Finite Group Databasebeta

← #32007710

Subgroups — #32007710

Order 18176.

1266 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C22⋊C8×D71C211yesmaximal, characteristic
2C2×#17985856C211yesmaximal, characteristic
3#380×D71C211yesmaximal, characteristic
4#22588535C211yesmaximal, characteristic, complemented
5#22588534C212yesmaximal, complemented
6#22588533C211yesmaximal, characteristic, complemented
7#22588532C211yesmaximal, characteristic, complemented
8#22588531C212yesmaximal, complemented
9#22588364C212yesmaximal, complemented
10#22588362C212yesmaximal, complemented
11C71×#1853C211yesmaximal, characteristic, complemented
12C23×C4×D71C2211yescharacteristic
13C22×C8×D71C2212yes—
14C22×C4⋊Dic71C2211yescharacteristic
15C2×C4⋊C4×D71C2211yescharacteristic
16C2×C4⋊C4×D71C2211yescharacteristic
17C22×(C2×D71)⋊C4—21no—
18C2×C22⋊C4×D71—21no—
19C2×#14556060C2212yescomplemented
20C2×#14556059C2212yescomplemented
21C2×#14556059C2212yescomplemented
22C2×C22⋊(C71⋊C8)C2211yescharacteristic
23C2×(C2×D71)⋊C8C2212yes—
24C22⋊C8×D71C418yescomplemented
25#17985859C2211yescharacteristic
26#17985858—21no—
27#17985857—21no—
28#17985856—21no—
29#17985856—21no—
30#17985856C2212yes—
31#17985856C2211yescharacteristic
32#17985856C2212yes—
33#17985856C2211yescharacteristic
34C2×C71⋊#95—21no—
35#95×D71—21no—
36#17985762C2212yescomplemented
37#17985761C2212yescomplemented
38#17985761C2212yescomplemented
39C22×C4⋊C4×C71C2211yescharacteristic
40C2×#106×C71C2212yescomplemented
41C2×C22⋊C8×C71C2211yescharacteristic
42#380×C71C2211yescharacteristic
43#340×C71C2212yescomplemented
44C24×D71D411yescharacteristic
45C23×Dic71—21no—
46C23×Dic71D411yescharacteristic
47C22×C4×D71—21no—
48C22×C4×D71—21no—
49C22×C4×D71—42no—
50C22×C4×D71—42no—
51C22×C4×D71C2×C414yes—
52C22×C4×D71D411yescharacteristic
53C22×C4×D71D411yescharacteristic
54C22×C4×D71Q814yes—
55C22×C4×D71C2311yescharacteristic
56C22×C4×D71C2311yescharacteristic
57C22×C71⋊C8C2312yes—
58C2×C8×D71—42no—
59C2×C8×D71—42no—
60C2×C8×D71C2×C418yescomplemented
61C2×C4⋊Dic71—21no—
62C2×C4⋊Dic71—42no—
63C2×C4⋊Dic71C2312yes—
64C2×C4⋊Dic71C2311yescharacteristic
65C2×C4⋊Dic71C2311yescharacteristic
66C2×Dic71⋊C4—21no—
67C2×Dic71⋊C4—21no—
68C4⋊C4×D71—42no—
69C4⋊C4×D71—42no—
70C2×C22⋊Dic71—21no—
71C2×(C2×D71)⋊C4—22no—
72C2×(C2×D71)⋊C4—22no—
73C2×(C2×D71)⋊C4—21no—
74C2×(C2×D71)⋊C4—21no—
75C2×(C2×D71)⋊C4—44no—
76C22⋊C4×D71—44no—
77#14556060—42no—
78#14556059—42no—
79#14556059—42no—
80C22⋊(C71⋊C8)C2×C414yescomplemented
81(C2×D71)⋊C8C2×C418yescomplemented
82C71⋊#95—21no—
83C71⋊#95—21no—
84C71⋊#95—22no—
85C71⋊#95—22no—
86C71⋊#95—21no—
87C23×C4×C71—21no—
88C23×C4×C71C2311yescharacteristic
89C22×C8×C71C2312yes—
90C2×C4⋊C4×C71—21no—
91C2×C4⋊C4×C71—42no—
92C2×C4⋊C4×C71C2311yescharacteristic
93C2×C4⋊C4×C71C2312yes—
94C2×C4⋊C4×C71C2311yescharacteristic
95C2×C22⋊C4×C71—21no—
96#106×C71—42no—
97C22⋊C8×C71C2×C414yescomplemented
98#95×C71—21no—
99C23×D71—41no—
100C23×D71—41no—
101C23×D71—41no—
102C23×D71—41no—
103C23×D71C4⋊C414yes—
104C23×D71SD1614yes—
105C23×D71SD1614yes—
106C23×D71C2×D411yescharacteristic
107C23×D71C2×D411yescharacteristic
108C22×Dic71—21no—
109C22×Dic71—21no—
110C22×Dic71—22no—
111C22×Dic71—41no—
112C22×Dic71—41no—
113C22×Dic71—42no—
114C22×Dic71—42no—
115C22×Dic71—42no—
116C22×Dic71C4⋊C414yes—
117C22×Dic71C2×D411yescharacteristic
118C22×Dic71C2×D411yescharacteristic
119C2×C4×D71—22no—
120C2×C4×D71—22no—
121C2×C4×D71—22no—
122C2×C4×D71—22no—
123C2×C4×D71—48no—
124C2×C4×D71—48no—
125C2×C4×D71—41no—
126C2×C4×D71—41no—
127C2×C4×D71—41no—
128C2×C4×D71—41no—
129C2×C4×D71—42no—
130C2×C4×D71—42no—
131C2×C4×D71—42no—
132C2×C4×D71—42no—
133C2×C4×D71—42no—
134C2×C4×D71—42no—
135C2×C4×D71C4⋊C414yes—
136C2×C4×D71C4⋊C414yes—
137C2×C4×D71C22×C412yes—
138C2×C4×D71C22×C412yes—
139C2×C4×D71C2×Q812yes—
140C2×C4×D71C2×Q812yes—
141C2×C71⋊C8—22no—
142C2×C71⋊C8—22no—
143C2×C71⋊C8C22×C414yes—
144C8×D71—48no—
145C4⋊Dic71—22no—
146C4⋊Dic71—22no—
147C4⋊Dic71—22no—
148C4⋊Dic71—22no—
149C4⋊Dic71—42no—
150Dic71⋊C4—42no—
151Dic71⋊C4—42no—
152C22⋊Dic71—42no—
153(C2×D71)⋊C4—42no—
154(C2×D71)⋊C4—42no—
155(C2×D71)⋊C4—44no—
156(C2×D71)⋊C4—44no—
157C24×C71C2×D411yescharacteristic
158C22×C4×C71—22no—
159C22×C4×C71—22no—
160C22×C4×C71—21no—
161C22×C4×C71—21no—
162C22×C4×C71—22no—
163C22×C4×C71—42no—
164C22×C4×C71—42no—
165C22×C4×C71—42no—
166C22×C4×C71C22×C412yes—
167C22×C4×C71C2×D411yescharacteristic
168C22×C4×C71C2×Q812yes—
169C22×C4×C71C2411yescharacteristic
170C2×C8×C71—22no—
171C2×C8×C71—22no—
172C2×C8×C71C22×C414yes—
173C4⋊C4×C71—22no—
174C4⋊C4×C71—22no—
175C4⋊C4×C71—22no—
176C4⋊C4×C71—22no—
177C4⋊C4×C71—42no—
178C22⋊C4×C71—42no—
179C23×C71—21no—
180C23×C71—21no—
181C23×C71—21no—
182C23×C71—21no—
183C23×C71C2×C4⋊C412yes—
184C23×C71C2×SD1612yes—
185C23×C71C2×SD1612yes—
186C23×C71C22×D411yescharacteristic
187C22×D71—21no—
188C22×D71—21no—
189C22×D71—21no—
190C22×D71—21no—
191C22×D71—21no—
192C22×D71—21no—
193C22×D71—21no—
194C22×D71—21no—
195C22×D71—44no—
196C22×D71—44no—
197C22×D71—44no—
198C22×D71—44no—
199C22×D71—44no—
200C22×D71—44no—
201C22×D71#106116yescomplemented
202C22×D71C2×C4⋊C412yes—
203C22×D71C2×C4⋊C412yes—
204C22×D71C2×SD1612yes—
205C22×D71C2×SD1612yes—
206C22×D71C2×SD1612yes—
207C22×D71C2×SD1612yes—
208C2×C4×C71—21no—
209C2×C4×C71—21no—
210C2×C4×C71—24no—
211C2×C4×C71—24no—
212C2×C4×C71—42no—
213C2×C4×C71—41no—
214C2×C4×C71—41no—
215C2×C4×C71—41no—
216C2×C4×C71—41no—
217C2×C4×C71—42no—
218C2×C4×C71—42no—
219C2×C4×C71—41no—
220C2×C4×C71—41no—
221C2×C4×C71—84no—
222C2×C4×C71C2×C4⋊C412yes—
223C2×C4×C71C23×C411yescharacteristic
224C2×C4×C71C22×Q811yescharacteristic
225C2×C4×C71H(5,𝔽2)12yes—
226C2×C4×C71H(5,𝔽2)12yes—
227C2×Dic71—21no—
228C2×Dic71—21no—
229C2×Dic71—21no—
230C2×Dic71—21no—
231C2×Dic71—44no—
232C2×Dic71—41no—
233C2×Dic71—41no—
234C2×Dic71—42no—
235C2×Dic71—42no—
236C2×Dic71—41no—
237C2×Dic71—41no—
238C2×Dic71—41no—
239C2×Dic71—41no—
240C2×Dic71—42no—
241C2×Dic71—84no—
242C2×Dic71C2×C4⋊C412yes—
243C2×Dic71C2×C4⋊C412yes—
244C4×D71—24no—
245C4×D71—24no—
246C4×D71—24no—
247C4×D71—24no—
248C4×D71—44no—
249C4×D71—44no—
250C4×D71—84no—
251C4×D71—84no—
252C8×C71—24no—
253C71⋊C8—24no—
254C22×C71—22no—
255C22×C71—22no—
256C22×C71—22no—
257C22×C71—22no—
258C22×C71—22no—
259C22×C71—22no—
260C22×C71C2×#10614yes—
261C22×C71C22×C4⋊C411yescharacteristic
262C22×C71#53411yescharacteristic
263C22×C71#53411yescharacteristic
264C22×C71C22×SD1611yescharacteristic
265C22×C71C22×SD1611yescharacteristic
266C22×C71#57511yescharacteristic
267C22×C71#57511yescharacteristic
268C2×D71—22no—
269C2×D71—22no—
270C2×D71—22no—
271C2×D71—22no—
272C2×D71—22no—
273C2×D71—22no—
274C2×D71—22no—
275C2×D71—22no—
276C2×D71—22no—
277C2×D71—22no—
278C2×D71—22no—
279C2×D71—22no—
280C2×D71—416no—
281C2×D71C2×#10614yes—
282C2×D71C2×#10614yes—
283C4×C71—22no—
284C4×C71—81no—
285C4×C71—81no—
286C4×C71—82no—
287C4×C71#51812yes—
288C4×C71#54312yes—
289Dic71—22no—
290Dic71—22no—
291Dic71—81no—
292Dic71—81no—
293Dic71—82no—
294#21938—711nomaximal, Sylow
295C2×C71—24no—
296C2×C71#231511yescharacteristic
297C2×C71C22×#10611yescharacteristic
298C2×C71#291111yescharacteristic
299C2×C71#318711yescharacteristic
300C2×C71#318711yescharacteristic
301C2×C71#318711yescharacteristic
302C2×C71#318711yescharacteristic
303D71—24no—
304D71—24no—
305C22×C22⋊C8—711no—
306C2×#380—711no—
307C2×#380—711no—
308#1853—712no—
309#1853—711no—
310#1853—711no—
311#1853—712no—
312#1853—711no—
313#1853D7111yescharacteristic, complemented
314#1807—712no—
315#1807—712no—
316C71#2193811yescharacteristic, complemented, Sylow
317C24×C4—711no—
318C23×C8—712no—
319C22×C4⋊C4—711no—
320C22×C4⋊C4—711no—
321C22×C4⋊C4—711no—
322C22×C4⋊C4C2×D7111yescharacteristic
323C22×C22⋊C4—1421no—
324C22×C22⋊C4—1421no—
325C2×#106—712no—
326C2×#106—712no—
327C2×#106—712no—
328C2×#106C2×D7112yescomplemented
329C2×C22⋊C8—711no—
330C2×C22⋊C8—712no—
331C2×C22⋊C8—718no—
332C2×C22⋊C8C2×D7111yescharacteristic
333#380—711no—
334#380—711no—
335#380—711no—
336#380—712no—
337#380—712no—
338#380—1421no—
339#380—1421no—
340#380—1421no—
341#380—1421no—
342#380C2×D7111yescharacteristic
343C2×#95—1421no—
344C2×#95—1421no—
345#340—712no—
346#340—712no—
347#340—712no—
348#340C2×D7112yescomplemented
349C25—711no—
350C23×C4—21no—
351C23×C4—2842no—
352C23×C4—2842no—
353C23×C4—711no—
354C23×C4—714no—
355C23×C4—711no—
356C23×C4—711no—
357C23×C4—714no—
358C23×C4—711no—
359C23×C4—711no—
360C23×C4—1421no—
361C23×C4—1421no—
362C23×C4—1421no—
363C23×C4C22×D7111yescharacteristic
364C22×C8—2842no—
365C22×C8—2842no—
366C22×C8—718no—
367C22×C8—712no—
368C22×C8C22×D7112yes—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—42no—
371C2×C4⋊C4—2842no—
372C2×C4⋊C4—2842no—
373C2×C4⋊C4—2842no—
374C2×C4⋊C4—711no—
375C2×C4⋊C4—712no—
376C2×C4⋊C4—711no—
377C2×C4⋊C4—1421no—
378C2×C4⋊C4—1421no—
379C2×C4⋊C4—1421no—
380C2×C4⋊C4C22×D7111yescharacteristic
381C2×C4⋊C4C22×D7111yescharacteristic
382C2×C4⋊C4C22×D7112yes—
383C2×C22⋊C4—21no—
384C2×C22⋊C4—2844no—
385C2×C22⋊C4—2844no—
386C2×C22⋊C4—1421no—
387C2×C22⋊C4—1421no—
388C2×C22⋊C4—1422no—
389C2×C22⋊C4—1422no—
390C2×C22⋊C4—1421no—
391#106—42no—
392#106—2842no—
393#106—2842no—
394#106—2842no—
395C22⋊C8—714no—
396C22⋊C8—718no—
397C22⋊C8C4×D7114yescomplemented
398#95—21no—
399#95—1421no—
400#95—1421no—
401#95—1422no—
402#95—1422no—
403#95—1421no—
404C24—2841no—
405C24—2841no—
406C24—2841no—
407C24—2841no—
408C24—714no—
409C24—714no—
410C24—714no—
411C24—711no—
412C24—711no—
413C24D4×D7111yescharacteristic
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—2841no—
423C22×C4—2841no—
424C22×C4—2841no—
425C22×C4—2841no—
426C22×C4—2848no—
427C22×C4—2848no—
428C22×C4—2842no—
429C22×C4—2842no—
430C22×C4—2842no—
431C22×C4—2842no—
432C22×C4—2842no—
433C22×C4—2842no—
434C22×C4—2841no—
435C22×C4—2841no—
436C22×C4—2842no—
437C22×C4—2842no—
438C22×C4—2842no—
439C22×C4—714no—
440C22×C4—714no—
441C22×C4—714no—
442C22×C4—711no—
443C22×C4—711no—
444C22×C4—712no—
445C22×C4—712no—
446C22×C4—712no—
447C22×C4—712no—
448C22×C4—1421no—
449C22×C4—1422no—
450C22×C4—1421no—
451C22×C4—1422no—
452C22×C4—1422no—
453C22×C4—1422no—
454C22×C4—1422no—
455C22×C4C2×C4×D7112yes—
456C22×C4D4×D7111yescharacteristic
457C22×C4Q8×D7112yes—
458C22×C4C23×D7111yescharacteristic
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—2848no—
462C2×C8—714no—
463C2×C8—1422no—
464C2×C8—1422no—
465C2×C8C2×C4×D7114yes—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—42no—
471C4⋊C4—2842no—
472C4⋊C4—2842no—
473C4⋊C4—2842no—
474C4⋊C4—1422no—
475C4⋊C4—1422no—
476C4⋊C4—1422no—
477C4⋊C4—1422no—
478C22⋊C4—42no—
479C22⋊C4—2842no—
480C22⋊C4—2842no—
481C22⋊C4—2844no—
482C22⋊C4—2844no—
483C22⋊C4—2842no—
484C23—21no—
485C23—21no—
486C23—21no—
487C23—21no—
488C23—2844no—
489C23—2844no—
490C23—2844no—
491C23—2844no—
492C23—2844no—
493C23—2844no—
494C23—7116no—
495C23—712no—
496C23—712no—
497C23—712no—
498C23—712no—
499C23—712no—
500C23—712no—
501C23—1421no—
502C23—1421no—
503C23—1421no—
504C23—1421no—
505C23—1421no—
506C23—1421no—
507C23—1421no—
508C23—1421no—
509C23C4⋊C4×D7112yes—
510C23SD16×D7112yes—
511C23SD16×D7112yes—
512C23C2×D4×D7111yescharacteristic
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—42no—
525C2×C4—41no—
526C2×C4—84no—
527C2×C4—2844no—
528C2×C4—2844no—
529C2×C4—2842no—
530C2×C4—2841no—
531C2×C4—2841no—
532C2×C4—2842no—
533C2×C4—2841no—
534C2×C4—2841no—
535C2×C4—2841no—
536C2×C4—2842no—
537C2×C4—2841no—
538C2×C4—2844no—
539C2×C4—5684no—
540C2×C4—5684no—
541C2×C4—5684no—
542C2×C4—712no—
543C2×C4—712no—
544C2×C4—1421no—
545C2×C4—1421no—
546C2×C4—1421no—
547C2×C4—1421no—
548C2×C4—1424no—
549C2×C4—1424no—
550C2×C4—1424no—
551C2×C4—1424no—
552C2×C4C4⋊C4×D7112yes—
553C2×C4C22×C4×D7111yescharacteristic
554C2×C4C2×Q8×D7111yescharacteristic
555C2×C4#1455617312yes—
556C2×C4#1455617412yes—
557C8—24no—
558C8—1424no—
559C22—22no—
560C22—22no—
561C22—22no—
562C22—22no—
563C22—22no—
564C22—22no—
565C22—28416no—
566C22—714no—
567C22—714no—
568C22—1422no—
569C22—1422no—
570C22—1422no—
571C22—1422no—
572C22—1422no—
573C22—1422no—
574C22—1422no—
575C22—1422no—
576C22—1422no—
577C22—1422no—
578C22—1422no—
579C22—1422no—
580C22#106×D7114yes—
581C22C2×C4⋊C4×D7111yescharacteristic
582C22#1798655911yescharacteristic
583C22#1798656011yescharacteristic
584C22C2×SD16×D7111yescharacteristic
585C22C2×SD16×D7111yescharacteristic
586C22#1798679511yescharacteristic
587C22#1798679811yescharacteristic
588C4—22no—
589C4—82no—
590C4—81no—
591C4—81no—
592C4—5681no—
593C4—5682no—
594C4—5681no—
595C4—1422no—
596C4—1422no—
597C4#1798647112yes—
598C4#1798662412yes—
599C2—24no—
600C2—1424no—
601C2—1424no—
602C2#2259095111yescharacteristic
603C2C2×#106×D7111yescharacteristic
604C2#2259692711yescharacteristic
605C2#2259909411yescharacteristic
606C2#2259909511yescharacteristic
607C2#2259909611yescharacteristic
608C2#2259909911yescharacteristic
subgroupG/H
centerC23C2×D4×D71
commutatorC2×C4×C71C23×C4
FrattiniC22×C4C23×D71
FittingC71×#1853C2
socleC23×C71C22×D4
prime radicalC24×D71D4
cosocleC22×C4×C71C24
hypercenter#1853D71
hypocenterC71#21938
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1853

Complements

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

NH ≅ G/Ntype
4. #22588535C2semidirect
5. #22588534C2semidirect
6. #22588533C2semidirect
7. #22588532C2semidirect
8. #22588531C2semidirect
9. #22588364C2semidirect
10. #22588362C2semidirect
11. C71×#1853C2semidirect
19. C2×#14556060C22semidirect
20. C2×#14556059C22semidirect
21. C2×#14556059C22semidirect
24. C22⋊C8×D71C4semidirect
36. #17985762C22semidirect
37. #17985761C22semidirect
38. #17985761C22semidirect
40. C2×#106×C71C22semidirect
43. #340×C71C22semidirect
60. C2×C8×D71C2×C4semidirect
80. C22⋊(C71⋊C8)C2×C4semidirect
81. (C2×D71)⋊C8C2×C4semidirect
97. C22⋊C8×C71C2×C4semidirect
201. C22×D71#106semidirect
313. #1853D71semidirect
316. C71#21938semidirect
328. C2×#106C2×D71semidirect
348. #340C2×D71semidirect
397. C22⋊C8C4×D71semidirect

Maximal subgroups

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

subgroupindex
C2×C22⋊C8×D712
C2×#179858562
#380×D712
#225885352
#225885342
#225885332
#225885322
#225885312
#225883642
#225883622
C71×#18532
#2193871