Finite Group Databasebeta

← C2×#22593251 #32168715

Subgroups — C2×#22593251 #32168715

Order 18176.

3526 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14556108C211yesmaximal, characteristic, complemented
2C22×#14556107C211yesmaximal, characteristic, complemented
3C22×#14556093C211yesmaximal, characteristic, complemented
4C22×C4×C22⋊D71C211yesmaximal, characteristic
5#22593251C2116yesmaximal, direct product factor
6C2×#17985933C212yesmaximal, complemented
7C2×#17985914C212yesmaximal
8C2×#17985892C212yesmaximal, complemented
9C2×#17985888C212yesmaximal, complemented
10C2×#17985864C211yesmaximal, characteristic, complemented
11C2×#17985856C211yesmaximal, characteristic, complemented
12C2×#380×C71C211yesmaximal, characteristic, complemented
13C23×Dic142C2211yescharacteristic
14C23×C22⋊D71C2211yescharacteristic
15C23×C4×D71C2211yescharacteristic
16C2×#14556108—28no—
17C2×#14556108C2218yescomplemented
18C2×#14556107—28no—
19C2×#14556107C2218yescomplemented
20C2×#14556093—28no—
21C2×#14556093C2218yescomplemented
22C2×C4×C22⋊D71—28no—
23C2×C4×C22⋊D71C2218yes—
24C22×C4⋊Dic71C2211yescharacteristic
25C22×C4⋊Dic71C2212yes—
26C22×Dic71⋊C4C2211yescharacteristic
27C22×Dic71⋊C4C2212yes—
28C22×Dic71⋊C4C2212yes—
29C22×C22⋊Dic71C2211yescharacteristic
30C22×C22⋊Dic71C2212yes—
31C22×(C2×D71)⋊C4C2211yescharacteristic
32C22×(C2×D71)⋊C4C2212yes—
33C22×(C2×D71)⋊C4C2212yes—
34C22×C4×Dic71C2212yes—
35C22×C4×Dic71C2211yescharacteristic
36#17985933C22116yescomplemented
37#17985914C22116yes—
38#17985892C22116yescomplemented
39#17985888C22116yescomplemented
40#17985864C2218yescomplemented
41#17985856C2218yescomplemented
42C2×C71⋊#95C2212yes—
43C2×C71⋊#95C2212yes—
44C2×C71⋊#95C2212yes—
45C24×C4×C71C2211yescharacteristic
46C22×C4⋊C4×C71C2211yescharacteristic
47C22×C4⋊C4×C71C2211yescharacteristic
48C22×C22⋊C4×C71C2212yes—
49#380×C71C2218yescomplemented
50C2×#95×C71C2212yes—
51C24×D71C2311yescharacteristic
52C22×Dic142—24no—
53C22×Dic142—24no—
54C22×Dic142D418yescomplemented
55C22×Dic142C2314yes—
56C22×C22⋊D71—24no—
57C22×C22⋊D71—24no—
58C22×C22⋊D71—24no—
59C22×C22⋊D71C2314yes—
60C23×Dic71C2311yescharacteristic
61C23×Dic71C2312yes—
62C23×Dic71C2311yescharacteristic
63C23×Dic71C2312yes—
64C23×Dic71C2311yescharacteristic
65C22×C4×D71—24no—
66C22×C4×D71—24no—
67C22×C4×D71—24no—
68C22×C4×D71C2314yes—
69#14556108—232no—
70#14556107—232no—
71#14556093—232no—
72C4×C22⋊D71—232no—
73C2×C4⋊Dic71—28no—
74C2×C4⋊Dic71—24no—
75C2×C4⋊Dic71C2314yes—
76C2×C4⋊Dic71C2318yes—
77C2×Dic71⋊C4—24no—
78C2×Dic71⋊C4—28no—
79C2×Dic71⋊C4—28no—
80C2×Dic71⋊C4C2318yes—
81C2×Dic71⋊C4C2318yes—
82C2×Dic71⋊C4C2314yes—
83C2×C22⋊Dic71—24no—
84C2×C22⋊Dic71—28no—
85C2×C22⋊Dic71C2314yes—
86C2×C22⋊Dic71C2318yes—
87C2×(C2×D71)⋊C4—28no—
88C2×(C2×D71)⋊C4—24no—
89C2×(C2×D71)⋊C4—28no—
90C2×(C2×D71)⋊C4C2318yes—
91C2×(C2×D71)⋊C4C2314yes—
92C2×(C2×D71)⋊C4C2318yes—
93C2×C4×Dic71—28no—
94C2×C4×Dic71D418yes—
95C2×C4×Dic71C2318yes—
96C2×C4×Dic71C2314yes—
97C71⋊#95C2318yes—
98C71⋊#95C2318yes—
99C71⋊#95C2318yes—
100C25×C71C2311yescharacteristic
101C23×C4×C71—24no—
102C23×C4×C71—24no—
103C23×C4×C71—24no—
104C23×C4×C71C2312yes—
105C23×C4×C71C2312yes—
106C23×C4×C71C2311yescharacteristic
107C23×C4×C71C2314yes—
108C23×C4×C71C2311yescharacteristic
109C2×C4⋊C4×C71—24no—
110C2×C4⋊C4×C71—24no—
111C2×C4⋊C4×C71C2314yes—
112C2×C4⋊C4×C71C2314yes—
113C2×C22⋊C4×C71—28no—
114C2×C22⋊C4×C71C2318yes—
115#95×C71C2318yes—
116C23×D71—22no—
117C23×D71—22no—
118C23×D71—22no—
119C23×D71—22no—
120C23×D71—22no—
121C23×D71—22no—
122C23×D71#4114yes—
123C23×D71C2412yes—
124C2×Dic142—232no—
125C2×Dic142—28no—
126C2×Dic142—28no—
127C2×Dic142C2×D4116yescomplemented
128C2×C22⋊D71—232no—
129C2×C22⋊D71—28no—
130C2×C22⋊D71—28no—
131C2×C22⋊D71—28no—
132C22×Dic71—24no—
133C22×Dic71—22no—
134C22×Dic71—22no—
135C22×Dic71—24no—
136C22×Dic71—22no—
137C22×Dic71—22no—
138C22×Dic71—22no—
139C22×Dic71—22no—
140C22×Dic71—24no—
141C22×Dic71—22no—
142C22×Dic71—24no—
143C22×Dic71—24no—
144C22×Dic71C2×D414yes—
145C22×Dic71C2×D418yes—
146C22×Dic71C2×D414yes—
147C22×Dic71C2412yes—
148C22×Dic71C2414yes—
149C22×Dic71C2412yes—
150C22×Dic71C2414yes—
151C22×Dic71C2412yes—
152C2×C4×D71—28no—
153C2×C4×D71—28no—
154C2×C4×D71—28no—
155C2×C4×D71—416no—
156C4⋊Dic71—28no—
157C4⋊Dic71—216no—
158Dic71⋊C4—28no—
159Dic71⋊C4—216no—
160Dic71⋊C4—216no—
161C22⋊Dic71—28no—
162C22⋊Dic71—216no—
163(C2×D71)⋊C4—216no—
164(C2×D71)⋊C4—28no—
165(C2×D71)⋊C4—216no—
166C4×Dic71—216no—
167C4×Dic71C2×D4116yes—
168C24×C71—22no—
169C24×C71—22no—
170C24×C71—22no—
171C24×C71—22no—
172C24×C71—22no—
173C24×C71—22no—
174C24×C71#4114yes—
175C24×C71C2412yes—
176C24×C71C2411yescharacteristic
177C22×C4×C71—24no—
178C22×C4×C71—24no—
179C22×C4×C71—22no—
180C22×C4×C71—22no—
181C22×C4×C71—24no—
182C22×C4×C71—24no—
183C22×C4×C71—22no—
184C22×C4×C71—24no—
185C22×C4×C71—24no—
186C22×C4×C71—28no—
187C22×C4×C71—28no—
188C22×C4×C71—28no—
189C22×C4×C71—416no—
190C22×C4×C71C2×D414yes—
191C22×C4×C71C2×D414yes—
192C22×C4×C71#4114yes—
193C22×C4×C71C2412yes—
194C22×C4×C71C2414yes—
195C22×C4×C71C2414yes—
196C22×C4×C71C2412yes—
197C4⋊C4×C71—28no—
198C4⋊C4×C71—28no—
199C22⋊C4×C71—216no—
200C23×C71—28no—
201C23×C71—28no—
202C23×C71—28no—
203C23×C71—22no—
204C23×C71—22no—
205C23×C71—22no—
206C23×C71—22no—
207C23×C71—22no—
208C23×C71—22no—
209C23×C71—44no—
210C23×C71—44no—
211C23×C71—44no—
212C23×C71—44no—
213C23×C71C22×D412yes—
214C23×C71C22×D412yes—
215C23×C71C2×#4112yes—
216C23×C71C2×#4114yes—
217C23×C71C2×#4112yes—
218C23×C71C2×#4112yes—
219C23×C712−1+412yes—
220C23×C712−1+412yes—
221C23×C71C2511yescharacteristic
222C22×D71—28no—
223C22×D71—28no—
224C22×D71—28no—
225C22×D71—22no—
226C22×D71—22no—
227C22×D71—22no—
228C22×D71—22no—
229C22×D71—22no—
230C22×D71—22no—
231C22×D71—44no—
232C22×D71—44no—
233C22×D71—44no—
234C22×D71—44no—
235C22×D71C2×#4114yes—
236C2×C4×C71—24no—
237C2×C4×C71—24no—
238C2×C4×C71—22no—
239C2×C4×C71—22no—
240C2×C4×C71—24no—
241C2×C4×C71—24no—
242C2×C4×C71—22no—
243C2×C4×C71—24no—
244C2×C4×C71—24no—
245C2×C4×C71—44no—
246C2×C4×C71—48no—
247C2×C4×C71—48no—
248C2×C4×C71—416no—
249C2×C4×C71#121116yes—
250C2×C4×C71C22×D414yes—
251C2×C4×C71C22×D414yes—
252C2×C4×C71C2×#4114yes—
253C22⋊D71—232no—
254C2×Dic71—216no—
255C2×Dic71—28no—
256C2×Dic71—28no—
257C2×Dic71—24no—
258C2×Dic71—22no—
259C2×Dic71—22no—
260C2×Dic71—24no—
261C2×Dic71—22no—
262C2×Dic71—22no—
263C2×Dic71—24no—
264C2×Dic71—22no—
265C2×Dic71—22no—
266C2×Dic71—24no—
267C2×Dic71—22no—
268C2×Dic71—24no—
269C2×Dic71—44no—
270C2×Dic71—48no—
271C2×Dic71C22×D414yes—
272C2×Dic71C22×D414yes—
273C2×Dic71C22×D418yes—
274C4×D71—416no—
275Dic142—232no—
276C22×C71—24no—
277C22×C71—24no—
278C22×C71—24no—
279C22×C71—42no—
280C22×C71—42no—
281C22×C71—42no—
282C22×C71—42no—
283C22×C71—416no—
284C22×C71C2×#12114yes—
285C22×C71#52514yes—
286C22×C71#53614yes—
287C22×C71#54814yes—
288C22×C71#54814yes—
289C22×C71#55214yes—
290C22×C71#55214yes—
291C22×C71C23×D411yescharacteristic
292C22×C71C23×D411yescharacteristic
293C22×C71C22×#4111yescharacteristic
294C22×C71C22×#4111yescharacteristic
295C22×C71C22×#4111yescharacteristic
296C22×C71C2×2−1+411yescharacteristic
297C22×C71C2×2−1+411yescharacteristic
298C2×D71—24no—
299C2×D71—24no—
300C2×D71—24no—
301C2×D71—42no—
302C2×D71—42no—
303C2×D71—42no—
304C2×D71—42no—
305C2×D71—416no—
306C4×C71—42no—
307C4×C71—44no—
308C4×C71—44no—
309C4×C71C2×#12118yes—
310Dic71—28no—
311Dic71—24no—
312Dic71—24no—
313Dic71—42no—
314Dic71—44no—
315C2×#2536—711nomaximal, Sylow
316C2×C71—44no—
317C2×C71#253618yescomplemented
318C2×C71C22×#12111yescharacteristic
319C2×C71C2×#52511yescharacteristic
320C2×C71C2×#53611yescharacteristic
321C2×C71C2×#54811yescharacteristic
322C2×C71C2×#54811yescharacteristic
323C2×C71C2×#55211yescharacteristic
324C2×C71C2×#55211yescharacteristic
325D71—44no—
326C22×C22⋊Q8—711no—
327C22×C22⋊Q8—711no—
328C22×C4×Q8—711no—
329C22×C4×D4—711no—
330#2536—7116no—
331C2×#398—712no—
332C2×#394—712no—
333C2×#388—712no—
334C2×#388—712no—
335C2×#382—711no—
336C2×#380—711no—
337C2×#380D7111yescharacteristic, complemented
338C71C2×#253611yescharacteristic, complemented, Sylow
339C23×Q8—711no—
340C23×D4—711no—
341C24×C4—711no—
342C24×C4C2×D7111yescharacteristic
343C2×C22⋊Q8—718no—
344C2×C22⋊Q8—718no—
345C2×C22⋊Q8—1428no—
346C2×C22⋊Q8—1428no—
347C2×C4×Q8—718no—
348C2×C4×Q8—1428no—
349C2×C4×D4—718no—
350C2×C4×D4—1428no—
351C22×C4⋊C4—712no—
352C22×C4⋊C4—711no—
353C22×C4⋊C4—712no—
354C22×C4⋊C4—712no—
355C22×C4⋊C4—711no—
356C22×C4⋊C4C2×D7111yescharacteristic
357C22×C4⋊C4C2×D7111yescharacteristic
358C22×C22⋊C4—712no—
359C22×C22⋊C4—711no—
360C22×C22⋊C4—712no—
361C22×C22⋊C4—712no—
362C22×C22⋊C4—711no—
363C22×C22⋊C4C2×D7112yes—
364C22×C42—711no—
365C22×C42—712no—
366#398—7116no—
367#394—7116no—
368#388—7116no—
369#388—7116no—
370#382—718no—
371#380—718no—
372#380C2×D7118yescomplemented
373C2×#95—712no—
374C2×#95—712no—
375C2×#95—712no—
376C2×#95C2×D7112yes—
377C25—711no—
378C25C22×D7111yescharacteristic
379C22×Q8—718no—
380C22×Q8—714no—
381C22×Q8—1424no—
382C22×Q8—1424no—
383C22×D4—714no—
384C22×D4—1424no—
385C22×D4—1424no—
386C22×D4—1424no—
387C23×C4—24no—
388C23×C4—24no—
389C23×C4—24no—
390C23×C4—712no—
391C23×C4—711no—
392C23×C4—711no—
393C23×C4—712no—
394C23×C4—711no—
395C23×C4—714no—
396C23×C4—1424no—
397C23×C4—1424no—
398C23×C4—1424no—
399C23×C4C22×D7114yes—
400C23×C4C22×D7112yes—
401C23×C4C22×D7112yes—
402C23×C4C22×D7111yescharacteristic
403C23×C4C22×D7111yescharacteristic
404C22⋊Q8—14232no—
405C22⋊Q8—14232no—
406C4×Q8—14232no—
407C4×D4—14232no—
408C2×C4⋊C4—24no—
409C2×C4⋊C4—24no—
410C2×C4⋊C4—714no—
411C2×C4⋊C4—718no—
412C2×C4⋊C4—714no—
413C2×C4⋊C4—718no—
414C2×C4⋊C4—718no—
415C2×C4⋊C4—1424no—
416C2×C4⋊C4—1428no—
417C2×C4⋊C4—1428no—
418C2×C4⋊C4—1428no—
419C2×C4⋊C4—1424no—
420C2×C4⋊C4C22×D7114yes—
421C2×C4⋊C4C22×D7114yes—
422C2×C22⋊C4—28no—
423C2×C22⋊C4—714no—
424C2×C22⋊C4—718no—
425C2×C22⋊C4—714no—
426C2×C22⋊C4—718no—
427C2×C22⋊C4—718no—
428C2×C22⋊C4—1424no—
429C2×C22⋊C4—1428no—
430C2×C22⋊C4—1428no—
431C2×C22⋊C4—1424no—
432C2×C22⋊C4—1428no—
433C2×C22⋊C4C22×D7118yes—
434C2×C42—718no—
435C2×C42—714no—
436C2×C42—718no—
437C2×C42—1428no—
438#95—718no—
439#95—718no—
440#95—718no—
441#95C22×D7118yes—
442C24—22no—
443C24—22no—
444C24—22no—
445C24—22no—
446C24—22no—
447C24—22no—
448C24—714no—
449C24—712no—
450C24—1422no—
451C24—1422no—
452C24—1422no—
453C24—1422no—
454C24—1422no—
455C24—1422no—
456C24#1176586414yes—
457C24C23×D7112yes—
458C24C23×D7111yescharacteristic
459C2×Q8—7116no—
460C2×Q8—1428no—
461C2×Q8—1428no—
462C2×Q8—14232no—
463C2×D4—1428no—
464C2×D4—1428no—
465C2×D4—14232no—
466C2×D4—1428no—
467C22×C4—24no—
468C22×C4—24no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—24no—
472C22×C4—24no—
473C22×C4—24no—
474C22×C4—22no—
475C22×C4—24no—
476C22×C4—28no—
477C22×C4—28no—
478C22×C4—28no—
479C22×C4—416no—
480C22×C4—28416no—
481C22×C4—714no—
482C22×C4—714no—
483C22×C4—718no—
484C22×C4—712no—
485C22×C4—712no—
486C22×C4—714no—
487C22×C4—714no—
488C22×C4—712no—
489C22×C4—1424no—
490C22×C4—1422no—
491C22×C4—1422no—
492C22×C4—1424no—
493C22×C4—1422no—
494C22×C4—1424no—
495C22×C4—1428no—
496C22×C4—1428no—
497C22×C4—1428no—
498C22×C4—1424no—
499C22×C4—1422no—
500C22×C4—1422no—
501C22×C4—1424no—
502C22×C4—1422no—
503C22×C4—1422no—
504C22×C4D4×D7114yes—
505C22×C4D4×D7114yes—
506C22×C4#1176586514yes—
507C22×C4C23×D7114yes—
508C22×C4C23×D7114yes—
509C22×C4C23×D7112yes—
510C22×C4C23×D7112yes—
511C4⋊C4—28no—
512C4⋊C4—28no—
513C4⋊C4—1428no—
514C4⋊C4—14216no—
515C4⋊C4—14216no—
516C4⋊C4—14216no—
517C4⋊C4—1428no—
518C22⋊C4—216no—
519C22⋊C4—1428no—
520C22⋊C4—14216no—
521C22⋊C4—14216no—
522C22⋊C4—1428no—
523C22⋊C4—14216no—
524C42—7116no—
525C42—14216no—
526C23—22no—
527C23—22no—
528C23—22no—
529C23—22no—
530C23—22no—
531C23—22no—
532C23—28no—
533C23—28no—
534C23—28no—
535C23—44no—
536C23—44no—
537C23—44no—
538C23—44no—
539C23—2844no—
540C23—2844no—
541C23—2844no—
542C23—2844no—
543C23—714no—
544C23—1422no—
545C23—1422no—
546C23—1422no—
547C23—1422no—
548C23—1422no—
549C23—1422no—
550C23—1428no—
551C23—1428no—
552C23—1428no—
553C23C2×D4×D7112yes—
554C23C2×D4×D7112yes—
555C23#41×D7112yes—
556C23C2×#1176586414yes—
557C23C2×#1176586412yes—
558C23C2×#1176586512yes—
559C23#1455617512yes—
560C23#1455617612yes—
561C23C24×D7111yescharacteristic
562Q8—14232no—
563D4—14232no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—22no—
572C2×C4—24no—
573C2×C4—48no—
574C2×C4—44no—
575C2×C4—48no—
576C2×C4—416no—
577C2×C4—2848no—
578C2×C4—2844no—
579C2×C4—28416no—
580C2×C4—714no—
581C2×C4—714no—
582C2×C4—718no—
583C2×C4—1424no—
584C2×C4—1422no—
585C2×C4—1422no—
586C2×C4—1424no—
587C2×C4—1422no—
588C2×C4—1424no—
589C2×C4—1424no—
590C2×C4—1422no—
591C2×C4—1422no—
592C2×C4—1424no—
593C2×C4—1428no—
594C2×C4—1422no—
595C2×C4—1422no—
596C2×C4—14216no—
597C2×C4—1428no—
598C2×C4#14556101116yes—
599C2×C4C2×D4×D7114yes—
600C2×C4C2×D4×D7114yes—
601C2×C4C2×#1176586514yes—
602C22—24no—
603C22—24no—
604C22—24no—
605C22—42no—
606C22—42no—
607C22—42no—
608C22—42no—
609C22—416no—
610C22—2842no—
611C22—2842no—
612C22—2842no—
613C22—2842no—
614C22—28416no—
615C22—1424no—
616C22—1424no—
617C22—1424no—
618C22C2×#1455610114yes—
619C22#1798650914yes—
620C22#1798657214yes—
621C22#1798665514yes—
622C22#1798665914yes—
623C22#1798668414yes—
624C22#1798668814yes—
625C22C22×D4×D7111yescharacteristic
626C22C22×D4×D7111yescharacteristic
627C22C2×#41×D7111yescharacteristic
628C22C22×#1176586411yescharacteristic
629C22C22×#1176586511yescharacteristic
630C22C2×#1455617511yescharacteristic
631C22C2×#1455617611yescharacteristic
632C4—44no—
633C4—42no—
634C4—44no—
635C4—2844no—
636C4—2842no—
637C4—1428no—
638C4—1424no—
639C4—1424no—
640C4C2×#1455610118yes—
641C2—44no—
642C2—2844no—
643C2#2259325118yesdirect product factor
644C2C22×#1455610111yescharacteristic
645C2C2×#1798650911yescharacteristic
646C2C2×#1798657211yescharacteristic
647C2C2×#1798665511yescharacteristic
648C2C2×#1798665911yescharacteristic
649C2C2×#1798668411yescharacteristic
650C2C2×#1798668811yescharacteristic
subgroupG/H
centerC24C23×D71
commutator, cosocleC23×C71C25
FrattiniC23C24×D71
FittingC2×#380×C71C2
socleC24×C71C24
prime radicalC23×C22⋊D71C22
hypercenterC2×#380D71
hypocenterC71C2×#2536
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C24 ◁ C2×#380

Complements

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

NH ≅ G/Ntype
1. C22×#14556108C2semidirect
2. C22×#14556107C2semidirect
3. C22×#14556093C2semidirect
5. #22593251C2direct (N × H)
6. C2×#17985933C2semidirect
8. C2×#17985892C2semidirect
9. C2×#17985888C2semidirect
10. C2×#17985864C2semidirect
11. C2×#17985856C2semidirect
12. C2×#380×C71C2semidirect
17. C2×#14556108C22semidirect
19. C2×#14556107C22semidirect
21. C2×#14556093C22semidirect
36. #17985933C22semidirect
38. #17985892C22semidirect
39. #17985888C22semidirect
40. #17985864C22semidirect
41. #17985856C22semidirect
49. #380×C71C22semidirect
54. C22×Dic142D4semidirect
127. C2×Dic142C2×D4semidirect
317. C2×C71#2536semidirect
337. C2×#380D71semidirect
338. C71C2×#2536semidirect
372. #380C2×D71semidirect
643. C2#22593251direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145561082
C22×#145561072
C22×#145560932
C22×C4×C22⋊D712
#225932512
C2×#179859332
C2×#179859142
C2×#179858922
C2×#179858882
C2×#179858642
C2×#179858562
C2×#380×C712
C2×#253671