Finite Group Databasebeta

← #13372164

Subgroups — #13372164

Order 1792.

982 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×Dic7⋊C4C211yesmaximal, characteristic, complemented
2C2×#89798C211yesmaximal, characteristic
3C2×#89589C211yesmaximal, characteristic
4C2×#89560C211yesmaximal, characteristic
5#11733648C211yesmaximal, characteristic
6C7×#1865C211yesmaximal, characteristic, complemented
7#11728338C211yesmaximal, characteristic
8#11728310C211yesmaximal, characteristic, complemented
9#11727990C211yesmaximal, characteristic, complemented
10#11727976C211yesmaximal, characteristic, complemented
11#11727374C211yesmaximal, characteristic, complemented
12#11726646C211yesmaximal, characteristic
13#11726613C211yesmaximal, characteristic, complemented
14#11726424C211yesmaximal, characteristic
15#11726408C211yesmaximal, characteristic, complemented
16C2×C4×C7×Q8C2211yescharacteristic
17C2×Q8×Dic7C2211yescharacteristic
18C2×Q8×Dic7C2211yescharacteristic
19C22×#1143C2211yescharacteristic
20C2×#6456C2211yescharacteristic
21C2×#6456C2211yescharacteristic
22C2×C4×D28C2211yescharacteristic
23C2×C7×#106C2211yescharacteristic
24C2×C7×#103C2211yescharacteristic
25C2×C7×#103C2211yescharacteristic
26C7×#389C2211yescharacteristic
27C2×#6409C2211yescharacteristic
28C2×#6395C2211yescharacteristic
29C2×#6395C2211yescharacteristic
30C2×#6392C2211yescharacteristic
31C2×#6389C2211yescharacteristic
32C2×Dic7⋊C8C2211yescharacteristic
33C2×C8×Dic7C2211yescharacteristic
34#89798—24no—
35#89761—21no—
36#89759—21no—
37#89757C2211yescharacteristic
38#89749C2211yescharacteristic
39#89748—21no—
40#89748C2211yescharacteristic
41#89745—21no—
42#89745C2211yescharacteristic
43C2×#6383C2211yescharacteristic
44C2×#6382C2211yescharacteristic
45C2×#6381C2211yescharacteristic
46C2×#6381C2211yescharacteristic
47C4×Dic7⋊C4—21no—
48C4×Dic7⋊C4C2211yescharacteristic
49#89700—21no—
50#89700C2211yescharacteristic
51C2×#6377C2211yescharacteristic
52#89589C418yescomplemented
53#89560—24no—
54#89440—21no—
55D7×#95—21no—
56C7×#340C2211yescharacteristic
57C7×#336C2211yescharacteristic
58#89346C2211yescharacteristic
59#89345C2211yescharacteristic
60#89324C2211yescharacteristic
61#89324C2211yescharacteristic
62#22678—71nomaximal, Sylow
63C22×C7×Q8C2311yescharacteristic
64C22×D28C2311yescharacteristic
65C22×C4×D7—21no—
66C22×C4×D7—21no—
67C22×C4×D7—21no—
68C22×C7×C8C2311yescharacteristic
69C22×C7×C8C2311yescharacteristic
70C4×C7×Q8—22no—
71C4×C7×Q8—41no—
72C2×C7×C4⋊C4—21no—
73C2×C7×C4⋊C4—21no—
74C2×C7×C4⋊C4C2311yescharacteristic
75C2×C7×C4⋊C4C2311yescharacteristic
76C2×C7×C4⋊C4C2311yescharacteristic
77C2×C42×C7—21no—
78C2×C42×C7C2311yescharacteristic
79Q8×Dic7—22no—
80Q8×Dic7—22no—
81Q8×Dic7—41no—
82Q8×Dic7C2×C414yescomplemented
83C2×#1143C2×C414yes—
84C2×#1143D414yes—
85C2×#1143Q814yes—
86C2×(C2×D7)⋊C4—21no—
87C2×(C2×D7)⋊C4—21no—
88C2×(C2×D7)⋊C4—21no—
89C2×C4⋊Dic7—21no—
90C2×C4⋊Dic7—21no—
91C2×C4⋊Dic7—21no—
92C2×C4⋊Dic7C2311yescharacteristic
93C2×C4⋊Dic7C2311yescharacteristic
94C2×C4⋊Dic7C2311yescharacteristic
95C2×Dic7⋊C4—21no—
96C2×Dic7⋊C4—21no—
97C2×Dic7⋊C4—21no—
98C2×Dic7⋊C4—21no—
99C2×Dic7⋊C4D411yescharacteristic
100C2×Dic7⋊C4D411yescharacteristic
101C2×C4×Dic7—21no—
102C2×C4×Dic7—21no—
103C2×C4×Dic7C2311yescharacteristic
104C2×C4×Dic7C2311yescharacteristic
105C22×C7⋊C8C2311yescharacteristic
106C22×C7⋊C8C2311yescharacteristic
107#6456—22no—
108#6456—22no—
109#6456—41no—
110#6456C2×C414yes—
111C4×D28—22no—
112C4×D28—41no—
113C7×#106—41no—
114C7×#103—22no—
115C7×#103C2×C414yescomplemented
116C7×#95—21no—
117#6409—22no—
118C7⋊#95—21no—
119C7⋊#95—21no—
120C7⋊#95—21no—
121C7⋊#95—21no—
122C7⋊#95—21no—
123C7⋊#95D411yescharacteristic
124C7⋊#95D411yescharacteristic
125#6395—22no—
126#6395C2×C414yes—
127#6392—41no—
128#6389C2×C414yescomplemented
129Dic7⋊C8—22no—
130C8×Dic7—22no—
131#6383—22no—
132#6382—22no—
133#6381—22no—
134#6381C2×C414yescomplemented
135#6377—22no—
136C2×#440—71no—
137C2×#440—71no—
138C2×C4×SD16—71no—
139Q8×C4⋊C4—71no—
140#2349—71no—
141#1943—71no—
142#1942—71no—
143#1865D711yescharacteristic, complemented
144#1864—71no—
145#1863—71no—
146#1862—71no—
147#1847—71no—
148#1805—71no—
149#1802—71no—
150#1771—71no—
151C23×D7—21no—
152C2×C7×Q8—21no—
153C2×C7×Q8—21no—
154C2×C7×Q8—21no—
155C2×C7×Q8C22×C412yes—
156C2×C7×Q8C2×D412yes—
157C2×C7×Q8C2×Q812yes—
158C22×C4×C7—21no—
159C22×C4×C7—21no—
160C22×C4×C7—21no—
161C22×C4×C7—21no—
162C22×C4×C7C2×D411yescharacteristic
163C22×C4×C7C2×D411yescharacteristic
164C22×C4×C7C2411yescharacteristic
165C22×Dic7—21no—
166C22×Dic7—21no—
167C22×Dic7—21no—
168C22×Dic7C2×D411yescharacteristic
169C22×Dic7C2×D411yescharacteristic
170C22×Dic7C2×D411yescharacteristic
171C22×Dic7C2×D411yescharacteristic
172C2×D28—21no—
173C2×D28—21no—
174C2×D28—21no—
175C2×D28C22×C412yes—
176C2×D28C2×D412yes—
177C2×D28C2×Q812yes—
178C2×C4×D7—24no—
179C2×C4×D7—42no—
180C2×C4×D7—42no—
181C2×C4×D7—42no—
182C2×C4×D7—42no—
183C2×C4×D7—81no—
184C2×C4×D7—81no—
185C2×C4×D7—81no—
186C2×C4×D7—81no—
187C2×C7×C8—21no—
188C2×C7×C8—21no—
189C2×C7×C8—21no—
190C2×C7×C8—21no—
191C2×C7×C8—21no—
192C2×C7×C8C22×C412yes—
193C7×C4⋊C4—21no—
194C7×C4⋊C4—21no—
195C7×C4⋊C4—21no—
196C7×C4⋊C4—21no—
197C7×C4⋊C4—21no—
198C7×C4⋊C4—42no—
199C7×C4⋊C4—81no—
200C7×C4⋊C4C22×C412yes—
201C42×C7—21no—
202C42×C7—21no—
203C42×C7—42no—
204#1143C4⋊C4116yescomplemented
205(C2×D7)⋊C4—24no—
206(C2×D7)⋊C4—42no—
207(C2×D7)⋊C4—42no—
208C4⋊Dic7—21no—
209C4⋊Dic7—24no—
210C4⋊Dic7—21no—
211C4⋊Dic7—21no—
212C4⋊Dic7—21no—
213C4⋊Dic7—21no—
214C4⋊Dic7—42no—
215C4⋊Dic7—81no—
216C4⋊Dic7C22×C412yes—
217Dic7⋊C4—22no—
218Dic7⋊C4—41no—
219Dic7⋊C4—42no—
220Dic7⋊C4—81no—
221Dic7⋊C4—81no—
222Dic7⋊C4—81no—
223C4×Dic7—21no—
224C4×Dic7—21no—
225C4×Dic7—24no—
226C4×Dic7—42no—
227C4×Dic7C22×C412yes—
228C4×Dic7#4112yes—
229C2×C7⋊C8—21no—
230C2×C7⋊C8—21no—
231C2×C7⋊C8—21no—
232C2×C7⋊C8C22×C412yes—
233C2×C7⋊C8C2×D412yes—
234C2×C7⋊C8C2×Q812yes—
235C22×SD16—71no—
236C2×C4×Q8—71no—
237C2×C4×Q8—71no—
238C2×C4×Q8C2×D711yescharacteristic
239C2×C4×D4—71no—
240C2×C4×D4—71no—
241C2×C4×D4—71no—
242#440—78no—
243#440—144no—
244C4×SD16—144no—
245C2×#107—71no—
246C2×#107—71no—
247C2×#107—71no—
248C2×#106—71no—
249C2×#106C2×D711yescharacteristic
250C2×C4⋊C8—71no—
251C2×#103—71no—
252C2×#103C2×D711yescharacteristic
253C2×#103C2×D711yescharacteristic
254C2×#102—71no—
255C2×#102—71no—
256C2×#102—71no—
257C2×#97—71no—
258C2×#97—71no—
259C2×C4×C8—71no—
260#389—71no—
261#389—71no—
262#389—71no—
263#389—141no—
264#389—141no—
265#389C2×D711yescharacteristic
266#385—141no—
267#385—141no—
268#384—71no—
269#384—141no—
270#382—71no—
271#380—141no—
272C4×C4⋊C4—71no—
273C4×C4⋊C4—141no—
274C2×#95—141no—
275#340—71no—
276#340—71no—
277#340—71no—
278#340C2×D711yescharacteristic
279#336—71no—
280#336C2×D711yescharacteristic
281C23×C7C22×D411yescharacteristic
282C22×D7—22no—
283C22×D7—22no—
284C22×D7—22no—
285C22×D7—41no—
286C22×D7—41no—
287C22×D7—41no—
288C22×D7—41no—
289C7×Q8—22no—
290C7×Q8—22no—
291C7×Q8—22no—
292C7×Q8C2×C4⋊C414yes—
293C2×C4×C7—22no—
294C2×C4×C7—22no—
295C2×C4×C7—22no—
296C2×C4×C7—22no—
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—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—41no—
311C2×C4×C7C23×C411yescharacteristic
312C2×C4×C7C22×D411yescharacteristic
313C2×C4×C7C22×Q811yescharacteristic
314C2×C4×C7C2×#4111yescharacteristic
315C2×C4×C7H(5,𝔽2)11yescharacteristic
316C2×C4×C72−1+411yescharacteristic
317C2×Dic7—21no—
318C2×Dic7—21no—
319C2×Dic7—21no—
320C2×Dic7—21no—
321C2×Dic7—21no—
322C2×Dic7—21no—
323C2×Dic7—22no—
324C2×Dic7—21no—
325C2×Dic7—21no—
326C2×Dic7—21no—
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—
336C2×Dic7C4×D412yes—
337C2×Dic7#13712yes—
338D28—22no—
339D28—22no—
340D28—22no—
341D28C2×C4⋊C414yes—
342C4×D7—84no—
343C4×D7—84no—
344C4×D7—84no—
345C7×C8—22no—
346C7×C8—41no—
347C7⋊C8—41no—
348C7⋊C8C2×C4⋊C414yes—
349C22×Q8C22×D711yescharacteristic
350C22×D4—71no—
351C23×C4—141no—
352C23×C4—141no—
353C23×C4—141no—
354C2×SD16—74no—
355C2×SD16—74no—
356C2×SD16—74no—
357C22×C8—71no—
358C22×C8—71no—
359C22×C8C22×D711yescharacteristic
360C22×C8C22×D711yescharacteristic
361C4×Q8—22no—
362C4×Q8—41no—
363C4×Q8—74no—
364C4×Q8—142no—
365C4×Q8—142no—
366C4×Q8—281no—
367C4×D4—74no—
368C4×D4—142no—
369C4×D4—142no—
370C4×D4—142no—
371C4×D4—281no—
372C4×D4—281no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4—21no—
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⋊C4C22×D711yescharacteristic
388C2×C4⋊C4C22×D711yescharacteristic
389C2×C4⋊C4C22×D711yescharacteristic
390C2×C22⋊C4—141no—
391C2×C22⋊C4—141no—
392C2×C22⋊C4—141no—
393C2×C42—21no—
394C2×C42—71no—
395C2×C42—71no—
396C2×C42—141no—
397C2×C42—141no—
398C2×C42C22×D711yescharacteristic
399#107—74no—
400#107—142no—
401#107—281no—
402#106—41no—
403#106—142no—
404C4⋊C8—142no—
405#103—22no—
406#103—142no—
407#103C4×D714yescomplemented
408#102—74no—
409#102—142no—
410#102—142no—
411#97—74no—
412#97—142no—
413C4×C8—142no—
414#95—21no—
415#95—71no—
416#95—71no—
417#95—141no—
418#95—141no—
419#95—141no—
420#95—141no—
421#95—141no—
422C22×C7C2×C4×D411yescharacteristic
423C22×C7D4211yescharacteristic
424C22×C7D4×Q811yescharacteristic
425C22×C7C2×#13711yescharacteristic
426C22×C7#57511yescharacteristic
427C22×C7#57611yescharacteristic
428C22×C7#57711yescharacteristic
429C2×D7—24no—
430C2×D7—42no—
431C2×D7—42no—
432C2×D7—42no—
433C2×D7—42no—
434C2×D7—42no—
435C2×D7—42no—
436C4×C7—24no—
437C4×C7—42no—
438C4×C7—41no—
439C4×C7—41no—
440C4×C7—81no—
441C4×C7—81no—
442C4×C7C22×C4⋊C411yescharacteristic
443C4×C7#52011yescharacteristic
444C4×C7#54711yescharacteristic
445C4×C7#55411yescharacteristic
446Dic7—22no—
447Dic7—22no—
448Dic7—22no—
449Dic7—42no—
450Dic7—41no—
451Dic7—81no—
452Dic7—81no—
453C24—141no—
454C2×Q8—21no—
455C2×Q8—21no—
456C2×Q8—21no—
457C2×Q8C2×C4×D712yes—
458C2×Q8D4×D712yes—
459C2×Q8Q8×D712yes—
460C2×D4—72no—
461C2×D4—72no—
462C2×D4—72no—
463C2×D4—141no—
464C2×D4—141no—
465C2×D4—141no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—71no—
471C22×C4—71no—
472C22×C4—71no—
473C22×C4—71no—
474C22×C4—144no—
475C22×C4—141no—
476C22×C4—141no—
477C22×C4—141no—
478C22×C4—282no—
479C22×C4—282no—
480C22×C4—282no—
481C22×C4—282no—
482C22×C4—561no—
483C22×C4—561no—
484C22×C4—561no—
485C22×C4—561no—
486C22×C4D4×D711yescharacteristic
487C22×C4D4×D711yescharacteristic
488C22×C4C23×D711yescharacteristic
489SD16—716no—
490C2×C8—21no—
491C2×C8—21no—
492C2×C8—21no—
493C2×C8—21no—
494C2×C8—21no—
495C2×C8—72no—
496C2×C8—72no—
497C2×C8—72no—
498C2×C8—141no—
499C2×C8—141no—
500C2×C8—141no—
501C2×C8C2×C4×D712yes—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—42no—
508C4⋊C4—72no—
509C4⋊C4—81no—
510C4⋊C4—142no—
511C4⋊C4—144no—
512C4⋊C4—141no—
513C4⋊C4—141no—
514C4⋊C4—141no—
515C4⋊C4—141no—
516C4⋊C4—141no—
517C4⋊C4—281no—
518C4⋊C4—282no—
519C4⋊C4—282no—
520C4⋊C4—561no—
521C4⋊C4—561no—
522C4⋊C4—561no—
523C4⋊C4—561no—
524C4⋊C4C2×C4×D712yes—
525C22⋊C4—144no—
526C22⋊C4—282no—
527C22⋊C4—282no—
528C42—21no—
529C42—21no—
530C42—42no—
531C42—72no—
532C42—72no—
533C42—144no—
534C42—141no—
535C42—141no—
536C42—282no—
537C2×C7D4×C4⋊C411yescharacteristic
538C2×C7C4×#13711yescharacteristic
539C2×C7#294611yescharacteristic
540C2×C7#326611yescharacteristic
541C2×C7#326811yescharacteristic
542C2×C7#338511yescharacteristic
543C2×C7#338811yescharacteristic
544D7—44no—
545C23—142no—
546C23—142no—
547C23—142no—
548C23—281no—
549C23—281no—
550C23—281no—
551C23—281no—
552C23C2×D4×D711yescharacteristic
553Q8—22no—
554Q8—22no—
555Q8—22no—
556Q8D7×C4⋊C414yes—
557D4—74no—
558D4—142no—
559D4—142no—
560D4—142no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—22no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—72no—
580C2×C4—72no—
581C2×C4—72no—
582C2×C4—141no—
583C2×C4—141no—
584C2×C4—141no—
585C2×C4—141no—
586C2×C4—141no—
587C2×C4—141no—
588C2×C4—141no—
589C2×C4—141no—
590C2×C4—142no—
591C2×C4—141no—
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×C4—564no—
603C2×C4C22×C4×D711yescharacteristic
604C2×C4C2×D4×D711yescharacteristic
605C2×C4#654811yescharacteristic
606C2×C4C2×Q8×D711yescharacteristic
607C2×C4D7×#4111yescharacteristic
608C2×C4#655411yescharacteristic
609C8—22no—
610C8—41no—
611C8—74no—
612C8—281no—
613C7#2267811yescharacteristic, complemented, Sylow
614C22—144no—
615C22—282no—
616C22—282no—
617C22—282no—
618C22—282no—
619C22—282no—
620C22—282no—
621C22C4×D4×D711yescharacteristic
622C22#9028111yescharacteristic
623C22#9032311yescharacteristic
624C22#9045311yescharacteristic
625C22#9046011yescharacteristic
626C22D7×#13711yescharacteristic
627C22#9046811yescharacteristic
628C4—24no—
629C4—42no—
630C4—41no—
631C4—41no—
632C4—81no—
633C4—81no—
634C4—142no—
635C4—142no—
636C4—142no—
637C4—282no—
638C4—281no—
639C4—561no—
640C4—561no—
641C4C2×D7×C4⋊C411yescharacteristic
642C4#9023711yescharacteristic
643C4#9032711yescharacteristic
644C4#9034611yescharacteristic
645C2—284no—
646C2#1173183811yescharacteristic
647C2#1173709211yescharacteristic
648C2#1173727411yescharacteristic
649C2#1173750611yescharacteristic
650C2#1173771111yescharacteristic
651C2#1173845311yescharacteristic
652C2#1173866811yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1865C2
socleC23×C7C22×D4
prime radicalC22×D28C23
cosocleC22×C4×C7C24
hypercenter#1865D7
hypocenterC7#22678
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
1. Q8×Dic7⋊C4C2semidirect
6. C7×#1865C2semidirect
8. #11728310C2semidirect
9. #11727990C2semidirect
10. #11727976C2semidirect
11. #11727374C2semidirect
13. #11726613C2semidirect
15. #11726408C2semidirect
52. #89589C4semidirect
82. Q8×Dic7C2×C4semidirect
115. C7×#103C2×C4semidirect
128. #6389C2×C4semidirect
134. #6381C2×C4semidirect
143. #1865D7semidirect
204. #1143C4⋊C4semidirect
407. #103C4×D7semidirect
613. C7#22678semidirect

Maximal subgroups

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

subgroupindex
Q8×Dic7⋊C42
C2×#897982
C2×#895892
C2×#895602
#117336482
C7×#18652
#117283382
#117283102
#117279902
#117279762
#117273742
#117266462
#117266132
#117264242
#117264082
#226787