Finite Group Databasebeta

← #31998458

Subgroups — #31998458

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×#17986480C211yesmaximal, characteristic
2C2×#17986039C211yesmaximal, characteristic, complemented
3C2×#17985998C211yesmaximal, characteristic
4C2×#17985996C211yesmaximal, characteristic
5#22590958C211yesmaximal, characteristic, complemented
6#22588609C212yesmaximal, complemented
7#22588602C212yesmaximal, complemented
8#22588595C212yesmaximal, complemented
9#22588588C212yesmaximal, complemented
10#22588036C211yesmaximal, characteristic, complemented
11C2×#340×C71C211yesmaximal, characteristic, complemented
12C22×#11765865C2211yescharacteristic
13C22×#11765849C2211yescharacteristic
14#17986480—24no—
15#17986480—24no—
16C2×#14556093C2212yes—
17C2×#14556090C2212yes—
18C2×C4×C22⋊D71—22no—
19C2×#14556087C2211yescharacteristic
20C2×#14556087C2211yescharacteristic
21C2×#14556086—21no—
22C2×C4⋊C4×D71—21no—
23#17986039—24no—
24C2×#14556061C2212yes—
25C2×#14556059C2212yes—
26#17985998C418yescomplemented
27#17985996—24no—
28C2×#14556053C2211yescharacteristic
29C2×#14556053C2211yescharacteristic
30C2×#14556051C2212yes—
31C2×#14556050C2211yescharacteristic
32C2×#14556050C2211yescharacteristic
33C2×#14556048C2212yes—
34#17985898—21no—
35#17985898—21no—
36#17985898C2212yes—
37#17985898C2212yes—
38#17985859—22no—
39#17985753C2212yes—
40#17985753C2212yes—
41C23×C8×C71C2211yescharacteristic
42C22×C4⋊C4×C71C2211yescharacteristic
43C22×C4⋊C4×C71C2211yescharacteristic
44#340×C71C2212yes—
45#340×C71C2212yes—
46C2×#11765865—22no—
47C2×#11765865—22no—
48C2×#11765865—22no—
49C2×#11765865C2×C414yes—
50C2×#11765865D414yes—
51C2×#11765865Q814yes—
52C22×Dic142C2311yescharacteristic
53C22×D284C2311yescharacteristic
54C22×C22⋊D71—21no—
55C22×C4×D71—22no—
56C22×C4×D71—21no—
57C2×#11765849—22no—
58C2×#11765849—22no—
59C2×#11765849C2×C414yescomplemented
60C22×C71⋊C8C2312yes—
61#14556093—24no—
62#14556093—24no—
63#14556090—24no—
64#14556090—24no—
65C4×C22⋊D71—216no—
66#14556087—22no—
67#14556087—22no—
68#14556087—22no—
69#14556087—22no—
70#14556086—28no—
71C2×C4⋊Dic71—22no—
72C2×C4⋊Dic71—22no—
73C2×C4⋊Dic71D411yescharacteristic
74C2×C4⋊Dic71D411yescharacteristic
75C2×C4⋊Dic71C2311yescharacteristic
76C2×C4⋊Dic71C2311yescharacteristic
77C2×Dic71⋊C4—22no—
78C4⋊C4×D71—28no—
79C2×C22⋊Dic71—21no—
80C2×C22⋊Dic71D412yes—
81C2×(C2×D71)⋊C4—22no—
82C2×C4×Dic71—22no—
83C2×C4×Dic71—21no—
84C2×C4×Dic71C2312yes—
85#14556061—24no—
86#14556059—24no—
87#14556053—22no—
88#14556053—22no—
89#14556051C2×C418yescomplemented
90#14556050—22no—
91#14556050—22no—
92#14556048C2×C418yescomplemented
93C71⋊#95—22no—
94C71⋊#95—22no—
95C23×C4×C71—21no—
96C23×C4×C71—21no—
97C23×C4×C71C2311yescharacteristic
98C22×C8×C71—22no—
99C22×C8×C71—22no—
100C22×C8×C71—22no—
101C22×C8×C71C2311yescharacteristic
102C22×C8×C71C2311yescharacteristic
103C2×C4⋊C4×C71—21no—
104C2×C4⋊C4×C71—21no—
105C2×C4⋊C4×C71—22no—
106C2×C4⋊C4×C71—22no—
107C2×C4⋊C4×C71C2×C414yescomplemented
108C2×C4⋊C4×C71D414yes—
109C2×C4⋊C4×C71C2312yes—
110C2×C4⋊C4×C71C2312yes—
111C23×D71—21no—
112#11765865—28no—
113#11765865—28no—
114#11765865—28no—
115#11765865C4⋊C4116yescomplemented
116C2×Dic142—21no—
117C2×Dic142—21no—
118C2×Dic142—21no—
119C2×Dic142C22×C412yes—
120C2×Dic142C2×D412yes—
121C2×Dic142C2×Q812yes—
122C2×D284—21no—
123C2×D284—21no—
124C2×D284—21no—
125C2×D284C22×C412yes—
126C2×D284C2×D412yes—
127C2×D284C2×Q812yes—
128C2×C22⋊D71—24no—
129C2×C22⋊D71—24no—
130C2×C22⋊D71—24no—
131C22×Dic71—21no—
132C22×Dic71—21no—
133C22×Dic71—21no—
134C22×Dic71C2×D412yes—
135C22×Dic71C2×D412yes—
136C2×C4×D71—24no—
137C2×C4×D71—24no—
138C2×C4×D71—24no—
139C2×C4×D71—28no—
140C2×C4×D71—82no—
141C2×C4×D71—82no—
142#11765849—28no—
143C2×C71⋊C8—22no—
144C2×C71⋊C8—22no—
145C2×C71⋊C8C22×C414yes—
146C4⋊Dic71—22no—
147C4⋊Dic71—22no—
148C4⋊Dic71—21no—
149C4⋊Dic71—21no—
150C4⋊Dic71—21no—
151C4⋊Dic71—21no—
152C4⋊Dic71—82no—
153C4⋊Dic71—82no—
154Dic71⋊C4—28no—
155C22⋊Dic71—24no—
156C22⋊Dic71—42no—
157(C2×D71)⋊C4—28no—
158C4×Dic71—28no—
159C4×Dic71—22no—
160C4×Dic71—22no—
161C4×Dic71—42no—
162C24×C71C2×D411yescharacteristic
163C22×C4×C71—22no—
164C22×C4×C71—22no—
165C22×C4×C71—24no—
166C22×C4×C71—24no—
167C22×C4×C71—81no—
168C22×C4×C71—81no—
169C22×C4×C71—81no—
170C22×C4×C71—81no—
171C22×C4×C71C22×C412yes—
172C22×C4×C71C2×D412yes—
173C22×C4×C71C2×D411yescharacteristic
174C22×C4×C71C2×D412yes—
175C22×C4×C71C2×Q812yes—
176C22×C4×C71#4112yes—
177C22×C4×C71#4112yes—
178C22×C4×C71C2411yescharacteristic
179C2×C8×C71—21no—
180C2×C8×C71—21no—
181C2×C8×C71—21no—
182C2×C8×C71—21no—
183C2×C8×C71—21no—
184C2×C8×C71—21no—
185C2×C8×C71—44no—
186C4⋊C4×C71—24no—
187C4⋊C4×C71—24no—
188C4⋊C4×C71—22no—
189C4⋊C4×C71—22no—
190C4⋊C4×C71C22×C414yes—
191C4⋊C4×C71C2×D414yes—
192C23×C71—21no—
193C23×C71—21no—
194C23×C71—21no—
195C23×C71—21no—
196C23×C71C4×D412yes—
197C23×C71#12012yes—
198C23×C71C22⋊Q812yes—
199C23×C71C22×D411yescharacteristic
200C22×D71—22no—
201C22×D71—22no—
202C22×D71—22no—
203C22×D71—41no—
204C22×D71—41no—
205C22×D71—41no—
206C22×D71—41no—
207C2×C4×C71—24no—
208C2×C4×C71—24no—
209C2×C4×C71—42no—
210C2×C4×C71—42no—
211C2×C4×C71—42no—
212C2×C4×C71—42no—
213C2×C4×C71—84no—
214C2×C4×C71—84no—
215C2×C4×C71C2×C4⋊C414yes—
216C2×C4×C71C4×D412yes—
217C2×C4×C71C4×D414yes—
218C2×C4×C71#12012yes—
219C2×C4×C71C22⋊Q812yes—
220C2×C4×C71C22⋊Q814yes—
221C2×C4×C71#12314yes—
222C2×C4×C71C23×C411yescharacteristic
223C2×C4×C71C22×D411yescharacteristic
224C2×C4×C71C22×D411yescharacteristic
225C2×C4×C71C22×Q811yescharacteristic
226C2×C4×C71C2×#4111yescharacteristic
227C2×C4×C71C2×#4111yescharacteristic
228C22⋊D71—216no—
229C2×Dic71—22no—
230C2×Dic71—22no—
231C2×Dic71—22no—
232C2×Dic71—22no—
233C2×Dic71—22no—
234C2×Dic71—22no—
235C2×Dic71—22no—
236C2×Dic71—22no—
237C2×Dic71—22no—
238C2×Dic71—41no—
239C2×Dic71—41no—
240C2×Dic71—41no—
241C2×Dic71—41no—
242C2×Dic71—41no—
243C2×Dic71—41no—
244D284—22no—
245D284—22no—
246D284—22no—
247D284C2×C4⋊C414yes—
248C4×D71—216no—
249C4×D71—88no—
250Dic142—22no—
251Dic142—22no—
252Dic142—22no—
253Dic142C2×C4⋊C414yes—
254C8×C71—41no—
255C8×C71—41no—
256C71⋊C8—24no—
257C22×C71—22no—
258C22×C71—22no—
259C22×C71—22no—
260C22×C71—22no—
261C22×C71—22no—
262C22×C71—22no—
263C22×C71#38514yes—
264C22×C71C2×C4×D411yescharacteristic
265C22×C71C2×#12011yescharacteristic
266C22×C71C2×C22⋊Q811yescharacteristic
267C22×C71#57611yescharacteristic
268C22×C71#57711yescharacteristic
269C22×C71#57711yescharacteristic
270C22×C71#57811yescharacteristic
271C2×D71—24no—
272C2×D71—42no—
273C2×D71—42no—
274C2×D71—42no—
275C2×D71—42no—
276C2×D71—42no—
277C2×D71—42no—
278C4×C71—44no—
279C4×C71—44no—
280C4×C71#38514yes—
281C4×C71C22×C4⋊C411yescharacteristic
282C4×C71C2×C4×D411yescharacteristic
283C4×C71C2×C22⋊Q811yescharacteristic
284C4×C71C2×#12311yescharacteristic
285Dic71—24no—
286Dic71—42no—
287Dic71—42no—
288Dic71—81no—
289Dic71—81no—
290#20878—711nomaximal, Sylow
291C2×C71—24no—
292C2×C71C2×#38511yescharacteristic
293C2×C71#294611yescharacteristic
294C2×C71#294711yescharacteristic
295C2×C71#301211yescharacteristic
296C2×C71#301311yescharacteristic
297C2×C71#308311yescharacteristic
298C2×C71#308411yescharacteristic
299D71—44no—
300C2×#520—711no—
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×#340D7111yescharacteristic, complemented
310#1729—711no—
311C71#2087811yescharacteristic, complemented, Sylow
312C22×#41—711no—
313C22×M16—711no—
314C23×C8C2×D7111yescharacteristic
315#520—1424no—
316#520—1424no—
317C2×C4×Q8—712no—
318C2×C4×D4—712no—
319C2×C4×D4—1422no—
320C2×#117—711no—
321C2×#117—711no—
322C2×#117—1421no—
323C22×C4⋊C4—1421no—
324C22×C4⋊C4C2×D7111yescharacteristic
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#340C2×D7112yes—
343#340C2×D7112yes—
344#336—712no—
345#336—712no—
346C2×#41—714no—
347C2×#41—714no—
348C2×#41—714no—
349C2×#41—1422no—
350C2×#41—1422no—
351C2×#41—1422no—
352C22×Q8—711no—
353C22×D4—711no—
354C22×D4—1421no—
355C23×C4—21no—
356C23×C4—21no—
357C23×C4—1422no—
358C23×C4—1421no—
359C23×C4C22×D7111yescharacteristic
360C2×M16—714no—
361C2×M16—1422no—
362C2×M16—1422no—
363C22×C8—22no—
364C22×C8—22no—
365C22×C8—22no—
366C22×C8—712no—
367C22×C8C22×D7111yescharacteristic
368C22×C8C22×D7111yescharacteristic
369C4×Q8—1424no—
370C4×Q8—1424no—
371C4×D4—1424no—
372C4×D4—14216no—
373C4×D4—1424no—
374#117—1422no—
375#117—1422no—
376#117—1428no—
377#117—1422no—
378#117—1422no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—22no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—22no—
383C2×C4⋊C4—711no—
384C2×C4⋊C4—711no—
385C2×C4⋊C4—711no—
386C2×C4⋊C4—711no—
387C2×C4⋊C4—1428no—
388C2×C4⋊C4—1422no—
389C2×C4⋊C4—1422no—
390C2×C4⋊C4—1422no—
391C2×C4⋊C4C4×D7114yescomplemented
392C2×C4⋊C4D28414yes—
393C2×C4⋊C4C22×D7112yes—
394C2×C4⋊C4C22×D7112yes—
395C2×C22⋊C4—712no—
396C2×C22⋊C4—1421no—
397C2×C22⋊C4—1422no—
398C2×C42—712no—
399C2×C42—1421no—
400C2×C42—1422no—
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—1421no—
412C24D4×D7111yescharacteristic
413#41—7116no—
414#41—1428no—
415#41—1428no—
416#41—1428no—
417C2×Q8—712no—
418C2×Q8—712no—
419C2×Q8—712no—
420C2×Q8—1421no—
421C2×Q8—1421no—
422C2×Q8—1421no—
423C2×D4—712no—
424C2×D4—712no—
425C2×D4—712no—
426C2×D4—1424no—
427C2×D4—1424no—
428C2×D4—1424no—
429C2×D4—1421no—
430C2×D4—1421no—
431C2×D4—1421no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—81no—
437C22×C4—81no—
438C22×C4—81no—
439C22×C4—81no—
440C22×C4—5682no—
441C22×C4—5682no—
442C22×C4—712no—
443C22×C4—712no—
444C22×C4—1428no—
445C22×C4—1421no—
446C22×C4—1421no—
447C22×C4—1421no—
448C22×C4—1424no—
449C22×C4—1424no—
450C22×C4—1424no—
451C22×C4C2×C4×D7112yes—
452C22×C4D4×D7112yes—
453C22×C4D4×D7111yescharacteristic
454C22×C4C2×D28412yes—
455C22×C4Q8×D7112yes—
456C22×C4#1176586412yes—
457C22×C4#1176586512yes—
458C22×C4C23×D7111yescharacteristic
459M16—1428no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—44no—
467C2×C8—714no—
468C2×C8—1422no—
469C2×C8—1422no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—24no—
473C4⋊C4—24no—
474C4⋊C4—5682no—
475C4⋊C4—5682no—
476C4⋊C4—1428no—
477C4⋊C4—1421no—
478C4⋊C4—1421no—
479C4⋊C4—1421no—
480C4⋊C4—1421no—
481C4⋊C4—1422no—
482C4⋊C4—1422no—
483C4⋊C4C2×C4×D7114yes—
484C4⋊C4C2×D28414yes—
485C22⋊C4—2842no—
486C22⋊C4—1428no—
487C22⋊C4—1424no—
488C42—2842no—
489C42—1428no—
490C42—1422no—
491C42—1422no—
492C23—21no—
493C23—21no—
494C23—21no—
495C23—21no—
496C23—2841no—
497C23—2841no—
498C23—2841no—
499C23—2841no—
500C23—1422no—
501C23—1422no—
502C23—1422no—
503C23(C22⋊D71)⋊C412yes—
504C23#1455609812yes—
505C23#1455610612yes—
506C23C2×D4×D7111yescharacteristic
507Q8—714no—
508Q8—1422no—
509Q8—1422no—
510Q8—1422no—
511D4—714no—
512D4—14216no—
513D4—1422no—
514D4—1422no—
515D4—1422no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—84no—
523C2×C4—84no—
524C2×C4—2841no—
525C2×C4—2841no—
526C2×C4—2841no—
527C2×C4—2841no—
528C2×C4—2841no—
529C2×C4—2841no—
530C2×C4—5688no—
531C2×C4—1422no—
532C2×C4—1422no—
533C2×C4—1422no—
534C2×C4—1422no—
535C2×C4—1422no—
536C2×C4—1422no—
537C2×C4—1422no—
538C2×C4—1422no—
539C2×C4—1422no—
540C2×C4—14216no—
541C2×C4C4⋊C4×D7114yes—
542C2×C4(C22⋊D71)⋊C412yes—
543C2×C4C4×D28414yes—
544C2×C4#1455609812yes—
545C2×C4#1455610612yes—
546C2×C4#1455610814yes—
547C2×C4#1455611014yes—
548C2×C4C22×C4×D7111yescharacteristic
549C2×C4C2×D4×D7111yescharacteristic
550C2×C4C22×D28411yescharacteristic
551C2×C4C2×Q8×D7111yescharacteristic
552C2×C4C2×#1176586411yescharacteristic
553C2×C4C2×#1176586511yescharacteristic
554C8—41no—
555C8—41no—
556C8—1424no—
557C22—22no—
558C22—22no—
559C22—22no—
560C22—22no—
561C22—22no—
562C22—22no—
563C22—2842no—
564C22—2842no—
565C22—2842no—
566C22—2842no—
567C22—2842no—
568C22—2842no—
569C22—1424no—
570C22#1798587314yes—
571C22C2×(C22⋊D71)⋊C411yescharacteristic
572C22C2×#1455609811yescharacteristic
573C22C2×#1455610611yescharacteristic
574C22#1798680011yescharacteristic
575C22#1798680511yescharacteristic
576C22#1798680511yescharacteristic
577C22#1798681211yescharacteristic
578C4—44no—
579C4—44no—
580C4—2842no—
581C4—2842no—
582C4—5681no—
583C4—5681no—
584C4—1424no—
585C4#1798587314yes—
586C4C2×C4⋊C4×D7111yescharacteristic
587C4C2×C4×D28411yescharacteristic
588C4C2×#1455610811yescharacteristic
589C4C2×#1455611011yescharacteristic
590C2—24no—
591C2—2844no—
592C2C2×#1798587311yescharacteristic
593C2#2259716611yescharacteristic
594C2#2259717711yescharacteristic
595C2#2259764911yescharacteristic
596C2#2259766011yescharacteristic
597C2#2259819611yescharacteristic
598C2#2259820711yescharacteristic
subgroupG/H
centerC23C2×D4×D71
commutatorC2×C4×C71C23×C4
FrattiniC22×C4C23×D71
FittingC2×#340×C71C2
socleC23×C71C22×D4
prime radicalC22×#11765865C22
cosocleC22×C4×C71C24
hypercenterC2×#340D71
hypocenterC71#20878
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#340

Complements

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

NH ≅ G/Ntype
2. C2×#17986039C2semidirect
5. #22590958C2semidirect
6. #22588609C2semidirect
7. #22588602C2semidirect
8. #22588595C2semidirect
9. #22588588C2semidirect
10. #22588036C2semidirect
11. C2×#340×C71C2semidirect
26. #17985998C4semidirect
59. C2×#11765849C2×C4semidirect
89. #14556051C2×C4semidirect
92. #14556048C2×C4semidirect
107. C2×C4⋊C4×C71C2×C4semidirect
115. #11765865C4⋊C4semidirect
309. C2×#340D71semidirect
311. C71#20878semidirect
391. C2×C4⋊C4C4×D71semidirect

Maximal subgroups

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

subgroupindex
C2×#179864802
C2×#179860392
C2×#179859982
C2×#179859962
#225909582
#225886092
#225886022
#225885952
#225885882
#225880362
C2×#340×C712
#2087871