Finite Group Databasebeta

← #32000100

Subgroups — #32000100

Order 18176.

1122 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17985998C211yesmaximal, characteristic
2C2×#17985993C211yesmaximal, characteristic
3C2×#17985955C211yesmaximal, characteristic, complemented
4#22590928C211yesmaximal, characteristic
5#22589036C212yesmaximal, complemented
6#22589029C212yesmaximal, complemented
7#22589022C212yesmaximal, complemented
8#22589015C212yesmaximal, complemented
9#22588061C211yesmaximal, characteristic, complemented
10C2×#17985761C211yesmaximal, characteristic, complemented
11C71×#2317C211yesmaximal, characteristic, complemented
12C22×#11765865C2211yescharacteristic
13C22×#11765849C2211yescharacteristic
14C23×C71⋊C8C2211yescharacteristic
15C22×C4⋊Dic71C2211yescharacteristic
16#17985998—24no—
17#17985993C418yescomplemented
18C2×#14556051C2212yes—
19C2×#14556051C2211yescharacteristic
20C2×#14556051C2211yescharacteristic
21C2×#14556048C2212yes—
22C2×#14556048C2211yescharacteristic
23C2×#14556048C2211yescharacteristic
24#17985955—24no—
25C2×#14556039C2212yes—
26C2×C4×C71⋊C8C2212yes—
27#17985905C2212yes—
28#17985901C2212yes—
29#17985876—22no—
30#17985856—21no—
31#17985856—21no—
32#17985761C2212yes—
33#17985761C2212yes—
34#17985761C2212yes—
35#17985761C2212yes—
36C2×#117×C71C2211yescharacteristic
37C2×#117×C71C2211yescharacteristic
38C22×C4⋊C4×C71C2211yescharacteristic
39#389×C71C2212yes—
40#389×C71C2212yes—
41#385×C71—22no—
42#384×C71—21no—
43#382×C71—21no—
44C2×#11765865—22no—
45C2×#11765865—22no—
46C2×#11765865—22no—
47C2×#11765865—22no—
48C2×#11765865—22no—
49C2×#11765865C2×C414yes—
50C22×Dic142C2311yescharacteristic
51C22×D284C2311yescharacteristic
52C22×C22⋊D71—21no—
53C23×Dic71—21no—
54C22×C4×D71—21no—
55C2×#11765849—22no—
56C2×#11765849—22no—
57C2×#11765849D414yes—
58C22×C71⋊C8—22no—
59C22×C71⋊C8—22no—
60C22×C71⋊C8C2×C414yes—
61C22×C71⋊C8C2311yescharacteristic
62C22×C71⋊C8C2311yescharacteristic
63C22×C71⋊C8C2312yes—
64C2×C4⋊Dic71—21no—
65C2×C4⋊Dic71—22no—
66C2×C4⋊Dic71—22no—
67C2×C4⋊Dic71C2312yes—
68C2×(C2×D71)⋊C4—22no—
69C2×(C2×D71)⋊C4—22no—
70#14556051—24no—
71#14556051C2×C414yescomplemented
72#14556051C2×C414yescomplemented
73#14556048—24no—
74#14556048C2×C414yescomplemented
75#14556048C2×C414yescomplemented
76#14556039—24no—
77C4×C71⋊C8—24no—
78C71⋊#95—22no—
79C71⋊#95—22no—
80C23×C4×C71—21no—
81C23×C4×C71C2311yescharacteristic
82#117×C71—22no—
83#117×C71—22no—
84#117×C71—22no—
85#117×C71C2×C414yes—
86C2×C4⋊C4×C71—22no—
87C2×C4⋊C4×C71—22no—
88C2×C4⋊C4×C71—21no—
89C2×C4⋊C4×C71—22no—
90C2×C4⋊C4×C71—22no—
91C2×C4⋊C4×C71D411yescharacteristic
92C2×C4⋊C4×C71D411yescharacteristic
93C2×C4⋊C4×C71C2312yes—
94C2×C4⋊C4×C71C2311yescharacteristic
95C2×C4⋊C4×C71C2311yescharacteristic
96C2×C22⋊C4×C71—21no—
97C2×C22⋊C4×C71D412yes—
98C2×C42×C71—21no—
99C2×C42×C71C2312yes—
100#95×C71—22no—
101#95×C71—22no—
102C23×D71—21no—
103#11765865—28no—
104#11765865—28no—
105#11765865—44no—
106#11765865—44no—
107C2×Dic142—21no—
108C2×Dic142—21no—
109C2×Dic142—21no—
110C2×Dic142—21no—
111C2×Dic142—21no—
112C2×Dic142C22×C412yes—
113C2×D284—21no—
114C2×D284—21no—
115C2×D284—21no—
116C2×D284—21no—
117C2×D284—21no—
118C2×D284C22×C412yes—
119C2×C22⋊D71—24no—
120C2×C22⋊D71—42no—
121C2×C22⋊D71—42no—
122C22×Dic71—21no—
123C22×Dic71—22no—
124C22×Dic71—42no—
125C22×Dic71—81no—
126C22×Dic71—81no—
127C2×C4×D71—24no—
128C2×C4×D71—42no—
129C2×C4×D71—42no—
130#11765849—28no—
131C2×C71⋊C8—28no—
132C2×C71⋊C8—21no—
133C2×C71⋊C8—21no—
134C2×C71⋊C8—21no—
135C2×C71⋊C8—21no—
136C2×C71⋊C8—22no—
137C2×C71⋊C8—22no—
138C2×C71⋊C8C22×C412yes—
139C2×C71⋊C8C22×C412yes—
140C2×C71⋊C8C2×D414yes—
141C4⋊Dic71—22no—
142C4⋊Dic71—22no—
143C4⋊Dic71—42no—
144(C2×D71)⋊C4—82no—
145(C2×D71)⋊C4—82no—
146C24×C71C2×D411yescharacteristic
147C22×C4×C71—21no—
148C22×C4×C71—21no—
149C22×C4×C71—22no—
150C22×C4×C71—42no—
151C22×C4×C71—81no—
152C22×C4×C71—81no—
153C22×C4×C71C22×C412yes—
154C22×C4×C71C2×D412yes—
155C22×C4×C71C2×D412yes—
156C22×C4×C71C2×D412yes—
157C22×C4×C71C2×D411yescharacteristic
158C22×C4×C71#4112yes—
159C22×C4×C71#4112yes—
160C22×C4×C71#4112yes—
161C22×C4×C71#4112yes—
162C22×C4×C71C2411yescharacteristic
163C4⋊C4×C71—22no—
164C4⋊C4×C71—22no—
165C4⋊C4×C71—21no—
166C4⋊C4×C71—21no—
167C4⋊C4×C71—22no—
168C4⋊C4×C71—22no—
169C4⋊C4×C71—42no—
170C4⋊C4×C71—82no—
171C4⋊C4×C71—82no—
172C4⋊C4×C71C22×C412yes—
173C4⋊C4×C71C22×C412yes—
174C22⋊C4×C71—24no—
175C22⋊C4×C71—24no—
176C42×C71—24no—
177C42×C71—22no—
178C42×C71—22no—
179C23×C71—21no—
180C23×C71—21no—
181C23×C71—21no—
182C23×C71—21no—
183C23×C71C4×D412yes—
184C23×C71#12112yes—
185C23×C71#12412yes—
186C23×C71C22×D411yescharacteristic
187C22×D71—22no—
188C22×D71—41no—
189C22×D71—41no—
190C22×D71—41no—
191C22×D71—41no—
192C22×D71—41no—
193C22×D71—41no—
194C2×C4×C71—22no—
195C2×C4×C71—22no—
196C2×C4×C71—22no—
197C2×C4×C71—22no—
198C2×C4×C71—22no—
199C2×C4×C71—22no—
200C2×C4×C71—22no—
201C2×C4×C71—22no—
202C2×C4×C71—41no—
203C2×C4×C71—41no—
204C2×C4×C71—41no—
205C2×C4×C71—41no—
206C2×C4×C71—42no—
207C2×C4×C71—42no—
208C2×C4×C71—42no—
209C2×C4×C71—84no—
210C2×C4×C71#11714yes—
211C2×C4×C71C4×D412yes—
212C2×C4×C71C4×D414yes—
213C2×C4×C71#12112yes—
214C2×C4×C71#12314yes—
215C2×C4×C71#12412yes—
216C2×C4×C71#12614yes—
217C2×C4×C71C23×C411yescharacteristic
218C2×C4×C71C22×D411yescharacteristic
219C2×C4×C71C2×#4111yescharacteristic
220C2×C4×C71C2×#4111yescharacteristic
221C2×C4×C71C2×#4111yescharacteristic
222C2×C4×C71C2×#4111yescharacteristic
223C22⋊D71—48no—
224C2×Dic71—22no—
225C2×Dic71—41no—
226C2×Dic71—41no—
227C2×Dic71—42no—
228C2×Dic71—42no—
229C2×Dic71—42no—
230C2×Dic71—84no—
231D284—22no—
232D284—22no—
233D284—41no—
234D284—41no—
235C4×D71—48no—
236Dic142—22no—
237Dic142—22no—
238Dic142—41no—
239Dic142—41no—
240C71⋊C8—22no—
241C71⋊C8—22no—
242C71⋊C8—24no—
243C22×C71—22no—
244C22×C71—22no—
245C22×C71—22no—
246C22×C71—22no—
247C22×C71—22no—
248C22×C71—22no—
249C22×C71#38814yes—
250C22×C71C2×C4×D411yescharacteristic
251C22×C71C2×#12111yescharacteristic
252C22×C71C2×#12411yescharacteristic
253C22×C71#57611yescharacteristic
254C22×C71#57711yescharacteristic
255C22×C71#57711yescharacteristic
256C22×C71#57811yescharacteristic
257C2×D71—42no—
258C2×D71—42no—
259C2×D71—42no—
260C2×D71—81no—
261C2×D71—81no—
262C2×D71—81no—
263C2×D71—81no—
264C4×C71—42no—
265C4×C71—42no—
266C4×C71—42no—
267C4×C71—42no—
268C4×C71—82no—
269C4×C71#38814yes—
270C4×C71C2×#11711yescharacteristic
271C4×C71C2×C4×D411yescharacteristic
272C4×C71C2×#12311yescharacteristic
273C4×C71C2×#12611yescharacteristic
274Dic71—42no—
275Dic71—82no—
276#21087—711nomaximal, Sylow
277C2×C71—24no—
278C2×C71C2×#38811yescharacteristic
279C2×C71#295111yescharacteristic
280C2×C71#295211yescharacteristic
281C2×C71#306211yescharacteristic
282C2×C71#306311yescharacteristic
283C2×C71#313611yescharacteristic
284C2×C71#313711yescharacteristic
285D71—82no—
286C2×#417—711no—
287C2×#416—711no—
288C2×#405—711no—
289#2317D7111yescharacteristic, complemented
290#2311—711no—
291#1936—712no—
292#1935—712no—
293#1934—712no—
294#1933—712no—
295#1736—711no—
296C2×#340—711no—
297C71#2108711yescharacteristic, complemented, Sylow
298C22×#41—711no—
299C22×M16—711no—
300C23×C8—711no—
301C2×#117C2×D7111yescharacteristic
302C2×#117C2×D7111yescharacteristic
303C22×C4⋊C4—711no—
304C22×C4⋊C4C2×D7111yescharacteristic
305#417—1424no—
306#416—718no—
307C2×#103—711no—
308C2×#103—711no—
309C2×#103—712no—
310C2×#102—711no—
311C2×#102—711no—
312C2×#102—712no—
313#405—1424no—
314C2×#97—712no—
315C2×C4×C8—712no—
316#391—712no—
317#390—712no—
318#389C2×D7112yes—
319#389C2×D7112yes—
320#386—1422no—
321#385—22no—
322#384—21no—
323#382—21no—
324#380—1421no—
325#380—1421no—
326#340—712no—
327#340—712no—
328#340—712no—
329#340—712no—
330C2×#41—714no—
331C2×#41—1422no—
332C2×#41—1422no—
333C2×#41—1422no—
334C2×#41—1422no—
335C2×#41—1422no—
336C22×Q8—711no—
337C22×D4—711no—
338C22×D4—1421no—
339C23×C4—21no—
340C23×C4—1421no—
341C23×C4—1421no—
342C23×C4C22×D7111yescharacteristic
343C2×M16—714no—
344C2×M16—1422no—
345C2×M16—1422no—
346C22×C8—714no—
347C22×C8—711no—
348C22×C8—711no—
349C22×C8—712no—
350C22×C8—1422no—
351C22×C8—1422no—
352#117—22no—
353#117—22no—
354#117—22no—
355#117C4×D7114yes—
356C2×C4⋊C4—22no—
357C2×C4⋊C4—22no—
358C2×C4⋊C4—21no—
359C2×C4⋊C4—22no—
360C2×C4⋊C4—22no—
361C2×C4⋊C4—712no—
362C2×C4⋊C4—1421no—
363C2×C4⋊C4—1422no—
364C2×C4⋊C4—1422no—
365C2×C4⋊C4C22⋊D7111yescharacteristic
366C2×C4⋊C4C22⋊D7111yescharacteristic
367C2×C4⋊C4C22×D7112yes—
368C2×C4⋊C4C22×D7111yescharacteristic
369C2×C4⋊C4C22×D7111yescharacteristic
370C2×C22⋊C4—21no—
371C2×C22⋊C4—1422no—
372C2×C22⋊C4—1422no—
373C2×C22⋊C4C22⋊D7112yes—
374C2×C42—21no—
375C2×C42C22×D7112yes—
376#103—714no—
377#103—714no—
378#103—1424no—
379#102—714no—
380#102—714no—
381#102—1424no—
382#97—1424no—
383C4×C8—1424no—
384#95—22no—
385#95—22no—
386#95—1422no—
387#95—1422no—
388C24—1421no—
389C24C2×C22⋊D7111yescharacteristic
390#41—2844no—
391#41—2844no—
392#41—1428no—
393#41—1428no—
394C2×Q8—712no—
395C2×Q8—1421no—
396C2×Q8—1421no—
397C2×Q8—1421no—
398C2×Q8—1421no—
399C2×Q8—1421no—
400C2×D4—2842no—
401C2×D4—2842no—
402C2×D4—712no—
403C2×D4—1424no—
404C2×D4—1421no—
405C2×D4—1421no—
406C2×D4—1421no—
407C2×D4—1421no—
408C2×D4—1421no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—22no—
412C22×C4—42no—
413C22×C4—81no—
414C22×C4—81no—
415C22×C4—2842no—
416C22×C4—2842no—
417C22×C4—2842no—
418C22×C4—5681no—
419C22×C4—5681no—
420C22×C4—1422no—
421C22×C4—1421no—
422C22×C4—1424no—
423C22×C4C2×C4×D7112yes—
424C22×C4D4×D7112yes—
425C22×C4C2×C22⋊D7112yes—
426C22×C4C2×C22⋊D7112yes—
427C22×C4C2×C22⋊D7111yescharacteristic
428C22×C4#1176586412yes—
429C22×C4#1176586512yes—
430C22×C4#1176586612yes—
431C22×C4#1176586612yes—
432C22×C4C23×D7111yescharacteristic
433M16—1428no—
434C2×C8—712no—
435C2×C8—712no—
436C2×C8—714no—
437C2×C8—1421no—
438C2×C8—1421no—
439C2×C8—1422no—
440C2×C8—1421no—
441C2×C8—1421no—
442C2×C8—1422no—
443C2×C8—1428no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—42no—
451C4⋊C4—82no—
452C4⋊C4—82no—
453C4⋊C4—2842no—
454C4⋊C4—1422no—
455C4⋊C4—1422no—
456C4⋊C4C2×C4×D7112yes—
457C4⋊C4C2×C4×D7112yes—
458C22⋊C4—24no—
459C22⋊C4—24no—
460C22⋊C4—5682no—
461C22⋊C4—5682no—
462C42—22no—
463C42—22no—
464C42—24no—
465C23—21no—
466C23—21no—
467C23—21no—
468C23—21no—
469C23—2841no—
470C23—2841no—
471C23—2841no—
472C23—2841no—
473C23—2841no—
474C23—2841no—
475C23—1422no—
476C23C4×C22⋊D7112yes—
477C23#1455610012yes—
478C23#1455611712yes—
479C23C22×C22⋊D7111yescharacteristic
480Q8—2841no—
481Q8—2841no—
482Q8—1422no—
483Q8—1422no—
484D4—2848no—
485D4—2841no—
486D4—2841no—
487D4—1422no—
488D4—1422no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—84no—
505C2×C4—2841no—
506C2×C4—2841no—
507C2×C4—2848no—
508C2×C4—2842no—
509C2×C4—2842no—
510C2×C4—2842no—
511C2×C4—5684no—
512C2×C4—1422no—
513C2×C4#1455608614yes—
514C2×C4C4×C22⋊D7112yes—
515C2×C4#1455609014yes—
516C2×C4#1455610012yes—
517C2×C4#1455611314yes—
518C2×C4#1455611712yes—
519C2×C4#1455612314yes—
520C2×C4C22×C4×D7111yescharacteristic
521C2×C4C2×D4×D7111yescharacteristic
522C2×C4C2×#1176586411yescharacteristic
523C2×C4C2×#1176586511yescharacteristic
524C2×C4C2×#1176586611yescharacteristic
525C2×C4C2×#1176586611yescharacteristic
526C8—1422no—
527C8—1422no—
528C8—1424no—
529C22—22no—
530C22—22no—
531C22—22no—
532C22—22no—
533C22—22no—
534C22—22no—
535C22—2842no—
536C22—2842no—
537C22—2842no—
538C22—5681no—
539C22—5681no—
540C22—5681no—
541C22—5681no—
542C22#1798588914yes—
543C22C2×C4×C22⋊D7111yescharacteristic
544C22C2×#1455610011yescharacteristic
545C22C2×#1455611711yescharacteristic
546C22#1798680211yescharacteristic
547C22#1798680711yescharacteristic
548C22#1798681011yescharacteristic
549C22#1798681411yescharacteristic
550C4—42no—
551C4—42no—
552C4—42no—
553C4—42no—
554C4—82no—
555C4—2842no—
556C4—5682no—
557C4#1798588914yes—
558C4C2×#1455608611yescharacteristic
559C4C2×#1455609011yescharacteristic
560C4C2×#1455611311yescharacteristic
561C4C2×#1455612311yescharacteristic
562C2—24no—
563C2—5682no—
564C2C2×#1798588911yescharacteristic
565C2#2259721311yescharacteristic
566C2#2259722111yescharacteristic
567C2#2259801611yescharacteristic
568C2#2259802711yescharacteristic
569C2#2259869911yescharacteristic
570C2#2259871011yescharacteristic
subgroupG/H
centerC23C22×C22⋊D71
commutatorC2×C4×C71C23×C4
FrattiniC22×C4C23×D71
FittingC71×#2317C2
socleC23×C71C22×D4
prime radicalC22×#11765865C22
cosocleC22×C4×C71C24
hypercenter#2317D71
hypocenterC71#21087
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2317

Complements

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

NH ≅ G/Ntype
3. C2×#17985955C2semidirect
5. #22589036C2semidirect
6. #22589029C2semidirect
7. #22589022C2semidirect
8. #22589015C2semidirect
9. #22588061C2semidirect
10. C2×#17985761C2semidirect
11. C71×#2317C2semidirect
17. #17985993C4semidirect
71. #14556051C2×C4semidirect
72. #14556051C2×C4semidirect
74. #14556048C2×C4semidirect
75. #14556048C2×C4semidirect
289. #2317D71semidirect
297. C71#21087semidirect

Maximal subgroups

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

subgroupindex
C2×#179859982
C2×#179859932
C2×#179859552
#225909282
#225890362
#225890292
#225890222
#225890152
#225880612
C2×#179857612
C71×#23172
#2108771