Finite Group Databasebeta

← #2537×D71 #32169187

Subgroups — #2537×D71 #32169187

Order 18176.

4782 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#121×D71C214yesmaximal, complemented
2C2×C4×D4×D71C212yesmaximal, complemented
3#22593256C211yesmaximal, characteristic, complemented
4#22593255C214yesmaximal, complemented
5#22593254C214yesmaximal, complemented
6#22593253C212yesmaximal, complemented
7#22593252C214yesmaximal, complemented
8#394×D71C214yesmaximal, complemented
9#384×D71C211yesmaximal, characteristic, complemented
10#380×D71C214yesmaximal, complemented
11C71×#2537C211yesmaximal, characteristic, complemented
12C22×D4×D71C2214yes—
13C22×D4×D71C2212yes—
14C23×C4×D71C2214yes—
15C2×#14556103C2214yescomplemented
16C2×#14556102C2214yescomplemented
17C2×#14556101C2218yescomplemented
18C2×#14556100C2218yescomplemented
19C2×#14556099C2214yescomplemented
20#121×D71—232no—
21C2×C4×D284C2212yescomplemented
22C2×D4×Dic71C2212yescomplemented
23C2×#14556090C2212yescomplemented
24C2×C4×C22⋊D71C2214yescomplemented
25C2×(C22⋊D71)⋊C4C2214yescomplemented
26C4×D4×D71—216no—
27C2×C4⋊C4×D71C2214yes—
28C2×C4⋊C4×D71C2212yes—
29C2×C4⋊C4×D71C2212yes—
30C2×C22⋊C4×D71C2214yes—
31C2×C22⋊C4×D71C2218yes—
32C2×C42×D71C2211yescharacteristic
33#17985917C2214yescomplemented
34#17985916C2214yescomplemented
35#17985915C2218yescomplemented
36#17985914C2218yescomplemented
37#17985913C2214yescomplemented
38#17985869C2211yescharacteristic, complemented
39#17985868C2212yescomplemented
40#17985868C2214yescomplemented
41#17985859C2214yescomplemented
42#17985858C2218yescomplemented
43#17985857C2218yescomplemented
44#17985856C2214yescomplemented
45#17985856C2214yescomplemented
46#95×D71C2214yes—
47C2×#121×C71C2214yescomplemented
48C2×C4×D4×C71C2212yescomplemented
49#394×C71C2214yescomplemented
50#384×C71C2211yescharacteristic, complemented
51#380×C71C2214yescomplemented
52C24×D71C2314yes—
53C22×D284C2314yes—
54C22×D284C2312yes—
55C22×C22⋊D71C2318yes—
56C22×C22⋊D71C2314yes—
57C2×D4×D71—232no—
58C2×D4×D71—28no—
59C2×D4×D71—28no—
60C2×D4×D71—48no—
61C2×D4×D71D4116yescomplemented
62C23×Dic71C2314yes—
63C22×C4×D71—216no—
64C22×C4×D71—216no—
65C22×C4×D71—216no—
66C22×C4×D71C2314yes—
67C22×C4×D71C2314yes—
68C22×C4×D71C2314yes—
69C22×C4×D71C2314yes—
70C22×C4×D71C2311yescharacteristic
71C22×C4×D71C2312yes—
72#14556103—216no—
73#14556102—216no—
74#14556101—232no—
75#14556100—232no—
76#14556099—216no—
77C4×D284—28no—
78D4×Dic71—28no—
79#14556090—28no—
80C4×C22⋊D71—216no—
81(C22⋊D71)⋊C4—216no—
82C2×C4⋊Dic71C2312yes—
83C2×C4⋊Dic71C2314yes—
84C2×C4⋊Dic71C2312yes—
85C2×Dic71⋊C4C2314yes—
86C2×Dic71⋊C4C2318yes—
87C2×Dic71⋊C4C2314yes—
88C4⋊C4×D71—28no—
89C4⋊C4×D71—216no—
90C4⋊C4×D71—44no—
91C2×C22⋊Dic71C2318yes—
92C2×C22⋊Dic71C2314yes—
93C2×(C2×D71)⋊C4C2318yes—
94C2×(C2×D71)⋊C4C2318yes—
95C2×(C2×D71)⋊C4C2314yes—
96C2×(C2×D71)⋊C4C2314yes—
97C22⋊C4×D71—232no—
98C22⋊C4×D71—216no—
99C2×C4×Dic71C2311yescharacteristic
100C2×C4×Dic71C2312yes—
101C42×D71D418yescomplemented
102C71⋊#95C2314yes—
103C71⋊#95C2318yes—
104C22×D4×C71C2314yes—
105C22×D4×C71C2312yes—
106C23×C4×C71C2314yes—
107#121×C71—216no—
108C4×D4×C71—28no—
109C2×C4⋊C4×C71C2312yes—
110C2×C4⋊C4×C71C2314yes—
111C2×C4⋊C4×C71C2312yes—
112C2×C22⋊C4×C71C2318yes—
113C2×C22⋊C4×C71C2314yes—
114C2×C42×C71C2311yescharacteristic
115#95×C71C2314yes—
116C23×D71—28no—
117C23×D71—28no—
118C23×D71—28no—
119C23×D71—28no—
120C23×D71—28no—
121C23×D71—28no—
122C23×D71C2×D4116yes—
123C23×D71C2414yes—
124C23×D71C2411yescharacteristic
125C2×D284—24no—
126C2×D284—24no—
127C2×D284—216no—
128C2×D284—44no—
129C2×D284C2×D418yescomplemented
130C2×C22⋊D71—216no—
131C2×C22⋊D71—216no—
132C2×C22⋊D71—28no—
133C2×C22⋊D71—28no—
134C2×C22⋊D71—48no—
135C2×C22⋊D71C2×D4116yescomplemented
136D4×D71—264no—
137D4×D71—464no—
138C22×Dic71—28no—
139C22×Dic71—28no—
140C22×Dic71—28no—
141C22×Dic71C2414yes—
142C22×Dic71C2414yes—
143C22×Dic71C2411yescharacteristic
144C22×Dic71C2412yes—
145C2×C4×D71—22no—
146C2×C4×D71—22no—
147C2×C4×D71—28no—
148C2×C4×D71—28no—
149C2×C4×D71—216no—
150C2×C4×D71—28no—
151C2×C4×D71—28no—
152C2×C4×D71—28no—
153C2×C4×D71—28no—
154C2×C4×D71—28no—
155C2×C4×D71—28no—
156C2×C4×D71—28no—
157C2×C4×D71—28no—
158C2×C4×D71—464no—
159C2×C4×D71C2×D414yes—
160C2×C4×D71C2×D418yes—
161C2×C4×D71C2×D418yes—
162C2×C4×D71#4118yes—
163C4⋊Dic71—24no—
164C4⋊Dic71—28no—
165C4⋊Dic71—42no—
166Dic71⋊C4—28no—
167Dic71⋊C4—216no—
168Dic71⋊C4—44no—
169C22⋊Dic71—216no—
170C22⋊Dic71—28no—
171(C2×D71)⋊C4—216no—
172(C2×D71)⋊C4—216no—
173(C2×D71)⋊C4—28no—
174(C2×D71)⋊C4—28no—
175C4×Dic71C2×D414yescomplemented
176C4×Dic71C2×D418yescomplemented
177C24×C71C2414yes—
178C2×D4×C71—216no—
179C2×D4×C71—24no—
180C2×D4×C71—24no—
181C2×D4×C71—44no—
182C2×D4×C71C2×D418yescomplemented
183C22×C4×C71—28no—
184C22×C4×C71—28no—
185C22×C4×C71—28no—
186C22×C4×C71C2414yes—
187C22×C4×C71C2414yes—
188C22×C4×C71C2411yescharacteristic
189C22×C4×C71C2412yes—
190C4⋊C4×C71—24no—
191C4⋊C4×C71—28no—
192C4⋊C4×C71—42no—
193C22⋊C4×C71—216no—
194C22⋊C4×C71—28no—
195C42×C71C2×D414yescomplemented
196C23×C71—24no—
197C23×C71—24no—
198C23×C71—24no—
199C23×C71—24no—
200C23×C71—24no—
201C23×C71—24no—
202C23×C71C22×D418yes—
203C23×C71C2511yescharacteristic
204C22×D71—24no—
205C22×D71—24no—
206C22×D71—24no—
207C22×D71—24no—
208C22×D71—24no—
209C22×D71—24no—
210C22×D71—232no—
211C22×D71—232no—
212C22×D71—232no—
213C22×D71—416no—
214C22×D71—416no—
215C22×D71—416no—
216C22×D71—416no—
217C22×D71C22×D418yes—
218C22×D71C22×D412yes—
219C22×D71C22×D414yes—
220C22×D71C2×#4114yes—
221C22×D71H(5,𝔽2)12yes—
222C22×D712−1+412yes—
223D4×C71—216no—
224D4×C71—416no—
225C2×C4×C71—21no—
226C2×C4×C71—21no—
227C2×C4×C71—24no—
228C2×C4×C71—24no—
229C2×C4×C71—28no—
230C2×C4×C71—24no—
231C2×C4×C71—24no—
232C2×C4×C71—416no—
233C2×C4×C71C22×D412yes—
234C2×C4×C71C22×D414yes—
235C2×C4×C71C22×D414yes—
236C2×C4×C71C2×#4114yes—
237C22⋊D71—232no—
238C22⋊D71—432no—
239C2×Dic71—21no—
240C2×Dic71—21no—
241C2×Dic71—24no—
242C2×Dic71—24no—
243C2×Dic71—28no—
244C2×Dic71—24no—
245C2×Dic71—24no—
246C2×Dic71—416no—
247C2×Dic71C22×D412yes—
248C2×Dic71C22×D414yes—
249C2×Dic71C22×D414yes—
250C2×Dic71C2×#4114yes—
251D284—216no—
252D284—416no—
253C4×D71—28no—
254C4×D71—416no—
255C4×D71—416no—
256C4×D71—416no—
257C4×D71—416no—
258C4×D71#121132yescomplemented
259C22×C71—28no—
260C22×C71—28no—
261C22×C71—28no—
262C22×C71—44no—
263C22×C71—44no—
264C22×C71—44no—
265C22×C71—44no—
266C22×C71C23×D411yescharacteristic
267C22×C71C23×D412yes—
268C22×C71C22×#4112yes—
269C22×C71C2×H(5,𝔽2)11yescharacteristic
270C22×C71C2×2−1+411yescharacteristic
271C2×D71—28no—
272C2×D71—28no—
273C2×D71—28no—
274C2×D71—44no—
275C2×D71—44no—
276C2×D71—44no—
277C2×D71—44no—
278C2×D71—464no—
279C2×D71C2×#12118yes—
280C2×D71#53714yes—
281C2×D71D4214yes—
282C2×D71#54718yes—
283C2×D71#55514yes—
284C4×C71—22no—
285C4×C71—44no—
286C4×C71—44no—
287C4×C71C2×#12118yescomplemented
288Dic71—22no—
289Dic71—44no—
290Dic71—44no—
291Dic71C2×#12118yescomplemented
292C2×#2537—711nomaximal, Sylow
293C2×C71—48no—
294C2×C71C22×#12112yes—
295C2×C71C2×#53711yescharacteristic
296C2×C71C2×D4211yescharacteristic
297C2×C71C2×#54712yes—
298C2×C71C2×#55511yescharacteristic
299D71—48no—
300D71#253718yesdirect product factor
301C22×#121—714no—
302C22×C4×D4—712no—
303#2537—714no—
304#2537—711no—
305#2537—714no—
306#2537—714no—
307#2537—712no—
308#2537D7111yescharacteristic, direct product factor
309C2×#394—714no—
310C2×#384—711no—
311C2×#380—714no—
312C71C2×#253711yescharacteristic, complemented, Sylow
313C23×D4—714no—
314C23×D4—712no—
315C24×C4—714no—
316C2×#121—714no—
317C2×#121—718no—
318C2×#121—714no—
319C2×#121—714no—
320C2×#121—718no—
321C2×#121—14232no—
322C2×#121C2×D7114yescomplemented
323C2×C4×D4—712no—
324C2×C4×D4—714no—
325C2×C4×D4—712no—
326C2×C4×D4—714no—
327C2×C4×D4—712no—
328C2×C4×D4—14216no—
329C2×C4×D4C2×D7112yescomplemented
330C22×C4⋊C4—712no—
331C22×C4⋊C4—712no—
332C22×C4⋊C4—714no—
333C22×C22⋊C4—714no—
334C22×C22⋊C4—718no—
335C22×C42—711no—
336#394—714no—
337#394—714no—
338#394—718no—
339#394—718no—
340#394—714no—
341#394C2×D7114yescomplemented
342#384—714no—
343#384—712no—
344#384—711no—
345#384C2×D7111yescharacteristic, complemented
346#380—714no—
347#380—714no—
348#380—714no—
349#380—718no—
350#380—718no—
351#380C2×D7114yescomplemented
352C2×#95—714no—
353C25—714no—
354C22×D4—2848no—
355C22×D4—718no—
356C22×D4—714no—
357C22×D4—7116no—
358C22×D4—714no—
359C22×D4—712no—
360C22×D4—1428no—
361C22×D4—1428no—
362C22×D4—14232no—
363C22×D4C22×D7114yes—
364C22×D4C22×D7112yes—
365C23×C4—712no—
366C23×C4—711no—
367C23×C4—714no—
368C23×C4—714no—
369C23×C4—714no—
370C23×C4—714no—
371C23×C4—714no—
372C23×C4—14216no—
373C23×C4—14216no—
374C23×C4—14216no—
375C23×C4C22×D7114yes—
376#121—216no—
377#121—14232no—
378#121—14216no—
379#121—14216no—
380#121—14232no—
381#121—14216no—
382C4×D4—28no—
383C4×D4—14216no—
384C4×D4—1428no—
385C4×D4—14216no—
386C4×D4—1428no—
387C4×D4—1428no—
388C2×C4⋊C4—2844no—
389C2×C4⋊C4—714no—
390C2×C4⋊C4—712no—
391C2×C4⋊C4—714no—
392C2×C4⋊C4—712no—
393C2×C4⋊C4—718no—
394C2×C4⋊C4—714no—
395C2×C4⋊C4—1428no—
396C2×C4⋊C4—14216no—
397C2×C4⋊C4C22×D7112yes—
398C2×C4⋊C4C22×D7114yes—
399C2×C4⋊C4C22×D7112yes—
400C2×C22⋊C4—714no—
401C2×C22⋊C4—718no—
402C2×C22⋊C4—714no—
403C2×C22⋊C4—714no—
404C2×C22⋊C4—718no—
405C2×C22⋊C4—718no—
406C2×C22⋊C4—14216no—
407C2×C22⋊C4—14232no—
408C2×C22⋊C4C22×D7114yes—
409C2×C22⋊C4C22×D7118yes—
410C2×C42—711no—
411C2×C42—712no—
412C2×C42—718no—
413C2×C42C22×D7111yescharacteristic
414#95—714no—
415#95—718no—
416#95C22×D7114yes—
417C24—711no—
418C24—7116no—
419C24—714no—
420C24—1428no—
421C24—1428no—
422C24—1428no—
423C24—1428no—
424C24—1428no—
425C24—1428no—
426C24C23×D7114yes—
427C2×D4—216no—
428C2×D4—24no—
429C2×D4—24no—
430C2×D4—44no—
431C2×D4—2844no—
432C2×D4—28464no—
433C2×D4—2848no—
434C2×D4—7116no—
435C2×D4—718no—
436C2×D4—14216no—
437C2×D4—14264no—
438C2×D4—1428no—
439C2×D4—1428no—
440C2×D4—14216no—
441C2×D4—14216no—
442C2×D4—1424no—
443C2×D4—1424no—
444C2×D4D4×D7118yescomplemented
445C22×C4—28no—
446C22×C4—28no—
447C22×C4—28no—
448C22×C4—28464no—
449C22×C4—712no—
450C22×C4—718no—
451C22×C4—718no—
452C22×C4—718no—
453C22×C4—711no—
454C22×C4—714no—
455C22×C4—714no—
456C22×C4—714no—
457C22×C4—1428no—
458C22×C4—1428no—
459C22×C4—1428no—
460C22×C4—1428no—
461C22×C4—1428no—
462C22×C4—14216no—
463C22×C4—1428no—
464C22×C4—1428no—
465C22×C4—1422no—
466C22×C4—1422no—
467C22×C4—1428no—
468C22×C4—1428no—
469C22×C4—1428no—
470C22×C4—1428no—
471C22×C4—1428no—
472C22×C4—1428no—
473C22×C4C23×D7114yes—
474C22×C4C23×D7114yes—
475C22×C4C23×D7111yescharacteristic
476C22×C4C23×D7112yes—
477C4⋊C4—24no—
478C4⋊C4—28no—
479C4⋊C4—42no—
480C4⋊C4—2844no—
481C4⋊C4—2842no—
482C4⋊C4—1428no—
483C4⋊C4—14216no—
484C4⋊C4—1424no—
485C4⋊C4—1428no—
486C22⋊C4—216no—
487C22⋊C4—28no—
488C22⋊C4—1428no—
489C22⋊C4—14216no—
490C22⋊C4—14216no—
491C22⋊C4—1428no—
492C22⋊C4—14216no—
493C22⋊C4—1428no—
494C42—714no—
495C42—718no—
496C42D4×D7114yescomplemented
497C23—24no—
498C23—24no—
499C23—24no—
500C23—24no—
501C23—24no—
502C23—24no—
503C23—28416no—
504C23—28416no—
505C23—28416no—
506C23—28416no—
507C23—712no—
508C23—714no—
509C23—714no—
510C23—712no—
511C23—712no—
512C23—718no—
513C23—14232no—
514C23—14232no—
515C23—14232no—
516C23—1424no—
517C23—1424no—
518C23—1424no—
519C23—1424no—
520C23—1424no—
521C23—1424no—
522C23C2×D4×D7118yes—
523C23C24×D7111yescharacteristic
524D4—216no—
525D4—416no—
526D4—28416no—
527D4—28432no—
528D4—14232no—
529D4—14216no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—28no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—416no—
538C2×C4—28416no—
539C2×C4—28416no—
540C2×C4—28416no—
541C2×C4—28416no—
542C2×C4—28416no—
543C2×C4—714no—
544C2×C4—714no—
545C2×C4—714no—
546C2×C4—7132no—
547C2×C4—712no—
548C2×C4—1428no—
549C2×C4—1424no—
550C2×C4—1424no—
551C2×C4—1428no—
552C2×C4—1424no—
553C2×C4—1424no—
554C2×C4—1421no—
555C2×C4—1421no—
556C2×C4C2×D4×D7112yes—
557C2×C4C2×D4×D7114yes—
558C2×C4C2×D4×D7114yes—
559C2×C4#41×D7114yes—
560C22—28no—
561C22—28no—
562C22—28no—
563C22—44no—
564C22—44no—
565C22—44no—
566C22—44no—
567C22—28464no—
568C22—2844no—
569C22—2844no—
570C22—2844no—
571C22—2844no—
572C22—714no—
573C22—718no—
574C22—718no—
575C22—714no—
576C22—714no—
577C22—1428no—
578C22—1428no—
579C22—1428no—
580C22C22×D4×D7112yes—
581C22C22×D4×D7111yescharacteristic
582C22C2×#41×D7112yes—
583C22H(5,𝔽2)×D7111yescharacteristic
584C222−1+4×D7111yescharacteristic
585C4—22no—
586C4—44no—
587C4—44no—
588C4—2844no—
589C4—2844no—
590C4—718no—
591C4—1422no—
592C4#121×D7118yescomplemented
593C2—48no—
594C2—2848no—
595C2—718no—
596C2C2×#121×D7112yes—
597C2#537×D7111yescharacteristic
598C2D42×D7111yescharacteristic
599C2#547×D7112yes—
600C2#555×D7111yescharacteristic
subgroupG/H
center, FrattiniC23C24×D71
commutator, socle, cosocleC23×C71C25
FittingC71×#2537C2
hypercenter#2537D71
hypocenterC71C2×#2537
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C23 ◁ #2537

Complements

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

NH ≅ G/Ntype
1. C2×#121×D71C2semidirect
2. C2×C4×D4×D71C2semidirect
3. #22593256C2semidirect
4. #22593255C2semidirect
5. #22593254C2semidirect
6. #22593253C2semidirect
7. #22593252C2semidirect
8. #394×D71C2semidirect
9. #384×D71C2semidirect
10. #380×D71C2semidirect
11. C71×#2537C2semidirect
15. C2×#14556103C22semidirect
16. C2×#14556102C22semidirect
17. C2×#14556101C22semidirect
18. C2×#14556100C22semidirect
19. C2×#14556099C22semidirect
21. C2×C4×D284C22semidirect
22. C2×D4×Dic71C22semidirect
23. C2×#14556090C22semidirect
24. C2×C4×C22⋊D71C22semidirect
25. C2×(C22⋊D71)⋊C4C22semidirect
33. #17985917C22semidirect
34. #17985916C22semidirect
35. #17985915C22semidirect
36. #17985914C22semidirect
37. #17985913C22semidirect
38. #17985869C22semidirect
39. #17985868C22semidirect
40. #17985868C22semidirect
41. #17985859C22semidirect
42. #17985858C22semidirect
43. #17985857C22semidirect
44. #17985856C22semidirect
45. #17985856C22semidirect
47. C2×#121×C71C22semidirect
48. C2×C4×D4×C71C22semidirect
49. #394×C71C22semidirect
50. #384×C71C22semidirect
51. #380×C71C22semidirect
61. C2×D4×D71D4semidirect
101. C42×D71D4semidirect
129. C2×D284C2×D4semidirect
135. C2×C22⋊D71C2×D4semidirect
175. C4×Dic71C2×D4semidirect
176. C4×Dic71C2×D4semidirect
182. C2×D4×C71C2×D4semidirect
195. C42×C71C2×D4semidirect
258. C4×D71#121semidirect
287. C4×C71C2×#121semidirect
291. Dic71C2×#121semidirect
300. D71#2537direct (N × H)
308. #2537D71direct (N × H)
312. C71C2×#2537semidirect
322. C2×#121C2×D71semidirect
329. C2×C4×D4C2×D71semidirect
341. #394C2×D71semidirect
345. #384C2×D71semidirect
351. #380C2×D71semidirect
444. C2×D4D4×D71semidirect
496. C42D4×D71semidirect
592. C4#121×D71semidirect

Maximal subgroups

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

subgroupindex
C2×#121×D712
C2×C4×D4×D712
#225932562
#225932552
#225932542
#225932532
#225932522
#394×D712
#384×D712
#380×D712
C71×#25372
C2×#253771