Finite Group Databasebeta

← C2×#20242424 #26188500

Subgroups — C2×#20242424 #26188500

Order 12032.

1264 conjugacy classes of subgroups; only the 203 normal subgroups are shown.

normal subgroups only

This group has 1264 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×#15689704C211yesmaximal, characteristic
2C2×#15689441C212yesmaximal, complemented
3#20242424C218yesmaximal, direct product factor
4C2×#15688856C212yesmaximal, complemented
5C2×#15688849C211yesmaximal, characteristic, complemented
6C2×C47×#400C211yesmaximal, characteristic, complemented
7C22×#10625350—21no—
8#15689704—24no—
9#15689704—22no—
10#15689704C2214yes—
11C2×#13094004—22no—
12C2×#13093999—22no—
13C2×#13093998—21no—
14C2×#13093998C2212yes—
15#15689441—24no—
16#15689441C2218yescomplemented
17C2×#13093955—22no—
18C2×#13093944—22no—
19C2×C22⋊(C47⋊C8)C2212yes—
20#15688856C2218yescomplemented
21#15688849C2214yescomplemented
22C22×C4⋊C4×C47C2211yescharacteristic
23C22×C4⋊C4×C47C2211yescharacteristic
24C47×#400—22no—
25C47×#400C2214yescomplemented
26C2×#95×C47—21no—
27C2×#95×C47—21no—
28C2×#95×C47C2211yescharacteristic
29C2×#10625350—24no—
30C2×#10625350—24no—
31C2×#10625350—44no—
32C2×#10625350—44no—
33C22×Dic94—21no—
34C22×D188—21no—
35C22×D188—21no—
36C22×D188D412yes—
37C22×C22⋊D47—22no—
38C22×C4×D47—22no—
39C22×C47⋊C8—22no—
40#13094004—28no—
41#13094004—28no—
42#13093999—28no—
43#13093999—28no—
44#13093998—24no—
45#13093998—24no—
46#13093998—22no—
47#13093998—22no—
48#13093998D414yes—
49#13093998C2314yes—
50C2×C4⋊Dic47—21no—
51C2×C4⋊Dic47D412yes—
52C2×(C2×D47)⋊C4—22no—
53C2×(C2×D47)⋊C4—22no—
54#13093955—28no—
55#13093944—28no—
56C22⋊(C47⋊C8)—22no—
57C22⋊(C47⋊C8)C2314yes—
58C23×C4×C47—21no—
59C23×C4×C47—21no—
60C23×C4×C47D411yescharacteristic
61C23×C4×C47D411yescharacteristic
62C23×C4×C47C2311yescharacteristic
63C2×C4⋊C4×C47—22no—
64C2×C4⋊C4×C47—22no—
65C2×C4⋊C4×C47—22no—
66C2×C4⋊C4×C47—22no—
67C2×C4⋊C4×C47—21no—
68C2×C4⋊C4×C47D412yes—
69C2×C4⋊C4×C47C2312yes—
70C2×C4⋊C4×C47C2312yes—
71#95×C47—24no—
72#95×C47—24no—
73#95×C47—21no—
74#95×C47C2312yes—
75C24×C47C2×D411yescharacteristic
76C23×D47—22no—
77C22×C4×C47—21no—
78C22×C4×C47—21no—
79C22×C4×C47—22no—
80C22×C4×C47—21no—
81C22×C4×C47—21no—
82C22×C4×C47—22no—
83C22×C4×C47—22no—
84C22×C4×C47—42no—
85C22×C4×C47—42no—
86C22×C4×C47—42no—
87C22×C4×C47—42no—
88C22×C4×C47—44no—
89C22×C4×C47—41no—
90C22×C4×C47—41no—
91C22×C4×C47—42no—
92C22×C4×C47SD1614yes—
93C22×C4×C47C2×D412yes—
94C22×C4×C47C2×D412yes—
95C22×C4×C47C2×D412yes—
96C22×C4×C47C2×D411yescharacteristic
97C22×C4×C47C2×D412yes—
98C22×C4×C47C2411yescharacteristic
99C2×C22⋊D47—24no—
100C2×C22⋊D47—24no—
101C2×C22⋊D47—44no—
102C2×C22⋊D47—44no—
103C22×Dic47—22no—
104#10625350—48no—
105#10625350—416no—
106#10625350—48no—
107C2×D188—24no—
108C2×D188—24no—
109C2×D188—22no—
110C2×D188—44no—
111C2×D188—44no—
112C2×D188—82no—
113C2×D188C2×D414yes—
114C2×C4×D47—28no—
115C2×C4×D47—48no—
116C2×Dic94—24no—
117C2×Dic94—44no—
118C4⋊C4×C47—24no—
119C4⋊C4×C47—24no—
120C4⋊C4×C47—24no—
121C4⋊C4×C47—42no—
122(C2×D47)⋊C4—28no—
123(C2×D47)⋊C4—24no—
124(C2×D47)⋊C4—24no—
125C4⋊Dic47—24no—
126C4⋊Dic47—22no—
127C4⋊Dic47C2×D414yes—
128C2×C47⋊C8—24no—
129C2×C47⋊C8—44no—
130C23×C47—22no—
131C23×C47—22no—
132C23×C47#12011yescharacteristic
133C23×C47#12011yescharacteristic
134C23×C47C2×SD1612yes—
135C23×C47AGL(1,ℤ/8)12yes—
136C23×C47C22×D411yescharacteristic
137C22×D47—24no—
138C22×D47—44no—
139C22×D47—44no—
140C22×D47—44no—
141C2×C4×C47—22no—
142C2×C4×C47—22no—
143C2×C4×C47—24no—
144C2×C4×C47—24no—
145C2×C4×C47—24no—
146C2×C4×C47—22no—
147C2×C4×C47—42no—
148C2×C4×C47—41no—
149C2×C4×C47—41no—
150C2×C4×C47—42no—
151C2×C4×C47—42no—
152C2×C4×C47—42no—
153C2×C4×C47—42no—
154C2×C4×C47—44no—
155C2×C4×C47—42no—
156C2×C4×C47—42no—
157C2×C4×C47—81no—
158C2×C4×C47—84no—
159C2×C4×C47—82no—
160C2×C4×C47—82no—
161C2×C4×C47#12012yes—
162C2×C4×C47C2×SD1614yes—
163C2×C4×C47C22×D412yes—
164C22⋊D47—44no—
165C22⋊D47—44no—
166C22⋊D47—48no—
167C2×Dic47—24no—
168C2×Dic47—44no—
169D188—44no—
170D188—48no—
171D188—82no—
172D188—84no—
173C4×D47—416no—
174Dic94—48no—
175C47⋊C8—44no—
176C2×#1616—471nomaximal, Sylow
177C22×C47—24no—
178C22×C47—24no—
179C22×C47—21no—
180C22×C47—21no—
181C22×C47—21no—
182C22×C47—21no—
183C22×C47#45012yes—
184C22×C47#45012yes—
185C22×C47#45412yes—
186C22×C47#45812yes—
187C22×C47C2×#12011yescharacteristic
188C22×C47C22×SD1611yescharacteristic
189C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
190C2×D47—44no—
191C2×D47—44no—
192C2×D47—44no—
193C2×D47—88no—
194C4×C47—24no—
195C4×C47—22no—
196C4×C47—22no—
197C4×C47—42no—
198C4×C47—82no—
199C4×C47—82no—
200Dic47—44no—
201C2×#537—471no—
202C2×#481—472no—
203C2×#400D4711yescharacteristic, complemented
204#1616—478no—
205C2×#327—472no—
206C2×#324—471no—
207C2×C47—21no—
208C2×C47—21no—
209C2×C47—22no—
210C2×C47#161614yescomplemented
211C2×C47C2×#45011yescharacteristic
212C2×C47C2×#45411yescharacteristic
213C2×C47C2×#45811yescharacteristic
214D47—84no—
215C22×#41—941no—
216#537—474no—
217#537—944no—
218#537—942no—
219C2×C22⋊Q8—942no—
220C2×#121—472no—
221C2×#121—942no—
222C2×#121—941no—
223C22×C4⋊C4C2×D4711yescharacteristic
224C22×C4⋊C4C2×D4711yescharacteristic
225#481—478no—
226#481—944no—
227C2×#106—942no—
228C2×#102—942no—
229C2×C22⋊C8—472no—
230#400—22no—
231#400C2×D4714yescomplemented
232C2×#95—21no—
233C2×#95—21no—
234C2×#95C2×D4711yescharacteristic
235#327—478no—
236#324—474no—
237C47C2×#161611yescharacteristic, complemented, Sylow
238C2×#41—944no—
239C2×#41—944no—
240C2×#41—1884no—
241C2×#41—1884no—
242C22×Q8—941no—
243C22×D4—472no—
244C22×D4—941no—
245C22×D4—941no—
246C22×D4—942no—
247C23×C4—21no—
248C23×C4—21no—
249C23×C4—942no—
250C23×C4C22⋊D4711yescharacteristic
251C23×C4C22⋊D4711yescharacteristic
252C23×C4C22×D4711yescharacteristic
253C22×C8—942no—
254C22⋊Q8—948no—
255C22⋊Q8—948no—
256#121—474no—
257#121—474no—
258#121—948no—
259#121—948no—
260#121—944no—
261#121—942no—
262#121—942no—
263#121—944no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—472no—
270C2×C4⋊C4—941no—
271C2×C4⋊C4C22⋊D4712yes—
272C2×C4⋊C4C22×D4712yes—
273C2×C4⋊C4C22×D4712yes—
274C2×C22⋊C4—942no—
275C2×C22⋊C4—942no—
276#106—948no—
277#102—948no—
278C22⋊C8—474no—
279C22⋊C8—942no—
280#95—24no—
281#95—24no—
282#95—21no—
283#95C22×D4712yes—
284C24—942no—
285C24C2×C22⋊D4711yescharacteristic
286#41—18816no—
287#41—1888no—
288#41—1888no—
289C2×Q8—944no—
290C2×Q8—1884no—
291C2×D4—474no—
292C2×D4—944no—
293C2×D4—944no—
294C2×D4—944no—
295C2×D4—944no—
296C2×D4—942no—
297C2×D4—3762no—
298C2×D4—1884no—
299C2×D4—1884no—
300C2×D4—1884no—
301C2×D4—1884no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—22no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—44no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—42no—
317C22×C4—948no—
318C22×C4—942no—
319C22×C4—1888no—
320C22×C4#1062533614yes—
321C22×C4D4×D4712yes—
322C22×C4D4×D4712yes—
323C22×C4C2×C22⋊D4712yes—
324C22×C4C2×C22⋊D4712yes—
325C22×C4C2×C22⋊D4711yescharacteristic
326C22×C4C23×D4711yescharacteristic
327C2×C8—944no—
328C2×C8—1884no—
329C4⋊C4—24no—
330C4⋊C4—24no—
331C4⋊C4—24no—
332C4⋊C4—42no—
333C4⋊C4—474no—
334C4⋊C4—944no—
335C4⋊C4—942no—
336C22⋊C4—948no—
337C22⋊C4—944no—
338C22⋊C4—944no—
339C23—22no—
340C23—22no—
341C23—944no—
342C23—1884no—
343C23—1884no—
344C23—1884no—
345C23#1309399211yescharacteristic
346C23#1309399211yescharacteristic
347C23C2×#1062533612yes—
348C23#1309404912yes—
349C23C22×C22⋊D4711yescharacteristic
350Q8—1888no—
351D4—3762no—
352D4—3764no—
353D4—1884no—
354D4—1888no—
355D4—1888no—
356D4—1884no—
357D4—1884no—
358C2×C4—24no—
359C2×C4—24no—
360C2×C4—24no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—42no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—44no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—81no—
375C2×C4—84no—
376C2×C4—82no—
377C2×C4—82no—
378C2×C4—944no—
379C2×C4—18816no—
380C2×C4—1884no—
381C2×C4#1309399212yes—
382C2×C4C2×#1062533614yes—
383C2×C4C2×D4×D4712yes—
384C8—1884no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—24no—
390C22—24no—
391C22—3768no—
392C22—1884no—
393C22—1884no—
394C22—1884no—
395C22#1568927312yes—
396C22#1568927312yes—
397C22#1568929512yes—
398C22#1568931212yes—
399C22C2×#1309399211yescharacteristic
400C22C22×#1062533611yescharacteristic
401C22C2×#1309404911yescharacteristic
402C4—22no—
403C4—22no—
404C4—24no—
405C4—42no—
406C4—82no—
407C4—82no—
408C4—1884no—
409C2—21no—
410C2—22no—
411C2—21no—
412C2—3764no—
413C2#2024242414yesdirect product factor
414C2C2×#1568927311yescharacteristic
415C2C2×#1568929511yescharacteristic
416C2C2×#1568931211yescharacteristic
subgroupG/H
centerC23#13093992
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC2×C47×#400C2
socleC23×C47#120
prime radicalC2×#15689704C2
hypercenterC2×#400D47
hypocenterC47C2×#1616
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#400

Complements

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

NH ≅ G/Ntype
2. C2×#15689441C2semidirect
3. #20242424C2direct (N × H)
4. C2×#15688856C2semidirect
5. C2×#15688849C2semidirect
6. C2×C47×#400C2semidirect
16. #15689441C22semidirect
20. #15688856C22semidirect
21. #15688849C22semidirect
25. C47×#400C22semidirect
203. C2×#400D47semidirect
210. C2×C47#1616semidirect
231. #400C2×D47semidirect
237. C47C2×#1616semidirect
413. C2#20242424direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156897042
C2×#156894412
#202424242
C2×#156888562
C2×#156888492
C2×C47×#4002
C2×#161647