Finite Group Databasebeta

← C2×#20242415 #26188413

Subgroups — C2×#20242415 #26188413

Order 12032.

1180 conjugacy classes of subgroups; only the 203 normal subgroups are shown.

normal subgroups only

This group has 1180 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×#15689459C212yesmaximal, complemented
3C2×#15689059C211yesmaximal, characteristic
4#20242415C218yesmaximal, 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×Dic47⋊C4—21no—
16C22×C22⋊Dic47—21no—
17C22×C22⋊Dic47—21no—
18#15689459—24no—
19#15689459C2218yescomplemented
20C2×#13093958—22no—
21C2×#13093949—22no—
22#15689059—22no—
23#15689059C2214yes—
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×Dic47⋊C4—24no—
54C2×Dic47⋊C4—44no—
55C2×C22⋊Dic47—24no—
56C2×C22⋊Dic47—22no—
57C2×C22⋊Dic47—24no—
58C2×C22⋊Dic47—22no—
59C2×C22⋊Dic47—24no—
60C2×C22⋊Dic47—44no—
61#13093958—28no—
62#13093949—28no—
63C47⋊#95—21no—
64C47⋊#95C2312yes—
65C25×C47D411yescharacteristic
66C23×C4×C47D411yescharacteristic
67C23×C4×C47C2311yescharacteristic
68C22×C8×C47—22no—
69C2×C22⋊C4×C47—22no—
70C2×C22⋊C4×C47—22no—
71C2×C22⋊C4×C47D412yes—
72C2×C22⋊C4×C47C2312yes—
73C22⋊C8×C47—22no—
74C22⋊C8×C47C2314yes—
75C24×C47—21no—
76C24×C47—21no—
77C24×C47—22no—
78C24×C47—42no—
79C24×C47—42no—
80C24×C47Q1614yes—
81C24×C47C2×D411yescharacteristic
82C24×C47C2×D412yes—
83C22×C4×C47—21no—
84C22×C4×C47—21no—
85C22×C4×C47—42no—
86C22×C4×C47—42no—
87C22×C4×C47C2×D412yes—
88C22×C4×C47C2×D412yes—
89C22×C4×C47C2×D411yescharacteristic
90C22×C4×C47C2×D412yes—
91C22×C4×C47C2411yescharacteristic
92C22×Dic47—24no—
93C22×Dic47—22no—
94C22×Dic47—22no—
95C22×Dic47—22no—
96C22×Dic47—24no—
97C22×Dic47—24no—
98C22×Dic47—22no—
99C22×Dic47—42no—
100C22×Dic47—42no—
101C22×Dic47—42no—
102C22×Dic47—42no—
103C22×Dic47—41no—
104C22×Dic47—41no—
105C22×Dic47—82no—
106C22×Dic47—82no—
107C2×Dic94—24no—
108C2×Dic94—22no—
109C2×Dic94—44no—
110C2×Dic94—82no—
111C2×Dic94C2×D414yes—
112C2×C8×C47—24no—
113C2×C8×C47—44no—
114C22⋊C4×C47—24no—
115C22⋊C4×C47—24no—
116C22⋊Dic47—28no—
117C22⋊Dic47—24no—
118C22⋊Dic47—24no—
119C22⋊Dic47—48no—
120C4⋊Dic47—24no—
121C4⋊Dic47—22no—
122C4⋊Dic47—24no—
123C4⋊Dic47—42no—
124C4⋊Dic47C2×D414yes—
125Dic47⋊C4—28no—
126Dic47⋊C4—24no—
127Dic47⋊C4—24no—
128Dic47⋊C4—48no—
129C23×C47—22no—
130C23×C47—22no—
131C23×C47—22no—
132C23×C47—24no—
133C23×C47—22no—
134C23×C47—24no—
135C23×C47—42no—
136C23×C47—48no—
137C23×C47—42no—
138C23×C47—42no—
139C23×C47—42no—
140C23×C47—84no—
141C23×C47—81no—
142C23×C47—81no—
143C23×C47#12011yescharacteristic
144C23×C47#12011yescharacteristic
145C23×C47C2×Q1612yes—
146C23×C47C2×Q1614yes—
147C23×C47#13712yes—
148C23×C47C22×D411yescharacteristic
149C2×C4×C47—24no—
150C2×C4×C47—22no—
151C2×C4×C47—41no—
152C2×C4×C47—42no—
153C2×C4×C47—41no—
154C2×C4×C47—42no—
155C2×C4×C47—42no—
156C2×C4×C47—81no—
157C2×C4×C47#12012yes—
158C2×C4×C47C22×D412yes—
159C2×Dic47—24no—
160C2×Dic47—24no—
161C2×Dic47—24no—
162C2×Dic47—48no—
163C2×Dic47—42no—
164C2×Dic47—42no—
165C2×Dic47—44no—
166C2×Dic47—44no—
167C2×Dic47—42no—
168C2×Dic47—42no—
169C2×Dic47—42no—
170C2×Dic47—84no—
171C2×Dic47—84no—
172C2×Dic47—82no—
173C2×Dic47—82no—
174Dic94—44no—
175Dic94—82no—
176Dic94—84no—
177C8×C47—44no—
178C2×#1614—471nomaximal, Sylow
179C22×C47—24no—
180C22×C47—24no—
181C22×C47—21no—
182C22×C47—21no—
183C22×C47—21no—
184C22×C47—21no—
185C22×C47—24no—
186C22×C47—44no—
187C22×C47—48no—
188C22×C47—42no—
189C22×C47—42no—
190C22×C47—82no—
191C22×C47—84no—
192C22×C47#45112yes—
193C22×C47#45112yes—
194C22×C47#45312yes—
195C22×C47#45812yes—
196C22×C47C2×#12011yescharacteristic
197C22×C47C22×Q1611yescharacteristic
198C22×C47C2×#13711yescharacteristic
199C4×C47—22no—
200C4×C47—22no—
201C4×C47—42no—
202Dic47—44no—
203Dic47—44no—
204Dic47—44no—
205Dic47—82no—
206C2×#543—471no—
207C2×#484—472no—
208C2×#399—471no—
209#1614—478no—
210C2×#328—472no—
211C2×#323D4711yescharacteristic
212C2×C47—21no—
213C2×C47—21no—
214C2×C47—22no—
215C2×C47—44no—
216C2×C47#161414yescomplemented
217C2×C47C2×#45111yescharacteristic
218C2×C47C2×#45311yescharacteristic
219C2×C47C2×#45811yescharacteristic
220#543—474no—
221#543—944no—
222#543—942no—
223C2×C22⋊Q8—472no—
224C2×C22⋊Q8—942no—
225C2×C22⋊Q8—942no—
226C2×C22⋊Q8—941no—
227C22×C4⋊C4—471no—
228C22×C4⋊C4—941no—
229C22×C22⋊C4—941no—
230C22×C22⋊C4—941no—
231C22×C22⋊C4C2×D4711yescharacteristic
232#484—478no—
233#484—944no—
234C2×#107—942no—
235C2×#103—942no—
236C2×C22⋊C8C2×D4712yes—
237#399—474no—
238#399—942no—
239C2×#95—471no—
240#328—478no—
241#323C2×D4714yes—
242C47C2×#161411yescharacteristic, complemented, Sylow
243C25D18811yescharacteristic
244C22×Q8—472no—
245C22×Q8—941no—
246C23×C4—941no—
247C23×C4—941no—
248C23×C4—941no—
249C23×C4—941no—
250C23×C4D18811yescharacteristic
251C23×C4C22×D4711yescharacteristic
252C22×C8—22no—
253C22⋊Q8—474no—
254C22⋊Q8—474no—
255C22⋊Q8—948no—
256C22⋊Q8—948no—
257C22⋊Q8—948no—
258C22⋊Q8—948no—
259C22⋊Q8—944no—
260C22⋊Q8—942no—
261C22⋊Q8—944no—
262C22⋊Q8—942no—
263C2×C4⋊C4—472no—
264C2×C4⋊C4—472no—
265C2×C4⋊C4—942no—
266C2×C4⋊C4—941no—
267C2×C4⋊C4—942no—
268C2×C4⋊C4—942no—
269C2×C4⋊C4—942no—
270C2×C4⋊C4—941no—
271C2×C4⋊C4—944no—
272C2×C4⋊C4—1884no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—942no—
276C2×C22⋊C4—944no—
277C2×C22⋊C4—942no—
278C2×C22⋊C4—944no—
279C2×C22⋊C4—944no—
280C2×C22⋊C4—1884no—
281C2×C22⋊C4D18812yes—
282C2×C22⋊C4C22×D4712yes—
283#107—948no—
284#103—948no—
285C22⋊C8—22no—
286C22⋊C8C22×D4714yes—
287#95—472no—
288#95—941no—
289C24—21no—
290C24—21no—
291C24—22no—
292C24—42no—
293C24—42no—
294C24Dic18814yes—
295C24C2×D18811yescharacteristic
296C24C2×D18812yes—
297C2×Q8—474no—
298C2×Q8—944no—
299C2×Q8—942no—
300C2×Q8—3762no—
301C2×Q8—1884no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—944no—
307C22×C4—944no—
308C22×C4—944no—
309C22×C4—942no—
310C22×C4—942no—
311C22×C4—942no—
312C22×C4—942no—
313C22×C4—3762no—
314C22×C4—3762no—
315C22×C4—1882no—
316C22×C4—1882no—
317C22×C4—1881no—
318C22×C4—1881no—
319C22×C4—1882no—
320C22×C4—1882no—
321C22×C4C2×D18812yes—
322C22×C4C2×D18811yescharacteristic
323C22×C4D4×D4712yes—
324C22×C4D4×D4712yes—
325C22×C4C23×D4711yescharacteristic
326C2×C8—24no—
327C2×C8—44no—
328C4⋊C4—474no—
329C4⋊C4—948no—
330C4⋊C4—944no—
331C4⋊C4—944no—
332C4⋊C4—942no—
333C4⋊C4—944no—
334C4⋊C4—944no—
335C4⋊C4—1888no—
336C4⋊C4—1882no—
337C22⋊C4—24no—
338C22⋊C4—24no—
339C22⋊C4—948no—
340C22⋊C4—944no—
341C22⋊C4—944no—
342C22⋊C4—1888no—
343C23—22no—
344C23—22no—
345C23—24no—
346C23—22no—
347C23—24no—
348C23—22no—
349C23—42no—
350C23—48no—
351C23—42no—
352C23—42no—
353C23—42no—
354C23—81no—
355C23—81no—
356C23—84no—
357C23#1309399411yescharacteristic
358C23#1309399411yescharacteristic
359C23C2×Dic18812yes—
360C23C2×Dic18814yes—
361C23#1309405712yes—
362C23C22×D18811yescharacteristic
363Q8—3762no—
364Q8—3764no—
365Q8—1884no—
366C2×C4—24no—
367C2×C4—22no—
368C2×C4—41no—
369C2×C4—42no—
370C2×C4—41no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—81no—
374C2×C4—944no—
375C2×C4—944no—
376C2×C4—944no—
377C2×C4—3762no—
378C2×C4—3762no—
379C2×C4—3764no—
380C2×C4—3764no—
381C2×C4—1882no—
382C2×C4—1882no—
383C2×C4—1882no—
384C2×C4—1882no—
385C2×C4—1888no—
386C2×C4—1884no—
387C2×C4—1884no—
388C2×C4—1882no—
389C2×C4#1309399412yes—
390C2×C4C2×D4×D4712yes—
391C8—44no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—24no—
397C22—24no—
398C22—24no—
399C22—44no—
400C22—42no—
401C22—42no—
402C22—48no—
403C22—82no—
404C22—84no—
405C22#1568928112yes—
406C22#1568928112yes—
407C22#1568929312yes—
408C22#1568931412yes—
409C22C2×#1309399411yescharacteristic
410C22C22×Dic18811yescharacteristic
411C22C2×#1309405711yescharacteristic
412C4—22no—
413C4—22no—
414C4—42no—
415C4—3762no—
416C4—1884no—
417C4—1884no—
418C4—1884no—
419C2—21no—
420C2—21no—
421C2—22no—
422C2—44no—
423C2#2024241514yesdirect product factor
424C2C2×#1568928111yescharacteristic
425C2C2×#1568929311yescharacteristic
426C2C2×#1568931411yescharacteristic
subgroupG/H
centerC23#13093994
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC2×C47×#323C2
socleC23×C47#120
prime radicalC25×C47D4
hypercenterC2×#323D47
hypocenterC47C2×#1614
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×#15689459C2semidirect
4. #20242415C2direct (N × H)
5. C2×#15688861C2semidirect
19. #15689459C22semidirect
25. #15688861C22semidirect
216. C2×C47#1614semidirect
242. C47C2×#1614semidirect
423. C2#20242415direct (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×#156894592
C2×#156890592
#202424152
C2×#156888612
C2×C47×#3232
C2×#161447