Finite Group Databasebeta

← #31998456

Subgroups — #31998456

Order 18176.

1458 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17986039C211yesmaximal, characteristic, complemented
2C2×#17986000C211yesmaximal, characteristic
3C2×#17985995C211yesmaximal, characteristic
4#22590956C211yesmaximal, characteristic, complemented
5#22588608C212yesmaximal, complemented
6#22588601C212yesmaximal, complemented
7#22588594C212yesmaximal, complemented
8#22588587C212yesmaximal, complemented
9C2×#17985761C211yesmaximal, characteristic, complemented
10#22588037C211yesmaximal, characteristic, complemented
11C2×#520×C71C211yesmaximal, characteristic
12C22×#11765849C2211yescharacteristic
13C23×C71⋊C8C2211yescharacteristic
14C2×#14556087C2211yescharacteristic
15C22×C4⋊Dic71C2211yescharacteristic
16#17986039—24no—
17C2×#14556061C2212yes—
18C2×#14556059C2212yes—
19#17986000C418yescomplemented
20#17985995—24no—
21C2×#14556052C2211yescharacteristic
22C2×#14556052C2211yescharacteristic
23C2×#14556052C2212yes—
24C2×#14556049C2211yescharacteristic
25C2×#14556049C2211yescharacteristic
26C2×#14556049C2212yes—
27#17985897C2212yes—
28#17985897C2212yes—
29#17985895—21no—
30#17985895—21no—
31#17985857—22no—
32#17985761C2212yes—
33#17985761C2212yes—
34#17985754C2212yes—
35#17985754C2212yes—
36C22×#41×C71C2211yescharacteristic
37#520×C71—24no—
38#520×C71—24no—
39C2×C4×Q8×C71C2212yes—
40C2×C4×D4×C71—22no—
41C2×C4×D4×C71C2212yes—
42C2×#117×C71—21no—
43C2×#117×C71C2211yescharacteristic
44C22×C4⋊C4×C71—21no—
45C22×C4⋊C4×C71C2211yescharacteristic
46C23×Dic71—21no—
47C2×#11765849—22no—
48C2×#11765849—22no—
49C2×#11765849C2×C414yescomplemented
50C22×C71⋊C8—22no—
51C22×C71⋊C8—22no—
52C22×C71⋊C8—22no—
53C22×C71⋊C8C2311yescharacteristic
54C22×C71⋊C8C2311yescharacteristic
55C22×C71⋊C8C2312yes—
56#14556087—22no—
57#14556087—22no—
58C2×C4⋊Dic71—21no—
59C2×C4⋊Dic71—22no—
60C2×C4⋊Dic71C2×C414yescomplemented
61C2×C4⋊Dic71C2311yescharacteristic
62C2×C4⋊Dic71C2311yescharacteristic
63C2×C4⋊Dic71C2312yes—
64C2×Dic71⋊C4—22no—
65C2×Dic71⋊C4—22no—
66C2×C22⋊Dic71—21no—
67C2×C4×Dic71—21no—
68#14556061—24no—
69#14556059—24no—
70#14556052—22no—
71#14556052—22no—
72#14556052C2×C418yescomplemented
73#14556049—22no—
74#14556049—22no—
75#14556049C2×C418yescomplemented
76C71⋊#95—22no—
77C71⋊#95—22no—
78C2×#41×C71—22no—
79C2×#41×C71—22no—
80C2×#41×C71—22no—
81C2×#41×C71C2×C414yes—
82C2×#41×C71D414yes—
83C2×#41×C71Q814yes—
84C22×Q8×C71C2311yescharacteristic
85C22×D4×C71—21no—
86C22×D4×C71C2311yescharacteristic
87C23×C4×C71—21no—
88C23×C4×C71—21no—
89C23×C4×C71—22no—
90C23×C4×C71C2311yescharacteristic
91C4×Q8×C71—24no—
92C4×Q8×C71—24no—
93C4×D4×C71—216no—
94C4×D4×C71—24no—
95C4×D4×C71—24no—
96#117×C71—28no—
97#117×C71—22no—
98#117×C71—22no—
99C2×C4⋊C4×C71—28no—
100C2×C4⋊C4×C71—21no—
101C2×C4⋊C4×C71—22no—
102C2×C4⋊C4×C71—22no—
103C2×C4⋊C4×C71D411yescharacteristic
104C2×C4⋊C4×C71D411yescharacteristic
105C2×C4⋊C4×C71D414yes—
106C2×C4⋊C4×C71C2312yes—
107C2×C22⋊C4×C71—22no—
108C2×C22⋊C4×C71D412yes—
109C2×C42×C71—22no—
110C2×C42×C71C2312yes—
111C22×Dic71—22no—
112C22×Dic71—24no—
113C22×Dic71—21no—
114C22×Dic71—21no—
115C22×Dic71—81no—
116C22×Dic71—81no—
117#11765849—28no—
118C2×C71⋊C8—22no—
119C2×C71⋊C8—22no—
120C2×C71⋊C8—21no—
121C2×C71⋊C8—21no—
122C2×C71⋊C8—21no—
123C2×C71⋊C8—21no—
124C2×C71⋊C8—21no—
125C2×C71⋊C8—21no—
126C2×C71⋊C8—44no—
127C2×C71⋊C8C22×C414yes—
128C4⋊Dic71—24no—
129C4⋊Dic71—22no—
130C4⋊Dic71—21no—
131C4⋊Dic71—21no—
132C4⋊Dic71—21no—
133C4⋊Dic71—21no—
134C4⋊Dic71C22×C414yes—
135Dic71⋊C4—82no—
136Dic71⋊C4—82no—
137C22⋊Dic71—42no—
138C4×Dic71—42no—
139C24×C71—21no—
140C24×C71C2×D411yescharacteristic
141#41×C71—28no—
142#41×C71—28no—
143#41×C71—28no—
144#41×C71C4⋊C4116yescomplemented
145C2×Q8×C71—21no—
146C2×Q8×C71—21no—
147C2×Q8×C71—21no—
148C2×Q8×C71C22×C412yes—
149C2×Q8×C71C2×D412yes—
150C2×Q8×C71C2×Q812yes—
151C2×D4×C71—24no—
152C2×D4×C71—24no—
153C2×D4×C71—24no—
154C2×D4×C71—21no—
155C2×D4×C71—21no—
156C2×D4×C71—21no—
157C2×D4×C71C22×C412yes—
158C2×D4×C71C2×D412yes—
159C2×D4×C71C2×Q812yes—
160C22×C4×C71—21no—
161C22×C4×C71—24no—
162C22×C4×C71—24no—
163C22×C4×C71—24no—
164C22×C4×C71—22no—
165C22×C4×C71—28no—
166C22×C4×C71—24no—
167C22×C4×C71—82no—
168C22×C4×C71—82no—
169C22×C4×C71—81no—
170C22×C4×C71—81no—
171C22×C4×C71C22×C412yes—
172C22×C4×C71C2×D412yes—
173C22×C4×C71C2×D412yes—
174C22×C4×C71C2×D412yes—
175C22×C4×C71C2×D411yescharacteristic
176C22×C4×C71C2×D412yes—
177C22×C4×C71C2×Q812yes—
178C22×C4×C71#4112yes—
179C22×C4×C71#4112yes—
180C22×C4×C71C2411yescharacteristic
181C4⋊C4×C71—28no—
182C4⋊C4×C71—22no—
183C4⋊C4×C71—24no—
184C4⋊C4×C71—22no—
185C4⋊C4×C71—22no—
186C4⋊C4×C71C2×D414yes—
187C22⋊C4×C71—28no—
188C22⋊C4×C71—24no—
189C42×C71—28no—
190C42×C71—22no—
191C42×C71—22no—
192C23×C71—21no—
193C23×C71—21no—
194C23×C71—21no—
195C23×C71—21no—
196C23×C71—22no—
197C23×C71—22no—
198C23×C71—22no—
199C23×C71—41no—
200C23×C71—41no—
201C23×C71—41no—
202C23×C71—41no—
203C23×C71C4×D412yes—
204C23×C71#12012yes—
205C23×C71C22⋊Q812yes—
206C23×C71C22×D411yescharacteristic
207Q8×C71—22no—
208Q8×C71—22no—
209Q8×C71—22no—
210Q8×C71C2×C4⋊C414yes—
211D4×C71—216no—
212D4×C71—22no—
213D4×C71—22no—
214D4×C71—22no—
215D4×C71C2×C4⋊C414yes—
216C2×C4×C71—22no—
217C2×C4×C71—22no—
218C2×C4×C71—22no—
219C2×C4×C71—216no—
220C2×C4×C71—22no—
221C2×C4×C71—22no—
222C2×C4×C71—24no—
223C2×C4×C71—22no—
224C2×C4×C71—22no—
225C2×C4×C71—22no—
226C2×C4×C71—22no—
227C2×C4×C71—42no—
228C2×C4×C71—42no—
229C2×C4×C71—84no—
230C2×C4×C71—88no—
231C2×C4×C71C2×C4⋊C414yes—
232C2×C4×C71C4×D414yes—
233C2×C4×C71C4×D412yes—
234C2×C4×C71#12012yes—
235C2×C4×C71C22⋊Q814yes—
236C2×C4×C71C22⋊Q812yes—
237C2×C4×C71#12314yes—
238C2×C4×C71C23×C411yescharacteristic
239C2×C4×C71C22×D411yescharacteristic
240C2×C4×C71C22×D411yescharacteristic
241C2×C4×C71C22×Q811yescharacteristic
242C2×C4×C71C2×#4111yescharacteristic
243C2×C4×C71C2×#4111yescharacteristic
244C2×Dic71—24no—
245C2×Dic71—42no—
246C2×Dic71—42no—
247C2×Dic71—41no—
248C2×Dic71—41no—
249C2×Dic71—41no—
250C2×Dic71—41no—
251C2×Dic71—41no—
252C2×Dic71—41no—
253C2×Dic71—84no—
254C71⋊C8—24no—
255C71⋊C8—41no—
256C71⋊C8—41no—
257C22×C71—22no—
258C22×C71—22no—
259C22×C71—22no—
260C22×C71—22no—
261C22×C71—22no—
262C22×C71—22no—
263C22×C71—24no—
264C22×C71—42no—
265C22×C71—42no—
266C22×C71—42no—
267C22×C71—42no—
268C22×C71—42no—
269C22×C71—42no—
270C22×C71#38514yes—
271C22×C71C2×C4×D411yescharacteristic
272C22×C71C2×#12011yescharacteristic
273C22×C71C2×C22⋊Q811yescharacteristic
274C22×C71#57611yescharacteristic
275C22×C71#57711yescharacteristic
276C22×C71#57711yescharacteristic
277C22×C71#57811yescharacteristic
278C4×C71—24no—
279C4×C71—42no—
280C4×C71—44no—
281C4×C71—42no—
282C4×C71#38514yes—
283C4×C71C22×C4⋊C411yescharacteristic
284C4×C71C2×C4×D411yescharacteristic
285C4×C71C2×C22⋊Q811yescharacteristic
286C4×C71C2×#12311yescharacteristic
287Dic71—44no—
288Dic71—81no—
289Dic71—81no—
290#20878—711nomaximal, Sylow
291C2×C71—24no—
292C2×C71—44no—
293C2×C71C2×#38511yescharacteristic
294C2×C71#294611yescharacteristic
295C2×C71#294711yescharacteristic
296C2×C71#301211yescharacteristic
297C2×C71#301311yescharacteristic
298C2×C71#308311yescharacteristic
299C2×C71#308411yescharacteristic
300C2×#520D7111yescharacteristic
301C2×#428—711no—
302C2×#417—711no—
303C2×#416—711no—
304#2316—711no—
305#1865—712no—
306#1864—712no—
307#1863—712no—
308#1862—712no—
309C2×#340—711no—
310#1729—711no—
311C71#2087811yescharacteristic, complemented, Sylow
312C22×#41C2×D7111yescharacteristic
313C22×M16—711no—
314C23×C8—711no—
315#520—24no—
316#520—24no—
317C2×C4×Q8C2×D7112yes—
318C2×C4×D4—22no—
319C2×C4×D4C2×D7112yes—
320C2×#117—21no—
321C2×#117—711no—
322C2×#117C2×D7111yescharacteristic
323C22×C4⋊C4—21no—
324C22×C4⋊C4—711no—
325C22×C4⋊C4C2×D7111yescharacteristic
326#428—1424no—
327C2×#107—712no—
328C2×#106—712no—
329#417—718no—
330#416—1424no—
331C2×#103—712no—
332C2×#103—711no—
333C2×#103—711no—
334C2×#102—712no—
335C2×#102—711no—
336C2×#102—711no—
337#389—712no—
338#389—712no—
339#389—1421no—
340#389—1421no—
341#380—1422no—
342#340—712no—
343#340—712no—
344#336—712no—
345#336—712no—
346C2×#41—22no—
347C2×#41—22no—
348C2×#41—22no—
349C2×#41Dic14214yes—
350C2×#41C4×D7114yes—
351C2×#41C22⋊D7114yes—
352C22×Q8C22×D7111yescharacteristic
353C22×D4—21no—
354C22×D4C22×D7111yescharacteristic
355C23×C4—21no—
356C23×C4—21no—
357C23×C4—22no—
358C23×C4—1421no—
359C23×C4C22×D7111yescharacteristic
360C2×M16—714no—
361C2×M16—1422no—
362C2×M16—1422no—
363C22×C8—712no—
364C22×C8—711no—
365C22×C8—711no—
366C22×C8—1422no—
367C22×C8—1422no—
368C22×C8—1422no—
369C4×Q8—24no—
370C4×Q8—24no—
371C4×D4—216no—
372C4×D4—24no—
373C4×D4—24no—
374#117—28no—
375#117—22no—
376#117—22no—
377#117—1422no—
378#117—1422no—
379C2×C4⋊C4—28no—
380C2×C4⋊C4—21no—
381C2×C4⋊C4—22no—
382C2×C4⋊C4—22no—
383C2×C4⋊C4—714no—
384C2×C4⋊C4—712no—
385C2×C4⋊C4—711no—
386C2×C4⋊C4—711no—
387C2×C4⋊C4—1422no—
388C2×C4⋊C4—1422no—
389C2×C4⋊C4—1421no—
390C2×C4⋊C4—1422no—
391C2×C4⋊C4C22⋊D7111yescharacteristic
392C2×C4⋊C4C22⋊D7111yescharacteristic
393C2×C4⋊C4C22⋊D7114yes—
394C2×C4⋊C4C22×D7112yes—
395C2×C22⋊C4—22no—
396C2×C22⋊C4—1421no—
397C2×C22⋊C4C22⋊D7112yes—
398C2×C42—22no—
399C2×C42—1421no—
400C2×C42C22×D7112yes—
401#107—1424no—
402#106—1424no—
403#103—718no—
404#103—1422no—
405#103—1422no—
406#102—718no—
407#102—1422no—
408#102—1422no—
409#95—1422no—
410#95—1422no—
411C24—21no—
412C24C2×C22⋊D7111yescharacteristic
413#41—28no—
414#41—28no—
415#41—28no—
416#41Dic71⋊C4116yescomplemented
417C2×Q8—21no—
418C2×Q8—21no—
419C2×Q8—21no—
420C2×Q8C2×C4×D7112yes—
421C2×Q8C2×C22⋊D7112yes—
422C2×Q8C2×Dic14212yes—
423C2×D4—24no—
424C2×D4—24no—
425C2×D4—24no—
426C2×D4—21no—
427C2×D4—21no—
428C2×D4—21no—
429C2×D4C2×C4×D7112yes—
430C2×D4C2×C22⋊D7112yes—
431C2×D4C2×Dic14212yes—
432C22×C4—21no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—28no—
438C22×C4—24no—
439C22×C4—82no—
440C22×C4—82no—
441C22×C4—81no—
442C22×C4—81no—
443C22×C4—5681no—
444C22×C4—5681no—
445C22×C4—1421no—
446C22×C4—1421no—
447C22×C4—1422no—
448C22×C4—1424no—
449C22×C4C2×C4×D7112yes—
450C22×C4C2×C22⋊D7112yes—
451C22×C4C2×C22⋊D7112yes—
452C22×C4C2×C22⋊D7112yes—
453C22×C4C2×C22⋊D7112yes—
454C22×C4C2×C22⋊D7111yescharacteristic
455C22×C4C2×Dic14212yes—
456C22×C4#1176586512yes—
457C22×C4#1176586512yes—
458C22×C4C23×D7111yescharacteristic
459M16—1428no—
460C2×C8—2844no—
461C2×C8—714no—
462C2×C8—1421no—
463C2×C8—1421no—
464C2×C8—1421no—
465C2×C8—1421no—
466C2×C8—1421no—
467C2×C8—1421no—
468C2×C8—1422no—
469C2×C8—1422no—
470C4⋊C4—28no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—24no—
474C4⋊C4—22no—
475C4⋊C4—5682no—
476C4⋊C4—5682no—
477C4⋊C4—714no—
478C4⋊C4—1421no—
479C4⋊C4—1421no—
480C4⋊C4—1421no—
481C4⋊C4—1421no—
482C4⋊C4—1422no—
483C4⋊C4—1424no—
484C4⋊C4C2×C22⋊D7114yes—
485C22⋊C4—28no—
486C22⋊C4—24no—
487C22⋊C4—2842no—
488C42—28no—
489C42—22no—
490C42—22no—
491C42—2842no—
492C23—21no—
493C23—21no—
494C23—21no—
495C23—21no—
496C23—22no—
497C23—22no—
498C23—22no—
499C23—41no—
500C23—41no—
501C23—41no—
502C23—41no—
503C23C4×C22⋊D7112yes—
504C23#1455609712yes—
505C23#1455610712yes—
506C23C22×C22⋊D7111yescharacteristic
507Q8—22no—
508Q8—22no—
509Q8—22no—
510Q8C2×Dic71⋊C414yes—
511D4—216no—
512D4—22no—
513D4—22no—
514D4—22no—
515D4C2×Dic71⋊C414yes—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—216no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—24no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—84no—
530C2×C4—88no—
531C2×C4—2842no—
532C2×C4—2842no—
533C2×C4—2841no—
534C2×C4—2841no—
535C2×C4—2841no—
536C2×C4—2841no—
537C2×C4—2841no—
538C2×C4—2841no—
539C2×C4—5684no—
540C2×C4—1424no—
541C2×C4C2×Dic71⋊C414yes—
542C2×C4C4×C22⋊D7112yes—
543C2×C4C4×C22⋊D7114yes—
544C2×C4#1455609712yes—
545C2×C4#1455610712yes—
546C2×C4#1455610714yes—
547C2×C4#1455611114yes—
548C2×C4C22×C4×D7111yescharacteristic
549C2×C4C22×C22⋊D7111yescharacteristic
550C2×C4C22×C22⋊D7111yescharacteristic
551C2×C4C22×Dic14211yescharacteristic
552C2×C4C2×#1176586511yescharacteristic
553C2×C4C2×#1176586511yescharacteristic
554C8—2841no—
555C8—2841no—
556C8—1424no—
557C22—22no—
558C22—22no—
559C22—22no—
560C22—22no—
561C22—22no—
562C22—22no—
563C22—24no—
564C22—42no—
565C22—42no—
566C22—42no—
567C22—42no—
568C22—42no—
569C22—42no—
570C22#1798587214yes—
571C22C2×C4×C22⋊D7111yescharacteristic
572C22C2×#1455609711yescharacteristic
573C22C2×#1455610711yescharacteristic
574C22#1798680211yescharacteristic
575C22#1798680711yescharacteristic
576C22#1798681011yescharacteristic
577C22#1798681411yescharacteristic
578C4—24no—
579C4—42no—
580C4—44no—
581C4—42no—
582C4—2844no—
583C4—5681no—
584C4—5681no—
585C4#1798587214yes—
586C4C22×Dic71⋊C411yescharacteristic
587C4C2×C4×C22⋊D7111yescharacteristic
588C4C2×#1455610711yescharacteristic
589C4C2×#1455611111yescharacteristic
590C2—24no—
591C2—44no—
592C2C2×#1798587211yescharacteristic
593C2#2259716811yescharacteristic
594C2#2259717911yescharacteristic
595C2#2259764411yescharacteristic
596C2#2259765511yescharacteristic
597C2#2259819111yescharacteristic
598C2#2259820211yescharacteristic
subgroupG/H
centerC23C22×C22⋊D71
commutatorC2×C4×C71C23×C4
FrattiniC22×C4C23×D71
FittingC2×#520×C71C2
socleC23×C71C22×D4
prime radicalC22×#41×C71C22
cosocleC22×C4×C71C24
hypercenterC2×#520D71
hypocenterC71#20878
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#520

Complements

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

NH ≅ G/Ntype
1. C2×#17986039C2semidirect
4. #22590956C2semidirect
5. #22588608C2semidirect
6. #22588601C2semidirect
7. #22588594C2semidirect
8. #22588587C2semidirect
9. C2×#17985761C2semidirect
10. #22588037C2semidirect
19. #17986000C4semidirect
49. C2×#11765849C2×C4semidirect
60. C2×C4⋊Dic71C2×C4semidirect
72. #14556052C2×C4semidirect
75. #14556049C2×C4semidirect
144. #41×C71C4⋊C4semidirect
311. C71#20878semidirect
416. #41Dic71⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#179860392
C2×#179860002
C2×#179859952
#225909562
#225886082
#225886012
#225885942
#225885872
C2×#179857612
#225880372
C2×#520×C712
#2087871