Finite Group Databasebeta

← #31998669

Subgroups — #31998669

Order 18176.

1986 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17986510C211yesmaximal, characteristic
2C22×#14556087C211yesmaximal, characteristic, complemented
3C2×#17985996C212yesmaximal, complemented
4#22588660C212yesmaximal, complemented
5#22588655C212yesmaximal, complemented
6#22588650C212yesmaximal, complemented
7#22588645C212yesmaximal, complemented
8#22588057C212yesmaximal, complemented
9C22×C22⋊C8×C71C211yesmaximal, characteristic, complemented
10C22×#11765865C2211yescharacteristic
11#17986510—24no—
12#17986510C418yescomplemented
13C2×#14556111—21no—
14C2×#14556107C2211yescharacteristic
15C2×#14556107C2211yescharacteristic
16C2×#14556102C2211yescharacteristic
17C2×#14556102C2211yescharacteristic
18C2×#14556097—21no—
19C2×C4×C22⋊D71—21no—
20C2×C4×C22⋊D71—21no—
21C2×#14556087—28no—
22C2×#14556087—21no—
23C2×#14556087—21no—
24C2×#14556087C2212yes—
25C2×#14556087C2211yescharacteristic
26C2×#14556087C2212yes—
27C22×C4⋊Dic71C2212yes—
28C22×C22⋊Dic71—21no—
29C22×C4×Dic71—21no—
30#17985996—28no—
31C2×#14556053C2214yescomplemented
32C2×#14556050C2214yescomplemented
33#17985762C2214yescomplemented
34#17985762C2214yescomplemented
35C24×C4×C71C2211yescharacteristic
36C23×C8×C71C2212yes—
37C2×C22⋊C8×C71—24no—
38C2×C22⋊C8×C71C2212yes—
39C2×C22⋊C8×C71C2212yes—
40C2×#11765865—22no—
41C2×#11765865—24no—
42C2×#11765865—24no—
43C2×#11765865C2×C414yescomplemented
44C22×Dic142C2311yescharacteristic
45C22×D284C2311yescharacteristic
46C22×C22⋊D71—21no—
47C22×C22⋊D71—21no—
48C22×C22⋊D71—21no—
49C23×Dic71—22no—
50C22×C4×D71—21no—
51#14556111—28no—
52#14556107—22no—
53#14556107—22no—
54#14556107C2×C414yes—
55#14556107C2×C414yescomplemented
56#14556102—22no—
57#14556102—22no—
58#14556102C2×C414yes—
59#14556102C2×C414yescomplemented
60#14556097—28no—
61C4×C22⋊D71—28no—
62C4×C22⋊D71—28no—
63#14556087—28no—
64#14556087—28no—
65#14556087—28no—
66#14556087—24no—
67#14556087—22no—
68#14556087C2×C414yescomplemented
69#14556087D418yescomplemented
70C2×C4⋊Dic71—28no—
71C2×C4⋊Dic71—22no—
72C2×C4⋊Dic71C2311yescharacteristic
73C2×C4⋊Dic71C2311yescharacteristic
74C2×C4⋊Dic71C2314yes—
75C2×Dic71⋊C4—21no—
76C2×Dic71⋊C4—21no—
77C2×C22⋊Dic71—28no—
78C2×C22⋊Dic71—22no—
79C2×C22⋊Dic71—22no—
80C2×C22⋊Dic71—21no—
81C2×C22⋊Dic71—21no—
82C2×C22⋊Dic71—21no—
83C2×(C2×D71)⋊C4—21no—
84C2×(C2×D71)⋊C4—21no—
85C2×C4×Dic71—28no—
86C2×C4×Dic71—22no—
87C2×C4×Dic71—22no—
88C2×C4×Dic71—21no—
89#14556053—28no—
90#14556050—28no—
91C25×C71D411yescharacteristic
92C23×C4×C71—24no—
93C23×C4×C71—24no—
94C23×C4×C71C2×C414yes—
95C23×C4×C71D411yescharacteristic
96C23×C4×C71D411yescharacteristic
97C23×C4×C71D411yescharacteristic
98C23×C4×C71D414yes—
99C23×C4×C71C2311yescharacteristic
100C23×C4×C71C2311yescharacteristic
101C23×C4×C71C2311yescharacteristic
102C22×C8×C71—24no—
103C22×C8×C71—24no—
104C22×C8×C71—24no—
105C22×C8×C71C2314yes—
106C22⋊C8×C71—24no—
107C22⋊C8×C71—24no—
108C23×D71—21no—
109#11765865—216no—
110#11765865—48no—
111C2×Dic142—22no—
112C2×Dic142—22no—
113C2×Dic142—21no—
114C2×Dic142C22×C412yes—
115C2×D284—22no—
116C2×D284—22no—
117C2×D284—21no—
118C2×D284C22×C412yes—
119C2×C22⋊D71—24no—
120C2×C22⋊D71—24no—
121C2×C22⋊D71—24no—
122C2×C22⋊D71—44no—
123C2×C22⋊D71—42no—
124C2×C22⋊D71—42no—
125C2×C22⋊D71—44no—
126C22×Dic71—28no—
127C22×Dic71—21no—
128C22×Dic71—24no—
129C22×Dic71—21no—
130C22×Dic71—21no—
131C22×Dic71—44no—
132C22×Dic71—44no—
133C2×C4×D71—24no—
134C2×C4×D71—42no—
135C2×C4×D71—42no—
136C4⋊Dic71—28no—
137C4⋊Dic71—24no—
138C4⋊Dic71—21no—
139C4⋊Dic71—21no—
140C4⋊Dic71C22×C412yes—
141C4⋊Dic71C22×C412yes—
142C4⋊Dic71C2×D418yes—
143Dic71⋊C4—24no—
144Dic71⋊C4—24no—
145C22⋊Dic71—28no—
146C22⋊Dic71—24no—
147C22⋊Dic71—24no—
148C22⋊Dic71—24no—
149C22⋊Dic71—28no—
150(C2×D71)⋊C4—24no—
151(C2×D71)⋊C4—24no—
152C4×Dic71—28no—
153C4×Dic71—24no—
154C4×Dic71—28no—
155C24×C71—22no—
156C24×C71—22no—
157C24×C71—24no—
158C24×C71—42no—
159C24×C71C22⋊C414yes—
160C24×C71C2×D411yescharacteristic
161C24×C71C2×D411yescharacteristic
162C24×C71C2×D411yescharacteristic
163C22×C4×C71—22no—
164C22×C4×C71—22no—
165C22×C4×C71—216no—
166C22×C4×C71—22no—
167C22×C4×C71—22no—
168C22×C4×C71—216no—
169C22×C4×C71—22no—
170C22×C4×C71—22no—
171C22×C4×C71—22no—
172C22×C4×C71—22no—
173C22×C4×C71—22no—
174C22×C4×C71—22no—
175C22×C4×C71C22⋊C414yes—
176C22×C4×C71C22⋊C414yes—
177C22×C4×C71C22⋊C414yes—
178C22×C4×C71C22×C412yes—
179C22×C4×C71C22×C412yes—
180C22×C4×C71C22×C412yes—
181C22×C4×C71C2×D414yes—
182C22×C4×C71C2×D412yes—
183C22×C4×C71C2×D412yes—
184C22×C4×C71C2×D412yes—
185C22×C4×C71C2×D411yescharacteristic
186C22×C4×C71C2×D411yescharacteristic
187C22×C4×C71C2×D411yescharacteristic
188C22×C4×C71#4114yes—
189C22×C4×C71C2411yescharacteristic
190C2×C8×C71—24no—
191C2×C8×C71—24no—
192C2×C8×C71—24no—
193C2×C8×C71—48no—
194C23×C71—21no—
195C23×C71—21no—
196C23×C71—22no—
197C23×C71—22no—
198C23×C71—22no—
199C23×C71—21no—
200C23×C71—21no—
201C23×C71—22no—
202C23×C71—22no—
203C23×C71—28no—
204C23×C71—28no—
205C23×C71—44no—
206C23×C71—48no—
207C23×C71—48no—
208C23×C71C2×C22⋊C412yes—
209C23×C71C2×C22⋊C412yes—
210C23×C71C2×C22⋊C412yes—
211C23×C71#12012yes—
212C23×C71#12312yes—
213C23×C71#13514yes—
214C23×C71C22×D411yescharacteristic
215C22×D71—22no—
216C22×D71—41no—
217C22×D71—41no—
218C22×D71—42no—
219C22×D71—42no—
220C2×C4×C71—21no—
221C2×C4×C71—21no—
222C2×C4×C71—21no—
223C2×C4×C71—21no—
224C2×C4×C71—28no—
225C2×C4×C71—24no—
226C2×C4×C71—24no—
227C2×C4×C71—24no—
228C2×C4×C71—24no—
229C2×C4×C71—28no—
230C2×C4×C71—28no—
231C2×C4×C71C2×C22⋊C412yes—
232C2×C4×C71C2×C22⋊C412yes—
233C2×C4×C71C2×C22⋊C412yes—
234C2×C4×C71C4×D418yes—
235C2×C4×C71#12012yes—
236C2×C4×C71#12118yes—
237C2×C4×C71#12312yes—
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—28no—
249C2×Dic71—416no—
250C2×Dic71—41no—
251C2×Dic71—41no—
252C2×Dic71—44no—
253C2×Dic71—42no—
254C2×Dic71—44no—
255C2×Dic71—42no—
256D284—24no—
257D284—42no—
258C4×D71—48no—
259Dic142—24no—
260Dic142—42no—
261C8×C71—44no—
262C22×C71—24no—
263C22×C71—24no—
264C22×C71—22no—
265C22×C71—22no—
266C22×C71—22no—
267C22×C71—24no—
268C22×C71—24no—
269C22×C71—22no—
270C22×C71—22no—
271C22×C71—22no—
272C22×C71—22no—
273C22×C71—416no—
274C22×C71—42no—
275C22×C71—42no—
276C22×C71#38614yes—
277C22×C71#41614yescomplemented
278C22×C71C22×C22⋊C411yescharacteristic
279C22×C71C2×#12011yescharacteristic
280C22×C71C2×#12311yescharacteristic
281C22×C71C2×#13512yes—
282C22×C71#57512yes—
283C2×D71—42no—
284C2×D71—42no—
285C2×D71—42no—
286C2×D71—82no—
287C2×D71—82no—
288C4×C71—22no—
289C4×C71—22no—
290C4×C71#38614yes—
291C4×C71C2×C4×D412yes—
292C4×C71C2×#12112yes—
293Dic71—42no—
294Dic71—42no—
295Dic71—48no—
296Dic71—42no—
297#20908—711nomaximal, Sylow
298C2×C71—24no—
299C2×C71—24no—
300C2×C71—42no—
301C2×C71C2×#38611yescharacteristic
302C2×C71C2×#41611yescharacteristic
303C2×C71#289611yescharacteristic
304C2×C71#300212yes—
305C2×C71#309112yes—
306D71—82no—
307C2×#525—711no—
308C22×#117—711no—
309C2×#416—712no—
310C22×C22⋊C8D7111yescharacteristic, complemented
311#1875—712no—
312#1874—712no—
313#1873—712no—
314#1872—712no—
315#1735—712no—
316C71#2090811yescharacteristic, complemented, Sylow
317C22×#41—711no—
318C24×C4C2×D7111yescharacteristic
319C23×C8C2×D7112yes—
320#525—718no—
321#525—1424no—
322C2×#123—1421no—
323C2×C22⋊Q8—711no—
324C2×C22⋊Q8—711no—
325C2×#121—711no—
326C2×#121—711no—
327C2×#120—1421no—
328C2×C4×D4—1421no—
329C2×C4×D4—1421no—
330C2×#117—712no—
331C2×#117—711no—
332C2×#117—712no—
333C2×#117—1428no—
334C2×#117—1421no—
335C2×#117—1421no—
336C22×C4⋊C4—712no—
337C22×C22⋊C4—1421no—
338C22×C42—1421no—
339#416—1428no—
340C2×#103—714no—
341C2×#102—714no—
342C2×C22⋊C8—24no—
343C2×C22⋊C8C2×D7112yes—
344C2×C22⋊C8C2×D7112yes—
345#340—714no—
346#340—714no—
347C25D28411yescharacteristic
348C2×#41—714no—
349C2×#41—1424no—
350C2×#41—1424no—
351C2×#41—1422no—
352C22×Q8—711no—
353C22×D4—711no—
354C22×D4—1421no—
355C22×D4—1421no—
356C22×D4—1421no—
357C23×C4—24no—
358C23×C4—24no—
359C23×C4—1422no—
360C23×C4—1421no—
361C23×C4C4×D7114yes—
362C23×C4D28411yescharacteristic
363C23×C4D28411yescharacteristic
364C23×C4D28411yescharacteristic
365C23×C4C22⋊D7114yes—
366C23×C4C22×D7111yescharacteristic
367C23×C4C22×D7111yescharacteristic
368C23×C4C22×D7111yescharacteristic
369C22×C8—24no—
370C22×C8—24no—
371C22×C8—24no—
372C22×C8C22×D7114yes—
373#123—1428no—
374C22⋊Q8—714no—
375C22⋊Q8—714no—
376C22⋊Q8—1422no—
377C22⋊Q8—1422no—
378#121—714no—
379#121—714no—
380#121—1422no—
381#121—1422no—
382#120—1428no—
383C4×D4—1428no—
384C4×D4—1428no—
385#117—714no—
386#117—718no—
387#117—1428no—
388#117—1428no—
389#117—1428no—
390#117—1424no—
391#117—1422no—
392C2×C4⋊C4—714no—
393C2×C4⋊C4—711no—
394C2×C4⋊C4—711no—
395C2×C4⋊C4—1428no—
396C2×C4⋊C4—1422no—
397C2×C4⋊C4—1421no—
398C2×C4⋊C4—1421no—
399C2×C22⋊C4—1428no—
400C2×C22⋊C4—1422no—
401C2×C22⋊C4—1422no—
402C2×C22⋊C4—1421no—
403C2×C22⋊C4—1421no—
404C2×C22⋊C4—1421no—
405C2×C22⋊C4—1421no—
406C2×C22⋊C4—1421no—
407C2×C42—1428no—
408C2×C42—1422no—
409C2×C42—1422no—
410C2×C42—1421no—
411#103—1428no—
412#102—1428no—
413C22⋊C8—24no—
414C22⋊C8—24no—
415C24—22no—
416C24—22no—
417C24—24no—
418C24—42no—
419C24—1421no—
420C24(C2×D71)⋊C414yes—
421C24C2×D28411yescharacteristic
422C24C2×D28411yescharacteristic
423C24C2×D28411yescharacteristic
424#41—2848no—
425#41—14216no—
426C2×Q8—712no—
427C2×Q8—1422no—
428C2×Q8—1422no—
429C2×Q8—1421no—
430C2×D4—2844no—
431C2×D4—2842no—
432C2×D4—2842no—
433C2×D4—2844no—
434C2×D4—712no—
435C2×D4—1424no—
436C2×D4—1424no—
437C2×D4—1424no—
438C2×D4—1422no—
439C2×D4—1421no—
440C2×D4—1422no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—216no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—216no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—2844no—
454C22×C4—2844no—
455C22×C4—2842no—
456C22×C4—2842no—
457C22×C4—1428no—
458C22×C4—1424no—
459C22×C4—1421no—
460C22×C4—1421no—
461C22×C4—1421no—
462C22×C4—1424no—
463C22×C4(C2×D71)⋊C414yes—
464C22×C4(C2×D71)⋊C414yes—
465C22×C4(C2×D71)⋊C414yes—
466C22×C4C2×C4×D7112yes—
467C22×C4C2×C4×D7112yes—
468C22×C4C2×C4×D7112yes—
469C22×C4D4×D7114yes—
470C22×C4C2×C22⋊D7112yes—
471C22×C4C2×C22⋊D7112yes—
472C22×C4C2×C22⋊D7112yes—
473C22×C4C2×D28411yescharacteristic
474C22×C4C2×D28411yescharacteristic
475C22×C4C2×D28411yescharacteristic
476C22×C4#1176586414yes—
477C22×C4C23×D7111yescharacteristic
478C2×C8—24no—
479C2×C8—24no—
480C2×C8—24no—
481C2×C8—48no—
482C4⋊C4—712no—
483C4⋊C4—712no—
484C4⋊C4—718no—
485C4⋊C4—1428no—
486C4⋊C4—1424no—
487C4⋊C4—1421no—
488C4⋊C4—1421no—
489C4⋊C4—1424no—
490C4⋊C4—1424no—
491C22⋊C4—1428no—
492C22⋊C4—1424no—
493C22⋊C4—1424no—
494C22⋊C4—1424no—
495C22⋊C4—1428no—
496C22⋊C4—1424no—
497C22⋊C4—1424no—
498C42—1428no—
499C42—1424no—
500C42—1428no—
501C23—21no—
502C23—21no—
503C23—22no—
504C23—22no—
505C23—22no—
506C23—28no—
507C23—21no—
508C23—21no—
509C23—28no—
510C23—22no—
511C23—22no—
512C23—44no—
513C23—48no—
514C23—48no—
515C23—2841no—
516C23—2841no—
517C23—2842no—
518C23—2842no—
519C23—1422no—
520C23C2×(C2×D71)⋊C412yes—
521C23C2×(C2×D71)⋊C412yes—
522C23C2×(C2×D71)⋊C412yes—
523C23#1455609812yes—
524C23#1455611212yes—
525C23#1455615014yes—
526C23C22×D28411yescharacteristic
527Q8—2842no—
528Q8—1424no—
529D4—2848no—
530D4—2848no—
531D4—2848no—
532D4—2842no—
533D4—1424no—
534C2×C4—28no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—28no—
540C2×C4—28no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—28416no—
546C2×C4—2841no—
547C2×C4—2841no—
548C2×C4—2844no—
549C2×C4—2842no—
550C2×C4—2844no—
551C2×C4—2842no—
552C2×C4—2848no—
553C2×C4—1422no—
554C2×C4—1422no—
555C2×C4—1422no—
556C2×C4—1428no—
557C2×C4C2×(C2×D71)⋊C412yes—
558C2×C4C2×(C2×D71)⋊C412yes—
559C2×C4C2×(C2×D71)⋊C412yes—
560C2×C4(C22⋊D71)⋊C418yes—
561C2×C4#1455609812yes—
562C2×C4#1455610018yes—
563C2×C4#1455611212yes—
564C2×C4C22×C4×D7111yescharacteristic
565C2×C4C2×D4×D7112yes—
566C2×C4C22×C22⋊D7111yescharacteristic
567C2×C4C2×#1176586412yes—
568C8—44no—
569C22—22no—
570C22—24no—
571C22—24no—
572C22—24no—
573C22—24no—
574C22—22no—
575C22—22no—
576C22—22no—
577C22—22no—
578C22—22no—
579C22—22no—
580C22—416no—
581C22—42no—
582C22—42no—
583C22—2842no—
584C22—2842no—
585C22—2842no—
586C22—5682no—
587C22—5682no—
588C22#1798587614yes—
589C22#1798599614yescomplemented
590C22C22×(C2×D71)⋊C411yescharacteristic
591C22C2×#1455609811yescharacteristic
592C22C2×#1455611211yescharacteristic
593C22C2×#1455615012yes—
594C22#1798679912yes—
595C4—22no—
596C4—22no—
597C4—2842no—
598C4—2848no—
599C4—2842no—
600C4—2842no—
601C4#1798587614yes—
602C4C2×(C22⋊D71)⋊C412yes—
603C4C2×#1455610012yes—
604C2—24no—
605C2—24no—
606C2—42no—
607C2—5682no—
608C2C2×#1798587611yescharacteristic
609C2C2×#1798599611yescharacteristic
610C2#2259686011yescharacteristic
611C2#2259756212yes—
612C2#2259825512yes—
subgroupG/H
centerC22×C4C2×D284
commutatorC2×C4×C71C23×C4
FrattiniC22×C4C23×D71
FittingC22×C22⋊C8×C71C2
socleC23×C71C22×D4
prime radicalC2×#17986510C2
cosocleC22×C4×C71C24
hypercenterC22×C22⋊C8D71
hypocenterC71#20908
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22×C4 ◁ C24×C4 ◁ C22×C22⋊C8

Complements

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

NH ≅ G/Ntype
2. C22×#14556087C2semidirect
3. C2×#17985996C2semidirect
4. #22588660C2semidirect
5. #22588655C2semidirect
6. #22588650C2semidirect
7. #22588645C2semidirect
8. #22588057C2semidirect
9. C22×C22⋊C8×C71C2semidirect
12. #17986510C4semidirect
31. C2×#14556053C22semidirect
32. C2×#14556050C22semidirect
33. #17985762C22semidirect
34. #17985762C22semidirect
43. C2×#11765865C2×C4semidirect
55. #14556107C2×C4semidirect
59. #14556102C2×C4semidirect
68. #14556087C2×C4semidirect
69. #14556087D4semidirect
277. C22×C71#416semidirect
310. C22×C22⋊C8D71semidirect
316. C71#20908semidirect
589. C22#17985996semidirect

Maximal subgroups

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

subgroupindex
C2×#179865102
C22×#145560872
C2×#179859962
#225886602
#225886552
#225886502
#225886452
#225880572
C22×C22⋊C8×C712
#2090871