Finite Group Databasebeta

← #31995111

Subgroups — #31995111

Order 18176.

1606 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17986478C211yesmaximal, characteristic
2C2×#17985971C212yesmaximal
3#22590695C212yesmaximal, complemented
4#99×Dic71C212yesmaximal, complemented
5C4×#14556042C212yesmaximal, complemented
6C4×#14556043C211yesmaximal, characteristic, complemented
7#22588127C212yesmaximal, complemented
8#22588071C212yesmaximal, complemented
9C4×#99×C71C211yesmaximal, characteristic, complemented
10C22×#11765864C2211yescharacteristic
11#17986478—24no—
12#17986478—24no—
13C2×C4×Dic142C411yescharacteristic, complemented
14C2×#14556093C411yescharacteristic, complemented
15C2×D4×Dic71C2211yescharacteristic
16C2×C4×C22⋊D71C411yescharacteristic, complemented
17C2×C4×C22⋊D71C2211yescharacteristic
18C2×(C22⋊D71)⋊C4C411yescharacteristic, complemented
19C2×(C22⋊D71)⋊C4C2211yescharacteristic
20C2×#14556085C411yescharacteristic, complemented
21C2×#14556084C411yescharacteristic, complemented
22C2×#14556084C2214yes—
23C2×#14556084C2211yescharacteristic
24C22×Dic71⋊C4C411yescharacteristic, complemented
25C22×Dic71⋊C4C2211yescharacteristic
26C2×C4⋊C4×D71C411yescharacteristic, complemented
27#17985971C4116yescomplemented
28C2×#14556043C2212yes—
29C2×#14556042C2214yes—
30C4×Dic71⋊C4—24no—
31#17985850—24no—
32C22⋊C4×Dic71C2214yes—
33C4×C22⋊Dic71C2212yes—
34#17985766C2212yes—
35#17985765C2214yes—
36C2×C4×D4×C71C2211yescharacteristic
37C2×#99×C71C2212yes—
38C4×C22⋊C4×C71C2212yes—
39#342×C71C2212yes—
40C2×#11765864—24no—
41C2×#11765864—24no—
42C2×#11765864C2×C418yes—
43C22×Dic142C2×C411yescharacteristic
44C22×C22⋊D71C2×C411yescharacteristic
45C22×C22⋊D71C2311yescharacteristic
46C23×Dic71C2×C411yescharacteristic
47C23×Dic71C2×C411yescharacteristic
48C23×Dic71C2311yescharacteristic
49C23×Dic71C2311yescharacteristic
50C22×C4×D71C2×C411yescharacteristic
51C22×C4×D71C2×C411yescharacteristic
52C22×C4×D71C2×C411yescharacteristic
53C4×Dic142—22no—
54C4×Dic142—22no—
55#14556093—24no—
56D4×Dic71—24no—
57C4×C22⋊D71—22no—
58C4×C22⋊D71—22no—
59C4×C22⋊D71—22no—
60C4×C22⋊D71—22no—
61(C22⋊D71)⋊C4—24no—
62(C22⋊D71)⋊C4—24no—
63#14556085—22no—
64#14556085—22no—
65#14556084—24no—
66#14556084—28no—
67#14556084—24no—
68#14556084C2×C4116yescomplemented
69C2×C4⋊Dic71C2×C411yescharacteristic
70C2×Dic71⋊C4—24no—
71C2×Dic71⋊C4—22no—
72C2×Dic71⋊C4—22no—
73C2×Dic71⋊C4C2×C411yescharacteristic
74C2×Dic71⋊C4C2×C411yescharacteristic
75C2×Dic71⋊C4C2×C411yescharacteristic
76C2×Dic71⋊C4C2×C411yescharacteristic
77C2×Dic71⋊C4C2×C411yescharacteristic
78C2×Dic71⋊C4C2×C411yescharacteristic
79C2×Dic71⋊C4C2×C411yescharacteristic
80C2×Dic71⋊C4C2×C411yescharacteristic
81C2×Dic71⋊C4D414yes—
82C2×Dic71⋊C4D414yes—
83C4⋊C4×D71—24no—
84C2×C22⋊Dic71C2×C411yescharacteristic
85C2×C22⋊Dic71C2311yescharacteristic
86C2×C22⋊Dic71C2314yes—
87C2×(C2×D71)⋊C4C2×C411yescharacteristic
88C2×(C2×D71)⋊C4C2×C411yescharacteristic
89C2×C4×Dic71—24no—
90C2×C4×Dic71—24no—
91C2×C4×Dic71—22no—
92C2×C4×Dic71C2×C411yescharacteristic
93C2×C4×Dic71C2×C411yescharacteristic
94C2×C4×Dic71C2×C411yescharacteristic
95C2×C4×Dic71C2×C411yescharacteristic
96C2×C4×Dic71C2×C411yescharacteristic
97#14556043C2×C418yescomplemented
98#14556042C2×C4116yescomplemented
99C71⋊#95—24no—
100C71⋊#95—22no—
101C22×D4×C71C2311yescharacteristic
102C23×C4×C71C2×C411yescharacteristic
103C23×C4×C71C2311yescharacteristic
104C4×D4×C71—22no—
105C4×D4×C71—22no—
106C2×C4⋊C4×C71C2×C411yescharacteristic
107C2×C22⋊C4×C71C2×C411yescharacteristic
108C2×C22⋊C4×C71C2311yescharacteristic
109C2×C22⋊C4×C71C2314yes—
110C2×C42×C71—22no—
111C2×C42×C71C2×C411yescharacteristic
112#99×C71C2×C418yescomplemented
113#95×C71—22no—
114C23×D71C22×C411yescharacteristic
115#11765864—216no—
116#11765864—216no—
117C2×Dic142—22no—
118C2×Dic142—22no—
119C2×Dic142C4214yescomplemented
120C2×C22⋊D71—22no—
121C2×C22⋊D71—22no—
122C2×C22⋊D71—22no—
123C2×C22⋊D71—22no—
124C2×C22⋊D71C4214yescomplemented
125C2×C22⋊D71C22×C414yes—
126C22×Dic71—22no—
127C22×Dic71—22no—
128C22×Dic71—24no—
129C22×Dic71—22no—
130C22×Dic71—81no—
131C22×Dic71—81no—
132C22×Dic71C4214yescomplemented
133C22×Dic71C22×C411yescharacteristic
134C22×Dic71C22×C411yescharacteristic
135C22×Dic71C22×C411yescharacteristic
136C22×Dic71C22×C411yescharacteristic
137C22×Dic71C22×C411yescharacteristic
138C22×Dic71C22×C411yescharacteristic
139C22×Dic71C22×C411yescharacteristic
140C22×Dic71C22×C414yes—
141C22×Dic71C2×D414yes—
142C22×Dic71C2×D414yes—
143C22×Dic71#4114yes—
144C2×C4×D71—22no—
145C2×C4×D71—22no—
146C2×C4×D71—22no—
147C2×C4×D71—22no—
148C2×C4×D71—81no—
149C2×C4×D71—81no—
150C2×C4×D71C4214yescomplemented
151C4⋊Dic71—22no—
152Dic71⋊C4—21no—
153Dic71⋊C4—21no—
154Dic71⋊C4—22no—
155Dic71⋊C4—22no—
156Dic71⋊C4—22no—
157Dic71⋊C4—216no—
158Dic71⋊C4C22⋊C414yes—
159Dic71⋊C4C22⋊C412yes—
160Dic71⋊C4C22⋊C412yes—
161Dic71⋊C4C22⋊C414yes—
162Dic71⋊C4C22⋊C412yes—
163Dic71⋊C4C22⋊C412yes—
164C22⋊Dic71—22no—
165C22⋊Dic71—24no—
166C22⋊Dic71—22no—
167C22⋊Dic71C22×C418yes—
168(C2×D71)⋊C4—21no—
169(C2×D71)⋊C4—21no—
170(C2×D71)⋊C4—22no—
171C4×Dic71—22no—
172C4×Dic71—22no—
173C4×Dic71—22no—
174C4×Dic71—22no—
175C4×Dic71—22no—
176C4×Dic71—216no—
177C4×Dic71—84no—
178C4×Dic71—82no—
179C24×C71C22×C411yescharacteristic
180C24×C71C2411yescharacteristic
181C2×D4×C71—22no—
182C2×D4×C71—22no—
183C2×D4×C71C22×C414yes—
184C22×C4×C71—24no—
185C22×C4×C71—21no—
186C22×C4×C71—21no—
187C22×C4×C71—81no—
188C22×C4×C71—81no—
189C22×C4×C71C22×C411yescharacteristic
190C22×C4×C71C22×C411yescharacteristic
191C22×C4×C71C22×C411yescharacteristic
192C22×C4×C71C22×C411yescharacteristic
193C22×C4×C71C22×C411yescharacteristic
194C22×C4×C71C2×D412yes—
195C22×C4×C71C2×D412yes—
196C4⋊C4×C71—22no—
197C22⋊C4×C71—22no—
198C22⋊C4×C71—24no—
199C22⋊C4×C71—22no—
200C22⋊C4×C71C22×C418yes—
201C42×C71—21no—
202C42×C71—21no—
203C42×C71—82no—
204C23×C71—21no—
205C23×C71—21no—
206C23×C71—41no—
207C23×C71—41no—
208C23×C71—41no—
209C23×C71—41no—
210C23×C71C2×C4212yes—
211C23×C71C23×C411yescharacteristic
212C23×C71C23×C412yes—
213C23×C71C22×D412yes—
214C23×C71C2×#4112yes—
215C22×D71—21no—
216C22×D71—21no—
217C22×D71—41no—
218C22×D71—41no—
219C22×D71C2×C4212yes—
220D4×C71—28no—
221C2×C4×C71—21no—
222C2×C4×C71—21no—
223C2×C4×C71—21no—
224C2×C4×C71—21no—
225C2×C4×C71—28no—
226C2×C4×C71—42no—
227C2×C4×C71—41no—
228C2×C4×C71—41no—
229C2×C4×C71—42no—
230C2×C4×C71—44no—
231C2×C4×C71—44no—
232C2×C4×C71—41no—
233C2×C4×C71—41no—
234C2×C4×C71—81no—
235C2×C4×C71—81no—
236C2×C4×C71C2×C4212yes—
237C2×C4×C71C2×C22⋊C411yescharacteristic
238C2×C4×C71C2×C22⋊C411yescharacteristic
239C2×C4×C71C2×C22⋊C412yes—
240C22⋊D71—28no—
241C22⋊D71—28no—
242C2×Dic71—21no—
243C2×Dic71—21no—
244C2×Dic71—28no—
245C2×Dic71—21no—
246C2×Dic71—21no—
247C2×Dic71—28no—
248C2×Dic71—44no—
249C2×Dic71—41no—
250C2×Dic71—41no—
251C2×Dic71—44no—
252C2×Dic71—44no—
253C2×Dic71—41no—
254C2×Dic71—41no—
255C2×Dic71—82no—
256C2×Dic71C2×C4212yes—
257C2×Dic71C2×C4212yes—
258C2×Dic71C2×C4212yes—
259C2×Dic71C2×C22⋊C412yes—
260C2×Dic71C2×C22⋊C412yes—
261C2×Dic71C2×C22⋊C412yes—
262C2×Dic71C2×C22⋊C412yes—
263C2×Dic71#11712yes—
264C2×Dic71#11712yes—
265C2×Dic71C4×D418yes—
266C2×Dic71C4×D418yes—
267C4×D71—28no—
268C4×D71—82no—
269C4×D71—82no—
270Dic142—28no—
271C22×C71—22no—
272C22×C71—21no—
273C22×C71—22no—
274C22×C71—21no—
275C22×C71—22no—
276C22×C71—42no—
277C22×C71—42no—
278C22×C71—42no—
279C22×C71—42no—
280C22×C71—41no—
281C22×C71—41no—
282C22×C71C22×C4211yescharacteristic
283C22×C71C22×C22⋊C411yescharacteristic
284C22×C71C2×#11711yescharacteristic
285C22×C71C2×C4×D414yes—
286C2×D71—22no—
287C2×D71—42no—
288C2×D71—41no—
289C2×D71—41no—
290C2×D71—42no—
291C4×C71—22no—
292C4×C71—21no—
293C4×C71—21no—
294C4×C71—22no—
295C4×C71—48no—
296C4×C71—41no—
297C4×C71—41no—
298Dic71—22no—
299Dic71—22no—
300Dic71—22no—
301Dic71—48no—
302Dic71—41no—
303Dic71—41no—
304Dic71C4×C22⋊C414yes—
305Dic71C4×C22⋊C414yes—
306#20393—711nomaximal, Sylow
307C2×C71—24no—
308C2×C71—21no—
309C2×C71—21no—
310C2×C71—42no—
311C2×C71C2×C4×C22⋊C411yescharacteristic
312C2×C71#288311yescharacteristic
313C2×C71#288411yescharacteristic
314D71—42no—
315C2×#520—711no—
316C2×#410—712no—
317#2274—712no—
318C4×#99—712no—
319C4×#99—712no—
320C4×#99—712no—
321C4×#99—711no—
322C4×#99D7111yescharacteristic, complemented
323#1740—712no—
324C71#2039311yescharacteristic, complemented, Sylow
325C22×#41—711no—
326#520—1424no—
327#520—1424no—
328C2×C4×Q8—711no—
329C2×C4×Q8—711no—
330C2×C4×D4—711no—
331C2×C4×D4—711no—
332C2×C4×D4—711no—
333C2×C4×D4—711no—
334C2×C4×D4—711no—
335C2×C4×D4C2×D7111yescharacteristic
336C2×#117—711no—
337C2×#117—711no—
338C2×#117—711no—
339C2×#117—714no—
340C22×C4⋊C4—711no—
341C22×C4⋊C4—711no—
342C22×C4⋊C4—711no—
343#410—7116no—
344C2×#99—712no—
345C2×#99—714no—
346C2×#99C2×D7112yes—
347C4×C4⋊C4—1424no—
348C4×C4⋊C4—1424no—
349C4×C22⋊C4—712no—
350C4×C22⋊C4—714no—
351C4×C22⋊C4C2×D7112yes—
352#342—714no—
353#342—712no—
354#342C2×D7112yes—
355C2×#41—718no—
356C2×#41—1424no—
357C2×#41—1424no—
358C22×Q8—711no—
359C22×D4—711no—
360C22×D4—711no—
361C22×D4C22×D7111yescharacteristic
362C23×C4—711no—
363C23×C4—711no—
364C23×C4—711no—
365C23×C4—711no—
366C23×C4—711no—
367C23×C4—711no—
368C23×C4—711no—
369C23×C4C4×D7111yescharacteristic
370C23×C4C22×D7111yescharacteristic
371C4×Q8—1422no—
372C4×Q8—1422no—
373C4×Q8—1424no—
374C4×D4—22no—
375C4×D4—22no—
376C4×D4—1424no—
377C4×D4—1424no—
378C4×D4—1422no—
379C4×D4—1422no—
380C4×D4—1424no—
381C4×D4—1422no—
382C4×D4—1422no—
383#117—7116no—
384#117—1422no—
385#117—1422no—
386#117—1424no—
387#117—1424no—
388#117—1428no—
389C2×C4⋊C4—711no—
390C2×C4⋊C4—711no—
391C2×C4⋊C4—711no—
392C2×C4⋊C4—711no—
393C2×C4⋊C4—714no—
394C2×C4⋊C4—714no—
395C2×C4⋊C4—711no—
396C2×C4⋊C4—711no—
397C2×C4⋊C4—711no—
398C2×C4⋊C4—711no—
399C2×C4⋊C4—711no—
400C2×C4⋊C4—1424no—
401C2×C4⋊C4—1422no—
402C2×C4⋊C4—1422no—
403C2×C4⋊C4—1424no—
404C2×C4⋊C4C4×D7111yescharacteristic
405C2×C22⋊C4—714no—
406C2×C22⋊C4—711no—
407C2×C22⋊C4—711no—
408C2×C22⋊C4—711no—
409C2×C22⋊C4—711no—
410C2×C22⋊C4C4×D7111yescharacteristic
411C2×C22⋊C4C22×D7111yescharacteristic
412C2×C22⋊C4C22×D7114yes—
413C2×C42—22no—
414C2×C42—711no—
415C2×C42—711no—
416C2×C42—711no—
417C2×C42—711no—
418C2×C42—711no—
419C2×C42—1424no—
420C2×C42—1422no—
421C2×C42—1424no—
422C2×C42C4×D7111yescharacteristic
423#99—718no—
424#99—7116no—
425#99C4×D7118yescomplemented
426#95—22no—
427#95—1424no—
428#95—1422no—
429C24—711no—
430C24C2×C4×D7111yescharacteristic
431C24C23×D7111yescharacteristic
432#41—14216no—
433#41—14216no—
434C2×Q8—714no—
435C2×Q8—1422no—
436C2×Q8—1422no—
437C2×D4—22no—
438C2×D4—22no—
439C2×D4—714no—
440C2×D4—714no—
441C2×D4—1422no—
442C2×D4—1422no—
443C2×D4—1422no—
444C2×D4—1422no—
445C2×D4C2×C4×D7114yes—
446C22×C4—24no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—81no—
450C22×C4—81no—
451C22×C4—5681no—
452C22×C4—5681no—
453C22×C4—5681no—
454C22×C4—5681no—
455C22×C4—711no—
456C22×C4—711no—
457C22×C4—714no—
458C22×C4—711no—
459C22×C4—711no—
460C22×C4—711no—
461C22×C4—711no—
462C22×C4—714no—
463C22×C4—714no—
464C22×C4—714no—
465C22×C4—711no—
466C22×C4—714no—
467C22×C4—714no—
468C22×C4—1422no—
469C22×C4—1422no—
470C22×C4—1422no—
471C22×C4—1424no—
472C22×C4—1422no—
473C22×C4—1422no—
474C22×C4—1422no—
475C22×C4—1422no—
476C22×C4C2×C4×D7111yescharacteristic
477C22×C4C2×C4×D7111yescharacteristic
478C22×C4C2×C4×D7111yescharacteristic
479C22×C4C2×C4×D7111yescharacteristic
480C22×C4C2×C4×D7111yescharacteristic
481C22×C4D4×D7112yes—
482C22×C4D4×D7112yes—
483C4⋊C4—22no—
484C4⋊C4—712no—
485C4⋊C4—712no—
486C4⋊C4—714no—
487C4⋊C4—712no—
488C4⋊C4—712no—
489C4⋊C4—714no—
490C4⋊C4—1422no—
491C4⋊C4—1422no—
492C4⋊C4—1422no—
493C4⋊C4—14216no—
494C4⋊C4—1422no—
495C4⋊C4—1421no—
496C4⋊C4—1421no—
497C22⋊C4—22no—
498C22⋊C4—24no—
499C22⋊C4—22no—
500C22⋊C4—718no—
501C22⋊C4—1422no—
502C22⋊C4—1422no—
503C22⋊C4—1422no—
504C22⋊C4—1424no—
505C22⋊C4—1421no—
506C22⋊C4—1421no—
507C22⋊C4C2×C4×D7118yes—
508C42—21no—
509C42—21no—
510C42—82no—
511C42—5682no—
512C42—5684no—
513C42—1422no—
514C42—1422no—
515C42—1422no—
516C42—1422no—
517C42—14216no—
518C42—1422no—
519C23—21no—
520C23—21no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23—2841no—
526C23—2841no—
527C23—712no—
528C23—1421no—
529C23—1421no—
530C23C42×D7112yes—
531C23C22×C4×D7111yescharacteristic
532C23C22×C4×D7112yes—
533C23C2×D4×D7112yes—
534C23#41×D7112yes—
535Q8—1428no—
536D4—28no—
537D4—1428no—
538D4—1428no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—28no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—81no—
553C2×C4—81no—
554C2×C4—2844no—
555C2×C4—2844no—
556C2×C4—2841no—
557C2×C4—2841no—
558C2×C4—2844no—
559C2×C4—2841no—
560C2×C4—2841no—
561C2×C4—5682no—
562C2×C4—5682no—
563C2×C4—5682no—
564C2×C4—712no—
565C2×C4—712no—
566C2×C4—712no—
567C2×C4—712no—
568C2×C4—712no—
569C2×C4—712no—
570C2×C4—718no—
571C2×C4—718no—
572C2×C4—712no—
573C2×C4—712no—
574C2×C4—712no—
575C2×C4—1428no—
576C2×C4—1421no—
577C2×C4—1421no—
578C2×C4—1428no—
579C2×C4—1428no—
580C2×C4—1421no—
581C2×C4—1421no—
582C2×C4C42×D7112yes—
583C2×C4C22⋊C4×D7111yescharacteristic
584C2×C4C22⋊C4×D7111yescharacteristic
585C2×C4C22⋊C4×D7112yes—
586C22—22no—
587C22—21no—
588C22—22no—
589C22—21no—
590C22—22no—
591C22—42no—
592C22—42no—
593C22—42no—
594C22—42no—
595C22—41no—
596C22—41no—
597C22—2842no—
598C22—2842no—
599C22—2841no—
600C22—2841no—
601C22—1422no—
602C22C2×C42×D7111yescharacteristic
603C22C2×C22⋊C4×D7111yescharacteristic
604C22#117×D7111yescharacteristic
605C22C4×D4×D7114yes—
606C4—21no—
607C4—21no—
608C4—22no—
609C4—22no—
610C4—48no—
611C4—41no—
612C4—41no—
613C4—2848no—
614C4—2841no—
615C4—2841no—
616C4—714no—
617C4—714no—
618C4—1422no—
619C4—1422no—
620C4—1422no—
621C2—24no—
622C2—21no—
623C2—21no—
624C2—42no—
625C2—2842no—
626C2C4×C22⋊C4×D7111yescharacteristic
627C2#2259679811yescharacteristic
628C2#2259680211yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D71
commutatorC22×C71C22×C42
FrattiniC24C23×D71
FittingC4×#99×C71C2
socleC22×C71C22×C22⋊C4
prime radicalC22×#11765864C22
cosocleC24×C71C24
hypercenterC4×#99D71
hypocenterC71#20393
LCSG ▷ C22×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C23×C4 ◁ C4×#99

Complements

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

NH ≅ G/Ntype
3. #22590695C2semidirect
4. #99×Dic71C2semidirect
5. C4×#14556042C2semidirect
6. C4×#14556043C2semidirect
7. #22588127C2semidirect
8. #22588071C2semidirect
9. C4×#99×C71C2semidirect
13. C2×C4×Dic142C4semidirect
14. C2×#14556093C4semidirect
16. C2×C4×C22⋊D71C4semidirect
18. C2×(C22⋊D71)⋊C4C4semidirect
20. C2×#14556085C4semidirect
21. C2×#14556084C4semidirect
24. C22×Dic71⋊C4C4semidirect
26. C2×C4⋊C4×D71C4semidirect
27. #17985971C4semidirect
68. #14556084C2×C4semidirect
97. #14556043C2×C4semidirect
98. #14556042C2×C4semidirect
112. #99×C71C2×C4semidirect
119. C2×Dic142C42semidirect
124. C2×C22⋊D71C42semidirect
132. C22×Dic71C42semidirect
150. C2×C4×D71C42semidirect
322. C4×#99D71semidirect
324. C71#20393semidirect
425. #99C4×D71semidirect

Maximal subgroups

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

subgroupindex
C2×#179864782
C2×#179859712
#225906952
#99×Dic712
C4×#145560422
C4×#145560432
#225881272
#225880712
C4×#99×C712
#2039371