Finite Group Databasebeta

← C2×#20280113 #27554342

Subgroups — C2×#20280113 #27554342

Order 13568.

1966 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20280113C2116yesmaximal, direct product factor
2C2×#16810561C211yesmaximal, characteristic
3C2×#16810556C211yesmaximal, characteristic
4C2×#16810555C211yesmaximal, characteristic
5C2×#16810522C211yesmaximal, characteristic
6C2×#16810522C211yesmaximal, characteristic
7C2×#16810522C211yesmaximal, characteristic
8C2×#16810510C211yesmaximal, characteristic
9C2×#16810508C211yesmaximal, characteristic
10C2×#16810508C211yesmaximal, characteristic
11C2×#16810506C211yesmaximal, characteristic
12C2×#16810504C211yesmaximal, characteristic
13C2×#16810504C211yesmaximal, characteristic
14C2×C4⋊C4×Dic53C211yesmaximal, characteristic
15C2×#16810474C211yesmaximal, characteristic
16C2×C53×#398C211yesmaximal, characteristic
17C22×C4⋊Dic53C2211yescharacteristic
18C22×C4⋊Dic53C2211yescharacteristic
19C22×C4⋊Dic53C2211yescharacteristic
20C22×C4⋊Dic53C2211yescharacteristic
21C22×C4⋊Dic53C2211yescharacteristic
22C22×C4⋊Dic53C2211yescharacteristic
23C22×#11719919C2211yescharacteristic
24C22×#11719919C2211yescharacteristic
25C22×#11719919C2211yescharacteristic
26C22×#11719919C2211yescharacteristic
27C22×C4×Dic53C2211yescharacteristic
28C22×C4×Dic53C2211yescharacteristic
29C22×C4×Dic53C2211yescharacteristic
30C22×C4×Dic53C2211yescharacteristic
31C22×C4×Dic53C2211yescharacteristic
32#16810561C2218yes—
33#16810556C2218yes—
34#16810555C2218yes—
35#16810522C2218yes—
36#16810522C2218yes—
37#16810522C2218yes—
38#16810510C2218yes—
39#16810508C2218yes—
40#16810508C2218yes—
41#16810506C2218yes—
42#16810504C2218yes—
43#16810504C2218yes—
44C4⋊C4×Dic53C2218yes—
45#16810474C2218yes—
46C2×#13133124C2211yescharacteristic
47C2×#13133124C2211yescharacteristic
48C2×#13133124C2211yescharacteristic
49C2×#13133124C2211yescharacteristic
50C2×#13133124C2211yescharacteristic
51C2×#13133124C2211yescharacteristic
52C2×#13133124C2211yescharacteristic
53C2×#13133124C2211yescharacteristic
54C2×#13133124C2211yescharacteristic
55C2×#13133124C2211yescharacteristic
56C2×#13133124C2211yescharacteristic
57C2×#13133124C2211yescharacteristic
58C2×#13133124C2211yescharacteristic
59C22×C4⋊C4×C53C2211yescharacteristic
60C22×C4⋊C4×C53C2211yescharacteristic
61C22×C4⋊C4×C53C2211yescharacteristic
62C22×C4⋊C4×C53C2211yescharacteristic
63C53×#398C2218yes—
64C2×#95×C53C2211yescharacteristic
65C2×#95×C53C2211yescharacteristic
66C2×#95×C53C2211yescharacteristic
67C23×Dic53C2311yescharacteristic
68C23×Dic53C2311yescharacteristic
69C23×Dic53C2311yescharacteristic
70C23×Dic53C2311yescharacteristic
71C23×Dic53C2311yescharacteristic
72C23×Dic53C2311yescharacteristic
73C23×Dic53C2311yescharacteristic
74C23×Dic53C2311yescharacteristic
75C2×C4⋊Dic53—24no—
76C2×C4⋊Dic53—24no—
77C2×C4⋊Dic53—24no—
78C2×C4⋊Dic53—24no—
79C2×C4⋊Dic53—42no—
80C2×C4⋊Dic53—42no—
81C2×C4⋊Dic53C2314yes—
82C2×C4⋊Dic53C2314yes—
83C2×C4⋊Dic53C2314yes—
84C2×C4⋊Dic53C2314yes—
85C2×C4⋊Dic53C2314yes—
86C2×C4⋊Dic53C2314yes—
87C2×#11719919—24no—
88C2×#11719919—24no—
89C2×#11719919—42no—
90C2×#11719919Q818yes—
91C2×#11719919C2314yes—
92C2×#11719919C2314yes—
93C2×#11719919C2314yes—
94C2×#11719919C2314yes—
95C2×C4×Dic53—24no—
96C2×C4×Dic53—24no—
97C2×C4×Dic53—42no—
98C2×C4×Dic53—42no—
99C2×C4×Dic53—42no—
100C2×C4×Dic53C2314yes—
101C2×C4×Dic53C2314yes—
102C2×C4×Dic53C2314yes—
103C2×C4×Dic53C2314yes—
104C2×C4×Dic53C2314yes—
105#13133124C2314yes—
106#13133124C2314yes—
107#13133124C2314yes—
108#13133124C2314yes—
109#13133124C2314yes—
110#13133124C2314yes—
111#13133124C2314yes—
112#13133124C2314yes—
113#13133124C2314yes—
114#13133124C2314yes—
115#13133124C2314yes—
116#13133124C2314yes—
117#13133124C2314yes—
118C23×C4×C53C2311yescharacteristic
119C23×C4×C53C2311yescharacteristic
120C23×C4×C53C2311yescharacteristic
121C23×C4×C53C2311yescharacteristic
122C23×C4×C53C2311yescharacteristic
123C23×C4×C53C2311yescharacteristic
124C23×C4×C53C2311yescharacteristic
125C2×C4⋊C4×C53—24no—
126C2×C4⋊C4×C53—24no—
127C2×C4⋊C4×C53—24no—
128C2×C4⋊C4×C53—24no—
129C2×C4⋊C4×C53C2314yes—
130C2×C4⋊C4×C53C2314yes—
131C2×C4⋊C4×C53C2314yes—
132C2×C4⋊C4×C53C2314yes—
133#95×C53C2314yes—
134#95×C53C2314yes—
135#95×C53C2314yes—
136C24×C53C2411yescharacteristic
137C22×C4×C53—22no—
138C22×C4×C53—22no—
139C22×C4×C53—22no—
140C22×C4×C53—22no—
141C22×C4×C53—22no—
142C22×C4×C53—22no—
143C22×C4×C53—22no—
144C22×C4×C53—22no—
145C22×C4×C53—22no—
146C22×C4×C53—22no—
147C22×C4×C53—22no—
148C22×C4×C53—22no—
149C22×C4×C53—22no—
150C22×C4×C53—22no—
151C22×C4×C53—22no—
152C22×C4×C53—22no—
153C22×C4×C53—22no—
154C22×C4×C53C2×Q814yes—
155C22×C4×C53#4114yes—
156C22×C4×C53#4114yes—
157C22×C4×C53#4114yes—
158C22×C4×C53C2412yes—
159C22×C4×C53C2412yes—
160C22×C4×C53C2412yes—
161C22×C4×C53C2412yes—
162C22×C4×C53C2412yes—
163C22×C4×C53C2412yes—
164C22×C4×C53C2412yes—
165C22×Dic53—22no—
166C22×Dic53—22no—
167C22×Dic53—22no—
168C22×Dic53—22no—
169C22×Dic53—22no—
170C22×Dic53—22no—
171C22×Dic53—22no—
172C22×Dic53—22no—
173C22×Dic53—22no—
174C22×Dic53—22no—
175C22×Dic53—22no—
176C22×Dic53—22no—
177C22×Dic53—22no—
178C22×Dic53—22no—
179C22×Dic53—22no—
180C22×Dic53—22no—
181C22×Dic53—22no—
182C22×Dic53—22no—
183C22×Dic53—22no—
184C22×Dic53—22no—
185C22×Dic53—22no—
186C22×Dic53C2×Q814yes—
187C22×Dic53C2×Q814yes—
188C22×Dic53#4114yes—
189C22×Dic53C2412yes—
190C22×Dic53C2412yes—
191C22×Dic53C2412yes—
192C22×Dic53C2412yes—
193C22×Dic53C2412yes—
194C22×Dic53C2412yes—
195C22×Dic53C2412yes—
196C22×Dic53C2412yes—
197C4⋊C4×C53—28no—
198C4⋊C4×C53—28no—
199C4⋊C4×C53—28no—
200C4⋊C4×C53—28no—
201C4⋊Dic53—28no—
202C4⋊Dic53—28no—
203C4⋊Dic53—28no—
204C4⋊Dic53—28no—
205C4⋊Dic53—44no—
206C4⋊Dic53—44no—
207#11719919—28no—
208#11719919—28no—
209#11719919—44no—
210#11719919C2×Q8116yes—
211C4×Dic53—28no—
212C4×Dic53—28no—
213C4×Dic53—44no—
214C4×Dic53—44no—
215C4×Dic53—44no—
216C23×C53C22×Q812yes—
217C23×C53C2×#4112yes—
218C23×C53C2×#4112yes—
219C23×C53C2×#4112yes—
220C23×C53C2×#4112yes—
221C23×C53H(5,𝔽2)12yes—
222C23×C532−1+412yes—
223C23×C53C2511yescharacteristic
224C2×C4×C53—28no—
225C2×C4×C53—28no—
226C2×C4×C53—28no—
227C2×C4×C53—28no—
228C2×C4×C53—22no—
229C2×C4×C53—22no—
230C2×C4×C53—22no—
231C2×C4×C53—22no—
232C2×C4×C53—22no—
233C2×C4×C53—22no—
234C2×C4×C53—22no—
235C2×C4×C53—22no—
236C2×C4×C53—22no—
237C2×C4×C53—22no—
238C2×C4×C53—22no—
239C2×C4×C53—22no—
240C2×C4×C53—22no—
241C2×C4×C53—22no—
242C2×C4×C53—22no—
243C2×C4×C53—22no—
244C2×C4×C53—22no—
245C2×C4×C53—44no—
246C2×C4×C53—44no—
247C2×C4×C53—44no—
248C2×C4×C53C22×Q814yes—
249C2×C4×C53C2×#4114yes—
250C2×C4×C53C2×#4114yes—
251C2×C4×C53C2×#4114yes—
252C2×Dic53—28no—
253C2×Dic53—28no—
254C2×Dic53—28no—
255C2×Dic53—22no—
256C2×Dic53—22no—
257C2×Dic53—22no—
258C2×Dic53—22no—
259C2×Dic53—22no—
260C2×Dic53—22no—
261C2×Dic53—22no—
262C2×Dic53—22no—
263C2×Dic53—22no—
264C2×Dic53—22no—
265C2×Dic53—22no—
266C2×Dic53—22no—
267C2×Dic53—22no—
268C2×Dic53—22no—
269C2×Dic53—22no—
270C2×Dic53—22no—
271C2×Dic53—22no—
272C2×Dic53—22no—
273C2×Dic53—22no—
274C2×Dic53—22no—
275C2×Dic53—22no—
276C2×Dic53—44no—
277C2×Dic53—44no—
278C2×Dic53—44no—
279C2×Dic53—44no—
280C2×Dic53—44no—
281C2×Dic53C22×Q814yes—
282C2×Dic53C22×Q814yes—
283C2×Dic53C2×#4114yes—
284C2×#2585—531nomaximal, Sylow
285C22×C53#52914yes—
286C22×C53#53314yes—
287C22×C53#53914yes—
288C22×C53#55214yes—
289C22×C53#55314yes—
290C22×C53#55414yes—
291C22×C53#55714yes—
292C22×C53C23×Q811yescharacteristic
293C22×C53C22×#4111yescharacteristic
294C22×C53C22×#4111yescharacteristic
295C22×C53C22×#4111yescharacteristic
296C22×C53C22×#4111yescharacteristic
297C22×C53C2×H(5,𝔽2)11yescharacteristic
298C22×C53C2×2−1+411yescharacteristic
299C4×C53—24no—
300C4×C53—24no—
301C4×C53—24no—
302C4×C53—24no—
303C4×C53—42no—
304C4×C53—42no—
305C4×C53—42no—
306Dic53—24no—
307Dic53—24no—
308Dic53—24no—
309Dic53—42no—
310Dic53—42no—
311Dic53—42no—
312Dic53—42no—
313Dic53—42no—
314#2585—5316no—
315C2×#400—531no—
316C2×#398—531no—
317C2×#398—531no—
318C2×#398D5311yescharacteristic
319C2×#389—531no—
320C2×#389—531no—
321C2×#389—531no—
322C2×#387—531no—
323C2×#387—531no—
324C2×#387—531no—
325C2×#387—531no—
326C2×#387—531no—
327C2×#387—531no—
328C2×C4×C4⋊C4—531no—
329C2×C4×C4⋊C4—531no—
330C2×C53#258518yescomplemented
331C2×C53C2×#52911yescharacteristic
332C2×C53C2×#53311yescharacteristic
333C2×C53C2×#53911yescharacteristic
334C2×C53C2×#55211yescharacteristic
335C2×C53C2×#55311yescharacteristic
336C2×C53C2×#55411yescharacteristic
337C2×C53C2×#55711yescharacteristic
338C22×C4⋊C4—531no—
339C22×C4⋊C4—531no—
340C22×C4⋊C4—531no—
341C22×C4⋊C4—531no—
342C22×C4⋊C4—531no—
343C22×C4⋊C4—531no—
344C22×C4⋊C4—531no—
345C22×C4⋊C4—531no—
346C22×C4⋊C4—531no—
347C22×C4⋊C4—531no—
348C22×C4⋊C4C2×D5311yescharacteristic
349C22×C4⋊C4C2×D5311yescharacteristic
350C22×C4⋊C4C2×D5311yescharacteristic
351C22×C4⋊C4C2×D5311yescharacteristic
352C22×C42—531no—
353C22×C42—531no—
354C22×C42—531no—
355C22×C42—531no—
356C22×C42—531no—
357#400—538no—
358#398—538no—
359#398—538no—
360#398C2×D5318yes—
361#389—538no—
362#389—538no—
363#389—538no—
364#387—538no—
365#387—538no—
366#387—538no—
367#387—538no—
368#387—538no—
369#387—538no—
370C4×C4⋊C4—538no—
371C4×C4⋊C4—538no—
372C2×#95—531no—
373C2×#95—531no—
374C2×#95—531no—
375C2×#95—531no—
376C2×#95—531no—
377C2×#95—531no—
378C2×#95—531no—
379C2×#95—531no—
380C2×#95—531no—
381C2×#95—531no—
382C2×#95—531no—
383C2×#95—531no—
384C2×#95—531no—
385C2×#95C2×D5311yescharacteristic
386C2×#95C2×D5311yescharacteristic
387C2×#95C2×D5311yescharacteristic
388C53C2×#258511yescharacteristic, complemented, Sylow
389C23×C4—531no—
390C23×C4—531no—
391C23×C4—531no—
392C23×C4—531no—
393C23×C4—531no—
394C23×C4—531no—
395C23×C4—531no—
396C23×C4—531no—
397C23×C4C22×D5311yescharacteristic
398C23×C4C22×D5311yescharacteristic
399C23×C4C22×D5311yescharacteristic
400C23×C4C22×D5311yescharacteristic
401C23×C4C22×D5311yescharacteristic
402C23×C4C22×D5311yescharacteristic
403C23×C4C22×D5311yescharacteristic
404C2×C4⋊C4—24no—
405C2×C4⋊C4—24no—
406C2×C4⋊C4—24no—
407C2×C4⋊C4—24no—
408C2×C4⋊C4—538no—
409C2×C4⋊C4—534no—
410C2×C4⋊C4—534no—
411C2×C4⋊C4—534no—
412C2×C4⋊C4—534no—
413C2×C4⋊C4—534no—
414C2×C4⋊C4—534no—
415C2×C4⋊C4—534no—
416C2×C4⋊C4—534no—
417C2×C4⋊C4—534no—
418C2×C4⋊C4—534no—
419C2×C4⋊C4—1064no—
420C2×C4⋊C4—1064no—
421C2×C4⋊C4—1064no—
422C2×C4⋊C4—1064no—
423C2×C4⋊C4—1064no—
424C2×C4⋊C4—1064no—
425C2×C4⋊C4—2122no—
426C2×C4⋊C4—2122no—
427C2×C4⋊C4—2122no—
428C2×C4⋊C4C22×D5314yes—
429C2×C4⋊C4C22×D5314yes—
430C2×C4⋊C4C22×D5314yes—
431C2×C4⋊C4C22×D5314yes—
432C2×C42—534no—
433C2×C42—534no—
434C2×C42—534no—
435C2×C42—534no—
436C2×C42—534no—
437C2×C42—1064no—
438C2×C42—1064no—
439C2×C42—2122no—
440C2×C42—2122no—
441C2×C42—2122no—
442#95—534no—
443#95—534no—
444#95—534no—
445#95—534no—
446#95—534no—
447#95—534no—
448#95—534no—
449#95—534no—
450#95—534no—
451#95—534no—
452#95—534no—
453#95—534no—
454#95—534no—
455#95C22×D5314yes—
456#95C22×D5314yes—
457#95C22×D5314yes—
458C24C23×D5311yescharacteristic
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—534no—
477C22×C4—534no—
478C22×C4—534no—
479C22×C4—532no—
480C22×C4—532no—
481C22×C4—532no—
482C22×C4—532no—
483C22×C4—532no—
484C22×C4—532no—
485C22×C4—532no—
486C22×C4—532no—
487C22×C4—1062no—
488C22×C4—1062no—
489C22×C4—1062no—
490C22×C4—1062no—
491C22×C4—1062no—
492C22×C4—1062no—
493C22×C4—1062no—
494C22×C4—1062no—
495C22×C4—1062no—
496C22×C4—1062no—
497C22×C4—1062no—
498C22×C4—1062no—
499C22×C4—1062no—
500C22×C4—1062no—
501C22×C4—1062no—
502C22×C4—1062no—
503C22×C4—1062no—
504C22×C4—1062no—
505C22×C4—1062no—
506C22×C4—1062no—
507C22×C4—1062no—
508C22×C4#1171994714yes—
509C22×C4#1171994714yes—
510C22×C4Q8×D5314yes—
511C22×C4#1171994914yes—
512C22×C4C23×D5312yes—
513C22×C4C23×D5312yes—
514C22×C4C23×D5312yes—
515C22×C4C23×D5312yes—
516C22×C4C23×D5312yes—
517C22×C4C23×D5312yes—
518C22×C4C23×D5312yes—
519C4⋊C4—28no—
520C4⋊C4—28no—
521C4⋊C4—28no—
522C4⋊C4—28no—
523C4⋊C4—5316no—
524C4⋊C4—1068no—
525C4⋊C4—1068no—
526C4⋊C4—1068no—
527C4⋊C4—1068no—
528C4⋊C4—1068no—
529C4⋊C4—1068no—
530C4⋊C4—2124no—
531C4⋊C4—2124no—
532C4⋊C4—2124no—
533C42—1068no—
534C42—1068no—
535C42—2124no—
536C42—2124no—
537C42—2124no—
538C23C2×Q8×D5312yes—
539C23#41×D5312yes—
540C23C2×#1171994712yes—
541C23C2×#1171994712yes—
542C23C2×#1171994912yes—
543C23#1313326312yes—
544C23#1313326412yes—
545C23C24×D5311yescharacteristic
546C2×C4—28no—
547C2×C4—28no—
548C2×C4—28no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—28no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—534no—
571C2×C4—534no—
572C2×C4—534no—
573C2×C4—1068no—
574C2×C4—1068no—
575C2×C4—1068no—
576C2×C4—1062no—
577C2×C4—1062no—
578C2×C4—1062no—
579C2×C4—1062no—
580C2×C4—1062no—
581C2×C4—1062no—
582C2×C4—1062no—
583C2×C4—1062no—
584C2×C4—1062no—
585C2×C4—1062no—
586C2×C4—1062no—
587C2×C4—1062no—
588C2×C4—1062no—
589C2×C4—1062no—
590C2×C4—1062no—
591C2×C4—1062no—
592C2×C4—1062no—
593C2×C4—1062no—
594C2×C4—1062no—
595C2×C4—1062no—
596C2×C4—1062no—
597C2×C4—2124no—
598C2×C4—2124no—
599C2×C4—2124no—
600C2×C4—2124no—
601C2×C4—2124no—
602C2×C4C2×Q8×D5314yes—
603C2×C4C2×#1171994714yes—
604C2×C4C2×#1171994714yes—
605C2×C4C2×#1171994914yes—
606C22#1681115414yes—
607C22#1681117814yes—
608C22#1681121714yes—
609C22#1681131514yes—
610C22#1681132614yes—
611C22#1681133114yes—
612C22#1681135014yes—
613C22C22×Q8×D5311yescharacteristic
614C22C2×#41×D5311yescharacteristic
615C22C22×#1171994711yescharacteristic
616C22C22×#1171994711yescharacteristic
617C22C22×#1171994911yescharacteristic
618C22C2×#1313326311yescharacteristic
619C22C2×#1313326411yescharacteristic
620C4—24no—
621C4—24no—
622C4—24no—
623C4—24no—
624C4—42no—
625C4—42no—
626C4—42no—
627C4—1064no—
628C4—1064no—
629C4—1064no—
630C4—2122no—
631C4—2122no—
632C4—2122no—
633C4—2122no—
634C4—2122no—
635C2#2028011318yesdirect product factor
636C2C2×#1681115411yescharacteristic
637C2C2×#1681117811yescharacteristic
638C2C2×#1681121711yescharacteristic
639C2C2×#1681131511yescharacteristic
640C2C2×#1681132611yescharacteristic
641C2C2×#1681133111yescharacteristic
642C2C2×#1681135011yescharacteristic
subgroupG/H
centerC24C23×D53
commutator, cosocleC23×C53C25
FrattiniC23C24×D53
FittingC2×C53×#398C2
socle, prime radicalC24×C53C24
hypercenterC2×#398D53
hypocenterC53C2×#2585
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C24 ◁ C2×#398

Complements

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

NH ≅ G/Ntype
1. #20280113C2direct (N × H)
330. C2×C53#2585semidirect
388. C53C2×#2585semidirect
635. C2#20280113direct (N × H)

Maximal subgroups

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

subgroupindex
#202801132
C2×#168105612
C2×#168105562
C2×#168105552
C2×#168105222
C2×#168105102
C2×#168105082
C2×#168105062
C2×#168105042
C2×C4⋊C4×Dic532
C2×#168104742
C2×C53×#3982
C2×#258553