Finite Group Databasebeta

← #13355510

Subgroups — #13355510

Order 1792.

2238 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6501C211yesmaximal, characteristic
2C22×D7×M16C211yesmaximal, characteristic, complemented
3C2×C7×#418C211yesmaximal, characteristic, complemented
4C2×#89987C211yesmaximal, characteristic, complemented
5#11739712C211yesmaximal, characteristic, complemented
6#11729739C211yesmaximal, characteristic, complemented
7#11729335C211yesmaximal, characteristic, complemented
8#11727957C214yesmaximal, complemented
9#11727956C214yesmaximal, complemented
10C23×C7×D4C2211yescharacteristic
11C23×C22⋊D7—21no—
12C23×C4×D7C2211yescharacteristic
13C22×C7×M16C2211yescharacteristic
14C2×C7×#117C2211yescharacteristic
15C22×C22⋊Dic7—21no—
16C2×#6501—21no—
17C2×#6501—21no—
18C2×#6501—24no—
19C2×#6501C418yescomplemented
20C2×#6501C2212yes—
21C2×#6501C2212yes—
22C2×#6489C2211yescharacteristic
23C22×C4⋊Dic7C2211yescharacteristic
24C22×#1136C2211yescharacteristic
25C2×D7×M16—24no—
26C2×D7×M16—24no—
27C2×D7×M16—24no—
28C22×#1131C2212yescomplemented
29C22×C8×D7C2212yescomplemented
30C7×#418—24no—
31C2×C7×#102C2214yescomplemented
32#89987—24no—
33C2×#6406C2214yescomplemented
34#89759C2212yescomplemented
35#89759C2212yescomplemented
36#89727—21no—
37#89715—21no—
38C4×(C2×D7)⋊C4—21no—
39C4×(C2×D7)⋊C4—21no—
40#89346C2214yescomplemented
41#89324C2214yescomplemented
42#20973—71nomaximal, Sylow
43C25×C7—21no—
44C24×D7D411yescharacteristic
45C22×C7×D4—22no—
46C22×C7×D4—22no—
47C22×C7×D4—21no—
48C22×C7×D4—22no—
49C22×C7×D4—22no—
50C22×C7×D4—22no—
51C22×C7×D4C2×C414yes—
52C22×C7×D4C2312yes—
53C23×C4×C7C2311yescharacteristic
54C22×C22⋊D7—28no—
55C22×C22⋊D7—22no—
56C22×C22⋊D7—22no—
57C22×C22⋊D7—48no—
58C23×Dic7—21no—
59C23×Dic7D411yescharacteristic
60C22×C4×D7—22no—
61C22×C4×D7—22no—
62C22×C4×D7—22no—
63C22×C4×D7—22no—
64C22×C4×D7C2×C414yes—
65C22×C4×D7D411yescharacteristic
66C22×C4×D7D411yescharacteristic
67C22×C4×D7D414yes—
68C22×C4×D7C2311yescharacteristic
69C22×C4×D7C2311yescharacteristic
70C2×C7×M16—22no—
71C2×C7×M16—22no—
72C2×C7×M16D414yescomplemented
73C22×C7×C8C2312yes—
74C7×#117—22no—
75C7×#117—22no—
76C2×C7×C4⋊C4C2312yes—
77C2×C7×C22⋊C4—21no—
78C2×C42×C7—21no—
79C2×C22⋊Dic7—28no—
80C2×C22⋊Dic7—22no—
81C2×C22⋊Dic7—22no—
82C2×C22⋊Dic7—21no—
83#6501—28no—
84#6501—28no—
85#6501—24no—
86#6501—24no—
87#6501C2×C418yes—
88#6501C2×C418yes—
89C2×(C2×D7)⋊C4—22no—
90C2×(C2×D7)⋊C4—22no—
91#6489—22no—
92#6489—22no—
93C2×C4⋊Dic7—21no—
94C2×C4⋊Dic7—22no—
95C2×C4⋊Dic7C2×C414yes—
96C2×C4⋊Dic7C2312yes—
97C2×C4⋊Dic7C2312yes—
98C2×C4×Dic7—21no—
99C2×#1136—22no—
100C2×#1136—22no—
101C2×#1136D414yescomplemented
102C22×C7⋊C8C2312yes—
103D7×M16—232no—
104C2×#1131—24no—
105C2×#1131—24no—
106C2×#1131—24no—
107C2×C8×D7—24no—
108C2×C8×D7—24no—
109C2×C8×D7—24no—
110C7×#102—28no—
111#6406—28no—
112C7⋊#95—22no—
113C7⋊#95—22no—
114C23×M16—71no—
115C22×#121—71no—
116C2×#418—71no—
117C2×#418D711yescharacteristic, complemented
118#2280—71no—
119#1895—74no—
120#1893—74no—
121#1736—71no—
122#1736—71no—
123C24×C7—24no—
124C24×C7—22no—
125C24×C7—44no—
126C24×C7—44no—
127C24×C7—42no—
128C24×C7—42no—
129C24×C7C2×D411yescharacteristic
130C23×D7—22no—
131C23×D7—22no—
132C23×D7—24no—
133C23×D7—42no—
134C23×D7C22⋊C414yes—
135C23×D7C2×D411yescharacteristic
136C23×D7C2×D411yescharacteristic
137C2×C7×D4—24no—
138C2×C7×D4—28no—
139C2×C7×D4—28no—
140C2×C7×D4—22no—
141C2×C7×D4—22no—
142C2×C7×D4—22no—
143C2×C7×D4—22no—
144C2×C7×D4—22no—
145C2×C7×D4—44no—
146C2×C7×D4—42no—
147C2×C7×D4—42no—
148C2×C7×D4—44no—
149C2×C7×D4C22×C414yes—
150C22×C4×C7—22no—
151C22×C4×C7C22×C412yes—
152C22×C4×C7C2×D411yescharacteristic
153C22×C4×C7C2×D412yes—
154C22×C4×C7C2×D412yes—
155C22×C4×C7C2×D412yes—
156C22×C4×C7#4112yes—
157C22×C4×C7#4112yes—
158C22×C4×C7C2411yescharacteristic
159C2×C22⋊D7—28no—
160C2×C22⋊D7—28no—
161C2×C22⋊D7—432no—
162C2×C22⋊D7—48no—
163C2×C22⋊D7—48no—
164C22×Dic7—22no—
165C22×Dic7—22no—
166C22×Dic7—24no—
167C22×Dic7—22no—
168C22×Dic7—22no—
169C22×Dic7—44no—
170C22×Dic7C22⋊C414yes—
171C22×Dic7C2×D411yescharacteristic
172C22×Dic7C2×D411yescharacteristic
173C2×C4×D7—28no—
174C2×C4×D7—28no—
175C2×C4×D7—28no—
176C2×C4×D7—22no—
177C2×C4×D7—22no—
178C2×C4×D7—22no—
179C2×C4×D7—22no—
180C2×C4×D7—28no—
181C2×C4×D7—21no—
182C2×C4×D7—21no—
183C2×C4×D7—21no—
184C2×C4×D7—21no—
185C2×C4×D7—21no—
186C2×C4×D7—21no—
187C2×C4×D7—21no—
188C2×C4×D7—21no—
189C2×C4×D7C22⋊C414yes—
190C2×C4×D7C22⋊C414yes—
191C2×C4×D7C22×C412yes—
192C2×C4×D7C22×C412yes—
193C2×C4×D7C2×D412yes—
194C2×C4×D7C2×D412yes—
195C7×M16—28no—
196C2×C7×C8—22no—
197C2×C7×C8—22no—
198C2×C7×C8C2×D414yescomplemented
199C7×C4⋊C4—22no—
200C7×C4⋊C4—22no—
201C7×C22⋊C4—42no—
202C42×C7—42no—
203C22⋊Dic7—28no—
204C22⋊Dic7—28no—
205C22⋊Dic7—42no—
206(C2×D7)⋊C4—44no—
207(C2×D7)⋊C4—44no—
208C4⋊Dic7—24no—
209C4⋊Dic7—22no—
210C4⋊Dic7—22no—
211C4⋊Dic7—22no—
212C4⋊Dic7C22×C414yes—
213C4×Dic7—42no—
214#1136—28no—
215C2×C7⋊C8—22no—
216C2×C7⋊C8—22no—
217C2×C7⋊C8C2×D414yescomplemented
218#1131—216no—
219C8×D7—216no—
220C23×D4—141no—
221C23×D4C2×D711yescharacteristic
222C24×C4—71no—
223C22×M16—71no—
224C22×M16—72no—
225C22×M16—144no—
226C22×M16—144no—
227C22×M16—144no—
228C22×M16C2×D711yescharacteristic
229C23×C8—72no—
230C2×#121—78no—
231C2×#121—72no—
232C2×#121—72no—
233C2×#121—144no—
234C2×#121—141no—
235C2×#121—141no—
236C2×#117—71no—
237C2×#117C2×D711yescharacteristic
238C22×C4⋊C4—71no—
239C22×C22⋊C4—141no—
240#418—24no—
241#418—144no—
242C2×#102—74no—
243C2×#102C2×D714yescomplemented
244#381—141no—
245#380—72no—
246#380—72no—
247#379—141no—
248C4×C22⋊C4—141no—
249C4×C22⋊C4—141no—
250#340—74no—
251#340—74no—
252C23×C7—22no—
253C23×C7—22no—
254C23×C7—24no—
255C23×C7—48no—
256C23×C7—48no—
257C23×C7—42no—
258C23×C7—42no—
259C23×C7—44no—
260C23×C7—44no—
261C23×C7—48no—
262C23×C7—88no—
263C23×C7—88no—
264C23×C7C2×C22⋊C412yes—
265C23×C7#12112yes—
266C23×C7#12112yes—
267C23×C7C22×D411yescharacteristic
268C22×D7—21no—
269C22×D7—21no—
270C22×D7—22no—
271C22×D7—21no—
272C22×D7—21no—
273C22×D7—22no—
274C22×D7—22no—
275C22×D7—28no—
276C22×D7—28no—
277C22×D7—44no—
278C22×D7—48no—
279C22×D7—48no—
280C22×D7C2×C22⋊C412yes—
281C22×D7C2×C22⋊C412yes—
282C22×D7AGL(1,ℤ/8)14yes—
283C7×D4—24no—
284C7×D4—24no—
285C7×D4—42no—
286C7×D4—42no—
287C7×D4—48no—
288C2×C4×C7—42no—
289C2×C4×C7—42no—
290C2×C4×C7—42no—
291C2×C4×C7C2×C22⋊C412yes—
292C2×C4×C7C4×D414yes—
293C2×C4×C7C4×D414yes—
294C2×C4×C7#12014yes—
295C2×C4×C7#12112yes—
296C2×C4×C7#12112yes—
297C2×C4×C7#12314yes—
298C2×C4×C7C23×C411yescharacteristic
299C2×C4×C7C22×D411yescharacteristic
300C2×C4×C7C22×D411yescharacteristic
301C2×C4×C7C22×D411yescharacteristic
302C2×C4×C7C2×#4111yescharacteristic
303C2×C4×C7C2×#4111yescharacteristic
304C22⋊D7—416no—
305C22⋊D7—416no—
306C2×Dic7—21no—
307C2×Dic7—21no—
308C2×Dic7—21no—
309C2×Dic7—21no—
310C2×Dic7—28no—
311C2×Dic7—24no—
312C2×Dic7—48no—
313C2×Dic7—44no—
314C2×Dic7—42no—
315C2×Dic7—42no—
316C2×Dic7—42no—
317C2×Dic7C2×C22⋊C412yes—
318C2×Dic7C2×C22⋊C412yes—
319C4×D7—22no—
320C4×D7—22no—
321C4×D7—22no—
322C4×D7—22no—
323C4×D7—22no—
324C4×D7—22no—
325C4×D7—22no—
326C4×D7—22no—
327C4×D7—28no—
328C4×D7—28no—
329C7×C8—24no—
330C7⋊C8—24no—
331C25—21no—
332C25—71no—
333C22×D4—22no—
334C22×D4—22no—
335C22×D4—21no—
336C22×D4—22no—
337C22×D4—22no—
338C22×D4—22no—
339C22×D4—148no—
340C22×D4—142no—
341C22×D4—142no—
342C22×D4—288no—
343C22×D4C4×D714yes—
344C22×D4C22×D712yes—
345C23×C4—71no—
346C23×C4—74no—
347C23×C4—74no—
348C23×C4—71no—
349C23×C4—71no—
350C23×C4—71no—
351C23×C4—71no—
352C23×C4—142no—
353C23×C4—142no—
354C23×C4—142no—
355C23×C4—142no—
356C23×C4—141no—
357C23×C4C22×D711yescharacteristic
358C2×M16—22no—
359C2×M16—22no—
360C2×M16—74no—
361C2×M16—144no—
362C2×M16—144no—
363C2×M16—1432no—
364C2×M16—142no—
365C2×M16—142no—
366C2×M16—144no—
367C2×M16C22⋊D714yescomplemented
368C22×C8—72no—
369C22×C8—144no—
370C22×C8—144no—
371C22×C8—144no—
372C22×C8C22×D712yes—
373#121—78no—
374#121—78no—
375#121—148no—
376#121—148no—
377#121—144no—
378#121—144no—
379#117—22no—
380#117—22no—
381#117—142no—
382#117—142no—
383C2×C4⋊C4—74no—
384C2×C4⋊C4—72no—
385C2×C4⋊C4—72no—
386C2×C4⋊C4—141no—
387C2×C4⋊C4—142no—
388C2×C4⋊C4C22×D712yes—
389C2×C22⋊C4—21no—
390C2×C22⋊C4—148no—
391C2×C22⋊C4—142no—
392C2×C22⋊C4—142no—
393C2×C22⋊C4—142no—
394C2×C22⋊C4—142no—
395C2×C22⋊C4—141no—
396C2×C42—21no—
397C2×C42—141no—
398#102—28no—
399#102—148no—
400#95—142no—
401#95—142no—
402C22×C7—22no—
403C22×C7—22no—
404C22×C7—24no—
405C22×C7—24no—
406C22×C7—44no—
407C22×C7—44no—
408C22×C7—44no—
409C22×C7—816no—
410C22×C7—84no—
411C22×C7—84no—
412C22×C7#38614yes—
413C22×C7C22×C22⋊C411yescharacteristic
414C22×C7C2×#12111yescharacteristic
415C22×C7C2×#12111yescharacteristic
416C22×C7C2×AGL(1,ℤ/8)12yes—
417C22×C7#57512yes—
418C2×D7—22no—
419C2×D7—22no—
420C2×D7—24no—
421C2×D7—24no—
422C2×D7—22no—
423C2×D7—22no—
424C2×D7—22no—
425C2×D7—416no—
426C2×D7—42no—
427C2×D7—42no—
428C2×D7#41814yescomplemented
429C4×C7—82no—
430C4×C7#38614yes—
431C4×C7C2×C4×D411yescharacteristic
432C4×C7C2×C4×D411yescharacteristic
433C4×C7C2×#12011yescharacteristic
434C4×C7C2×#12311yescharacteristic
435Dic7—22no—
436Dic7—22no—
437Dic7—44no—
438Dic7—82no—
439C24—24no—
440C24—22no—
441C24—44no—
442C24—44no—
443C24—42no—
444C24—42no—
445C24—71no—
446C24—71no—
447C24—74no—
448C24—144no—
449C24—142no—
450C24—142no—
451C24—282no—
452C24D4×D711yescharacteristic
453C2×D4—24no—
454C2×D4—28no—
455C2×D4—28no—
456C2×D4—22no—
457C2×D4—22no—
458C2×D4—22no—
459C2×D4—22no—
460C2×D4—22no—
461C2×D4—44no—
462C2×D4—42no—
463C2×D4—44no—
464C2×D4—42no—
465C2×D4—148no—
466C2×D4—148no—
467C2×D4—288no—
468C2×D4—2832no—
469C2×D4—288no—
470C2×D4C2×C4×D714yes—
471C22×C4—22no—
472C22×C4—74no—
473C22×C4—71no—
474C22×C4—71no—
475C22×C4—74no—
476C22×C4—74no—
477C22×C4—72no—
478C22×C4—72no—
479C22×C4—72no—
480C22×C4—72no—
481C22×C4—142no—
482C22×C4—142no—
483C22×C4—148no—
484C22×C4—148no—
485C22×C4—142no—
486C22×C4—142no—
487C22×C4—148no—
488C22×C4—148no—
489C22×C4—142no—
490C22×C4—142no—
491C22×C4—141no—
492C22×C4—141no—
493C22×C4—141no—
494C22×C4—141no—
495C22×C4—141no—
496C22×C4—141no—
497C22×C4—141no—
498C22×C4—141no—
499C22×C4—144no—
500C22×C4—142no—
501C22×C4—142no—
502C22×C4—284no—
503C22×C4C2×C4×D712yes—
504C22×C4#115712yes—
505C22×C4D4×D711yescharacteristic
506C22×C4D4×D712yes—
507C22×C4D4×D712yes—
508C22×C4#115912yes—
509C22×C4C2×C22⋊D712yes—
510C22×C4C23×D711yescharacteristic
511M16—28no—
512M16—1416no—
513M16—148no—
514C2×C8—22no—
515C2×C8—22no—
516C2×C8—74no—
517C2×C8—1416no—
518C2×C8—142no—
519C2×C8—142no—
520C2×C8C2×C22⋊D714yescomplemented
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—74no—
524C4⋊C4—144no—
525C4⋊C4—142no—
526C4⋊C4—142no—
527C4⋊C4—142no—
528C22⋊C4—42no—
529C22⋊C4—148no—
530C22⋊C4—148no—
531C22⋊C4—284no—
532C22⋊C4—284no—
533C22⋊C4—282no—
534C42—42no—
535C42—282no—
536C2×C7—24no—
537C2×C7—84no—
538C2×C7C2×#38611yescharacteristic
539C2×C7C2×#41811yescharacteristic
540C2×C7#289611yescharacteristic
541C2×C7#305512yes—
542C2×C7#305512yes—
543D7—24no—
544D7—42no—
545C23—22no—
546C23—22no—
547C23—24no—
548C23—48no—
549C23—48no—
550C23—42no—
551C23—44no—
552C23—44no—
553C23—42no—
554C23—48no—
555C23—72no—
556C23—74no—
557C23—72no—
558C23—88no—
559C23—88no—
560C23—148no—
561C23—141no—
562C23—142no—
563C23—141no—
564C23—141no—
565C23—142no—
566C23—142no—
567C23—141no—
568C23—148no—
569C23—284no—
570C23—288no—
571C23—288no—
572C23D7×C22⋊C412yes—
573C23#644712yes—
574C23#650112yes—
575C23C2×D4×D711yescharacteristic
576D4—24no—
577D4—24no—
578D4—42no—
579D4—42no—
580D4—48no—
581D4—2816no—
582D4—2816no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—72no—
587C2×C4—72no—
588C2×C4—141no—
589C2×C4—141no—
590C2×C4—141no—
591C2×C4—141no—
592C2×C4—148no—
593C2×C4—142no—
594C2×C4—142no—
595C2×C4—142no—
596C2×C4—142no—
597C2×C4—142no—
598C2×C4—142no—
599C2×C4—142no—
600C2×C4—142no—
601C2×C4—148no—
602C2×C4—148no—
603C2×C4—144no—
604C2×C4—288no—
605C2×C4—284no—
606C2×C4—282no—
607C2×C4—282no—
608C2×C4—282no—
609C2×C4D7×C22⋊C412yes—
610C2×C4(C22⋊D7)⋊C414yes—
611C2×C4#644614yes—
612C2×C4#644712yes—
613C2×C4C4×C22⋊D714yes—
614C2×C4#650014yes—
615C2×C4#650112yes—
616C2×C4C22×C4×D711yescharacteristic
617C2×C4C2×#115711yescharacteristic
618C2×C4C2×D4×D711yescharacteristic
619C2×C4C2×D4×D711yescharacteristic
620C2×C4C2×#115911yescharacteristic
621C2×C4C22×C22⋊D711yescharacteristic
622C8—24no—
623C8—144no—
624C7#2097311yescharacteristic, complemented, Sylow
625C22—22no—
626C22—22no—
627C22—24no—
628C22—24no—
629C22—44no—
630C22—44no—
631C22—44no—
632C22—74no—
633C22—816no—
634C22—84no—
635C22—84no—
636C22—142no—
637C22—144no—
638C22—142no—
639C22—144no—
640C22—142no—
641C22—142no—
642C22—142no—
643C22—2816no—
644C22—282no—
645C22—282no—
646C22#8972814yes—
647C22C2×D7×C22⋊C411yescharacteristic
648C22C2×#644711yescharacteristic
649C22D7×AGL(1,ℤ/8)12yes—
650C22#9046412yes—
651C22C2×#650111yescharacteristic
652C4—82no—
653C4—142no—
654C4—142no—
655C4—284no—
656C4—562no—
657C4#8972814yes—
658C4C2×(C22⋊D7)⋊C411yescharacteristic
659C4C2×#644611yescharacteristic
660C4C2×C4×C22⋊D711yescharacteristic
661C4C2×#650011yescharacteristic
662C2—24no—
663C2—84no—
664C2—144no—
665C2—282no—
666C2D7×#41811yescharacteristic
667C2#1173636911yescharacteristic
668C2#1173637512yes—
669C2C2×#8972811yescharacteristic
670C2#1174139512yes—
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC2×C7×#418C2
socleC23×C7C22×D4
prime radicalC22×#6501C2
cosocleC22×C4×C7C24
hypercenterC2×#418D7
hypocenterC7#20973
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#418

Complements

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

NH ≅ G/Ntype
2. C22×D7×M16C2semidirect
3. C2×C7×#418C2semidirect
4. C2×#89987C2semidirect
5. #11739712C2semidirect
6. #11729739C2semidirect
7. #11729335C2semidirect
8. #11727957C2semidirect
9. #11727956C2semidirect
19. C2×#6501C4semidirect
28. C22×#1131C22semidirect
29. C22×C8×D7C22semidirect
31. C2×C7×#102C22semidirect
33. C2×#6406C22semidirect
34. #89759C22semidirect
35. #89759C22semidirect
40. #89346C22semidirect
41. #89324C22semidirect
72. C2×C7×M16D4semidirect
101. C2×#1136D4semidirect
117. C2×#418D7semidirect
198. C2×C7×C8C2×D4semidirect
217. C2×C7⋊C8C2×D4semidirect
243. C2×#102C2×D7semidirect
367. C2×M16C22⋊D7semidirect
428. C2×D7#418semidirect
520. C2×C8C2×C22⋊D7semidirect
624. C7#20973semidirect

Maximal subgroups

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

subgroupindex
C22×#65012
C22×D7×M162
C2×C7×#4182
C2×#899872
#117397122
#117297392
#117293352
#117279572
#117279562
#209737