Finite Group Databasebeta

← C2×#20242356 #26188449

Subgroups — C2×#20242356 #26188449

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×#15689704C211yesmaximal, characteristic
2C2×#15689262C212yesmaximal, complemented
3#20242356C218yesmaximal, direct product factor
4C2×#15688856C212yesmaximal, complemented
5C2×#15688849C211yesmaximal, characteristic, complemented
6C2×C47×#395C211yesmaximal, characteristic, complemented
7C22×#10625350—21no—
8C22×#10625336—22no—
9#15689704—24no—
10#15689704—22no—
11#15689704C2214yes—
12C2×#13094004—22no—
13C2×#13093999—22no—
14C2×#13093998—21no—
15C2×#13093998C2212yes—
16#15689262—24no—
17#15689262C2218yescomplemented
18C2×#13093948—22no—
19C2×C22⋊(C47⋊C8)C2212yes—
20#15688856C2218yescomplemented
21#15688849C2214yescomplemented
22C23×Q8×C47C2211yescharacteristic
23C22×C4⋊C4×C47—21no—
24C22×C4⋊C4×C47C2211yescharacteristic
25C47×#395—22no—
26C47×#395C2214yescomplemented
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—
39C2×#10625336—28no—
40C2×#10625336—48no—
41C22×C47⋊C8—22no—
42#13094004—28no—
43#13094004—28no—
44#13093999—28no—
45#13093999—28no—
46#13093998—24no—
47#13093998—24no—
48#13093998—22no—
49#13093998—22no—
50#13093998D414yes—
51#13093998C2314yes—
52C2×C4⋊Dic47—21no—
53C2×C4⋊Dic47D412yes—
54C2×(C2×D47)⋊C4—22no—
55C2×(C2×D47)⋊C4—22no—
56#13093948—28no—
57C22⋊(C47⋊C8)—22no—
58C22⋊(C47⋊C8)C2314yes—
59C22×Q8×C47—21no—
60C22×Q8×C47—22no—
61C22×Q8×C47—22no—
62C22×Q8×C47—44no—
63C22×Q8×C47C2312yes—
64C23×C4×C47—21no—
65C23×C4×C47—21no—
66C23×C4×C47D411yescharacteristic
67C23×C4×C47D411yescharacteristic
68C23×C4×C47C2311yescharacteristic
69C2×C4⋊C4×C47—22no—
70C2×C4⋊C4×C47—22no—
71C2×C4⋊C4×C47—24no—
72C2×C4⋊C4×C47—44no—
73C2×C4⋊C4×C47D412yes—
74C2×C4⋊C4×C47C2312yes—
75#95×C47—24no—
76#95×C47—21no—
77#95×C47C2312yes—
78C24×C47C2×D411yescharacteristic
79C23×D47—22no—
80C2×Q8×C47—28no—
81C2×Q8×C47—24no—
82C2×Q8×C47—44no—
83C2×Q8×C47—44no—
84C2×Q8×C47—44no—
85C2×Q8×C47—48no—
86C2×Q8×C47—42no—
87C22×C4×C47—21no—
88C22×C4×C47—21no—
89C22×C4×C47—22no—
90C22×C4×C47—24no—
91C22×C4×C47—24no—
92C22×C4×C47—21no—
93C22×C4×C47—22no—
94C22×C4×C47—22no—
95C22×C4×C47—21no—
96C22×C4×C47—42no—
97C22×C4×C47—42no—
98C22×C4×C47—42no—
99C22×C4×C47—41no—
100C22×C4×C47—41no—
101C22×C4×C47—42no—
102C22×C4×C47—42no—
103C22×C4×C47SD1614yescomplemented
104C22×C4×C47C2×D412yes—
105C22×C4×C47C2×D412yes—
106C22×C4×C47C2×D412yes—
107C22×C4×C47C2×D411yescharacteristic
108C22×C4×C47C2×D412yes—
109C22×C4×C47C2411yescharacteristic
110C2×C22⋊D47—24no—
111C2×C22⋊D47—24no—
112C2×C22⋊D47—44no—
113C2×C22⋊D47—44no—
114C22×Dic47—22no—
115#10625350—48no—
116#10625350—416no—
117#10625350—48no—
118C2×D188—24no—
119C2×D188—24no—
120C2×D188—22no—
121C2×D188—44no—
122C2×D188—44no—
123C2×D188—82no—
124C2×D188C2×D414yes—
125C2×C4×D47—28no—
126C2×C4×D47—48no—
127C2×Dic94—24no—
128C2×Dic94—44no—
129C4⋊C4×C47—24no—
130C4⋊C4×C47—24no—
131C4⋊C4×C47—48no—
132#10625336—416no—
133(C2×D47)⋊C4—28no—
134(C2×D47)⋊C4—24no—
135(C2×D47)⋊C4—24no—
136C4⋊Dic47—24no—
137C4⋊Dic47—22no—
138C4⋊Dic47C2×D414yes—
139C2×C47⋊C8—24no—
140C2×C47⋊C8—44no—
141C23×C47—22no—
142C23×C47—22no—
143C23×C47#12011yescharacteristic
144C23×C47#12011yescharacteristic
145C23×C47C2×SD1612yes—
146C23×C47AGL(1,ℤ/8)12yes—
147C23×C47C22×D411yescharacteristic
148C22×D47—24no—
149C22×D47—44no—
150C22×D47—44no—
151C22×D47—44no—
152Q8×C47—28no—
153Q8×C47—44no—
154Q8×C47—48no—
155C2×C4×C47—22no—
156C2×C4×C47—22no—
157C2×C4×C47—24no—
158C2×C4×C47—24no—
159C2×C4×C47—24no—
160C2×C4×C47—24no—
161C2×C4×C47—24no—
162C2×C4×C47—22no—
163C2×C4×C47—42no—
164C2×C4×C47—41no—
165C2×C4×C47—41no—
166C2×C4×C47—42no—
167C2×C4×C47—48no—
168C2×C4×C47—42no—
169C2×C4×C47—44no—
170C2×C4×C47—42no—
171C2×C4×C47—42no—
172C2×C4×C47—42no—
173C2×C4×C47—44no—
174C2×C4×C47—42no—
175C2×C4×C47—81no—
176C2×C4×C47#12012yes—
177C2×C4×C47C2×SD1614yescomplemented
178C2×C4×C47C22×D412yes—
179C22⋊D47—44no—
180C22⋊D47—44no—
181C22⋊D47—48no—
182C2×Dic47—24no—
183C2×Dic47—44no—
184D188—44no—
185D188—48no—
186D188—82no—
187D188—84no—
188C4×D47—416no—
189Dic94—48no—
190C47⋊C8—44no—
191C2×#1604—471nomaximal, Sylow
192C22×C47—24no—
193C22×C47—24no—
194C22×C47—21no—
195C22×C47—21no—
196C22×C47—21no—
197C22×C47—21no—
198C22×C47#45012yes—
199C22×C47#45012yes—
200C22×C47#45612yes—
201C22×C47B(4,𝔽2)12yes—
202C22×C47C2×#12011yescharacteristic
203C22×C47C22×SD1611yescharacteristic
204C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
205C2×D47—44no—
206C2×D47—44no—
207C2×D47—44no—
208C2×D47—88no—
209C4×C47—24no—
210C4×C47—22no—
211C4×C47—22no—
212C4×C47—44no—
213C4×C47—42no—
214C4×C47—44no—
215Dic47—44no—
216C2×#537—471no—
217C2×#448—472no—
218C2×#395D4711yescharacteristic, complemented
219#1604—478no—
220C2×#327—472no—
221C2×#324—471no—
222C2×C47—21no—
223C2×C47—21no—
224C2×C47—22no—
225C2×C47#160414yescomplemented
226C2×C47C2×#45011yescharacteristic
227C2×C47C2×#45611yescharacteristic
228C2×C47C2×B(4,𝔽2)11yescharacteristic
229D47—84no—
230C22×#41—941no—
231C23×Q8C2×D4711yescharacteristic
232C22×SD16—942no—
233#537—474no—
234#537—944no—
235#537—942no—
236C2×C22⋊Q8—942no—
237C2×#121—472no—
238C2×#121—942no—
239C2×#121—941no—
240C22×C4⋊C4—21no—
241C22×C4⋊C4C2×D4711yescharacteristic
242#448—478no—
243#448—944no—
244C2×#103—942no—
245C2×C22⋊C8—472no—
246#395—22no—
247#395C2×D4714yescomplemented
248C2×#95—21no—
249C2×#95C2×D4711yescharacteristic
250#327—478no—
251#324—474no—
252C47C2×#160411yescharacteristic, complemented, Sylow
253C2×#41—944no—
254C2×#41—944no—
255C2×#41—1884no—
256C2×#41—1884no—
257C22×Q8—21no—
258C22×Q8—22no—
259C22×Q8—22no—
260C22×Q8—44no—
261C22×Q8—941no—
262C22×Q8C22×D4712yes—
263C22×D4—472no—
264C22×D4—941no—
265C22×D4—941no—
266C22×D4—942no—
267C23×C4—21no—
268C23×C4—21no—
269C23×C4—942no—
270C23×C4C22⋊D4711yescharacteristic
271C23×C4C22⋊D4711yescharacteristic
272C23×C4C22×D4711yescharacteristic
273C2×SD16—948no—
274C2×SD16—1888no—
275C22×C8—942no—
276C22⋊Q8—948no—
277C22⋊Q8—948no—
278#121—474no—
279#121—474no—
280#121—948no—
281#121—948no—
282#121—944no—
283#121—942no—
284#121—942no—
285#121—944no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—44no—
290C2×C4⋊C4—472no—
291C2×C4⋊C4—941no—
292C2×C4⋊C4C22⋊D4712yes—
293C2×C4⋊C4C22×D4712yes—
294C2×C22⋊C4—942no—
295C2×C22⋊C4—942no—
296#103—948no—
297C22⋊C8—474no—
298C22⋊C8—942no—
299#95—24no—
300#95—21no—
301#95C22×D4712yes—
302C24—942no—
303C24C2×C22⋊D4711yescharacteristic
304#41—18816no—
305#41—1888no—
306#41—1888no—
307C2×Q8—28no—
308C2×Q8—24no—
309C2×Q8—44no—
310C2×Q8—44no—
311C2×Q8—44no—
312C2×Q8—48no—
313C2×Q8—42no—
314C2×Q8—944no—
315C2×Q8—1884no—
316C2×D4—474no—
317C2×D4—944no—
318C2×D4—944no—
319C2×D4—944no—
320C2×D4—944no—
321C2×D4—942no—
322C2×D4—3762no—
323C2×D4—1884no—
324C2×D4—1884no—
325C2×D4—1884no—
326C2×D4—1884no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—22no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—21no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—21no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—948no—
344C22×C4—942no—
345C22×C4—1888no—
346C22×C4#1062533614yescomplemented
347C22×C4D4×D4712yes—
348C22×C4D4×D4712yes—
349C22×C4C2×C22⋊D4712yes—
350C22×C4C2×C22⋊D4712yes—
351C22×C4C2×C22⋊D4711yescharacteristic
352C22×C4C23×D4711yescharacteristic
353SD16—18816no—
354C2×C8—944no—
355C2×C8—1884no—
356C4⋊C4—24no—
357C4⋊C4—24no—
358C4⋊C4—48no—
359C4⋊C4—474no—
360C4⋊C4—944no—
361C4⋊C4—942no—
362C22⋊C4—948no—
363C22⋊C4—944no—
364C22⋊C4—944no—
365C23—22no—
366C23—22no—
367C23—944no—
368C23—1884no—
369C23—1884no—
370C23—1884no—
371C23#1309399211yescharacteristic
372C23#1309399211yescharacteristic
373C23C2×#1062533612yes—
374C23#1309404912yes—
375C23C22×C22⋊D4711yescharacteristic
376Q8—28no—
377Q8—44no—
378Q8—48no—
379Q8—1888no—
380D4—3762no—
381D4—3764no—
382D4—1884no—
383D4—1888no—
384D4—1888no—
385D4—1884no—
386D4—1884no—
387C2×C4—24no—
388C2×C4—24no—
389C2×C4—24no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—24no—
393C2×C4—24no—
394C2×C4—22no—
395C2×C4—42no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—42no—
399C2×C4—48no—
400C2×C4—42no—
401C2×C4—44no—
402C2×C4—44no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—81no—
408C2×C4—944no—
409C2×C4—18816no—
410C2×C4—1884no—
411C2×C4#1309399212yes—
412C2×C4C2×#1062533614yescomplemented
413C2×C4C2×D4×D4712yes—
414C8—1884no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—24no—
420C22—24no—
421C22—3768no—
422C22—1884no—
423C22—1884no—
424C22—1884no—
425C22#1568927312yes—
426C22#1568927312yes—
427C22#1568930512yes—
428C22#1568930912yes—
429C22C2×#1309399211yescharacteristic
430C22C22×#1062533611yescharacteristic
431C22C2×#1309404911yescharacteristic
432C4—22no—
433C4—22no—
434C4—24no—
435C4—44no—
436C4—44no—
437C4—42no—
438C4—1884no—
439C2—21no—
440C2—22no—
441C2—21no—
442C2—3764no—
443C2#2024235614yesdirect product factor
444C2C2×#1568927311yescharacteristic
445C2C2×#1568930511yescharacteristic
446C2C2×#1568930911yescharacteristic
subgroupG/H
centerC23#13093992
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC2×C47×#395C2
socleC23×C47#120
prime radicalC2×#15689704C2
hypercenterC2×#395D47
hypocenterC47C2×#1604
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#395

Complements

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

NH ≅ G/Ntype
2. C2×#15689262C2semidirect
3. #20242356C2direct (N × H)
4. C2×#15688856C2semidirect
5. C2×#15688849C2semidirect
6. C2×C47×#395C2semidirect
17. #15689262C22semidirect
20. #15688856C22semidirect
21. #15688849C22semidirect
26. C47×#395C22semidirect
103. C22×C4×C47SD16semidirect
177. C2×C4×C47C2×SD16semidirect
218. C2×#395D47semidirect
225. C2×C47#1604semidirect
247. #395C2×D47semidirect
252. C47C2×#1604semidirect
346. C22×C4#10625336semidirect
412. C2×C4C2×#10625336semidirect
443. C2#20242356direct (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×#156892622
#202423562
C2×#156888562
C2×#156888492
C2×C47×#3952
C2×#160447