Finite Group Databasebeta

← C47×#12181 #26051879

Subgroups — C47×#12181 #26051879

Order 12032.

372 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 74 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
1C47×#2571C211yesmaximal, characteristic, complemented
2C47×#2039C211yesmaximal, characteristic
3C47×#1616C212yesmaximal
4C47×#1353C212yesmaximal, complemented
5C47×#1325C211yesmaximal, characteristic, complemented
6C47×#537C2211yescharacteristic
7C47×#481—22no—
8C2×#104×C47—21no—
9C47×#417—21no—
10C47×#400—21no—
11C47×#400C2212yes—
12C47×#389—22no—
13C47×#384C2211yescharacteristic
14C47×#382—22no—
15C4×C4⋊C4×C47—21no—
16C47×#343C2211yescharacteristic
17C47×#327—22no—
18C47×#324C2212yes—
19C2×#41×C47—21no—
20C2×M16×C47—21no—
21C22⋊Q8×C47—41no—
22#121×C47—22no—
23#121×C47—41no—
24C2×C4⋊C4×C47—21no—
25C2×C4⋊C4×C47—21no—
26C2×C4⋊C4×C47—22no—
27C2×C4⋊C4×C47—21no—
28C2×C4⋊C4×C47D412yes—
29C2×C4⋊C4×C47D411yescharacteristic
30C2×C4⋊C4×C47C2311yescharacteristic
31C2×C42×C47—22no—
32C2×C42×C47D411yescharacteristic
33#106×C47—42no—
34#104×C47—42no—
35#103×C47—41no—
36#102×C47—42no—
37#102×C47—41no—
38C22⋊C8×C47—22no—
39#95×C47—21no—
40#95×C47—22no—
41#95×C47—22no—
42#95×C47—22no—
43#95×C47D412yes—
44#41×C47—41no—
45#41×C47—41no—
46#41×C47—41no—
47#41×C47—41no—
48C2×Q8×C47—21no—
49C2×D4×C47—21no—
50C2×D4×C47—22no—
51C2×D4×C47—41no—
52C22×C4×C47—21no—
53C22×C4×C47—22no—
54C22×C4×C47—22no—
55C22×C4×C47—21no—
56C22×C4×C47—41no—
57C22×C4×C47C2×D411yescharacteristic
58C22×C4×C47C2×D412yes—
59M16×C47—41no—
60M16×C47—41no—
61C2×C8×C47—41no—
62C2×C8×C47—42no—
63C4⋊C4×C47—22no—
64C4⋊C4×C47—41no—
65C4⋊C4×C47—42no—
66C4⋊C4×C47—44no—
67C4⋊C4×C47—42no—
68C4⋊C4×C47—41no—
69C4⋊C4×C47—42no—
70C4⋊C4×C47—81no—
71C22⋊C4×C47—42no—
72C42×C47—21no—
73C42×C47—44no—
74C42×C47SD1612yes—
75C23×C47—41no—
76C23×C47#12011yescharacteristic
77Q8×C47—41no—
78Q8×C47—41no—
79D4×C47—41no—
80D4×C47—41no—
81D4×C47—82no—
82D4×C47—81no—
83D4×C47—81no—
84C2×C4×C47—22no—
85C2×C4×C47—22no—
86C2×C4×C47—22no—
87C2×C4×C47—21no—
88C2×C4×C47—21no—
89C2×C4×C47—22no—
90C2×C4×C47—42no—
91C2×C4×C47—42no—
92C2×C4×C47—42no—
93C2×C4×C47—42no—
94C2×C4×C47—42no—
95C2×C4×C47—42no—
96C2×C4×C47—42no—
97C2×C4×C47—41no—
98C2×C4×C47—41no—
99C2×C4×C47—42no—
100C2×C4×C47—81no—
101C2×C4×C47—81no—
102C2×C4×C47C2×SD1611yescharacteristic
103C2×C4×C47#13711yescharacteristic
104C8×C47—81no—
105C8×C47—82no—
106#12181C4711yesmaximal, characteristic, direct product factor, Sylow
107C22×C47—21no—
108C22×C47—21no—
109C22×C47—81no—
110C22×C47—81no—
111C22×C47—81no—
112C22×C47#44811yescharacteristic
113C22×C47#45411yescharacteristic
114C22×C47B(4,𝔽2)11yescharacteristic
115C4×C47—24no—
116C4×C47—21no—
117C4×C47—21no—
118C4×C47—42no—
119C4×C47—82no—
120C4×C47—81no—
121C4×C47—81no—
122C4×C47—82no—
123#2571C2×C4711yescharacteristic, complemented
124#2039C2×C4711yescharacteristic
125#1616C2×C4712yes—
126#1353C2×C4712yescomplemented
127#1325C2×C4711yescharacteristic, complemented
128C2×C47—21no—
129C2×C47—21no—
130C2×C47—161no—
131C2×C47#159911yescharacteristic
132C2×C47#219811yescharacteristic
133C2×C47#220411yescharacteristic
134#537C22×C4711yescharacteristic
135#481—22no—
136C2×#104—21no—
137#417—21no—
138#400—21no—
139#400C22×C4712yes—
140#389—22no—
141#384C22×C4711yescharacteristic
142#382—22no—
143C4×C4⋊C4—21no—
144#343C22×C4711yescharacteristic
145#327—22no—
146#324C22×C4712yes—
147C47#1218111yescharacteristic, direct product factor, Sylow
148C2×#41—21no—
149C2×M16—21no—
150C22⋊Q8—41no—
151#121—22no—
152#121—41no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4D4×C4711yescharacteristic
158C2×C4⋊C4D4×C4712yes—
159C2×C4⋊C4C23×C4711yescharacteristic
160C2×C42—22no—
161C2×C42D4×C4711yescharacteristic
162#106—42no—
163#104—42no—
164#103—41no—
165#102—41no—
166#102—42no—
167C22⋊C8—22no—
168#95—21no—
169#95—22no—
170#95—22no—
171#95—22no—
172#95D4×C4712yes—
173#41—41no—
174#41—41no—
175#41—41no—
176#41—41no—
177C2×Q8—21no—
178C2×D4—21no—
179C2×D4—22no—
180C2×D4—41no—
181C22×C4—21no—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—21no—
185C22×C4—41no—
186C22×C4C2×D4×C4711yescharacteristic
187C22×C4C2×D4×C4712yes—
188M16—41no—
189M16—41no—
190C2×C8—42no—
191C2×C8—41no—
192C4⋊C4—22no—
193C4⋊C4—41no—
194C4⋊C4—41no—
195C4⋊C4—42no—
196C4⋊C4—42no—
197C4⋊C4—44no—
198C4⋊C4—42no—
199C4⋊C4—81no—
200C22⋊C4—42no—
201C42—21no—
202C42—44no—
203C42SD16×C4712yes—
204C23—41no—
205C23#120×C4711yescharacteristic
206Q8—41no—
207Q8—41no—
208D4—41no—
209D4—41no—
210D4—82no—
211D4—81no—
212D4—81no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—42no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—42no—
229C2×C4—81no—
230C2×C4—81no—
231C2×C4C2×SD16×C4711yescharacteristic
232C2×C4#137×C4711yescharacteristic
233C8—81no—
234C8—82no—
235C22—21no—
236C22—21no—
237C22—81no—
238C22—81no—
239C22—81no—
240C22C47×#44811yescharacteristic
241C22C47×#45411yescharacteristic
242C22C47×B(4,𝔽2)11yescharacteristic
243C4—24no—
244C4—21no—
245C4—21no—
246C4—42no—
247C4—82no—
248C4—81no—
249C4—81no—
250C4—82no—
251C2—21no—
252C2—21no—
253C2—161no—
254C2C47×#159911yescharacteristic
255C2C47×#219811yescharacteristic
256C2C47×#220411yescharacteristic
subgroupG/H
center, socleC22×C47B(4,𝔽2)
commutator, Frattini, cosocleC2×C4⋊C4C23×C47
prime radicalC47×#537C22
LCSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C47 ◁ C23×C47 ◁ C2×C4⋊C4×C47 ◁ G

Complements

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

NH ≅ G/Ntype
1. C47×#2571C2semidirect
4. C47×#1353C2semidirect
5. C47×#1325C2semidirect
106. #12181C47direct (N × H)
123. #2571C2×C47semidirect
126. #1353C2×C47semidirect
127. #1325C2×C47semidirect
147. C47#12181direct (N × H)

Maximal subgroups

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

subgroupindex
C47×#25712
C47×#20392
C47×#16162
C47×#13532
C47×#13252
#1218147