Finite Group Databasebeta

← #31968852

Subgroups — #31968852

Order 18176.

566 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22598375C211yesmaximal, characteristic
2#22597442C211yesmaximal, characteristic
3#22597430C211yesmaximal, characteristic
4#22587246C211yesmaximal, characteristic, complemented
5#22587234C211yesmaximal, characteristic, complemented
6#22587214C211yesmaximal, characteristic, complemented
7#22587208C211yesmaximal, characteristic, complemented
8#22587193C211yesmaximal, characteristic, complemented
9#22587187C211yesmaximal, characteristic, complemented
10#22587156C211yesmaximal, characteristic, complemented
11#22587144C211yesmaximal, characteristic, complemented
12#22587121C211yesmaximal, characteristic
13#22586736C211yesmaximal, characteristic
14#22586640C211yesmaximal, characteristic
15C71×#1497C211yesmaximal, characteristic
16C2×C4×Dic142C2211yescharacteristic
17#17986332C411yescharacteristic
18#17986332C411yescharacteristic
19#17986291C411yescharacteristic
20#17986291C411yescharacteristic
21#17986206C411yescharacteristic
22#17986206C411yescharacteristic
23#17986154C411yescharacteristic
24#17986154C411yescharacteristic
25#17986130—21no—
26#17986130C2211yescharacteristic
27#17986130C2211yescharacteristic
28#17986127—21no—
29#17986127—21no—
30#17986126C2211yescharacteristic
31#17986126C2211yescharacteristic
32Q8×C71⋊C8—21no—
33Q8×C71⋊C8C2211yescharacteristic
34Q8×C71⋊C8C2211yescharacteristic
35#17986123C2211yescharacteristic
36#17986123C2211yescharacteristic
37C4×#11765855C2211yescharacteristic
38C4×#11765855C2211yescharacteristic
39C4×#11765855C2211yescharacteristic
40C4×#11765855C2211yescharacteristic
41#17986060—21no—
42#17986060—21no—
43#17986053—21no—
44#17986052C2211yescharacteristic
45#17986052C2211yescharacteristic
46#17986052C2211yescharacteristic
47#17986049—21no—
48#17985751C2211yescharacteristic
49#17985751C2211yescharacteristic
50#17985749C2211yescharacteristic
51#17985749C2211yescharacteristic
52#17985731C2211yescharacteristic
53#17985731C2211yescharacteristic
54#17985731C2211yescharacteristic
55#17985731C2211yescharacteristic
56#17985721C2211yescharacteristic
57#17985721C2211yescharacteristic
58#17985720C2211yescharacteristic
59C8×C71⋊C8C2211yescharacteristic
60#533×C71C2211yescharacteristic
61#432×C71C2211yescharacteristic
62#431×C71C2211yescharacteristic
63Q8⋊C8×C71C2211yescharacteristic
64Q8⋊C8×C71C2211yescharacteristic
65Q8⋊C8×C71C2211yescharacteristic
66Q8⋊C8×C71C2211yescharacteristic
67C22×Dic142C2×C411yescharacteristic
68C2×#11765855C2×C411yescharacteristic
69C2×#11765855C2×C411yescharacteristic
70C2×#11765855C2×C411yescharacteristic
71C2×#11765855C2×C411yescharacteristic
72C4×Dic142—21no—
73C4×Dic142—21no—
74C4×Dic142C2×C411yescharacteristic
75C4×Dic142C2×C411yescharacteristic
76C4×Dic142C2311yescharacteristic
77C4×Dic142C2311yescharacteristic
78C2×C4⋊Dic71C2×C411yescharacteristic
79C2×Dic71⋊C4—21no—
80C2×C4×Dic71—21no—
81#14556061C2×C411yescharacteristic
82#14556061C2×C411yescharacteristic
83#14556061C2×C411yescharacteristic
84#14556061C2×C411yescharacteristic
85C4⋊(C71⋊C8)—21no—
86C4⋊(C71⋊C8)—21no—
87C4⋊(C71⋊C8)—21no—
88C4⋊(C71⋊C8)—21no—
89C4⋊(C71⋊C8)C2×C411yescharacteristic
90C4⋊(C71⋊C8)C2×C411yescharacteristic
91C4⋊(C71⋊C8)D411yescharacteristic
92C4⋊(C71⋊C8)D411yescharacteristic
93C4⋊(C71⋊C8)C2311yescharacteristic
94C4⋊(C71⋊C8)C2311yescharacteristic
95Dic71⋊C8—21no—
96Dic71⋊C8—21no—
97Dic71⋊C8—21no—
98Dic71⋊C8—21no—
99#14556052C2×C411yescharacteristic
100#14556052C2×C411yescharacteristic
101#14556052C2×C411yescharacteristic
102#14556052C2×C411yescharacteristic
103#14556051C2×C411yescharacteristic
104#14556051C2×C411yescharacteristic
105#14556051C2×C411yescharacteristic
106#14556051C2×C411yescharacteristic
107C22⋊(C71⋊C8)—21no—
108C22⋊(C71⋊C8)C2×C411yescharacteristic
109C22⋊(C71⋊C8)C2×C411yescharacteristic
110C22⋊(C71⋊C8)D411yescharacteristic
111C22⋊(C71⋊C8)D411yescharacteristic
112#14556039—21no—
113#14556039—21no—
114#14556038—21no—
115#14556038—21no—
116C4×C71⋊C8—21no—
117C4×C71⋊C8—21no—
118C4×C71⋊C8—21no—
119C4×C71⋊C8C2311yescharacteristic
120C4×C71⋊C8C2311yescharacteristic
121C4×C71⋊C8C2311yescharacteristic
122C4×C71⋊C8C2311yescharacteristic
123C8×Dic71—21no—
124C8×Dic71—21no—
125C4⋊Q8×C71C2×C411yescharacteristic
126C4⋊Q8×C71C2×C411yescharacteristic
127#125×C71—21no—
128C22⋊Q8×C71—21no—
129C22⋊Q8×C71C2×C411yescharacteristic
130C22⋊Q8×C71C2×C411yescharacteristic
131C4×Q8×C71—21no—
132C4×Q8×C71C2311yescharacteristic
133C4×Q8×C71C2311yescharacteristic
134#117×C71—21no—
135C2×C42×C71C2311yescharacteristic
136C4⋊C8×C71—21no—
137C4⋊C8×C71C2311yescharacteristic
138C4⋊C8×C71C2311yescharacteristic
139C22⋊C8×C71—21no—
140C22⋊C8×C71—21no—
141#97×C71—21no—
142C4×C8×C71C2311yescharacteristic
143C4×C8×C71C2311yescharacteristic
144C2×Dic142—22no—
145C2×Dic142—21no—
146C2×Dic142—41no—
147C2×Dic142C22×C411yescharacteristic
148C2×Dic142C22×C411yescharacteristic
149C22×Dic71—21no—
150C22×Dic71—21no—
151#11765855—22no—
152#11765855—22no—
153#11765855—41no—
154#11765855—41no—
155C2×C71⋊C8—21no—
156C2×C71⋊C8—21no—
157C2×C71⋊C8C22×C411yescharacteristic
158C2×C71⋊C8C22×C411yescharacteristic
159C2×C71⋊C8C22×C411yescharacteristic
160C2×C71⋊C8C22×C411yescharacteristic
161C2×C71⋊C8C2×D411yescharacteristic
162C2×C71⋊C8C2×D411yescharacteristic
163C2×C71⋊C8C2×D411yescharacteristic
164C2×C71⋊C8C2×D411yescharacteristic
165C4⋊Dic71—21no—
166C4⋊Dic71C22×C411yescharacteristic
167C4⋊Dic71C22×C411yescharacteristic
168Dic71⋊C4—21no—
169Dic71⋊C4—21no—
170Dic71⋊C4—21no—
171Dic71⋊C4—21no—
172C4×Dic71—21no—
173C4×Dic71—21no—
174C4×Dic71—21no—
175C4×Dic71—21no—
176C2×Q8×C71—21no—
177C2×Q8×C71C22×C411yescharacteristic
178C2×Q8×C71C22×C411yescharacteristic
179C22×C4×C71C22×C411yescharacteristic
180C22×C4×C71C2×D411yescharacteristic
181C22×C4×C71#4111yescharacteristic
182C2×C8×C71—21no—
183C2×C8×C71—21no—
184C2×C8×C71—21no—
185C2×C8×C71—21no—
186C4⋊C4×C71—21no—
187C4⋊C4×C71—21no—
188C4⋊C4×C71—21no—
189C4⋊C4×C71—21no—
190C4⋊C4×C71—21no—
191C4⋊C4×C71—21no—
192C4⋊C4×C71—21no—
193C4⋊C4×C71C22×C411yescharacteristic
194C4⋊C4×C71C22×C411yescharacteristic
195C22⋊C4×C71—21no—
196C22⋊C4×C71—21no—
197C42×C71—21no—
198C42×C71—21no—
199C42×C71C22×C411yescharacteristic
200C42×C71C2×D411yescharacteristic
201C42×C71#4111yescharacteristic
202C42×C71C2411yescharacteristic
203C23×C71C4×D411yescharacteristic
204Q8×C71—21no—
205Q8×C71—21no—
206Q8×C71—21no—
207Q8×C71—21no—
208Q8×C71—42no—
209C2×C4×C71—21no—
210C2×C4×C71—21no—
211C2×C4×C71—21no—
212C2×C4×C71—21no—
213C2×C4×C71—21no—
214C2×C4×C71—21no—
215C2×C4×C71—21no—
216C2×C4×C71—21no—
217C2×C4×C71—41no—
218C2×C4×C71C4×D411yescharacteristic
219C2×C4×C71C4×D411yescharacteristic
220C2×C4×C71C4×D411yescharacteristic
221C2×C4×C71C23×C411yescharacteristic
222C2×C4×C71C22×D411yescharacteristic
223C2×C4×C71C2×#4111yescharacteristic
224C2×Dic71—21no—
225C2×Dic71—21no—
226C2×Dic71—21no—
227C2×Dic71—21no—
228C2×Dic71—42no—
229C2×Dic71—42no—
230Dic142—21no—
231Dic142—21no—
232Dic142—21no—
233Dic142—42no—
234Dic142#13112yes—
235C8×C71—41no—
236C8×C71—41no—
237C8×C71—41no—
238C8×C71—41no—
239C71⋊C8—21no—
240C71⋊C8—21no—
241C71⋊C8—21no—
242C71⋊C8—21no—
243C71⋊C8—21no—
244C71⋊C8—21no—
245C71⋊C8—21no—
246C71⋊C8—21no—
247C71⋊C8—41no—
248C71⋊C8—41no—
249C22×C71—21no—
250C22×C71—21no—
251C22×C71—21no—
252C22×C71C2×C4×D411yescharacteristic
253C4×C71—21no—
254C4×C71—21no—
255C4×C71—21no—
256C4×C71—21no—
257C4×C71—41no—
258C4×C71—41no—
259C4×C71—41no—
260C4×C71—41no—
261C4×C71C2×#13111yescharacteristic
262C4×C71#56811yescharacteristic
263C4×C71#57811yescharacteristic
264C4×C71#57811yescharacteristic
265Dic71—22no—
266Dic71—41no—
267Dic71—41no—
268Dic71—41no—
269#18032—711nomaximal, Sylow
270C2×C71—41no—
271C2×C71#293311yescharacteristic
272C2×C71#295011yescharacteristic
273C2×C71#295511yescharacteristic
274#3104—711no—
275#2986—711no—
276#2984—711no—
277#1593—711no—
278#1591—711no—
279#1587—711no—
280#1587—711no—
281#1584—711no—
282#1584—711no—
283#1579—711no—
284C8×Q16—711no—
285#1572—711no—
286#1497D7111yescharacteristic
287#1479—711no—
288#1454—711no—
289C71#1803211yescharacteristic, complemented, Sylow
290#533C2×D7111yescharacteristic
291C2×C4×Q8—711no—
292#484—711no—
293#484—711no—
294#477—711no—
295#477—711no—
296#462—711no—
297#462—711no—
298#451—711no—
299#451—711no—
300#446—711no—
301#446—711no—
302#446—711no—
303#446—711no—
304#446—1421no—
305#446—1421no—
306#446—1421no—
307C8×Q8—711no—
308C8×Q8—711no—
309C8×Q8—711no—
310C8×Q8—711no—
311C8×Q8—1421no—
312C4×Q16—711no—
313C4×Q16—711no—
314C4×Q16—711no—
315C4×Q16—711no—
316#433—1421no—
317#433—1421no—
318#432—1421no—
319#432C2×D7111yescharacteristic
320#431—711no—
321#431—711no—
322#431—711no—
323#431—1421no—
324#431C2×D7111yescharacteristic
325#335—711no—
326#335—711no—
327#334—711no—
328#334—711no—
329Q8⋊C8—711no—
330Q8⋊C8—711no—
331Q8⋊C8—711no—
332Q8⋊C8—711no—
333Q8⋊C8C2×D7111yescharacteristic
334Q8⋊C8C2×D7111yescharacteristic
335Q8⋊C8C2×D7111yescharacteristic
336Q8⋊C8C2×D7111yescharacteristic
337#322—711no—
338#322—711no—
339#322—711no—
340C82—711no—
341C22×Q8—711no—
342C2×Q16—711no—
343C2×Q16—711no—
344C2×Q16—711no—
345C2×Q16—711no—
346C4⋊Q8C4×D7111yescharacteristic
347C4⋊Q8C4×D7111yescharacteristic
348#125—21no—
349C22⋊Q8—21no—
350C22⋊Q8C4×D7111yescharacteristic
351C22⋊Q8C4×D7111yescharacteristic
352C4×Q8—21no—
353C4×Q8—711no—
354C4×Q8—711no—
355C4×Q8—711no—
356C4×Q8—711no—
357C4×Q8—1421no—
358C4×Q8—1421no—
359C4×Q8C22×D7111yescharacteristic
360C4×Q8C22×D7111yescharacteristic
361#117—21no—
362C2×C4⋊C4—711no—
363C2×C4⋊C4—1421no—
364C2×C42—1421no—
365C2×C42C22×D7111yescharacteristic
366#107—711no—
367#107—711no—
368#107—711no—
369#107—711no—
370C4⋊C8—21no—
371C4⋊C8—711no—
372C4⋊C8—711no—
373C4⋊C8—711no—
374C4⋊C8—711no—
375C4⋊C8—711no—
376C4⋊C8—711no—
377C4⋊C8—1421no—
378C4⋊C8—1421no—
379C4⋊C8—1421no—
380C4⋊C8—1421no—
381C4⋊C8—1421no—
382C4⋊C8—1421no—
383C4⋊C8—1421no—
384C4⋊C8—1421no—
385C4⋊C8C22×D7111yescharacteristic
386C4⋊C8C22×D7111yescharacteristic
387#103—711no—
388#103—711no—
389#103—711no—
390#103—711no—
391#103—711no—
392#103—711no—
393#103—711no—
394#103—711no—
395C22⋊C8—21no—
396C22⋊C8—21no—
397C22⋊C8—711no—
398C22⋊C8—711no—
399C22⋊C8—711no—
400C22⋊C8—711no—
401C22⋊C8—1421no—
402#97—21no—
403#97—1421no—
404#97—1421no—
405#97—1421no—
406#97—1421no—
407C4×C8—711no—
408C4×C8—711no—
409C4×C8—711no—
410C4×C8—711no—
411C4×C8—1421no—
412C4×C8—1421no—
413C4×C8—1421no—
414C4×C8—1421no—
415C4×C8—1421no—
416C4×C8C22×D7111yescharacteristic
417C4×C8C22×D7111yescharacteristic
418C2×Q8—21no—
419C2×Q8—2841no—
420C2×Q8—711no—
421C2×Q8—711no—
422C2×Q8—1422no—
423C2×Q8—1421no—
424C2×Q8C2×C4×D7111yescharacteristic
425C2×Q8C2×C4×D7111yescharacteristic
426C22×C4—1421no—
427C22×C4—1421no—
428C22×C4C2×C4×D7111yescharacteristic
429C22×C4D4×D7111yescharacteristic
430C22×C4#1176586411yescharacteristic
431Q16—2841no—
432Q16—2841no—
433Q16—1422no—
434Q16—1422no—
435C2×C8—21no—
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—711no—
440C2×C8—711no—
441C2×C8—711no—
442C2×C8—711no—
443C2×C8—711no—
444C2×C8—711no—
445C2×C8—711no—
446C2×C8—711no—
447C2×C8—1421no—
448C2×C8—1421no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—711no—
457C4⋊C4—711no—
458C4⋊C4—1421no—
459C4⋊C4—1421no—
460C4⋊C4—1421no—
461C4⋊C4—1421no—
462C4⋊C4—1421no—
463C4⋊C4C2×C4×D7111yescharacteristic
464C4⋊C4C2×C4×D7111yescharacteristic
465C22⋊C4—21no—
466C22⋊C4—21no—
467C42—21no—
468C42—21no—
469C42—1421no—
470C42—1421no—
471C42—1421no—
472C42—1421no—
473C42C2×C4×D7111yescharacteristic
474C42D4×D7111yescharacteristic
475C42#1176586411yescharacteristic
476C42C23×D7111yescharacteristic
477C23(C22⋊D71)⋊C411yescharacteristic
478Q8—21no—
479Q8—21no—
480Q8—21no—
481Q8—21no—
482Q8—42no—
483Q8—2842no—
484Q8—712no—
485Q8—1421no—
486Q8—1421no—
487Q8—1421no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—41no—
497C2×C4—2842no—
498C2×C4—2842no—
499C2×C4—1421no—
500C2×C4—1421no—
501C2×C4—1421no—
502C2×C4—1421no—
503C2×C4(C22⋊D71)⋊C411yescharacteristic
504C2×C4(C22⋊D71)⋊C411yescharacteristic
505C2×C4(C22⋊D71)⋊C411yescharacteristic
506C2×C4C22×C4×D7111yescharacteristic
507C2×C4C2×D4×D7111yescharacteristic
508C2×C4C2×#1176586411yescharacteristic
509C8—41no—
510C8—41no—
511C8—41no—
512C8—41no—
513C8—2841no—
514C8—2841no—
515C8—1421no—
516C8—1421no—
517C8—1421no—
518C8—1421no—
519C8—1421no—
520C8—1421no—
521C8—1421no—
522C8—1421no—
523C22—21no—
524C22—21no—
525C22—21no—
526C22C2×(C22⋊D71)⋊C411yescharacteristic
527C4—21no—
528C4—21no—
529C4—21no—
530C4—21no—
531C4—41no—
532C4—41no—
533C4—41no—
534C4—41no—
535C4—2841no—
536C4—2841no—
537C4—2841no—
538C4—1422no—
539C4#131×D7111yescharacteristic
540C4#1798676411yescharacteristic
541C4#1798681511yescharacteristic
542C4#1798681511yescharacteristic
543C2—41no—
544C2#2259707011yescharacteristic
545C2#2259720411yescharacteristic
546C2#2259723911yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D71
commutatorC2×C4×C71C23×C4
FrattiniC42C23×D71
FittingC71×#1497C2
socleC22×C71C2×C4×D4
prime radicalC23×C71C4×D4
cosocleC42×C71C24
hypercenter#1497D71
hypocenterC71#18032
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1497

Complements

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

NH ≅ G/Ntype
4. #22587246C2semidirect
5. #22587234C2semidirect
6. #22587214C2semidirect
7. #22587208C2semidirect
8. #22587193C2semidirect
9. #22587187C2semidirect
10. #22587156C2semidirect
11. #22587144C2semidirect
289. C71#18032semidirect

Maximal subgroups

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

subgroupindex
#225983752
#225974422
#225974302
#225872462
#225872342
#225872142
#225872082
#225871932
#225871872
#225871562
#225871442
#225871212
#225867362
#225866402
C71×#14972
#1803271