Finite Group Databasebeta

← C2×#22611101 #33113406

Subgroups — C2×#22611101 #33113406

Order 18688.

1234 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17989140C211yesmaximal, characteristic, complemented
2C22×#14556518C211yesmaximal, characteristic
3C2×#17988561C211yesmaximal, characteristic
4#22611101C218yesmaximal, direct product factor
5C2×#17988358C211yesmaximal, characteristic, complemented
6C2×#17988348C211yesmaximal, characteristic, complemented
7C2×#17988343C211yesmaximal, characteristic, complemented
8C2×#421×C73C211yesmaximal, characteristic
9C22×#11924012C2211yescharacteristic
10#17989140—22no—
11#17989140—22no—
12#17989140—22no—
13#17989140C2214yescomplemented
14C2×#14556590—21no—
15C2×#14556582—21no—
16C2×#14556568—22no—
17C2×C4×Dic146—22no—
18C2×#14556550C2211yescharacteristic
19C2×#14556550C2211yescharacteristic
20C22×C4×Dic73C2211yescharacteristic
21C2×#14556518—21no—
22C2×#14556518—22no—
23C2×#14556518—21no—
24C2×#14556518C2214yes—
25#17988561—22no—
26#17988561C2214yes—
27C2×#14556506—22no—
28#17988358C2214yescomplemented
29#17988348C2214yescomplemented
30#17988343C2214yescomplemented
31C22×#41×C73C2211yescharacteristic
32C22×M16×C73C2211yescharacteristic
33C2×#117×C73C2211yescharacteristic
34#421×C73—22no—
35#421×C73C2214yes—
36C2×#104×C73—22no—
37C22×Dic146—22no—
38C23×Dic73—21no—
39C2×#11924012—21no—
40C2×#11924012—21no—
41C2×#11924012—21no—
42C2×#11924012—21no—
43C2×#11924012—21no—
44C2×#11924012C2312yes—
45C22×#10596831—21no—
46#14556590—24no—
47#14556590—24no—
48#14556582—24no—
49#14556582—24no—
50#14556568—28no—
51#14556568—28no—
52C4×Dic146—28no—
53C4×Dic146—28no—
54#14556550—21no—
55#14556550—21no—
56#14556550—21no—
57#14556550—21no—
58#14556550—21no—
59#14556550D412yes—
60#14556550C2312yes—
61#14556550C2312yes—
62C2×C4⋊Dic73—22no—
63C2×C4⋊Dic73—21no—
64C2×C4⋊Dic73—21no—
65C2×#11924007—22no—
66C2×#11924007—22no—
67C2×C22⋊Dic73—21no—
68C2×C22⋊Dic73D412yes—
69C2×C4×Dic73—21no—
70C2×C4×Dic73—21no—
71C2×C4×Dic73—21no—
72C2×C4×Dic73—21no—
73C2×C4×Dic73—21no—
74C2×C4×Dic73D411yescharacteristic
75C2×C4×Dic73D412yes—
76C2×C4×Dic73D411yescharacteristic
77C2×C4×Dic73C2312yes—
78#14556518—24no—
79#14556518—24no—
80#14556506—28no—
81C2×#41×C73—21no—
82C2×#41×C73—21no—
83C2×#41×C73—21no—
84C2×#41×C73—21no—
85C2×#41×C73—21no—
86C2×#41×C73—21no—
87C2×#41×C73—21no—
88C2×#41×C73—21no—
89C2×#41×C73—21no—
90C2×#41×C73—21no—
91C2×#41×C73Q812yescomplemented
92C2×#41×C73C2312yes—
93C22×Q8×C73Q811yescharacteristic, complemented
94C22×D4×C73—21no—
95C22×D4×C73Q811yescharacteristic, complemented
96C23×C4×C73—21no—
97C23×C4×C73C2311yescharacteristic
98C2×M16×C73—21no—
99C2×M16×C73—21no—
100C2×M16×C73—21no—
101C2×M16×C73—21no—
102C2×M16×C73—21no—
103C2×M16×C73C2312yes—
104C22×C8×C73—21no—
105#117×C73—21no—
106#117×C73—21no—
107#117×C73D412yes—
108#117×C73C2312yes—
109C2×C4⋊C4×C73—21no—
110C2×C22⋊C4×C73—21no—
111C2×C42×C73D412yes—
112#104×C73—28no—
113C2×Dic146—28no—
114C2×Dic146—48no—
115C22×Dic73—21no—
116C22×Dic73—21no—
117C22×Dic73—22no—
118C22×Dic73—22no—
119C22×Dic73—22no—
120C22×Dic73—44no—
121C22×Dic73—41no—
122#11924012—22no—
123#11924012—22no—
124#11924012—41no—
125#11924012—41no—
126C2×#10596831—22no—
127C2×#10596831—41no—
128C2×#10596831—41no—
129C4⋊Dic73—28no—
130C4⋊Dic73—22no—
131C4⋊Dic73—22no—
132C4⋊Dic73—22no—
133C4⋊Dic73—41no—
134#11924007—28no—
135#11924007—28no—
136C22⋊Dic73—22no—
137C22⋊Dic73—22no—
138C22⋊Dic73—41no—
139C22⋊Dic73C2×D414yes—
140C4×Dic73—22no—
141C4×Dic73—21no—
142C4×Dic73—21no—
143C4×Dic73—22no—
144C4×Dic73—22no—
145C4×Dic73—22no—
146C4×Dic73—22no—
147C4×Dic73—41no—
148C4×Dic73C2×D412yes—
149C4×Dic73C2×D412yes—
150C24×C73—21no—
151C24×C73C2×D411yescharacteristic
152#41×C73—22no—
153#41×C73—22no—
154#41×C73—22no—
155#41×C73—22no—
156#41×C73—22no—
157#41×C73—22no—
158#41×C73—22no—
159#41×C73—22no—
160#41×C73—41no—
161#41×C73—41no—
162#41×C73—41no—
163#41×C73—41no—
164#41×C73—41no—
165#41×C73—41no—
166#41×C73—41no—
167#41×C73—41no—
168C2×Q8×C73—21no—
169C2×Q8×C73—21no—
170C2×Q8×C73—21no—
171C2×Q8×C73—21no—
172C2×Q8×C73—21no—
173C2×Q8×C73C2×Q812yescomplemented
174C2×D4×C73—22no—
175C2×D4×C73—22no—
176C2×D4×C73—21no—
177C2×D4×C73—21no—
178C2×D4×C73—21no—
179C2×D4×C73—21no—
180C2×D4×C73—21no—
181C2×D4×C73—41no—
182C2×D4×C73—41no—
183C2×D4×C73—41no—
184C2×D4×C73—41no—
185C2×D4×C73C2×Q812yescomplemented
186C22×C4×C73—22no—
187C22×C4×C73—21no—
188C22×C4×C73—22no—
189C22×C4×C73—22no—
190C22×C4×C73—41no—
191C22×C4×C73—41no—
192C22×C4×C73—41no—
193C22×C4×C73—41no—
194C22×C4×C73C2×D411yescharacteristic
195C22×C4×C73C2×D411yescharacteristic
196C22×C4×C73C2×D411yescharacteristic
197C22×C4×C73C2×D411yescharacteristic
198C22×C4×C73C2×D411yescharacteristic
199C22×C4×C73C2×Q811yescharacteristic
200C22×C4×C73C2×Q811yescharacteristic
201C22×C4×C73#4111yescharacteristic
202C22×C4×C73#4111yescharacteristic
203C22×C4×C73#4111yescharacteristic
204C22×C4×C73#4111yescharacteristic
205C22×C4×C73#4111yescharacteristic
206C22×C4×C73#4111yescharacteristic
207C22×C4×C73C2411yescharacteristic
208M16×C73—22no—
209M16×C73—22no—
210M16×C73—41no—
211M16×C73—41no—
212C2×C8×C73—22no—
213C2×C8×C73—41no—
214C2×C8×C73—41no—
215C4⋊C4×C73—22no—
216C4⋊C4×C73—22no—
217C22⋊C4×C73—22no—
218C22⋊C4×C73—22no—
219C42×C73—22no—
220C42×C73C2×D414yes—
221C23×C73—22no—
222C23×C73—42no—
223C23×C73—41no—
224C23×C73—41no—
225C23×C73—82no—
226C23×C73#12011yescharacteristic
227C23×C73#12011yescharacteristic
228C23×C73C22⋊Q811yescharacteristic
229C23×C73C22⋊Q811yescharacteristic
230C23×C73#12411yescharacteristic
231C23×C73#12411yescharacteristic
232C23×C73C22×D411yescharacteristic
233Q8×C73—22no—
234Q8×C73—22no—
235Q8×C73—41no—
236Q8×C73—41no—
237D4×C73—22no—
238D4×C73—22no—
239D4×C73—42no—
240D4×C73—42no—
241D4×C73—42no—
242D4×C73—42no—
243D4×C73—41no—
244D4×C73—41no—
245C2×C4×C73—24no—
246C2×C4×C73—22no—
247C2×C4×C73—44no—
248C2×C4×C73—41no—
249C2×C4×C73—41no—
250C2×C4×C73—42no—
251C2×C4×C73—42no—
252C2×C4×C73—42no—
253C2×C4×C73—42no—
254C2×C4×C73#12011yescharacteristic
255C2×C4×C73#12011yescharacteristic
256C2×C4×C73C22⋊Q811yescharacteristic
257C2×C4×C73C22⋊Q811yescharacteristic
258C2×C4×C73C22⋊Q811yescharacteristic
259C2×C4×C73C22⋊Q811yescharacteristic
260C2×C4×C73C22⋊Q811yescharacteristic
261C2×C4×C73C22⋊Q811yescharacteristic
262C2×C4×C73C22⋊Q811yescharacteristic
263C2×C4×C73C22⋊Q811yescharacteristic
264C2×C4×C73C22⋊Q811yescharacteristic
265C2×C4×C73C22⋊Q811yescharacteristic
266C2×C4×C73#12411yescharacteristic
267C2×C4×C73#12411yescharacteristic
268C2×C4×C73#12411yescharacteristic
269C2×C4×C73#12411yescharacteristic
270C2×C4×C73#12411yescharacteristic
271C2×C4×C73#12411yescharacteristic
272C2×C4×C73#12411yescharacteristic
273C2×C4×C73#12411yescharacteristic
274C2×C4×C73#12411yescharacteristic
275C2×C4×C73#12411yescharacteristic
276C2×C4×C73C22×D411yescharacteristic
277C2×C4×C73C22×D411yescharacteristic
278C2×C4×C73C22×Q811yescharacteristic
279C2×C4×C73C2×#4111yescharacteristic
280C2×C4×C73C2×#4111yescharacteristic
281C2×C4×C73C2×#4111yescharacteristic
282C2×Dic73—24no—
283C2×Dic73—22no—
284C2×Dic73—22no—
285C2×Dic73—24no—
286C2×Dic73—44no—
287C2×Dic73—44no—
288C2×Dic73—42no—
289C2×Dic73—44no—
290C2×Dic73—42no—
291C2×Dic73—82no—
292Dic146—416no—
293C8×C73—42no—
294#10596831—42no—
295C22×C73—22no—
296C22×C73—22no—
297C22×C73—22no—
298C22×C73—41no—
299C22×C73—41no—
300C22×C73—41no—
301C22×C73—41no—
302C22×C73—42no—
303C22×C73—82no—
304C22×C73—82no—
305C22×C73—82no—
306C22×C73#39311yescharacteristic
307C22×C73#39311yescharacteristic
308C22×C73#39311yescharacteristic
309C22×C73#39311yescharacteristic
310C22×C73C2×#12011yescharacteristic
311C22×C73C2×C22⋊Q811yescharacteristic
312C22×C73C2×#12411yescharacteristic
313C4×C73—44no—
314C4×C73—42no—
315C4×C73#39311yescharacteristic
316C4×C73#39311yescharacteristic
317C4×C73#39311yescharacteristic
318C4×C73#39311yescharacteristic
319C4×C73C2×C22⋊Q811yescharacteristic
320C4×C73C2×C22⋊Q811yescharacteristic
321C4×C73C2×#12411yescharacteristic
322C4×C73C2×#12411yescharacteristic
323Dic73—24no—
324Dic73—44no—
325Dic73—42no—
326Dic73—82no—
327C2×#2031—731nomaximal, Sylow
328C2×C73—21no—
329C2×C73—21no—
330C2×C73—21no—
331C2×C73—21no—
332C2×C73—21no—
333C2×C73—21no—
334C2×C73—82no—
335C2×C73#203112yescomplemented
336C2×C73C2×#39311yescharacteristic
337C2×#533—731no—
338C2×#421D7311yescharacteristic
339C22×#104—731no—
340C2×#410—731no—
341#2031—738no—
342C2×#344—731no—
343C2×#339—731no—
344C2×H(3,ℤ/4)—731no—
345C73C2×#203111yescharacteristic, complemented, Sylow
346C22×#41C2×D7311yescharacteristic
347C22×M16—731no—
348C22×M16C2×D7311yescharacteristic
349#533—734no—
350#533—1462no—
351#533—1462no—
352#533—1462no—
353C2×C4⋊Q8—1461no—
354C2×#125—1461no—
355C2×C22⋊Q8—1462no—
356C2×C4×Q8—1462no—
357C2×#117—731no—
358C2×#117—731no—
359C2×#117C2×D7311yescharacteristic
360C22×C42—731no—
361#421—22no—
362#421C2×D7314yes—
363C2×#104—22no—
364C2×#104—734no—
365C2×#104—1461no—
366C2×#104—1461no—
367C2×#104—1462no—
368#410—734no—
369#410—1462no—
370C2×#99—1462no—
371#344—734no—
372#339—734no—
373H(3,ℤ/4)—734no—
374C2×#41—21no—
375C2×#41—21no—
376C2×#41—21no—
377C2×#41—21no—
378C2×#41—21no—
379C2×#41—21no—
380C2×#41—21no—
381C2×#41—21no—
382C2×#41—21no—
383C2×#41—21no—
384C2×#41Dic14612yescomplemented
385C2×#41C22×D7312yes—
386C22×Q8—1462no—
387C22×Q8Dic14611yescharacteristic, complemented
388C22×D4—21no—
389C22×D4Dic14611yescharacteristic, complemented
390C23×C4—21no—
391C23×C4—1461no—
392C23×C4C22×D7311yescharacteristic
393C2×M16—21no—
394C2×M16—21no—
395C2×M16—21no—
396C2×M16—21no—
397C2×M16—21no—
398C2×M16—732no—
399C2×M16—1461no—
400C2×M16—1461no—
401C2×M16—1461no—
402C2×M16—1461no—
403C2×M16—1461no—
404C2×M16C22×D7312yes—
405C22×C8—21no—
406C22×C8—1461no—
407C4⋊Q8—1464no—
408C4⋊Q8—1464no—
409#125—1464no—
410#125—1464no—
411C22⋊Q8—1468no—
412C22⋊Q8—1468no—
413C4×Q8—1468no—
414C4×Q8—1468no—
415#117—21no—
416#117—21no—
417#117—732no—
418#117—732no—
419#117—732no—
420#117—1461no—
421#117—1461no—
422#117—1461no—
423#117—1461no—
424#117—1461no—
425#117C22⋊D7312yes—
426#117C22×D7312yes—
427C2×C4⋊C4—21no—
428C2×C4⋊C4—1461no—
429C2×C4⋊C4—1462no—
430C2×C4⋊C4—1462no—
431C2×C4⋊C4—1462no—
432C2×C4⋊C4—1461no—
433C2×C22⋊C4—21no—
434C2×C22⋊C4—732no—
435C2×C22⋊C4—1461no—
436C2×C42—731no—
437C2×C42—731no—
438C2×C42—732no—
439C2×C42—732no—
440C2×C42—1461no—
441C2×C42—1461no—
442C2×C42—1461no—
443C2×C42—1461no—
444C2×C42—1461no—
445C2×C42C22⋊D7312yes—
446#104—28no—
447#104—1464no—
448#104—1464no—
449#99—1468no—
450C24—21no—
451C24D4×D7311yescharacteristic
452#41—22no—
453#41—22no—
454#41—22no—
455#41—22no—
456#41—22no—
457#41—22no—
458#41—22no—
459#41—22no—
460#41—41no—
461#41—41no—
462#41—41no—
463#41—41no—
464#41—41no—
465#41—41no—
466#41—41no—
467#41—41no—
468C2×Q8—21no—
469C2×Q8—21no—
470C2×Q8—21no—
471C2×Q8—21no—
472C2×Q8—21no—
473C2×Q8—2928no—
474C2×Q8—1468no—
475C2×Q8C2×Dic14612yescomplemented
476C2×D4—22no—
477C2×D4—22no—
478C2×D4—21no—
479C2×D4—21no—
480C2×D4—21no—
481C2×D4—21no—
482C2×D4—21no—
483C2×D4—41no—
484C2×D4—41no—
485C2×D4—41no—
486C2×D4—41no—
487C2×D4C2×Dic14612yescomplemented
488C22×C4—22no—
489C22×C4—21no—
490C22×C4—22no—
491C22×C4—22no—
492C22×C4—41no—
493C22×C4—41no—
494C22×C4—41no—
495C22×C4—41no—
496C22×C4—2924no—
497C22×C4—2921no—
498C22×C4—1461no—
499C22×C4—1462no—
500C22×C4—1462no—
501C22×C4—1462no—
502C22×C4—1461no—
503C22×C4D4×D7311yescharacteristic
504C22×C4D4×D7311yescharacteristic
505C22×C4D4×D7311yescharacteristic
506C22×C4C2×C22⋊D7311yescharacteristic
507C22×C4C2×C22⋊D7311yescharacteristic
508C22×C4C2×Dic14611yescharacteristic
509C22×C4C2×Dic14611yescharacteristic
510C22×C4#1192402711yescharacteristic
511C22×C4#1192402711yescharacteristic
512C22×C4#1192402711yescharacteristic
513C22×C4#1192402711yescharacteristic
514C22×C4#1192402811yescharacteristic
515C22×C4#1192402811yescharacteristic
516C22×C4C23×D7311yescharacteristic
517M16—22no—
518M16—22no—
519M16—41no—
520M16—41no—
521M16—2921no—
522M16—2921no—
523M16—1462no—
524M16—1462no—
525C2×C8—22no—
526C2×C8—41no—
527C2×C8—41no—
528C2×C8—2921no—
529C2×C8—2921no—
530C2×C8—1462no—
531C4⋊C4—22no—
532C4⋊C4—22no—
533C4⋊C4—2921no—
534C4⋊C4—1468no—
535C4⋊C4—1468no—
536C4⋊C4—1468no—
537C4⋊C4—1462no—
538C4⋊C4—1462no—
539C4⋊C4—1462no—
540C22⋊C4—22no—
541C22⋊C4—22no—
542C22⋊C4—2921no—
543C22⋊C4—734no—
544C22⋊C4—1462no—
545C22⋊C4—1462no—
546C42—22no—
547C42—2921no—
548C42—732no—
549C42—732no—
550C42—1462no—
551C42—1462no—
552C42—1461no—
553C42—1461no—
554C42—1462no—
555C42—1462no—
556C42—1462no—
557C42C2×C22⋊D7314yes—
558C23—22no—
559C23—42no—
560C23—41no—
561C23—41no—
562C23—82no—
563C23#1455655911yescharacteristic
564C23#1455655911yescharacteristic
565C23#1455656811yescharacteristic
566C23#1455656811yescharacteristic
567C23#1455657711yescharacteristic
568C23#1455657711yescharacteristic
569C23C2×D4×D7311yescharacteristic
570Q8—22no—
571Q8—22no—
572Q8—41no—
573Q8—41no—
574Q8—29216no—
575D4—22no—
576D4—22no—
577D4—42no—
578D4—42no—
579D4—42no—
580D4—42no—
581D4—41no—
582D4—41no—
583C2×C4—24no—
584C2×C4—22no—
585C2×C4—44no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—2922no—
593C2×C4—2924no—
594C2×C4—2924no—
595C2×C4—2924no—
596C2×C4—2922no—
597C2×C4—5842no—
598C2×C4—1462no—
599C2×C4—1464no—
600C2×C4—1464no—
601C2×C4—1462no—
602C2×C4#1455655911yescharacteristic
603C2×C4#1455655911yescharacteristic
604C2×C4#1455656811yescharacteristic
605C2×C4#1455656811yescharacteristic
606C2×C4#1455656811yescharacteristic
607C2×C4#1455656811yescharacteristic
608C2×C4#1455656811yescharacteristic
609C2×C4#1455656811yescharacteristic
610C2×C4#1455657011yescharacteristic
611C2×C4#1455657011yescharacteristic
612C2×C4#1455657011yescharacteristic
613C2×C4#1455657011yescharacteristic
614C2×C4#1455657711yescharacteristic
615C2×C4#1455657711yescharacteristic
616C2×C4#1455657711yescharacteristic
617C2×C4#1455657711yescharacteristic
618C2×C4#1455657711yescharacteristic
619C2×C4#1455657711yescharacteristic
620C2×C4#1455657811yescharacteristic
621C2×C4#1455657811yescharacteristic
622C2×C4#1455657811yescharacteristic
623C2×C4#1455657811yescharacteristic
624C2×C4C2×D4×D7311yescharacteristic
625C2×C4C22×C22⋊D7311yescharacteristic
626C2×C4C22×Dic14611yescharacteristic
627C2×C4C2×#1192402711yescharacteristic
628C2×C4C2×#1192402711yescharacteristic
629C2×C4C2×#1192402811yescharacteristic
630C8—42no—
631C8—2922no—
632C22—22no—
633C22—22no—
634C22—22no—
635C22—41no—
636C22—41no—
637C22—41no—
638C22—41no—
639C22—42no—
640C22—82no—
641C22—82no—
642C22—82no—
643C22#1798849911yescharacteristic
644C22#1798849911yescharacteristic
645C22#1798849911yescharacteristic
646C22#1798849911yescharacteristic
647C22C2×#1455655911yescharacteristic
648C22C2×#1455656811yescharacteristic
649C22C2×#1455657711yescharacteristic
650C4—44no—
651C4—42no—
652C4—2922no—
653C4—2924no—
654C4—5842no—
655C4—1464no—
656C4#1798849911yescharacteristic
657C4#1798849911yescharacteristic
658C4#1798849911yescharacteristic
659C4#1798849911yescharacteristic
660C4C2×#1455656811yescharacteristic
661C4C2×#1455657011yescharacteristic
662C4C2×#1455657711yescharacteristic
663C4C2×#1455657811yescharacteristic
664C2—21no—
665C2—21no—
666C2—21no—
667C2—21no—
668C2—21no—
669C2—21no—
670C2—82no—
671C2#2261110112yesdirect product factor
672C2C2×#1798849911yescharacteristic
subgroupG/H
centerC2×C4#14556578
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC2×#421×C73C2
socleC22×C73#393
prime radicalC22×#41×C73C22
hypercenterC2×#421D73
hypocenterC73C2×#2031
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C2×#421

Complements

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

NH ≅ G/Ntype
1. C2×#17989140C2semidirect
4. #22611101C2direct (N × H)
5. C2×#17988358C2semidirect
6. C2×#17988348C2semidirect
7. C2×#17988343C2semidirect
13. #17989140C22semidirect
28. #17988358C22semidirect
29. #17988348C22semidirect
30. #17988343C22semidirect
91. C2×#41×C73Q8semidirect
93. C22×Q8×C73Q8semidirect
95. C22×D4×C73Q8semidirect
173. C2×Q8×C73C2×Q8semidirect
185. C2×D4×C73C2×Q8semidirect
335. C2×C73#2031semidirect
345. C73C2×#2031semidirect
384. C2×#41Dic146semidirect
387. C22×Q8Dic146semidirect
389. C22×D4Dic146semidirect
475. C2×Q8C2×Dic146semidirect
487. C2×D4C2×Dic146semidirect
671. C2#22611101direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#179891402
C22×#145565182
C2×#179885612
#226111012
C2×#179883582
C2×#179883482
C2×#179883432
C2×#421×C732
C2×#203173