Finite Group Databasebeta

← #13372197

Subgroups — #13372197

Order 1792.

1030 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D4×Dic7⋊C4C211yesmaximal, characteristic
2C2×C4×#1142C211yesmaximal, characteristic
3C2×#89530C211yesmaximal, characteristic
4C2×#89530C211yesmaximal, characteristic
5#11733426C211yesmaximal, characteristic, complemented
6C7×#1864C211yesmaximal, characteristic
7#11728342C211yesmaximal, characteristic, complemented
8#11728308C211yesmaximal, characteristic, complemented
9#11727924C211yesmaximal, characteristic
10#11727907C211yesmaximal, characteristic
11#11727372C211yesmaximal, characteristic, complemented
12#11726658C211yesmaximal, characteristic, complemented
13#11726612C211yesmaximal, characteristic, complemented
14#11726438C211yesmaximal, characteristic, complemented
15#11726405C211yesmaximal, characteristic, complemented
16C2×C4×C7×D4C2211yescharacteristic
17C2×D4×Dic7C2211yescharacteristic
18C2×D4×Dic7C2211yescharacteristic
19C22×#1142C2211yescharacteristic
20C22×Dic7⋊C4—21no—
21C2×#6450C2211yescharacteristic
22C2×#6450C2211yescharacteristic
23C2×C4×Dic14C2211yescharacteristic
24C2×C7×#106C2211yescharacteristic
25C2×C7×#102C2211yescharacteristic
26C2×C7×#102C2211yescharacteristic
27C7×#389C2211yescharacteristic
28C2×#6406C2211yescharacteristic
29#89982—21no—
30C2×#6391C2211yescharacteristic
31C2×#6390C2211yescharacteristic
32C2×#6390C2211yescharacteristic
33C2×Dic7⋊C8C2211yescharacteristic
34C2×C8×Dic7C2211yescharacteristic
35C2×C8×Dic7C2211yescharacteristic
36C4×#1142—24no—
37#89757C2211yescharacteristic
38#89748—21no—
39#89748C2211yescharacteristic
40#89748C2211yescharacteristic
41#89745C2211yescharacteristic
42#89744—21no—
43C2×#6384C2211yescharacteristic
44C2×#6382C2211yescharacteristic
45C2×#6382C2211yescharacteristic
46C2×#6382C2211yescharacteristic
47#89718—21no—
48#89714—21no—
49C4×Dic7⋊C4C2211yescharacteristic
50#89700C2211yescharacteristic
51C2×C4×C7⋊C8C2211yescharacteristic
52#89530—24no—
53#89530C418yescomplemented
54#89421—21no—
55#89419—21no—
56C7×#340C2211yescharacteristic
57C7×#336C2211yescharacteristic
58#89346C2211yescharacteristic
59#89345C2211yescharacteristic
60#89324C2211yescharacteristic
61#89324C2211yescharacteristic
62C4⋊C4×SD16—71nomaximal, Sylow
63C22×C7×D4C2311yescharacteristic
64C23×C4×C7—21no—
65C23×Dic7—21no—
66C23×Dic7—21no—
67C22×Dic14C2311yescharacteristic
68C22×C7×C8C2311yescharacteristic
69C22×C7×C8C2311yescharacteristic
70C4×C7×D4—22no—
71C4×C7×D4—41no—
72C2×C7×C4⋊C4—21no—
73C2×C7×C4⋊C4C2311yescharacteristic
74C2×C7×C4⋊C4C2311yescharacteristic
75C2×C7×C4⋊C4C2311yescharacteristic
76C2×C7×C22⋊C4—21no—
77C2×C42×C7C2311yescharacteristic
78C2×C22⋊Dic7—21no—
79C2×C22⋊Dic7—21no—
80D4×Dic7—22no—
81D4×Dic7—22no—
82D4×Dic7—41no—
83D4×Dic7C2×C414yes—
84C2×#1142C2×C414yes—
85C2×#1142D414yes—
86C2×#1142Q814yes—
87C2×C4⋊Dic7—21no—
88C2×C4⋊Dic7C2311yescharacteristic
89C2×C4⋊Dic7C2311yescharacteristic
90C2×C4⋊Dic7C2311yescharacteristic
91C2×Dic7⋊C4—21no—
92C2×Dic7⋊C4—21no—
93C2×Dic7⋊C4—21no—
94C2×Dic7⋊C4—21no—
95C2×Dic7⋊C4—21no—
96C2×Dic7⋊C4—21no—
97C2×Dic7⋊C4—21no—
98C2×Dic7⋊C4—21no—
99C2×Dic7⋊C4—21no—
100C2×Dic7⋊C4—44no—
101C2×Dic7⋊C4D411yescharacteristic
102C2×Dic7⋊C4D411yescharacteristic
103C2×C4×Dic7—21no—
104C2×C4×Dic7—21no—
105C2×C4×Dic7—21no—
106C2×C4×Dic7C2311yescharacteristic
107C2×C4×Dic7C2311yescharacteristic
108C22×C7⋊C8C2311yescharacteristic
109C22×C7⋊C8C2311yescharacteristic
110#6450—22no—
111#6450—22no—
112#6450—41no—
113#6450C2×C414yescomplemented
114C4×Dic14—22no—
115C4×Dic14—41no—
116C7×#106—41no—
117C7×#102—22no—
118C7×#102C2×C414yes—
119C7×#95—21no—
120#6406—22no—
121C7⋊#95—21no—
122C7⋊#95—21no—
123C7⋊#95—21no—
124C7⋊#95D411yescharacteristic
125C7⋊#95D411yescharacteristic
126#6391—41no—
127#6390—22no—
128#6390C2×C414yescomplemented
129Dic7⋊C8—22no—
130C8×Dic7—22no—
131C8×Dic7C2×C414yescomplemented
132#6384—22no—
133#6382—22no—
134#6382—22no—
135#6382C2×C414yescomplemented
136C4×C7⋊C8—22no—
137C2×C4×SD16—71no—
138C2×C4×SD16—71no—
139C2×C4×SD16—71no—
140Q8×C4⋊C4—71no—
141D4×C4⋊C4—71no—
142#1941—71no—
143#1941—71no—
144#1865—71no—
145#1865—71no—
146#1864—71no—
147#1864D711yescharacteristic
148#1844—71no—
149#1803—71no—
150#1802—71no—
151C8×C4⋊C4—71no—
152C24×C7—21no—
153C2×C7×D4—21no—
154C2×C7×D4—21no—
155C2×C7×D4—21no—
156C2×C7×D4C22×C412yes—
157C2×C7×D4C2×D412yes—
158C2×C7×D4C2×Q812yes—
159C22×C4×C7—21no—
160C22×C4×C7—21no—
161C22×C4×C7—21no—
162C22×C4×C7—42no—
163C22×C4×C7—42no—
164C22×C4×C7—81no—
165C22×C4×C7C2×D411yescharacteristic
166C22×C4×C7C2×D411yescharacteristic
167C22×C4×C7C2411yescharacteristic
168C22×Dic7—21no—
169C22×Dic7—21no—
170C22×Dic7—24no—
171C22×Dic7—21no—
172C22×Dic7—21no—
173C22×Dic7—42no—
174C22×Dic7—42no—
175C22×Dic7—42no—
176C22×Dic7—42no—
177C22×Dic7—81no—
178C22×Dic7C2×D411yescharacteristic
179C22×Dic7C2×D411yescharacteristic
180C22×Dic7C2×D411yescharacteristic
181C22×Dic7C2×D411yescharacteristic
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×C8C22×C412yes—
194C7×C4⋊C4—21no—
195C7×C4⋊C4—21no—
196C7×C4⋊C4—21no—
197C7×C4⋊C4—21no—
198C7×C4⋊C4—21no—
199C7×C4⋊C4—81no—
200C7×C4⋊C4C22×C412yes—
201C7×C22⋊C4—42no—
202C42×C7—21no—
203C42×C7—21no—
204C22⋊Dic7—24no—
205C22⋊Dic7—42no—
206#1142C4⋊C4116yescomplemented
207C4⋊Dic7—21no—
208C4⋊Dic7—21no—
209C4⋊Dic7—21no—
210C4⋊Dic7—21no—
211C4⋊Dic7—21no—
212C4⋊Dic7—81no—
213C4⋊Dic7C22×C412yes—
214Dic7⋊C4—24no—
215Dic7⋊C4—22no—
216Dic7⋊C4—42no—
217Dic7⋊C4—42no—
218Dic7⋊C4—42no—
219Dic7⋊C4—42no—
220Dic7⋊C4—42no—
221Dic7⋊C4—42no—
222Dic7⋊C4—81no—
223Dic7⋊C4—81no—
224Dic7⋊C4SD1614yescomplemented
225C4×Dic7—21no—
226C4×Dic7—21no—
227C4×Dic7—24no—
228C4×Dic7—42no—
229C4×Dic7—42no—
230C4×Dic7C22×C412yes—
231C4×Dic7#4112yes—
232C2×C7⋊C8—21no—
233C2×C7⋊C8—21no—
234C2×C7⋊C8—21no—
235C2×C7⋊C8C22×C412yes—
236C2×C7⋊C8C2×D412yes—
237C2×C7⋊C8C2×Q812yes—
238C22×SD16—71no—
239C2×C4×Q8—71no—
240C2×C4×Q8—71no—
241C2×C4×Q8—71no—
242C2×C4×D4—71no—
243C2×C4×D4—71no—
244C2×C4×D4C2×D711yescharacteristic
245C22×C4⋊C4—141no—
246C4×SD16—78no—
247C4×SD16—144no—
248C4×SD16—144no—
249C2×#106—71no—
250C2×#106—71no—
251C2×#106—71no—
252C2×#106—71no—
253C2×#106C2×D711yescharacteristic
254C2×C4⋊C8—71no—
255C2×#103—71no—
256C2×#103—71no—
257C2×#103—71no—
258C2×#102—71no—
259C2×#102C2×D711yescharacteristic
260C2×#102C2×D711yescharacteristic
261C2×C4×C8—71no—
262C2×C4×C8—71no—
263C2×C4×C8—71no—
264#389—71no—
265#389—71no—
266#389—71no—
267#389—141no—
268#389—141no—
269#389C2×D711yescharacteristic
270#385—141no—
271#385—141no—
272#384—71no—
273#384—71no—
274#384—141no—
275#380—141no—
276C4×C4⋊C4—71no—
277C4×C4⋊C4—141no—
278#340—71no—
279#340—71no—
280#340—71no—
281#340C2×D711yescharacteristic
282#336—71no—
283#336C2×D711yescharacteristic
284C23×C7—22no—
285C23×C7—22no—
286C23×C7—22no—
287C23×C7—41no—
288C23×C7—41no—
289C23×C7—41no—
290C23×C7—41no—
291C23×C7C22×D411yescharacteristic
292C7×D4—22no—
293C7×D4—22no—
294C7×D4—22no—
295C7×D4C2×C4⋊C414yes—
296C2×C4×C7—21no—
297C2×C4×C7—21no—
298C2×C4×C7—22no—
299C2×C4×C7—21no—
300C2×C4×C7—21no—
301C2×C4×C7—21no—
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×C7—41no—
310C2×C4×C7—84no—
311C2×C4×C7C2×SD1612yes—
312C2×C4×C7C23×C411yescharacteristic
313C2×C4×C7C22×D411yescharacteristic
314C2×C4×C7C22×Q811yescharacteristic
315C2×C4×C7C2×#4111yescharacteristic
316C2×C4×C7H(5,𝔽2)11yescharacteristic
317C2×C4×C72−1+411yescharacteristic
318C2×Dic7—22no—
319C2×Dic7—21no—
320C2×Dic7—22no—
321C2×Dic7—22no—
322C2×Dic7—21no—
323C2×Dic7—21no—
324C2×Dic7—21no—
325C2×Dic7—22no—
326C2×Dic7—21no—
327C2×Dic7—21no—
328C2×Dic7—21no—
329C2×Dic7—41no—
330C2×Dic7—41no—
331C2×Dic7—48no—
332C2×Dic7—41no—
333C2×Dic7—41no—
334C2×Dic7—41no—
335C2×Dic7—41no—
336C2×Dic7—41no—
337C2×Dic7—41no—
338C2×Dic7—84no—
339C2×Dic7C4×D412yes—
340C2×Dic7C4×D412yes—
341C2×Dic7C2×SD1612yes—
342C2×Dic7C2×SD1612yes—
343C2×Dic7#13512yes—
344Dic14—22no—
345Dic14—22no—
346Dic14—22no—
347Dic14C2×C4⋊C414yescomplemented
348C7×C8—22no—
349C7×C8—41no—
350C7⋊C8—41no—
351C7⋊C8C2×C4⋊C414yescomplemented
352C22×Q8—71no—
353C22×D4C22×D711yescharacteristic
354C23×C4—21no—
355C23×C4—141no—
356C23×C4—141no—
357C2×SD16—74no—
358C2×SD16—74no—
359C2×SD16—74no—
360C22×C8—71no—
361C22×C8—71no—
362C22×C8C22×D711yescharacteristic
363C22×C8C22×D711yescharacteristic
364C4×Q8—74no—
365C4×Q8—142no—
366C4×Q8—142no—
367C4×Q8—142no—
368C4×Q8—281no—
369C4×Q8—281no—
370C4×D4—22no—
371C4×D4—41no—
372C4×D4—74no—
373C4×D4—142no—
374C4×D4—142no—
375C4×D4—281no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4—71no—
378C2×C4⋊C4—71no—
379C2×C4⋊C4—71no—
380C2×C4⋊C4—71no—
381C2×C4⋊C4—71no—
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—141no—
390C2×C4⋊C4—141no—
391C2×C4⋊C4—141no—
392C2×C4⋊C4—284no—
393C2×C4⋊C4C22×D711yescharacteristic
394C2×C4⋊C4C22×D711yescharacteristic
395C2×C4⋊C4C22×D711yescharacteristic
396C2×C22⋊C4—21no—
397C2×C22⋊C4—141no—
398C2×C22⋊C4—141no—
399C2×C42—71no—
400C2×C42—71no—
401C2×C42—141no—
402C2×C42—141no—
403C2×C42—141no—
404C2×C42C22×D711yescharacteristic
405#106—41no—
406#106—74no—
407#106—142no—
408#106—142no—
409#106—281no—
410C4⋊C8—142no—
411#103—74no—
412#103—142no—
413#103—142no—
414#102—22no—
415#102—142no—
416#102C4×D714yes—
417C4×C8—74no—
418C4×C8—142no—
419C4×C8—142no—
420#95—21no—
421#95—71no—
422#95—71no—
423#95—141no—
424#95—141no—
425#95—141no—
426C22×C7—24no—
427C22×C7—42no—
428C22×C7—42no—
429C22×C7—42no—
430C22×C7—42no—
431C22×C7—42no—
432C22×C7—42no—
433C22×C7C2×C4×D411yescharacteristic
434C22×C7D4211yescharacteristic
435C22×C7D4×Q811yescharacteristic
436C22×C7C22×SD1611yescharacteristic
437C22×C7C2×#13511yescharacteristic
438C22×C7#57711yescharacteristic
439C22×C7#57711yescharacteristic
440C4×C7—22no—
441C4×C7—42no—
442C4×C7—41no—
443C4×C7—81no—
444C4×C7—81no—
445C4×C7C22×C4⋊C411yescharacteristic
446C4×C7#52011yescharacteristic
447C4×C7#54711yescharacteristic
448C4×C7#55411yescharacteristic
449Dic7—24no—
450Dic7—22no—
451Dic7—22no—
452Dic7—42no—
453Dic7—41no—
454Dic7—81no—
455Dic7—81no—
456Dic7C4×SD1614yescomplemented
457C24—21no—
458C2×Q8—72no—
459C2×Q8—72no—
460C2×Q8—72no—
461C2×Q8—141no—
462C2×Q8—141no—
463C2×Q8—141no—
464C2×D4—21no—
465C2×D4—21no—
466C2×D4—21no—
467C2×D4C2×C4×D712yes—
468C2×D4D4×D712yes—
469C2×D4Q8×D712yes—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—42no—
474C22×C4—42no—
475C22×C4—71no—
476C22×C4—71no—
477C22×C4—71no—
478C22×C4—71no—
479C22×C4—81no—
480C22×C4—144no—
481C22×C4—141no—
482C22×C4—141no—
483C22×C4—141no—
484C22×C4—141no—
485C22×C4—282no—
486C22×C4—282no—
487C22×C4—282no—
488C22×C4—282no—
489C22×C4—561no—
490C22×C4D4×D711yescharacteristic
491C22×C4D4×D711yescharacteristic
492C22×C4C23×D711yescharacteristic
493SD16—716no—
494C2×C8—21no—
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—21no—
498C2×C8—21no—
499C2×C8—72no—
500C2×C8—72no—
501C2×C8—72no—
502C2×C8—141no—
503C2×C8—141no—
504C2×C8—141no—
505C2×C8C2×C4×D712yes—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—74no—
512C4⋊C4—72no—
513C4⋊C4—81no—
514C4⋊C4—142no—
515C4⋊C4—144no—
516C4⋊C4—141no—
517C4⋊C4—141no—
518C4⋊C4—141no—
519C4⋊C4—141no—
520C4⋊C4—141no—
521C4⋊C4—282no—
522C4⋊C4—282no—
523C4⋊C4—282no—
524C4⋊C4—282no—
525C4⋊C4—282no—
526C4⋊C4—282no—
527C4⋊C4—561no—
528C4⋊C4—561no—
529C4⋊C4—561no—
530C4⋊C4C2×C4×D712yes—
531C22⋊C4—42no—
532C22⋊C4—144no—
533C22⋊C4—282no—
534C42—21no—
535C42—21no—
536C42—72no—
537C42—72no—
538C42—144no—
539C42—141no—
540C42—141no—
541C42—282no—
542C42—282no—
543C2×C7—44no—
544C2×C7D4×C4⋊C411yescharacteristic
545C2×C7C2×C4×SD1611yescharacteristic
546C2×C7#295211yescharacteristic
547C2×C7D4×SD1611yescharacteristic
548C2×C7#328411yescharacteristic
549C2×C7Q8×SD1611yescharacteristic
550C2×C7#338111yescharacteristic
551C23—22no—
552C23—22no—
553C23—22no—
554C23—41no—
555C23—41no—
556C23—41no—
557C23—41no—
558C23C2×D4×D711yescharacteristic
559Q8—74no—
560Q8—142no—
561Q8—142no—
562Q8—142no—
563D4—22no—
564D4—22no—
565D4—22no—
566D4D7×C4⋊C414yes—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—22no—
572C2×C4—21no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—72no—
582C2×C4—72no—
583C2×C4—72no—
584C2×C4—72no—
585C2×C4—72no—
586C2×C4—84no—
587C2×C4—141no—
588C2×C4—141no—
589C2×C4—141no—
590C2×C4—142no—
591C2×C4—141no—
592C2×C4—142no—
593C2×C4—142no—
594C2×C4—141no—
595C2×C4—141no—
596C2×C4—142no—
597C2×C4—141no—
598C2×C4—288no—
599C2×C4—281no—
600C2×C4—281no—
601C2×C4—281no—
602C2×C4—281no—
603C2×C4—281no—
604C2×C4—281no—
605C2×C4—281no—
606C2×C4—281no—
607C2×C4—564no—
608C2×C4D7×SD1612yes—
609C2×C4C22×C4×D711yescharacteristic
610C2×C4C2×D4×D711yescharacteristic
611C2×C4#654811yescharacteristic
612C2×C4C2×Q8×D711yescharacteristic
613C2×C4D7×#4111yescharacteristic
614C2×C4#655411yescharacteristic
615C8—22no—
616C8—41no—
617C8—74no—
618C8—281no—
619C7C4⋊C4×SD1611yescharacteristic, complemented, Sylow
620C22—24no—
621C22—42no—
622C22—42no—
623C22—42no—
624C22—42no—
625C22—42no—
626C22—42no—
627C22C4×D4×D711yescharacteristic
628C22#9028111yescharacteristic
629C22#9032311yescharacteristic
630C22C2×D7×SD1611yescharacteristic
631C22D7×#13511yescharacteristic
632C22#9046611yescharacteristic
633C22#9046611yescharacteristic
634C4—22no—
635C4—42no—
636C4—41no—
637C4—74no—
638C4—81no—
639C4—81no—
640C4—142no—
641C4—144no—
642C4—142no—
643C4—282no—
644C4—281no—
645C4—561no—
646C4—561no—
647C4C2×D7×C4⋊C411yescharacteristic
648C4#9023711yescharacteristic
649C4#9032711yescharacteristic
650C4#9034611yescharacteristic
651C2—44no—
652C2#1173183811yescharacteristic
653C2C4×D7×SD1611yescharacteristic
654C2#1173731311yescharacteristic
655C2#1173757511yescharacteristic
656C2#1173761611yescharacteristic
657C2#1173849811yescharacteristic
658C2#1173860211yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1864C2
socleC23×C7C22×D4
prime radicalC22×C7×D4C23
cosocleC22×C4×C7C24
hypercenter#1864D7
hypocenterC7C4⋊C4×SD16
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1864

Complements

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

NH ≅ G/Ntype
5. #11733426C2semidirect
7. #11728342C2semidirect
8. #11728308C2semidirect
11. #11727372C2semidirect
12. #11726658C2semidirect
13. #11726612C2semidirect
14. #11726438C2semidirect
15. #11726405C2semidirect
53. #89530C4semidirect
113. #6450C2×C4semidirect
128. #6390C2×C4semidirect
131. C8×Dic7C2×C4semidirect
135. #6382C2×C4semidirect
206. #1142C4⋊C4semidirect
224. Dic7⋊C4SD16semidirect
347. Dic14C2×C4⋊C4semidirect
351. C7⋊C8C2×C4⋊C4semidirect
456. Dic7C4×SD16semidirect
619. C7C4⋊C4×SD16semidirect

Maximal subgroups

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

subgroupindex
D4×Dic7⋊C42
C2×C4×#11422
C2×#895302
#117334262
C7×#18642
#117283422
#117283082
#117279242
#117279072
#117273722
#117266582
#117266122
#117264382
#117264052
C4⋊C4×SD167