Finite Group Databasebeta

← #26225446

Subgroups — #26225446

Order 12032.

1270 conjugacy classes of subgroups; only the 231 normal subgroups are shown.

normal subgroups only

This group has 1270 conjugacy classes of subgroups — too many to draw. Only its 231 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
1#20257032C211yesmaximal, characteristic
2#20251864C212yesmaximal, complemented
3#20243937C214yesmaximal, complemented
4#20243803C214yesmaximal, complemented
5#20243497C212yesmaximal, complemented
6#20243316C211yesmaximal, characteristic, complemented
7C47×#1785C211yesmaximal, characteristic, complemented
8C2×#13094069C2211yescharacteristic
9#15689889C2214yes—
10#15689772—24no—
11#15689761—24no—
12#15689697C2212yes—
13#15689682C414yes—
14#15689634C2212yes—
15C2×#13094007C414yes—
16#15689598C2211yescharacteristic
17#15689105C2214yes—
18#15689104—24no—
19C2×#13093943C2212yescomplemented
20C2×#13093941C2212yes—
21#100×D47—24no—
22#15689087C2212yescomplemented
23#15688895C2214yescomplemented
24#15688894C2214yescomplemented
25C22×#41×C47C2211yescharacteristic
26C2×#101×C47C2212yescomplemented
27C2×#100×C47C2212yes—
28C47×#407C2212yescomplemented
29#13094069—42no—
30#13094069—42no—
31C2×#10625350C2312yes—
32C2×#10625352D414yes—
33C22×C22⋊D47C2×C414yes—
34C2×D4×D47D414yes—
35#13094032—44no—
36#13094030—44no—
37C2×#10625329C2314yes—
38M16×D47—28no—
39#13094022—22no—
40#13094013—22no—
41#13094007—28no—
42#13094007C2×C414yes—
43#13094007C2×C418yes—
44#13094005—24no—
45#13094003—22no—
46#13094000—24no—
47#13093998—22no—
48#13093996—28no—
49#13093993C2×C414yes—
50#13093992—24no—
51D4×Dic47—24no—
52C4×C22⋊D47—24no—
53#13093983C2312yes—
54C2×Dic47⋊C4—24no—
55C2×C22⋊Dic47C2×C414yes—
56C2×(C2×D47)⋊C4—24no—
57#13093943—42no—
58#13093942—24no—
59#13093941—24no—
60#13093940—24no—
61C22⋊(C47⋊C8)—42no—
62C2×#41×C47—24no—
63C2×#41×C47—42no—
64C2×#41×C47—42no—
65C22×Q8×C47C2311yescharacteristic
66C22×D4×C47C2×C412yes—
67C22×D4×C47C2311yescharacteristic
68C23×C4×C47C2×C412yes—
69C23×C4×C47C2311yescharacteristic
70C2×M16×C47C2314yes—
71#101×C47—42no—
72#100×C47—24no—
73C22⋊C8×C47—42no—
74C24×C47C22×C411yescharacteristic
75C24×C47C22×C412yes—
76C23×D47C22⋊C414yes—
77#41×C47—416no—
78C2×Q8×C47—22no—
79C2×Q8×C47—22no—
80C2×Q8×C47—22no—
81C2×D4×C47—24no—
82C2×D4×C47—22no—
83C2×D4×C47—22no—
84C2×D4×C47—42no—
85C2×D4×C47—42no—
86C2×D4×C47C2×D414yes—
87C22×C4×C47—24no—
88C22×C4×C47—22no—
89C22×C4×C47—22no—
90C22×C4×C47—22no—
91C22×C4×C47—42no—
92C22×C4×C47—42no—
93C22×C4×C47C22×C411yescharacteristic
94C22×C4×C47C22×C412yes—
95C22×C4×C47C2411yescharacteristic
96C2×C22⋊D47—24no—
97C2×C22⋊D47—44no—
98C2×C22⋊D47—44no—
99C2×C22⋊D47C22⋊C418yes—
100C2×C22⋊D47C22×C414yes—
101C22×Dic47—24no—
102C22×Dic47C22⋊C414yes—
103#10625352—48no—
104D4×D47—48no—
105#10625350—24no—
106#10625350—24no—
107C2×D188C2×D412yes—
108C2×C4×D47C2×D414yes—
109C2×Dic94C2×D412yes—
110M16×C47—24no—
111M16×C47—24no—
112C2×C8×C47—24no—
113C22⋊Dic47—24no—
114C22⋊Dic47—24no—
115C22⋊Dic47C22×C414yes—
116(C2×D47)⋊C4—24no—
117(C2×D47)⋊C4—28no—
118C4⋊Dic47—22no—
119Dic47⋊C4—24no—
120Dic47⋊C4—28no—
121C4×Dic47—22no—
122#10625329—24no—
123#10625329—24no—
124C2×C47⋊C8—24no—
125#10625324—48no—
126C8×D47—48no—
127C23×C47—22no—
128C23×C47—22no—
129C23×C47—21no—
130C23×C47—21no—
131C23×C47—21no—
132C23×C47—81no—
133C23×C47—82no—
134C23×C47C2×C22⋊C414yes—
135C23×C47C23×C411yescharacteristic
136C22×D47—44no—
137C22×D47—84no—
138C22×D47C2×C22⋊C414yes—
139Q8×C47—44no—
140D4×C47—28no—
141D4×C47—48no—
142C2×C4×C47—21no—
143C2×C4×C47—22no—
144C2×C4×C47—21no—
145C2×C4×C47—21no—
146C2×C4×C47—22no—
147C2×C4×C47—22no—
148C2×C4×C47—48no—
149C2×C4×C47—44no—
150C2×C4×C47C22×D412yes—
151C2×C4×C47H(5,𝔽2)12yes—
152C2×C4×C47H(5,𝔽2)12yes—
153C22⋊D47—24no—
154C22⋊D47—24no—
155C22⋊D47—48no—
156C22⋊D47—84no—
157C2×Dic47—24no—
158C2×Dic47—44no—
159C2×Dic47—84no—
160C2×Dic47C2×C22⋊C414yes—
161D188—24no—
162C4×D47—28no—
163Dic94—24no—
164C8×C47—44no—
165C47⋊C8—44no—
166#21779—471nomaximal, Sylow
167C22×C47—24no—
168C22×C47—41no—
169C22×C47—44no—
170C22×C47—42no—
171C22×C47—42no—
172C22×C47—42no—
173C22×C47—41no—
174C22×C47—82no—
175C22×C47—82no—
176C22×C47C22×C22⋊C411yescharacteristic
177C22×C47#51811yescharacteristic
178C22×C47#51811yescharacteristic
179C2×D47—24no—
180C2×D47—44no—
181C2×D47—164no—
182C4×C47—41no—
183C4×C47—42no—
184C4×C47#53414yes—
185Dic47—24no—
186Dic47—44no—
187#3485—471no—
188#2888—472no—
189#1915—474no—
190#1891—474no—
191#1829—472no—
192#1785—471no—
193#1785D4711yescharacteristic, complemented
194C2×C47—21no—
195C2×C47—21no—
196C2×C47—21no—
197C2×C47—44no—
198C2×C47—81no—
199C2×C47#230811yescharacteristic
200D47—44no—
201C2×H(5,𝔽2)—471no—
202C22×#41C2×D4711yescharacteristic
203#568—474no—
204#547—944no—
205#546—944no—
206#536—472no—
207#534—474no—
208#525—472no—
209C2×#123—474no—
210#518—471no—
211#413—474no—
212#413—944no—
213C2×#101—472no—
214C2×#101C2×D4712yescomplemented
215C2×#100—472no—
216C2×#100—944no—
217C2×#100C2×D4712yes—
218#407—472no—
219#407C2×D4712yescomplemented
220#344—474no—
221#344—474no—
222C47#2177911yescharacteristic, complemented, Sylow
223H(5,𝔽2)—1882no—
224H(5,𝔽2)—1882no—
225C2×#41—24no—
226C2×#41—42no—
227C2×#41—42no—
228C2×#41—474no—
229C2×#41—472no—
230C22×Q8C22×D4711yescharacteristic
231C22×D4—474no—
232C22×D4—474no—
233C22×D4C4×D4712yes—
234C22×D4C22×D4711yescharacteristic
235C23×C4C4×D4712yes—
236C23×C4C22×D4711yescharacteristic
237#131—1884no—
238#131—1884no—
239C2×M16—474no—
240C2×M16—948no—
241C2×M16C22×D4714yes—
242C4⋊Q8—942no—
243#124—942no—
244#123—474no—
245#123—478no—
246#123—948no—
247#123—944no—
248C22⋊Q8—942no—
249C22⋊Q8—944no—
250#121—942no—
251#121—948no—
252#120—474no—
253#120—944no—
254C4×D4—944no—
255C4×D4—944no—
256#117—472no—
257C2×C4⋊C4—944no—
258C2×C22⋊C4—474no—
259C2×C22⋊C4—944no—
260#101—42no—
261#101—944no—
262#101—1882no—
263#100—24no—
264#100—944no—
265#100—944no—
266C22⋊C8—42no—
267C22⋊C8—1882no—
268C24—474no—
269C24C2×C4×D4711yescharacteristic
270C24C2×C4×D4712yes—
271#41—416no—
272#41—944no—
273#41—944no—
274#41—1888no—
275C2×Q8—22no—
276C2×Q8—22no—
277C2×Q8—22no—
278C2×Q8—472no—
279C2×D4—24no—
280C2×D4—22no—
281C2×D4—22no—
282C2×D4—42no—
283C2×D4—42no—
284C2×D4—474no—
285C2×D4—478no—
286C2×D4—472no—
287C2×D4—944no—
288C2×D4—1888no—
289C2×D4—1884no—
290C2×D4—1884no—
291C2×D4D4×D4714yes—
292C22×C4—24no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—474no—
299C22×C4—474no—
300C22×C4—944no—
301C22×C4C2×C4×D4711yescharacteristic
302C22×C4C2×C4×D4712yes—
303C22×C4C23×D4711yescharacteristic
304M16—24no—
305M16—24no—
306M16—944no—
307M16—944no—
308M16—1888no—
309C2×C8—24no—
310C2×C8—944no—
311C2×C8—1888no—
312C4⋊C4—944no—
313C4⋊C4—948no—
314C4⋊C4—942no—
315C22⋊C4—474no—
316C22⋊C4—944no—
317C22⋊C4—948no—
318C22⋊C4—944no—
319C22⋊C4—944no—
320C42—942no—
321C23—22no—
322C23—22no—
323C23—21no—
324C23—21no—
325C23—21no—
326C23—81no—
327C23—82no—
328C23—474no—
329C23—3764no—
330C23—1884no—
331C23C22⋊C4×D4714yes—
332C23C22×C4×D4711yescharacteristic
333Q8—44no—
334Q8—944no—
335D4—28no—
336D4—48no—
337D4—944no—
338D4—944no—
339D4—944no—
340D4—3764no—
341D4—1888no—
342C2×C4—21no—
343C2×C4—22no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—48no—
349C2×C4—44no—
350C2×C4—474no—
351C2×C4—948no—
352C2×C4—944no—
353C2×C4—3764no—
354C2×C4—1884no—
355C2×C4C2×D4×D4712yes—
356C2×C4#1309406912yes—
357C2×C4#1309406912yes—
358C8—44no—
359C8—1884no—
360C22—24no—
361C22—41no—
362C22—44no—
363C22—42no—
364C22—42no—
365C22—42no—
366C22—41no—
367C22—82no—
368C22—82no—
369C22—944no—
370C22—1884no—
371C22—7524no—
372C22C2×C22⋊C4×D4711yescharacteristic
373C22#1568959611yescharacteristic
374C22#1568959611yescharacteristic
375C4—41no—
376C4—42no—
377C4—944no—
378C4—1884no—
379C4#1568968314yes—
380C2—21no—
381C2—21no—
382C2—21no—
383C2—44no—
384C2—81no—
385C2—1884no—
386C2#2024596411yescharacteristic
subgroupG/H
centerC2#20245964
commutatorC23×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1785C2
socleC2×C47#2308
prime radical#20257032C2
cosocleC22×C4×C47C24
hypercenter#1785D47
hypocenterC47#21779
LCSG ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2 ◁ C22×C4 ◁ #1785

Complements

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

NH ≅ G/Ntype
2. #20251864C2semidirect
3. #20243937C2semidirect
4. #20243803C2semidirect
5. #20243497C2semidirect
6. #20243316C2semidirect
7. C47×#1785C2semidirect
19. C2×#13093943C22semidirect
22. #15689087C22semidirect
23. #15688895C22semidirect
24. #15688894C22semidirect
26. C2×#101×C47C22semidirect
28. C47×#407C22semidirect
193. #1785D47semidirect
214. C2×#101C2×D47semidirect
219. #407C2×D47semidirect
222. C47#21779semidirect

Maximal subgroups

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

subgroupindex
#202570322
#202518642
#202439372
#202438032
#202434972
#202433162
C47×#17852
#2177947