Finite Group Databasebeta

← #31923992

Subgroups — #31923992

Order 18176.

1926 conjugacy classes of subgroups.

The subgroup lattice has 682 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17985873C212yesmaximal
2C2×#17985858C211yesmaximal, characteristic
3C2×#17985856C211yesmaximal, characteristic
4C2×#95×D71C211yesmaximal, characteristic
5#22586605C211yesmaximal, characteristic, complemented
6#22586604C211yesmaximal, characteristic, complemented
7#22586599C211yesmaximal, characteristic, complemented
8#22586594C212yesmaximal, complemented
9#22586591C212yesmaximal, complemented
10(C2×D71)⋊#95C212yesmaximal
11C71×#1442C211yesmaximal, characteristic, complemented
12C23×C4×D71C2211yescharacteristic
13C23×C4×D71C2211yescharacteristic
14C23×C4×D71C2211yescharacteristic
15C22×C4⋊Dic71C2212yes—
16C22×C4⋊Dic71C2211yescharacteristic
17C22×Dic71⋊C4C2212yes—
18C22×Dic71⋊C4C2212yes—
19C22×(C2×D71)⋊C4C2212yes—
20C22×(C2×D71)⋊C4C2212yes—
21#17985873—28no—
22#17985858C418yescomplemented
23#17985856—24no—
24C2×C71⋊#95C2212yes—
25C2×C71⋊#95C2211yescharacteristic
26C2×C71⋊#95C2211yescharacteristic
27C2×C71⋊#95C2211yescharacteristic
28C2×C71⋊#95C2212yes—
29C2×C71⋊#95C2212yes—
30C2×C71⋊#95C2212yes—
31C2×C71⋊#95C2212yes—
32C2×C71⋊#95C2211yescharacteristic
33#95×D71—24no—
34C22×C4⋊C4×C71C2211yescharacteristic
35C2×#95×C71C2212yes—
36C2×#95×C71C2211yescharacteristic
37C2×#95×C71C2212yes—
38C2×#95×C71C2211yescharacteristic
39C24×D71C2311yescharacteristic
40C23×Dic71C2312yes—
41C23×Dic71C2312yes—
42C23×Dic71C2311yescharacteristic
43C23×Dic71C2311yescharacteristic
44C23×Dic71C2311yescharacteristic
45C22×C4×D71—22no—
46C22×C4×D71—22no—
47C22×C4×D71—22no—
48C22×C4×D71—22no—
49C22×C4×D71—42no—
50C22×C4×D71—42no—
51C22×C4×D71—42no—
52C22×C4×D71—42no—
53C22×C4×D71—42no—
54C22×C4×D71—42no—
55C22×C4×D71C2×C414yes—
56C22×C4×D71D414yes—
57C22×C4×D71D414yes—
58C22×C4×D71Q814yes—
59C22×C4×D71Q814yes—
60C2×C4⋊Dic71—24no—
61C2×C4⋊Dic71—22no—
62C2×C4⋊Dic71—42no—
63C2×C4⋊Dic71—42no—
64C2×C4⋊Dic71—42no—
65C2×Dic71⋊C4—24no—
66C2×Dic71⋊C4—42no—
67C2×Dic71⋊C4—42no—
68C2×Dic71⋊C4—42no—
69C2×Dic71⋊C4—42no—
70C2×Dic71⋊C4C2×C418yescomplemented
71C2×(C2×D71)⋊C4—24no—
72C2×(C2×D71)⋊C4—24no—
73C2×(C2×D71)⋊C4—24no—
74C2×(C2×D71)⋊C4—24no—
75C2×(C2×D71)⋊C4—24no—
76C2×(C2×D71)⋊C4C2×C418yes—
77C71⋊#95—22no—
78C71⋊#95—22no—
79C71⋊#95—24no—
80C71⋊#95—24no—
81C71⋊#95—22no—
82C71⋊#95—42no—
83C71⋊#95—42no—
84C71⋊#95—42no—
85C71⋊#95C2×C414yescomplemented
86C23×C4×C71C2312yes—
87C23×C4×C71C2312yes—
88C23×C4×C71C2311yescharacteristic
89C23×C4×C71C2311yescharacteristic
90C23×C4×C71C2311yescharacteristic
91C2×C4⋊C4×C71—22no—
92C2×C4⋊C4×C71—42no—
93#95×C71—22no—
94#95×C71—24no—
95#95×C71—42no—
96#95×C71C2×C414yescomplemented
97C23×D71—22no—
98C23×D71—22no—
99C23×D71—22no—
100C23×D71—22no—
101C23×D71C22×C412yes—
102C23×D71C2×D412yes—
103C23×D71C2×D412yes—
104C23×D71C2×Q812yes—
105C23×D71C2×Q812yes—
106C23×D71#4112yes—
107C23×D71#4112yes—
108C22×Dic71—22no—
109C22×Dic71—22no—
110C22×Dic71—21no—
111C22×Dic71—21no—
112C22×Dic71—22no—
113C22×Dic71—21no—
114C22×Dic71—21no—
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×Dic71—22no—
124C22×Dic71—22no—
125C22×Dic71—22no—
126C22×Dic71—22no—
127C22×Dic71—22no—
128C22×Dic71—22no—
129C22×Dic71—22no—
130C22×Dic71—22no—
131C22×Dic71C22×C414yes—
132C22×Dic71C22×C412yes—
133C22×Dic71C2×D412yes—
134C22×Dic71C2×D412yes—
135C22×Dic71C2×Q812yes—
136C22×Dic71C2×Q812yes—
137C2×C4×D71—28no—
138C2×C4×D71—28no—
139C2×C4×D71—48no—
140C2×C4×D71—44no—
141C2×C4×D71—44no—
142C2×C4×D71—48no—
143C2×C4×D71—48no—
144C2×C4×D71—48no—
145C2×C4×D71—44no—
146C2×C4×D71—44no—
147C2×C4×D71—44no—
148C2×C4×D71—44no—
149C2×C4×D71—48no—
150C2×C4×D71—48no—
151C2×C4×D71C4⋊C4116yescomplemented
152C4⋊Dic71—48no—
153C4⋊Dic71—44no—
154Dic71⋊C4—48no—
155Dic71⋊C4—48no—
156(C2×D71)⋊C4—216no—
157(C2×D71)⋊C4—48no—
158C24×C71C2411yescharacteristic
159C22×C4×C71—22no—
160C22×C4×C71—22no—
161C22×C4×C71—21no—
162C22×C4×C71—21no—
163C22×C4×C71—22no—
164C22×C4×C71—21no—
165C22×C4×C71—21no—
166C22×C4×C71—22no—
167C22×C4×C71—22no—
168C22×C4×C71—22no—
169C22×C4×C71—22no—
170C22×C4×C71—22no—
171C22×C4×C71—22no—
172C22×C4×C71—22no—
173C22×C4×C71—22no—
174C22×C4×C71—22no—
175C22×C4×C71—22no—
176C22×C4×C71—22no—
177C22×C4×C71—22no—
178C22×C4×C71—22no—
179C22×C4×C71—22no—
180C22×C4×C71—22no—
181C22×C4×C71—22no—
182C22×C4×C71C22×C414yes—
183C22×C4×C71C22×C412yes—
184C22×C4×C71C2×D412yes—
185C22×C4×C71C2×D412yes—
186C22×C4×C71C2×Q812yes—
187C22×C4×C71C2×Q812yes—
188C4⋊C4×C71—44no—
189C23×C71C23×C411yescharacteristic
190C23×C71C22×D411yescharacteristic
191C23×C71C22×D411yescharacteristic
192C23×C71C22×Q811yescharacteristic
193C23×C71C22×Q811yescharacteristic
194C23×C71C2×#4111yescharacteristic
195C23×C71C2×#4111yescharacteristic
196C23×C71H(5,𝔽2)12yes—
197C23×C71H(5,𝔽2)12yes—
198C23×C71H(5,𝔽2)12yes—
199C23×C712−1+412yes—
200C22×D71—24no—
201C22×D71—24no—
202C22×D71—22no—
203C22×D71—22no—
204C22×D71—24no—
205C22×D71—22no—
206C22×D71—22no—
207C22×D71—24no—
208C22×D71—24no—
209C22×D71—24no—
210C22×D71—24no—
211C22×D71—24no—
212C22×D71—24no—
213C22×D71—24no—
214C22×D71—24no—
215C22×D71—24no—
216C22×D71C2×C4⋊C414yes—
217C22×D71C4×D414yes—
218C22×D71C4×Q814yes—
219C22×D71#12114yes—
220C22×D71C22⋊Q814yes—
221C22×D71#12514yes—
222C22×D71C4⋊Q814yes—
223C2×C4×C71—24no—
224C2×C4×C71—22no—
225C2×C4×C71—22no—
226C2×C4×C71—24no—
227C2×C4×C71—24no—
228C2×C4×C71—24no—
229C2×C4×C71—24no—
230C2×C4×C71—22no—
231C2×C4×C71—22no—
232C2×C4×C71—24no—
233C2×C4×C71—24no—
234C2×C4×C71—42no—
235C2×C4×C71—41no—
236C2×C4×C71—41no—
237C2×C4×C71—41no—
238C2×C4×C71—41no—
239C2×C4×C71—42no—
240C2×C4×C71—42no—
241C2×C4×C71—42no—
242C2×C4×C71—42no—
243C2×C4×C71—42no—
244C2×C4×C71—42no—
245C2×C4×C71—42no—
246C2×C4×C71—42no—
247C2×C4×C71—42no—
248C2×C4×C71—42no—
249C2×C4×C71—42no—
250C2×C4×C71—42no—
251C2×C4×C71C2×C4⋊C414yes—
252C2×Dic71—24no—
253C2×Dic71—22no—
254C2×Dic71—22no—
255C2×Dic71—24no—
256C2×Dic71—24no—
257C2×Dic71—24no—
258C2×Dic71—24no—
259C2×Dic71—22no—
260C2×Dic71—22no—
261C2×Dic71—24no—
262C2×Dic71—24no—
263C2×Dic71—42no—
264C2×Dic71—41no—
265C2×Dic71—41no—
266C2×Dic71—41no—
267C2×Dic71—41no—
268C2×Dic71—42no—
269C2×Dic71—42no—
270C2×Dic71—42no—
271C2×Dic71—42no—
272C2×Dic71—42no—
273C2×Dic71—42no—
274C2×Dic71—42no—
275C2×Dic71—42no—
276C2×Dic71—42no—
277C2×Dic71—42no—
278C2×Dic71—42no—
279C2×Dic71—42no—
280C2×Dic71C2×C4⋊C414yes—
281C4×D71—416no—
282C4×D71—416no—
283C4×D71—416no—
284C22×C71C22×C4⋊C411yescharacteristic
285C22×C71C2×C4×D411yescharacteristic
286C22×C71C2×C4×Q811yescharacteristic
287C22×C71#51812yes—
288C22×C71#52012yes—
289C22×C71C2×#12111yescharacteristic
290C22×C71C2×C22⋊Q811yescharacteristic
291C22×C71C2×#12511yescharacteristic
292C22×C71C2×C4⋊Q811yescharacteristic
293C22×C71#53411yescharacteristic
294C22×C71#53412yes—
295C22×C71#53411yescharacteristic
296C22×C71#53511yescharacteristic
297C22×C71#53611yescharacteristic
298C22×C71#53712yes—
299C22×C71#53812yes—
300C22×C71#53812yes—
301C22×C71#53912yes—
302C22×C71#53912yes—
303C22×C71#54312yes—
304C22×C71#54312yes—
305C22×C71#54412yes—
306C22×C71#54412yes—
307C2×D71—24no—
308C2×D71—24no—
309C2×D71—28no—
310C2×D71—28no—
311C2×D71—24no—
312C2×D71—24no—
313C2×D71—24no—
314C2×D71—24no—
315C2×D71—28no—
316C2×D71—28no—
317C2×D71#38918yes—
318C4×C71—24no—
319C4×C71—44no—
320C4×C71—42no—
321C4×C71—42no—
322C4×C71—82no—
323Dic71—24no—
324Dic71—44no—
325Dic71—42no—
326Dic71—42no—
327Dic71—82no—
328#14275—711nomaximal, Sylow
329C2×C71C2×#38911yescharacteristic
330C2×C71#231511yescharacteristic
331C2×C71#231711yescharacteristic
332C2×C71#232512yes—
333C2×C71#232912yes—
334C2×C71#261711yescharacteristic
335C2×C71#261811yescharacteristic
336C2×C71#262611yescharacteristic
337C2×C71#262711yescharacteristic
338C2×C71#264612yes—
339C2×C71#265912yes—
340D71—28no—
341C2×#385—712no—
342C2×#380—711no—
343C2×#380—711no—
344C22×#95—711no—
345#1444—711no—
346#1444—711no—
347#1442—711no—
348#1442D7111yescharacteristic, complemented
349#1441—712no—
350#1441—712no—
351C22⋊#95—712no—
352C71#1427511yescharacteristic, complemented, Sylow
353C24×C4—711no—
354C24×C4—711no—
355C24×C4—711no—
356C22×C4⋊C4—712no—
357C22×C4⋊C4—712no—
358C22×C4⋊C4—712no—
359C22×C4⋊C4—711no—
360C22×C4⋊C4C2×D7111yescharacteristic
361C22×C22⋊C4—712no—
362C22×C22⋊C4—712no—
363#385—1428no—
364#380—718no—
365#380—1424no—
366C2×#95—712no—
367C2×#95—712no—
368C2×#95—712no—
369C2×#95—711no—
370C2×#95—711no—
371C2×#95—711no—
372C2×#95—712no—
373C2×#95—712no—
374C2×#95—711no—
375C2×#95—1424no—
376C2×#95C2×D7112yes—
377C2×#95C2×D7111yescharacteristic
378C2×#95C2×D7112yes—
379C2×#95C2×D7111yescharacteristic
380C25—711no—
381C23×C4—2842no—
382C23×C4—2842no—
383C23×C4—2842no—
384C23×C4—2842no—
385C23×C4—2842no—
386C23×C4—2842no—
387C23×C4—714no—
388C23×C4—714no—
389C23×C4—714no—
390C23×C4—714no—
391C23×C4—714no—
392C23×C4—711no—
393C23×C4—712no—
394C23×C4—711no—
395C23×C4—711no—
396C23×C4—712no—
397C23×C4—1422no—
398C23×C4—1422no—
399C23×C4—1422no—
400C23×C4—1422no—
401C23×C4C22×D7112yes—
402C23×C4C22×D7112yes—
403C23×C4C22×D7111yescharacteristic
404C23×C4C22×D7111yescharacteristic
405C23×C4C22×D7111yescharacteristic
406C2×C4⋊C4—22no—
407C2×C4⋊C4—42no—
408C2×C4⋊C4—2842no—
409C2×C4⋊C4—2842no—
410C2×C4⋊C4—2842no—
411C2×C4⋊C4—2842no—
412C2×C4⋊C4—2842no—
413C2×C4⋊C4—2842no—
414C2×C4⋊C4—2842no—
415C2×C4⋊C4—718no—
416C2×C4⋊C4—1424no—
417C2×C4⋊C4—1422no—
418C2×C4⋊C4—1424no—
419C2×C22⋊C4—718no—
420C2×C22⋊C4—1424no—
421C2×C22⋊C4—1424no—
422C2×C22⋊C4—1424no—
423C2×C22⋊C4—1424no—
424C2×C22⋊C4—1424no—
425#95—22no—
426#95—24no—
427#95—42no—
428#95—2842no—
429#95—2842no—
430#95—2842no—
431#95—714no—
432#95—1422no—
433#95—1422no—
434#95—1424no—
435#95—1424no—
436#95—1422no—
437#95C4×D7114yescomplemented
438C24—712no—
439C24—712no—
440C24—712no—
441C24—712no—
442C24—712no—
443C24—712no—
444C24—712no—
445C24—1422no—
446C24—1422no—
447C24—1422no—
448C24—1422no—
449C24C23×D7111yescharacteristic
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—2844no—
474C22×C4—2848no—
475C22×C4—2848no—
476C22×C4—2844no—
477C22×C4—2848no—
478C22×C4—2848no—
479C22×C4—2844no—
480C22×C4—2844no—
481C22×C4—2844no—
482C22×C4—2848no—
483C22×C4—2848no—
484C22×C4—2844no—
485C22×C4—7116no—
486C22×C4—712no—
487C22×C4—712no—
488C22×C4—712no—
489C22×C4—714no—
490C22×C4—712no—
491C22×C4—712no—
492C22×C4—1428no—
493C22×C4—1428no—
494C22×C4—1422no—
495C22×C4—1422no—
496C22×C4—1422no—
497C22×C4—1422no—
498C22×C4—1422no—
499C22×C4—1422no—
500C22×C4—1422no—
501C22×C4—1421no—
502C22×C4—1421no—
503C22×C4—1422no—
504C22×C4—1422no—
505C22×C4—1421no—
506C22×C4—1421no—
507C22×C4—1422no—
508C22×C4—1422no—
509C22×C4—1422no—
510C22×C4—1422no—
511C22×C4—1422no—
512C22×C4—1422no—
513C22×C4—1422no—
514C22×C4—1422no—
515C22×C4—1422no—
516C22×C4—1422no—
517C22×C4C2×C4×D7114yes—
518C22×C4C2×C4×D7112yes—
519C22×C4D4×D7112yes—
520C22×C4D4×D7112yes—
521C22×C4Q8×D7112yes—
522C22×C4Q8×D7112yes—
523C4⋊C4—44no—
524C4⋊C4—2848no—
525C4⋊C4—2844no—
526C4⋊C4—2848no—
527C4⋊C4—2848no—
528C22⋊C4—2848no—
529C22⋊C4—14216no—
530C23—714no—
531C23—714no—
532C23—714no—
533C23—714no—
534C23—714no—
535C23—714no—
536C23—714no—
537C23—1422no—
538C23—1424no—
539C23—1422no—
540C23—1422no—
541C23—1424no—
542C23—1424no—
543C23—1424no—
544C23—1424no—
545C23—1424no—
546C23—1424no—
547C23—1424no—
548C23—1424no—
549C23—1424no—
550C23—1424no—
551C23—1424no—
552C23—1422no—
553C23C22×C4×D7111yescharacteristic
554C23C2×D4×D7111yescharacteristic
555C23C2×D4×D7111yescharacteristic
556C23C2×Q8×D7111yescharacteristic
557C23C2×Q8×D7111yescharacteristic
558C23#41×D7111yescharacteristic
559C23#41×D7111yescharacteristic
560C23#1455617312yes—
561C23#1455617412yes—
562C23#1455617412yes—
563C23#1455617512yes—
564C2×C4—22no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—22no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—42no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—28416no—
593C2×C4—28416no—
594C2×C4—28416no—
595C2×C4—2842no—
596C2×C4—2841no—
597C2×C4—2841no—
598C2×C4—2841no—
599C2×C4—2842no—
600C2×C4—2842no—
601C2×C4—2842no—
602C2×C4—2842no—
603C2×C4—2842no—
604C2×C4—2841no—
605C2×C4—2842no—
606C2×C4—2842no—
607C2×C4—2842no—
608C2×C4—2842no—
609C2×C4—2842no—
610C2×C4—2842no—
611C2×C4—2842no—
612C2×C4—714no—
613C2×C4—1424no—
614C2×C4—1424no—
615C2×C4—1424no—
616C2×C4—1424no—
617C2×C4—1424no—
618C2×C4—1422no—
619C2×C4—1422no—
620C2×C4—1422no—
621C2×C4—1424no—
622C2×C4—1424no—
623C2×C4—1422no—
624C2×C4C4⋊C4×D7114yes—
625C22—718no—
626C22—1428no—
627C22—1424no—
628C22—1424no—
629C22—1428no—
630C22—1428no—
631C22—1424no—
632C22—1424no—
633C22—1428no—
634C22—1424no—
635C22—1424no—
636C22C2×C4⋊C4×D7111yescharacteristic
637C22C4×D4×D7111yescharacteristic
638C22C4×Q8×D7111yescharacteristic
639C22#1798647112yes—
640C22#1798647912yes—
641C22#121×D7111yescharacteristic
642C22C22⋊Q8×D7111yescharacteristic
643C22#125×D7111yescharacteristic
644C22C4⋊Q8×D7111yescharacteristic
645C22#1798655911yescharacteristic
646C22#1798656012yes—
647C22#1798656011yescharacteristic
648C22#1798656711yescharacteristic
649C22#1798656911yescharacteristic
650C22#1798657812yes—
651C22#1798658312yes—
652C22#1798658812yes—
653C22#1798659012yes—
654C22#1798659212yes—
655C22#1798662412yes—
656C22#1798662412yes—
657C22#1798662912yes—
658C22#1798662912yes—
659C4—24no—
660C4—42no—
661C4—44no—
662C4—42no—
663C4—82no—
664C4—2842no—
665C4—2844no—
666C4—2842no—
667C4—5682no—
668C4—1424no—
669C2—1428no—
670C2#389×D7111yescharacteristic
671C2#2259095111yescharacteristic
672C2#2259096511yescharacteristic
673C2#2259104412yes—
674C2#2259106712yes—
675C2#2259407511yescharacteristic
676C2#2259408111yescharacteristic
677C2#2259414411yescharacteristic
678C2#2259415411yescharacteristic
679C2#2259431112yes—
680C2#2259445512yes—
subgroupG/H
center, FrattiniC24C23×D71
commutatorC23×C71C23×C4
FittingC71×#1442C2
socle, cosocleC24×C71C24
prime radicalC24×D71C23
hypercenter#1442D71
hypocenterC71#14275
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C24 ◁ #1442

Complements

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

NH ≅ G/Ntype
5. #22586605C2semidirect
6. #22586604C2semidirect
7. #22586599C2semidirect
8. #22586594C2semidirect
9. #22586591C2semidirect
11. C71×#1442C2semidirect
22. #17985858C4semidirect
70. C2×Dic71⋊C4C2×C4semidirect
85. C71⋊#95C2×C4semidirect
96. #95×C71C2×C4semidirect
151. C2×C4×D71C4⋊C4semidirect
348. #1442D71semidirect
352. C71#14275semidirect
437. #95C4×D71semidirect

Maximal subgroups

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

subgroupindex
C2×#179858732
C2×#179858582
C2×#179858562
C2×#95×D712
#225866052
#225866042
#225865992
#225865942
#225865912
(C2×D71)⋊#952
C71×#14422
#1427571