Finite Group Databasebeta

← #26248245

Subgroups — #26248245

Order 12032.

1158 conjugacy classes of subgroups; only the 195 normal subgroups are shown.

normal subgroups only

This group has 1158 conjugacy classes of subgroups — too many to draw. Only its 195 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×#15689735C211yesmaximal, characteristic
2C2×#15689279C212yesmaximal
3#20244539C212yesmaximal, complemented
4#20244487C212yesmaximal, complemented
5#20243732C214yesmaximal, complemented
6#20243700C212yesmaximal
7#20243459C211yesmaximal, characteristic
8C47×#2310C211yesmaximal, characteristic
9C22×#10625337C2214yes—
10#15689735—28no—
11C2×#13094024C2211yescharacteristic
12C2×#13094017—22no—
13C2×#13094016C2211yescharacteristic
14C2×#13094014—22no—
15C2×#13094003C2212yes—
16C2×C4×Dic94C2212yes—
17#15689279—28no—
18C2×#13093948C2214yes—
19C2×#13093947C2214yes—
20C2×C22⋊(C47⋊C8)C2212yes—
21#15688885C2214yes—
22#15688878C2212yes—
23#15688878C2212yes—
24C23×Q8×C47C2211yescharacteristic
25C2×#117×C47C2211yescharacteristic
26C2×#117×C47C2211yescharacteristic
27C47×#391—22no—
28C47×#391C2212yes—
29C47×#391C2212yes—
30C47×#381—22no—
31C22×Dic94C2312yes—
32C2×#10625337—28no—
33C2×#10625337—28no—
34C2×#10625337—44no—
35C22×C47⋊C8C2314yes—
36#13094024—24no—
37#13094017—216no—
38#13094016—24no—
39#13094014—216no—
40#13094003—24no—
41#13094003D418yes—
42C4×Dic94—24no—
43C4×Dic94—24no—
44C2×C4⋊Dic47—22no—
45C2×C4⋊Dic47C2312yes—
46C2×Dic47⋊C4—22no—
47C2×Dic47⋊C4—22no—
48C2×Dic47⋊C4—22no—
49C2×C22⋊Dic47—22no—
50C2×C4×Dic47—22no—
51#13093948—28no—
52#13093947—44no—
53C22⋊(C47⋊C8)—24no—
54C22×Q8×C47—24no—
55C22×Q8×C47—21no—
56C22×Q8×C47—24no—
57C22×Q8×C47—42no—
58C22×Q8×C47C2312yes—
59C23×C4×C47—21no—
60C23×C4×C47C2311yescharacteristic
61#117×C47—24no—
62#117×C47—42no—
63C2×C4⋊C4×C47D412yes—
64C2×C4⋊C4×C47C2312yes—
65C2×C22⋊C4×C47—21no—
66C2×C22⋊C4×C47D412yes—
67C2×C42×C47—21no—
68C2×C42×C47C2311yescharacteristic
69C2×C42×C47C2311yescharacteristic
70#95×C47—24no—
71#95×C47—24no—
72C24×C47C2×D411yescharacteristic
73C2×Q8×C47—24no—
74C2×Q8×C47—22no—
75C2×Q8×C47—24no—
76C2×Q8×C47—24no—
77C2×Q8×C47—416no—
78C2×Q8×C47—42no—
79C2×Q8×C47—42no—
80C2×Q8×C47#4114yes—
81C22×C4×C47—22no—
82C22×C4×C47—22no—
83C22×C4×C47—24no—
84C22×C4×C47—22no—
85C22×C4×C47—22no—
86C22×C4×C47—42no—
87C22×C4×C47—42no—
88C22×C4×C47C2×D412yes—
89C22×C4×C47C2×D412yes—
90C22×C4×C47C2×D411yescharacteristic
91C22×C4×C47C2×D414yes—
92C22×C4×C47C2411yescharacteristic
93C22×Dic47—22no—
94C22×Dic47—22no—
95C2×Dic94—24no—
96C2×Dic94—24no—
97C2×Dic94—22no—
98C2×Dic94C2×D414yes—
99C4⋊C4×C47—24no—
100C4⋊C4×C47—24no—
101C22⋊C4×C47—24no—
102C22⋊C4×C47—42no—
103C42×C47—22no—
104C42×C47—22no—
105C42×C47—42no—
106C22⋊Dic47—28no—
107#10625337—416no—
108C4⋊Dic47—28no—
109C4⋊Dic47—22no—
110C4⋊Dic47C2×D414yes—
111Dic47⋊C4—28no—
112Dic47⋊C4—28no—
113Dic47⋊C4—28no—
114C4×Dic47—28no—
115C2×C47⋊C8—24no—
116C2×C47⋊C8—24no—
117C2×C47⋊C8—24no—
118C23×C47—21no—
119C23×C47—21no—
120C23×C47—24no—
121C23×C47#12012yes—
122C23×C47C22×D411yescharacteristic
123Q8×C47—28no—
124Q8×C47—48no—
125Q8×C47—44no—
126C2×C4×C47—21no—
127C2×C4×C47—21no—
128C2×C4×C47—28no—
129C2×C4×C47—22no—
130C2×C4×C47—22no—
131C2×C4×C47—24no—
132C2×C4×C47—24no—
133C2×C4×C47—24no—
134C2×C4×C47—48no—
135C2×C4×C47—42no—
136C2×C4×C47—42no—
137C2×C4×C47—42no—
138C2×C4×C47—44no—
139C2×C4×C47#12012yes—
140C2×C4×C47C22×D412yes—
141C2×C4×C47C2×#4112yes—
142C2×C4×C47H(5,𝔽2)12yes—
143C2×Dic47—24no—
144C2×Dic47—24no—
145C2×Dic47—42no—
146C2×Dic47—42no—
147C2×Dic47—44no—
148Dic94—28no—
149Dic94—44no—
150C47⋊C8—44no—
151#23943—471nomaximal, Sylow
152C22×C47—22no—
153C22×C47—24no—
154C22×C47—42no—
155C22×C47—42no—
156C22×C47C2×#12011yescharacteristic
157C22×C47#52511yescharacteristic
158C22×C47#53511yescharacteristic
159C22×C47#57814yes—
160C4×C47—22no—
161C4×C47—44no—
162C4×C47—42no—
163C4×C47—42no—
164C4×C47—82no—
165C4×C47#54614yes—
166Dic47—44no—
167Dic47—44no—
168C2×#541—471no—
169C2×#451—472no—
170#2310D4711yescharacteristic
171#2026—472no—
172#2016—472no—
173#1878—474no—
174#1873—472no—
175#1818—471no—
176C2×C47—42no—
177C2×C47#240411yescharacteristic
178C2×C47#301112yes—
179C2×C47#312712yes—
180C2×C47#322312yes—
181C23×Q8C2×D4711yescharacteristic
182C22×Q16—474no—
183#541—948no—
184C2×C4⋊Q8—471no—
185C2×#126—942no—
186C2×#125—471no—
187C2×#125—942no—
188C2×C22⋊Q8—472no—
189C2×C4×Q8—472no—
190C2×#117C2×D4711yescharacteristic
191C2×#117C2×D4711yescharacteristic
192#451—948no—
193C2×#103—474no—
194C2×#103—474no—
195C2×C22⋊C8—472no—
196#391—22no—
197#391C2×D4712yes—
198#391C2×D4712yes—
199#381—22no—
200#340—474no—
201#336—472no—
202#336—472no—
203C47#2394311yescharacteristic, complemented, Sylow
204C22×Q8—24no—
205C22×Q8—21no—
206C22×Q8—24no—
207C22×Q8—42no—
208C22×Q8—472no—
209C22×Q8C22×D4712yes—
210C23×C4—21no—
211C23×C4C22×D4711yescharacteristic
212C2×Q16—948no—
213C2×Q16—948no—
214C2×Q16—1884no—
215C22×C8—474no—
216C4⋊Q8—944no—
217#126—9416no—
218#125—944no—
219#125—9416no—
220C22⋊Q8—478no—
221C22⋊Q8—944no—
222C4×Q8—944no—
223C4×Q8—944no—
224#117—24no—
225#117—42no—
226C2×C4⋊C4—472no—
227C2×C4⋊C4—942no—
228C2×C4⋊C4—942no—
229C2×C4⋊C4—942no—
230C2×C4⋊C4—942no—
231C2×C4⋊C4C22⋊D4712yes—
232C2×C4⋊C4C22×D4712yes—
233C2×C22⋊C4—21no—
234C2×C22⋊C4—942no—
235C2×C22⋊C4C22⋊D4712yes—
236C2×C42—21no—
237C2×C42—942no—
238C2×C42C22×D4711yescharacteristic
239C2×C42C22×D4711yescharacteristic
240#103—948no—
241#103—1884no—
242C22⋊C8—944no—
243#95—24no—
244#95—24no—
245C24C2×C22⋊D4711yescharacteristic
246C2×Q8—24no—
247C2×Q8—22no—
248C2×Q8—24no—
249C2×Q8—24no—
250C2×Q8—416no—
251C2×Q8—42no—
252C2×Q8—42no—
253C2×Q8—474no—
254C2×Q8—942no—
255C2×Q8—944no—
256C2×Q8—944no—
257C2×Q8#1062535014yes—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—24no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—942no—
266C22×C4—942no—
267C22×C4D4×D4714yes—
268C22×C4C2×C22⋊D4712yes—
269C22×C4C2×C22⋊D4712yes—
270C22×C4C2×C22⋊D4711yescharacteristic
271C22×C4C23×D4711yescharacteristic
272Q16—18816no—
273C2×C8—944no—
274C2×C8—944no—
275C2×C8—944no—
276C4⋊C4—24no—
277C4⋊C4—24no—
278C4⋊C4—474no—
279C4⋊C4—948no—
280C4⋊C4—948no—
281C4⋊C4—948no—
282C4⋊C4—948no—
283C4⋊C4—942no—
284C22⋊C4—24no—
285C22⋊C4—42no—
286C22⋊C4—948no—
287C42—22no—
288C42—22no—
289C42—42no—
290C42—948no—
291C23—21no—
292C23—24no—
293C23—21no—
294C23#1309399212yes—
295C23C22×C22⋊D4711yescharacteristic
296Q8—28no—
297Q8—48no—
298Q8—44no—
299Q8—948no—
300Q8—1884no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—28no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—24no—
307C2×C4—24no—
308C2×C4—24no—
309C2×C4—48no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—44no—
313C2×C4—42no—
314C2×C4—944no—
315C2×C4—944no—
316C2×C4—1882no—
317C2×C4—1884no—
318C2×C4—1882no—
319C2×C4#1309399212yes—
320C2×C4C2×D4×D4712yes—
321C2×C4C2×#1062535012yes—
322C2×C4#1309406912yes—
323C8—1884no—
324C22—24no—
325C22—22no—
326C22—42no—
327C22—42no—
328C22C2×#1309399211yescharacteristic
329C22#1568963411yescharacteristic
330C22#1568969011yescharacteristic
331C22#1568993814yes—
332C4—22no—
333C4—42no—
334C4—42no—
335C4—44no—
336C4—82no—
337C4—1884no—
338C4—1884no—
339C4#1568976314yes—
340C2—42no—
341C2#2024670711yescharacteristic
342C2#2025268112yes—
343C2#2025368812yes—
344C2#2025447112yes—
subgroupG/H
centerC23C22×C22⋊D47
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC47×#2310C2
socleC23×C47C22×D4
prime radicalC24×C47C2×D4
hypercenter#2310D47
hypocenterC47#23943
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2310

Complements

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

NH ≅ G/Ntype
3. #20244539C2semidirect
4. #20244487C2semidirect
5. #20243732C2semidirect
203. C47#23943semidirect

Maximal subgroups

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

subgroupindex
C2×#156897352
C2×#156892792
#202445392
#202444872
#202437322
#202437002
#202434592
C47×#23102
#2394347