Finite Group Databasebeta

← #13372213

Subgroups — #13372213

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#11739906C211yesmaximal, characteristic, complemented
2C2×C4×#1143C211yesmaximal, characteristic
3C2×#89560C211yesmaximal, characteristic
4C2×#89560C211yesmaximal, characteristic
5#11733648C211yesmaximal, characteristic
6C7×#1863C211yesmaximal, characteristic, complemented
7#11728392C211yesmaximal, characteristic
8#11728361C211yesmaximal, characteristic, complemented
9#11727990C211yesmaximal, characteristic, complemented
10#11727976C211yesmaximal, characteristic, complemented
11#11727372C211yesmaximal, characteristic, complemented
12#11726647C211yesmaximal, characteristic
13#11726612C211yesmaximal, characteristic, complemented
14#11726424C211yesmaximal, characteristic
15#11726405C211yesmaximal, characteristic, complemented
16C2×C4×C7×Q8C2211yescharacteristic
17C2×Q8×Dic7C2211yescharacteristic
18C2×Q8×Dic7C2211yescharacteristic
19C22×#1143C2211yescharacteristic
20C2×#6456C2211yescharacteristic
21C2×#6456C2211yescharacteristic
22C2×C4×D28C2211yescharacteristic
23C2×C7×#107C2211yescharacteristic
24C2×C7×#103C2211yescharacteristic
25C2×C7×#103C2211yescharacteristic
26C7×#389C2211yescharacteristic
27C2×#6409C2211yescharacteristic
28C2×#6395C2211yescharacteristic
29C2×#6395C2211yescharacteristic
30C2×#6392C2211yescharacteristic
31C2×Dic7⋊C8C2211yescharacteristic
32C2×C8×Dic7C2211yescharacteristic
33C2×C8×Dic7C2211yescharacteristic
34C4×#1143—24no—
35#89761—21no—
36#89759—21no—
37#89757C2211yescharacteristic
38#89749—21no—
39#89749C2211yescharacteristic
40#89749C2211yescharacteristic
41#89745—21no—
42#89745C2211yescharacteristic
43C2×#6383C2211yescharacteristic
44C2×#6382C2211yescharacteristic
45C2×#6382C2211yescharacteristic
46C2×#6382C2211yescharacteristic
47C4×Dic7⋊C4—21no—
48C4×Dic7⋊C4C2211yescharacteristic
49#89700—21no—
50#89700C2211yescharacteristic
51C2×C4×C7⋊C8C2211yescharacteristic
52#89560—24no—
53#89560C418yescomplemented
54#89440—21no—
55D7×#95—21no—
56C7×#340C2211yescharacteristic
57C7×#336C2211yescharacteristic
58#89346C2211yescharacteristic
59#89345C2211yescharacteristic
60#89324C2211yescharacteristic
61#89324C2211yescharacteristic
62#22683—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⋊C4D411yescharacteristic
98C2×Dic7⋊C4D411yescharacteristic
99C2×C4×Dic7—21no—
100C2×C4×Dic7—21no—
101C2×C4×Dic7C2311yescharacteristic
102C2×C4×Dic7C2311yescharacteristic
103C22×C7⋊C8C2311yescharacteristic
104C22×C7⋊C8C2311yescharacteristic
105#6456—22no—
106#6456—22no—
107#6456—41no—
108#6456C2×C414yes—
109C4×D28—22no—
110C4×D28—41no—
111C7×#107—41no—
112C7×#103—22no—
113C7×#103C2×C414yescomplemented
114C7×#95—21no—
115#6409—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#6395—22no—
126#6395C2×C414yes—
127#6392—41no—
128Dic7⋊C8—22no—
129C8×Dic7—22no—
130C8×Dic7C2×C414yescomplemented
131#6383—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#1863D711yescharacteristic, complemented
145#1862—71no—
146#1862—71no—
147#1846—71no—
148#1805—71no—
149#1804—71no—
150C8×C4⋊C4—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—22no—
219Dic7⋊C4—42no—
220Dic7⋊C4—42no—
221C4×Dic7—21no—
222C4×Dic7—21no—
223C4×Dic7—24no—
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×Q8C2×D711yescharacteristic
237C2×C4×D4—71no—
238C2×C4×D4—71no—
239C2×C4×D4—71no—
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×#103C2×D711yescharacteristic
251C2×#103C2×D711yescharacteristic
252C2×#102—71no—
253C2×#102—71no—
254C2×#102—71no—
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×C7C22×D411yescharacteristic
280C22×D7—22no—
281C22×D7—22no—
282C22×D7—22no—
283C22×D7—41no—
284C22×D7—41no—
285C22×D7—41no—
286C22×D7—41no—
287C7×Q8—22no—
288C7×Q8—22no—
289C7×Q8—22no—
290C7×Q8C2×C4⋊C414yes—
291C2×C4×C7—22no—
292C2×C4×C7—22no—
293C2×C4×C7—21no—
294C2×C4×C7—22no—
295C2×C4×C7—21no—
296C2×C4×C7—22no—
297C2×C4×C7—21no—
298C2×C4×C7—21no—
299C2×C4×C7—21no—
300C2×C4×C7—41no—
301C2×C4×C7—41no—
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#13512yes—
309C2×C4×C7C23×C411yescharacteristic
310C2×C4×C7C22×D411yescharacteristic
311C2×C4×C7C22×Q811yescharacteristic
312C2×C4×C7C2×#4111yescharacteristic
313C2×C4×C7H(5,𝔽2)11yescharacteristic
314C2×C4×C72−1+411yescharacteristic
315C2×Dic7—21no—
316C2×Dic7—21no—
317C2×Dic7—21no—
318C2×Dic7—21no—
319C2×Dic7—21no—
320C2×Dic7—21no—
321C2×Dic7—22no—
322C2×Dic7—21no—
323C2×Dic7—21no—
324C2×Dic7—21no—
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×Dic7C4×D412yes—
334C2×Dic7C4×D412yes—
335C2×Dic7#13512yes—
336D28—22no—
337D28—22no—
338D28—22no—
339D28C2×C4⋊C414yes—
340C4×D7—84no—
341C4×D7—84no—
342C4×D7—84no—
343C7×C8—22no—
344C7×C8—41no—
345C7⋊C8—41no—
346C7⋊C8C2×C4⋊C414yes—
347C22×Q8C22×D711yescharacteristic
348C22×D4—71no—
349C23×C4—141no—
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—22no—
360C4×Q8—41no—
361C4×Q8—74no—
362C4×Q8—142no—
363C4×Q8—142no—
364C4×Q8—281no—
365C4×D4—74no—
366C4×D4—142no—
367C4×D4—142no—
368C4×D4—142no—
369C4×D4—281no—
370C4×D4—281no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4—71no—
374C2×C4⋊C4—71no—
375C2×C4⋊C4—71no—
376C2×C4⋊C4—71no—
377C2×C4⋊C4—71no—
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—141no—
387C2×C22⋊C4—141no—
388C2×C22⋊C4—141no—
389C2×C42—21no—
390C2×C42—71no—
391C2×C42—71no—
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—22no—
402#103—142no—
403#103C4×D714yescomplemented
404#102—74no—
405#102—142no—
406#102—142no—
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×C7C2×C4×D411yescharacteristic
421C22×C7D4211yescharacteristic
422C22×C7D4×Q811yescharacteristic
423C22×C7C2×#13511yescharacteristic
424C22×C7C2×#13511yescharacteristic
425C22×C7#57711yescharacteristic
426C22×C7#57711yescharacteristic
427C2×D7—24no—
428C2×D7—42no—
429C2×D7—42no—
430C2×D7—42no—
431C2×D7—42no—
432C2×D7—42no—
433C2×D7—42no—
434C4×C7—24no—
435C4×C7—22no—
436C4×C7—42no—
437C4×C7—41no—
438C4×C7—81no—
439C4×C7—81no—
440C4×C7C22×C4⋊C411yescharacteristic
441C4×C7#52011yescharacteristic
442C4×C7#54711yescharacteristic
443C4×C7#55411yescharacteristic
444Dic7—22no—
445Dic7—22no—
446Dic7—22no—
447Dic7—42no—
448Dic7—41no—
449Dic7—81no—
450Dic7—81no—
451C24—141no—
452C2×Q8—21no—
453C2×Q8—21no—
454C2×Q8—21no—
455C2×Q8C2×C4×D712yes—
456C2×Q8D4×D712yes—
457C2×Q8Q8×D712yes—
458C2×D4—72no—
459C2×D4—72no—
460C2×D4—72no—
461C2×D4—141no—
462C2×D4—141no—
463C2×D4—141no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—71no—
469C22×C4—71no—
470C22×C4—71no—
471C22×C4—71no—
472C22×C4—144no—
473C22×C4—141no—
474C22×C4—141no—
475C22×C4—141no—
476C22×C4—282no—
477C22×C4—282no—
478C22×C4—282no—
479C22×C4—282no—
480C22×C4—561no—
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—81no—
508C4⋊C4—142no—
509C4⋊C4—144no—
510C4⋊C4—142no—
511C4⋊C4—141no—
512C4⋊C4—141no—
513C4⋊C4—141no—
514C4⋊C4—141no—
515C4⋊C4—141no—
516C4⋊C4—282no—
517C4⋊C4—282no—
518C4⋊C4—282no—
519C4⋊C4—561no—
520C4⋊C4C2×C4×D712yes—
521C22⋊C4—144no—
522C22⋊C4—282no—
523C22⋊C4—282no—
524C42—21no—
525C42—21no—
526C42—42no—
527C42—72no—
528C42—72no—
529C42—144no—
530C42—141no—
531C42—141no—
532C42—282no—
533C2×C7D4×C4⋊C411yescharacteristic
534C2×C7C4×#13511yescharacteristic
535C2×C7#294911yescharacteristic
536C2×C7#328211yescharacteristic
537C2×C7#328211yescharacteristic
538C2×C7#338111yescharacteristic
539C2×C7#338111yescharacteristic
540D7—44no—
541C23—142no—
542C23—142no—
543C23—142no—
544C23—281no—
545C23—281no—
546C23—281no—
547C23—281no—
548C23C2×D4×D711yescharacteristic
549Q8—22no—
550Q8—22no—
551Q8—22no—
552Q8D7×C4⋊C414yes—
553D4—74no—
554D4—142no—
555D4—142no—
556D4—142no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—21no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—22no—
565C2×C4—21no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—72no—
575C2×C4—72no—
576C2×C4—72no—
577C2×C4—141no—
578C2×C4—141no—
579C2×C4—141no—
580C2×C4—141no—
581C2×C4—141no—
582C2×C4—141no—
583C2×C4—141no—
584C2×C4—141no—
585C2×C4—142no—
586C2×C4—141no—
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—564no—
596C2×C4—564no—
597C2×C4—564no—
598C2×C4#648212yes—
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—144no—
611C22—282no—
612C22—282no—
613C22—282no—
614C22—282no—
615C22—282no—
616C22—282no—
617C22C4×D4×D711yescharacteristic
618C22#9028111yescharacteristic
619C22#9032311yescharacteristic
620C22C2×#648211yescharacteristic
621C22D7×#13511yescharacteristic
622C22#9046611yescharacteristic
623C22#9046911yescharacteristic
624C4—22no—
625C4—24no—
626C4—42no—
627C4—41no—
628C4—81no—
629C4—81no—
630C4—142no—
631C4—142no—
632C4—142no—
633C4—281no—
634C4—282no—
635C4—561no—
636C4—561no—
637C4C2×D7×C4⋊C411yescharacteristic
638C4#9023711yescharacteristic
639C4#9032711yescharacteristic
640C4#9034611yescharacteristic
641C2—284no—
642C2#1173183811yescharacteristic
643C2C4×#648211yescharacteristic
644C2#1173722211yescharacteristic
645C2#1173757111yescharacteristic
646C2#1173765211yescharacteristic
647C2#1173858111yescharacteristic
648C2#1173866611yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1863C2
socleC23×C7C22×D4
prime radicalC22×D28C23
cosocleC22×C4×C7C24
hypercenter#1863D7
hypocenterC7#22683
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1863

Complements

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

NH ≅ G/Ntype
1. #11739906C2semidirect
6. C7×#1863C2semidirect
8. #11728361C2semidirect
9. #11727990C2semidirect
10. #11727976C2semidirect
11. #11727372C2semidirect
13. #11726612C2semidirect
15. #11726405C2semidirect
53. #89560C4semidirect
82. Q8×Dic7C2×C4semidirect
113. C7×#103C2×C4semidirect
130. C8×Dic7C2×C4semidirect
134. #6382C2×C4semidirect
144. #1863D7semidirect
204. #1143C4⋊C4semidirect
403. #103C4×D7semidirect
609. C7#22683semidirect

Maximal subgroups

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

subgroupindex
#117399062
C2×C4×#11432
C2×#895602
#117336482
C7×#18632
#117283922
#117283612
#117279902
#117279762
#117273722
#117266472
#117266122
#117264242
#117264052
#226837