Finite Group Databasebeta

← C2×#22598290 #32865260

Subgroups — C2×#22598290 #32865260

Order 18176.

2326 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14556108C211yesmaximal, characteristic
2C2×#117×D71C211yesmaximal, characteristic
3#22598290C2116yesmaximal, direct product factor
4C2×#17986336C212yesmaximal, complemented
5C2×#17986343C212yesmaximal, complemented
6C2×#17986342C212yesmaximal, complemented
7C2×#17986331C212yesmaximal, complemented
8C2×#17986039C211yesmaximal, characteristic, complemented
9C2×#17986009C211yesmaximal, characteristic, complemented
10C2×#17986007C211yesmaximal, characteristic, complemented
11C2×#17985962C211yesmaximal, characteristic
12C2×#428×C71C211yesmaximal, characteristic, complemented
13C23×Dic142C2211yescharacteristic
14C23×C4×D71C2211yescharacteristic
15C22×#11765849C2211yescharacteristic
16C2×#14556108—21no—
17C2×#14556108—24no—
18C2×#14556108—21no—
19C2×#14556108C2212yes—
20C2×#14556108C2212yes—
21C2×#14556108C2218yes—
22C2×#14556087C2211yescharacteristic
23C2×#14556086C2211yescharacteristic
24C2×#14556086C2211yescharacteristic
25C2×#14556085—21no—
26C2×#14556084—21no—
27#117×D71—24no—
28#117×D71C2218yes—
29C22×C4⋊Dic71—21no—
30C2×C4⋊C4×D71C2211yescharacteristic
31C2×C4⋊C4×D71C2211yescharacteristic
32C22×(C2×D71)⋊C4—21no—
33C2×C22⋊C4×D71—21no—
34C2×C42×D71—21no—
35#17986343C22116yescomplemented
36#17986342C22116yescomplemented
37#17986336C22116yescomplemented
38#17986331C22116yescomplemented
39#17986039C2218yescomplemented
40C2×#14556061C2212yescomplemented
41C2×#14556059C2212yescomplemented
42#17986009C2218yescomplemented
43#17986007C2218yescomplemented
44C2×#14556053C2212yescomplemented
45C2×#14556053C2212yescomplemented
46C2×#14556051C2212yescomplemented
47C2×#14556051C2212yescomplemented
48#17985962C2218yes—
49C2×(C2×D71)⋊C8C2212yes—
50C2×(C2×D71)⋊C8C2212yes—
51C22×M16×C71C2211yescharacteristic
52C2×#117×C71C2211yescharacteristic
53C22×C4⋊C4×C71C2211yescharacteristic
54#428×C71C2218yescomplemented
55C2×#107×C71C2212yescomplemented
56C2×#106×C71C2212yescomplemented
57C24×D71D411yescharacteristic
58C22×Dic142—21no—
59C22×Dic142—22no—
60C22×Dic142—24no—
61C22×Dic142—24no—
62C22×Dic142C2312yes—
63C22×Dic142C2314yes—
64C23×Dic71—21no—
65C23×Dic71D411yescharacteristic
66C22×C4×D71—21no—
67C22×C4×D71—21no—
68C22×C4×D71—24no—
69C22×C4×D71—24no—
70C22×C4×D71D411yescharacteristic
71C22×C4×D71D414yes—
72C22×C4×D71D411yescharacteristic
73C22×C4×D71C2311yescharacteristic
74C22×C4×D71C2314yes—
75C22×C4×D71C2311yescharacteristic
76C2×#11765849—24no—
77C2×#11765849C2314yes—
78C22×C71⋊C8C2312yes—
79#14556108—28no—
80#14556108—24no—
81#14556108—28no—
82#14556108—24no—
83#14556108C2318yes—
84#14556108C2318yes—
85#14556087—22no—
86#14556087C2314yes—
87#14556086—22no—
88#14556086—22no—
89#14556086C2314yes—
90#14556086C2314yes—
91#14556085—28no—
92#14556084—28no—
93C2×C4⋊Dic71—22no—
94C2×C4⋊Dic71—22no—
95C2×C4⋊Dic71—28no—
96C2×C4⋊Dic71C2311yescharacteristic
97C2×C4⋊Dic71C2311yescharacteristic
98C2×Dic71⋊C4—21no—
99C2×Dic71⋊C4—21no—
100C4⋊C4×D71—22no—
101C4⋊C4×D71—22no—
102C4⋊C4×D71C2314yes—
103C4⋊C4×D71C2314yes—
104C2×C22⋊Dic71—21no—
105C2×(C2×D71)⋊C4—21no—
106C2×(C2×D71)⋊C4—21no—
107C2×(C2×D71)⋊C4—22no—
108C2×(C2×D71)⋊C4—28no—
109C2×(C2×D71)⋊C4—22no—
110C22⋊C4×D71—28no—
111C2×C4×Dic71—21no—
112C2×C4×Dic71—21no—
113C2×C4×Dic71—21no—
114C42×D71—28no—
115#14556061C2318yescomplemented
116#14556059C2318yescomplemented
117#14556053C2318yescomplemented
118#14556053C2318yescomplemented
119#14556051C2318yescomplemented
120#14556051C2318yescomplemented
121(C2×D71)⋊C8C2318yes—
122(C2×D71)⋊C8C2318yes—
123C23×C4×C71—21no—
124C23×C4×C71C2311yescharacteristic
125C2×M16×C71—24no—
126C2×M16×C71C2314yes—
127C22×C8×C71C2312yes—
128#117×C71—22no—
129#117×C71C2314yes—
130C2×C4⋊C4×C71—21no—
131C2×C4⋊C4×C71—22no—
132C2×C4⋊C4×C71C2311yescharacteristic
133C2×C4⋊C4×C71C2312yes—
134C2×C4⋊C4×C71C2311yescharacteristic
135C2×C4⋊C4×C71C2314yes—
136C2×C22⋊C4×C71—21no—
137C2×C42×C71—21no—
138#107×C71C2318yescomplemented
139#106×C71C2318yescomplemented
140C23×D71—24no—
141C23×D71—24no—
142C23×D71—42no—
143C23×D71C2×D411yescharacteristic
144C23×D71C2×D414yes—
145C23×D71C2×D411yescharacteristic
146C2×Dic142—24no—
147C2×Dic142—28no—
148C2×Dic142—24no—
149C2×Dic142—28no—
150C2×Dic142—24no—
151C2×Dic142—22no—
152C2×Dic142—48no—
153C2×Dic142—44no—
154C2×Dic142C2414yes—
155C22×Dic71—24no—
156C22×Dic71—21no—
157C22×Dic71—21no—
158C22×Dic71—24no—
159C22×Dic71—22no—
160C22×Dic71—44no—
161C22×Dic71C2×D411yescharacteristic
162C22×Dic71C2×D411yescharacteristic
163C22×Dic71C2×D414yes—
164C2×C4×D71—24no—
165C2×C4×D71—216no—
166C2×C4×D71—24no—
167C2×C4×D71—22no—
168C2×C4×D71—22no—
169C2×C4×D71—28no—
170C2×C4×D71—22no—
171C2×C4×D71—22no—
172C2×C4×D71—28no—
173C2×C4×D71—24no—
174C2×C4×D71—24no—
175C2×C4×D71—44no—
176C2×C4×D71—44no—
177C2×C4×D71C2×D414yes—
178C2×C4×D71C2×D412yes—
179C2×C4×D71C2×D414yes—
180C2×C4×D71C2×D412yes—
181C2×C4×D71C2412yes—
182C2×C4×D71C2412yes—
183#11765849—28no—
184C2×C71⋊C8—24no—
185C2×C71⋊C8C2414yes—
186C4⋊Dic71—21no—
187C4⋊Dic71—21no—
188C4⋊Dic71—28no—
189C4⋊Dic71—28no—
190C4⋊Dic71C2412yes—
191C4⋊Dic71C2412yes—
192Dic71⋊C4—24no—
193Dic71⋊C4—24no—
194C22⋊Dic71—24no—
195(C2×D71)⋊C4—24no—
196(C2×D71)⋊C4—24no—
197(C2×D71)⋊C4—28no—
198(C2×D71)⋊C4—28no—
199C4×Dic71—24no—
200C4×Dic71—24no—
201C4×Dic71—24no—
202C24×C71C2×D411yescharacteristic
203C22×C4×C71—21no—
204C22×C4×C71—21no—
205C22×C4×C71—24no—
206C22×C4×C71—22no—
207C22×C4×C71—44no—
208C22×C4×C71C2×D411yescharacteristic
209C22×C4×C71C2×D412yes—
210C22×C4×C71#4114yes—
211C22×C4×C71#4114yes—
212C22×C4×C71C2411yescharacteristic
213C22×C4×C71C2412yes—
214M16×C71—28no—
215C2×C8×C71—24no—
216C2×C8×C71C2414yes—
217C4⋊C4×C71—24no—
218C4⋊C4×C71—22no—
219C4⋊C4×C71—21no—
220C4⋊C4×C71—21no—
221C4⋊C4×C71C2412yes—
222C4⋊C4×C71C2414yes—
223C4⋊C4×C71C2412yes—
224C22⋊C4×C71—24no—
225C42×C71—24no—
226C23×C71—22no—
227C23×C71—22no—
228C23×C71#12314yes—
229C23×C71C22×D412yes—
230C23×C71C22×D411yescharacteristic
231C22×D71—22no—
232C22×D71—22no—
233C22×D71—28no—
234C22×D71—22no—
235C22×D71—22no—
236C22×D71—22no—
237C22×D71—28no—
238C22×D71—44no—
239C22×D71—416no—
240C22×D71#13714yes—
241C22×D71C22×D412yes—
242C22×D71C22×D412yes—
243C2×C4×C71—22no—
244C2×C4×C71—22no—
245C2×C4×C71—24no—
246C2×C4×C71—48no—
247C2×C4×C71—42no—
248C2×C4×C71—44no—
249C2×C4×C71—42no—
250C2×C4×C71#12318yes—
251C2×C4×C71#12314yes—
252C2×C4×C71C22×D412yes—
253C2×C4×C71C22×D411yescharacteristic
254C2×C4×C71C2×#4112yes—
255C2×C4×C71C2×#4114yes—
256C2×C4×C71C2×#4112yes—
257C2×C4×C71C2×#4114yes—
258C2×C4×C71C2511yescharacteristic
259C2×Dic71—28no—
260C2×Dic71—22no—
261C2×Dic71—22no—
262C2×Dic71—22no—
263C2×Dic71—22no—
264C2×Dic71—24no—
265C2×Dic71—48no—
266C2×Dic71—44no—
267C2×Dic71—42no—
268C2×Dic71—42no—
269C2×Dic71C22×D412yes—
270C2×Dic71C22×D412yes—
271C4×D71—28no—
272C4×D71—24no—
273C4×D71—28no—
274C4×D71—24no—
275C4×D71—22no—
276C4×D71—22no—
277C4×D71—22no—
278C4×D71—22no—
279C4×D71—48no—
280C4×D71—48no—
281Dic142—24no—
282Dic142—24no—
283Dic142—48no—
284Dic142—44no—
285C8×C71—24no—
286C71⋊C8—24no—
287C22×C71—28no—
288C22×C71—22no—
289C22×C71—22no—
290C22×C71C2×#12314yes—
291C22×C71C2×#12312yes—
292C22×C71C2×#13712yes—
293C22×C71#57512yes—
294C22×C71C23×D411yescharacteristic
295C2×D71—28no—
296C2×D71—22no—
297C2×D71—22no—
298C2×D71—22no—
299C2×D71—22no—
300C2×D71—22no—
301C2×D71—416no—
302C2×D71—44no—
303C2×D71C2×#13714yes—
304C4×C71—42no—
305C4×C71—42no—
306C4×C71—44no—
307C4×C71C2×#12314yes—
308C4×C71C2×#12312yes—
309C4×C71C22×#4111yescharacteristic
310C4×C71C22×#4111yescharacteristic
311Dic71—22no—
312Dic71—22no—
313Dic71—44no—
314Dic71—42no—
315Dic71—42no—
316C2×#3095—711nomaximal, Sylow
317C2×C71—24no—
318C2×C71#309514yescomplemented
319C2×C71C22×#12311yescharacteristic
320C2×C71C22×#13711yescharacteristic
321C2×C71C2×#57511yescharacteristic
322D71—24no—
323D71—42no—
324C22×C22⋊Q8—711no—
325C22×#117—711no—
326#3095—7116no—
327C2×#484—712no—
328C2×#485—712no—
329C2×#485—712no—
330C2×#483—712no—
331C2×#428—711no—
332C2×#428D7111yescharacteristic, complemented
333C2×#419—711no—
334C2×#419—711no—
335C2×#407—711no—
336C71C2×#309511yescharacteristic, complemented, Sylow
337C23×Q8—711no—
338C24×C4—711no—
339C22×M16—711no—
340C22×M16C2×D7111yescharacteristic
341C2×C22⋊Q8—712no—
342C2×C22⋊Q8—712no—
343C2×C22⋊Q8—718no—
344C2×C22⋊Q8—1424no—
345C2×C22⋊Q8—1421no—
346C2×C22⋊Q8—1421no—
347C2×#117—711no—
348C2×#117—718no—
349C2×#117—711no—
350C2×#117—711no—
351C2×#117—1421no—
352C2×#117—1421no—
353C2×#117—1424no—
354C2×#117C2×D7111yescharacteristic
355C22×C4⋊C4—711no—
356C22×C4⋊C4—711no—
357C22×C4⋊C4—1421no—
358C22×C4⋊C4C2×D7111yescharacteristic
359C22×C22⋊C4—1421no—
360C22×C22⋊C4—1421no—
361C22×C42—1421no—
362#485—7116no—
363#485—7116no—
364#484—7116no—
365#483—7116no—
366#428—718no—
367#428C2×D7118yescomplemented
368C2×#107—712no—
369C2×#107C2×D7112yescomplemented
370C2×#106—712no—
371C2×#106C2×D7112yescomplemented
372#419—718no—
373#419—718no—
374C2×#103—712no—
375C2×#103—712no—
376C2×#103—712no—
377C2×#103—712no—
378#407—718no—
379C2×C22⋊C8—712no—
380C2×C22⋊C8—712no—
381C25—711no—
382C22×Q8—712no—
383C22×Q8—714no—
384C22×Q8—1424no—
385C22×Q8—1424no—
386C22×Q8—1422no—
387C22×Q8—1421no—
388C23×C4—21no—
389C23×C4—711no—
390C23×C4—714no—
391C23×C4—711no—
392C23×C4—711no—
393C23×C4—711no—
394C23×C4—711no—
395C23×C4—714no—
396C23×C4—1421no—
397C23×C4—1421no—
398C23×C4—1421no—
399C23×C4—1424no—
400C23×C4—1424no—
401C23×C4C22×D7111yescharacteristic
402C2×M16—24no—
403C2×M16—714no—
404C2×M16—1424no—
405C2×M16C22×D7114yes—
406C22×C8—712no—
407C22×C8C22×D7112yes—
408C22⋊Q8—718no—
409C22⋊Q8—718no—
410C22⋊Q8—1428no—
411C22⋊Q8—1424no—
412C22⋊Q8—1424no—
413C22⋊Q8—1428no—
414#117—22no—
415#117—714no—
416#117—714no—
417#117—714no—
418#117—1428no—
419#117—1428no—
420#117—1422no—
421#117—1422no—
422#117—1422no—
423#117C22×D7114yes—
424C2×C4⋊C4—22no—
425C2×C4⋊C4—21no—
426C2×C4⋊C4—711no—
427C2×C4⋊C4—711no—
428C2×C4⋊C4—714no—
429C2×C4⋊C4—714no—
430C2×C4⋊C4—1422no—
431C2×C4⋊C4—1422no—
432C2×C4⋊C4—1421no—
433C2×C4⋊C4—1421no—
434C2×C4⋊C4—1428no—
435C2×C4⋊C4—1422no—
436C2×C4⋊C4—1422no—
437C2×C4⋊C4C22×D7111yescharacteristic
438C2×C4⋊C4C22×D7111yescharacteristic
439C2×C4⋊C4C22×D7112yes—
440C2×C4⋊C4C22×D7114yes—
441C2×C22⋊C4—21no—
442C2×C22⋊C4—1428no—
443C2×C22⋊C4—1421no—
444C2×C22⋊C4—1421no—
445C2×C22⋊C4—1428no—
446C2×C22⋊C4—1422no—
447C2×C22⋊C4—1422no—
448C2×C22⋊C4—1421no—
449C2×C42—21no—
450C2×C42—1428no—
451C2×C42—1421no—
452C2×C42—1421no—
453C2×C42—1421no—
454#107—718no—
455#107C22×D7118yescomplemented
456#106—718no—
457#106C22×D7118yescomplemented
458#103—718no—
459#103—718no—
460#103—718no—
461#103—718no—
462C22⋊C8—718no—
463C22⋊C8—718no—
464C24—2842no—
465C24—714no—
466C24—711no—
467C24—711no—
468C24—1424no—
469C24—1424no—
470C24D4×D7111yescharacteristic
471C2×Q8—2848no—
472C2×Q8—2844no—
473C2×Q8—714no—
474C2×Q8—1424no—
475C2×Q8—1428no—
476C2×Q8—1424no—
477C2×Q8—1428no—
478C2×Q8—1424no—
479C2×Q8—1422no—
480C22×C4—24no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—22no—
484C22×C4—44no—
485C22×C4—2844no—
486C22×C4—2844no—
487C22×C4—2844no—
488C22×C4—714no—
489C22×C4—711no—
490C22×C4—711no—
491C22×C4—714no—
492C22×C4—712no—
493C22×C4—712no—
494C22×C4—714no—
495C22×C4—712no—
496C22×C4—712no—
497C22×C4—1424no—
498C22×C4—1424no—
499C22×C4—1424no—
500C22×C4—1421no—
501C22×C4—1421no—
502C22×C4—1422no—
503C22×C4—1424no—
504C22×C4—1422no—
505C22×C4—1422no—
506C22×C4—1428no—
507C22×C4—1422no—
508C22×C4—1422no—
509C22×C4—1428no—
510C22×C4—14216no—
511C22×C4—1424no—
512C22×C4—1424no—
513C22×C4D4×D7112yes—
514C22×C4D4×D7111yescharacteristic
515C22×C4#1176586514yes—
516C22×C4#1176586614yes—
517C22×C4C23×D7112yes—
518C22×C4C23×D7111yescharacteristic
519M16—28no—
520M16—1428no—
521C2×C8—24no—
522C2×C8—714no—
523C2×C8—1424no—
524C2×C8C23×D7114yes—
525C4⋊C4—24no—
526C4⋊C4—22no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—712no—
530C4⋊C4—712no—
531C4⋊C4—1424no—
532C4⋊C4—1424no—
533C4⋊C4—1428no—
534C4⋊C4—1428no—
535C4⋊C4—1421no—
536C4⋊C4—1421no—
537C4⋊C4C23×D7112yes—
538C4⋊C4C23×D7112yes—
539C4⋊C4C23×D7114yes—
540C22⋊C4—24no—
541C22⋊C4—1424no—
542C22⋊C4—1424no—
543C22⋊C4—1428no—
544C22⋊C4—1428no—
545C22⋊C4—1424no—
546C42—24no—
547C42—1424no—
548C42—1424no—
549C42—1424no—
550C23—22no—
551C23—22no—
552C23—28416no—
553C23—2844no—
554C23—712no—
555C23—714no—
556C23—712no—
557C23—1428no—
558C23—1428no—
559C23—1422no—
560C23—1422no—
561C23—1422no—
562C23—1422no—
563C23—1422no—
564C23#1455611314yes—
565C23C2×D4×D7112yes—
566C23C2×D4×D7111yescharacteristic
567Q8—2848no—
568Q8—2844no—
569Q8—1424no—
570Q8—1424no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—24no—
574C2×C4—48no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—2848no—
579C2×C4—2848no—
580C2×C4—2848no—
581C2×C4—2844no—
582C2×C4—2842no—
583C2×C4—2842no—
584C2×C4—712no—
585C2×C4—712no—
586C2×C4—1422no—
587C2×C4—1422no—
588C2×C4—1424no—
589C2×C4—1428no—
590C2×C4—1422no—
591C2×C4—1422no—
592C2×C4—1422no—
593C2×C4—1422no—
594C2×C4—1422no—
595C2×C4—1422no—
596C2×C4—1424no—
597C2×C4—1428no—
598C2×C4—1428no—
599C2×C4—1424no—
600C2×C4#1455611318yes—
601C2×C4#1455611314yes—
602C2×C4C2×D4×D7112yes—
603C2×C4C2×D4×D7111yescharacteristic
604C2×C4C2×#1176586512yes—
605C2×C4C2×#1176586514yes—
606C2×C4C2×#1176586612yes—
607C2×C4C2×#1176586614yes—
608C2×C4C24×D7111yescharacteristic
609C8—24no—
610C8—1424no—
611C22—22no—
612C22—22no—
613C22—28no—
614C22—28416no—
615C22—2844no—
616C22—714no—
617C22—1422no—
618C22—1422no—
619C22—1422no—
620C22—1422no—
621C22—1422no—
622C22—1428no—
623C22C2×#1455611314yes—
624C22C2×#1455611312yes—
625C22#137×D7112yes—
626C22#1798679512yes—
627C22C22×D4×D7111yescharacteristic
628C4—42no—
629C4—44no—
630C4—42no—
631C4—2842no—
632C4—2842no—
633C4—2844no—
634C4—1422no—
635C4—1422no—
636C4C2×#1455611314yes—
637C4C2×#1455611312yes—
638C4C22×#1176586511yescharacteristic
639C4C22×#1176586611yescharacteristic
640C2—24no—
641C2—2842no—
642C2—1424no—
643C2#2259829014yesdirect product factor
644C2C22×#1455611311yescharacteristic
645C2C2×#137×D7111yescharacteristic
646C2C2×#1798679511yescharacteristic
subgroupG/H
centerC23C2×D4×D71
commutator, cosocleC2×C4×C71C25
FrattiniC2×C4C24×D71
FittingC2×#428×C71C2
socleC23×C71C22×D4
prime radicalC24×D71D4
hypercenterC2×#428D71
hypocenterC71C2×#3095
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#428

Complements

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

NH ≅ G/Ntype
3. #22598290C2direct (N × H)
4. C2×#17986336C2semidirect
5. C2×#17986343C2semidirect
6. C2×#17986342C2semidirect
7. C2×#17986331C2semidirect
8. C2×#17986039C2semidirect
9. C2×#17986009C2semidirect
10. C2×#17986007C2semidirect
12. C2×#428×C71C2semidirect
35. #17986343C22semidirect
36. #17986342C22semidirect
37. #17986336C22semidirect
38. #17986331C22semidirect
39. #17986039C22semidirect
40. C2×#14556061C22semidirect
41. C2×#14556059C22semidirect
42. #17986009C22semidirect
43. #17986007C22semidirect
44. C2×#14556053C22semidirect
45. C2×#14556053C22semidirect
46. C2×#14556051C22semidirect
47. C2×#14556051C22semidirect
54. #428×C71C22semidirect
55. C2×#107×C71C22semidirect
56. C2×#106×C71C22semidirect
115. #14556061C23semidirect
116. #14556059C23semidirect
117. #14556053C23semidirect
118. #14556053C23semidirect
119. #14556051C23semidirect
120. #14556051C23semidirect
138. #107×C71C23semidirect
139. #106×C71C23semidirect
318. C2×C71#3095semidirect
332. C2×#428D71semidirect
336. C71C2×#3095semidirect
367. #428C2×D71semidirect
369. C2×#107C2×D71semidirect
371. C2×#106C2×D71semidirect
455. #107C22×D71semidirect
457. #106C22×D71semidirect
643. C2#22598290direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145561082
C2×#117×D712
#225982902
C2×#179863362
C2×#179863432
C2×#179863422
C2×#179863312
C2×#179860392
C2×#179860092
C2×#179860072
C2×#179859622
C2×#428×C712
C2×#309571