Finite Group Databasebeta

← C2×#20242381 #26188407

Subgroups — C2×#20242381 #26188407

Order 12032.

1136 conjugacy classes of subgroups; only the 203 normal subgroups are shown.

normal subgroups only

This group has 1136 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×#15689749C211yesmaximal, characteristic
2C2×#15689454C212yesmaximal, complemented
3C2×#15689048C211yesmaximal, characteristic
4#20242381C218yesmaximal, direct product factor
5C2×#15688861C212yesmaximal, complemented
6C2×C47×#323C211yesmaximal, characteristic
7#15689749—24no—
8#15689749—22no—
9#15689749C2214yes—
10C2×#13094003—21no—
11C2×#13094003C2212yes—
12C2×#13094001—22no—
13C2×#13094001—22no—
14C22×C4⋊Dic47C2211yescharacteristic
15C22×C22⋊Dic47—21no—
16C22×C22⋊Dic47—21no—
17#15689454—24no—
18#15689454C2218yescomplemented
19C2×#13093956—22no—
20C2×#13093949—22no—
21#15689048—22no—
22#15689048C2214yes—
23C2×C47⋊#95—21no—
24C2×C47⋊#95C2211yescharacteristic
25#15688861C2218yescomplemented
26C22×C22⋊C4×C47C2211yescharacteristic
27C2×C22⋊C8×C47C2212yes—
28C47×#323C2214yes—
29C22×Dic94—21no—
30C22×Dic94D412yes—
31C23×Dic47—21no—
32C23×Dic47—21no—
33C23×Dic47—21no—
34C23×Dic47—21no—
35#13094003—24no—
36#13094003—22no—
37#13094003—22no—
38#13094003—24no—
39#13094003D414yes—
40#13094003C2314yes—
41#13094001—28no—
42#13094001—28no—
43#13094001—28no—
44#13094001—28no—
45C2×C4⋊Dic47—21no—
46C2×C4⋊Dic47—22no—
47C2×C4⋊Dic47—22no—
48C2×C4⋊Dic47—21no—
49C2×C4⋊Dic47D412yes—
50C2×C4⋊Dic47C2312yes—
51C2×Dic47⋊C4—22no—
52C2×Dic47⋊C4—22no—
53C2×C22⋊Dic47—24no—
54C2×C22⋊Dic47—22no—
55C2×C22⋊Dic47—24no—
56C2×C22⋊Dic47—22no—
57C2×C22⋊Dic47—24no—
58C2×C22⋊Dic47—44no—
59#13093956—28no—
60#13093949—28no—
61C47⋊#95—24no—
62C47⋊#95—21no—
63C47⋊#95C2312yes—
64C25×C47D411yescharacteristic
65C23×C4×C47D411yescharacteristic
66C23×C4×C47C2311yescharacteristic
67C22×C8×C47—22no—
68C2×C22⋊C4×C47—22no—
69C2×C22⋊C4×C47—22no—
70C2×C22⋊C4×C47D412yes—
71C2×C22⋊C4×C47C2312yes—
72C22⋊C8×C47—22no—
73C22⋊C8×C47C2314yes—
74C24×C47—21no—
75C24×C47—21no—
76C24×C47—22no—
77C24×C47—42no—
78C24×C47—42no—
79C24×C47SD1614yes—
80C24×C47C2×D411yescharacteristic
81C24×C47C2×D412yes—
82C22×C4×C47—21no—
83C22×C4×C47—21no—
84C22×C4×C47—42no—
85C22×C4×C47—42no—
86C22×C4×C47C2×D412yes—
87C22×C4×C47C2×D412yes—
88C22×C4×C47C2×D411yescharacteristic
89C22×C4×C47C2×D412yes—
90C22×C4×C47C2411yescharacteristic
91C22×Dic47—22no—
92C22×Dic47—22no—
93C22×Dic47—24no—
94C22×Dic47—24no—
95C22×Dic47—22no—
96C22×Dic47—22no—
97C22×Dic47—42no—
98C22×Dic47—42no—
99C22×Dic47—41no—
100C22×Dic47—41no—
101C22×Dic47—42no—
102C22×Dic47—42no—
103C22×Dic47—42no—
104C22×Dic47—82no—
105C22×Dic47—82no—
106C2×Dic94—24no—
107C2×Dic94—22no—
108C2×Dic94—44no—
109C2×Dic94—82no—
110C2×Dic94C2×D414yes—
111C2×C8×C47—24no—
112C2×C8×C47—44no—
113C22⋊C4×C47—24no—
114C22⋊C4×C47—24no—
115C22⋊Dic47—28no—
116C22⋊Dic47—24no—
117C22⋊Dic47—24no—
118C22⋊Dic47—48no—
119C4⋊Dic47—24no—
120C4⋊Dic47—22no—
121C4⋊Dic47—24no—
122C4⋊Dic47—42no—
123C4⋊Dic47C2×D414yes—
124Dic47⋊C4—28no—
125Dic47⋊C4—24no—
126Dic47⋊C4—24no—
127C23×C47—22no—
128C23×C47—22no—
129C23×C47—22no—
130C23×C47—24no—
131C23×C47—22no—
132C23×C47—24no—
133C23×C47—48no—
134C23×C47—42no—
135C23×C47—42no—
136C23×C47—42no—
137C23×C47—42no—
138C23×C47—84no—
139C23×C47—81no—
140C23×C47—81no—
141C23×C47#12011yescharacteristic
142C23×C47#12011yescharacteristic
143C23×C47C2×SD1612yes—
144C23×C47C2×SD1614yes—
145C23×C47#13712yes—
146C23×C47C22×D411yescharacteristic
147C2×C4×C47—24no—
148C2×C4×C47—22no—
149C2×C4×C47—41no—
150C2×C4×C47—42no—
151C2×C4×C47—41no—
152C2×C4×C47—42no—
153C2×C4×C47—42no—
154C2×C4×C47—81no—
155C2×C4×C47#12012yes—
156C2×C4×C47C22×D412yes—
157C2×Dic47—24no—
158C2×Dic47—24no—
159C2×Dic47—44no—
160C2×Dic47—44no—
161C2×Dic47—42no—
162C2×Dic47—42no—
163C2×Dic47—42no—
164C2×Dic47—42no—
165C2×Dic47—42no—
166C2×Dic47—42no—
167C2×Dic47—84no—
168C2×Dic47—84no—
169C2×Dic47—84no—
170C2×Dic47—82no—
171C2×Dic47—82no—
172Dic94—44no—
173Dic94—82no—
174Dic94—84no—
175C8×C47—44no—
176C2×#1608—471nomaximal, Sylow
177C22×C47—24no—
178C22×C47—24no—
179C22×C47—21no—
180C22×C47—21no—
181C22×C47—21no—
182C22×C47—21no—
183C22×C47—24no—
184C22×C47—44no—
185C22×C47—48no—
186C22×C47—42no—
187C22×C47—42no—
188C22×C47—82no—
189C22×C47—84no—
190C22×C47#44812yes—
191C22×C47#44812yes—
192C22×C47#45512yes—
193C22×C47#45812yes—
194C22×C47C2×#12011yescharacteristic
195C22×C47C22×SD1611yescharacteristic
196C22×C47C2×#13711yescharacteristic
197C4×C47—22no—
198C4×C47—22no—
199C4×C47—42no—
200Dic47—44no—
201Dic47—44no—
202Dic47—82no—
203Dic47—82no—
204C2×#543—471no—
205C2×#483—472no—
206C2×#397—471no—
207#1608—478no—
208C2×#328—472no—
209C2×#323D4711yescharacteristic
210C2×C47—21no—
211C2×C47—21no—
212C2×C47—22no—
213C2×C47—44no—
214C2×C47#160814yescomplemented
215C2×C47C2×#44811yescharacteristic
216C2×C47C2×#45511yescharacteristic
217C2×C47C2×#45811yescharacteristic
218#543—474no—
219#543—944no—
220#543—942no—
221C2×C22⋊Q8—472no—
222C2×C22⋊Q8—942no—
223C2×C22⋊Q8—942no—
224C2×C22⋊Q8—941no—
225C22×C4⋊C4—471no—
226C22×C22⋊C4—941no—
227C22×C22⋊C4—941no—
228C22×C22⋊C4C2×D4711yescharacteristic
229#483—478no—
230#483—944no—
231C2×#106—942no—
232C2×#103—942no—
233C2×C22⋊C8C2×D4712yes—
234#397—474no—
235#397—942no—
236C2×#95—471no—
237C2×#95—941no—
238#328—478no—
239#323C2×D4714yes—
240C47C2×#160811yescharacteristic, complemented, Sylow
241C25D18811yescharacteristic
242C22×Q8—472no—
243C22×Q8—941no—
244C23×C4—941no—
245C23×C4—941no—
246C23×C4—941no—
247C23×C4—941no—
248C23×C4D18811yescharacteristic
249C23×C4C22×D4711yescharacteristic
250C22×C8—22no—
251C22⋊Q8—474no—
252C22⋊Q8—474no—
253C22⋊Q8—948no—
254C22⋊Q8—948no—
255C22⋊Q8—948no—
256C22⋊Q8—948no—
257C22⋊Q8—944no—
258C22⋊Q8—942no—
259C22⋊Q8—944no—
260C22⋊Q8—942no—
261C2×C4⋊C4—472no—
262C2×C4⋊C4—472no—
263C2×C4⋊C4—942no—
264C2×C4⋊C4—941no—
265C2×C4⋊C4—942no—
266C2×C4⋊C4—942no—
267C2×C4⋊C4—942no—
268C2×C4⋊C4—941no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—942no—
272C2×C22⋊C4—944no—
273C2×C22⋊C4—942no—
274C2×C22⋊C4—944no—
275C2×C22⋊C4—944no—
276C2×C22⋊C4—1884no—
277C2×C22⋊C4D18812yes—
278C2×C22⋊C4C22×D4712yes—
279#106—948no—
280#103—948no—
281C22⋊C8—22no—
282C22⋊C8C22×D4714yes—
283#95—472no—
284#95—941no—
285#95—944no—
286C24—21no—
287C24—21no—
288C24—22no—
289C24—42no—
290C24—42no—
291C24#1062532514yes—
292C24C2×D18811yescharacteristic
293C24C2×D18812yes—
294C2×Q8—474no—
295C2×Q8—944no—
296C2×Q8—942no—
297C2×Q8—3762no—
298C2×Q8—1884no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—944no—
304C22×C4—944no—
305C22×C4—942no—
306C22×C4—942no—
307C22×C4—942no—
308C22×C4—942no—
309C22×C4—3762no—
310C22×C4—3762no—
311C22×C4—1882no—
312C22×C4—1882no—
313C22×C4—1881no—
314C22×C4—1881no—
315C22×C4—1882no—
316C22×C4—1882no—
317C22×C4—1882no—
318C22×C4C2×D18812yes—
319C22×C4C2×D18811yescharacteristic
320C22×C4D4×D4712yes—
321C22×C4D4×D4712yes—
322C22×C4C23×D4711yescharacteristic
323C2×C8—24no—
324C2×C8—44no—
325C4⋊C4—474no—
326C4⋊C4—948no—
327C4⋊C4—944no—
328C4⋊C4—944no—
329C4⋊C4—942no—
330C4⋊C4—944no—
331C4⋊C4—944no—
332C4⋊C4—1882no—
333C22⋊C4—24no—
334C22⋊C4—24no—
335C22⋊C4—948no—
336C22⋊C4—944no—
337C22⋊C4—944no—
338C22⋊C4—1888no—
339C23—22no—
340C23—22no—
341C23—24no—
342C23—22no—
343C23—24no—
344C23—22no—
345C23—48no—
346C23—42no—
347C23—42no—
348C23—42no—
349C23—42no—
350C23—81no—
351C23—81no—
352C23—84no—
353C23#1309399411yescharacteristic
354C23#1309399411yescharacteristic
355C23C2×#1062532512yes—
356C23C2×#1062532514yes—
357C23#1309405712yes—
358C23C22×D18811yescharacteristic
359Q8—3762no—
360Q8—3764no—
361Q8—1884no—
362C2×C4—24no—
363C2×C4—22no—
364C2×C4—41no—
365C2×C4—42no—
366C2×C4—41no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—81no—
370C2×C4—944no—
371C2×C4—944no—
372C2×C4—3764no—
373C2×C4—3762no—
374C2×C4—3762no—
375C2×C4—3764no—
376C2×C4—3764no—
377C2×C4—1882no—
378C2×C4—1882no—
379C2×C4—1882no—
380C2×C4—1882no—
381C2×C4—1882no—
382C2×C4—1882no—
383C2×C4—1884no—
384C2×C4—1884no—
385C2×C4#1309399412yes—
386C2×C4C2×D4×D4712yes—
387C8—44no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—24no—
393C22—24no—
394C22—24no—
395C22—44no—
396C22—42no—
397C22—42no—
398C22—48no—
399C22—84no—
400C22—82no—
401C22#1568926412yes—
402C22#1568926412yes—
403C22#1568930312yes—
404C22#1568931412yes—
405C22C2×#1309399411yescharacteristic
406C22C22×#1062532511yescharacteristic
407C22C2×#1309405711yescharacteristic
408C4—22no—
409C4—22no—
410C4—42no—
411C4—3762no—
412C4—3762no—
413C4—1884no—
414C4—1884no—
415C2—21no—
416C2—22no—
417C2—21no—
418C2—44no—
419C2#2024238114yesdirect product factor
420C2C2×#1568926411yescharacteristic
421C2C2×#1568930311yescharacteristic
422C2C2×#1568931411yescharacteristic
subgroupG/H
centerC23#13093994
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC2×C47×#323C2
socleC23×C47#120
prime radicalC25×C47D4
hypercenterC2×#323D47
hypocenterC47C2×#1608
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#323

Complements

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

NH ≅ G/Ntype
2. C2×#15689454C2semidirect
4. #20242381C2direct (N × H)
5. C2×#15688861C2semidirect
18. #15689454C22semidirect
25. #15688861C22semidirect
214. C2×C47#1608semidirect
240. C47C2×#1608semidirect
419. C2#20242381direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156897492
C2×#156894542
C2×#156890482
#202423812
C2×#156888612
C2×C47×#3232
C2×#160847