Finite Group Databasebeta

← #26117266

Subgroups — #26117266

Order 12032.

528 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 105 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#20253392C212yesmaximal
2#20244186C211yesmaximal, characteristic
3#20241791C212yesmaximal, complemented
4#20241219C211yesmaximal, characteristic, complemented
5C47×#3162C211yesmaximal, characteristic, complemented
6C2×C4×D188C2211yescharacteristic
7#15689451—22no—
8#15689410—22no—
9#15689325—22no—
10#15689273—22no—
11C4×#10625336—24no—
12#15689176C2212yes—
13C2×C4⋊(C47⋊C8)C2211yescharacteristic
14C2×(C2×D47)⋊C8—22no—
15Q8⋊(C47⋊C8)—24no—
16C47×#533C2212yes—
17C47×#501—22no—
18C47×#501—22no—
19C47×#427—22no—
20C2×#97×C47C2211yescharacteristic
21C22×D188D411yescharacteristic
22C22×C4×D47—21no—
23C2×#10625336—24no—
24C22×C47⋊C8—21no—
25C4×D188—22no—
26C4×D188—22no—
27C2×C4⋊Dic47D411yescharacteristic
28C2×(C2×D47)⋊C4—21no—
29#13093955—24no—
30C4⋊(C47⋊C8)—22no—
31C4⋊(C47⋊C8)—22no—
32#13093948—24no—
33#13093944—24no—
34C22⋊(C47⋊C8)—22no—
35(C2×D47)⋊C8—44no—
36C4×C47⋊C8D414yes—
37C22×C8×C47D412yes—
38C4⋊Q8×C47D412yes—
39C4⋊Q8×C47D412yes—
40#125×C47—22no—
41C22⋊Q8×C47—22no—
42C22⋊Q8×C47—22no—
43C4×Q8×C47—22no—
44C4×Q8×C47—22no—
45#117×C47—22no—
46C2×C42×C47C2311yescharacteristic
47#107×C47—22no—
48#107×C47—22no—
49#106×C47—24no—
50#97×C47—22no—
51C23×D47—21no—
52C2×Q8×C47—22no—
53C2×Q8×C47—22no—
54C22×C4×C47C2×D411yescharacteristic
55C22×C4×C47C2×D411yescharacteristic
56C22×C4×C47#4111yescharacteristic
57C22×Dic47—21no—
58C2×D188—22no—
59C2×D188—42no—
60C2×C4×D47—41no—
61C2×C4×D47—41no—
62C2×C4×D47—42no—
63C2×C4×D47—42no—
64C2×C8×C47—22no—
65C2×C8×C47—22no—
66C2×C8×C47SD1614yescomplemented
67C4⋊C4×C47—22no—
68C4⋊C4×C47—22no—
69C4⋊C4×C47—22no—
70C4⋊C4×C47—22no—
71C4⋊C4×C47—42no—
72C4⋊C4×C47—42no—
73C22⋊C4×C47—42no—
74C42×C47—42no—
75C42×C47C2×D412yes—
76C42×C47C2×D412yes—
77#10625336—48no—
78(C2×D47)⋊C4—42no—
79C4⋊Dic47—22no—
80C2×C47⋊C8—24no—
81C2×C47⋊C8—41no—
82C2×C47⋊C8—42no—
83C23×C47#12111yescharacteristic
84C22×D47—22no—
85C22×D47—41no—
86C22×D47—42no—
87C22×D47—41no—
88C22×D47—42no—
89Q8×C47—44no—
90Q8×C47—44no—
91C2×C4×C47—41no—
92C2×C4×C47—41no—
93C2×C4×C47—42no—
94C2×C4×C47—42no—
95C2×C4×C47#12012yes—
96C2×C4×C47#12011yescharacteristic
97C2×C4×C47#12112yes—
98C2×C4×C47#12711yescharacteristic
99C2×C4×C47C2×SD1612yes—
100C2×C4×C47#13712yes—
101C2×Dic47—41no—
102C2×Dic47—42no—
103D188—24no—
104D188—42no—
105C4×D47—82no—
106C4×D47—81no—
107C4×D47—81no—
108C8×C47—24no—
109C47⋊C8—44no—
110C47⋊C8—81no—
111#10036—471nomaximal, Sylow
112C22×C47—21no—
113C22×C47—21no—
114C22×C47#39211yescharacteristic
115C22×C47#46112yes—
116C2×D47—44no—
117C2×D47—42no—
118C2×D47—82no—
119C2×D47—81no—
120C2×D47—81no—
121C4×C47—22no—
122C4×C47—22no—
123C4×C47—82no—
124C4×C47—82no—
125C4×C47#44812yes—
126C4×C47#49211yescharacteristic
127C4×C47#49711yescharacteristic
128Dic47—81no—
129#3162D4711yescharacteristic, complemented
130#3101—472no—
131#1955—471no—
132#1481—472no—
133#1275—471no—
134C2×C47—22no—
135C2×C47#201011yescharacteristic
136C2×C47#201311yescharacteristic
137C2×C47#201711yescharacteristic
138D47—82no—
139#533C2×D4712yes—
140C2×C4×D4—471no—
141#501—22no—
142#501—22no—
143#483—942no—
144#476—942no—
145#461—942no—
146#450—942no—
147C4×SD16—944no—
148#431—472no—
149#427—22no—
150C2×C4⋊C8—471no—
151C2×C22⋊C8—942no—
152C2×#97C2×D4711yescharacteristic
153Q8⋊C8—944no—
154C47#1003611yescharacteristic, complemented, Sylow
155C22×D4—471no—
156C23×C4—941no—
157C2×SD16—944no—
158C22×C8—941no—
159C22×C8C22⋊D4712yes—
160C4⋊Q8C22⋊D4712yes—
161C4⋊Q8C22⋊D4712yes—
162#125—22no—
163C22⋊Q8—22no—
164C22⋊Q8—22no—
165C4×Q8—22no—
166C4×Q8—22no—
167C4×D4—942no—
168C4×D4—942no—
169#117—22no—
170C2×C4⋊C4—471no—
171C2×C22⋊C4—941no—
172C2×C42C22×D4711yescharacteristic
173#107—22no—
174#107—22no—
175#106—24no—
176#106—944no—
177C4⋊C8—942no—
178C4⋊C8—942no—
179#103—944no—
180#102—944no—
181C22⋊C8—942no—
182C22⋊C8—1884no—
183#97—22no—
184C4×C8—474no—
185C24—941no—
186C2×Q8—22no—
187C2×Q8—22no—
188C2×D4—942no—
189C2×D4—1882no—
190C22×C4—941no—
191C22×C4—1881no—
192C22×C4—1881no—
193C22×C4—1882no—
194C22×C4—1882no—
195C22×C4D4×D4711yescharacteristic
196C22×C4#1062535211yescharacteristic
197C22×C4C2×C22⋊D4711yescharacteristic
198SD16—1888no—
199C2×C8—22no—
200C2×C8—22no—
201C2×C8—944no—
202C2×C8—1881no—
203C2×C8—1882no—
204C2×C8#1062533614yescomplemented
205C4⋊C4—22no—
206C4⋊C4—22no—
207C4⋊C4—22no—
208C4⋊C4—22no—
209C4⋊C4—42no—
210C4⋊C4—42no—
211C4⋊C4—942no—
212C22⋊C4—42no—
213C22⋊C4—1882no—
214C42—42no—
215C42D4×D4712yes—
216C42C2×C22⋊D4712yes—
217C23—942no—
218C23—1881no—
219C23—1882no—
220C23—1881no—
221C23—1882no—
222C23#1309399511yescharacteristic
223Q8—44no—
224Q8—44no—
225D4—944no—
226D4—1882no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—3761no—
232C2×C4—3761no—
233C2×C4—3762no—
234C2×C4—1881no—
235C2×C4—1882no—
236C2×C4#1309399212yes—
237C2×C4#1309399311yescharacteristic
238C2×C4#1309399612yes—
239C2×C4#1309402011yescharacteristic
240C2×C4C2×#1062533612yes—
241C2×C4#1309405512yes—
242C8—24no—
243C8—3761no—
244C8—1884no—
245C22—21no—
246C22—21no—
247C22—3762no—
248C22—3761no—
249C22—3761no—
250C22—1884no—
251C22—1882no—
252C22#1568903111yescharacteristic
253C22#1568932512yes—
254C4—22no—
255C4—22no—
256C4—82no—
257C4—82no—
258C4—3761no—
259C4#1568926212yes—
260C4#1568949711yescharacteristic
261C4#1568951511yescharacteristic
262C2—22no—
263C2—3762no—
264C2#2024445211yescharacteristic
265C2#2024447111yescharacteristic
266C2#2024449111yescharacteristic
subgroupG/H
centerC2×C4#13094020
commutator, cosocleC2×C42×C47C23
FrattiniC2×C42C22×D47
FittingC47×#3162C2
socleC22×C47#392
prime radicalC22×D188D4
hypercenter#3162D47
hypocenterC47#10036
LCSG ▷ C2×C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3162

Complements

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

NH ≅ G/Ntype
3. #20241791C2semidirect
4. #20241219C2semidirect
5. C47×#3162C2semidirect
66. C2×C8×C47SD16semidirect
129. #3162D47semidirect
154. C47#10036semidirect
204. C2×C8#10625336semidirect

Maximal subgroups

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

subgroupindex
#202533922
#202441862
#202417912
#202412192
C47×#31622
#1003647