Finite Group Databasebeta

← #32156550

Subgroups — #32156550

Order 18176.

2518 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17986604C211yesmaximal, characteristic
2#22591136C214yesmaximal, complemented
3#22591103C214yesmaximal, complemented
4#22591079C211yesmaximal, characteristic, complemented
5#22591063C211yesmaximal, characteristic, complemented
6#22591029C212yesmaximal, complemented
7#22590978C212yesmaximal, complemented
8#22590902C211yesmaximal, characteristic, complemented
9#22590895C211yesmaximal, characteristic, complemented
10#22590861C214yesmaximal, complemented
11#22590852C214yesmaximal, complemented
12#22590846C212yesmaximal, complemented
13C4×#14556090C212yesmaximal, complemented
14#117×Dic71C211yesmaximal, characteristic, complemented
15C71×#2314C211yesmaximal, characteristic, complemented
16#17986604C4116yescomplemented
17C2×#14556125C2211yescharacteristic
18C2×#14556120C2211yescharacteristic
19C2×#14556113C2214yes—
20C2×#14556102C2212yes—
21C2×#14556101C2214yes—
22C2×C4×D284C2212yes—
23C2×#14556090C2212yes—
24C2×#14556090C2212yes—
25C2×#14556090C2214yes—
26C2×(C22⋊D71)⋊C4C2214yes—
27C2×#14556087C2211yescharacteristic
28C22×C4×Dic71C2211yescharacteristic
29C2×C42×D71C2214yes—
30#17985900C2212yes—
31#17985900C2212yes—
32#17985900C2214yes—
33#17985899C2214yes—
34#17985895C2212yes—
35#17985895C2212yes—
36#17985894C2214yes—
37#17985891C2214yes—
38#17985890C2214yes—
39#17985890C2214yes—
40#17985888C2214yes—
41#17985881C2214yes—
42#17985880C2214yes—
43#17985877C2214yes—
44#17985877C2214yes—
45#17985876C2214yes—
46#17985875C2214yes—
47#17985873C2214yes—
48#17985861C2214yes—
49#17985858C2214yes—
50C4⋊C4×Dic71C2212yes—
51C4⋊C4×Dic71C2212yes—
52C4⋊C4×Dic71C2212yes—
53C4⋊C4×Dic71C2212yes—
54#17985850C2214yes—
55C22⋊C4×Dic71C2212yes—
56C22⋊C4×Dic71C2212yes—
57C4×(C2×D71)⋊C4C2214yes—
58#17985846C2214yes—
59#17985846C2214yes—
60#17985846C2214yes—
61#17985845C2211yescharacteristic
62#17985845C2211yescharacteristic
63C42×Dic71C2211yescharacteristic
64C42×Dic71C2211yescharacteristic
65C2×#117×C71C2211yescharacteristic
66#388×C71C2214yes—
67#387×C71C2214yes—
68#384×C71C2211yescharacteristic
69#382×C71C2211yescharacteristic
70#380×C71C2212yes—
71C4×C4⋊C4×C71C2212yes—
72C22×D284C2312yes—
73C22×D284C2314yes—
74C22×C22⋊D71C2314yes—
75C23×Dic71C2311yescharacteristic
76C23×Dic71C2311yescharacteristic
77C22×C4×D71C2314yes—
78C22×C4×D71C2314yes—
79C22×C4×D71C2314yes—
80#14556125C2×C418yescomplemented
81#14556120C2×C418yescomplemented
82#14556113C2×C4132yescomplemented
83#14556102C2×C4116yescomplemented
84#14556101C2×C4132yescomplemented
85C4×D284C2×C4116yescomplemented
86#14556090—28no—
87#14556090—28no—
88#14556090—48no—
89(C22⋊D71)⋊C4—48no—
90#14556087—24no—
91C2×C4⋊Dic71C2312yes—
92C2×C4⋊Dic71C2312yes—
93C2×Dic71⋊C4C2314yes—
94C2×Dic71⋊C4C2314yes—
95C2×Dic71⋊C4C2314yes—
96C2×C22⋊Dic71C2312yes—
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×C4×Dic71—24no—
106C2×C4×Dic71C2311yescharacteristic
107C2×C4×Dic71C2312yes—
108C2×C4×Dic71C2312yes—
109C2×C4×Dic71C2311yescharacteristic
110C2×C4×Dic71C2311yescharacteristic
111C2×C4×Dic71C2314yes—
112C2×C4×Dic71C2311yescharacteristic
113C2×C4×Dic71C2312yes—
114C2×C4×Dic71C2312yes—
115C2×C4×Dic71C2314yes—
116C2×C4×Dic71C2311yescharacteristic
117C2×C4×Dic71C2311yescharacteristic
118C42×D71—48no—
119C71⋊#95C2312yes—
120C71⋊#95C2312yes—
121C71⋊#95C2312yes—
122C71⋊#95C2312yes—
123C71⋊#95C2314yes—
124C71⋊#95C2314yes—
125C71⋊#95C2314yes—
126C71⋊#95C2314yes—
127C23×C4×C71C2311yescharacteristic
128#117×C71C2×C418yescomplemented
129C2×C4⋊C4×C71C2312yes—
130C2×C4⋊C4×C71C2312yes—
131C2×C4⋊C4×C71C2314yes—
132C2×C4⋊C4×C71C2312yes—
133C2×C22⋊C4×C71C2314yes—
134C2×C22⋊C4×C71C2312yes—
135C2×C42×C71C2314yes—
136C2×C42×C71C2311yescharacteristic
137C2×C42×C71C2311yescharacteristic
138#95×C71C2314yes—
139#95×C71C2314yes—
140#95×C71C2314yes—
141C23×D71C2414yes—
142C2×D284—24no—
143C2×D284—24no—
144C2×D284—44no—
145C2×D284—44no—
146C2×D284C22×C418yes—
147C2×D284C22×C4116yes—
148C2×C22⋊D71—44no—
149C2×C22⋊D71—44no—
150C2×C22⋊D71C22×C4116yes—
151C22×Dic71—22no—
152C22×Dic71—22no—
153C22×Dic71—24no—
154C22×Dic71—42no—
155C22×Dic71C2411yescharacteristic
156C22×Dic71C2411yescharacteristic
157C22×Dic71C2412yes—
158C22×Dic71C2414yes—
159C22×Dic71C2412yes—
160C22×Dic71C2411yescharacteristic
161C22×Dic71C2411yescharacteristic
162C2×C4×D71—28no—
163C2×C4×D71—28no—
164C2×C4×D71—28no—
165C2×C4×D71—28no—
166C2×C4×D71—44no—
167C2×C4×D71—44no—
168C2×C4×D71—44no—
169C2×C4×D71—44no—
170C2×C4×D71C22×C4116yes—
171C4⋊Dic71—24no—
172C4⋊Dic71C22×C418yes—
173Dic71⋊C4—44no—
174Dic71⋊C4—44no—
175Dic71⋊C4C22×C4116yes—
176C22⋊Dic71—24no—
177(C2×D71)⋊C4—28no—
178(C2×D71)⋊C4—28no—
179(C2×D71)⋊C4—44no—
180(C2×D71)⋊C4—44no—
181(C2×D71)⋊C4—44no—
182(C2×D71)⋊C4—44no—
183(C2×D71)⋊C4C22×C4116yes—
184(C2×D71)⋊C4C22×C4116yes—
185C4×Dic71—22no—
186C4×Dic71—22no—
187C4×Dic71—22no—
188C4×Dic71—22no—
189C4×Dic71—24no—
190C4×Dic71—28no—
191C4×Dic71—24no—
192C4×Dic71—44no—
193C4×Dic71—42no—
194C4×Dic71—42no—
195C4×Dic71#4114yes—
196C4×Dic71#4114yes—
197C24×C71C2411yescharacteristic
198C22×C4×C71—22no—
199C22×C4×C71—22no—
200C22×C4×C71C22×C414yes—
201C22×C4×C71C2414yes—
202C22×C4×C71C2414yes—
203C22×C4×C71C2412yes—
204C22×C4×C71C2412yes—
205C22×C4×C71C2411yescharacteristic
206C22×C4×C71C2411yescharacteristic
207C4⋊C4×C71—24no—
208C4⋊C4×C71—24no—
209C4⋊C4×C71—44no—
210C4⋊C4×C71C22×C418yes—
211C22⋊C4×C71—44no—
212C22⋊C4×C71C22×C418yes—
213C42×C71—44no—
214C42×C71C22×C414yes—
215C42×C71C22×C414yes—
216C23×C71—21no—
217C23×C71—21no—
218C23×C71—22no—
219C23×C71—22no—
220C23×C71C23×C412yes—
221C23×C71C2511yescharacteristic
222C22×D71—24no—
223C22×D71—24no—
224C22×D71—24no—
225C22×D71—24no—
226C22×D71—24no—
227C22×D71—24no—
228C22×D71C23×C418yes—
229C2×C4×C71—21no—
230C2×C4×C71—21no—
231C2×C4×C71—22no—
232C2×C4×C71—22no—
233C2×C4×C71—22no—
234C2×C4×C71—22no—
235C2×C4×C71—28no—
236C2×C4×C71—24no—
237C2×C4×C71—24no—
238C2×C4×C71—24no—
239C2×C4×C71—24no—
240C2×C4×C71—24no—
241C2×C4×C71—24no—
242C2×C4×C71C23×C414yes—
243C2×C4×C71C23×C414yes—
244C2×C4×C71C23×C412yes—
245C2×C4×C71C23×C412yes—
246C2×C4×C71C2×#4112yes—
247C2×C4×C71C2×#4112yes—
248C22⋊D71—416no—
249C2×Dic71—22no—
250C2×Dic71—24no—
251C2×Dic71—22no—
252C2×Dic71—22no—
253C2×Dic71—24no—
254C2×Dic71—21no—
255C2×Dic71—21no—
256C2×Dic71—22no—
257C2×Dic71—22no—
258C2×Dic71—22no—
259C2×Dic71—22no—
260C2×Dic71—22no—
261C2×Dic71—44no—
262C2×Dic71—44no—
263C2×Dic71C23×C418yes—
264C2×Dic71C2×#4112yes—
265C2×Dic71C2×#4114yes—
266C2×Dic71C2×#4112yes—
267C2×Dic71H(5,𝔽2)14yes—
268D284—216no—
269D284—416no—
270C4×D71—416no—
271C4×D71—416no—
272C4×D71—416no—
273C22×C71—22no—
274C22×C71—24no—
275C22×C71—41no—
276C22×C71—41no—
277C22×C71—42no—
278C22×C71C24×C411yescharacteristic
279C22×C71C22×#4111yescharacteristic
280C22×C71C22×#4111yescharacteristic
281C22×C71C2×H(5,𝔽2)12yes—
282C22×C71#58512yes—
283C2×D71—28no—
284C2×D71—28no—
285C2×D71—28no—
286C2×D71—44no—
287C2×D71—44no—
288C2×D71—44no—
289C2×D71—44no—
290C4×C71—22no—
291C4×C71—24no—
292C4×C71—24no—
293C4×C71—44no—
294C4×C71—44no—
295C4×C71C4×#4114yes—
296Dic71—28no—
297Dic71—22no—
298Dic71—24no—
299Dic71—41no—
300Dic71—41no—
301Dic71—42no—
302Dic71#54014yescomplemented
303C4×#540—711nomaximal, Sylow
304C2×C71—42no—
305C2×C71C2×C4×#4111yescharacteristic
306C2×C71C4×H(5,𝔽2)12yes—
307C2×C71C2×#54011yescharacteristic
308C2×C71#345511yescharacteristic
309C2×C71#346012yes—
310D71—48no—
311C2×#540—711no—
312#2335—714no—
313#2333—714no—
314#2330—711no—
315#2328—711no—
316#2324—712no—
317#2319—712no—
318#2314D7111yescharacteristic, complemented
319C4×#127—711no—
320C4×#125—711no—
321C4×#123—714no—
322C4×#121—714no—
323C4×#121—712no—
324C42×D4—712no—
325C4×#117—711no—
326C71C4×#54011yescharacteristic, complemented, Sylow
327#540—7116no—
328C2×#127—711no—
329C2×#125—711no—
330C2×#123—714no—
331C2×#121—712no—
332C2×#121—714no—
333C2×C4×D4—714no—
334C2×C4×D4—712no—
335C2×C4×D4—712no—
336C2×C4×D4—714no—
337C2×C4×D4—712no—
338C2×#117—711no—
339C2×#117C2×D7111yescharacteristic
340C22×C42—714no—
341C22×C42—711no—
342#390—712no—
343#390—714no—
344#390—712no—
345#390—714no—
346#389—712no—
347#389—712no—
348#389—714no—
349#388—714no—
350#388—714no—
351#388—714no—
352#388—714no—
353#388C2×D7114yes—
354#387—714no—
355#387—714no—
356#387C2×D7114yes—
357#386—714no—
358#386—714no—
359#386—714no—
360#386—714no—
361#385—714no—
362#384C2×D7111yescharacteristic
363#382C2×D7111yescharacteristic
364#381—714no—
365#380—714no—
366#380C2×D7112yes—
367C4×C4⋊C4—712no—
368C4×C4⋊C4—712no—
369C4×C4⋊C4—712no—
370C4×C4⋊C4—712no—
371C4×C4⋊C4—714no—
372C4×C4⋊C4C2×D7112yes—
373C4×C22⋊C4—712no—
374C4×C22⋊C4—712no—
375C4×C22⋊C4—714no—
376C4×C22⋊C4—714no—
377C4×C22⋊C4—714no—
378C4×C22⋊C4—714no—
379#376—711no—
380#376—711no—
381C43—711no—
382C43—711no—
383C22×D4—714no—
384C22×D4—714no—
385C22×D4—712no—
386C23×C4—714no—
387C23×C4—711no—
388C23×C4—714no—
389C23×C4—711no—
390C23×C4—714no—
391C23×C4C22×D7111yescharacteristic
392#127—718no—
393#125—718no—
394#123—7132no—
395#121—7116no—
396#121—7132no—
397C4×D4—2848no—
398C4×D4—2848no—
399C4×D4—7116no—
400C4×D4—1428no—
401C4×D4—1428no—
402#117—1424no—
403#117C4×D7118yescomplemented
404C2×C4⋊C4—714no—
405C2×C4⋊C4—714no—
406C2×C4⋊C4—712no—
407C2×C4⋊C4—714no—
408C2×C4⋊C4—712no—
409C2×C4⋊C4C22×D7112yes—
410C2×C4⋊C4C22×D7112yes—
411C2×C4⋊C4C22×D7114yes—
412C2×C4⋊C4C22×D7112yes—
413C2×C22⋊C4—714no—
414C2×C22⋊C4—714no—
415C2×C22⋊C4—714no—
416C2×C22⋊C4—714no—
417C2×C22⋊C4—714no—
418C2×C22⋊C4—714no—
419C2×C22⋊C4—712no—
420C2×C22⋊C4—714no—
421C2×C22⋊C4—714no—
422C2×C22⋊C4C22×D7114yes—
423C2×C22⋊C4C22×D7112yes—
424C2×C42—2848no—
425C2×C42—712no—
426C2×C42—712no—
427C2×C42—714no—
428C2×C42—711no—
429C2×C42—711no—
430C2×C42—711no—
431C2×C42—712no—
432C2×C42—714no—
433C2×C42—712no—
434C2×C42—711no—
435C2×C42—711no—
436C2×C42—711no—
437C2×C42—1424no—
438C2×C42C22×D7114yes—
439C2×C42C22×D7111yescharacteristic
440C2×C42C22×D7111yescharacteristic
441#95—712no—
442#95—712no—
443#95—712no—
444#95—712no—
445#95—714no—
446#95—714no—
447#95—714no—
448#95—714no—
449#95C22×D7114yes—
450#95C22×D7114yes—
451#95C22×D7114yes—
452C24—714no—
453C24C23×D7111yescharacteristic
454C2×D4—2844no—
455C2×D4—2844no—
456C2×D4—2844no—
457C2×D4—2844no—
458C2×D4—7116no—
459C2×D4—7116no—
460C2×D4—718no—
461C2×D4—1424no—
462C2×D4—1424no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—2844no—
466C22×C4—2844no—
467C22×C4—2844no—
468C22×C4—2844no—
469C22×C4—2842no—
470C22×C4—7116no—
471C22×C4—711no—
472C22×C4—711no—
473C22×C4—712no—
474C22×C4—714no—
475C22×C4—711no—
476C22×C4—712no—
477C22×C4—711no—
478C22×C4—1428no—
479C22×C4—1428no—
480C22×C4—1428no—
481C22×C4—1428no—
482C22×C4—1422no—
483C22×C4—1422no—
484C22×C4—1424no—
485C22×C4C2×C4×D7114yes—
486C22×C4C23×D7114yes—
487C22×C4C23×D7114yes—
488C22×C4C23×D7112yes—
489C22×C4C23×D7112yes—
490C22×C4C23×D7111yescharacteristic
491C22×C4C23×D7111yescharacteristic
492C4⋊C4—24no—
493C4⋊C4—24no—
494C4⋊C4—44no—
495C4⋊C4—2844no—
496C4⋊C4—2844no—
497C4⋊C4—7116no—
498C4⋊C4—718no—
499C4⋊C4—1424no—
500C4⋊C4C2×C4×D7118yes—
501C22⋊C4—44no—
502C22⋊C4—2844no—
503C22⋊C4—2844no—
504C22⋊C4—2844no—
505C22⋊C4—2844no—
506C22⋊C4—7116no—
507C22⋊C4—7116no—
508C22⋊C4—1428no—
509C22⋊C4—1428no—
510C22⋊C4—1424no—
511C22⋊C4C2×C4×D7118yes—
512C42—44no—
513C42—2844no—
514C42—2842no—
515C42—2842no—
516C42—714no—
517C42—714no—
518C42—1424no—
519C42—1428no—
520C42—1422no—
521C42—1422no—
522C42—1424no—
523C42—1422no—
524C42—1422no—
525C42C2×C4×D7114yes—
526C42C2×C4×D7114yes—
527C23—21no—
528C23—21no—
529C23—22no—
530C23—22no—
531C23—718no—
532C23—1424no—
533C23—1424no—
534C23—1424no—
535C23—1424no—
536C23—1424no—
537C23—1424no—
538C23C22×C4×D7112yes—
539C23C24×D7111yescharacteristic
540D4—28416no—
541D4—28416no—
542D4—14216no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—28no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—28416no—
557C2×C4—28416no—
558C2×C4—28416no—
559C2×C4—2844no—
560C2×C4—2844no—
561C2×C4—712no—
562C2×C4—714no—
563C2×C4—718no—
564C2×C4—712no—
565C2×C4—714no—
566C2×C4—1421no—
567C2×C4—1421no—
568C2×C4—1422no—
569C2×C4—1422no—
570C2×C4—1422no—
571C2×C4—1422no—
572C2×C4—1422no—
573C2×C4—1422no—
574C2×C4—1424no—
575C2×C4—1422no—
576C2×C4—1422no—
577C2×C4—1424no—
578C2×C4C22×C4×D7114yes—
579C2×C4C22×C4×D7114yes—
580C2×C4C22×C4×D7112yes—
581C2×C4C22×C4×D7112yes—
582C2×C4#41×D7112yes—
583C2×C4#41×D7112yes—
584C22—22no—
585C22—24no—
586C22—41no—
587C22—42no—
588C22—41no—
589C22—2844no—
590C22—2844no—
591C22—2844no—
592C22—2844no—
593C22—1428no—
594C22—1428no—
595C22—1428no—
596C22C23×C4×D7111yescharacteristic
597C22C2×#41×D7111yescharacteristic
598C22C2×#41×D7111yescharacteristic
599C22H(5,𝔽2)×D7112yes—
600C22#1798683612yes—
601C4—22no—
602C4—24no—
603C4—24no—
604C4—44no—
605C4—44no—
606C4—2841no—
607C4—2842no—
608C4—2841no—
609C4—714no—
610C4—1422no—
611C4—1424no—
612C4—1428no—
613C4#1798646714yes—
614C2—42no—
615C2—2848no—
616C2C2×#1798646711yescharacteristic
617C2#2260154612yes—
618C2#540×D7111yescharacteristic
619C2#2260173111yescharacteristic
620C2#2260176012yes—
subgroupG/H
center, FrattiniC23C24×D71
commutatorC22×C71C24×C4
FittingC71×#2314C2
socle, cosocleC23×C71C25
prime radicalC2×#17986604C2
hypercenter#2314D71
hypocenterC71C4×#540
LCSG ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ #2314

Complements

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

NH ≅ G/Ntype
2. #22591136C2semidirect
3. #22591103C2semidirect
4. #22591079C2semidirect
5. #22591063C2semidirect
6. #22591029C2semidirect
7. #22590978C2semidirect
8. #22590902C2semidirect
9. #22590895C2semidirect
10. #22590861C2semidirect
11. #22590852C2semidirect
12. #22590846C2semidirect
13. C4×#14556090C2semidirect
14. #117×Dic71C2semidirect
15. C71×#2314C2semidirect
16. #17986604C4semidirect
80. #14556125C2×C4semidirect
81. #14556120C2×C4semidirect
82. #14556113C2×C4semidirect
83. #14556102C2×C4semidirect
84. #14556101C2×C4semidirect
85. C4×D284C2×C4semidirect
128. #117×C71C2×C4semidirect
302. Dic71#540semidirect
318. #2314D71semidirect
326. C71C4×#540semidirect
403. #117C4×D71semidirect

Maximal subgroups

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

subgroupindex
C2×#179866042
#225911362
#225911032
#225910792
#225910632
#225910292
#225909782
#225909022
#225908952
#225908612
#225908522
#225908462
C4×#145560902
#117×Dic712
C71×#23142
C4×#54071