Finite Group Databasebeta

← #31998331

Subgroups — #31998331

Order 18176.

1422 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22596929C211yesmaximal, characteristic, complemented
2C2×#17986016C211yesmaximal, characteristic
3#22596859C211yesmaximal, characteristic
4#22590964C211yesmaximal, characteristic, complemented
5#22588573C214yesmaximal, complemented
6#22588566C214yesmaximal, complemented
7C2×#17985768C211yesmaximal, characteristic, complemented
8#22588049C211yesmaximal, characteristic, complemented
9C2×#520×C71C211yesmaximal, characteristic
10C22×#11765849C2211yescharacteristic
11C22×#11765849C2211yescharacteristic
12C2×#14556087C2211yescharacteristic
13C2×#14556087C2211yescharacteristic
14#17986042C2212yescomplemented
15#17986042C2212yescomplemented
16#17986039—21no—
17#17986039—21no—
18#17986035—22no—
19#17986016C418yescomplemented
20C2×#14556055C2214yes—
21#17986008—21no—
22#17986004—21no—
23#17986000C2212yes—
24#17986000C2212yes—
25#17985995—22no—
26#17985895—22no—
27#17985872—22no—
28#17985868C2212yescomplemented
29#17985864C2212yescomplemented
30#17985768C2214yescomplemented
31#17985758C2214yescomplemented
32C22×#41×C71C2211yescharacteristic
33#520×C71—22no—
34#520×C71—22no—
35#520×C71—22no—
36#520×C71C2214yes—
37C2×C4×Q8×C71—21no—
38C2×C4×D4×C71—21no—
39C2×C4×D4×C71—22no—
40C2×#117×C71—21no—
41C2×#117×C71C2211yescharacteristic
42C22×C4⋊C4×C71—21no—
43C22×C4⋊C4×C71C2211yescharacteristic
44C2×#11765849—21no—
45C2×#11765849—21no—
46C2×#11765849—22no—
47C2×#11765849—22no—
48C2×#11765849—22no—
49C2×#11765849C2×C414yescomplemented
50C2×#11765849C2312yes—
51C2×#11765849C2312yes—
52C22×C71⋊C8—21no—
53C22×C71⋊C8—21no—
54#14556087—21no—
55#14556087—21no—
56#14556087—21no—
57#14556087—21no—
58#14556087—22no—
59#14556087C2×C414yescomplemented
60C2×C4⋊Dic71—21no—
61C2×C4⋊Dic71C2311yescharacteristic
62C2×C4⋊Dic71C2311yescharacteristic
63C2×Dic71⋊C4—24no—
64C2×C22⋊Dic71—21no—
65C2×C22⋊Dic71—21no—
66C2×C4×Dic71—21no—
67C2×C4×Dic71C2312yes—
68#14556061—22no—
69#14556061—22no—
70#14556059—22no—
71#14556059—22no—
72#14556055C2×C4116yescomplemented
73#14556052—22no—
74#14556052—22no—
75#14556049—22no—
76#14556049—22no—
77C71⋊#95—24no—
78C2×#41×C71—21no—
79C2×#41×C71—21no—
80C2×#41×C71—22no—
81C2×#41×C71—22no—
82C2×#41×C71C2×C414yes—
83C2×#41×C71D412yes—
84C2×#41×C71Q814yes—
85C2×#41×C71C2312yes—
86C22×Q8×C71D411yescharacteristic
87C22×D4×C71—21no—
88C22×D4×C71D411yescharacteristic
89C23×C4×C71—21no—
90C23×C4×C71—22no—
91C23×C4×C71D412yes—
92C23×C4×C71C2311yescharacteristic
93C4×Q8×C71—24no—
94C4×Q8×C71—24no—
95C4×D4×C71—28no—
96C4×D4×C71—24no—
97C4×D4×C71—24no—
98C4×D4×C71—28no—
99#117×C71—24no—
100#117×C71—21no—
101#117×C71—21no—
102#117×C71—24no—
103#117×C71D412yes—
104#117×C71C2312yes—
105C2×C4⋊C4×C71—22no—
106C2×C4⋊C4×C71—24no—
107C2×C4⋊C4×C71—21no—
108C2×C4⋊C4×C71—21no—
109C2×C4⋊C4×C71—21no—
110C2×C4⋊C4×C71—21no—
111C2×C4⋊C4×C71—24no—
112C2×C4⋊C4×C71D412yes—
113C2×C4⋊C4×C71D411yescharacteristic
114C2×C4⋊C4×C71D411yescharacteristic
115C2×C4⋊C4×C71C2312yes—
116C2×C22⋊C4×C71—22no—
117C2×C22⋊C4×C71—21no—
118C2×C42×C71—22no—
119C2×C42×C71—21no—
120C22×Dic71—22no—
121C22×Dic71—21no—
122C22×Dic71—21no—
123#11765849—24no—
124#11765849—22no—
125#11765849—22no—
126#11765849—22no—
127#11765849—42no—
128#11765849C22×C414yescomplemented
129C2×C71⋊C8—24no—
130C2×C71⋊C8—22no—
131C2×C71⋊C8—22no—
132C2×C71⋊C8—42no—
133C4⋊Dic71—24no—
134C4⋊Dic71—21no—
135C4⋊Dic71—21no—
136C4⋊Dic71—21no—
137C4⋊Dic71—21no—
138Dic71⋊C4—84no—
139C22⋊Dic71—24no—
140C22⋊Dic71—41no—
141C22⋊Dic71—41no—
142C4×Dic71—22no—
143C4×Dic71—41no—
144C4×Dic71—41no—
145C4×Dic71C22×C414yescomplemented
146C24×C71—21no—
147C24×C71C2×D411yescharacteristic
148#41×C71—24no—
149#41×C71—24no—
150#41×C71—24no—
151#41×C71—24no—
152#41×C71—24no—
153#41×C71—22no—
154#41×C71—22no—
155#41×C71C4⋊C418yescomplemented
156#41×C71C22×C414yes—
157#41×C71C2×Q814yes—
158C2×Q8×C71—21no—
159C2×Q8×C71—22no—
160C2×Q8×C71C4⋊C414yescomplemented
161C2×Q8×C71C2×D412yes—
162C2×D4×C71—24no—
163C2×D4×C71—22no—
164C2×D4×C71—24no—
165C2×D4×C71—22no—
166C2×D4×C71—21no—
167C2×D4×C71—22no—
168C2×D4×C71C4⋊C414yescomplemented
169C2×D4×C71C2×D412yes—
170C22×C4×C71—21no—
171C22×C4×C71—24no—
172C22×C4×C71—22no—
173C22×C4×C71—24no—
174C22×C4×C71—22no—
175C22×C4×C71—21no—
176C22×C4×C71—22no—
177C22×C4×C71—21no—
178C22×C4×C71—28no—
179C22×C4×C71—22no—
180C22×C4×C71—82no—
181C22×C4×C71—84no—
182C22×C4×C71C22×C412yes—
183C22×C4×C71C2×D414yes—
184C22×C4×C71C2×D411yescharacteristic
185C22×C4×C71C2×D411yescharacteristic
186C22×C4×C71C2×D411yescharacteristic
187C22×C4×C71C2×D411yescharacteristic
188C22×C4×C71C2×D411yescharacteristic
189C22×C4×C71C2×Q812yes—
190C22×C4×C71#4112yes—
191C22×C4×C71#4112yes—
192C22×C4×C71C2411yescharacteristic
193C4⋊C4×C71—28no—
194C4⋊C4×C71—22no—
195C4⋊C4×C71—21no—
196C4⋊C4×C71—22no—
197C4⋊C4×C71—21no—
198C4⋊C4×C71—22no—
199C4⋊C4×C71—22no—
200C4⋊C4×C71C2×D412yes—
201C4⋊C4×C71C2×D412yes—
202C22⋊C4×C71—28no—
203C22⋊C4×C71—22no—
204C22⋊C4×C71—22no—
205C42×C71—28no—
206C42×C71—22no—
207C42×C71—22no—
208C23×C71—22no—
209C23×C71—24no—
210C23×C71—41no—
211C23×C71—41no—
212C23×C71—42no—
213C23×C71—42no—
214C23×C71C4×D412yes—
215C23×C71#12011yescharacteristic
216C23×C71#12011yescharacteristic
217C23×C71C22⋊Q812yes—
218C23×C71C22×D411yescharacteristic
219Q8×C71—22no—
220Q8×C71—24no—
221Q8×C71C2×C4⋊C414yes—
222D4×C71—24no—
223D4×C71—24no—
224D4×C71—28no—
225D4×C71—22no—
226D4×C71—24no—
227D4×C71C2×C4⋊C414yes—
228C2×C4×C71—22no—
229C2×C4×C71—24no—
230C2×C4×C71—24no—
231C2×C4×C71—24no—
232C2×C4×C71—28no—
233C2×C4×C71—22no—
234C2×C4×C71—24no—
235C2×C4×C71—22no—
236C2×C4×C71—42no—
237C2×C4×C71—44no—
238C2×C4×C71—42no—
239C2×C4×C71—44no—
240C2×C4×C71—82no—
241C2×C4×C71—88no—
242C2×C4×C71C2×C4⋊C412yes—
243C2×C4×C71C2×C4⋊C412yes—
244C2×C4×C71C4×D412yes—
245C2×C4×C71C4×D412yes—
246C2×C4×C71C4×D412yes—
247C2×C4×C71#12011yescharacteristic
248C2×C4×C71#12011yescharacteristic
249C2×C4×C71C22⋊Q812yes—
250C2×C4×C71C22⋊Q812yes—
251C2×C4×C71C22⋊Q812yes—
252C2×C4×C71#12312yes—
253C2×C4×C71#12312yes—
254C2×C4×C71C23×C411yescharacteristic
255C2×C4×C71C22×D411yescharacteristic
256C2×C4×C71C22×D411yescharacteristic
257C2×C4×C71C22×Q811yescharacteristic
258C2×C4×C71C2×#4111yescharacteristic
259C2×C4×C71C2×#4111yescharacteristic
260C2×Dic71—24no—
261C2×Dic71—42no—
262C2×Dic71—42no—
263C2×Dic71—42no—
264C2×Dic71—42no—
265C2×Dic71—41no—
266C2×Dic71—41no—
267C71⋊C8—24no—
268C71⋊C8—42no—
269C22×C71—21no—
270C22×C71—21no—
271C22×C71—21no—
272C22×C71—21no—
273C22×C71—21no—
274C22×C71—21no—
275C22×C71—24no—
276C22×C71—42no—
277C22×C71—42no—
278C22×C71—48no—
279C22×C71—42no—
280C22×C71—42no—
281C22×C71#38512yes—
282C22×C71#38512yes—
283C22×C71C2×C4×D411yescharacteristic
284C22×C71C2×#12011yescharacteristic
285C22×C71C2×C22⋊Q811yescharacteristic
286C4×C71—24no—
287C4×C71—44no—
288C4×C71—42no—
289C4×C71—42no—
290C4×C71#38512yes—
291C4×C71#38512yes—
292C4×C71C22×C4⋊C411yescharacteristic
293C4×C71C2×C4×D411yescharacteristic
294C4×C71C2×C22⋊Q811yescharacteristic
295C4×C71C2×#12311yescharacteristic
296Dic71—44no—
297Dic71—81no—
298Dic71—81no—
299#20864—711nomaximal, Sylow
300C2×C71—21no—
301C2×C71—22no—
302C2×C71—21no—
303C2×C71—22no—
304C2×C71—44no—
305C2×C71C2×#38511yescharacteristic
306C2×C71#301911yescharacteristic
307C2×C71#302011yescharacteristic
308C2×#520D7111yescharacteristic
309#2911—711no—
310C2×#421—711no—
311#2896—711no—
312#2317—711no—
313#1860—714no—
314#1859—714no—
315C2×#343—711no—
316#1733—711no—
317C71#2086411yescharacteristic, complemented, Sylow
318C22×#41C2×D7111yescharacteristic
319C22×M16—711no—
320C22×M16—711no—
321#520—22no—
322#520—22no—
323#520—22no—
324#520C2×D7114yes—
325C2×C4×Q8—21no—
326C2×C4×D4—21no—
327C2×C4×D4—22no—
328C2×#117—21no—
329C2×#117—711no—
330C2×#117—711no—
331C2×#117C2×D7111yescharacteristic
332C22×C4⋊C4—21no—
333C22×C4⋊C4C2×D7111yescharacteristic
334#428—712no—
335#428—712no—
336#428—1421no—
337#428—1421no—
338#427—1422no—
339#421—718no—
340C2×#104—714no—
341#419—1421no—
342#418—1421no—
343#417—712no—
344#417—712no—
345#416—1422no—
346#389—1422no—
347#385—1422no—
348#384—712no—
349#382—712no—
350#343—714no—
351#339—714no—
352C2×#41—21no—
353C2×#41—21no—
354C2×#41—22no—
355C2×#41—22no—
356C2×#41Dic14214yes—
357C2×#41C4×D7114yes—
358C2×#41C22⋊D7112yes—
359C2×#41C22×D7112yes—
360C22×Q8C22⋊D7111yescharacteristic
361C22×D4—21no—
362C22×D4C22⋊D7111yescharacteristic
363C23×C4—21no—
364C23×C4—22no—
365C23×C4C22⋊D7112yes—
366C23×C4C22×D7111yescharacteristic
367C2×M16—714no—
368C2×M16—712no—
369C2×M16—712no—
370C2×M16—1422no—
371C2×M16—1422no—
372C2×M16—1421no—
373C2×M16—1421no—
374C2×M16—1422no—
375C22×C8—1421no—
376C22×C8—1421no—
377C4×Q8—24no—
378C4×Q8—24no—
379C4×D4—28no—
380C4×D4—24no—
381C4×D4—24no—
382C4×D4—28no—
383#117—24no—
384#117—21no—
385#117—21no—
386#117—24no—
387#117—714no—
388#117—1421no—
389#117—1421no—
390#117—1421no—
391#117—1421no—
392#117—1422no—
393#117C22⋊D7112yes—
394#117C22×D7112yes—
395C2×C4⋊C4—22no—
396C2×C4⋊C4—24no—
397C2×C4⋊C4—21no—
398C2×C4⋊C4—21no—
399C2×C4⋊C4—21no—
400C2×C4⋊C4—21no—
401C2×C4⋊C4—24no—
402C2×C4⋊C4—711no—
403C2×C4⋊C4—711no—
404C2×C4⋊C4—1424no—
405C2×C4⋊C4—1421no—
406C2×C4⋊C4C22⋊D7112yes—
407C2×C4⋊C4C22⋊D7111yescharacteristic
408C2×C4⋊C4C22⋊D7111yescharacteristic
409C2×C4⋊C4C22×D7112yes—
410C2×C22⋊C4—22no—
411C2×C22⋊C4—21no—
412C2×C22⋊C4—1421no—
413C2×C22⋊C4—1421no—
414C2×C42—22no—
415C2×C42—21no—
416C2×C42—712no—
417C2×C42—1421no—
418#107—1422no—
419#107—1422no—
420#106—1422no—
421#106—1422no—
422#104—7116no—
423#103—1422no—
424#103—1422no—
425#102—1422no—
426#102—1422no—
427#95—1424no—
428C24—21no—
429C24C2×C22⋊D7111yescharacteristic
430#41—22no—
431#41—22no—
432#41—24no—
433#41—24no—
434#41—24no—
435#41—24no—
436#41—24no—
437#41Dic71⋊C418yescomplemented
438#41C2×C4×D7114yes—
439#41C2×Dic14214yes—
440C2×Q8—21no—
441C2×Q8—22no—
442C2×Q8Dic71⋊C414yescomplemented
443C2×Q8C2×C22⋊D7112yes—
444C2×D4—22no—
445C2×D4—24no—
446C2×D4—22no—
447C2×D4—24no—
448C2×D4—21no—
449C2×D4—22no—
450C2×D4Dic71⋊C414yescomplemented
451C2×D4C2×C22⋊D7112yes—
452C22×C4—21no—
453C22×C4—22no—
454C22×C4—24no—
455C22×C4—22no—
456C22×C4—24no—
457C22×C4—21no—
458C22×C4—22no—
459C22×C4—21no—
460C22×C4—28no—
461C22×C4—22no—
462C22×C4—82no—
463C22×C4—84no—
464C22×C4—1421no—
465C22×C4—1421no—
466C22×C4—1422no—
467C22×C4C2×C4×D7112yes—
468C22×C4C2×C22⋊D7114yes—
469C22×C4C2×C22⋊D7111yescharacteristic
470C22×C4C2×C22⋊D7111yescharacteristic
471C22×C4C2×C22⋊D7111yescharacteristic
472C22×C4C2×C22⋊D7111yescharacteristic
473C22×C4C2×C22⋊D7111yescharacteristic
474C22×C4C2×Dic14212yes—
475C22×C4#1176586512yes—
476C22×C4#1176586512yes—
477C22×C4C23×D7111yescharacteristic
478M16—2842no—
479M16—714no—
480M16—1422no—
481M16—1422no—
482M16—1422no—
483M16—1424no—
484C2×C8—2842no—
485C2×C8—1422no—
486C2×C8—1422no—
487C2×C8—1424no—
488C4⋊C4—28no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—21no—
493C4⋊C4—22no—
494C4⋊C4—21no—
495C4⋊C4—5684no—
496C4⋊C4—1421no—
497C4⋊C4—1421no—
498C4⋊C4—1421no—
499C4⋊C4—1421no—
500C4⋊C4—1424no—
501C4⋊C4C2×C22⋊D7112yes—
502C4⋊C4C2×C22⋊D7112yes—
503C22⋊C4—28no—
504C22⋊C4—22no—
505C22⋊C4—22no—
506C22⋊C4—2841no—
507C22⋊C4—2841no—
508C22⋊C4—1424no—
509C42—28no—
510C42—22no—
511C42—22no—
512C42—2841no—
513C42—2841no—
514C42—714no—
515C42—1422no—
516C23—24no—
517C23—22no—
518C23—41no—
519C23—41no—
520C23—42no—
521C23—42no—
522C23C4×C22⋊D7112yes—
523C23#1455609711yescharacteristic
524C23#1455609711yescharacteristic
525C23#1455610712yes—
526C23C22×C22⋊D7111yescharacteristic
527Q8—24no—
528Q8—22no—
529Q8C2×Dic71⋊C414yes—
530D4—28no—
531D4—24no—
532D4—24no—
533D4—24no—
534D4—22no—
535D4C2×Dic71⋊C414yes—
536C2×C4—24no—
537C2×C4—22no—
538C2×C4—28no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—22no—
542C2×C4—24no—
543C2×C4—22no—
544C2×C4—44no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—44no—
548C2×C4—82no—
549C2×C4—88no—
550C2×C4—2842no—
551C2×C4—2842no—
552C2×C4—2842no—
553C2×C4—2842no—
554C2×C4—2841no—
555C2×C4—2841no—
556C2×C4—1424no—
557C2×C4C2×Dic71⋊C412yes—
558C2×C4C2×Dic71⋊C412yes—
559C2×C4C4×C22⋊D7112yes—
560C2×C4C4×C22⋊D7112yes—
561C2×C4C4×C22⋊D7112yes—
562C2×C4#1455609711yescharacteristic
563C2×C4#1455609711yescharacteristic
564C2×C4#1455610712yes—
565C2×C4#1455610712yes—
566C2×C4#1455610712yes—
567C2×C4#1455611112yes—
568C2×C4#1455611112yes—
569C2×C4C22×C4×D7111yescharacteristic
570C2×C4C22×C22⋊D7111yescharacteristic
571C2×C4C22×C22⋊D7111yescharacteristic
572C2×C4C22×Dic14211yescharacteristic
573C2×C4C2×#1176586511yescharacteristic
574C2×C4C2×#1176586511yescharacteristic
575C8—2842no—
576C8—1424no—
577C22—21no—
578C22—21no—
579C22—21no—
580C22—21no—
581C22—21no—
582C22—21no—
583C22—24no—
584C22—42no—
585C22—42no—
586C22—48no—
587C22—42no—
588C22—42no—
589C22#1798587212yes—
590C22#1798587212yes—
591C22C2×C4×C22⋊D7111yescharacteristic
592C22C2×#1455609711yescharacteristic
593C22C2×#1455610711yescharacteristic
594C4—24no—
595C4—44no—
596C4—42no—
597C4—42no—
598C4—2844no—
599C4—5681no—
600C4—5681no—
601C4#1798587212yes—
602C4#1798587212yes—
603C4C22×Dic71⋊C411yescharacteristic
604C4C2×C4×C22⋊D7111yescharacteristic
605C4C2×#1455610711yescharacteristic
606C4C2×#1455611111yescharacteristic
607C2—21no—
608C2—22no—
609C2—22no—
610C2—21no—
611C2—44no—
612C2C2×#1798587211yescharacteristic
613C2#2259769711yescharacteristic
614C2#2259770411yescharacteristic
subgroupG/H
centerC22C2×#14556097
commutatorC2×C4×C71C23×C4
FrattiniC22×C4C23×D71
FittingC2×#520×C71C2
socleC22×C71C2×#120
prime radicalC22×#41×C71C22
cosocleC22×C4×C71C24
hypercenterC2×#520D71
hypocenterC71#20864
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#520

Complements

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

NH ≅ G/Ntype
1. #22596929C2semidirect
4. #22590964C2semidirect
5. #22588573C2semidirect
6. #22588566C2semidirect
7. C2×#17985768C2semidirect
8. #22588049C2semidirect
14. #17986042C22semidirect
15. #17986042C22semidirect
19. #17986016C4semidirect
28. #17985868C22semidirect
29. #17985864C22semidirect
30. #17985768C22semidirect
31. #17985758C22semidirect
49. C2×#11765849C2×C4semidirect
59. #14556087C2×C4semidirect
72. #14556055C2×C4semidirect
128. #11765849C22×C4semidirect
145. C4×Dic71C22×C4semidirect
155. #41×C71C4⋊C4semidirect
160. C2×Q8×C71C4⋊C4semidirect
168. C2×D4×C71C4⋊C4semidirect
317. C71#20864semidirect
437. #41Dic71⋊C4semidirect
442. C2×Q8Dic71⋊C4semidirect
450. C2×D4Dic71⋊C4semidirect

Maximal subgroups

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

subgroupindex
#225969292
C2×#179860162
#225968592
#225909642
#225885732
#225885662
C2×#179857682
#225880492
C2×#520×C712
#2086471