Finite Group Databasebeta

← #26105507

Subgroups — #26105507

Order 12032.

410 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 87 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#20250420C211yesmaximal, characteristic
2#20244865C212yesmaximal
3#20241214C211yesmaximal, characteristic
4#20241212C212yesmaximal
5C47×#2096C211yesmaximal, characteristic
6C2×#13094022—21no—
7C2×#13093955—22no—
8#15689053—21no—
9#15689044—22no—
10#15689021C2212yes—
11#15689008—22no—
12#15688885—22no—
13#15688878C2212yes—
14C2×#106×C47—21no—
15C2×#103×C47—21no—
16C47×#391C2211yescharacteristic
17C47×#384C2211yescharacteristic
18C47×#336C2211yescharacteristic
19C22×Dic94—21no—
20C22×C47⋊C8—22no—
21#13094022—42no—
22#13094022—42no—
23C2×C4⋊Dic47—21no—
24C2×C4⋊Dic47D412yes—
25C2×C4⋊Dic47D412yes—
26C2×Dic47⋊C4—22no—
27C2×Dic47⋊C4—22no—
28C2×C4×Dic47—21no—
29#13093955—44no—
30C47⋊#95—21no—
31C47⋊#95—22no—
32C47⋊#95—22no—
33C22×Q8×C47D411yescharacteristic
34C22×C8×C47—21no—
35C2×C4⋊C4×C47—22no—
36C2×C4⋊C4×C47—21no—
37C2×C4⋊C4×C47D411yescharacteristic
38C2×C42×C47C2311yescharacteristic
39#106×C47—42no—
40#103×C47—42no—
41#95×C47—22no—
42C2×Q8×C47—21no—
43C2×Q8×C47—41no—
44C2×Q8×C47—41no—
45C2×Q8×C47SD1612yes—
46C22×C4×C47—21no—
47C22×C4×C47—21no—
48C22×C4×C47—41no—
49C22×C4×C47C2×D411yescharacteristic
50C22×C4×C47C2×D412yes—
51C22×Dic47—22no—
52C22×Dic47—22no—
53C2×Dic94—42no—
54C2×Dic94—81no—
55C2×Dic94—81no—
56C2×C8×C47—22no—
57C2×C8×C47—41no—
58C2×C8×C47—41no—
59C4⋊C4×C47—41no—
60C4⋊C4×C47—41no—
61C4⋊C4×C47—41no—
62C4⋊C4×C47—42no—
63C4⋊C4×C47—42no—
64C42×C47—41no—
65C4⋊Dic47—22no—
66C4⋊Dic47—42no—
67C4⋊Dic47—81no—
68C4⋊Dic47SD1614yes—
69Dic47⋊C4—44no—
70Dic47⋊C4—42no—
71Dic47⋊C4—42no—
72C4×Dic47—42no—
73C2×C47⋊C8—24no—
74C2×C47⋊C8—42no—
75C2×C47⋊C8—42no—
76C23×C47#12011yescharacteristic
77Q8×C47—42no—
78Q8×C47—81no—
79C2×C4×C47—21no—
80C2×C4×C47—22no—
81C2×C4×C47—22no—
82C2×C4×C47—21no—
83C2×C4×C47—22no—
84C2×C4×C47—41no—
85C2×C4×C47—41no—
86C2×C4×C47—41no—
87C2×C4×C47—42no—
88C2×C4×C47—82no—
89C2×C4×C47—81no—
90C2×C4×C47C2×SD1612yes—
91C2×C4×C47C2×SD1611yescharacteristic
92C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
93C2×C4×C47#13712yes—
94C2×Dic47—24no—
95C2×Dic47—42no—
96C2×Dic47—42no—
97C2×Dic47—42no—
98C2×Dic47—42no—
99C2×Dic47—42no—
100Dic94—84no—
101C8×C47—42no—
102C47⋊C8—44no—
103#7817—471nomaximal, Sylow
104C22×C47#44812yes—
105C22×C47#45011yescharacteristic
106C22×C47#45311yescharacteristic
107C22×C47#45612yes—
108C22×C47#45811yescharacteristic
109C4×C47—22no—
110C4×C47—22no—
111C4×C47—21no—
112C4×C47—21no—
113C4×C47—42no—
114C4×C47—81no—
115C4×C47—161no—
116Dic47—44no—
117Dic47—82no—
118#2734—471no—
119#2096D4711yescharacteristic
120#2084—472no—
121#1272—471no—
122#1271—472no—
123C2×C47#161112yes—
124C2×C47#161311yescharacteristic
125C2×C47#162311yescharacteristic
126C2×C47#165112yes—
127C2×C47#166011yescharacteristic
128C2×C4⋊Q8—941no—
129C2×#106—21no—
130C2×#106—942no—
131C2×#103—21no—
132#398—941no—
133#395—942no—
134#391C2×D4711yescharacteristic
135#389—472no—
136#387—942no—
137#384C2×D4711yescharacteristic
138#340—942no—
139#336—472no—
140#336C2×D4711yescharacteristic
141C47#781711yescharacteristic, complemented, Sylow
142C22×Q8—941no—
143C22×Q8C22⋊D4711yescharacteristic
144C22×C8—21no—
145C22×C8—942no—
146C4⋊Q8—1882no—
147C4⋊Q8—1882no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—472no—
151C2×C4⋊C4—472no—
152C2×C4⋊C4—941no—
153C2×C4⋊C4—942no—
154C2×C4⋊C4—942no—
155C2×C4⋊C4C22⋊D4711yescharacteristic
156C2×C42—941no—
157C2×C42C22×D4711yescharacteristic
158#106—42no—
159#106—1884no—
160#103—42no—
161#95—22no—
162#95—941no—
163#95—942no—
164#95—942no—
165C2×Q8—21no—
166C2×Q8—41no—
167C2×Q8—41no—
168C2×Q8—3761no—
169C2×Q8—3761no—
170C2×Q8—1882no—
171C2×Q8#1062533612yes—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—41no—
175C22×C4—942no—
176C22×C4—942no—
177C22×C4D4×D4712yes—
178C22×C4C2×C22⋊D4711yescharacteristic
179C2×C8—22no—
180C2×C8—41no—
181C2×C8—41no—
182C2×C8—944no—
183C2×C8—1882no—
184C2×C8—1882no—
185C4⋊C4—41no—
186C4⋊C4—41no—
187C4⋊C4—41no—
188C4⋊C4—42no—
189C4⋊C4—42no—
190C4⋊C4—474no—
191C4⋊C4—942no—
192C4⋊C4—3761no—
193C4⋊C4—1882no—
194C4⋊C4—1884no—
195C4⋊C4—1882no—
196C4⋊C4—1882no—
197C42—41no—
198C42—1882no—
199C23#1309399211yescharacteristic
200Q8—42no—
201Q8—81no—
202Q8—3764no—
203C2×C4—21no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—21no—
207C2×C4—22no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—42no—
212C2×C4—82no—
213C2×C4—81no—
214C2×C4—944no—
215C2×C4—1882no—
216C2×C4—1882no—
217C2×C4—1882no—
218C2×C4—1882no—
219C2×C4—1882no—
220C2×C4SD16×D4712yes—
221C2×C4C2×#1062533611yescharacteristic
222C2×C4#1309404911yescharacteristic
223C2×C4#1309405312yes—
224C8—42no—
225C8—1884no—
226C22#1568926012yes—
227C22#1568927311yescharacteristic
228C22#1568929011yescharacteristic
229C22#1568930412yes—
230C22#1568931211yescharacteristic
231C4—22no—
232C4—22no—
233C4—21no—
234C4—21no—
235C4—42no—
236C4—81no—
237C4—161no—
238C4—3762no—
239C4—1884no—
240C2#2024239412yes—
241C2#2024240711yescharacteristic
242C2#2024246411yescharacteristic
243C2#2024265012yes—
244C2#2024269711yescharacteristic
subgroupG/H
centerC23#13093992
commutator, cosocleC2×C42×C47C23
FrattiniC2×C42C22×D47
FittingC47×#2096C2
socle, prime radicalC23×C47#120
hypercenter#2096D47
hypocenterC47#7817
LCSG ▷ C2×C42×C47 ▷ C23×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2096

Complements

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

NH ≅ G/Ntype
141. C47#7817semidirect

Maximal subgroups

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

subgroupindex
#202504202
#202448652
#202412142
#202412122
C47×#20962
#781747