Finite Group Databasebeta

← #11740147

Subgroups — #11740147

Order 896.

1246 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×D4×D7C211yesmaximal, characteristic, complemented
2C2×C7×#121C211yesmaximal, characteristic, complemented
3C2×#6502C211yesmaximal, characteristic, complemented
4C2×#6500C211yesmaximal, characteristic, complemented
5C2×#6500C211yesmaximal, characteristic, complemented
6C22×(C2×D7)⋊C4C211yesmaximal, characteristic, complemented
7#89995C211yesmaximal, characteristic, complemented
8#89992C211yesmaximal, characteristic, complemented
9#89985C211yesmaximal, characteristic, complemented
10#89763C211yesmaximal, characteristic, complemented
11#89761C211yesmaximal, characteristic, complemented
12#89732C211yesmaximal, characteristic, complemented
13#89731C211yesmaximal, characteristic, complemented
14#89728C211yesmaximal, characteristic, complemented
15#89727C211yesmaximal, characteristic, complemented
16C24×D7C2211yescharacteristic
17C24×D7C2211yescharacteristic
18C22×C7×D4C2211yescharacteristic
19C22×C7×D4C2211yescharacteristic, complemented
20C22×C7×D4C2211yescharacteristic, complemented
21C23×C4×C7C2211yescharacteristic, complemented
22C22×C22⋊D7C2211yescharacteristic
23C22×C22⋊D7C2211yescharacteristic
24C22×C22⋊D7C2211yescharacteristic
25C22×C22⋊D7C2211yescharacteristic, complemented
26C22×C22⋊D7C2211yescharacteristic, complemented
27C23×Dic7C2211yescharacteristic, complemented
28C2×D4×D7—24no—
29C2×D4×D7—24no—
30C2×D4×D7—24no—
31C22×D28C2211yescharacteristic
32C22×C4×D7C2211yescharacteristic
33C7×#121—24no—
34C2×C7×C4⋊C4C2211yescharacteristic, complemented
35C2×C7×C22⋊C4C2211yescharacteristic, complemented
36C2×C7×C22⋊C4C2211yescharacteristic, complemented
37C2×C22⋊Dic7C2211yescharacteristic, complemented
38C2×C22⋊Dic7C2211yescharacteristic, complemented
39C2×C22⋊Dic7C2211yescharacteristic, complemented
40C2×C22⋊Dic7C2211yescharacteristic, complemented
41#6502—24no—
42#6500—24no—
43#6500—24no—
44C2×(C2×D7)⋊C4—24no—
45C2×(C2×D7)⋊C4C2211yescharacteristic
46C2×(C2×D7)⋊C4C2211yescharacteristic
47C2×(C2×D7)⋊C4C2211yescharacteristic
48C2×(C2×D7)⋊C4C2211yescharacteristic, complemented
49C2×(C2×D7)⋊C4C2211yescharacteristic, complemented
50C2×(C2×D7)⋊C4C2211yescharacteristic, complemented
51C2×(C2×D7)⋊C4C2211yescharacteristic, complemented
52C2×(C2×D7)⋊C4C2211yescharacteristic, complemented
53C2×(C2×D7)⋊C4C2211yescharacteristic, complemented
54C2×Dic7⋊C4C2211yescharacteristic, complemented
55C7⋊#95C2211yescharacteristic, complemented
56C7⋊#95C2211yescharacteristic, complemented
57C7⋊#95C2211yescharacteristic, complemented
58C7⋊#95C2211yescharacteristic, complemented
59#2410—71nomaximal, Sylow
60C24×C7C2311yescharacteristic
61C24×C7C2311yescharacteristic
62C24×C7C2311yescharacteristic
63C23×D7—22no—
64C23×D7—22no—
65C23×D7—22no—
66C23×D7—22no—
67C23×D7—22no—
68C23×D7—22no—
69C23×D7—22no—
70C23×D7—22no—
71C23×D7—22no—
72C23×D7—41no—
73C23×D7—41no—
74C23×D7—41no—
75C23×D7—41no—
76C23×D7D414yescomplemented
77C23×D7C2311yescharacteristic
78C23×D7C2311yescharacteristic
79C23×D7C2311yescharacteristic
80C2×C7×D4—22no—
81C2×C7×D4—22no—
82C2×C7×D4—22no—
83C2×C7×D4—22no—
84C2×C7×D4—22no—
85C2×C7×D4—22no—
86C2×C7×D4—22no—
87C2×C7×D4—41no—
88C2×C7×D4D414yescomplemented
89C22×C4×C7—22no—
90C22×C4×C7—22no—
91C22×C4×C7—41no—
92C22×C4×C7C2311yescharacteristic
93C22×C4×C7C2311yescharacteristic
94C22×C4×C7C2311yescharacteristic
95C22×C4×C7C2311yescharacteristic
96C2×C22⋊D7—22no—
97C2×C22⋊D7—22no—
98C2×C22⋊D7—22no—
99C2×C22⋊D7—22no—
100C2×C22⋊D7—22no—
101C2×C22⋊D7—22no—
102C2×C22⋊D7—22no—
103C2×C22⋊D7—22no—
104C2×C22⋊D7—22no—
105C2×C22⋊D7—22no—
106C2×C22⋊D7—22no—
107C2×C22⋊D7—41no—
108C2×C22⋊D7—41no—
109C2×C22⋊D7—41no—
110C2×C22⋊D7D414yescomplemented
111C22×Dic7—22no—
112C22×Dic7—22no—
113C22×Dic7—41no—
114C22×Dic7C2311yescharacteristic
115C22×Dic7C2311yescharacteristic
116C22×Dic7C2311yescharacteristic
117C22×Dic7C2311yescharacteristic
118C22×Dic7C2311yescharacteristic
119D4×D7—232no—
120C2×D28—22no—
121C2×D28—22no—
122C2×D28—22no—
123C2×C4×D7—22no—
124C2×C4×D7—22no—
125C2×C4×D7—22no—
126C7×C4⋊C4—22no—
127C7×C22⋊C4—22no—
128C7×C22⋊C4—22no—
129C22⋊Dic7—22no—
130C22⋊Dic7—22no—
131C22⋊Dic7—22no—
132C22⋊Dic7—41no—
133(C2×D7)⋊C4—22no—
134(C2×D7)⋊C4—22no—
135(C2×D7)⋊C4—22no—
136(C2×D7)⋊C4—22no—
137(C2×D7)⋊C4—22no—
138(C2×D7)⋊C4—22no—
139(C2×D7)⋊C4—22no—
140(C2×D7)⋊C4—41no—
141(C2×D7)⋊C4D414yescomplemented
142Dic7⋊C4—22no—
143C23×D4—71no—
144C2×#121—71no—
145C2×#121D711yescharacteristic, complemented
146C2×#120—71no—
147C2×#120—71no—
148C22×C22⋊C4—71no—
149#394—71no—
150#394—71no—
151#392—71no—
152#392—71no—
153#386—71no—
154#386—71no—
155#386—71no—
156#385—71no—
157#379—71no—
158C23×C7—21no—
159C23×C7—21no—
160C23×C7—21no—
161C23×C7—21no—
162C23×C7—21no—
163C23×C7—21no—
164C23×C7—21no—
165C23×C7—21no—
166C23×C7—21no—
167C23×C7—21no—
168C23×C7—21no—
169C23×C7—21no—
170C23×C7—21no—
171C23×C7—21no—
172C23×C7—21no—
173C23×C7—21no—
174C23×C7—21no—
175C23×C7C2×D412yes—
176C23×C7C2×D412yes—
177C23×C7C2×D412yes—
178C23×C7C2×D412yes—
179C23×C7C2411yescharacteristic
180C22×D7—21no—
181C22×D7—21no—
182C22×D7—21no—
183C22×D7—21no—
184C22×D7—28no—
185C22×D7—28no—
186C22×D7—21no—
187C22×D7—21no—
188C22×D7—21no—
189C22×D7—21no—
190C22×D7—21no—
191C22×D7—21no—
192C22×D7—21no—
193C22×D7—21no—
194C22×D7—21no—
195C22×D7—21no—
196C22×D7—21no—
197C22×D7—21no—
198C22×D7—28no—
199C22×D7—44no—
200C22×D7—44no—
201C22×D7—44no—
202C22×D7—44no—
203C22×D7—44no—
204C22×D7—44no—
205C22×D7—44no—
206C22×D7—44no—
207C22×D7—44no—
208C22×D7—44no—
209C22×D7—44no—
210C22×D7C2×D412yes—
211C22×D7C2×D412yes—
212C22×D7C2×D412yes—
213C22×D7C2×D412yes—
214C22×D7#4112yes—
215C7×D4—28no—
216C7×D4—44no—
217C7×D4—44no—
218C2×C4×C7—21no—
219C2×C4×C7—21no—
220C2×C4×C7—21no—
221C2×C4×C7—21no—
222C2×C4×C7—21no—
223C2×C4×C7—21no—
224C2×C4×C7—21no—
225C2×C4×C7—21no—
226C2×C4×C7—21no—
227C2×C4×C7—21no—
228C2×C4×C7—44no—
229C2×C4×C7C2×D412yes—
230C2×C4×C7C2×D412yescomplemented
231C22⋊D7—28no—
232C22⋊D7—28no—
233C22⋊D7—44no—
234C22⋊D7—44no—
235C22⋊D7—44no—
236C2×Dic7—21no—
237C2×Dic7—21no—
238C2×Dic7—21no—
239C2×Dic7—21no—
240C2×Dic7—21no—
241C2×Dic7—21no—
242C2×Dic7—21no—
243C2×Dic7—21no—
244C2×Dic7—21no—
245C2×Dic7—21no—
246C2×Dic7—21no—
247C2×Dic7—21no—
248C2×Dic7—21no—
249C2×Dic7—44no—
250C2×Dic7C2×D412yes—
251C2×Dic7C2×D412yescomplemented
252D28—28no—
253C4×D7—28no—
254C25—71no—
255C25—71no—
256C22×D4—71no—
257C22×D4—71no—
258C22×D4—71no—
259C22×D4—71no—
260C22×D4—71no—
261C22×D4—71no—
262C22×D4—144no—
263C22×D4—144no—
264C22×D4—144no—
265C22×D4C2×D711yescharacteristic
266C22×D4C2×D711yescharacteristic, complemented
267C22×D4C2×D711yescharacteristic, complemented
268C23×C4—71no—
269C23×C4—71no—
270C23×C4C2×D711yescharacteristic, complemented
271#121—24no—
272#121—144no—
273#120—144no—
274#120—144no—
275C2×C4⋊C4—71no—
276C2×C4⋊C4C2×D711yescharacteristic, complemented
277C2×C22⋊C4—71no—
278C2×C22⋊C4—71no—
279C2×C22⋊C4—71no—
280C2×C22⋊C4—71no—
281C2×C22⋊C4—71no—
282C2×C22⋊C4—71no—
283C2×C22⋊C4—71no—
284C2×C22⋊C4—71no—
285C2×C22⋊C4—71no—
286C2×C22⋊C4—71no—
287C2×C22⋊C4—71no—
288C2×C22⋊C4—71no—
289C2×C22⋊C4—71no—
290C2×C22⋊C4—144no—
291C2×C22⋊C4C2×D711yescharacteristic, complemented
292C2×C22⋊C4C2×D711yescharacteristic, complemented
293#95—71no—
294#95—71no—
295#95—71no—
296#95—71no—
297C22×C7—22no—
298C22×C7—22no—
299C22×C7—22no—
300C22×C7—22no—
301C22×C7—22no—
302C22×C7—22no—
303C22×C7—22no—
304C22×C7—22no—
305C22×C7—22no—
306C22×C7—41no—
307C22×C7—41no—
308C22×C7—41no—
309C22×C7—41no—
310C22×C7—41no—
311C22×C7—41no—
312C22×C7—41no—
313C22×C7—41no—
314C22×C7—41no—
315C22×C7—41no—
316C22×C7—41no—
317C22×C7#12014yescomplemented
318C22×C7C22×D411yescharacteristic
319C22×C7C22×D411yescharacteristic
320C22×C7C22×D411yescharacteristic
321C22×C7C22×D411yescharacteristic
322C22×C7C2×#4111yescharacteristic
323C22×C7H(5,𝔽2)11yescharacteristic
324C22×C7H(5,𝔽2)11yescharacteristic
325C2×D7—22no—
326C2×D7—22no—
327C2×D7—22no—
328C2×D7—22no—
329C2×D7—22no—
330C2×D7—22no—
331C2×D7—22no—
332C2×D7—22no—
333C2×D7—22no—
334C2×D7—22no—
335C2×D7—22no—
336C2×D7—22no—
337C2×D7—416no—
338C2×D7—416no—
339C2×D7—41no—
340C2×D7—41no—
341C2×D7—41no—
342C2×D7—41no—
343C2×D7—41no—
344C2×D7—41no—
345C2×D7—41no—
346C2×D7—41no—
347C2×D7#12114yescomplemented
348C4×C7—22no—
349C4×C7—22no—
350C4×C7—41no—
351C4×C7—41no—
352Dic7—22no—
353Dic7—22no—
354Dic7—41no—
355Dic7—41no—
356Dic7—41no—
357C24—71no—
358C24—71no—
359C24—71no—
360C24—74no—
361C24—142no—
362C24—142no—
363C24—142no—
364C24—142no—
365C24—142no—
366C24—142no—
367C24—142no—
368C24—142no—
369C24—142no—
370C24—281no—
371C24—281no—
372C24—281no—
373C24—281no—
374C24C22×D711yescharacteristic
375C24C22×D711yescharacteristic
376C24C22×D711yescharacteristic
377C2×D4—22no—
378C2×D4—22no—
379C2×D4—22no—
380C2×D4—22no—
381C2×D4—22no—
382C2×D4—22no—
383C2×D4—22no—
384C2×D4—41no—
385C2×D4—74no—
386C2×D4—142no—
387C2×D4—142no—
388C2×D4—142no—
389C2×D4—142no—
390C2×D4—1432no—
391C2×D4—142no—
392C2×D4—142no—
393C2×D4—142no—
394C2×D4—142no—
395C2×D4—142no—
396C2×D4—142no—
397C2×D4—142no—
398C2×D4—142no—
399C2×D4—142no—
400C2×D4—142no—
401C2×D4—281no—
402C2×D4—281no—
403C2×D4—281no—
404C2×D4C22⋊D714yescomplemented
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—41no—
408C22×C4—71no—
409C22×C4—71no—
410C22×C4—71no—
411C22×C4—71no—
412C22×C4—71no—
413C22×C4—142no—
414C22×C4—142no—
415C22×C4—142no—
416C22×C4—142no—
417C22×C4—142no—
418C22×C4—281no—
419C22×C4C22×D711yescharacteristic
420C22×C4C22×D711yescharacteristic
421C22×C4C22×D711yescharacteristic
422C22×C4C22×D711yescharacteristic
423C4⋊C4—22no—
424C4⋊C4—142no—
425C22⋊C4—22no—
426C22⋊C4—22no—
427C22⋊C4—74no—
428C22⋊C4—142no—
429C22⋊C4—142no—
430C22⋊C4—142no—
431C22⋊C4—142no—
432C22⋊C4—142no—
433C22⋊C4—142no—
434C22⋊C4—142no—
435C22⋊C4—142no—
436C22⋊C4—142no—
437C22⋊C4—142no—
438C22⋊C4—281no—
439C22⋊C4—281no—
440C2×C7—24no—
441C2×C7—42no—
442C2×C7—81no—
443C2×C7C2×#12011yescharacteristic
444C2×C7C2×#12111yescharacteristic
445C2×C7#53411yescharacteristic
446C2×C7D4211yescharacteristic
447C2×C7D4211yescharacteristic
448C2×C7#54611yescharacteristic
449C2×C7#54611yescharacteristic
450D7—24no—
451D7—42no—
452D7—42no—
453C23—21no—
454C23—21no—
455C23—21no—
456C23—21no—
457C23—21no—
458C23—21no—
459C23—21no—
460C23—21no—
461C23—21no—
462C23—21no—
463C23—21no—
464C23—21no—
465C23—21no—
466C23—21no—
467C23—21no—
468C23—21no—
469C23—21no—
470C23—72no—
471C23—72no—
472C23—72no—
473C23—72no—
474C23—72no—
475C23—148no—
476C23—148no—
477C23—141no—
478C23—141no—
479C23—141no—
480C23—141no—
481C23—148no—
482C23—141no—
483C23—141no—
484C23—141no—
485C23—141no—
486C23—141no—
487C23—141no—
488C23—141no—
489C23—141no—
490C23—141no—
491C23—141no—
492C23—141no—
493C23—141no—
494C23—284no—
495C23—284no—
496C23—284no—
497C23—284no—
498C23—284no—
499C23—284no—
500C23—284no—
501C23—284no—
502C23—284no—
503C23—284no—
504C23—284no—
505C23D4×D712yes—
506C23D4×D712yes—
507C23C2×C22⋊D712yes—
508C23C2×C22⋊D712yes—
509C23C23×D711yescharacteristic
510D4—28no—
511D4—44no—
512D4—44no—
513D4—148no—
514D4—148no—
515D4—148no—
516D4—284no—
517D4—284no—
518D4—284no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—44no—
530C2×C4—72no—
531C2×C4—72no—
532C2×C4—148no—
533C2×C4—141no—
534C2×C4—141no—
535C2×C4—141no—
536C2×C4—141no—
537C2×C4—141no—
538C2×C4—141no—
539C2×C4—141no—
540C2×C4—141no—
541C2×C4—141no—
542C2×C4—141no—
543C2×C4—141no—
544C2×C4—141no—
545C2×C4—141no—
546C2×C4—284no—
547C2×C4D4×D712yescomplemented
548C2×C4C2×C22⋊D712yes—
549C7#241011yescharacteristic, complemented, Sylow
550C22—22no—
551C22—22no—
552C22—22no—
553C22—22no—
554C22—22no—
555C22—22no—
556C22—22no—
557C22—22no—
558C22—22no—
559C22—41no—
560C22—41no—
561C22—41no—
562C22—41no—
563C22—41no—
564C22—41no—
565C22—41no—
566C22—41no—
567C22—41no—
568C22—41no—
569C22—41no—
570C22—74no—
571C22—142no—
572C22—142no—
573C22—142no—
574C22—142no—
575C22—142no—
576C22—142no—
577C22—142no—
578C22—142no—
579C22—142no—
580C22—142no—
581C22—142no—
582C22—142no—
583C22—2816no—
584C22—2816no—
585C22—281no—
586C22—281no—
587C22—281no—
588C22—281no—
589C22—281no—
590C22—281no—
591C22—281no—
592C22—281no—
593C22#650014yescomplemented
594C22C2×D4×D711yescharacteristic
595C22C2×D4×D711yescharacteristic
596C22C2×D4×D711yescharacteristic
597C22#654811yescharacteristic
598C22D7×#4111yescharacteristic
599C22#655311yescharacteristic
600C22C22×C22⋊D711yescharacteristic
601C4—22no—
602C4—22no—
603C4—41no—
604C4—41no—
605C4—142no—
606C4—142no—
607C4—281no—
608C4—281no—
609C4—281no—
610C2—24no—
611C2—42no—
612C2—81no—
613C2—144no—
614C2—282no—
615C2—282no—
616C2D7×#12111yescharacteristic
617C2#9029611yescharacteristic
618C2#9030011yescharacteristic
619C2#9030111yescharacteristic
620C2C2×#650011yescharacteristic
621C2D4×C22⋊D711yescharacteristic
622C2#9052011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC2×C7×#121C2
hypercenterC2×#121D7
hypocenterC7#2410
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C22×D4×D7C2semidirect
2. C2×C7×#121C2semidirect
3. C2×#6502C2semidirect
4. C2×#6500C2semidirect
5. C2×#6500C2semidirect
6. C22×(C2×D7)⋊C4C2semidirect
7. #89995C2semidirect
8. #89992C2semidirect
9. #89985C2semidirect
10. #89763C2semidirect
11. #89761C2semidirect
12. #89732C2semidirect
13. #89731C2semidirect
14. #89728C2semidirect
15. #89727C2semidirect
19. C22×C7×D4C22semidirect
20. C22×C7×D4C22semidirect
21. C23×C4×C7C22semidirect
25. C22×C22⋊D7C22semidirect
26. C22×C22⋊D7C22semidirect
27. C23×Dic7C22semidirect
34. C2×C7×C4⋊C4C22semidirect
35. C2×C7×C22⋊C4C22semidirect
36. C2×C7×C22⋊C4C22semidirect
37. C2×C22⋊Dic7C22semidirect
38. C2×C22⋊Dic7C22semidirect
39. C2×C22⋊Dic7C22semidirect
40. C2×C22⋊Dic7C22semidirect
48. C2×(C2×D7)⋊C4C22semidirect
49. C2×(C2×D7)⋊C4C22semidirect
50. C2×(C2×D7)⋊C4C22semidirect
51. C2×(C2×D7)⋊C4C22semidirect
52. C2×(C2×D7)⋊C4C22semidirect
53. C2×(C2×D7)⋊C4C22semidirect
54. C2×Dic7⋊C4C22semidirect
55. C7⋊#95C22semidirect
56. C7⋊#95C22semidirect
57. C7⋊#95C22semidirect
58. C7⋊#95C22semidirect
76. C23×D7D4semidirect
88. C2×C7×D4D4semidirect
110. C2×C22⋊D7D4semidirect
141. (C2×D7)⋊C4D4semidirect
145. C2×#121D7semidirect
230. C2×C4×C7C2×D4semidirect
251. C2×Dic7C2×D4semidirect
266. C22×D4C2×D7semidirect
267. C22×D4C2×D7semidirect
270. C23×C4C2×D7semidirect
276. C2×C4⋊C4C2×D7semidirect
291. C2×C22⋊C4C2×D7semidirect
292. C2×C22⋊C4C2×D7semidirect
317. C22×C7#120semidirect
347. C2×D7#121semidirect
404. C2×D4C22⋊D7semidirect
547. C2×C4D4×D7semidirect
549. C7#2410semidirect
593. C22#6500semidirect

Maximal subgroups

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

subgroupindex
C22×D4×D72
C2×C7×#1212
C2×#65022
C2×#65002
C22×(C2×D7)⋊C42
#899952
#899922
#899852
#897632
#897612
#897322
#897312
#897282
#897272
#24107