Finite Group Databasebeta

← #13357185

Subgroups — #13357185

Order 1792.

1058 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89653C211yesmaximal, characteristic
2C2×#89633C211yesmaximal, characteristic
3C2×#89479C211yesmaximal, characteristic
4D7×#389C211yesmaximal, characteristic
5#11733408C211yesmaximal, characteristic
6C7×#1944C211yesmaximal, characteristic, complemented
7#11728365C211yesmaximal, characteristic, complemented
8#11728314C211yesmaximal, characteristic, complemented
9#11727573C211yesmaximal, characteristic, complemented
10#11727375C211yesmaximal, characteristic, complemented
11#11727374C211yesmaximal, characteristic, complemented
12#11726621C211yesmaximal, characteristic
13#11726621C211yesmaximal, characteristic
14#11726613C211yesmaximal, characteristic, complemented
15#11726612C211yesmaximal, characteristic, complemented
16C22×#1131C2211yescharacteristic
17C22×#1131C2211yescharacteristic
18C2×#6455C2211yescharacteristic
19C2×#6455C2211yescharacteristic
20C2×D7×C4⋊C4—21no—
21C2×D7×C4⋊C4C2211yescharacteristic
22C2×D7×C4⋊C4C2211yescharacteristic
23C2×C42×D7C2211yescharacteristic
24C2×C7×#107C2211yescharacteristic
25C2×C7×#106C2211yescharacteristic
26C2×C7×#97C2211yescharacteristic
27C7×#389C2211yescharacteristic
28C7×#389C2211yescharacteristic
29C2×#6392C2211yescharacteristic
30C2×#6391C2211yescharacteristic
31C2×C8×Dic7C2211yescharacteristic
32C2×C8×Dic7C2211yescharacteristic
33#89761—21no—
34#89757C2211yescharacteristic
35#89757C2211yescharacteristic
36C4⋊C4×Dic7—21no—
37#89745C2211yescharacteristic
38#89745C2211yescharacteristic
39#89744—21no—
40C2×#6382C2211yescharacteristic
41C2×#6382C2211yescharacteristic
42C2×#6381C2211yescharacteristic
43C2×#6381C2211yescharacteristic
44#89700C2211yescharacteristic
45#89700C2211yescharacteristic
46C2×#6377C2211yescharacteristic
47#89653—24no—
48#89633C418yescomplemented
49#89479—24no—
50#89440—21no—
51D7×#95—21no—
52#89421—21no—
53#89419—21no—
54C7×#340C2211yescharacteristic
55C7×#340C2211yescharacteristic
56#89346C2211yescharacteristic
57#89346C2211yescharacteristic
58#89324C2211yescharacteristic
59#89324C2211yescharacteristic
60#89324C2211yescharacteristic
61#89324C2211yescharacteristic
62#21159—71nomaximal, Sylow
63C22×C4×D7—21no—
64C22×C4×D7—21no—
65C22×C4×D7D411yescharacteristic
66C22×C4×D7D411yescharacteristic
67C22×C4×D7C2311yescharacteristic
68C22×C7×C8C2311yescharacteristic
69C22×C7×C8C2311yescharacteristic
70C2×C7×C4⋊C4—21no—
71C2×C7×C4⋊C4—21no—
72C2×C7×C4⋊C4C2311yescharacteristic
73C2×C7×C4⋊C4C2311yescharacteristic
74C2×C7×C4⋊C4C2311yescharacteristic
75C2×C7×C4⋊C4C2311yescharacteristic
76C2×C42×C7C2311yescharacteristic
77C2×(C2×D7)⋊C4—21no—
78C2×(C2×D7)⋊C4—21no—
79C2×C4⋊Dic7—21no—
80C2×C4⋊Dic7—21no—
81C2×C4⋊Dic7C2311yescharacteristic
82C2×C4⋊Dic7C2311yescharacteristic
83C2×C4⋊Dic7C2311yescharacteristic
84C2×C4⋊Dic7C2311yescharacteristic
85C2×Dic7⋊C4—21no—
86C2×Dic7⋊C4—21no—
87C2×Dic7⋊C4—21no—
88C2×Dic7⋊C4—21no—
89C2×Dic7⋊C4—21no—
90C2×Dic7⋊C4—21no—
91C2×C4×Dic7—21no—
92C2×C4×Dic7—21no—
93C2×C4×Dic7D411yescharacteristic
94C2×C4×Dic7D411yescharacteristic
95C2×C4×Dic7C2311yescharacteristic
96C22×C7⋊C8C2311yescharacteristic
97C22×C7⋊C8C2311yescharacteristic
98C2×#1131—22no—
99C2×#1131—22no—
100C2×#1131C2×C414yes—
101C2×#1131D414yes—
102C2×#1131Q814yes—
103C2×#1131Q814yes—
104#6455—22no—
105#6455—22no—
106#6455—22no—
107#6455C2×C414yes—
108D7×C4⋊C4—22no—
109D7×C4⋊C4—22no—
110D7×C4⋊C4—22no—
111D7×C4⋊C4—44no—
112D7×C4⋊C4C2×C414yes—
113C42×D7—22no—
114C42×D7—22no—
115C7×#107—22no—
116C7×#106C2×C414yescomplemented
117C7×#97—22no—
118C7×#95—21no—
119C7×#95—21no—
120C7⋊#95—21no—
121C7⋊#95—21no—
122C7⋊#95—21no—
123C7⋊#95—21no—
124C7⋊#95—21no—
125C7⋊#95—21no—
126#6392C2×C414yescomplemented
127#6391—22no—
128C8×Dic7—22no—
129C8×Dic7—22no—
130#6382—22no—
131#6382C2×C414yescomplemented
132#6381—22no—
133#6381C2×C414yescomplemented
134#6377—22no—
135C2×#428—71no—
136C2×#428—71no—
137C2×C4×M16—71no—
138#2296—71no—
139C2×#389—71no—
140#1944—71no—
141#1944D711yescharacteristic, complemented
142#1943—71no—
143#1943—71no—
144#1942—71no—
145#1942—71no—
146#1941—71no—
147#1941—71no—
148#1736—71no—
149#1736—71no—
150C23×D7C2×D411yescharacteristic
151C22×C4×C7—21no—
152C22×C4×C7—21no—
153C22×C4×C7—21no—
154C22×C4×C7—21no—
155C22×C4×C7C2×D411yescharacteristic
156C22×C4×C7C2×D411yescharacteristic
157C22×C4×C7C2411yescharacteristic
158C22×Dic7—21no—
159C22×Dic7—21no—
160C22×Dic7—21no—
161C22×Dic7—21no—
162C22×Dic7C2×D411yescharacteristic
163C22×Dic7C2×D411yescharacteristic
164C22×Dic7C2×D411yescharacteristic
165C2×C4×D7—24no—
166C2×C4×D7—22no—
167C2×C4×D7—22no—
168C2×C4×D7—22no—
169C2×C4×D7—22no—
170C2×C4×D7—41no—
171C2×C4×D7—41no—
172C2×C4×D7—42no—
173C2×C4×D7—42no—
174C2×C4×D7—42no—
175C2×C4×D7—42no—
176C2×C4×D7—42no—
177C2×C4×D7C22×C412yes—
178C2×C4×D7C2×D412yes—
179C2×C4×D7C2×Q812yes—
180C2×C4×D7C2×Q812yes—
181C2×C4×D7#4112yes—
182C2×C4×D7#4112yes—
183C2×C7×C8—21no—
184C2×C7×C8—21no—
185C2×C7×C8C22×C412yes—
186C2×C7×C8C2×D412yes—
187C2×C7×C8C2×Q812yes—
188C2×C7×C8C2×Q812yes—
189C7×C4⋊C4—21no—
190C7×C4⋊C4—21no—
191C7×C4⋊C4—21no—
192C7×C4⋊C4—21no—
193C7×C4⋊C4—21no—
194C7×C4⋊C4—21no—
195C7×C4⋊C4—42no—
196C7×C4⋊C4—42no—
197C7×C4⋊C4C22×C412yes—
198C7×C4⋊C4C22×C412yes—
199C42×C7—21no—
200C42×C7—21no—
201(C2×D7)⋊C4—24no—
202(C2×D7)⋊C4—42no—
203C4⋊Dic7—21no—
204C4⋊Dic7—21no—
205C4⋊Dic7—21no—
206C4⋊Dic7—21no—
207C4⋊Dic7—21no—
208C4⋊Dic7—21no—
209C4⋊Dic7—42no—
210C4⋊Dic7—42no—
211C4⋊Dic7C22×C412yes—
212C4⋊Dic7C22×C412yes—
213Dic7⋊C4—24no—
214Dic7⋊C4—42no—
215Dic7⋊C4—42no—
216Dic7⋊C4—42no—
217Dic7⋊C4—42no—
218Dic7⋊C4—42no—
219C4×Dic7—21no—
220C4×Dic7—21no—
221C4×Dic7—24no—
222C4×Dic7—22no—
223C4×Dic7—22no—
224C4×Dic7—42no—
225C2×C7⋊C8—21no—
226C2×C7⋊C8—21no—
227C2×C7⋊C8C22×C412yes—
228C2×C7⋊C8C2×D412yes—
229C2×C7⋊C8C2×Q812yes—
230C2×C7⋊C8C2×Q812yes—
231#1131—28no—
232#1131C4⋊C4116yescomplemented
233C22×M16—71no—
234C22×M16—71no—
235C2×#117—71no—
236C2×#117—71no—
237C22×C4⋊C4—71no—
238C22×C4⋊C4—71no—
239C22×C4⋊C4—141no—
240C22×C42—71no—
241#428—78no—
242#428—144no—
243C2×#107—71no—
244C2×#107—71no—
245C2×#107—71no—
246C2×#107C2×D711yescharacteristic
247C2×#106—71no—
248C2×#106—71no—
249C2×#106—71no—
250C2×#106C2×D711yescharacteristic
251C4×M16—144no—
252C2×#97—71no—
253C2×#97C2×D711yescharacteristic
254C2×C4×C8—71no—
255C2×C4×C8—71no—
256#389—71no—
257#389—71no—
258#389—71no—
259#389—71no—
260#389—71no—
261#389—71no—
262#389—141no—
263#389—141no—
264#389C2×D711yescharacteristic
265#389C2×D711yescharacteristic
266#385—141no—
267#385—141no—
268C4×C4⋊C4—141no—
269C4×C4⋊C4—141no—
270C2×#95—141no—
271#340—71no—
272#340—71no—
273#340—71no—
274#340—71no—
275#340—71no—
276#340—71no—
277#340C2×D711yescharacteristic
278#340C2×D711yescharacteristic
279C23×C7C22×D411yescharacteristic
280C22×D7—21no—
281C22×D7—21no—
282C22×D7—21no—
283C22×D7—21no—
284C22×D7C4×D412yes—
285C22×D7#12112yes—
286C22×D7C4⋊Q812yes—
287C2×C4×C7—22no—
288C2×C4×C7—22no—
289C2×C4×C7—21no—
290C2×C4×C7—21no—
291C2×C4×C7—21no—
292C2×C4×C7—21no—
293C2×C4×C7—21no—
294C2×C4×C7—21no—
295C2×C4×C7—41no—
296C2×C4×C7—41no—
297C2×C4×C7—41no—
298C2×C4×C7—41no—
299C2×C4×C7—41no—
300C2×C4×C7—41no—
301C2×C4×C7—41no—
302C2×C4×C7—41no—
303C2×C4×C7—41no—
304C2×C4×C7—41no—
305C2×C4×C7C23×C411yescharacteristic
306C2×C4×C7C22×D411yescharacteristic
307C2×C4×C7C22×Q811yescharacteristic
308C2×C4×C7C22×Q811yescharacteristic
309C2×C4×C7C2×#4111yescharacteristic
310C2×C4×C7C2×#4111yescharacteristic
311C2×Dic7—21no—
312C2×Dic7—21no—
313C2×Dic7—21no—
314C2×Dic7—21no—
315C2×Dic7—22no—
316C2×Dic7—22no—
317C2×Dic7—41no—
318C2×Dic7—41no—
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×Dic7C4×D412yes—
328C2×Dic7#12112yes—
329C2×Dic7C4⋊Q812yes—
330C2×Dic7AGL(1,ℤ/8)12yes—
331C2×Dic7#13712yes—
332C4×D7—44no—
333C4×D7—44no—
334C4×D7—48no—
335C4×D7—84no—
336C4×D7C2×C4⋊C414yes—
337C4×D7C4×Q814yes—
338C4×D7C22⋊Q814yes—
339C4×D7#12514yes—
340C7×C8—22no—
341C7×C8C2×C4⋊C414yescomplemented
342C7⋊C8—22no—
343C7⋊C8C2×C4⋊C414yescomplemented
344C23×C4—71no—
345C23×C4—71no—
346C23×C4—71no—
347C23×C4—141no—
348C23×C4—141no—
349C2×M16—74no—
350C2×M16—74no—
351C2×M16—74no—
352C2×M16—74no—
353C2×M16—142no—
354C2×M16—142no—
355C22×C8—71no—
356C22×C8—71no—
357C22×C8C22×D711yescharacteristic
358C22×C8C22×D711yescharacteristic
359#117—74no—
360#117—142no—
361#117—142no—
362#117—142no—
363C2×C4⋊C4—21no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—71no—
366C2×C4⋊C4—71no—
367C2×C4⋊C4—74no—
368C2×C4⋊C4—71no—
369C2×C4⋊C4—71no—
370C2×C4⋊C4—142no—
371C2×C4⋊C4—142no—
372C2×C4⋊C4—142no—
373C2×C4⋊C4—141no—
374C2×C4⋊C4—141no—
375C2×C4⋊C4—141no—
376C2×C4⋊C4—141no—
377C2×C4⋊C4—141no—
378C2×C4⋊C4—141no—
379C2×C4⋊C4—141no—
380C2×C4⋊C4—141no—
381C2×C4⋊C4—284no—
382C2×C4⋊C4C22×D711yescharacteristic
383C2×C4⋊C4C22×D711yescharacteristic
384C2×C4⋊C4C22×D711yescharacteristic
385C2×C4⋊C4C22×D711yescharacteristic
386C2×C22⋊C4—141no—
387C2×C22⋊C4—141no—
388C2×C42—71no—
389C2×C42—71no—
390C2×C42—71no—
391C2×C42—142no—
392C2×C42—142no—
393C2×C42—141no—
394C2×C42—141no—
395C2×C42C22×D711yescharacteristic
396#107—22no—
397#107—74no—
398#107—74no—
399#107—142no—
400#106—74no—
401#106—142no—
402#106—142no—
403#106C4×D714yescomplemented
404#97—22no—
405#97—142no—
406C4×C8—142no—
407C4×C8—142no—
408#95—21no—
409#95—21no—
410#95—141no—
411#95—141no—
412#95—141no—
413#95—141no—
414#95—141no—
415#95—141no—
416C22×C7C2×C4×D411yescharacteristic
417C22×C7C2×#12111yescharacteristic
418C22×C7C2×C4⋊Q811yescharacteristic
419C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
420C22×C7C2×#13711yescharacteristic
421C22×C7#57511yescharacteristic
422C22×C7#57511yescharacteristic
423C2×D7—22no—
424C2×D7—22no—
425C2×D7—22no—
426C2×D7—22no—
427C2×D7—22no—
428C2×D7—22no—
429C2×D7#38914yes—
430C4×C7—42no—
431C4×C7—42no—
432C4×C7—41no—
433C4×C7—41no—
434C4×C7—81no—
435C4×C7—81no—
436C4×C7C22×C4⋊C411yescharacteristic
437C4×C7C2×C4×Q811yescharacteristic
438C4×C7C2×C22⋊Q811yescharacteristic
439C4×C7C2×#12511yescharacteristic
440Dic7—22no—
441Dic7—22no—
442Dic7—42no—
443Dic7—42no—
444Dic7—81no—
445Dic7—81no—
446Dic7#38914yes—
447C24—71no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—71no—
453C22×C4—72no—
454C22×C4—72no—
455C22×C4—72no—
456C22×C4—72no—
457C22×C4—72no—
458C22×C4—72no—
459C22×C4—71no—
460C22×C4—71no—
461C22×C4—142no—
462C22×C4—142no—
463C22×C4—142no—
464C22×C4—142no—
465C22×C4—144no—
466C22×C4—141no—
467C22×C4—141no—
468C22×C4—141no—
469C22×C4—141no—
470C22×C4—281no—
471C22×C4—281no—
472C22×C4—282no—
473C22×C4—282no—
474C22×C4—282no—
475C22×C4—282no—
476C22×C4—282no—
477C22×C4D4×D711yescharacteristic
478C22×C4D4×D711yescharacteristic
479C22×C4C23×D711yescharacteristic
480M16—716no—
481M16—148no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—72no—
485C2×C8—72no—
486C2×C8—72no—
487C2×C8—72no—
488C2×C8—141no—
489C2×C8—141no—
490C2×C8C2×C4×D712yes—
491C2×C8D4×D712yes—
492C2×C8Q8×D712yes—
493C2×C8Q8×D712yes—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—42no—
501C4⋊C4—42no—
502C4⋊C4—72no—
503C4⋊C4—72no—
504C4⋊C4—144no—
505C4⋊C4—141no—
506C4⋊C4—141no—
507C4⋊C4—141no—
508C4⋊C4—141no—
509C4⋊C4—141no—
510C4⋊C4—141no—
511C4⋊C4—282no—
512C4⋊C4—282no—
513C4⋊C4—282no—
514C4⋊C4—282no—
515C4⋊C4—282no—
516C4⋊C4—282no—
517C4⋊C4—282no—
518C4⋊C4C2×C4×D712yes—
519C4⋊C4C2×C4×D712yes—
520C22⋊C4—144no—
521C22⋊C4—282no—
522C42—21no—
523C42—21no—
524C42—142no—
525C42—142no—
526C42—144no—
527C42—141no—
528C42—141no—
529C42—282no—
530C2×C7C2×#38911yescharacteristic
531C2×C7C4×AGL(1,ℤ/8)11yescharacteristic
532C2×C7C4×#13711yescharacteristic
533C2×C7#305511yescharacteristic
534C2×C7#305611yescharacteristic
535C2×C7#316511yescharacteristic
536C2×C7#316611yescharacteristic
537D7—24no—
538C23—72no—
539C23—72no—
540C23—72no—
541C23—141no—
542C23—141no—
543C23—141no—
544C23—141no—
545C23C2×D4×D711yescharacteristic
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—72no—
565C2×C4—72no—
566C2×C4—72no—
567C2×C4—74no—
568C2×C4—74no—
569C2×C4—74no—
570C2×C4—74no—
571C2×C4—72no—
572C2×C4—72no—
573C2×C4—141no—
574C2×C4—141no—
575C2×C4—141no—
576C2×C4—141no—
577C2×C4—142no—
578C2×C4—142no—
579C2×C4—284no—
580C2×C4—284no—
581C2×C4—288no—
582C2×C4—281no—
583C2×C4—281no—
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—564no—
593C2×C4C22×C4×D711yescharacteristic
594C2×C4C2×D4×D711yescharacteristic
595C2×C4C2×Q8×D711yescharacteristic
596C2×C4C2×Q8×D711yescharacteristic
597C2×C4D7×#4111yescharacteristic
598C2×C4D7×#4111yescharacteristic
599C8—22no—
600C8—74no—
601C8—142no—
602C8D7×C4⋊C414yescomplemented
603C7#2115911yescharacteristic, complemented, Sylow
604C22—74no—
605C22—142no—
606C22—142no—
607C22—142no—
608C22—142no—
609C22—142no—
610C22—142no—
611C22C4×D4×D711yescharacteristic
612C22D7×#12111yescharacteristic
613C22D7×C4⋊Q811yescharacteristic
614C22D7×AGL(1,ℤ/8)11yescharacteristic
615C22#9046411yescharacteristic
616C22D7×#13711yescharacteristic
617C22#9046811yescharacteristic
618C4—42no—
619C4—42no—
620C4—41no—
621C4—41no—
622C4—74no—
623C4—81no—
624C4—81no—
625C4—142no—
626C4—142no—
627C4—282no—
628C4—282no—
629C4—561no—
630C4—561no—
631C4C2×D7×C4⋊C411yescharacteristic
632C4C4×Q8×D711yescharacteristic
633C4D7×C22⋊Q811yescharacteristic
634C4D7×#12511yescharacteristic
635C2—144no—
636C2D7×#38911yescharacteristic
637C2#1173721811yescharacteristic
638C2#1173721911yescharacteristic
639C2#1173830911yescharacteristic
640C2#1173834911yescharacteristic
641C2#1173942111yescharacteristic
642C2#1173942211yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1944C2
socleC23×C7C22×D4
prime radicalC23×D7C2×D4
cosocleC22×C4×C7C24
hypercenter#1944D7
hypocenterC7#21159
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1944

Complements

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

NH ≅ G/Ntype
6. C7×#1944C2semidirect
7. #11728365C2semidirect
8. #11728314C2semidirect
9. #11727573C2semidirect
10. #11727375C2semidirect
11. #11727374C2semidirect
14. #11726613C2semidirect
15. #11726612C2semidirect
48. #89633C4semidirect
116. C7×#106C2×C4semidirect
126. #6392C2×C4semidirect
131. #6382C2×C4semidirect
133. #6381C2×C4semidirect
141. #1944D7semidirect
232. #1131C4⋊C4semidirect
341. C7×C8C2×C4⋊C4semidirect
343. C7⋊C8C2×C4⋊C4semidirect
403. #106C4×D7semidirect
602. C8D7×C4⋊C4semidirect
603. C7#21159semidirect

Maximal subgroups

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

subgroupindex
C2×#896532
C2×#896332
C2×#894792
D7×#3892
#117334082
C7×#19442
#117283652
#117283142
#117275732
#117273752
#117273742
#117266212
#117266132
#117266122
#211597