Finite Group Databasebeta

← C4×#89926 #13362336

Subgroups — C4×#89926 #13362336

Order 1792.

1262 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C7×D8C211yesmaximal, characteristic, complemented
2C2×#89926C211yesmaximal, characteristic
3C4×#6503C211yesmaximal, characteristic, complemented
4C4×#6499C211yesmaximal, characteristic, complemented
5C2×C4×#1142C211yesmaximal, characteristic, complemented
6C2×C4×#1141C211yesmaximal, characteristic, complemented
7C2×C4×#1132C211yesmaximal, characteristic, complemented
8#11729793C211yesmaximal, characteristic, complemented
9#11728383C211yesmaximal, characteristic, complemented
10#11728362C211yesmaximal, characteristic, complemented
11#11727946C211yesmaximal, characteristic, complemented
12#11727945C211yesmaximal, characteristic, complemented
13#11727914C211yesmaximal, characteristic, complemented
14#11727913C211yesmaximal, characteristic, complemented
15C4×#6389C211yesmaximal, characteristic, complemented
16C22×C7×D8C2211yescharacteristic
17C2×C4×C7×D4C2211yescharacteristic
18C2×C4×C7×D4C2211yescharacteristic
19C2×#6503C2211yescharacteristic
20C2×#6499C2211yescharacteristic
21C22×#1142C2211yescharacteristic
22C22×#1141C2211yescharacteristic
23C2×C4×C22⋊D7—22no—
24C22×#1132C2211yescharacteristic
25C2×C4×D28C2211yescharacteristic
26C2×C4×Dic14C2211yescharacteristic
27C4×C7×D8—24no—
28C2×C7×#107C2211yescharacteristic
29C2×C7×#102C2211yescharacteristic
30C2×C7×#102C2211yescharacteristic
31C2×C4×C7×C8C2211yescharacteristic
32#89993C2211yescharacteristic
33#89993C2211yescharacteristic
34C2×#6406C2211yescharacteristic
35C2×#6406C2211yescharacteristic
36C4×C22⋊Dic7—22no—
37#89926C418yesdirect product factor
38C2×#6395C2211yescharacteristic
39C2×#6391C2211yescharacteristic
40C2×#6390C2211yescharacteristic
41C2×#6389C2211yescharacteristic
42C2×#6389C2211yescharacteristic
43C2×#6389C2211yescharacteristic
44C2×#6389C2211yescharacteristic
45C4×#1142—24no—
46C4×#1141—24no—
47#89760C2211yescharacteristic
48#89749C2211yescharacteristic
49#89748C2211yescharacteristic
50#89746C2211yescharacteristic
51C2×#6384C2211yescharacteristic
52C2×#6383C2211yescharacteristic
53C2×#6382C2211yescharacteristic
54C2×#6381C2211yescharacteristic
55#89729—22no—
56#89717—22no—
57C42×Dic7C2211yescharacteristic
58C2×C4×C7⋊C8C2211yescharacteristic
59C4×#1132—24no—
60C4×#496—71nomaximal, Sylow
61C22×C7×D4C2311yescharacteristic
62C22×C7×D4C2311yescharacteristic
63C23×C4×C7—21no—
64C23×C4×C7—21no—
65C22×C22⋊D7—22no—
66C22×D28C2311yescharacteristic
67C22×C4×D7—21no—
68C22×Dic14C2311yescharacteristic
69C2×C7×D8—22no—
70C2×C7×D8—22no—
71C2×C7×D8C2×C414yescomplemented
72C22×C7×C8C2311yescharacteristic
73C22×C7×C8C2311yescharacteristic
74C4×C7×D4—22no—
75C4×C7×D4—22no—
76C4×C7×D4—22no—
77C4×C7×D4—22no—
78C2×C7×C4⋊C4C2311yescharacteristic
79C2×C7×C4⋊C4C2311yescharacteristic
80C2×C7×C22⋊C4—21no—
81C2×C7×C22⋊C4—21no—
82C2×C42×C7C2311yescharacteristic
83C2×C22⋊Dic7—22no—
84C2×C22⋊Dic7—22no—
85C2×C22⋊Dic7—22no—
86#6503—22no—
87#6503C2×C414yescomplemented
88#6499—22no—
89#6499C2×C414yescomplemented
90C2×#1142—22no—
91C2×#1142—22no—
92C2×#1142C2×C414yescomplemented
93C2×#1141—22no—
94C2×#1141—22no—
95C2×#1141C2×C414yescomplemented
96C4×C22⋊D7—48no—
97C2×(C2×D7)⋊C4—21no—
98C2×(C2×D7)⋊C4—22no—
99C2×C4⋊Dic7C2311yescharacteristic
100C2×C4⋊Dic7C2311yescharacteristic
101C2×Dic7⋊C4—21no—
102C2×Dic7⋊C4—22no—
103C2×C4×Dic7—21no—
104C2×C4×Dic7D412yes—
105C2×C4×Dic7D412yes—
106C2×C4×Dic7C2311yescharacteristic
107C2×C4×Dic7C2311yescharacteristic
108C22×C7⋊C8C2311yescharacteristic
109C22×C7⋊C8C2311yescharacteristic
110C2×#1132—22no—
111C2×#1132—22no—
112C2×#1132C2×C414yescomplemented
113C4×D28—22no—
114C4×D28—22no—
115C4×Dic14—22no—
116C4×Dic14—22no—
117C7×#107—22no—
118C7×#102—22no—
119C7×#102—22no—
120C4×C7×C8D414yescomplemented
121#6406—22no—
122#6406—22no—
123C7⋊#95—22no—
124C7⋊#95—22no—
125#6395—22no—
126#6391—22no—
127#6390—22no—
128#6389—22no—
129#6389—22no—
130#6389—22no—
131#6389C2×C414yescomplemented
132#6384—22no—
133#6383—22no—
134#6382—22no—
135#6381—22no—
136C4×C7⋊C8D414yescomplemented
137C2×#496—71no—
138C2×C4×SD16—71no—
139C2×C4×SD16—71no—
140C2×C4×D8—71no—
141C2×C4×D8D711yescharacteristic, complemented
142C4×#127—71no—
143C4×#124—71no—
144#1973—71no—
145#1973—71no—
146#1972—71no—
147#1972—71no—
148#1961—71no—
149#1959—71no—
150#1847—71no—
151C4×#97—71no—
152C24×C7—21no—
153C24×C7—21no—
154C23×D7—21no—
155C2×C7×D4—21no—
156C2×C7×D4—21no—
157C2×C7×D4—21no—
158C2×C7×D4—21no—
159C2×C7×D4—21no—
160C2×C7×D4—21no—
161C2×C7×D4—21no—
162C2×C7×D4—21no—
163C2×C7×D4—21no—
164C2×C7×D4—21no—
165C2×C7×D4C22×C412yes—
166C2×C7×D4C22×C412yes—
167C22×C4×C7—21no—
168C22×C4×C7—21no—
169C22×C4×C7—42no—
170C22×C4×C7—42no—
171C22×C4×C7—44no—
172C22×C4×C7—44no—
173C22×C4×C7C2×D411yescharacteristic
174C22×C4×C7C2×D411yescharacteristic
175C22×C4×C7C2411yescharacteristic
176C2×C22⋊D7—28no—
177C2×C22⋊D7—44no—
178C2×C22⋊D7—44no—
179C22×Dic7—21no—
180C22×Dic7—21no—
181C22×Dic7—21no—
182C22×Dic7C2×D412yes—
183C22×Dic7C2×D412yes—
184C2×D28—21no—
185C2×D28—21no—
186C2×D28—21no—
187C2×D28—21no—
188C2×D28—21no—
189C2×D28C22×C412yes—
190C2×C4×D7—42no—
191C2×C4×D7—44no—
192C2×Dic14—21no—
193C2×Dic14—21no—
194C2×Dic14—21no—
195C2×Dic14—21no—
196C2×Dic14—21no—
197C2×Dic14C22×C412yes—
198C7×D8—28no—
199C2×C7×C8—21no—
200C2×C7×C8—21no—
201C2×C7×C8C22×C412yes—
202C2×C7×C8C2×D412yes—
203C2×C7×C8C2×D412yes—
204C2×C7×C8#4112yes—
205C7×C4⋊C4—21no—
206C7×C4⋊C4—21no—
207C7×C4⋊C4—21no—
208C7×C4⋊C4—21no—
209C7×C22⋊C4—42no—
210C7×C22⋊C4—42no—
211C42×C7C2×D412yes—
212C42×C7C2×D412yes—
213C22⋊Dic7—28no—
214C22⋊Dic7—44no—
215C22⋊Dic7—44no—
216#1142—28no—
217#1141—28no—
218(C2×D7)⋊C4—44no—
219(C2×D7)⋊C4—42no—
220C4⋊Dic7—21no—
221C4⋊Dic7—21no—
222C4⋊Dic7—21no—
223C4⋊Dic7—21no—
224Dic7⋊C4—44no—
225Dic7⋊C4—42no—
226C4×Dic7—24no—
227C4×Dic7—21no—
228C4×Dic7—21no—
229C4×Dic7—42no—
230C4×Dic7—42no—
231C4×Dic7C22×C412yes—
232C4×Dic7#4112yes—
233C2×C7⋊C8—21no—
234C2×C7⋊C8—21no—
235C2×C7⋊C8C22×C412yes—
236C2×C7⋊C8C2×D412yes—
237C2×C7⋊C8C2×D412yes—
238C2×C7⋊C8#4112yes—
239#1132—28no—
240C22×SD16—71no—
241C22×SD16—71no—
242C22×D8—71no—
243C22×D8C2×D711yescharacteristic
244C2×#127—71no—
245C2×#124—71no—
246C2×C4×Q8—71no—
247C2×C4×D4—71no—
248C2×C4×D4—142no—
249C2×C4×D4C2×D711yescharacteristic
250C2×C4×D4C2×D711yescharacteristic
251#496—78no—
252C4×SD16—144no—
253C4×SD16—144no—
254C4×D8—24no—
255C4×D8—144no—
256C2×#107—71no—
257C2×#107C2×D711yescharacteristic
258C2×#106—71no—
259C2×#106—71no—
260C2×#103—71no—
261C2×#103—71no—
262C2×#102—71no—
263C2×#102—71no—
264C2×#102—71no—
265C2×#102—71no—
266C2×#102C2×D711yescharacteristic
267C2×#102C2×D711yescharacteristic
268C2×#97—71no—
269C2×#97—71no—
270C2×#97—71no—
271C2×#97—71no—
272C2×C4×C8—71no—
273C2×C4×C8C2×D711yescharacteristic
274#391—71no—
275#390—71no—
276#390—71no—
277#390—71no—
278#390—142no—
279#388—142no—
280#384—71no—
281#382—71no—
282C4×C22⋊C4—142no—
283C43—71no—
284C23×C7—22no—
285C23×C7—22no—
286C23×C7—41no—
287C23×C7—41no—
288C23×C7—41no—
289C23×C7—41no—
290C23×C7—42no—
291C23×C7—42no—
292C23×C7—42no—
293C23×C7—42no—
294C23×C7C22×D411yescharacteristic
295C22×D7—22no—
296C22×D7—41no—
297C22×D7—41no—
298C22×D7—42no—
299C22×D7—42no—
300C7×D4—22no—
301C7×D4—22no—
302C7×D4—22no—
303C7×D4—22no—
304C7×D4—41no—
305C7×D4—41no—
306C7×D4—41no—
307C7×D4—41no—
308C2×C4×C7—22no—
309C2×C4×C7—41no—
310C2×C4×C7—41no—
311C2×C4×C7—42no—
312C2×C4×C7—42no—
313C2×C4×C7—84no—
314C2×C4×C7—84no—
315C2×C4×C7#12712yes—
316C2×C4×C7#12712yes—
317C2×C4×C7AGL(1,ℤ/8)14yes—
318C2×C4×C7C23×C411yescharacteristic
319C2×C4×C7C22×D411yescharacteristic
320C2×C4×C7C22×D411yescharacteristic
321C2×C4×C7C2×#4111yescharacteristic
322C2×C4×C7C2×#4111yescharacteristic
323C2×C4×C7C2×#4111yescharacteristic
324C22⋊D7—416no—
325C2×Dic7—22no—
326C2×Dic7—22no—
327C2×Dic7—22no—
328C2×Dic7—22no—
329C2×Dic7—22no—
330C2×Dic7—22no—
331C2×Dic7—41no—
332C2×Dic7—41no—
333C2×Dic7—41no—
334C2×Dic7—41no—
335C2×Dic7—42no—
336C2×Dic7—42no—
337C2×Dic7C4×D414yes—
338D28—22no—
339D28—22no—
340D28—41no—
341D28—41no—
342C4×D7—84no—
343Dic14—22no—
344Dic14—22no—
345Dic14—41no—
346Dic14—41no—
347C7×C8—22no—
348C7×C8C4×D414yescomplemented
349C7⋊C8—22no—
350C7⋊C8C4×D414yescomplemented
351C22×Q8—71no—
352C22×D4—71no—
353C22×D4—142no—
354C22×D4C22×D711yescharacteristic
355C22×D4C22×D711yescharacteristic
356C23×C4—21no—
357C23×C4—21no—
358C23×C4—141no—
359C2×SD16—74no—
360C2×SD16—74no—
361C2×SD16—142no—
362C2×SD16—142no—
363C2×SD16—142no—
364C2×SD16—142no—
365C2×D8—22no—
366C2×D8—22no—
367C2×D8—74no—
368C2×D8—142no—
369C2×D8—142no—
370C2×D8C4×D714yescomplemented
371C22×C8—71no—
372C22×C8—71no—
373C22×C8C22×D711yescharacteristic
374C22×C8C22×D711yescharacteristic
375#127—74no—
376#127—142no—
377#124—74no—
378#124—142no—
379C4×Q8—142no—
380C4×Q8—142no—
381C4×D4—22no—
382C4×D4—22no—
383C4×D4—22no—
384C4×D4—22no—
385C4×D4—142no—
386C4×D4—142no—
387C4×D4—288no—
388C2×C4⋊C4—71no—
389C2×C4⋊C4—71no—
390C2×C4⋊C4—142no—
391C2×C4⋊C4—141no—
392C2×C4⋊C4C22×D711yescharacteristic
393C2×C4⋊C4C22×D711yescharacteristic
394C2×C22⋊C4—21no—
395C2×C22⋊C4—21no—
396C2×C22⋊C4—142no—
397C2×C22⋊C4—142no—
398C2×C22⋊C4—142no—
399C2×C22⋊C4—142no—
400C2×C22⋊C4—141no—
401C2×C42—71no—
402C2×C42—72no—
403C2×C42—71no—
404C2×C42—72no—
405C2×C42—141no—
406C2×C42C22×D711yescharacteristic
407#107—22no—
408#107—142no—
409#106—142no—
410#106—142no—
411#103—142no—
412#103—142no—
413#102—22no—
414#102—22no—
415#102—142no—
416#102—142no—
417#102—142no—
418#102—142no—
419#97—74no—
420#97—142no—
421#97—142no—
422#97—142no—
423C4×C8—74no—
424C4×C8C22⋊D714yescomplemented
425#95—142no—
426#95—142no—
427C22×C7—42no—
428C22×C7—42no—
429C22×C7—44no—
430C22×C7—44no—
431C22×C7—81no—
432C22×C7—81no—
433C22×C7—81no—
434C22×C7—81no—
435C22×C7—82no—
436C22×C7—82no—
437C22×C7C2×C4×D411yescharacteristic
438C22×C7C2×#12711yescharacteristic
439C22×C7#53211yescharacteristic
440C22×C7C2×AGL(1,ℤ/8)12yes—
441C22×C7#57512yes—
442C2×D7—42no—
443C2×D7—44no—
444C2×D7—81no—
445C2×D7—81no—
446C2×D7—82no—
447C4×C7—22no—
448C4×C7—81no—
449C4×C7—81no—
450C4×C7#49614yescomplemented
451C4×C7C2×C4×D411yescharacteristic
452C4×C7C2×C4×D411yescharacteristic
453C4×C7#53211yescharacteristic
454C4×C7#53211yescharacteristic
455Dic7—24no—
456Dic7—42no—
457Dic7—42no—
458Dic7—81no—
459Dic7—81no—
460C24—21no—
461C24—21no—
462C24—141no—
463C2×Q8—72no—
464C2×Q8—141no—
465C2×Q8—141no—
466C2×Q8—141no—
467C2×Q8—141no—
468C2×Q8—141no—
469C2×D4—21no—
470C2×D4—21no—
471C2×D4—21no—
472C2×D4—21no—
473C2×D4—21no—
474C2×D4—21no—
475C2×D4—21no—
476C2×D4—21no—
477C2×D4—21no—
478C2×D4—21no—
479C2×D4—72no—
480C2×D4—148no—
481C2×D4—141no—
482C2×D4—141no—
483C2×D4—141no—
484C2×D4—141no—
485C2×D4—141no—
486C2×D4—284no—
487C2×D4—284no—
488C2×D4C2×C4×D712yes—
489C2×D4C2×C4×D712yes—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—44no—
493C22×C4—44no—
494C22×C4—42no—
495C22×C4—42no—
496C22×C4—72no—
497C22×C4—72no—
498C22×C4—141no—
499C22×C4—141no—
500C22×C4—141no—
501C22×C4—284no—
502C22×C4—282no—
503C22×C4D4×D711yescharacteristic
504C22×C4D4×D711yescharacteristic
505C22×C4C23×D711yescharacteristic
506SD16—148no—
507SD16—148no—
508D8—28no—
509D8—148no—
510C2×C8—21no—
511C2×C8—21no—
512C2×C8—72no—
513C2×C8—72no—
514C2×C8—72no—
515C2×C8—72no—
516C2×C8—141no—
517C2×C8—141no—
518C2×C8C2×C4×D712yes—
519C2×C8#115712yes—
520C2×C8C2×C22⋊D712yes—
521C2×C8C2×C22⋊D712yes—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—141no—
527C4⋊C4—141no—
528C4⋊C4—141no—
529C4⋊C4—141no—
530C4⋊C4—284no—
531C4⋊C4—282no—
532C22⋊C4—42no—
533C22⋊C4—42no—
534C22⋊C4—148no—
535C22⋊C4—284no—
536C22⋊C4—284no—
537C22⋊C4—284no—
538C22⋊C4—282no—
539C42—72no—
540C42—72no—
541C42—144no—
542C42—141no—
543C42—141no—
544C42—282no—
545C42—282no—
546C42D4×D712yes—
547C42C2×C22⋊D712yes—
548C2×C7—82no—
549C2×C7—82no—
550C2×C7C4×#12711yescharacteristic
551C2×C7C4×AGL(1,ℤ/8)12yes—
552C2×C7C2×#49611yescharacteristic
553C2×C7#315011yescharacteristic
554C2×C7#317112yes—
555D7—82no—
556C23—22no—
557C23—22no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—41no—
563C23—41no—
564C23—41no—
565C23—41no—
566C23—142no—
567C23—282no—
568C23—282no—
569C23—281no—
570C23—281no—
571C23C2×D4×D711yescharacteristic
572Q8—142no—
573Q8—142no—
574Q8—281no—
575Q8—281no—
576D4—22no—
577D4—22no—
578D4—22no—
579D4—22no—
580D4—41no—
581D4—41no—
582D4—41no—
583D4—41no—
584D4—142no—
585D4—142no—
586D4—2816no—
587D4—281no—
588D4—281no—
589C2×C4—22no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—74no—
595C2×C4—84no—
596C2×C4—84no—
597C2×C4—142no—
598C2×C4—142no—
599C2×C4—142no—
600C2×C4—142no—
601C2×C4—142no—
602C2×C4—142no—
603C2×C4—281no—
604C2×C4—281no—
605C2×C4—282no—
606C2×C4—282no—
607C2×C4—281no—
608C2×C4—281no—
609C2×C4—564no—
610C2×C4#647414yes—
611C2×C4#650312yes—
612C2×C4#650312yes—
613C2×C4C22×C4×D711yescharacteristic
614C2×C4C2×#115711yescharacteristic
615C2×C4C2×D4×D711yescharacteristic
616C2×C4D7×#4111yescharacteristic
617C2×C4D7×#4111yescharacteristic
618C2×C4C22×C22⋊D711yescharacteristic
619C8—22no—
620C8—74no—
621C8—142no—
622C8C4×C22⋊D714yescomplemented
623C7C4×#49611yescharacteristic, complemented, Sylow
624C22—42no—
625C22—42no—
626C22—44no—
627C22—44no—
628C22—81no—
629C22—81no—
630C22—81no—
631C22—81no—
632C22—82no—
633C22—82no—
634C22—282no—
635C22—284no—
636C22—561no—
637C22—561no—
638C22—562no—
639C22C4×D4×D711yescharacteristic
640C22#9023911yescharacteristic
641C22C2×#647412yes—
642C22#9045912yes—
643C22C2×#650311yescharacteristic
644C4—22no—
645C4—81no—
646C4—81no—
647C4—144no—
648C4—282no—
649C4—282no—
650C4—561no—
651C4—561no—
652C4#8992614yesdirect product factor
653C4C4×D4×D711yescharacteristic
654C4#9023911yescharacteristic
655C4C2×C4×C22⋊D711yescharacteristic
656C4#9052311yescharacteristic
657C2—82no—
658C2—82no—
659C2—562no—
660C2C4×#647412yes—
661C2#1173704412yes—
662C2C4×#650311yescharacteristic
663C2C2×#8992611yescharacteristic
664C2#1174134911yescharacteristic
subgroupG/H
centerC22×C4D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC2×C4×C7×D8C2
socleC23×C7C22×D4
prime radicalC2×#89926C2
cosocleC22×C4×C7C24
hypercenterC2×C4×D8D7
hypocenterC7C4×#496
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22×C4 ◁ C2×C42 ◁ C2×C4×D8

Complements

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

NH ≅ G/Ntype
1. C2×C4×C7×D8C2semidirect
3. C4×#6503C2semidirect
4. C4×#6499C2semidirect
5. C2×C4×#1142C2semidirect
6. C2×C4×#1141C2semidirect
7. C2×C4×#1132C2semidirect
8. #11729793C2semidirect
9. #11728383C2semidirect
10. #11728362C2semidirect
11. #11727946C2semidirect
12. #11727945C2semidirect
13. #11727914C2semidirect
14. #11727913C2semidirect
15. C4×#6389C2semidirect
37. #89926C4direct (N × H)
71. C2×C7×D8C2×C4semidirect
87. #6503C2×C4semidirect
89. #6499C2×C4semidirect
92. C2×#1142C2×C4semidirect
95. C2×#1141C2×C4semidirect
112. C2×#1132C2×C4semidirect
120. C4×C7×C8D4semidirect
131. #6389C2×C4semidirect
136. C4×C7⋊C8D4semidirect
141. C2×C4×D8D7semidirect
348. C7×C8C4×D4semidirect
350. C7⋊C8C4×D4semidirect
370. C2×D8C4×D7semidirect
424. C4×C8C22⋊D7semidirect
450. C4×C7#496semidirect
622. C8C4×C22⋊D7semidirect
623. C7C4×#496semidirect
652. C4#89926direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C4×C7×D82
C2×#899262
C4×#65032
C4×#64992
C2×C4×#11422
C2×C4×#11412
C2×C4×#11322
#117297932
#117283832
#117283622
#117279462
#117279452
#117279142
#117279132
C4×#63892
C4×#4967