Finite Group Databasebeta

← C2×#20273649 #27356257

Subgroups — C2×#20273649 #27356257

Order 13568.

1652 conjugacy classes of subgroups; only the 203 normal subgroups are shown.

normal subgroups only

This group has 1652 conjugacy classes of subgroups — too many to draw. Only its 203 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811204C211yesmaximal, characteristic, complemented
2C2×#16810757C212yesmaximal, complemented
3#20273649C218yesmaximal, direct product factor
4C2×#16810356C212yesmaximal, complemented
5C2×#16810349C211yesmaximal, characteristic, complemented
6C2×C53×#394C211yesmaximal, characteristic, complemented
7C22×#11719945—21no—
8C22×#11719922—22no—
9#16811204—24no—
10#16811204—22no—
11#16811204C2214yescomplemented
12C2×#13133197—22no—
13C2×#13133192—22no—
14C2×#13133191—21no—
15C2×#13133191C2212yes—
16#16810757—24no—
17#16810757C2218yescomplemented
18C2×#13133138—22no—
19C2×C22⋊#88322C2212yes—
20#16810356C2218yescomplemented
21#16810349C2214yescomplemented
22C23×D4×C53C2211yescharacteristic
23C22×C4⋊C4×C53C2211yescharacteristic
24C22×C22⋊C4×C53—21no—
25C22×C22⋊C4×C53—21no—
26C53×#394—22no—
27C53×#394C2214yescomplemented
28C2×#95×C53C2211yescharacteristic
29C2×#11719945—24no—
30C2×#11719945—24no—
31C2×#11719945—44no—
32C2×#11719945—44no—
33C22×Dic106—21no—
34C22×D212—21no—
35C22×D212—21no—
36C22×D212D412yes—
37C22×C22⋊D53—22no—
38C22×C4×D53—22no—
39C2×#11719922—28no—
40C2×#11719922—48no—
41C22×#88322—22no—
42#13133197—28no—
43#13133197—28no—
44#13133192—28no—
45#13133192—28no—
46#13133191—24no—
47#13133191—24no—
48#13133191—22no—
49#13133191—22no—
50#13133191D414yes—
51#13133191C2314yes—
52C2×C4⋊Dic53—21no—
53C2×C4⋊Dic53D412yes—
54C2×#11719921—22no—
55C2×#11719921—22no—
56#13133138—28no—
57C22⋊#88322—22no—
58C22⋊#88322C2314yes—
59C25×C53—21no—
60C22×D4×C53—22no—
61C22×D4×C53—22no—
62C22×D4×C53—21no—
63C22×D4×C53—44no—
64C22×D4×C53C2312yes—
65C23×C4×C53—21no—
66C23×C4×C53D411yescharacteristic
67C23×C4×C53D411yescharacteristic
68C23×C4×C53C2311yescharacteristic
69C2×C4⋊C4×C53—22no—
70C2×C4⋊C4×C53—22no—
71C2×C4⋊C4×C53D412yes—
72C2×C4⋊C4×C53C2312yes—
73C2×C22⋊C4×C53—24no—
74C2×C22⋊C4×C53—24no—
75C2×C22⋊C4×C53—44no—
76C2×C22⋊C4×C53—44no—
77#95×C53—21no—
78#95×C53C2312yes—
79C24×C53—24no—
80C24×C53—22no—
81C24×C53—42no—
82C24×C53—42no—
83C24×C53—42no—
84C24×C53—41no—
85C24×C53—41no—
86C24×C53—42no—
87C24×C53—42no—
88C24×C53C2×D411yescharacteristic
89C23×D53—22no—
90C2×D4×C53—28no—
91C2×D4×C53—24no—
92C2×D4×C53—44no—
93C2×D4×C53—44no—
94C2×D4×C53—48no—
95C2×D4×C53—44no—
96C2×D4×C53—42no—
97C22×C4×C53—21no—
98C22×C4×C53—21no—
99C22×C4×C53—22no—
100C22×C4×C53—21no—
101C22×C4×C53—21no—
102C22×C4×C53—22no—
103C22×C4×C53—42no—
104C22×C4×C53—42no—
105C22×C4×C53—42no—
106C22×C4×C53—42no—
107C22×C4×C53—42no—
108C22×C4×C53D814yescomplemented
109C22×C4×C53C2×D412yes—
110C22×C4×C53C2×D412yes—
111C22×C4×C53C2×D412yes—
112C22×C4×C53C2×D411yescharacteristic
113C22×C4×C53C2×D412yes—
114C22×C4×C53C2411yescharacteristic
115C2×C22⋊D53—24no—
116C2×C22⋊D53—24no—
117C2×C22⋊D53—44no—
118C2×C22⋊D53—44no—
119C22×Dic53—22no—
120#11719945—48no—
121#11719945—416no—
122#11719945—48no—
123C2×D212—24no—
124C2×D212—24no—
125C2×D212—22no—
126C2×D212—44no—
127C2×D212—44no—
128C2×D212—82no—
129C2×D212C2×D414yes—
130C2×C4×D53—28no—
131C2×C4×D53—48no—
132C2×Dic106—24no—
133C2×Dic106—44no—
134C4⋊C4×C53—24no—
135C4⋊C4×C53—24no—
136C22⋊C4×C53—48no—
137C22⋊C4×C53—48no—
138#11719922—416no—
139#11719921—28no—
140#11719921—24no—
141#11719921—24no—
142C4⋊Dic53—24no—
143C4⋊Dic53—22no—
144C4⋊Dic53C2×D414yes—
145C2×#88322—24no—
146C2×#88322—44no—
147C23×C53—22no—
148C23×C53—22no—
149C23×C53—24no—
150C23×C53—42no—
151C23×C53—44no—
152C23×C53—48no—
153C23×C53—48no—
154C23×C53—42no—
155C23×C53—42no—
156C23×C53—42no—
157C23×C53—42no—
158C23×C53—42no—
159C23×C53—44no—
160C23×C53—84no—
161C23×C53—84no—
162C23×C53—82no—
163C23×C53—82no—
164C23×C53—82no—
165C23×C53—82no—
166C23×C53#12011yescharacteristic
167C23×C53#12011yescharacteristic
168C23×C53C2×D812yes—
169C23×C53AGL(1,ℤ/8)12yes—
170C23×C53C22×D411yescharacteristic
171C22×D53—24no—
172C22×D53—44no—
173C22×D53—44no—
174C22×D53—44no—
175D4×C53—28no—
176D4×C53—44no—
177D4×C53—48no—
178C2×C4×C53—22no—
179C2×C4×C53—22no—
180C2×C4×C53—24no—
181C2×C4×C53—24no—
182C2×C4×C53—24no—
183C2×C4×C53—22no—
184C2×C4×C53—42no—
185C2×C4×C53—41no—
186C2×C4×C53—41no—
187C2×C4×C53—42no—
188C2×C4×C53—42no—
189C2×C4×C53—42no—
190C2×C4×C53—42no—
191C2×C4×C53—42no—
192C2×C4×C53—81no—
193C2×C4×C53—84no—
194C2×C4×C53#12012yes—
195C2×C4×C53C2×D814yescomplemented
196C2×C4×C53C22×D412yes—
197C22⋊D53—44no—
198C22⋊D53—44no—
199C22⋊D53—48no—
200C2×Dic53—24no—
201C2×Dic53—44no—
202D212—44no—
203D212—48no—
204D212—82no—
205D212—84no—
206C4×D53—416no—
207Dic106—48no—
208#88322—44no—
209C2×#1598—531nomaximal, Sylow
210C22×C53—24no—
211C22×C53—24no—
212C22×C53—21no—
213C22×C53—21no—
214C22×C53—21no—
215C22×C53—21no—
216C22×C53—44no—
217C22×C53—44no—
218C22×C53—44no—
219C22×C53—82no—
220C22×C53—82no—
221C22×C53—88no—
222C22×C53—88no—
223C22×C53—82no—
224C22×C53—82no—
225C22×C53#44712yes—
226C22×C53#44712yes—
227C22×C53#45412yes—
228C22×C53B(4,𝔽2)12yes—
229C22×C53C2×#12011yescharacteristic
230C22×C53C22×D811yescharacteristic
231C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
232C2×D53—44no—
233C2×D53—44no—
234C2×D53—44no—
235C2×D53—88no—
236C4×C53—24no—
237C4×C53—22no—
238C4×C53—22no—
239C4×C53—42no—
240C4×C53—82no—
241Dic53—44no—
242C2×#537—531no—
243C2×#447—532no—
244C2×#394D5311yescharacteristic, complemented
245#1598—538no—
246C2×#327—532no—
247C2×#324—531no—
248C2×C53—21no—
249C2×C53—21no—
250C2×C53—22no—
251C2×C53—84no—
252C2×C53#159814yescomplemented
253C2×C53C2×#44711yescharacteristic
254C2×C53C2×#45411yescharacteristic
255C2×C53C2×B(4,𝔽2)11yescharacteristic
256D53—84no—
257C22×#41—1061no—
258C23×D4C2×D5311yescharacteristic
259C22×D8—1062no—
260#537—534no—
261#537—1064no—
262#537—1062no—
263C2×C22⋊Q8—1062no—
264C2×#121—532no—
265C2×#121—1062no—
266C2×#121—1061no—
267C22×C4⋊C4C2×D5311yescharacteristic
268C22×C22⋊C4—21no—
269C22×C22⋊C4—21no—
270#447—538no—
271#447—1064no—
272C2×#102—1062no—
273C2×C22⋊C8—532no—
274#394—22no—
275#394C2×D5314yescomplemented
276C2×#95C2×D5311yescharacteristic
277#327—538no—
278#324—534no—
279C53C2×#159811yescharacteristic, complemented, Sylow
280C25—21no—
281C2×#41—1064no—
282C2×#41—1064no—
283C2×#41—2124no—
284C2×#41—2124no—
285C22×Q8—1061no—
286C22×D4—22no—
287C22×D4—22no—
288C22×D4—21no—
289C22×D4—44no—
290C22×D4—532no—
291C22×D4—1061no—
292C22×D4—1061no—
293C22×D4—1062no—
294C22×D4C22×D5312yes—
295C23×C4—21no—
296C23×C4—1062no—
297C23×C4C22⋊D5311yescharacteristic
298C23×C4C22⋊D5311yescharacteristic
299C23×C4C22×D5311yescharacteristic
300C2×D8—1068no—
301C2×D8—2128no—
302C22×C8—1062no—
303C22⋊Q8—1068no—
304C22⋊Q8—1068no—
305#121—534no—
306#121—534no—
307#121—1068no—
308#121—1068no—
309#121—1064no—
310#121—1062no—
311#121—1062no—
312#121—1064no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—532no—
316C2×C4⋊C4—1061no—
317C2×C4⋊C4C22⋊D5312yes—
318C2×C4⋊C4C22×D5312yes—
319C2×C22⋊C4—24no—
320C2×C22⋊C4—24no—
321C2×C22⋊C4—44no—
322C2×C22⋊C4—44no—
323C2×C22⋊C4—1062no—
324C2×C22⋊C4—1062no—
325#102—1068no—
326C22⋊C8—534no—
327C22⋊C8—1062no—
328#95—21no—
329#95C22×D5312yes—
330C24—24no—
331C24—22no—
332C24—42no—
333C24—42no—
334C24—42no—
335C24—41no—
336C24—41no—
337C24—42no—
338C24—42no—
339C24—1062no—
340C24C2×C22⋊D5311yescharacteristic
341#41—21216no—
342#41—2128no—
343#41—2128no—
344C2×Q8—1064no—
345C2×Q8—2124no—
346C2×D4—28no—
347C2×D4—24no—
348C2×D4—44no—
349C2×D4—44no—
350C2×D4—48no—
351C2×D4—44no—
352C2×D4—42no—
353C2×D4—534no—
354C2×D4—1064no—
355C2×D4—1064no—
356C2×D4—1064no—
357C2×D4—1064no—
358C2×D4—1062no—
359C2×D4—4242no—
360C2×D4—2124no—
361C2×D4—2124no—
362C2×D4—2124no—
363C2×D4—2124no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—22no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—22no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—1068no—
376C22×C4—1062no—
377C22×C4—2128no—
378C22×C4#1171992214yescomplemented
379C22×C4D4×D5312yes—
380C22×C4D4×D5312yes—
381C22×C4C2×C22⋊D5312yes—
382C22×C4C2×C22⋊D5312yes—
383C22×C4C2×C22⋊D5311yescharacteristic
384C22×C4C23×D5311yescharacteristic
385D8—21216no—
386C2×C8—1064no—
387C2×C8—2124no—
388C4⋊C4—24no—
389C4⋊C4—24no—
390C4⋊C4—534no—
391C4⋊C4—1064no—
392C4⋊C4—1062no—
393C22⋊C4—48no—
394C22⋊C4—48no—
395C22⋊C4—1068no—
396C22⋊C4—1064no—
397C22⋊C4—1064no—
398C23—22no—
399C23—22no—
400C23—24no—
401C23—42no—
402C23—44no—
403C23—42no—
404C23—42no—
405C23—48no—
406C23—48no—
407C23—44no—
408C23—42no—
409C23—42no—
410C23—42no—
411C23—82no—
412C23—84no—
413C23—84no—
414C23—82no—
415C23—82no—
416C23—82no—
417C23—1064no—
418C23—2124no—
419C23—2124no—
420C23—2124no—
421C23#1313318511yescharacteristic
422C23#1313318511yescharacteristic
423C23C2×#1171992212yes—
424C23#1313324212yes—
425C23C22×C22⋊D5311yescharacteristic
426Q8—2128no—
427D4—28no—
428D4—44no—
429D4—48no—
430D4—4242no—
431D4—4244no—
432D4—2124no—
433D4—2128no—
434D4—2128no—
435D4—2124no—
436D4—2124no—
437C2×C4—24no—
438C2×C4—24no—
439C2×C4—24no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—42no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—81no—
452C2×C4—84no—
453C2×C4—1064no—
454C2×C4—21216no—
455C2×C4—2124no—
456C2×C4#1313318512yes—
457C2×C4C2×#1171992214yescomplemented
458C2×C4C2×D4×D5312yes—
459C8—2124no—
460C22—21no—
461C22—21no—
462C22—21no—
463C22—21no—
464C22—24no—
465C22—24no—
466C22—44no—
467C22—44no—
468C22—44no—
469C22—82no—
470C22—82no—
471C22—88no—
472C22—82no—
473C22—88no—
474C22—82no—
475C22—4248no—
476C22—2124no—
477C22—2124no—
478C22—2124no—
479C22#1681075612yes—
480C22#1681075612yes—
481C22#1681079512yes—
482C22#1681080912yes—
483C22C2×#1313318511yescharacteristic
484C22C22×#1171992211yescharacteristic
485C22C2×#1313324211yescharacteristic
486C4—22no—
487C4—22no—
488C4—24no—
489C4—42no—
490C4—82no—
491C4—2124no—
492C2—21no—
493C2—21no—
494C2—22no—
495C2—84no—
496C2—4244no—
497C2#2027364914yesdirect product factor
498C2C2×#1681075611yescharacteristic
499C2C2×#1681079511yescharacteristic
500C2C2×#1681080911yescharacteristic
subgroupG/H
centerC23#13133185
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#394C2
socleC23×C53#120
hypercenterC2×#394D53
hypocenterC53C2×#1598
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#394

Complements

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

NH ≅ G/Ntype
1. C2×#16811204C2semidirect
2. C2×#16810757C2semidirect
3. #20273649C2direct (N × H)
4. C2×#16810356C2semidirect
5. C2×#16810349C2semidirect
6. C2×C53×#394C2semidirect
11. #16811204C22semidirect
17. #16810757C22semidirect
20. #16810356C22semidirect
21. #16810349C22semidirect
27. C53×#394C22semidirect
108. C22×C4×C53D8semidirect
195. C2×C4×C53C2×D8semidirect
244. C2×#394D53semidirect
252. C2×C53#1598semidirect
275. #394C2×D53semidirect
279. C53C2×#1598semidirect
378. C22×C4#11719922semidirect
457. C2×C4C2×#11719922semidirect
497. C2#20273649direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168112042
C2×#168107572
#202736492
C2×#168103562
C2×#168103492
C2×C53×#3942
C2×#159853