Finite Group Databasebeta

← #13411042

Subgroups — #13411042

Order 1792.

990 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C7×#103C211yesmaximal, characteristic
2Q8×C22⋊Dic7C211yesmaximal, characteristic
3#11739616C211yesmaximal, characteristic
4C2×#89566C211yesmaximal, characteristic, complemented
5#11729886C211yesmaximal, characteristic
6#11729728C211yesmaximal, characteristic
7#11729725C211yesmaximal, characteristic
8#11729309C211yesmaximal, characteristic
9#11728016C211yesmaximal, characteristic, complemented
10#11728003C211yesmaximal, characteristic, complemented
11#11728002C211yesmaximal, characteristic, complemented
12#11727990C211yesmaximal, characteristic, complemented
13#11727989C211yesmaximal, characteristic, complemented
14#11727985C211yesmaximal, characteristic, complemented
15#11727975C211yesmaximal, characteristic, complemented
16C23×C7×Q8C2211yescharacteristic
17C23×C7×C8C2211yescharacteristic
18C22×C7×C4⋊C4C2211yescharacteristic
19C2×Q8×Dic7C2211yescharacteristic
20C2×Q8×Dic7C2211yescharacteristic
21C2×#6487—21no—
22C2×#6451C2211yescharacteristic
23C2×C7×#103—24no—
24C2×C7×#103C2211yescharacteristic
25C2×C7×#103C2211yescharacteristic
26C2×C7×#103C2211yescharacteristic
27C2×C7×#103C2211yescharacteristic
28#90002—21no—
29#90002—21no—
30#90002C2211yescharacteristic
31#90002C2211yescharacteristic
32C2×#6409C2211yescharacteristic
33C2×#6409C2211yescharacteristic
34C2×#6409C2211yescharacteristic
35C2×#6409C2211yescharacteristic
36#89984—21no—
37#89984C2211yescharacteristic
38#89984C2211yescharacteristic
39#89982—21no—
40C4×C22⋊Dic7—21no—
41C4×C22⋊Dic7C2211yescharacteristic
42C2×C22⋊(C7⋊C8)C2211yescharacteristic
43C2×#6392C2211yescharacteristic
44C2×#6392C2211yescharacteristic
45C2×Dic7⋊C8C2211yescharacteristic
46C2×Dic7⋊C8C2211yescharacteristic
47#89757C2211yescharacteristic
48#89756—21no—
49#89751—21no—
50#89748C2211yescharacteristic
51#89745C2211yescharacteristic
52C2×#6381C2211yescharacteristic
53C2×#6381C2211yescharacteristic
54#89566—24no—
55#89346C2211yescharacteristic
56#89346C2211yescharacteristic
57#89345C2211yescharacteristic
58#89345C2211yescharacteristic
59#89324C2211yescharacteristic
60#89324C2211yescharacteristic
61#25963—71nomaximal, Sylow
62C22×C7×Q8—21no—
63C22×C7×Q8—22no—
64C22×C7×Q8—22no—
65C22×C7×Q8—22no—
66C22×C7×Q8—41no—
67C22×C7×Q8—41no—
68C22×C7×Q8Q814yescomplemented
69C22×C7×Q8C2311yescharacteristic
70C22×C7×Q8C2311yescharacteristic
71C23×C4×C7—21no—
72C23×C4×C7—21no—
73C23×C4×C7C2311yescharacteristic
74C22×Dic14—21no—
75C22×C7×C8—22no—
76C22×C7×C8—22no—
77C22×C7×C8—41no—
78C22×C7×C8C2311yescharacteristic
79C22×C7×C8C2311yescharacteristic
80C2×C7×C4⋊C4—22no—
81C2×C7×C4⋊C4—21no—
82C2×C7×C4⋊C4—41no—
83C2×C7×C4⋊C4C2311yescharacteristic
84C2×C7×C4⋊C4C2311yescharacteristic
85C2×C22⋊Dic7—21no—
86C2×C22⋊Dic7—21no—
87C2×C22⋊Dic7D411yescharacteristic
88C2×C22⋊Dic7D411yescharacteristic
89Q8×Dic7—22no—
90Q8×Dic7—22no—
91Q8×Dic7—41no—
92Q8×Dic7—41no—
93C2×C4⋊Dic7—21no—
94C2×C4⋊Dic7—21no—
95C2×C4⋊Dic7—21no—
96C2×C4⋊Dic7C2311yescharacteristic
97C2×C4⋊Dic7C2311yescharacteristic
98C2×C4⋊Dic7C2311yescharacteristic
99C2×C4⋊Dic7C2311yescharacteristic
100#6487—44no—
101C2×Dic7⋊C4—21no—
102C2×Dic7⋊C4—21no—
103C2×Dic7⋊C4—21no—
104C2×Dic7⋊C4—21no—
105C2×C4×Dic7—21no—
106C2×C4×Dic7—21no—
107C2×C4×Dic7C2311yescharacteristic
108C2×C4×Dic7C2311yescharacteristic
109C22×C7⋊C8C2311yescharacteristic
110C22×C7⋊C8C2311yescharacteristic
111#6451—22no—
112#6451—41no—
113C7×#103—22no—
114C7×#103—22no—
115C7×#103—22no—
116C7×#103D414yes—
117#6409—22no—
118#6409—22no—
119#6409—41no—
120#6409—41no—
121C7⋊#95—21no—
122C7⋊#95—21no—
123C7⋊#95—21no—
124C7⋊#95—21no—
125C7⋊#95—21no—
126C7⋊#95—21no—
127C7⋊#95—21no—
128C7⋊#95—21no—
129C7⋊#95D411yescharacteristic
130C7⋊#95D411yescharacteristic
131C22⋊(C7⋊C8)—22no—
132#6392—22no—
133#6392—41no—
134Dic7⋊C8—22no—
135Dic7⋊C8—41no—
136#6381—22no—
137#6381—41no—
138C2×#477—71no—
139C22×#103D711yescharacteristic
140#2440—71no—
141Q8×C22⋊C4—71no—
142#2095—71no—
143#2068—71no—
144#2068—71no—
145#2024—71no—
146#1863—71no—
147#1863—71no—
148#1852—71no—
149#1850—71no—
150#1808—71no—
151#1803—71no—
152#1787—71no—
153C24×C7C2×D411yescharacteristic
154C2×C7×Q8—24no—
155C2×C7×Q8—21no—
156C2×C7×Q8—21no—
157C2×C7×Q8—21no—
158C2×C7×Q8—21no—
159C2×C7×Q8—28no—
160C2×C7×Q8—21no—
161C2×C7×Q8—21no—
162C2×C7×Q8—21no—
163C2×C7×Q8—21no—
164C2×C7×Q8—42no—
165C2×C7×Q8—42no—
166C2×C7×Q8—44no—
167C2×C7×Q8—44no—
168C2×C7×Q8—44no—
169C2×C7×Q8C2×D412yes—
170C2×C7×Q8C2×Q812yes—
171C2×C7×Q8C2×Q812yes—
172C2×C7×Q8#4112yes—
173C22×C4×C7—21no—
174C22×C4×C7—21no—
175C22×C4×C7—21no—
176C22×C4×C7—21no—
177C22×C4×C7—21no—
178C22×C4×C7—24no—
179C22×C4×C7—21no—
180C22×C4×C7—21no—
181C22×C4×C7—21no—
182C22×C4×C7—42no—
183C22×C4×C7—42no—
184C22×C4×C7—42no—
185C22×C4×C7—42no—
186C22×C4×C7—81no—
187C22×C4×C7C2×D411yescharacteristic
188C22×C4×C7C2×Q812yes—
189C22×C4×C7#4112yes—
190C22×C4×C7C2411yescharacteristic
191C22×Dic7—21no—
192C22×Dic7—21no—
193C22×Dic7—21no—
194C22×Dic7—21no—
195C22×Dic7C2×D411yescharacteristic
196C22×Dic7C2×D411yescharacteristic
197C22×Dic7C2×D411yescharacteristic
198C22×Dic7C2×D411yescharacteristic
199C2×Dic14—42no—
200C2×Dic14—42no—
201C2×Dic14—81no—
202C2×C7×C8—21no—
203C2×C7×C8—21no—
204C2×C7×C8—21no—
205C2×C7×C8—21no—
206C2×C7×C8—21no—
207C2×C7×C8—44no—
208C2×C7×C8C2×D412yes—
209C7×C4⋊C4—21no—
210C7×C4⋊C4—21no—
211C7×C4⋊C4—21no—
212C7×C4⋊C4—42no—
213C7×C4⋊C4C2×D412yes—
214C22⋊Dic7—22no—
215C22⋊Dic7—42no—
216C22⋊Dic7—42no—
217C22⋊Dic7Q1614yes—
218C4⋊Dic7—21no—
219C4⋊Dic7—21no—
220C4⋊Dic7—21no—
221C4⋊Dic7—21no—
222C4⋊Dic7—21no—
223C4⋊Dic7—21no—
224C4⋊Dic7—21no—
225C4⋊Dic7—21no—
226C4⋊Dic7—42no—
227C4⋊Dic7—42no—
228C4⋊Dic7—42no—
229Dic7⋊C4—42no—
230Dic7⋊C4—42no—
231Dic7⋊C4—42no—
232Dic7⋊C4—81no—
233C4×Dic7—21no—
234C4×Dic7—21no—
235C4×Dic7—21no—
236C4×Dic7—21no—
237C4×Dic7—42no—
238C4×Dic7—42no—
239C2×C7⋊C8—21no—
240C2×C7⋊C8—21no—
241C2×C7⋊C8—21no—
242C2×C7⋊C8—21no—
243C2×C7⋊C8—21no—
244C2×C7⋊C8—21no—
245C23×Q8C2×D711yescharacteristic
246C23×C8C2×D711yescharacteristic
247C2×C4⋊Q8—71no—
248C2×C22⋊Q8—141no—
249C2×C4×Q8—71no—
250C2×C4×Q8—71no—
251C22×C4⋊C4C2×D711yescharacteristic
252#477—144no—
253C2×#107—71no—
254C2×#107—71no—
255C2×#107—71no—
256C2×#107—71no—
257C2×C4⋊C8—71no—
258C2×C4⋊C8—71no—
259C2×#103—24no—
260C2×#103—71no—
261C2×#103—71no—
262C2×#103—71no—
263C2×#103—71no—
264C2×#103C2×D711yescharacteristic
265C2×#103C2×D711yescharacteristic
266C2×#103C2×D711yescharacteristic
267C2×#103C2×D711yescharacteristic
268C2×C22⋊C8—71no—
269#398—141no—
270#395—141no—
271#391—71no—
272#391—71no—
273#391—141no—
274#391—141no—
275#389—71no—
276#389—71no—
277#385—141no—
278#384—71no—
279#380—71no—
280#380—71no—
281#380—141no—
282C4×C22⋊C4—71no—
283C4×C22⋊C4—141no—
284#340—71no—
285#340—71no—
286#340—71no—
287#340—71no—
288#336—71no—
289#336—71no—
290C23×C7—21no—
291C23×C7—21no—
292C23×C7—21no—
293C23×C7—21no—
294C23×C7C22⋊Q812yes—
295C23×C7C2×Q1612yes—
296C23×C7AGL(1,ℤ/8)12yes—
297C23×C7C22×D411yescharacteristic
298C7×Q8—22no—
299C7×Q8—22no—
300C7×Q8—22no—
301C7×Q8—22no—
302C7×Q8—22no—
303C7×Q8—48no—
304C7×Q8—41no—
305C7×Q8—41no—
306C7×Q8C22⋊Q814yescomplemented
307C2×C4×C7—22no—
308C2×C4×C7—22no—
309C2×C4×C7—21no—
310C2×C4×C7—22no—
311C2×C4×C7—21no—
312C2×C4×C7—22no—
313C2×C4×C7—22no—
314C2×C4×C7—22no—
315C2×C4×C7—22no—
316C2×C4×C7—22no—
317C2×C4×C7—22no—
318C2×C4×C7—48no—
319C2×C4×C7—41no—
320C2×C4×C7—41no—
321C2×C4×C7—41no—
322C2×C4×C7—41no—
323C2×C4×C7—41no—
324C2×C4×C7—41no—
325C2×C4×C7—41no—
326C2×C4×C7—84no—
327C2×C4×C7C22⋊Q812yes—
328C2×C4×C7C22×D411yescharacteristic
329C2×C4×C7C22×Q811yescharacteristic
330C2×C4×C7C2×#4111yescharacteristic
331C2×C4×C7C2×#4111yescharacteristic
332C2×C4×C7H(5,𝔽2)11yescharacteristic
333C2×C4×C7H(5,𝔽2)11yescharacteristic
334C2×Dic7—21no—
335C2×Dic7—21no—
336C2×Dic7—21no—
337C2×Dic7—21no—
338C2×Dic7—21no—
339C2×Dic7—21no—
340C2×Dic7—21no—
341C2×Dic7—21no—
342C2×Dic7—21no—
343C2×Dic7—21no—
344C2×Dic7—41no—
345C2×Dic7—41no—
346C2×Dic7—41no—
347C2×Dic7—41no—
348C2×Dic7—41no—
349C2×Dic7—41no—
350C2×Dic7—41no—
351C2×Dic7—41no—
352C2×Dic7—41no—
353C2×Dic7—41no—
354C2×Dic7—41no—
355C2×Dic7—41no—
356C2×Dic7C2×Q1612yes—
357C2×Dic7C2×Q1612yes—
358Dic14—84no—
359C7×C8—22no—
360C7×C8—41no—
361C7⋊C8—41no—
362C7⋊C8—41no—
363C22×Q8—21no—
364C22×Q8—22no—
365C22×Q8—22no—
366C22×Q8—22no—
367C22×Q8—41no—
368C22×Q8—41no—
369C22×Q8—141no—
370C22×Q8Dic1414yescomplemented
371C22×Q8C22×D711yescharacteristic
372C22×Q8C22×D711yescharacteristic
373C23×C4—21no—
374C23×C4—21no—
375C23×C4C22×D711yescharacteristic
376C22×C8—22no—
377C22×C8—22no—
378C22×C8—41no—
379C22×C8—71no—
380C22×C8—71no—
381C22×C8C22×D711yescharacteristic
382C22×C8C22×D711yescharacteristic
383C4⋊Q8—142no—
384C4⋊Q8—281no—
385C22⋊Q8—284no—
386C4×Q8—142no—
387C4×Q8—142no—
388C4×Q8—281no—
389C4×Q8—281no—
390C2×C4⋊C4—22no—
391C2×C4⋊C4—21no—
392C2×C4⋊C4—41no—
393C2×C4⋊C4—71no—
394C2×C4⋊C4—71no—
395C2×C4⋊C4—71no—
396C2×C4⋊C4—71no—
397C2×C4⋊C4—141no—
398C2×C4⋊C4—141no—
399C2×C4⋊C4—141no—
400C2×C4⋊C4—141no—
401C2×C4⋊C4—141no—
402C2×C4⋊C4—141no—
403C2×C4⋊C4—141no—
404C2×C4⋊C4C22×D711yescharacteristic
405C2×C4⋊C4C22×D711yescharacteristic
406C2×C22⋊C4—71no—
407C2×C22⋊C4—71no—
408C2×C22⋊C4—141no—
409C2×C22⋊C4—141no—
410C2×C42—71no—
411C2×C42—71no—
412C2×C42—141no—
413C2×C42—141no—
414#107—142no—
415#107—142no—
416#107—281no—
417#107—281no—
418C4⋊C8—142no—
419C4⋊C8—281no—
420#103—22no—
421#103—22no—
422#103—22no—
423#103—142no—
424#103—142no—
425#103—281no—
426#103—281no—
427#103C22⋊D714yes—
428C22⋊C8—142no—
429#95—71no—
430#95—71no—
431#95—141no—
432#95—141no—
433#95—141no—
434#95—141no—
435#95—141no—
436#95—141no—
437#95—141no—
438#95—141no—
439C22×C7—22no—
440C22×C7—22no—
441C22×C7—22no—
442C22×C7—22no—
443C22×C7—22no—
444C22×C7—22no—
445C22×C7#47714yescomplemented
446C22×C7C2×C22⋊Q811yescharacteristic
447C22×C7D4211yescharacteristic
448C22×C7#54611yescharacteristic
449C22×C7C22×Q1611yescharacteristic
450C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
451C22×C7#57711yescharacteristic
452C22×C7#57811yescharacteristic
453C4×C7—22no—
454C4×C7—24no—
455C4×C7—42no—
456C4×C7—42no—
457C4×C7—81no—
458C4×C7C2×C22⋊Q811yescharacteristic
459C4×C7#54311yescharacteristic
460C4×C7#54611yescharacteristic
461C4×C7#55111yescharacteristic
462Dic7—22no—
463Dic7—22no—
464Dic7—41no—
465Dic7—41no—
466Dic7—81no—
467Dic7—81no—
468Dic7—81no—
469Dic7—81no—
470C24D4×D711yescharacteristic
471C2×Q8—24no—
472C2×Q8—21no—
473C2×Q8—21no—
474C2×Q8—21no—
475C2×Q8—21no—
476C2×Q8—28no—
477C2×Q8—21no—
478C2×Q8—21no—
479C2×Q8—21no—
480C2×Q8—21no—
481C2×Q8—42no—
482C2×Q8—42no—
483C2×Q8—44no—
484C2×Q8—44no—
485C2×Q8—44no—
486C2×Q8—282no—
487C2×Q8—282no—
488C2×Q8—561no—
489C2×Q8C2×Dic1412yes—
490C2×Q8C2×Dic1412yes—
491C2×Q8#115712yes—
492C2×Q8C2×C22⋊D712yes—
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—21no—
496C22×C4—21no—
497C22×C4—21no—
498C22×C4—24no—
499C22×C4—21no—
500C22×C4—21no—
501C22×C4—21no—
502C22×C4—42no—
503C22×C4—42no—
504C22×C4—42no—
505C22×C4—42no—
506C22×C4—71no—
507C22×C4—71no—
508C22×C4—71no—
509C22×C4—71no—
510C22×C4—81no—
511C22×C4—141no—
512C22×C4—141no—
513C22×C4—141no—
514C22×C4—141no—
515C22×C4C2×Dic1412yes—
516C22×C4D4×D711yescharacteristic
517C22×C4#115912yes—
518C22×C4C23×D711yescharacteristic
519C2×C8—21no—
520C2×C8—21no—
521C2×C8—21no—
522C2×C8—21no—
523C2×C8—21no—
524C2×C8—44no—
525C2×C8—141no—
526C2×C8—141no—
527C2×C8—141no—
528C2×C8—141no—
529C2×C8—141no—
530C2×C8—141no—
531C2×C8C2×C22⋊D712yes—
532C4⋊C4—21no—
533C4⋊C4—21no—
534C4⋊C4—21no—
535C4⋊C4—42no—
536C4⋊C4—141no—
537C4⋊C4—141no—
538C4⋊C4—141no—
539C4⋊C4—141no—
540C4⋊C4—141no—
541C4⋊C4—141no—
542C4⋊C4—141no—
543C4⋊C4—141no—
544C4⋊C4—282no—
545C4⋊C4—282no—
546C4⋊C4—282no—
547C4⋊C4—282no—
548C4⋊C4—282no—
549C4⋊C4—282no—
550C4⋊C4—561no—
551C4⋊C4C2×C22⋊D712yes—
552C22⋊C4—74no—
553C22⋊C4—142no—
554C22⋊C4—282no—
555C22⋊C4—282no—
556C42—141no—
557C42—141no—
558C42—141no—
559C42—141no—
560C42—282no—
561C42—282no—
562C2×C7—24no—
563C2×C7#243711yescharacteristic
564C2×C7C2×#47711yescharacteristic
565C2×C7#308211yescharacteristic
566C2×C7#326511yescharacteristic
567C2×C7D4×Q1611yescharacteristic
568C2×C7#329911yescharacteristic
569C2×C7#330811yescharacteristic
570C23—21no—
571C23—21no—
572C23—21no—
573C23—21no—
574C23#644112yes—
575C23#647712yes—
576C23D7×Q1612yes—
577C23C2×D4×D711yescharacteristic
578Q8—22no—
579Q8—22no—
580Q8—22no—
581Q8—22no—
582Q8—22no—
583Q8—48no—
584Q8—41no—
585Q8—41no—
586Q8—564no—
587Q8#648714yescomplemented
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—21no—
591C2×C4—22no—
592C2×C4—21no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—48no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—72no—
608C2×C4—72no—
609C2×C4—84no—
610C2×C4—141no—
611C2×C4—141no—
612C2×C4—141no—
613C2×C4—141no—
614C2×C4—141no—
615C2×C4—141no—
616C2×C4—141no—
617C2×C4—141no—
618C2×C4—141no—
619C2×C4—141no—
620C2×C4—281no—
621C2×C4—281no—
622C2×C4—281no—
623C2×C4—281no—
624C2×C4—281no—
625C2×C4—281no—
626C2×C4—281no—
627C2×C4—281no—
628C2×C4—281no—
629C2×C4—281no—
630C2×C4—281no—
631C2×C4—281no—
632C2×C4#644112yes—
633C2×C4C22×Dic1411yescharacteristic
634C2×C4C2×#115711yescharacteristic
635C2×C4C2×#115911yescharacteristic
636C2×C4#654811yescharacteristic
637C2×C4#654811yescharacteristic
638C2×C4C22×C22⋊D711yescharacteristic
639C8—22no—
640C8—41no—
641C8—281no—
642C8—281no—
643C7#2596311yescharacteristic, complemented, Sylow
644C22—22no—
645C22—22no—
646C22—22no—
647C22—22no—
648C22—22no—
649C22—22no—
650C22#8956614yescomplemented
651C22C2×#644111yescharacteristic
652C22#9018111yescharacteristic
653C22C2×#647711yescharacteristic
654C22#9045311yescharacteristic
655C22C2×D7×Q1611yescharacteristic
656C22#9045711yescharacteristic
657C22D4×C22⋊D711yescharacteristic
658C4—22no—
659C4—24no—
660C4—42no—
661C4—42no—
662C4—81no—
663C4—142no—
664C4—142no—
665C4—281no—
666C4—281no—
667C4—561no—
668C4—561no—
669C4—561no—
670C4—561no—
671C4#9017411yescharacteristic
672C4#9018611yescharacteristic
673C4C2×#648711yescharacteristic
674C4#9049711yescharacteristic
675C2—24no—
676C2C2×#8956611yescharacteristic
677C2#1173622411yescharacteristic
678C2#1173625111yescharacteristic
679C2#1173625411yescharacteristic
680C2#1173955511yescharacteristic
681C2#1174130911yescharacteristic
682C2Q16×C22⋊D711yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC22×C7×#103C2
socleC23×C7C22×D4
prime radicalC24×C7C2×D4
hypercenterC22×#103D7
hypocenterC7#25963
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#103

Complements

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

NH ≅ G/Ntype
4. C2×#89566C2semidirect
9. #11728016C2semidirect
10. #11728003C2semidirect
11. #11728002C2semidirect
12. #11727990C2semidirect
13. #11727989C2semidirect
14. #11727985C2semidirect
15. #11727975C2semidirect
68. C22×C7×Q8Q8semidirect
306. C7×Q8C22⋊Q8semidirect
370. C22×Q8Dic14semidirect
445. C22×C7#477semidirect
587. Q8#6487semidirect
643. C7#25963semidirect
650. C22#89566semidirect

Maximal subgroups

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

subgroupindex
C22×C7×#1032
Q8×C22⋊Dic72
#117396162
C2×#895662
#117298862
#117297282
#117297252
#117293092
#117280162
#117280032
#117280022
#117279902
#117279892
#117279852
#117279752
#259637