23×D7—22no—
118C23×D7—21no—
119C23×D7—44no—
120C23×D7C22⋊C414yes—
121C23×D7C2×D412yes—
122C22×C4×C7—22no—
123C22×C4×C7—22no—
124C22×C4×C7C22×C412yes—
125C22×C4×C7C2×D411yescharacteristic
126C22×C4×C7C2×D412yes—
127C22×C4×C7C2×D412yes—
128C22×C4×C7C2×D412yes—
129C22×C4×C7#4112yes—
130C22×C4×C7#4112yes—
131C22×C4×C7C2411yescharacteristic
132C2×C22⋊D7—24no—
133C2×C22⋊D7—42no—
134C2×C22⋊D7—42no—
135C22×Dic7—22no—
136C22×Dic7—22no—
137C22×Dic7—21no—
138C22×Dic7—22no—
139C22×Dic7C22⋊C414yes—
140C22×Dic7C2×D412yes—
141#1157—28no—
142#1157—28no—
143#1157—44no—
144#1157—44no—
145C2×D28—21no—
146C2×D28—21no—
147C2×D28—21no—
148C2×D28—21no—
149C2×D28—28no—
150C2×D28—21no—
151C2×D28—48no—
152C2×D28C22×C412yes—
153C2×C4×D7—28no—
154C2×C4×D7—28no—
155C2×C4×D7—28no—
156C2×C4×D7—24no—
157C2×C4×D7—24no—
158C2×C4×D7—28no—
159C2×C4×D7—22no—
160C2×C4×D7—22no—
161C2×C4×D7—22no—
162C2×C4×D7—22no—
163C2×C4×D7—24no—
164C2×C4×D7—42no—
165C2×C4×D7—42no—
166C2×C4×D7C22⋊C418yes—
167C2×C4×D7C22×C414yes—
168C2×C4×D7C2×D414yes—
169C2×Dic14—21no—
170C2×Dic14—21no—
171C2×Dic14—21no—
172C2×Dic14—21no—
173C2×Dic14—21no—
174C2×Dic14C22×C412yes—
175C2×C7×C8—28no—
176C2×C7×C8—22no—
177C2×C7×C8—22no—
178C2×C7×C8C2×D414yes—
179C7×C4⋊C4—22no—
180C7×C4⋊C4—22no—
181C7×C4⋊C4—22no—
182C7×C4⋊C4C22×C414yes—
183C7×C22⋊C4—24no—
184C7×C22⋊C4—42no—
185C42×C7—24no—
186C42×C7—42no—
187C22⋊Dic7—42no—
188(C2×D7)⋊C4—28no—
189(C2×D7)⋊C4—28no—
190(C2×D7)⋊C4—48no—
191C4⋊Dic7—28no—
192C4⋊Dic7—22no—
193C4⋊Dic7—22no—
194C4×Dic7—42no—
195C2×C7⋊C8—28no—
196C2×C7⋊C8—22no—
197C2×C7⋊C8—22no—
198C2×C7⋊C8C2×D414yes—
199#1131—232no—
200C22×#41—71no—
201C24×C4—71no—
202C22×M16—74no—
203C22×M16—144no—
204C22×M16—144no—
205C22×M16—144no—
206C23×C8—71no—
207C23×C8C2×D711yescharacteristic
208#525—78no—
209#525—144no—
210C2×#123—141no—
211C2×C22⋊Q8—72no—
212C2×#121—72no—
213C2×#120—141no—
214C2×C4×D4—142no—
215C2×#117—71no—
216C2×#117C2×D711yescharacteristic
217C2×#117C2×D711yescharacteristic
218#416—144no—
219#416—144no—
220C2×#103—72no—
221C2×#103—72no—
222C2×#102—72no—
223C2×#102—72no—
224#381—141no—
225#380—74no—
226#379—141no—
227C4×C22⋊C4—142no—
228#340—74no—
229#340C2×D714yescomplemented
230C23×C7—24no—
231C23×C7C2×C22⋊C412yes—
232C23×C7#12112yes—
233C23×C7#12112yes—
234C23×C7C22×D411yescharacteristic
235C22×D7—22no—
236C22×D7—22no—
237C22×D7—24no—
238C22×D7—24no—
239C22×D7—28no—
240C22×D7—22no—
241C22×D7—48no—
242C22×D7—48no—
243C22×D7—48no—
244C22×D7—41no—
245C22×D7—41no—
246C22×D7—44no—
247C22×D7C2×C22⋊C414yes—
248C2×C4×C7—24no—
249C2×C4×C7—44no—
250C2×C4×C7—42no—
251C2×C4×C7—44no—
252C2×C4×C7C2×C22⋊C412yes—
253C2×C4×C7C4×D414yes—
254C2×C4×C7C4×D414yes—
255C2×C4×C7#12014yes—
256C2×C4×C7#12112yes—
257C2×C4×C7#12112yes—
258C2×C4×C7#12314yes—
259C2×C4×C7C23×C411yescharacteristic
260C2×C4×C7C22×D411yescharacteristic
261C2×C4×C7C22×D411yescharacteristic
262C2×C4×C7C22×D411yescharacteristic
263C2×C4×C7C2×#4111yescharacteristic
264C2×C4×C7C2×#4111yescharacteristic
265C22⋊D7—48no—
266C2×Dic7—22no—
267C2×Dic7—22no—
268C2×Dic7—28no—
269C2×Dic7—22no—
270C2×Dic7—41no—
271C2×Dic7—41no—
272C2×Dic7—42no—
273C2×Dic7—44no—
274C2×Dic7C2×C22⋊C414yes—
275D28—22no—
276D28—22no—
277D28—416no—
278D28—41no—
279D28—41no—
280C4×D7—24no—
281C4×D7—24no—
282C4×D7—24no—
283C4×D7—24no—
284C4×D7—216no—
285C4×D7—48no—
286Dic14—22no—
287Dic14—22no—
288Dic14—41no—
289Dic14—41no—
290C7×C8—24no—
291C7⋊C8—24no—
292C25—71no—
293C2×#41—74no—
294C2×#41—142no—
295C2×#41—142no—
296C2×#41—142no—
297C2×#41—142no—
298C2×#41—142no—
299C22×Q8—71no—
300C22×D4—71no—
301C22×D4—142no—
302C22×D4—141no—
303C23×C4—71no—
304C23×C4—74no—
305C23×C4—74no—
306C23×C4—72no—
307C23×C4—72no—
308C23×C4—142no—
309C23×C4—142no—
310C23×C4—142no—
311C23×C4—142no—
312C23×C4—141no—
313C23×C4C22×D711yescharacteristic
314C2×M16—148no—
315C2×M16—148no—
316C2×M16—1432no—
317C2×M16—148no—
318C22×C8—22no—
319C22×C8—22no—
320C22×C8—74no—
321C22×C8—72no—
322C22×C8—142no—
323C22×C8—142no—
324C22×C8D2814yescomplemented
325C22×C8C22×D712yes—
326#123—148no—
327C22⋊Q8—78no—
328C22⋊Q8—144no—
329#121—78no—
330#121—144no—
331#120—148no—
332C4×D4—1416no—
333#117—22no—
334#117—22no—
335#117—22no—
336#117—142no—
337#117—142no—
338#117C4×D714yes—
339C2×C4⋊C4—72no—
340C2×C4⋊C4—142no—
341C2×C4⋊C4C22×D712yes—
342C2×C4⋊C4C22×D712yes—
343C2×C22⋊C4—21no—
344C2×C22⋊C4—21no—
345C2×C22⋊C4—142no—
346C2×C22⋊C4—142no—
347C2×C22⋊C4—144no—
348C2×C22⋊C4—141no—
349C2×C42—21no—
350C2×C42—21no—
351C2×C42—141no—
352#103—144no—
353#103—144no—
354#102—144no—
355#102—144no—
356#95—144no—
357C22×C7—24no—
358C22×C7—24no—
359C22×C7—24no—
360C22×C7#38614yes—
361C22×C7C22×C22⋊C411yescharacteristic
362C22×C7C2×#12111yescharacteristic
363C22×C7C2×#12111yescharacteristic
364C22×C7#57514yes—
365C2×D7—24no—
366C2×D7—24no—
367C2×D7—24no—
368C2×D7—416no—
369C2×D7—44no—
370C2×D7—44no—
371C2×D7—44no—
372C2×D7—42no—
373C2×D7—82no—
374C2×D7—82no—
375C4×C7—44no—
376C4×C7—82no—
377C4×C7#38614yes—
378C4×C7C2×C4×D411yescharacteristic
379C4×C7C2×C4×D411yescharacteristic
380C4×C7C2×#12011yescharacteristic
381C4×C7C2×#12311yescharacteristic
382Dic7—24no—
383Dic7—42no—
384Dic7—82no—
385C24—72no—
386C24—74no—
387C24—142no—
388C24—142no—
389C24—141no—
390C24—284no—
391C24D4×D711yescharacteristic
392#41—148no—
393#41—148no—
394#41—284no—
395#41—284no—
396C2×Q8—72no—
397C2×Q8—141no—
398C2×Q8—141no—
399C2×Q8—141no—
400C2×Q8—141no—
401C2×Q8—141no—
402C2×D4—72no—
403C2×D4—148no—
404C2×D4—144no—
405C2×D4—141no—
406C2×D4—141no—
407C2×D4—141no—
408C2×D4—141no—
409C2×D4—141no—
410C2×D4—288no—
411C2×D4—282no—
412C2×D4—282no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—74no—
416C22×C4—72no—
417C22×C4—78no—
418C22×C4—74no—
419C22×C4—74no—
420C22×C4—142no—
421C22×C4—142no—
422C22×C4—148no—
423C22×C4—148no—
424C22×C4—144no—
425C22×C4—148no—
426C22×C4—148no—
427C22×C4—144no—
428C22×C4—142no—
429C22×C4—142no—
430C22×C4—142no—
431C22×C4—142no—
432C22×C4—141no—
433C22×C4—144no—
434C22×C4—142no—
435C22×C4—282no—
436C22×C4—282no—
437C22×C4C2×C4×D712yes—
438C22×C4C2×D2812yes—
439C22×C4#115712yes—
440C22×C4D4×D712yes—
441C22×C4D4×D712yes—
442C22×C4D4×D711yescharacteristic
443C22×C4#116112yes—
444C22×C4C23×D711yescharacteristic
445M16—1432no—
446C2×C8—28no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—74no—
450C2×C8—148no—
451C2×C8—142no—
452C2×C8—142no—
453C2×C8C2×D2814yes—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—148no—
458C4⋊C4—142no—
459C4⋊C4—142no—
460C4⋊C4C2×C4×D714yes—
461C22⋊C4—24no—
462C22⋊C4—42no—
463C22⋊C4—148no—
464C22⋊C4—148no—
465C22⋊C4—288no—
466C22⋊C4—282no—
467C42—24no—
468C42—42no—
469C42—282no—
470C2×C7—24no—
471C2×C7C2×#38611yescharacteristic
472C2×C7#289612yes—
473C2×C7#305412yes—
474C2×C7#305412yes—
475D7—44no—
476D7—82no—
477C23—24no—
478C23—74no—
479C23—142no—
480C23—142no—
481C23—144no—
482C23—144no—
483C23—148no—
484C23—142no—
485C23—288no—
486C23—288no—
487C23—288no—
488C23—281no—
489C23—281no—
490C23—284no—
491C23D7×C22⋊C412yes—
492C23#644712yes—
493C23#645812yes—
494C23C2×D4×D711yescharacteristic
495Q8—142no—
496Q8—142no—
497Q8—281no—
498Q8—281no—
499D4—142no—
500D4—142no—
501D4—2816no—
502D4—288no—
503D4—281no—
504D4—281no—
505C2×C4—24no—
506C2×C4—44no—
507C2×C4—42no—
508C2×C4—44no—
509C2×C4—74no—
510C2×C4—142no—
511C2×C4—142no—
512C2×C4—148no—
513C2×C4—144no—
514C2×C4—144no—
515C2×C4—144no—
516C2×C4—144no—
517C2×C4—1416no—
518C2×C4—142no—
519C2×C4—281no—
520C2×C4—281no—
521C2×C4—288no—
522C2×C4—282no—
523C2×C4—284no—
524C2×C4C4×D2814yes—
525C2×C4D7×C22⋊C412yes—
526C2×C4#644514yes—
527C2×C4#644712yes—
528C2×C4#645614yes—
529C2×C4#645714yes—
530C2×C4#645812yes—
531C2×C4C22×C4×D711yescharacteristic
532C2×C4C22×D2811yescharacteristic
533C2×C4C2×#115711yescharacteristic
534C2×C4C2×D4×D711yescharacteristic
535C2×C4C2×D4×D711yescharacteristic
536C2×C4C2×#116111yescharacteristic
537C8—24no—
538C8—144no—
539C7#2097011yescharacteristic, complemented, Sylow
540C22—24no—
541C22—24no—
542C22—24no—
543C22—144no—
544C22—144no—
545C22—144no—
546C22—2816no—
547C22—284no—
548C22—284no—
549C22—282no—
550C22—284no—
551C22—562no—
552C22—562no—
553C22#8943714yes—
554C22C2×D7×C22⋊C411yescharacteristic
555C22C2×#644711yescharacteristic
556C22C2×#645811yescharacteristic
557C22#9045914yes—
558C4—44no—
559C4—82no—
560C4—144no—
561C4—282no—
562C4—562no—
563C4#8943714yes—
564C4C2×C4×D2811yescharacteristic
565C4C2×#644511yescharacteristic
566C4C2×#645611yescharacteristic
567C4C2×#645711yescharacteristic
568C2—24no—
569C2—284no—
570C2—562no—
571C2C2×#8943711yescharacteristic
572C2#1173627512yes—
573C2#1173628412yes—
574C2#1173662612yes—
subgroupG/H
centerC22×C4D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1735C2
socleC23×C7C22×D4
prime radicalC2×#90215C2
cosocleC22×C4×C7C24
hypercenter#1735D7
hypocenterC7#20970
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22×C4 ◁ C23×C4 ◁ #1735

Complements

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

NH ≅ G/Ntype
1. C23×#1131C2semidirect
3. C2×#89885C2semidirect
4. #11739712C2semidirect
5. C2×#89774C2semidirect
6. C7×#1735C2semidirect
7. #11729334C2semidirect
8. #11726639C2semidirect
9. #11726638C2semidirect
22. #90215C4semidirect
29. C2×#6395C22semidirect
30. C2×#6390C22semidirect
33. C2×#6384C22semidirect
34. C2×#6383C22semidirect
38. C7×#340C22semidirect
39. #89324C22semidirect
65. C22×C7×C8D4semidirect
88. C22×C7⋊C8D4semidirect
114. #1735D7semidirect
229. #340C2×D7semidirect
324. C22×C8D28semidirect
539. C7#20970semidirect

Maximal subgroups

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

subgroupindex
C23×#11312
C2×#902152
C2×#898852
#117397122
C2×#897742
C7×#17352
#117293342
#117266392
#117266382
#209707