Finite Group Databasebeta

← #13355017

Subgroups — #13355017

Order 1792.

1986 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C7×#117C211yesmaximal, characteristic, complemented
2C2×#90482C211yesmaximal, characteristic
3C22×C22⋊(C7⋊C8)C211yesmaximal, characteristic, complemented
4C2×#89774C212yesmaximal, complemented
5#11729340C212yesmaximal, complemented
6#11729339C212yesmaximal, complemented
7#11729334C212yesmaximal, complemented
8#11729314C212yesmaximal, complemented
9#11729313C212yesmaximal, complemented
10C24×C4×C7C2211yescharacteristic
11C22×#1157C2211yescharacteristic
12C2×C7×#117—21no—
13C2×C7×#117—21no—
14C2×C7×#117—28no—
15C2×C7×#117C2212yes—
16C2×C7×#117C2212yes—
17C22×C7×C4⋊C4C2212yes—
18C22×C7×C22⋊C4—21no—
19C22×C42×C7—21no—
20C2×#6516—21no—
21#90482—24no—
22#90482C418yescomplemented
23C2×#6493C2211yescharacteristic
24C2×#6493C2211yescharacteristic
25C2×#6492—21no—
26C2×C4×C22⋊D7—21no—
27C2×C4×C22⋊D7—21no—
28C2×#6489C2211yescharacteristic
29C2×#6487C2211yescharacteristic
30C2×#6487C2211yescharacteristic
31C23×C7⋊C8C2212yes—
32C2×C22⋊(C7⋊C8)—24no—
33C2×C22⋊(C7⋊C8)C2212yes—
34C2×C22⋊(C7⋊C8)C2212yes—
35#89774—28no—
36C2×#6384C2214yescomplemented
37C2×#6383C2214yescomplemented
38#89324C2214yescomplemented
39#89324C2214yescomplemented
40#20908—71nomaximal, Sylow
41C25×C7D411yescharacteristic
42C23×C4×C7—24no—
43C23×C4×C7—24no—
44C23×C4×C7—22no—
45C23×C4×C7C2×C414yes—
46C23×C4×C7D414yes—
47C23×C4×C7D411yescharacteristic
48C23×C4×C7D411yescharacteristic
49C23×C4×C7D411yescharacteristic
50C23×C4×C7C2311yescharacteristic
51C23×C4×C7C2311yescharacteristic
52C23×C4×C7C2311yescharacteristic
53C22×C22⋊D7—21no—
54C22×C22⋊D7—21no—
55C22×C22⋊D7—21no—
56C2×#1157—22no—
57C2×#1157—24no—
58C2×#1157—24no—
59C2×#1157C2×C414yescomplemented
60C22×D28C2311yescharacteristic
61C22×C4×D7—21no—
62C22×Dic14C2311yescharacteristic
63C7×#117—28no—
64C7×#117—28no—
65C7×#117—24no—
66C7×#117—28no—
67C7×#117D418yescomplemented
68C2×C7×C4⋊C4—22no—
69C2×C7×C4⋊C4—28no—
70C2×C7×C4⋊C4C2314yes—
71C2×C7×C22⋊C4—28no—
72C2×C7×C22⋊C4—22no—
73C2×C7×C22⋊C4—22no—
74C2×C42×C7—28no—
75C2×C42×C7—22no—
76C2×C42×C7—22no—
77#6516—28no—
78C2×C22⋊Dic7—21no—
79C2×C22⋊Dic7—21no—
80C2×C22⋊Dic7—21no—
81#6493—22no—
82#6493—22no—
83#6493C2×C414yes—
84#6493C2×C414yescomplemented
85#6492—28no—
86C4×C22⋊D7—28no—
87C4×C22⋊D7—28no—
88C2×(C2×D7)⋊C4—21no—
89C2×(C2×D7)⋊C4—21no—
90#6489—22no—
91#6489C2×C414yescomplemented
92C2×C4⋊Dic7C2311yescharacteristic
93C2×C4⋊Dic7C2311yescharacteristic
94#6487—22no—
95#6487—22no—
96#6487C2×C414yes—
97#6487C2×C414yescomplemented
98C2×Dic7⋊C4—21no—
99C2×Dic7⋊C4—21no—
100C2×C4×Dic7—21no—
101C22×C7⋊C8—24no—
102C22×C7⋊C8—24no—
103C22×C7⋊C8—24no—
104C22×C7⋊C8C2314yes—
105C22⋊(C7⋊C8)—24no—
106C22⋊(C7⋊C8)—24no—
107#6384—28no—
108#6383—28no—
109C2×#525—71no—
110C22×#117D711yescharacteristic, complemented
111C2×#416—72no—
112C22×C22⋊C8—71no—
113#1875—72no—
114#1874—72no—
115#1873—72no—
116#1872—72no—
117#1735—72no—
118C24×C7—22no—
119C24×C7—22no—
120C24×C7—24no—
121C24×C7—42no—
122C24×C7C22⋊C414yes—
123C24×C7C2×D411yescharacteristic
124C24×C7C2×D411yescharacteristic
125C24×C7C2×D411yescharacteristic
126C23×D7—21no—
127C22×C4×C7—22no—
128C22×C4×C7—22no—
129C22×C4×C7—216no—
130C22×C4×C7—22no—
131C22×C4×C7—22no—
132C22×C4×C7—216no—
133C22×C4×C7—22no—
134C22×C4×C7—22no—
135C22×C4×C7—22no—
136C22×C4×C7—22no—
137C22×C4×C7—22no—
138C22×C4×C7—22no—
139C22×C4×C7—28no—
140C22×C4×C7—24no—
141C22×C4×C7—44no—
142C22×C4×C7—44no—
143C22×C4×C7C22⋊C414yes—
144C22×C4×C7C22⋊C414yes—
145C22×C4×C7C22⋊C414yes—
146C22×C4×C7C22×C412yes—
147C22×C4×C7C22×C412yes—
148C22×C4×C7C22×C412yes—
149C22×C4×C7C2×D411yescharacteristic
150C22×C4×C7C2×D411yescharacteristic
151C22×C4×C7C2×D412yes—
152C22×C4×C7C2×D412yes—
153C22×C4×C7C2×D411yescharacteristic
154C22×C4×C7C2×D412yes—
155C22×C4×C7C2×D414yes—
156C22×C4×C7#4114yes—
157C22×C4×C7C2411yescharacteristic
158C2×C22⋊D7—24no—
159C2×C22⋊D7—24no—
160C2×C22⋊D7—24no—
161C2×C22⋊D7—44no—
162C2×C22⋊D7—42no—
163C2×C22⋊D7—42no—
164C2×C22⋊D7—44no—
165C22×Dic7—21no—
166C22×Dic7—21no—
167C22×Dic7—21no—
168#1157—216no—
169#1157—48no—
170C2×D28—22no—
171C2×D28—22no—
172C2×D28—21no—
173C2×D28C22×C412yes—
174C2×C4×D7—24no—
175C2×C4×D7—42no—
176C2×C4×D7—42no—
177C2×Dic14—22no—
178C2×Dic14—22no—
179C2×Dic14—21no—
180C2×Dic14C22×C412yes—
181C7×C4⋊C4—28no—
182C7×C4⋊C4—24no—
183C7×C4⋊C4C2×D418yes—
184C7×C22⋊C4—28no—
185C7×C22⋊C4—28no—
186C42×C7—28no—
187C42×C7—28no—
188C22⋊Dic7—24no—
189C22⋊Dic7—24no—
190C22⋊Dic7—24no—
191(C2×D7)⋊C4—24no—
192(C2×D7)⋊C4—24no—
193C4⋊Dic7—21no—
194C4⋊Dic7—21no—
195C4⋊Dic7C22×C412yes—
196C4⋊Dic7C22×C412yes—
197Dic7⋊C4—24no—
198Dic7⋊C4—24no—
199C4×Dic7—24no—
200C2×C7⋊C8—24no—
201C2×C7⋊C8—24no—
202C2×C7⋊C8—24no—
203C2×C7⋊C8—48no—
204C22×#41—71no—
205C24×C4C2×D711yescharacteristic
206C23×C8—72no—
207#525—78no—
208#525—144no—
209C2×#123—141no—
210C2×C22⋊Q8—71no—
211C2×C22⋊Q8—71no—
212C2×#121—71no—
213C2×#121—71no—
214C2×#120—141no—
215C2×C4×D4—141no—
216C2×C4×D4—141no—
217C2×#117—21no—
218C2×#117—21no—
219C2×#117—28no—
220C2×#117—71no—
221C2×#117C2×D712yes—
222C2×#117C2×D712yes—
223C22×C4⋊C4C2×D712yes—
224C22×C22⋊C4—21no—
225C22×C42—21no—
226#416—148no—
227C2×#103—74no—
228C2×#102—74no—
229C2×C22⋊C8—72no—
230C2×C22⋊C8—72no—
231C2×C22⋊C8—144no—
232#340—74no—
233#340—74no—
234C23×C7—21no—
235C23×C7—21no—
236C23×C7—22no—
237C23×C7—22no—
238C23×C7—22no—
239C23×C7—21no—
240C23×C7—21no—
241C23×C7—22no—
242C23×C7—22no—
243C23×C7—28no—
244C23×C7—28no—
245C23×C7—44no—
246C23×C7—48no—
247C23×C7—48no—
248C23×C7C2×C22⋊C412yes—
249C23×C7C2×C22⋊C412yes—
250C23×C7C2×C22⋊C412yes—
251C23×C7#12012yes—
252C23×C7#12312yes—
253C23×C7#13514yes—
254C23×C7C22×D411yescharacteristic
255C22×D7—22no—
256C22×D7—41no—
257C22×D7—41no—
258C22×D7—42no—
259C22×D7—42no—
260C2×C4×C7—21no—
261C2×C4×C7—21no—
262C2×C4×C7—21no—
263C2×C4×C7—21no—
264C2×C4×C7—28no—
265C2×C4×C7—24no—
266C2×C4×C7—24no—
267C2×C4×C7—24no—
268C2×C4×C7—24no—
269C2×C4×C7—28no—
270C2×C4×C7—28no—
271C2×C4×C7—28no—
272C2×C4×C7—416no—
273C2×C4×C7—44no—
274C2×C4×C7—44no—
275C2×C4×C7C2×C22⋊C412yes—
276C2×C4×C7C2×C22⋊C412yes—
277C2×C4×C7C2×C22⋊C412yes—
278C2×C4×C7C4×D418yes—
279C2×C4×C7#12012yes—
280C2×C4×C7#12118yes—
281C2×C4×C7#12312yes—
282C2×C4×C7C23×C411yescharacteristic
283C2×C4×C7C22×D411yescharacteristic
284C2×C4×C7C22×D412yes—
285C2×C4×C7C2×#4112yes—
286C22⋊D7—48no—
287C22⋊D7—48no—
288C22⋊D7—48no—
289C2×Dic7—22no—
290C2×Dic7—22no—
291C2×Dic7—22no—
292C2×Dic7—41no—
293C2×Dic7—41no—
294C2×Dic7—42no—
295C2×Dic7—42no—
296D28—24no—
297D28—42no—
298C4×D7—48no—
299Dic14—24no—
300Dic14—42no—
301C7⋊C8—44no—
302C25C22⋊D711yescharacteristic
303C2×#41—74no—
304C2×#41—142no—
305C2×#41—144no—
306C2×#41—144no—
307C22×Q8—71no—
308C22×D4—71no—
309C22×D4—141no—
310C22×D4—141no—
311C22×D4—141no—
312C23×C4—24no—
313C23×C4—24no—
314C23×C4—22no—
315C23×C4—141no—
316C23×C4C4×D714yes—
317C23×C4D2814yes—
318C23×C4C22⋊D711yescharacteristic
319C23×C4C22⋊D711yescharacteristic
320C23×C4C22⋊D711yescharacteristic
321C23×C4C22×D711yescharacteristic
322C23×C4C22×D711yescharacteristic
323C23×C4C22×D711yescharacteristic
324C22×C8—74no—
325C22×C8—144no—
326C22×C8—144no—
327C22×C8—144no—
328#123—148no—
329C22⋊Q8—74no—
330C22⋊Q8—74no—
331C22⋊Q8—142no—
332C22⋊Q8—142no—
333#121—74no—
334#121—74no—
335#121—142no—
336#121—142no—
337#120—148no—
338C4×D4—148no—
339C4×D4—148no—
340#117—28no—
341#117—28no—
342#117—24no—
343#117—28no—
344#117—74no—
345#117—142no—
346#117C22⋊D718yescomplemented
347C2×C4⋊C4—22no—
348C2×C4⋊C4—28no—
349C2×C4⋊C4—71no—
350C2×C4⋊C4—71no—
351C2×C4⋊C4—141no—
352C2×C4⋊C4—141no—
353C2×C4⋊C4C22×D714yes—
354C2×C22⋊C4—28no—
355C2×C22⋊C4—22no—
356C2×C22⋊C4—22no—
357C2×C22⋊C4—141no—
358C2×C22⋊C4—141no—
359C2×C22⋊C4—141no—
360C2×C22⋊C4—141no—
361C2×C22⋊C4—141no—
362C2×C42—28no—
363C2×C42—22no—
364C2×C42—22no—
365C2×C42—141no—
366#103—148no—
367#102—148no—
368C22⋊C8—144no—
369C22⋊C8—144no—
370C22×C7—24no—
371C22×C7—24no—
372C22×C7—22no—
373C22×C7—22no—
374C22×C7—22no—
375C22×C7—24no—
376C22×C7—24no—
377C22×C7—22no—
378C22×C7—22no—
379C22×C7—22no—
380C22×C7—22no—
381C22×C7—416no—
382C22×C7—42no—
383C22×C7—42no—
384C22×C7#38614yes—
385C22×C7#41614yescomplemented
386C22×C7C22×C22⋊C411yescharacteristic
387C22×C7C2×#12011yescharacteristic
388C22×C7C2×#12311yescharacteristic
389C22×C7C2×#13512yes—
390C22×C7#57512yes—
391C2×D7—42no—
392C2×D7—42no—
393C2×D7—42no—
394C2×D7—82no—
395C2×D7—82no—
396C4×C7—22no—
397C4×C7—22no—
398C4×C7—48no—
399C4×C7#38614yes—
400C4×C7C2×C4×D412yes—
401C4×C7C2×#12112yes—
402Dic7—42no—
403Dic7—42no—
404Dic7—42no—
405C24—22no—
406C24—24no—
407C24—22no—
408C24—42no—
409C24—141no—
410C24(C2×D7)⋊C414yes—
411C24C2×C22⋊D711yescharacteristic
412C24C2×C22⋊D711yescharacteristic
413C24C2×C22⋊D711yescharacteristic
414#41—1416no—
415#41—288no—
416C2×Q8—72no—
417C2×Q8—141no—
418C2×Q8—142no—
419C2×Q8—142no—
420C2×D4—72no—
421C2×D4—144no—
422C2×D4—144no—
423C2×D4—144no—
424C2×D4—141no—
425C2×D4—142no—
426C2×D4—142no—
427C2×D4—284no—
428C2×D4—282no—
429C2×D4—284no—
430C2×D4—282no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—216no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—216no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—28no—
444C22×C4—24no—
445C22×C4—44no—
446C22×C4—44no—
447C22×C4—141no—
448C22×C4—141no—
449C22×C4—141no—
450C22×C4—144no—
451C22×C4—282no—
452C22×C4—282no—
453C22×C4(C2×D7)⋊C414yes—
454C22×C4(C2×D7)⋊C414yes—
455C22×C4(C2×D7)⋊C414yes—
456C22×C4C2×C4×D712yes—
457C22×C4C2×C4×D712yes—
458C22×C4C2×C4×D712yes—
459C22×C4C2×D2812yes—
460C22×C4C2×D2812yes—
461C22×C4C2×D2812yes—
462C22×C4#115714yes—
463C22×C4C2×C22⋊D711yescharacteristic
464C22×C4C2×C22⋊D711yescharacteristic
465C22×C4C2×C22⋊D714yes—
466C22×C4C2×C22⋊D711yescharacteristic
467C22×C4C23×D711yescharacteristic
468C2×C8—144no—
469C2×C8—144no—
470C2×C8—144no—
471C2×C8—288no—
472C4⋊C4—28no—
473C4⋊C4—24no—
474C4⋊C4—72no—
475C4⋊C4—72no—
476C4⋊C4—144no—
477C4⋊C4—144no—
478C4⋊C4—141no—
479C4⋊C4—141no—
480C4⋊C4C2×C22⋊D718yes—
481C22⋊C4—28no—
482C22⋊C4—28no—
483C22⋊C4—144no—
484C22⋊C4—144no—
485C22⋊C4—144no—
486C22⋊C4—144no—
487C22⋊C4—144no—
488C42—28no—
489C42—28no—
490C42—144no—
491C2×C7—24no—
492C2×C7—24no—
493C2×C7—42no—
494C2×C7C2×#38611yescharacteristic
495C2×C7C2×#41611yescharacteristic
496C2×C7#289611yescharacteristic
497C2×C7#300212yes—
498C2×C7#309112yes—
499D7—82no—
500C23—21no—
501C23—22no—
502C23—22no—
503C23—22no—
504C23—21no—
505C23—21no—
506C23—28no—
507C23—22no—
508C23—22no—
509C23—21no—
510C23—28no—
511C23—44no—
512C23—48no—
513C23—48no—
514C23—142no—
515C23—281no—
516C23—282no—
517C23—282no—
518C23—281no—
519C23C2×(C2×D7)⋊C412yes—
520C23C2×(C2×D7)⋊C412yes—
521C23C2×(C2×D7)⋊C412yes—
522C23#649212yes—
523C23#651314yes—
524C23#651612yes—
525C23C22×C22⋊D711yescharacteristic
526Q8—144no—
527Q8—282no—
528D4—144no—
529D4—288no—
530D4—288no—
531D4—288no—
532D4—282no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—28no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—28no—
543C2×C4—28no—
544C2×C4—28no—
545C2×C4—416no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—142no—
549C2×C4—142no—
550C2×C4—142no—
551C2×C4—281no—
552C2×C4—282no—
553C2×C4—282no—
554C2×C4—281no—
555C2×C4—288no—
556C2×C4C2×(C2×D7)⋊C412yes—
557C2×C4C2×(C2×D7)⋊C412yes—
558C2×C4C2×(C2×D7)⋊C412yes—
559C2×C4C4×C22⋊D718yes—
560C2×C4#649212yes—
561C2×C4#649318yes—
562C2×C4#651612yes—
563C2×C4C22×C4×D711yescharacteristic
564C2×C4C22×D2811yescharacteristic
565C2×C4C2×#115712yes—
566C2×C4C22×C22⋊D712yes—
567C8—284no—
568C7#2090811yescharacteristic, complemented, Sylow
569C22—22no—
570C22—22no—
571C22—22no—
572C22—24no—
573C22—24no—
574C22—22no—
575C22—24no—
576C22—24no—
577C22—22no—
578C22—22no—
579C22—22no—
580C22—416no—
581C22—42no—
582C22—42no—
583C22—282no—
584C22—282no—
585C22—282no—
586C22—562no—
587C22—562no—
588C22#8977414yescomplemented
589C22#8998514yes—
590C22C22×(C2×D7)⋊C411yescharacteristic
591C22C2×#649211yescharacteristic
592C22C2×#651312yes—
593C22#9051312yes—
594C22C2×#651611yescharacteristic
595C4—22no—
596C4—22no—
597C4—48no—
598C4—282no—
599C4—282no—
600C4—282no—
601C4#8998514yes—
602C4C2×C4×C22⋊D712yes—
603C4C2×#649312yes—
604C2—24no—
605C2—24no—
606C2—42no—
607C2—562no—
608C2C2×#8977411yescharacteristic
609C2#1173965811yescharacteristic
610C2#1173966012yes—
611C2C2×#8998511yescharacteristic
612C2#1174152512yes—
subgroupG/H
centerC22×C4C2×C22⋊D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC22×C7×#117C2
socleC23×C7C22×D4
prime radicalC2×#90482C2
cosocleC22×C4×C7C24
hypercenterC22×#117D7
hypocenterC7#20908
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22×C4 ◁ C24×C4 ◁ C22×#117

Complements

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

NH ≅ G/Ntype
1. C22×C7×#117C2semidirect
3. C22×C22⋊(C7⋊C8)C2semidirect
4. C2×#89774C2semidirect
5. #11729340C2semidirect
6. #11729339C2semidirect
7. #11729334C2semidirect
8. #11729314C2semidirect
9. #11729313C2semidirect
22. #90482C4semidirect
36. C2×#6384C22semidirect
37. C2×#6383C22semidirect
38. #89324C22semidirect
39. #89324C22semidirect
59. C2×#1157C2×C4semidirect
67. C7×#117D4semidirect
84. #6493C2×C4semidirect
91. #6489C2×C4semidirect
97. #6487C2×C4semidirect
110. C22×#117D7semidirect
346. #117C22⋊D7semidirect
385. C22×C7#416semidirect
568. C7#20908semidirect
588. C22#89774semidirect

Maximal subgroups

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

subgroupindex
C22×C7×#1172
C2×#904822
C22×C22⋊(C7⋊C8)2
C2×#897742
#117293402
#117293392
#117293342
#117293142
#117293132
#209087