Finite Group Databasebeta

← #31999152

Subgroups — #31999152

Order 18176.

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
1C23×#11765849C211yesmaximal, characteristic, complemented
2C2×#17986510C211yesmaximal, characteristic
3C2×#17985998C212yesmaximal, complemented
4#22588776C212yesmaximal, complemented
5#22588771C212yesmaximal, complemented
6#22588766C212yesmaximal, complemented
7#22588761C212yesmaximal, complemented
8#22588059C212yesmaximal, complemented
9C2×#380×C71C211yesmaximal, characteristic, complemented
10C22×#11765865C2211yescharacteristic
11C22×#11765849—28no—
12C22×#11765849—24no—
13C22×#11765849C2212yes—
14C22×#11765849C2212yes—
15C23×C71⋊C8C2212yes—
16#17986510—24no—
17#17986510C418yescomplemented
18C2×#14556111—21no—
19C2×#14556107C2211yescharacteristic
20C2×#14556107C2211yescharacteristic
21C2×#14556102C2211yescharacteristic
22C2×#14556102C2211yescharacteristic
23C2×#14556097—21no—
24C2×C4×C22⋊D71—21no—
25C2×C4×C22⋊D71—21no—
26C2×#14556087C2211yescharacteristic
27#17985998—28no—
28C2×#14556051C2214yescomplemented
29C2×#14556048C2214yescomplemented
30#17985761C2214yescomplemented
31#17985761C2214yescomplemented
32C24×C4×C71C2211yescharacteristic
33C22×C4⋊C4×C71C2212yes—
34C22×C22⋊C4×C71—21no—
35#380×C71—21no—
36#380×C71—21no—
37#380×C71C2212yes—
38#380×C71C2212yes—
39C2×#95×C71—21no—
40C2×#11765865—22no—
41C2×#11765865—24no—
42C2×#11765865—24no—
43C2×#11765865C2×C414yes—
44C22×Dic142C2311yescharacteristic
45C22×D284C2311yescharacteristic
46C22×C22⋊D71—21no—
47C22×C22⋊D71—21no—
48C22×C22⋊D71—21no—
49C22×C4×D71—21no—
50C2×#11765849—232no—
51C2×#11765849—28no—
52C2×#11765849—24no—
53C2×#11765849—24no—
54C2×#11765849—24no—
55C2×#11765849D418yescomplemented
56C22×C71⋊C8—28no—
57C22×C71⋊C8—24no—
58C22×C71⋊C8C2314yes—
59#14556111—28no—
60#14556107—22no—
61#14556107—22no—
62#14556107C2×C414yescomplemented
63#14556107C2×C414yescomplemented
64#14556102—22no—
65#14556102—22no—
66#14556102C2×C414yescomplemented
67#14556102C2×C414yescomplemented
68#14556097—28no—
69C4×C22⋊D71—28no—
70C4×C22⋊D71—28no—
71#14556087—22no—
72#14556087C2×C414yes—
73C2×C4⋊Dic71C2311yescharacteristic
74C2×C4⋊Dic71C2311yescharacteristic
75C2×Dic71⋊C4—21no—
76C2×Dic71⋊C4—21no—
77C2×C22⋊Dic71—21no—
78C2×C22⋊Dic71—21no—
79C2×C22⋊Dic71—21no—
80C2×(C2×D71)⋊C4—21no—
81C2×(C2×D71)⋊C4—21no—
82C2×C4×Dic71—21no—
83#14556051—28no—
84#14556048—28no—
85C25×C71D411yescharacteristic
86C23×C4×C71—24no—
87C23×C4×C71—24no—
88C23×C4×C71—22no—
89C23×C4×C71C2×C414yes—
90C23×C4×C71D411yescharacteristic
91C23×C4×C71D411yescharacteristic
92C23×C4×C71D411yescharacteristic
93C23×C4×C71D414yes—
94C23×C4×C71C2311yescharacteristic
95C23×C4×C71C2311yescharacteristic
96C23×C4×C71C2311yescharacteristic
97C2×C4⋊C4×C71—24no—
98C2×C4⋊C4×C71—24no—
99C2×C4⋊C4×C71—22no—
100C2×C4⋊C4×C71C2314yes—
101C2×C22⋊C4×C71—22no—
102C2×C22⋊C4×C71—22no—
103C2×C22⋊C4×C71—44no—
104#95×C71—22no—
105#95×C71—22no—
106C23×D71—21no—
107#11765865—216no—
108#11765865—44no—
109#11765865—44no—
110C2×Dic142—22no—
111C2×Dic142—22no—
112C2×Dic142—21no—
113C2×Dic142C22×C412yes—
114C2×D284—22no—
115C2×D284—22no—
116C2×D284—21no—
117C2×D284C22×C412yes—
118C2×C22⋊D71—24no—
119C2×C22⋊D71—24no—
120C2×C22⋊D71—24no—
121C2×C22⋊D71—44no—
122C2×C22⋊D71—44no—
123C2×C22⋊D71—42no—
124C2×C22⋊D71—42no—
125C22×Dic71—21no—
126C22×Dic71—21no—
127C22×Dic71—21no—
128C2×C4×D71—24no—
129C2×C4×D71—44no—
130#11765849—216no—
131#11765849—216no—
132C2×C71⋊C8—216no—
133C2×C71⋊C8—24no—
134C2×C71⋊C8—24no—
135C2×C71⋊C8C2×D418yes—
136C4⋊Dic71—21no—
137C4⋊Dic71—21no—
138C4⋊Dic71C22×C412yes—
139C4⋊Dic71C22×C412yes—
140Dic71⋊C4—24no—
141Dic71⋊C4—24no—
142C22⋊Dic71—24no—
143C22⋊Dic71—24no—
144C22⋊Dic71—24no—
145(C2×D71)⋊C4—24no—
146(C2×D71)⋊C4—24no—
147C4×Dic71—24no—
148C24×C71—24no—
149C24×C71—22no—
150C24×C71—22no—
151C24×C71—42no—
152C24×C71C22⋊C414yes—
153C24×C71C2×D411yescharacteristic
154C24×C71C2×D411yescharacteristic
155C24×C71C2×D411yescharacteristic
156C22×C4×C71—24no—
157C22×C4×C71—216no—
158C22×C4×C71—28no—
159C22×C4×C71—24no—
160C22×C4×C71—24no—
161C22×C4×C71—28no—
162C22×C4×C71—22no—
163C22×C4×C71—22no—
164C22×C4×C71—22no—
165C22×C4×C71—22no—
166C22×C4×C71—24no—
167C22×C4×C71—44no—
168C22×C4×C71—44no—
169C22×C4×C71—44no—
170C22×C4×C71C22⋊C414yes—
171C22×C4×C71C22⋊C414yes—
172C22×C4×C71C22⋊C414yes—
173C22×C4×C71C22×C412yes—
174C22×C4×C71C22×C412yes—
175C22×C4×C71C22×C412yes—
176C22×C4×C71C2×D414yes—
177C22×C4×C71C2×D411yescharacteristic
178C22×C4×C71C2×D411yescharacteristic
179C22×C4×C71C2×D411yescharacteristic
180C22×C4×C71C2×D412yes—
181C22×C4×C71C2×D412yes—
182C22×C4×C71C2×D412yes—
183C22×C4×C71#4114yes—
184C22×C4×C71C2411yescharacteristic
185C4⋊C4×C71—24no—
186C4⋊C4×C71—24no—
187C4⋊C4×C71—44no—
188C22⋊C4×C71—44no—
189C22⋊C4×C71—44no—
190C23×C71—22no—
191C23×C71—21no—
192C23×C71—21no—
193C23×C71—22no—
194C23×C71—22no—
195C23×C71—22no—
196C23×C71—21no—
197C23×C71—21no—
198C23×C71—22no—
199C23×C71—28no—
200C23×C71—28no—
201C23×C71—44no—
202C23×C71—48no—
203C23×C71—48no—
204C23×C71C2×C22⋊C412yes—
205C23×C71C2×C22⋊C412yes—
206C23×C71C2×C22⋊C412yes—
207C23×C71#12114yes—
208C23×C71AGL(1,ℤ/8)12yes—
209C23×C71#13712yes—
210C23×C71C22×D411yescharacteristic
211C22×D71—22no—
212C22×D71—42no—
213C22×D71—41no—
214C22×D71—41no—
215C22×D71—42no—
216C2×C4×C71—22no—
217C2×C4×C71—22no—
218C2×C4×C71—28no—
219C2×C4×C71—24no—
220C2×C4×C71—24no—
221C2×C4×C71—22no—
222C2×C4×C71—22no—
223C2×C4×C71—22no—
224C2×C4×C71—22no—
225C2×C4×C71—28no—
226C2×C4×C71—28no—
227C2×C4×C71—44no—
228C2×C4×C71—44no—
229C2×C4×C71—44no—
230C2×C4×C71—88no—
231C2×C4×C71C2×C22⋊C412yes—
232C2×C4×C71C2×C22⋊C412yes—
233C2×C4×C71C2×C22⋊C412yes—
234C2×C4×C71C4×D418yes—
235C2×C4×C71#12014yes—
236C2×C4×C71#12114yes—
237C2×C4×C71#12314yes—
238C2×C4×C71C23×C411yescharacteristic
239C2×C4×C71C22×D412yes—
240C2×C4×C71C22×D411yescharacteristic
241C2×C4×C71C2×#4112yes—
242C22⋊D71—48no—
243C22⋊D71—48no—
244C22⋊D71—48no—
245C2×Dic71—22no—
246C2×Dic71—22no—
247C2×Dic71—22no—
248C2×Dic71—42no—
249C2×Dic71—41no—
250C2×Dic71—41no—
251C2×Dic71—42no—
252D284—24no—
253D284—41no—
254D284—41no—
255C4×D71—48no—
256Dic142—24no—
257Dic142—41no—
258Dic142—41no—
259C71⋊C8—28no—
260C22×C71—22no—
261C22×C71—22no—
262C22×C71—22no—
263C22×C71—22no—
264C22×C71—24no—
265C22×C71—24no—
266C22×C71—24no—
267C22×C71—24no—
268C22×C71—22no—
269C22×C71—22no—
270C22×C71—22no—
271C22×C71—416no—
272C22×C71—42no—
273C22×C71—42no—
274C22×C71#38614yes—
275C22×C71#41714yescomplemented
276C22×C71C22×C22⋊C411yescharacteristic
277C22×C71C2×#12112yes—
278C22×C71C2×AGL(1,ℤ/8)11yescharacteristic
279C22×C71C2×#13711yescharacteristic
280C22×C71#57512yes—
281C2×D71—42no—
282C2×D71—42no—
283C2×D71—42no—
284C2×D71—82no—
285C2×D71—82no—
286C4×C71—22no—
287C4×C71—22no—
288C4×C71—84no—
289C4×C71#38614yes—
290C4×C71C2×C4×D412yes—
291C4×C71C2×#12011yescharacteristic
292C4×C71C2×#12311yescharacteristic
293Dic71—42no—
294Dic71—42no—
295Dic71—42no—
296#20971—711nomaximal, Sylow
297C2×C71—24no—
298C2×C71—24no—
299C2×C71—42no—
300C2×C71C2×#38611yescharacteristic
301C2×C71C2×#41711yescharacteristic
302C2×C71#289611yescharacteristic
303C2×C71#305512yes—
304C2×C71#305612yes—
305D71—82no—
306C23×M16—711no—
307C2×#525—711no—
308C2×#417—712no—
309C2×#380D7111yescharacteristic, complemented
310#1895—712no—
311#1894—712no—
312#1893—712no—
313#1892—712no—
314#1736—712no—
315C71#2097111yescharacteristic, complemented, Sylow
316C22×#41—711no—
317C24×C4C2×D7111yescharacteristic
318C22×M16—712no—
319C22×M16—712no—
320C22×M16—1428no—
321C22×M16—1424no—
322C23×C8—712no—
323#525—718no—
324#525—1424no—
325C2×#123—1421no—
326C2×C22⋊Q8—711no—
327C2×C22⋊Q8—711no—
328C2×#121—711no—
329C2×#121—711no—
330C2×#120—1421no—
331C2×C4×D4—1421no—
332C2×C4×D4—1421no—
333C2×#117—711no—
334C22×C4⋊C4C2×D7112yes—
335C22×C22⋊C4—21no—
336#417—1428no—
337C2×#103—714no—
338C2×#102—714no—
339#380—21no—
340#380—21no—
341#380C2×D7112yes—
342#380C2×D7112yes—
343C2×#95—21no—
344#340—714no—
345#340—714no—
346C25C22⋊D7111yescharacteristic
347C2×#41—714no—
348C2×#41—1424no—
349C2×#41—1424no—
350C2×#41—1422no—
351C22×Q8—711no—
352C22×D4—711no—
353C22×D4—1421no—
354C22×D4—1421no—
355C22×D4—1421no—
356C23×C4—24no—
357C23×C4—24no—
358C23×C4—22no—
359C23×C4—1421no—
360C23×C4C4×D7114yes—
361C23×C4D28414yes—
362C23×C4C22⋊D7111yescharacteristic
363C23×C4C22⋊D7111yescharacteristic
364C23×C4C22⋊D7111yescharacteristic
365C23×C4C22×D7111yescharacteristic
366C23×C4C22×D7111yescharacteristic
367C23×C4C22×D7111yescharacteristic
368C2×M16—718no—
369C2×M16—1428no—
370C2×M16—14232no—
371C2×M16—1424no—
372C2×M16—1424no—
373C2×M16—1424no—
374C22×C8—714no—
375C22×C8—1428no—
376C22×C8—1424no—
377#123—1428no—
378C22⋊Q8—714no—
379C22⋊Q8—714no—
380C22⋊Q8—1422no—
381C22⋊Q8—1422no—
382#121—714no—
383#121—714no—
384#121—1422no—
385#121—1422no—
386#120—1428no—
387C4×D4—1428no—
388C4×D4—1428no—
389#117—714no—
390#117—1422no—
391C2×C4⋊C4—24no—
392C2×C4⋊C4—24no—
393C2×C4⋊C4—22no—
394C2×C4⋊C4—711no—
395C2×C4⋊C4—711no—
396C2×C4⋊C4—1421no—
397C2×C4⋊C4—1421no—
398C2×C4⋊C4C22×D7114yes—
399C2×C22⋊C4—22no—
400C2×C22⋊C4—22no—
401C2×C22⋊C4—44no—
402C2×C22⋊C4—1421no—
403C2×C22⋊C4—1421no—
404C2×C22⋊C4—1421no—
405C2×C22⋊C4—1421no—
406C2×C22⋊C4—1421no—
407C2×C42—1421no—
408#103—1428no—
409#102—1428no—
410#95—22no—
411#95—22no—
412C24—24no—
413C24—22no—
414C24—22no—
415C24—42no—
416C24—1421no—
417C24(C2×D71)⋊C414yes—
418C24C2×C22⋊D7111yescharacteristic
419C24C2×C22⋊D7111yescharacteristic
420C24C2×C22⋊D7111yescharacteristic
421#41—2844no—
422#41—2844no—
423#41—14216no—
424C2×Q8—712no—
425C2×Q8—1422no—
426C2×Q8—1422no—
427C2×Q8—1421no—
428C2×D4—2844no—
429C2×D4—2844no—
430C2×D4—2842no—
431C2×D4—2842no—
432C2×D4—712no—
433C2×D4—1424no—
434C2×D4—1424no—
435C2×D4—1424no—
436C2×D4—1422no—
437C2×D4—1421no—
438C2×D4—1422no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—216no—
445C22×C4—28no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—28no—
449C22×C4—24no—
450C22×C4—44no—
451C22×C4—44no—
452C22×C4—44no—
453C22×C4—2844no—
454C22×C4—1421no—
455C22×C4—1421no—
456C22×C4—1421no—
457C22×C4—1424no—
458C22×C4(C2×D71)⋊C414yes—
459C22×C4(C2×D71)⋊C414yes—
460C22×C4(C2×D71)⋊C414yes—
461C22×C4C2×C4×D7112yes—
462C22×C4C2×C4×D7112yes—
463C22×C4C2×C4×D7112yes—
464C22×C4D4×D7114yes—
465C22×C4C2×C22⋊D7111yescharacteristic
466C22×C4C2×C22⋊D7111yescharacteristic
467C22×C4C2×C22⋊D7111yescharacteristic
468C22×C4C2×D28412yes—
469C22×C4C2×D28412yes—
470C22×C4C2×D28412yes—
471C22×C4#1176586414yes—
472C22×C4C23×D7111yescharacteristic
473M16—14216no—
474M16—14216no—
475C2×C8—718no—
476C2×C8—14216no—
477C2×C8—1424no—
478C2×C8—1424no—
479C4⋊C4—24no—
480C4⋊C4—24no—
481C4⋊C4—44no—
482C4⋊C4—712no—
483C4⋊C4—712no—
484C4⋊C4—1421no—
485C4⋊C4—1421no—
486C4⋊C4—1424no—
487C4⋊C4—1424no—
488C22⋊C4—44no—
489C22⋊C4—44no—
490C22⋊C4—1424no—
491C22⋊C4—1424no—
492C22⋊C4—1424no—
493C22⋊C4—1424no—
494C22⋊C4—1424no—
495C42—1424no—
496C23—22no—
497C23—21no—
498C23—21no—
499C23—22no—
500C23—22no—
501C23—28no—
502C23—22no—
503C23—28no—
504C23—21no—
505C23—21no—
506C23—22no—
507C23—44no—
508C23—48no—
509C23—48no—
510C23—2842no—
511C23—2841no—
512C23—2841no—
513C23—2842no—
514C23—1422no—
515C23C2×(C2×D71)⋊C412yes—
516C23C2×(C2×D71)⋊C412yes—
517C23C2×(C2×D71)⋊C412yes—
518C23#1455610014yes—
519C23#1455615412yes—
520C23#1455615912yes—
521C23C22×C22⋊D7111yescharacteristic
522Q8—2841no—
523Q8—2841no—
524Q8—1424no—
525D4—2848no—
526D4—2848no—
527D4—2848no—
528D4—2841no—
529D4—2841no—
530D4—1424no—
531C2×C4—28no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—28no—
535C2×C4—28no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—88no—
546C2×C4—2842no—
547C2×C4—2841no—
548C2×C4—2841no—
549C2×C4—2842no—
550C2×C4—2848no—
551C2×C4—1422no—
552C2×C4—1422no—
553C2×C4—1422no—
554C2×C4C2×(C2×D71)⋊C412yes—
555C2×C4C2×(C2×D71)⋊C412yes—
556C2×C4C2×(C2×D71)⋊C412yes—
557C2×C4(C22⋊D71)⋊C418yes—
558C2×C4#1455609814yes—
559C2×C4#1455610014yes—
560C2×C4#1455611214yes—
561C2×C4C22×C4×D7111yescharacteristic
562C2×C4C2×D4×D7112yes—
563C2×C4C22×D28411yescharacteristic
564C2×C4C2×#1176586412yes—
565C8—1428no—
566C22—22no—
567C22—24no—
568C22—24no—
569C22—24no—
570C22—24no—
571C22—22no—
572C22—22no—
573C22—22no—
574C22—22no—
575C22—22no—
576C22—22no—
577C22—416no—
578C22—42no—
579C22—42no—
580C22—2842no—
581C22—2842no—
582C22—2842no—
583C22—5682no—
584C22—5682no—
585C22#1798587614yes—
586C22#1798599814yescomplemented
587C22C22×(C2×D71)⋊C411yescharacteristic
588C22C2×#1455610012yes—
589C22C2×#1455615411yescharacteristic
590C22C2×#1455615911yescharacteristic
591C22#1798679712yes—
592C4—22no—
593C4—22no—
594C4—84no—
595C4—2842no—
596C4—2842no—
597C4—2842no—
598C4#1798587614yes—
599C4C2×(C22⋊D71)⋊C412yes—
600C4C2×#1455609811yescharacteristic
601C4C2×#1455611211yescharacteristic
602C2—24no—
603C2—24no—
604C2—42no—
605C2—5682no—
606C2C2×#1798587611yescharacteristic
607C2C2×#1798599811yescharacteristic
608C2#2259685711yescharacteristic
609C2#2259795512yes—
610C2#2259796612yes—
subgroupG/H
centerC23C22×C22⋊D71
commutatorC2×C4×C71C23×C4
FrattiniC22×C4C23×D71
FittingC2×#380×C71C2
socleC23×C71C22×D4
prime radicalC2×#17986510C2
cosocleC22×C4×C71C24
hypercenterC2×#380D71
hypocenterC71#20971
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C24×C4 ◁ C2×#380

Complements

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

NH ≅ G/Ntype
1. C23×#11765849C2semidirect
3. C2×#17985998C2semidirect
4. #22588776C2semidirect
5. #22588771C2semidirect
6. #22588766C2semidirect
7. #22588761C2semidirect
8. #22588059C2semidirect
9. C2×#380×C71C2semidirect
17. #17986510C4semidirect
28. C2×#14556051C22semidirect
29. C2×#14556048C22semidirect
30. #17985761C22semidirect
31. #17985761C22semidirect
55. C2×#11765849D4semidirect
62. #14556107C2×C4semidirect
63. #14556107C2×C4semidirect
66. #14556102C2×C4semidirect
67. #14556102C2×C4semidirect
275. C22×C71#417semidirect
309. C2×#380D71semidirect
315. C71#20971semidirect
586. C22#17985998semidirect

Maximal subgroups

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

subgroupindex
C23×#117658492
C2×#179865102
C2×#179859982
#225887762
#225887712
#225887662
#225887612
#225880592
C2×#380×C712
#2097171