Finite Group Databasebeta

← C2×#20242373 #26188392

Subgroups — C2×#20242373 #26188392

Order 12032.

1480 conjugacy classes of subgroups; only the 203 normal subgroups are shown.

normal subgroups only

This group has 1480 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×#15689685C211yesmaximal, characteristic
2C2×#15689436C212yesmaximal, complemented
3#20242373C218yesmaximal, direct product factor
4C2×#15688856C212yesmaximal, complemented
5C2×#15688847C211yesmaximal, characteristic, complemented
6C2×C47×#397C211yesmaximal, characteristic, complemented
7C23×C22⋊D47—21no—
8#15689685—24no—
9#15689685—22no—
10#15689685C2214yes—
11C2×#13094008—22no—
12C2×#13093998—21no—
13C2×#13093998C2212yes—
14C2×#13093994—22no—
15#15689436—24no—
16#15689436C2218yescomplemented
17C2×#13093957—22no—
18C2×#13093944—22no—
19C2×C22⋊(C47⋊C8)C2212yescomplemented
20#15688856C2218yescomplemented
21#15688847C2214yescomplemented
22C22×C4⋊C4×C47C2211yescharacteristic, complemented
23C22×C22⋊C4×C47—21no—
24C22×C22⋊C4×C47C2211yescharacteristic
25C47×#397—22no—
26C47×#397C2214yescomplemented
27C2×#95×C47—21no—
28C2×#95×C47C2211yescharacteristic, complemented
29C24×D47—21no—
30C22×D188—21no—
31C22×D188D412yes—
32C22×C22⋊D47—22no—
33C22×C22⋊D47—24no—
34C22×C22⋊D47—22no—
35C22×C22⋊D47—24no—
36C22×C22⋊D47—44no—
37C22×C22⋊D47—44no—
38C23×Dic47—21no—
39C22×C47⋊C8—22no—
40#13094008—28no—
41#13094008—28no—
42#13093998—24no—
43#13093998—24no—
44#13093998—22no—
45#13093998—22no—
46#13093998D414yes—
47#13093998C2314yes—
48#13093994—28no—
49#13093994—28no—
50C2×C4⋊Dic47—21no—
51C2×C4⋊Dic47D412yes—
52C2×(C2×D47)⋊C4—22no—
53C2×(C2×D47)⋊C4—22no—
54#13093957—28no—
55#13093944—28no—
56C22⋊(C47⋊C8)—22no—
57C22⋊(C47⋊C8)C2314yescomplemented
58C25×C47D411yescharacteristic
59C23×C4×C47—21no—
60C23×C4×C47—21no—
61C23×C4×C47D411yescharacteristic
62C23×C4×C47C2311yescharacteristic
63C2×C4⋊C4×C47—22no—
64C2×C4⋊C4×C47—22no—
65C2×C4⋊C4×C47—21no—
66C2×C4⋊C4×C47C2312yescomplemented
67C2×C22⋊C4×C47—22no—
68C2×C22⋊C4×C47—22no—
69C2×C22⋊C4×C47—24no—
70C2×C22⋊C4×C47—44no—
71C2×C22⋊C4×C47D412yes—
72C2×C22⋊C4×C47C2312yes—
73#95×C47—24no—
74#95×C47—21no—
75#95×C47C2312yescomplemented
76C24×C47—21no—
77C24×C47—21no—
78C24×C47—22no—
79C24×C47—42no—
80C24×C47—42no—
81C24×C47D814yes—
82C24×C47C2×D411yescharacteristic
83C24×C47C2×D412yes—
84C23×D47—22no—
85C23×D47—24no—
86C23×D47—44no—
87C23×D47—82no—
88C23×D47—82no—
89C22×C4×C47—21no—
90C22×C4×C47—21no—
91C22×C4×C47—22no—
92C22×C4×C47—22no—
93C22×C4×C47—42no—
94C22×C4×C47—42no—
95C22×C4×C47—42no—
96C22×C4×C47—42no—
97C22×C4×C47—41no—
98C22×C4×C47—41no—
99C22×C4×C47—42no—
100C22×C4×C47—42no—
101C22×C4×C47—42no—
102C22×C4×C47C2×D412yes—
103C22×C4×C47C2×D412yes—
104C22×C4×C47C2×D411yescharacteristic
105C22×C4×C47C2×D412yes—
106C22×C4×C47C2411yescharacteristic
107C2×C22⋊D47—28no—
108C2×C22⋊D47—24no—
109C2×C22⋊D47—24no—
110C2×C22⋊D47—48no—
111C2×C22⋊D47—416no—
112C2×C22⋊D47—48no—
113C2×C22⋊D47—48no—
114C2×C22⋊D47—44no—
115C2×C22⋊D47—44no—
116C22×Dic47—24no—
117C22×Dic47—22no—
118C22×Dic47—44no—
119C2×D188—24no—
120C2×D188—22no—
121C2×D188—44no—
122C2×D188—82no—
123C2×D188C2×D414yes—
124C4⋊C4×C47—24no—
125C4⋊C4×C47—42no—
126C22⋊C4×C47—24no—
127C22⋊C4×C47—24no—
128C22⋊C4×C47—48no—
129(C2×D47)⋊C4—28no—
130(C2×D47)⋊C4—24no—
131(C2×D47)⋊C4—24no—
132C4⋊Dic47—24no—
133C4⋊Dic47—22no—
134C4⋊Dic47C2×D414yes—
135C2×C47⋊C8—24no—
136C2×C47⋊C8—44no—
137C23×C47—22no—
138C23×C47—22no—
139C23×C47—22no—
140C23×C47—24no—
141C23×C47—22no—
142C23×C47—24no—
143C23×C47—48no—
144C23×C47—42no—
145C23×C47—42no—
146C23×C47—42no—
147C23×C47—42no—
148C23×C47—84no—
149C23×C47—81no—
150C23×C47—81no—
151C23×C47#12011yescharacteristic
152C23×C47#12011yescharacteristic
153C23×C47C2×D812yes—
154C23×C47C2×D814yes—
155C23×C47AGL(1,ℤ/8)12yes—
156C23×C47C22×D411yescharacteristic
157C22×D47—24no—
158C22×D47—48no—
159C22×D47—44no—
160C22×D47—44no—
161C22×D47—44no—
162C22×D47—84no—
163C22×D47—84no—
164C22×D47—816no—
165C2×C4×C47—24no—
166C2×C4×C47—22no—
167C2×C4×C47—42no—
168C2×C4×C47—41no—
169C2×C4×C47—41no—
170C2×C4×C47—42no—
171C2×C4×C47—42no—
172C2×C4×C47—42no—
173C2×C4×C47—42no—
174C2×C4×C47—42no—
175C2×C4×C47—42no—
176C2×C4×C47—42no—
177C2×C4×C47—42no—
178C2×C4×C47—81no—
179C2×C4×C47—84no—
180C2×C4×C47—82no—
181C2×C4×C47—82no—
182C2×C4×C47#12012yes—
183C2×C4×C47C22×D412yes—
184C22⋊D47—416no—
185C22⋊D47—44no—
186C22⋊D47—44no—
187C22⋊D47—48no—
188C2×Dic47—24no—
189C2×Dic47—48no—
190C2×Dic47—44no—
191D188—44no—
192D188—82no—
193D188—84no—
194C47⋊C8—44no—
195C2×#1607—471nomaximal, Sylow
196C22×C47—24no—
197C22×C47—24no—
198C22×C47—21no—
199C22×C47—21no—
200C22×C47—21no—
201C22×C47—21no—
202C22×C47—24no—
203C22×C47—44no—
204C22×C47—48no—
205C22×C47—42no—
206C22×C47—42no—
207C22×C47—82no—
208C22×C47—84no—
209C22×C47#44712yes—
210C22×C47#44712yes—
211C22×C47#45512yes—
212C22×C47#45812yes—
213C22×C47C2×#12011yescharacteristic
214C22×C47C22×D811yescharacteristic
215C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
216C2×D47—44no—
217C2×D47—44no—
218C2×D47—44no—
219C2×D47—816no—
220C2×D47—88no—
221C4×C47—22no—
222C4×C47—22no—
223C4×C47—42no—
224C4×C47—82no—
225C4×C47—82no—
226Dic47—44no—
227C2×#534—471no—
228C2×#480—472no—
229C2×#397D4711yescharacteristic, complemented
230#1607—478no—
231C2×#327—472no—
232C2×#323—471no—
233C2×C47—21no—
234C2×C47—21no—
235C2×C47—22no—
236C2×C47—44no—
237C2×C47#160714yescomplemented
238C2×C47C2×#44711yescharacteristic
239C2×C47C2×#45511yescharacteristic
240C2×C47C2×#45811yescharacteristic
241D47—84no—
242C23×D4—941no—
243#534—474no—
244#534—944no—
245#534—942no—
246C2×#123—942no—
247C2×#121—472no—
248C2×#121—941no—
249C2×#120—942no—
250C22×C4⋊C4C2×D4711yescharacteristic, complemented
251C22×C22⋊C4—21no—
252C22×C22⋊C4C2×D4711yescharacteristic
253#480—478no—
254#480—944no—
255C2×#107—942no—
256C2×#102—942no—
257C2×C22⋊C8—472no—
258#397—22no—
259#397C2×D4714yescomplemented
260C2×#95—21no—
261C2×#95C2×D4711yescharacteristic, complemented
262#327—478no—
263#323—474no—
264C47C2×#160711yescharacteristic, complemented, Sylow
265C25—941no—
266C25C22⋊D4711yescharacteristic
267C22×D4—472no—
268C22×D4—941no—
269C22×D4—942no—
270C22×D4—944no—
271C22×D4—942no—
272C22×D4—944no—
273C22×D4—1884no—
274C22×D4—1884no—
275C23×C4—21no—
276C23×C4—21no—
277C23×C4—941no—
278C23×C4C22⋊D4711yescharacteristic
279C23×C4C22×D4711yescharacteristic
280C22×C8—942no—
281#123—948no—
282#123—948no—
283#121—474no—
284#121—474no—
285#121—944no—
286#121—942no—
287#121—942no—
288#121—944no—
289#120—948no—
290#120—948no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—472no—
295C2×C4⋊C4—941no—
296C2×C4⋊C4C22×D4712yescomplemented
297C2×C22⋊C4—22no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4—942no—
302C2×C22⋊C4—942no—
303C2×C22⋊C4C22⋊D4712yes—
304C2×C22⋊C4C22×D4712yes—
305#107—948no—
306#102—948no—
307C22⋊C8—474no—
308C22⋊C8—942no—
309#95—24no—
310#95—21no—
311#95C22×D4712yescomplemented
312C24—21no—
313C24—21no—
314C24—22no—
315C24—42no—
316C24—42no—
317C24—944no—
318C24—942no—
319C24—3762no—
320C24—3762no—
321C24—1884no—
322C24#1062533414yes—
323C24C2×C22⋊D4711yescharacteristic
324C24C2×C22⋊D4712yes—
325C2×D4—474no—
326C2×D4—944no—
327C2×D4—948no—
328C2×D4—944no—
329C2×D4—944no—
330C2×D4—942no—
331C2×D4—3762no—
332C2×D4—1884no—
333C2×D4—18816no—
334C2×D4—1888no—
335C2×D4—1888no—
336C2×D4—1888no—
337C2×D4—1884no—
338C2×D4—1884no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—944no—
353C22×C4—942no—
354C22×C4—1884no—
355C22×C4D4×D4712yes—
356C22×C4D4×D4712yes—
357C22×C4C2×C22⋊D4712yes—
358C22×C4C2×C22⋊D4711yescharacteristic
359C22×C4C23×D4711yescharacteristic
360C2×C8—944no—
361C2×C8—1884no—
362C4⋊C4—24no—
363C4⋊C4—42no—
364C4⋊C4—474no—
365C4⋊C4—944no—
366C4⋊C4—942no—
367C22⋊C4—24no—
368C22⋊C4—24no—
369C22⋊C4—48no—
370C22⋊C4—948no—
371C22⋊C4—944no—
372C22⋊C4—944no—
373C23—22no—
374C23—22no—
375C23—24no—
376C23—22no—
377C23—24no—
378C23—22no—
379C23—48no—
380C23—42no—
381C23—42no—
382C23—42no—
383C23—42no—
384C23—81no—
385C23—81no—
386C23—84no—
387C23—944no—
388C23—37616no—
389C23—3764no—
390C23—3764no—
391C23—1888no—
392C23—1884no—
393C23—1884no—
394C23—1884no—
395C23#1309399211yescharacteristic
396C23#1309399211yescharacteristic
397C23C2×#1062533412yes—
398C23C2×#1062533414yes—
399C23#1309404912yes—
400C23C22×C22⋊D4711yescharacteristic
401D4—3762no—
402D4—3764no—
403D4—1884no—
404D4—18816no—
405D4—1888no—
406D4—1884no—
407D4—1884no—
408C2×C4—24no—
409C2×C4—22no—
410C2×C4—42no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—81no—
422C2×C4—84no—
423C2×C4—82no—
424C2×C4—82no—
425C2×C4—944no—
426C2×C4—1888no—
427C2×C4—1884no—
428C2×C4#1309399212yes—
429C2×C4C2×D4×D4712yes—
430C8—1884no—
431C22—21no—
432C22—21no—
433C22—21no—
434C22—21no—
435C22—24no—
436C22—24no—
437C22—24no—
438C22—44no—
439C22—42no—
440C22—42no—
441C22—48no—
442C22—84no—
443C22—82no—
444C22—37616no—
445C22—3768no—
446C22—1884no—
447C22—1884no—
448C22—1884no—
449C22#1568925612yes—
450C22#1568925612yes—
451C22#1568930012yes—
452C22#1568931212yes—
453C22C2×#1309399211yescharacteristic
454C22C22×#1062533411yescharacteristic
455C22C2×#1309404911yescharacteristic
456C4—22no—
457C4—22no—
458C4—42no—
459C4—82no—
460C4—82no—
461C4—1884no—
462C2—21no—
463C2—22no—
464C2—21no—
465C2—44no—
466C2—3764no—
467C2#2024237314yesdirect product factor
468C2C2×#1568925611yescharacteristic
469C2C2×#1568930011yescharacteristic
470C2C2×#1568931211yescharacteristic
subgroupG/H
centerC23#13093992
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC2×C47×#397C2
socleC23×C47#120
prime radicalC2×#15689685C2
hypercenterC2×#397D47
hypocenterC47C2×#1607
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#397

Complements

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

NH ≅ G/Ntype
2. C2×#15689436C2semidirect
3. #20242373C2direct (N × H)
4. C2×#15688856C2semidirect
5. C2×#15688847C2semidirect
6. C2×C47×#397C2semidirect
16. #15689436C22semidirect
19. C2×C22⋊(C47⋊C8)C22semidirect
20. #15688856C22semidirect
21. #15688847C22semidirect
22. C22×C4⋊C4×C47C22semidirect
26. C47×#397C22semidirect
28. C2×#95×C47C22semidirect
57. C22⋊(C47⋊C8)C23semidirect
66. C2×C4⋊C4×C47C23semidirect
75. #95×C47C23semidirect
229. C2×#397D47semidirect
237. C2×C47#1607semidirect
250. C22×C4⋊C4C2×D47semidirect
259. #397C2×D47semidirect
261. C2×#95C2×D47semidirect
264. C47C2×#1607semidirect
296. C2×C4⋊C4C22×D47semidirect
311. #95C22×D47semidirect
467. C2#20242373direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156896852
C2×#156894362
#202423732
C2×#156888562
C2×#156888472
C2×C47×#3972
C2×#160747