Finite Group Databasebeta

← #32007005

Subgroups — #32007005

Order 18176.

970 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17986627C211yesmaximal, characteristic
2#22590949C211yesmaximal, characteristic
3#22589313C211yesmaximal, characteristic, complemented
4#22589309C211yesmaximal, characteristic, complemented
5#22589305C211yesmaximal, characteristic, complemented
6#22589301C211yesmaximal, characteristic, complemented
7#22589295C211yesmaximal, characteristic, complemented
8#22589295C211yesmaximal, characteristic, complemented
9#22589291C211yesmaximal, characteristic, complemented
10#22589287C211yesmaximal, characteristic, complemented
11#22588436C211yesmaximal, characteristic
12#22588436C211yesmaximal, characteristic
13#22588431C211yesmaximal, characteristic
14#22588427C211yesmaximal, characteristic
15C2×#433×C71C211yesmaximal, characteristic
16#17986627—24no—
17#17986627C418yescomplemented
18C2×#14556128C2211yescharacteristic
19C2×#14556128C2211yescharacteristic
20C2×#14556127—21no—
21C2×#14556120C2211yescharacteristic
22C2×#14556120C2211yescharacteristic
23C2×#14556119—21no—
24C2×#14556105—21no—
25C2×#14556105—21no—
26C2×#14556087C2211yescharacteristic
27C22×C4⋊Dic71C2211yescharacteristic
28#17985892—21no—
29#17985892—21no—
30#17985885—21no—
31#17985885—21no—
32#17985869C2211yescharacteristic
33#17985865C2211yescharacteristic
34#17985859C2211yescharacteristic
35#17985859C2211yescharacteristic
36C4×C4⋊Dic71C2211yescharacteristic
37C4×C4⋊Dic71C2211yescharacteristic
38#17985762C2211yescharacteristic
39#17985762C2211yescharacteristic
40#17985762C2211yescharacteristic
41#17985762C2211yescharacteristic
42#17985762C2211yescharacteristic
43#17985762C2211yescharacteristic
44#17985762C2211yescharacteristic
45#17985762C2211yescharacteristic
46#17985762C2211yescharacteristic
47#17985762C2211yescharacteristic
48#17985762C2211yescharacteristic
49#17985762C2211yescharacteristic
50#17985762C2211yescharacteristic
51#17985762C2211yescharacteristic
52#17985762C2211yescharacteristic
53#17985762C2211yescharacteristic
54C2×#117×C71C2211yescharacteristic
55#433×C71—24no—
56C2×C4⋊C8×C71C2211yescharacteristic
57C2×C4⋊C8×C71C2211yescharacteristic
58C2×C22⋊C8×C71C2211yescharacteristic
59C2×C22⋊C8×C71C2211yescharacteristic
60C2×#97×C71C2211yescharacteristic
61C2×C4×C8×C71C2211yescharacteristic
62C22×Dic142—21no—
63C22×Dic142—21no—
64C23×Dic71—21no—
65#14556128—22no—
66#14556128—22no—
67#14556128C2×C414yes—
68#14556128C2×C414yes—
69#14556127—28no—
70#14556120—22no—
71#14556120—22no—
72#14556120C2×C414yes—
73#14556120C2×C414yes—
74#14556119—28no—
75#14556105—28no—
76#14556105—28no—
77#14556087—22no—
78#14556087C2×C414yes—
79C2×C4⋊Dic71—22no—
80C2×C4⋊Dic71—21no—
81C2×C4⋊Dic71—21no—
82C2×C4⋊Dic71—21no—
83C2×C4⋊Dic71—21no—
84C2×C4⋊Dic71—21no—
85C2×C4⋊Dic71C2×C414yes—
86C2×C4⋊Dic71C2311yescharacteristic
87C2×C4⋊Dic71C2311yescharacteristic
88C2×C4⋊Dic71C2311yescharacteristic
89C2×C4⋊Dic71C2311yescharacteristic
90C2×C4⋊Dic71C2311yescharacteristic
91C2×C4⋊Dic71C2311yescharacteristic
92C2×C4⋊Dic71C2311yescharacteristic
93C2×C4⋊Dic71C2311yescharacteristic
94C2×Dic71⋊C4—21no—
95C2×Dic71⋊C4—21no—
96C2×Dic71⋊C4—21no—
97C2×Dic71⋊C4—21no—
98C2×C22⋊Dic71—21no—
99C2×C22⋊Dic71—21no—
100C2×C22⋊Dic71—21no—
101C2×C4×Dic71—21no—
102C2×C4×Dic71—21no—
103C2×C4×Dic71—21no—
104C71⋊#95—21no—
105C71⋊#95—21no—
106C71⋊#95—21no—
107C71⋊#95—21no—
108C71⋊#95—21no—
109C71⋊#95—21no—
110C23×C4×C71C2311yescharacteristic
111C22×C8×C71C2311yescharacteristic
112C22×C8×C71C2311yescharacteristic
113C22×C8×C71C2311yescharacteristic
114C22×C8×C71C2311yescharacteristic
115#117×C71C2×C414yes—
116#117×C71D414yes—
117C2×C4⋊C4×C71D411yescharacteristic
118C2×C4⋊C4×C71D411yescharacteristic
119C2×C22⋊C4×C71D411yescharacteristic
120C2×C22⋊C4×C71D411yescharacteristic
121C2×C42×C71C2311yescharacteristic
122C2×C42×C71C2311yescharacteristic
123C4⋊C8×C71—22no—
124C4⋊C8×C71—22no—
125C22⋊C8×C71—22no—
126C22⋊C8×C71—22no—
127#97×C71—22no—
128C4×C8×C71—22no—
129C2×Dic142—24no—
130C2×Dic142—24no—
131C2×Dic142—81no—
132C2×Dic142—81no—
133C2×Dic142—81no—
134C2×Dic142—81no—
135C22×Dic71—21no—
136C22×Dic71—21no—
137C22×Dic71—21no—
138C22×Dic71—21no—
139C22×Dic71—24no—
140C22×Dic71—21no—
141C22×Dic71—21no—
142C22×Dic71—21no—
143C22×Dic71—21no—
144C22×Dic71—81no—
145C22×Dic71—81no—
146C4⋊Dic71—24no—
147C4⋊Dic71—24no—
148C4⋊Dic71—24no—
149C4⋊Dic71—21no—
150C4⋊Dic71—21no—
151C4⋊Dic71—21no—
152C4⋊Dic71—21no—
153C4⋊Dic71—21no—
154C4⋊Dic71—21no—
155C4⋊Dic71—21no—
156C4⋊Dic71—21no—
157C4⋊Dic71—21no—
158C4⋊Dic71—21no—
159C4⋊Dic71—21no—
160C4⋊Dic71—21no—
161C4⋊Dic71—81no—
162C4⋊Dic71—81no—
163C4⋊Dic71C22×C412yes—
164C4⋊Dic71C22×C412yes—
165C4⋊Dic71C22×C412yes—
166C4⋊Dic71C22×C412yes—
167Dic71⋊C4—24no—
168Dic71⋊C4—24no—
169Dic71⋊C4—24no—
170Dic71⋊C4—24no—
171C22⋊Dic71—24no—
172C22⋊Dic71—81no—
173C22⋊Dic71—81no—
174C4×Dic71—24no—
175C4×Dic71—81no—
176C4×Dic71—81no—
177C24×C71C2×D411yescharacteristic
178C22×C4×C71—21no—
179C22×C4×C71—21no—
180C22×C4×C71—21no—
181C22×C4×C71—21no—
182C22×C4×C71C22×C412yes—
183C22×C4×C71C2×D412yes—
184C22×C4×C71C2×D411yescharacteristic
185C22×C4×C71C2×D411yescharacteristic
186C22×C4×C71C2×D411yescharacteristic
187C22×C4×C71C2×D411yescharacteristic
188C22×C4×C71C2×D411yescharacteristic
189C22×C4×C71C2411yescharacteristic
190C2×C8×C71—21no—
191C2×C8×C71—21no—
192C2×C8×C71—21no—
193C2×C8×C71—21no—
194C2×C8×C71—21no—
195C2×C8×C71—21no—
196C2×C8×C71—21no—
197C2×C8×C71—21no—
198C2×C8×C71—21no—
199C2×C8×C71—21no—
200C2×C8×C71—21no—
201C2×C8×C71—21no—
202C4⋊C4×C71C22⋊C414yes—
203C4⋊C4×C71C22⋊C414yes—
204C22⋊C4×C71C22⋊C414yes—
205C22⋊C4×C71C22⋊C414yes—
206C42×C71C22×C412yes—
207C42×C71C22×C412yes—
208C42×C71C2×D412yes—
209C42×C71C2×D412yes—
210C23×C71—21no—
211C23×C71—21no—
212C23×C71—21no—
213C23×C71—21no—
214C23×C71—21no—
215C23×C71—21no—
216C23×C71C2×C22⋊C412yes—
217C23×C71C22×D411yescharacteristic
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—21no—
225C2×C4×C71—21no—
226C2×C4×C71—21no—
227C2×C4×C71—21no—
228C2×C4×C71—22no—
229C2×C4×C71—22no—
230C2×C4×C71—22no—
231C2×C4×C71—22no—
232C2×C4×C71C2×C22⋊C412yes—
233C2×C4×C71C2×C22⋊C412yes—
234C2×C4×C71C2×C22⋊C412yes—
235C2×C4×C71C2×C22⋊C412yes—
236C2×C4×C71C2×C22⋊C412yes—
237C2×C4×C71C23×C411yescharacteristic
238C2×C4×C71C22×D411yescharacteristic
239C2×C4×C712−1+411yescharacteristic
240C2×C4×C712−1+411yescharacteristic
241C2×C4×C712−1+411yescharacteristic
242C2×C4×C712−1+411yescharacteristic
243C2×Dic71—22no—
244C2×Dic71—22no—
245C2×Dic71—22no—
246C2×Dic71—22no—
247C2×Dic71—41no—
248C2×Dic71—41no—
249C2×Dic71—41no—
250C2×Dic71—41no—
251C2×Dic71—41no—
252C2×Dic71—41no—
253C2×Dic71—41no—
254C2×Dic71—41no—
255C2×Dic71—41no—
256C2×Dic71—41no—
257C2×Dic71—41no—
258C2×Dic71—41no—
259C2×Dic71—41no—
260C2×Dic71—41no—
261C2×Dic71—41no—
262C2×Dic71—41no—
263C2×Dic71—41no—
264C2×Dic71—41no—
265C2×Dic71—41no—
266C2×Dic71—41no—
267C2×Dic71—84no—
268Dic142—84no—
269Dic142—84no—
270C8×C71—41no—
271C8×C71—41no—
272C8×C71—41no—
273C8×C71—41no—
274C22×C71—22no—
275C22×C71—22no—
276C22×C71—22no—
277C22×C71—41no—
278C22×C71—41no—
279C22×C71—41no—
280C22×C71—41no—
281C22×C71C22×C22⋊C411yescharacteristic
282C22×C71#53611yescharacteristic
283C22×C71#53611yescharacteristic
284C22×C71#57611yescharacteristic
285C22×C71#57711yescharacteristic
286C22×C71#57711yescharacteristic
287C22×C71#57811yescharacteristic
288C4×C71—22no—
289C4×C71—22no—
290C4×C71—22no—
291C4×C71—22no—
292C4×C71—22no—
293C4×C71#51911yescharacteristic
294C4×C71#51911yescharacteristic
295C4×C71#53611yescharacteristic
296C4×C71#53611yescharacteristic
297Dic71—42no—
298Dic71—42no—
299Dic71—42no—
300Dic71—42no—
301Dic71—81no—
302Dic71—81no—
303Dic71—81no—
304Dic71—81no—
305#21843—711nomaximal, Sylow
306C2×C71—42no—
307C2×C71#231011yescharacteristic
308C2×C71#289711yescharacteristic
309C2×C71#289811yescharacteristic
310C2×C71#324111yescharacteristic
311C2×C71#324211yescharacteristic
312C2×C71#324311yescharacteristic
313C2×C71#324411yescharacteristic
314C2×#544—711no—
315C2×#433D7111yescharacteristic
316#2314—711no—
317#1993—711no—
318#1992—711no—
319#1991—711no—
320#1990—711no—
321#1989—711no—
322#1989—711no—
323#1988—711no—
324#1987—711no—
325#1825—711no—
326#1825—711no—
327#1824—711no—
328#1823—711no—
329C71#2184311yescharacteristic, complemented, Sylow
330#544—718no—
331#544—1424no—
332C2×C4⋊Q8—711no—
333C2×C4⋊Q8—711no—
334C2×C4⋊Q8—1421no—
335C2×#125—711no—
336C2×#125—711no—
337C2×#125—1421no—
338C2×C22⋊Q8—1421no—
339C2×C22⋊Q8—1421no—
340C2×#117—711no—
341C2×#117C2×D7111yescharacteristic
342C22×C4⋊C4—711no—
343#433—24no—
344C2×C4⋊C8C2×D7111yescharacteristic
345C2×C4⋊C8C2×D7111yescharacteristic
346C2×C22⋊C8C2×D7111yescharacteristic
347C2×C22⋊C8C2×D7111yescharacteristic
348C2×#97C2×D7111yescharacteristic
349C2×C4×C8C2×D7111yescharacteristic
350#388—1421no—
351#388—1421no—
352#387—1421no—
353#387—1421no—
354#384—711no—
355#382—711no—
356#380—711no—
357#380—711no—
358C4×C4⋊C4—711no—
359C4×C4⋊C4—711no—
360#340—711no—
361#340—711no—
362#340—711no—
363#340—711no—
364#340—711no—
365#340—711no—
366#340—711no—
367#340—711no—
368#340—711no—
369#340—711no—
370#340—711no—
371#340—711no—
372#340—711no—
373#340—711no—
374#340—711no—
375#340—711no—
376C22×Q8—1421no—
377C22×Q8—1421no—
378C23×C4—1421no—
379C23×C4C22×D7111yescharacteristic
380C22×C8C22×D7111yescharacteristic
381C22×C8C22×D7111yescharacteristic
382C22×C8C22×D7111yescharacteristic
383C22×C8C22×D7111yescharacteristic
384C4⋊Q8—714no—
385C4⋊Q8—714no—
386C4⋊Q8—1428no—
387C4⋊Q8—1422no—
388C4⋊Q8—1422no—
389#125—714no—
390#125—714no—
391#125—1428no—
392#125—1422no—
393#125—1422no—
394C22⋊Q8—1428no—
395C22⋊Q8—1428no—
396#117—714no—
397#117—1422no—
398#117C4×D7114yes—
399#117C22⋊D7114yes—
400C2×C4⋊C4—714no—
401C2×C4⋊C4—711no—
402C2×C4⋊C4—711no—
403C2×C4⋊C4—711no—
404C2×C4⋊C4—711no—
405C2×C4⋊C4—711no—
406C2×C4⋊C4—711no—
407C2×C4⋊C4—711no—
408C2×C4⋊C4—711no—
409C2×C4⋊C4—1421no—
410C2×C4⋊C4—1421no—
411C2×C4⋊C4—1422no—
412C2×C4⋊C4—1421no—
413C2×C4⋊C4—1421no—
414C2×C4⋊C4—1421no—
415C2×C4⋊C4—1421no—
416C2×C4⋊C4—1421no—
417C2×C4⋊C4—1421no—
418C2×C4⋊C4—1421no—
419C2×C4⋊C4D28411yescharacteristic
420C2×C4⋊C4D28411yescharacteristic
421C2×C22⋊C4—1421no—
422C2×C22⋊C4—1421no—
423C2×C22⋊C4—1421no—
424C2×C22⋊C4D28411yescharacteristic
425C2×C22⋊C4D28411yescharacteristic
426C2×C42—1421no—
427C2×C42—1421no—
428C2×C42—1421no—
429C2×C42C22×D7111yescharacteristic
430C2×C42C22×D7111yescharacteristic
431C4⋊C8—22no—
432C4⋊C8—22no—
433C22⋊C8—22no—
434C22⋊C8—22no—
435#97—22no—
436C4×C8—22no—
437#95—1421no—
438#95—1421no—
439#95—1421no—
440#95—1421no—
441#95—1421no—
442#95—1421no—
443C24C2×D28411yescharacteristic
444C2×Q8—5681no—
445C2×Q8—5681no—
446C2×Q8—5681no—
447C2×Q8—5681no—
448C2×Q8—1424no—
449C2×Q8—1424no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—5681no—
455C22×C4—5681no—
456C22×C4—1421no—
457C22×C4—1421no—
458C22×C4—1421no—
459C22×C4—1421no—
460C22×C4—1421no—
461C22×C4—1421no—
462C22×C4—1421no—
463C22×C4—1421no—
464C22×C4—1424no—
465C22×C4C2×C4×D7112yes—
466C22×C4C2×C22⋊D7112yes—
467C22×C4C2×D28411yescharacteristic
468C22×C4C2×D28411yescharacteristic
469C22×C4C2×D28411yescharacteristic
470C22×C4C2×D28411yescharacteristic
471C22×C4C2×D28411yescharacteristic
472C22×C4C23×D7111yescharacteristic
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—21no—
485C4⋊C4—5681no—
486C4⋊C4—5681no—
487C4⋊C4—712no—
488C4⋊C4—712no—
489C4⋊C4—712no—
490C4⋊C4—712no—
491C4⋊C4—1421no—
492C4⋊C4—1421no—
493C4⋊C4—1421no—
494C4⋊C4—1421no—
495C4⋊C4—1421no—
496C4⋊C4—1421no—
497C4⋊C4—1421no—
498C4⋊C4—1421no—
499C4⋊C4—1421no—
500C4⋊C4—1421no—
501C4⋊C4—1421no—
502C4⋊C4—1421no—
503C4⋊C4—1424no—
504C4⋊C4—1424no—
505C4⋊C4—1424no—
506C4⋊C4—1424no—
507C4⋊C4—1424no—
508C4⋊C4—1424no—
509C4⋊C4—1424no—
510C4⋊C4(C2×D71)⋊C414yes—
511C4⋊C4(C2×D71)⋊C414yes—
512C22⋊C4—5681no—
513C22⋊C4—5681no—
514C22⋊C4—1424no—
515C22⋊C4(C2×D71)⋊C414yes—
516C22⋊C4(C2×D71)⋊C414yes—
517C42—5681no—
518C42—5681no—
519C42—1424no—
520C42C2×C4×D7112yes—
521C42C2×C4×D7112yes—
522C42C2×C22⋊D7112yes—
523C42C2×C22⋊D7112yes—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23C2×(C2×D71)⋊C412yes—
531C23C22×D28411yescharacteristic
532Q8—5684no—
533Q8—5684no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—2841no—
549C2×C4—2841no—
550C2×C4—2841no—
551C2×C4—2841no—
552C2×C4—2841no—
553C2×C4—2841no—
554C2×C4—2841no—
555C2×C4—2841no—
556C2×C4—2841no—
557C2×C4—2841no—
558C2×C4—2841no—
559C2×C4—2841no—
560C2×C4—2841no—
561C2×C4—2841no—
562C2×C4—2841no—
563C2×C4—2841no—
564C2×C4—2841no—
565C2×C4—2841no—
566C2×C4—2841no—
567C2×C4—2841no—
568C2×C4—5684no—
569C2×C4—1422no—
570C2×C4—1422no—
571C2×C4—1422no—
572C2×C4—1422no—
573C2×C4C2×(C2×D71)⋊C412yes—
574C2×C4C2×(C2×D71)⋊C412yes—
575C2×C4C2×(C2×D71)⋊C412yes—
576C2×C4C2×(C2×D71)⋊C412yes—
577C2×C4C2×(C2×D71)⋊C412yes—
578C2×C4C22×C4×D7111yescharacteristic
579C2×C4C22×C22⋊D7111yescharacteristic
580C2×C4#1455617511yescharacteristic
581C2×C4#1455617511yescharacteristic
582C2×C4#1455617511yescharacteristic
583C2×C4#1455617511yescharacteristic
584C8—41no—
585C8—41no—
586C8—41no—
587C8—41no—
588C22—22no—
589C22—22no—
590C22—22no—
591C22—41no—
592C22—41no—
593C22—41no—
594C22—41no—
595C22C22×(C2×D71)⋊C411yescharacteristic
596C22#1798657511yescharacteristic
597C22#1798657511yescharacteristic
598C22#1798680411yescharacteristic
599C22#1798681111yescharacteristic
600C22#1798681111yescharacteristic
601C22#1798681611yescharacteristic
602C4—22no—
603C4—22no—
604C4—22no—
605C4—22no—
606C4—22no—
607C4—2842no—
608C4—2842no—
609C4—2842no—
610C4—2842no—
611C4—5681no—
612C4—5681no—
613C4—5681no—
614C4—5681no—
615C4#1798647611yescharacteristic
616C4#1798647611yescharacteristic
617C4#1798657011yescharacteristic
618C4#1798657011yescharacteristic
619C2—42no—
620C2#2259092411yescharacteristic
621C2#2259686711yescharacteristic
622C2#2259687411yescharacteristic
623C2#2259959511yescharacteristic
624C2#2259961011yescharacteristic
625C2#2259962511yescharacteristic
626C2#2259964011yescharacteristic
subgroupG/H
centerC23C22×D284
commutatorC2×C4×C71C23×C4
FrattiniC22×C4C23×D71
FittingC2×#433×C71C2
socleC23×C71C22×D4
prime radicalC24×C71C2×D4
cosocleC22×C4×C71C24
hypercenterC2×#433D71
hypocenterC71#21843
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#433

Complements

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

NH ≅ G/Ntype
3. #22589313C2semidirect
4. #22589309C2semidirect
5. #22589305C2semidirect
6. #22589301C2semidirect
7. #22589295C2semidirect
8. #22589295C2semidirect
9. #22589291C2semidirect
10. #22589287C2semidirect
17. #17986627C4semidirect
329. C71#21843semidirect

Maximal subgroups

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

subgroupindex
C2×#179866272
#225909492
#225893132
#225893092
#225893052
#225893012
#225892952
#225892912
#225892872
#225884362
#225884312
#225884272
C2×#433×C712
#2184371