Finite Group Databasebeta

← C2×#20273217 #27351230

Subgroups — C2×#20273217 #27351230

Order 13568.

1834 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811134C211yesmaximal, characteristic
2C2×#16810587C211yesmaximal, characteristic, complemented
3#20273217C218yesmaximal, direct product factor
4C2×#16810359C212yesmaximal, complemented
5C2×#16810356C212yesmaximal, complemented
6C22×#95×C53C211yesmaximal, characteristic, complemented
7C22×#11719945C411yescharacteristic, complemented
8C22×#11719917—21no—
9#16811134—22no—
10#16811134—22no—
11#16811134C414yescomplemented
12#16811134C2214yes—
13C2×#13133200—21no—
14C2×#13133196C411yescharacteristic, complemented
15C2×#13133196C2211yescharacteristic
16C2×#13133191C411yescharacteristic, complemented
17C2×#13133191C2211yescharacteristic
18C2×#13133186—21no—
19C2×C4×C22⋊D53—21no—
20C2×C4×C22⋊D53—21no—
21C2×#13133176C411yescharacteristic, complemented
22#16810587—22no—
23#16810587C2214yescomplemented
24C2×C22⋊#88322—21no—
25C2×C22⋊#88322C2212yes—
26#16810359C2218yescomplemented
27#16810356C2218yescomplemented
28C24×C4×C53—21no—
29C24×C4×C53C2211yescharacteristic
30C2×#95×C53—22no—
31C2×#95×C53—21no—
32C2×#95×C53C2212yes—
33C2×#95×C53C2214yescomplemented
34C2×#11719945—22no—
35C2×#11719945—44no—
36C2×#11719945C2×C414yescomplemented
37C22×Dic106C2×C411yescharacteristic, complemented
38C22×D212C2×C411yescharacteristic, complemented
39C22×C22⋊D53—21no—
40C22×C22⋊D53—21no—
41C22×C22⋊D53—21no—
42C22×C4×D53—21no—
43C2×#11719917—24no—
44C2×#11719917—44no—
45C22×#88322—22no—
46#13133200—24no—
47#13133200—24no—
48#13133196—22no—
49#13133196—21no—
50#13133196—21no—
51#13133196C2×C414yescomplemented
52#13133196C2×C412yescomplemented
53#13133196C2312yes—
54#13133191—22no—
55#13133191—21no—
56#13133191—21no—
57#13133191C2×C414yescomplemented
58#13133191C2×C412yescomplemented
59#13133191C2312yes—
60#13133186—24no—
61#13133186—24no—
62C4×C22⋊D53—24no—
63C4×C22⋊D53—24no—
64C4×C22⋊D53—24no—
65C4×C22⋊D53—24no—
66#13133176—22no—
67#13133176C2×C414yescomplemented
68C2×C4⋊Dic53C2×C411yescharacteristic, complemented
69C2×C4⋊Dic53C2×C411yescharacteristic, complemented
70C2×#11719919—21no—
71C2×#11719919—21no—
72C2×C22⋊Dic53—21no—
73C2×C22⋊Dic53—21no—
74C2×C22⋊Dic53—21no—
75C2×#11719921—21no—
76C2×#11719921—21no—
77C2×C4×Dic53—21no—
78C22⋊#88322—24no—
79C22⋊#88322—22no—
80C22⋊#88322C2314yes—
81C25×C53D411yescharacteristic
82C23×C4×C53—28no—
83C23×C4×C53—22no—
84C23×C4×C53—24no—
85C23×C4×C53—44no—
86C23×C4×C53—44no—
87C23×C4×C53—44no—
88C23×C4×C53C2×C411yescharacteristic
89C23×C4×C53C2×C411yescharacteristic
90C23×C4×C53C2×C412yes—
91C23×C4×C53D411yescharacteristic
92C23×C4×C53D411yescharacteristic
93C23×C4×C53D412yes—
94C23×C4×C53D411yescharacteristic
95C23×C4×C53D412yes—
96C23×C4×C53C2311yescharacteristic
97C23×C4×C53C2312yes—
98#95×C53—22no—
99#95×C53—24no—
100#95×C53C2314yes—
101C24×C53—24no—
102C24×C53—24no—
103C24×C53—22no—
104C24×C53—22no—
105C24×C53C22⋊C412yes—
106C24×C53C22⋊C411yescharacteristic
107C24×C53C22⋊C411yescharacteristic
108C24×C53C2×D412yes—
109C24×C53C2×D411yescharacteristic
110C23×D53—21no—
111C22×C4×C53—24no—
112C22×C4×C53—24no—
113C22×C4×C53—28no—
114C22×C4×C53—216no—
115C22×C4×C53—24no—
116C22×C4×C53—42no—
117C22×C4×C53—44no—
118C22×C4×C53—42no—
119C22×C4×C53—42no—
120C22×C4×C53—44no—
121C22×C4×C53—44no—
122C22×C4×C53—42no—
123C22×C4×C53—44no—
124C22×C4×C53—432no—
125C22×C4×C53—44no—
126C22×C4×C53—48no—
127C22×C4×C53—48no—
128C22×C4×C53—44no—
129C22×C4×C53C22⋊C411yescharacteristic
130C22×C4×C53C22⋊C411yescharacteristic
131C22×C4×C53C22⋊C412yes—
132C22×C4×C53C22⋊C412yes—
133C22×C4×C53C22⋊C412yes—
134C22×C4×C53C22⋊C412yes—
135C22×C4×C53C22⋊C412yes—
136C22×C4×C53C22×C411yescharacteristic
137C22×C4×C53C22×C412yes—
138C22×C4×C53C22×C411yescharacteristic
139C22×C4×C53C22×C412yes—
140C22×C4×C53C2×D411yescharacteristic
141C22×C4×C53C2×D412yes—
142C22×C4×C53C2×D412yes—
143C22×C4×C53C2×D411yescharacteristic
144C22×C4×C53C2×D412yes—
145C22×C4×C53C2×D411yescharacteristic
146C22×C4×C53C2411yescharacteristic
147C2×C22⋊D53—22no—
148C2×C22⋊D53—24no—
149C2×C22⋊D53—24no—
150C2×C22⋊D53—22no—
151C2×C22⋊D53—44no—
152C2×C22⋊D53—44no—
153C2×C22⋊D53—42no—
154C2×C22⋊D53—42no—
155C22×Dic53—21no—
156C22×Dic53—21no—
157C22×Dic53—21no—
158#11719945—48no—
159#11719945—48no—
160C2×D212—21no—
161C2×D212—42no—
162C2×D212C22×C412yescomplemented
163C2×C4×D53—24no—
164C2×C4×D53—44no—
165C2×Dic106—21no—
166C2×Dic106—42no—
167C2×Dic106C22×C412yescomplemented
168C22⋊Dic53—24no—
169C22⋊Dic53—22no—
170C22⋊Dic53—24no—
171C22⋊Dic53—22no—
172#11719921—24no—
173#11719921—22no—
174#11719921—22no—
175C4⋊Dic53—21no—
176C4⋊Dic53—21no—
177C4⋊Dic53C22×C412yescomplemented
178C4⋊Dic53C22×C412yescomplemented
179#11719919—24no—
180#11719919—22no—
181#11719919—22no—
182C4×Dic53—24no—
183#11719917—48no—
184C2×#88322—24no—
185C2×#88322—44no—
186C23×C53—216no—
187C23×C53—216no—
188C23×C53—22no—
189C23×C53—22no—
190C23×C53—24no—
191C23×C53—22no—
192C23×C53—24no—
193C23×C53—22no—
194C23×C53—24no—
195C23×C53—22no—
196C23×C53—22no—
197C23×C53—24no—
198C23×C53—22no—
199C23×C53—24no—
200C23×C53—22no—
201C23×C53#9914yes—
202C23×C53#10414yes—
203C23×C53C2×C22⋊C412yes—
204C23×C53C2×C22⋊C411yescharacteristic
205C23×C53C2×C22⋊C412yes—
206C23×C53C2×C22⋊C411yescharacteristic
207C23×C53AGL(1,ℤ/8)12yes—
208C23×C53#13712yes—
209C23×C53C22×D411yescharacteristic
210C22×D53—22no—
211C22×D53—42no—
212C22×D53—42no—
213C22×D53—42no—
214C2×C4×C53—22no—
215C2×C4×C53—22no—
216C2×C4×C53—24no—
217C2×C4×C53—24no—
218C2×C4×C53—24no—
219C2×C4×C53—22no—
220C2×C4×C53—22no—
221C2×C4×C53—24no—
222C2×C4×C53—28no—
223C2×C4×C53—42no—
224C2×C4×C53—42no—
225C2×C4×C53—42no—
226C2×C4×C53—41no—
227C2×C4×C53—42no—
228C2×C4×C53—41no—
229C2×C4×C53—42no—
230C2×C4×C53—41no—
231C2×C4×C53—42no—
232C2×C4×C53—41no—
233C2×C4×C53—432no—
234C2×C4×C53—44no—
235C2×C4×C53—416no—
236C2×C4×C53—44no—
237C2×C4×C53C2×C22⋊C411yescharacteristic
238C2×C4×C53C2×C22⋊C412yes—
239C2×C4×C53C2×C22⋊C412yes—
240C2×C4×C53C2×C22⋊C411yescharacteristic
241C2×C4×C53C23×C411yescharacteristic
242C2×C4×C53C22×D411yescharacteristic
243C22⋊D53—42no—
244C22⋊D53—42no—
245C22⋊D53—48no—
246C22⋊D53—44no—
247C22⋊D53—48no—
248C2×Dic53—22no—
249C2×Dic53—22no—
250C2×Dic53—22no—
251C2×Dic53—42no—
252C2×Dic53—42no—
253C2×Dic53—42no—
254D212—42no—
255D212—42no—
256C4×D53—48no—
257Dic106—42no—
258Dic106—42no—
259#88322—44no—
260C2×#1504—531nomaximal, Sylow
261C22×C53—28no—
262C22×C53—28no—
263C22×C53—216no—
264C22×C53—21no—
265C22×C53—21no—
266C22×C53—22no—
267C22×C53—22no—
268C22×C53—22no—
269C22×C53—21no—
270C22×C53—22no—
271C22×C53—21no—
272C22×C53—24no—
273C22×C53—28no—
274C22×C53—28no—
275C22×C53—24no—
276C22×C53C2×#9914yes—
277C22×C53C2×#9912yes—
278C22×C53#41712yes—
279C22×C53#41712yes—
280C22×C53C2×#10414yes—
281C22×C53C2×#10412yes—
282C22×C53C22×C22⋊C411yescharacteristic
283C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
284C22×C53C2×#13711yescharacteristic
285C2×D53—42no—
286C2×D53—42no—
287C2×D53—42no—
288C2×D53—84no—
289C4×C53—21no—
290C4×C53—21no—
291C4×C53—21no—
292C4×C53—22no—
293C4×C53—21no—
294C4×C53—22no—
295C4×C53—48no—
296Dic53—42no—
297Dic53—42no—
298Dic53—42no—
299C2×#525—531no—
300C2×#407—531no—
301C22×#95D5311yescharacteristic, complemented
302#1504—538no—
303C2×#328—532no—
304C2×#327—532no—
305C2×C53—21no—
306C2×C53—21no—
307C2×C53—24no—
308C2×C53—22no—
309C2×C53—24no—
310C2×C53#150414yescomplemented
311C2×C53C22×#9911yescharacteristic
312C2×C53C2×#41711yescharacteristic
313C2×C53C22×#10411yescharacteristic
314D53—82no—
315C22×#41—531no—
316C24×C4—21no—
317C24×C4C2×D5311yescharacteristic
318C22×M16—1061no—
319#525—534no—
320#525—534no—
321#525—1062no—
322#525—1062no—
323C2×#123—1061no—
324C2×C22⋊Q8—531no—
325C2×C22⋊Q8—531no—
326C2×#121—531no—
327C2×#121—531no—
328C2×#120—1061no—
329C2×C4×D4—1061no—
330C2×C4×D4—1061no—
331C2×#117—531no—
332#407—534no—
333#407—1062no—
334C2×C22⋊C8—532no—
335C2×C22⋊C8—1061no—
336C2×#95—22no—
337C2×#95—21no—
338C2×#95C2×D5312yes—
339C2×#95C2×D5314yescomplemented
340#328—538no—
341#327—538no—
342C53C2×#150411yescharacteristic, complemented, Sylow
343C25C22⋊D5311yescharacteristic
344C2×#41—534no—
345C2×#41—1062no—
346C2×#41—2124no—
347C22×Q8—531no—
348C22×D4—531no—
349C22×D4—1061no—
350C22×D4—1061no—
351C22×D4—1061no—
352C23×C4—28no—
353C23×C4—22no—
354C23×C4—24no—
355C23×C4—44no—
356C23×C4—44no—
357C23×C4—44no—
358C23×C4—1061no—
359C23×C4C4×D5311yescharacteristic
360C23×C4C4×D5311yescharacteristic
361C23×C4C4×D5312yes—
362C23×C4D21211yescharacteristic
363C23×C4D21211yescharacteristic
364C23×C4D21212yes—
365C23×C4C22⋊D5311yescharacteristic
366C23×C4C22⋊D5312yes—
367C23×C4C22×D5311yescharacteristic
368C23×C4C22×D5312yes—
369C2×M16—1064no—
370C2×M16—2124no—
371C22×C8—1062no—
372#123—1064no—
373#123—1064no—
374C22⋊Q8—534no—
375C22⋊Q8—532no—
376C22⋊Q8—532no—
377C22⋊Q8—1061no—
378C22⋊Q8—1061no—
379C22⋊Q8—1062no—
380#121—534no—
381#121—532no—
382#121—532no—
383#121—1061no—
384#121—1061no—
385#121—1062no—
386#120—1064no—
387#120—1064no—
388C4×D4—1064no—
389C4×D4—1064no—
390C4×D4—1064no—
391C4×D4—1064no—
392#117—534no—
393#117—1062no—
394C2×C4⋊C4—531no—
395C2×C4⋊C4—531no—
396C2×C4⋊C4—1061no—
397C2×C4⋊C4—1061no—
398C2×C22⋊C4—1061no—
399C2×C22⋊C4—1061no—
400C2×C22⋊C4—1061no—
401C2×C22⋊C4—1061no—
402C2×C22⋊C4—1061no—
403C2×C42—1061no—
404C22⋊C8—534no—
405C22⋊C8—1062no—
406C22⋊C8—1064no—
407#95—22no—
408#95—24no—
409#95C22×D5314yes—
410C24—24no—
411C24—24no—
412C24—22no—
413C24—22no—
414C24—1061no—
415C24#1171992112yes—
416C24#1171992111yescharacteristic
417C24#1171992111yescharacteristic
418C24C2×C22⋊D5312yes—
419C24C2×C22⋊D5311yescharacteristic
420#41—2128no—
421#41—2128no—
422C2×Q8—532no—
423C2×Q8—1061no—
424C2×Q8—2122no—
425C2×D4—532no—
426C2×D4—1062no—
427C2×D4—1064no—
428C2×D4—1064no—
429C2×D4—1062no—
430C2×D4—1061no—
431C2×D4—2122no—
432C2×D4—2124no—
433C2×D4—2124no—
434C2×D4—2122no—
435C2×D4—2122no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—28no—
439C22×C4—216no—
440C22×C4—24no—
441C22×C4—42no—
442C22×C4—44no—
443C22×C4—42no—
444C22×C4—44no—
445C22×C4—42no—
446C22×C4—44no—
447C22×C4—42no—
448C22×C4—44no—
449C22×C4—432no—
450C22×C4—44no—
451C22×C4—48no—
452C22×C4—48no—
453C22×C4—44no—
454C22×C4—1061no—
455C22×C4—1061no—
456C22×C4—1061no—
457C22×C4—1064no—
458C22×C4—2124no—
459C22×C4#1171992112yes—
460C22×C4#1171992112yes—
461C22×C4#1171992111yescharacteristic
462C22×C4#1171992111yescharacteristic
463C22×C4#1171992112yes—
464C22×C4#1171992112yes—
465C22×C4#1171992112yes—
466C22×C4C2×C4×D5312yes—
467C22×C4C2×C4×D5311yescharacteristic
468C22×C4C2×C4×D5311yescharacteristic
469C22×C4C2×C4×D5312yes—
470C22×C4C2×D21212yes—
471C22×C4C2×D21211yescharacteristic
472C22×C4C2×D21212yes—
473C22×C4C2×D21211yescharacteristic
474C22×C4C2×C22⋊D5312yes—
475C22×C4C2×C22⋊D5311yescharacteristic
476C22×C4C23×D5311yescharacteristic
477M16—2128no—
478C2×C8—1064no—
479C2×C8—2124no—
480C4⋊C4—532no—
481C4⋊C4—532no—
482C4⋊C4—1061no—
483C4⋊C4—1061no—
484C4⋊C4—1062no—
485C4⋊C4—1064no—
486C4⋊C4—1062no—
487C22⋊C4—1064no—
488C22⋊C4—1062no—
489C22⋊C4—1064no—
490C22⋊C4—1062no—
491C22⋊C4—1062no—
492C22⋊C4—1064no—
493C22⋊C4—1062no—
494C42—1064no—
495C23—216no—
496C23—216no—
497C23—24no—
498C23—22no—
499C23—24no—
500C23—22no—
501C23—24no—
502C23—22no—
503C23—22no—
504C23—24no—
505C23—22no—
506C23—24no—
507C23—22no—
508C23—22no—
509C23—22no—
510C23—1062no—
511C23—2122no—
512C23—2122no—
513C23—2122no—
514C23#1313313114yes—
515C23#1313314514yes—
516C23C2×#1171992112yes—
517C23C2×#1171992111yescharacteristic
518C23C2×#1171992112yes—
519C23C2×#1171992111yescharacteristic
520C23#1313324312yes—
521C23#1313324812yes—
522C23C22×C22⋊D5311yescharacteristic
523Q8—2122no—
524Q8—2122no—
525D4—2122no—
526D4—2122no—
527D4—2128no—
528D4—2124no—
529D4—2128no—
530D4—2122no—
531D4—2122no—
532C2×C4—22no—
533C2×C4—24no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—22no—
539C2×C4—24no—
540C2×C4—28no—
541C2×C4—41no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—41no—
545C2×C4—42no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—432no—
552C2×C4—44no—
553C2×C4—416no—
554C2×C4—44no—
555C2×C4—1062no—
556C2×C4—1062no—
557C2×C4—1062no—
558C2×C4—2128no—
559C2×C4—2122no—
560C2×C4—2122no—
561C2×C4—2122no—
562C2×C4C2×#1171992112yes—
563C2×C4C2×#1171992112yes—
564C2×C4C2×#1171992111yescharacteristic
565C2×C4C2×#1171992111yescharacteristic
566C2×C4C22×C4×D5311yescharacteristic
567C2×C4C22×D21211yescharacteristic
568C8—2124no—
569C22—28no—
570C22—216no—
571C22—28no—
572C22—21no—
573C22—21no—
574C22—22no—
575C22—24no—
576C22—28no—
577C22—28no—
578C22—24no—
579C22—22no—
580C22—22no—
581C22—21no—
582C22—22no—
583C22—21no—
584C22—4244no—
585C22—2122no—
586C22—2122no—
587C22—2122no—
588C22C2×#1313313114yes—
589C22C2×#1313313112yes—
590C22#1681062212yes—
591C22#1681062212yes—
592C22C2×#1313314514yes—
593C22C2×#1313314512yes—
594C22C22×#1171992111yescharacteristic
595C22C2×#1313324311yescharacteristic
596C22C2×#1313324811yescharacteristic
597C4—21no—
598C4—22no—
599C4—21no—
600C4—21no—
601C4—22no—
602C4—21no—
603C4—48no—
604C4—2122no—
605C4—2122no—
606C4—2122no—
607C2—21no—
608C2—24no—
609C2—22no—
610C2—24no—
611C2—21no—
612C2—4242no—
613C2#2027321714yesdirect product factor
614C2C22×#1313313111yescharacteristic
615C2C2×#1681062211yescharacteristic
616C2C22×#1313314511yescharacteristic
subgroupG/H
centerC23C2×#11719921
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC22×#95×C53C2
socleC23×C53C2×C22⋊C4
prime radicalC2×#16811134C2
cosocleC22×C4×C53C24
hypercenterC22×#95D53
hypocenterC53C2×#1504
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C24×C4 ◁ C22×#95

Complements

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

NH ≅ G/Ntype
2. C2×#16810587C2semidirect
3. #20273217C2direct (N × H)
4. C2×#16810359C2semidirect
5. C2×#16810356C2semidirect
6. C22×#95×C53C2semidirect
7. C22×#11719945C4semidirect
11. #16811134C4semidirect
14. C2×#13133196C4semidirect
16. C2×#13133191C4semidirect
21. C2×#13133176C4semidirect
23. #16810587C22semidirect
26. #16810359C22semidirect
27. #16810356C22semidirect
33. C2×#95×C53C22semidirect
36. C2×#11719945C2×C4semidirect
37. C22×Dic106C2×C4semidirect
38. C22×D212C2×C4semidirect
51. #13133196C2×C4semidirect
52. #13133196C2×C4semidirect
57. #13133191C2×C4semidirect
58. #13133191C2×C4semidirect
67. #13133176C2×C4semidirect
68. C2×C4⋊Dic53C2×C4semidirect
69. C2×C4⋊Dic53C2×C4semidirect
162. C2×D212C22×C4semidirect
167. C2×Dic106C22×C4semidirect
177. C4⋊Dic53C22×C4semidirect
178. C4⋊Dic53C22×C4semidirect
301. C22×#95D53semidirect
310. C2×C53#1504semidirect
339. C2×#95C2×D53semidirect
342. C53C2×#1504semidirect
613. C2#20273217direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168111342
C2×#168105872
#202732172
C2×#168103592
C2×#168103562
C22×#95×C532
C2×#150453