Finite Group Databasebeta

← #13385980

Subgroups — #13385980

Order 1792.

1500 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#553C211yesmaximal, characteristic, complemented
2C2×#89808C211yesmaximal, characteristic, complemented
3#11739848C211yesmaximal, characteristic, complemented
4C22×#6383C211yesmaximal, characteristic, complemented
5#11729497C212yesmaximal, complemented
6#11729438C212yesmaximal, complemented
7#11729435C211yesmaximal, characteristic, complemented
8#11729367C212yesmaximal, complemented
9#11729364C211yesmaximal, characteristic, complemented
10#11729340C212yesmaximal, complemented
11#11729308C211yesmaximal, characteristic, complemented
12C23×D28C2211yescharacteristic
13C7×#553—24no—
14C7×#553—24no—
15C2×C7×#126—21no—
16C2×C7×#125C2211yescharacteristic, complemented
17C2×C7×#123—21no—
18C2×C7×#121—21no—
19C2×C7×#121C2211yescharacteristic
20C2×C7×#121C2211yescharacteristic, complemented
21C2×C4×C7×D4—21no—
22C2×C4×C7×D4C2211yescharacteristic
23C2×C4×C7×D4C2211yescharacteristic, complemented
24C2×C7×#117C2211yescharacteristic, complemented
25C22×C7×C4⋊C4C2211yescharacteristic
26C2×#6500—21no—
27C22×#1141C2212yes—
28C2×#6493C2212yes—
29C23×C7⋊C8C2211yescharacteristic
30C2×#6406C2212yescomplemented
31#89984C2211yescharacteristic
32C2×C22⋊(C7⋊C8)C2211yescharacteristic
33C2×C22⋊(C7⋊C8)C2212yescomplemented
34#89808—24no—
35#89763—21no—
36C2×#6383—24no—
37C2×#6383C2212yes—
38C2×#6383C2212yes—
39C2×#6383C2212yes—
40#89732—21no—
41#89727—21no—
42#89711C2212yes—
43#89324C2212yescomplemented
44#89324C2212yescomplemented
45#89324C2212yescomplemented
46#89319C2212yescomplemented
47#23963—71nomaximal, Sylow
48C24×D7—21no—
49C22×C7×D4—21no—
50C22×C7×D4—21no—
51C22×C7×D4D411yescharacteristic
52C22×C7×D4C2311yescharacteristic
53C23×C4×C7—21no—
54C23×C4×C7—21no—
55C23×C4×C7C2311yescharacteristic
56C23×C4×C7C2311yescharacteristic
57C22×C22⋊D7—22no—
58C22×D28—22no—
59C22×D28—24no—
60C22×D28—22no—
61C22×D28—21no—
62C22×D28—42no—
63C22×D28C2312yes—
64C7×#126—28no—
65C7×#125—22no—
66C7×#125—22no—
67C7×#123—28no—
68C7×#121—28no—
69C7×#121—22no—
70C7×#121—22no—
71C7×#121—22no—
72C7×#121D414yescomplemented
73C4×C7×D4—28no—
74C4×C7×D4—22no—
75C4×C7×D4—22no—
76C4×C7×D4—22no—
77C4×C7×D4—22no—
78C7×#117—22no—
79C7×#117—22no—
80C2×C7×C4⋊C4—21no—
81C2×C7×C4⋊C4—22no—
82C2×C7×C4⋊C4—21no—
83C2×C7×C4⋊C4—21no—
84C2×C7×C4⋊C4D411yescharacteristic
85C2×C7×C4⋊C4D414yescomplemented
86C2×C7×C4⋊C4C2311yescharacteristic
87C2×C7×C4⋊C4C2311yescharacteristic
88C2×C7×C4⋊C4C2311yescharacteristic
89C2×C7×C22⋊C4—21no—
90C2×C7×C22⋊C4—21no—
91C2×C7×C22⋊C4—21no—
92C2×C7×C22⋊C4—21no—
93C2×C7×C22⋊C4D411yescharacteristic
94C2×C7×C22⋊C4D411yescharacteristic
95C2×C42×C7—21no—
96C2×C42×C7C2311yescharacteristic
97C2×C22⋊Dic7—21no—
98#6500—44no—
99C2×#1141—24no—
100C2×#1141—24no—
101C2×#1141—42no—
102#6493—24no—
103#6493—42no—
104C2×(C2×D7)⋊C4—22no—
105C2×(C2×D7)⋊C4—22no—
106C2×(C2×D7)⋊C4—22no—
107C2×C4⋊Dic7C2312yes—
108C22×C7⋊C8—22no—
109C22×C7⋊C8—42no—
110C22×C7⋊C8C2312yes—
111C22×C7⋊C8C2312yes—
112#6406—42no—
113C7⋊#95—21no—
114C22⋊(C7⋊C8)—22no—
115C22⋊(C7⋊C8)—42no—
116#6383—24no—
117#6383—24no—
118#6383—24no—
119C2×#553D711yescharacteristic, complemented
120C2×#447—71no—
121C22×#102—71no—
122#2408—71no—
123#2025—72no—
124#1999—72no—
125#1893—71no—
126#1879—72no—
127#1875—72no—
128#1869—71no—
129#1824—71no—
130C24×C7—21no—
131C24×C7C2×D411yescharacteristic
132C24×C7C2×D411yescharacteristic
133C23×D7—22no—
134C23×D7—44no—
135C23×D7—42no—
136C23×D7—42no—
137C23×D7—42no—
138C23×D7—82no—
139C2×C7×D4—22no—
140C2×C7×D4—24no—
141C2×C7×D4—22no—
142C2×C7×D4—22no—
143C2×C7×D4—24no—
144C2×C7×D4—21no—
145C2×C7×D4—22no—
146C2×C7×D4—22no—
147C2×C7×D4—42no—
148C2×C7×D4—42no—
149C2×C7×D4—82no—
150C2×C7×D4C2×D412yes—
151C22×C4×C7—22no—
152C22×C4×C7—22no—
153C22×C4×C7—21no—
154C22×C4×C7—21no—
155C22×C4×C7—21no—
156C22×C4×C7—24no—
157C22×C4×C7—24no—
158C22×C4×C7—21no—
159C22×C4×C7—21no—
160C22×C4×C7—22no—
161C22×C4×C7—22no—
162C22×C4×C7—82no—
163C22×C4×C7—82no—
164C22×C4×C7C2×D411yescharacteristic
165C22×C4×C7C2×D411yescharacteristic
166C22×C4×C7C2×D411yescharacteristic
167C22×C4×C7C2×D411yescharacteristic
168C22×C4×C7C2×D412yes—
169C22×C4×C7C2×D412yes—
170C22×C4×C7C2411yescharacteristic
171C2×C22⋊D7—44no—
172C2×C22⋊D7—44no—
173C2×C22⋊D7—82no—
174C22×Dic7—22no—
175C2×D28—28no—
176C2×D28—22no—
177C2×D28—22no—
178C2×D28—22no—
179C2×D28—22no—
180C2×D28—22no—
181C2×D28—48no—
182C2×D28—44no—
183C2×D28—44no—
184C2×D28—42no—
185C2×D28#4114yes—
186C7×C4⋊C4—24no—
187C7×C4⋊C4—24no—
188C7×C4⋊C4—24no—
189C7×C4⋊C4—22no—
190C7×C4⋊C4—21no—
191C7×C4⋊C4—21no—
192C7×C4⋊C4—21no—
193C7×C4⋊C4C2×D412yes—
194C7×C4⋊C4C2×D412yes—
195C7×C4⋊C4C2×D412yes—
196C7×C22⋊C4—24no—
197C7×C22⋊C4—24no—
198C7×C22⋊C4—24no—
199C7×C22⋊C4—22no—
200C7×C22⋊C4—24no—
201C7×C22⋊C4—22no—
202C42×C7—24no—
203C42×C7—21no—
204C42×C7—21no—
205C22⋊Dic7—42no—
206#1141—48no—
207(C2×D7)⋊C4—44no—
208(C2×D7)⋊C4—44no—
209(C2×D7)⋊C4—82no—
210C4⋊Dic7—22no—
211C4⋊Dic7—22no—
212C2×C7⋊C8—22no—
213C2×C7⋊C8—22no—
214C2×C7⋊C8—22no—
215C2×C7⋊C8—22no—
216C2×C7⋊C8—22no—
217C2×C7⋊C8—22no—
218C2×C7⋊C8—44no—
219C23×D4—71no—
220C22×D8—72no—
221C23×C8—71no—
222#553—24no—
223#553—24no—
224C2×#126—21no—
225C2×#125C2×D711yescharacteristic, complemented
226C2×#123—21no—
227C2×#121—21no—
228C2×#121—72no—
229C2×#121C2×D711yescharacteristic
230C2×#121C2×D711yescharacteristic, complemented
231C2×#120—141no—
232C2×C4×D4—21no—
233C2×C4×D4C2×D711yescharacteristic
234C2×C4×D4C2×D711yescharacteristic, complemented
235C2×#117C2×D711yescharacteristic, complemented
236C22×C4⋊C4C2×D711yescharacteristic
237#447—144no—
238C2×#102—72no—
239C2×#102—72no—
240C2×#102—72no—
241C2×#102—72no—
242C2×#102—144no—
243C2×C22⋊C8—71no—
244C2×C22⋊C8—72no—
245#394—141no—
246#392—141no—
247#390—72no—
248#380—71no—
249#379—141no—
250#340—72no—
251#340—72no—
252#340—72no—
253#336—72no—
254C23×C7—22no—
255C23×C7—22no—
256C23×C7—21no—
257C23×C7—22no—
258C23×C7—22no—
259C23×C7—21no—
260C23×C7—21no—
261C23×C7—22no—
262C23×C7—42no—
263C23×C7—41no—
264C23×C7—41no—
265C23×C7—42no—
266C23×C7#12012yes—
267C23×C7#13512yes—
268C23×C7AGL(1,ℤ/8)12yes—
269C23×C7C22×D411yescharacteristic
270C22×D7—24no—
271C22×D7—48no—
272C22×D7—42no—
273C22×D7—42no—
274C22×D7—42no—
275C22×D7—42no—
276C22×D7—42no—
277C22×D7—42no—
278C22×D7—88no—
279C22×D7—88no—
280C22×D7—84no—
281C22×D7—84no—
282C7×D4—24no—
283C7×D4—44no—
284C7×D4—48no—
285C7×D4—41no—
286C7×D4—41no—
287C7×D4—84no—
288C2×C4×C7—22no—
289C2×C4×C7—22no—
290C2×C4×C7—22no—
291C2×C4×C7—24no—
292C2×C4×C7—21no—
293C2×C4×C7—21no—
294C2×C4×C7—22no—
295C2×C4×C7—21no—
296C2×C4×C7—22no—
297C2×C4×C7—22no—
298C2×C4×C7—22no—
299C2×C4×C7—21no—
300C2×C4×C7—21no—
301C2×C4×C7—22no—
302C2×C4×C7—41no—
303C2×C4×C7—41no—
304C2×C4×C7—42no—
305C2×C4×C7—42no—
306C2×C4×C7—41no—
307C2×C4×C7—41no—
308C2×C4×C7—42no—
309C2×C4×C7—84no—
310C2×C4×C7—84no—
311C2×C4×C7#12012yes—
312C2×C4×C7C22×D411yescharacteristic
313C2×C4×C7C22×D411yescharacteristic
314C2×C4×C7C2×#4112yes—
315C2×C4×C7H(5,𝔽2)12yes—
316C22⋊D7—88no—
317C2×Dic7—42no—
318C2×Dic7—42no—
319C2×Dic7—42no—
320D28—24no—
321D28—24no—
322D28—48no—
323D28—42no—
324D28—42no—
325C7⋊C8—42no—
326C7⋊C8—42no—
327C25—141no—
328C22×D4—21no—
329C22×D4—21no—
330C22×D4—72no—
331C22×D4—144no—
332C22×D4—142no—
333C22×D4—142no—
334C22×D4—142no—
335C22×D4—141no—
336C22×D4—282no—
337C22×D4C22⋊D711yescharacteristic
338C22×D4C22×D711yescharacteristic
339C23×C4—21no—
340C23×C4—21no—
341C23×C4C22×D711yescharacteristic
342C23×C4C22×D711yescharacteristic
343C2×D8—144no—
344C2×D8—144no—
345C2×D8—282no—
346C22×C8—72no—
347C22×C8—72no—
348C22×C8—142no—
349C22×C8—282no—
350#126—28no—
351#125—22no—
352#125—22no—
353#123—28no—
354#121—28no—
355#121—22no—
356#121—22no—
357#121—22no—
358#121—144no—
359#121—282no—
360#121C22⋊D714yescomplemented
361#120—284no—
362C4×D4—28no—
363C4×D4—22no—
364C4×D4—22no—
365C4×D4—22no—
366C4×D4—22no—
367#117—22no—
368#117—22no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—22no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4—72no—
374C2×C4⋊C4C22⋊D711yescharacteristic
375C2×C4⋊C4C22⋊D714yescomplemented
376C2×C4⋊C4C22×D711yescharacteristic
377C2×C4⋊C4C22×D711yescharacteristic
378C2×C4⋊C4C22×D711yescharacteristic
379C2×C22⋊C4—21no—
380C2×C22⋊C4—21no—
381C2×C22⋊C4—21no—
382C2×C22⋊C4—21no—
383C2×C22⋊C4—141no—
384C2×C22⋊C4—142no—
385C2×C22⋊C4—142no—
386C2×C22⋊C4—142no—
387C2×C22⋊C4C22⋊D711yescharacteristic
388C2×C22⋊C4C22⋊D711yescharacteristic
389C2×C42—21no—
390C2×C42C22×D711yescharacteristic
391#102—144no—
392#102—144no—
393#102—144no—
394#102—282no—
395C22⋊C8—142no—
396C22⋊C8—282no—
397#95—141no—
398C22×C7—22no—
399C22×C7—22no—
400C22×C7—24no—
401C22×C7—24no—
402C22×C7—41no—
403C22×C7—42no—
404C22×C7—41no—
405C22×C7—42no—
406C22×C7—42no—
407C22×C7—41no—
408C22×C7—42no—
409C22×C7—42no—
410C22×C7—82no—
411C22×C7—82no—
412C22×C7#44914yes—
413C22×C7C2×#12011yescharacteristic
414C22×C7#54612yes—
415C22×C7C2×#13511yescharacteristic
416C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
417C22×C7#57612yes—
418C2×D7—44no—
419C2×D7—44no—
420C2×D7—44no—
421C2×D7—816no—
422C2×D7—82no—
423C2×D7—82no—
424C2×D7—82no—
425C2×D7—82no—
426C4×C7—22no—
427C4×C7—41no—
428C4×C7—41no—
429C4×C7—42no—
430C4×C7—41no—
431C4×C7—42no—
432C4×C7—42no—
433C4×C7#52511yescharacteristic
434C4×C7#53511yescharacteristic
435C4×C7#54612yes—
436Dic7—82no—
437C24—21no—
438C24—142no—
439C24—284no—
440C24—282no—
441C24—282no—
442C24—282no—
443C24—562no—
444C24C2×C22⋊D711yescharacteristic
445C24C2×C22⋊D711yescharacteristic
446C2×D4—22no—
447C2×D4—22no—
448C2×D4—22no—
449C2×D4—24no—
450C2×D4—24no—
451C2×D4—22no—
452C2×D4—22no—
453C2×D4—21no—
454C2×D4—42no—
455C2×D4—42no—
456C2×D4—74no—
457C2×D4—82no—
458C2×D4—148no—
459C2×D4—142no—
460C2×D4—142no—
461C2×D4—142no—
462C2×D4—142no—
463C2×D4—142no—
464C2×D4—288no—
465C2×D4—284no—
466C2×D4—284no—
467C2×D4—284no—
468C2×D4—284no—
469C2×D4—282no—
470C2×D4—562no—
471C2×D4C2×C22⋊D712yes—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—24no—
478C22×C4—24no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—82no—
484C22×C4—82no—
485C22×C4—142no—
486C22×C4C2×C22⋊D711yescharacteristic
487C22×C4C2×C22⋊D711yescharacteristic
488C22×C4C2×C22⋊D711yescharacteristic
489C22×C4C2×C22⋊D711yescharacteristic
490C22×C4C2×C22⋊D712yes—
491C22×C4C2×C22⋊D712yes—
492C22×C4C23×D711yescharacteristic
493D8—288no—
494C2×C8—142no—
495C2×C8—142no—
496C2×C8—142no—
497C2×C8—142no—
498C2×C8—142no—
499C2×C8—142no—
500C2×C8—284no—
501C4⋊C4—24no—
502C4⋊C4—24no—
503C4⋊C4—24no—
504C4⋊C4—22no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—142no—
509C4⋊C4—142no—
510C4⋊C4C2×C22⋊D712yes—
511C4⋊C4C2×C22⋊D712yes—
512C4⋊C4C2×C22⋊D712yes—
513C22⋊C4—24no—
514C22⋊C4—24no—
515C22⋊C4—24no—
516C22⋊C4—22no—
517C22⋊C4—24no—
518C22⋊C4—22no—
519C22⋊C4—284no—
520C22⋊C4—284no—
521C22⋊C4—282no—
522C22⋊C4—562no—
523C42—24no—
524C42—21no—
525C42—21no—
526C2×C7—24no—
527C2×C7—81no—
528C2×C7—82no—
529C2×C7#240411yescharacteristic
530C2×C7C2×#44911yescharacteristic
531C2×C7#300811yescharacteristic
532C2×C7#329712yes—
533C2×C7#331812yes—
534D7—84no—
535C23—22no—
536C23—22no—
537C23—21no—
538C23—21no—
539C23—22no—
540C23—22no—
541C23—21no—
542C23—22no—
543C23—41no—
544C23—41no—
545C23—42no—
546C23—42no—
547C23—144no—
548C23—288no—
549C23—282no—
550C23—282no—
551C23—282no—
552C23—282no—
553C23—282no—
554C23—282no—
555C23—568no—
556C23—568no—
557C23—564no—
558C23—564no—
559C23#651212yes—
560C23#651312yes—
561C23#651612yes—
562C23C22×C22⋊D711yescharacteristic
563D4—24no—
564D4—44no—
565D4—48no—
566D4—41no—
567D4—41no—
568D4—84no—
569D4—144no—
570D4—144no—
571D4—288no—
572D4—282no—
573D4—282no—
574D4—568no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—24no—
578C2×C4—22no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—22no—
586C2×C4—21no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—41no—
594C2×C4—42no—
595C2×C4—41no—
596C2×C4—84no—
597C2×C4—84no—
598C2×C4—282no—
599C2×C4—282no—
600C2×C4—282no—
601C2×C4#651612yes—
602C2×C4D7×#4112yes—
603C2×C4#655312yes—
604C2×C4C22×C22⋊D711yescharacteristic
605C2×C4C22×C22⋊D711yescharacteristic
606C8—282no—
607C8—282no—
608C7#2396311yescharacteristic, complemented, Sylow
609C22—22no—
610C22—22no—
611C22—24no—
612C22—24no—
613C22—41no—
614C22—42no—
615C22—41no—
616C22—42no—
617C22—42no—
618C22—41no—
619C22—42no—
620C22—42no—
621C22—82no—
622C22—82no—
623C22—284no—
624C22—284no—
625C22—284no—
626C22—5616no—
627C22—562no—
628C22—562no—
629C22—562no—
630C22—562no—
631C22#9001014yes—
632C22C2×#651211yescharacteristic
633C22C2×#651311yescharacteristic
634C22#9052012yes—
635C22#9052612yes—
636C22C2×#651611yescharacteristic
637C4—22no—
638C4—41no—
639C4—42no—
640C4—41no—
641C4—41no—
642C4—42no—
643C4—42no—
644C4—562no—
645C4#9051911yescharacteristic
646C4#9052012yes—
647C4#9052111yescharacteristic
648C2—24no—
649C2—81no—
650C2—82no—
651C2—564no—
652C2#1174105912yes—
653C2#1174106112yes—
654C2C2×#9001011yescharacteristic
655C2#1174154111yescharacteristic
656C2#1174154611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC2×C7×#553C2
socleC23×C7C22×D4
hypercenterC2×#553D7
hypocenterC7#23963
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#553

Complements

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

NH ≅ G/Ntype
1. C2×C7×#553C2semidirect
2. C2×#89808C2semidirect
3. #11739848C2semidirect
4. C22×#6383C2semidirect
5. #11729497C2semidirect
6. #11729438C2semidirect
7. #11729435C2semidirect
8. #11729367C2semidirect
9. #11729364C2semidirect
10. #11729340C2semidirect
11. #11729308C2semidirect
16. C2×C7×#125C22semidirect
20. C2×C7×#121C22semidirect
23. C2×C4×C7×D4C22semidirect
24. C2×C7×#117C22semidirect
30. C2×#6406C22semidirect
33. C2×C22⋊(C7⋊C8)C22semidirect
43. #89324C22semidirect
44. #89324C22semidirect
45. #89324C22semidirect
46. #89319C22semidirect
72. C7×#121D4semidirect
85. C2×C7×C4⋊C4D4semidirect
119. C2×#553D7semidirect
225. C2×#125C2×D7semidirect
230. C2×#121C2×D7semidirect
234. C2×C4×D4C2×D7semidirect
235. C2×#117C2×D7semidirect
360. #121C22⋊D7semidirect
375. C2×C4⋊C4C22⋊D7semidirect
608. C7#23963semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#5532
C2×#898082
#117398482
C22×#63832
#117294972
#117294382
#117294352
#117293672
#117293642
#117293402
#117293082
#239637