Finite Group Databasebeta

← C2×#22591835 #32165586

Subgroups — C2×#22591835 #32165586

Order 18176.

3438 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
1C22×#14556103C211yesmaximal, characteristic
2C22×#14556090C211yesmaximal, characteristic, complemented
3#22591835C2116yesmaximal, direct product factor
4C2×#17985928C212yesmaximal, complemented
5C2×#17985917C212yesmaximal, complemented
6C2×#17985901C211yesmaximal, characteristic, complemented
7C2×#17985895C211yesmaximal, characteristic, complemented
8C2×#17985877C212yesmaximal, complemented
9C2×#17985861C212yesmaximal, complemented
10C2×#17985856C211yesmaximal, characteristic, complemented
11C2×C4×(C2×D71)⋊C4C211yesmaximal, characteristic, complemented
12C2×#382×C71C211yesmaximal, characteristic, complemented
13C23×D284C2212yes—
14C23×D284C2211yescharacteristic
15C23×C4×D71C2211yescharacteristic
16C23×C4×D71C2212yes—
17C2×#14556103—28no—
18C2×#14556103C2218yes—
19C2×#14556090—28no—
20C2×#14556090C2218yescomplemented
21C22×C4⋊Dic71C2211yescharacteristic
22C22×Dic71⋊C4C2212yes—
23C22×(C2×D71)⋊C4C2211yescharacteristic
24C22×(C2×D71)⋊C4C2212yes—
25C22×(C2×D71)⋊C4C2211yescharacteristic
26C22×(C2×D71)⋊C4C2212yes—
27C22×(C2×D71)⋊C4C2212yes—
28C22×(C2×D71)⋊C4C2212yes—
29C22×(C2×D71)⋊C4C2212yes—
30C22×C4×Dic71C2211yescharacteristic
31#17985928C22116yescomplemented
32#17985917C22116yescomplemented
33#17985901C2218yescomplemented
34#17985895C2218yescomplemented
35#17985877C22116yescomplemented
36#17985861C22116yescomplemented
37#17985856C2218yescomplemented
38C4×(C2×D71)⋊C4C2218yescomplemented
39C2×C71⋊#95C2212yes—
40C2×C71⋊#95C2211yescharacteristic
41C2×C71⋊#95C2212yes—
42C2×C71⋊#95C2211yescharacteristic
43C22×C4⋊C4×C71C2211yescharacteristic
44C22×C4⋊C4×C71C2211yescharacteristic
45C22×C42×C71C2211yescharacteristic
46#382×C71C2218yescomplemented
47C2×#95×C71C2212yes—
48C2×#95×C71C2212yes—
49C24×D71C2312yes—
50C24×D71C2311yescharacteristic
51C22×D284—28no—
52C22×D284—28no—
53C22×D284—24no—
54C22×D284—24no—
55C22×D284—44no—
56C22×D284D418yes—
57C22×D284C2318yes—
58C22×D284C2314yes—
59C23×Dic71C2311yescharacteristic
60C23×Dic71C2312yes—
61C23×Dic71C2311yescharacteristic
62C23×Dic71C2311yescharacteristic
63C22×C4×D71—28no—
64C22×C4×D71—28no—
65C22×C4×D71—24no—
66C22×C4×D71—24no—
67C22×C4×D71—24no—
68C22×C4×D71—44no—
69C22×C4×D71C2314yes—
70C22×C4×D71C2318yes—
71#14556103—232no—
72#14556090—232no—
73C2×C4⋊Dic71—24no—
74C2×C4⋊Dic71C2314yes—
75C2×Dic71⋊C4—44no—
76C2×Dic71⋊C4C2318yes—
77C2×(C2×D71)⋊C4—28no—
78C2×(C2×D71)⋊C4—28no—
79C2×(C2×D71)⋊C4—24no—
80C2×(C2×D71)⋊C4—24no—
81C2×(C2×D71)⋊C4—44no—
82C2×(C2×D71)⋊C4—44no—
83C2×(C2×D71)⋊C4—44no—
84C2×(C2×D71)⋊C4C2314yes—
85C2×(C2×D71)⋊C4C2314yes—
86C2×(C2×D71)⋊C4C2318yes—
87C2×(C2×D71)⋊C4C2318yes—
88C2×(C2×D71)⋊C4C2318yes—
89C2×(C2×D71)⋊C4C2318yes—
90C2×(C2×D71)⋊C4C2318yes—
91C2×C4×Dic71—24no—
92C2×C4×Dic71C2314yes—
93C71⋊#95C2314yes—
94C71⋊#95C2314yes—
95C71⋊#95C2318yes—
96C71⋊#95C2318yes—
97C23×C4×C71C2312yes—
98C23×C4×C71C2312yes—
99C23×C4×C71C2311yescharacteristic
100C23×C4×C71C2311yescharacteristic
101C23×C4×C71C2311yescharacteristic
102C2×C4⋊C4×C71—24no—
103C2×C4⋊C4×C71—24no—
104C2×C4⋊C4×C71C2314yes—
105C2×C4⋊C4×C71C2314yes—
106C2×C42×C71—24no—
107C2×C42×C71C2314yes—
108#95×C71C2318yes—
109#95×C71C2318yes—
110C23×D71—22no—
111C23×D71—24no—
112C23×D71—22no—
113C23×D71—22no—
114C23×D71—24no—
115C23×D71—24no—
116C23×D71—24no—
117C23×D71—22no—
118C23×D71—24no—
119C23×D71—22no—
120C23×D71—24no—
121C23×D71—22no—
122C23×D71C2×D418yes—
123C23×D71#4114yes—
124C23×D71C2414yes—
125C23×D71C2412yes—
126C2×D284—232no—
127C2×D284—28no—
128C2×D284—28no—
129C2×D284—216no—
130C2×D284—216no—
131C2×D284—48no—
132C2×D284—432no—
133C2×D284C2×D4116yes—
134C22×Dic71—22no—
135C22×Dic71—22no—
136C22×Dic71—22no—
137C22×Dic71—22no—
138C22×Dic71—22no—
139C22×Dic71—22no—
140C22×Dic71—22no—
141C22×Dic71—24no—
142C22×Dic71—24no—
143C22×Dic71—24no—
144C22×Dic71—22no—
145C22×Dic71#4114yes—
146C22×Dic71C2412yes—
147C22×Dic71C2414yes—
148C22×Dic71C2412yes—
149C22×Dic71C2412yes—
150C2×C4×D71—216no—
151C2×C4×D71—28no—
152C2×C4×D71—28no—
153C2×C4×D71—28no—
154C2×C4×D71—216no—
155C2×C4×D71—48no—
156C2×C4×D71—432no—
157C2×C4×D71—416no—
158C4⋊Dic71—28no—
159Dic71⋊C4—48no—
160(C2×D71)⋊C4—216no—
161(C2×D71)⋊C4—216no—
162(C2×D71)⋊C4—28no—
163(C2×D71)⋊C4—28no—
164(C2×D71)⋊C4—48no—
165(C2×D71)⋊C4—48no—
166(C2×D71)⋊C4—48no—
167C4×Dic71—28no—
168C24×C71C2411yescharacteristic
169C22×C4×C71—24no—
170C22×C4×C71—24no—
171C22×C4×C71—24no—
172C22×C4×C71—24no—
173C22×C4×C71—22no—
174C22×C4×C71—22no—
175C22×C4×C71—22no—
176C22×C4×C71—22no—
177C22×C4×C71—24no—
178C22×C4×C71—24no—
179C22×C4×C71—22no—
180C22×C4×C71—22no—
181C22×C4×C71C2×D414yes—
182C22×C4×C71#4114yes—
183C22×C4×C71#4114yes—
184C22×C4×C71C2414yes—
185C22×C4×C71C2414yes—
186C22×C4×C71C2412yes—
187C22×C4×C71C2412yes—
188C22×C4×C71C2412yes—
189C4⋊C4×C71—28no—
190C4⋊C4×C71—28no—
191C42×C71—28no—
192C23×C71C22×D412yes—
193C23×C71C2×#4112yes—
194C23×C71C2×#4112yes—
195C23×C71C2×#4112yes—
196C23×C71C2×#4112yes—
197C23×C71H(5,𝔽2)12yes—
198C23×C71H(5,𝔽2)12yes—
199C23×C71C2511yescharacteristic
200C22×D71—216no—
201C22×D71—28no—
202C22×D71—216no—
203C22×D71—28no—
204C22×D71—216no—
205C22×D71—28no—
206C22×D71—22no—
207C22×D71—22no—
208C22×D71—24no—
209C22×D71—22no—
210C22×D71—22no—
211C22×D71—24no—
212C22×D71—24no—
213C22×D71—24no—
214C22×D71—22no—
215C22×D71—24no—
216C22×D71—24no—
217C22×D71—22no—
218C22×D71—44no—
219C22×D71—48no—
220C22×D71—44no—
221C22×D71—48no—
222C22×D71—44no—
223C22×D71—48no—
224C22×D71—48no—
225C22×D71—44no—
226C22×D71C22×D418yes—
227C22×D71C2×#4114yes—
228C2×C4×C71—24no—
229C2×C4×C71—24no—
230C2×C4×C71—22no—
231C2×C4×C71—22no—
232C2×C4×C71—22no—
233C2×C4×C71—22no—
234C2×C4×C71—24no—
235C2×C4×C71—24no—
236C2×C4×C71—22no—
237C2×C4×C71—22no—
238C2×C4×C71—24no—
239C2×C4×C71—24no—
240C2×C4×C71—48no—
241C2×C4×C71—48no—
242C2×C4×C71—44no—
243C2×C4×C71—44no—
244C2×C4×C71#123116yes—
245C2×C4×C71C22×D414yes—
246C2×C4×C71C2×#4114yes—
247C2×C4×C71C2×#4114yes—
248C2×Dic71—28no—
249C2×Dic71—22no—
250C2×Dic71—22no—
251C2×Dic71—22no—
252C2×Dic71—22no—
253C2×Dic71—22no—
254C2×Dic71—24no—
255C2×Dic71—22no—
256C2×Dic71—22no—
257C2×Dic71—24no—
258C2×Dic71—24no—
259C2×Dic71—22no—
260C2×Dic71—44no—
261C2×Dic71—48no—
262C2×Dic71—44no—
263C2×Dic71C2×#4114yes—
264D284—232no—
265D284—432no—
266C4×D71—432no—
267C4×D71—416no—
268C22×C71C2×#12314yes—
269C22×C71#52914yes—
270C22×C71#54014yes—
271C22×C71#54614yes—
272C22×C71#54614yes—
273C22×C71#55314yes—
274C22×C71#55514yes—
275C22×C71C23×D411yescharacteristic
276C22×C71C22×#4111yescharacteristic
277C22×C71C22×#4111yescharacteristic
278C22×C71C22×#4111yescharacteristic
279C22×C71C22×#4111yescharacteristic
280C22×C71C2×H(5,𝔽2)11yescharacteristic
281C22×C71C2×H(5,𝔽2)11yescharacteristic
282C2×D71—28no—
283C2×D71—24no—
284C2×D71—28no—
285C2×D71—24no—
286C2×D71—28no—
287C2×D71—24no—
288C2×D71—42no—
289C2×D71—44no—
290C2×D71—432no—
291C2×D71—416no—
292C2×D71—42no—
293C2×D71—44no—
294C2×D71—42no—
295C2×D71—44no—
296C2×D71—42no—
297C2×D71—44no—
298C4×C71—44no—
299C4×C71—44no—
300C4×C71—42no—
301C4×C71—42no—
302C4×C71C2×#12318yes—
303Dic71—24no—
304Dic71—42no—
305Dic71—44no—
306Dic71—42no—
307C2×#2420—711nomaximal, Sylow
308C2×C71#242018yescomplemented
309C2×C71C22×#12311yescharacteristic
310C2×C71C2×#52911yescharacteristic
311C2×C71C2×#54011yescharacteristic
312C2×C71C2×#54611yescharacteristic
313C2×C71C2×#54611yescharacteristic
314C2×C71C2×#55311yescharacteristic
315C2×C71C2×#55511yescharacteristic
316D71—48no—
317D71—44no—
318C22×#121—711no—
319C22×C4×D4—711no—
320#2420—7116no—
321C2×#397—712no—
322C2×#394—712no—
323C2×#390—711no—
324C2×#389—711no—
325C2×#386—712no—
326C2×#382D7111yescharacteristic, complemented
327C2×#381—712no—
328C2×#380—711no—
329C2×C4×C22⋊C4—711no—
330C71C2×#242011yescharacteristic, complemented, Sylow
331C23×D4—712no—
332C23×D4—711no—
333C24×C4—712no—
334C24×C4—711no—
335C2×#121—718no—
336C2×#121—1428no—
337C2×C4×D4—718no—
338C2×C4×D4—1428no—
339C22×C4⋊C4—712no—
340C22×C4⋊C4—711no—
341C22×C4⋊C4C2×D7111yescharacteristic
342C22×C4⋊C4C2×D7111yescharacteristic
343C22×C22⋊C4—712no—
344C22×C22⋊C4—712no—
345C22×C22⋊C4—712no—
346C22×C22⋊C4—711no—
347C22×C22⋊C4—712no—
348C22×C22⋊C4—712no—
349C22×C22⋊C4—711no—
350C22×C42—711no—
351C22×C42C2×D7111yescharacteristic
352#397—7116no—
353#394—7116no—
354#390—718no—
355#389—718no—
356#386—7116no—
357#382C2×D7118yescomplemented
358#381—7116no—
359#380—718no—
360C4×C22⋊C4—718no—
361C2×#95—711no—
362C2×#95—712no—
363C2×#95—711no—
364C2×#95—712no—
365C2×#95C2×D7112yes—
366C2×#95C2×D7112yes—
367C25—712no—
368C25—711no—
369C22×D4—2844no—
370C22×D4—718no—
371C22×D4—718no—
372C22×D4—714no—
373C22×D4—1428no—
374C22×D4—1428no—
375C22×D4—1424no—
376C22×D4—1424no—
377C23×C4—2844no—
378C23×C4—718no—
379C23×C4—711no—
380C23×C4—712no—
381C23×C4—711no—
382C23×C4—711no—
383C23×C4—714no—
384C23×C4—1428no—
385C23×C4—1428no—
386C23×C4—1424no—
387C23×C4—1424no—
388C23×C4—1424no—
389C23×C4C22×D7111yescharacteristic
390C23×C4C22×D7111yescharacteristic
391C23×C4C22×D7112yes—
392C23×C4C22×D7112yes—
393C23×C4C22×D7111yescharacteristic
394#121—14232no—
395C4×D4—14232no—
396C2×C4⋊C4—24no—
397C2×C4⋊C4—24no—
398C2×C4⋊C4—2844no—
399C2×C4⋊C4—718no—
400C2×C4⋊C4—714no—
401C2×C4⋊C4—1424no—
402C2×C4⋊C4C22×D7114yes—
403C2×C4⋊C4C22×D7114yes—
404C2×C22⋊C4—2844no—
405C2×C22⋊C4—2844no—
406C2×C22⋊C4—2844no—
407C2×C22⋊C4—718no—
408C2×C22⋊C4—718no—
409C2×C22⋊C4—718no—
410C2×C22⋊C4—718no—
411C2×C22⋊C4—718no—
412C2×C22⋊C4—714no—
413C2×C22⋊C4—714no—
414C2×C22⋊C4—1424no—
415C2×C22⋊C4—1428no—
416C2×C22⋊C4—1428no—
417C2×C22⋊C4—1424no—
418C2×C42—24no—
419C2×C42—714no—
420C2×C42—1424no—
421C2×C42C22×D7114yes—
422#95—714no—
423#95—714no—
424#95—718no—
425#95—718no—
426#95C22×D7118yes—
427#95C22×D7118yes—
428C24—718no—
429C24—714no—
430C24—714no—
431C24—712no—
432C24—1422no—
433C24—1422no—
434C24—1424no—
435C24—1424no—
436C24—1422no—
437C24—1424no—
438C24—1422no—
439C24—1424no—
440C24—1422no—
441C24—1424no—
442C24—1424no—
443C24—1422no—
444C24C23×D7111yescharacteristic
445C2×D4—2848no—
446C2×D4—28432no—
447C2×D4—7116no—
448C2×D4—14216no—
449C2×D4—14216no—
450C2×D4—1428no—
451C2×D4—1428no—
452C2×D4—14232no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—2848no—
466C22×C4—28432no—
467C22×C4—28416no—
468C22×C4—714no—
469C22×C4—712no—
470C22×C4—712no—
471C22×C4—714no—
472C22×C4—712no—
473C22×C4—14216no—
474C22×C4—14216no—
475C22×C4—1422no—
476C22×C4—1422no—
477C22×C4—1422no—
478C22×C4—1422no—
479C22×C4—1424no—
480C22×C4—1422no—
481C22×C4—1424no—
482C22×C4—1424no—
483C22×C4—1422no—
484C22×C4—1428no—
485C22×C4—1428no—
486C22×C4—1428no—
487C22×C4—1422no—
488C22×C4—1422no—
489C22×C4D4×D7114yes—
490C22×C4#1176586514yes—
491C22×C4#1176586614yes—
492C22×C4C23×D7112yes—
493C22×C4C23×D7112yes—
494C22×C4C23×D7114yes—
495C22×C4C23×D7114yes—
496C22×C4C23×D7112yes—
497C4⋊C4—28no—
498C4⋊C4—28no—
499C4⋊C4—2848no—
500C4⋊C4—1428no—
501C22⋊C4—2848no—
502C22⋊C4—2848no—
503C22⋊C4—2848no—
504C22⋊C4—1428no—
505C22⋊C4—1428no—
506C22⋊C4—14216no—
507C22⋊C4—14216no—
508C42—28no—
509C42—1428no—
510C23—2844no—
511C23—2848no—
512C23—2844no—
513C23—2848no—
514C23—2844no—
515C23—2848no—
516C23—2844no—
517C23—2848no—
518C23—718no—
519C23—714no—
520C23—1422no—
521C23—1422no—
522C23—1424no—
523C23—1424no—
524C23—1422no—
525C23—1424no—
526C23—1422no—
527C23—1424no—
528C23—1422no—
529C23—1424no—
530C23—1424no—
531C23—1422no—
532C23—14216no—
533C23—1428no—
534C23—14216no—
535C23—1428no—
536C23—14216no—
537C23—1428no—
538C23C2×D4×D7112yes—
539C23#41×D7112yes—
540C23#41×D7112yes—
541C23C2×#1176586512yes—
542C23C2×#1176586612yes—
543C23#1455617412yes—
544C23#1455617412yes—
545C23C24×D7111yescharacteristic
546D4—28432no—
547D4—14232no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—48no—
561C2×C4—48no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—28432no—
565C2×C4—2848no—
566C2×C4—2844no—
567C2×C4—2844no—
568C2×C4—28416no—
569C2×C4—714no—
570C2×C4—1422no—
571C2×C4—1422no—
572C2×C4—1422no—
573C2×C4—1422no—
574C2×C4—1424no—
575C2×C4—1422no—
576C2×C4—1424no—
577C2×C4—1424no—
578C2×C4—1422no—
579C2×C4—1422no—
580C2×C4—1422no—
581C2×C4—1428no—
582C2×C4#14556113116yes—
583C2×C4C2×D4×D7114yes—
584C2×C4C2×#1176586514yes—
585C2×C4C2×#1176586614yes—
586C22—2842no—
587C22—28416no—
588C22—28432no—
589C22—2842no—
590C22—2844no—
591C22—2844no—
592C22—2842no—
593C22—2844no—
594C22—2842no—
595C22—2844no—
596C22—1428no—
597C22—1424no—
598C22—1428no—
599C22—1428no—
600C22—1424no—
601C22—1424no—
602C22C2×#1455611314yes—
603C22#1798653414yes—
604C22#1798660514yes—
605C22#1798664214yes—
606C22#1798664214yes—
607C22#1798669314yes—
608C22#1798671214yes—
609C22C22×D4×D7111yescharacteristic
610C22C2×#41×D7111yescharacteristic
611C22C2×#41×D7111yescharacteristic
612C22C22×#1176586511yescharacteristic
613C22C22×#1176586611yescharacteristic
614C22C2×#1455617411yescharacteristic
615C22C2×#1455617411yescharacteristic
616C4—44no—
617C4—44no—
618C4—42no—
619C4—42no—
620C4—2842no—
621C4—2842no—
622C4—2844no—
623C4—1424no—
624C4C2×#1455611318yes—
625C2—2844no—
626C2—2848no—
627C2#2259183518yesdirect product factor
628C2C22×#1455611311yescharacteristic
629C2C2×#1798653411yescharacteristic
630C2C2×#1798660511yescharacteristic
631C2C2×#1798664211yescharacteristic
632C2C2×#1798664211yescharacteristic
633C2C2×#1798669311yescharacteristic
634C2C2×#1798671211yescharacteristic
subgroupG/H
centerC24C23×D71
commutator, cosocleC23×C71C25
FrattiniC23C24×D71
FittingC2×#382×C71C2
socleC24×C71C24
prime radicalC22×#14556103C2
hypercenterC2×#382D71
hypocenterC71C2×#2420
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C24 ◁ C2×#382

Complements

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

NH ≅ G/Ntype
2. C22×#14556090C2semidirect
3. #22591835C2direct (N × H)
4. C2×#17985928C2semidirect
5. C2×#17985917C2semidirect
6. C2×#17985901C2semidirect
7. C2×#17985895C2semidirect
8. C2×#17985877C2semidirect
9. C2×#17985861C2semidirect
10. C2×#17985856C2semidirect
11. C2×C4×(C2×D71)⋊C4C2semidirect
12. C2×#382×C71C2semidirect
20. C2×#14556090C22semidirect
31. #17985928C22semidirect
32. #17985917C22semidirect
33. #17985901C22semidirect
34. #17985895C22semidirect
35. #17985877C22semidirect
36. #17985861C22semidirect
37. #17985856C22semidirect
38. C4×(C2×D71)⋊C4C22semidirect
46. #382×C71C22semidirect
308. C2×C71#2420semidirect
326. C2×#382D71semidirect
330. C71C2×#2420semidirect
357. #382C2×D71semidirect
627. C2#22591835direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145561032
C22×#145560902
#225918352
C2×#179859282
C2×#179859172
C2×#179859012
C2×#179858952
C2×#179858772
C2×#179858612
C2×#179858562
C2×C4×(C2×D71)⋊C42
C2×#382×C712
C2×#242071