Finite Group Databasebeta

← #31920280

Subgroups — #31920280

Order 18176.

1918 conjugacy classes of subgroups.

The subgroup lattice has 692 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17985860C211yesmaximal, characteristic
2C2×#17985860C211yesmaximal, characteristic
3C2×C4×C22⋊Dic71C211yesmaximal, characteristic
4C22×C71⋊#95C211yesmaximal, characteristic
5#22586601C212yesmaximal, complemented
6#22586599C212yesmaximal, complemented
7#22586572C212yesmaximal, complemented
8#22586566C212yesmaximal, complemented
9#22586562C211yesmaximal, characteristic
10#22586562C211yesmaximal, characteristic
11C2×#385×C71C211yesmaximal, characteristic
12C24×Dic71C2211yescharacteristic
13C24×Dic71C2211yescharacteristic
14C22×C22⋊Dic71C2211yescharacteristic
15C22×C22⋊Dic71C2211yescharacteristic
16C22×C4×Dic71C2212yes—
17#17985860—24no—
18#17985860C418yescomplemented
19C4×C22⋊Dic71—24no—
20C2×C71⋊#95—24no—
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⋊#95C2212yes—
30C2×C71⋊#95C2212yes—
31C2×C71⋊#95C2212yes—
32C24×C4×C71C2211yescharacteristic
33C22×C4⋊C4×C71C2211yescharacteristic
34C22×C4⋊C4×C71C2211yescharacteristic
35C22×C22⋊C4×C71C2211yescharacteristic
36C22×C22⋊C4×C71C2211yescharacteristic
37#385×C71—24no—
38C2×#95×C71C2211yescharacteristic
39C2×#95×C71C2211yescharacteristic
40C23×Dic71—24no—
41C23×Dic71—22no—
42C23×Dic71—22no—
43C23×Dic71—22no—
44C23×Dic71—22no—
45C23×Dic71—22no—
46C23×Dic71—22no—
47C23×Dic71—22no—
48C23×Dic71—22no—
49C23×Dic71—22no—
50C23×Dic71—42no—
51C23×Dic71C2×C414yes—
52C23×Dic71C2312yes—
53C23×Dic71C2312yes—
54C23×Dic71C2312yes—
55C23×Dic71C2312yes—
56C2×C22⋊Dic71—22no—
57C2×C22⋊Dic71—22no—
58C2×C22⋊Dic71—22no—
59C2×C22⋊Dic71—22no—
60C2×C22⋊Dic71C2×C414yes—
61C2×C22⋊Dic71D414yes—
62C2×C4×Dic71—24no—
63C2×C4×Dic71—24no—
64C2×C4×Dic71—42no—
65C71⋊#95—24no—
66C71⋊#95—24no—
67C71⋊#95—24no—
68C71⋊#95—24no—
69C71⋊#95—24no—
70C71⋊#95—24no—
71C71⋊#95—24no—
72C71⋊#95—42no—
73C71⋊#95—42no—
74C71⋊#95C2×C418yescomplemented
75C71⋊#95C2×C418yescomplemented
76C25×C71C2311yescharacteristic
77C23×C4×C71—22no—
78C23×C4×C71—22no—
79C23×C4×C71—24no—
80C23×C4×C71—42no—
81C23×C4×C71D414yes—
82C23×C4×C71C2311yescharacteristic
83C23×C4×C71C2311yescharacteristic
84C23×C4×C71C2311yescharacteristic
85C23×C4×C71C2311yescharacteristic
86C23×C4×C71C2311yescharacteristic
87C23×C4×C71C2311yescharacteristic
88C2×C4⋊C4×C71—22no—
89C2×C4⋊C4×C71—22no—
90C2×C4⋊C4×C71—24no—
91C2×C4⋊C4×C71—42no—
92C2×C22⋊C4×C71—22no—
93C2×C22⋊C4×C71—22no—
94C2×C22⋊C4×C71—22no—
95C2×C22⋊C4×C71—22no—
96C2×C22⋊C4×C71—22no—
97C2×C22⋊C4×C71C2×C414yes—
98#95×C71—22no—
99#95×C71—22no—
100C22×Dic71—28no—
101C22×Dic71—28no—
102C22×Dic71—28no—
103C22×Dic71—28no—
104C22×Dic71—22no—
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×Dic71—22no—
124C22×Dic71—22no—
125C22×Dic71—22no—
126C22×Dic71—22no—
127C22×Dic71—22no—
128C22×Dic71—48no—
129C22×Dic71—48no—
130C22×Dic71—44no—
131C22×Dic71—44no—
132C22×Dic71—44no—
133C22×Dic71—44no—
134C22×Dic71—44no—
135C22×Dic71—44no—
136C22×Dic71C22×C414yes—
137C22×Dic71C22×C414yes—
138C22×Dic71C2×D414yes—
139C22×Dic71#4114yes—
140C22⋊Dic71—28no—
141C22⋊Dic71—28no—
142C4×Dic71—48no—
143C24×C71—22no—
144C24×C71—22no—
145C24×C71—22no—
146C24×C71—22no—
147C24×C71C22×C412yes—
148C24×C71C2×D412yes—
149C24×C71C2×D412yes—
150C24×C71#4112yes—
151C24×C71#4112yes—
152C24×C71#4112yes—
153C24×C71#4112yes—
154C24×C71C2411yescharacteristic
155C22×C4×C71—21no—
156C22×C4×C71—21no—
157C22×C4×C71—22no—
158C22×C4×C71—28no—
159C22×C4×C71—21no—
160C22×C4×C71—21no—
161C22×C4×C71—21no—
162C22×C4×C71—21no—
163C22×C4×C71—21no—
164C22×C4×C71—22no—
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—28no—
172C22×C4×C71—21no—
173C22×C4×C71—21no—
174C22×C4×C71—21no—
175C22×C4×C71—21no—
176C22×C4×C71—21no—
177C22×C4×C71—21no—
178C22×C4×C71—22no—
179C22×C4×C71—22no—
180C22×C4×C71—22no—
181C22×C4×C71—22no—
182C22×C4×C71—44no—
183C22×C4×C71—48no—
184C22×C4×C71—48no—
185C22×C4×C71C22×C412yes—
186C22×C4×C71C22×C412yes—
187C22×C4×C71C2×D412yes—
188C22×C4×C71C2×D412yes—
189C22×C4×C71#4114yes—
190C4⋊C4×C71—44no—
191C4⋊C4×C71—44no—
192C22⋊C4×C71—28no—
193C22⋊C4×C71—44no—
194C23×C71—24no—
195C23×C71—24no—
196C23×C71—22no—
197C23×C71—22no—
198C23×C71—24no—
199C23×C71—22no—
200C23×C71—22no—
201C23×C71—24no—
202C23×C71—24no—
203C23×C71—24no—
204C23×C71—24no—
205C23×C71—24no—
206C23×C71—24no—
207C23×C71—24no—
208C23×C71—24no—
209C23×C71—24no—
210C23×C71#11714yes—
211C23×C71C4×D414yes—
212C23×C71C4×D414yes—
213C23×C71#12114yes—
214C23×C71#12314yes—
215C23×C71#12414yes—
216C23×C71#12614yes—
217C23×C71C23×C411yescharacteristic
218C23×C71C22×D411yescharacteristic
219C23×C71C22×D411yescharacteristic
220C23×C71C2×#4112yes—
221C23×C71C2×#4111yescharacteristic
222C23×C71C2×#4112yes—
223C23×C71C2×#4111yescharacteristic
224C23×C71C2×#4111yescharacteristic
225C23×C71C2×#4111yescharacteristic
226C23×C71H(5,𝔽2)12yes—
227C23×C712−1+412yes—
228C2×C4×C71—22no—
229C2×C4×C71—22no—
230C2×C4×C71—24no—
231C2×C4×C71—24no—
232C2×C4×C71—22no—
233C2×C4×C71—22no—
234C2×C4×C71—24no—
235C2×C4×C71—24no—
236C2×C4×C71—22no—
237C2×C4×C71—22no—
238C2×C4×C71—22no—
239C2×C4×C71—416no—
240C2×C4×C71—41no—
241C2×C4×C71—41no—
242C2×C4×C71—42no—
243C2×C4×C71—41no—
244C2×C4×C71—41no—
245C2×C4×C71—42no—
246C2×C4×C71—42no—
247C2×C4×C71—42no—
248C2×C4×C71—41no—
249C2×C4×C71—41no—
250C2×C4×C71—41no—
251C2×C4×C71—41no—
252C2×C4×C71—41no—
253C2×C4×C71—41no—
254C2×C4×C71—42no—
255C2×C4×C71—42no—
256C2×C4×C71—42no—
257C2×C4×C71—42no—
258C2×C4×C71#12314yes—
259C2×Dic71—24no—
260C2×Dic71—24no—
261C2×Dic71—24no—
262C2×Dic71—24no—
263C2×Dic71—24no—
264C2×Dic71—24no—
265C2×Dic71—24no—
266C2×Dic71—24no—
267C2×Dic71—24no—
268C2×Dic71—24no—
269C2×Dic71—24no—
270C2×Dic71—24no—
271C2×Dic71—416no—
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×Dic71—42no—
281C2×Dic71—42no—
282C2×Dic71—42no—
283C2×Dic71—42no—
284C2×Dic71—42no—
285C2×Dic71—42no—
286C2×Dic71—42no—
287C2×Dic71—42no—
288C2×Dic71—416no—
289C22×C71—24no—
290C22×C71—24no—
291C22×C71—28no—
292C22×C71—28no—
293C22×C71—24no—
294C22×C71—24no—
295C22×C71—28no—
296C22×C71—28no—
297C22×C71—24no—
298C22×C71—24no—
299C22×C71#38818yes—
300C22×C71C2×#11711yescharacteristic
301C22×C71C2×C4×D411yescharacteristic
302C22×C71C2×C4×D411yescharacteristic
303C22×C71C4×#4112yes—
304C22×C71#52012yes—
305C22×C71C2×#12111yescharacteristic
306C22×C71C2×#12311yescharacteristic
307C22×C71C2×#12311yescharacteristic
308C22×C71#52511yescharacteristic
309C22×C71C2×#12411yescharacteristic
310C22×C71C2×#12611yescharacteristic
311C22×C71#52912yes—
312C22×C71#53411yescharacteristic
313C22×C71#53611yescharacteristic
314C22×C71#53912yes—
315C22×C71#54612yes—
316C22×C71#54712yes—
317C22×C71#55112yes—
318C22×C71#55112yes—
319C22×C71#55112yes—
320C22×C71#55212yes—
321C22×C71#55212yes—
322C22×C71#55212yes—
323C4×C71—24no—
324C4×C71—42no—
325C4×C71—42no—
326C4×C71—42no—
327C4×C71—81no—
328C4×C71—81no—
329Dic71—44no—
330Dic71—44no—
331Dic71—44no—
332Dic71—44no—
333#13972—711nomaximal, Sylow
334C2×C71—28no—
335C2×C71C2×#38811yescharacteristic
336C2×C71C4×#12311yescharacteristic
337C2×C71#232211yescharacteristic
338C2×C71#234312yes—
339C2×C71#235912yes—
340C2×C71#241211yescharacteristic
341C2×C71#241311yescharacteristic
342C2×C71#241912yes—
343C2×C71#249312yes—
344C2×C71#250311yescharacteristic
345C2×C71#256111yescharacteristic
346C2×#385D7111yescharacteristic
347C2×#381—711no—
348C2×#381—711no—
349C2×C4×C22⋊C4—711no—
350C22×#95—711no—
351#1443—712no—
352#1442—712no—
353#1436—712no—
354#1434—712no—
355C22⋊#95—711no—
356C22⋊#95—711no—
357C71#1397211yescharacteristic, complemented, Sylow
358C24×C4—711no—
359C24×C4—711no—
360C24×C4C2×D7111yescharacteristic
361C22×C4⋊C4C2×D7111yescharacteristic
362C22×C4⋊C4C2×D7111yescharacteristic
363C22×C22⋊C4—711no—
364C22×C22⋊C4—711no—
365C22×C22⋊C4C2×D7111yescharacteristic
366C22×C22⋊C4C2×D7111yescharacteristic
367C22×C42—712no—
368#385—24no—
369#381—718no—
370#381—1424no—
371C4×C22⋊C4—1424no—
372C2×#95—712no—
373C2×#95—712no—
374C2×#95—712no—
375C2×#95—712no—
376C2×#95—712no—
377C2×#95—712no—
378C2×#95—712no—
379C2×#95—712no—
380C2×#95—712no—
381C2×#95—712no—
382C2×#95—712no—
383C2×#95—1424no—
384C2×#95C2×D7111yescharacteristic
385C2×#95C2×D7111yescharacteristic
386C25C22×D7111yescharacteristic
387C23×C4—22no—
388C23×C4—22no—
389C23×C4—24no—
390C23×C4—42no—
391C23×C4—2842no—
392C23×C4—714no—
393C23×C4—712no—
394C23×C4—712no—
395C23×C4—712no—
396C23×C4—712no—
397C23×C4—1422no—
398C23×C4—1422no—
399C23×C4—1424no—
400C23×C4—1422no—
401C23×C4—1422no—
402C23×C4—1422no—
403C23×C4—1422no—
404C23×C4—1422no—
405C23×C4—1422no—
406C23×C4—1422no—
407C23×C4C22⋊D7114yes—
408C23×C4C22×D7111yescharacteristic
409C23×C4C22×D7111yescharacteristic
410C23×C4C22×D7111yescharacteristic
411C23×C4C22×D7111yescharacteristic
412C23×C4C22×D7111yescharacteristic
413C23×C4C22×D7111yescharacteristic
414C2×C4⋊C4—22no—
415C2×C4⋊C4—22no—
416C2×C4⋊C4—24no—
417C2×C4⋊C4—42no—
418C2×C22⋊C4—22no—
419C2×C22⋊C4—22no—
420C2×C22⋊C4—22no—
421C2×C22⋊C4—22no—
422C2×C22⋊C4—22no—
423C2×C22⋊C4—714no—
424C2×C22⋊C4—714no—
425C2×C22⋊C4—1422no—
426C2×C22⋊C4—1422no—
427C2×C22⋊C4—1422no—
428C2×C22⋊C4—1422no—
429C2×C22⋊C4C4×D7114yes—
430C2×C42—2842no—
431C2×C42—1424no—
432C2×C42—1424no—
433#95—22no—
434#95—22no—
435#95—2842no—
436#95—2842no—
437#95—718no—
438#95—718no—
439#95—1424no—
440#95—1424no—
441#95—1424no—
442#95—1424no—
443#95—1424no—
444#95—1424no—
445#95—1424no—
446C24—22no—
447C24—22no—
448C24—22no—
449C24—22no—
450C24C2×C4×D7112yes—
451C24D4×D7112yes—
452C24C2×C22⋊D7112yes—
453C24#1176586412yes—
454C24#1176586512yes—
455C24#1176586612yes—
456C24#1176586612yes—
457C24C23×D7111yescharacteristic
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—22no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—21no—
467C22×C4—22no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—28no—
474C22×C4—28no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—22no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—44no—
486C22×C4—48no—
487C22×C4—48no—
488C22×C4—2844no—
489C22×C4—2844no—
490C22×C4—2844no—
491C22×C4—2844no—
492C22×C4—2844no—
493C22×C4—2844no—
494C22×C4—2848no—
495C22×C4—2848no—
496C22×C4—714no—
497C22×C4—714no—
498C22×C4—714no—
499C22×C4—714no—
500C22×C4—1428no—
501C22×C4—1422no—
502C22×C4—1422no—
503C22×C4—1422no—
504C22×C4—1422no—
505C22×C4—1422no—
506C22×C4—1422no—
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×C4—1422no—
518C22×C4—1422no—
519C22×C4—1422no—
520C22×C4—1422no—
521C22×C4—1422no—
522C22×C4—1422no—
523C22×C4—1422no—
524C22×C4—1422no—
525C22×C4—1428no—
526C22×C4—1428no—
527C22×C4—1428no—
528C22×C4C2×C4×D7112yes—
529C22×C4C2×C4×D7112yes—
530C22×C4C2×C22⋊D7112yes—
531C22×C4C2×C22⋊D7112yes—
532C22×C4#1176586414yes—
533C4⋊C4—44no—
534C4⋊C4—44no—
535C22⋊C4—28no—
536C22⋊C4—44no—
537C22⋊C4—1428no—
538C22⋊C4—1428no—
539C42—2848no—
540C23—22no—
541C23—24no—
542C23—24no—
543C23—24no—
544C23—24no—
545C23—24no—
546C23—24no—
547C23—24no—
548C23—24no—
549C23—24no—
550C23—24no—
551C23—24no—
552C23—22no—
553C23—22no—
554C23—24no—
555C23—22no—
556C23#1455608614yes—
557C23C4×C22⋊D7114yes—
558C23#1455609014yes—
559C23#1455610014yes—
560C23#1455611314yes—
561C23#1455611714yes—
562C23#1455612314yes—
563C23C22×C4×D7111yescharacteristic
564C23C2×D4×D7111yescharacteristic
565C23C22×C22⋊D7111yescharacteristic
566C23#41×D7112yes—
567C23C2×#1176586412yes—
568C23C2×#1176586411yescharacteristic
569C23C2×#1176586511yescharacteristic
570C23C2×#1176586611yescharacteristic
571C23C2×#1176586611yescharacteristic
572C23#1455617312yes—
573C23#1455617512yes—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—416no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—42no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—28416no—
605C2×C4—2842no—
606C2×C4—2842no—
607C2×C4—2842no—
608C2×C4—28416no—
609C2×C4—2842no—
610C2×C4—2842no—
611C2×C4—2842no—
612C2×C4—2842no—
613C2×C4—2842no—
614C2×C4—2842no—
615C2×C4—2842no—
616C2×C4—2842no—
617C2×C4—2842no—
618C2×C4—2842no—
619C2×C4—2842no—
620C2×C4—2842no—
621C2×C4—2842no—
622C2×C4—1424no—
623C2×C4—1424no—
624C2×C4—1424no—
625C2×C4—1424no—
626C2×C4—1424no—
627C2×C4—1424no—
628C2×C4—1424no—
629C2×C4—1424no—
630C2×C4—1424no—
631C2×C4—1424no—
632C2×C4—1424no—
633C2×C4—1424no—
634C2×C4#1455610914yes—
635C22—24no—
636C22—24no—
637C22—24no—
638C22—28no—
639C22—28no—
640C22—24no—
641C22—28no—
642C22—24no—
643C22—28no—
644C22—24no—
645C22#1798588918yes—
646C22C2×#1455608611yescharacteristic
647C22C2×C4×C22⋊D7111yescharacteristic
648C22C2×#1455609011yescharacteristic
649C22C4×#1176586412yes—
650C22#1798647812yes—
651C22C2×#1455610011yescharacteristic
652C22C2×#1455610911yescharacteristic
653C22C2×#1455611311yescharacteristic
654C22#1798650611yescharacteristic
655C22C2×#1455611711yescharacteristic
656C22C2×#1455612311yescharacteristic
657C22#1798652912yes—
658C22#1798655811yescharacteristic
659C22#1798657011yescharacteristic
660C22#1798659112yes—
661C22#1798663612yes—
662C22#1798664712yes—
663C22#1798667612yes—
664C22#1798667612yes—
665C22#1798668012yes—
666C22#1798668612yes—
667C22#1798669112yes—
668C22#1798669112yes—
669C4—24no—
670C4—42no—
671C4—42no—
672C4—42no—
673C4—81no—
674C4—81no—
675C4—2844no—
676C4—2844no—
677C4—2844no—
678C4—2844no—
679C2—28no—
680C2C2×#1798588911yescharacteristic
681C2C4×#1455610911yescharacteristic
682C2#2259100511yescharacteristic
683C2#2259119812yes—
684C2#2259133012yes—
685C2#2259173711yescharacteristic
686C2#2259174811yescharacteristic
687C2#2259181912yes—
688C2#2259272712yes—
689C2#2259282611yescharacteristic
690C2#2259349911yescharacteristic
subgroupG/H
center, FrattiniC24C23×D71
commutatorC23×C71C23×C4
FittingC2×#385×C71C2
socle, cosocleC24×C71C24
prime radicalC25×C71C23
hypercenterC2×#385D71
hypocenterC71#13972
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C24 ◁ C2×#385

Complements

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

NH ≅ G/Ntype
5. #22586601C2semidirect
6. #22586599C2semidirect
7. #22586572C2semidirect
8. #22586566C2semidirect
18. #17985860C4semidirect
74. C71⋊#95C2×C4semidirect
75. C71⋊#95C2×C4semidirect
357. C71#13972semidirect

Maximal subgroups

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

subgroupindex
C2×#179858602
C2×C4×C22⋊Dic712
C22×C71⋊#952
#225866012
#225865992
#225865722
#225865662
#225865622
C2×#385×C712
#1397271