Finite Group Databasebeta

← #26108837

Subgroups — #26108837

Order 12032.

728 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 728 conjugacy classes of subgroups — too many to draw. Only its 139 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#20244366C212yesmaximal, complemented
2#20244217C211yesmaximal, characteristic
3#20242126C212yesmaximal, complemented
4#20242123C211yesmaximal, characteristic
5C47×#3550C211yesmaximal, characteristic
6#15689147C2212yes—
7#15689147C2211yescharacteristic
8C2×#13093951—22no—
9C42×Dic47C2211yescharacteristic
10#15688882—24no—
11#15688872—21no—
12#15688869—22no—
13#15688869—22no—
14#15688867—21no—
15C47×#563C414yescomplemented
16C47×#559—21no—
17Q82×C47—21no—
18C47×#556—22no—
19D4×Q8×C47—22no—
20C47×#542—24no—
21C47×#536—22no—
22C47×#533C2212yes—
23C47×#532C2211yescharacteristic
24C2×#10625329—21no—
25C2×#10625329—22no—
26C2×C4×Dic47—21no—
27C2×C4×Dic47—22no—
28C4⋊(C47⋊C8)—22no—
29C4⋊(C47⋊C8)—22no—
30C4⋊(C47⋊C8)—22no—
31#13093951—44no—
32C2×#41×C47D412yes—
33C22×Q8×C47—22no—
34C4⋊Q8×C47—24no—
35C4⋊Q8×C47—24no—
36C4⋊Q8×C47D411yescharacteristic
37C4⋊Q8×C47D412yes—
38C4⋊Q8×C47D412yes—
39#127×C47D411yescharacteristic
40#126×C47—28no—
41#125×C47C2×C414yes—
42#124×C47—24no—
43#124×C47—24no—
44#124×C47C2×C412yes—
45#123×C47—24no—
46#123×C47—24no—
47C22⋊Q8×C47—24no—
48C22⋊Q8×C47—22no—
49C22⋊Q8×C47—22no—
50C22⋊Q8×C47—28no—
51#121×C47—22no—
52C4×Q8×C47—24no—
53C4×Q8×C47—22no—
54C4×Q8×C47—22no—
55C4×D4×C47—24no—
56C4×D4×C47—22no—
57#117×C47—22no—
58#117×C47D414yes—
59C2×C42×C47C2311yescharacteristic
60#41×C47—42no—
61#41×C47—42no—
62C2×Q8×C47—24no—
63C2×Q8×C47—22no—
64C2×Q8×C47—22no—
65C2×Q8×C47—82no—
66C2×Q8×C47—82no—
67C2×Q8×C47C22⋊C412yes—
68C2×D4×C47—22no—
69C2×D4×C47—22no—
70C2×D4×C47C22⋊C412yes—
71C22×C4×C47—24no—
72C22×C4×C47—22no—
73C22×C4×C47C2×D412yes—
74C22×C4×C47C2×D411yescharacteristic
75C22×Dic47—41no—
76C4⋊C4×C47—28no—
77C4⋊C4×C47—24no—
78C4⋊C4×C47—24no—
79C4⋊C4×C47—22no—
80C4⋊C4×C47—22no—
81C4⋊C4×C47—22no—
82C4⋊C4×C47C22⋊C414yes—
83C4⋊C4×C47C22⋊C414yes—
84C22⋊C4×C47—24no—
85C22⋊C4×C47—28no—
86C22⋊C4×C47—24no—
87C22⋊C4×C47—24no—
88C42×C47—24no—
89C42×C47—24no—
90C42×C47C22×C411yescharacteristic
91C42×C47C2×D412yes—
92C42×C47C2×D411yescharacteristic
93C4×Dic47—22no—
94C4×Dic47—24no—
95C4×Dic47—42no—
96C4×Dic47—41no—
97#10625329—42no—
98#10625329—44no—
99C2×C47⋊C8—41no—
100C2×C47⋊C8—42no—
101C23×C47—22no—
102C23×C47#12011yescharacteristic
103Q8×C47—42no—
104Q8×C47—42no—
105Q8×C47—42no—
106Q8×C47—42no—
107Q8×C47—42no—
108Q8×C47—84no—
109D4×C47—42no—
110D4×C47—42no—
111D4×C47—42no—
112D4×C47—82no—
113C2×C4×C47—21no—
114C2×C4×C47—22no—
115C2×C4×C47—21no—
116C2×C4×C47—22no—
117C2×C4×C47—24no—
118C2×C4×C47—22no—
119C2×C4×C47—24no—
120C2×C4×C47—42no—
121C2×C4×C47—42no—
122C2×C4×C47—84no—
123C2×C4×C47C2×C22⋊C412yes—
124C2×C4×C47C2×C22⋊C411yescharacteristic
125C2×C4×C47#12011yescharacteristic
126C2×C4×C47#12012yes—
127C2×Dic47—42no—
128C2×Dic47—44no—
129C47⋊C8—81no—
130C47⋊C8—82no—
131#8531—471nomaximal, Sylow
132C22×C47—21no—
133C22×C47—22no—
134C22×C47—42no—
135C22×C47—44no—
136C22×C47#37911yescharacteristic
137C4×C47—21no—
138C4×C47—22no—
139C4×C47—42no—
140C4×C47—44no—
141C4×C47—44no—
142C4×C47#45311yescharacteristic
143C4×C47#45412yes—
144C4×C47#45511yescharacteristic
145C4×C47#45612yes—
146Dic47—44no—
147#3550D4711yescharacteristic
148#1994—472no—
149#1962—471no—
150#1561—472no—
151#1560—471no—
152C2×C47—41no—
153C2×C47—82no—
154C2×C47#179211yescharacteristic
155C2×C47#179312yes—
156#563Dic4714yescomplemented
157#559—21no—
158Q82—21no—
159#556—22no—
160D4×Q8—22no—
161#542—24no—
162#536—22no—
163#533C2×D4712yes—
164#532C2×D4711yescharacteristic
165#423—472no—
166#423—471no—
167C2×#104—942no—
168C43—471no—
169#339—944no—
170#333—941no—
171#332—942no—
172#332—942no—
173#331—941no—
174C47#853111yescharacteristic, complemented, Sylow
175C2×#41C22⋊D4712yes—
176C22×Q8—22no—
177C2×M16—942no—
178C2×M16—941no—
179C4⋊Q8—24no—
180C4⋊Q8—24no—
181C4⋊Q8C22⋊D4711yescharacteristic
182C4⋊Q8C22⋊D4712yes—
183C4⋊Q8C22⋊D4712yes—
184#127C22⋊D4711yescharacteristic
185#126—28no—
186#125C2×Dic4714yes—
187#124—24no—
188#124—24no—
189#124C2×Dic4712yes—
190#123—24no—
191#123—24no—
192C22⋊Q8—24no—
193C22⋊Q8—22no—
194C22⋊Q8—22no—
195C22⋊Q8—28no—
196#121—22no—
197C4×Q8—24no—
198C4×Q8—22no—
199C4×Q8—22no—
200C4×D4—24no—
201C4×D4—22no—
202#117—22no—
203#117C22⋊D4714yes—
204C2×C42—942no—
205C2×C42—941no—
206C2×C42C22×D4711yescharacteristic
207C4⋊C8—942no—
208C4⋊C8—942no—
209C4⋊C8—942no—
210#104—1884no—
211#41—42no—
212#41—42no—
213C2×Q8—24no—
214C2×Q8—22no—
215C2×Q8—22no—
216C2×Q8—82no—
217C2×Q8—82no—
218C2×Q8C22⋊Dic4712yes—
219C2×D4—22no—
220C2×D4—22no—
221C2×D4C22⋊Dic4712yes—
222C22×C4—24no—
223C22×C4—22no—
224C22×C4—1881no—
225C22×C4C2×C22⋊D4712yes—
226C22×C4C2×C22⋊D4711yescharacteristic
227M16—1882no—
228M16—1884no—
229C2×C8—1881no—
230C2×C8—1882no—
231C4⋊C4—28no—
232C4⋊C4—24no—
233C4⋊C4—24no—
234C4⋊C4—22no—
235C4⋊C4—22no—
236C4⋊C4—22no—
237C4⋊C4C22⋊Dic4714yes—
238C4⋊C4C22⋊Dic4714yes—
239C22⋊C4—24no—
240C22⋊C4—28no—
241C22⋊C4—24no—
242C22⋊C4—24no—
243C42—24no—
244C42—24no—
245C42—942no—
246C42—944no—
247C42—1882no—
248C42—1881no—
249C42C22×Dic4711yescharacteristic
250C42C2×C22⋊D4712yes—
251C42C2×C22⋊D4711yescharacteristic
252C23—22no—
253C23#1309399311yescharacteristic
254Q8—42no—
255Q8—42no—
256Q8—42no—
257Q8—42no—
258Q8—42no—
259Q8—84no—
260D4—42no—
261D4—42no—
262D4—42no—
263D4—82no—
264C2×C4—21no—
265C2×C4—22no—
266C2×C4—21no—
267C2×C4—22no—
268C2×C4—24no—
269C2×C4—22no—
270C2×C4—24no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—84no—
274C2×C4—1882no—
275C2×C4—1884no—
276C2×C4C2×C22⋊Dic4712yes—
277C2×C4C2×C22⋊Dic4711yescharacteristic
278C2×C4#1309399311yescharacteristic
279C2×C4#1309399312yes—
280C8—3761no—
281C8—3762no—
282C22—21no—
283C22—22no—
284C22—42no—
285C22—44no—
286C22#1568897911yescharacteristic
287C4—21no—
288C4—22no—
289C4—42no—
290C4—44no—
291C4—44no—
292C4—1884no—
293C4#1568929111yescharacteristic
294C4#1568929612yes—
295C4#1568930111yescharacteristic
296C4#1568930612yes—
297C2—41no—
298C2—82no—
299C2#2024334211yescharacteristic
300C2#2024334712yes—
subgroupG/H
centerC22#15688979
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#3550C2
socleC22×C47#379
prime radicalC47×#532C22
cosocleC2×C42×C47C23
hypercenter#3550D47
hypocenterC47#8531
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3550

Complements

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

NH ≅ G/Ntype
1. #20244366C2semidirect
3. #20242126C2semidirect
15. C47×#563C4semidirect
156. #563Dic47semidirect
174. C47#8531semidirect

Maximal subgroups

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

subgroupindex
#202443662
#202442172
#202421262
#202421232
C47×#35502
#853147