Finite Group Databasebeta

← #13373187

Subgroups — #13373187

Order 1792.

894 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#440C211yesmaximal, characteristic
2#11739908C211yesmaximal, characteristic, complemented
3D4×C4⋊Dic7C211yesmaximal, characteristic
4#11729801C211yesmaximal, characteristic
5#11729801C211yesmaximal, characteristic
6#11729380C211yesmaximal, characteristic, complemented
7#11729359C211yesmaximal, characteristic
8#11728364C211yesmaximal, characteristic, complemented
9#11728363C211yesmaximal, characteristic, complemented
10#11728009C211yesmaximal, characteristic, complemented
11#11728007C211yesmaximal, characteristic, complemented
12#11727906C211yesmaximal, characteristic
13Dic7×#102C211yesmaximal, characteristic
14#11727569C211yesmaximal, characteristic, complemented
15#11727564C211yesmaximal, characteristic, complemented
16C22×C7×SD16C2211yescharacteristic
17C2×C4×C7×Q8C2211yescharacteristic
18C2×C4×C7×D4C2211yescharacteristic
19C2×D4×Dic7C2211yescharacteristic
20C2×D4×Dic7C2211yescharacteristic
21C22×C4⋊Dic7—21no—
22C7×#440C418yescomplemented
23C2×C7×#107C2211yescharacteristic
24C2×C7×#103C2211yescharacteristic
25C2×C7×#102C2211yescharacteristic
26C2×C7×#97C2211yescharacteristic
27#90002C2211yescharacteristic
28#90002C2211yescharacteristic
29C2×#6409C2211yescharacteristic
30C2×#6409C2211yescharacteristic
31C2×#6406C2211yescharacteristic
32C2×#6406C2211yescharacteristic
33#89984—21no—
34C2×#6389C2211yescharacteristic
35C2×C8×Dic7C2211yescharacteristic
36#89757C2211yescharacteristic
37#89757C2211yescharacteristic
38#89753—21no—
39#89753—21no—
40C4⋊C4×Dic7C2211yescharacteristic
41C4⋊C4×Dic7C2211yescharacteristic
42#89722—21no—
43#89722—21no—
44#89709—21no—
45#89708C2211yescharacteristic
46#89707C2211yescharacteristic
47C4×C4⋊Dic7C2211yescharacteristic
48#89704—21no—
49C2×C4⋊(C7⋊C8)C2211yescharacteristic
50#89346C2211yescharacteristic
51#89346C2211yescharacteristic
52#89346C2211yescharacteristic
53#89346C2211yescharacteristic
54#89345C2211yescharacteristic
55#89345C2211yescharacteristic
56#89324C2211yescharacteristic
57#89324C2211yescharacteristic
58#89324C2211yescharacteristic
59#89324C2211yescharacteristic
60#22778—71nomaximal, Sylow
61C22×C7×Q8C2311yescharacteristic
62C22×C7×D4C2311yescharacteristic
63C23×C4×C7—21no—
64C23×Dic7—21no—
65C23×Dic7—21no—
66C2×C7×SD16—22no—
67C2×C7×SD16—22no—
68C2×C7×SD16C2×C414yes—
69C22×C7×C8C2311yescharacteristic
70C22×C7×C8C2311yescharacteristic
71C4×C7×Q8—22no—
72C4×C7×Q8C2×C414yescomplemented
73C4×C7×D4—22no—
74C4×C7×D4C2×C414yes—
75C2×C7×C4⋊C4—21no—
76C2×C7×C4⋊C4C2311yescharacteristic
77C2×C7×C4⋊C4C2311yescharacteristic
78C2×C7×C22⋊C4—21no—
79C2×C42×C7—21no—
80C2×C42×C7C2311yescharacteristic
81C2×C22⋊Dic7—21no—
82C2×C22⋊Dic7—21no—
83D4×Dic7—22no—
84D4×Dic7—22no—
85D4×Dic7—41no—
86D4×Dic7—41no—
87C2×C4⋊Dic7—21no—
88C2×C4⋊Dic7—21no—
89C2×C4⋊Dic7—21no—
90C2×C4⋊Dic7—21no—
91C2×C4⋊Dic7—44no—
92C2×C4⋊Dic7D411yescharacteristic
93C2×C4⋊Dic7D411yescharacteristic
94C2×C4⋊Dic7C2311yescharacteristic
95C2×C4⋊Dic7C2311yescharacteristic
96C2×C4⋊Dic7C2311yescharacteristic
97C2×C4⋊Dic7C2311yescharacteristic
98C2×C4×Dic7—21no—
99C2×C4×Dic7—21no—
100C2×C4×Dic7C2311yescharacteristic
101C2×C4×Dic7C2311yescharacteristic
102C22×C7⋊C8C2311yescharacteristic
103C22×C7⋊C8C2311yescharacteristic
104C7×#107C2×C414yescomplemented
105C7×#103C2×C414yescomplemented
106C7×#102C2×C414yes—
107C7×#97C2×C414yescomplemented
108#6409—41no—
109#6409—41no—
110#6406—22no—
111#6406—22no—
112C7⋊#95—21no—
113C7⋊#95—21no—
114C7⋊#95—21no—
115C7⋊#95—21no—
116C7⋊#95—21no—
117C7⋊#95—21no—
118C7⋊#95—21no—
119C7⋊#95—21no—
120C7⋊#95—21no—
121C7⋊#95—21no—
122C7⋊#95D411yescharacteristic
123C7⋊#95D411yescharacteristic
124#6389—41no—
125C8×Dic7—41no—
126C4⋊(C7⋊C8)—22no—
127C2×#440D711yescharacteristic
128#2356—71no—
129D4×C4⋊C4—71no—
130#1935—71no—
131#1933—71no—
132#1921—71no—
133#1918—71no—
134#1877—71no—
135#1877—71no—
136#1836—71no—
137#1804—71no—
138#1803—71no—
139#1771—71no—
140#1762—71no—
141C4×#102—71no—
142C24×C7—21no—
143C2×C7×Q8—21no—
144C2×C7×Q8—21no—
145C2×C7×Q8—21no—
146C2×C7×Q8—21no—
147C2×C7×Q8—21no—
148C2×C7×Q8C22×C412yes—
149C2×C7×D4—21no—
150C2×C7×D4—21no—
151C2×C7×D4—21no—
152C2×C7×D4C22×C412yes—
153C2×C7×D4#4112yes—
154C2×C7×D4#4112yes—
155C22×C4×C7—21no—
156C22×C4×C7—21no—
157C22×C4×C7—21no—
158C22×C4×C7—24no—
159C22×C4×C7—42no—
160C22×C4×C7—42no—
161C22×C4×C7C2×D411yescharacteristic
162C22×C4×C7C2×D411yescharacteristic
163C22×C4×C7C2411yescharacteristic
164C22×Dic7—21no—
165C22×Dic7—21no—
166C22×Dic7—21no—
167C22×Dic7—21no—
168C22×Dic7—42no—
169C22×Dic7—42no—
170C22×Dic7—42no—
171C22×Dic7—42no—
172C22×Dic7—81no—
173C22×Dic7—81no—
174C22×Dic7C2×D411yescharacteristic
175C22×Dic7C2×D411yescharacteristic
176C22×Dic7C2×D411yescharacteristic
177C22×Dic7C2×D411yescharacteristic
178C7×SD16—28no—
179C2×C7×C8—21no—
180C2×C7×C8—21no—
181C2×C7×C8—21no—
182C2×C7×C8—21no—
183C2×C7×C8C22×C412yes—
184C2×C7×C8C22×C412yes—
185C7×C4⋊C4—24no—
186C7×C4⋊C4—21no—
187C7×C4⋊C4—21no—
188C7×C4⋊C4C22×C412yes—
189C7×C4⋊C4C22×C412yes—
190C7×C22⋊C4—24no—
191C42×C7—24no—
192C42×C7C22×C412yes—
193C42×C7#4112yes—
194C22⋊Dic7—42no—
195C22⋊Dic7—42no—
196C4⋊Dic7—22no—
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—41no—
206C4⋊Dic7—42no—
207C4⋊Dic7—42no—
208C4⋊Dic7—42no—
209C4⋊Dic7—81no—
210C4×Dic7—21no—
211C4×Dic7—21no—
212C4×Dic7—21no—
213C4×Dic7—21no—
214C4×Dic7—42no—
215C4×Dic7—42no—
216C2×C7⋊C8—21no—
217C2×C7⋊C8—21no—
218C2×C7⋊C8—21no—
219C2×C7⋊C8—21no—
220C2×C7⋊C8—21no—
221C2×C7⋊C8—21no—
222C22×SD16C2×D711yescharacteristic
223C2×C4×Q8C2×D711yescharacteristic
224C2×C4×D4—71no—
225C2×C4×D4—71no—
226C2×C4×D4C2×D711yescharacteristic
227C22×C4⋊C4—141no—
228#440Dic718yescomplemented
229C2×#107C2×D711yescharacteristic
230C2×C4⋊C8—71no—
231C2×#103—71no—
232C2×#103—71no—
233C2×#103C2×D711yescharacteristic
234C2×#102—71no—
235C2×#102—71no—
236C2×#102C2×D711yescharacteristic
237C2×#97—71no—
238C2×#97C2×D711yescharacteristic
239C2×C4×C8—71no—
240#391—71no—
241#391—71no—
242#389—71no—
243#389—71no—
244#387—141no—
245#387—141no—
246#385—141no—
247#385—141no—
248#384—71no—
249#383—141no—
250#382—71no—
251#380—141no—
252C4×C4⋊C4—71no—
253C4×C4⋊C4—71no—
254C4×C4⋊C4—71no—
255#376—141no—
256#340—71no—
257#340—71no—
258#340—71no—
259#340—71no—
260#340—71no—
261#340—71no—
262#340—71no—
263#340—71no—
264#336—71no—
265#336—71no—
266C23×C7—22no—
267C23×C7—22no—
268C23×C7—22no—
269C23×C7—41no—
270C23×C7—41no—
271C23×C7—41no—
272C23×C7—41no—
273C23×C7C22×D411yescharacteristic
274C7×Q8—22no—
275C7×Q8—22no—
276C7×Q8—41no—
277C7×Q8—41no—
278C7×D4—22no—
279C7×D4—22no—
280C7×D4—22no—
281C7×D4#11714yes—
282C2×C4×C7—21no—
283C2×C4×C7—21no—
284C2×C4×C7—21no—
285C2×C4×C7—21no—
286C2×C4×C7—22no—
287C2×C4×C7—22no—
288C2×C4×C7—22no—
289C2×C4×C7—48no—
290C2×C4×C7—41no—
291C2×C4×C7—41no—
292C2×C4×C7—41no—
293C2×C4×C7—41no—
294C2×C4×C7—41no—
295C2×C4×C7—41no—
296C2×C4×C7C4×D412yes—
297C2×C4×C7C4×D412yes—
298C2×C4×C7C23×C411yescharacteristic
299C2×C4×C7C2×#4111yescharacteristic
300C2×C4×C7C2×#4111yescharacteristic
301C2×C4×C7C2×#4111yescharacteristic
302C2×C4×C7H(5,𝔽2)11yescharacteristic
303C2×C4×C72−1+411yescharacteristic
304C2×Dic7—21no—
305C2×Dic7—21no—
306C2×Dic7—21no—
307C2×Dic7—21no—
308C2×Dic7—21no—
309C2×Dic7—21no—
310C2×Dic7—21no—
311C2×Dic7—21no—
312C2×Dic7—21no—
313C2×Dic7—21no—
314C2×Dic7—21no—
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—41no—
326C2×Dic7—41no—
327C2×Dic7—84no—
328C2×Dic7—84no—
329C2×Dic7AGL(1,ℤ/8)12yes—
330C7×C8—22no—
331C7×C8—22no—
332C7⋊C8—41no—
333C7⋊C8—41no—
334C22×Q8C22×D711yescharacteristic
335C22×D4C22×D711yescharacteristic
336C23×C4—21no—
337C23×C4—141no—
338C23×C4—141no—
339C2×SD16—22no—
340C2×SD16—22no—
341C2×SD16C2×Dic714yes—
342C22×C8—71no—
343C22×C8—71no—
344C22×C8C22×D711yescharacteristic
345C22×C8C22×D711yescharacteristic
346C4×Q8—22no—
347C4×Q8C2×Dic714yescomplemented
348C4×D4—22no—
349C4×D4—142no—
350C4×D4—142no—
351C4×D4—281no—
352C4×D4—281no—
353C4×D4C2×Dic714yes—
354C2×C4⋊C4—21no—
355C2×C4⋊C4—71no—
356C2×C4⋊C4—71no—
357C2×C4⋊C4—71no—
358C2×C4⋊C4—71no—
359C2×C4⋊C4—71no—
360C2×C4⋊C4—71no—
361C2×C4⋊C4—141no—
362C2×C4⋊C4—141no—
363C2×C4⋊C4—141no—
364C2×C4⋊C4—141no—
365C2×C4⋊C4—284no—
366C2×C4⋊C4C22×D711yescharacteristic
367C2×C4⋊C4C22×D711yescharacteristic
368C2×C22⋊C4—21no—
369C2×C22⋊C4—141no—
370C2×C22⋊C4—141no—
371C2×C42—21no—
372C2×C42—71no—
373C2×C42—71no—
374C2×C42—141no—
375C2×C42—141no—
376C2×C42C22×D711yescharacteristic
377#107C2×Dic714yescomplemented
378C4⋊C8—142no—
379#103—281no—
380#103—281no—
381#103C2×Dic714yescomplemented
382#102—142no—
383#102—142no—
384#102C2×Dic714yes—
385#97—281no—
386#97C2×Dic714yescomplemented
387C4×C8—281no—
388#95—71no—
389#95—71no—
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#95—141no—
399#95—141no—
400C22×C7—24no—
401C22×C7—42no—
402C22×C7—42no—
403C22×C7—42no—
404C22×C7—42no—
405C22×C7—42no—
406C22×C7—42no—
407C22×C7C2×C4×D411yescharacteristic
408C22×C7#54611yescharacteristic
409C22×C7#54811yescharacteristic
410C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
411C22×C7#57511yescharacteristic
412C22×C7#57711yescharacteristic
413C22×C7#57811yescharacteristic
414C4×C7—22no—
415C4×C7—22no—
416C4×C7—42no—
417C4×C7—42no—
418C4×C7—42no—
419C4×C7C2×#11711yescharacteristic
420C4×C7#52011yescharacteristic
421C4×C7#55211yescharacteristic
422C4×C7#55311yescharacteristic
423Dic7—22no—
424Dic7—41no—
425Dic7—41no—
426Dic7—41no—
427Dic7—81no—
428Dic7—81no—
429Dic7—81no—
430Dic7—81no—
431C24—21no—
432C2×Q8—21no—
433C2×Q8—21no—
434C2×Q8—21no—
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8C22×Dic712yes—
438C2×D4—21no—
439C2×D4—21no—
440C2×D4—21no—
441C2×D4#115712yes—
442C2×D4#115712yes—
443C2×D4C22×Dic712yes—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—24no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—71no—
451C22×C4—71no—
452C22×C4—71no—
453C22×C4—71no—
454C22×C4—141no—
455C22×C4—141no—
456C22×C4—141no—
457C22×C4—141no—
458C22×C4—282no—
459C22×C4—282no—
460C22×C4—282no—
461C22×C4—282no—
462C22×C4—561no—
463C22×C4—561no—
464C22×C4D4×D711yescharacteristic
465C22×C4D4×D711yescharacteristic
466C22×C4C23×D711yescharacteristic
467SD16—28no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—141no—
473C2×C8—141no—
474C2×C8—141no—
475C2×C8—141no—
476C2×C8—141no—
477C2×C8—141no—
478C2×C8C22×Dic712yes—
479C2×C8C22×Dic712yes—
480C4⋊C4—24no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—142no—
484C4⋊C4—141no—
485C4⋊C4—141no—
486C4⋊C4—141no—
487C4⋊C4—141no—
488C4⋊C4—141no—
489C4⋊C4—141no—
490C4⋊C4—141no—
491C4⋊C4—141no—
492C4⋊C4—281no—
493C4⋊C4—282no—
494C4⋊C4—282no—
495C4⋊C4—282no—
496C4⋊C4—561no—
497C4⋊C4C22×Dic712yes—
498C4⋊C4C22×Dic712yes—
499C22⋊C4—24no—
500C22⋊C4—282no—
501C22⋊C4—282no—
502C42—24no—
503C42—141no—
504C42—141no—
505C42—141no—
506C42—141no—
507C42—282no—
508C42—282no—
509C42#115912yes—
510C42C22×Dic712yes—
511C2×C7—44no—
512C2×C7#235211yescharacteristic
513C2×C7C4×AGL(1,ℤ/8)11yescharacteristic
514C2×C7#294711yescharacteristic
515C2×C7#331011yescharacteristic
516C2×C7#331511yescharacteristic
517C2×C7#336711yescharacteristic
518C2×C7#337611yescharacteristic
519C23—22no—
520C23—22no—
521C23—22no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23—41no—
526C23C2×D4×D711yescharacteristic
527Q8—22no—
528Q8—22no—
529Q8—41no—
530Q8—41no—
531D4—22no—
532D4—22no—
533D4—22no—
534D4#648914yes—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—48no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—72no—
550C2×C4—141no—
551C2×C4—141no—
552C2×C4—141no—
553C2×C4—141no—
554C2×C4—141no—
555C2×C4—141no—
556C2×C4—141no—
557C2×C4—141no—
558C2×C4—141no—
559C2×C4—141no—
560C2×C4—141no—
561C2×C4—281no—
562C2×C4—281no—
563C2×C4—281no—
564C2×C4—281no—
565C2×C4—281no—
566C2×C4—281no—
567C2×C4—281no—
568C2×C4—281no—
569C2×C4—281no—
570C2×C4—281no—
571C2×C4—281no—
572C2×C4—281no—
573C2×C4—564no—
574C2×C4—564no—
575C2×C4D4×Dic712yes—
576C2×C4D4×Dic712yes—
577C2×C4C2×#115711yescharacteristic
578C2×C4C2×#115711yescharacteristic
579C2×C4C2×#115911yescharacteristic
580C2×C4#655311yescharacteristic
581C2×C4#655411yescharacteristic
582C2×C4C23×Dic711yescharacteristic
583C8—22no—
584C8—22no—
585C8—281no—
586C8—281no—
587C7#2277811yescharacteristic, complemented, Sylow
588C22—24no—
589C22—42no—
590C22—42no—
591C22—42no—
592C22—42no—
593C22—42no—
594C22—42no—
595C22#9024111yescharacteristic
596C22#9024311yescharacteristic
597C22D7×AGL(1,ℤ/8)11yescharacteristic
598C22#9046411yescharacteristic
599C22#9046911yescharacteristic
600C22#9047011yescharacteristic
601C22C2×D4×Dic711yescharacteristic
602C4—22no—
603C4—22no—
604C4—42no—
605C4—42no—
606C4—42no—
607C4—142no—
608C4—281no—
609C4—281no—
610C4—281no—
611C4—561no—
612C4—561no—
613C4—561no—
614C4—561no—
615C4#9023211yescharacteristic
616C4#9025611yescharacteristic
617C4C2×#648911yescharacteristic
618C4#9051811yescharacteristic
619C2—44no—
620C2#1173719611yescharacteristic
621C2#1173720211yescharacteristic
622C2#1173724511yescharacteristic
623C2#1173724811yescharacteristic
624C2#1173978011yescharacteristic
625C2Dic7×AGL(1,ℤ/8)11yescharacteristic
626C2#1174141011yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC2×C7×#440C2
socleC23×C7C22×D4
prime radicalC22×C7×D4C23
cosocleC22×C4×C7C24
hypercenterC2×#440D7
hypocenterC7#22778
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#440

Complements

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

NH ≅ G/Ntype
2. #11739908C2semidirect
6. #11729380C2semidirect
8. #11728364C2semidirect
9. #11728363C2semidirect
10. #11728009C2semidirect
11. #11728007C2semidirect
14. #11727569C2semidirect
15. #11727564C2semidirect
22. C7×#440C4semidirect
72. C4×C7×Q8C2×C4semidirect
104. C7×#107C2×C4semidirect
105. C7×#103C2×C4semidirect
107. C7×#97C2×C4semidirect
228. #440Dic7semidirect
347. C4×Q8C2×Dic7semidirect
377. #107C2×Dic7semidirect
381. #103C2×Dic7semidirect
386. #97C2×Dic7semidirect
587. C7#22778semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#4402
#117399082
D4×C4⋊Dic72
#117298012
#117293802
#117293592
#117283642
#117283632
#117280092
#117280072
#117279062
Dic7×#1022
#117275692
#117275642
#227787