Finite Group Databasebeta

← #13355496

Subgroups — #13355496

Order 1792.

1986 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#90482C211yesmaximal, characteristic
2C23×#1136C211yesmaximal, characteristic, complemented
3C2×C7×#380C211yesmaximal, characteristic, complemented
4C2×#89738C212yesmaximal, complemented
5#11729312C212yesmaximal, complemented
6#11727271C212yesmaximal, complemented
7#11727270C212yesmaximal, complemented
8#11727269C212yesmaximal, complemented
9#11727268C212yesmaximal, complemented
10C24×C4×C7C2211yescharacteristic
11C22×#1157C2211yescharacteristic
12C22×C7×C4⋊C4C2212yes—
13C22×C7×C22⋊C4—21no—
14C2×#6516—21no—
15#90482—24no—
16#90482C418yescomplemented
17C2×#6493C2211yescharacteristic
18C2×#6493C2211yescharacteristic
19C2×#6492—21no—
20C2×C4×C22⋊D7—21no—
21C2×C4×C22⋊D7—21no—
22C2×#6489C2211yescharacteristic
23C2×#6487C2211yescharacteristic
24C2×#6487C2211yescharacteristic
25C22×#1136—24no—
26C22×#1136—28no—
27C22×#1136C2212yes—
28C22×#1136C2212yes—
29C23×C7⋊C8C2212yes—
30C7×#380—21no—
31C7×#380—21no—
32C7×#380C2212yes—
33C7×#380C2212yes—
34C2×C7×#95—21no—
35C2×#6384C2214yescomplemented
36#89738—28no—
37C2×#6383C2214yescomplemented
38#89324C2214yescomplemented
39#89324C2214yescomplemented
40#20971—71nomaximal, Sylow
41C25×C7D411yescharacteristic
42C23×C4×C7—24no—
43C23×C4×C7—24no—
44C23×C4×C7—22no—
45C23×C4×C7C2×C414yes—
46C23×C4×C7D414yes—
47C23×C4×C7D411yescharacteristic
48C23×C4×C7D411yescharacteristic
49C23×C4×C7D411yescharacteristic
50C23×C4×C7C2311yescharacteristic
51C23×C4×C7C2311yescharacteristic
52C23×C4×C7C2311yescharacteristic
53C22×C22⋊D7—21no—
54C22×C22⋊D7—21no—
55C22×C22⋊D7—21no—
56C2×#1157—22no—
57C2×#1157—24no—
58C2×#1157—24no—
59C2×#1157C2×C414yes—
60C22×D28C2311yescharacteristic
61C22×C4×D7—21no—
62C22×Dic14C2311yescharacteristic
63C2×C7×C4⋊C4—24no—
64C2×C7×C4⋊C4—24no—
65C2×C7×C4⋊C4—22no—
66C2×C7×C4⋊C4C2314yes—
67C2×C7×C22⋊C4—22no—
68C2×C7×C22⋊C4—22no—
69C2×C7×C22⋊C4—44no—
70#6516—28no—
71C2×C22⋊Dic7—21no—
72C2×C22⋊Dic7—21no—
73C2×C22⋊Dic7—21no—
74#6493—22no—
75#6493—22no—
76#6493C2×C414yescomplemented
77#6493C2×C414yescomplemented
78#6492—28no—
79C4×C22⋊D7—28no—
80C4×C22⋊D7—28no—
81C2×(C2×D7)⋊C4—21no—
82C2×(C2×D7)⋊C4—21no—
83#6489—22no—
84#6489C2×C414yes—
85C2×C4⋊Dic7C2311yescharacteristic
86C2×C4⋊Dic7C2311yescharacteristic
87#6487—22no—
88#6487—22no—
89#6487C2×C414yescomplemented
90#6487C2×C414yescomplemented
91C2×Dic7⋊C4—21no—
92C2×Dic7⋊C4—21no—
93C2×C4×Dic7—21no—
94C2×#1136—232no—
95C2×#1136—24no—
96C2×#1136—24no—
97C2×#1136—24no—
98C2×#1136—28no—
99C2×#1136D418yescomplemented
100C22×C7⋊C8—24no—
101C22×C7⋊C8—28no—
102C22×C7⋊C8C2314yes—
103C7×#95—22no—
104C7×#95—22no—
105#6384—28no—
106#6383—28no—
107C23×M16—71no—
108C2×#525—71no—
109C2×#417—72no—
110C2×#380D711yescharacteristic, complemented
111#1895—72no—
112#1894—72no—
113#1893—72no—
114#1892—72no—
115#1736—72no—
116C24×C7—24no—
117C24×C7—22no—
118C24×C7—22no—
119C24×C7—42no—
120C24×C7C22⋊C414yes—
121C24×C7C2×D411yescharacteristic
122C24×C7C2×D411yescharacteristic
123C24×C7C2×D411yescharacteristic
124C23×D7—21no—
125C22×C4×C7—24no—
126C22×C4×C7—216no—
127C22×C4×C7—28no—
128C22×C4×C7—24no—
129C22×C4×C7—24no—
130C22×C4×C7—28no—
131C22×C4×C7—22no—
132C22×C4×C7—22no—
133C22×C4×C7—22no—
134C22×C4×C7—22no—
135C22×C4×C7—24no—
136C22×C4×C7—44no—
137C22×C4×C7—44no—
138C22×C4×C7—44no—
139C22×C4×C7C22⋊C414yes—
140C22×C4×C7C22⋊C414yes—
141C22×C4×C7C22⋊C414yes—
142C22×C4×C7C22×C412yes—
143C22×C4×C7C22×C412yes—
144C22×C4×C7C22×C412yes—
145C22×C4×C7C2×D411yescharacteristic
146C22×C4×C7C2×D411yescharacteristic
147C22×C4×C7C2×D412yes—
148C22×C4×C7C2×D412yes—
149C22×C4×C7C2×D411yescharacteristic
150C22×C4×C7C2×D412yes—
151C22×C4×C7C2×D414yes—
152C22×C4×C7#4114yes—
153C22×C4×C7C2411yescharacteristic
154C2×C22⋊D7—24no—
155C2×C22⋊D7—24no—
156C2×C22⋊D7—24no—
157C2×C22⋊D7—44no—
158C2×C22⋊D7—44no—
159C2×C22⋊D7—42no—
160C2×C22⋊D7—42no—
161C22×Dic7—21no—
162C22×Dic7—21no—
163C22×Dic7—21no—
164#1157—216no—
165#1157—44no—
166#1157—44no—
167C2×D28—22no—
168C2×D28—22no—
169C2×D28—21no—
170C2×D28C22×C412yes—
171C2×C4×D7—24no—
172C2×C4×D7—44no—
173C2×Dic14—22no—
174C2×Dic14—22no—
175C2×Dic14—21no—
176C2×Dic14C22×C412yes—
177C7×C4⋊C4—24no—
178C7×C4⋊C4—24no—
179C7×C4⋊C4—44no—
180C7×C22⋊C4—44no—
181C7×C22⋊C4—44no—
182C22⋊Dic7—24no—
183C22⋊Dic7—24no—
184C22⋊Dic7—24no—
185(C2×D7)⋊C4—24no—
186(C2×D7)⋊C4—24no—
187C4⋊Dic7—21no—
188C4⋊Dic7—21no—
189C4⋊Dic7C22×C412yes—
190C4⋊Dic7C22×C412yes—
191Dic7⋊C4—24no—
192Dic7⋊C4—24no—
193C4×Dic7—24no—
194#1136—216no—
195#1136—216no—
196C2×C7⋊C8—216no—
197C2×C7⋊C8—24no—
198C2×C7⋊C8—24no—
199C2×C7⋊C8C2×D418yes—
200C22×#41—71no—
201C24×C4C2×D711yescharacteristic
202C22×M16—72no—
203C22×M16—72no—
204C22×M16—148no—
205C22×M16—144no—
206C23×C8—72no—
207#525—78no—
208#525—144no—
209C2×#123—141no—
210C2×C22⋊Q8—71no—
211C2×C22⋊Q8—71no—
212C2×#121—71no—
213C2×#121—71no—
214C2×#120—141no—
215C2×C4×D4—141no—
216C2×C4×D4—141no—
217C2×#117—71no—
218C22×C4⋊C4C2×D712yes—
219C22×C22⋊C4—21no—
220#417—148no—
221C2×#103—74no—
222C2×#102—74no—
223#380—21no—
224#380—21no—
225#380C2×D712yes—
226#380C2×D712yes—
227C2×#95—21no—
228#340—74no—
229#340—74no—
230C23×C7—22no—
231C23×C7—21no—
232C23×C7—21no—
233C23×C7—22no—
234C23×C7—22no—
235C23×C7—22no—
236C23×C7—21no—
237C23×C7—21no—
238C23×C7—22no—
239C23×C7—28no—
240C23×C7—28no—
241C23×C7—44no—
242C23×C7—48no—
243C23×C7—48no—
244C23×C7C2×C22⋊C412yes—
245C23×C7C2×C22⋊C412yes—
246C23×C7C2×C22⋊C412yes—
247C23×C7#12114yes—
248C23×C7AGL(1,ℤ/8)12yes—
249C23×C7#13712yes—
250C23×C7C22×D411yescharacteristic
251C22×D7—22no—
252C22×D7—42no—
253C22×D7—41no—
254C22×D7—41no—
255C22×D7—42no—
256C2×C4×C7—22no—
257C2×C4×C7—22no—
258C2×C4×C7—28no—
259C2×C4×C7—24no—
260C2×C4×C7—24no—
261C2×C4×C7—22no—
262C2×C4×C7—22no—
263C2×C4×C7—22no—
264C2×C4×C7—22no—
265C2×C4×C7—28no—
266C2×C4×C7—28no—
267C2×C4×C7—44no—
268C2×C4×C7—44no—
269C2×C4×C7—44no—
270C2×C4×C7—88no—
271C2×C4×C7C2×C22⋊C412yes—
272C2×C4×C7C2×C22⋊C412yes—
273C2×C4×C7C2×C22⋊C412yes—
274C2×C4×C7C4×D418yes—
275C2×C4×C7#12014yes—
276C2×C4×C7#12114yes—
277C2×C4×C7#12314yes—
278C2×C4×C7C23×C411yescharacteristic
279C2×C4×C7C22×D411yescharacteristic
280C2×C4×C7C22×D412yes—
281C2×C4×C7C2×#4112yes—
282C22⋊D7—48no—
283C22⋊D7—48no—
284C22⋊D7—48no—
285C2×Dic7—22no—
286C2×Dic7—22no—
287C2×Dic7—22no—
288C2×Dic7—42no—
289C2×Dic7—41no—
290C2×Dic7—41no—
291C2×Dic7—42no—
292D28—24no—
293D28—41no—
294D28—41no—
295C4×D7—48no—
296Dic14—24no—
297Dic14—41no—
298Dic14—41no—
299C7⋊C8—28no—
300C25C22⋊D711yescharacteristic
301C2×#41—74no—
302C2×#41—142no—
303C2×#41—144no—
304C2×#41—144no—
305C22×Q8—71no—
306C22×D4—71no—
307C22×D4—141no—
308C22×D4—141no—
309C22×D4—141no—
310C23×C4—24no—
311C23×C4—24no—
312C23×C4—22no—
313C23×C4—141no—
314C23×C4C4×D714yes—
315C23×C4D2814yes—
316C23×C4C22⋊D711yescharacteristic
317C23×C4C22⋊D711yescharacteristic
318C23×C4C22⋊D711yescharacteristic
319C23×C4C22×D711yescharacteristic
320C23×C4C22×D711yescharacteristic
321C23×C4C22×D711yescharacteristic
322C2×M16—78no—
323C2×M16—148no—
324C2×M16—1432no—
325C2×M16—144no—
326C2×M16—144no—
327C2×M16—144no—
328C22×C8—74no—
329C22×C8—148no—
330C22×C8—144no—
331#123—148no—
332C22⋊Q8—74no—
333C22⋊Q8—74no—
334C22⋊Q8—142no—
335C22⋊Q8—142no—
336#121—74no—
337#121—74no—
338#121—142no—
339#121—142no—
340#120—148no—
341C4×D4—148no—
342C4×D4—148no—
343#117—74no—
344#117—142no—
345C2×C4⋊C4—24no—
346C2×C4⋊C4—24no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—71no—
349C2×C4⋊C4—71no—
350C2×C4⋊C4—141no—
351C2×C4⋊C4—141no—
352C2×C4⋊C4C22×D714yes—
353C2×C22⋊C4—22no—
354C2×C22⋊C4—22no—
355C2×C22⋊C4—44no—
356C2×C22⋊C4—141no—
357C2×C22⋊C4—141no—
358C2×C22⋊C4—141no—
359C2×C22⋊C4—141no—
360C2×C22⋊C4—141no—
361C2×C42—141no—
362#103—148no—
363#102—148no—
364#95—22no—
365#95—22no—
366C22×C7—22no—
367C22×C7—22no—
368C22×C7—22no—
369C22×C7—22no—
370C22×C7—24no—
371C22×C7—24no—
372C22×C7—24no—
373C22×C7—24no—
374C22×C7—22no—
375C22×C7—22no—
376C22×C7—22no—
377C22×C7—416no—
378C22×C7—42no—
379C22×C7—42no—
380C22×C7#38614yes—
381C22×C7#41714yescomplemented
382C22×C7C22×C22⋊C411yescharacteristic
383C22×C7C2×#12112yes—
384C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
385C22×C7C2×#13711yescharacteristic
386C22×C7#57512yes—
387C2×D7—42no—
388C2×D7—42no—
389C2×D7—42no—
390C2×D7—82no—
391C2×D7—82no—
392C4×C7—22no—
393C4×C7—22no—
394C4×C7—84no—
395C4×C7#38614yes—
396C4×C7C2×C4×D412yes—
397C4×C7C2×#12011yescharacteristic
398C4×C7C2×#12311yescharacteristic
399Dic7—42no—
400Dic7—42no—
401Dic7—42no—
402C24—22no—
403C24—24no—
404C24—22no—
405C24—42no—
406C24—141no—
407C24(C2×D7)⋊C414yes—
408C24C2×C22⋊D711yescharacteristic
409C24C2×C22⋊D711yescharacteristic
410C24C2×C22⋊D711yescharacteristic
411#41—1416no—
412#41—284no—
413#41—284no—
414C2×Q8—72no—
415C2×Q8—141no—
416C2×Q8—142no—
417C2×Q8—142no—
418C2×D4—72no—
419C2×D4—144no—
420C2×D4—144no—
421C2×D4—144no—
422C2×D4—141no—
423C2×D4—142no—
424C2×D4—142no—
425C2×D4—284no—
426C2×D4—282no—
427C2×D4—284no—
428C2×D4—282no—
429C22×C4—24no—
430C22×C4—216no—
431C22×C4—28no—
432C22×C4—28no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—44no—
441C22×C4—44no—
442C22×C4—44no—
443C22×C4—141no—
444C22×C4—141no—
445C22×C4—141no—
446C22×C4—144no—
447C22×C4—284no—
448C22×C4(C2×D7)⋊C414yes—
449C22×C4(C2×D7)⋊C414yes—
450C22×C4(C2×D7)⋊C414yes—
451C22×C4C2×C4×D712yes—
452C22×C4C2×C4×D712yes—
453C22×C4C2×C4×D712yes—
454C22×C4C2×D2812yes—
455C22×C4C2×D2812yes—
456C22×C4C2×D2812yes—
457C22×C4D4×D714yes—
458C22×C4#115914yes—
459C22×C4C2×C22⋊D711yescharacteristic
460C22×C4C2×C22⋊D711yescharacteristic
461C22×C4C2×C22⋊D711yescharacteristic
462C22×C4C23×D711yescharacteristic
463M16—1416no—
464M16—1416no—
465C2×C8—78no—
466C2×C8—1416no—
467C2×C8—144no—
468C2×C8—144no—
469C4⋊C4—24no—
470C4⋊C4—24no—
471C4⋊C4—44no—
472C4⋊C4—72no—
473C4⋊C4—72no—
474C4⋊C4—144no—
475C4⋊C4—144no—
476C4⋊C4—141no—
477C4⋊C4—141no—
478C22⋊C4—44no—
479C22⋊C4—44no—
480C22⋊C4—144no—
481C22⋊C4—144no—
482C22⋊C4—144no—
483C22⋊C4—144no—
484C22⋊C4—144no—
485C42—144no—
486C2×C7—24no—
487C2×C7—24no—
488C2×C7—42no—
489C2×C7C2×#38611yescharacteristic
490C2×C7C2×#41711yescharacteristic
491C2×C7#289611yescharacteristic
492C2×C7#305512yes—
493C2×C7#305612yes—
494D7—82no—
495C23—21no—
496C23—22no—
497C23—21no—
498C23—22no—
499C23—22no—
500C23—21no—
501C23—22no—
502C23—28no—
503C23—21no—
504C23—28no—
505C23—22no—
506C23—44no—
507C23—48no—
508C23—48no—
509C23—142no—
510C23—281no—
511C23—282no—
512C23—281no—
513C23—282no—
514C23C2×(C2×D7)⋊C412yes—
515C23C2×(C2×D7)⋊C412yes—
516C23C2×(C2×D7)⋊C412yes—
517C23#649512yes—
518C23#650214yes—
519C23#650512yes—
520C23C22×C22⋊D711yescharacteristic
521Q8—144no—
522Q8—281no—
523Q8—281no—
524D4—144no—
525D4—288no—
526D4—288no—
527D4—288no—
528D4—281no—
529D4—281no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—28no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—28no—
540C2×C4—28no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—88no—
545C2×C4—142no—
546C2×C4—142no—
547C2×C4—142no—
548C2×C4—281no—
549C2×C4—282no—
550C2×C4—281no—
551C2×C4—282no—
552C2×C4—288no—
553C2×C4(C22⋊D7)⋊C418yes—
554C2×C4#644514yes—
555C2×C4#644914yes—
556C2×C4C2×(C2×D7)⋊C412yes—
557C2×C4C2×(C2×D7)⋊C412yes—
558C2×C4C2×(C2×D7)⋊C412yes—
559C2×C4#650214yes—
560C2×C4C22×C4×D711yescharacteristic
561C2×C4C22×D2811yescharacteristic
562C2×C4C2×D4×D712yes—
563C2×C4C2×#115912yes—
564C8—148no—
565C7#2097111yescharacteristic, complemented, Sylow
566C22—22no—
567C22—22no—
568C22—24no—
569C22—24no—
570C22—22no—
571C22—24no—
572C22—24no—
573C22—22no—
574C22—22no—
575C22—22no—
576C22—22no—
577C22—416no—
578C22—42no—
579C22—42no—
580C22—282no—
581C22—282no—
582C22—282no—
583C22—562no—
584C22—562no—
585C22#8973014yes—
586C22#8973814yescomplemented
587C22C22×(C2×D7)⋊C411yescharacteristic
588C22C2×#649511yescharacteristic
589C22C2×#650212yes—
590C22C2×#650511yescharacteristic
591C22#9051312yes—
592C4—22no—
593C4—22no—
594C4—84no—
595C4—282no—
596C4—282no—
597C4—282no—
598C4#8973014yes—
599C4C2×(C22⋊D7)⋊C412yes—
600C4C2×#644511yescharacteristic
601C4C2×#644911yescharacteristic
602C2—24no—
603C2—24no—
604C2—42no—
605C2—562no—
606C2C2×#8973011yescharacteristic
607C2C2×#8973811yescharacteristic
608C2#1173965811yescharacteristic
609C2#1174006612yes—
610C2#1174017712yes—
subgroupG/H
centerC23C22×C22⋊D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC2×C7×#380C2
socleC23×C7C22×D4
prime radicalC2×#90482C2
cosocleC22×C4×C7C24
hypercenterC2×#380D7
hypocenterC7#20971
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C24×C4 ◁ C2×#380

Complements

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

NH ≅ G/Ntype
2. C23×#1136C2semidirect
3. C2×C7×#380C2semidirect
4. C2×#89738C2semidirect
5. #11729312C2semidirect
6. #11727271C2semidirect
7. #11727270C2semidirect
8. #11727269C2semidirect
9. #11727268C2semidirect
16. #90482C4semidirect
35. C2×#6384C22semidirect
37. C2×#6383C22semidirect
38. #89324C22semidirect
39. #89324C22semidirect
76. #6493C2×C4semidirect
77. #6493C2×C4semidirect
89. #6487C2×C4semidirect
90. #6487C2×C4semidirect
99. C2×#1136D4semidirect
110. C2×#380D7semidirect
381. C22×C7#417semidirect
565. C7#20971semidirect
586. C22#89738semidirect

Maximal subgroups

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

subgroupindex
C2×#904822
C23×#11362
C2×C7×#3802
C2×#897382
#117293122
#117272712
#117272702
#117272692
#117272682
#209717