Finite Group Databasebeta

← #32164613

Subgroups — #32164613

Order 18176.

1918 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22594379C214yesmaximal, complemented
2#22594334C214yesmaximal, complemented
3#22594319C211yesmaximal, characteristic, complemented
4#22594303C212yesmaximal, complemented
5#22594286C211yesmaximal, characteristic, complemented
6#22594171C211yesmaximal, characteristic, complemented
7#22594134C212yesmaximal, complemented
8#22594093C211yesmaximal, characteristic
9#22591603C214yesmaximal, complemented
10#22591580C214yesmaximal, complemented
11C4×#14556125C211yesmaximal, characteristic, complemented
12C4×#14556103C212yesmaximal, complemented
13#22590801C212yesmaximal, complemented
14C4×#14556085C211yesmaximal, characteristic, complemented
15C4×#125×C71C211yesmaximal, characteristic, complemented
16C2×#14556125C2211yescharacteristic
17C2×#14556103C2212yes—
18C2×C4×D284C2212yes—
19C2×C4×D284C2214yes—
20C2×C4×D284C2214yes—
21C2×#14556085C2211yescharacteristic
22C2×#14556085C2211yescharacteristic
23C2×#14556085C2214yes—
24C2×C42×D71C2211yescharacteristic
25#17985941C2214yes—
26#17985934C2214yes—
27#17985932C2214yes—
28#17985931C2214yes—
29#17985928C2214yes—
30#17985928C2214yes—
31#17985926C2214yes—
32#17985925C2214yes—
33#17985921C2214yes—
34#17985921C2214yes—
35#17985917C2214yes—
36#17985917C2214yes—
37#17985916C2214yes—
38#17985916C2214yes—
39#17985916C2214yes—
40#17985909C2214yes—
41#17985901C2214yes—
42#17985901C2212yes—
43#17985900C2212yes—
44#17985900C2212yes—
45#17985900C2212yes—
46#17985898C2212yes—
47#17985895C2212yes—
48#17985895C2212yes—
49#17985891C2214yes—
50#17985888C2214yes—
51#17985888C2214yes—
52#17985877C2214yes—
53#17985869C2211yescharacteristic
54#17985856C2212yes—
55C4×Dic71⋊C4C2212yes—
56C4×Dic71⋊C4C2212yes—
57C4×(C2×D71)⋊C4C2212yes—
58C4×(C2×D71)⋊C4C2214yes—
59C4×(C2×D71)⋊C4C2212yes—
60C4×(C2×D71)⋊C4C2214yes—
61#17985845C2211yescharacteristic
62#17985845C2211yescharacteristic
63C42×Dic71C2211yescharacteristic
64C2×#125×C71C2211yescharacteristic
65#389×C71C2212yes—
66#387×C71C2214yes—
67#382×C71C2211yescharacteristic
68C4×C4⋊C4×C71C2212yes—
69C4×C4⋊C4×C71C2214yes—
70C43×C71C2211yescharacteristic
71C22×D284C2312yes—
72C22×D284C2314yes—
73C22×D284C2314yes—
74C22×C4×D71C2311yescharacteristic
75C22×C4×D71C2314yes—
76C22×C4×D71C2311yescharacteristic
77C22×C4×D71C2311yescharacteristic
78#14556125—24no—
79#14556103—28no—
80C4×D284—28no—
81C4×D284—48no—
82C4×D284—48no—
83#14556085—24no—
84#14556085—24no—
85#14556085—48no—
86C2×C4⋊Dic71C2312yes—
87C2×C4⋊Dic71C2314yes—
88C2×C4⋊Dic71C2314yes—
89C2×Dic71⋊C4C2312yes—
90C2×Dic71⋊C4C2312yes—
91C2×Dic71⋊C4C2314yes—
92C2×Dic71⋊C4C2314yes—
93C2×(C2×D71)⋊C4C2312yes—
94C2×(C2×D71)⋊C4C2312yes—
95C2×(C2×D71)⋊C4C2314yes—
96C2×(C2×D71)⋊C4C2314yes—
97C2×(C2×D71)⋊C4C2314yes—
98C2×(C2×D71)⋊C4C2314yes—
99C2×(C2×D71)⋊C4C2314yes—
100C2×(C2×D71)⋊C4C2314yes—
101C2×(C2×D71)⋊C4C2314yes—
102C2×(C2×D71)⋊C4C2314yes—
103C2×(C2×D71)⋊C4C2314yes—
104C2×(C2×D71)⋊C4C2314yes—
105C2×(C2×D71)⋊C4C2314yes—
106C2×C4×Dic71C2311yescharacteristic
107C2×C4×Dic71C2311yescharacteristic
108C2×C4×Dic71C2312yes—
109C2×C4×Dic71C2311yescharacteristic
110C2×C4×Dic71C2311yescharacteristic
111C2×C4×Dic71C2311yescharacteristic
112C2×C4×Dic71C2312yes—
113C42×D71—24no—
114C71⋊#95C2312yes—
115C71⋊#95C2312yes—
116C71⋊#95C2312yes—
117C71⋊#95C2312yes—
118C71⋊#95C2314yes—
119C71⋊#95C2314yes—
120C71⋊#95C2314yes—
121C71⋊#95C2314yes—
122#125×C71—24no—
123C2×C4⋊C4×C71C2312yes—
124C2×C4⋊C4×C71C2312yes—
125C2×C4⋊C4×C71C2314yes—
126C2×C4⋊C4×C71C2314yes—
127C2×C42×C71C2311yescharacteristic
128C2×C42×C71C2314yes—
129C2×C42×C71C2312yes—
130C2×C42×C71C2312yes—
131C2×C42×C71C2311yescharacteristic
132C2×C42×C71C2311yescharacteristic
133#95×C71C2314yes—
134#95×C71C2314yes—
135#95×C71C2314yes—
136C23×D71C2414yes—
137C23×D71C2411yescharacteristic
138C2×D284—28no—
139C2×D284—24no—
140C2×D284—24no—
141C2×D284—28no—
142C2×D284—24no—
143C2×D284—44no—
144C2×D284—44no—
145C2×D284—44no—
146C2×D284—44no—
147C22×Dic71C2411yescharacteristic
148C22×Dic71C2411yescharacteristic
149C22×Dic71C2414yes—
150C22×Dic71C2412yes—
151C22×Dic71C2412yes—
152C22×Dic71C2411yescharacteristic
153C2×C4×D71—22no—
154C2×C4×D71—22no—
155C2×C4×D71—22no—
156C2×C4×D71—22no—
157C2×C4×D71—28no—
158C2×C4×D71—28no—
159C2×C4×D71—24no—
160C2×C4×D71—28no—
161C2×C4×D71—42no—
162C2×C4×D71#4114yes—
163C4⋊Dic71—24no—
164C4⋊Dic71—44no—
165C4⋊Dic71—44no—
166Dic71⋊C4—24no—
167Dic71⋊C4—24no—
168Dic71⋊C4—44no—
169Dic71⋊C4—44no—
170(C2×D71)⋊C4—24no—
171(C2×D71)⋊C4—24no—
172(C2×D71)⋊C4—28no—
173(C2×D71)⋊C4—28no—
174(C2×D71)⋊C4—44no—
175(C2×D71)⋊C4—44no—
176(C2×D71)⋊C4—44no—
177(C2×D71)⋊C4—44no—
178(C2×D71)⋊C4—44no—
179(C2×D71)⋊C4—44no—
180(C2×D71)⋊C4—44no—
181(C2×D71)⋊C4—44no—
182(C2×D71)⋊C4—44no—
183C4×Dic71—22no—
184C4×Dic71—22no—
185C4×Dic71—22no—
186C4×Dic71—22no—
187C4×Dic71—22no—
188C4×Dic71—24no—
189C4×Dic71—42no—
190C22×C4×C71C2414yes—
191C22×C4×C71C2412yes—
192C22×C4×C71C2414yes—
193C22×C4×C71C2411yescharacteristic
194C22×C4×C71C2412yes—
195C22×C4×C71C2411yescharacteristic
196C22×C4×C71C2411yescharacteristic
197C4⋊C4×C71—28no—
198C4⋊C4×C71—24no—
199C4⋊C4×C71—24no—
200C4⋊C4×C71—44no—
201C42×C71—22no—
202C42×C71—28no—
203C42×C71—24no—
204C42×C71—42no—
205C42×C71#4114yes—
206C42×C71#4114yes—
207C23×C71C2511yescharacteristic
208C22×D71—21no—
209C22×D71—21no—
210C22×D71—22no—
211C22×D71—22no—
212C22×D71—24no—
213C22×D71—24no—
214C22×D71—24no—
215C22×D71—24no—
216C22×D71—24no—
217C22×D71—24no—
218C22×D71—24no—
219C22×D71C2×#4112yes—
220C2×C4×C71—22no—
221C2×C4×C71—22no—
222C2×C4×C71—22no—
223C2×C4×C71—22no—
224C2×C4×C71—24no—
225C2×C4×C71—24no—
226C2×C4×C71—22no—
227C2×C4×C71—24no—
228C2×C4×C71—24no—
229C2×C4×C71—22no—
230C2×C4×C71—24no—
231C2×C4×C71—24no—
232C2×C4×C71C2×#4112yes—
233C2×C4×C71C2×#4114yes—
234C2×C4×C71C2×#4112yes—
235C2×C4×C71C2×#4112yes—
236C2×C4×C71C2×#4112yes—
237C2×C4×C71C2×#4112yes—
238C2×C4×C71H(5,𝔽2)14yes—
239C2×Dic71—21no—
240C2×Dic71—21no—
241C2×Dic71—21no—
242C2×Dic71—21no—
243C2×Dic71—22no—
244C2×Dic71—22no—
245C2×Dic71—24no—
246C2×Dic71—22no—
247C2×Dic71—22no—
248C2×Dic71—22no—
249C2×Dic71—24no—
250C2×Dic71—24no—
251C2×Dic71—22no—
252C2×Dic71—22no—
253C2×Dic71—22no—
254C2×Dic71C2×#4112yes—
255D284—416no—
256D284—416no—
257D284—48no—
258C4×D71—28no—
259C4×D71—44no—
260C4×D71—44no—
261C4×D71—416no—
262C22×C71C22×#4111yescharacteristic
263C22×C71C22×#4111yescharacteristic
264C22×C71C22×#4111yescharacteristic
265C22×C71C2×H(5,𝔽2)12yes—
266C22×C71#58512yes—
267C2×D71—22no—
268C2×D71—24no—
269C2×D71—41no—
270C2×D71—41no—
271C2×D71—42no—
272C2×D71—44no—
273C2×D71—44no—
274C2×D71—44no—
275C2×D71—44no—
276C2×D71—44no—
277C2×D71—44no—
278C2×D71—44no—
279C4×C71—22no—
280C4×C71—24no—
281C4×C71—44no—
282C4×C71—44no—
283C4×C71—42no—
284C4×C71#52914yes—
285C4×C71#54014yes—
286Dic71—22no—
287Dic71—41no—
288Dic71—44no—
289Dic71—41no—
290Dic71—42no—
291Dic71—42no—
292#34621—711nomaximal, Sylow
293C2×C71C2×#52911yescharacteristic
294C2×C71C2×#54011yescharacteristic
295C2×C71#345511yescharacteristic
296C2×C71#346012yes—
297C2×C71#346012yes—
298D71—42no—
299D71—84no—
300#2651—714no—
301#2648—714no—
302#2647—711no—
303#2644—712no—
304#2641—711no—
305#2629—711no—
306#2624—712no—
307#2619—711no—
308#2395—714no—
309#2392—714no—
310C4×#127—711no—
311C4×#125D7111yescharacteristic, complemented
312C4×#121—712no—
313#2293—712no—
314C4×#117—711no—
315C71#3462111yescharacteristic, complemented, Sylow
316C2×#127—711no—
317C2×#125C2×D7111yescharacteristic
318C2×#121—712no—
319C2×C4×D4—714no—
320C2×C4×D4—714no—
321C2×C4×D4—712no—
322C2×#117—714no—
323C2×#117—711no—
324C2×#117—711no—
325C22×C42—711no—
326#400—714no—
327#399—714no—
328#398—714no—
329#398—714no—
330#397—714no—
331#397—714no—
332#397—714no—
333#397—714no—
334#396—714no—
335#396—714no—
336#394—714no—
337#394—714no—
338#394—714no—
339#394—714no—
340#394—714no—
341#392—714no—
342#390—712no—
343#390—712no—
344#390—712no—
345#390—712no—
346#390—714no—
347#389—712no—
348#389—712no—
349#389—712no—
350#389C2×D7112yes—
351#388—714no—
352#388—714no—
353#388—714no—
354#387C2×D7114yes—
355#386—714no—
356#384—711no—
357#382C2×D7111yescharacteristic
358#380—712no—
359C4×C4⋊C4—712no—
360C4×C4⋊C4—712no—
361C4×C4⋊C4C2×D7114yes—
362C4×C4⋊C4C2×D7112yes—
363C4×C22⋊C4—712no—
364C4×C22⋊C4—714no—
365C4×C22⋊C4—712no—
366C4×C22⋊C4—714no—
367#376—711no—
368#376—711no—
369C43—711no—
370C43C2×D7111yescharacteristic
371C22×D4—714no—
372C22×D4—714no—
373C22×D4—712no—
374C23×C4—711no—
375C23×C4—711no—
376C23×C4—714no—
377C23×C4—711no—
378#127—1424no—
379#125—24no—
380#121—1428no—
381C4×D4—2848no—
382C4×D4—2848no—
383C4×D4—1428no—
384#117—2848no—
385#117—1424no—
386#117—1424no—
387C2×C4⋊C4—714no—
388C2×C4⋊C4—714no—
389C2×C4⋊C4—714no—
390C2×C4⋊C4—714no—
391C2×C4⋊C4—712no—
392C2×C4⋊C4—712no—
393C2×C4⋊C4—712no—
394C2×C4⋊C4C22×D7112yes—
395C2×C4⋊C4C22×D7112yes—
396C2×C4⋊C4C22×D7114yes—
397C2×C4⋊C4C22×D7114yes—
398C2×C22⋊C4—714no—
399C2×C22⋊C4—714no—
400C2×C22⋊C4—714no—
401C2×C22⋊C4—714no—
402C2×C22⋊C4—714no—
403C2×C22⋊C4—714no—
404C2×C22⋊C4—714no—
405C2×C22⋊C4—714no—
406C2×C22⋊C4—714no—
407C2×C22⋊C4—712no—
408C2×C22⋊C4—712no—
409C2×C22⋊C4—714no—
410C2×C22⋊C4—714no—
411C2×C42—712no—
412C2×C42—711no—
413C2×C42—711no—
414C2×C42—711no—
415C2×C42—711no—
416C2×C42—712no—
417C2×C42—711no—
418C2×C42—1424no—
419C2×C42C22×D7111yescharacteristic
420C2×C42C22×D7114yes—
421C2×C42C22×D7112yes—
422C2×C42C22×D7112yes—
423C2×C42C22×D7111yescharacteristic
424C2×C42C22×D7111yescharacteristic
425#95—712no—
426#95—712no—
427#95—712no—
428#95—712no—
429#95—714no—
430#95—714no—
431#95—714no—
432#95—714no—
433#95C22×D7114yes—
434#95C22×D7114yes—
435#95C22×D7114yes—
436C24—711no—
437C24—714no—
438C2×D4—2844no—
439C2×D4—2844no—
440C2×D4—2844no—
441C2×D4—2844no—
442C2×D4—1428no—
443C2×D4—1424no—
444C2×D4—1428no—
445C2×D4—1424no—
446C2×D4—1424no—
447C22×C4—2842no—
448C22×C4—711no—
449C22×C4—714no—
450C22×C4—711no—
451C22×C4—712no—
452C22×C4—712no—
453C22×C4—711no—
454C22×C4—714no—
455C22×C4—1422no—
456C22×C4—1422no—
457C22×C4—1428no—
458C22×C4—1428no—
459C22×C4—1428no—
460C22×C4—1424no—
461C22×C4—1422no—
462C22×C4—1422no—
463C22×C4C23×D7114yes—
464C22×C4C23×D7112yes—
465C22×C4C23×D7114yes—
466C22×C4C23×D7112yes—
467C22×C4C23×D7111yescharacteristic
468C22×C4C23×D7111yescharacteristic
469C22×C4C23×D7111yescharacteristic
470C4⋊C4—24no—
471C4⋊C4—28no—
472C4⋊C4—24no—
473C4⋊C4—44no—
474C4⋊C4—2844no—
475C4⋊C4—2844no—
476C4⋊C4—2844no—
477C4⋊C4—2844no—
478C4⋊C4—1424no—
479C4⋊C4—1424no—
480C4⋊C4—1424no—
481C22⋊C4—2844no—
482C22⋊C4—2844no—
483C22⋊C4—2844no—
484C22⋊C4—2844no—
485C22⋊C4—2844no—
486C22⋊C4—2844no—
487C22⋊C4—2844no—
488C22⋊C4—2844no—
489C22⋊C4—2844no—
490C22⋊C4—1428no—
491C22⋊C4—1424no—
492C22⋊C4—1424no—
493C22⋊C4—1428no—
494C42—22no—
495C42—28no—
496C42—24no—
497C42—42no—
498C42—2842no—
499C42—1422no—
500C42—1422no—
501C42—1422no—
502C42—1422no—
503C42—1422no—
504C42—1424no—
505C42#1176586514yes—
506C42#1176586614yes—
507C23—712no—
508C23—1424no—
509C23—1424no—
510C23—1424no—
511C23—1424no—
512C23—1424no—
513C23—1424no—
514C23—1424no—
515C23—1421no—
516C23—1421no—
517C23—1422no—
518C23—1422no—
519C23C24×D7111yescharacteristic
520D4—28416no—
521D4—28416no—
522D4—2848no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—22no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—22no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—2844no—
536C2×C4—2844no—
537C2×C4—28416no—
538C2×C4—712no—
539C2×C4—1422no—
540C2×C4—1422no—
541C2×C4—1421no—
542C2×C4—1421no—
543C2×C4—1424no—
544C2×C4—1422no—
545C2×C4—1422no—
546C2×C4—1422no—
547C2×C4—1424no—
548C2×C4—1424no—
549C2×C4—1422no—
550C2×C4—1422no—
551C2×C4—1422no—
552C2×C4—1421no—
553C2×C4—1421no—
554C2×C4—1428no—
555C2×C4#41×D7112yes—
556C2×C4C2×#1176586514yes—
557C2×C4C2×#1176586512yes—
558C2×C4C2×#1176586612yes—
559C2×C4C2×#1176586612yes—
560C2×C4C2×#1176586612yes—
561C2×C4#1455617414yes—
562C22—2841no—
563C22—2842no—
564C22—2841no—
565C22—2844no—
566C22—2844no—
567C22—2844no—
568C22—2844no—
569C22—2844no—
570C22—2844no—
571C22—2844no—
572C22—1424no—
573C22—1422no—
574C22C2×#41×D7111yescharacteristic
575C22C22×#1176586511yescharacteristic
576C22C22×#1176586611yescharacteristic
577C22C2×#1455617412yes—
578C22#1798683512yes—
579C4—22no—
580C4—24no—
581C4—44no—
582C4—44no—
583C4—42no—
584C4—2841no—
585C4—2842no—
586C4—2842no—
587C4—2841no—
588C4—2844no—
589C4—1422no—
590C4#1798653614yes—
591C4#1798660514yes—
592C2—2842no—
593C2—5684no—
594C2C2×#1798653611yescharacteristic
595C2C2×#1798660511yescharacteristic
596C2#2260173211yescharacteristic
597C2#2260175612yes—
598C2#2260176112yes—
subgroupG/H
centerC22×C4C23×D71
commutator, socle, cosocleC23×C71C25
FrattiniC23C24×D71
FittingC4×#125×C71C2
prime radical#22594093C2
hypercenterC4×#125D71
hypocenterC71#34621
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C22×C4 ◁ C4×#125

Complements

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

NH ≅ G/Ntype
1. #22594379C2semidirect
2. #22594334C2semidirect
3. #22594319C2semidirect
4. #22594303C2semidirect
5. #22594286C2semidirect
6. #22594171C2semidirect
7. #22594134C2semidirect
9. #22591603C2semidirect
10. #22591580C2semidirect
11. C4×#14556125C2semidirect
12. C4×#14556103C2semidirect
13. #22590801C2semidirect
14. C4×#14556085C2semidirect
15. C4×#125×C71C2semidirect
311. C4×#125D71semidirect
315. C71#34621semidirect

Maximal subgroups

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

subgroupindex
#225943792
#225943342
#225943192
#225943032
#225942862
#225941712
#225941342
#225940932
#225916032
#225915802
C4×#145561252
C4×#145561032
#225908012
C4×#145560852
C4×#125×C712
#3462171