Finite Group Databasebeta

← #13412593

Subgroups — #13412593

Order 1792.

826 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89893C211yesmaximal, characteristic
2C2×#89454C211yesmaximal, characteristic, complemented
3#11730938C211yesmaximal, characteristic
4#11730937C211yesmaximal, characteristic
5C7×#1753C211yesmaximal, characteristic
6#11729742C211yesmaximal, characteristic
7#11727683C211yesmaximal, characteristic
8#11727681C211yesmaximal, characteristic
9#11727573C211yesmaximal, characteristic, complemented
10#11727570C211yesmaximal, characteristic, complemented
11#11727523C211yesmaximal, characteristic, complemented
12#11726395C211yesmaximal, characteristic, complemented
13#11726393C211yesmaximal, characteristic, complemented
14#11726387C211yesmaximal, characteristic, complemented
15#11726386C211yesmaximal, characteristic, complemented
16C22×C7×M16C2211yescharacteristic
17C2×#6489C2211yescharacteristic
18C22×C4⋊Dic7C2211yescharacteristic
19C2×#6441—21no—
20C2×#6432C2211yescharacteristic
21C2×#6432C2211yescharacteristic
22C2×C7×C22⋊C8C2211yescharacteristic
23C2×C7×#97C2211yescharacteristic
24C2×C4×C7×C8C2211yescharacteristic
25C4×C7×C22⋊C4C2211yescharacteristic
26#89984C2211yescharacteristic
27#89984C2211yescharacteristic
28#89893—24no—
29C2×#6392C2211yescharacteristic
30C2×#6392C2211yescharacteristic
31C2×#6392C2211yescharacteristic
32C2×#6392C2211yescharacteristic
33C2×#6392C2211yescharacteristic
34C2×#6392C2211yescharacteristic
35C2×#6391C2211yescharacteristic
36C2×#6391C2211yescharacteristic
37C2×#6391C2211yescharacteristic
38C2×#6391C2211yescharacteristic
39#89722—21no—
40#89720—21no—
41#89719—21no—
42#89708C2211yescharacteristic
43#89707C2211yescharacteristic
44#89700C2211yescharacteristic
45#89700C2211yescharacteristic
46#89700C2211yescharacteristic
47#89700C2211yescharacteristic
48#89454—24no—
49#89433—21no—
50#89428—21no—
51#89427—21no—
52#89418—21no—
53C7×#336C2211yescharacteristic
54C7×#336C2211yescharacteristic
55#89346C2211yescharacteristic
56#89346C2211yescharacteristic
57#89346C2211yescharacteristic
58#89346C2211yescharacteristic
59#89346C2211yescharacteristic
60#89346C2211yescharacteristic
61#26074—71nomaximal, Sylow
62C23×C4×C7C2311yescharacteristic
63C23×Dic7—21no—
64C22×Dic14—21no—
65C22×Dic14—21no—
66C2×C7×M16—22no—
67C2×C7×M16—22no—
68C2×C7×M16Q814yescomplemented
69C22×C7×C8C2311yescharacteristic
70C22×C7×C8C2311yescharacteristic
71C22×C7×C8C2311yescharacteristic
72C22×C7×C8C2311yescharacteristic
73C2×C7×C22⋊C4D411yescharacteristic
74C2×C7×C22⋊C4D411yescharacteristic
75C2×C42×C7C2311yescharacteristic
76C2×C42×C7C2311yescharacteristic
77C2×C22⋊Dic7—21no—
78C2×C22⋊Dic7—21no—
79C2×C22⋊Dic7—21no—
80#6489—22no—
81#6489—41no—
82C2×C4⋊Dic7—22no—
83C2×C4⋊Dic7—21no—
84C2×C4⋊Dic7—21no—
85C2×C4⋊Dic7—21no—
86C2×C4⋊Dic7—21no—
87C2×C4⋊Dic7—21no—
88C2×C4⋊Dic7—41no—
89C2×C4⋊Dic7C2311yescharacteristic
90C2×C4⋊Dic7C2311yescharacteristic
91C2×C4⋊Dic7C2311yescharacteristic
92C2×C4⋊Dic7C2311yescharacteristic
93C2×C4⋊Dic7C2311yescharacteristic
94C2×C4⋊Dic7C2311yescharacteristic
95C2×C4⋊Dic7C2311yescharacteristic
96C2×C4⋊Dic7C2311yescharacteristic
97C2×Dic7⋊C4—21no—
98C2×Dic7⋊C4—21no—
99C2×Dic7⋊C4—21no—
100C2×Dic7⋊C4—21no—
101C2×C4×Dic7—21no—
102#6441—44no—
103#6432—22no—
104#6432—22no—
105#6432—41no—
106#6432—41no—
107C7×C22⋊C8Q814yescomplemented
108C7×#97—22no—
109C4×C7×C8—22no—
110C7×#95D411yescharacteristic
111C7×#95D411yescharacteristic
112C7⋊#95—21no—
113C7⋊#95—21no—
114C7⋊#95—21no—
115C7⋊#95—21no—
116C7⋊#95—21no—
117C7⋊#95—21no—
118C7⋊#95—21no—
119C7⋊#95—21no—
120#6392—22no—
121#6392—22no—
122#6392—22no—
123#6392—22no—
124#6392—41no—
125#6392D414yes—
126#6391—22no—
127#6391—22no—
128#6391—22no—
129#6391—41no—
130C2×#500—71no—
131C2×#428—71no—
132#2440—71no—
133#2439—71no—
134#2085—71no—
135#2083—71no—
136#2058—71no—
137#2057—71no—
138#1943—71no—
139#1941—71no—
140#1854—71no—
141#1852—71no—
142#1847—71no—
143#1809—71no—
144#1753D711yescharacteristic
145C24×C7C2×D411yescharacteristic
146C22×C4×C7—21no—
147C22×C4×C7—21no—
148C22×C4×C7—21no—
149C22×C4×C7—21no—
150C22×C4×C7C2×D411yescharacteristic
151C22×C4×C7C2×D411yescharacteristic
152C22×C4×C7C2×D411yescharacteristic
153C22×C4×C7C2×D411yescharacteristic
154C22×C4×C7C2×D411yescharacteristic
155C22×C4×C7C2×Q812yes—
156C22×C4×C7C2×Q812yes—
157C22×C4×C7C2411yescharacteristic
158C22×Dic7—21no—
159C22×Dic7—21no—
160C22×Dic7—21no—
161C22×Dic7—21no—
162C22×Dic7—21no—
163C22×Dic7—21no—
164C22×Dic7—21no—
165C22×Dic7—21no—
166C22×Dic7—42no—
167C22×Dic7—42no—
168C22×Dic7—81no—
169C2×Dic14—42no—
170C2×Dic14—42no—
171C2×Dic14—42no—
172C2×Dic14—42no—
173C2×Dic14—81no—
174C2×Dic14—81no—
175C7×M16—28no—
176C2×C7×C8—21no—
177C2×C7×C8—21no—
178C2×C7×C8—21no—
179C2×C7×C8—21no—
180C2×C7×C8—21no—
181C2×C7×C8—21no—
182C2×C7×C8C2×D412yes—
183C2×C7×C8C2×Q812yes—
184C2×C7×C8C2×Q812yes—
185C2×C7×C8C2×Q812yes—
186C2×C7×C8C2×Q812yes—
187C2×C7×C8#4112yes—
188C7×C22⋊C4—22no—
189C7×C22⋊C4—22no—
190C42×C7—21no—
191C42×C7—21no—
192C42×C7—21no—
193C42×C7—21no—
194C22⋊Dic7—42no—
195C22⋊Dic7—42no—
196C22⋊Dic7—81no—
197C4⋊Dic7—21no—
198C4⋊Dic7—21no—
199C4⋊Dic7—21no—
200C4⋊Dic7—21no—
201C4⋊Dic7—21no—
202C4⋊Dic7—21no—
203C4⋊Dic7—21no—
204C4⋊Dic7—21no—
205C4⋊Dic7—21no—
206C4⋊Dic7—21no—
207C4⋊Dic7—21no—
208C4⋊Dic7—21no—
209C4⋊Dic7—21no—
210C4⋊Dic7—21no—
211C4⋊Dic7—42no—
212C4⋊Dic7—42no—
213C4⋊Dic7—42no—
214C4⋊Dic7—42no—
215C4⋊Dic7—81no—
216C4⋊Dic7C2×D412yes—
217C4⋊Dic7C2×D412yes—
218Dic7⋊C4—42no—
219Dic7⋊C4—42no—
220Dic7⋊C4—42no—
221Dic7⋊C4—81no—
222C4×Dic7—42no—
223C22×M16C2×D711yescharacteristic
224C2×C4⋊Q8—71no—
225C2×C4⋊Q8—71no—
226C2×C22⋊Q8—141no—
227C2×#117—71no—
228C22×C4⋊C4—71no—
229#500—144no—
230#428—144no—
231C2×#107—71no—
232C2×#107—71no—
233C2×#107—71no—
234C2×#107—71no—
235C2×#107—71no—
236C2×#107—71no—
237C2×#106—71no—
238C2×#106—71no—
239C2×#106—71no—
240C2×#106—71no—
241C2×C22⋊C8C2×D711yescharacteristic
242C2×#97C2×D711yescharacteristic
243C2×C4×C8C2×D711yescharacteristic
244#398—141no—
245#396—141no—
246#395—141no—
247#393—141no—
248#391—141no—
249#389—71no—
250#389—71no—
251#389—71no—
252#389—71no—
253#387—141no—
254#385—141no—
255#384—71no—
256#382—71no—
257#380—71no—
258#380—71no—
259C4×C22⋊C4C2×D711yescharacteristic
260#340—71no—
261#340—71no—
262#340—71no—
263#340—71no—
264#340—71no—
265#340—71no—
266#336C2×D711yescharacteristic
267#336C2×D711yescharacteristic
268C23×C7—21no—
269C23×C7—21no—
270C23×C7—21no—
271C23×C7—21no—
272C23×C7—21no—
273C23×C7—21no—
274C23×C7C4⋊Q812yes—
275C23×C7C22×D411yescharacteristic
276C2×C4×C7—21no—
277C2×C4×C7—21no—
278C2×C4×C7—21no—
279C2×C4×C7—21no—
280C2×C4×C7—21no—
281C2×C4×C7—21no—
282C2×C4×C7—22no—
283C2×C4×C7—22no—
284C2×C4×C7—22no—
285C2×C4×C7—22no—
286C2×C4×C7—21no—
287C2×C4×C7—21no—
288C2×C4×C7—21no—
289C2×C4×C7—21no—
290C2×C4×C7—21no—
291C2×C4×C7—21no—
292C2×C4×C7C4⋊Q812yes—
293C2×C4×C7AGL(1,ℤ/8)12yes—
294C2×C4×C7#13712yes—
295C2×C4×C7C22×D411yescharacteristic
296C2×C4×C7C22×Q811yescharacteristic
297C2×C4×C7C22×Q811yescharacteristic
298C2×C4×C7C2×#4111yescharacteristic
299C2×C4×C7H(5,𝔽2)11yescharacteristic
300C2×C4×C72−1+411yescharacteristic
301C2×Dic7—22no—
302C2×Dic7—22no—
303C2×Dic7—41no—
304C2×Dic7—41no—
305C2×Dic7—41no—
306C2×Dic7—41no—
307C2×Dic7—41no—
308C2×Dic7—41no—
309C2×Dic7—41no—
310C2×Dic7—41no—
311C2×Dic7—41no—
312C2×Dic7—41no—
313C2×Dic7—41no—
314C2×Dic7—41no—
315C2×Dic7—41no—
316C2×Dic7—41no—
317C2×Dic7—41no—
318C2×Dic7—41no—
319C2×Dic7—41no—
320C2×Dic7—41no—
321C2×Dic7—41no—
322C2×Dic7—41no—
323C2×Dic7—41no—
324C2×Dic7—41no—
325C2×Dic7—84no—
326Dic14—84no—
327Dic14—84no—
328C7×C8—22no—
329C7×C8—22no—
330C7×C8—22no—
331C7×C8C22⋊Q814yescomplemented
332C22×Q8—141no—
333C22×Q8—141no—
334C23×C4—141no—
335C23×C4C22×D711yescharacteristic
336C2×M16—22no—
337C2×M16—22no—
338C2×M16Dic1414yescomplemented
339C22×C8C22×D711yescharacteristic
340C22×C8C22×D711yescharacteristic
341C22×C8C22×D711yescharacteristic
342C22×C8C22×D711yescharacteristic
343C4⋊Q8—142no—
344C4⋊Q8—142no—
345C4⋊Q8—281no—
346C4⋊Q8—281no—
347C22⋊Q8—284no—
348#117—142no—
349#117—281no—
350C2×C4⋊C4—71no—
351C2×C4⋊C4—71no—
352C2×C4⋊C4—71no—
353C2×C4⋊C4—71no—
354C2×C4⋊C4—71no—
355C2×C4⋊C4—71no—
356C2×C4⋊C4—71no—
357C2×C4⋊C4—71no—
358C2×C4⋊C4—142no—
359C2×C4⋊C4—141no—
360C2×C4⋊C4—141no—
361C2×C4⋊C4—141no—
362C2×C4⋊C4—141no—
363C2×C4⋊C4—141no—
364C2×C4⋊C4—141no—
365C2×C4⋊C4—141no—
366C2×C4⋊C4—141no—
367C2×C4⋊C4—141no—
368C2×C4⋊C4—281no—
369C2×C22⋊C4—141no—
370C2×C22⋊C4—141no—
371C2×C22⋊C4—141no—
372C2×C22⋊C4D2811yescharacteristic
373C2×C22⋊C4D2811yescharacteristic
374C2×C42—141no—
375C2×C42C22×D711yescharacteristic
376C2×C42C22×D711yescharacteristic
377#107—74no—
378#107—142no—
379#107—142no—
380#107—142no—
381#107—142no—
382#107—281no—
383#106—142no—
384#106—142no—
385#106—142no—
386#106—281no—
387C22⋊C8Dic1414yescomplemented
388#97—22no—
389C4×C8—22no—
390#95—141no—
391#95—141no—
392#95—141no—
393#95—141no—
394#95—141no—
395#95—141no—
396#95—141no—
397#95—141no—
398#95D2811yescharacteristic
399#95D2811yescharacteristic
400C22×C7—22no—
401C22×C7—22no—
402C22×C7—22no—
403C22×C7—41no—
404C22×C7—41no—
405C22×C7—41no—
406C22×C7—41no—
407C22×C7C2×C4⋊Q811yescharacteristic
408C22×C7D4211yescharacteristic
409C22×C7#54711yescharacteristic
410C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
411C22×C7C2×#13711yescharacteristic
412C22×C7#57611yescharacteristic
413C22×C7#57811yescharacteristic
414C4×C7—22no—
415C4×C7—22no—
416C4×C7—22no—
417C4×C7—41no—
418C4×C7—41no—
419C4×C7C2×C22⋊Q811yescharacteristic
420C4×C7#54411yescharacteristic
421C4×C7D4×Q811yescharacteristic
422C4×C7#55411yescharacteristic
423Dic7—42no—
424Dic7—42no—
425Dic7—81no—
426Dic7—81no—
427Dic7—81no—
428Dic7—81no—
429Dic7—81no—
430Dic7—81no—
431C24C2×D2811yescharacteristic
432C2×Q8—282no—
433C2×Q8—282no—
434C2×Q8—282no—
435C2×Q8—282no—
436C2×Q8—561no—
437C2×Q8—561no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—141no—
443C22×C4—141no—
444C22×C4—141no—
445C22×C4—141no—
446C22×C4—141no—
447C22×C4—141no—
448C22×C4—141no—
449C22×C4—141no—
450C22×C4—282no—
451C22×C4—282no—
452C22×C4—561no—
453C22×C4C2×Dic1412yes—
454C22×C4C2×Dic1412yes—
455C22×C4C2×D2811yescharacteristic
456C22×C4C2×D2811yescharacteristic
457C22×C4C2×D2811yescharacteristic
458C22×C4C2×D2811yescharacteristic
459C22×C4C2×D2811yescharacteristic
460C22×C4C23×D711yescharacteristic
461M16—28no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8C2×Dic1412yes—
469C2×C8C2×Dic1412yes—
470C2×C8C2×Dic1412yes—
471C2×C8C2×Dic1412yes—
472C2×C8D4×D712yes—
473C2×C8#115912yes—
474C4⋊C4—72no—
475C4⋊C4—72no—
476C4⋊C4—141no—
477C4⋊C4—141no—
478C4⋊C4—141no—
479C4⋊C4—141no—
480C4⋊C4—141no—
481C4⋊C4—141no—
482C4⋊C4—141no—
483C4⋊C4—141no—
484C4⋊C4—141no—
485C4⋊C4—141no—
486C4⋊C4—141no—
487C4⋊C4—141no—
488C4⋊C4—141no—
489C4⋊C4—141no—
490C4⋊C4—282no—
491C4⋊C4—282no—
492C4⋊C4—282no—
493C4⋊C4—282no—
494C4⋊C4—282no—
495C4⋊C4—282no—
496C4⋊C4—282no—
497C4⋊C4—561no—
498C4⋊C4—561no—
499C22⋊C4—22no—
500C22⋊C4—22no—
501C22⋊C4—282no—
502C22⋊C4—282no—
503C22⋊C4—561no—
504C42—21no—
505C42—21no—
506C42—21no—
507C42—21no—
508C42—282no—
509C2×C7—42no—
510C2×C7#244011yescharacteristic
511C2×C7#316411yescharacteristic
512C2×C7#316511yescharacteristic
513C2×C7#326411yescharacteristic
514C2×C7#327011yescharacteristic
515C2×C7#335311yescharacteristic
516C2×C7#335711yescharacteristic
517C23—21no—
518C23—21no—
519C23—21no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23#643212yes—
524C23C22×D2811yescharacteristic
525Q8—564no—
526Q8—564no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—142no—
544C2×C4—142no—
545C2×C4—281no—
546C2×C4—281no—
547C2×C4—281no—
548C2×C4—281no—
549C2×C4—281no—
550C2×C4—281no—
551C2×C4—281no—
552C2×C4—281no—
553C2×C4—281no—
554C2×C4—281no—
555C2×C4—281no—
556C2×C4—281no—
557C2×C4—281no—
558C2×C4—281no—
559C2×C4—281no—
560C2×C4—281no—
561C2×C4—281no—
562C2×C4—281no—
563C2×C4—281no—
564C2×C4—281no—
565C2×C4—281no—
566C2×C4—281no—
567C2×C4—564no—
568C2×C4#643212yes—
569C2×C4#647112yes—
570C2×C4#647212yes—
571C2×C4C22×Dic1411yescharacteristic
572C2×C4C22×Dic1411yescharacteristic
573C2×C4C2×D4×D711yescharacteristic
574C2×C4C2×#115911yescharacteristic
575C2×C4#655311yescharacteristic
576C2×C4#655411yescharacteristic
577C8—22no—
578C8—22no—
579C8—22no—
580C8#644114yescomplemented
581C7#2607411yescharacteristic, complemented, Sylow
582C22—22no—
583C22—22no—
584C22—22no—
585C22—41no—
586C22—41no—
587C22—41no—
588C22—41no—
589C22C2×#643211yescharacteristic
590C22D4×D2811yescharacteristic
591C22#9024611yescharacteristic
592C22C2×#647111yescharacteristic
593C22C2×#647211yescharacteristic
594C22#9044311yescharacteristic
595C22#9044411yescharacteristic
596C4—22no—
597C4—22no—
598C4—22no—
599C4—41no—
600C4—41no—
601C4—282no—
602C4—282no—
603C4—561no—
604C4—561no—
605C4—561no—
606C4—561no—
607C4—561no—
608C4—561no—
609C4C2×#644111yescharacteristic
610C4#9021011yescharacteristic
611C4D4×Dic1411yescharacteristic
612C4#9023411yescharacteristic
613C2—42no—
614C2#1173093811yescharacteristic
615C2#1173595111yescharacteristic
616C2#1173599111yescharacteristic
617C2#1173693811yescharacteristic
618C2#1173695211yescharacteristic
619C2#1173697611yescharacteristic
620C2#1173698011yescharacteristic
subgroupG/H
centerC23C22×D28
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#1753C2
socleC23×C7C22×D4
prime radicalC24×C7C2×D4
hypercenter#1753D7
hypocenterC7#26074
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #1753

Complements

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

NH ≅ G/Ntype
2. C2×#89454C2semidirect
9. #11727573C2semidirect
10. #11727570C2semidirect
11. #11727523C2semidirect
12. #11726395C2semidirect
13. #11726393C2semidirect
14. #11726387C2semidirect
15. #11726386C2semidirect
68. C2×C7×M16Q8semidirect
107. C7×C22⋊C8Q8semidirect
331. C7×C8C22⋊Q8semidirect
338. C2×M16Dic14semidirect
387. C22⋊C8Dic14semidirect
580. C8#6441semidirect
581. C7#26074semidirect

Maximal subgroups

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

subgroupindex
C2×#898932
C2×#894542
#117309382
#117309372
C7×#17532
#117297422
#117276832
#117276812
#117275732
#117275702
#117275232
#117263952
#117263932
#117263872
#117263862
#260747