Finite Group Databasebeta

← #31918717

Subgroups — #31918717

Order 18176.

1710 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17985861C211yesmaximal, characteristic
2C2×#17985858C211yesmaximal, characteristic
3C2×C4×(C2×D71)⋊C4C211yesmaximal, characteristic
4#22586578C212yesmaximal
5#22586576C212yesmaximal
6#22586573C212yesmaximal, complemented
7#22586571C212yesmaximal, complemented
8#22586565C211yesmaximal, characteristic, complemented
9#22586565C211yesmaximal, characteristic, complemented
10C4×C71⋊#95C211yesmaximal, characteristic, complemented
11C4×#95×C71C211yesmaximal, characteristic, complemented
12C23×C4×D71C2211yescharacteristic
13C22×Dic71⋊C4C2212yes—
14C22×(C2×D71)⋊C4C2211yescharacteristic
15C22×(C2×D71)⋊C4C2211yescharacteristic
16C22×(C2×D71)⋊C4C2212yes—
17C22×(C2×D71)⋊C4C2212yes—
18C22×C4×Dic71C2212yes—
19C22×C4×Dic71C2211yescharacteristic
20#17985861C418yescomplemented
21#17985858C418yescomplemented
22C4×(C2×D71)⋊C4C418yescomplemented
23C2×C71⋊#95C2212yes—
24C2×C71⋊#95C2212yes—
25C2×C71⋊#95C2212yes—
26C2×C71⋊#95C2212yes—
27C2×C71⋊#95C2212yes—
28C2×C71⋊#95C2211yescharacteristic
29C2×C71⋊#95C2211yescharacteristic
30C2×C71⋊#95C2211yescharacteristic
31C2×C71⋊#95C2212yes—
32C2×C71⋊#95C2211yescharacteristic
33C22×C42×C71C2212yes—
34C22×C42×C71C2211yescharacteristic
35C2×#95×C71C2212yes—
36C2×#95×C71C2211yescharacteristic
37C2×#95×C71C2211yescharacteristic
38C24×D71C2311yescharacteristic
39C23×Dic71C2312yes—
40C23×Dic71C2311yescharacteristic
41C23×Dic71C2311yescharacteristic
42C23×Dic71C2312yes—
43C23×Dic71C2311yescharacteristic
44C22×C4×D71—42no—
45C22×C4×D71—42no—
46C22×C4×D71C2×C414yes—
47C22×C4×D71C2×C414yes—
48C22×C4×D71C2×C414yes—
49C2×Dic71⋊C4—42no—
50C2×Dic71⋊C4—42no—
51C2×Dic71⋊C4C2×C418yes—
52C2×(C2×D71)⋊C4—24no—
53C2×(C2×D71)⋊C4—24no—
54C2×(C2×D71)⋊C4—42no—
55C2×(C2×D71)⋊C4—42no—
56C2×(C2×D71)⋊C4—42no—
57C2×(C2×D71)⋊C4—42no—
58C2×(C2×D71)⋊C4C2×C414yes—
59C2×(C2×D71)⋊C4C2×C414yes—
60C2×(C2×D71)⋊C4C2×C418yes—
61C2×(C2×D71)⋊C4C2×C418yes—
62C2×C4×Dic71—24no—
63C2×C4×Dic71—42no—
64C2×C4×Dic71—42no—
65C2×C4×Dic71—42no—
66C2×C4×Dic71C2×C414yes—
67C71⋊#95—24no—
68C71⋊#95—42no—
69C71⋊#95—42no—
70C71⋊#95—42no—
71C71⋊#95—42no—
72C71⋊#95C2×C414yes—
73C71⋊#95C2×C414yes—
74C71⋊#95C2×C414yes—
75C71⋊#95C2×C418yes—
76C71⋊#95C2×C414yes—
77C23×C4×C71C2312yes—
78C23×C4×C71C2312yes—
79C23×C4×C71C2311yescharacteristic
80C23×C4×C71C2311yescharacteristic
81C23×C4×C71C2311yescharacteristic
82C2×C42×C71—24no—
83C2×C42×C71—42no—
84C2×C42×C71—42no—
85C2×C42×C71—42no—
86C2×C42×C71C2×C414yes—
87#95×C71—24no—
88#95×C71C2×C414yes—
89#95×C71C2×C414yes—
90C23×D71—22no—
91C23×D71—22no—
92C23×D71—22no—
93C23×D71—22no—
94C23×D71—21no—
95C23×D71—21no—
96C23×D71—21no—
97C23×D71—21no—
98C23×D71C22×C412yes—
99C23×D71C22×C412yes—
100C23×D71C22×C412yes—
101C22×Dic71—22no—
102C22×Dic71—22no—
103C22×Dic71—22no—
104C22×Dic71—22no—
105C22×Dic71—22no—
106C22×Dic71—22no—
107C22×Dic71—22no—
108C22×Dic71—22no—
109C22×Dic71—21no—
110C22×Dic71—21no—
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—21no—
121C22×Dic71—22no—
122C22×Dic71—21no—
123C22×Dic71C22×C414yes—
124C22×Dic71C22×C412yes—
125C22×Dic71C22×C412yes—
126C22×Dic71C22×C414yes—
127C22×Dic71C22×C412yes—
128C22×Dic71C22×C412yes—
129C22×Dic71C22×C412yes—
130C2×C4×D71—48no—
131C2×C4×D71—44no—
132C2×C4×D71—44no—
133C2×C4×D71—48no—
134C2×C4×D71C42116yescomplemented
135Dic71⋊C4—48no—
136(C2×D71)⋊C4—28no—
137(C2×D71)⋊C4—28no—
138(C2×D71)⋊C4—48no—
139(C2×D71)⋊C4—48no—
140C4×Dic71—44no—
141C4×Dic71—48no—
142C24×C71C2411yescharacteristic
143C22×C4×C71—22no—
144C22×C4×C71—22no—
145C22×C4×C71—22no—
146C22×C4×C71—21no—
147C22×C4×C71—22no—
148C22×C4×C71—21no—
149C22×C4×C71—21no—
150C22×C4×C71—21no—
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×C71—22no—
158C22×C4×C71—22no—
159C22×C4×C71—22no—
160C22×C4×C71—22no—
161C22×C4×C71—22no—
162C22×C4×C71—22no—
163C22×C4×C71C22×C414yes—
164C22×C4×C71C22×C414yes—
165C22×C4×C71C22×C412yes—
166C22×C4×C71C22×C412yes—
167C22×C4×C71C22×C412yes—
168C22×C4×C71C22×C412yes—
169C22×C4×C71C22×C412yes—
170C22×C4×C71#4114yes—
171C22×C4×C71#4114yes—
172C42×C71—44no—
173C42×C71—48no—
174C23×C71C23×C411yescharacteristic
175C23×C71C23×C411yescharacteristic
176C23×C71C23×C411yescharacteristic
177C23×C71C2×#4112yes—
178C23×C71C2×#4112yes—
179C23×C71H(5,𝔽2)12yes—
180C23×C71H(5,𝔽2)11yescharacteristic
181C23×C71H(5,𝔽2)12yes—
182C23×C71H(5,𝔽2)11yescharacteristic
183C23×C712−1+411yescharacteristic
184C23×C712−1+411yescharacteristic
185C22×D71—22no—
186C22×D71—22no—
187C22×D71—24no—
188C22×D71—24no—
189C22×D71—22no—
190C22×D71—24no—
191C22×D71—24no—
192C22×D71—22no—
193C22×D71—22no—
194C22×D71—22no—
195C22×D71—22no—
196C22×D71—22no—
197C22×D71—22no—
198C22×D71—22no—
199C22×D71—42no—
200C22×D71—42no—
201C22×D71—42no—
202C22×D71—42no—
203C22×D71—42no—
204C22×D71—42no—
205C22×D71—42no—
206C22×D71—42no—
207C22×D71C2×C4214yes—
208C2×C4×C71—22no—
209C2×C4×C71—22no—
210C2×C4×C71—22no—
211C2×C4×C71—22no—
212C2×C4×C71—24no—
213C2×C4×C71—24no—
214C2×C4×C71—22no—
215C2×C4×C71—22no—
216C2×C4×C71—24no—
217C2×C4×C71—24no—
218C2×C4×C71—24no—
219C2×C4×C71—24no—
220C2×C4×C71—24no—
221C2×C4×C71—24no—
222C2×C4×C71—24no—
223C2×C4×C71—24no—
224C2×C4×C71—24no—
225C2×C4×C71—24no—
226C2×C4×C71—42no—
227C2×C4×C71—42no—
228C2×C4×C71—42no—
229C2×C4×C71—42no—
230C2×C4×C71—42no—
231C2×C4×C71—42no—
232C2×C4×C71—42no—
233C2×C4×C71—42no—
234C2×C4×C71C2×C4214yes—
235C2×C4×C71#11714yes—
236C2×C4×C71#11714yes—
237C2×Dic71—24no—
238C2×Dic71—24no—
239C2×Dic71—22no—
240C2×Dic71—22no—
241C2×Dic71—22no—
242C2×Dic71—22no—
243C2×Dic71—24no—
244C2×Dic71—24no—
245C2×Dic71—22no—
246C2×Dic71—22no—
247C2×Dic71—24no—
248C2×Dic71—24no—
249C2×Dic71—24no—
250C2×Dic71—22no—
251C2×Dic71—22no—
252C2×Dic71—24no—
253C2×Dic71—42no—
254C2×Dic71—42no—
255C2×Dic71—42no—
256C2×Dic71—42no—
257C2×Dic71—42no—
258C2×Dic71—42no—
259C2×Dic71—42no—
260C2×Dic71—42no—
261C2×Dic71—42no—
262C2×Dic71—42no—
263C2×Dic71—42no—
264C2×Dic71—42no—
265C2×Dic71C2×C4214yes—
266C4×D71—416no—
267C22×C71C22×C4211yescharacteristic
268C22×C71C2×#11711yescharacteristic
269C22×C71C2×#11711yescharacteristic
270C22×C71C4×#4112yes—
271C22×C71C4×#4112yes—
272C22×C71#51812yes—
273C22×C71#51811yescharacteristic
274C22×C71#51812yes—
275C22×C71#51811yescharacteristic
276C22×C71#51811yescharacteristic
277C22×C71#51911yescharacteristic
278C22×C71#52011yescharacteristic
279C22×C71#52011yescharacteristic
280C22×C71#52011yescharacteristic
281C22×C71#52011yescharacteristic
282C22×C71#53812yes—
283C22×C71#53812yes—
284C22×C71#53912yes—
285C22×C71#53912yes—
286C22×C71#54012yes—
287C22×C71#54012yes—
288C22×C71#54212yes—
289C22×C71#54212yes—
290C2×D71—24no—
291C2×D71—24no—
292C2×D71—24no—
293C2×D71—42no—
294C2×D71—42no—
295C2×D71—42no—
296C2×D71—42no—
297C2×D71—44no—
298C2×D71—44no—
299C2×D71—44no—
300C2×D71—44no—
301C4×C71—24no—
302C4×C71—24no—
303C4×C71—24no—
304C4×C71—44no—
305C4×C71—44no—
306Dic71—24no—
307Dic71—42no—
308Dic71—42no—
309Dic71—44no—
310Dic71—44no—
311#13780—711nomaximal, Sylow
312C2×C71C4×#11711yescharacteristic
313C2×C71#227311yescharacteristic
314C2×C71#227411yescharacteristic
315C2×C71#231811yescharacteristic
316C2×C71#231811yescharacteristic
317C2×C71#231911yescharacteristic
318C2×C71#231911yescharacteristic
319C2×C71#233212yes—
320C2×C71#233312yes—
321C2×C71#233512yes—
322C2×C71#233512yes—
323D71—44no—
324C2×#381—711no—
325C2×#380—711no—
326C2×C4×C22⋊C4—711no—
327#1437—712no—
328#1437—712no—
329#1436—712no—
330#1436—712no—
331#1434—711no—
332#1434—711no—
333C4×#95—711no—
334C4×#95D7111yescharacteristic, complemented
335C71#1378011yescharacteristic, complemented, Sylow
336C24×C4—711no—
337C22×C4⋊C4—712no—
338C22×C22⋊C4—712no—
339C22×C22⋊C4—712no—
340C22×C22⋊C4—711no—
341C22×C22⋊C4—711no—
342C22×C42—711no—
343C22×C42—712no—
344C22×C42C2×D7112yes—
345C22×C42C2×D7111yescharacteristic
346#381—718no—
347#380—718no—
348C4×C22⋊C4—718no—
349C2×#95—712no—
350C2×#95—712no—
351C2×#95—712no—
352C2×#95—712no—
353C2×#95—711no—
354C2×#95—711no—
355C2×#95—711no—
356C2×#95—711no—
357C2×#95—712no—
358C2×#95—712no—
359C2×#95C2×D7112yes—
360C2×#95C2×D7111yescharacteristic
361C2×#95C2×D7111yescharacteristic
362C25—711no—
363C23×C4—2842no—
364C23×C4—2842no—
365C23×C4—714no—
366C23×C4—714no—
367C23×C4—714no—
368C23×C4—711no—
369C23×C4—712no—
370C23×C4—712no—
371C23×C4—711no—
372C23×C4—711no—
373C23×C4C22×D7112yes—
374C23×C4C22×D7112yes—
375C23×C4C22×D7111yescharacteristic
376C23×C4C22×D7111yescharacteristic
377C23×C4C22×D7111yescharacteristic
378C2×C4⋊C4—2842no—
379C2×C4⋊C4—2842no—
380C2×C4⋊C4—718no—
381C2×C22⋊C4—2842no—
382C2×C22⋊C4—2842no—
383C2×C22⋊C4—2842no—
384C2×C22⋊C4—2842no—
385C2×C22⋊C4—718no—
386C2×C22⋊C4—718no—
387C2×C22⋊C4—714no—
388C2×C22⋊C4—714no—
389C2×C22⋊C4—1424no—
390C2×C22⋊C4—1424no—
391C2×C42—24no—
392C2×C42—42no—
393C2×C42—42no—
394C2×C42—42no—
395C2×C42—2842no—
396C2×C42—2842no—
397C2×C42—2842no—
398C2×C42—714no—
399C2×C42—1424no—
400C2×C42C4×D7114yes—
401#95—24no—
402#95—2842no—
403#95—2842no—
404#95—2842no—
405#95—2842no—
406#95—714no—
407#95—714no—
408#95—714no—
409#95—718no—
410#95—714no—
411#95—1424no—
412#95C4×D7114yes—
413#95C4×D7114yes—
414C24—712no—
415C24—712no—
416C24—712no—
417C24—1422no—
418C24—1422no—
419C24—1421no—
420C24—1422no—
421C24—1421no—
422C24—1422no—
423C24—1421no—
424C24—1421no—
425C24C23×D7111yescharacteristic
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—2848no—
447C22×C4—2848no—
448C22×C4—2844no—
449C22×C4—2844no—
450C22×C4—7116no—
451C22×C4—712no—
452C22×C4—712no—
453C22×C4—712no—
454C22×C4—714no—
455C22×C4—712no—
456C22×C4—712no—
457C22×C4—714no—
458C22×C4—1422no—
459C22×C4—1422no—
460C22×C4—1422no—
461C22×C4—1422no—
462C22×C4—1422no—
463C22×C4—1422no—
464C22×C4—1421no—
465C22×C4—1422no—
466C22×C4—1422no—
467C22×C4—1422no—
468C22×C4—1421no—
469C22×C4—1422no—
470C22×C4—1422no—
471C22×C4—1422no—
472C22×C4—1422no—
473C22×C4—1422no—
474C22×C4—1422no—
475C22×C4—1421no—
476C22×C4—1422no—
477C22×C4—1421no—
478C22×C4—1422no—
479C22×C4—1422no—
480C22×C4C2×C4×D7114yes—
481C22×C4C2×C4×D7114yes—
482C22×C4C2×C4×D7112yes—
483C22×C4C2×C4×D7112yes—
484C22×C4C2×C4×D7112yes—
485C22×C4C2×C4×D7112yes—
486C22×C4C2×C4×D7112yes—
487C22×C4#1176586514yes—
488C22×C4#1176586514yes—
489C4⋊C4—2848no—
490C22⋊C4—2848no—
491C22⋊C4—2848no—
492C22⋊C4—1428no—
493C22⋊C4—1428no—
494C42—44no—
495C42—48no—
496C42—2848no—
497C42—2844no—
498C23—2842no—
499C23—2842no—
500C23—2842no—
501C23—2842no—
502C23—2842no—
503C23—2842no—
504C23—2842no—
505C23—2842no—
506C23—714no—
507C23—1422no—
508C23—1422no—
509C23—1424no—
510C23—1424no—
511C23—1422no—
512C23—1424no—
513C23—1424no—
514C23—1422no—
515C23—1422no—
516C23—1422no—
517C23—1422no—
518C23—1422no—
519C23—1422no—
520C23—1422no—
521C23C22×C4×D7111yescharacteristic
522C23C22×C4×D7111yescharacteristic
523C23C22×C4×D7111yescharacteristic
524C23C2×#1176586512yes—
525C23C2×#1176586512yes—
526C23#1455617312yes—
527C23#1455617311yescharacteristic
528C23#1455617411yescharacteristic
529C23#1455617412yes—
530C23#1455617511yescharacteristic
531C23#1455617611yescharacteristic
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—28416no—
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—2842no—
569C2×C4—2842no—
570C2×C4—2842no—
571C2×C4—714no—
572C2×C4—1422no—
573C2×C4—1422no—
574C2×C4—1424no—
575C2×C4—1424no—
576C2×C4—1424no—
577C2×C4—1424no—
578C2×C4—1424no—
579C2×C4—1422no—
580C2×C4—1422no—
581C2×C4—1422no—
582C2×C4—1422no—
583C2×C4—1424no—
584C2×C4—1424no—
585C2×C4—1424no—
586C2×C4—1422no—
587C2×C4—1422no—
588C2×C4C42×D7114yes—
589C2×C4#1455608514yes—
590C2×C4#1455608514yes—
591C22—2842no—
592C22—2842no—
593C22—2844no—
594C22—2844no—
595C22—2844no—
596C22—2844no—
597C22—2842no—
598C22—2842no—
599C22—1424no—
600C22—1424no—
601C22—1424no—
602C22C2×C42×D7111yescharacteristic
603C22C2×#1455608511yescharacteristic
604C22C2×#1455608511yescharacteristic
605C22C4×#1176586512yes—
606C22C4×#1176586512yes—
607C22#1798647112yes—
608C22#1798647111yescharacteristic
609C22#1798647112yes—
610C22#1798647211yescharacteristic
611C22#1798647311yescharacteristic
612C22#1798647511yescharacteristic
613C22#1798647811yescharacteristic
614C22#1798647911yescharacteristic
615C22#1798648011yescharacteristic
616C22#1798648111yescharacteristic
617C22#1798658212yes—
618C22#1798658712yes—
619C22#1798659112yes—
620C22#1798659612yes—
621C22#1798660012yes—
622C22#1798660412yes—
623C22#1798662012yes—
624C22#1798662112yes—
625C4—24no—
626C4—24no—
627C4—24no—
628C4—44no—
629C4—44no—
630C4—2842no—
631C4—2842no—
632C4—2844no—
633C4—2844no—
634C4—1424no—
635C2—2844no—
636C2C4×#1455608511yescharacteristic
637C2#2259069211yescharacteristic
638C2#2259069411yescharacteristic
639C2#2259096711yescharacteristic
640C2#2259097411yescharacteristic
641C2#2259097811yescharacteristic
642C2#2259098411yescharacteristic
643C2#2259109112yes—
644C2#2259110012yes—
645C2#2259112412yes—
646C2#2259113212yes—
subgroupG/H
center, FrattiniC24C23×D71
commutatorC22×C71C22×C42
FittingC4×#95×C71C2
socle, cosocleC24×C71C24
prime radicalC24×D71C23
hypercenterC4×#95D71
hypocenterC71#13780
LCSG ▷ C22×C71 ▷ C71
UCSC1 ◁ C24 ◁ C4×#95

Complements

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

NH ≅ G/Ntype
6. #22586573C2semidirect
7. #22586571C2semidirect
8. #22586565C2semidirect
9. #22586565C2semidirect
10. C4×C71⋊#95C2semidirect
11. C4×#95×C71C2semidirect
20. #17985861C4semidirect
21. #17985858C4semidirect
22. C4×(C2×D71)⋊C4C4semidirect
134. C2×C4×D71C42semidirect
334. C4×#95D71semidirect
335. C71#13780semidirect

Maximal subgroups

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

subgroupindex
C2×#179858612
C2×#179858582
C2×C4×(C2×D71)⋊C42
#225865782
#225865762
#225865732
#225865712
#225865652
C4×C71⋊#952
C4×#95×C712
#1378071