Finite Group Databasebeta

← #32015180

Subgroups — #32015180

Order 18176.

782 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17986041C211yesmaximal, characteristic
2#22591223C211yesmaximal, characteristic
3#22591215C211yesmaximal, characteristic
4#22589078C211yesmaximal, characteristic, complemented
5#22589073C211yesmaximal, characteristic, complemented
6#22588933C211yesmaximal, characteristic, complemented
7#22588926C211yesmaximal, characteristic, complemented
8#22588919C211yesmaximal, characteristic, complemented
9#22588912C211yesmaximal, characteristic, complemented
10#22588436C211yesmaximal, characteristic
11#22588427C211yesmaximal, characteristic
12#22588373C211yesmaximal, characteristic
13#22588237C211yesmaximal, characteristic, complemented
14C4×#14556060C211yesmaximal, characteristic, complemented
15C71×#1753C211yesmaximal, characteristic
16C2×#14556087C2211yescharacteristic
17C22×C4⋊Dic71C2211yescharacteristic
18#17986041C418yescomplemented
19C2×#14556062C2211yescharacteristic
20C2×#14556062C2211yescharacteristic
21C2×#14556062C2211yescharacteristic
22C2×#14556060C2211yescharacteristic
23C2×#14556060C2211yescharacteristic
24C2×#14556060C2211yescharacteristic
25#17985897C2211yescharacteristic
26#17985897C2211yescharacteristic
27#17985897C2211yescharacteristic
28#17985897C2211yescharacteristic
29#17985887—21no—
30#17985887—21no—
31#17985882—21no—
32#17985882—21no—
33#17985863—21no—
34#17985859C2211yescharacteristic
35#17985859C2211yescharacteristic
36#17985857—21no—
37C4×C4⋊Dic71C2211yescharacteristic
38C4×C4⋊Dic71C2211yescharacteristic
39C4×C4⋊Dic71C2211yescharacteristic
40C4×C4⋊Dic71C2211yescharacteristic
41C22⋊C4×Dic71—21no—
42#17985844—21no—
43#17985762C2211yescharacteristic
44#17985762C2211yescharacteristic
45#17985762C2211yescharacteristic
46#17985762C2211yescharacteristic
47#17985762C2211yescharacteristic
48#17985762C2211yescharacteristic
49#17985762C2211yescharacteristic
50#17985762C2211yescharacteristic
51#17985762C2211yescharacteristic
52#17985762C2211yescharacteristic
53C22×M16×C71C2211yescharacteristic
54C2×C22⋊C8×C71C2211yescharacteristic
55C2×#97×C71C2211yescharacteristic
56C2×C4×C8×C71C2211yescharacteristic
57C4×C22⋊C4×C71C2211yescharacteristic
58#336×C71C2211yescharacteristic
59#336×C71C2211yescharacteristic
60C23×Dic71—21no—
61#14556087—22no—
62#14556087C2×C414yes—
63C2×C4⋊Dic71—21no—
64C2×C4⋊Dic71—22no—
65C2×C4⋊Dic71C2×C414yes—
66C2×C4⋊Dic71C2311yescharacteristic
67C2×C4⋊Dic71C2311yescharacteristic
68C2×C4⋊Dic71C2311yescharacteristic
69C2×C4⋊Dic71C2311yescharacteristic
70C2×C4⋊Dic71C2311yescharacteristic
71C2×C4⋊Dic71C2311yescharacteristic
72C2×C4⋊Dic71C2311yescharacteristic
73C2×C4⋊Dic71C2311yescharacteristic
74C2×Dic71⋊C4—21no—
75C2×Dic71⋊C4—21no—
76C2×Dic71⋊C4—21no—
77C2×Dic71⋊C4—21no—
78C2×C22⋊Dic71—21no—
79C2×C22⋊Dic71—21no—
80C2×C22⋊Dic71—21no—
81C2×C4×Dic71—21no—
82C2×C4×Dic71—21no—
83C2×C4×Dic71—21no—
84C2×C4×Dic71—21no—
85C2×C4×Dic71—21no—
86#14556062—22no—
87#14556062C2×C414yes—
88#14556062C2×C414yes—
89#14556060—22no—
90#14556060C2×C414yes—
91#14556060C2×C414yes—
92C71⋊#95—21no—
93C71⋊#95—21no—
94C71⋊#95—21no—
95C71⋊#95—21no—
96C71⋊#95—21no—
97C71⋊#95—21no—
98C71⋊#95—21no—
99C71⋊#95—21no—
100C71⋊#95—21no—
101C71⋊#95—21no—
102C23×C4×C71C2311yescharacteristic
103C2×M16×C71—22no—
104C2×M16×C71—22no—
105C2×M16×C71C2×C414yes—
106C22×C8×C71C2311yescharacteristic
107C22×C8×C71C2311yescharacteristic
108C22×C8×C71C2311yescharacteristic
109C22×C8×C71C2311yescharacteristic
110C2×C22⋊C4×C71D411yescharacteristic
111C2×C22⋊C4×C71D411yescharacteristic
112C2×C42×C71C2311yescharacteristic
113C2×C42×C71C2311yescharacteristic
114C22⋊C8×C71—22no—
115#97×C71—22no—
116C4×C8×C71—22no—
117#95×C71D411yescharacteristic
118#95×C71D411yescharacteristic
119C22×Dic71—21no—
120C22×Dic71—21no—
121C22×Dic71—21no—
122C22×Dic71—21no—
123C22×Dic71—24no—
124C22×Dic71—21no—
125C22×Dic71—21no—
126C22×Dic71—21no—
127C22×Dic71—21no—
128C22×Dic71—81no—
129C22×Dic71—81no—
130C4⋊Dic71—24no—
131C4⋊Dic71—21no—
132C4⋊Dic71—21no—
133C4⋊Dic71—21no—
134C4⋊Dic71—21no—
135C4⋊Dic71—21no—
136C4⋊Dic71—21no—
137C4⋊Dic71—21no—
138C4⋊Dic71—21no—
139C4⋊Dic71—21no—
140C4⋊Dic71—21no—
141C4⋊Dic71—21no—
142C4⋊Dic71—21no—
143C4⋊Dic71C22×C412yes—
144C4⋊Dic71C22×C412yes—
145C4⋊Dic71C22×C412yes—
146C4⋊Dic71C22×C412yes—
147Dic71⋊C4—81no—
148Dic71⋊C4—81no—
149Dic71⋊C4—81no—
150Dic71⋊C4—81no—
151C22⋊Dic71—24no—
152C22⋊Dic71—42no—
153C22⋊Dic71—81no—
154C4×Dic71—24no—
155C4×Dic71—42no—
156C4×Dic71—42no—
157C4×Dic71—81no—
158C4×Dic71—81no—
159C24×C71C2×D411yescharacteristic
160C22×C4×C71—21no—
161C22×C4×C71—21no—
162C22×C4×C71—21no—
163C22×C4×C71—21no—
164C22×C4×C71C22×C412yes—
165C22×C4×C71C2×D411yescharacteristic
166C22×C4×C71C2×D411yescharacteristic
167C22×C4×C71C2×D411yescharacteristic
168C22×C4×C71C2×D411yescharacteristic
169C22×C4×C71C2×D411yescharacteristic
170C22×C4×C71#4112yes—
171C22×C4×C71C2411yescharacteristic
172M16×C71—28no—
173C2×C8×C71—21no—
174C2×C8×C71—21no—
175C2×C8×C71—21no—
176C2×C8×C71—21no—
177C2×C8×C71—21no—
178C2×C8×C71—21no—
179C2×C8×C71—21no—
180C2×C8×C71—21no—
181C2×C8×C71C22×C412yes—
182C2×C8×C71C22×C412yes—
183C2×C8×C71#4112yes—
184C2×C8×C71#4112yes—
185C22⋊C4×C71—22no—
186C22⋊C4×C71—41no—
187C42×C71—21no—
188C42×C71—21no—
189C42×C71—21no—
190C42×C71—21no—
191C23×C71—21no—
192C23×C71—21no—
193C23×C71—21no—
194C23×C71—21no—
195C23×C71—21no—
196C23×C71—21no—
197C23×C71C4×D412yes—
198C23×C71C22×D411yescharacteristic
199C2×C4×C71—21no—
200C2×C4×C71—21no—
201C2×C4×C71—21no—
202C2×C4×C71—21no—
203C2×C4×C71—22no—
204C2×C4×C71—22no—
205C2×C4×C71—22no—
206C2×C4×C71—22no—
207C2×C4×C71—21no—
208C2×C4×C71—21no—
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×C71C4×D412yes—
217C2×C4×C71#13712yes—
218C2×C4×C71C23×C411yescharacteristic
219C2×C4×C71C2×#4111yescharacteristic
220C2×C4×C71C2×#4111yescharacteristic
221C2×C4×C71C2×#4111yescharacteristic
222C2×C4×C712−1+411yescharacteristic
223C2×C4×C712−1+411yescharacteristic
224C2×Dic71—22no—
225C2×Dic71—22no—
226C2×Dic71—22no—
227C2×Dic71—22no—
228C2×Dic71—41no—
229C2×Dic71—41no—
230C2×Dic71—41no—
231C2×Dic71—41no—
232C2×Dic71—41no—
233C2×Dic71—41no—
234C2×Dic71—41no—
235C2×Dic71—41no—
236C2×Dic71—41no—
237C2×Dic71—41no—
238C2×Dic71—41no—
239C2×Dic71—41no—
240C2×Dic71—41no—
241C2×Dic71—41no—
242C2×Dic71—41no—
243C2×Dic71—41no—
244C2×Dic71—41no—
245C2×Dic71—41no—
246C2×Dic71—41no—
247C2×Dic71—41no—
248C2×Dic71—84no—
249C8×C71—22no—
250C8×C71—41no—
251C8×C71—41no—
252C8×C71#11714yes—
253C22×C71—22no—
254C22×C71—22no—
255C22×C71—22no—
256C22×C71—41no—
257C22×C71—41no—
258C22×C71—41no—
259C22×C71—41no—
260C22×C71C2×C4×D411yescharacteristic
261C22×C71#54711yescharacteristic
262C22×C71#54711yescharacteristic
263C22×C71C2×#13711yescharacteristic
264C22×C71#57511yescharacteristic
265C22×C71#57611yescharacteristic
266C22×C71#57711yescharacteristic
267C4×C71—22no—
268C4×C71—22no—
269C4×C71—41no—
270C4×C71—41no—
271C4×C71—41no—
272C4×C71C2×#11711yescharacteristic
273C4×C71#51911yescharacteristic
274C4×C71#55211yescharacteristic
275C4×C71#55211yescharacteristic
276Dic71—42no—
277Dic71—42no—
278Dic71—42no—
279Dic71—42no—
280Dic71—81no—
281Dic71—81no—
282Dic71—81no—
283Dic71—81no—
284#22621—711nomaximal, Sylow
285C2×C71—42no—
286C2×C71#234411yescharacteristic
287C2×C71C4×#13711yescharacteristic
288C2×C71#294611yescharacteristic
289C2×C71#334411yescharacteristic
290C2×C71#334711yescharacteristic
291C2×C71#335111yescharacteristic
292C2×C71#335711yescharacteristic
293C2×#428—711no—
294#2346—711no—
295#2345—711no—
296#1943—711no—
297#1942—711no—
298#1921—711no—
299#1920—711no—
300#1919—711no—
301#1918—711no—
302#1825—711no—
303#1823—711no—
304#1809—711no—
305#1776—711no—
306C4×#106—711no—
307#1753D7111yescharacteristic
308C71#2262111yescharacteristic, complemented, Sylow
309C22×M16C2×D7111yescharacteristic
310C2×#117—711no—
311C22×C4⋊C4—711no—
312#428—718no—
313C2×#107—711no—
314C2×#107—711no—
315C2×#107—711no—
316C2×#106—711no—
317C2×#106—711no—
318C2×#106—711no—
319C2×C22⋊C8C2×D7111yescharacteristic
320C2×#97C2×D7111yescharacteristic
321C2×C4×C8C2×D7111yescharacteristic
322#389—711no—
323#389—711no—
324#389—711no—
325#389—711no—
326#388—1421no—
327#388—1421no—
328#387—1421no—
329#387—1421no—
330#382—1421no—
331#380—711no—
332#380—711no—
333#380—1421no—
334C4×C4⋊C4—711no—
335C4×C4⋊C4—711no—
336C4×C4⋊C4—711no—
337C4×C4⋊C4—711no—
338C4×C22⋊C4—1421no—
339C4×C22⋊C4C2×D7111yescharacteristic
340#376—1421no—
341#340—711no—
342#340—711no—
343#340—711no—
344#340—711no—
345#340—711no—
346#340—711no—
347#340—711no—
348#340—711no—
349#340—711no—
350#340—711no—
351#336C2×D7111yescharacteristic
352#336C2×D7111yescharacteristic
353C23×C4—1421no—
354C23×C4C22×D7111yescharacteristic
355C2×M16—22no—
356C2×M16—22no—
357C2×M16C4×D7114yes—
358C22×C8C22×D7111yescharacteristic
359C22×C8C22×D7111yescharacteristic
360C22×C8C22×D7111yescharacteristic
361C22×C8C22×D7111yescharacteristic
362#117—714no—
363#117—1422no—
364C2×C4⋊C4—714no—
365C2×C4⋊C4—711no—
366C2×C4⋊C4—711no—
367C2×C4⋊C4—711no—
368C2×C4⋊C4—711no—
369C2×C4⋊C4—711no—
370C2×C4⋊C4—711no—
371C2×C4⋊C4—711no—
372C2×C4⋊C4—711no—
373C2×C4⋊C4—1421no—
374C2×C4⋊C4—1421no—
375C2×C4⋊C4—1421no—
376C2×C4⋊C4—1421no—
377C2×C4⋊C4—1421no—
378C2×C4⋊C4—1422no—
379C2×C22⋊C4—1421no—
380C2×C22⋊C4—1421no—
381C2×C22⋊C4—1421no—
382C2×C22⋊C4D28411yescharacteristic
383C2×C22⋊C4D28411yescharacteristic
384C2×C42—1421no—
385C2×C42—1421no—
386C2×C42—1421no—
387C2×C42—1421no—
388C2×C42—1421no—
389C2×C42C22×D7111yescharacteristic
390C2×C42C22×D7111yescharacteristic
391#107—714no—
392#107—714no—
393#107—1422no—
394#106—714no—
395#106—714no—
396#106—1422no—
397C22⋊C8—22no—
398#97—22no—
399C4×C8—22no—
400#95—1421no—
401#95—1421no—
402#95—1421no—
403#95—1421no—
404#95—1421no—
405#95—1421no—
406#95—1421no—
407#95—1421no—
408#95—1421no—
409#95—1421no—
410#95D28411yescharacteristic
411#95D28411yescharacteristic
412C24C2×D28411yescharacteristic
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—5681no—
418C22×C4—5681no—
419C22×C4—1421no—
420C22×C4—1421no—
421C22×C4—1421no—
422C22×C4—1421no—
423C22×C4—1421no—
424C22×C4—1421no—
425C22×C4—1421no—
426C22×C4—1421no—
427C22×C4—1424no—
428C22×C4C2×C4×D7112yes—
429C22×C4C2×D28411yescharacteristic
430C22×C4C2×D28411yescharacteristic
431C22×C4C2×D28411yescharacteristic
432C22×C4C2×D28411yescharacteristic
433C22×C4C2×D28411yescharacteristic
434C22×C4#1176586512yes—
435C22×C4C23×D7111yescharacteristic
436M16—28no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8C2×C4×D7112yes—
446C2×C8C2×C4×D7112yes—
447C2×C8#1176586412yes—
448C2×C8#1176586412yes—
449C4⋊C4—5681no—
450C4⋊C4—5681no—
451C4⋊C4—5681no—
452C4⋊C4—5681no—
453C4⋊C4—712no—
454C4⋊C4—712no—
455C4⋊C4—712no—
456C4⋊C4—712no—
457C4⋊C4—1421no—
458C4⋊C4—1421no—
459C4⋊C4—1421no—
460C4⋊C4—1421no—
461C4⋊C4—1421no—
462C4⋊C4—1421no—
463C4⋊C4—1421no—
464C4⋊C4—1421no—
465C4⋊C4—1421no—
466C4⋊C4—1421no—
467C4⋊C4—1421no—
468C4⋊C4—1421no—
469C4⋊C4—1424no—
470C22⋊C4—22no—
471C22⋊C4—41no—
472C22⋊C4—2842no—
473C22⋊C4—5681no—
474C22⋊C4—1424no—
475C42—21no—
476C42—21no—
477C42—21no—
478C42—21no—
479C42—2842no—
480C42—2842no—
481C42—5681no—
482C42—5681no—
483C42—1424no—
484C23—21no—
485C23—21no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23C4×D28412yes—
491C23C22×D28411yescharacteristic
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—2841no—
510C2×C4—2841no—
511C2×C4—2841no—
512C2×C4—2841no—
513C2×C4—2841no—
514C2×C4—2841no—
515C2×C4—2841no—
516C2×C4—2841no—
517C2×C4—2841no—
518C2×C4—2841no—
519C2×C4—2841no—
520C2×C4—2841no—
521C2×C4—2841no—
522C2×C4—2841no—
523C2×C4—2841no—
524C2×C4—2841no—
525C2×C4—2841no—
526C2×C4—2841no—
527C2×C4—2841no—
528C2×C4—2841no—
529C2×C4—5684no—
530C2×C4—1422no—
531C2×C4—1422no—
532C2×C4—1422no—
533C2×C4—1422no—
534C2×C4C4×D28412yes—
535C2×C4#1455616112yes—
536C2×C4C22×C4×D7111yescharacteristic
537C2×C4C2×#1176586411yescharacteristic
538C2×C4C2×#1176586411yescharacteristic
539C2×C4C2×#1176586511yescharacteristic
540C2×C4#1455617511yescharacteristic
541C2×C4#1455617511yescharacteristic
542C8—22no—
543C8—41no—
544C8—41no—
545C8#1455608414yes—
546C22—22no—
547C22—22no—
548C22—22no—
549C22—41no—
550C22—41no—
551C22—41no—
552C22—41no—
553C22C2×C4×D28411yescharacteristic
554C22#1798665311yescharacteristic
555C22#1798665311yescharacteristic
556C22C2×#1455616111yescharacteristic
557C22#1798679911yescharacteristic
558C22#1798680411yescharacteristic
559C22#1798681111yescharacteristic
560C4—22no—
561C4—22no—
562C4—41no—
563C4—41no—
564C4—41no—
565C4—2842no—
566C4—2842no—
567C4—2842no—
568C4—2842no—
569C4—5681no—
570C4—5681no—
571C4—5681no—
572C4—5681no—
573C4C2×#1455608411yescharacteristic
574C4#1798647611yescharacteristic
575C4#1798668911yescharacteristic
576C4#1798668911yescharacteristic
577C2—42no—
578C2#2259120711yescharacteristic
579C2C4×#1455616111yescharacteristic
580C2#2259717411yescharacteristic
581C2#2260079711yescharacteristic
582C2#2260084211yescharacteristic
583C2#2260090211yescharacteristic
584C2#2260096711yescharacteristic
subgroupG/H
centerC23C22×D284
commutatorC2×C4×C71C23×C4
FrattiniC22×C4C23×D71
FittingC71×#1753C2
socleC23×C71C22×D4
prime radicalC24×C71C2×D4
cosocleC22×C4×C71C24
hypercenter#1753D71
hypocenterC71#22621
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #1753

Complements

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

NH ≅ G/Ntype
4. #22589078C2semidirect
5. #22589073C2semidirect
6. #22588933C2semidirect
7. #22588926C2semidirect
8. #22588919C2semidirect
9. #22588912C2semidirect
13. #22588237C2semidirect
14. C4×#14556060C2semidirect
18. #17986041C4semidirect
308. C71#22621semidirect

Maximal subgroups

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

subgroupindex
C2×#179860412
#225912232
#225912152
#225890782
#225890732
#225889332
#225889262
#225889192
#225889122
#225884362
#225884272
#225883732
#225882372
C4×#145560602
C71×#17532
#2262171