Finite Group Databasebeta

← #26125603

Subgroups — #26125603

Order 12032.

364 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 91 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#20252096C211yesmaximal, characteristic
2#20245184C212yesmaximal, complemented
3#20241314C211yesmaximal, characteristic, complemented
4#20241313C212yesmaximal, complemented
5C47×#1315C211yesmaximal, characteristic, complemented
6C2×#13094032C411yescharacteristic
7C2×#13094031C411yescharacteristic
8#15689669C411yescharacteristic
9#15689198—22no—
10(C22⋊D47)⋊C8—22no—
11#15689146C411yescharacteristic
12#15689089C2211yescharacteristic
13C2×C8×Dic47C2211yescharacteristic
14Dic47⋊C16—22no—
15C22⋊(C47⋊C16)C2212yes—
16C16×Dic47—22no—
17C47×#408C2211yescharacteristic
18C47×C22⋊C16C2212yes—
19C2×#10625350C2×C411yescharacteristic
20#13094032—42no—
21#13094031—42no—
22C2×#10625329—21no—
23C2×#10625329C2×C411yescharacteristic
24C2×#10625324—21no—
25C22×C47⋊C8—21no—
26C22×C47⋊C8C2×C411yescharacteristic
27C2×C8×D47—21no—
28#13094022C811yescharacteristic
29#13094020C811yescharacteristic
30#13094013C811yescharacteristic
31#13094011C811yescharacteristic
32#13093996—22no—
33C4×C22⋊D47—22no—
34C2×C4×Dic47C2×C411yescharacteristic
35C2×C47⋊C16—22no—
36Dic47⋊C8—22no—
37(C2×D47)⋊C8—22no—
38C8×Dic47D412yes—
39C8×Dic47D412yes—
40C2×#41×C47C2×C411yescharacteristic
41C2×M16×C47C2×C411yescharacteristic
42C22×C8×C47C2311yescharacteristic
43C2×C16×C47—22no—
44C22⋊C8×C47—22no—
45#41×C47—41no—
46#41×C47—41no—
47C2×Q8×C47C2×C811yescharacteristic
48C2×D4×C47—21no—
49C2×D4×C47C2×C811yescharacteristic
50C22×C4×C47—21no—
51C22×C4×C47C22×C411yescharacteristic
52C2×C22⋊D47—21no—
53C2×C22⋊D47—22no—
54C22×Dic47M1612yes—
55#10625350—41no—
56#10625350—41no—
57C2×D188C2×C811yescharacteristic
58C2×C4×D47—21no—
59C2×Dic94C2×C811yescharacteristic
60M16×C47—41no—
61C2×C8×C47—21no—
62C2×C8×C47—41no—
63C2×C8×C47C2×D412yes—
64C22⋊Dic47—22no—
65(C2×D47)⋊C4—22no—
66Dic47⋊C4—22no—
67C4×Dic47C22⋊C412yes—
68C4×Dic47C2×C811yescharacteristic
69C4×Dic47C2×C811yescharacteristic
70#10625329—42no—
71#10625329—41no—
72C2×C47⋊C8—22no—
73C2×C47⋊C8—21no—
74C2×C47⋊C8—42no—
75C2×C47⋊C8C22⋊C411yescharacteristic
76C2×C47⋊C8C22⋊C411yescharacteristic
77#10625324—42no—
78C8×D47—42no—
79C16×C47—42no—
80C47⋊C16—42no—
81C23×C47—21no—
82C23×C47C2×M1611yescharacteristic
83C22×D47—21no—
84Q8×C47—41no—
85D4×C47—41no—
86D4×C47—42no—
87C2×C4×C47—21no—
88C2×C4×C47—21no—
89C2×C4×C47—21no—
90C2×C4×C47—41no—
91C2×C4×C47—41no—
92C2×C4×C47C2×C22⋊C411yescharacteristic
93C2×C4×C47C22×C811yescharacteristic
94C22⋊D47—42no—
95C22⋊D47—44no—
96C2×Dic47—24no—
97C2×Dic47—21no—
98C2×Dic47C22⋊C812yes—
99C2×Dic47C22⋊C812yes—
100D188—41no—
101C4×D47—41no—
102C4×D47—41no—
103Dic94—41no—
104C8×C47—22no—
105C8×C47—41no—
106C47⋊C8—21no—
107C47⋊C8—41no—
108C47⋊C8#10012yes—
109#11490—471nomaximal, Sylow
110C22×C47—22no—
111C22×C47—21no—
112C22×C47—42no—
113C22×C47—41no—
114C22×C47C2×C22⋊C811yescharacteristic
115C2×D47—42no—
116C2×D47—41no—
117C4×C47—21no—
118C4×C47—41no—
119C4×C47#41011yescharacteristic
120C4×C47C2×#10011yescharacteristic
121Dic47—22no—
122Dic47—41no—
123Dic47#35014yes—
124#2931—471no—
125#2162—472no—
126#1315—471no—
127#1315—472no—
128#1315D4711yescharacteristic, complemented
129C2×C47—41no—
130C2×C47—81no—
131C2×C47#146011yescharacteristic
132C2×C47C2×#35012yes—
133D47—81no—
134C2×#131—471no—
135C2×#131—471no—
136#532—471no—
137#436—942no—
138C8×D4—942no—
139#423—471no—
140#408—471no—
141#408C2×D4711yescharacteristic
142C2×C4×C8—471no—
143C4⋊C16—942no—
144C22⋊C16—472no—
145C22⋊C16C2×D4712yes—
146C4×C16—942no—
147C47#1149011yescharacteristic, complemented, Sylow
148C2×#41—471no—
149C2×#41C4×D4711yescharacteristic
150#131—1882no—
151#131—1882no—
152C2×M16—471no—
153C2×M16—941no—
154C2×M16—941no—
155C2×M16C4×D4711yescharacteristic
156C22×C8—471no—
157C22×C8—941no—
158C22×C8—941no—
159C22×C8C22×D4711yescharacteristic
160C4⋊Q8—471no—
161#127—471no—
162#124—471no—
163#124—471no—
164#121—942no—
165C4×D4—942no—
166C2×C42—471no—
167C2×C16—22no—
168C2×C16—942no—
169C4⋊C8—942no—
170C22⋊C8—22no—
171C22⋊C8—942no—
172C4×C8—472no—
173C4×C8—472no—
174#41—41no—
175#41—41no—
176#41—1881no—
177#41—1881no—
178C2×Q8—471no—
179C2×Q8C8×D4711yescharacteristic
180C2×D4—21no—
181C2×D4—471no—
182C2×D4—942no—
183C2×D4—941no—
184C2×D4C8×D4711yescharacteristic
185C22×C4—21no—
186C22×C4—472no—
187C22×C4—941no—
188C22×C4C2×C4×D4711yescharacteristic
189M16—41no—
190M16—1882no—
191M16—1882no—
192M16—1881no—
193C2×C8—21no—
194C2×C8—41no—
195C2×C8—471no—
196C2×C8—471no—
197C2×C8—942no—
198C2×C8—941no—
199C2×C8—1882no—
200C2×C8—1882no—
201C2×C8D4×D4712yes—
202C4⋊C4—942no—
203C22⋊C4—942no—
204C22⋊C4—942no—
205C42—472no—
206C42—471no—
207C42—471no—
208C16—42no—
209C16—1882no—
210C23—21no—
211C23—941no—
212C23M16×D4711yescharacteristic
213Q8—41no—
214Q8—1881no—
215D4—41no—
216D4—42no—
217D4—1881no—
218D4—1884no—
219D4—1882no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—472no—
226C2×C4—472no—
227C2×C4—944no—
228C2×C4—941no—
229C2×C4—1881no—
230C2×C4—1881no—
231C2×C4C22⋊C4×D4711yescharacteristic
232C2×C4C2×C8×D4711yescharacteristic
233C8—22no—
234C8—41no—
235C8—472no—
236C8—941no—
237C8—1881no—
238C22—22no—
239C22—21no—
240C22—42no—
241C22—41no—
242C22—1882no—
243C22—1881no—
244C22C22⋊C8×D4711yescharacteristic
245C4—21no—
246C4—41no—
247C4—474no—
248C4—942no—
249C4—1881no—
250C4#1568909511yescharacteristic
251C4#100×D4711yescharacteristic
252C2—41no—
253C2—81no—
254C2—3761no—
255C2#2024170711yescharacteristic
256C2#350×D4712yes—
subgroupG/H
centerC2×C4C22⋊C4×D47
commutatorC2×C4×C47C22×C8
FrattiniC22×C8C22×D47
FittingC47×#1315C2
socleC22×C47C2×C22⋊C8
prime radical#15689669C4
cosocleC22×C8×C47C23
hypercenter#1315D47
hypocenterC47#11490
LCSG ▷ C2×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ #1315

Complements

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

NH ≅ G/Ntype
2. #20245184C2semidirect
3. #20241314C2semidirect
4. #20241313C2semidirect
5. C47×#1315C2semidirect
128. #1315D47semidirect
147. C47#11490semidirect

Maximal subgroups

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

subgroupindex
#202520962
#202451842
#202413142
#202413132
C47×#13152
#1149047