Finite Group Databasebeta

← C2×#21423018 #28560764

Subgroups — C2×#21423018 #28560764

Order 15104.

1696 conjugacy classes of subgroups; only the 203 normal subgroups are shown.

normal subgroups only

This group has 1696 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×#16838028C211yesmaximal, characteristic, complemented
2C2×#16837553C212yesmaximal, complemented
3#21423018C218yesmaximal, direct product factor
4C2×#16837138C212yesmaximal, complemented
5C2×#16837126C211yesmaximal, characteristic, complemented
6C2×C59×#392C211yesmaximal, characteristic
7C22×#11745670—22no—
8#16838028—24no—
9#16838028—22no—
10#16838028C2214yescomplemented
11C2×#14273706—21no—
12C2×#14273706C2212yes—
13C2×#14273704—22no—
14C2×#14273704—22no—
15C22×C22⋊Dic59—21no—
16#16837553—24no—
17#16837553C2218yescomplemented
18C2×#14273648—22no—
19C2×C22⋊(C59⋊C8)C2212yes—
20#16837138C2218yescomplemented
21#16837126C2214yescomplemented
22C23×D4×C59—21no—
23C23×D4×C59C2211yescharacteristic
24C22×C22⋊C4×C59—21no—
25C22×C22⋊C4×C59C2211yescharacteristic
26C59×#392—22no—
27C59×#392C2214yes—
28C2×#95×C59C2211yescharacteristic
29C22×Dic118—21no—
30C22×Dic118D412yescomplemented
31C23×Dic59—21no—
32C23×Dic59—21no—
33C2×#11745670—28no—
34C2×#11745670—48no—
35C22×C59⋊C8—22no—
36#14273706—24no—
37#14273706—24no—
38#14273706—22no—
39#14273706—22no—
40#14273706D414yescomplemented
41#14273706C2314yes—
42#14273704—28no—
43#14273704—28no—
44#14273704—28no—
45#14273704—28no—
46C2×C4⋊Dic59—21no—
47C2×C4⋊Dic59D412yescomplemented
48C2×Dic59⋊C4—22no—
49C2×Dic59⋊C4—22no—
50C2×C22⋊Dic59—22no—
51C2×C22⋊Dic59—24no—
52C2×C22⋊Dic59—22no—
53C2×C22⋊Dic59—24no—
54#14273648—28no—
55C22⋊(C59⋊C8)—22no—
56C22⋊(C59⋊C8)C2314yes—
57C25×C59—21no—
58C25×C59D411yescharacteristic
59C22×D4×C59—24no—
60C22×D4×C59—22no—
61C22×D4×C59—22no—
62C22×D4×C59—21no—
63C22×D4×C59—44no—
64C22×D4×C59—44no—
65C22×D4×C59—44no—
66C22×D4×C59—44no—
67C22×D4×C59C2312yes—
68C23×C4×C59—21no—
69C23×C4×C59D411yescharacteristic
70C23×C4×C59C2311yescharacteristic
71C2×C22⋊C4×C59—22no—
72C2×C22⋊C4×C59—22no—
73C2×C22⋊C4×C59—24no—
74C2×C22⋊C4×C59—44no—
75C2×C22⋊C4×C59D412yes—
76C2×C22⋊C4×C59C2312yes—
77#95×C59—21no—
78#95×C59C2312yes—
79C24×C59—21no—
80C24×C59—21no—
81C24×C59—22no—
82C24×C59—24no—
83C24×C59—22no—
84C24×C59—42no—
85C24×C59—42no—
86C24×C59—42no—
87C24×C59—42no—
88C24×C59—42no—
89C24×C59—41no—
90C24×C59—41no—
91C24×C59—42no—
92C24×C59—42no—
93C24×C59SD1614yescomplemented
94C24×C59C2×D411yescharacteristic
95C24×C59C2×D412yes—
96C2×D4×C59—28no—
97C2×D4×C59—24no—
98C2×D4×C59—44no—
99C2×D4×C59—48no—
100C2×D4×C59—44no—
101C2×D4×C59—48no—
102C2×D4×C59—44no—
103C2×D4×C59—48no—
104C2×D4×C59—432no—
105C2×D4×C59—48no—
106C2×D4×C59—42no—
107C22×C4×C59—24no—
108C22×C4×C59—21no—
109C22×C4×C59—22no—
110C22×C4×C59—21no—
111C22×C4×C59—42no—
112C22×C4×C59—42no—
113C22×C4×C59—42no—
114C22×C4×C59—42no—
115C22×C4×C59C2×D412yes—
116C22×C4×C59C2×D412yes—
117C22×C4×C59C2×D411yescharacteristic
118C22×C4×C59C2×D412yes—
119C22×C4×C59C2411yescharacteristic
120C22×Dic59—22no—
121C22×Dic59—22no—
122C22×Dic59—24no—
123C22×Dic59—24no—
124C22×Dic59—82no—
125C22×Dic59—82no—
126C2×Dic118—24no—
127C2×Dic118—22no—
128C2×Dic118—44no—
129C2×Dic118—82no—
130C2×Dic118C2×D414yescomplemented
131C22⋊C4×C59—24no—
132C22⋊C4×C59—24no—
133C22⋊C4×C59—48no—
134C22⋊Dic59—28no—
135C22⋊Dic59—24no—
136C22⋊Dic59—24no—
137#11745670—416no—
138C4⋊Dic59—24no—
139C4⋊Dic59—22no—
140C4⋊Dic59C2×D414yescomplemented
141Dic59⋊C4—28no—
142Dic59⋊C4—24no—
143Dic59⋊C4—24no—
144C2×C59⋊C8—24no—
145C2×C59⋊C8—44no—
146C23×C59—22no—
147C23×C59—22no—
148C23×C59—22no—
149C23×C59—24no—
150C23×C59—22no—
151C23×C59—24no—
152C23×C59—24no—
153C23×C59—48no—
154C23×C59—42no—
155C23×C59—42no—
156C23×C59—42no—
157C23×C59—48no—
158C23×C59—42no—
159C23×C59—42no—
160C23×C59—44no—
161C23×C59—48no—
162C23×C59—42no—
163C23×C59—42no—
164C23×C59—42no—
165C23×C59—42no—
166C23×C59—42no—
167C23×C59—44no—
168C23×C59—84no—
169C23×C59—81no—
170C23×C59—81no—
171C23×C59—84no—
172C23×C59—84no—
173C23×C59—82no—
174C23×C59—82no—
175C23×C59—82no—
176C23×C59—82no—
177C23×C59#12011yescharacteristic
178C23×C59#12011yescharacteristic
179C23×C59C2×SD1612yes—
180C23×C59C2×SD1614yescomplemented
181C23×C59#13712yes—
182C23×C59C22×D411yescharacteristic
183D4×C59—28no—
184D4×C59—44no—
185D4×C59—48no—
186D4×C59—432no—
187C2×C4×C59—24no—
188C2×C4×C59—24no—
189C2×C4×C59—22no—
190C2×C4×C59—42no—
191C2×C4×C59—41no—
192C2×C4×C59—41no—
193C2×C4×C59—48no—
194C2×C4×C59—42no—
195C2×C4×C59—42no—
196C2×C4×C59—42no—
197C2×C4×C59—42no—
198C2×C4×C59—81no—
199C2×C4×C59#12012yes—
200C2×C4×C59C22×D412yes—
201C2×Dic59—24no—
202C2×Dic59—24no—
203C2×Dic59—44no—
204C2×Dic59—44no—
205C2×Dic59—84no—
206C2×Dic59—84no—
207Dic118—44no—
208Dic118—82no—
209Dic118—84no—
210C59⋊C8—44no—
211C2×#1596—591nomaximal, Sylow
212C22×C59—24no—
213C22×C59—24no—
214C22×C59—21no—
215C22×C59—21no—
216C22×C59—21no—
217C22×C59—21no—
218C22×C59—24no—
219C22×C59—44no—
220C22×C59—44no—
221C22×C59—44no—
222C22×C59—48no—
223C22×C59—42no—
224C22×C59—42no—
225C22×C59—44no—
226C22×C59—82no—
227C22×C59—84no—
228C22×C59—82no—
229C22×C59—82no—
230C22×C59—88no—
231C22×C59—88no—
232C22×C59—82no—
233C22×C59—82no—
234C22×C59#44812yes—
235C22×C59#44812yes—
236C22×C59#45312yes—
237C22×C59B(4,𝔽2)12yes—
238C22×C59C2×#12011yescharacteristic
239C22×C59C22×SD1611yescharacteristic
240C22×C59C2×#13711yescharacteristic
241C4×C59—22no—
242C4×C59—22no—
243C4×C59—44no—
244C4×C59—42no—
245Dic59—44no—
246Dic59—44no—
247C2×#543—591no—
248C2×#450—592no—
249C2×#392D5911yescharacteristic
250#1596—598no—
251C2×#328—592no—
252C2×#323—591no—
253C2×C59—21no—
254C2×C59—21no—
255C2×C59—22no—
256C2×C59—44no—
257C2×C59—84no—
258C2×C59#159614yescomplemented
259C2×C59C2×#44811yescharacteristic
260C2×C59C2×#45311yescharacteristic
261C2×C59C2×B(4,𝔽2)11yescharacteristic
262C23×D4—21no—
263C23×D4C2×D5911yescharacteristic
264C22×SD16—1182no—
265#543—594no—
266#543—1184no—
267#543—1182no—
268C2×C22⋊Q8—592no—
269C2×C22⋊Q8—1182no—
270C2×C22⋊Q8—1182no—
271C2×C22⋊Q8—1181no—
272C22×C22⋊C4—21no—
273C22×C22⋊C4—1181no—
274C22×C22⋊C4C2×D5911yescharacteristic
275#450—598no—
276#450—1184no—
277C2×#102—1182no—
278C2×C22⋊C8—592no—
279#392—22no—
280#392C2×D5914yes—
281C2×#95C2×D5911yescharacteristic
282#328—598no—
283#323—594no—
284C59C2×#159611yescharacteristic, complemented, Sylow
285C25—21no—
286C25C22⋊D5911yescharacteristic
287C22×Q8—592no—
288C22×Q8—1181no—
289C22×D4—24no—
290C22×D4—22no—
291C22×D4—22no—
292C22×D4—21no—
293C22×D4—44no—
294C22×D4—44no—
295C22×D4—44no—
296C22×D4—44no—
297C22×D4C22×D5912yes—
298C23×C4—21no—
299C23×C4—1181no—
300C23×C4—1181no—
301C23×C4C22⋊D5911yescharacteristic
302C23×C4C22×D5911yescharacteristic
303C2×SD16—1188no—
304C2×SD16—2368no—
305C22×C8—1182no—
306C22⋊Q8—594no—
307C22⋊Q8—594no—
308C22⋊Q8—1188no—
309C22⋊Q8—1188no—
310C22⋊Q8—1188no—
311C22⋊Q8—1188no—
312C22⋊Q8—1184no—
313C22⋊Q8—1182no—
314C22⋊Q8—1182no—
315C22⋊Q8—1184no—
316C2×C4⋊C4—592no—
317C2×C4⋊C4—1182no—
318C2×C4⋊C4—1181no—
319C2×C4⋊C4—1182no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—44no—
324C2×C22⋊C4—1182no—
325C2×C22⋊C4—1184no—
326C2×C22⋊C4—1182no—
327C2×C22⋊C4—1184no—
328C2×C22⋊C4C22⋊D5912yes—
329C2×C22⋊C4C22×D5912yes—
330#102—1188no—
331C22⋊C8—594no—
332C22⋊C8—1182no—
333#95—21no—
334#95C22×D5912yes—
335C24—21no—
336C24—21no—
337C24—22no—
338C24—24no—
339C24—22no—
340C24—42no—
341C24—42no—
342C24—42no—
343C24—42no—
344C24—42no—
345C24—41no—
346C24—41no—
347C24—42no—
348C24—42no—
349C24#1174567014yescomplemented
350C24C2×C22⋊D5911yescharacteristic
351C24C2×C22⋊D5912yes—
352C2×Q8—594no—
353C2×Q8—1184no—
354C2×Q8—1182no—
355C2×Q8—4722no—
356C2×Q8—2364no—
357C2×D4—28no—
358C2×D4—24no—
359C2×D4—44no—
360C2×D4—48no—
361C2×D4—44no—
362C2×D4—48no—
363C2×D4—44no—
364C2×D4—48no—
365C2×D4—432no—
366C2×D4—48no—
367C2×D4—42no—
368C22×C4—24no—
369C22×C4—21no—
370C22×C4—22no—
371C22×C4—21no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—1184no—
377C22×C4—1184no—
378C22×C4—1182no—
379C22×C4—1182no—
380C22×C4—4722no—
381C22×C4—4722no—
382C22×C4D4×D5912yes—
383C22×C4D4×D5912yes—
384C22×C4C2×C22⋊D5912yes—
385C22×C4C2×C22⋊D5911yescharacteristic
386C22×C4C23×D5911yescharacteristic
387SD16—23616no—
388C2×C8—1184no—
389C2×C8—2364no—
390C4⋊C4—594no—
391C4⋊C4—1188no—
392C4⋊C4—1184no—
393C4⋊C4—1182no—
394C4⋊C4—1184no—
395C4⋊C4—1184no—
396C22⋊C4—24no—
397C22⋊C4—24no—
398C22⋊C4—48no—
399C22⋊C4—1188no—
400C22⋊C4—1184no—
401C22⋊C4—1184no—
402C23—22no—
403C23—22no—
404C23—24no—
405C23—22no—
406C23—22no—
407C23—24no—
408C23—24no—
409C23—48no—
410C23—42no—
411C23—42no—
412C23—42no—
413C23—42no—
414C23—42no—
415C23—44no—
416C23—48no—
417C23—48no—
418C23—42no—
419C23—42no—
420C23—42no—
421C23—42no—
422C23—44no—
423C23—42no—
424C23—81no—
425C23—81no—
426C23—84no—
427C23—82no—
428C23—84no—
429C23—84no—
430C23—82no—
431C23—82no—
432C23—82no—
433C23#1427369511yescharacteristic
434C23#1427369511yescharacteristic
435C23C2×#1174567012yes—
436C23C2×#1174567014yescomplemented
437C23#1427375712yes—
438C23C22×C22⋊D5911yescharacteristic
439Q8—4722no—
440Q8—4724no—
441Q8—2364no—
442D4—28no—
443D4—44no—
444D4—48no—
445D4—432no—
446C2×C4—24no—
447C2×C4—24no—
448C2×C4—22no—
449C2×C4—42no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—42no—
453C2×C4—48no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—81no—
458C2×C4—1184no—
459C2×C4—1184no—
460C2×C4—4724no—
461C2×C4—4724no—
462C2×C4—2364no—
463C2×C4—2364no—
464C2×C4#1427369512yes—
465C2×C4C2×D4×D5912yes—
466C8—2364no—
467C22—21no—
468C22—21no—
469C22—21no—
470C22—21no—
471C22—24no—
472C22—24no—
473C22—24no—
474C22—44no—
475C22—44no—
476C22—44no—
477C22—42no—
478C22—42no—
479C22—48no—
480C22—44no—
481C22—82no—
482C22—84no—
483C22—82no—
484C22—82no—
485C22—88no—
486C22—82no—
487C22—88no—
488C22—82no—
489C22#1683754012yes—
490C22#1683754012yes—
491C22#1683756912yes—
492C22#1683758812yes—
493C22C2×#1427369511yescharacteristic
494C22C22×#1174567011yescharacteristic
495C22C2×#1427375711yescharacteristic
496C4—22no—
497C4—22no—
498C4—44no—
499C4—42no—
500C4—2364no—
501C4—2364no—
502C2—21no—
503C2—21no—
504C2—22no—
505C2—44no—
506C2—84no—
507C2#2142301814yesdirect product factor
508C2C2×#1683754011yescharacteristic
509C2C2×#1683756911yescharacteristic
510C2C2×#1683758811yescharacteristic
subgroupG/H
centerC23#14273695
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
Fitting, prime radicalC2×C59×#392C2
socleC23×C59#120
hypercenterC2×#392D59
hypocenterC59C2×#1596
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#392

Complements

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

NH ≅ G/Ntype
1. C2×#16838028C2semidirect
2. C2×#16837553C2semidirect
3. #21423018C2direct (N × H)
4. C2×#16837138C2semidirect
5. C2×#16837126C2semidirect
10. #16838028C22semidirect
17. #16837553C22semidirect
20. #16837138C22semidirect
21. #16837126C22semidirect
30. C22×Dic118D4semidirect
40. #14273706D4semidirect
47. C2×C4⋊Dic59D4semidirect
93. C24×C59SD16semidirect
130. C2×Dic118C2×D4semidirect
140. C4⋊Dic59C2×D4semidirect
180. C23×C59C2×SD16semidirect
258. C2×C59#1596semidirect
284. C59C2×#1596semidirect
349. C24#11745670semidirect
436. C23C2×#11745670semidirect
507. C2#21423018direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168380282
C2×#168375532
#214230182
C2×#168371382
C2×#168371262
C2×C59×#3922
C2×#159659