Finite Group Databasebeta

← #13372216

Subgroups — #13372216

Order 1792.

986 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11739779C211yesmaximal, characteristic
2C2×C4×#1142C211yesmaximal, characteristic
3C2×#89530C211yesmaximal, characteristic
4C2×#89530C211yesmaximal, characteristic
5#11733427C211yesmaximal, characteristic, complemented
6C7×#1862C211yesmaximal, characteristic
7#11728396C211yesmaximal, characteristic, complemented
8#11728361C211yesmaximal, characteristic, complemented
9#11727922C211yesmaximal, characteristic
10#11727908C211yesmaximal, characteristic
11#11727372C211yesmaximal, characteristic, complemented
12#11726659C211yesmaximal, characteristic, complemented
13#11726612C211yesmaximal, characteristic, complemented
14#11726437C211yesmaximal, characteristic, complemented
15#11726405C211yesmaximal, characteristic, complemented
16C2×C4×C7×D4C2211yescharacteristic
17C2×D4×Dic7C2211yescharacteristic
18C2×D4×Dic7C2211yescharacteristic
19C22×#1142C2211yescharacteristic
20C2×#6450C2211yescharacteristic
21C2×#6450C2211yescharacteristic
22C2×C4×Dic14C2211yescharacteristic
23C2×C7×#107C2211yescharacteristic
24C2×C7×#102C2211yescharacteristic
25C2×C7×#102C2211yescharacteristic
26C7×#389C2211yescharacteristic
27C2×#6406C2211yescharacteristic
28#89982—21no—
29C2×C7⋊#95—21no—
30C2×#6392C2211yescharacteristic
31C2×#6390C2211yescharacteristic
32C2×#6390C2211yescharacteristic
33C2×Dic7⋊C8C2211yescharacteristic
34C2×C8×Dic7C2211yescharacteristic
35C2×C8×Dic7C2211yescharacteristic
36C4×#1142—24no—
37#89757C2211yescharacteristic
38#89749C2211yescharacteristic
39#89749C2211yescharacteristic
40C4⋊C4×Dic7—21no—
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#89422—21no—
55#89419—21no—
56C7×#340C2211yescharacteristic
57C7×#336C2211yescharacteristic
58#89346C2211yescharacteristic
59#89345C2211yescharacteristic
60#89324C2211yescharacteristic
61#89324C2211yescharacteristic
62#22683—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⋊C4D411yescharacteristic
97C2×Dic7⋊C4D411yescharacteristic
98C2×C4×Dic7—21no—
99C2×C4×Dic7—21no—
100C2×C4×Dic7—21no—
101C2×C4×Dic7C2311yescharacteristic
102C2×C4×Dic7C2311yescharacteristic
103C22×C7⋊C8C2311yescharacteristic
104C22×C7⋊C8C2311yescharacteristic
105#6450—22no—
106#6450—22no—
107#6450—41no—
108#6450C2×C414yescomplemented
109C4×Dic14—22no—
110C4×Dic14—41no—
111C7×#107—41no—
112C7×#102—22no—
113C7×#102C2×C414yes—
114C7×#95—21no—
115#6406—22no—
116C7⋊#95—21no—
117C7⋊#95—21no—
118C7⋊#95—21no—
119C7⋊#95—21no—
120C7⋊#95—21no—
121C7⋊#95—21no—
122C7⋊#95—21no—
123C7⋊#95D411yescharacteristic
124C7⋊#95D411yescharacteristic
125#6392—41no—
126#6390—22no—
127#6390C2×C414yescomplemented
128Dic7⋊C8—22no—
129C8×Dic7—22no—
130C8×Dic7C2×C414yescomplemented
131#6384—22no—
132#6382—22no—
133#6382—22no—
134#6382C2×C414yescomplemented
135C4×C7⋊C8—22no—
136C2×C4×SD16—71no—
137C2×C4×SD16—71no—
138C2×C4×SD16—71no—
139#2351—71no—
140#2349—71no—
141#1941—71no—
142#1941—71no—
143#1863—71no—
144#1863—71no—
145#1862—71no—
146#1862D711yescharacteristic
147#1846—71no—
148#1805—71no—
149#1804—71no—
150C8×C4⋊C4—71no—
151C24×C7—21no—
152C2×C7×D4—21no—
153C2×C7×D4—21no—
154C2×C7×D4—21no—
155C2×C7×D4C22×C412yes—
156C2×C7×D4C2×D412yes—
157C2×C7×D4C2×Q812yes—
158C22×C4×C7—21no—
159C22×C4×C7—21no—
160C22×C4×C7—21no—
161C22×C4×C7—42no—
162C22×C4×C7—42no—
163C22×C4×C7—81no—
164C22×C4×C7C2×D411yescharacteristic
165C22×C4×C7C2×D411yescharacteristic
166C22×C4×C7C2411yescharacteristic
167C22×Dic7—21no—
168C22×Dic7—21no—
169C22×Dic7—24no—
170C22×Dic7—21no—
171C22×Dic7—21no—
172C22×Dic7—42no—
173C22×Dic7—42no—
174C22×Dic7—81no—
175C22×Dic7—81no—
176C22×Dic7—81no—
177C22×Dic7C2×D411yescharacteristic
178C22×Dic7C2×D411yescharacteristic
179C22×Dic7C2×D411yescharacteristic
180C22×Dic7C2×D411yescharacteristic
181C2×Dic14—21no—
182C2×Dic14—21no—
183C2×Dic14—21no—
184C2×Dic14C22×C412yes—
185C2×Dic14C2×D412yes—
186C2×Dic14C2×Q812yes—
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⋊C4C22×C412yes—
200C7×C22⋊C4—42no—
201C42×C7—21no—
202C42×C7—21no—
203C22⋊Dic7—24no—
204C22⋊Dic7—42no—
205#1142C4⋊C4116yescomplemented
206C4⋊Dic7—21no—
207C4⋊Dic7—21no—
208C4⋊Dic7—21no—
209C4⋊Dic7—21no—
210C4⋊Dic7—21no—
211C4⋊Dic7—42no—
212C4⋊Dic7C22×C412yes—
213Dic7⋊C4—22no—
214Dic7⋊C4—24no—
215Dic7⋊C4—22no—
216Dic7⋊C4—42no—
217Dic7⋊C4—42no—
218Dic7⋊C4—81no—
219Dic7⋊C4—81no—
220C4×Dic7—21no—
221C4×Dic7—21no—
222C4×Dic7—24no—
223C4×Dic7—42no—
224C4×Dic7—42no—
225C4×Dic7C22×C412yes—
226C4×Dic7#4112yes—
227C2×C7⋊C8—21no—
228C2×C7⋊C8—21no—
229C2×C7⋊C8—21no—
230C2×C7⋊C8C22×C412yes—
231C2×C7⋊C8C2×D412yes—
232C2×C7⋊C8C2×Q812yes—
233C22×SD16—71no—
234C2×C4×Q8—71no—
235C2×C4×Q8—71no—
236C2×C4×Q8—71no—
237C2×C4×D4—71no—
238C2×C4×D4—71no—
239C2×C4×D4C2×D711yescharacteristic
240C4×SD16—78no—
241C4×SD16—144no—
242C4×SD16—144no—
243C2×#107—71no—
244C2×#107C2×D711yescharacteristic
245C2×#106—71no—
246C2×#106—71no—
247C2×#106—71no—
248C2×C4⋊C8—71no—
249C2×#103—71no—
250C2×#103—71no—
251C2×#103—71no—
252C2×#102—71no—
253C2×#102C2×D711yescharacteristic
254C2×#102C2×D711yescharacteristic
255C2×C4×C8—71no—
256C2×C4×C8—71no—
257C2×C4×C8—71no—
258#389—71no—
259#389—71no—
260#389—71no—
261#389—141no—
262#389—141no—
263#389C2×D711yescharacteristic
264#385—141no—
265#385—141no—
266#382—71no—
267#382—71no—
268#382—141no—
269#380—141no—
270C4×C4⋊C4—71no—
271C4×C4⋊C4—141no—
272C2×#95—141no—
273#340—71no—
274#340—71no—
275#340—71no—
276#340C2×D711yescharacteristic
277#336—71no—
278#336C2×D711yescharacteristic
279C23×C7—22no—
280C23×C7—22no—
281C23×C7—22no—
282C23×C7—41no—
283C23×C7—41no—
284C23×C7—41no—
285C23×C7—41no—
286C23×C7C22×D411yescharacteristic
287C7×D4—22no—
288C7×D4—22no—
289C7×D4—22no—
290C7×D4C2×C4⋊C414yes—
291C2×C4×C7—21no—
292C2×C4×C7—21no—
293C2×C4×C7—22no—
294C2×C4×C7—21no—
295C2×C4×C7—21no—
296C2×C4×C7—21no—
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×C7—84no—
306C2×C4×C7#13512yes—
307C2×C4×C7C23×C411yescharacteristic
308C2×C4×C7C22×D411yescharacteristic
309C2×C4×C7C22×Q811yescharacteristic
310C2×C4×C7C2×#4111yescharacteristic
311C2×C4×C7H(5,𝔽2)11yescharacteristic
312C2×C4×C72−1+411yescharacteristic
313C2×Dic7—22no—
314C2×Dic7—21no—
315C2×Dic7—22no—
316C2×Dic7—22no—
317C2×Dic7—21no—
318C2×Dic7—21no—
319C2×Dic7—21no—
320C2×Dic7—21no—
321C2×Dic7—21no—
322C2×Dic7—22no—
323C2×Dic7—21no—
324C2×Dic7—21no—
325C2×Dic7—21no—
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—84no—
335C2×Dic7—84no—
336C2×Dic7C4×D412yes—
337C2×Dic7C4×D412yes—
338C2×Dic7#13512yes—
339Dic14—22no—
340Dic14—22no—
341Dic14—22no—
342Dic14C2×C4⋊C414yes—
343C7×C8—22no—
344C7×C8—41no—
345C7⋊C8—41no—
346C7⋊C8C2×C4⋊C414yes—
347C22×Q8—71no—
348C22×D4C22×D711yescharacteristic
349C23×C4—21no—
350C23×C4—141no—
351C23×C4—141no—
352C2×SD16—74no—
353C2×SD16—74no—
354C2×SD16—74no—
355C22×C8—71no—
356C22×C8—71no—
357C22×C8C22×D711yescharacteristic
358C22×C8C22×D711yescharacteristic
359C4×Q8—74no—
360C4×Q8—142no—
361C4×Q8—142no—
362C4×Q8—142no—
363C4×Q8—281no—
364C4×Q8—281no—
365C4×D4—22no—
366C4×D4—41no—
367C4×D4—74no—
368C4×D4—142no—
369C4×D4—142no—
370C4×D4—281no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—71no—
373C2×C4⋊C4—71no—
374C2×C4⋊C4—71no—
375C2×C4⋊C4—71no—
376C2×C4⋊C4—71no—
377C2×C4⋊C4—141no—
378C2×C4⋊C4—141no—
379C2×C4⋊C4—141no—
380C2×C4⋊C4—141no—
381C2×C4⋊C4—141no—
382C2×C4⋊C4—141no—
383C2×C4⋊C4C22×D711yescharacteristic
384C2×C4⋊C4C22×D711yescharacteristic
385C2×C4⋊C4C22×D711yescharacteristic
386C2×C22⋊C4—21no—
387C2×C22⋊C4—141no—
388C2×C22⋊C4—141no—
389C2×C42—71no—
390C2×C42—71no—
391C2×C42—141no—
392C2×C42—141no—
393C2×C42—141no—
394C2×C42C22×D711yescharacteristic
395#107—41no—
396#107—281no—
397#106—74no—
398#106—142no—
399#106—142no—
400C4⋊C8—142no—
401#103—74no—
402#103—142no—
403#103—142no—
404#102—22no—
405#102—142no—
406#102C4×D714yes—
407C4×C8—74no—
408C4×C8—142no—
409C4×C8—142no—
410#95—21no—
411#95—71no—
412#95—71no—
413#95—141no—
414#95—141no—
415#95—141no—
416#95—141no—
417#95—141no—
418#95—141no—
419#95—141no—
420C22×C7—24no—
421C22×C7—42no—
422C22×C7—42no—
423C22×C7—42no—
424C22×C7—42no—
425C22×C7—42no—
426C22×C7—42no—
427C22×C7C2×C4×D411yescharacteristic
428C22×C7D4211yescharacteristic
429C22×C7D4×Q811yescharacteristic
430C22×C7C2×#13511yescharacteristic
431C22×C7C2×#13511yescharacteristic
432C22×C7#57711yescharacteristic
433C22×C7#57711yescharacteristic
434C4×C7—22no—
435C4×C7—42no—
436C4×C7—41no—
437C4×C7—81no—
438C4×C7—81no—
439C4×C7C22×C4⋊C411yescharacteristic
440C4×C7#52011yescharacteristic
441C4×C7#54711yescharacteristic
442C4×C7#55411yescharacteristic
443Dic7—24no—
444Dic7—22no—
445Dic7—22no—
446Dic7—22no—
447Dic7—42no—
448Dic7—41no—
449Dic7—81no—
450Dic7—81no—
451C24—21no—
452C2×Q8—72no—
453C2×Q8—72no—
454C2×Q8—72no—
455C2×Q8—141no—
456C2×Q8—141no—
457C2×Q8—141no—
458C2×D4—21no—
459C2×D4—21no—
460C2×D4—21no—
461C2×D4C2×C4×D712yes—
462C2×D4D4×D712yes—
463C2×D4Q8×D712yes—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—71no—
470C22×C4—71no—
471C22×C4—71no—
472C22×C4—71no—
473C22×C4—81no—
474C22×C4—144no—
475C22×C4—141no—
476C22×C4—141no—
477C22×C4—141no—
478C22×C4—141no—
479C22×C4—282no—
480C22×C4—282no—
481C22×C4—561no—
482C22×C4—561no—
483C22×C4—561no—
484C22×C4D4×D711yescharacteristic
485C22×C4D4×D711yescharacteristic
486C22×C4C23×D711yescharacteristic
487SD16—716no—
488C2×C8—21no—
489C2×C8—21no—
490C2×C8—21no—
491C2×C8—21no—
492C2×C8—21no—
493C2×C8—72no—
494C2×C8—72no—
495C2×C8—72no—
496C2×C8—141no—
497C2×C8—141no—
498C2×C8—141no—
499C2×C8C2×C4×D712yes—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—42no—
506C4⋊C4—72no—
507C4⋊C4—142no—
508C4⋊C4—144no—
509C4⋊C4—142no—
510C4⋊C4—141no—
511C4⋊C4—141no—
512C4⋊C4—141no—
513C4⋊C4—141no—
514C4⋊C4—141no—
515C4⋊C4—282no—
516C4⋊C4—282no—
517C4⋊C4—282no—
518C4⋊C4—561no—
519C4⋊C4—561no—
520C4⋊C4C2×C4×D712yes—
521C22⋊C4—42no—
522C22⋊C4—144no—
523C22⋊C4—282no—
524C42—21no—
525C42—21no—
526C42—72no—
527C42—72no—
528C42—144no—
529C42—141no—
530C42—141no—
531C42—282no—
532C42—282no—
533C2×C7—44no—
534C2×C7D4×C4⋊C411yescharacteristic
535C2×C7C4×#13511yescharacteristic
536C2×C7#294911yescharacteristic
537C2×C7#328211yescharacteristic
538C2×C7#328211yescharacteristic
539C2×C7#338111yescharacteristic
540C2×C7#338111yescharacteristic
541C23—22no—
542C23—22no—
543C23—22no—
544C23—41no—
545C23—41no—
546C23—41no—
547C23—41no—
548C23C2×D4×D711yescharacteristic
549Q8—74no—
550Q8—142no—
551Q8—142no—
552Q8—142no—
553D4—22no—
554D4—22no—
555D4—22no—
556D4D7×C4⋊C414yes—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—22no—
562C2×C4—21no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—72no—
572C2×C4—72no—
573C2×C4—72no—
574C2×C4—84no—
575C2×C4—141no—
576C2×C4—141no—
577C2×C4—141no—
578C2×C4—141no—
579C2×C4—141no—
580C2×C4—142no—
581C2×C4—141no—
582C2×C4—142no—
583C2×C4—142no—
584C2×C4—141no—
585C2×C4—141no—
586C2×C4—142no—
587C2×C4—141no—
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—564no—
597C2×C4—564no—
598C2×C4#647512yes—
599C2×C4C22×C4×D711yescharacteristic
600C2×C4C2×D4×D711yescharacteristic
601C2×C4#654811yescharacteristic
602C2×C4C2×Q8×D711yescharacteristic
603C2×C4D7×#4111yescharacteristic
604C2×C4#655411yescharacteristic
605C8—22no—
606C8—41no—
607C8—74no—
608C8—281no—
609C7#2268311yescharacteristic, complemented, Sylow
610C22—24no—
611C22—42no—
612C22—42no—
613C22—42no—
614C22—42no—
615C22—42no—
616C22—42no—
617C22C4×D4×D711yescharacteristic
618C22#9028111yescharacteristic
619C22#9032311yescharacteristic
620C22C2×#647511yescharacteristic
621C22D7×#13511yescharacteristic
622C22#9046611yescharacteristic
623C22#9046911yescharacteristic
624C4—22no—
625C4—41no—
626C4—42no—
627C4—81no—
628C4—81no—
629C4—142no—
630C4—144no—
631C4—142no—
632C4—142no—
633C4—281no—
634C4—282no—
635C4—561no—
636C4—561no—
637C4C2×D7×C4⋊C411yescharacteristic
638C4#9023711yescharacteristic
639C4#9032711yescharacteristic
640C4#9034611yescharacteristic
641C2—44no—
642C2#1173183811yescharacteristic
643C2C4×#647511yescharacteristic
644C2#1173722211yescharacteristic
645C2#1173747211yescharacteristic
646C2#1173768911yescharacteristic
647C2#1173840311yescharacteristic
648C2#1173860211yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1862C2
socleC23×C7C22×D4
prime radicalC22×C7×D4C23
cosocleC22×C4×C7C24
hypercenter#1862D7
hypocenterC7#22683
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1862

Complements

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

NH ≅ G/Ntype
5. #11733427C2semidirect
7. #11728396C2semidirect
8. #11728361C2semidirect
11. #11727372C2semidirect
12. #11726659C2semidirect
13. #11726612C2semidirect
14. #11726437C2semidirect
15. #11726405C2semidirect
53. #89530C4semidirect
108. #6450C2×C4semidirect
127. #6390C2×C4semidirect
130. C8×Dic7C2×C4semidirect
134. #6382C2×C4semidirect
205. #1142C4⋊C4semidirect
609. C7#22683semidirect

Maximal subgroups

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

subgroupindex
#117397792
C2×C4×#11422
C2×#895302
#117334272
C7×#18622
#117283962
#117283612
#117279222
#117279082
#117273722
#117266592
#117266122
#117264372
#117264052
#226837