Finite Group Databasebeta

← #26129121

Subgroups — #26129121

Order 12032.

380 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 63 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#20244549C211yesmaximal, characteristic
2#20242361C212yesmaximal
3#20241418C212yesmaximal
4#20241322C211yesmaximal, characteristic
5C47×#2811C211yesmaximal, characteristic
6#15689754C2211yescharacteristic
7#15689279—22no—
8#15689132—21no—
9#15688892C2211yescharacteristic
10#15688883—21no—
11#15688859—22no—
12#15688849C2212yes—
13C47×#395—22no—
14C47×#395C2212yes—
15C47×#391—22no—
16C47×#391—21no—
17C47×#391—21no—
18C47×#391C2211yescharacteristic
19C2×#10625337—42no—
20C2×#10625329—21no—
21#13094023—41no—
22#13094015—41no—
23#13094003—22no—
24#13093983—21no—
25#13093948—42no—
26#13093948—41no—
27#13093948—41no—
28C22⋊(C47⋊C8)—22no—
29C22×Q8×C47—21no—
30C22×Q8×C47D411yescharacteristic
31C22×Q8×C47D412yes—
32C2×C4⋊C4×C47—21no—
33C2×C4⋊C4×C47—21no—
34C2×C4⋊C4×C47—22no—
35C2×C4⋊C4×C47C2311yescharacteristic
36C2×C42×C47—21no—
37C2×C42×C47D411yescharacteristic
38#95×C47—22no—
39#95×C47—22no—
40#95×C47—22no—
41#95×C47—22no—
42#95×C47D412yes—
43C2×Q8×C47—24no—
44C2×Q8×C47—21no—
45C2×Q8×C47—41no—
46C2×Q8×C47—41no—
47C2×Q8×C47—42no—
48C2×Q8×C47—82no—
49C2×Q8×C47—81no—
50C2×Q8×C47—81no—
51C2×Q8×C47Q1612yescomplemented
52C22×C4×C47—22no—
53C22×C4×C47—21no—
54C22×C4×C47—21no—
55C22×C4×C47—22no—
56C22×C4×C47C2×D411yescharacteristic
57C22×C4×C47C2×D412yes—
58C2×Dic94—22no—
59C4⋊C4×C47—41no—
60C4⋊C4×C47—82no—
61C4⋊C4×C47—81no—
62C4⋊C4×C47—81no—
63C42×C47—41no—
64C42×C47—81no—
65C22⋊Dic47—41no—
66#10625337—84no—
67C4⋊Dic47—21no—
68C4⋊Dic47—21no—
69C4⋊Dic47—22no—
70Dic47⋊C4—42no—
71C4×Dic47—41no—
72#10625329—41no—
73#10625329—41no—
74C2×C47⋊C8—42no—
75C2×C47⋊C8—41no—
76C23×C47#12011yescharacteristic
77Q8×C47—42no—
78Q8×C47—44no—
79Q8×C47—81no—
80Q8×C47—84no—
81Q8×C47—82no—
82C2×C4×C47—21no—
83C2×C4×C47—21no—
84C2×C4×C47—24no—
85C2×C4×C47—22no—
86C2×C4×C47—22no—
87C2×C4×C47—22no—
88C2×C4×C47—42no—
89C2×C4×C47—41no—
90C2×C4×C47—41no—
91C2×C4×C47—41no—
92C2×C4×C47—42no—
93C2×C4×C47—41no—
94C2×C4×C47—42no—
95C2×C4×C47—42no—
96C2×C4×C47—42no—
97C2×C4×C47—42no—
98C2×C4×C47C2×Q1611yescharacteristic
99C2×C4×C47#13711yescharacteristic
100C2×Dic47—41no—
101C2×Dic47—41no—
102Dic94—82no—
103C47⋊C8—81no—
104C47⋊C8—82no—
105#12224—471nomaximal, Sylow
106C22×C47—21no—
107C22×C47—21no—
108C22×C47#45111yescharacteristic
109C22×C47#45511yescharacteristic
110C22×C47B(4,𝔽2)11yescharacteristic
111C4×C47—21no—
112C4×C47—21no—
113C4×C47—42no—
114C4×C47—44no—
115C4×C47—42no—
116C4×C47—42no—
117C4×C47—82no—
118Dic47—81no—
119Dic47—81no—
120#2811D4711yescharacteristic
121#2028—471no—
122#1605—472no—
123#1354—472no—
124#1319—471no—
125C2×C47—21no—
126C2×C47—21no—
127C2×C47#160211yescharacteristic
128C2×C47#219911yescharacteristic
129C2×C47#220511yescharacteristic
130#544—471no—
131#451—942no—
132#419—941no—
133#395—22no—
134#395C2×D4712yes—
135#391—21no—
136#391—21no—
137#391—22no—
138#391C2×D4711yescharacteristic
139#343—471no—
140#339—941no—
141#328—942no—
142#324—472no—
143C47#1222411yescharacteristic, complemented, Sylow
144C22×Q8—21no—
145C22×Q8C22⋊D4711yescharacteristic
146C22×Q8C22⋊D4712yes—
147C2×Q16—1882no—
148C2×M16—941no—
149C4⋊Q8—1881no—
150#125—1881no—
151C22⋊Q8—942no—
152#117—941no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4C22×D4711yescharacteristic
157C2×C42—21no—
158C2×C42C22⋊D4711yescharacteristic
159#103—1881no—
160#103—1881no—
161#103—1882no—
162C22⋊C8—942no—
163#95—22no—
164#95—22no—
165#95—22no—
166#95—22no—
167#95C22⋊D4712yes—
168C2×Q8—24no—
169C2×Q8—21no—
170C2×Q8—41no—
171C2×Q8—42no—
172C2×Q8—41no—
173C2×Q8—82no—
174C2×Q8—81no—
175C2×Q8—81no—
176C2×Q8—942no—
177C2×Q8#1062533712yescomplemented
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—22no—
181C22×C4—22no—
182C22×C4C2×C22⋊D4711yescharacteristic
183C22×C4C2×C22⋊D4712yes—
184Q16—3764no—
185M16—1881no—
186M16—1881no—
187C2×C8—1882no—
188C2×C8—1881no—
189C4⋊C4—41no—
190C4⋊C4—82no—
191C4⋊C4—81no—
192C4⋊C4—81no—
193C4⋊C4—941no—
194C4⋊C4—941no—
195C4⋊C4—942no—
196C4⋊C4—1882no—
197C22⋊C4—1881no—
198C42—41no—
199C42—81no—
200C42—1881no—
201C23#1309399311yescharacteristic
202Q8—42no—
203Q8—44no—
204Q8—81no—
205Q8—84no—
206Q8—82no—
207Q8—3762no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—22no—
211C2×C4—24no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—41no—
215C2×C4—42no—
216C2×C4—41no—
217C2×C4—41no—
218C2×C4—42no—
219C2×C4—41no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—1881no—
225C2×C4—1881no—
226C2×C4C2×#1062533711yescharacteristic
227C2×C4#1309405511yescharacteristic
228C8—3761no—
229C8—3762no—
230C22—21no—
231C22—21no—
232C22#1568927911yescharacteristic
233C22#1568930111yescharacteristic
234C22#1568931011yescharacteristic
235C4—21no—
236C4—21no—
237C4—42no—
238C4—44no—
239C4—42no—
240C4—42no—
241C4—82no—
242C4—3761no—
243C4—3761no—
244C2—21no—
245C2—21no—
246C2#2024234511yescharacteristic
247C2#2024538911yescharacteristic
248C2#2024541811yescharacteristic
subgroupG/H
centerC22#15689310
commutator, cosocleC2×C4⋊C4×C47C23
FrattiniC2×C4⋊C4C22×D47
FittingC47×#2811C2
socleC22×C47B(4,𝔽2)
prime radicalC23×C47#120
hypercenter#2811D47
hypocenterC47#12224
LCSG ▷ C2×C4⋊C4×C47 ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2811

Complements

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

NH ≅ G/Ntype
51. C2×Q8×C47Q16semidirect
143. C47#12224semidirect
177. C2×Q8#10625337semidirect

Maximal subgroups

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

subgroupindex
#202445492
#202423612
#202414182
#202413222
C47×#28112
#1222447