Finite Group Databasebeta

← #32004709

Subgroups — #32004709

Order 18176.

900 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17986317C211yesmaximal, characteristic
2#22591033C211yesmaximal, characteristic
3#22588967C211yesmaximal, characteristic, complemented
4#22588960C211yesmaximal, characteristic, complemented
5#22588925C211yesmaximal, characteristic, complemented
6#22588918C211yesmaximal, characteristic, complemented
7#22588662C211yesmaximal, characteristic
8#22588539C211yesmaximal, characteristic, complemented
9#22588426C211yesmaximal, characteristic, complemented
10#22588238C211yesmaximal, characteristic, complemented
11C4×#14556059C211yesmaximal, characteristic, complemented
12#22588168C211yesmaximal, characteristic, complemented
13C4×#14556048C211yesmaximal, characteristic, complemented
14#22588106C211yesmaximal, characteristic, complemented
15C71×#2274C211yesmaximal, characteristic, complemented
16C2×#14556102C2211yescharacteristic
17C2×C4×D284C2211yescharacteristic
18#17986317C418yescomplemented
19C2×#14556061C2211yescharacteristic
20C2×#14556059C2211yescharacteristic
21C2×#14556059C2211yescharacteristic
22C2×#14556059C2211yescharacteristic
23C2×#14556048C2211yescharacteristic
24C2×#14556048C2211yescharacteristic
25C2×#14556048C2211yescharacteristic
26C2×#14556048C2211yescharacteristic
27C2×C22⋊(C71⋊C8)C2211yescharacteristic
28C2×C22⋊(C71⋊C8)C2211yescharacteristic
29C2×#14556039C2211yescharacteristic
30C2×C4×C71⋊C8C2211yescharacteristic
31#17985901C2211yescharacteristic
32#17985899—21no—
33#17985897C2211yescharacteristic
34#17985889—21no—
35#17985875—21no—
36#17985870—21no—
37#17985859C2211yescharacteristic
38C4×C4⋊Dic71C2211yescharacteristic
39#17985761C2211yescharacteristic
40#17985761C2211yescharacteristic
41#17985761C2211yescharacteristic
42#17985761C2211yescharacteristic
43#17985761C2211yescharacteristic
44#17985761C2211yescharacteristic
45#17985761C2211yescharacteristic
46#17985761C2211yescharacteristic
47#17985754C2211yescharacteristic
48#17985754C2211yescharacteristic
49C2×#117×C71C2211yescharacteristic
50C2×#117×C71C2211yescharacteristic
51C22×C4⋊C4×C71C2211yescharacteristic
52C4×C4⋊C4×C71—21no—
53C4×C4⋊C4×C71—21no—
54C4×C4⋊C4×C71C2211yescharacteristic
55C4×C4⋊C4×C71C2211yescharacteristic
56C4×C4⋊C4×C71C2211yescharacteristic
57C4×C4⋊C4×C71C2211yescharacteristic
58C4×C22⋊C4×C71—21no—
59C4×C22⋊C4×C71—21no—
60C22×D284C2311yescharacteristic
61C22×C22⋊D71—21no—
62C22×C4×D71—21no—
63C22×C71⋊C8C2311yescharacteristic
64C22×C71⋊C8C2311yescharacteristic
65C22×C71⋊C8C2311yescharacteristic
66C22×C71⋊C8C2311yescharacteristic
67#14556102—22no—
68#14556102C2×C414yes—
69C4×D284—22no—
70C4×D284—22no—
71C2×C4⋊Dic71C2311yescharacteristic
72C2×C4⋊Dic71C2311yescharacteristic
73C2×C4⋊Dic71C2311yescharacteristic
74C2×Dic71⋊C4—21no—
75C2×C22⋊Dic71—21no—
76C2×(C2×D71)⋊C4—21no—
77C2×(C2×D71)⋊C4—21no—
78C2×(C2×D71)⋊C4—21no—
79C2×(C2×D71)⋊C4—21no—
80C2×C4×Dic71—21no—
81#14556061—22no—
82#14556059—22no—
83#14556059C2×C414yescomplemented
84#14556059C2×C414yescomplemented
85#14556048—22no—
86#14556048—22no—
87#14556048C2×C414yes—
88#14556048C2×C414yes—
89C22⋊(C71⋊C8)—22no—
90C22⋊(C71⋊C8)C2×C414yescomplemented
91#14556039—22no—
92C4×C71⋊C8—22no—
93C71⋊#95—21no—
94C71⋊#95—21no—
95C71⋊#95—21no—
96C23×C4×C71—21no—
97C23×C4×C71C2311yescharacteristic
98#117×C71—22no—
99#117×C71—22no—
100#117×C71—22no—
101#117×C71—22no—
102C2×C4⋊C4×C71—21no—
103C2×C4⋊C4×C71—22no—
104C2×C4⋊C4×C71C2×C414yescomplemented
105C2×C4⋊C4×C71D411yescharacteristic
106C2×C4⋊C4×C71D411yescharacteristic
107C2×C4⋊C4×C71C2311yescharacteristic
108C2×C4⋊C4×C71C2311yescharacteristic
109C2×C4⋊C4×C71C2311yescharacteristic
110C2×C4⋊C4×C71C2311yescharacteristic
111C2×C22⋊C4×C71—21no—
112C2×C22⋊C4×C71D411yescharacteristic
113C2×C22⋊C4×C71D411yescharacteristic
114C2×C42×C71—21no—
115C2×C42×C71—21no—
116C2×C42×C71—21no—
117C2×C42×C71—21no—
118C2×C42×C71—21no—
119C2×C42×C71C2311yescharacteristic
120C2×C42×C71C2311yescharacteristic
121#95×C71—21no—
122#95×C71—21no—
123#95×C71—21no—
124#95×C71—21no—
125C23×D71—21no—
126C2×D284—21no—
127C2×D284—21no—
128C2×D284—21no—
129C2×D284—21no—
130C2×D284—21no—
131C2×D284C22×C412yes—
132C2×C22⋊D71—24no—
133C2×C22⋊D71—81no—
134C2×C22⋊D71—81no—
135C22×Dic71—21no—
136C22×Dic71—21no—
137C22×Dic71—21no—
138C2×C4×D71—42no—
139C2×C4×D71—81no—
140C2×C4×D71—81no—
141C2×C71⋊C8—21no—
142C2×C71⋊C8—21no—
143C2×C71⋊C8—21no—
144C2×C71⋊C8—21no—
145C2×C71⋊C8—21no—
146C2×C71⋊C8—21no—
147C2×C71⋊C8—21no—
148C2×C71⋊C8—21no—
149C2×C71⋊C8—21no—
150C2×C71⋊C8—21no—
151C2×C71⋊C8C22×C412yes—
152C2×C71⋊C8C22×C412yes—
153C4⋊Dic71—21no—
154C4⋊Dic71—21no—
155C4⋊Dic71—21no—
156C4⋊Dic71—21no—
157C4⋊Dic71—21no—
158C4⋊Dic71C22×C412yes—
159Dic71⋊C4—81no—
160C22⋊Dic71—81no—
161(C2×D71)⋊C4—24no—
162(C2×D71)⋊C4—42no—
163(C2×D71)⋊C4—81no—
164(C2×D71)⋊C4—81no—
165C4×Dic71—81no—
166C24×C71C2×D411yescharacteristic
167C22×C4×C71—21no—
168C22×C4×C71—21no—
169C22×C4×C71—21no—
170C22×C4×C71—21no—
171C22×C4×C71—21no—
172C22×C4×C71—21no—
173C22×C4×C71—21no—
174C22×C4×C71—21no—
175C22×C4×C71—24no—
176C22×C4×C71—81no—
177C22×C4×C71—81no—
178C22×C4×C71C22×C412yes—
179C22×C4×C71C2×D411yescharacteristic
180C22×C4×C71C2×D411yescharacteristic
181C22×C4×C71C2×D411yescharacteristic
182C22×C4×C71C2×D411yescharacteristic
183C22×C4×C71C2×D411yescharacteristic
184C22×C4×C71#4112yes—
185C22×C4×C71C2411yescharacteristic
186C4⋊C4×C71—24no—
187C4⋊C4×C71—22no—
188C4⋊C4×C71—22no—
189C4⋊C4×C71—21no—
190C4⋊C4×C71—21no—
191C4⋊C4×C71—21no—
192C4⋊C4×C71—21no—
193C4⋊C4×C71C22×C412yes—
194C4⋊C4×C71C22×C412yes—
195C4⋊C4×C71#4112yes—
196C4⋊C4×C71#4112yes—
197C22⋊C4×C71—24no—
198C22⋊C4×C71—22no—
199C22⋊C4×C71—22no—
200C42×C71—24no—
201C42×C71—21no—
202C42×C71—21no—
203C42×C71—81no—
204C42×C71—81no—
205C42×C71—81no—
206C42×C71—81no—
207C42×C71#4112yes—
208C42×C71#4112yes—
209C23×C71—21no—
210C23×C71—21no—
211C23×C71—21no—
212C23×C71—21no—
213C23×C71—21no—
214C23×C71—21no—
215C23×C71C4×D412yes—
216C23×C71C22×D411yescharacteristic
217C22×D71—22no—
218C22×D71—41no—
219C22×D71—41no—
220C22×D71—41no—
221C22×D71—41no—
222C22×D71—41no—
223C22×D71—41no—
224C2×C4×C71—21no—
225C2×C4×C71—21no—
226C2×C4×C71—21no—
227C2×C4×C71—21no—
228C2×C4×C71—21no—
229C2×C4×C71—21no—
230C2×C4×C71—22no—
231C2×C4×C71—22no—
232C2×C4×C71—22no—
233C2×C4×C71—22no—
234C2×C4×C71—22no—
235C2×C4×C71—22no—
236C2×C4×C71—21no—
237C2×C4×C71—21no—
238C2×C4×C71—22no—
239C2×C4×C71—22no—
240C2×C4×C71—21no—
241C2×C4×C71—21no—
242C2×C4×C71—21no—
243C2×C4×C71—21no—
244C2×C4×C71—41no—
245C2×C4×C71—41no—
246C2×C4×C71—41no—
247C2×C4×C71—41no—
248C2×C4×C71—41no—
249C2×C4×C71—41no—
250C2×C4×C71—41no—
251C2×C4×C71—41no—
252C2×C4×C71—84no—
253C2×C4×C71C4×D412yes—
254C2×C4×C71#12312yes—
255C2×C4×C71#12312yes—
256C2×C4×C71C23×C411yescharacteristic
257C2×C4×C71C2×#4111yescharacteristic
258C2×C4×C71C2×#4111yescharacteristic
259C2×C4×C71C2×#4111yescharacteristic
260C2×C4×C71C2×#4111yescharacteristic
261C2×C4×C71C2×#4111yescharacteristic
262C22⋊D71—84no—
263C2×Dic71—22no—
264C2×Dic71—41no—
265C2×Dic71—41no—
266C2×Dic71—41no—
267C2×Dic71—41no—
268C2×Dic71—41no—
269C2×Dic71—41no—
270C2×Dic71—41no—
271C2×Dic71—41no—
272D284—22no—
273D284—22no—
274D284—41no—
275D284—41no—
276C4×D71—84no—
277C71⋊C8—22no—
278C71⋊C8—22no—
279C71⋊C8—41no—
280C71⋊C8—41no—
281C22×C71—22no—
282C22×C71—22no—
283C22×C71—22no—
284C22×C71—41no—
285C22×C71—41no—
286C22×C71—41no—
287C22×C71—41no—
288C22×C71C2×C4×D411yescharacteristic
289C22×C71C2×#12311yescharacteristic
290C22×C71#52511yescharacteristic
291C22×C71#57611yescharacteristic
292C22×C71#57711yescharacteristic
293C22×C71#57711yescharacteristic
294C22×C71#57711yescharacteristic
295C2×D71—42no—
296C2×D71—42no—
297C2×D71—42no—
298C2×D71—81no—
299C2×D71—81no—
300C2×D71—81no—
301C2×D71—81no—
302C4×C71—22no—
303C4×C71—22no—
304C4×C71—22no—
305C4×C71—41no—
306C4×C71—41no—
307C4×C71—42no—
308C4×C71—42no—
309C4×C71—42no—
310C4×C71—42no—
311C4×C71C4×#4111yescharacteristic
312C4×C71C4×#4111yescharacteristic
313C4×C71#52911yescharacteristic
314C4×C71#52911yescharacteristic
315Dic71—42no—
316Dic71—81no—
317Dic71—81no—
318#21584—711nomaximal, Sylow
319C2×C71—42no—
320C2×C71C4×#12311yescharacteristic
321C2×C71#294611yescharacteristic
322C2×C71#294911yescharacteristic
323C2×C71#309611yescharacteristic
324C2×C71#309911yescharacteristic
325C2×C71#311911yescharacteristic
326C2×C71#312511yescharacteristic
327D71—82no—
328C2×#481—711no—
329#2324—711no—
330#2274D7111yescharacteristic, complemented
331#1926—711no—
332#1925—711no—
333#1920—711no—
334#1919—711no—
335#1875—711no—
336#1854—711no—
337#1823—711no—
338#1776—711no—
339C4×#106—711no—
340#1762—711no—
341C4×#102—711no—
342#1750—711no—
343C71#2158411yescharacteristic, complemented, Sylow
344C2×#121—711no—
345C2×C4×D4—711no—
346C2×#117C2×D7111yescharacteristic
347C2×#117C2×D7111yescharacteristic
348C22×C4⋊C4C2×D7111yescharacteristic
349#481—718no—
350C2×#107—711no—
351C2×#106—711no—
352C2×#106—711no—
353C2×#106—711no—
354C2×#102—711no—
355C2×#102—711no—
356C2×#102—711no—
357C2×#102—711no—
358C2×C22⋊C8—711no—
359C2×C22⋊C8—711no—
360C2×#97—711no—
361C2×C4×C8—711no—
362#390—711no—
363#390—1421no—
364#389—711no—
365#388—1421no—
366#386—1421no—
367#385—1421no—
368#380—711no—
369C4×C4⋊C4—21no—
370C4×C4⋊C4—21no—
371C4×C4⋊C4—711no—
372C4×C4⋊C4C2×D7111yescharacteristic
373C4×C4⋊C4C2×D7111yescharacteristic
374C4×C4⋊C4C2×D7111yescharacteristic
375C4×C4⋊C4C2×D7111yescharacteristic
376C4×C22⋊C4—21no—
377C4×C22⋊C4—21no—
378#340—711no—
379#340—711no—
380#340—711no—
381#340—711no—
382#340—711no—
383#340—711no—
384#340—711no—
385#340—711no—
386#336—711no—
387#336—711no—
388C22×D4—711no—
389C22×D4—1421no—
390C23×C4—21no—
391C23×C4—1421no—
392C23×C4C22×D7111yescharacteristic
393C22×C8—711no—
394C22×C8—711no—
395C22×C8—711no—
396C22×C8—711no—
397#121—714no—
398#121—1422no—
399C4×D4—1422no—
400C4×D4—1422no—
401#117—22no—
402#117—22no—
403#117—22no—
404#117—22no—
405C2×C4⋊C4—21no—
406C2×C4⋊C4—22no—
407C2×C4⋊C4—711no—
408C2×C4⋊C4—711no—
409C2×C4⋊C4—711no—
410C2×C4⋊C4—1421no—
411C2×C4⋊C4C4×D7114yescomplemented
412C2×C4⋊C4C22⋊D7111yescharacteristic
413C2×C4⋊C4C22⋊D7111yescharacteristic
414C2×C4⋊C4C22×D7111yescharacteristic
415C2×C4⋊C4C22×D7111yescharacteristic
416C2×C4⋊C4C22×D7111yescharacteristic
417C2×C4⋊C4C22×D7111yescharacteristic
418C2×C22⋊C4—21no—
419C2×C22⋊C4—1421no—
420C2×C22⋊C4—1421no—
421C2×C22⋊C4—1421no—
422C2×C22⋊C4—1421no—
423C2×C22⋊C4—1421no—
424C2×C22⋊C4C22⋊D7111yescharacteristic
425C2×C22⋊C4C22⋊D7111yescharacteristic
426C2×C42—21no—
427C2×C42—21no—
428C2×C42—21no—
429C2×C42—21no—
430C2×C42—21no—
431C2×C42—1421no—
432C2×C42C22×D7111yescharacteristic
433C2×C42C22×D7111yescharacteristic
434#107—1422no—
435#106—714no—
436#106—714no—
437#106—1422no—
438#102—714no—
439#102—714no—
440#102—1422no—
441#102—1422no—
442C22⋊C8—714no—
443C22⋊C8—1422no—
444#97—1422no—
445C4×C8—1422no—
446#95—21no—
447#95—21no—
448#95—21no—
449#95—21no—
450#95—1421no—
451#95—1421no—
452#95—1421no—
453C24—1421no—
454C24C2×C22⋊D7111yescharacteristic
455C2×D4—5681no—
456C2×D4—5681no—
457C2×D4—712no—
458C2×D4—1424no—
459C2×D4—1421no—
460C2×D4—1421no—
461C2×D4—1421no—
462C2×D4—1421no—
463C2×D4—1421no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—24no—
473C22×C4—81no—
474C22×C4—81no—
475C22×C4—2842no—
476C22×C4—5681no—
477C22×C4—5681no—
478C22×C4—1421no—
479C22×C4—1421no—
480C22×C4—1421no—
481C22×C4C2×C4×D7112yes—
482C22×C4C2×C22⋊D7111yescharacteristic
483C22×C4C2×C22⋊D7111yescharacteristic
484C22×C4C2×C22⋊D7111yescharacteristic
485C22×C4C2×C22⋊D7111yescharacteristic
486C22×C4C2×C22⋊D7111yescharacteristic
487C22×C4#1176586512yes—
488C22×C4C23×D7111yescharacteristic
489C2×C8—712no—
490C2×C8—712no—
491C2×C8—1421no—
492C2×C8—1421no—
493C2×C8—1421no—
494C2×C8—1421no—
495C2×C8—1421no—
496C2×C8—1421no—
497C2×C8—1421no—
498C2×C8—1421no—
499C2×C8—1421no—
500C2×C8—1421no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—24no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—5681no—
509C4⋊C4—712no—
510C4⋊C4—1421no—
511C4⋊C4—1421no—
512C4⋊C4—1421no—
513C4⋊C4—1421no—
514C4⋊C4—1421no—
515C4⋊C4C2×C4×D7112yes—
516C4⋊C4C2×C4×D7112yes—
517C4⋊C4#1176586512yes—
518C4⋊C4#1176586512yes—
519C22⋊C4—24no—
520C22⋊C4—22no—
521C22⋊C4—22no—
522C22⋊C4—2842no—
523C22⋊C4—5681no—
524C22⋊C4—5681no—
525C22⋊C4—5681no—
526C22⋊C4—1424no—
527C42—24no—
528C42—21no—
529C42—21no—
530C42—81no—
531C42—81no—
532C42—81no—
533C42—81no—
534C42—5681no—
535C42#1176586512yes—
536C42#1176586512yes—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—21no—
541C23—21no—
542C23—21no—
543C23—2841no—
544C23—2841no—
545C23—2841no—
546C23—2841no—
547C23—2841no—
548C23—2841no—
549C23—1422no—
550C23C4×C22⋊D7112yes—
551C23C22×C22⋊D7111yescharacteristic
552D4—2841no—
553D4—2841no—
554D4—5684no—
555D4—1422no—
556D4—1422no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—84no—
586C2×C4—2841no—
587C2×C4—2841no—
588C2×C4—2841no—
589C2×C4—2841no—
590C2×C4—2841no—
591C2×C4—2841no—
592C2×C4—2841no—
593C2×C4—2841no—
594C2×C4—5684no—
595C2×C4—1422no—
596C2×C4C4×C22⋊D7112yes—
597C2×C4#1455611112yes—
598C2×C4#1455611112yes—
599C2×C4C22×C4×D7111yescharacteristic
600C2×C4C2×#1176586511yescharacteristic
601C2×C4C2×#1176586511yescharacteristic
602C2×C4C2×#1176586511yescharacteristic
603C2×C4C2×#1176586511yescharacteristic
604C2×C4C2×#1176586511yescharacteristic
605C8—2841no—
606C8—2841no—
607C8—1422no—
608C8—1422no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—41no—
613C22—41no—
614C22—41no—
615C22—41no—
616C22—2842no—
617C22—2842no—
618C22—2842no—
619C22—5681no—
620C22—5681no—
621C22—5681no—
622C22—5681no—
623C22C2×C4×C22⋊D7111yescharacteristic
624C22C2×#1455611111yescharacteristic
625C22#1798651011yescharacteristic
626C22#1798680211yescharacteristic
627C22#1798680711yescharacteristic
628C22#1798680711yescharacteristic
629C22#1798681011yescharacteristic
630C4—22no—
631C4—22no—
632C4—22no—
633C4—42no—
634C4—42no—
635C4—42no—
636C4—41no—
637C4—41no—
638C4—42no—
639C4—2842no—
640C4—5681no—
641C4—5681no—
642C4C4×#1176586511yescharacteristic
643C4C4×#1176586511yescharacteristic
644C4#1798653311yescharacteristic
645C4#1798653311yescharacteristic
646C2—42no—
647C2—5682no—
648C2C4×#1455611111yescharacteristic
649C2#2259717211yescharacteristic
650C2#2259719911yescharacteristic
651C2#2259829811yescharacteristic
652C2#2259832911yescharacteristic
653C2#2259853911yescharacteristic
654C2#2259861711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D71
commutatorC2×C4×C71C23×C4
FrattiniC22×C4C23×D71
FittingC71×#2274C2
socleC23×C71C22×D4
prime radicalC2×#14556102C22
cosocleC22×C4×C71C24
hypercenter#2274D71
hypocenterC71#21584
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2274

Complements

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

NH ≅ G/Ntype
3. #22588967C2semidirect
4. #22588960C2semidirect
5. #22588925C2semidirect
6. #22588918C2semidirect
8. #22588539C2semidirect
9. #22588426C2semidirect
10. #22588238C2semidirect
11. C4×#14556059C2semidirect
12. #22588168C2semidirect
13. C4×#14556048C2semidirect
14. #22588106C2semidirect
15. C71×#2274C2semidirect
18. #17986317C4semidirect
83. #14556059C2×C4semidirect
84. #14556059C2×C4semidirect
90. C22⋊(C71⋊C8)C2×C4semidirect
104. C2×C4⋊C4×C71C2×C4semidirect
330. #2274D71semidirect
343. C71#21584semidirect
411. C2×C4⋊C4C4×D71semidirect

Maximal subgroups

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

subgroupindex
C2×#179863172
#225910332
#225889672
#225889602
#225889252
#225889182
#225886622
#225885392
#225884262
#225882382
C4×#145560592
#225881682
C4×#145560482
#225881062
C71×#22742
#2158471