Finite Group Databasebeta

← #31919987

Subgroups — #31919987

Order 18176.

1862 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14556086C211yesmaximal, characteristic
2C2×#17985856C211yesmaximal, characteristic
3C2×C4×(C2×D71)⋊C4C211yesmaximal, characteristic
4#22586588C214yesmaximal
5#22586570C211yesmaximal, characteristic, complemented
6#22586569C211yesmaximal, characteristic, complemented
7#22586569C211yesmaximal, characteristic, complemented
8#22586566C214yesmaximal, complemented
9C71×C4⋊#95C211yesmaximal, characteristic, complemented
10C23×C4×D71C2211yescharacteristic
11C2×#14556086—24no—
12C2×#14556086—24no—
13C2×#14556086C418yescomplemented
14C22×C4⋊Dic71C2211yescharacteristic
15C22×C4⋊Dic71C2211yescharacteristic
16C22×(C2×D71)⋊C4C2212yes—
17C22×(C2×D71)⋊C4C2212yes—
18C22×(C2×D71)⋊C4C2212yes—
19C22×C4×Dic71C2211yescharacteristic
20C22×C4×Dic71C2212yes—
21#17985856—24no—
22C4×(C2×D71)⋊C4—24no—
23C2×C71⋊#95C2212yes—
24C2×C71⋊#95C2212yes—
25C2×C71⋊#95C2214yes—
26C2×C71⋊#95C2214yes—
27C2×C71⋊#95C2214yes—
28C22×C4⋊C4×C71C2211yescharacteristic
29C22×C4⋊C4×C71C2211yescharacteristic
30C22×C42×C71C2211yescharacteristic
31C2×#95×C71C2214yes—
32C24×D71C2311yescharacteristic
33C23×Dic71C2312yes—
34C23×Dic71C2312yes—
35C23×Dic71C2312yes—
36C23×Dic71C2311yescharacteristic
37C22×C4×D71—22no—
38C22×C4×D71—22no—
39C22×C4×D71—22no—
40C22×C4×D71—22no—
41C22×C4×D71C2×C414yes—
42C22×C4×D71D414yes—
43C22×C4×D71D414yes—
44#14556086—232no—
45C2×C4⋊Dic71—22no—
46C2×C4⋊Dic71—22no—
47C2×C4⋊Dic71—22no—
48C2×C4⋊Dic71—22no—
49C2×C4⋊Dic71—22no—
50C2×C4⋊Dic71C2×C414yes—
51C2×(C2×D71)⋊C4—24no—
52C2×(C2×D71)⋊C4—24no—
53C2×(C2×D71)⋊C4—24no—
54C2×(C2×D71)⋊C4—24no—
55C2×(C2×D71)⋊C4—24no—
56C2×(C2×D71)⋊C4—24no—
57C2×(C2×D71)⋊C4—44no—
58C2×(C2×D71)⋊C4C2×C418yes—
59C2×C4×Dic71—24no—
60C2×C4×Dic71—24no—
61C2×C4×Dic71—22no—
62C2×C4×Dic71—22no—
63C2×C4×Dic71—22no—
64C2×C4×Dic71C2×C418yes—
65C71⋊#95—24no—
66C71⋊#95—24no—
67C71⋊#95—28no—
68C71⋊#95—28no—
69C71⋊#95—44no—
70C23×C4×C71C2312yes—
71C23×C4×C71C2312yes—
72C23×C4×C71C2312yes—
73C23×C4×C71C2311yescharacteristic
74C2×C4⋊C4×C71—22no—
75C2×C4⋊C4×C71—22no—
76C2×C4⋊C4×C71—22no—
77C2×C4⋊C4×C71—22no—
78C2×C4⋊C4×C71—22no—
79C2×C4⋊C4×C71C2×C414yes—
80C2×C42×C71—22no—
81C2×C42×C71—22no—
82C2×C42×C71—22no—
83#95×C71—44no—
84C23×D71—21no—
85C23×D71—21no—
86C23×D71—21no—
87C23×D71—21no—
88C23×D71—22no—
89C23×D71—22no—
90C23×D71—22no—
91C23×D71—22no—
92C23×D71C22×C412yes—
93C23×D71C2×D412yes—
94C23×D71C2×D412yes—
95C22×Dic71—21no—
96C22×Dic71—21no—
97C22×Dic71—21no—
98C22×Dic71—21no—
99C22×Dic71—22no—
100C22×Dic71—22no—
101C22×Dic71—22no—
102C22×Dic71—24no—
103C22×Dic71—24no—
104C22×Dic71—22no—
105C22×Dic71—22no—
106C22×Dic71—22no—
107C22×Dic71—22no—
108C22×Dic71—22no—
109C22×Dic71—22no—
110C22×Dic71—24no—
111C22×Dic71—22no—
112C22×Dic71—22no—
113C22×Dic71—24no—
114C22×Dic71C22×C414yes—
115C22×Dic71C22×C412yes—
116C22×Dic71C2×D412yes—
117C22×Dic71C2×D412yes—
118C22×Dic71#4114yes—
119C2×C4×D71—28no—
120C2×C4×D71—28no—
121C2×C4×D71—28no—
122C2×C4×D71—28no—
123C2×C4×D71—28no—
124C2×C4×D71—28no—
125C2×C4×D71C22⋊C4116yes—
126C4⋊Dic71—28no—
127C4⋊Dic71—44no—
128(C2×D71)⋊C4—216no—
129(C2×D71)⋊C4—216no—
130(C2×D71)⋊C4—48no—
131C4×Dic71—216no—
132C4×Dic71—44no—
133C24×C71C2411yescharacteristic
134C22×C4×C71—22no—
135C22×C4×C71—22no—
136C22×C4×C71—22no—
137C22×C4×C71—22no—
138C22×C4×C71—22no—
139C22×C4×C71—22no—
140C22×C4×C71—22no—
141C22×C4×C71—24no—
142C22×C4×C71—24no—
143C22×C4×C71—22no—
144C22×C4×C71—22no—
145C22×C4×C71—22no—
146C22×C4×C71—22no—
147C22×C4×C71—24no—
148C22×C4×C71—24no—
149C22×C4×C71C22×C414yes—
150C22×C4×C71C22×C412yes—
151C22×C4×C71C2×D412yes—
152C22×C4×C71C2×D412yes—
153C22×C4×C71#4112yes—
154C22×C4×C71#4112yes—
155C22×C4×C71#4112yes—
156C22×C4×C71#4114yes—
157C22×C4×C71#4112yes—
158C4⋊C4×C71—28no—
159C4⋊C4×C71—44no—
160C42×C71—44no—
161C23×C71C23×C411yescharacteristic
162C23×C71C22×D411yescharacteristic
163C23×C71C22×D411yescharacteristic
164C23×C71C2×#4112yes—
165C23×C71C2×#4111yescharacteristic
166C23×C71C2×#4111yescharacteristic
167C23×C71C2×#4111yescharacteristic
168C23×C71C2×#4111yescharacteristic
169C23×C71C2×#4112yes—
170C23×C71H(5,𝔽2)12yes—
171C23×C712−1+412yes—
172C22×D71—22no—
173C22×D71—22no—
174C22×D71—24no—
175C22×D71—24no—
176C22×D71—22no—
177C22×D71—22no—
178C22×D71—22no—
179C22×D71—22no—
180C22×D71—24no—
181C22×D71—22no—
182C22×D71—22no—
183C22×D71—22no—
184C22×D71—22no—
185C22×D71—24no—
186C22×D71—42no—
187C22×D71—42no—
188C22×D71—42no—
189C22×D71—42no—
190C22×D71—42no—
191C22×D71—42no—
192C22×D71—42no—
193C22×D71—42no—
194C22×D71C2×C22⋊C414yes—
195C2×C4×C71—24no—
196C2×C4×C71—28no—
197C2×C4×C71—24no—
198C2×C4×C71—24no—
199C2×C4×C71—24no—
200C2×C4×C71—42no—
201C2×C4×C71—42no—
202C2×C4×C71—42no—
203C2×C4×C71—44no—
204C2×C4×C71—42no—
205C2×C4×C71—42no—
206C2×C4×C71—42no—
207C2×C4×C71—42no—
208C2×C4×C71—42no—
209C2×C4×C71—42no—
210C2×C4×C71—44no—
211C2×C4×C71—44no—
212C2×C4×C71C2×C22⋊C414yes—
213C2×C4×C71#11714yes—
214C2×C4×C71#11714yes—
215C2×C4×C71#12314yes—
216C2×C4×C71#12314yes—
217C2×C4×C71#12314yes—
218C2×C4×C71#12314yes—
219C2×Dic71—22no—
220C2×Dic71—22no—
221C2×Dic71—22no—
222C2×Dic71—22no—
223C2×Dic71—22no—
224C2×Dic71—22no—
225C2×Dic71—24no—
226C2×Dic71—28no—
227C2×Dic71—24no—
228C2×Dic71—24no—
229C2×Dic71—24no—
230C2×Dic71—42no—
231C2×Dic71—44no—
232C2×Dic71—42no—
233C2×Dic71—42no—
234C2×Dic71—42no—
235C2×Dic71—42no—
236C2×Dic71—42no—
237C2×Dic71—42no—
238C2×Dic71—42no—
239C2×Dic71—42no—
240C2×Dic71—44no—
241C2×Dic71—44no—
242C2×Dic71C2×C22⋊C414yes—
243C4×D71—232no—
244C22×C71C22×C22⋊C411yescharacteristic
245C22×C71C2×#11711yescharacteristic
246C22×C71C2×#11711yescharacteristic
247C22×C71C4×#4112yes—
248C22×C71#52012yes—
249C22×C71C2×#12311yescharacteristic
250C22×C71C2×#12311yescharacteristic
251C22×C71C2×#12311yescharacteristic
252C22×C71C2×#12311yescharacteristic
253C22×C71#52512yes—
254C22×C71#53211yescharacteristic
255C22×C71#53211yescharacteristic
256C22×C71#53511yescharacteristic
257C22×C71#53611yescharacteristic
258C22×C71#53712yes—
259C22×C71#55212yes—
260C22×C71#55212yes—
261C22×C71#55312yes—
262C22×C71#55312yes—
263C22×C71#55512yes—
264C22×C71#55512yes—
265C22×C71#55612yes—
266C22×C71#55612yes—
267C2×D71—24no—
268C2×D71—24no—
269C2×D71—24no—
270C2×D71—44no—
271C2×D71—44no—
272C2×D71—44no—
273C2×D71—44no—
274C2×D71—42no—
275C2×D71—42no—
276C2×D71—42no—
277C2×D71—42no—
278C4×C71—44no—
279C4×C71—44no—
280C4×C71—82no—
281C4×C71#38118yes—
282Dic71—24no—
283Dic71—44no—
284Dic71—44no—
285Dic71—82no—
286#13939—711nomaximal, Sylow
287C2×C71C2×#38111yescharacteristic
288C2×C71#229711yescharacteristic
289C2×C71#231111yescharacteristic
290C2×C71#234612yes—
291C2×C71#234712yes—
292C2×C71#248512yes—
293C2×C71#248512yes—
294C2×C71#249911yescharacteristic
295C2×C71#249911yescharacteristic
296C2×C71#250011yescharacteristic
297C2×C71#250011yescharacteristic
298D71—44no—
299C22×#117—711no—
300C2×#380—711no—
301C2×C4×C22⋊C4—711no—
302#1440—714no—
303C4⋊#95—711no—
304C4⋊#95—711no—
305C4⋊#95—711no—
306C4⋊#95D7111yescharacteristic, complemented
307#1434—714no—
308C71#1393911yescharacteristic, complemented, Sylow
309C24×C4—711no—
310C2×#117—718no—
311C2×#117—1424no—
312C2×#117—1424no—
313C22×C4⋊C4—711no—
314C22×C4⋊C4—711no—
315C22×C4⋊C4C2×D7111yescharacteristic
316C22×C4⋊C4C2×D7111yescharacteristic
317C22×C22⋊C4—712no—
318C22×C22⋊C4—712no—
319C22×C22⋊C4—712no—
320C22×C42—712no—
321C22×C42—711no—
322C22×C42C2×D7111yescharacteristic
323#380—1424no—
324C4×C22⋊C4—1424no—
325C2×#95—714no—
326C2×#95—712no—
327C2×#95—712no—
328C2×#95—714no—
329C2×#95—714no—
330C2×#95C2×D7114yes—
331C25—711no—
332C23×C4—711no—
333C23×C4—714no—
334C23×C4—714no—
335C23×C4—714no—
336C23×C4—712no—
337C23×C4—712no—
338C23×C4—712no—
339C23×C4—1422no—
340C23×C4—1422no—
341C23×C4—1422no—
342C23×C4—1422no—
343C23×C4C22×D7112yes—
344C23×C4C22×D7112yes—
345C23×C4C22×D7112yes—
346C23×C4C22×D7111yescharacteristic
347#117—14232no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4—22no—
350C2×C4⋊C4—22no—
351C2×C4⋊C4—22no—
352C2×C4⋊C4—22no—
353C2×C4⋊C4—714no—
354C2×C4⋊C4—1422no—
355C2×C4⋊C4—1422no—
356C2×C4⋊C4—1422no—
357C2×C4⋊C4—1422no—
358C2×C4⋊C4—1422no—
359C2×C4⋊C4C4×D7114yes—
360C2×C22⋊C4—2844no—
361C2×C22⋊C4—718no—
362C2×C22⋊C4—1424no—
363C2×C22⋊C4—1424no—
364C2×C22⋊C4—1424no—
365C2×C22⋊C4—1424no—
366C2×C22⋊C4—1424no—
367C2×C22⋊C4—1424no—
368C2×C42—22no—
369C2×C42—22no—
370C2×C42—22no—
371C2×C42—718no—
372C2×C42—1422no—
373C2×C42—1422no—
374C2×C42—1422no—
375C2×C42—1424no—
376C2×C42—1424no—
377#95—44no—
378#95—2844no—
379#95—1424no—
380#95—1424no—
381#95—1428no—
382#95—1428no—
383C24—712no—
384C24—712no—
385C24—712no—
386C24—1422no—
387C24—1421no—
388C24—1421no—
389C24—1421no—
390C24—1422no—
391C24—1421no—
392C24—1422no—
393C24—1422no—
394C24C23×D7111yescharacteristic
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—712no—
411C22×C4—712no—
412C22×C4—712no—
413C22×C4—7116no—
414C22×C4—714no—
415C22×C4—714no—
416C22×C4—1422no—
417C22×C4—1422no—
418C22×C4—1424no—
419C22×C4—1422no—
420C22×C4—1424no—
421C22×C4—1422no—
422C22×C4—1422no—
423C22×C4—1422no—
424C22×C4—1424no—
425C22×C4—1422no—
426C22×C4—1424no—
427C22×C4—1422no—
428C22×C4—1422no—
429C22×C4—1422no—
430C22×C4—1422no—
431C22×C4—1421no—
432C22×C4—1421no—
433C22×C4—1421no—
434C22×C4—1421no—
435C22×C4—1428no—
436C22×C4—1428no—
437C22×C4—1428no—
438C22×C4—1428no—
439C22×C4—1428no—
440C22×C4—1428no—
441C22×C4C2×C4×D7114yes—
442C22×C4C2×C4×D7112yes—
443C22×C4D4×D7112yes—
444C22×C4D4×D7112yes—
445C22×C4#1176586412yes—
446C22×C4#1176586512yes—
447C22×C4#1176586512yes—
448C22×C4#1176586614yes—
449C22×C4#1176586612yes—
450C4⋊C4—28no—
451C4⋊C4—44no—
452C4⋊C4—2844no—
453C4⋊C4—1428no—
454C22⋊C4—2848no—
455C22⋊C4—14216no—
456C22⋊C4—14216no—
457C42—44no—
458C42—2844no—
459C42—14216no—
460C23—2842no—
461C23—2842no—
462C23—2842no—
463C23—2842no—
464C23—2842no—
465C23—2842no—
466C23—2842no—
467C23—2842no—
468C23—714no—
469C23—1422no—
470C23—1422no—
471C23—1424no—
472C23—1422no—
473C23—1422no—
474C23—1424no—
475C23—1424no—
476C23—1422no—
477C23—1422no—
478C23—1422no—
479C23—1422no—
480C23—1424no—
481C23—1422no—
482C23—1422no—
483C23C22×C4×D7111yescharacteristic
484C23C2×D4×D7111yescharacteristic
485C23C2×D4×D7111yescharacteristic
486C23#41×D7112yes—
487C23C2×#1176586411yescharacteristic
488C23C2×#1176586511yescharacteristic
489C23C2×#1176586511yescharacteristic
490C23C2×#1176586611yescharacteristic
491C23C2×#1176586612yes—
492C23#1455617312yes—
493C23#1455617512yes—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—28no—
498C2×C4—24no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—44no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—2842no—
512C2×C4—2842no—
513C2×C4—2842no—
514C2×C4—2842no—
515C2×C4—2844no—
516C2×C4—2842no—
517C2×C4—2842no—
518C2×C4—2842no—
519C2×C4—2844no—
520C2×C4—2844no—
521C2×C4—2842no—
522C2×C4—2842no—
523C2×C4—714no—
524C2×C4—1424no—
525C2×C4—1424no—
526C2×C4—1424no—
527C2×C4—1428no—
528C2×C4—1424no—
529C2×C4—1422no—
530C2×C4—1422no—
531C2×C4—1422no—
532C2×C4—1422no—
533C2×C4—1422no—
534C2×C4—1422no—
535C2×C4—14232no—
536C2×C4C22⋊C4×D7114yes—
537C2×C4#1455608514yes—
538C2×C4#1455608614yes—
539C2×C4#1455611014yes—
540C2×C4#1455611014yes—
541C2×C4#1455611314yes—
542C2×C4#1455611314yes—
543C22—2842no—
544C22—2842no—
545C22—2842no—
546C22—2842no—
547C22—2844no—
548C22—2844no—
549C22—2844no—
550C22—2844no—
551C22—1424no—
552C22—1424no—
553C22—1424no—
554C22C2×C22⋊C4×D7111yescharacteristic
555C22C2×#1455608511yescharacteristic
556C22C2×#1455608611yescharacteristic
557C22C4×#1176586612yes—
558C22#1798647812yes—
559C22C2×#1455611011yescharacteristic
560C22C2×#1455611011yescharacteristic
561C22C2×#1455611311yescharacteristic
562C22C2×#1455611311yescharacteristic
563C22#1798651212yes—
564C22#1798654811yescharacteristic
565C22#1798655011yescharacteristic
566C22#1798656211yescharacteristic
567C22#1798656911yescharacteristic
568C22#1798657812yes—
569C22#1798668712yes—
570C22#1798669112yes—
571C22#1798669712yes—
572C22#1798669912yes—
573C22#1798671312yes—
574C22#1798671512yes—
575C22#1798671712yes—
576C22#1798672212yes—
577C4—44no—
578C4—44no—
579C4—82no—
580C4—2844no—
581C4—2844no—
582C4—5682no—
583C4—1424no—
584C4#1798586118yes—
585C2—2844no—
586C2C2×#1798586111yescharacteristic
587C2#2259082911yescharacteristic
588C2#2259093011yescharacteristic
589C2#2259122412yes—
590C2#2259122912yes—
591C2#2259263912yes—
592C2#2259264012yes—
593C2#2259279511yescharacteristic
594C2#2259279611yescharacteristic
595C2#2259279911yescharacteristic
596C2#2259280111yescharacteristic
subgroupG/H
center, FrattiniC24C23×D71
commutatorC23×C71C23×C4
FittingC71×C4⋊#95C2
socle, cosocleC24×C71C24
prime radicalC24×D71C23
hypercenterC4⋊#95D71
hypocenterC71#13939
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C24 ◁ C4⋊#95

Complements

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

NH ≅ G/Ntype
5. #22586570C2semidirect
6. #22586569C2semidirect
7. #22586569C2semidirect
8. #22586566C2semidirect
9. C71×C4⋊#95C2semidirect
13. C2×#14556086C4semidirect
306. C4⋊#95D71semidirect
308. C71#13939semidirect

Maximal subgroups

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

subgroupindex
C22×#145560862
C2×#179858562
C2×C4×(C2×D71)⋊C42
#225865882
#225865702
#225865692
#225865662
C71×C4⋊#952
#1393971