Finite Group Databasebeta

← #13370902

Subgroups — #13370902

Order 1792.

962 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×(C2×D7)⋊C4C211yesmaximal, characteristic
2C2×D7×#103C211yesmaximal, characteristic
3#11733646C211yesmaximal, characteristic
4C7×#1865C211yesmaximal, characteristic, complemented
5#11729385C211yesmaximal, characteristic, complemented
6#11728331C211yesmaximal, characteristic
7#11728324C211yesmaximal, characteristic, complemented
8#11728020C211yesmaximal, characteristic
9#11727992C211yesmaximal, characteristic, complemented
10#11727974C211yesmaximal, characteristic, complemented
11#11727370C211yesmaximal, characteristic, complemented
12#11726626C211yesmaximal, characteristic
13#11726609C211yesmaximal, characteristic, complemented
14#11726402C211yesmaximal, characteristic
15#11726381C211yesmaximal, characteristic, complemented
16C22×Q8×D7C2211yescharacteristic
17C2×C4×C7×Q8C2211yescharacteristic
18C2×Q8×Dic7C2211yescharacteristic
19C22×C8×D7C2211yescharacteristic
20C2×D7×C4⋊C4C2211yescharacteristic
21C2×C7×#106C2211yescharacteristic
22C2×C7×#103C2211yescharacteristic
23C2×C7×#103C2211yescharacteristic
24C7×#389C2211yescharacteristic
25C2×#6409C2211yescharacteristic
26C2×#6409C2211yescharacteristic
27C2×(C2×D7)⋊C8C2211yescharacteristic
28C2×#6391C2211yescharacteristic
29C2×#6390C2211yescharacteristic
30C2×#6390C2211yescharacteristic
31C2×#6389C2211yescharacteristic
32#89759—21no—
33#89759—21no—
34#89759C2211yescharacteristic
35#89759C2211yescharacteristic
36#89757C2211yescharacteristic
37#89750—21no—
38C4⋊C4×Dic7C2211yescharacteristic
39#89746—21no—
40#89746C2211yescharacteristic
41C2×#6384C2211yescharacteristic
42C2×#6384C2211yescharacteristic
43C4×(C2×D7)⋊C4—21no—
44C4×(C2×D7)⋊C4C2211yescharacteristic
45C4×C4⋊Dic7C2211yescharacteristic
46#89701—21no—
47#89701C2211yescharacteristic
48C2×#6377C2211yescharacteristic
49D7×#103C418yescomplemented
50#89438—21no—
51#89427—21no—
52C7×#340C2211yescharacteristic
53C7×#336C2211yescharacteristic
54#89346C2211yescharacteristic
55#89345C2211yescharacteristic
56#89324C2211yescharacteristic
57#89324C2211yescharacteristic
58#89324C2211yescharacteristic
59#89324C2211yescharacteristic
60#22567—71nomaximal, Sylow
61C22×C7×Q8C2311yescharacteristic
62C2×Q8×D7—22no—
63C2×Q8×D7—22no—
64C2×Q8×D7—22no—
65C2×Q8×D7—41no—
66C2×Q8×D7—41no—
67C2×Q8×D7C2×C414yes—
68C22×C4×D7—21no—
69C22×C4×D7—21no—
70C22×C4×D7C2311yescharacteristic
71C22×Dic14—21no—
72C22×Dic14C2311yescharacteristic
73C22×C7×C8C2311yescharacteristic
74C22×C7×C8C2311yescharacteristic
75C4×C7×Q8—22no—
76C4×C7×Q8—41no—
77C2×C7×C4⋊C4—21no—
78C2×C7×C4⋊C4—21no—
79C2×C7×C4⋊C4C2311yescharacteristic
80C2×C7×C4⋊C4C2311yescharacteristic
81C2×C7×C4⋊C4C2311yescharacteristic
82C2×C42×C7—21no—
83C2×C42×C7C2311yescharacteristic
84Q8×Dic7—22no—
85Q8×Dic7—41no—
86C2×(C2×D7)⋊C4—21no—
87C2×(C2×D7)⋊C4—21no—
88C2×(C2×D7)⋊C4D411yescharacteristic
89C2×(C2×D7)⋊C4D411yescharacteristic
90C2×C4⋊Dic7—21no—
91C2×C4⋊Dic7—21no—
92C2×C4⋊Dic7C2311yescharacteristic
93C2×C4⋊Dic7C2311yescharacteristic
94C2×C4⋊Dic7C2311yescharacteristic
95C2×Dic7⋊C4—21no—
96C2×C4×Dic7—21no—
97C2×C4×Dic7—21no—
98C2×C4×Dic7—21no—
99C2×C4×Dic7C2311yescharacteristic
100C22×C7⋊C8C2311yescharacteristic
101C22×C7⋊C8C2311yescharacteristic
102C2×C8×D7—41no—
103C2×C8×D7—41no—
104C2×C8×D7C2×C414yes—
105D7×C4⋊C4—22no—
106D7×C4⋊C4C2×C414yes—
107C7×#106—41no—
108C7×#103—22no—
109C7×#103C2×C414yes—
110C7×#95—21no—
111#6409—22no—
112#6409C2×C414yes—
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(C2×D7)⋊C8—22no—
125#6391—41no—
126#6390—41no—
127#6390C2×C414yes—
128#6389—41no—
129#6384—41no—
130#6384C2×C414yes—
131#6377—41no—
132C22×#103—71no—
133#2345—71no—
134Q8×C22⋊C4—71no—
135#1935—71no—
136#1935—71no—
137#1927—71no—
138#1927—71no—
139#1865—71no—
140#1865D711yescharacteristic, complemented
141#1824—71no—
142#1807—71no—
143#1787—71no—
144#1763—71no—
145#1763—71no—
146#1752—71no—
147C23×D7C2×D411yescharacteristic
148C2×C7×Q8—21no—
149C2×C7×Q8—21no—
150C2×C7×Q8—21no—
151C2×C7×Q8C22×C412yes—
152C2×C7×Q8#4112yes—
153C2×C7×Q8#4112yes—
154C22×C4×C7—21no—
155C22×C4×C7—21no—
156C22×C4×C7—21no—
157C22×C4×C7—21no—
158C22×C4×C7C2×D411yescharacteristic
159C22×C4×C7C2×D411yescharacteristic
160C22×C4×C7C2411yescharacteristic
161C22×Dic7—21no—
162C22×Dic7—21no—
163C22×Dic7—21no—
164C22×Dic7—21no—
165C22×Dic7C2×D411yescharacteristic
166C22×Dic7C2×D411yescharacteristic
167C22×Dic7C2×D411yescharacteristic
168Q8×D7—28no—
169Q8×D7—44no—
170Q8×D7—44no—
171Q8×D7—44no—
172C2×C4×D7—21no—
173C2×C4×D7—21no—
174C2×C4×D7—21no—
175C2×C4×D7—21no—
176C2×C4×D7—24no—
177C2×C4×D7—24no—
178C2×C4×D7—42no—
179C2×C4×D7—42no—
180C2×C4×D7—81no—
181C2×C4×D7—81no—
182C2×C4×D7C22×C412yes—
183C2×C4×D7#4112yes—
184C2×Dic14—21no—
185C2×Dic14—21no—
186C2×Dic14—21no—
187C2×Dic14—21no—
188C2×Dic14—21no—
189C2×Dic14—24no—
190C2×Dic14—42no—
191C2×Dic14—42no—
192C2×Dic14C22×C412yes—
193C2×C7×C8—21no—
194C2×C7×C8—21no—
195C2×C7×C8—21no—
196C2×C7×C8—21no—
197C2×C7×C8—21no—
198C2×C7×C8C22×C412yes—
199C7×C4⋊C4—21no—
200C7×C4⋊C4—21no—
201C7×C4⋊C4—21no—
202C7×C4⋊C4—21no—
203C7×C4⋊C4—21no—
204C7×C4⋊C4—42no—
205C7×C4⋊C4—81no—
206C7×C4⋊C4C22×C412yes—
207C42×C7—21no—
208C42×C7—21no—
209C42×C7—42no—
210(C2×D7)⋊C4—22no—
211(C2×D7)⋊C4—22no—
212(C2×D7)⋊C4—42no—
213(C2×D7)⋊C4—42no—
214C4⋊Dic7—21no—
215C4⋊Dic7—21no—
216C4⋊Dic7—21no—
217C4⋊Dic7—21no—
218C4⋊Dic7—21no—
219C4⋊Dic7—42no—
220C4⋊Dic7—81no—
221C4⋊Dic7C22×C412yes—
222Dic7⋊C4—24no—
223C4×Dic7—21no—
224C4×Dic7—21no—
225C4×Dic7—42no—
226C4×Dic7—81no—
227C4×Dic7—81no—
228C2×C7⋊C8—21no—
229C2×C7⋊C8—21no—
230C2×C7⋊C8—21no—
231C2×C7⋊C8—21no—
232C2×C7⋊C8—21no—
233C2×C7⋊C8C22×C412yes—
234C8×D7—44no—
235C23×Q8—71no—
236C23×C8—71no—
237C2×C4×Q8—71no—
238C2×C4×Q8C2×D711yescharacteristic
239C22×C4⋊C4—71no—
240C2×#106—71no—
241C2×#106C2×D711yescharacteristic
242C2×#103—71no—
243C2×#103—71no—
244C2×#103—71no—
245C2×#103—78no—
246C2×#103—71no—
247C2×#103—71no—
248C2×#103—71no—
249C2×#103C2×D711yescharacteristic
250C2×#103C2×D711yescharacteristic
251C2×C22⋊C8—71no—
252C2×#97—71no—
253C2×#97—71no—
254#391—71no—
255#391—71no—
256#391—141no—
257#391—141no—
258#389—71no—
259#389C2×D711yescharacteristic
260#387—141no—
261#387—141no—
262#380—71no—
263#380—71no—
264#380—141no—
265#380—141no—
266C4×C4⋊C4—71no—
267C4×C4⋊C4—71no—
268C4×C22⋊C4—71no—
269C4×C22⋊C4—141no—
270C4×C22⋊C4—141no—
271#340—71no—
272#340—71no—
273#340—71no—
274#340—71no—
275#340—71no—
276#340C2×D711yescharacteristic
277#336—71no—
278#336C2×D711yescharacteristic
279C23×C7C22×D411yescharacteristic
280C22×D7—21no—
281C22×D7—21no—
282C22×D7—21no—
283C22×D7—21no—
284C22×D7C4×D412yes—
285C22×D7#13712yes—
286C22×D7#13712yes—
287C7×Q8—22no—
288C7×Q8—22no—
289C7×Q8—22no—
290C7×Q8#11714yes—
291C2×C4×C7—22no—
292C2×C4×C7—22no—
293C2×C4×C7—22no—
294C2×C4×C7—22no—
295C2×C4×C7—21no—
296C2×C4×C7—21no—
297C2×C4×C7—21no—
298C2×C4×C7—21no—
299C2×C4×C7—21no—
300C2×C4×C7—21no—
301C2×C4×C7—41no—
302C2×C4×C7—41no—
303C2×C4×C7—41no—
304C2×C4×C7—41no—
305C2×C4×C7—41no—
306C2×C4×C7—41no—
307C2×C4×C7—41no—
308C2×C4×C7—41no—
309C2×C4×C7C23×C411yescharacteristic
310C2×C4×C7C2×#4111yescharacteristic
311C2×C4×C7C2×#4111yescharacteristic
312C2×C4×C7C2×#4111yescharacteristic
313C2×C4×C7H(5,𝔽2)11yescharacteristic
314C2×C4×C7H(5,𝔽2)11yescharacteristic
315C2×Dic7—21no—
316C2×Dic7—21no—
317C2×Dic7—22no—
318C2×Dic7—22no—
319C2×Dic7—21no—
320C2×Dic7—21no—
321C2×Dic7—21no—
322C2×Dic7—21no—
323C2×Dic7—21no—
324C2×Dic7—21no—
325C2×Dic7—41no—
326C2×Dic7—41no—
327C2×Dic7—41no—
328C2×Dic7—41no—
329C2×Dic7—41no—
330C2×Dic7—41no—
331C2×Dic7—41no—
332C2×Dic7—41no—
333C2×Dic7—41no—
334C2×Dic7—41no—
335C2×Dic7C4×D412yes—
336C4×D7—22no—
337C4×D7—22no—
338C4×D7—22no—
339C4×D7—22no—
340C4×D7—48no—
341C4×D7—84no—
342Dic14—22no—
343Dic14—22no—
344Dic14—48no—
345Dic14—41no—
346Dic14—41no—
347C7×C8—22no—
348C7×C8—41no—
349C7⋊C8—22no—
350C7⋊C8—41no—
351C22×Q8—74no—
352C22×Q8—71no—
353C22×Q8—141no—
354C22×Q8—142no—
355C22×Q8—142no—
356C22×Q8—142no—
357C22×Q8—281no—
358C22×Q8—281no—
359C22×Q8C22×D711yescharacteristic
360C23×C4—71no—
361C23×C4—141no—
362C23×C4—141no—
363C22×C8—71no—
364C22×C8—74no—
365C22×C8—71no—
366C22×C8—281no—
367C22×C8—281no—
368C22×C8C22×D711yescharacteristic
369C22×C8C22×D711yescharacteristic
370C4×Q8—22no—
371C4×Q8—41no—
372C4×Q8—142no—
373C4×Q8—281no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—71no—
377C2×C4⋊C4—74no—
378C2×C4⋊C4—71no—
379C2×C4⋊C4—71no—
380C2×C4⋊C4—142no—
381C2×C4⋊C4—141no—
382C2×C4⋊C4—141no—
383C2×C4⋊C4—141no—
384C2×C4⋊C4C22×D711yescharacteristic
385C2×C4⋊C4C22×D711yescharacteristic
386C2×C4⋊C4C22×D711yescharacteristic
387C2×C22⋊C4—71no—
388C2×C22⋊C4—71no—
389C2×C22⋊C4—141no—
390C2×C22⋊C4—141no—
391C2×C42—21no—
392C2×C42—71no—
393C2×C42—141no—
394C2×C42—141no—
395C2×C42—141no—
396C2×C42C22×D711yescharacteristic
397#106—41no—
398#106—281no—
399#103—22no—
400#103—74no—
401#103—74no—
402#103—74no—
403#103—142no—
404#103—281no—
405#103—281no—
406#103C4×D714yes—
407C22⋊C8—142no—
408#97—281no—
409#97—281no—
410#95—21no—
411#95—71no—
412#95—71no—
413#95—141no—
414#95—141no—
415#95—141no—
416#95—141no—
417#95—141no—
418#95—141no—
419#95—141no—
420#95—141no—
421#95—141no—
422C22×C7C2×C4×D411yescharacteristic
423C22×C7#54611yescharacteristic
424C22×C7#54611yescharacteristic
425C22×C7C2×#13711yescharacteristic
426C22×C7C2×#13711yescharacteristic
427C22×C7#57711yescharacteristic
428C22×C7#57711yescharacteristic
429C2×D7—22no—
430C2×D7—22no—
431C2×D7—22no—
432C2×D7—22no—
433C2×D7—22no—
434C2×D7—22no—
435C2×D7#44114yes—
436C4×C7—24no—
437C4×C7—42no—
438C4×C7—41no—
439C4×C7—41no—
440C4×C7—81no—
441C4×C7—81no—
442C4×C7C2×#11711yescharacteristic
443C4×C7#51811yescharacteristic
444C4×C7#55111yescharacteristic
445C4×C7#55111yescharacteristic
446Dic7—22no—
447Dic7—42no—
448Dic7—42no—
449Dic7—41no—
450Dic7—41no—
451Dic7—81no—
452Dic7—81no—
453C24—71no—
454C2×Q8—21no—
455C2×Q8—21no—
456C2×Q8—21no—
457C2×Q8—72no—
458C2×Q8—144no—
459C2×Q8—141no—
460C2×Q8—148no—
461C2×Q8—141no—
462C2×Q8—141no—
463C2×Q8—141no—
464C2×Q8—141no—
465C2×Q8—282no—
466C2×Q8—282no—
467C2×Q8—284no—
468C2×Q8—284no—
469C2×Q8—284no—
470C2×Q8C2×C4×D712yes—
471C2×Q8#115912yes—
472C2×Q8#116112yes—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—71no—
478C22×C4—72no—
479C22×C4—72no—
480C22×C4—71no—
481C22×C4—71no—
482C22×C4—144no—
483C22×C4—144no—
484C22×C4—141no—
485C22×C4—141no—
486C22×C4—141no—
487C22×C4—141no—
488C22×C4—141no—
489C22×C4—141no—
490C22×C4—141no—
491C22×C4—141no—
492C22×C4—282no—
493C22×C4—282no—
494C22×C4—561no—
495C22×C4—561no—
496C22×C4D4×D711yescharacteristic
497C22×C4D4×D711yescharacteristic
498C22×C4C23×D711yescharacteristic
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—72no—
505C2×C8—141no—
506C2×C8—141no—
507C2×C8—141no—
508C2×C8—141no—
509C2×C8—141no—
510C2×C8—284no—
511C2×C8C2×C4×D712yes—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—42no—
518C4⋊C4—72no—
519C4⋊C4—81no—
520C4⋊C4—144no—
521C4⋊C4—141no—
522C4⋊C4—141no—
523C4⋊C4—141no—
524C4⋊C4—141no—
525C4⋊C4—141no—
526C4⋊C4—282no—
527C4⋊C4—561no—
528C4⋊C4C2×C4×D712yes—
529C22⋊C4—142no—
530C22⋊C4—142no—
531C22⋊C4—282no—
532C22⋊C4—282no—
533C42—21no—
534C42—21no—
535C42—42no—
536C42—141no—
537C42—141no—
538C42—282no—
539C42—561no—
540C42—561no—
541C2×C7#234211yescharacteristic
542C2×C7C2×#44111yescharacteristic
543C2×C7#294911yescharacteristic
544C2×C7#331611yescharacteristic
545C2×C7#331611yescharacteristic
546C2×C7#332011yescharacteristic
547C2×C7#332011yescharacteristic
548D7—24no—
549C23—72no—
550C23—72no—
551C23—72no—
552C23—141no—
553C23—141no—
554C23—141no—
555C23—141no—
556C23C2×D4×D711yescharacteristic
557Q8—22no—
558Q8—22no—
559Q8—22no—
560Q8—142no—
561Q8—142no—
562Q8—288no—
563Q8—281no—
564Q8—281no—
565Q8#645514yes—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—22no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—72no—
585C2×C4—141no—
586C2×C4—141no—
587C2×C4—142no—
588C2×C4—142no—
589C2×C4—142no—
590C2×C4—142no—
591C2×C4—141no—
592C2×C4—141no—
593C2×C4—141no—
594C2×C4—141no—
595C2×C4—142no—
596C2×C4—142no—
597C2×C4—141no—
598C2×C4—141no—
599C2×C4—288no—
600C2×C4—281no—
601C2×C4—281no—
602C2×C4—281no—
603C2×C4—281no—
604C2×C4—281no—
605C2×C4—281no—
606C2×C4—281no—
607C2×C4—281no—
608C2×C4—281no—
609C2×C4—281no—
610C2×C4—564no—
611C2×C4C22×C4×D711yescharacteristic
612C2×C4C2×#115911yescharacteristic
613C2×C4#654811yescharacteristic
614C2×C4C2×#116111yescharacteristic
615C2×C4D7×#4111yescharacteristic
616C2×C4#655311yescharacteristic
617C8—22no—
618C8—41no—
619C8—142no—
620C8—281no—
621C7#2256711yescharacteristic, complemented, Sylow
622C22—74no—
623C22—142no—
624C22—142no—
625C22—142no—
626C22—142no—
627C22—142no—
628C22—142no—
629C22C4×D4×D711yescharacteristic
630C22#9028211yescharacteristic
631C22#9032111yescharacteristic
632C22#9045311yescharacteristic
633C22#9046111yescharacteristic
634C22D7×#13711yescharacteristic
635C22D7×#13711yescharacteristic
636C4—24no—
637C4—42no—
638C4—41no—
639C4—41no—
640C4—81no—
641C4—81no—
642C4—142no—
643C4—282no—
644C4—281no—
645C4—282no—
646C4—281no—
647C4—561no—
648C4—561no—
649C4C2×#645511yescharacteristic
650C4#9023611yescharacteristic
651C4#9032811yescharacteristic
652C4#9034911yescharacteristic
653C2—144no—
654C2#1173204011yescharacteristic
655C2#1173709411yescharacteristic
656C2D7×#44111yescharacteristic
657C2#1173749811yescharacteristic
658C2#1173770011yescharacteristic
659C2#1173846911yescharacteristic
660C2#1173869511yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1865C2
socleC23×C7C22×D4
prime radicalC23×D7C2×D4
cosocleC22×C4×C7C24
hypercenter#1865D7
hypocenterC7#22567
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1865

Complements

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

NH ≅ G/Ntype
4. C7×#1865C2semidirect
5. #11729385C2semidirect
7. #11728324C2semidirect
9. #11727992C2semidirect
10. #11727974C2semidirect
11. #11727370C2semidirect
13. #11726609C2semidirect
15. #11726381C2semidirect
49. D7×#103C4semidirect
140. #1865D7semidirect
621. C7#22567semidirect

Maximal subgroups

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

subgroupindex
Q8×(C2×D7)⋊C42
C2×D7×#1032
#117336462
C7×#18652
#117293852
#117283312
#117283242
#117280202
#117279922
#117279742
#117273702
#117266262
#117266092
#117264022
#117263812
#225677