Finite Group Databasebeta

← #32009609

Subgroups — #32009609

Order 18176.

784 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17986299C211yesmaximal, characteristic
2C4×#14556120C211yesmaximal, characteristic, complemented
3#22589297C211yesmaximal, characteristic, complemented
4#22589108C211yesmaximal, characteristic, complemented
5#22589080C211yesmaximal, characteristic, complemented
6#22589075C211yesmaximal, characteristic, complemented
7#22589040C211yesmaximal, characteristic
8#22589040C211yesmaximal, characteristic
9#22588964C211yesmaximal, characteristic
10#22588957C211yesmaximal, characteristic
11#22588915C211yesmaximal, characteristic, complemented
12#22588908C211yesmaximal, characteristic, complemented
13#22588350C211yesmaximal, characteristic
14#22588193C211yesmaximal, characteristic, complemented
15C71×#2293C211yesmaximal, characteristic
16C2×#14556120C2211yescharacteristic
17#17986299C418yescomplemented
18C2×#14556061C2211yescharacteristic
19C2×#14556061C2211yescharacteristic
20C2×#14556059C2211yescharacteristic
21C2×#14556059C2211yescharacteristic
22C2×C4⋊(C71⋊C8)C2211yescharacteristic
23C2×C4⋊(C71⋊C8)C2211yescharacteristic
24C2×#14556049C2211yescharacteristic
25C2×#14556049C2211yescharacteristic
26C2×#14556049C2211yescharacteristic
27C2×#14556049C2211yescharacteristic
28C2×#14556039C2211yescharacteristic
29C2×#14556039C2211yescharacteristic
30#17985897C2211yescharacteristic
31#17985897C2211yescharacteristic
32#17985883—21no—
33#17985883—21no—
34#17985865C2211yescharacteristic
35C4×Dic71⋊C4—21no—
36C4×Dic71⋊C4—21no—
37C4×C4⋊Dic71C2211yescharacteristic
38C4×C4⋊Dic71C2211yescharacteristic
39#17985761C2211yescharacteristic
40#17985761C2211yescharacteristic
41#17985761C2211yescharacteristic
42#17985761C2211yescharacteristic
43#17985761C2211yescharacteristic
44#17985761C2211yescharacteristic
45#17985761C2211yescharacteristic
46#17985761C2211yescharacteristic
47#17985754C2211yescharacteristic
48#17985754C2211yescharacteristic
49C2×C4×D4×C71C2211yescharacteristic
50C2×#117×C71—21no—
51#390×C71C2211yescharacteristic
52#390×C71C2211yescharacteristic
53#389×C71C2211yescharacteristic
54#389×C71C2211yescharacteristic
55#388×C71—21no—
56#388×C71—21no—
57C4×C4⋊C4×C71C2211yescharacteristic
58C4×C22⋊C4×C71—21no—
59C43×C71C2211yescharacteristic
60C22×C71⋊C8C2311yescharacteristic
61C22×C71⋊C8C2311yescharacteristic
62C22×C71⋊C8C2311yescharacteristic
63C22×C71⋊C8C2311yescharacteristic
64#14556120—22no—
65#14556120C2×C414yes—
66C2×C4⋊Dic71C2311yescharacteristic
67C2×C4⋊Dic71C2311yescharacteristic
68C2×C4⋊Dic71C2311yescharacteristic
69C2×C4⋊Dic71C2311yescharacteristic
70C2×Dic71⋊C4—21no—
71C2×Dic71⋊C4—21no—
72C2×Dic71⋊C4—21no—
73C2×Dic71⋊C4—21no—
74C2×C4×Dic71—21no—
75C2×C4×Dic71—21no—
76#14556061—41no—
77#14556061C2×C414yes—
78#14556059—41no—
79#14556059C2×C414yes—
80C4⋊(C71⋊C8)—22no—
81C4⋊(C71⋊C8)C2×C414yes—
82#14556049—41no—
83#14556049—41no—
84#14556049C2×C414yes—
85#14556049C2×C414yes—
86#14556039—41no—
87#14556039—41no—
88C71⋊#95—21no—
89C71⋊#95—21no—
90C71⋊#95—21no—
91C71⋊#95—21no—
92C71⋊#95—21no—
93C71⋊#95—21no—
94C22×D4×C71C2311yescharacteristic
95C23×C4×C71—21no—
96C4×D4×C71—22no—
97C4×D4×C71C2×C414yes—
98#117×C71—44no—
99C2×C4⋊C4×C71—21no—
100C2×C4⋊C4×C71—21no—
101C2×C4⋊C4×C71C2311yescharacteristic
102C2×C4⋊C4×C71C2311yescharacteristic
103C2×C4⋊C4×C71C2311yescharacteristic
104C2×C22⋊C4×C71—21no—
105C2×C22⋊C4×C71—21no—
106C2×C22⋊C4×C71—21no—
107C2×C22⋊C4×C71—21no—
108C2×C22⋊C4×C71—21no—
109C2×C42×C71—21no—
110C2×C42×C71D411yescharacteristic
111C2×C42×C71D411yescharacteristic
112C2×C42×C71D411yescharacteristic
113C2×C42×C71D411yescharacteristic
114C2×C42×C71C2311yescharacteristic
115C2×C42×C71C2311yescharacteristic
116C2×C42×C71C2311yescharacteristic
117#95×C71—21no—
118#95×C71—21no—
119#95×C71—21no—
120C22×Dic71—21no—
121C22×Dic71—21no—
122C22×Dic71—21no—
123C22×Dic71—21no—
124C2×C71⋊C8—21no—
125C2×C71⋊C8—21no—
126C2×C71⋊C8—21no—
127C2×C71⋊C8—21no—
128C2×C71⋊C8—21no—
129C2×C71⋊C8—21no—
130C2×C71⋊C8—21no—
131C2×C71⋊C8—21no—
132C2×C71⋊C8—21no—
133C2×C71⋊C8—21no—
134C2×C71⋊C8C22×C412yes—
135C2×C71⋊C8C22×C412yes—
136C4⋊Dic71—21no—
137C4⋊Dic71—21no—
138C4⋊Dic71—21no—
139C4⋊Dic71—21no—
140C4⋊Dic71—21no—
141C4⋊Dic71—21no—
142C4⋊Dic71C22×C412yes—
143C4⋊Dic71C22×C412yes—
144Dic71⋊C4—24no—
145Dic71⋊C4—24no—
146Dic71⋊C4—42no—
147Dic71⋊C4—42no—
148C4×Dic71—42no—
149C4×Dic71—42no—
150C24×C71—21no—
151C2×D4×C71—21no—
152C2×D4×C71—21no—
153C2×D4×C71—21no—
154C2×D4×C71—21no—
155C2×D4×C71—21no—
156C2×D4×C71C22×C412yes—
157C22×C4×C71—21no—
158C22×C4×C71—24no—
159C22×C4×C71—21no—
160C22×C4×C71—21no—
161C22×C4×C71—42no—
162C22×C4×C71—42no—
163C22×C4×C71C2×D411yescharacteristic
164C22×C4×C71C2×D411yescharacteristic
165C22×C4×C71C2×D411yescharacteristic
166C22×C4×C71C2×D411yescharacteristic
167C22×C4×C71C2×D411yescharacteristic
168C22×C4×C71C2×D411yescharacteristic
169C22×C4×C71C2411yescharacteristic
170C4⋊C4×C71—21no—
171C4⋊C4×C71—21no—
172C4⋊C4×C71—21no—
173C4⋊C4×C71—21no—
174C4⋊C4×C71—21no—
175C4⋊C4×C71—42no—
176C4⋊C4×C71—42no—
177C4⋊C4×C71C22×C412yes—
178C22⋊C4×C71—24no—
179C22⋊C4×C71—42no—
180C22⋊C4×C71—42no—
181C22⋊C4×C71—42no—
182C22⋊C4×C71—42no—
183C42×C71—22no—
184C42×C71—21no—
185C42×C71—21no—
186C42×C71—21no—
187C42×C71—21no—
188C42×C71—41no—
189C42×C71—41no—
190C42×C71—41no—
191C42×C71—42no—
192C42×C71C22×C412yes—
193C42×C71#4112yes—
194C23×C71—22no—
195C23×C71—41no—
196C23×C71—41no—
197C23×C71—41no—
198C23×C71—41no—
199C23×C71—41no—
200C23×C71—41no—
201C23×C71C22×D411yescharacteristic
202D4×C71—22no—
203D4×C71—22no—
204D4×C71—41no—
205D4×C71—41no—
206C2×C4×C71—21no—
207C2×C4×C71—21no—
208C2×C4×C71—21no—
209C2×C4×C71—21no—
210C2×C4×C71—21no—
211C2×C4×C71—21no—
212C2×C4×C71—22no—
213C2×C4×C71—21no—
214C2×C4×C71—21no—
215C2×C4×C71—21no—
216C2×C4×C71—21no—
217C2×C4×C71—21no—
218C2×C4×C71—21no—
219C2×C4×C71—21no—
220C2×C4×C71—21no—
221C2×C4×C71—21no—
222C2×C4×C71—48no—
223C2×C4×C71—41no—
224C2×C4×C71—41no—
225C2×C4×C71—41no—
226C2×C4×C71—41no—
227C2×C4×C71—41no—
228C2×C4×C71—41no—
229C2×C4×C71—41no—
230C2×C4×C71—41no—
231C2×C4×C71C4×D412yes—
232C2×C4×C71C4×D412yes—
233C2×C4×C71AGL(1,ℤ/8)12yes—
234C2×C4×C71C23×C411yescharacteristic
235C2×C4×C71C2×#4111yescharacteristic
236C2×C4×C71H(5,𝔽2)11yescharacteristic
237C2×C4×C712−1+411yescharacteristic
238C2×C4×C712−1+411yescharacteristic
239C2×C4×C712−1+411yescharacteristic
240C2×Dic71—22no—
241C2×Dic71—22no—
242C2×Dic71—41no—
243C2×Dic71—41no—
244C2×Dic71—41no—
245C2×Dic71—41no—
246C2×Dic71—41no—
247C2×Dic71—41no—
248C2×Dic71—41no—
249C2×Dic71—41no—
250C2×Dic71—41no—
251C2×Dic71—41no—
252C71⋊C8—22no—
253C71⋊C8—22no—
254C71⋊C8—41no—
255C71⋊C8—41no—
256C22×C71—42no—
257C22×C71—42no—
258C22×C71—42no—
259C22×C71—81no—
260C22×C71—81no—
261C22×C71—81no—
262C22×C71—81no—
263C22×C71C2×C4×D411yescharacteristic
264C22×C71#53611yescharacteristic
265C22×C71#53711yescharacteristic
266C22×C71C2×AGL(1,ℤ/8)11yescharacteristic
267C22×C71#57511yescharacteristic
268C22×C71#57511yescharacteristic
269C22×C71#57511yescharacteristic
270C4×C71—22no—
271C4×C71—22no—
272C4×C71—22no—
273C4×C71—42no—
274C4×C71—41no—
275C4×C71—41no—
276C4×C71—41no—
277C4×C71—81no—
278C4×C71—81no—
279C4×C71#51911yescharacteristic
280C4×C71#52011yescharacteristic
281C4×C71#54111yescharacteristic
282C4×C71#54211yescharacteristic
283Dic71—42no—
284Dic71—42no—
285Dic71—81no—
286Dic71—81no—
287#22120—711nomaximal, Sylow
288C2×C71—82no—
289C2×C71#232311yescharacteristic
290C2×C71#294311yescharacteristic
291C2×C71C4×AGL(1,ℤ/8)11yescharacteristic
292C2×C71#323411yescharacteristic
293C2×C71#324011yescharacteristic
294C2×C71#325111yescharacteristic
295C2×C71#325611yescharacteristic
296C2×#478—711no—
297C4×#125—711no—
298#2293D7111yescharacteristic
299#1989—711no—
300#1949—711no—
301#1944—711no—
302#1943—711no—
303#1936—711no—
304#1936—711no—
305#1926—711no—
306#1925—711no—
307#1919—711no—
308#1918—711no—
309#1804—711no—
310#1767—711no—
311C71#2212011yescharacteristic, complemented, Sylow
312C2×#125—711no—
313C2×C4×D4C2×D7111yescharacteristic
314C2×#117—21no—
315#478—718no—
316C2×#107—711no—
317C2×#107—711no—
318C2×#106—711no—
319C2×#106—711no—
320C2×C4⋊C8—711no—
321C2×C4⋊C8—711no—
322C2×#102—711no—
323C2×#102—711no—
324C2×#102—711no—
325C2×#102—711no—
326C2×#97—711no—
327C2×#97—711no—
328#390C2×D7111yescharacteristic
329#390C2×D7111yescharacteristic
330#389—711no—
331#389—711no—
332#389C2×D7111yescharacteristic
333#389C2×D7111yescharacteristic
334#388—21no—
335#388—21no—
336#387—1421no—
337#387—1421no—
338#382—711no—
339C4×C4⋊C4—711no—
340C4×C4⋊C4—711no—
341C4×C4⋊C4—1421no—
342C4×C4⋊C4—1421no—
343C4×C4⋊C4C2×D7111yescharacteristic
344C4×C22⋊C4—21no—
345C43C2×D7111yescharacteristic
346#340—711no—
347#340—711no—
348#340—711no—
349#340—711no—
350#340—711no—
351#340—711no—
352#340—711no—
353#340—711no—
354#336—711no—
355#336—711no—
356C22×D4C22×D7111yescharacteristic
357C23×C4—21no—
358C22×C8—711no—
359C22×C8—711no—
360C22×C8—711no—
361C22×C8—711no—
362#125—714no—
363#125—1422no—
364C4×D4—22no—
365C4×D4C4×D7114yes—
366#117—44no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—21no—
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⋊C4C22×D7111yescharacteristic
378C2×C4⋊C4C22×D7111yescharacteristic
379C2×C4⋊C4C22×D7111yescharacteristic
380C2×C22⋊C4—21no—
381C2×C22⋊C4—21no—
382C2×C22⋊C4—21no—
383C2×C22⋊C4—21no—
384C2×C22⋊C4—21no—
385C2×C42—21no—
386C2×C42—1421no—
387C2×C42—1421no—
388C2×C42C22⋊D7111yescharacteristic
389C2×C42C22⋊D7111yescharacteristic
390C2×C42C22⋊D7111yescharacteristic
391C2×C42C22⋊D7111yescharacteristic
392C2×C42C22×D7111yescharacteristic
393C2×C42C22×D7111yescharacteristic
394C2×C42C22×D7111yescharacteristic
395#107—2841no—
396#107—714no—
397#106—2841no—
398#106—714no—
399C4⋊C8—714no—
400C4⋊C8—1422no—
401#102—2841no—
402#102—2841no—
403#102—714no—
404#102—714no—
405#97—2841no—
406#97—2841no—
407#95—21no—
408#95—21no—
409#95—21no—
410#95—1421no—
411#95—1421no—
412#95—1421no—
413#95—1421no—
414#95—1421no—
415#95—1421no—
416C24—21no—
417C2×D4—21no—
418C2×D4—21no—
419C2×D4—21no—
420C2×D4—21no—
421C2×D4—21no—
422C2×D4C2×C4×D7112yes—
423C22×C4—21no—
424C22×C4—24no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—1421no—
430C22×C4—1421no—
431C22×C4—1421no—
432C22×C4—1421no—
433C22×C4C2×C22⋊D7111yescharacteristic
434C22×C4C2×C22⋊D7111yescharacteristic
435C22×C4C2×C22⋊D7111yescharacteristic
436C22×C4C2×C22⋊D7111yescharacteristic
437C22×C4C2×C22⋊D7111yescharacteristic
438C22×C4C2×C22⋊D7111yescharacteristic
439C22×C4C23×D7111yescharacteristic
440C2×C8—712no—
441C2×C8—712no—
442C2×C8—1421no—
443C2×C8—1421no—
444C2×C8—1421no—
445C2×C8—1421no—
446C2×C8—1421no—
447C2×C8—1421no—
448C2×C8—1421no—
449C2×C8—1421no—
450C2×C8—1421no—
451C2×C8—1421no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—42no—
458C4⋊C4—42no—
459C4⋊C4—2842no—
460C4⋊C4—2842no—
461C4⋊C4—712no—
462C4⋊C4—712no—
463C4⋊C4—1421no—
464C4⋊C4—1421no—
465C4⋊C4—1421no—
466C4⋊C4—1421no—
467C4⋊C4—1421no—
468C4⋊C4—1421no—
469C4⋊C4—1424no—
470C4⋊C4—1424no—
471C4⋊C4C2×C4×D7112yes—
472C22⋊C4—24no—
473C22⋊C4—42no—
474C22⋊C4—42no—
475C22⋊C4—42no—
476C22⋊C4—42no—
477C42—21no—
478C42—21no—
479C42—21no—
480C42—21no—
481C42—22no—
482C42—41no—
483C42—41no—
484C42—41no—
485C42—42no—
486C42—2842no—
487C42—2842no—
488C42C2×C4×D7112yes—
489C42#1176586512yes—
490C23—22no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23C22×C22⋊D7111yescharacteristic
498D4—22no—
499D4—22no—
500D4—41no—
501D4—41no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—22no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—48no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—2841no—
528C2×C4—2841no—
529C2×C4—2841no—
530C2×C4—2841no—
531C2×C4—2841no—
532C2×C4—2841no—
533C2×C4—2841no—
534C2×C4—2841no—
535C2×C4—2841no—
536C2×C4—2841no—
537C2×C4—1422no—
538C2×C4—1422no—
539C2×C4C4×C22⋊D7112yes—
540C2×C4C4×C22⋊D7112yes—
541C2×C4#1455615412yes—
542C2×C4C22×C4×D7111yescharacteristic
543C2×C4C2×#1176586511yescharacteristic
544C2×C4#1455617411yescharacteristic
545C2×C4#1455617511yescharacteristic
546C2×C4#1455617511yescharacteristic
547C2×C4#1455617511yescharacteristic
548C8—2841no—
549C8—2841no—
550C8—1422no—
551C8—1422no—
552C22—42no—
553C22—42no—
554C22—42no—
555C22—81no—
556C22—81no—
557C22—81no—
558C22—81no—
559C22C2×C4×C22⋊D7111yescharacteristic
560C22#1798657011yescharacteristic
561C22#1798657711yescharacteristic
562C22C2×#1455615411yescharacteristic
563C22#1798679711yescharacteristic
564C22#1798679711yescharacteristic
565C22#1798679711yescharacteristic
566C4—22no—
567C4—22no—
568C4—22no—
569C4—41no—
570C4—42no—
571C4—41no—
572C4—41no—
573C4—81no—
574C4—81no—
575C4—2842no—
576C4—2842no—
577C4—5681no—
578C4—5681no—
579C4#1798647611yescharacteristic
580C4#1798647911yescharacteristic
581C4#1798661111yescharacteristic
582C4#1798662011yescharacteristic
583C2—82no—
584C2#2259102211yescharacteristic
585C2#2259713811yescharacteristic
586C2C4×#1455615411yescharacteristic
587C2#2259952611yescharacteristic
588C2#2259957611yescharacteristic
589C2#2259969611yescharacteristic
590C2#2259973611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D71
commutatorC2×C4×C71C23×C4
FrattiniC22×C4C23×D71
FittingC71×#2293C2
socleC23×C71C22×D4
prime radicalC22×D4×C71C23
cosocleC22×C4×C71C24
hypercenter#2293D71
hypocenterC71#22120
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C43 ◁ #2293

Complements

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

NH ≅ G/Ntype
2. C4×#14556120C2semidirect
3. #22589297C2semidirect
4. #22589108C2semidirect
5. #22589080C2semidirect
6. #22589075C2semidirect
11. #22588915C2semidirect
12. #22588908C2semidirect
14. #22588193C2semidirect
17. #17986299C4semidirect
311. C71#22120semidirect

Maximal subgroups

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

subgroupindex
C2×#179862992
C4×#145561202
#225892972
#225891082
#225890802
#225890752
#225890402
#225889642
#225889572
#225889152
#225889082
#225883502
#225881932
C71×#22932
#2212071