Finite Group Databasebeta

← C2×#20242327 #26188455

Subgroups — C2×#20242327 #26188455

Order 12032.

1372 conjugacy classes of subgroups; only the 203 normal subgroups are shown.

normal subgroups only

This group has 1372 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×#15689754C211yesmaximal, characteristic, complemented
2C2×#15689274C212yesmaximal
3#20242327C218yesmaximal, direct product factor
4C2×#15688859C212yesmaximal, complemented
5C2×#15688849C211yesmaximal, characteristic, complemented
6C2×C47×#394C211yesmaximal, characteristic
7C22×#10625335—22no—
8#15689754—24no—
9#15689754—22no—
10#15689754C2214yescomplemented
11C2×#13094023—22no—
12C2×#13094015—22no—
13C2×#13094003—21no—
14C2×#13094003C2212yes—
15C2×#13093983—21no—
16#15689274—24no—
17#15689274C2218yes—
18C2×#13093945—22no—
19C2×C22⋊(C47⋊C8)C2212yes—
20#15688859C2218yescomplemented
21#15688849C2214yescomplemented
22C23×D4×C47C2211yescharacteristic
23C22×C4⋊C4×C47C2211yescharacteristic
24C22×C22⋊C4×C47—21no—
25C22×C22⋊C4×C47—21no—
26C47×#394—22no—
27C47×#394C2214yes—
28C2×#95×C47C2211yescharacteristic
29C22×Dic94—21no—
30C22×Dic94D412yes—
31C2×#10625335—28no—
32C2×#10625335—48no—
33C22×C47⋊C8—22no—
34#13094023—28no—
35#13094023—28no—
36#13094015—28no—
37#13094015—28no—
38#13094003—24no—
39#13094003—22no—
40#13094003—22no—
41#13094003—24no—
42#13094003D414yes—
43#13094003C2314yes—
44#13093983—24no—
45#13093983—24no—
46C2×C4⋊Dic47—21no—
47C2×C4⋊Dic47—21no—
48C2×C4⋊Dic47—21no—
49C2×C4⋊Dic47D412yes—
50C2×Dic47⋊C4—22no—
51C2×Dic47⋊C4—22no—
52C2×C22⋊Dic47—22no—
53C2×C4×Dic47—22no—
54#13093945—28no—
55C22⋊(C47⋊C8)—22no—
56C22⋊(C47⋊C8)C2314yes—
57C25×C47—21no—
58C22×D4×C47—22no—
59C22×D4×C47—22no—
60C22×D4×C47—21no—
61C22×D4×C47—44no—
62C22×D4×C47C2312yes—
63C23×C4×C47—21no—
64C23×C4×C47D411yescharacteristic
65C23×C4×C47D411yescharacteristic
66C23×C4×C47C2311yescharacteristic
67C2×C4⋊C4×C47—22no—
68C2×C4⋊C4×C47—22no—
69C2×C4⋊C4×C47D412yes—
70C2×C4⋊C4×C47C2312yes—
71C2×C22⋊C4×C47—24no—
72C2×C22⋊C4×C47—24no—
73C2×C22⋊C4×C47—44no—
74C2×C22⋊C4×C47—44no—
75#95×C47—21no—
76#95×C47C2312yes—
77C24×C47—24no—
78C24×C47—22no—
79C24×C47—42no—
80C24×C47—42no—
81C24×C47—42no—
82C24×C47—41no—
83C24×C47—41no—
84C24×C47—42no—
85C24×C47—42no—
86C24×C47C2×D411yescharacteristic
87C2×D4×C47—28no—
88C2×D4×C47—24no—
89C2×D4×C47—44no—
90C2×D4×C47—44no—
91C2×D4×C47—48no—
92C2×D4×C47—44no—
93C2×D4×C47—42no—
94C22×C4×C47—21no—
95C22×C4×C47—21no—
96C22×C4×C47—22no—
97C22×C4×C47—21no—
98C22×C4×C47—21no—
99C22×C4×C47—22no—
100C22×C4×C47—42no—
101C22×C4×C47—42no—
102C22×C4×C47—42no—
103C22×C4×C47—42no—
104C22×C4×C47—42no—
105C22×C4×C47SD1614yescomplemented
106C22×C4×C47C2×D412yes—
107C22×C4×C47C2×D412yes—
108C22×C4×C47C2×D412yes—
109C22×C4×C47C2×D411yescharacteristic
110C22×C4×C47C2×D412yes—
111C22×C4×C47C2411yescharacteristic
112C22×Dic47—22no—
113C22×Dic47—22no—
114C2×Dic94—24no—
115C2×Dic94—22no—
116C2×Dic94—44no—
117C2×Dic94—82no—
118C2×Dic94C2×D414yes—
119C4⋊C4×C47—24no—
120C4⋊C4×C47—24no—
121C22⋊C4×C47—48no—
122C22⋊C4×C47—48no—
123C22⋊Dic47—24no—
124C22⋊Dic47—24no—
125#10625335—416no—
126C4⋊Dic47—24no—
127C4⋊Dic47—24no—
128C4⋊Dic47—24no—
129C4⋊Dic47—22no—
130C4⋊Dic47C2×D414yes—
131Dic47⋊C4—28no—
132Dic47⋊C4—24no—
133Dic47⋊C4—24no—
134C4×Dic47—28no—
135C2×C47⋊C8—24no—
136C2×C47⋊C8—44no—
137C23×C47—22no—
138C23×C47—22no—
139C23×C47—24no—
140C23×C47—42no—
141C23×C47—44no—
142C23×C47—48no—
143C23×C47—48no—
144C23×C47—42no—
145C23×C47—42no—
146C23×C47—42no—
147C23×C47—42no—
148C23×C47—42no—
149C23×C47—44no—
150C23×C47—84no—
151C23×C47—84no—
152C23×C47—82no—
153C23×C47—82no—
154C23×C47—82no—
155C23×C47—82no—
156C23×C47#12011yescharacteristic
157C23×C47#12011yescharacteristic
158C23×C47C2×SD1612yes—
159C23×C47#13712yes—
160C23×C47C22×D411yescharacteristic
161D4×C47—28no—
162D4×C47—44no—
163D4×C47—48no—
164C2×C4×C47—22no—
165C2×C4×C47—22no—
166C2×C4×C47—24no—
167C2×C4×C47—24no—
168C2×C4×C47—24no—
169C2×C4×C47—22no—
170C2×C4×C47—42no—
171C2×C4×C47—41no—
172C2×C4×C47—41no—
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#12012yes—
181C2×C4×C47C2×SD1614yescomplemented
182C2×C4×C47C22×D412yes—
183C2×Dic47—24no—
184C2×Dic47—24no—
185C2×Dic47—44no—
186C2×Dic47—44no—
187Dic94—44no—
188Dic94—82no—
189Dic94—84no—
190C47⋊C8—44no—
191C2×#1599—471nomaximal, Sylow
192C22×C47—24no—
193C22×C47—24no—
194C22×C47—21no—
195C22×C47—21no—
196C22×C47—21no—
197C22×C47—21no—
198C22×C47—44no—
199C22×C47—44no—
200C22×C47—44no—
201C22×C47—82no—
202C22×C47—82no—
203C22×C47—88no—
204C22×C47—88no—
205C22×C47—82no—
206C22×C47—82no—
207C22×C47#44812yes—
208C22×C47#44812yes—
209C22×C47#45412yes—
210C22×C47B(4,𝔽2)12yes—
211C22×C47C2×#12011yescharacteristic
212C22×C47C22×SD1611yescharacteristic
213C22×C47C2×#13711yescharacteristic
214C4×C47—24no—
215C4×C47—22no—
216C4×C47—22no—
217C4×C47—42no—
218C4×C47—82no—
219Dic47—44no—
220Dic47—44no—
221C2×#544—471no—
222C2×#450—472no—
223C2×#394D4711yescharacteristic
224#1599—478no—
225C2×#328—472no—
226C2×#324—471no—
227C2×C47—21no—
228C2×C47—21no—
229C2×C47—22no—
230C2×C47—84no—
231C2×C47#159914yescomplemented
232C2×C47C2×#44811yescharacteristic
233C2×C47C2×#45411yescharacteristic
234C2×C47C2×B(4,𝔽2)11yescharacteristic
235C23×D4C2×D4711yescharacteristic
236C22×SD16—942no—
237#544—474no—
238#544—944no—
239#544—942no—
240C2×C4⋊Q8—942no—
241C2×#125—942no—
242C2×C22⋊Q8—472no—
243C2×C22⋊Q8—941no—
244C2×#117—941no—
245C22×C4⋊C4C2×D4711yescharacteristic
246C22×C22⋊C4—21no—
247C22×C22⋊C4—21no—
248#450—478no—
249#450—944no—
250C2×#102—942no—
251C2×C22⋊C8—472no—
252#394—22no—
253#394C2×D4714yes—
254C2×#95C2×D4711yescharacteristic
255#328—478no—
256#324—474no—
257C47C2×#159911yescharacteristic, complemented, Sylow
258C25—21no—
259C22×Q8—472no—
260C22×Q8—941no—
261C22×D4—22no—
262C22×D4—22no—
263C22×D4—21no—
264C22×D4—44no—
265C22×D4C22×D4712yes—
266C23×C4—21no—
267C23×C4C22⋊D4711yescharacteristic
268C23×C4C22⋊D4711yescharacteristic
269C23×C4C22×D4711yescharacteristic
270C2×SD16—948no—
271C2×SD16—1888no—
272C22×C8—942no—
273C4⋊Q8—948no—
274C4⋊Q8—948no—
275#125—948no—
276#125—948no—
277C22⋊Q8—474no—
278C22⋊Q8—474no—
279C22⋊Q8—944no—
280C22⋊Q8—942no—
281C22⋊Q8—942no—
282C22⋊Q8—944no—
283#117—944no—
284#117—944no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—472no—
288C2×C4⋊C4—941no—
289C2×C4⋊C4—941no—
290C2×C4⋊C4—941no—
291C2×C4⋊C4—942no—
292C2×C4⋊C4—942no—
293C2×C4⋊C4C22⋊D4712yes—
294C2×C4⋊C4C22×D4712yes—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—44no—
298C2×C22⋊C4—44no—
299C2×C22⋊C4—942no—
300C2×C42—942no—
301#102—948no—
302C22⋊C8—474no—
303C22⋊C8—942no—
304#95—21no—
305#95C22×D4712yes—
306C24—24no—
307C24—22no—
308C24—42no—
309C24—42no—
310C24—42no—
311C24—41no—
312C24—41no—
313C24—42no—
314C24—42no—
315C24C2×C22⋊D4711yescharacteristic
316C2×Q8—474no—
317C2×Q8—944no—
318C2×Q8—942no—
319C2×Q8—3762no—
320C2×Q8—1884no—
321C2×D4—28no—
322C2×D4—24no—
323C2×D4—44no—
324C2×D4—44no—
325C2×D4—48no—
326C2×D4—44no—
327C2×D4—42no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—22no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—22no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—942no—
340C22×C4—942no—
341C22×C4#1062533514yescomplemented
342C22×C4D4×D4712yes—
343C22×C4D4×D4712yes—
344C22×C4C2×C22⋊D4712yes—
345C22×C4C2×C22⋊D4712yes—
346C22×C4C2×C22⋊D4711yescharacteristic
347C22×C4C23×D4711yescharacteristic
348SD16—18816no—
349C2×C8—944no—
350C2×C8—1884no—
351C4⋊C4—24no—
352C4⋊C4—24no—
353C4⋊C4—474no—
354C4⋊C4—948no—
355C4⋊C4—944no—
356C4⋊C4—944no—
357C4⋊C4—944no—
358C4⋊C4—944no—
359C4⋊C4—942no—
360C4⋊C4—944no—
361C22⋊C4—48no—
362C22⋊C4—48no—
363C22⋊C4—944no—
364C22⋊C4—944no—
365C42—948no—
366C23—22no—
367C23—22no—
368C23—24no—
369C23—42no—
370C23—44no—
371C23—42no—
372C23—42no—
373C23—48no—
374C23—48no—
375C23—44no—
376C23—42no—
377C23—42no—
378C23—42no—
379C23—82no—
380C23—84no—
381C23—84no—
382C23—82no—
383C23—82no—
384C23—82no—
385C23#1309399211yescharacteristic
386C23#1309399211yescharacteristic
387C23C2×#1062533512yes—
388C23#1309405412yes—
389C23C22×C22⋊D4711yescharacteristic
390Q8—3762no—
391Q8—3764no—
392Q8—1884no—
393D4—28no—
394D4—44no—
395D4—48no—
396C2×C4—24no—
397C2×C4—24no—
398C2×C4—24no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—42no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—81no—
411C2×C4—84no—
412C2×C4—944no—
413C2×C4—944no—
414C2×C4—1884no—
415C2×C4—1884no—
416C2×C4#1309399212yes—
417C2×C4C2×#1062533514yescomplemented
418C2×C4C2×D4×D4712yes—
419C8—1884no—
420C22—21no—
421C22—21no—
422C22—21no—
423C22—21no—
424C22—24no—
425C22—24no—
426C22—44no—
427C22—44no—
428C22—44no—
429C22—82no—
430C22—82no—
431C22—88no—
432C22—82no—
433C22—88no—
434C22—82no—
435C22#1568926112yes—
436C22#1568926112yes—
437C22#1568929512yes—
438C22#1568930912yes—
439C22C2×#1309399211yescharacteristic
440C22C22×#1062533511yescharacteristic
441C22C2×#1309405411yescharacteristic
442C4—22no—
443C4—22no—
444C4—24no—
445C4—42no—
446C4—82no—
447C4—1884no—
448C4—1884no—
449C2—21no—
450C2—21no—
451C2—22no—
452C2—84no—
453C2#2024232714yesdirect product factor
454C2C2×#1568926111yescharacteristic
455C2C2×#1568929511yescharacteristic
456C2C2×#1568930911yescharacteristic
subgroupG/H
centerC23#13093992
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC2×C47×#394C2
socleC23×C47#120
prime radicalC23×D4×C47C22
hypercenterC2×#394D47
hypocenterC47C2×#1599
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#394

Complements

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

NH ≅ G/Ntype
1. C2×#15689754C2semidirect
3. #20242327C2direct (N × H)
4. C2×#15688859C2semidirect
5. C2×#15688849C2semidirect
10. #15689754C22semidirect
20. #15688859C22semidirect
21. #15688849C22semidirect
105. C22×C4×C47SD16semidirect
181. C2×C4×C47C2×SD16semidirect
231. C2×C47#1599semidirect
257. C47C2×#1599semidirect
341. C22×C4#10625335semidirect
417. C2×C4C2×#10625335semidirect
453. C2#20242327direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156897542
C2×#156892742
#202423272
C2×#156888592
C2×#156888492
C2×C47×#3942
C2×#159947