Finite Group Databasebeta

← #26114152

Subgroups — #26114152

Order 12032.

522 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 522 conjugacy classes of subgroups — too many to draw. Only its 115 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#20248293C211yesmaximal, characteristic
2#20243384C211yesmaximal, characteristic, complemented
3#20241814C212yesmaximal
4#20241247C212yesmaximal, complemented
5C47×#2983C211yesmaximal, characteristic, complemented
6C2×#13094000—21no—
7C2×C4×D188C2211yescharacteristic
8#15689147C2212yes—
9C2×#13093943—21no—
10#15689053C412yes—
11#15689040C412yes—
12#15689014—22no—
13#15688989C2211yescharacteristic
14#15688980—21no—
15#15688892—21no—
16#15688852—24no—
17C2×C4×Q8×C47C2211yescharacteristic
18C47×#446—24no—
19C47×#432—22no—
20C47×#431C2212yes—
21C22×D188C2×C411yescharacteristic
22C22×C4×D47—21no—
23C2×#10625329—22no—
24#13094000—44no—
25C4×D188—22no—
26C4×D188—22no—
27C2×C4⋊Dic47—21no—
28C2×C4⋊Dic47C2×C411yescharacteristic
29C2×Dic47⋊C4—22no—
30C2×(C2×D47)⋊C4—22no—
31C2×(C2×D47)⋊C4—22no—
32C2×(C2×D47)⋊C4—21no—
33C4⋊(C47⋊C8)—22no—
34C4⋊(C47⋊C8)—22no—
35#13093943—42no—
36C47⋊#95—22no—
37C47⋊#95—21no—
38C47⋊#95C2×C412yes—
39C22×Q8×C47D411yescharacteristic
40C4×Q8×C47—22no—
41C4×Q8×C47—22no—
42C2×C4⋊C4×C47C2×C412yes—
43C2×C4⋊C4×C47D411yescharacteristic
44C2×C42×C47D412yes—
45C2×C42×C47C2311yescharacteristic
46C4⋊C8×C47—24no—
47C4⋊C8×C47—22no—
48C22⋊C8×C47—22no—
49#97×C47—24no—
50C4×C8×C47D414yes—
51C23×D47—21no—
52C2×Q8×C47—22no—
53C2×Q8×C47—21no—
54C2×Q8×C47—22no—
55C2×Q8×C47—21no—
56C22×C4×C47C22⋊C412yes—
57C22×C4×C47C22⋊C412yes—
58C22×C4×C47C22×C411yescharacteristic
59C22×C4×C47C2×D411yescharacteristic
60C22×C4×C47C2×D411yescharacteristic
61C22×Dic47—22no—
62C22×Dic47—21no—
63C2×D188—22no—
64C2×D188—42no—
65C2×C4×D47—42no—
66C2×C4×D47—42no—
67C2×C4×D47—41no—
68C2×C4×D47—41no—
69C2×C8×C47—24no—
70C4⋊C4×C47—24no—
71C4⋊C4×C47—22no—
72C42×C47—22no—
73C42×C47—22no—
74C42×C47C2×D412yes—
75C42×C47#4112yes—
76(C2×D47)⋊C4—44no—
77(C2×D47)⋊C4—44no—
78(C2×D47)⋊C4—41no—
79(C2×D47)⋊C4—41no—
80C4⋊Dic47—22no—
81C4⋊Dic47—42no—
82Dic47⋊C4—44no—
83#10625329—42no—
84#10625329—42no—
85C2×C47⋊C8—42no—
86C23×C47C2×C22⋊C411yescharacteristic
87C22×D47—22no—
88C22×D47—41no—
89C22×D47—41no—
90C22×D47—41no—
91C22×D47—41no—
92C22×D47—41no—
93C22×D47—41no—
94Q8×C47—42no—
95Q8×C47—42no—
96C2×C4×C47—22no—
97C2×C4×C47—22no—
98C2×C4×C47—22no—
99C2×C4×C47—22no—
100C2×C4×C47—41no—
101C2×C4×C47—41no—
102C2×C4×C47—42no—
103C2×C4×C47C4×D412yes—
104C2×C4×C47#12011yescharacteristic
105C2×C4×C47#12112yes—
106C2×C4×C47#12311yescharacteristic
107C2×C4×C47AGL(1,ℤ/8)12yes—
108C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
109C2×C4×C47#13711yescharacteristic
110C2×Dic47—42no—
111C2×Dic47—42no—
112C2×Dic47—41no—
113C2×Dic47—41no—
114C2×Dic47—41no—
115C2×Dic47—42no—
116D188—24no—
117D188—42no—
118C4×D47—82no—
119C4×D47—82no—
120C8×C47—44no—
121C47⋊C8—82no—
122#9490—471nomaximal, Sylow
123C22×C47—21no—
124C22×C47—21no—
125C22×C47#38611yescharacteristic
126C22×C47#41711yescharacteristic
127C22×C47#41811yescharacteristic
128C2×D47—42no—
129C2×D47—42no—
130C2×D47—42no—
131C2×D47—81no—
132C2×D47—81no—
133C2×D47—81no—
134C2×D47—81no—
135C4×C47—22no—
136C4×C47—21no—
137C4×C47—21no—
138C4×C47—42no—
139C4×C47—42no—
140C4×C47#46812yes—
141C4×C47#46912yes—
142Dic47—82no—
143Dic47—81no—
144#2983D4711yescharacteristic, complemented
145#2536—471no—
146#1801—471no—
147#1486—472no—
148#1286—472no—
149C2×C47—21no—
150C2×C47—21no—
151C2×C47#189111yescharacteristic
152C2×C47#189511yescharacteristic
153C2×C47#191211yescharacteristic
154D47—82no—
155C2×C22⋊Q8—941no—
156C2×C4×Q8C2×D4711yescharacteristic
157C2×C4×D4—471no—
158#446—24no—
159#432—22no—
160#431C2×D4712yes—
161#423—472no—
162C2×#101—941no—
163#398—472no—
164#394—472no—
165#388—942no—
166#382—471no—
167#380—941no—
168#343—941no—
169D4⋊C8—944no—
170C47#949011yescharacteristic, complemented, Sylow
171C22×Q8D18811yescharacteristic
172C22×D4—471no—
173C23×C4—941no—
174C2×M16—942no—
175C22⋊Q8—1884no—
176C4×Q8—22no—
177C4×Q8—22no—
178C4×D4—942no—
179C4×D4—942no—
180C2×C4⋊C4—471no—
181C2×C4⋊C4—942no—
182C2×C4⋊C4—941no—
183C2×C4⋊C4C4×D4712yes—
184C2×C4⋊C4D18811yescharacteristic
185C2×C22⋊C4—942no—
186C2×C22⋊C4—942no—
187C2×C22⋊C4—941no—
188C2×C42C22⋊D4712yes—
189C2×C42C22×D4711yescharacteristic
190C4⋊C8—24no—
191C4⋊C8—22no—
192C4⋊C8—942no—
193C4⋊C8—942no—
194#101—1882no—
195C22⋊C8—22no—
196#97—24no—
197C4×C8C22⋊D4714yes—
198#95—472no—
199#95—942no—
200#95—941no—
201C24—941no—
202C2×Q8—22no—
203C2×Q8—21no—
204C2×Q8—22no—
205C2×Q8—21no—
206C2×D4—942no—
207C2×D4—1882no—
208C22×C4—942no—
209C22×C4—941no—
210C22×C4—1882no—
211C22×C4—1882no—
212C22×C4—1881no—
213C22×C4—1881no—
214C22×C4(C2×D47)⋊C412yes—
215C22×C4(C2×D47)⋊C412yes—
216C22×C4C2×C4×D4711yescharacteristic
217C22×C4C2×D18811yescharacteristic
218C22×C4C2×C22⋊D4711yescharacteristic
219M16—1882no—
220M16—1882no—
221C2×C8—24no—
222C2×C8—1882no—
223C4⋊C4—24no—
224C4⋊C4—22no—
225C4⋊C4—942no—
226C4⋊C4—1884no—
227C4⋊C4—1882no—
228C22⋊C4—1884no—
229C22⋊C4—1884no—
230C22⋊C4—1881no—
231C22⋊C4—1881no—
232C42—22no—
233C42—22no—
234C42#1062535012yes—
235C42C2×C22⋊D4712yes—
236C23—942no—
237C23—1881no—
238C23—1881no—
239C23—1881no—
240C23—1881no—
241C23—1881no—
242C23—1881no—
243C23C2×(C2×D47)⋊C411yescharacteristic
244Q8—42no—
245Q8—42no—
246D4—944no—
247D4—1882no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—42no—
255C2×C4—3762no—
256C2×C4—3762no—
257C2×C4—1882no—
258C2×C4—1882no—
259C2×C4—1881no—
260C2×C4—1881no—
261C2×C4—1882no—
262C2×C4—1881no—
263C2×C4C4×C22⋊D4712yes—
264C2×C4#1309399311yescharacteristic
265C2×C4#1309399812yes—
266C2×C4#1309400711yescharacteristic
267C2×C4#1309405212yes—
268C2×C4#1309405211yescharacteristic
269C2×C4#1309405711yescharacteristic
270C8—44no—
271C8—3762no—
272C22—21no—
273C22—21no—
274C22—3761no—
275C22—3761no—
276C22—3761no—
277C22—3761no—
278C22—1882no—
279C22—1882no—
280C22—1882no—
281C22#1568900211yescharacteristic
282C22#1568912511yescharacteristic
283C22#1568912911yescharacteristic
284C4—22no—
285C4—21no—
286C4—21no—
287C4—42no—
288C4—42no—
289C4—3762no—
290C4—3761no—
291C4#1568936912yes—
292C4#1568937612yes—
293C2—21no—
294C2—21no—
295C2—3762no—
296C2#2024380611yescharacteristic
297C2#2024382611yescharacteristic
298C2#2024392511yescharacteristic
subgroupG/H
centerC22#15689002
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#2983C2
socleC22×C47#386
prime radicalC22×D188C2×C4
cosocleC2×C42×C47C23
hypercenter#2983D47
hypocenterC47#9490
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2983

Complements

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

NH ≅ G/Ntype
2. #20243384C2semidirect
4. #20241247C2semidirect
5. C47×#2983C2semidirect
144. #2983D47semidirect
170. C47#9490semidirect

Maximal subgroups

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

subgroupindex
#202482932
#202433842
#202418142
#202412472
C47×#29832
#949047