Finite Group Databasebeta

← #26104059

Subgroups — #26104059

Order 12032.

294 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 59 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#20245053C211yesmaximal, characteristic
2#20244217C211yesmaximal, characteristic
3C47×#2126C211yesmaximal, characteristic
4#15689147C411yescharacteristic, complemented
5C2×#13093951—21no—
6C2×#13093951—21no—
7#15689097—21no—
8C42×Dic47C411yescharacteristic, complemented
9#15688918C412yescomplemented
10#15688916—21no—
11C47×#532C2211yescharacteristic
12C47×#410—21no—
13C47×#354C412yescomplemented
14C47×#353—21no—
15C2×#10625329—21no—
16#13093983—41no—
17C2×C4×Dic47—21no—
18C2×C4×Dic47—41no—
19C4⋊(C47⋊C8)—41no—
20#13093951—42no—
21#13093951—42no—
22#13093939—22no—
23#13093939—41no—
24C2×#41×C47D411yescharacteristic
25C2×#41×C47D411yescharacteristic
26C4⋊Q8×C47Q811yescharacteristic
27#127×C47D411yescharacteristic
28#124×C47C2×C411yescharacteristic
29#124×C47C2×C411yescharacteristic
30#121×C47—41no—
31C4×D4×C47—41no—
32#117×C47—41no—
33C2×C42×C47C2×C411yescharacteristic
34#99×C47—22no—
35#99×C47—41no—
36#41×C47—41no—
37#41×C47—41no—
38#41×C47—41no—
39#41×C47—41no—
40C2×Q8×C47C4⋊C411yescharacteristic
41C2×Q8×C47C4⋊C411yescharacteristic
42C2×D4×C47—21no—
43C2×D4×C47—21no—
44C2×D4×C47—41no—
45C2×D4×C47C22⋊C411yescharacteristic
46C2×D4×C47C22⋊C411yescharacteristic
47C22×C4×C47—21no—
48C22×C4×C47—21no—
49C22×C4×C47—21no—
50C22×C4×C47C22⋊C411yescharacteristic
51C22×Dic47—41no—
52C4⋊C4×C47—41no—
53C4⋊C4×C47—81no—
54C22⋊C4×C47—41no—
55C22⋊C4×C47—41no—
56C22⋊C4×C47—42no—
57C42×C47—21no—
58C42×C47—81no—
59C42×C47C4211yescharacteristic
60C42×C47C4⋊C411yescharacteristic
61C22⋊Dic47—42no—
62C4⋊Dic47—81no—
63C4×Dic47—22no—
64C4×Dic47—41no—
65C4×Dic47—44no—
66C4×Dic47—81no—
67#10625329—41no—
68#10625329—41no—
69C2×C47⋊C8—41no—
70C23×C47—21no—
71C23×C47—21no—
72C23×C47#9911yescharacteristic
73Q8×C47—41no—
74Q8×C47—41no—
75D4×C47—41no—
76D4×C47—41no—
77D4×C47—82no—
78D4×C47—81no—
79D4×C47—81no—
80C2×C4×C47—21no—
81C2×C4×C47—21no—
82C2×C4×C47—21no—
83C2×C4×C47—21no—
84C2×C4×C47—21no—
85C2×C4×C47—21no—
86C2×C4×C47—42no—
87C2×C4×C47—41no—
88C2×C4×C47—41no—
89C2×C4×C47—41no—
90C2×C4×C47—41no—
91C2×C4×C47—82no—
92C2×C4×C47#9511yescharacteristic
93C2×C4×C47#9911yescharacteristic
94C2×Dic47—42no—
95C2×Dic47—44no—
96C2×Dic47—82no—
97C47⋊C8—81no—
98#7500—471nomaximal, Sylow
99C22×C47—21no—
100C22×C47—41no—
101C22×C47—41no—
102C22×C47—41no—
103C22×C47—81no—
104C22×C47—81no—
105C22×C47#34211yescharacteristic
106C4×C47—22no—
107C4×C47—21no—
108C4×C47—41no—
109C4×C47—41no—
110C4×C47—41no—
111C4×C47—162no—
112C4×C47#35311yescharacteristic
113C4×C47#35411yescharacteristic
114Dic47—44no—
115Dic47—162no—
116#2126—471no—
117#2126D4711yescharacteristic
118#1962—471no—
119C2×C47—21no—
120C2×C47—41no—
121C2×C47—81no—
122C2×C47—81no—
123C2×C47#139311yescharacteristic
124#532C2×D4711yescharacteristic
125#423—471no—
126C2×#104—941no—
127C2×#104—941no—
128#410—21no—
129#410—941no—
130C43—471no—
131#354—472no—
132#354Dic4712yescomplemented
133#353—21no—
134#353—941no—
135C47#750011yescharacteristic, complemented, Sylow
136C2×#41D18811yescharacteristic
137C2×#41C22⋊D4711yescharacteristic
138C2×M16—941no—
139C4⋊Q8Dic9411yescharacteristic
140#127C22⋊D4711yescharacteristic
141#124C4×D4711yescharacteristic
142#124C2×Dic4711yescharacteristic
143#121—41no—
144C4×D4—41no—
145#117—41no—
146#117—1881no—
147C2×C42—941no—
148C2×C42—1881no—
149C2×C42C4×D4711yescharacteristic
150C4⋊C8—1881no—
151#104—1882no—
152#104—1882no—
153#99—22no—
154#99—41no—
155#99—942no—
156#99—1881no—
157#41—41no—
158#41—41no—
159#41—41no—
160#41—41no—
161C2×Q8Dic47⋊C411yescharacteristic
162C2×Q8C4⋊Dic4711yescharacteristic
163C2×D4—21no—
164C2×D4—21no—
165C2×D4—41no—
166C2×D4(C2×D47)⋊C411yescharacteristic
167C2×D4C22⋊Dic4711yescharacteristic
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—1881no—
172C22×C4(C2×D47)⋊C411yescharacteristic
173M16—1881no—
174M16—1881no—
175C2×C8—1881no—
176C4⋊C4—41no—
177C4⋊C4—81no—
178C4⋊C4—3761no—
179C22⋊C4—41no—
180C22⋊C4—41no—
181C22⋊C4—42no—
182C22⋊C4—1882no—
183C42—21no—
184C42—81no—
185C42—942no—
186C42—3761no—
187C42—1881no—
188C42—1884no—
189C42C4×Dic4711yescharacteristic
190C42Dic47⋊C411yescharacteristic
191C23—21no—
192C23—21no—
193C23#1309393811yescharacteristic
194Q8—41no—
195Q8—41no—
196D4—41no—
197D4—41no—
198D4—82no—
199D4—81no—
200D4—81no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—42no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4—82no—
213C2×C4—3762no—
214C2×C4—1882no—
215C2×C4—1884no—
216C2×C4C47⋊#9511yescharacteristic
217C2×C4#1309393811yescharacteristic
218C8—3761no—
219C22—21no—
220C22—41no—
221C22—41no—
222C22—41no—
223C22—81no—
224C22—81no—
225C22#1568888911yescharacteristic
226C4—22no—
227C4—21no—
228C4—41no—
229C4—41no—
230C4—41no—
231C4—162no—
232C4—1884no—
233C4—7522no—
234C4#1568891511yescharacteristic
235C4#1568891711yescharacteristic
236C2—21no—
237C2—41no—
238C2—81no—
239C2—81no—
240C2#2024151211yescharacteristic
subgroupG/H
centerC4#15688917
commutatorC42×C47C42
Frattini#532C2×D47
FittingC47×#2126C2
socleC2×C47#1393
prime radical, cosocleC47×#532C22
hypercenter#2126D47
hypocenterC47#7500
LCSG ▷ C42×C47 ▷ C2×C4×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C4 ◁ C2×C4 ◁ C22×C4 ◁ #532 ◁ #2126

Complements

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

NH ≅ G/Ntype
4. #15689147C4semidirect
8. C42×Dic47C4semidirect
9. #15688918C4semidirect
13. C47×#354C4semidirect
132. #354Dic47semidirect
135. C47#7500semidirect

Maximal subgroups

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

subgroupindex
#202450532
#202442172
C47×#21262
#750047