Finite Group Databasebeta

← C2×#22589509 #32024873

Subgroups — C2×#22589509 #32024873

Order 18176.

1234 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17986017C211yesmaximal, characteristic
2C22×#14556056C211yesmaximal, characteristic
3C2×#17985972C211yesmaximal, characteristic
4#22589509C218yesmaximal, direct product factor
5C2×#17985769C211yesmaximal, characteristic, complemented
6C2×#17985759C211yesmaximal, characteristic, complemented
7C2×#17985755C211yesmaximal, characteristic, complemented
8C2×#533×C71C211yesmaximal, characteristic, complemented
9C22×#11765865C2211yescharacteristic
10C22×#11765849C2211yescharacteristic
11C22×#11765849C2211yescharacteristic
12C2×#14556087C2211yescharacteristic
13#17986017—22no—
14#17986017C2214yes—
15C2×#14556056—21no—
16C2×#14556056—21no—
17C2×#14556056—22no—
18C2×#14556056—22no—
19C2×#14556056C2214yes—
20#17985972—22no—
21#17985972C2214yes—
22C2×#14556042—22no—
23#17985769C2214yescomplemented
24#17985759C2214yescomplemented
25#17985755C2214yescomplemented
26#533×C71—22no—
27#533×C71—22no—
28#533×C71—22no—
29#533×C71C2214yescomplemented
30C2×C4⋊Q8×C71—21no—
31C2×#125×C71—21no—
32C2×C22⋊Q8×C71—22no—
33C2×C4×Q8×C71—22no—
34C2×#117×C71C2211yescharacteristic
35C2×#117×C71C2211yescharacteristic
36C22×C42×C71C2211yescharacteristic
37C2×#11765865—21no—
38C2×#11765865—21no—
39C2×#11765865—21no—
40C2×#11765865—21no—
41C2×#11765865—21no—
42C2×#11765865—21no—
43C2×#11765865—21no—
44C2×#11765865—21no—
45C2×#11765865—21no—
46C2×#11765865—21no—
47C2×#11765865Q812yescomplemented
48C2×#11765865C2312yes—
49C22×Dic142Q811yescharacteristic, complemented
50C22×D284Q811yescharacteristic, complemented
51C22×C22⋊D71—21no—
52C22×C4×D71—21no—
53C2×#11765849—21no—
54C2×#11765849—21no—
55C2×#11765849—21no—
56C2×#11765849—21no—
57C2×#11765849—21no—
58C2×#11765849—21no—
59C2×#11765849—21no—
60C2×#11765849—21no—
61C2×#11765849—21no—
62C2×#11765849—21no—
63C2×#11765849C2312yes—
64C2×#11765849C2312yes—
65C22×C71⋊C8—21no—
66C22×C71⋊C8—21no—
67#14556087—21no—
68#14556087—21no—
69#14556087—21no—
70#14556087C2312yes—
71C2×C4⋊Dic71—21no—
72C2×C22⋊Dic71—21no—
73C2×C4×Dic71—21no—
74#14556056—24no—
75#14556056—24no—
76#14556056—28no—
77#14556042—28no—
78C22×Q8×C71—22no—
79C23×C4×C71—21no—
80C23×C4×C71C2311yescharacteristic
81C4⋊Q8×C71—24no—
82C4⋊Q8×C71—24no—
83#125×C71—24no—
84#125×C71—24no—
85C22⋊Q8×C71—28no—
86C22⋊Q8×C71—28no—
87C4×Q8×C71—28no—
88C4×Q8×C71—28no—
89#117×C71—21no—
90#117×C71—21no—
91#117×C71—21no—
92#117×C71—21no—
93#117×C71D412yes—
94#117×C71D412yes—
95#117×C71C2312yes—
96#117×C71C2312yes—
97C2×C4⋊C4×C71—24no—
98C2×C4⋊C4×C71—22no—
99C2×C4⋊C4×C71—21no—
100C2×C4⋊C4×C71—21no—
101C2×C22⋊C4×C71—21no—
102C2×C22⋊C4×C71D412yes—
103C2×C42×C71—21no—
104C2×C42×C71—21no—
105C2×C42×C71—21no—
106C2×C42×C71—21no—
107C2×C42×C71D411yescharacteristic
108C2×C42×C71D411yescharacteristic
109C2×C42×C71D412yes—
110C2×C42×C71D412yes—
111C2×C42×C71C2312yes—
112C23×D71—21no—
113#11765865—22no—
114#11765865—22no—
115#11765865—22no—
116#11765865—22no—
117#11765865—22no—
118#11765865—22no—
119#11765865—22no—
120#11765865—22no—
121#11765865—41no—
122#11765865—41no—
123#11765865—41no—
124#11765865—41no—
125#11765865—41no—
126#11765865—41no—
127#11765865—41no—
128#11765865—41no—
129C2×Dic142—21no—
130C2×Dic142—21no—
131C2×Dic142—21no—
132C2×Dic142—21no—
133C2×Dic142—21no—
134C2×Dic142C2×Q812yescomplemented
135C2×D284—21no—
136C2×D284—21no—
137C2×D284—21no—
138C2×D284—21no—
139C2×D284—21no—
140C2×D284C2×Q812yescomplemented
141C2×C22⋊D71—22no—
142C2×C22⋊D71—22no—
143C2×C22⋊D71—41no—
144C2×C22⋊D71—41no—
145C2×C22⋊D71—41no—
146C2×C22⋊D71—41no—
147C22×Dic71—21no—
148C22×Dic71—41no—
149C2×C4×D71—22no—
150C2×C4×D71—22no—
151C2×C4×D71—41no—
152C2×C4×D71—41no—
153C2×C4×D71—41no—
154C2×C4×D71—41no—
155#11765849—22no—
156#11765849—22no—
157#11765849—22no—
158#11765849—22no—
159#11765849—41no—
160#11765849—41no—
161#11765849—41no—
162#11765849—41no—
163C2×C71⋊C8—22no—
164C2×C71⋊C8—22no—
165C2×C71⋊C8—41no—
166C2×C71⋊C8—41no—
167C2×C71⋊C8—41no—
168C2×C71⋊C8—41no—
169C4⋊Dic71—22no—
170C4⋊Dic71—41no—
171C22⋊Dic71—22no—
172C22⋊Dic71—41no—
173C4×Dic71—22no—
174C4×Dic71—41no—
175C24×C71C2×D411yescharacteristic
176C2×Q8×C71—28no—
177C2×Q8×C71—48no—
178C22×C4×C71—21no—
179C22×C4×C71—21no—
180C22×C4×C71—22no—
181C22×C4×C71—22no—
182C22×C4×C71—22no—
183C22×C4×C71—22no—
184C22×C4×C71—44no—
185C22×C4×C71C2×D411yescharacteristic
186C22×C4×C71C2×D411yescharacteristic
187C22×C4×C71C2×D411yescharacteristic
188C22×C4×C71C2×D411yescharacteristic
189C22×C4×C71C2×D411yescharacteristic
190C22×C4×C71C2×Q811yescharacteristic
191C22×C4×C71C2×Q811yescharacteristic
192C22×C4×C71#4111yescharacteristic
193C22×C4×C71#4111yescharacteristic
194C22×C4×C71#4111yescharacteristic
195C22×C4×C71#4111yescharacteristic
196C22×C4×C71#4111yescharacteristic
197C22×C4×C71#4111yescharacteristic
198C22×C4×C71C2411yescharacteristic
199C4⋊C4×C71—216no—
200C4⋊C4×C71—28no—
201C4⋊C4×C71—22no—
202C4⋊C4×C71—22no—
203C4⋊C4×C71—22no—
204C4⋊C4×C71—22no—
205C22⋊C4×C71—22no—
206C22⋊C4×C71—22no—
207C22⋊C4×C71—22no—
208C22⋊C4×C71C2×D414yes—
209C42×C71—22no—
210C42×C71—21no—
211C42×C71—21no—
212C42×C71—22no—
213C42×C71—22no—
214C42×C71—22no—
215C42×C71—22no—
216C42×C71C2×D412yes—
217C42×C71C2×D412yes—
218C42×C71C2×D414yes—
219C23×C71—42no—
220C23×C71#12011yescharacteristic
221C23×C71#12011yescharacteristic
222C23×C71C22⋊Q811yescharacteristic
223C23×C71C22⋊Q811yescharacteristic
224C23×C71#12411yescharacteristic
225C23×C71#12411yescharacteristic
226C23×C71C22×D411yescharacteristic
227C22×D71—22no—
228C22×D71—41no—
229C22×D71—41no—
230C22×D71—82no—
231Q8×C71—416no—
232C2×C4×C71—24no—
233C2×C4×C71—22no—
234C2×C4×C71—22no—
235C2×C4×C71—24no—
236C2×C4×C71—24no—
237C2×C4×C71—44no—
238C2×C4×C71—44no—
239C2×C4×C71—42no—
240C2×C4×C71—44no—
241C2×C4×C71—44no—
242C2×C4×C71#12011yescharacteristic
243C2×C4×C71#12011yescharacteristic
244C2×C4×C71C22⋊Q811yescharacteristic
245C2×C4×C71C22⋊Q811yescharacteristic
246C2×C4×C71C22⋊Q811yescharacteristic
247C2×C4×C71C22⋊Q811yescharacteristic
248C2×C4×C71C22⋊Q811yescharacteristic
249C2×C4×C71C22⋊Q811yescharacteristic
250C2×C4×C71C22⋊Q811yescharacteristic
251C2×C4×C71C22⋊Q811yescharacteristic
252C2×C4×C71C22⋊Q811yescharacteristic
253C2×C4×C71C22⋊Q811yescharacteristic
254C2×C4×C71#12411yescharacteristic
255C2×C4×C71#12411yescharacteristic
256C2×C4×C71#12411yescharacteristic
257C2×C4×C71#12411yescharacteristic
258C2×C4×C71#12411yescharacteristic
259C2×C4×C71#12411yescharacteristic
260C2×C4×C71#12411yescharacteristic
261C2×C4×C71#12411yescharacteristic
262C2×C4×C71#12411yescharacteristic
263C2×C4×C71#12411yescharacteristic
264C2×C4×C71C22×D411yescharacteristic
265C2×C4×C71C22×D411yescharacteristic
266C2×C4×C71C22×Q811yescharacteristic
267C2×C4×C71C2×#4111yescharacteristic
268C2×C4×C71C2×#4111yescharacteristic
269C2×C4×C71C2×#4111yescharacteristic
270C22⋊D71—42no—
271C22⋊D71—42no—
272C22⋊D71—42no—
273C22⋊D71—42no—
274C2×Dic71—22no—
275C2×Dic71—41no—
276C2×Dic71—41no—
277C2×Dic71—42no—
278C2×Dic71—82no—
279D284—22no—
280D284—22no—
281D284—41no—
282D284—41no—
283C4×D71—42no—
284C4×D71—42no—
285C4×D71—42no—
286C4×D71—42no—
287Dic142—22no—
288Dic142—22no—
289Dic142—41no—
290Dic142—41no—
291C71⋊C8—42no—
292C71⋊C8—42no—
293C22×C71—22no—
294C22×C71—22no—
295C22×C71—22no—
296C22×C71—41no—
297C22×C71—41no—
298C22×C71—41no—
299C22×C71—41no—
300C22×C71#39311yescharacteristic
301C22×C71#39311yescharacteristic
302C22×C71#39311yescharacteristic
303C22×C71#39311yescharacteristic
304C22×C71C2×#12011yescharacteristic
305C22×C71C2×C22⋊Q811yescharacteristic
306C22×C71C2×#12411yescharacteristic
307C2×D71—42no—
308C2×D71—82no—
309C2×D71—82no—
310C2×D71—82no—
311C4×C71—24no—
312C4×C71—42no—
313C4×C71—44no—
314C4×C71—44no—
315C4×C71#39311yescharacteristic
316C4×C71#39311yescharacteristic
317C4×C71#39311yescharacteristic
318C4×C71#39311yescharacteristic
319C4×C71C2×C22⋊Q811yescharacteristic
320C4×C71C2×C22⋊Q811yescharacteristic
321C4×C71C2×#12411yescharacteristic
322C4×C71C2×#12411yescharacteristic
323Dic71—42no—
324Dic71—82no—
325C2×#2031—711nomaximal, Sylow
326C2×C71—21no—
327C2×C71—21no—
328C2×C71—21no—
329C2×C71—21no—
330C2×C71—21no—
331C2×C71—21no—
332C2×C71#203112yescomplemented
333C2×C71C2×#39311yescharacteristic
334D71—82no—
335C2×#533D7111yescharacteristic, complemented
336C2×#421—711no—
337C22×#104—711no—
338C2×#410—711no—
339#2031—718no—
340C2×#344—711no—
341C2×#339—711no—
342C2×H(3,ℤ/4)—711no—
343C71C2×#203111yescharacteristic, complemented, Sylow
344C22×#41—711no—
345C22×M16—711no—
346C22×M16—711no—
347#533—22no—
348#533—22no—
349#533—22no—
350#533C2×D7114yescomplemented
351C2×C4⋊Q8—21no—
352C2×#125—21no—
353C2×C22⋊Q8—22no—
354C2×C4×Q8—22no—
355C2×#117—711no—
356C2×#117C2×D7111yescharacteristic
357C2×#117C2×D7111yescharacteristic
358C22×C42C2×D7111yescharacteristic
359#421—714no—
360#421—1422no—
361C2×#104—714no—
362C2×#104—1421no—
363C2×#104—1421no—
364C2×#104—1422no—
365C2×#104—1422no—
366#410—714no—
367#410—1422no—
368C2×#99—1422no—
369#344—714no—
370#339—714no—
371H(3,ℤ/4)—714no—
372C2×#41—712no—
373C2×#41—712no—
374C2×#41—1421no—
375C2×#41—1421no—
376C2×#41—1421no—
377C2×#41—1421no—
378C2×#41—1421no—
379C2×#41—1421no—
380C2×#41—1421no—
381C2×#41—1421no—
382C2×#41—1421no—
383C2×#41—1421no—
384C22×Q8—22no—
385C22×Q8—711no—
386C22×D4—711no—
387C22×D4—1421no—
388C23×C4—21no—
389C23×C4—1421no—
390C23×C4C22×D7111yescharacteristic
391C2×M16—712no—
392C2×M16—712no—
393C2×M16—1421no—
394C2×M16—1421no—
395C2×M16—1421no—
396C2×M16—1421no—
397C2×M16—1421no—
398C2×M16—1421no—
399C2×M16—1421no—
400C2×M16—1421no—
401C2×M16—1421no—
402C2×M16—1421no—
403C22×C8—1421no—
404C22×C8—1421no—
405C4⋊Q8—24no—
406C4⋊Q8—24no—
407#125—24no—
408#125—24no—
409C22⋊Q8—28no—
410C22⋊Q8—28no—
411C4×Q8—28no—
412C4×Q8—28no—
413#117—21no—
414#117—21no—
415#117—21no—
416#117—21no—
417#117—712no—
418#117—1421no—
419#117—1421no—
420#117—1421no—
421#117C22⋊D7112yes—
422#117C22⋊D7112yes—
423#117C22×D7112yes—
424#117C22×D7112yes—
425C2×C4⋊C4—24no—
426C2×C4⋊C4—22no—
427C2×C4⋊C4—21no—
428C2×C4⋊C4—21no—
429C2×C4⋊C4—1421no—
430C2×C22⋊C4—21no—
431C2×C22⋊C4—1421no—
432C2×C22⋊C4C22⋊D7112yes—
433C2×C42—21no—
434C2×C42—21no—
435C2×C42—21no—
436C2×C42—21no—
437C2×C42—1421no—
438C2×C42C22⋊D7112yes—
439C2×C42C22⋊D7111yescharacteristic
440C2×C42C22⋊D7111yescharacteristic
441C2×C42C22⋊D7112yes—
442C2×C42C22×D7112yes—
443#104—1424no—
444#104—1424no—
445#104—1428no—
446#99—1428no—
447C24—1421no—
448C24C2×C22⋊D7111yescharacteristic
449#41—2841no—
450#41—2841no—
451#41—2841no—
452#41—2841no—
453#41—2841no—
454#41—2841no—
455#41—2841no—
456#41—2841no—
457#41—1422no—
458#41—1422no—
459#41—1422no—
460#41—1422no—
461#41—1422no—
462#41—1422no—
463#41—1422no—
464#41—1422no—
465C2×Q8—28no—
466C2×Q8—48no—
467C2×Q8—712no—
468C2×Q8—1421no—
469C2×Q8—1421no—
470C2×Q8—1421no—
471C2×Q8—1421no—
472C2×Q8—1421no—
473C2×D4—2841no—
474C2×D4—2841no—
475C2×D4—2841no—
476C2×D4—2841no—
477C2×D4—712no—
478C2×D4—1422no—
479C2×D4—1422no—
480C2×D4—1421no—
481C2×D4—1421no—
482C2×D4—1421no—
483C2×D4—1421no—
484C2×D4—1421no—
485C22×C4—21no—
486C22×C4—22no—
487C22×C4—21no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—44no—
492C22×C4—2841no—
493C22×C4—2841no—
494C22×C4—2841no—
495C22×C4—2841no—
496C22×C4—2841no—
497C22×C4—1421no—
498C22×C4—1422no—
499C22×C4—1422no—
500C22×C4C2×C22⋊D7111yescharacteristic
501C22×C4C2×C22⋊D7111yescharacteristic
502C22×C4C2×C22⋊D7111yescharacteristic
503C22×C4C2×C22⋊D7111yescharacteristic
504C22×C4C2×C22⋊D7111yescharacteristic
505C22×C4Q8×D7111yescharacteristic
506C22×C4Q8×D7111yescharacteristic
507C22×C4#1176586611yescharacteristic
508C22×C4#1176586611yescharacteristic
509C22×C4#1176586611yescharacteristic
510C22×C4#1176586611yescharacteristic
511C22×C4#1176586611yescharacteristic
512C22×C4#1176586611yescharacteristic
513C22×C4C23×D7111yescharacteristic
514M16—2841no—
515M16—2841no—
516M16—2841no—
517M16—2841no—
518M16—1422no—
519M16—1422no—
520M16—1422no—
521M16—1422no—
522C2×C8—2841no—
523C2×C8—2841no—
524C2×C8—2841no—
525C2×C8—2841no—
526C2×C8—1422no—
527C2×C8—1422no—
528C4⋊C4—216no—
529C4⋊C4—28no—
530C4⋊C4—22no—
531C4⋊C4—22no—
532C4⋊C4—22no—
533C4⋊C4—22no—
534C4⋊C4—2841no—
535C4⋊C4—1422no—
536C22⋊C4—22no—
537C22⋊C4—22no—
538C22⋊C4—22no—
539C22⋊C4—2841no—
540C22⋊C4—1422no—
541C22⋊C4C2×C22⋊D7114yes—
542C42—22no—
543C42—22no—
544C42—22no—
545C42—21no—
546C42—21no—
547C42—22no—
548C42—22no—
549C42—2841no—
550C42—1422no—
551C42C2×C22⋊D7112yes—
552C42C2×C22⋊D7112yes—
553C42C2×C22⋊D7114yes—
554C23—42no—
555C23—2841no—
556C23—2841no—
557C23—5682no—
558C23—1422no—
559C23#1455609711yescharacteristic
560C23#1455609711yescharacteristic
561C23#1455610411yescharacteristic
562C23#1455610411yescharacteristic
563C23#1455611711yescharacteristic
564C23#1455611711yescharacteristic
565C23C22×C22⋊D7111yescharacteristic
566Q8—416no—
567Q8—2841no—
568Q8—2841no—
569Q8—1422no—
570Q8—1422no—
571D4—2842no—
572D4—2842no—
573D4—2842no—
574D4—2842no—
575D4—2841no—
576D4—2841no—
577D4—1422no—
578D4—1422no—
579C2×C4—24no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—2841no—
590C2×C4—2841no—
591C2×C4—2842no—
592C2×C4—2842no—
593C2×C4—2842no—
594C2×C4—2842no—
595C2×C4—2842no—
596C2×C4—5682no—
597C2×C4—1422no—
598C2×C4#1455609711yescharacteristic
599C2×C4#1455609711yescharacteristic
600C2×C4#1455610411yescharacteristic
601C2×C4#1455610411yescharacteristic
602C2×C4#1455610411yescharacteristic
603C2×C4#1455610411yescharacteristic
604C2×C4#1455610411yescharacteristic
605C2×C4#1455610411yescharacteristic
606C2×C4#1455610411yescharacteristic
607C2×C4#1455610411yescharacteristic
608C2×C4#1455610411yescharacteristic
609C2×C4#1455610411yescharacteristic
610C2×C4#1455611711yescharacteristic
611C2×C4#1455611711yescharacteristic
612C2×C4#1455611711yescharacteristic
613C2×C4#1455611711yescharacteristic
614C2×C4#1455611711yescharacteristic
615C2×C4#1455611711yescharacteristic
616C2×C4#1455611711yescharacteristic
617C2×C4#1455611711yescharacteristic
618C2×C4#1455611711yescharacteristic
619C2×C4#1455611711yescharacteristic
620C2×C4C22×C22⋊D7111yescharacteristic
621C2×C4C22×C22⋊D7111yescharacteristic
622C2×C4C2×Q8×D7111yescharacteristic
623C2×C4C2×#1176586611yescharacteristic
624C2×C4C2×#1176586611yescharacteristic
625C2×C4C2×#1176586611yescharacteristic
626C8—2842no—
627C8—2842no—
628C22—22no—
629C22—22no—
630C22—22no—
631C22—41no—
632C22—41no—
633C22—41no—
634C22—41no—
635C22—2842no—
636C22—5682no—
637C22—5682no—
638C22—5682no—
639C22#1798591011yescharacteristic
640C22#1798591011yescharacteristic
641C22#1798591011yescharacteristic
642C22#1798591011yescharacteristic
643C22C2×#1455609711yescharacteristic
644C22C2×#1455610411yescharacteristic
645C22C2×#1455611711yescharacteristic
646C4—24no—
647C4—42no—
648C4—44no—
649C4—44no—
650C4—2842no—
651C4—5682no—
652C4#1798591011yescharacteristic
653C4#1798591011yescharacteristic
654C4#1798591011yescharacteristic
655C4#1798591011yescharacteristic
656C4C2×#1455610411yescharacteristic
657C4C2×#1455610411yescharacteristic
658C4C2×#1455611711yescharacteristic
659C4C2×#1455611711yescharacteristic
660C2—21no—
661C2—21no—
662C2—21no—
663C2—21no—
664C2—21no—
665C2—21no—
666C2—5682no—
667C2#2258950912yesdirect product factor
668C2C2×#1798591011yescharacteristic
subgroupG/H
centerC2×C4#14556117
commutator, cosocleC22×C4×C71C24
FrattiniC22×C4C23×D71
FittingC2×#533×C71C2
socleC22×C71#393
prime radicalC22×#11765865C22
hypercenterC2×#533D71
hypocenterC71C2×#2031
LCSG ▷ C22×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C2×#533

Complements

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

NH ≅ G/Ntype
4. #22589509C2direct (N × H)
5. C2×#17985769C2semidirect
6. C2×#17985759C2semidirect
7. C2×#17985755C2semidirect
8. C2×#533×C71C2semidirect
23. #17985769C22semidirect
24. #17985759C22semidirect
25. #17985755C22semidirect
29. #533×C71C22semidirect
47. C2×#11765865Q8semidirect
49. C22×Dic142Q8semidirect
50. C22×D284Q8semidirect
134. C2×Dic142C2×Q8semidirect
140. C2×D284C2×Q8semidirect
332. C2×C71#2031semidirect
335. C2×#533D71semidirect
343. C71C2×#2031semidirect
350. #533C2×D71semidirect
667. C2#22589509direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#179860172
C22×#145560562
C2×#179859722
#225895092
C2×#179857692
C2×#179857592
C2×#179857552
C2×#533×C712
C2×#203171