Finite Group Databasebeta

← #31994858

Subgroups — #31994858

Order 18176.

2270 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×M16×D71C211yesmaximal, characteristic
2C22×#14556086C211yesmaximal, characteristic
3C2×#117×D71C211yesmaximal, characteristic
4#22588061C211yesmaximal, characteristic, complemented
5#22588060C211yesmaximal, characteristic, complemented
6#22588059C211yesmaximal, characteristic, complemented
7#22588058C214yesmaximal, complemented
8#22588055C214yesmaximal, complemented
9C71×#1736C211yesmaximal, characteristic, complemented
10C23×C4×D71C2211yescharacteristic
11C22×#11765849C2211yescharacteristic
12C22×#11765848C2212yes—
13C2×M16×D71—24no—
14C2×M16×D71—24no—
15C2×M16×D71C418yescomplemented
16C22×C8×D71C2212yes—
17C2×#14556087C2211yescharacteristic
18C2×#14556086—21no—
19C2×#14556086—21no—
20C2×#14556086—24no—
21C2×#14556086C418yescomplemented
22C2×#14556086C2212yes—
23C2×#14556086C2212yes—
24C2×#14556086C2212yes—
25C2×#14556085—21no—
26C2×#14556084—21no—
27#117×D71—24no—
28#117×D71C418yescomplemented
29C22×C4⋊Dic71C2211yescharacteristic
30C2×C4⋊C4×D71C2212yes—
31C22×(C2×D71)⋊C4—21no—
32C2×C22⋊C4×D71—21no—
33C22×C4×Dic71—21no—
34C2×C42×D71—21no—
35#17985762C2214yescomplemented
36#17985761C2214yescomplemented
37#17985761C2214yescomplemented
38C22×M16×C71C2211yescharacteristic
39C2×#117×C71C2211yescharacteristic
40C22×C4⋊C4×C71C2211yescharacteristic
41#340×C71C2214yescomplemented
42C24×D71D411yescharacteristic
43C23×Dic71—21no—
44C23×Dic71D411yescharacteristic
45C22×C4×D71—22no—
46C22×C4×D71C2×C414yes—
47C22×C4×D71C2×C414yes—
48C22×C4×D71C2×C414yes—
49C22×C4×D71D411yescharacteristic
50C22×C4×D71D411yescharacteristic
51C22×C4×D71D414yes—
52C22×C4×D71D414yes—
53C22×C4×D71Q814yes—
54C22×C4×D71C2311yescharacteristic
55C22×C4×D71C2311yescharacteristic
56C2×#11765849—22no—
57C2×#11765849—22no—
58C2×#11765849C2×C414yescomplemented
59C2×#11765848—24no—
60C2×#11765848—24no—
61C2×#11765848C2×C418yescomplemented
62M16×D71—232no—
63C22×C71⋊C8C2312yes—
64C2×C8×D71—24no—
65C2×C8×D71—24no—
66C2×C8×D71C2×C418yescomplemented
67#14556087—22no—
68#14556087C2×C414yes—
69#14556086—28no—
70#14556086—28no—
71#14556086—24no—
72#14556086—24no—
73#14556086—24no—
74#14556086C2×C418yes—
75#14556086C2×C418yescomplemented
76#14556086C2×C418yescomplemented
77#14556085—28no—
78#14556084—28no—
79C2×C4⋊Dic71—21no—
80C2×C4⋊Dic71—22no—
81C2×C4⋊Dic71C2×C414yescomplemented
82C2×C4⋊Dic71C2312yes—
83C2×C4⋊Dic71C2312yes—
84C2×Dic71⋊C4—22no—
85C4⋊C4×D71—24no—
86C4⋊C4×D71C2×C418yescomplemented
87C2×C22⋊Dic71—21no—
88C2×(C2×D71)⋊C4—28no—
89C2×(C2×D71)⋊C4—22no—
90C2×(C2×D71)⋊C4—22no—
91C2×(C2×D71)⋊C4—22no—
92C22⋊C4×D71—28no—
93C2×C4×Dic71—28no—
94C2×C4×Dic71—22no—
95C2×C4×Dic71—22no—
96C2×C4×Dic71—22no—
97C2×C4×Dic71—21no—
98C42×D71—28no—
99C23×C4×C71—21no—
100C23×C4×C71C2311yescharacteristic
101C2×M16×C71—22no—
102C2×M16×C71—22no—
103C2×M16×C71C2×C414yescomplemented
104C22×C8×C71C2312yes—
105#117×C71—22no—
106#117×C71C2×C414yes—
107C2×C4⋊C4×C71—21no—
108C2×C4⋊C4×C71—22no—
109C2×C4⋊C4×C71C2×C414yescomplemented
110C2×C4⋊C4×C71C2312yes—
111C2×C4⋊C4×C71C2312yes—
112C2×C22⋊C4×C71—21no—
113C2×C42×C71—21no—
114C23×D71—42no—
115C23×D71—42no—
116C23×D71C22⋊C414yes—
117C23×D71C22⋊C414yes—
118C23×D71C4⋊C414yes—
119C23×D71C2×D411yescharacteristic
120C23×D71C2×D411yescharacteristic
121C22×Dic71—24no—
122C22×Dic71—22no—
123C22×Dic71—22no—
124C22×Dic71—44no—
125C22×Dic71C22⋊C414yes—
126C22×Dic71C22⋊C414yes—
127C22×Dic71C4⋊C414yes—
128C22×Dic71C2×D411yescharacteristic
129C22×Dic71C2×D411yescharacteristic
130C2×C4×D71—28no—
131C2×C4×D71—44no—
132C2×C4×D71—44no—
133C2×C4×D71C42116yes—
134C2×C4×D71C22⋊C414yes—
135C2×C4×D71C22⋊C414yes—
136C2×C4×D71C22⋊C414yes—
137C2×C4×D71C22⋊C414yes—
138C2×C4×D71C22⋊C4116yes—
139C2×C4×D71C4⋊C414yes—
140C2×C4×D71C4⋊C414yes—
141C2×C4×D71C4⋊C4116yes—
142C2×C4×D71C4⋊C4116yes—
143C2×C4×D71C22×C412yes—
144C2×C4×D71C22×C412yes—
145C2×C4×D71C22×C412yes—
146C2×C4×D71C22×C412yes—
147C2×C4×D71C22×C412yes—
148C2×C4×D71C22×C412yes—
149C2×C4×D71C2×D412yes—
150C2×C4×D71C2×D412yes—
151C2×C4×D71C2×D412yes—
152C2×C4×D71C2×D412yes—
153C2×C4×D71C2×Q812yes—
154C2×C4×D71C2×Q812yes—
155#11765849—28no—
156#11765848—216no—
157C2×C71⋊C8—22no—
158C2×C71⋊C8—22no—
159C2×C71⋊C8C22×C414yes—
160C8×D71—216no—
161C4⋊Dic71—24no—
162C4⋊Dic71—22no—
163C4⋊Dic71—22no—
164C4⋊Dic71C22×C414yes—
165C4⋊Dic71C22×C414yes—
166Dic71⋊C4—28no—
167C22⋊Dic71—24no—
168(C2×D71)⋊C4—28no—
169(C2×D71)⋊C4—28no—
170(C2×D71)⋊C4—28no—
171C4×Dic71—24no—
172C4×Dic71—28no—
173C4×Dic71—28no—
174C4×Dic71—28no—
175C24×C71C2×D411yescharacteristic
176C22×C4×C71—24no—
177C22×C4×C71—22no—
178C22×C4×C71—22no—
179C22×C4×C71—44no—
180C22×C4×C71C22×C412yes—
181C22×C4×C71C22×C412yes—
182C22×C4×C71C22×C412yes—
183C22×C4×C71C2×D411yescharacteristic
184C22×C4×C71C2×D412yes—
185C22×C4×C71C2×D412yes—
186C22×C4×C71C2×Q812yes—
187C22×C4×C71C2411yescharacteristic
188M16×C71—28no—
189C2×C8×C71—22no—
190C2×C8×C71—22no—
191C2×C8×C71C22×C414yes—
192C4⋊C4×C71—24no—
193C4⋊C4×C71—22no—
194C4⋊C4×C71—22no—
195C4⋊C4×C71C22×C414yes—
196C4⋊C4×C71C22×C414yes—
197C22⋊C4×C71—24no—
198C42×C71—24no—
199C23×C71—22no—
200C23×C71—22no—
201C23×C71C2×C22⋊C412yes—
202C23×C71C2×C22⋊C412yes—
203C23×C71C2×C4⋊C412yes—
204C23×C71C22×D411yescharacteristic
205C22×D71—22no—
206C22×D71—22no—
207C22×D71—22no—
208C22×D71—22no—
209C22×D71—48no—
210C22×D71—44no—
211C22×D71—44no—
212C22×D71—48no—
213C22×D71#95116yes—
214C22×D71C2×C22⋊C412yes—
215C22×D71C2×C22⋊C412yes—
216C22×D71C2×C22⋊C412yes—
217C22×D71C2×C22⋊C412yes—
218C22×D71C2×C4⋊C412yes—
219C22×D71C2×C4⋊C412yes—
220C2×C4×C71—24no—
221C2×C4×C71—24no—
222C2×C4×C71—48no—
223C2×C4×C71—44no—
224C2×C4×C71—42no—
225C2×C4×C71—42no—
226C2×C4×C71C2×C4214yes—
227C2×C4×C71C2×C22⋊C414yes—
228C2×C4×C71C2×C22⋊C412yes—
229C2×C4×C71C2×C22⋊C412yes—
230C2×C4×C71C2×C4⋊C414yes—
231C2×C4×C71C2×C4⋊C414yes—
232C2×C4×C71C2×C4⋊C412yes—
233C2×C4×C71C23×C411yescharacteristic
234C2×C4×C71C23×C411yescharacteristic
235C2×C4×C71C23×C411yescharacteristic
236C2×C4×C71C22×D411yescharacteristic
237C2×C4×C71C22×D411yescharacteristic
238C2×C4×C71C22×Q811yescharacteristic
239C2×Dic71—24no—
240C2×Dic71—24no—
241C2×Dic71—48no—
242C2×Dic71—42no—
243C2×Dic71—42no—
244C2×Dic71—44no—
245C2×Dic71#95116yes—
246C2×Dic71C2×C22⋊C412yes—
247C2×Dic71C2×C22⋊C412yes—
248C2×Dic71C2×C22⋊C412yes—
249C2×Dic71C2×C22⋊C412yes—
250C2×Dic71C2×C4⋊C412yes—
251C2×Dic71C2×C4⋊C412yes—
252C4×D71—416no—
253C4×D71#95116yes—
254C4×D71#95116yes—
255C4×D71C2×C4214yes—
256C4×D71C2×C4214yes—
257C4×D71C2×C22⋊C414yes—
258C4×D71C2×C22⋊C414yes—
259C4×D71C2×C4⋊C414yes—
260C4×D71C2×C4⋊C414yes—
261C4×D71C2×C4⋊C414yes—
262C4×D71C2×C4⋊C414yes—
263C8×C71—24no—
264C71⋊C8—24no—
265C22×C71—22no—
266C22×C71—22no—
267C22×C71—24no—
268C22×C71—24no—
269C22×C71C2×#9514yes—
270C22×C71C22×C22⋊C411yescharacteristic
271C22×C71C22×C22⋊C411yescharacteristic
272C22×C71C22×C4⋊C411yescharacteristic
273C22×C71#57512yes—
274C22×C71#57512yes—
275C2×D71—24no—
276C2×D71—22no—
277C2×D71—22no—
278C2×D71—24no—
279C2×D71—22no—
280C2×D71—22no—
281C2×D71—24no—
282C2×D71—24no—
283C2×D71—416no—
284C2×D71C2×#9514yes—
285C2×D71C2×#9514yes—
286C4×C71—44no—
287C4×C71—44no—
288C4×C71C2×#9514yes—
289C4×C71C22×C4211yescharacteristic
290C4×C71C22×C22⋊C411yescharacteristic
291C4×C71C22×C4⋊C411yescharacteristic
292C4×C71C22×C4⋊C411yescharacteristic
293Dic71—44no—
294Dic71—44no—
295Dic71C2×#9514yes—
296Dic71C2×#9514yes—
297#20347—711nomaximal, Sylow
298C2×C71—24no—
299C2×C71C22×#9511yescharacteristic
300C2×C71#289611yescharacteristic
301C2×C71#289612yes—
302C2×C71#289611yescharacteristic
303C2×C71#291112yes—
304D71—24no—
305D71—24no—
306C23×M16—711no—
307C22×#117—711no—
308C22×#117—711no—
309#1736—714no—
310#1736—711no—
311#1736—711no—
312#1736—711no—
313#1736D7111yescharacteristic, complemented
314#1735—714no—
315C71#2034711yescharacteristic, complemented, Sylow
316C24×C4—711no—
317C22×M16—711no—
318C22×M16—718no—
319C22×M16—712no—
320C22×M16—1424no—
321C22×M16—1424no—
322C22×M16C2×D7111yescharacteristic
323C23×C8—712no—
324C2×#117—711no—
325C2×#117—718no—
326C2×#117—718no—
327C2×#117—712no—
328C2×#117—712no—
329C2×#117—712no—
330C2×#117—1421no—
331C2×#117—1421no—
332C2×#117—1421no—
333C2×#117—1424no—
334C2×#117—1421no—
335C2×#117—1424no—
336C2×#117C2×D7111yescharacteristic
337C22×C4⋊C4—711no—
338C22×C4⋊C4—712no—
339C22×C4⋊C4C2×D7111yescharacteristic
340C22×C22⋊C4—1421no—
341C22×C22⋊C4—1421no—
342C22×C42—1421no—
343C22×C42—1421no—
344#340—714no—
345#340—714no—
346#340—714no—
347#340C2×D7114yescomplemented
348C25—711no—
349C23×C4—21no—
350C23×C4—711no—
351C23×C4—714no—
352C23×C4—714no—
353C23×C4—714no—
354C23×C4—711no—
355C23×C4—714no—
356C23×C4—714no—
357C23×C4—711no—
358C23×C4—714no—
359C23×C4—711no—
360C23×C4—711no—
361C23×C4—1422no—
362C23×C4—1421no—
363C23×C4C22×D7111yescharacteristic
364C2×M16—22no—
365C2×M16—22no—
366C2×M16—714no—
367C2×M16—718no—
368C2×M16—1424no—
369C2×M16—1424no—
370C2×M16—14232no—
371C2×M16—1422no—
372C2×M16—1422no—
373C2×M16C4×D7114yescomplemented
374C22×C8—712no—
375C22×C8—718no—
376C22×C8—1424no—
377C22×C8—1424no—
378C22×C8C22×D7112yes—
379#117—22no—
380#117—714no—
381#117—718no—
382#117—718no—
383#117—718no—
384#117—1428no—
385#117—1428no—
386#117—1424no—
387#117—1424no—
388#117—1424no—
389#117—1428no—
390#117—1428no—
391#117—1422no—
392#117C4×D7114yes—
393C2×C4⋊C4—22no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—714no—
396C2×C4⋊C4—712no—
397C2×C4⋊C4—712no—
398C2×C4⋊C4—718no—
399C2×C4⋊C4—1424no—
400C2×C4⋊C4—1422no—
401C2×C4⋊C4—1422no—
402C2×C4⋊C4—1421no—
403C2×C4⋊C4C4×D7114yescomplemented
404C2×C4⋊C4C22×D7112yes—
405C2×C4⋊C4C22×D7112yes—
406C2×C22⋊C4—21no—
407C2×C22⋊C4—1428no—
408C2×C22⋊C4—1422no—
409C2×C22⋊C4—1422no—
410C2×C22⋊C4—1422no—
411C2×C22⋊C4—1428no—
412C2×C22⋊C4—1421no—
413C2×C42—21no—
414C2×C42—1428no—
415C2×C42—1422no—
416C2×C42—1422no—
417C2×C42—1422no—
418C2×C42—1428no—
419C2×C42—1421no—
420C24—2842no—
421C24—2842no—
422C24—714no—
423C24—714no—
424C24—714no—
425C24—711no—
426C24—711no—
427C24D4×D7111yescharacteristic
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—44no—
432C22×C4—2844no—
433C22×C4—2844no—
434C22×C4—2844no—
435C22×C4—711no—
436C22×C4—711no—
437C22×C4—714no—
438C22×C4—714no—
439C22×C4—714no—
440C22×C4—712no—
441C22×C4—712no—
442C22×C4—712no—
443C22×C4—712no—
444C22×C4—712no—
445C22×C4—712no—
446C22×C4—712no—
447C22×C4—712no—
448C22×C4—712no—
449C22×C4—712no—
450C22×C4—712no—
451C22×C4—712no—
452C22×C4—7116no—
453C22×C4—714no—
454C22×C4—714no—
455C22×C4—714no—
456C22×C4—714no—
457C22×C4—7116no—
458C22×C4—7116no—
459C22×C4—7116no—
460C22×C4—714no—
461C22×C4—714no—
462C22×C4—1428no—
463C22×C4—1422no—
464C22×C4—1422no—
465C22×C4—1424no—
466C22×C4C2×C4×D7112yes—
467C22×C4C2×C4×D7112yes—
468C22×C4C2×C4×D7112yes—
469C22×C4D4×D7112yes—
470C22×C4D4×D7112yes—
471C22×C4D4×D7111yescharacteristic
472C22×C4Q8×D7112yes—
473C22×C4C23×D7111yescharacteristic
474M16—28no—
475M16—14216no—
476M16—1428no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—714no—
480C2×C8—14216no—
481C2×C8—1422no—
482C2×C8—1422no—
483C2×C8C2×C4×D7114yes—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—24no—
487C4⋊C4—714no—
488C4⋊C4—714no—
489C4⋊C4—1428no—
490C4⋊C4—1422no—
491C4⋊C4—1422no—
492C4⋊C4—1424no—
493C4⋊C4C2×C4×D7114yes—
494C4⋊C4C2×C4×D7114yes—
495C22⋊C4—24no—
496C22⋊C4—1428no—
497C22⋊C4—1428no—
498C22⋊C4—1428no—
499C22⋊C4—1424no—
500C42—24no—
501C42—1428no—
502C42—1428no—
503C42—1428no—
504C42—1424no—
505C23—22no—
506C23—22no—
507C23—2848no—
508C23—2844no—
509C23—2848no—
510C23—2844no—
511C23—7116no—
512C23—712no—
513C23—712no—
514C23—712no—
515C23—712no—
516C23—712no—
517C23—712no—
518C23—1422no—
519C23—1422no—
520C23—1422no—
521C23—1422no—
522C23C22⋊C4×D7112yes—
523C23C22⋊C4×D7112yes—
524C23C4⋊C4×D7112yes—
525C23C2×D4×D7111yescharacteristic
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—48no—
529C2×C4—44no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—28416no—
533C2×C4—2848no—
534C2×C4—2842no—
535C2×C4—2842no—
536C2×C4—2844no—
537C2×C4—7116no—
538C2×C4—712no—
539C2×C4—712no—
540C2×C4—712no—
541C2×C4—712no—
542C2×C4—712no—
543C2×C4—712no—
544C2×C4—7116no—
545C2×C4—7116no—
546C2×C4—714no—
547C2×C4—714no—
548C2×C4—714no—
549C2×C4—714no—
550C2×C4—714no—
551C2×C4—714no—
552C2×C4—714no—
553C2×C4—714no—
554C2×C4—1424no—
555C2×C4—1424no—
556C2×C4C42×D7114yes—
557C2×C4C22⋊C4×D7112yes—
558C2×C4C22⋊C4×D7112yes—
559C2×C4C22⋊C4×D7114yes—
560C2×C4C4⋊C4×D7112yes—
561C2×C4C4⋊C4×D7114yes—
562C2×C4C4⋊C4×D7114yes—
563C2×C4C22×C4×D7111yescharacteristic
564C2×C4C22×C4×D7111yescharacteristic
565C2×C4C22×C4×D7111yescharacteristic
566C2×C4C2×D4×D7111yescharacteristic
567C2×C4C2×D4×D7111yescharacteristic
568C2×C4C2×Q8×D7111yescharacteristic
569C8—24no—
570C8—1424no—
571C22—24no—
572C22—22no—
573C22—24no—
574C22—22no—
575C22—28416no—
576C22—714no—
577C22—714no—
578C22—1424no—
579C22—1424no—
580C22—1422no—
581C22—1422no—
582C22—1424no—
583C22—1424no—
584C22—1422no—
585C22—1422no—
586C22#95×D7114yes—
587C22C2×C22⋊C4×D7111yescharacteristic
588C22C2×C22⋊C4×D7111yescharacteristic
589C22C2×C4⋊C4×D7111yescharacteristic
590C22#1798679512yes—
591C22#1798679812yes—
592C4—44no—
593C4—44no—
594C4—2844no—
595C4—2844no—
596C4—714no—
597C4—714no—
598C4#95×D7114yes—
599C4C2×C42×D7111yescharacteristic
600C4C2×C22⋊C4×D7111yescharacteristic
601C4C2×C4⋊C4×D7111yescharacteristic
602C4C2×C4⋊C4×D7111yescharacteristic
603C2—24no—
604C2—1424no—
605C2—1424no—
606C2C2×#95×D7111yescharacteristic
607C2#2259685411yescharacteristic
608C2#2259685612yes—
609C2#2259685811yescharacteristic
610C2#2259692712yes—
subgroupG/H
centerC23C2×D4×D71
commutatorC4×C71C22×C42
FrattiniC22×C4C23×D71
FittingC71×#1736C2
socleC23×C71C22×D4
prime radicalC24×D71D4
cosocleC22×C4×C71C24
hypercenter#1736D71
hypocenterC71#20347
LCSG ▷ C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1736

Complements

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

NH ≅ G/Ntype
4. #22588061C2semidirect
5. #22588060C2semidirect
6. #22588059C2semidirect
7. #22588058C2semidirect
8. #22588055C2semidirect
9. C71×#1736C2semidirect
15. C2×M16×D71C4semidirect
21. C2×#14556086C4semidirect
28. #117×D71C4semidirect
35. #17985762C22semidirect
36. #17985761C22semidirect
37. #17985761C22semidirect
41. #340×C71C22semidirect
58. C2×#11765849C2×C4semidirect
61. C2×#11765848C2×C4semidirect
66. C2×C8×D71C2×C4semidirect
75. #14556086C2×C4semidirect
76. #14556086C2×C4semidirect
81. C2×C4⋊Dic71C2×C4semidirect
86. C4⋊C4×D71C2×C4semidirect
103. C2×M16×C71C2×C4semidirect
109. C2×C4⋊C4×C71C2×C4semidirect
313. #1736D71semidirect
315. C71#20347semidirect
347. #340C2×D71semidirect
373. C2×M16C4×D71semidirect
403. C2×C4⋊C4C4×D71semidirect

Maximal subgroups

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

subgroupindex
C22×M16×D712
C22×#145560862
C2×#117×D712
#225880612
#225880602
#225880592
#225880582
#225880552
C71×#17362
#2034771