Finite Group Databasebeta

← #31920103

Subgroups — #31920103

Order 18176.

2006 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14556085C211yesmaximal, characteristic
2C2×#17985894C212yesmaximal, complemented
3C2×#17985858C211yesmaximal, characteristic
4C2×#17985846C211yesmaximal, characteristic
5#22586584C212yesmaximal
6#22586582C212yesmaximal
7Dic71⋊#95C212yesmaximal, complemented
8Dic71⋊#95C211yesmaximal, characteristic, complemented
9#22586567C211yesmaximal, characteristic, complemented
10#22586566C211yesmaximal, characteristic, complemented
11C71×C4⋊#95C211yesmaximal, characteristic, complemented
12C23×C4×D71C2211yescharacteristic
13C2×#14556085—28no—
14C2×#14556085C418yescomplemented
15C22×Dic71⋊C4C2212yes—
16C22×Dic71⋊C4C2212yes—
17C22×Dic71⋊C4C2211yescharacteristic
18C22×Dic71⋊C4C2211yescharacteristic
19C22×(C2×D71)⋊C4C2212yes—
20C22×(C2×D71)⋊C4C2211yescharacteristic
21C22×(C2×D71)⋊C4C2211yescharacteristic
22C22×(C2×D71)⋊C4C2212yes—
23C22×C4×Dic71C2212yes—
24C22×C4×Dic71C2211yescharacteristic
25#17985894—28no—
26#17985858—24no—
27#17985846—24no—
28C2×C71⋊#95C2212yes—
29C2×C71⋊#95C2212yes—
30C2×C71⋊#95C2211yescharacteristic
31C2×C71⋊#95C2212yes—
32C2×C71⋊#95C2212yes—
33C2×C71⋊#95C2211yescharacteristic
34C2×C71⋊#95C2212yes—
35C22×C4⋊C4×C71C2212yes—
36C22×C42×C71C2211yescharacteristic
37C2×#95×C71C2211yescharacteristic
38C2×#95×C71C2211yescharacteristic
39C2×#95×C71C2212yes—
40C24×D71C2311yescharacteristic
41C23×Dic71C2312yes—
42C23×Dic71C2311yescharacteristic
43C23×Dic71C2312yes—
44C23×Dic71C2311yescharacteristic
45C23×Dic71C2311yescharacteristic
46C22×C4×D71—24no—
47C22×C4×D71—24no—
48C22×C4×D71C2×C414yes—
49C22×C4×D71D414yes—
50C22×C4×D71Q814yes—
51#14556085—232no—
52C2×Dic71⋊C4—24no—
53C2×Dic71⋊C4—24no—
54C2×Dic71⋊C4—24no—
55C2×Dic71⋊C4—24no—
56C2×Dic71⋊C4—24no—
57C2×Dic71⋊C4—24no—
58C2×Dic71⋊C4—42no—
59C2×Dic71⋊C4C2×C414yes—
60C2×Dic71⋊C4C2×C414yes—
61C2×Dic71⋊C4D418yes—
62C2×(C2×D71)⋊C4—24no—
63C2×(C2×D71)⋊C4—24no—
64C2×(C2×D71)⋊C4—24no—
65C2×(C2×D71)⋊C4—24no—
66C2×(C2×D71)⋊C4—24no—
67C2×(C2×D71)⋊C4—24no—
68C2×(C2×D71)⋊C4—42no—
69C2×(C2×D71)⋊C4—42no—
70C2×(C2×D71)⋊C4C2×C414yes—
71C2×(C2×D71)⋊C4C2×C414yes—
72C2×C4×Dic71—24no—
73C2×C4×Dic71—24no—
74C2×C4×Dic71—24no—
75C2×C4×Dic71—24no—
76C2×C4×Dic71C2×C414yes—
77C71⋊#95—24no—
78C71⋊#95—24no—
79C71⋊#95—24no—
80C71⋊#95—22no—
81C71⋊#95—22no—
82C71⋊#95—24no—
83C71⋊#95—42no—
84C23×C4×C71C2312yes—
85C23×C4×C71C2311yescharacteristic
86C23×C4×C71C2312yes—
87C23×C4×C71C2311yescharacteristic
88C23×C4×C71C2311yescharacteristic
89C2×C4⋊C4×C71—24no—
90C2×C4⋊C4×C71—24no—
91C2×C4⋊C4×C71—42no—
92C2×C42×C71—24no—
93C2×C42×C71C2×C414yes—
94#95×C71—22no—
95#95×C71—22no—
96#95×C71—42no—
97C23×D71—22no—
98C23×D71—22no—
99C23×D71—22no—
100C23×D71—22no—
101C23×D71—21no—
102C23×D71—21no—
103C23×D71—21no—
104C23×D71—21no—
105C23×D71C22×C412yes—
106C23×D71C2×D412yes—
107C23×D71C2×Q812yes—
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—21no—
118C22×Dic71—21no—
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×Dic71C22×C412yes—
129C22×Dic71C22×C412yes—
130C22×Dic71C22×C412yes—
131C22×Dic71C2×D414yes—
132C22×Dic71C2×D414yes—
133C22×Dic71C2×D412yes—
134C22×Dic71C2×Q812yes—
135C22×Dic71#4114yes—
136C22×Dic71#4114yes—
137C2×C4×D71—28no—
138C2×C4×D71—28no—
139C2×C4×D71—216no—
140C2×C4×D71—216no—
141C2×C4×D71C4⋊C4116yescomplemented
142Dic71⋊C4—28no—
143Dic71⋊C4—48no—
144Dic71⋊C4—48no—
145Dic71⋊C4C22⋊C4116yescomplemented
146(C2×D71)⋊C4—216no—
147(C2×D71)⋊C4—28no—
148(C2×D71)⋊C4—28no—
149(C2×D71)⋊C4—48no—
150C4×Dic71—28no—
151C4×Dic71—48no—
152C24×C71C2411yescharacteristic
153C22×C4×C71—22no—
154C22×C4×C71—22no—
155C22×C4×C71—22no—
156C22×C4×C71—21no—
157C22×C4×C71—21no—
158C22×C4×C71—22no—
159C22×C4×C71—22no—
160C22×C4×C71—22no—
161C22×C4×C71—22no—
162C22×C4×C71—22no—
163C22×C4×C71—22no—
164C22×C4×C71—21no—
165C22×C4×C71—22no—
166C22×C4×C71—22no—
167C22×C4×C71—21no—
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×C71C22×C412yes—
176C22×C4×C71C22×C412yes—
177C22×C4×C71C22×C412yes—
178C22×C4×C71C2×D414yes—
179C22×C4×C71C2×D412yes—
180C22×C4×C71C2×Q812yes—
181C22×C4×C71#4114yes—
182C4⋊C4×C71—48no—
183C42×C71—28no—
184C23×C71C23×C411yescharacteristic
185C23×C71C22×D412yes—
186C23×C71C22×D411yescharacteristic
187C23×C71C22×Q811yescharacteristic
188C23×C71C2×#4112yes—
189C23×C71C2×#4112yes—
190C23×C71C2×#4112yes—
191C23×C71H(5,𝔽2)11yescharacteristic
192C23×C71H(5,𝔽2)11yescharacteristic
193C23×C712−1+411yescharacteristic
194C23×C712−1+411yescharacteristic
195C22×D71—22no—
196C22×D71—22no—
197C22×D71—22no—
198C22×D71—22no—
199C22×D71—22no—
200C22×D71—22no—
201C22×D71—24no—
202C22×D71—24no—
203C22×D71—24no—
204C22×D71—24no—
205C22×D71—22no—
206C22×D71—22no—
207C22×D71—22no—
208C22×D71—22no—
209C22×D71—42no—
210C22×D71—42no—
211C22×D71—42no—
212C22×D71—42no—
213C22×D71—42no—
214C22×D71—42no—
215C22×D71—42no—
216C22×D71—42no—
217C22×D71C2×C4⋊C414yes—
218C2×C4×C71—22no—
219C2×C4×C71—22no—
220C2×C4×C71—22no—
221C2×C4×C71—22no—
222C2×C4×C71—24no—
223C2×C4×C71—24no—
224C2×C4×C71—24no—
225C2×C4×C71—24no—
226C2×C4×C71—22no—
227C2×C4×C71—22no—
228C2×C4×C71—22no—
229C2×C4×C71—24no—
230C2×C4×C71—24no—
231C2×C4×C71—24no—
232C2×C4×C71—42no—
233C2×C4×C71—42no—
234C2×C4×C71—42no—
235C2×C4×C71—42no—
236C2×C4×C71—42no—
237C2×C4×C71—42no—
238C2×C4×C71—42no—
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×C71C2×C22⋊C414yes—
246C2×C4×C71C2×C4⋊C414yes—
247C2×Dic71—22no—
248C2×Dic71—22no—
249C2×Dic71—24no—
250C2×Dic71—24no—
251C2×Dic71—22no—
252C2×Dic71—22no—
253C2×Dic71—22no—
254C2×Dic71—24no—
255C2×Dic71—24no—
256C2×Dic71—24no—
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×Dic71—42no—
266C2×Dic71—42no—
267C2×Dic71—42no—
268C2×Dic71—42no—
269C2×Dic71—42no—
270C2×Dic71C2×C22⋊C414yes—
271C2×Dic71C2×C22⋊C414yes—
272C2×Dic71C2×C4⋊C414yes—
273C2×Dic71#11714yes—
274C2×Dic71#12118yes—
275C2×Dic71C22⋊Q818yes—
276C4×D71—232no—
277C22×C71C22×C22⋊C411yescharacteristic
278C22×C71C22×C4⋊C411yescharacteristic
279C22×C71C2×#11711yescharacteristic
280C22×C71C4×#4111yescharacteristic
281C22×C71C4×#4111yescharacteristic
282C22×C71#52011yescharacteristic
283C22×C71#52011yescharacteristic
284C22×C71C2×#12112yes—
285C22×C71C2×C22⋊Q812yes—
286C22×C71#52512yes—
287C22×C71#53312yes—
288C22×C71#53511yescharacteristic
289C22×C71#53611yescharacteristic
290C22×C71#54411yescharacteristic
291C22×C71#54411yescharacteristic
292C22×C71#54612yes—
293C22×C71#54712yes—
294C22×C71#54812yes—
295C22×C71#55012yes—
296C22×C71#55212yes—
297C22×C71#55312yes—
298C22×C71#55512yes—
299C22×C71#55612yes—
300C2×D71—24no—
301C2×D71—24no—
302C2×D71—24no—
303C2×D71—42no—
304C2×D71—42no—
305C2×D71—42no—
306C2×D71—42no—
307C2×D71—44no—
308C2×D71—44no—
309C2×D71—44no—
310C2×D71—44no—
311C4×C71—24no—
312C4×C71—24no—
313C4×C71—42no—
314C4×C71—44no—
315C4×C71—82no—
316Dic71—24no—
317Dic71—42no—
318Dic71—44no—
319Dic71—82no—
320Dic71#38018yes—
321#13951—711nomaximal, Sylow
322C2×C71C2×#38011yescharacteristic
323C2×C71#229611yescharacteristic
324C2×C71#231711yescharacteristic
325C2×C71#233911yescharacteristic
326C2×C71#234111yescharacteristic
327C2×C71#234611yescharacteristic
328C2×C71#234711yescharacteristic
329C2×C71#241412yes—
330C2×C71#241512yes—
331C2×C71#243912yes—
332C2×C71#244112yes—
333D71—44no—
334C22×#117—711no—
335C2×#389—712no—
336C2×#380—711no—
337C2×C4×C22⋊C4—711no—
338#1439—712no—
339#1439—712no—
340C4⋊#95—711no—
341C4⋊#95—712no—
342C4⋊#95D7111yescharacteristic, complemented
343#1434—711no—
344#1434—711no—
345C71#1395111yescharacteristic, complemented, Sylow
346C24×C4—711no—
347C2×#117—718no—
348C2×#117—1428no—
349C22×C4⋊C4—712no—
350C22×C4⋊C4—711no—
351C22×C4⋊C4—711no—
352C22×C4⋊C4—712no—
353C22×C4⋊C4C2×D7112yes—
354C22×C22⋊C4—712no—
355C22×C22⋊C4—711no—
356C22×C22⋊C4—711no—
357C22×C22⋊C4—712no—
358C22×C42—711no—
359C22×C42—712no—
360C22×C42C2×D7111yescharacteristic
361#389—1428no—
362#380—1424no—
363C4×C22⋊C4—1424no—
364C2×#95—712no—
365C2×#95—711no—
366C2×#95—712no—
367C2×#95—712no—
368C2×#95—711no—
369C2×#95—712no—
370C2×#95—712no—
371C2×#95C2×D7111yescharacteristic
372C2×#95C2×D7111yescharacteristic
373C2×#95C2×D7112yes—
374C25—711no—
375C23×C4—711no—
376C23×C4—711no—
377C23×C4—714no—
378C23×C4—714no—
379C23×C4—714no—
380C23×C4—711no—
381C23×C4—712no—
382C23×C4—712no—
383C23×C4—1424no—
384C23×C4—1424no—
385C23×C4C22×D7112yes—
386C23×C4C22×D7111yescharacteristic
387C23×C4C22×D7112yes—
388C23×C4C22×D7111yescharacteristic
389C23×C4C22×D7111yescharacteristic
390#117—14232no—
391C2×C4⋊C4—24no—
392C2×C4⋊C4—24no—
393C2×C4⋊C4—42no—
394C2×C4⋊C4—2842no—
395C2×C4⋊C4—714no—
396C2×C4⋊C4—718no—
397C2×C4⋊C4—714no—
398C2×C4⋊C4—1424no—
399C2×C4⋊C4—1424no—
400C2×C4⋊C4—1424no—
401C2×C4⋊C4—1424no—
402C2×C4⋊C4—1424no—
403C2×C4⋊C4—1424no—
404C2×C22⋊C4—2842no—
405C2×C22⋊C4—2842no—
406C2×C22⋊C4—714no—
407C2×C22⋊C4—714no—
408C2×C22⋊C4—1424no—
409C2×C22⋊C4—1424no—
410C2×C22⋊C4—1424no—
411C2×C22⋊C4—1424no—
412C2×C22⋊C4—1424no—
413C2×C22⋊C4—1424no—
414C2×C42—24no—
415C2×C42—714no—
416C2×C42—1424no—
417C2×C42—1424no—
418C2×C42—1424no—
419C2×C42—1424no—
420C2×C42C4×D7114yes—
421#95—22no—
422#95—22no—
423#95—42no—
424#95—2842no—
425#95—1424no—
426#95—1424no—
427#95—1424no—
428#95—1422no—
429#95—1422no—
430#95—1424no—
431C24—712no—
432C24—712no—
433C24—712no—
434C24—1422no—
435C24—1422no—
436C24—1422no—
437C24—1422no—
438C24—1421no—
439C24—1421no—
440C24—1421no—
441C24—1421no—
442C24C23×D7111yescharacteristic
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—21no—
455C22×C4—22no—
456C22×C4—22no—
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—712no—
466C22×C4—712no—
467C22×C4—714no—
468C22×C4—712no—
469C22×C4—714no—
470C22×C4—712no—
471C22×C4—714no—
472C22×C4—7116no—
473C22×C4—712no—
474C22×C4—714no—
475C22×C4—1428no—
476C22×C4—1428no—
477C22×C4—14216no—
478C22×C4—14216no—
479C22×C4—1422no—
480C22×C4—1422no—
481C22×C4—1422no—
482C22×C4—1422no—
483C22×C4—1422no—
484C22×C4—1422no—
485C22×C4—1422no—
486C22×C4—1422no—
487C22×C4—1422no—
488C22×C4—1422no—
489C22×C4—1422no—
490C22×C4—1422no—
491C22×C4—1422no—
492C22×C4—1421no—
493C22×C4—1422no—
494C22×C4—1422no—
495C22×C4—1421no—
496C22×C4—1422no—
497C22×C4—1421no—
498C22×C4—1421no—
499C22×C4C2×C4×D7112yes—
500C22×C4C2×C4×D7112yes—
501C22×C4C2×C4×D7112yes—
502C22×C4D4×D7114yes—
503C22×C4D4×D7112yes—
504C22×C4Q8×D7112yes—
505C22×C4#1176586514yes—
506C4⋊C4—48no—
507C4⋊C4—2848no—
508C4⋊C4—2848no—
509C4⋊C4—7116no—
510C4⋊C4—1428no—
511C22⋊C4—2848no—
512C22⋊C4—1428no—
513C22⋊C4—14216no—
514C22⋊C4—1428no—
515C42—28no—
516C42—2848no—
517C42—1428no—
518C23—2842no—
519C23—2842no—
520C23—2842no—
521C23—2842no—
522C23—2842no—
523C23—2842no—
524C23—2842no—
525C23—2842no—
526C23—714no—
527C23—1422no—
528C23—1422no—
529C23—1422no—
530C23—1422no—
531C23—1422no—
532C23—1422no—
533C23—1422no—
534C23—1422no—
535C23—1422no—
536C23—1422no—
537C23—1424no—
538C23—1424no—
539C23—1424no—
540C23—1424no—
541C23C22×C4×D7111yescharacteristic
542C23C2×D4×D7111yescharacteristic
543C23C2×D4×D7112yes—
544C23C2×Q8×D7111yescharacteristic
545C23#41×D7112yes—
546C23#41×D7112yes—
547C23C2×#1176586512yes—
548C23#1455617311yescharacteristic
549C23#1455617311yescharacteristic
550C23#1455617611yescharacteristic
551C23#1455617611yescharacteristic
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—24no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—2842no—
580C2×C4—2842no—
581C2×C4—2842no—
582C2×C4—2842no—
583C2×C4—2842no—
584C2×C4—2842no—
585C2×C4—2842no—
586C2×C4—2842no—
587C2×C4—2842no—
588C2×C4—2842no—
589C2×C4—2842no—
590C2×C4—2842no—
591C2×C4—2842no—
592C2×C4—714no—
593C2×C4—714no—
594C2×C4—714no—
595C2×C4—718no—
596C2×C4—718no—
597C2×C4—714no—
598C2×C4—14232no—
599C2×C4—1424no—
600C2×C4—1424no—
601C2×C4—1422no—
602C2×C4—1422no—
603C2×C4—1422no—
604C2×C4—1424no—
605C2×C4—1424no—
606C2×C4—1424no—
607C2×C4—1422no—
608C2×C4—1422no—
609C2×C4C22⋊C4×D7114yes—
610C2×C4C4⋊C4×D7114yes—
611C22—2842no—
612C22—2844no—
613C22—2844no—
614C22—2844no—
615C22—2844no—
616C22—2842no—
617C22—2842no—
618C22—2842no—
619C22—1424no—
620C22—1424no—
621C22—1424no—
622C22C2×C22⋊C4×D7111yescharacteristic
623C22C2×C4⋊C4×D7111yescharacteristic
624C22#117×D7111yescharacteristic
625C22#1798646711yescharacteristic
626C22C4×#1176586511yescharacteristic
627C22#1798648011yescharacteristic
628C22#1798648011yescharacteristic
629C22#121×D7112yes—
630C22C22⋊Q8×D7112yes—
631C22#1798650812yes—
632C22#1798655312yes—
633C22#1798656211yescharacteristic
634C22#1798657411yescharacteristic
635C22#1798662611yescharacteristic
636C22#1798662611yescharacteristic
637C22#1798663912yes—
638C22#1798665012yes—
639C22#1798665912yes—
640C22#1798666712yes—
641C22#1798668112yes—
642C22#1798669812yes—
643C22#1798671112yes—
644C22#1798672112yes—
645C4—24no—
646C4—24no—
647C4—44no—
648C4—42no—
649C4—82no—
650C4—2844no—
651C4—2842no—
652C4—5682no—
653C4—718no—
654C4—1424no—
655C2—2844no—
656C2#380×D7111yescharacteristic
657C2#2259082011yescharacteristic
658C2#2259096111yescharacteristic
659C2#2259116711yescharacteristic
660C2#2259117711yescharacteristic
661C2#2259122011yescharacteristic
662C2#2259123411yescharacteristic
663C2#2259175512yes—
664C2#2259176812yes—
665C2#2259202812yes—
666C2#2259205012yes—
subgroupG/H
center, FrattiniC24C23×D71
commutatorC23×C71C23×C4
FittingC71×C4⋊#95C2
socle, cosocleC24×C71C24
prime radicalC24×D71C23
hypercenterC4⋊#95D71
hypocenterC71#13951
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C24 ◁ C4⋊#95

Complements

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

NH ≅ G/Ntype
2. C2×#17985894C2semidirect
7. Dic71⋊#95C2semidirect
8. Dic71⋊#95C2semidirect
9. #22586567C2semidirect
10. #22586566C2semidirect
11. C71×C4⋊#95C2semidirect
14. C2×#14556085C4semidirect
141. C2×C4×D71C4⋊C4semidirect
145. Dic71⋊C4C22⋊C4semidirect
342. C4⋊#95D71semidirect
345. C71#13951semidirect

Maximal subgroups

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

subgroupindex
C22×#145560852
C2×#179858942
C2×#179858582
C2×#179858462
#225865842
#225865822
Dic71⋊#952
#225865672
#225865662
C71×C4⋊#952
#1395171