Finite Group Databasebeta

← C2×#20276173 #27413027

Subgroups — C2×#20276173 #27413027

Order 13568.

1006 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810629C211yesmaximal, characteristic
2C2×#16810629C211yesmaximal, characteristic
3C22×#13133133C211yesmaximal, characteristic
4#20276173C218yesmaximal, direct product factor
5C2×#16810395C211yesmaximal, characteristic, complemented
6C2×#16810394C211yesmaximal, characteristic, complemented
7C2×#16810380C211yesmaximal, characteristic, complemented
8C2×C53×#430C211yesmaximal, characteristic, complemented
9C23×D212C2211yescharacteristic
10C22×#11719917C2211yescharacteristic
11C2×#13133176C2211yescharacteristic
12C2×#13133176C2211yescharacteristic
13#16810629—22no—
14#16810629—22no—
15#16810629C2214yes—
16#16810629C2214yes—
17C2×#13133139—22no—
18C2×#13133139—22no—
19C2×#13133133—22no—
20C2×#13133133—22no—
21C2×#13133133C2214yes—
22#16810395C2214yescomplemented
23#16810394C2214yescomplemented
24#16810380C2214yescomplemented
25C22×M16×C53C2211yescharacteristic
26C22×M16×C53C2211yescharacteristic
27C22×M16×C53C2211yescharacteristic
28C53×#430—22no—
29C53×#430C2214yescomplemented
30C2×#108×C53—22no—
31C24×D53—21no—
32C22×D212—21no—
33C22×D212—21no—
34C22×D212—21no—
35C22×D212—21no—
36C22×D212—21no—
37C22×D212—21no—
38C22×D212—21no—
39C22×D212—21no—
40C22×D212—21no—
41C22×D212—21no—
42C22×D212—21no—
43C22×D212D412yes—
44C22×D212D412yes—
45C22×D212C2312yes—
46C2×#11719917—21no—
47C2×#11719917—21no—
48C2×#11719917—21no—
49C2×#11719917—21no—
50C2×#11719917—21no—
51C2×#11719917C2312yes—
52C22×#88322—21no—
53#13133176—21no—
54#13133176—21no—
55#13133176—21no—
56#13133176—21no—
57#13133176—21no—
58#13133176—21no—
59#13133176C2312yes—
60#13133176C2312yes—
61C2×C4⋊Dic53—21no—
62C2×C4⋊Dic53—21no—
63C2×C22⋊Dic53—21no—
64C2×C22⋊Dic53—21no—
65C2×C4×Dic53—21no—
66C2×C4×Dic53—21no—
67#13133139—28no—
68#13133139—28no—
69#13133133—28no—
70C23×C4×C53C2311yescharacteristic
71C2×M16×C53—21no—
72C2×M16×C53—21no—
73C2×M16×C53—21no—
74C2×M16×C53—21no—
75C2×M16×C53—21no—
76C2×M16×C53—21no—
77C2×M16×C53—21no—
78C2×M16×C53—21no—
79C2×M16×C53—21no—
80C2×M16×C53—21no—
81C2×M16×C53—21no—
82C2×M16×C53—21no—
83C2×M16×C53—21no—
84C2×M16×C53—21no—
85C2×M16×C53D412yes—
86C2×M16×C53C2312yes—
87C2×M16×C53C2312yes—
88C2×M16×C53C2312yes—
89C22×C8×C53—21no—
90C22×C8×C53—21no—
91C22×C8×C53D412yes—
92#108×C53—28no—
93C24×C53#4111yescharacteristic
94C23×D53—22no—
95C23×D53—22no—
96C23×D53—22no—
97C23×D53—41no—
98C23×D53—41no—
99C23×D53—41no—
100C23×D53—41no—
101C23×D53—84no—
102C22×C4×C53C2×D411yescharacteristic
103C22×C4×C53C2×D411yescharacteristic
104C22×C4×C53C2×D411yescharacteristic
105C22×C4×C53C2×D411yescharacteristic
106C22×C4×C53#4111yescharacteristic
107C22×C4×C53#4111yescharacteristic
108C22×C4×C53#4111yescharacteristic
109C22×C4×C53#4111yescharacteristic
110C22×C4×C53#4111yescharacteristic
111C22×C4×C53#4111yescharacteristic
112C22×C4×C53#4111yescharacteristic
113C22×C4×C53#4111yescharacteristic
114C22×C4×C53#4111yescharacteristic
115C22×C4×C53C2411yescharacteristic
116C22×Dic53—41no—
117C22×Dic53—41no—
118C2×D212—22no—
119C2×D212—22no—
120C2×D212—22no—
121C2×D212—22no—
122C2×D212—22no—
123C2×D212—22no—
124C2×D212—22no—
125C2×D212—22no—
126C2×D212—22no—
127C2×D212—22no—
128C2×D212—22no—
129C2×D212—22no—
130C2×D212—22no—
131C2×D212—22no—
132C2×D212—22no—
133C2×D212—41no—
134C2×D212—41no—
135C2×D212—41no—
136C2×D212—41no—
137C2×D212—41no—
138C2×D212—41no—
139C2×D212—41no—
140C2×D212—41no—
141C2×D212—41no—
142C2×D212—41no—
143C2×D212—41no—
144C2×D212—41no—
145C2×D212C2×D414yes—
146M16×C53—22no—
147M16×C53—22no—
148M16×C53—22no—
149M16×C53—22no—
150M16×C53—22no—
151M16×C53—22no—
152M16×C53—22no—
153M16×C53—22no—
154M16×C53—41no—
155M16×C53—41no—
156M16×C53—41no—
157M16×C53—41no—
158C2×C8×C53—22no—
159C2×C8×C53—22no—
160C2×C8×C53—22no—
161C2×C8×C53—22no—
162C2×C8×C53—41no—
163C2×C8×C53—41no—
164C2×C8×C53—41no—
165C2×C8×C53—41no—
166C2×C8×C53C2×D414yes—
167C22⋊Dic53—22no—
168C22⋊Dic53—22no—
169C22⋊Dic53—41no—
170C22⋊Dic53—41no—
171C4⋊Dic53—22no—
172C4⋊Dic53—22no—
173C4⋊Dic53—41no—
174C4⋊Dic53—41no—
175C4×Dic53—22no—
176C4×Dic53—22no—
177C4×Dic53—41no—
178C4×Dic53—41no—
179#11719917—22no—
180#11719917—22no—
181#11719917—41no—
182#11719917—41no—
183C2×#88322—22no—
184C2×#88322—41no—
185C2×#88322—41no—
186C23×C53—42no—
187C23×C53#12311yescharacteristic
188C23×C53#12311yescharacteristic
189C23×C53#12411yescharacteristic
190C23×C53#12411yescharacteristic
191C23×C53#12611yescharacteristic
192C23×C53#12611yescharacteristic
193C23×C53C2×#4111yescharacteristic
194C22×D53—24no—
195C22×D53—42no—
196C22×D53—42no—
197C22×D53—42no—
198C22×D53—42no—
199C22×D53—42no—
200C22×D53—42no—
201C22×D53—84no—
202C22×D53—84no—
203C22×D53—84no—
204C22×D53—84no—
205C22×D53—84no—
206C22×D53—84no—
207C22×D53—84no—
208C2×C4×C53#12111yescharacteristic
209C2×C4×C53#12111yescharacteristic
210C2×C4×C53#12111yescharacteristic
211C2×C4×C53#12111yescharacteristic
212C2×C4×C53#12311yescharacteristic
213C2×C4×C53#12311yescharacteristic
214C2×C4×C53#12311yescharacteristic
215C2×C4×C53#12311yescharacteristic
216C2×C4×C53#12311yescharacteristic
217C2×C4×C53#12311yescharacteristic
218C2×C4×C53#12311yescharacteristic
219C2×C4×C53#12311yescharacteristic
220C2×C4×C53#12311yescharacteristic
221C2×C4×C53#12311yescharacteristic
222C2×C4×C53#12411yescharacteristic
223C2×C4×C53#12411yescharacteristic
224C2×C4×C53#12611yescharacteristic
225C2×C4×C53#12611yescharacteristic
226C2×C4×C53#12611yescharacteristic
227C2×C4×C53#12611yescharacteristic
228C2×C4×C53#12611yescharacteristic
229C2×C4×C53#12611yescharacteristic
230C2×C4×C53C22×D411yescharacteristic
231C2×C4×C53C22×D411yescharacteristic
232C2×C4×C53C2×#4111yescharacteristic
233C2×C4×C53C2×#4111yescharacteristic
234C2×C4×C53C2×#4111yescharacteristic
235C2×C4×C53C2×#4111yescharacteristic
236C2×Dic53—42no—
237C2×Dic53—42no—
238C2×Dic53—82no—
239C2×Dic53—82no—
240D212—24no—
241D212—24no—
242D212—42no—
243D212—42no—
244D212—42no—
245D212—42no—
246D212—42no—
247D212—42no—
248C8×C53—24no—
249C8×C53—42no—
250C8×C53—42no—
251#88322—42no—
252C2×#2081—531nomaximal, Sylow
253C22×C53—22no—
254C22×C53—22no—
255C22×C53—22no—
256C22×C53—41no—
257C22×C53—41no—
258C22×C53—41no—
259C22×C53—41no—
260C22×C53#39711yescharacteristic
261C22×C53#39711yescharacteristic
262C22×C53#39711yescharacteristic
263C22×C53#39711yescharacteristic
264C22×C53C2×#12311yescharacteristic
265C22×C53C2×#12411yescharacteristic
266C22×C53C2×#12611yescharacteristic
267C2×D53—44no—
268C2×D53—84no—
269C2×D53—84no—
270C2×D53—84no—
271C2×D53—84no—
272C2×D53—84no—
273C2×D53—84no—
274C2×D53—84no—
275C4×C53#39711yescharacteristic
276C4×C53#39711yescharacteristic
277C4×C53#39711yescharacteristic
278C4×C53#39711yescharacteristic
279C4×C53C2×#12111yescharacteristic
280C4×C53C2×#12311yescharacteristic
281C4×C53C2×#12311yescharacteristic
282C4×C53C2×#12611yescharacteristic
283Dic53—82no—
284Dic53—82no—
285C2×#430D5311yescharacteristic, complemented
286C2×#418—531no—
287C2×#418—531no—
288C22×#100—531no—
289#2081—538no—
290C2×#344—531no—
291C2×#344—531no—
292C2×H(3,ℤ/4)—531no—
293C2×C53—21no—
294C2×C53—21no—
295C2×C53—21no—
296C2×C53—21no—
297C2×C53—21no—
298C2×C53—21no—
299C2×C53#208112yescomplemented
300C2×C53C2×#39711yescharacteristic
301D53—84no—
302C23×D4—531no—
303C22×M16—531no—
304C22×M16C2×D5311yescharacteristic
305C22×M16C2×D5311yescharacteristic
306C22×M16C2×D5311yescharacteristic
307C2×#117—531no—
308C2×#117—531no—
309#430—22no—
310#430C2×D5314yescomplemented
311C2×#108—22no—
312#418—534no—
313#418—534no—
314#418—1062no—
315#418—1062no—
316C2×#102—1062no—
317C2×#102—1062no—
318C2×#100—534no—
319C2×#100—1062no—
320C2×#100—1062no—
321#344—534no—
322#344—534no—
323H(3,ℤ/4)—534no—
324C53C2×#208111yescharacteristic, complemented, Sylow
325C25—1061no—
326C22×D4—532no—
327C22×D4—532no—
328C22×D4—532no—
329C22×D4—1061no—
330C22×D4—1061no—
331C22×D4—1061no—
332C22×D4—1061no—
333C22×D4—1061no—
334C22×D4—1061no—
335C22×D4—1061no—
336C22×D4—1061no—
337C22×D4—1061no—
338C22×D4—1061no—
339C22×D4—1061no—
340C23×C4C22×D5311yescharacteristic
341C2×M16—21no—
342C2×M16—21no—
343C2×M16—21no—
344C2×M16—21no—
345C2×M16—21no—
346C2×M16—21no—
347C2×M16—21no—
348C2×M16—21no—
349C2×M16—21no—
350C2×M16—21no—
351C2×M16—21no—
352C2×M16—21no—
353C2×M16—21no—
354C2×M16—21no—
355C2×M16—532no—
356C2×M16—1061no—
357C2×M16—1061no—
358C2×M16—1061no—
359C2×M16—1061no—
360C2×M16—1061no—
361C2×M16C22⋊D5312yes—
362C2×M16C22×D5312yes—
363C2×M16C22×D5312yes—
364C2×M16C22×D5312yes—
365C22×C8—21no—
366C22×C8—21no—
367C22×C8—1061no—
368C22×C8C22⋊D5312yes—
369#117—532no—
370#117—532no—
371#117—1061no—
372#117—1061no—
373#117—1061no—
374#117—1061no—
375#117—1061no—
376#117—1061no—
377C2×C4⋊C4—1061no—
378C2×C4⋊C4—1061no—
379C2×C22⋊C4—1061no—
380C2×C22⋊C4—1061no—
381C2×C42—1061no—
382C2×C42—1061no—
383#108—28no—
384#102—1068no—
385#102—1068no—
386#100—1068no—
387C24—1062no—
388C24—1062no—
389C24—1062no—
390C24—4244no—
391C24—2121no—
392C24—2121no—
393C24—2121no—
394C24—2121no—
395C24#1171994911yescharacteristic
396C2×D4—534no—
397C2×D4—1062no—
398C2×D4—1062no—
399C2×D4—1062no—
400C2×D4—1062no—
401C2×D4—1062no—
402C2×D4—1062no—
403C2×D4—1062no—
404C2×D4—1062no—
405C2×D4—1062no—
406C2×D4—1062no—
407C2×D4—1062no—
408C2×D4—1062no—
409C2×D4—1062no—
410C2×D4—1062no—
411C2×D4—1062no—
412C2×D4—2121no—
413C2×D4—2121no—
414C2×D4—2121no—
415C2×D4—2121no—
416C2×D4—2121no—
417C2×D4—2121no—
418C2×D4—2121no—
419C2×D4—2121no—
420C2×D4—2121no—
421C2×D4—2121no—
422C2×D4—2121no—
423C2×D4—2121no—
424C22×C4—2121no—
425C22×C4—2121no—
426C22×C4#1171994511yescharacteristic
427C22×C4#1171994511yescharacteristic
428C22×C4#1171994511yescharacteristic
429C22×C4#1171994511yescharacteristic
430C22×C4D4×D5311yescharacteristic
431C22×C4D4×D5311yescharacteristic
432C22×C4#1171994711yescharacteristic
433C22×C4#1171994711yescharacteristic
434C22×C4#1171994911yescharacteristic
435C22×C4#1171994911yescharacteristic
436C22×C4#1171994911yescharacteristic
437C22×C4C2×C22⋊D5311yescharacteristic
438C22×C4C2×C22⋊D5311yescharacteristic
439C22×C4C23×D5311yescharacteristic
440M16—22no—
441M16—22no—
442M16—22no—
443M16—22no—
444M16—22no—
445M16—22no—
446M16—22no—
447M16—22no—
448M16—41no—
449M16—41no—
450M16—41no—
451M16—41no—
452M16—1062no—
453M16—1062no—
454M16—2121no—
455M16—2121no—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—41no—
461C2×C8—41no—
462C2×C8—41no—
463C2×C8—41no—
464C2×C8—1062no—
465C2×C8—2121no—
466C2×C8—2121no—
467C2×C8C2×C22⋊D5314yes—
468C4⋊C4—1062no—
469C4⋊C4—1062no—
470C4⋊C4—2121no—
471C4⋊C4—2121no—
472C22⋊C4—1062no—
473C22⋊C4—1062no—
474C22⋊C4—2121no—
475C22⋊C4—2121no—
476C42—1062no—
477C42—1062no—
478C42—2121no—
479C42—2121no—
480C23—42no—
481C23—1064no—
482C23—4244no—
483C23—4244no—
484C23—4244no—
485C23—4244no—
486C23—4244no—
487C23—4244no—
488C23—4244no—
489C23—2122no—
490C23—2122no—
491C23—2122no—
492C23—2122no—
493C23—2122no—
494C23—2122no—
495C23#1313320211yescharacteristic
496C23#1313320211yescharacteristic
497C23#1313320611yescharacteristic
498C23#1313320611yescharacteristic
499C23#1313321211yescharacteristic
500C23#1313321211yescharacteristic
501C23C2×#1171994911yescharacteristic
502D4—1064no—
503D4—1064no—
504D4—2122no—
505D4—2122no—
506D4—2122no—
507D4—2122no—
508D4—2122no—
509D4—2122no—
510C2×C4—4242no—
511C2×C4—4242no—
512C2×C4—2122no—
513C2×C4—2122no—
514C2×C4#1313318811yescharacteristic
515C2×C4#1313318811yescharacteristic
516C2×C4#1313318811yescharacteristic
517C2×C4#1313318811yescharacteristic
518C2×C4#1313320011yescharacteristic
519C2×C4#1313320011yescharacteristic
520C2×C4#1313320011yescharacteristic
521C2×C4#1313320011yescharacteristic
522C2×C4#1313320211yescharacteristic
523C2×C4#1313320211yescharacteristic
524C2×C4#1313320211yescharacteristic
525C2×C4#1313320211yescharacteristic
526C2×C4#1313320211yescharacteristic
527C2×C4#1313320211yescharacteristic
528C2×C4#1313320611yescharacteristic
529C2×C4#1313320611yescharacteristic
530C2×C4#1313321211yescharacteristic
531C2×C4#1313321211yescharacteristic
532C2×C4#1313321211yescharacteristic
533C2×C4#1313321211yescharacteristic
534C2×C4#1313321211yescharacteristic
535C2×C4#1313321211yescharacteristic
536C2×C4C2×D4×D5311yescharacteristic
537C2×C4C22×C22⋊D5311yescharacteristic
538C2×C4C2×#1171994711yescharacteristic
539C2×C4C2×#1171994511yescharacteristic
540C2×C4C2×#1171994511yescharacteristic
541C2×C4C2×#1171994911yescharacteristic
542C8—24no—
543C8—42no—
544C8—42no—
545C8—2122no—
546C22—22no—
547C22—22no—
548C22—22no—
549C22—41no—
550C22—41no—
551C22—41no—
552C22—41no—
553C22—4244no—
554C22—4244no—
555C22—4244no—
556C22—4244no—
557C22—4244no—
558C22—4244no—
559C22—4244no—
560C22—2124no—
561C22#1681054911yescharacteristic
562C22#1681054911yescharacteristic
563C22#1681054911yescharacteristic
564C22#1681054911yescharacteristic
565C22C2×#1313320211yescharacteristic
566C22C2×#1313320611yescharacteristic
567C22C2×#1313321211yescharacteristic
568C4—4242no—
569C4—4242no—
570C4#1681054911yescharacteristic
571C4#1681054911yescharacteristic
572C4#1681054911yescharacteristic
573C4#1681054911yescharacteristic
574C4C2×#1313318811yescharacteristic
575C4C2×#1313320011yescharacteristic
576C4C2×#1313320211yescharacteristic
577C4C2×#1313321211yescharacteristic
578C2—21no—
579C2—21no—
580C2—21no—
581C2—21no—
582C2—21no—
583C2—21no—
584C2—4244no—
585C2#2027617312yesdirect product factor
586C2C2×#1681054911yescharacteristic
subgroupG/H
centerC22#16810549
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#430C2
socleC22×C53#397
prime radicalC23×D212C22
hypercenterC2×#430D53
hypocenterC53C2×#2081
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#430

Complements

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

NH ≅ G/Ntype
4. #20276173C2direct (N × H)
5. C2×#16810395C2semidirect
6. C2×#16810394C2semidirect
7. C2×#16810380C2semidirect
8. C2×C53×#430C2semidirect
22. #16810395C22semidirect
23. #16810394C22semidirect
24. #16810380C22semidirect
29. C53×#430C22semidirect
285. C2×#430D53semidirect
299. C2×C53#2081semidirect
310. #430C2×D53semidirect
324. C53C2×#2081semidirect
585. C2#20276173direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168106292
C22×#131331332
#202761732
C2×#168103952
C2×#168103942
C2×#168103802
C2×C53×#4302
C2×#208153