Finite Group Databasebeta

← C2×#22591284 #32156796

Subgroups — C2×#22591284 #32156796

Order 18176.

3678 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×C22⋊D71C211yesmaximal, characteristic
2C22×#14556087C211yesmaximal, characteristic, complemented
3C22×#14556086C211yesmaximal, characteristic, complemented
4#22591284C2116yesmaximal, direct product factor
5C2×#17985899C212yesmaximal
6C2×#17985893C212yesmaximal, complemented
7C2×#17985888C212yesmaximal, complemented
8C2×#17985868C211yesmaximal, characteristic, complemented
9C2×#17985856C211yesmaximal, characteristic, complemented
10C2×C4⋊C4×Dic71C211yesmaximal, characteristic, complemented
11C2×#17985850C212yesmaximal, complemented
12C2×#380×C71C211yesmaximal, characteristic, complemented
13C23×C22⋊D71C2211yescharacteristic
14C23×C4×D71C2211yescharacteristic
15C2×C4×C22⋊D71C4116yescomplemented
16C2×C4×C22⋊D71C2218yes—
17C2×#14556087—28no—
18C2×#14556087C2218yescomplemented
19C2×#14556086—28no—
20C2×#14556086C2218yescomplemented
21C22×C4⋊Dic71C2211yescharacteristic
22C22×C4⋊Dic71C2211yescharacteristic
23C22×Dic71⋊C4C2212yes—
24C22×Dic71⋊C4C2212yes—
25C22×Dic71⋊C4C2211yescharacteristic
26C22×C22⋊Dic71C2212yes—
27C22×C22⋊Dic71C2211yescharacteristic
28C22×(C2×D71)⋊C4C2212yes—
29C22×(C2×D71)⋊C4C2212yes—
30C22×(C2×D71)⋊C4C2211yescharacteristic
31C22×C4×Dic71C2212yes—
32C22×C4×Dic71C2212yes—
33C22×C4×Dic71C2212yes—
34C22×C4×Dic71C2211yescharacteristic
35#17985899C22116yes—
36#17985893C22116yescomplemented
37#17985888C22116yescomplemented
38#17985868C2218yescomplemented
39#17985856C2218yescomplemented
40C4⋊C4×Dic71C2218yescomplemented
41#17985850C22116yescomplemented
42C2×C71⋊#95C2212yes—
43C2×C71⋊#95C2212yes—
44C24×C4×C71C2211yescharacteristic
45C22×C4⋊C4×C71C2211yescharacteristic
46C22×C4⋊C4×C71C2211yescharacteristic
47C22×C22⋊C4×C71C2212yes—
48#380×C71C2218yescomplemented
49C2×#95×C71C2212yes—
50C24×D71C2311yescharacteristic
51C22×C22⋊D71—24no—
52C22×C22⋊D71—24no—
53C22×C22⋊D71C2×C418yes—
54C22×C22⋊D71C2314yes—
55C23×Dic71C2312yes—
56C23×Dic71C2311yescharacteristic
57C23×Dic71C2311yescharacteristic
58C23×Dic71C2312yes—
59C23×Dic71C2311yescharacteristic
60C22×C4×D71—24no—
61C22×C4×D71—24no—
62C22×C4×D71C2×C418yes—
63C22×C4×D71C2314yes—
64C4×C22⋊D71C2×C4164yescomplemented
65#14556087—232no—
66#14556086—232no—
67C2×C4⋊Dic71—24no—
68C2×C4⋊Dic71D418yescomplemented
69C2×C4⋊Dic71C2314yes—
70C2×C4⋊Dic71C2314yes—
71C2×Dic71⋊C4—28no—
72C2×Dic71⋊C4—44no—
73C2×Dic71⋊C4C2×C418yes—
74C2×Dic71⋊C4C2318yes—
75C2×Dic71⋊C4C2318yes—
76C2×Dic71⋊C4C2314yes—
77C2×C22⋊Dic71—28no—
78C2×C22⋊Dic71C2×C418yes—
79C2×C22⋊Dic71C2318yes—
80C2×C22⋊Dic71C2314yes—
81C2×(C2×D71)⋊C4—28no—
82C2×(C2×D71)⋊C4—44no—
83C2×(C2×D71)⋊C4C2×C418yes—
84C2×(C2×D71)⋊C4C2318yes—
85C2×(C2×D71)⋊C4C2318yes—
86C2×(C2×D71)⋊C4C2314yes—
87C2×C4×Dic71—28no—
88C2×C4×Dic71—28no—
89C2×C4×Dic71—28no—
90C2×C4×Dic71C2×C418yes—
91C2×C4×Dic71C2318yes—
92C2×C4×Dic71C2318yes—
93C2×C4×Dic71C2318yes—
94C2×C4×Dic71C2314yes—
95C71⋊#95C2318yes—
96C71⋊#95C2318yes—
97C25×C71C2311yescharacteristic
98C23×C4×C71—24no—
99C23×C4×C71—24no—
100C23×C4×C71C2×C418yes—
101C23×C4×C71C2312yes—
102C23×C4×C71C2312yes—
103C23×C4×C71C2311yescharacteristic
104C23×C4×C71C2314yes—
105C23×C4×C71C2311yescharacteristic
106C2×C4⋊C4×C71—24no—
107C2×C4⋊C4×C71—24no—
108C2×C4⋊C4×C71C2314yes—
109C2×C4⋊C4×C71C2314yes—
110C2×C22⋊C4×C71—44no—
111C2×C22⋊C4×C71C2318yes—
112#95×C71C2318yes—
113C23×D71—22no—
114C23×D71—22no—
115C23×D71—22no—
116C23×D71—22no—
117C23×D71—22no—
118C23×D71—22no—
119C23×D71C22×C414yes—
120C23×D71C2412yes—
121C2×C22⋊D71—232no—
122C2×C22⋊D71—28no—
123C2×C22⋊D71—28no—
124C2×C22⋊D71C22×C4116yes—
125C22×Dic71—22no—
126C22×Dic71—22no—
127C22×Dic71—24no—
128C22×Dic71—22no—
129C22×Dic71—22no—
130C22×Dic71—24no—
131C22×Dic71—24no—
132C22×Dic71—24no—
133C22×Dic71—24no—
134C22×Dic71C22×C414yes—
135C22×Dic71C22×C414yes—
136C22×Dic71C22×C414yes—
137C22×Dic71C2×D418yes—
138C22×Dic71#4114yes—
139C22×Dic71#4114yes—
140C22×Dic71C2414yes—
141C22×Dic71C2412yes—
142C22×Dic71C2412yes—
143C22×Dic71C2414yes—
144C22×Dic71C2412yes—
145C2×C4×D71—232no—
146C2×C4×D71—28no—
147C2×C4×D71—28no—
148C2×C4×D71C22×C4116yes—
149C4⋊Dic71—28no—
150C4⋊Dic71C2×D4116yescomplemented
151Dic71⋊C4—216no—
152Dic71⋊C4—48no—
153Dic71⋊C4C22×C4116yes—
154C22⋊Dic71—216no—
155C22⋊Dic71C22×C4116yes—
156(C2×D71)⋊C4—216no—
157(C2×D71)⋊C4—48no—
158(C2×D71)⋊C4C22×C4116yes—
159C4×Dic71—216no—
160C4×Dic71—216no—
161C4×Dic71—216no—
162C4×Dic71C22×C4116yes—
163C24×C71—22no—
164C24×C71—22no—
165C24×C71—22no—
166C24×C71—22no—
167C24×C71—22no—
168C24×C71—22no—
169C24×C71C22×C414yes—
170C24×C71C2412yes—
171C24×C71C2411yescharacteristic
172C22×C4×C71—22no—
173C22×C4×C71—22no—
174C22×C4×C71—232no—
175C22×C4×C71—28no—
176C22×C4×C71—28no—
177C22×C4×C71—24no—
178C22×C4×C71—24no—
179C22×C4×C71—24no—
180C22×C4×C71—24no—
181C22×C4×C71—24no—
182C22×C4×C71—24no—
183C22×C4×C71C22×C414yes—
184C22×C4×C71C22×C4116yes—
185C22×C4×C71C22×C414yes—
186C22×C4×C71C2×D414yes—
187C22×C4×C71#4114yes—
188C22×C4×C71C2414yes—
189C22×C4×C71C2414yes—
190C22×C4×C71C2412yes—
191C22×C4×C71C2412yes—
192C4⋊C4×C71—28no—
193C4⋊C4×C71—28no—
194C22⋊C4×C71—48no—
195C23×C71—28no—
196C23×C71—28no—
197C23×C71—28no—
198C23×C71—22no—
199C23×C71—22no—
200C23×C71—22no—
201C23×C71—22no—
202C23×C71—22no—
203C23×C71—22no—
204C23×C71—44no—
205C23×C71—44no—
206C23×C71—44no—
207C23×C71—44no—
208C23×C71C23×C414yes—
209C23×C71C23×C412yes—
210C23×C71C22×D412yes—
211C23×C71C2×#4112yes—
212C23×C71C2×#4112yes—
213C23×C71C2×#4112yes—
214C23×C71H(5,𝔽2)12yes—
215C23×C712−1+412yes—
216C23×C71C2511yescharacteristic
217C22×D71—28no—
218C22×D71—28no—
219C22×D71—28no—
220C22×D71—22no—
221C22×D71—22no—
222C22×D71—22no—
223C22×D71—22no—
224C22×D71—22no—
225C22×D71—22no—
226C22×D71—44no—
227C22×D71—44no—
228C22×D71—44no—
229C22×D71—44no—
230C22×D71C23×C414yes—
231C2×C4×C71—28no—
232C2×C4×C71—22no—
233C2×C4×C71—22no—
234C2×C4×C71—232no—
235C2×C4×C71—24no—
236C2×C4×C71—24no—
237C2×C4×C71—24no—
238C2×C4×C71—24no—
239C2×C4×C71—24no—
240C2×C4×C71—24no—
241C2×C4×C71—48no—
242C2×C4×C71—48no—
243C2×C4×C71C4×D4116yes—
244C2×C4×C71C23×C414yes—
245C2×C4×C71C23×C414yes—
246C2×C4×C71C22×D414yes—
247C2×C4×C71C2×#4114yes—
248C22⋊D71—232no—
249C2×Dic71—28no—
250C2×Dic71—28no—
251C2×Dic71—216no—
252C2×Dic71—22no—
253C2×Dic71—24no—
254C2×Dic71—22no—
255C2×Dic71—22no—
256C2×Dic71—22no—
257C2×Dic71—24no—
258C2×Dic71—24no—
259C2×Dic71—24no—
260C2×Dic71—24no—
261C2×Dic71—48no—
262C2×Dic71#117116yescomplemented
263C2×Dic71C23×C414yes—
264C2×Dic71C23×C414yes—
265C2×Dic71C23×C414yes—
266C2×Dic71C22×D418yes—
267C2×Dic71C2×#4114yes—
268C2×Dic71C2×#4114yes—
269C4×D71—232no—
270C22×C71—24no—
271C22×C71—24no—
272C22×C71—24no—
273C22×C71—416no—
274C22×C71—42no—
275C22×C71—42no—
276C22×C71—42no—
277C22×C71—42no—
278C22×C71C2×#11714yes—
279C22×C71C2×C4×D414yes—
280C22×C71#52014yes—
281C22×C71#54714yes—
282C22×C71#55014yes—
283C22×C71#55514yes—
284C22×C71#55614yes—
285C22×C71C24×C411yescharacteristic
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—416no—
296C2×D71—42no—
297C2×D71—42no—
298C2×D71—42no—
299C2×D71—42no—
300C4×C71—24no—
301C4×C71—44no—
302C4×C71—44no—
303C4×C71C2×C4×D418yes—
304Dic71—24no—
305Dic71—24no—
306Dic71—28no—
307Dic71—44no—
308Dic71C2×#11718yescomplemented
309C2×#2354—711nomaximal, Sylow
310C2×C71—44no—
311C2×C71#235418yescomplemented
312C2×C71C22×#11711yescharacteristic
313C2×C71C22×C4×D411yescharacteristic
314C2×C71C2×#52011yescharacteristic
315C2×C71C2×#54711yescharacteristic
316C2×C71C2×#55011yescharacteristic
317C2×C71C2×#55511yescharacteristic
318C2×C71C2×#55611yescharacteristic
319D71—44no—
320C22×C4×D4—711no—
321C22×#117—711no—
322C22×#117—711no—
323#2354—7116no—
324C2×#390—712no—
325C2×#388—712no—
326C2×#388—712no—
327C2×#384—711no—
328C2×#380—711no—
329C2×#380D7111yescharacteristic, complemented
330C2×C4×C4⋊C4—712no—
331C2×C4×C4⋊C4—711no—
332C71C2×#235411yescharacteristic, complemented, Sylow
333C23×D4—711no—
334C24×C4—711no—
335C24×C4C2×D7111yescharacteristic
336C2×C4×D4—7116no—
337C2×C4×D4—718no—
338C2×#117—718no—
339C2×#117—718no—
340C2×#117—1428no—
341C2×#117—1428no—
342C22×C4⋊C4—712no—
343C22×C4⋊C4—711no—
344C22×C4⋊C4—711no—
345C22×C4⋊C4—711no—
346C22×C4⋊C4—712no—
347C22×C4⋊C4C2×D7111yescharacteristic
348C22×C4⋊C4C2×D7111yescharacteristic
349C22×C22⋊C4—712no—
350C22×C22⋊C4—711no—
351C22×C22⋊C4—711no—
352C22×C22⋊C4—712no—
353C22×C22⋊C4—712no—
354C22×C22⋊C4C2×D7112yes—
355C22×C42—711no—
356C22×C42—712no—
357C22×C42—712no—
358C22×C42—712no—
359#390—7116no—
360#388—7116no—
361#388—7116no—
362#384—718no—
363#380—718no—
364#380C2×D7118yescomplemented
365C4×C4⋊C4—718no—
366C4×C4⋊C4—7116no—
367C2×#95—712no—
368C2×#95—712no—
369C2×#95C2×D7112yes—
370C25—711no—
371C25C22×D7111yescharacteristic
372C22×D4—718no—
373C22×D4—714no—
374C22×D4—1424no—
375C22×D4—1424no—
376C23×C4—24no—
377C23×C4—24no—
378C23×C4—711no—
379C23×C4—711no—
380C23×C4—712no—
381C23×C4—711no—
382C23×C4—718no—
383C23×C4—714no—
384C23×C4—712no—
385C23×C4—1424no—
386C23×C4—1424no—
387C23×C4C4×D7118yes—
388C23×C4C22×D7112yes—
389C23×C4C22×D7112yes—
390C23×C4C22×D7114yes—
391C23×C4C22×D7111yescharacteristic
392C23×C4C22×D7111yescharacteristic
393C4×D4—7164no—
394#117—14232no—
395#117—14232no—
396C2×C4⋊C4—24no—
397C2×C4⋊C4—24no—
398C2×C4⋊C4—2844no—
399C2×C4⋊C4—718no—
400C2×C4⋊C4—718no—
401C2×C4⋊C4—718no—
402C2×C4⋊C4—714no—
403C2×C4⋊C4—714no—
404C2×C4⋊C4—714no—
405C2×C4⋊C4—718no—
406C2×C4⋊C4—1428no—
407C2×C4⋊C4—1424no—
408C2×C4⋊C4C22×D7114yes—
409C2×C4⋊C4C22×D7114yes—
410C2×C22⋊C4—44no—
411C2×C22⋊C4—2844no—
412C2×C22⋊C4—718no—
413C2×C22⋊C4—718no—
414C2×C22⋊C4—714no—
415C2×C22⋊C4—718no—
416C2×C22⋊C4—714no—
417C2×C22⋊C4—718no—
418C2×C22⋊C4—718no—
419C2×C22⋊C4—1428no—
420C2×C22⋊C4—1428no—
421C2×C22⋊C4C22×D7118yes—
422C2×C42—718no—
423C2×C42—714no—
424C2×C42—718no—
425C2×C42—718no—
426C2×C42—718no—
427C2×C42—1428no—
428C2×C42—1428no—
429C2×C42—1428no—
430#95—718no—
431#95—718no—
432#95C22×D7118yes—
433C24—22no—
434C24—22no—
435C24—22no—
436C24—22no—
437C24—22no—
438C24—22no—
439C24—714no—
440C24—712no—
441C24—1422no—
442C24—1422no—
443C24—1422no—
444C24—1422no—
445C24—1422no—
446C24—1422no—
447C24C2×C4×D7114yes—
448C24C23×D7112yes—
449C24C23×D7111yescharacteristic
450C2×D4—7116no—
451C2×D4—1428no—
452C2×D4—1428no—
453C2×D4—14232no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—28no—
457C22×C4—28no—
458C22×C4—232no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—714no—
466C22×C4—714no—
467C22×C4—714no—
468C22×C4—712no—
469C22×C4—714no—
470C22×C4—714no—
471C22×C4—718no—
472C22×C4—712no—
473C22×C4—712no—
474C22×C4—714no—
475C22×C4—7116no—
476C22×C4—714no—
477C22×C4—1424no—
478C22×C4—1424no—
479C22×C4—1424no—
480C22×C4—1422no—
481C22×C4—1424no—
482C22×C4—1422no—
483C22×C4—1422no—
484C22×C4—1424no—
485C22×C4—1422no—
486C22×C4—1428no—
487C22×C4—1428no—
488C22×C4—14232no—
489C22×C4C2×C4×D7114yes—
490C22×C4C2×C4×D71116yes—
491C22×C4C2×C4×D7114yes—
492C22×C4D4×D7114yes—
493C22×C4#1176586414yes—
494C22×C4C23×D7114yes—
495C22×C4C23×D7114yes—
496C22×C4C23×D7112yes—
497C22×C4C23×D7112yes—
498C4⋊C4—28no—
499C4⋊C4—28no—
500C4⋊C4—2848no—
501C4⋊C4—7116no—
502C4⋊C4—7116no—
503C4⋊C4—14216no—
504C4⋊C4—1428no—
505C22⋊C4—48no—
506C22⋊C4—2848no—
507C22⋊C4—7116no—
508C22⋊C4—7116no—
509C22⋊C4—14216no—
510C22⋊C4—14216no—
511C42—7116no—
512C42—14216no—
513C42—14216no—
514C42—14216no—
515C23—22no—
516C23—22no—
517C23—22no—
518C23—22no—
519C23—22no—
520C23—22no—
521C23—28no—
522C23—28no—
523C23—28no—
524C23—44no—
525C23—44no—
526C23—44no—
527C23—44no—
528C23—2844no—
529C23—2844no—
530C23—2844no—
531C23—2844no—
532C23—714no—
533C23—1422no—
534C23—1422no—
535C23—1422no—
536C23—1422no—
537C23—1422no—
538C23—1422no—
539C23—1428no—
540C23—1428no—
541C23—1428no—
542C23C22×C4×D7114yes—
543C23C22×C4×D7112yes—
544C23C2×D4×D7112yes—
545C23#41×D7112yes—
546C23#41×D7112yes—
547C23C2×#1176586412yes—
548C23#1455617312yes—
549C23#1455617612yes—
550C23C24×D7111yescharacteristic
551D4—14232no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—28no—
555C2×C4—232no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—48no—
563C2×C4—48no—
564C2×C4—2848no—
565C2×C4—714no—
566C2×C4—714no—
567C2×C4—714no—
568C2×C4—7116no—
569C2×C4—714no—
570C2×C4—714no—
571C2×C4—718no—
572C2×C4—1424no—
573C2×C4—1424no—
574C2×C4—1424no—
575C2×C4—1422no—
576C2×C4—1424no—
577C2×C4—1422no—
578C2×C4—1422no—
579C2×C4—1422no—
580C2×C4—1424no—
581C2×C4—14216no—
582C2×C4—1428no—
583C2×C4—1428no—
584C2×C4—14232no—
585C2×C4(C22⋊D71)⋊C4116yes—
586C2×C4C22×C4×D7114yes—
587C2×C4C22×C4×D7114yes—
588C2×C4C2×D4×D7114yes—
589C2×C4C2×#1176586414yes—
590C22—24no—
591C22—24no—
592C22—24no—
593C22—416no—
594C22—42no—
595C22—42no—
596C22—42no—
597C22—42no—
598C22—28416no—
599C22—2842no—
600C22—2842no—
601C22—2842no—
602C22—2842no—
603C22—1424no—
604C22—1424no—
605C22—1424no—
606C22#117×D7114yes—
607C22C2×(C22⋊D71)⋊C414yes—
608C22#1798648014yes—
609C22#1798665114yes—
610C22#1798666714yes—
611C22#1798671314yes—
612C22#1798672014yes—
613C22C23×C4×D7111yescharacteristic
614C22C22×D4×D7111yescharacteristic
615C22C2×#41×D7111yescharacteristic
616C22C2×#41×D7111yescharacteristic
617C22C22×#1176586411yescharacteristic
618C22C2×#1455617311yescharacteristic
619C22C2×#1455617611yescharacteristic
620C4—24no—
621C4—44no—
622C4—44no—
623C4—2844no—
624C4—718no—
625C4—1424no—
626C4—1424no—
627C4—1428no—
628C4C2×(C22⋊D71)⋊C418yes—
629C2—44no—
630C2—2844no—
631C2#2259128418yesdirect product factor
632C2C2×#117×D7111yescharacteristic
633C2C22×(C22⋊D71)⋊C411yescharacteristic
634C2C2×#1798648011yescharacteristic
635C2C2×#1798665111yescharacteristic
636C2C2×#1798666711yescharacteristic
637C2C2×#1798671311yescharacteristic
638C2C2×#1798672011yescharacteristic
subgroupG/H
centerC24C23×D71
commutatorC22×C71C24×C4
FrattiniC23C24×D71
FittingC2×#380×C71C2
socleC24×C71C24
prime radicalC23×C22⋊D71C22
cosocleC23×C71C25
hypercenterC2×#380D71
hypocenterC71C2×#2354
LCSG ▷ C22×C71 ▷ C71
UCSC1 ◁ C24 ◁ C2×#380

Complements

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

NH ≅ G/Ntype
2. C22×#14556087C2semidirect
3. C22×#14556086C2semidirect
4. #22591284C2direct (N × H)
6. C2×#17985893C2semidirect
7. C2×#17985888C2semidirect
8. C2×#17985868C2semidirect
9. C2×#17985856C2semidirect
10. C2×C4⋊C4×Dic71C2semidirect
11. C2×#17985850C2semidirect
12. C2×#380×C71C2semidirect
15. C2×C4×C22⋊D71C4semidirect
18. C2×#14556087C22semidirect
20. C2×#14556086C22semidirect
36. #17985893C22semidirect
37. #17985888C22semidirect
38. #17985868C22semidirect
39. #17985856C22semidirect
40. C4⋊C4×Dic71C22semidirect
41. #17985850C22semidirect
48. #380×C71C22semidirect
64. C4×C22⋊D71C2×C4semidirect
68. C2×C4⋊Dic71D4semidirect
150. C4⋊Dic71C2×D4semidirect
262. C2×Dic71#117semidirect
308. Dic71C2×#117semidirect
311. C2×C71#2354semidirect
329. C2×#380D71semidirect
332. C71C2×#2354semidirect
364. #380C2×D71semidirect
631. C2#22591284direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4×C22⋊D712
C22×#145560872
C22×#145560862
#225912842
C2×#179858992
C2×#179858932
C2×#179858882
C2×#179858682
C2×#179858562
C2×C4⋊C4×Dic712
C2×#179858502
C2×#380×C712
C2×#235471