Finite Group Databasebeta

← #31968857

Subgroups — #31968857

Order 18176.

598 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22598352C211yesmaximal, characteristic
2#22597469C211yesmaximal, characteristic, complemented
3#22587240C211yesmaximal, characteristic, complemented
4#22587223C211yesmaximal, characteristic, complemented
5#22587218C211yesmaximal, characteristic, complemented
6#22587216C211yesmaximal, characteristic, complemented
7#22587188C211yesmaximal, characteristic, complemented
8#22587169C211yesmaximal, characteristic, complemented
9#22587152C211yesmaximal, characteristic, complemented
10C8×#11765855C211yesmaximal, characteristic, complemented
11#22587114C211yesmaximal, characteristic, complemented
12#22586805C211yesmaximal, characteristic, complemented
13#22586725C211yesmaximal, characteristic, complemented
14#22586640C211yesmaximal, characteristic, complemented
15C71×#2994C211yesmaximal, characteristic, complemented
16C4×#11765865C2211yescharacteristic
17#17986337C411yescharacteristic
18#17986320C411yescharacteristic
19#17986303C411yescharacteristic
20#17986296C411yescharacteristic
21#17986218C411yescharacteristic
22#17986211C411yescharacteristic
23#17986159C411yescharacteristic
24#17986142C411yescharacteristic
25C8×Dic142C2211yescharacteristic
26C8×Dic142C2211yescharacteristic
27C4×#11765855C2211yescharacteristic
28C4×#11765852C2211yescharacteristic
29C4×#11765853C2211yescharacteristic
30C4×#11765850C2211yescharacteristic
31#17986078C2211yescharacteristic
32#17986078C2211yescharacteristic
33#17986074—21no—
34#17986074—21no—
35#17986052C2211yescharacteristic
36#17986052C2211yescharacteristic
37#17986051—21no—
38#17986023C2211yescharacteristic
39#17986021—21no—
40#17985751C2211yescharacteristic
41#17985751C2211yescharacteristic
42#17985751C2211yescharacteristic
43#17985751C2211yescharacteristic
44Q8⋊(C71⋊C8)C2211yescharacteristic
45Q8⋊(C71⋊C8)C2211yescharacteristic
46#17985729C2211yescharacteristic
47#17985729C2211yescharacteristic
48D4⋊(C71⋊C8)C2211yescharacteristic
49D4⋊(C71⋊C8)C2211yescharacteristic
50#17985726C2211yescharacteristic
51#17985726C2211yescharacteristic
52#17985721C2211yescharacteristic
53#17985721C2211yescharacteristic
54#17985720C2211yescharacteristic
55C8×C71⋊C8C2211yescharacteristic
56#529×C71C2211yescharacteristic
57#446×C71C2211yescharacteristic
58C8×Q8×C71C2211yescharacteristic
59#436×C71C2211yescharacteristic
60#435×C71—21no—
61C8×D4×C71—21no—
62C8×D4×C71C2211yescharacteristic
63#433×C71—21no—
64#431×C71C2211yescharacteristic
65#424×C71—21no—
66C2×C4⋊C8×C71C2211yescharacteristic
67C2×#11765865C2×C411yescharacteristic
68C2×#11765855C2×C411yescharacteristic
69C2×#11765853C2×C411yescharacteristic
70C2×#11765852C2×C411yescharacteristic
71C2×#11765850C2×C411yescharacteristic
72C2×#11765849—21no—
73C2×#11765848—21no—
74C2×#11765848—21no—
75C4×Dic142C2×C411yescharacteristic
76C4×Dic142C2311yescharacteristic
77C4×D284C2×C411yescharacteristic
78C4×D284C2311yescharacteristic
79C4×C22⋊D71—21no—
80C4×C22⋊D71—21no—
81#14556087C2×C411yescharacteristic
82#14556085—21no—
83C42×D71—21no—
84#14556061C2×C411yescharacteristic
85#14556061C2×C411yescharacteristic
86#14556059C2×C411yescharacteristic
87#14556059C2×C411yescharacteristic
88C4⋊(C71⋊C8)—21no—
89C4⋊(C71⋊C8)—21no—
90C4⋊(C71⋊C8)C2×C411yescharacteristic
91C4⋊(C71⋊C8)C2×C411yescharacteristic
92C4⋊(C71⋊C8)C2311yescharacteristic
93C4⋊(C71⋊C8)C2311yescharacteristic
94Dic71⋊C8—21no—
95Dic71⋊C8—21no—
96#14556052C2×C411yescharacteristic
97#14556052C2×C411yescharacteristic
98#14556051C2×C411yescharacteristic
99#14556051C2×C411yescharacteristic
100#14556049C2×C411yescharacteristic
101#14556049C2×C411yescharacteristic
102#14556048C2×C411yescharacteristic
103#14556048C2×C411yescharacteristic
104C22⋊(C71⋊C8)—21no—
105C22⋊(C71⋊C8)C2×C411yescharacteristic
106C22⋊(C71⋊C8)C2×C411yescharacteristic
107(C2×D71)⋊C8—21no—
108(C2×D71)⋊C8—21no—
109C4×C71⋊C8C2311yescharacteristic
110C4×C71⋊C8C2311yescharacteristic
111C4×C71⋊C8C2311yescharacteristic
112C4×C71⋊C8C2311yescharacteristic
113C8×Dic71—21no—
114C8×Dic71—21no—
115C2×M16×C71—21no—
116C22×C8×C71—21no—
117C22×C8×C71—21no—
118#126×C71—21no—
119#125×C71C2×C411yescharacteristic
120#124×C71C2×C411yescharacteristic
121#123×C71—21no—
122C22⋊Q8×C71C2×C411yescharacteristic
123#121×C71C2×C411yescharacteristic
124C4×Q8×C71C2311yescharacteristic
125C4×D4×C71—21no—
126C4×D4×C71C2311yescharacteristic
127#117×C71—21no—
128C2×C42×C71C2311yescharacteristic
129C4⋊C8×C71—21no—
130C4⋊C8×C71—21no—
131C4⋊C8×C71—21no—
132C4⋊C8×C71D411yescharacteristic
133C4⋊C8×C71D411yescharacteristic
134C4⋊C8×C71C2311yescharacteristic
135C4⋊C8×C71C2311yescharacteristic
136C22⋊C8×C71—21no—
137C22⋊C8×C71—21no—
138C22⋊C8×C71D411yescharacteristic
139C22⋊C8×C71D411yescharacteristic
140#97×C71—21no—
141C4×C8×C71—21no—
142C4×C8×C71C2311yescharacteristic
143C4×C8×C71C2311yescharacteristic
144#11765865—22no—
145#11765865—41no—
146C2×Dic142C22×C411yescharacteristic
147C2×D284C22×C411yescharacteristic
148C2×C22⋊D71—21no—
149C2×C4×D71—21no—
150C2×C4×D71—21no—
151C2×C4×D71—21no—
152#11765855—22no—
153#11765853—22no—
154#11765852—41no—
155#11765850—41no—
156#11765849—81no—
157#11765848—81no—
158#11765848—81no—
159C2×C71⋊C8—21no—
160C2×C71⋊C8—21no—
161C2×C71⋊C8—21no—
162C2×C71⋊C8—21no—
163C2×C71⋊C8C22×C411yescharacteristic
164C2×C71⋊C8C22×C411yescharacteristic
165C2×C71⋊C8C22×C411yescharacteristic
166C2×C71⋊C8C22×C411yescharacteristic
167C4⋊Dic71C22×C411yescharacteristic
168C4⋊Dic71C22×C411yescharacteristic
169Dic71⋊C4—21no—
170Dic71⋊C4—21no—
171C22⋊Dic71—21no—
172(C2×D71)⋊C4—21no—
173(C2×D71)⋊C4—21no—
174C4×Dic71—21no—
175C4×Dic71—21no—
176C4×Dic71—21no—
177C2×Q8×C71C22×C411yescharacteristic
178C2×D4×C71—21no—
179C2×D4×C71C22×C411yescharacteristic
180C22×C4×C71—21no—
181C22×C4×C71C22×C411yescharacteristic
182C22×C4×C71C2×D411yescharacteristic
183C22×C4×C71#4111yescharacteristic
184M16×C71—81no—
185C2×C8×C71—21no—
186C2×C8×C71—21no—
187C2×C8×C71—81no—
188C2×C8×C71—81no—
189C2×C8×C71C2×D411yescharacteristic
190C2×C8×C71C2×D411yescharacteristic
191C2×C8×C71C2×D411yescharacteristic
192C2×C8×C71C2×D411yescharacteristic
193C4⋊C4×C71—21no—
194C4⋊C4×C71—21no—
195C4⋊C4×C71—21no—
196C4⋊C4×C71—21no—
197C4⋊C4×C71C22×C411yescharacteristic
198C4⋊C4×C71C22×C411yescharacteristic
199C22⋊C4×C71—21no—
200C22⋊C4×C71—21no—
201C22⋊C4×C71—21no—
202C22⋊C4×C71—21no—
203C22⋊C4×C71—21no—
204C42×C71—21no—
205C42×C71C22×C411yescharacteristic
206C42×C71C2×D411yescharacteristic
207C42×C71#4111yescharacteristic
208C42×C71C2411yescharacteristic
209C23×C71—21no—
210C23×C71C4×D411yescharacteristic
211C22×D71—21no—
212Q8×C71—21no—
213Q8×C71—21no—
214D4×C71—21no—
215D4×C71—21no—
216D4×C71—42no—
217C2×C4×C71—21no—
218C2×C4×C71—21no—
219C2×C4×C71—21no—
220C2×C4×C71—21no—
221C2×C4×C71—21no—
222C2×C4×C71—21no—
223C2×C4×C71—21no—
224C2×C4×C71—41no—
225C2×C4×C71—41no—
226C2×C4×C71—41no—
227C2×C4×C71C4×D411yescharacteristic
228C2×C4×C71C4×D411yescharacteristic
229C2×C4×C71C4×D411yescharacteristic
230C2×C4×C71C23×C411yescharacteristic
231C2×C4×C71C22×D411yescharacteristic
232C2×C4×C71C2×#4111yescharacteristic
233C22⋊D71—42no—
234C2×Dic71—21no—
235C2×Dic71—21no—
236C2×Dic71—21no—
237D284—21no—
238D284—21no—
239C4×D71—42no—
240C4×D71—42no—
241C4×D71—41no—
242C4×D71—41no—
243Dic142—21no—
244Dic142#13112yes—
245C8×C71—21no—
246C8×C71—21no—
247C8×C71—21no—
248C8×C71—21no—
249C8×C71—41no—
250C8×C71—41no—
251C71⋊C8—21no—
252C71⋊C8—21no—
253C71⋊C8—21no—
254C71⋊C8—21no—
255C71⋊C8—41no—
256C71⋊C8—41no—
257C71⋊C8—41no—
258C71⋊C8—41no—
259C22×C71—21no—
260C22×C71—21no—
261C22×C71—21no—
262C22×C71—41no—
263C22×C71—41no—
264C22×C71—41no—
265C22×C71C2×C4×D411yescharacteristic
266C2×D71—41no—
267C2×D71—41no—
268C2×D71—41no—
269C4×C71—21no—
270C4×C71—21no—
271C4×C71—21no—
272C4×C71—21no—
273C4×C71—41no—
274C4×C71—41no—
275C4×C71—41no—
276C4×C71C2×#13111yescharacteristic
277C4×C71#56811yescharacteristic
278C4×C71#57611yescharacteristic
279C4×C71#57811yescharacteristic
280Dic71—22no—
281Dic71—41no—
282Dic71—41no—
283#18033—711nomaximal, Sylow
284C2×C71—41no—
285C2×C71—81no—
286C2×C71#293311yescharacteristic
287C2×C71#295111yescharacteristic
288C2×C71#295511yescharacteristic
289D71—81no—
290#3102—711no—
291#2994D7111yescharacteristic, complemented
292#2991—711no—
293#1592—711no—
294#1589—711no—
295#1588—711no—
296#1588—711no—
297#1584—711no—
298#1581—711no—
299SD16⋊C8—711no—
300C8×Q16—711no—
301#1571—711no—
302#1495—711no—
303#1477—711no—
304#1454—711no—
305C71#1803311yescharacteristic, complemented, Sylow
306#529C2×D7111yescharacteristic
307C4×#41—711no—
308#485—711no—
309#482—711no—
310#479—711no—
311#478—711no—
312#464—711no—
313#463—711no—
314#452—711no—
315#449—711no—
316#446C2×D7111yescharacteristic
317C8×Q8—711no—
318C8×Q8—711no—
319C8×Q8C2×D7111yescharacteristic
320C4×Q16—711no—
321C4×SD16—711no—
322C4×SD16—711no—
323C4×D8—711no—
324#436—711no—
325#436—711no—
326#436—1421no—
327#436—1421no—
328#436C2×D7111yescharacteristic
329#435—21no—
330C8×D4—21no—
331C8×D4C2×D7111yescharacteristic
332#433—21no—
333#431—711no—
334#431—711no—
335#431—1421no—
336#431C2×D7111yescharacteristic
337#424—21no—
338#423—711no—
339#423—1421no—
340C2×C4⋊C8C2×D7111yescharacteristic
341#335—711no—
342#335—711no—
343#335—711no—
344#335—711no—
345Q8⋊C8—711no—
346Q8⋊C8—711no—
347Q8⋊C8—711no—
348Q8⋊C8—711no—
349D4⋊C8—711no—
350D4⋊C8—711no—
351D4⋊C8—711no—
352D4⋊C8—711no—
353#322—711no—
354#322—711no—
355#322—711no—
356C82—711no—
357C2×#41—711no—
358C2×Q16—711no—
359C2×SD16—711no—
360C2×SD16—711no—
361C2×D8—711no—
362C2×M16—21no—
363C2×M16—1421no—
364C2×M16—1421no—
365C2×M16—1421no—
366C22×C8—21no—
367C22×C8—21no—
368#126—21no—
369#125C4×D7111yescharacteristic
370#124C4×D7111yescharacteristic
371#123—21no—
372C22⋊Q8C4×D7111yescharacteristic
373#121C4×D7111yescharacteristic
374C4×Q8—711no—
375C4×Q8—711no—
376C4×Q8C22×D7111yescharacteristic
377C4×D4—21no—
378C4×D4—711no—
379C4×D4—711no—
380C4×D4—1421no—
381C4×D4—1421no—
382C4×D4C22×D7111yescharacteristic
383#117—21no—
384#117—711no—
385#117—1421no—
386C2×C42—1421no—
387C2×C42C22×D7111yescharacteristic
388#107—711no—
389#107—711no—
390#106—711no—
391#106—711no—
392C4⋊C8—21no—
393C4⋊C8—21no—
394C4⋊C8—21no—
395C4⋊C8—711no—
396C4⋊C8—711no—
397C4⋊C8—711no—
398C4⋊C8—711no—
399C4⋊C8—1421no—
400C4⋊C8—1421no—
401C4⋊C8—1421no—
402C4⋊C8—1421no—
403C4⋊C8C22⋊D7111yescharacteristic
404C4⋊C8C22⋊D7111yescharacteristic
405C4⋊C8C22×D7111yescharacteristic
406C4⋊C8C22×D7111yescharacteristic
407#103—711no—
408#103—711no—
409#103—711no—
410#103—711no—
411#102—711no—
412#102—711no—
413#102—711no—
414#102—711no—
415C22⋊C8—21no—
416C22⋊C8—21no—
417C22⋊C8—711no—
418C22⋊C8—711no—
419C22⋊C8—1421no—
420C22⋊C8—1421no—
421C22⋊C8—1421no—
422C22⋊C8C22⋊D7111yescharacteristic
423C22⋊C8C22⋊D7111yescharacteristic
424#97—21no—
425C4×C8—21no—
426C4×C8—711no—
427C4×C8—711no—
428C4×C8—711no—
429C4×C8—711no—
430C4×C8—1421no—
431C4×C8—1421no—
432C4×C8C22×D7111yescharacteristic
433C4×C8C22×D7111yescharacteristic
434#41—2841no—
435#41—1422no—
436C2×Q8—711no—
437C2×Q8C2×C4×D7111yescharacteristic
438C2×D4—21no—
439C2×D4—711no—
440C2×D4—1421no—
441C2×D4C2×C4×D7111yescharacteristic
442C22×C4—21no—
443C22×C4—1421no—
444C22×C4—1421no—
445C22×C4—1421no—
446C22×C4C2×C4×D7111yescharacteristic
447C22×C4C2×C22⋊D7111yescharacteristic
448C22×C4#1176586511yescharacteristic
449Q16—1422no—
450SD16—2841no—
451SD16—1422no—
452D8—2841no—
453M16—81no—
454M16—5681no—
455M16—5681no—
456M16—5681no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—81no—
460C2×C8—81no—
461C2×C8—711no—
462C2×C8—711no—
463C2×C8—711no—
464C2×C8—711no—
465C2×C8—1421no—
466C2×C8—1421no—
467C2×C8—1421no—
468C2×C8—1421no—
469C2×C8C2×C22⋊D7111yescharacteristic
470C2×C8C2×C22⋊D7111yescharacteristic
471C2×C8C2×C22⋊D7111yescharacteristic
472C2×C8C2×C22⋊D7111yescharacteristic
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—711no—
478C4⋊C4—711no—
479C4⋊C4—1421no—
480C4⋊C4—1421no—
481C4⋊C4C2×C4×D7111yescharacteristic
482C4⋊C4C2×C4×D7111yescharacteristic
483C22⋊C4—21no—
484C22⋊C4—21no—
485C22⋊C4—21no—
486C22⋊C4—21no—
487C22⋊C4—21no—
488C22⋊C4—1421no—
489C22⋊C4—1421no—
490C22⋊C4—1421no—
491C42—21no—
492C42—1421no—
493C42—1421no—
494C42—1421no—
495C42C2×C4×D7111yescharacteristic
496C42C2×C22⋊D7111yescharacteristic
497C42#1176586511yescharacteristic
498C42C23×D7111yescharacteristic
499C23—21no—
500C23—1421no—
501C23C4×C22⋊D7111yescharacteristic
502Q8—21no—
503Q8—21no—
504Q8—712no—
505Q8—1421no—
506D4—21no—
507D4—21no—
508D4—42no—
509D4—2842no—
510D4—1421no—
511D4—1421no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—2842no—
523C2×C4—2842no—
524C2×C4—2841no—
525C2×C4—2841no—
526C2×C4—1421no—
527C2×C4—1421no—
528C2×C4—1421no—
529C2×C4C4×C22⋊D7111yescharacteristic
530C2×C4C4×C22⋊D7111yescharacteristic
531C2×C4C4×C22⋊D7111yescharacteristic
532C2×C4C22×C4×D7111yescharacteristic
533C2×C4C22×C22⋊D7111yescharacteristic
534C2×C4C2×#1176586511yescharacteristic
535C8—21no—
536C8—21no—
537C8—21no—
538C8—21no—
539C8—41no—
540C8—41no—
541C8—2841no—
542C8—2841no—
543C8—2841no—
544C8—2841no—
545C8—1421no—
546C8—1421no—
547C8—1421no—
548C8—1421no—
549C22—21no—
550C22—21no—
551C22—21no—
552C22—41no—
553C22—41no—
554C22—41no—
555C22—2841no—
556C22—2841no—
557C22—2841no—
558C22C2×C4×C22⋊D7111yescharacteristic
559C4—21no—
560C4—21no—
561C4—21no—
562C4—21no—
563C4—41no—
564C4—41no—
565C4—41no—
566C4—2841no—
567C4—2841no—
568C4—1422no—
569C4#131×D7111yescharacteristic
570C4#1798676411yescharacteristic
571C4#1798680211yescharacteristic
572C4#1798681411yescharacteristic
573C2—41no—
574C2—81no—
575C2—5681no—
576C2#2259706211yescharacteristic
577C2#2259721311yescharacteristic
578C2#2259724111yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D71
commutatorC2×C4×C71C23×C4
FrattiniC42C23×D71
FittingC71×#2994C2
socleC22×C71C2×C4×D4
prime radical#17986142C4
cosocleC42×C71C24
hypercenter#2994D71
hypocenterC71#18033
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C2×C4 ◁ #431 ◁ #2994

Complements

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

NH ≅ G/Ntype
2. #22597469C2semidirect
3. #22587240C2semidirect
4. #22587223C2semidirect
5. #22587218C2semidirect
6. #22587216C2semidirect
7. #22587188C2semidirect
8. #22587169C2semidirect
9. #22587152C2semidirect
10. C8×#11765855C2semidirect
11. #22587114C2semidirect
12. #22586805C2semidirect
13. #22586725C2semidirect
14. #22586640C2semidirect
15. C71×#2994C2semidirect
291. #2994D71semidirect
305. C71#18033semidirect

Maximal subgroups

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

subgroupindex
#225983522
#225974692
#225872402
#225872232
#225872182
#225872162
#225871882
#225871692
#225871522
C8×#117658552
#225871142
#225868052
#225867252
#225866402
C71×#29942
#1803371