Finite Group Databasebeta

← C2×#22591904 #32165705

Subgroups — C2×#22591904 #32165705

Order 18176.

3678 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14556116C211yesmaximal, characteristic, complemented
2C22×#14556108C211yesmaximal, characteristic, complemented
3C22×#14556103C211yesmaximal, characteristic
4C22×#14556093C211yesmaximal, characteristic, complemented
5#22591904C2116yesmaximal, direct product factor
6C2×#17985921C212yesmaximal, complemented
7C2×#17985917C212yesmaximal, complemented
8C2×#17985900C211yesmaximal, characteristic, complemented
9C2×#17985900C212yesmaximal, complemented
10C2×#17985891C212yesmaximal, complemented
11C2×C4×(C2×D71)⋊C4C211yesmaximal, characteristic, complemented
12C2×#389×C71C211yesmaximal, characteristic, complemented
13C23×Dic142C2211yescharacteristic
14C23×D284C2211yescharacteristic
15C23×D284C2212yes—
16C23×C4×D71C2211yescharacteristic
17C2×#14556116—28no—
18C2×#14556116C2218yescomplemented
19C2×#14556108—28no—
20C2×#14556108C2218yescomplemented
21C2×#14556103—28no—
22C2×#14556103C2218yes—
23C2×#14556093—28no—
24C2×#14556093C2218yescomplemented
25C22×C4⋊Dic71C2212yes—
26C22×Dic71⋊C4C2212yes—
27C22×(C2×D71)⋊C4C2212yes—
28C22×(C2×D71)⋊C4C2212yes—
29C22×(C2×D71)⋊C4C2212yes—
30C22×(C2×D71)⋊C4C2211yescharacteristic
31C22×(C2×D71)⋊C4C2212yes—
32C22×(C2×D71)⋊C4C2212yes—
33C22×(C2×D71)⋊C4C2211yescharacteristic
34C22×(C2×D71)⋊C4C2212yes—
35C22×C4×Dic71C2212yes—
36C22×C4×Dic71C2211yescharacteristic
37#17985921C22116yescomplemented
38#17985917C22116yescomplemented
39#17985900C2218yescomplemented
40#17985900C22116yescomplemented
41#17985891C22116yescomplemented
42C4×(C2×D71)⋊C4C2218yescomplemented
43C2×C71⋊#95C2211yescharacteristic
44C2×C71⋊#95C2211yescharacteristic
45C22×C4⋊C4×C71C2211yescharacteristic
46C22×C4⋊C4×C71C2212yes—
47C22×C4⋊C4×C71C2211yescharacteristic
48C22×C42×C71C2211yescharacteristic
49#389×C71C2218yescomplemented
50C2×#95×C71C2212yes—
51C24×D71C2312yes—
52C24×D71C2311yescharacteristic
53C22×Dic142—24no—
54C22×Dic142—24no—
55C22×Dic142D418yescomplemented
56C22×Dic142C2314yes—
57C22×D284—28no—
58C22×D284—28no—
59C22×D284—24no—
60C22×D284—24no—
61C22×D284—24no—
62C22×D284—44no—
63C22×D284C2314yes—
64C22×D284C2318yes—
65C23×Dic71C2311yescharacteristic
66C23×Dic71C2311yescharacteristic
67C23×Dic71C2312yes—
68C23×Dic71C2311yescharacteristic
69C22×C4×D71—24no—
70C22×C4×D71—24no—
71C22×C4×D71—24no—
72C22×C4×D71C2314yes—
73#14556116—232no—
74#14556108—232no—
75#14556103—232no—
76#14556093—232no—
77C2×C4⋊Dic71—28no—
78C2×C4⋊Dic71C2318yes—
79C2×Dic71⋊C4—28no—
80C2×Dic71⋊C4C2318yes—
81C2×(C2×D71)⋊C4—24no—
82C2×(C2×D71)⋊C4—28no—
83C2×(C2×D71)⋊C4—28no—
84C2×(C2×D71)⋊C4—28no—
85C2×(C2×D71)⋊C4—24no—
86C2×(C2×D71)⋊C4—28no—
87C2×(C2×D71)⋊C4—28no—
88C2×(C2×D71)⋊C4—44no—
89C2×(C2×D71)⋊C4C2314yes—
90C2×(C2×D71)⋊C4C2314yes—
91C2×(C2×D71)⋊C4C2318yes—
92C2×(C2×D71)⋊C4C2318yes—
93C2×(C2×D71)⋊C4C2318yes—
94C2×(C2×D71)⋊C4C2318yes—
95C2×(C2×D71)⋊C4C2318yes—
96C2×(C2×D71)⋊C4C2318yes—
97C2×C4×Dic71—28no—
98C2×C4×Dic71—24no—
99C2×C4×Dic71C2314yes—
100C2×C4×Dic71C2318yes—
101C71⋊#95C2314yes—
102C71⋊#95C2314yes—
103C23×C4×C71C2311yescharacteristic
104C23×C4×C71C2311yescharacteristic
105C23×C4×C71C2312yes—
106C23×C4×C71C2312yes—
107C23×C4×C71C2311yescharacteristic
108C2×C4⋊C4×C71—24no—
109C2×C4⋊C4×C71—44no—
110C2×C4⋊C4×C71D418yes—
111C2×C4⋊C4×C71C2314yes—
112C2×C4⋊C4×C71C2318yes—
113C2×C4⋊C4×C71C2314yes—
114C2×C42×C71—24no—
115C2×C42×C71C2314yes—
116#95×C71C2318yes—
117C23×D71—22no—
118C23×D71—22no—
119C23×D71—22no—
120C23×D71—22no—
121C23×D71—22no—
122C23×D71—22no—
123C23×D71—24no—
124C23×D71—24no—
125C23×D71—24no—
126C23×D71—24no—
127C23×D71—24no—
128C23×D71—24no—
129C23×D71—24no—
130C23×D71#4114yes—
131C23×D71C2414yes—
132C23×D71C2412yes—
133C2×Dic142—232no—
134C2×Dic142—28no—
135C2×Dic142—28no—
136C2×Dic142C2×D4116yescomplemented
137C2×D284—216no—
138C2×D284—216no—
139C2×D284—28no—
140C2×D284—28no—
141C2×D284—28no—
142C2×D284—48no—
143C2×D284—432no—
144C2×D284—416no—
145C22×Dic71—24no—
146C22×Dic71—22no—
147C22×Dic71—24no—
148C22×Dic71—22no—
149C22×Dic71—22no—
150C22×Dic71—22no—
151C22×Dic71—22no—
152C22×Dic71—22no—
153C22×Dic71—22no—
154C22×Dic71—22no—
155C22×Dic71C2×D418yes—
156C22×Dic71#4114yes—
157C22×Dic71C2412yes—
158C22×Dic71C2412yes—
159C22×Dic71C2414yes—
160C22×Dic71C2412yes—
161C2×C4×D71—28no—
162C2×C4×D71—28no—
163C2×C4×D71—28no—
164C2×C4×D71—416no—
165C4⋊Dic71—216no—
166Dic71⋊C4—216no—
167(C2×D71)⋊C4—28no—
168(C2×D71)⋊C4—216no—
169(C2×D71)⋊C4—216no—
170(C2×D71)⋊C4—216no—
171(C2×D71)⋊C4—216no—
172(C2×D71)⋊C4—28no—
173(C2×D71)⋊C4—216no—
174(C2×D71)⋊C4—48no—
175C4×Dic71—216no—
176C4×Dic71—28no—
177C24×C71C2411yescharacteristic
178C22×C4×C71—24no—
179C22×C4×C71—24no—
180C22×C4×C71—22no—
181C22×C4×C71—22no—
182C22×C4×C71—24no—
183C22×C4×C71—22no—
184C22×C4×C71—22no—
185C22×C4×C71—24no—
186C22×C4×C71—22no—
187C22×C4×C71—24no—
188C22×C4×C71—22no—
189C22×C4×C71C2×D414yes—
190C22×C4×C71C2×D418yes—
191C22×C4×C71C2×D414yes—
192C22×C4×C71#4114yes—
193C22×C4×C71C2412yes—
194C22×C4×C71C2414yes—
195C22×C4×C71C2412yes—
196C22×C4×C71C2414yes—
197C22×C4×C71C2412yes—
198C4⋊C4×C71—28no—
199C4⋊C4×C71—48no—
200C4⋊C4×C71C2×D4116yes—
201C42×C71—28no—
202C23×C71C22×D412yes—
203C23×C71C22×D412yes—
204C23×C71C2×#4112yes—
205C23×C71C2×#4112yes—
206C23×C71C2×#4112yes—
207C23×C71H(5,𝔽2)12yes—
208C23×C712−1+412yes—
209C23×C71C2511yescharacteristic
210C22×D71—28no—
211C22×D71—28no—
212C22×D71—28no—
213C22×D71—22no—
214C22×D71—22no—
215C22×D71—22no—
216C22×D71—22no—
217C22×D71—22no—
218C22×D71—22no—
219C22×D71—24no—
220C22×D71—24no—
221C22×D71—24no—
222C22×D71—24no—
223C22×D71—24no—
224C22×D71—24no—
225C22×D71—24no—
226C22×D71—44no—
227C22×D71—44no—
228C22×D71—44no—
229C22×D71—44no—
230C22×D71—48no—
231C22×D71—48no—
232C22×D71—48no—
233C22×D71—48no—
234C22×D71—48no—
235C22×D71—48no—
236C22×D71—48no—
237C22×D71C2×#4114yes—
238C2×C4×C71—216no—
239C2×C4×C71—24no—
240C2×C4×C71—24no—
241C2×C4×C71—22no—
242C2×C4×C71—22no—
243C2×C4×C71—24no—
244C2×C4×C71—22no—
245C2×C4×C71—24no—
246C2×C4×C71—22no—
247C2×C4×C71—22no—
248C2×C4×C71—24no—
249C2×C4×C71—22no—
250C2×C4×C71—44no—
251C2×C4×C71—44no—
252C2×C4×C71—48no—
253C2×C4×C71#121116yes—
254C2×C4×C71C22×D414yes—
255C2×C4×C71C22×D418yes—
256C2×C4×C71C22×D414yes—
257C2×C4×C71C2×#4114yes—
258C2×Dic71—216no—
259C2×Dic71—28no—
260C2×Dic71—22no—
261C2×Dic71—24no—
262C2×Dic71—24no—
263C2×Dic71—22no—
264C2×Dic71—22no—
265C2×Dic71—22no—
266C2×Dic71—22no—
267C2×Dic71—22no—
268C2×Dic71—22no—
269C2×Dic71—22no—
270C2×Dic71—44no—
271C2×Dic71—44no—
272C2×Dic71C22×D418yes—
273C2×Dic71C2×#4114yes—
274D284—432no—
275D284—416no—
276C4×D71—416no—
277Dic142—232no—
278C22×C71C2×#12114yes—
279C22×C71#52914yes—
280C22×C71#54214yes—
281C22×C71#54614yes—
282C22×C71#54814yes—
283C22×C71#54814yes—
284C22×C71#55014yes—
285C22×C71C23×D411yescharacteristic
286C22×C71C23×D411yescharacteristic
287C22×C71C22×#4111yescharacteristic
288C22×C71C22×#4111yescharacteristic
289C22×C71C22×#4111yescharacteristic
290C22×C71C2×H(5,𝔽2)11yescharacteristic
291C22×C71C2×2−1+411yescharacteristic
292C2×D71—24no—
293C2×D71—24no—
294C2×D71—24no—
295C2×D71—42no—
296C2×D71—416no—
297C2×D71—42no—
298C2×D71—42no—
299C2×D71—42no—
300C2×D71—44no—
301C2×D71—44no—
302C2×D71—44no—
303C2×D71—44no—
304C2×D71—44no—
305C2×D71—44no—
306C2×D71—44no—
307C2×D71—816no—
308C4×C71—28no—
309C4×C71—42no—
310C4×C71—42no—
311C4×C71—44no—
312C4×C71C2×#12118yes—
313Dic71—28no—
314Dic71—24no—
315Dic71—42no—
316Dic71—42no—
317C2×#2426—711nomaximal, Sylow
318C2×C71#242618yescomplemented
319C2×C71C22×#12111yescharacteristic
320C2×C71C2×#52911yescharacteristic
321C2×C71C2×#54211yescharacteristic
322C2×C71C2×#54611yescharacteristic
323C2×C71C2×#54811yescharacteristic
324C2×C71C2×#54811yescharacteristic
325C2×C71C2×#55011yescharacteristic
326D71—44no—
327D71—84no—
328C22×#124—711no—
329C22×C22⋊Q8—711no—
330C22×#121—711no—
331C22×C4×Q8—711no—
332#2426—7116no—
333C2×#396—712no—
334C2×#394—712no—
335C2×#390—711no—
336C2×#390—712no—
337C2×#389D7111yescharacteristic, complemented
338C2×#388—712no—
339C2×C4×C22⋊C4—711no—
340C71C2×#242611yescharacteristic, complemented, Sylow
341C23×Q8—711no—
342C23×D4—712no—
343C23×D4—711no—
344C24×C4—711no—
345C2×#124—718no—
346C2×#124—1428no—
347C2×C22⋊Q8—718no—
348C2×C22⋊Q8—1428no—
349C2×#121—718no—
350C2×#121—1428no—
351C2×C4×Q8—718no—
352C2×C4×Q8—1428no—
353C22×C4⋊C4—712no—
354C22×C4⋊C4—712no—
355C22×C4⋊C4C2×D7111yescharacteristic
356C22×C4⋊C4C2×D7112yes—
357C22×C4⋊C4C2×D7111yescharacteristic
358C22×C22⋊C4—712no—
359C22×C22⋊C4—711no—
360C22×C22⋊C4—712no—
361C22×C22⋊C4—712no—
362C22×C22⋊C4—712no—
363C22×C22⋊C4—712no—
364C22×C22⋊C4—711no—
365C22×C22⋊C4—712no—
366C22×C42—711no—
367C22×C42—712no—
368C22×C42C2×D7111yescharacteristic
369#396—7116no—
370#394—7116no—
371#390—718no—
372#390—7116no—
373#389C2×D7118yescomplemented
374#388—7116no—
375C4×C22⋊C4—718no—
376C2×#95—711no—
377C2×#95—711no—
378C2×#95C2×D7112yes—
379C25—711no—
380C25—712no—
381C22×Q8—718no—
382C22×Q8—714no—
383C22×Q8—1424no—
384C22×Q8—1424no—
385C22×D4—2844no—
386C22×D4—718no—
387C22×D4—714no—
388C22×D4—1428no—
389C22×D4—1424no—
390C22×D4—1428no—
391C22×D4—1424no—
392C22×D4—1424no—
393C23×C4—712no—
394C23×C4—711no—
395C23×C4—711no—
396C23×C4—711no—
397C23×C4—714no—
398C23×C4—1424no—
399C23×C4—1424no—
400C23×C4—1424no—
401C23×C4C22×D7112yes—
402C23×C4C22×D7111yescharacteristic
403C23×C4C22×D7111yescharacteristic
404C23×C4C22×D7112yes—
405C23×C4C22×D7111yescharacteristic
406#124—14232no—
407C22⋊Q8—14232no—
408#121—14232no—
409C4×Q8—14232no—
410C2×C4⋊C4—24no—
411C2×C4⋊C4—44no—
412C2×C4⋊C4—718no—
413C2×C4⋊C4—718no—
414C2×C4⋊C4—1428no—
415C2×C4⋊C4—1428no—
416C2×C4⋊C4D28418yes—
417C2×C4⋊C4C22×D7114yes—
418C2×C4⋊C4C22×D7118yes—
419C2×C4⋊C4C22×D7114yes—
420C2×C22⋊C4—2844no—
421C2×C22⋊C4—718no—
422C2×C22⋊C4—718no—
423C2×C22⋊C4—718no—
424C2×C22⋊C4—718no—
425C2×C22⋊C4—718no—
426C2×C22⋊C4—714no—
427C2×C22⋊C4—714no—
428C2×C22⋊C4—718no—
429C2×C22⋊C4—1428no—
430C2×C22⋊C4—1428no—
431C2×C22⋊C4—1428no—
432C2×C22⋊C4—1428no—
433C2×C22⋊C4—1424no—
434C2×C22⋊C4—1424no—
435C2×C22⋊C4—1428no—
436C2×C42—24no—
437C2×C42—714no—
438C2×C42—718no—
439C2×C42—1424no—
440C2×C42—1428no—
441C2×C42C22×D7114yes—
442#95—714no—
443#95—714no—
444#95C22×D7118yes—
445C24—714no—
446C24—712no—
447C24—714no—
448C24—1424no—
449C24—1424no—
450C24—1424no—
451C24—1424no—
452C24—1424no—
453C24—1424no—
454C24—1424no—
455C24—1422no—
456C24—1422no—
457C24—1422no—
458C24—1422no—
459C24—1422no—
460C24—1422no—
461C24C23×D7111yescharacteristic
462C2×Q8—7116no—
463C2×Q8—1428no—
464C2×Q8—1428no—
465C2×Q8—14232no—
466C2×D4—28432no—
467C2×D4—2848no—
468C2×D4—28416no—
469C2×D4—14216no—
470C2×D4—1428no—
471C2×D4—14216no—
472C2×D4—1428no—
473C2×D4—1428no—
474C22×C4—24no—
475C22×C4—24no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—24no—
479C22×C4—22no—
480C22×C4—24no—
481C22×C4—22no—
482C22×C4—24no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—28416no—
486C22×C4—718no—
487C22×C4—714no—
488C22×C4—712no—
489C22×C4—714no—
490C22×C4—712no—
491C22×C4—712no—
492C22×C4—1422no—
493C22×C4—1422no—
494C22×C4—1422no—
495C22×C4—1422no—
496C22×C4—1422no—
497C22×C4—1422no—
498C22×C4—1428no—
499C22×C4—1428no—
500C22×C4—1428no—
501C22×C4—1422no—
502C22×C4—1424no—
503C22×C4—1424no—
504C22×C4—1422no—
505C22×C4D4×D7114yes—
506C22×C4C2×D28418yes—
507C22×C4C2×D28414yes—
508C22×C4#1176586614yes—
509C22×C4C23×D7112yes—
510C22×C4C23×D7112yes—
511C22×C4C23×D7114yes—
512C22×C4C23×D7114yes—
513C22×C4C23×D7112yes—
514C4⋊C4—28no—
515C4⋊C4—48no—
516C4⋊C4—14216no—
517C4⋊C4—14216no—
518C4⋊C4C2×D284116yes—
519C22⋊C4—2848no—
520C22⋊C4—14216no—
521C22⋊C4—1428no—
522C22⋊C4—14216no—
523C22⋊C4—14216no—
524C22⋊C4—14216no—
525C22⋊C4—1428no—
526C22⋊C4—14216no—
527C42—28no—
528C42—1428no—
529C42—14216no—
530C23—2844no—
531C23—2844no—
532C23—2844no—
533C23—2844no—
534C23—2848no—
535C23—2848no—
536C23—2848no—
537C23—2848no—
538C23—2848no—
539C23—2848no—
540C23—2848no—
541C23—714no—
542C23—1424no—
543C23—1424no—
544C23—1424no—
545C23—1424no—
546C23—1424no—
547C23—1424no—
548C23—1424no—
549C23—1422no—
550C23—1422no—
551C23—1422no—
552C23—1422no—
553C23—1422no—
554C23—1422no—
555C23—1428no—
556C23—1428no—
557C23—1428no—
558C23C2×D4×D7112yes—
559C23C22×D28412yes—
560C23#41×D7112yes—
561C23#41×D7112yes—
562C23C2×#1176586612yes—
563C23#1455617312yes—
564C23#1455617612yes—
565C23C24×D7111yescharacteristic
566Q8—14232no—
567D4—28432no—
568D4—28416no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—216no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—24no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—48no—
584C2×C4—2844no—
585C2×C4—2844no—
586C2×C4—28416no—
587C2×C4—718no—
588C2×C4—714no—
589C2×C4—1422no—
590C2×C4—1422no—
591C2×C4—1422no—
592C2×C4—1422no—
593C2×C4—1422no—
594C2×C4—1422no—
595C2×C4—1422no—
596C2×C4—1424no—
597C2×C4—1424no—
598C2×C4—1422no—
599C2×C4—1428no—
600C2×C4—14216no—
601C2×C4#14556103116yes—
602C2×C4C2×D4×D7114yes—
603C2×C4C22×D28418yes—
604C2×C4C22×D28414yes—
605C2×C4C2×#1176586614yes—
606C22—2842no—
607C22—2842no—
608C22—2842no—
609C22—28416no—
610C22—2842no—
611C22—2844no—
612C22—2844no—
613C22—2844no—
614C22—2844no—
615C22—2844no—
616C22—2844no—
617C22—2844no—
618C22—56816no—
619C22—1424no—
620C22—1424no—
621C22—1424no—
622C22C2×#1455610314yes—
623C22#1798653414yes—
624C22#1798662314yes—
625C22#1798664414yes—
626C22#1798665914yes—
627C22#1798666114yes—
628C22#1798666714yes—
629C22C22×D4×D7111yescharacteristic
630C22C23×D28411yescharacteristic
631C22C2×#41×D7111yescharacteristic
632C22C2×#41×D7111yescharacteristic
633C22C22×#1176586611yescharacteristic
634C22C2×#1455617311yescharacteristic
635C22C2×#1455617611yescharacteristic
636C4—28no—
637C4—42no—
638C4—42no—
639C4—44no—
640C4—2842no—
641C4—2842no—
642C4—1424no—
643C4—1428no—
644C4C2×#1455610318yes—
645C2—2844no—
646C2—5684no—
647C2#2259190418yesdirect product factor
648C2C22×#1455610311yescharacteristic
649C2C2×#1798653411yescharacteristic
650C2C2×#1798662311yescharacteristic
651C2C2×#1798664411yescharacteristic
652C2C2×#1798665911yescharacteristic
653C2C2×#1798666111yescharacteristic
654C2C2×#1798666711yescharacteristic
subgroupG/H
centerC24C23×D71
commutator, cosocleC23×C71C25
FrattiniC23C24×D71
FittingC2×#389×C71C2
socleC24×C71C24
prime radicalC22×#14556103C2
hypercenterC2×#389D71
hypocenterC71C2×#2426
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C24 ◁ C2×#389

Complements

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

NH ≅ G/Ntype
1. C22×#14556116C2semidirect
2. C22×#14556108C2semidirect
4. C22×#14556093C2semidirect
5. #22591904C2direct (N × H)
6. C2×#17985921C2semidirect
7. C2×#17985917C2semidirect
8. C2×#17985900C2semidirect
9. C2×#17985900C2semidirect
10. C2×#17985891C2semidirect
11. C2×C4×(C2×D71)⋊C4C2semidirect
12. C2×#389×C71C2semidirect
18. C2×#14556116C22semidirect
20. C2×#14556108C22semidirect
24. C2×#14556093C22semidirect
37. #17985921C22semidirect
38. #17985917C22semidirect
39. #17985900C22semidirect
40. #17985900C22semidirect
41. #17985891C22semidirect
42. C4×(C2×D71)⋊C4C22semidirect
49. #389×C71C22semidirect
55. C22×Dic142D4semidirect
136. C2×Dic142C2×D4semidirect
318. C2×C71#2426semidirect
337. C2×#389D71semidirect
340. C71C2×#2426semidirect
373. #389C2×D71semidirect
647. C2#22591904direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145561162
C22×#145561082
C22×#145561032
C22×#145560932
#225919042
C2×#179859212
C2×#179859172
C2×#179859002
C2×#179858912
C2×C4×(C2×D71)⋊C42
C2×#389×C712
C2×#242671