Finite Group Databasebeta

← #32160139

Subgroups — #32160139

Order 18176.

2568 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17986740C211yesmaximal, characteristic
2#22591533C211yesmaximal, characteristic, complemented
3#22591516C211yesmaximal, characteristic, complemented
4#22591496C213yesmaximal, complemented
5#22591495C213yesmaximal, complemented
6#22591485C213yesmaximal, complemented
7#22591480C213yesmaximal, complemented
8#22590902C211yesmaximal, characteristic, complemented
9#126×Dic71C211yesmaximal, characteristic, complemented
10#22590888C211yesmaximal, characteristic, complemented
11#22590861C213yesmaximal, complemented
12#22590852C213yesmaximal, complemented
13#22590848C213yesmaximal, complemented
14#22590841C213yesmaximal, complemented
15C71×#2383C211yesmaximal, characteristic, complemented
16C23×C4×D71C2211yescharacteristic
17#17986740C4116yescomplemented
18C2×#14556125C2211yescharacteristic
19C2×#14556122C2211yescharacteristic
20C2×#14556113C2213yes—
21C2×#14556103C2213yes—
22C2×#14556101C2213yes—
23C2×#14556098C2213yes—
24C2×#14556090C2213yes—
25C2×#14556090C2213yes—
26C2×#14556090C2213yescomplemented
27C2×(C22⋊D71)⋊C4C2213yes—
28C2×C42×D71C2213yescomplemented
29#17985901C2213yes—
30#17985900C2213yes—
31#17985900C2213yescomplemented
32#17985899C2213yes—
33#17985894C2213yescomplemented
34#17985892C2213yes—
35#17985891C2213yes—
36#17985890C2213yes—
37#17985890C2213yescomplemented
38#17985889C2213yes—
39#17985889C2213yescomplemented
40#17985888C2213yes—
41#17985885C2213yescomplemented
42#17985881C2213yescomplemented
43#17985880C2213yescomplemented
44#17985878C2213yes—
45#17985877C2213yes—
46#17985877C2213yes—
47#17985877C2213yes—
48#17985877C2213yes—
49#17985877C2213yescomplemented
50#17985877C2213yescomplemented
51#17985875C2213yes—
52#17985873C2213yescomplemented
53#17985870C2213yes—
54#17985870C2213yes—
55#17985867C2211yescharacteristic, complemented
56#17985861C2213yescomplemented
57#17985858C2213yescomplemented
58#17985856C2213yescomplemented
59#17985854C2213yes—
60C4⋊C4×Dic71C2213yescomplemented
61#17985850C2213yescomplemented
62C22⋊C4×Dic71C2213yes—
63#17985846C2213yes—
64#17985846C2213yes—
65#17985846C2213yescomplemented
66C42×Dic71C2211yescharacteristic, complemented
67C2×#126×C71C2211yescharacteristic
68#389×C71C2213yescomplemented
69#388×C71C2213yes—
70#387×C71C2213yescomplemented
71#385×C71C2213yes—
72#384×C71C2211yescharacteristic, complemented
73#383×C71C2211yescharacteristic, complemented
74C24×D71C2311yescharacteristic
75C22×D284C2313yes—
76C22×D284C2313yes—
77C22×D284C2313yes—
78C22×C22⋊D71C2313yes—
79C23×Dic71C2311yescharacteristic
80C22×C4×D71—46no—
81C22×C4×D71C2311yescharacteristic
82C22×C4×D71C2311yescharacteristic
83C22×C4×D71C2313yes—
84C22×C4×D71C2311yescharacteristic
85C22×C4×D71C2311yescharacteristic
86C22×C4×D71C2313yes—
87C22×C4×D71C2313yes—
88#14556125C2×C418yescomplemented
89#14556122C2×C418yescomplemented
90#14556113C2×C4124yescomplemented
91#14556103C2×C4124yescomplemented
92#14556101C2×C4124yescomplemented
93#14556098C2×C4124yescomplemented
94#14556090—46no—
95#14556090—46no—
96#14556090—46no—
97(C22⋊D71)⋊C4—46no—
98C2×C4⋊Dic71C2313yes—
99C2×Dic71⋊C4C2313yes—
100C2×Dic71⋊C4C2313yes—
101C2×Dic71⋊C4C2313yes—
102C2×C22⋊Dic71C2313yes—
103C2×(C2×D71)⋊C4C2313yes—
104C2×(C2×D71)⋊C4C2313yes—
105C2×(C2×D71)⋊C4C2313yes—
106C2×(C2×D71)⋊C4C2313yes—
107C2×(C2×D71)⋊C4C2313yes—
108C2×(C2×D71)⋊C4C2313yes—
109C2×(C2×D71)⋊C4C2313yes—
110C2×(C2×D71)⋊C4C2313yes—
111C2×(C2×D71)⋊C4C2313yes—
112C2×(C2×D71)⋊C4C2313yes—
113C2×(C2×D71)⋊C4C2313yes—
114C2×(C2×D71)⋊C4C2313yes—
115C2×(C2×D71)⋊C4C2313yes—
116C2×C4×Dic71C2313yes—
117C2×C4×Dic71C2313yes—
118C2×C4×Dic71C2313yes—
119C2×C4×Dic71C2313yes—
120C2×C4×Dic71C2313yes—
121C42×D71—46no—
122C71⋊#95C2313yes—
123C71⋊#95C2313yes—
124C71⋊#95C2313yes—
125C71⋊#95C2313yes—
126C71⋊#95C2313yes—
127C71⋊#95C2313yes—
128C71⋊#95C2313yes—
129C71⋊#95C2313yes—
130C23×C4×C71C2311yescharacteristic
131#126×C71C2×C418yescomplemented
132C2×C4⋊C4×C71C2313yes—
133C2×C4⋊C4×C71C2313yes—
134C2×C4⋊C4×C71C2313yes—
135C2×C4⋊C4×C71C2313yes—
136C2×C22⋊C4×C71C2313yes—
137C2×C22⋊C4×C71C2313yes—
138C2×C42×C71C2313yes—
139C2×C42×C71C2311yescharacteristic
140#95×C71C2313yes—
141#95×C71C2313yes—
142#95×C71C2313yes—
143#95×C71C2313yes—
144C23×D71—43no—
145C23×D71—43no—
146C23×D71C22×C414yes—
147C23×D71C2413yes—
148C23×D71C2411yescharacteristic
149C23×D71C2411yescharacteristic
150C2×D284—43no—
151C2×D284—43no—
152C2×D284—43no—
153C2×D284—43no—
154C2×D284—43no—
155C2×D284—43no—
156C2×D284C22×C4112yes—
157C2×D284C22×C4112yes—
158C2×D284C22×C4112yescomplemented
159C2×C22⋊D71—43no—
160C2×C22⋊D71—43no—
161C2×C22⋊D71C22×C4112yes—
162C22×Dic71—26no—
163C22×Dic71C2411yescharacteristic
164C22×Dic71C2413yes—
165C22×Dic71C2413yes—
166C22×Dic71C2413yes—
167C22×Dic71C2411yescharacteristic
168C2×C4×D71—26no—
169C2×C4×D71—26no—
170C2×C4×D71—26no—
171C2×C4×D71—26no—
172C2×C4×D71—26no—
173C2×C4×D71—43no—
174C2×C4×D71—43no—
175C2×C4×D71—43no—
176C2×C4×D71—43no—
177C2×C4×D71—43no—
178C2×C4×D71—43no—
179C2×C4×D71—43no—
180C2×C4×D71—416no—
181C2×C4×D71C22×C4112yescomplemented
182C4⋊Dic71—43no—
183Dic71⋊C4—43no—
184Dic71⋊C4—43no—
185Dic71⋊C4C22×C4112yescomplemented
186C22⋊Dic71—43no—
187(C2×D71)⋊C4—43no—
188(C2×D71)⋊C4—43no—
189(C2×D71)⋊C4—43no—
190(C2×D71)⋊C4—43no—
191(C2×D71)⋊C4—43no—
192(C2×D71)⋊C4—43no—
193(C2×D71)⋊C4—43no—
194(C2×D71)⋊C4—43no—
195(C2×D71)⋊C4—43no—
196(C2×D71)⋊C4—43no—
197(C2×D71)⋊C4C22×C4112yes—
198(C2×D71)⋊C4C22×C4112yes—
199(C2×D71)⋊C4C22×C4112yescomplemented
200C4×Dic71—26no—
201C4×Dic71—26no—
202C4×Dic71—26no—
203C4×Dic71—43no—
204C4×Dic71—43no—
205C24×C71C2411yescharacteristic
206C22×C4×C71—43no—
207C22×C4×C71C2411yescharacteristic
208C22×C4×C71C2411yescharacteristic
209C22×C4×C71C2413yes—
210C22×C4×C71C2413yes—
211C22×C4×C71C2413yes—
212C22×C4×C71C2413yes—
213C4⋊C4×C71—43no—
214C4⋊C4×C71—43no—
215C4⋊C4×C71—43no—
216C4⋊C4×C71C22×C4112yescomplemented
217C22⋊C4×C71—43no—
218C22⋊C4×C71C22×C4112yes—
219C42×C71—43no—
220C42×C71C22×C414yescomplemented
221C23×C71—23no—
222C23×C71—23no—
223C23×C71C23×C412yes—
224C23×C71C2511yescharacteristic
225C22×D71—23no—
226C22×D71—23no—
227C22×D71—23no—
228C22×D71—23no—
229C22×D71—23no—
230C22×D71—23no—
231C22×D71—23no—
232C22×D71—23no—
233C22×D71—23no—
234C22×D71—23no—
235C22×D71—412no—
236C22×D71—44no—
237C22×D71—412no—
238C22×D71C23×C412yes—
239C22×D71C23×C412yes—
240C22×D71C23×C416yes—
241C2×C4×C71—23no—
242C2×C4×C71—23no—
243C2×C4×C71—23no—
244C2×C4×C71—23no—
245C2×C4×C71—23no—
246C2×C4×C71—23no—
247C2×C4×C71—23no—
248C2×C4×C71—23no—
249C2×C4×C71—23no—
250C2×C4×C71—23no—
251C2×C4×C71—23no—
252C2×C4×C71—23no—
253C2×C4×C71—44no—
254C2×C4×C71C23×C416yes—
255C2×C4×C71C23×C416yes—
256C22⋊D71—412no—
257C2×Dic71—23no—
258C2×Dic71—23no—
259C2×Dic71—23no—
260C2×Dic71—23no—
261C2×Dic71—23no—
262C2×Dic71—23no—
263C2×Dic71—23no—
264C2×Dic71—23no—
265C2×Dic71—23no—
266C2×Dic71—44no—
267C2×Dic71C23×C416yes—
268C2×Dic71H(5,𝔽2)16yes—
269D284—412no—
270D284—412no—
271D284—412no—
272C4×D71—412no—
273C4×D71—412no—
274C4×D71—44no—
275C4×D71—44no—
276C4×D71—412no—
277C4×D71—44no—
278C4×D71—44no—
279C22×C71—26no—
280C22×C71—41no—
281C22×C71—43no—
282C22×C71C24×C411yescharacteristic
283C22×C71C2×H(5,𝔽2)13yes—
284C22×C71#58513yes—
285C2×D71—26no—
286C2×D71—26no—
287C2×D71—26no—
288C2×D71—26no—
289C2×D71—26no—
290C2×D71—416no—
291C2×D71—41no—
292C2×D71—41no—
293C2×D71—43no—
294C2×D71—43no—
295C2×D71—43no—
296C2×D71—43no—
297C2×D71—43no—
298C2×D71—43no—
299C4×C71—26no—
300C4×C71—26no—
301C4×C71—41no—
302C4×C71—41no—
303C4×C71—43no—
304C4×C71—43no—
305Dic71—26no—
306Dic71—41no—
307Dic71—43no—
308Dic71—43no—
309Dic71#56014yescomplemented
310C4×#560—711nomaximal, Sylow
311C2×C71—42no—
312C2×C71C4×H(5,𝔽2)13yes—
313C2×C71C2×#56011yescharacteristic
314C2×C71#352913yes—
315D71—46no—
316D71—42no—
317D71—42no—
318C2×#560—711no—
319#2383—711no—
320#2383D7111yescharacteristic, complemented
321#2380—711no—
322#2377—713no—
323#2377—713no—
324#2376—713no—
325#2375—713no—
326C4×#127—711no—
327C4×#126—711no—
328C4×#126—711no—
329C4×#123—713no—
330C4×#121—713no—
331C4×#121—713no—
332C4×#120—713no—
333C71C4×#56011yescharacteristic, complemented, Sylow
334C24×C4—711no—
335#560—7116no—
336C2×#127—711no—
337C2×#126—711no—
338C2×#126C2×D7111yescharacteristic
339C2×#123—713no—
340C2×#121—713no—
341C2×#121—713no—
342C2×#120—713no—
343C2×C4×D4—713no—
344C2×C4×D4—713no—
345C2×C4×D4—713no—
346C2×C4×D4—713no—
347C22×C42—713no—
348#390—713no—
349#390—713no—
350#390—713no—
351#390—713no—
352#389—713no—
353#389C2×D7113yescomplemented
354#388—713no—
355#388—713no—
356#388—713no—
357#388—713no—
358#388—713no—
359#388—713no—
360#388—713no—
361#388C2×D7113yes—
362#387—713no—
363#387—713no—
364#387—713no—
365#387C2×D7113yescomplemented
366#386—713no—
367#386—713no—
368#386—713no—
369#386—713no—
370#386—713no—
371#386—713no—
372#386—713no—
373#386—713no—
374#385—713no—
375#385—713no—
376#385—713no—
377#385C2×D7113yes—
378#384C2×D7111yescharacteristic, complemented
379#383—711no—
380#383C2×D7111yescharacteristic, complemented
381#381—713no—
382#380—713no—
383#380—713no—
384#379—713no—
385C4×C4⋊C4—713no—
386C4×C4⋊C4—713no—
387C4×C22⋊C4—713no—
388C4×C22⋊C4—713no—
389C4×C22⋊C4—713no—
390C4×C22⋊C4—713no—
391C43—711no—
392C25—711no—
393C22×D4—713no—
394C22×D4—713no—
395C22×D4—713no—
396C22×D4—713no—
397C23×C4—2846no—
398C23×C4—711no—
399C23×C4—711no—
400C23×C4—713no—
401C23×C4—713no—
402C23×C4—711no—
403C23×C4—713no—
404C23×C4—711no—
405C23×C4—711no—
406C23×C4C22×D7111yescharacteristic
407#127—718no—
408#126—718no—
409#126C4×D7118yescomplemented
410#123—7124no—
411#121—7124no—
412#121—7124no—
413#120—7124no—
414C4×D4—2846no—
415C4×D4—2846no—
416C4×D4—2846no—
417C4×D4—2846no—
418C2×C4⋊C4—713no—
419C2×C4⋊C4—713no—
420C2×C4⋊C4—713no—
421C2×C4⋊C4—713no—
422C2×C4⋊C4C22×D7113yes—
423C2×C4⋊C4C22×D7113yes—
424C2×C4⋊C4C22×D7113yes—
425C2×C4⋊C4C22×D7113yes—
426C2×C22⋊C4—713no—
427C2×C22⋊C4—713no—
428C2×C22⋊C4—713no—
429C2×C22⋊C4—713no—
430C2×C22⋊C4—713no—
431C2×C22⋊C4—713no—
432C2×C22⋊C4—713no—
433C2×C22⋊C4—713no—
434C2×C22⋊C4—713no—
435C2×C22⋊C4—713no—
436C2×C22⋊C4—713no—
437C2×C22⋊C4—713no—
438C2×C22⋊C4—713no—
439C2×C22⋊C4—713no—
440C2×C22⋊C4C22×D7113yes—
441C2×C22⋊C4C22×D7113yes—
442C2×C42—2846no—
443C2×C42—713no—
444C2×C42—713no—
445C2×C42—713no—
446C2×C42—713no—
447C2×C42—713no—
448C2×C42C22×D7113yes—
449C2×C42C22×D7111yescharacteristic
450#95—713no—
451#95—713no—
452#95—713no—
453#95—713no—
454#95—713no—
455#95—713no—
456#95—713no—
457#95—713no—
458#95C22×D7113yes—
459#95C22×D7113yes—
460#95C22×D7113yes—
461#95C22×D7113yes—
462C24—2843no—
463C24—2843no—
464C24—714no—
465C24—711no—
466C24—711no—
467C24—713no—
468C24C23×D7111yescharacteristic
469C2×D4—2843no—
470C2×D4—2843no—
471C2×D4—2843no—
472C2×D4—2843no—
473C2×D4—2843no—
474C2×D4—2843no—
475C2×D4—2843no—
476C2×D4—2843no—
477C2×D4—7112no—
478C2×D4—7112no—
479C2×D4—7112no—
480C2×D4—7112no—
481C22×C4—43no—
482C22×C4—2843no—
483C22×C4—2843no—
484C22×C4—2843no—
485C22×C4—2843no—
486C22×C4—2843no—
487C22×C4—2843no—
488C22×C4—2843no—
489C22×C4—28416no—
490C22×C4—7112no—
491C22×C4—711no—
492C22×C4—713no—
493C22×C4—713no—
494C22×C4—711no—
495C22×C4—713no—
496C22×C4—1426no—
497C22×C4—1426no—
498C22×C4—1426no—
499C22×C4—1426no—
500C22×C4—1426no—
501C22×C4—1426no—
502C22×C4C23×D7111yescharacteristic
503C22×C4C23×D7111yescharacteristic
504C22×C4C23×D7113yes—
505C22×C4C23×D7113yes—
506C22×C4C23×D7113yes—
507C22×C4C23×D7113yes—
508C4⋊C4—43no—
509C4⋊C4—43no—
510C4⋊C4—43no—
511C4⋊C4—2843no—
512C4⋊C4—2843no—
513C4⋊C4—2843no—
514C4⋊C4—7112no—
515C4⋊C4C2×C4×D71112yescomplemented
516C22⋊C4—43no—
517C22⋊C4—2843no—
518C22⋊C4—2843no—
519C22⋊C4—2843no—
520C22⋊C4—2843no—
521C22⋊C4—2843no—
522C22⋊C4—2843no—
523C22⋊C4—2843no—
524C22⋊C4—2843no—
525C22⋊C4—2843no—
526C22⋊C4—2843no—
527C22⋊C4—2843no—
528C22⋊C4—7112no—
529C22⋊C4—7112no—
530C22⋊C4—7112no—
531C22⋊C4C2×C4×D71112yes—
532C42—43no—
533C42—2843no—
534C42—2843no—
535C42—1426no—
536C42—1426no—
537C42—1426no—
538C42C2×C4×D7114yescomplemented
539C23—23no—
540C23—23no—
541C23—28412no—
542C23—2844no—
543C23—28412no—
544C23—712no—
545C23—712no—
546C23—716no—
547C23—1423no—
548C23—1423no—
549C23—1423no—
550C23—1423no—
551C23—1423no—
552C23—1423no—
553C23—1423no—
554C23—1423no—
555C23—1423no—
556C23—1423no—
557C23C22×C4×D7112yes—
558C23C24×D7111yescharacteristic
559D4—28412no—
560D4—28412no—
561D4—28412no—
562D4—28412no—
563C2×C4—23no—
564C2×C4—23no—
565C2×C4—23no—
566C2×C4—23no—
567C2×C4—23no—
568C2×C4—23no—
569C2×C4—23no—
570C2×C4—23no—
571C2×C4—23no—
572C2×C4—23no—
573C2×C4—23no—
574C2×C4—23no—
575C2×C4—44no—
576C2×C4—28412no—
577C2×C4—2844no—
578C2×C4—2844no—
579C2×C4—28412no—
580C2×C4—2844no—
581C2×C4—2844no—
582C2×C4—28412no—
583C2×C4—2844no—
584C2×C4—716no—
585C2×C4—716no—
586C2×C4—1423no—
587C2×C4—1423no—
588C2×C4—1423no—
589C2×C4—1423no—
590C2×C4—1423no—
591C2×C4—1423no—
592C2×C4—1423no—
593C2×C4—1423no—
594C2×C4—1423no—
595C2×C4C22×C4×D7116yes—
596C2×C4C22×C4×D7116yes—
597C22—26no—
598C22—41no—
599C22—43no—
600C22—28416no—
601C22—2841no—
602C22—2841no—
603C22—2843no—
604C22—2843no—
605C22—2843no—
606C22—2843no—
607C22—2843no—
608C22—2843no—
609C22—1426no—
610C22—1426no—
611C22—1426no—
612C22—1426no—
613C22—1426no—
614C22C23×C4×D7111yescharacteristic
615C22H(5,𝔽2)×D7113yes—
616C22#1798683613yes—
617C4—26no—
618C4—26no—
619C4—41no—
620C4—41no—
621C4—43no—
622C4—43no—
623C4—2841no—
624C4—2843no—
625C4—2843no—
626C4—714no—
627C4—1426no—
628C2—42no—
629C2—2842no—
630C2—2842no—
631C2—2846no—
632C2#2260154613yes—
633C2#560×D7111yescharacteristic
634C2#2260248113yes—
subgroupG/H
center, FrattiniC23C24×D71
commutatorC22×C71C24×C4
FittingC71×#2383C2
socle, cosocleC23×C71C25
prime radicalC2×#17986740C2
hypercenter#2383D71
hypocenterC71C4×#560
LCSG ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ #2383

Complements

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

NH ≅ G/Ntype
2. #22591533C2semidirect
3. #22591516C2semidirect
4. #22591496C2semidirect
5. #22591495C2semidirect
6. #22591485C2semidirect
7. #22591480C2semidirect
8. #22590902C2semidirect
9. #126×Dic71C2semidirect
10. #22590888C2semidirect
11. #22590861C2semidirect
12. #22590852C2semidirect
13. #22590848C2semidirect
14. #22590841C2semidirect
15. C71×#2383C2semidirect
17. #17986740C4semidirect
26. C2×#14556090C22semidirect
28. C2×C42×D71C22semidirect
31. #17985900C22semidirect
33. #17985894C22semidirect
37. #17985890C22semidirect
39. #17985889C22semidirect
41. #17985885C22semidirect
42. #17985881C22semidirect
43. #17985880C22semidirect
49. #17985877C22semidirect
50. #17985877C22semidirect
52. #17985873C22semidirect
55. #17985867C22semidirect
56. #17985861C22semidirect
57. #17985858C22semidirect
58. #17985856C22semidirect
60. C4⋊C4×Dic71C22semidirect
61. #17985850C22semidirect
65. #17985846C22semidirect
66. C42×Dic71C22semidirect
68. #389×C71C22semidirect
70. #387×C71C22semidirect
72. #384×C71C22semidirect
73. #383×C71C22semidirect
88. #14556125C2×C4semidirect
89. #14556122C2×C4semidirect
90. #14556113C2×C4semidirect
91. #14556103C2×C4semidirect
92. #14556101C2×C4semidirect
93. #14556098C2×C4semidirect
131. #126×C71C2×C4semidirect
158. C2×D284C22×C4semidirect
181. C2×C4×D71C22×C4semidirect
185. Dic71⋊C4C22×C4semidirect
199. (C2×D71)⋊C4C22×C4semidirect
216. C4⋊C4×C71C22×C4semidirect
220. C42×C71C22×C4semidirect
309. Dic71#560semidirect
320. #2383D71semidirect
333. C71C4×#560semidirect
353. #389C2×D71semidirect
365. #387C2×D71semidirect
378. #384C2×D71semidirect
380. #383C2×D71semidirect
409. #126C4×D71semidirect
515. C4⋊C4C2×C4×D71semidirect
538. C42C2×C4×D71semidirect

Maximal subgroups

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

subgroupindex
C2×#179867402
#225915332
#225915162
#225914962
#225914952
#225914852
#225914802
#225909022
#126×Dic712
#225908882
#225908612
#225908522
#225908482
#225908412
C71×#23832
C4×#56071