Finite Group Databasebeta

← #13362292

Subgroups — #13362292

Order 1792.

1126 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C7×SD16C211yesmaximal, characteristic, complemented
2C2×#89947C211yesmaximal, characteristic
3#11739901C211yesmaximal, characteristic, complemented
4C2×#89798C211yesmaximal, characteristic, complemented
5#11739774C211yesmaximal, characteristic
6C2×#89790C211yesmaximal, characteristic, complemented
7C2×C4×#1132C211yesmaximal, characteristic, complemented
8#11729803C211yesmaximal, characteristic, complemented
9#11728328C211yesmaximal, characteristic, complemented
10#11728309C211yesmaximal, characteristic, complemented
11#11728014C211yesmaximal, characteristic, complemented
12#11727982C211yesmaximal, characteristic, complemented
13#11727946C211yesmaximal, characteristic, complemented
14#11727914C211yesmaximal, characteristic
15#11727503C211yesmaximal, characteristic, complemented
16C22×C7×SD16C2211yescharacteristic
17C2×C4×C7×Q8C2211yescharacteristic
18C2×C4×C7×D4C2211yescharacteristic
19C2×#6507C2211yescharacteristic
20C22×#1143C2211yescharacteristic
21C2×#6503C2211yescharacteristic
22C22×#1142C2211yescharacteristic
23C22×#1132C2211yescharacteristic
24C2×C4×D28C2211yescharacteristic
25C2×C4×Dic14C2211yescharacteristic
26C4×C7×SD16—24no—
27C2×C7×#106C2211yescharacteristic
28C2×C7×#103C2211yescharacteristic
29C2×C7×#102C2211yescharacteristic
30C2×C4×C7×C8C2211yescharacteristic
31#90002C2211yescharacteristic
32C2×#6409C2211yescharacteristic
33#89993C2211yescharacteristic
34C2×#6406C2211yescharacteristic
35#89985—22no—
36#89947C418yescomplemented
37C2×#6395C2211yescharacteristic
38C2×#6391C2211yescharacteristic
39C2×#6390C2211yescharacteristic
40C2×#6389C2211yescharacteristic
41C2×#6389C2211yescharacteristic
42C2×C8×Dic7C2211yescharacteristic
43C2×C8×Dic7C2211yescharacteristic
44#89798—24no—
45#89790—24no—
46#89760C2211yescharacteristic
47#89750—22no—
48#89749C2211yescharacteristic
49#89749C2211yescharacteristic
50#89746C2211yescharacteristic
51C2×#6384C2211yescharacteristic
52C2×#6383C2211yescharacteristic
53C2×#6381C2211yescharacteristic
54C2×#6381C2211yescharacteristic
55#89728—22no—
56#89705—22no—
57#89704C2211yescharacteristic
58C2×#6377C2211yescharacteristic
59C4×#1132—24no—
60#21695—71nomaximal, Sylow
61C22×C7×Q8C2311yescharacteristic
62C22×C7×D4C2311yescharacteristic
63C23×C4×C7—21no—
64C22×C22⋊D7—22no—
65C22×D28C2311yescharacteristic
66C22×C4×D7—21no—
67C22×Dic14C2311yescharacteristic
68C2×C7×SD16—22no—
69C2×C7×SD16—22no—
70C2×C7×SD16C2×C414yescomplemented
71C22×C7×C8C2311yescharacteristic
72C22×C7×C8C2311yescharacteristic
73C4×C7×Q8—22no—
74C4×C7×Q8—22no—
75C4×C7×D4—22no—
76C4×C7×D4—22no—
77C2×C7×C4⋊C4—21no—
78C2×C7×C4⋊C4C2311yescharacteristic
79C2×C7×C4⋊C4C2311yescharacteristic
80C2×C7×C22⋊C4—21no—
81C2×C42×C7—21no—
82C2×C42×C7C2311yescharacteristic
83C2×C22⋊Dic7—22no—
84#6507—22no—
85#6507C2×C414yescomplemented
86C2×#1143—22no—
87C2×#1143—22no—
88C2×#1143C2×C414yescomplemented
89#6503—22no—
90#6503C2×C414yes—
91C2×#1142—22no—
92C2×#1142—22no—
93C2×#1142C2×C414yescomplemented
94C2×(C2×D7)⋊C4—21no—
95C2×(C2×D7)⋊C4—22no—
96C2×C4⋊Dic7C2311yescharacteristic
97C2×C4⋊Dic7C2311yescharacteristic
98C2×Dic7⋊C4—22no—
99C2×Dic7⋊C4—21no—
100C2×C4×Dic7—21no—
101C2×C4×Dic7C2311yescharacteristic
102C2×C4×Dic7C2311yescharacteristic
103C22×C7⋊C8C2311yescharacteristic
104C22×C7⋊C8C2311yescharacteristic
105C2×#1132—22no—
106C2×#1132—22no—
107C2×#1132C2×C414yescomplemented
108C4×D28—22no—
109C4×D28—22no—
110C4×Dic14—22no—
111C4×Dic14—22no—
112C7×#106—22no—
113C7×#103—22no—
114C7×#102—22no—
115C4×C7×C8D414yescomplemented
116#6409—22no—
117#6406—22no—
118C7⋊#95—22no—
119C7⋊#95—22no—
120C7⋊#95—22no—
121C7⋊#95—22no—
122C7⋊#95D412yes—
123C7⋊#95D412yes—
124#6395—22no—
125#6391—22no—
126#6390—22no—
127#6389—22no—
128#6389—22no—
129C8×Dic7—22no—
130C8×Dic7C2×C414yescomplemented
131#6384—22no—
132#6383—22no—
133#6381—22no—
134#6381—22no—
135#6377D414yescomplemented
136C2×#492—71no—
137C2×#440—71no—
138C2×#440—71no—
139C2×C4×SD16—71no—
140C2×C4×SD16D711yescharacteristic, complemented
141#2328—71no—
142#2327—71no—
143#1973—71no—
144#1973—71no—
145#1968—71no—
146#1968—71no—
147#1961—71no—
148#1960—71no—
149#1846—71no—
150#1726—71no—
151C24×C7—21no—
152C23×D7—21no—
153C2×C7×Q8—21no—
154C2×C7×Q8—21no—
155C2×C7×Q8—21no—
156C2×C7×Q8—21no—
157C2×C7×Q8—21no—
158C2×C7×Q8C22×C412yes—
159C2×C7×D4—21no—
160C2×C7×D4—21no—
161C2×C7×D4—21no—
162C2×C7×D4—21no—
163C2×C7×D4—21no—
164C2×C7×D4C22×C412yes—
165C22×C4×C7—21no—
166C22×C4×C7—21no—
167C22×C4×C7—21no—
168C22×C4×C7—42no—
169C22×C4×C7—82no—
170C22×C4×C7C2×D411yescharacteristic
171C22×C4×C7C2×D411yescharacteristic
172C22×C4×C7C2411yescharacteristic
173C2×C22⋊D7—28no—
174C2×C22⋊D7—82no—
175C2×C22⋊D7—82no—
176C22×Dic7—21no—
177C22×Dic7—21no—
178C22×Dic7—21no—
179C22×Dic7C2×D412yes—
180C22×Dic7C2×D412yes—
181C2×D28—21no—
182C2×D28—21no—
183C2×D28—21no—
184C2×D28—21no—
185C2×D28—21no—
186C2×D28C22×C412yes—
187C2×C4×D7—42no—
188C2×C4×D7—82no—
189C2×Dic14—21no—
190C2×Dic14—21no—
191C2×Dic14—21no—
192C2×Dic14—21no—
193C2×Dic14—21no—
194C2×Dic14C22×C412yes—
195C7×SD16—28no—
196C2×C7×C8—21no—
197C2×C7×C8—21no—
198C2×C7×C8C22×C412yes—
199C2×C7×C8C2×D412yes—
200C2×C7×C8C2×D412yes—
201C2×C7×C8#4112yes—
202C7×C4⋊C4—21no—
203C7×C4⋊C4—21no—
204C7×C4⋊C4—21no—
205C7×C4⋊C4—21no—
206C7×C4⋊C4—42no—
207C7×C22⋊C4—42no—
208C42×C7—42no—
209C42×C7C2×D412yes—
210C42×C7C2×D412yes—
211C22⋊Dic7—82no—
212#1143—28no—
213#1142—28no—
214(C2×D7)⋊C4—42no—
215(C2×D7)⋊C4—82no—
216C4⋊Dic7—21no—
217C4⋊Dic7—21no—
218C4⋊Dic7—21no—
219C4⋊Dic7—21no—
220Dic7⋊C4—28no—
221Dic7⋊C4—42no—
222C4×Dic7—21no—
223C4×Dic7—21no—
224C4×Dic7—42no—
225C4×Dic7C22×C412yes—
226C4×Dic7#4112yes—
227C2×C7⋊C8—21no—
228C2×C7⋊C8—21no—
229C2×C7⋊C8C22×C412yes—
230C2×C7⋊C8C2×D412yes—
231C2×C7⋊C8C2×D412yes—
232C2×C7⋊C8#4112yes—
233#1132—28no—
234C22×SD16—71no—
235C22×SD16—71no—
236C22×SD16—71no—
237C22×SD16C2×D711yescharacteristic
238C2×C4⋊Q8—71no—
239C2×#127—71no—
240C2×C4×Q8—71no—
241C2×C4×Q8C2×D711yescharacteristic
242C2×C4×D4—71no—
243C2×C4×D4C2×D711yescharacteristic
244#492—78no—
245#440—144no—
246#440—144no—
247C4×SD16—24no—
248C4×SD16—144no—
249C2×#107—71no—
250C2×#107—71no—
251C2×#106—71no—
252C2×#106C2×D711yescharacteristic
253C2×#103—71no—
254C2×#103—71no—
255C2×#103—71no—
256C2×#103C2×D711yescharacteristic
257C2×#102—71no—
258C2×#102—71no—
259C2×#102—71no—
260C2×#102C2×D711yescharacteristic
261C2×#97—71no—
262C2×#97—71no—
263C2×#97—71no—
264C2×C4×C8—71no—
265C2×C4×C8—71no—
266C2×C4×C8C2×D711yescharacteristic
267#391—71no—
268#391—71no—
269#390—71no—
270#390—71no—
271#387—142no—
272#387—142no—
273#386—142no—
274#386—142no—
275#382—71no—
276#382—71no—
277#376—71no—
278C23×C7—22no—
279C23×C7—41no—
280C23×C7—41no—
281C23×C7—42no—
282C23×C7—42no—
283C23×C7C22×D411yescharacteristic
284C22×D7—22no—
285C22×D7—41no—
286C22×D7—41no—
287C22×D7—42no—
288C22×D7—42no—
289C7×Q8—22no—
290C7×Q8—22no—
291C7×Q8—41no—
292C7×Q8—41no—
293C7×D4—22no—
294C7×D4—22no—
295C7×D4—41no—
296C7×D4—41no—
297C2×C4×C7—22no—
298C2×C4×C7—22no—
299C2×C4×C7—22no—
300C2×C4×C7—41no—
301C2×C4×C7—41no—
302C2×C4×C7—41no—
303C2×C4×C7—41no—
304C2×C4×C7—42no—
305C2×C4×C7—42no—
306C2×C4×C7—84no—
307C2×C4×C7#12712yes—
308C2×C4×C7#12712yes—
309C2×C4×C7C23×C411yescharacteristic
310C2×C4×C7C22×D411yescharacteristic
311C2×C4×C7C22×D411yescharacteristic
312C2×C4×C7C2×#4111yescharacteristic
313C2×C4×C7C2×#4111yescharacteristic
314C2×C4×C7C2×#4111yescharacteristic
315C22⋊D7—88no—
316C2×Dic7—22no—
317C2×Dic7—22no—
318C2×Dic7—22no—
319C2×Dic7—22no—
320C2×Dic7—22no—
321C2×Dic7—22no—
322C2×Dic7—41no—
323C2×Dic7—41no—
324C2×Dic7—41no—
325C2×Dic7—41no—
326C2×Dic7—42no—
327C2×Dic7—42no—
328C2×Dic7C4×D414yes—
329D28—22no—
330D28—22no—
331D28—41no—
332D28—41no—
333C4×D7—84no—
334Dic14—22no—
335Dic14—22no—
336Dic14—41no—
337Dic14—41no—
338C7×C8—22no—
339C7×C8C4×D414yes—
340C7⋊C8—22no—
341C7⋊C8C4×D414yes—
342C22×Q8—71no—
343C22×Q8C22×D711yescharacteristic
344C22×D4—71no—
345C22×D4—142no—
346C22×D4C22×D711yescharacteristic
347C23×C4—21no—
348C23×C4—141no—
349C2×SD16—22no—
350C2×SD16—22no—
351C2×SD16—74no—
352C2×SD16—74no—
353C2×SD16—74no—
354C2×SD16—142no—
355C2×SD16—142no—
356C2×SD16—142no—
357C2×SD16—142no—
358C2×SD16—142no—
359C2×SD16—142no—
360C2×SD16C4×D714yescomplemented
361C22×C8—71no—
362C22×C8—71no—
363C22×C8C22×D711yescharacteristic
364C22×C8C22×D711yescharacteristic
365C4⋊Q8—74no—
366C4⋊Q8—142no—
367#127—74no—
368#127—142no—
369C4×Q8—22no—
370C4×Q8—22no—
371C4×Q8—142no—
372C4×Q8—142no—
373C4×D4—22no—
374C4×D4—22no—
375C4×D4—142no—
376C4×D4—142no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4—71no—
379C2×C4⋊C4—71no—
380C2×C4⋊C4—142no—
381C2×C4⋊C4—141no—
382C2×C4⋊C4C22×D711yescharacteristic
383C2×C4⋊C4C22×D711yescharacteristic
384C2×C22⋊C4—21no—
385C2×C22⋊C4—141no—
386C2×C22⋊C4—142no—
387C2×C22⋊C4—142no—
388C2×C42—21no—
389C2×C42—71no—
390C2×C42—71no—
391C2×C42—141no—
392C2×C42C22×D711yescharacteristic
393#107—142no—
394#107—142no—
395#106—22no—
396#106—142no—
397#103—22no—
398#103—142no—
399#103—142no—
400#103—142no—
401#102—22no—
402#102—142no—
403#102—142no—
404#102—142no—
405#97—74no—
406#97—142no—
407#97—142no—
408C4×C8—74no—
409C4×C8—142no—
410C4×C8C22⋊D714yescomplemented
411#95—72no—
412#95—72no—
413#95—142no—
414#95—142no—
415#95—142no—
416#95—142no—
417C22×C7—42no—
418C22×C7—44no—
419C22×C7—81no—
420C22×C7—81no—
421C22×C7—82no—
422C22×C7C2×C4×D411yescharacteristic
423C22×C7C2×#12711yescharacteristic
424C22×C7#53211yescharacteristic
425C22×C7#57712yes—
426C22×C7#57712yes—
427C2×D7—42no—
428C2×D7—44no—
429C2×D7—81no—
430C2×D7—81no—
431C2×D7—82no—
432C4×C7—22no—
433C4×C7—22no—
434C4×C7—42no—
435C4×C7—81no—
436C4×C7—81no—
437C4×C7C2×C4×D411yescharacteristic
438C4×C7C2×C4×D411yescharacteristic
439C4×C7#53211yescharacteristic
440C4×C7#53211yescharacteristic
441Dic7—24no—
442Dic7—42no—
443Dic7—42no—
444Dic7—81no—
445Dic7—81no—
446C24—21no—
447C24—141no—
448C2×Q8—21no—
449C2×Q8—21no—
450C2×Q8—21no—
451C2×Q8—21no—
452C2×Q8—21no—
453C2×Q8—72no—
454C2×Q8—141no—
455C2×Q8—141no—
456C2×Q8—141no—
457C2×Q8—141no—
458C2×Q8—141no—
459C2×Q8C2×C4×D712yes—
460C2×D4—21no—
461C2×D4—21no—
462C2×D4—21no—
463C2×D4—21no—
464C2×D4—21no—
465C2×D4—72no—
466C2×D4—148no—
467C2×D4—141no—
468C2×D4—141no—
469C2×D4—141no—
470C2×D4—141no—
471C2×D4—141no—
472C2×D4—562no—
473C2×D4—562no—
474C2×D4C2×C4×D712yes—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—42no—
479C22×C4—72no—
480C22×C4—72no—
481C22×C4—82no—
482C22×C4—141no—
483C22×C4—141no—
484C22×C4—141no—
485C22×C4—282no—
486C22×C4—562no—
487C22×C4D4×D711yescharacteristic
488C22×C4D4×D711yescharacteristic
489C22×C4C23×D711yescharacteristic
490SD16—28no—
491SD16—148no—
492SD16—148no—
493SD16—148no—
494C2×C8—21no—
495C2×C8—21no—
496C2×C8—72no—
497C2×C8—72no—
498C2×C8—72no—
499C2×C8—72no—
500C2×C8—141no—
501C2×C8—141no—
502C2×C8C2×C4×D712yes—
503C2×C8#115712yes—
504C2×C8C2×C22⋊D712yes—
505C2×C8C2×C22⋊D712yes—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—42no—
511C4⋊C4—148no—
512C4⋊C4—141no—
513C4⋊C4—141no—
514C4⋊C4—141no—
515C4⋊C4—141no—
516C4⋊C4—282no—
517C22⋊C4—42no—
518C22⋊C4—282no—
519C22⋊C4—562no—
520C22⋊C4—562no—
521C42—42no—
522C42—72no—
523C42—72no—
524C42—141no—
525C42—141no—
526C42—282no—
527C42D4×D712yes—
528C42C2×C22⋊D712yes—
529C2×C7—82no—
530C2×C7C4×#12711yescharacteristic
531C2×C7#294912yes—
532C2×C7#315411yescharacteristic
533C2×C7#315411yescharacteristic
534C2×C7#317412yes—
535D7—82no—
536C23—22no—
537C23—42no—
538C23—42no—
539C23—41no—
540C23—41no—
541C23—142no—
542C23—282no—
543C23—282no—
544C23—281no—
545C23—281no—
546C23C2×D4×D711yescharacteristic
547Q8—22no—
548Q8—22no—
549Q8—41no—
550Q8—41no—
551Q8—142no—
552Q8—142no—
553Q8—281no—
554Q8—281no—
555D4—22no—
556D4—22no—
557D4—41no—
558D4—41no—
559D4—142no—
560D4—142no—
561D4—281no—
562D4—281no—
563D4—568no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—74no—
574C2×C4—84no—
575C2×C4—142no—
576C2×C4—142no—
577C2×C4—142no—
578C2×C4—142no—
579C2×C4—142no—
580C2×C4—142no—
581C2×C4—281no—
582C2×C4—281no—
583C2×C4—282no—
584C2×C4—282no—
585C2×C4—281no—
586C2×C4—281no—
587C2×C4—564no—
588C2×C4#650312yes—
589C2×C4#650312yes—
590C2×C4C22×C4×D711yescharacteristic
591C2×C4C2×#115711yescharacteristic
592C2×C4C2×D4×D711yescharacteristic
593C2×C4D7×#4111yescharacteristic
594C2×C4D7×#4111yescharacteristic
595C2×C4C22×C22⋊D711yescharacteristic
596C8—22no—
597C8—74no—
598C8—142no—
599C8C4×C22⋊D714yes—
600C7#2169511yescharacteristic, complemented, Sylow
601C22—44no—
602C22—42no—
603C22—82no—
604C22—81no—
605C22—81no—
606C22—284no—
607C22—282no—
608C22—562no—
609C22—561no—
610C22—561no—
611C22C4×D4×D711yescharacteristic
612C22#9023911yescharacteristic
613C22#9045312yes—
614C22#9046112yes—
615C22C2×#650311yescharacteristic
616C4—22no—
617C4—22no—
618C4—42no—
619C4—81no—
620C4—81no—
621C4—144no—
622C4—282no—
623C4—282no—
624C4—561no—
625C4—561no—
626C4C4×D4×D711yescharacteristic
627C4#9023911yescharacteristic
628C4C2×C4×C22⋊D711yescharacteristic
629C4#9052311yescharacteristic
630C2—82no—
631C2—562no—
632C2#1173709412yes—
633C2#1173711212yes—
634C2C4×#650311yescharacteristic
635C2#1174129911yescharacteristic
636C2#1174135011yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC2×C4×C7×SD16C2
socleC23×C7C22×D4
prime radicalC2×#6503C22
cosocleC22×C4×C7C24
hypercenterC2×C4×SD16D7
hypocenterC7#21695
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×SD16

Complements

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

NH ≅ G/Ntype
1. C2×C4×C7×SD16C2semidirect
3. #11739901C2semidirect
4. C2×#89798C2semidirect
6. C2×#89790C2semidirect
7. C2×C4×#1132C2semidirect
8. #11729803C2semidirect
9. #11728328C2semidirect
10. #11728309C2semidirect
11. #11728014C2semidirect
12. #11727982C2semidirect
13. #11727946C2semidirect
15. #11727503C2semidirect
36. #89947C4semidirect
70. C2×C7×SD16C2×C4semidirect
85. #6507C2×C4semidirect
88. C2×#1143C2×C4semidirect
93. C2×#1142C2×C4semidirect
107. C2×#1132C2×C4semidirect
115. C4×C7×C8D4semidirect
130. C8×Dic7C2×C4semidirect
135. #6377D4semidirect
140. C2×C4×SD16D7semidirect
360. C2×SD16C4×D7semidirect
410. C4×C8C22⋊D7semidirect
600. C7#21695semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C7×SD162
C2×#899472
#117399012
C2×#897982
#117397742
C2×#897902
C2×C4×#11322
#117298032
#117283282
#117283092
#117280142
#117279822
#117279462
#117279142
#117275032
#216957