Finite Group Databasebeta

← #31920058

Subgroups — #31920058

Order 18176.

1486 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×(C2×D71)⋊C4C211yesmaximal, characteristic
2#22586601C212yesmaximal, complemented
3#22586588C212yesmaximal
4#22586578C212yesmaximal
5#22586567C211yesmaximal, characteristic, complemented
6#22586567C212yesmaximal, complemented
7#22586566C211yesmaximal, characteristic, complemented
8#22586566C211yesmaximal, characteristic, complemented
9(C2×D71)⋊#95C211yesmaximal, characteristic
10(C2×D71)⋊#95C211yesmaximal, characteristic
11C4×#95×C71C211yesmaximal, characteristic, complemented
12C23×C4×D71C2211yescharacteristic
13C22×(C2×D71)⋊C4C2212yes—
14C22×(C2×D71)⋊C4C2212yes—
15C22×(C2×D71)⋊C4C2211yescharacteristic
16C22×(C2×D71)⋊C4C2211yescharacteristic
17C22×C4×Dic71C2211yescharacteristic
18C4×(C2×D71)⋊C4C418yescomplemented
19C2×C71⋊#95C2211yescharacteristic
20C2×C71⋊#95C2211yescharacteristic
21C2×C71⋊#95C2212yes—
22C2×C71⋊#95C2212yes—
23C2×C71⋊#95C2212yes—
24C2×C71⋊#95C2212yes—
25C2×C71⋊#95C2212yes—
26C2×C71⋊#95C2212yes—
27C2×C71⋊#95C2212yes—
28C2×C71⋊#95C2212yes—
29C2×C71⋊#95C2211yescharacteristic
30C2×C71⋊#95C2211yescharacteristic
31C22×C42×C71C2212yes—
32C22×C42×C71C2211yescharacteristic
33C2×#95×C71C2211yescharacteristic
34C2×#95×C71C2211yescharacteristic
35C2×#95×C71C2212yes—
36C24×D71C2311yescharacteristic
37C23×Dic71C2312yes—
38C23×Dic71C2312yes—
39C23×Dic71C2311yescharacteristic
40C23×Dic71C2311yescharacteristic
41C23×Dic71C2311yescharacteristic
42C22×C4×D71—22no—
43C22×C4×D71—22no—
44C22×C4×D71—42no—
45C22×C4×D71—42no—
46C22×C4×D71C2×C414yes—
47C2×(C2×D71)⋊C4—22no—
48C2×(C2×D71)⋊C4—22no—
49C2×(C2×D71)⋊C4—24no—
50C2×(C2×D71)⋊C4—24no—
51C2×(C2×D71)⋊C4—22no—
52C2×(C2×D71)⋊C4—22no—
53C2×(C2×D71)⋊C4—24no—
54C2×(C2×D71)⋊C4—24no—
55C2×(C2×D71)⋊C4—42no—
56C2×(C2×D71)⋊C4—42no—
57C2×(C2×D71)⋊C4C2×C414yes—
58C2×(C2×D71)⋊C4C2×C414yes—
59C2×C4×Dic71—42no—
60C2×C4×Dic71C2×C414yes—
61C71⋊#95—24no—
62C71⋊#95—24no—
63C71⋊#95—22no—
64C71⋊#95—22no—
65C71⋊#95—24no—
66C71⋊#95—42no—
67C71⋊#95—42no—
68C71⋊#95—42no—
69C71⋊#95—42no—
70C71⋊#95—42no—
71C71⋊#95C2×C414yes—
72C71⋊#95C2×C414yes—
73C23×C4×C71C2311yescharacteristic
74C23×C4×C71C2312yes—
75C23×C4×C71C2312yes—
76C23×C4×C71C2311yescharacteristic
77C23×C4×C71C2311yescharacteristic
78C2×C42×C71—24no—
79C2×C42×C71—24no—
80C2×C42×C71—42no—
81C2×C42×C71—42no—
82C2×C42×C71C2×C414yes—
83#95×C71—24no—
84#95×C71—22no—
85#95×C71—22no—
86C23×D71—21no—
87C23×D71—21no—
88C23×D71—21no—
89C23×D71—22no—
90C23×D71—22no—
91C23×D71—21no—
92C23×D71—22no—
93C23×D71—22no—
94C23×D71C22×C412yes—
95C23×D71#4112yes—
96C23×D71#4112yes—
97C22×Dic71—21no—
98C22×Dic71—21no—
99C22×Dic71—22no—
100C22×Dic71—22no—
101C22×Dic71—21no—
102C22×Dic71—21no—
103C22×Dic71—21no—
104C22×Dic71—21no—
105C22×Dic71—22no—
106C22×Dic71—22no—
107C22×Dic71—22no—
108C22×Dic71—22no—
109C22×Dic71—22no—
110C22×Dic71—22no—
111C22×Dic71—22no—
112C22×Dic71—22no—
113C22×Dic71—22no—
114C22×Dic71—22no—
115C22×Dic71—22no—
116C22×Dic71—22no—
117C22×Dic71—22no—
118C22×Dic71—22no—
119C22×Dic71—22no—
120C22×Dic71—22no—
121C22×Dic71—22no—
122C22×Dic71—22no—
123C22×Dic71C22×C412yes—
124C22×Dic71C22×C412yes—
125C22×Dic71C22×C412yes—
126C2×C4×D71—28no—
127C2×C4×D71—44no—
128C2×C4×D71—44no—
129C2×C4×D71—48no—
130C2×C4×D71—48no—
131(C2×D71)⋊C4—28no—
132(C2×D71)⋊C4—28no—
133(C2×D71)⋊C4—48no—
134(C2×D71)⋊C4—48no—
135C4×Dic71—44no—
136C24×C71C2411yescharacteristic
137C22×C4×C71—22no—
138C22×C4×C71—22no—
139C22×C4×C71—22no—
140C22×C4×C71—22no—
141C22×C4×C71—22no—
142C22×C4×C71—22no—
143C22×C4×C71—22no—
144C22×C4×C71—22no—
145C22×C4×C71—22no—
146C22×C4×C71—22no—
147C22×C4×C71—22no—
148C22×C4×C71—21no—
149C22×C4×C71—21no—
150C22×C4×C71—22no—
151C22×C4×C71—22no—
152C22×C4×C71—22no—
153C22×C4×C71—22no—
154C22×C4×C71—22no—
155C22×C4×C71—22no—
156C22×C4×C71—22no—
157C22×C4×C71C22×C412yes—
158C22×C4×C71C22×C412yes—
159C22×C4×C71C22×C412yes—
160C22×C4×C71#4112yes—
161C22×C4×C71#4114yes—
162C22×C4×C71#4114yes—
163C22×C4×C71#4112yes—
164C22×C4×C71#4112yes—
165C22×C4×C71#4112yes—
166C42×C71—44no—
167C42×C71—48no—
168C23×C71C23×C411yescharacteristic
169C23×C71C2×#4111yescharacteristic
170C23×C71C2×#4111yescharacteristic
171C23×C71C2×#4111yescharacteristic
172C23×C71C2×#4111yescharacteristic
173C23×C71C2×#4111yescharacteristic
174C23×C71C2×#4112yes—
175C23×C71C2×#4112yes—
176C23×C71C2×#4111yescharacteristic
177C23×C71H(5,𝔽2)12yes—
178C23×C71H(5,𝔽2)12yes—
179C22×D71—22no—
180C22×D71—22no—
181C22×D71—22no—
182C22×D71—22no—
183C22×D71—22no—
184C22×D71—22no—
185C22×D71—22no—
186C22×D71—22no—
187C22×D71—22no—
188C22×D71—22no—
189C22×D71—24no—
190C22×D71—24no—
191C22×D71—24no—
192C22×D71—24no—
193C22×D71—42no—
194C22×D71—42no—
195C22×D71—42no—
196C22×D71—42no—
197C22×D71—42no—
198C22×D71—42no—
199C22×D71—42no—
200C22×D71—42no—
201C22×D71#11714yes—
202C2×C4×C71—22no—
203C2×C4×C71—22no—
204C2×C4×C71—22no—
205C2×C4×C71—22no—
206C2×C4×C71—24no—
207C2×C4×C71—24no—
208C2×C4×C71—24no—
209C2×C4×C71—24no—
210C2×C4×C71—22no—
211C2×C4×C71—22no—
212C2×C4×C71—22no—
213C2×C4×C71—24no—
214C2×C4×C71—24no—
215C2×C4×C71—24no—
216C2×C4×C71—24no—
217C2×C4×C71—24no—
218C2×C4×C71—24no—
219C2×C4×C71—42no—
220C2×C4×C71—42no—
221C2×C4×C71—42no—
222C2×C4×C71—42no—
223C2×C4×C71—42no—
224C2×C4×C71—42no—
225C2×C4×C71—42no—
226C2×C4×C71—42no—
227C2×C4×C71—42no—
228C2×C4×C71—42no—
229C2×C4×C71#11714yes—
230C2×C4×C71#11714yes—
231C2×Dic71—22no—
232C2×Dic71—22no—
233C2×Dic71—22no—
234C2×Dic71—22no—
235C2×Dic71—22no—
236C2×Dic71—22no—
237C2×Dic71—22no—
238C2×Dic71—22no—
239C2×Dic71—22no—
240C2×Dic71—42no—
241C2×Dic71—42no—
242C2×Dic71—42no—
243C2×Dic71—42no—
244C2×Dic71—42no—
245C2×Dic71—42no—
246C2×Dic71—42no—
247C2×Dic71—42no—
248C2×Dic71—42no—
249C2×Dic71—42no—
250C2×Dic71—42no—
251C2×Dic71—42no—
252C2×Dic71—42no—
253C2×Dic71—42no—
254C2×Dic71—42no—
255C2×Dic71—42no—
256C2×Dic71—42no—
257C2×Dic71—42no—
258C2×Dic71—42no—
259C2×Dic71—42no—
260C4×D71—416no—
261C22×C71C2×#11711yescharacteristic
262C22×C71C2×#11711yescharacteristic
263C22×C71C2×#11711yescharacteristic
264C22×C71C4×#4111yescharacteristic
265C22×C71C4×#4111yescharacteristic
266C22×C71#51811yescharacteristic
267C22×C71#51811yescharacteristic
268C22×C71#52911yescharacteristic
269C22×C71#52911yescharacteristic
270C22×C71#52912yes—
271C22×C71#52912yes—
272C22×C71#52911yescharacteristic
273C22×C71#52911yescharacteristic
274C22×C71#53812yes—
275C22×C71#53812yes—
276C22×C71#55112yes—
277C22×C71#55112yes—
278C22×C71#55112yes—
279C22×C71#55312yes—
280C22×C71#55312yes—
281C22×C71#55312yes—
282C22×C71#55312yes—
283C22×C71#55912yes—
284C2×D71—24no—
285C2×D71—24no—
286C2×D71—24no—
287C2×D71—42no—
288C2×D71—44no—
289C2×D71—44no—
290C2×D71—44no—
291C2×D71—44no—
292C2×D71—42no—
293C2×D71—42no—
294C2×D71—42no—
295C4×C71—24no—
296C4×C71—24no—
297C4×C71—44no—
298C4×C71—44no—
299C4×C71—42no—
300Dic71—42no—
301Dic71—42no—
302Dic71—42no—
303Dic71—82no—
304Dic71—82no—
305#13947—711nomaximal, Sylow
306C2×C71#228311yescharacteristic
307C2×C71#229811yescharacteristic
308C2×C71#231211yescharacteristic
309C2×C71#235311yescharacteristic
310C2×C71#235311yescharacteristic
311C2×C71#235311yescharacteristic
312C2×C71#235711yescharacteristic
313C2×C71#257612yes—
314C2×C71#257812yes—
315C2×C71#259512yes—
316C2×C71#259712yes—
317D71—44no—
318C2×C4×C22⋊C4—711no—
319#1443—712no—
320#1440—712no—
321#1437—712no—
322#1434—712no—
323#1434—711no—
324#1434—711no—
325#1434—711no—
326C22⋊#95—711no—
327C22⋊#95—711no—
328C4×#95D7111yescharacteristic, complemented
329C71#1394711yescharacteristic, complemented, Sylow
330C24×C4—711no—
331C22×C22⋊C4—712no—
332C22×C22⋊C4—712no—
333C22×C22⋊C4—711no—
334C22×C22⋊C4—711no—
335C22×C42—711no—
336C22×C42C2×D7112yes—
337C22×C42C2×D7111yescharacteristic
338C4×C22⋊C4—718no—
339C2×#95—712no—
340C2×#95—712no—
341C2×#95—712no—
342C2×#95—712no—
343C2×#95—712no—
344C2×#95—711no—
345C2×#95—711no—
346C2×#95—711no—
347C2×#95—711no—
348C2×#95—712no—
349C2×#95—712no—
350C2×#95—712no—
351C2×#95C2×D7111yescharacteristic
352C2×#95C2×D7111yescharacteristic
353C2×#95C2×D7112yes—
354C25—711no—
355C23×C4—2842no—
356C23×C4—2842no—
357C23×C4—714no—
358C23×C4—711no—
359C23×C4—711no—
360C23×C4—711no—
361C23×C4—712no—
362C23×C4—712no—
363C23×C4—1422no—
364C23×C4—1422no—
365C23×C4C22×D7111yescharacteristic
366C23×C4C22×D7112yes—
367C23×C4C22×D7112yes—
368C23×C4C22×D7111yescharacteristic
369C23×C4C22×D7111yescharacteristic
370C2×C22⋊C4—2842no—
371C2×C22⋊C4—2842no—
372C2×C22⋊C4—714no—
373C2×C22⋊C4—714no—
374C2×C22⋊C4—1424no—
375C2×C22⋊C4—1424no—
376C2×C22⋊C4—1424no—
377C2×C22⋊C4—1424no—
378C2×C22⋊C4—1422no—
379C2×C22⋊C4—1422no—
380C2×C22⋊C4—1422no—
381C2×C22⋊C4—1422no—
382C2×C42—24no—
383C2×C42—24no—
384C2×C42—42no—
385C2×C42—42no—
386C2×C42—2842no—
387C2×C42—714no—
388C2×C42C4×D7114yes—
389#95—24no—
390#95—22no—
391#95—22no—
392#95—2842no—
393#95—2842no—
394#95—2842no—
395#95—2842no—
396#95—2842no—
397#95—714no—
398#95—714no—
399#95—1422no—
400#95—1422no—
401#95—1424no—
402#95—1424no—
403#95—1424no—
404C24—712no—
405C24—712no—
406C24—712no—
407C24—1421no—
408C24—1422no—
409C24—1422no—
410C24—1421no—
411C24—1421no—
412C24—1421no—
413C24—1422no—
414C24—1422no—
415C24C23×D7111yescharacteristic
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—2848no—
437C22×C4—2848no—
438C22×C4—2844no—
439C22×C4—2844no—
440C22×C4—712no—
441C22×C4—712no—
442C22×C4—712no—
443C22×C4—1422no—
444C22×C4—1422no—
445C22×C4—1422no—
446C22×C4—1422no—
447C22×C4—1422no—
448C22×C4—1422no—
449C22×C4—1422no—
450C22×C4—1422no—
451C22×C4—1422no—
452C22×C4—1422no—
453C22×C4—1422no—
454C22×C4—1422no—
455C22×C4—1422no—
456C22×C4—1422no—
457C22×C4—1421no—
458C22×C4—1421no—
459C22×C4—1421no—
460C22×C4—1422no—
461C22×C4—1422no—
462C22×C4—1421no—
463C22×C4—1421no—
464C22×C4—1421no—
465C22×C4—1422no—
466C22×C4—1422no—
467C22×C4—1422no—
468C22×C4—1422no—
469C22×C4—1428no—
470C22×C4C2×C4×D7112yes—
471C22×C4C2×C4×D7112yes—
472C22×C4C2×C4×D7112yes—
473C22×C4#1176586412yes—
474C22×C4#1176586514yes—
475C22×C4#1176586514yes—
476C22×C4#1176586512yes—
477C22×C4#1176586512yes—
478C22×C4#1176586612yes—
479C22⋊C4—2848no—
480C22⋊C4—2848no—
481C22⋊C4—1428no—
482C22⋊C4—1428no—
483C42—44no—
484C42—48no—
485C42—2844no—
486C23—2842no—
487C23—2842no—
488C23—2842no—
489C23—2842no—
490C23—2842no—
491C23—2842no—
492C23—2842no—
493C23—2842no—
494C23—714no—
495C23—1424no—
496C23—1424no—
497C23—1424no—
498C23—1424no—
499C23—1422no—
500C23—1422no—
501C23—1422no—
502C23—1422no—
503C23—1422no—
504C23—1422no—
505C23—1422no—
506C23—1422no—
507C23—1422no—
508C23—1422no—
509C23C22×C4×D7111yescharacteristic
510C23#41×D7111yescharacteristic
511C23#41×D7111yescharacteristic
512C23C2×#1176586411yescharacteristic
513C23C2×#1176586511yescharacteristic
514C23C2×#1176586511yescharacteristic
515C23C2×#1176586512yes—
516C23C2×#1176586512yes—
517C23C2×#1176586611yescharacteristic
518C23#1455617312yes—
519C23#1455617412yes—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—28416no—
548C2×C4—2842no—
549C2×C4—2842no—
550C2×C4—2842no—
551C2×C4—2842no—
552C2×C4—2842no—
553C2×C4—2842no—
554C2×C4—2842no—
555C2×C4—2842no—
556C2×C4—2842no—
557C2×C4—2842no—
558C2×C4—2842no—
559C2×C4—2842no—
560C2×C4—2842no—
561C2×C4—2842no—
562C2×C4—2842no—
563C2×C4—2842no—
564C2×C4—2842no—
565C2×C4—2842no—
566C2×C4—2842no—
567C2×C4—2842no—
568C2×C4—1422no—
569C2×C4—1422no—
570C2×C4—1422no—
571C2×C4—1422no—
572C2×C4—1422no—
573C2×C4—1422no—
574C2×C4—1422no—
575C2×C4—1422no—
576C2×C4—1422no—
577C2×C4#1455608514yes—
578C2×C4#1455608614yes—
579C22—2842no—
580C22—2844no—
581C22—2842no—
582C22—2844no—
583C22—2844no—
584C22—2842no—
585C22—2844no—
586C22—2842no—
587C22—1424no—
588C22—1424no—
589C22—1424no—
590C22#117×D7111yescharacteristic
591C22C2×#1455608511yescharacteristic
592C22C2×#1455608611yescharacteristic
593C22C4×#1176586511yescharacteristic
594C22C4×#1176586511yescharacteristic
595C22#1798647211yescharacteristic
596C22#1798647311yescharacteristic
597C22#1798653111yescharacteristic
598C22#1798653111yescharacteristic
599C22#1798653312yes—
600C22#1798653312yes—
601C22#1798653611yescharacteristic
602C22#1798653611yescharacteristic
603C22#1798658212yes—
604C22#1798658212yes—
605C22#1798667312yes—
606C22#1798667512yes—
607C22#1798667812yes—
608C22#1798669312yes—
609C22#1798669512yes—
610C22#1798669712yes—
611C22#1798669912yes—
612C22#1798673412yes—
613C4—24no—
614C4—24no—
615C4—42no—
616C4—44no—
617C4—44no—
618C4—2842no—
619C4—2842no—
620C4—2842no—
621C4—5682no—
622C4—5682no—
623C2—2844no—
624C2#2259073611yescharacteristic
625C2#2259083711yescharacteristic
626C2#2259093511yescharacteristic
627C2#2259127611yescharacteristic
628C2#2259128111yescharacteristic
629C2#2259128311yescharacteristic
630C2#2259131111yescharacteristic
631C2#2259364412yes—
632C2#2259367412yes—
633C2#2259386112yes—
634C2#2259389112yes—
subgroupG/H
center, FrattiniC24C23×D71
commutatorC23×C71C23×C4
FittingC4×#95×C71C2
socle, cosocleC24×C71C24
prime radicalC24×D71C23
hypercenterC4×#95D71
hypocenterC71#13947
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C24 ◁ C4×#95

Complements

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

NH ≅ G/Ntype
2. #22586601C2semidirect
5. #22586567C2semidirect
6. #22586567C2semidirect
7. #22586566C2semidirect
8. #22586566C2semidirect
11. C4×#95×C71C2semidirect
18. C4×(C2×D71)⋊C4C4semidirect
328. C4×#95D71semidirect
329. C71#13947semidirect

Maximal subgroups

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

subgroupindex
C2×C4×(C2×D71)⋊C42
#225866012
#225865882
#225865782
#225865672
#225865662
(C2×D71)⋊#952
C4×#95×C712
#1394771