Finite Group Databasebeta

← #13372151

Subgroups — #13372151

Order 1792.

1010 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89482C211yesmaximal, characteristic
2C2×#89482C211yesmaximal, characteristic
3C2×C4×#1132C211yesmaximal, characteristic
4C4⋊C4×D28C211yesmaximal, characteristic
5#11731082C211yesmaximal, characteristic, complemented
6C7×#1771C211yesmaximal, characteristic, complemented
7#11728239C211yesmaximal, characteristic, complemented
8#11728234C211yesmaximal, characteristic
9#11727573C211yesmaximal, characteristic, complemented
10#11727570C211yesmaximal, characteristic, complemented
11#11727524C211yesmaximal, characteristic, complemented
12#11726438C211yesmaximal, characteristic, complemented
13#11726437C211yesmaximal, characteristic, complemented
14#11726425C211yesmaximal, characteristic
15#11726424C211yesmaximal, characteristic
16C22×#1132C2211yescharacteristic
17C2×D7×C4⋊C4—21no—
18C2×C4×D28C2211yescharacteristic
19C2×C4×D28C2211yescharacteristic
20C2×C4×D28C2211yescharacteristic
21C2×C4×Dic14C2211yescharacteristic
22C2×C4×Dic14C2211yescharacteristic
23C2×C4×Dic14C2211yescharacteristic
24C2×C7×C4⋊C8C2211yescharacteristic
25C2×C7×#97C2211yescharacteristic
26C2×C7×#97C2211yescharacteristic
27C2×C4×C7×C8C2211yescharacteristic
28C4×C7×C4⋊C4C2211yescharacteristic
29C2×#6395C2211yescharacteristic
30C2×#6395C2211yescharacteristic
31C2×#6395C2211yescharacteristic
32C2×#6392C2211yescharacteristic
33C2×#6392C2211yescharacteristic
34C2×#6392C2211yescharacteristic
35C2×#6391C2211yescharacteristic
36C2×#6391C2211yescharacteristic
37C2×#6390C2211yescharacteristic
38C2×#6390C2211yescharacteristic
39C2×#6390C2211yescharacteristic
40#89759—21no—
41#89744—21no—
42#89744—21no—
43#89708C2211yescharacteristic
44#89707C2211yescharacteristic
45#89700C2211yescharacteristic
46#89700C2211yescharacteristic
47#89700C2211yescharacteristic
48#89700C2211yescharacteristic
49#89482—24no—
50#89482C418yescomplemented
51C4×#1132—24no—
52#89440—21no—
53#89440—21no—
54#89422—21no—
55#89421—21no—
56C7×#336C2211yescharacteristic
57C7×#336C2211yescharacteristic
58#89346C2211yescharacteristic
59#89346C2211yescharacteristic
60#89346C2211yescharacteristic
61#89346C2211yescharacteristic
62#22677—71nomaximal, Sylow
63C22×D28C2311yescharacteristic
64C22×C4×D7—21no—
65C22×C4×D7—21no—
66C22×C4×D7—21no—
67C22×Dic14C2311yescharacteristic
68C22×C7×C8C2311yescharacteristic
69C22×C7×C8C2311yescharacteristic
70C22×C7×C8C2311yescharacteristic
71C22×C7×C8C2311yescharacteristic
72C2×C7×C4⋊C4D411yescharacteristic
73C2×C7×C4⋊C4D411yescharacteristic
74C2×C42×C7C2311yescharacteristic
75C2×C42×C7C2311yescharacteristic
76C2×C42×C7C2311yescharacteristic
77C2×(C2×D7)⋊C4—21no—
78C2×(C2×D7)⋊C4—21no—
79C2×(C2×D7)⋊C4—21no—
80C2×C4⋊Dic7—21no—
81C2×C4⋊Dic7—21no—
82C2×C4⋊Dic7C2311yescharacteristic
83C2×C4⋊Dic7C2311yescharacteristic
84C2×C4⋊Dic7C2311yescharacteristic
85C2×C4⋊Dic7C2311yescharacteristic
86C2×C4⋊Dic7C2311yescharacteristic
87C2×C4⋊Dic7C2311yescharacteristic
88C2×Dic7⋊C4—21no—
89C2×Dic7⋊C4—21no—
90C2×Dic7⋊C4—21no—
91C2×Dic7⋊C4—21no—
92C2×Dic7⋊C4—21no—
93C2×Dic7⋊C4—21no—
94C2×Dic7⋊C4—21no—
95C2×C4×Dic7—21no—
96C2×C4×Dic7—21no—
97C2×C4×Dic7—21no—
98C2×#1132C2×C414yes—
99C2×#1132D414yes—
100C2×#1132Q814yes—
101D7×C4⋊C4—44no—
102C4×D28—22no—
103C4×D28—22no—
104C4×D28—22no—
105C4×D28—41no—
106C4×D28—41no—
107C4×D28C2×C414yes—
108C4×Dic14—22no—
109C4×Dic14—22no—
110C4×Dic14—22no—
111C4×Dic14—41no—
112C4×Dic14—41no—
113C4×Dic14C2×C414yescomplemented
114C7×C4⋊C8—22no—
115C7×#97—22no—
116C7×#97C2×C414yescomplemented
117C4×C7×C8—22no—
118C7×#95D411yescharacteristic
119C7×#95D411yescharacteristic
120C7⋊#95—21no—
121C7⋊#95—21no—
122C7⋊#95—21no—
123C7⋊#95—21no—
124C7⋊#95—21no—
125C7⋊#95—21no—
126#6395—22no—
127#6395—22no—
128#6395C2×C414yes—
129#6392—22no—
130#6392—41no—
131#6392C2×C414yescomplemented
132#6391—22no—
133#6391—41no—
134#6390—22no—
135#6390—22no—
136#6390C2×C414yescomplemented
137C2×#440—71no—
138C2×#440—71no—
139C2×C4×SD16—71no—
140#2351—71no—
141D4×C4⋊C4—71no—
142#1943—71no—
143#1942—71no—
144#1865—71no—
145#1864—71no—
146#1863—71no—
147#1862—71no—
148#1847—71no—
149#1804—71no—
150#1803—71no—
151#1771D711yescharacteristic, complemented
152C23×D7—21no—
153C22×C4×C7C2×D411yescharacteristic
154C22×C4×C7C2×D411yescharacteristic
155C22×C4×C7C2×D411yescharacteristic
156C22×C4×C7C2×D411yescharacteristic
157C22×C4×C7C2×D411yescharacteristic
158C22×C4×C7C2×D411yescharacteristic
159C22×C4×C7C2411yescharacteristic
160C22×Dic7—21no—
161C22×Dic7—21no—
162C22×Dic7—21no—
163C22×Dic7—21no—
164C22×Dic7—21no—
165C22×Dic7—21no—
166C22×Dic7—21no—
167C2×D28—21no—
168C2×D28—21no—
169C2×D28—21no—
170C2×D28C22×C412yes—
171C2×D28C2×D412yes—
172C2×D28C2×Q812yes—
173C2×C4×D7—24no—
174C2×C4×D7—42no—
175C2×C4×D7—42no—
176C2×C4×D7—42no—
177C2×C4×D7—42no—
178C2×C4×D7—42no—
179C2×C4×D7—42no—
180C2×C4×D7—81no—
181C2×C4×D7—81no—
182C2×Dic14—21no—
183C2×Dic14—21no—
184C2×Dic14—21no—
185C2×Dic14C22×C412yes—
186C2×Dic14C2×D412yes—
187C2×Dic14C2×Q812yes—
188C2×C7×C8—21no—
189C2×C7×C8—21no—
190C2×C7×C8—21no—
191C2×C7×C8—21no—
192C2×C7×C8—21no—
193C2×C7×C8—21no—
194C2×C7×C8—21no—
195C2×C7×C8—21no—
196C2×C7×C8C22×C412yes—
197C2×C7×C8C22×C412yes—
198C2×C7×C8C2×D412yes—
199C2×C7×C8C2×Q812yes—
200C7×C4⋊C4—22no—
201C7×C4⋊C4—41no—
202C42×C7—21no—
203C42×C7—21no—
204C42×C7—21no—
205C42×C7—21no—
206C42×C7C22×C412yes—
207C42×C7#4112yes—
208(C2×D7)⋊C4—24no—
209(C2×D7)⋊C4—42no—
210(C2×D7)⋊C4—42no—
211C4⋊Dic7—21no—
212C4⋊Dic7—21no—
213C4⋊Dic7—21no—
214C4⋊Dic7—21no—
215C4⋊Dic7—21no—
216C4⋊Dic7—21no—
217C4⋊Dic7—21no—
218C4⋊Dic7—21no—
219C4⋊Dic7—21no—
220C4⋊Dic7—21no—
221C4⋊Dic7—42no—
222C4⋊Dic7—81no—
223C4⋊Dic7C22×C412yes—
224C4⋊Dic7C22×C412yes—
225Dic7⋊C4—24no—
226Dic7⋊C4—42no—
227Dic7⋊C4—42no—
228Dic7⋊C4—42no—
229Dic7⋊C4—42no—
230Dic7⋊C4—42no—
231Dic7⋊C4—42no—
232C4×Dic7—24no—
233C4×Dic7—42no—
234C4×Dic7—42no—
235#1132C4⋊C4116yescomplemented
236C22×SD16—71no—
237C2×C4×Q8—71no—
238C2×C4×Q8—71no—
239C2×C4×Q8—71no—
240C2×C4×D4—71no—
241C2×C4×D4—71no—
242C2×C4×D4—71no—
243C22×C4⋊C4—141no—
244#440—78no—
245#440—144no—
246C4×SD16—144no—
247C2×#107—71no—
248C2×#107—71no—
249C2×#107—71no—
250C2×#106—71no—
251C2×#106—71no—
252C2×C4⋊C8C2×D711yescharacteristic
253C2×#103—71no—
254C2×#103—71no—
255C2×#103—71no—
256C2×#102—71no—
257C2×#102—71no—
258C2×#102—71no—
259C2×#97C2×D711yescharacteristic
260C2×#97C2×D711yescharacteristic
261C2×C4×C8C2×D711yescharacteristic
262#389—71no—
263#389—71no—
264#389—71no—
265#389—71no—
266#389—141no—
267#389—141no—
268#385—141no—
269#385—141no—
270#384—71no—
271#382—71no—
272#382—141no—
273#380—141no—
274C4×C4⋊C4—141no—
275C4×C4⋊C4C2×D711yescharacteristic
276#340—71no—
277#340—71no—
278#340—71no—
279#340—71no—
280#336C2×D711yescharacteristic
281#336C2×D711yescharacteristic
282C23×C7C22×D411yescharacteristic
283C22×D7—22no—
284C22×D7—22no—
285C22×D7—22no—
286C22×D7—41no—
287C22×D7—41no—
288C22×D7—41no—
289C22×D7—41no—
290C2×C4×C7—21no—
291C2×C4×C7—21no—
292C2×C4×C7—21no—
293C2×C4×C7—21no—
294C2×C4×C7—21no—
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—21no—
302C2×C4×C7—21no—
303C2×C4×C7—21no—
304C2×C4×C7—21no—
305C2×C4×C7C4×D412yes—
306C2×C4×C7C4×D412yes—
307C2×C4×C7AGL(1,ℤ/8)12yes—
308C2×C4×C7C23×C411yescharacteristic
309C2×C4×C7C22×D411yescharacteristic
310C2×C4×C7C22×Q811yescharacteristic
311C2×C4×C7C2×#4111yescharacteristic
312C2×C4×C7H(5,𝔽2)11yescharacteristic
313C2×C4×C72−1+411yescharacteristic
314C2×Dic7—22no—
315C2×Dic7—22no—
316C2×Dic7—22no—
317C2×Dic7—22no—
318C2×Dic7—22no—
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—41no—
328C2×Dic7—41no—
329C2×Dic7—41no—
330C2×Dic7—41no—
331C2×Dic7—41no—
332C2×Dic7—41no—
333C2×Dic7—41no—
334C2×Dic7—41no—
335D28—22no—
336D28—22no—
337D28—22no—
338D28C2×C4⋊C414yes—
339C4×D7—48no—
340C4×D7—84no—
341C4×D7—84no—
342Dic14—22no—
343Dic14—22no—
344Dic14—22no—
345Dic14C2×C4⋊C414yescomplemented
346C7×C8—22no—
347C7×C8—41no—
348C7×C8—41no—
349C7×C8C2×C4⋊C414yescomplemented
350C22×Q8—71no—
351C22×D4—71no—
352C23×C4—141no—
353C23×C4—141no—
354C23×C4—141no—
355C2×SD16—74no—
356C2×SD16—74no—
357C2×SD16—74no—
358C22×C8C22×D711yescharacteristic
359C22×C8C22×D711yescharacteristic
360C22×C8C22×D711yescharacteristic
361C22×C8C22×D711yescharacteristic
362C4×Q8—74no—
363C4×Q8—142no—
364C4×Q8—142no—
365C4×Q8—142no—
366C4×Q8—281no—
367C4×Q8—281no—
368C4×D4—74no—
369C4×D4—142no—
370C4×D4—142no—
371C4×D4—142no—
372C4×D4—281no—
373C4×D4—281no—
374C2×C4⋊C4—71no—
375C2×C4⋊C4—71no—
376C2×C4⋊C4—71no—
377C2×C4⋊C4—71no—
378C2×C4⋊C4—71no—
379C2×C4⋊C4—71no—
380C2×C4⋊C4—141no—
381C2×C4⋊C4—141no—
382C2×C4⋊C4—141no—
383C2×C4⋊C4—141no—
384C2×C4⋊C4—141no—
385C2×C4⋊C4—141no—
386C2×C4⋊C4—141no—
387C2×C4⋊C4—141no—
388C2×C4⋊C4—141no—
389C2×C4⋊C4—284no—
390C2×C4⋊C4D2811yescharacteristic
391C2×C4⋊C4D2811yescharacteristic
392C2×C22⋊C4—141no—
393C2×C22⋊C4—141no—
394C2×C22⋊C4—141no—
395C2×C42—141no—
396C2×C42—141no—
397C2×C42—141no—
398C2×C42C22×D711yescharacteristic
399C2×C42C22×D711yescharacteristic
400C2×C42C22×D711yescharacteristic
401#107—74no—
402#107—142no—
403#107—281no—
404#106—142no—
405#106—281no—
406C4⋊C8—22no—
407#103—74no—
408#103—142no—
409#103—142no—
410#102—74no—
411#102—142no—
412#102—142no—
413#97—22no—
414#97C4×D714yescomplemented
415C4×C8—22no—
416#95—141no—
417#95—141no—
418#95—141no—
419#95—141no—
420#95—141no—
421#95—141no—
422#95D2811yescharacteristic
423#95D2811yescharacteristic
424C22×C7C2×C4×D411yescharacteristic
425C22×C7D4211yescharacteristic
426C22×C7D4×Q811yescharacteristic
427C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
428C22×C7#57511yescharacteristic
429C22×C7#57711yescharacteristic
430C22×C7#57811yescharacteristic
431C2×D7—24no—
432C2×D7—42no—
433C2×D7—42no—
434C2×D7—42no—
435C2×D7—42no—
436C2×D7—42no—
437C2×D7—42no—
438C4×C7—22no—
439C4×C7—22no—
440C4×C7—22no—
441C4×C7—41no—
442C4×C7—41no—
443C4×C7—41no—
444C4×C7C22×C4⋊C411yescharacteristic
445C4×C7#52011yescharacteristic
446C4×C7#54711yescharacteristic
447C4×C7#55411yescharacteristic
448Dic7—24no—
449Dic7—42no—
450Dic7—42no—
451Dic7—81no—
452Dic7—81no—
453Dic7—81no—
454Dic7—81no—
455C24—141no—
456C2×Q8—72no—
457C2×Q8—72no—
458C2×Q8—72no—
459C2×Q8—141no—
460C2×Q8—141no—
461C2×Q8—141no—
462C2×D4—72no—
463C2×D4—72no—
464C2×D4—72no—
465C2×D4—141no—
466C2×D4—141no—
467C2×D4—141no—
468C22×C4—141no—
469C22×C4—141no—
470C22×C4—141no—
471C22×C4—144no—
472C22×C4—141no—
473C22×C4—141no—
474C22×C4—141no—
475C22×C4—141no—
476C22×C4—282no—
477C22×C4—282no—
478C22×C4—282no—
479C22×C4—282no—
480C22×C4—282no—
481C22×C4—282no—
482C22×C4—561no—
483C22×C4—561no—
484C22×C4C2×D2811yescharacteristic
485C22×C4C2×D2811yescharacteristic
486C22×C4C2×D2811yescharacteristic
487C22×C4C2×D2811yescharacteristic
488C22×C4C2×D2811yescharacteristic
489C22×C4C2×D2811yescharacteristic
490C22×C4C23×D711yescharacteristic
491SD16—716no—
492C2×C8—21no—
493C2×C8—21no—
494C2×C8—21no—
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—21no—
498C2×C8—21no—
499C2×C8—21no—
500C2×C8C2×C4×D712yes—
501C2×C8C2×C4×D712yes—
502C2×C8D4×D712yes—
503C2×C8Q8×D712yes—
504C4⋊C4—22no—
505C4⋊C4—41no—
506C4⋊C4—72no—
507C4⋊C4—72no—
508C4⋊C4—144no—
509C4⋊C4—141no—
510C4⋊C4—141no—
511C4⋊C4—141no—
512C4⋊C4—141no—
513C4⋊C4—141no—
514C4⋊C4—141no—
515C4⋊C4—141no—
516C4⋊C4—141no—
517C4⋊C4—141no—
518C4⋊C4—141no—
519C4⋊C4—282no—
520C4⋊C4—282no—
521C4⋊C4—282no—
522C4⋊C4—282no—
523C4⋊C4—282no—
524C4⋊C4—282no—
525C4⋊C4—282no—
526C4⋊C4—561no—
527C22⋊C4—144no—
528C22⋊C4—282no—
529C22⋊C4—282no—
530C42—21no—
531C42—21no—
532C42—21no—
533C42—21no—
534C42—144no—
535C42—282no—
536C42—282no—
537C42C2×C4×D712yes—
538C42#115712yes—
539C2×C7D4×C4⋊C411yescharacteristic
540C2×C7C4×AGL(1,ℤ/8)11yescharacteristic
541C2×C7#294711yescharacteristic
542C2×C7#326511yescharacteristic
543C2×C7#326911yescharacteristic
544C2×C7#338611yescharacteristic
545C2×C7#338811yescharacteristic
546D7—44no—
547C23—142no—
548C23—142no—
549C23—142no—
550C23—281no—
551C23—281no—
552C23—281no—
553C23—281no—
554C23C22×D2811yescharacteristic
555Q8—74no—
556Q8—142no—
557Q8—142no—
558Q8—142no—
559D4—74no—
560D4—142no—
561D4—142no—
562D4—142no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—142no—
579C2×C4—142no—
580C2×C4—142no—
581C2×C4—142no—
582C2×C4—142no—
583C2×C4—288no—
584C2×C4—281no—
585C2×C4—281no—
586C2×C4—281no—
587C2×C4—281no—
588C2×C4—281no—
589C2×C4—281no—
590C2×C4—281no—
591C2×C4—281no—
592C2×C4—281no—
593C2×C4—281no—
594C2×C4—281no—
595C2×C4—281no—
596C2×C4—281no—
597C2×C4—281no—
598C2×C4—281no—
599C2×C4—281no—
600C2×C4—564no—
601C2×C4—564no—
602C2×C4C4×D2812yes—
603C2×C4C4×D2812yes—
604C2×C4#647112yes—
605C2×C4C22×C4×D711yescharacteristic
606C2×C4C2×#115711yescharacteristic
607C2×C4C2×D4×D711yescharacteristic
608C2×C4C2×Q8×D711yescharacteristic
609C2×C4#655311yescharacteristic
610C2×C4#655411yescharacteristic
611C8—22no—
612C8—41no—
613C8—41no—
614C8D7×C4⋊C414yescomplemented
615C7#2267711yescharacteristic, complemented, Sylow
616C22—144no—
617C22—282no—
618C22—282no—
619C22—282no—
620C22—282no—
621C22—282no—
622C22—282no—
623C22C2×C4×D2811yescharacteristic
624C22D4×D2811yescharacteristic
625C22Q8×D2811yescharacteristic
626C22C2×#647111yescharacteristic
627C22#9043911yescharacteristic
628C22#9044211yescharacteristic
629C22#9044411yescharacteristic
630C4—22no—
631C4—22no—
632C4—22no—
633C4—41no—
634C4—41no—
635C4—41no—
636C4—144no—
637C4—282no—
638C4—282no—
639C4—561no—
640C4—561no—
641C4—561no—
642C4—561no—
643C4C2×D7×C4⋊C411yescharacteristic
644C4#9021411yescharacteristic
645C4#9024211yescharacteristic
646C4#9027011yescharacteristic
647C2—284no—
648C2C4⋊C4×D2811yescharacteristic
649C2C4×#647111yescharacteristic
650C2#1173601611yescharacteristic
651C2#1173693911yescharacteristic
652C2#1173694311yescharacteristic
653C2#1173742211yescharacteristic
654C2#1173742311yescharacteristic
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#22677
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
5. #11731082C2semidirect
6. C7×#1771C2semidirect
7. #11728239C2semidirect
9. #11727573C2semidirect
10. #11727570C2semidirect
11. #11727524C2semidirect
12. #11726438C2semidirect
13. #11726437C2semidirect
50. #89482C4semidirect
113. C4×Dic14C2×C4semidirect
116. C7×#97C2×C4semidirect
131. #6392C2×C4semidirect
136. #6390C2×C4semidirect
151. #1771D7semidirect
235. #1132C4⋊C4semidirect
345. Dic14C2×C4⋊C4semidirect
349. C7×C8C2×C4⋊C4semidirect
414. #97C4×D7semidirect
614. C8D7×C4⋊C4semidirect
615. C7#22677semidirect

Maximal subgroups

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

subgroupindex
C2×#894822
C2×C4×#11322
C4⋊C4×D282
#117310822
C7×#17712
#117282392
#117282342
#117275732
#117275702
#117275242
#117264382
#117264372
#117264252
#117264242
#226777