Finite Group Databasebeta

← #26114140

Subgroups — #26114140

Order 12032.

454 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 111 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#20252473C211yesmaximal, characteristic
2#20248750C211yesmaximal, characteristic
3#20241816C212yesmaximal
4#20241248C212yesmaximal
5C47×#1801C211yesmaximal, characteristic
6C2×C4×Dic94C2211yescharacteristic
7#15689254—24no—
8#15689180—22no—
9#15689176C2212yes—
10#15689062C412yes—
11#15689043C412yes—
12#15689030—21no—
13#15689019—21no—
14#15689007—22no—
15#15688989C2211yescharacteristic
16#15688853—24no—
17C2×C4×Q8×C47C2211yescharacteristic
18C47×#423C2212yes—
19C2×#101×C47—21no—
20C47×#343—21no—
21C22×Dic94C2×C411yescharacteristic
22C4×Dic94—22no—
23C4×Dic94—22no—
24C2×C4⋊Dic47—21no—
25C2×C4⋊Dic47C2×C411yescharacteristic
26C2×Dic47⋊C4—22no—
27C2×Dic47⋊C4—21no—
28C2×C4×Dic47—21no—
29C4⋊(C47⋊C8)—24no—
30C4⋊(C47⋊C8)—22no—
31C22⋊(C47⋊C8)—22no—
32#13093935—24no—
33C4×C47⋊C8D414yes—
34C47⋊#95—22no—
35C47⋊#95—22no—
36C47⋊#95—22no—
37C47⋊#95—21no—
38C47⋊#95C2×C412yes—
39C22×Q8×C47D411yescharacteristic
40C2×M16×C47—22no—
41C4×Q8×C47—22no—
42C4×Q8×C47—22no—
43C2×C4⋊C4×C47C2×C412yes—
44C2×C4⋊C4×C47D411yescharacteristic
45C2×C42×C47D412yes—
46C2×C42×C47C2311yescharacteristic
47C4⋊C8×C47—22no—
48C4⋊C8×C47—22no—
49#101×C47—42no—
50C2×Q8×C47—22no—
51C2×Q8×C47—21no—
52C2×Q8×C47—21no—
53C2×Q8×C47—22no—
54C22×C4×C47C22⋊C412yes—
55C22×C4×C47C22⋊C412yes—
56C22×C4×C47C22×C411yescharacteristic
57C22×C4×C47C2×D411yescharacteristic
58C22×C4×C47C2×D411yescharacteristic
59C22×Dic47—21no—
60C22×Dic47—22no—
61C22×Dic47—21no—
62C2×Dic94—22no—
63C2×Dic94—42no—
64M16×C47—42no—
65M16×C47—42no—
66C2×C8×C47—42no—
67C4⋊C4×C47—24no—
68C4⋊C4×C47—22no—
69C42×C47—22no—
70C42×C47—22no—
71C42×C47C2×D412yes—
72C42×C47#4112yes—
73C4⋊Dic47—22no—
74C4⋊Dic47—41no—
75C4⋊Dic47—41no—
76Dic47⋊C4—44no—
77Dic47⋊C4—41no—
78Dic47⋊C4—41no—
79C4×Dic47—42no—
80C2×C47⋊C8—24no—
81C23×C47C2×C22⋊C411yescharacteristic
82Q8×C47—42no—
83Q8×C47—42no—
84C2×C4×C47—22no—
85C2×C4×C47—22no—
86C2×C4×C47—22no—
87C2×C4×C47—22no—
88C2×C4×C47—41no—
89C2×C4×C47—41no—
90C2×C4×C47—42no—
91C2×C4×C47C4×D412yes—
92C2×C4×C47#12011yescharacteristic
93C2×C4×C47#12112yes—
94C2×C4×C47#12311yescharacteristic
95C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
96C2×C4×C47#13712yes—
97C2×C4×C47#13711yescharacteristic
98C2×Dic47—22no—
99C2×Dic47—41no—
100C2×Dic47—41no—
101C2×Dic47—42no—
102C2×Dic47—42no—
103C2×Dic47—42no—
104C2×Dic47—41no—
105C2×Dic47—42no—
106Dic94—24no—
107Dic94—42no—
108C8×C47—82no—
109C47⋊C8—44no—
110#9488—471nomaximal, Sylow
111C22×C47—21no—
112C22×C47—21no—
113C22×C47#38611yescharacteristic
114C22×C47#41711yescharacteristic
115C22×C47#41911yescharacteristic
116C4×C47—22no—
117C4×C47—21no—
118C4×C47—21no—
119C4×C47—42no—
120C4×C47—42no—
121C4×C47#46812yes—
122C4×C47#47012yes—
123Dic47—42no—
124Dic47—82no—
125Dic47—81no—
126#2983—471no—
127#2581—471no—
128#1801D4711yescharacteristic
129#1487—472no—
130#1286—472no—
131C2×C47—21no—
132C2×C47—21no—
133C2×C47#189111yescharacteristic
134C2×C47#189411yescharacteristic
135C2×C47#191211yescharacteristic
136C2×C4×Q8—471no—
137C2×C4×Q8C2×D4711yescharacteristic
138#446—944no—
139#432—942no—
140#431—472no—
141#423C2×D4712yes—
142C2×#101—21no—
143#400—472no—
144#395—472no—
145#391—941no—
146#389—941no—
147#387—942no—
148#382—471no—
149#343—21no—
150Q8⋊C8—944no—
151C47#948811yescharacteristic, complemented, Sylow
152C22×Q8—471no—
153C22×Q8C22⋊D4711yescharacteristic
154C2×M16—22no—
155C4×Q8—22no—
156C4×Q8—22no—
157C4×Q8—942no—
158C4×Q8—942no—
159C2×C4⋊C4—471no—
160C2×C4⋊C4—942no—
161C2×C4⋊C4—941no—
162C2×C4⋊C4—941no—
163C2×C4⋊C4C4×D4712yes—
164C2×C4⋊C4C22⋊D4711yescharacteristic
165C2×C42—941no—
166C2×C42D18812yes—
167C2×C42C22×D4711yescharacteristic
168C4⋊C8—22no—
169C4⋊C8—22no—
170C4⋊C8—944no—
171C4⋊C8—942no—
172#101—42no—
173C22⋊C8—942no—
174#97—944no—
175C4×C8—474no—
176#95—472no—
177#95—942no—
178#95—942no—
179#95—942no—
180#95—941no—
181C2×Q8—22no—
182C2×Q8—21no—
183C2×Q8—22no—
184C2×Q8—21no—
185C2×Q8—942no—
186C2×Q8—1882no—
187C22×C4—941no—
188C22×C4—942no—
189C22×C4—941no—
190C22×C4(C2×D47)⋊C412yes—
191C22×C4(C2×D47)⋊C412yes—
192C22×C4C2×C4×D4711yescharacteristic
193C22×C4C2×D18811yescharacteristic
194C22×C4C2×C22⋊D4711yescharacteristic
195M16—42no—
196M16—42no—
197C2×C8—42no—
198C2×C8—944no—
199C4⋊C4—24no—
200C4⋊C4—22no—
201C4⋊C4—942no—
202C4⋊C4—1884no—
203C4⋊C4—1881no—
204C4⋊C4—1881no—
205C4⋊C4—1881no—
206C4⋊C4—1881no—
207C42—22no—
208C42—22no—
209C42—1882no—
210C42D4×D4712yes—
211C42#1062535212yes—
212C23C2×(C2×D47)⋊C411yescharacteristic
213Q8—42no—
214Q8—42no—
215Q8—944no—
216Q8—1882no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4—42no—
224C2×C4—942no—
225C2×C4—1882no—
226C2×C4—1882no—
227C2×C4—1882no—
228C2×C4—1881no—
229C2×C4—1882no—
230C2×C4—1881no—
231C2×C4—1881no—
232C2×C4(C22⋊D47)⋊C412yes—
233C2×C4#1309399411yescharacteristic
234C2×C4#1309399612yes—
235C2×C4#1309400811yescharacteristic
236C2×C4#1309405011yescharacteristic
237C2×C4#1309405412yes—
238C2×C4#1309405511yescharacteristic
239C8—82no—
240C8—1884no—
241C22—21no—
242C22—21no—
243C22#1568900011yescharacteristic
244C22#1568912211yescharacteristic
245C22#1568913111yescharacteristic
246C4—22no—
247C4—21no—
248C4—21no—
249C4—42no—
250C4—42no—
251C4—3762no—
252C4—3761no—
253C4—1882no—
254C4#1568936712yes—
255C4#1568937912yes—
256C2—21no—
257C2—21no—
258C2#2024380411yescharacteristic
259C2#2024381911yescharacteristic
260C2#2024392311yescharacteristic
subgroupG/H
centerC22#15689000
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1801C2
socleC22×C47#386
prime radicalC23×C47C2×C22⋊C4
cosocleC2×C42×C47C23
hypercenter#1801D47
hypocenterC47#9488
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1801

Complements

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

NH ≅ G/Ntype
151. C47#9488semidirect

Maximal subgroups

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

subgroupindex
#202524732
#202487502
#202418162
#202412482
C47×#18012
#948847