Finite Group Databasebeta

← #13373192

Subgroups — #13373192

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×#89482C211yesmaximal, characteristic
2C4⋊C4×D28C211yesmaximal, characteristic
3#11731088C211yesmaximal, characteristic, complemented
4C7×#1771C211yesmaximal, characteristic, complemented
5#11728239C211yesmaximal, characteristic, complemented
6#11728234C211yesmaximal, characteristic
7#11727571C211yesmaximal, characteristic, complemented
8#11727569C211yesmaximal, characteristic, complemented
9#11727566C211yesmaximal, characteristic
10#11727521C211yesmaximal, characteristic, complemented
11C4×#6395C211yesmaximal, characteristic
12#11726426C211yesmaximal, characteristic
13#11726426C211yesmaximal, characteristic
14#11726378C211yesmaximal, characteristic, complemented
15#11726375C211yesmaximal, characteristic, complemented
16C22×#1132C2211yescharacteristic
17C2×D7×C4⋊C4—21no—
18C2×C4×D28C2211yescharacteristic
19C2×C4×D28C2211yescharacteristic
20C2×C4×D28C2211yescharacteristic
21C2×C4×Dic14C2211yescharacteristic
22C2×C7×C4⋊C8C2211yescharacteristic
23C2×C7×#97C2211yescharacteristic
24C2×C7×#97C2211yescharacteristic
25C2×C4×C7×C8C2211yescharacteristic
26C4×C7×C4⋊C4C2211yescharacteristic
27C2×#6395C2211yescharacteristic
28C2×#6395C2211yescharacteristic
29C2×#6395C2211yescharacteristic
30C2×#6392C2211yescharacteristic
31C2×#6390C2211yescharacteristic
32C2×#6390C2211yescharacteristic
33C2×#6390C2211yescharacteristic
34#89759—21no—
35#89750—21no—
36#89750—21no—
37#89708C2211yescharacteristic
38#89707C2211yescharacteristic
39C4×C4⋊Dic7C2211yescharacteristic
40C4×C4⋊Dic7C2211yescharacteristic
41#89701C2211yescharacteristic
42#89701C2211yescharacteristic
43#89700C2211yescharacteristic
44#89700C2211yescharacteristic
45#89482C418yescomplemented
46#89440—21no—
47#89440—21no—
48#89423—21no—
49#89420—21no—
50C7×#336C2211yescharacteristic
51C7×#336C2211yescharacteristic
52#89346C2211yescharacteristic
53#89346C2211yescharacteristic
54#89346C2211yescharacteristic
55#89346C2211yescharacteristic
56#89346C2211yescharacteristic
57#89346C2211yescharacteristic
58#89346C2211yescharacteristic
59#89346C2211yescharacteristic
60#22778—71nomaximal, Sylow
61C22×D28C2311yescharacteristic
62C22×C4×D7—21no—
63C22×C4×D7—21no—
64C22×C4×D7—21no—
65C22×Dic14C2311yescharacteristic
66C22×C7×C8C2311yescharacteristic
67C22×C7×C8C2311yescharacteristic
68C22×C7×C8C2311yescharacteristic
69C22×C7×C8C2311yescharacteristic
70C2×C7×C4⋊C4D411yescharacteristic
71C2×C7×C4⋊C4D411yescharacteristic
72C2×C42×C7C2311yescharacteristic
73C2×C42×C7C2311yescharacteristic
74C2×C42×C7C2311yescharacteristic
75C2×(C2×D7)⋊C4—21no—
76C2×(C2×D7)⋊C4—21no—
77C2×(C2×D7)⋊C4—21no—
78C2×C4⋊Dic7—21no—
79C2×C4⋊Dic7—21no—
80C2×C4⋊Dic7C2311yescharacteristic
81C2×C4⋊Dic7C2311yescharacteristic
82C2×C4⋊Dic7C2311yescharacteristic
83C2×C4⋊Dic7C2311yescharacteristic
84C2×C4⋊Dic7C2311yescharacteristic
85C2×C4⋊Dic7C2311yescharacteristic
86C2×Dic7⋊C4—21no—
87C2×Dic7⋊C4—21no—
88C2×Dic7⋊C4—21no—
89C2×C4×Dic7—21no—
90C2×C4×Dic7—21no—
91C2×C4×Dic7—21no—
92C2×#1132—22no—
93C2×#1132—22no—
94C2×#1132C2×C414yes—
95D7×C4⋊C4—44no—
96C4×D28—22no—
97C4×D28—22no—
98C4×D28—22no—
99C4×D28—41no—
100C4×D28—41no—
101C4×D28C2×C414yes—
102C4×Dic14—22no—
103C4×Dic14C2×C414yescomplemented
104C7×C4⋊C8—22no—
105C7×#97—41no—
106C7×#97C2×C414yescomplemented
107C4×C7×C8—41no—
108C7×#95D411yescharacteristic
109C7×#95D411yescharacteristic
110C7⋊#95—21no—
111C7⋊#95—21no—
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#6395—22no—
121#6395—22no—
122#6395C2×C414yes—
123#6392C2×C414yescomplemented
124#6390—41no—
125#6390—41no—
126#6390C2×C414yescomplemented
127C2×#440—71no—
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#1771D711yescharacteristic, complemented
140#1762—71no—
141C4×#102—71no—
142C23×D7—21no—
143C22×C4×C7C2×D411yescharacteristic
144C22×C4×C7C2×D411yescharacteristic
145C22×C4×C7C2×D411yescharacteristic
146C22×C4×C7C2×D411yescharacteristic
147C22×C4×C7C2×D411yescharacteristic
148C22×C4×C7C2×D411yescharacteristic
149C22×C4×C7C2411yescharacteristic
150C22×Dic7—21no—
151C22×Dic7—21no—
152C22×Dic7—21no—
153C22×Dic7—21no—
154C22×Dic7—21no—
155C22×Dic7—21no—
156C22×Dic7—21no—
157C2×D28—21no—
158C2×D28—21no—
159C2×D28—21no—
160C2×D28C22×C412yes—
161C2×D28#4112yes—
162C2×D28#4112yes—
163C2×C4×D7—24no—
164C2×C4×D7—42no—
165C2×C4×D7—42no—
166C2×C4×D7—42no—
167C2×C4×D7—42no—
168C2×C4×D7—42no—
169C2×C4×D7—42no—
170C2×C4×D7—81no—
171C2×C4×D7—81no—
172C2×Dic14—21no—
173C2×Dic14—21no—
174C2×Dic14—21no—
175C2×Dic14—21no—
176C2×Dic14—21no—
177C2×Dic14C22×C412yes—
178C2×C7×C8—21no—
179C2×C7×C8—21no—
180C2×C7×C8—21no—
181C2×C7×C8—21no—
182C2×C7×C8—21no—
183C2×C7×C8—21no—
184C2×C7×C8—21no—
185C2×C7×C8—21no—
186C2×C7×C8—21no—
187C2×C7×C8—21no—
188C2×C7×C8C22×C412yes—
189C2×C7×C8C22×C412yes—
190C7×C4⋊C4—22no—
191C7×C4⋊C4—41no—
192C42×C7—21no—
193C42×C7—21no—
194C42×C7—21no—
195C42×C7—21no—
196C42×C7C22×C412yes—
197C42×C7#4112yes—
198(C2×D7)⋊C4—24no—
199(C2×D7)⋊C4—42no—
200(C2×D7)⋊C4—42no—
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—81no—
213C4⋊Dic7C22×C412yes—
214C4⋊Dic7C22×C412yes—
215Dic7⋊C4—24no—
216Dic7⋊C4—42no—
217Dic7⋊C4—42no—
218C4×Dic7—24no—
219C4×Dic7—42no—
220C4×Dic7—42no—
221#1132—28no—
222C22×SD16—71no—
223C2×C4×Q8—71no—
224C2×C4×D4—71no—
225C2×C4×D4—71no—
226C2×C4×D4—71no—
227C22×C4⋊C4—141no—
228#440—78no—
229C2×#107—71no—
230C2×C4⋊C8C2×D711yescharacteristic
231C2×#103—71no—
232C2×#103—71no—
233C2×#103—71no—
234C2×#102—71no—
235C2×#102—71no—
236C2×#102—71no—
237C2×#97C2×D711yescharacteristic
238C2×#97C2×D711yescharacteristic
239C2×C4×C8C2×D711yescharacteristic
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⋊C4C2×D711yescharacteristic
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#336C2×D711yescharacteristic
265#336C2×D711yescharacteristic
266C23×C7C22×D411yescharacteristic
267C22×D7—22no—
268C22×D7—22no—
269C22×D7—22no—
270C22×D7—41no—
271C22×D7—41no—
272C22×D7—41no—
273C22×D7—41no—
274C2×C4×C7—21no—
275C2×C4×C7—21no—
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—21no—
283C2×C4×C7—21no—
284C2×C4×C7—21no—
285C2×C4×C7—21no—
286C2×C4×C7—21no—
287C2×C4×C7—21no—
288C2×C4×C7—21no—
289C2×C4×C7C4×D412yes—
290C2×C4×C7C4×D412yes—
291C2×C4×C7AGL(1,ℤ/8)12yes—
292C2×C4×C7C23×C411yescharacteristic
293C2×C4×C7C2×#4111yescharacteristic
294C2×C4×C7C2×#4111yescharacteristic
295C2×C4×C7C2×#4111yescharacteristic
296C2×C4×C7H(5,𝔽2)11yescharacteristic
297C2×C4×C72−1+411yescharacteristic
298C2×Dic7—22no—
299C2×Dic7—22no—
300C2×Dic7—22no—
301C2×Dic7—41no—
302C2×Dic7—41no—
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—
319D28—22no—
320D28—22no—
321D28—22no—
322D28#11714yes—
323C4×D7—48no—
324C4×D7—84no—
325C4×D7—84no—
326Dic14—22no—
327Dic14—22no—
328Dic14—41no—
329Dic14—41no—
330C7×C8—22no—
331C7×C8—22no—
332C7×C8—41no—
333C7×C8—41no—
334C22×Q8—71no—
335C22×D4—71no—
336C23×C4—141no—
337C23×C4—141no—
338C23×C4—141no—
339C2×SD16—74no—
340C2×SD16—142no—
341C2×SD16—142no—
342C22×C8C22×D711yescharacteristic
343C22×C8C22×D711yescharacteristic
344C22×C8C22×D711yescharacteristic
345C22×C8C22×D711yescharacteristic
346C4×Q8—74no—
347C4×Q8—142no—
348C4×D4—74no—
349C4×D4—142no—
350C4×D4—142no—
351C4×D4—142no—
352C4×D4—281no—
353C4×D4—281no—
354C2×C4⋊C4—71no—
355C2×C4⋊C4—71no—
356C2×C4⋊C4—71no—
357C2×C4⋊C4—71no—
358C2×C4⋊C4—71no—
359C2×C4⋊C4—71no—
360C2×C4⋊C4—141no—
361C2×C4⋊C4—141no—
362C2×C4⋊C4—141no—
363C2×C4⋊C4—141no—
364C2×C4⋊C4—141no—
365C2×C4⋊C4—284no—
366C2×C4⋊C4D2811yescharacteristic
367C2×C4⋊C4D2811yescharacteristic
368C2×C22⋊C4—141no—
369C2×C22⋊C4—141no—
370C2×C22⋊C4—141no—
371C2×C42—141no—
372C2×C42—141no—
373C2×C42—141no—
374C2×C42C22×D711yescharacteristic
375C2×C42C22×D711yescharacteristic
376C2×C42C22×D711yescharacteristic
377#107—74no—
378C4⋊C8—22no—
379#103—74no—
380#103—281no—
381#103—281no—
382#102—74no—
383#102—142no—
384#102—142no—
385#97—41no—
386#97C4×D714yescomplemented
387C4×C8—41no—
388#95—141no—
389#95—141no—
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×C7C2×C4×D411yescharacteristic
401C22×C7#54611yescharacteristic
402C22×C7#54811yescharacteristic
403C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
404C22×C7#57511yescharacteristic
405C22×C7#57711yescharacteristic
406C22×C7#57811yescharacteristic
407C2×D7—24no—
408C2×D7—42no—
409C2×D7—42no—
410C2×D7—42no—
411C2×D7—42no—
412C2×D7—42no—
413C2×D7—42no—
414C4×C7—22no—
415C4×C7—22no—
416C4×C7—22no—
417C4×C7—41no—
418C4×C7—41no—
419C4×C7—41no—
420C4×C7C2×#11711yescharacteristic
421C4×C7#52011yescharacteristic
422C4×C7#55211yescharacteristic
423C4×C7#55311yescharacteristic
424Dic7—42no—
425Dic7—42no—
426Dic7—42no—
427Dic7—81no—
428Dic7—81no—
429Dic7—81no—
430Dic7—81no—
431C24—141no—
432C2×Q8—72no—
433C2×Q8—141no—
434C2×Q8—141no—
435C2×Q8—141no—
436C2×Q8—141no—
437C2×Q8—141no—
438C2×D4—72no—
439C2×D4—72no—
440C2×D4—72no—
441C2×D4—141no—
442C2×D4—141no—
443C2×D4—141no—
444C22×C4—141no—
445C22×C4—141no—
446C22×C4—141no—
447C22×C4—144no—
448C22×C4—141no—
449C22×C4—141no—
450C22×C4—141no—
451C22×C4—141no—
452C22×C4—282no—
453C22×C4—282no—
454C22×C4—282no—
455C22×C4—282no—
456C22×C4—282no—
457C22×C4—282no—
458C22×C4—561no—
459C22×C4—561no—
460C22×C4C2×D2811yescharacteristic
461C22×C4C2×D2811yescharacteristic
462C22×C4C2×D2811yescharacteristic
463C22×C4C2×D2811yescharacteristic
464C22×C4C2×D2811yescharacteristic
465C22×C4C2×D2811yescharacteristic
466C22×C4C23×D711yescharacteristic
467SD16—148no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8C2×C4×D712yes—
479C2×C8C2×C4×D712yes—
480C4⋊C4—22no—
481C4⋊C4—41no—
482C4⋊C4—72no—
483C4⋊C4—72no—
484C4⋊C4—144no—
485C4⋊C4—141no—
486C4⋊C4—141no—
487C4⋊C4—141no—
488C4⋊C4—141no—
489C4⋊C4—141no—
490C4⋊C4—141no—
491C4⋊C4—141no—
492C4⋊C4—141no—
493C4⋊C4—141no—
494C4⋊C4—141no—
495C4⋊C4—282no—
496C4⋊C4—282no—
497C4⋊C4—282no—
498C4⋊C4—561no—
499C22⋊C4—144no—
500C22⋊C4—282no—
501C22⋊C4—282no—
502C42—21no—
503C42—21no—
504C42—21no—
505C42—21no—
506C42—144no—
507C42—282no—
508C42—282no—
509C42C2×C4×D712yes—
510C42#115712yes—
511C2×C7#235211yescharacteristic
512C2×C7C4×AGL(1,ℤ/8)11yescharacteristic
513C2×C7#294711yescharacteristic
514C2×C7#331011yescharacteristic
515C2×C7#331511yescharacteristic
516C2×C7#336711yescharacteristic
517C2×C7#337611yescharacteristic
518D7—44no—
519C23—142no—
520C23—142no—
521C23—142no—
522C23—281no—
523C23—281no—
524C23—281no—
525C23—281no—
526C23C22×D2811yescharacteristic
527Q8—142no—
528Q8—142no—
529Q8—281no—
530Q8—281no—
531D4—74no—
532D4—142no—
533D4—142no—
534D4—142no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—142no—
551C2×C4—142no—
552C2×C4—142no—
553C2×C4—288no—
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—281no—
568C2×C4—281no—
569C2×C4—281no—
570C2×C4—281no—
571C2×C4—281no—
572C2×C4—564no—
573C2×C4—564no—
574C2×C4C4×D2812yes—
575C2×C4C4×D2812yes—
576C2×C4#647112yes—
577C2×C4C22×C4×D711yescharacteristic
578C2×C4C2×#115711yescharacteristic
579C2×C4#654811yescharacteristic
580C2×C4#655111yescharacteristic
581C2×C4D7×#4111yescharacteristic
582C2×C4D7×#4111yescharacteristic
583C8—22no—
584C8—22no—
585C8—41no—
586C8—41no—
587C7#2277811yescharacteristic, complemented, Sylow
588C22—144no—
589C22—282no—
590C22—282no—
591C22—282no—
592C22—282no—
593C22—282no—
594C22—282no—
595C22C2×C4×D2811yescharacteristic
596C22#9024511yescharacteristic
597C22#9026711yescharacteristic
598C22C2×#647111yescharacteristic
599C22#9043911yescharacteristic
600C22#9044211yescharacteristic
601C22#9044411yescharacteristic
602C4—22no—
603C4—22no—
604C4—22no—
605C4—41no—
606C4—41no—
607C4—41no—
608C4—282no—
609C4—282no—
610C4—282no—
611C4—561no—
612C4—561no—
613C4—561no—
614C4—561no—
615C4#9019511yescharacteristic
616C4D7×#11711yescharacteristic
617C4#9024911yescharacteristic
618C4#9027211yescharacteristic
619C2—284no—
620C2#1173121611yescharacteristic
621C2C4×#647111yescharacteristic
622C2#1173601611yescharacteristic
623C2#1173694611yescharacteristic
624C2#1173694911yescharacteristic
625C2#1173741511yescharacteristic
626C2#1173741911yescharacteristic
subgroupG/H
centerC23C22×D28
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1771C2
socleC23×C7C22×D4
prime radicalC22×D28C23
cosocleC22×C4×C7C24
hypercenter#1771D7
hypocenterC7#22778
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #1771

Complements

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

NH ≅ G/Ntype
3. #11731088C2semidirect
4. C7×#1771C2semidirect
5. #11728239C2semidirect
7. #11727571C2semidirect
8. #11727569C2semidirect
10. #11727521C2semidirect
14. #11726378C2semidirect
15. #11726375C2semidirect
45. #89482C4semidirect
103. C4×Dic14C2×C4semidirect
106. C7×#97C2×C4semidirect
123. #6392C2×C4semidirect
126. #6390C2×C4semidirect
139. #1771D7semidirect
386. #97C4×D7semidirect
587. C7#22778semidirect

Maximal subgroups

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

subgroupindex
C2×#894822
C4⋊C4×D282
#117310882
C7×#17712
#117282392
#117282342
#117275712
#117275692
#117275662
#117275212
C4×#63952
#117264262
#117263782
#117263752
#227787