Finite Group Databasebeta

← #26112289

Subgroups — #26112289

Order 12032.

504 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 127 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#20251952C211yesmaximal, characteristic
2#20249209C211yesmaximal, characteristic
3#20242072C212yesmaximal, complemented
4#20241247C212yesmaximal, complemented
5C47×#1458C211yesmaximal, characteristic
6C22×#10625329—21no—
7C2×#13094024C2211yescharacteristic
8#15689150—24no—
9#15689147—21no—
10#15689147C2212yes—
11#15689062C412yes—
12#15689059C412yes—
13#15689043—21no—
14#15689035—21no—
15#15689021—21no—
16#15688989C2211yescharacteristic
17#15688983—21no—
18#15688870—24no—
19C2×#117×C47C2211yescharacteristic
20C47×#431C2212yes—
21C47×#324—21no—
22C47×#323—21no—
23C22×Dic94—21no—
24C2×#10625329—22no—
25C2×#10625329—22no—
26C2×#10625329—42no—
27C2×#10625329—42no—
28C22×C47⋊C8—22no—
29#13094024—22no—
30#13094024—22no—
31C2×C4⋊Dic47—21no—
32C2×C4⋊Dic47—21no—
33C2×C4⋊Dic47C2×C412yes—
34C2×Dic47⋊C4—22no—
35C2×C22⋊Dic47—22no—
36C4⋊(C47⋊C8)—24no—
37C4⋊(C47⋊C8)—22no—
38C4⋊(C47⋊C8)D414yes—
39C47⋊#95—22no—
40C47⋊#95—22no—
41C47⋊#95—21no—
42C47⋊#95C2×C412yes—
43C23×C4×C47D411yescharacteristic
44#117×C47—22no—
45#117×C47—22no—
46C2×C4⋊C4×C47C2×C411yescharacteristic
47C2×C4⋊C4×C47D411yescharacteristic
48C2×C22⋊C4×C47C2×C411yescharacteristic
49C2×C22⋊C4×C47D411yescharacteristic
50C2×C42×C47D411yescharacteristic
51C2×C42×C47C2311yescharacteristic
52C4⋊C8×C47—22no—
53C22⋊C8×C47—22no—
54C4×C8×C47—22no—
55C24×C47C22⋊C411yescharacteristic
56C22×C4×C47—21no—
57C22×C4×C47—21no—
58C22×C4×C47—21no—
59C22×C4×C47—21no—
60C22×C4×C47C22⋊C411yescharacteristic
61C22×C4×C47C22⋊C411yescharacteristic
62C22×C4×C47C22⋊C411yescharacteristic
63C22×C4×C47SD1612yes—
64C22×C4×C47Q1612yes—
65C22×C4×C47C22×C411yescharacteristic
66C22×C4×C47C2×D411yescharacteristic
67C22×C4×C47C2×D411yescharacteristic
68C22×Dic47—22no—
69C22×Dic47—22no—
70C2×Dic94—42no—
71C2×Dic94—82no—
72C2×C8×C47—42no—
73C4⋊C4×C47—22no—
74C4⋊C4×C47—22no—
75C22⋊C4×C47—22no—
76C22⋊C4×C47—22no—
77C42×C47—21no—
78C42×C47—21no—
79C42×C47C2×D412yes—
80C42×C47#4112yes—
81C22⋊Dic47—82no—
82C4⋊Dic47—24no—
83C4⋊Dic47—41no—
84C4⋊Dic47—41no—
85C4⋊Dic47—41no—
86C4⋊Dic47—41no—
87Dic47⋊C4—82no—
88#10625329—44no—
89#10625329—42no—
90#10625329—42no—
91C2×C47⋊C8—24no—
92C2×C47⋊C8—44no—
93C23×C47—21no—
94C23×C47—21no—
95C23×C47—41no—
96C23×C47—41no—
97C23×C47#10312yes—
98C23×C47C2×C22⋊C411yescharacteristic
99C2×C4×C47—21no—
100C2×C4×C47—21no—
101C2×C4×C47—22no—
102C2×C4×C47—22no—
103C2×C4×C47—22no—
104C2×C4×C47—22no—
105C2×C4×C47—21no—
106C2×C4×C47—21no—
107C2×C4×C47—41no—
108C2×C4×C47—41no—
109C2×C4×C47—41no—
110C2×C4×C47—41no—
111C2×C4×C47#10312yes—
112C2×C4×C47C4×D412yes—
113C2×C4×C47#12011yescharacteristic
114C2×C4×C47#12112yes—
115C2×C4×C47#12311yescharacteristic
116C2×C4×C47C2×SD1611yescharacteristic
117C2×C4×C47C2×Q1611yescharacteristic
118C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
119C2×C4×C47#13711yescharacteristic
120C2×Dic47—42no—
121C2×Dic47—42no—
122C2×Dic47—42no—
123C2×Dic47—42no—
124C2×Dic47—42no—
125C2×Dic47—42no—
126Dic94—84no—
127C8×C47—82no—
128C47⋊C8—44no—
129#9193—471nomaximal, Sylow
130C22×C47—21no—
131C22×C47—21no—
132C22×C47—22no—
133C22×C47—42no—
134C22×C47—41no—
135C22×C47—41no—
136C22×C47—81no—
137C22×C47#38611yescharacteristic
138C22×C47C2×#10311yescharacteristic
139C22×C47#41711yescharacteristic
140C4×C47—22no—
141C4×C47—22no—
142C4×C47—22no—
143C4×C47—41no—
144C4×C47—41no—
145C4×C47#45011yescharacteristic
146C4×C47#45111yescharacteristic
147C4×C47#48311yescharacteristic
148C4×C47#48411yescharacteristic
149Dic47—82no—
150Dic47—82no—
151#2905—471no—
152#2628—471no—
153#1549—472no—
154#1458D4711yescharacteristic
155#1286—472no—
156C2×C47—21no—
157C2×C47—21no—
158C2×C47—42no—
159C2×C47#187211yescharacteristic
160C2×C47#191011yescharacteristic
161C2×C47#191511yescharacteristic
162C22×M16—941no—
163C2×C4⋊Q8—471no—
164C2×#117C2×D4711yescharacteristic
165#431C2×D4712yes—
166#424—944no—
167#423—472no—
168#423—941no—
169#400—472no—
170#399—472no—
171#395—941no—
172#393—941no—
173#389—941no—
174#382—471no—
175#380—941no—
176#332—944no—
177#324—21no—
178#323—21no—
179C47#919311yescharacteristic, complemented, Sylow
180C22×Q8—941no—
181C23×C4D18811yescharacteristic
182C2×M16—942no—
183C2×M16—942no—
184C2×M16—1882no—
185C2×M16—1882no—
186C22×C8—942no—
187C4⋊Q8—942no—
188C4⋊Q8—942no—
189#117—22no—
190#117—22no—
191C2×C4⋊C4—472no—
192C2×C4⋊C4—942no—
193C2×C4⋊C4—941no—
194C2×C4⋊C4—941no—
195C2×C4⋊C4C4×D4711yescharacteristic
196C2×C4⋊C4C22⋊D4711yescharacteristic
197C2×C22⋊C4—942no—
198C2×C22⋊C4C4×D4711yescharacteristic
199C2×C22⋊C4C22⋊D4711yescharacteristic
200C2×C42D18811yescharacteristic
201C2×C42C22×D4711yescharacteristic
202C4⋊C8—22no—
203C4⋊C8—474no—
204C4⋊C8—944no—
205C4⋊C8—942no—
206C22⋊C8—22no—
207C4×C8—22no—
208#95—472no—
209#95—942no—
210#95—942no—
211#95—941no—
212C24(C2×D47)⋊C411yescharacteristic
213C2×Q8—3762no—
214C2×Q8—1882no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—942no—
220C22×C4—942no—
221C22×C4#1062532512yes—
222C22×C4Dic18812yes—
223C22×C4(C2×D47)⋊C411yescharacteristic
224C22×C4(C2×D47)⋊C411yescharacteristic
225C22×C4(C2×D47)⋊C411yescharacteristic
226C22×C4C2×C4×D4711yescharacteristic
227C22×C4C2×D18811yescharacteristic
228C22×C4C2×C22⋊D4711yescharacteristic
229M16—1884no—
230M16—1882no—
231M16—1882no—
232C2×C8—42no—
233C2×C8—944no—
234C2×C8—1884no—
235C4⋊C4—22no—
236C4⋊C4—22no—
237C4⋊C4—944no—
238C4⋊C4—3762no—
239C4⋊C4—1881no—
240C4⋊C4—1881no—
241C4⋊C4—1881no—
242C4⋊C4—1881no—
243C22⋊C4—22no—
244C22⋊C4—22no—
245C22⋊C4—3762no—
246C42—21no—
247C42—21no—
248C42D4×D4712yes—
249C42#1062535212yes—
250C23—21no—
251C23—21no—
252C23—41no—
253C23—41no—
254C23#1309394912yes—
255C23C2×(C2×D47)⋊C411yescharacteristic
256Q8—3764no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—1882no—
270C2×C4—1882no—
271C2×C4—1882no—
272C2×C4—1882no—
273C2×C4—1882no—
274C2×C4—1882no—
275C2×C4#1309394912yes—
276C2×C4(C22⋊D47)⋊C412yes—
277C2×C4#1309399411yescharacteristic
278C2×C4#1309399612yes—
279C2×C4#1309400811yescharacteristic
280C2×C4C2×#1062532511yescharacteristic
281C2×C4C2×Dic18811yescharacteristic
282C2×C4#1309405211yescharacteristic
283C2×C4#1309405711yescharacteristic
284C8—82no—
285C8—1884no—
286C22—21no—
287C22—21no—
288C22—22no—
289C22—42no—
290C22—41no—
291C22—41no—
292C22—81no—
293C22#1568900011yescharacteristic
294C22C2×#1309394911yescharacteristic
295C22#1568912511yescharacteristic
296C4—22no—
297C4—22no—
298C4—22no—
299C4—41no—
300C4—41no—
301C4—3762no—
302C4—3762no—
303C4#1568927611yescharacteristic
304C4#1568928111yescharacteristic
305C4#1568945411yescharacteristic
306C4#1568945911yescharacteristic
307C2—21no—
308C2—21no—
309C2—42no—
310C2#2024369311yescharacteristic
311C2#2024391311yescharacteristic
312C2#2024393811yescharacteristic
subgroupG/H
centerC22#15689000
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1458C2
socleC22×C47#386
prime radicalC24×C47C22⋊C4
cosocleC2×C42×C47C23
hypercenter#1458D47
hypocenterC47#9193
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1458

Complements

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

NH ≅ G/Ntype
3. #20242072C2semidirect
4. #20241247C2semidirect
179. C47#9193semidirect

Maximal subgroups

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

subgroupindex
#202519522
#202492092
#202420722
#202412472
C47×#14582
#919347