Finite Group Databasebeta

← #26105375

Subgroups — #26105375

Order 12032.

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

normal subgroups only

This group has 478 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#20249760C211yesmaximal, characteristic
2#20244860C212yesmaximal, complemented
3#20241210C212yesmaximal, complemented
4#20241209C211yesmaximal, characteristic
5C47×#2093C211yesmaximal, characteristic
6C2×#13094001—22no—
7C2×#13093957—22no—
8#15689046—22no—
9#15689043—21no—
10#15689021C2212yes—
11#15689003—21no—
12#15688885—22no—
13#15688878C2212yes—
14C2×#107×C47—21no—
15C2×#102×C47—21no—
16C47×#390C2211yescharacteristic
17C47×#384C2211yescharacteristic
18C47×#336C2211yescharacteristic
19C22×Dic94—21no—
20C23×Dic47—21no—
21C22×C47⋊C8—22no—
22#13094001—44no—
23#13094001—44no—
24C2×C4⋊Dic47D412yes—
25C2×C4⋊Dic47D412yes—
26C2×Dic47⋊C4—22no—
27C2×Dic47⋊C4—22no—
28C2×C22⋊Dic47—22no—
29C2×C22⋊Dic47—21no—
30#13093957—44no—
31C47⋊#95—21no—
32C47⋊#95—22no—
33C22×D4×C47D411yescharacteristic
34C22×C8×C47—21no—
35C2×C4⋊C4×C47—22no—
36C2×C4⋊C4×C47—21no—
37C2×C4⋊C4×C47D411yescharacteristic
38C2×C22⋊C4×C47—22no—
39C2×C42×C47C2311yescharacteristic
40#107×C47—42no—
41#102×C47—42no—
42C24×C47—21no—
43C2×D4×C47—21no—
44C2×D4×C47—41no—
45C2×D4×C47—41no—
46C2×D4×C47Q1612yes—
47C22×C4×C47—21no—
48C22×C4×C47—41no—
49C22×C4×C47C2×D412yes—
50C22×C4×C47C2×D411yescharacteristic
51C22×Dic47—22no—
52C22×Dic47—22no—
53C22×Dic47—44no—
54C22×Dic47—81no—
55C2×Dic94—44no—
56C2×Dic94—81no—
57C2×C8×C47—22no—
58C2×C8×C47—41no—
59C2×C8×C47—41no—
60C4⋊C4×C47—41no—
61C4⋊C4×C47—41no—
62C4⋊C4×C47—41no—
63C4⋊C4×C47—42no—
64C4⋊C4×C47—42no—
65C22⋊C4×C47—42no—
66C22⋊C4×C47—42no—
67C42×C47—41no—
68C22⋊Dic47—44no—
69C22⋊Dic47—81no—
70C4⋊Dic47—22no—
71C4⋊Dic47—42no—
72C4⋊Dic47D814yes—
73Dic47⋊C4—44no—
74Dic47⋊C4—42no—
75Dic47⋊C4—42no—
76C2×C47⋊C8—24no—
77C2×C47⋊C8—42no—
78C2×C47⋊C8—42no—
79C23×C47—22no—
80C23×C47—42no—
81C23×C47—41no—
82C23×C47—42no—
83C23×C47—41no—
84C23×C47#12011yescharacteristic
85D4×C47—42no—
86D4×C47—81no—
87C2×C4×C47—21no—
88C2×C4×C47—22no—
89C2×C4×C47—22no—
90C2×C4×C47—21no—
91C2×C4×C47—42no—
92C2×C4×C47—41no—
93C2×C4×C47—82no—
94C2×C4×C47—81no—
95C2×C4×C47C2×D812yes—
96C2×C4×C47C2×Q1611yescharacteristic
97C2×C4×C47AGL(1,ℤ/8)12yes—
98C2×C4×C47#13711yescharacteristic
99C2×Dic47—24no—
100C2×Dic47—42no—
101C2×Dic47—42no—
102C2×Dic47—42no—
103C2×Dic47—42no—
104C2×Dic47—42no—
105C2×Dic47—84no—
106Dic94—84no—
107C8×C47—42no—
108C47⋊C8—44no—
109#7797—471nomaximal, Sylow
110C22×C47—42no—
111C22×C47—44no—
112C22×C47—82no—
113C22×C47—81no—
114C22×C47—81no—
115C22×C47#44712yes—
116C22×C47#45111yescharacteristic
117C22×C47#45311yescharacteristic
118C22×C47#45612yes—
119C22×C47#45811yescharacteristic
120C4×C47—22no—
121C4×C47—22no—
122C4×C47—21no—
123C4×C47—21no—
124C4×C47—81no—
125C4×C47—161no—
126Dic47—44no—
127Dic47—82no—
128#2683—471no—
129#2093D4711yescharacteristic
130#2083—472no—
131#1271—472no—
132#1270—471no—
133C2×C47—82no—
134C2×C47#161012yes—
135C2×C47#161411yescharacteristic
136C2×C47#161911yescharacteristic
137C2×C47#164912yes—
138C2×C47#166011yescharacteristic
139C2×C22⋊Q8—942no—
140C2×#107—21no—
141C2×#107—942no—
142C2×#102—21no—
143#396—942no—
144#395—941no—
145#390C2×D4711yescharacteristic
146#389—472no—
147#386—941no—
148#384C2×D4711yescharacteristic
149#340—942no—
150#336—472no—
151#336C2×D4711yescharacteristic
152C47#779711yescharacteristic, complemented, Sylow
153C22×Q8—941no—
154C22×D4C22⋊D4711yescharacteristic
155C23×C4—941no—
156C22×C8—21no—
157C22×C8—942no—
158C22⋊Q8—1884no—
159C22⋊Q8—1884no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—472no—
163C2×C4⋊C4—472no—
164C2×C4⋊C4—942no—
165C2×C4⋊C4—942no—
166C2×C4⋊C4C22⋊D4711yescharacteristic
167C2×C22⋊C4—22no—
168C2×C22⋊C4—941no—
169C2×C22⋊C4—942no—
170C2×C42C22×D4711yescharacteristic
171#107—42no—
172#107—1884no—
173#102—42no—
174#95—941no—
175#95—942no—
176C24—21no—
177C2×Q8—3761no—
178C2×Q8—1884no—
179C2×D4—21no—
180C2×D4—41no—
181C2×D4—41no—
182C2×D4#1062533712yes—
183C22×C4—21no—
184C22×C4—41no—
185C22×C4—942no—
186C22×C4—942no—
187C22×C4—3761no—
188C22×C4—1884no—
189C22×C4D4×D4712yes—
190C22×C4C2×C22⋊D4711yescharacteristic
191C2×C8—22no—
192C2×C8—41no—
193C2×C8—41no—
194C2×C8—944no—
195C2×C8—1882no—
196C2×C8—1882no—
197C4⋊C4—41no—
198C4⋊C4—41no—
199C4⋊C4—41no—
200C4⋊C4—42no—
201C4⋊C4—42no—
202C4⋊C4—474no—
203C4⋊C4—942no—
204C4⋊C4—1882no—
205C4⋊C4—1884no—
206C4⋊C4—1882no—
207C4⋊C4—1882no—
208C22⋊C4—42no—
209C22⋊C4—42no—
210C22⋊C4—3761no—
211C22⋊C4—1884no—
212C42—41no—
213C23—22no—
214C23—42no—
215C23—41no—
216C23—42no—
217C23—41no—
218C23#1309399211yescharacteristic
219Q8—3764no—
220D4—42no—
221D4—81no—
222C2×C4—21no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—21no—
226C2×C4—42no—
227C2×C4—41no—
228C2×C4—82no—
229C2×C4—81no—
230C2×C4—944no—
231C2×C4—3764no—
232C2×C4—1882no—
233C2×C4—1882no—
234C2×C4—1882no—
235C2×C4—1882no—
236C2×C4—1882no—
237C2×C4D8×D4712yes—
238C2×C4C2×#1062533711yescharacteristic
239C2×C4#1309404812yes—
240C2×C4#1309405411yescharacteristic
241C8—42no—
242C8—1884no—
243C22—44no—
244C22—42no—
245C22—81no—
246C22—81no—
247C22—82no—
248C22#1568925512yes—
249C22#1568927811yescharacteristic
250C22#1568929011yescharacteristic
251C22#1568930412yes—
252C22#1568931211yescharacteristic
253C4—22no—
254C4—22no—
255C4—21no—
256C4—21no—
257C4—81no—
258C4—161no—
259C4—3762no—
260C4—1884no—
261C2—82no—
262C2#2024238912yes—
263C2#2024241211yescharacteristic
264C2#2024244011yescharacteristic
265C2#2024263612yes—
266C2#2024269711yescharacteristic
subgroupG/H
centerC23#13093992
commutator, cosocleC2×C42×C47C23
FrattiniC2×C42C22×D47
FittingC47×#2093C2
socleC23×C47#120
prime radicalC22×D4×C47D4
hypercenter#2093D47
hypocenterC47#7797
LCSG ▷ C2×C42×C47 ▷ C23×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2093

Complements

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

NH ≅ G/Ntype
2. #20244860C2semidirect
3. #20241210C2semidirect
152. C47#7797semidirect

Maximal subgroups

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

subgroupindex
#202497602
#202448602
#202412102
#202412092
C47×#20932
#779747