Finite Group Databasebeta

← #26104052

Subgroups — #26104052

Order 12032.

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

normal subgroups only

This group has 290 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#20245073C211yesmaximal, characteristic
2#20244217C211yesmaximal, characteristic
3C47×#2126C211yesmaximal, characteristic
4#15689147C411yescharacteristic, complemented
5C2×#13093951—21no—
6C2×#13093951—21no—
7#15689107—21no—
8C42×Dic47C411yescharacteristic, complemented
9#15688922C412yescomplemented
10#15688920—21no—
11C47×#532C2211yescharacteristic
12C47×#410—21no—
13C47×#354—21no—
14C47×#353C412yescomplemented
15C2×#10625329—21no—
16C2×#10625329—41no—
17C2×C4×Dic47—21no—
18C2×C4×Dic47—41no—
19C4⋊(C47⋊C8)—41no—
20#13093951—42no—
21#13093951—42no—
22#13093943—22no—
23#13093941—41no—
24C2×#41×C47D411yescharacteristic
25C2×#41×C47D411yescharacteristic
26C4⋊Q8×C47C2×C411yescharacteristic
27#127×C47C2×C411yescharacteristic
28#124×C47D411yescharacteristic
29#124×C47Q811yescharacteristic
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×C47C22⋊C411yescharacteristic
41C2×Q8×C47C4⋊C411yescharacteristic
42C2×D4×C47—21no—
43C2×D4×C47—21no—
44C2×D4×C47—41no—
45C2×D4×C47C22⋊C411yescharacteristic
46C2×D4×C47C4⋊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
61C4×Dic47—22no—
62C4×Dic47—41no—
63C4×Dic47—44no—
64#10625329—42no—
65#10625329—41no—
66#10625329—41no—
67#10625329—81no—
68C2×C47⋊C8—41no—
69C2×C47⋊C8—81no—
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—
96C47⋊C8—82no—
97C47⋊C8—81no—
98#7497—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—
115#2132—471no—
116#2126D4711yescharacteristic
117#1962—471no—
118C2×C47—21no—
119C2×C47—41no—
120C2×C47—81no—
121C2×C47—81no—
122C2×C47#139311yescharacteristic
123#532C2×D4711yescharacteristic
124#423—471no—
125C2×#104—941no—
126C2×#104—941no—
127#413—941no—
128#410—21no—
129C43—471no—
130#356—472no—
131#355—941no—
132#354—21no—
133#353Dic4712yescomplemented
134C47#749711yescharacteristic, complemented, Sylow
135C2×#41D18811yescharacteristic
136C2×#41C22⋊D4711yescharacteristic
137C2×M16—941no—
138C2×M16—1881no—
139C4⋊Q8C4×D4711yescharacteristic
140#127C2×Dic4711yescharacteristic
141#124Dic9411yescharacteristic
142#124C22⋊D4711yescharacteristic
143#121—41no—
144C4×D4—41no—
145#117—41no—
146C2×C42—941no—
147C2×C42—1881no—
148C2×C42C4×D4711yescharacteristic
149C4⋊C8—1881no—
150#104—1882no—
151#104—1882no—
152#101—942no—
153#100—1881no—
154#99—22no—
155#99—41no—
156#41—41no—
157#41—41no—
158#41—41no—
159#41—41no—
160C2×Q8Dic47⋊C411yescharacteristic
161C2×Q8(C2×D47)⋊C411yescharacteristic
162C2×D4—21no—
163C2×D4—21no—
164C2×D4—41no—
165C2×D4C4⋊Dic4711yescharacteristic
166C2×D4C22⋊Dic4711yescharacteristic
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—1881no—
171C22×C4(C2×D47)⋊C411yescharacteristic
172M16—3761no—
173M16—1882no—
174M16—1881no—
175M16—1881no—
176C2×C8—3761no—
177C2×C8—1881no—
178C4⋊C4—41no—
179C4⋊C4—81no—
180C22⋊C4—41no—
181C22⋊C4—41no—
182C22⋊C4—42no—
183C42—21no—
184C42—81no—
185C42—942no—
186C42—1881no—
187C42—1884no—
188C42C4×Dic4711yescharacteristic
189C42Dic47⋊C411yescharacteristic
190C23—21no—
191C23—21no—
192C23#1309393811yescharacteristic
193Q8—41no—
194Q8—41no—
195D4—41no—
196D4—41no—
197D4—82no—
198D4—81no—
199D4—81no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—42no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—82no—
212C2×C4—1882no—
213C2×C4—1884no—
214C2×C4C47⋊#9511yescharacteristic
215C2×C4#1309393811yescharacteristic
216C8—3762no—
217C8—3761no—
218C22—21no—
219C22—41no—
220C22—41no—
221C22—41no—
222C22—81no—
223C22—81no—
224C22#1568888911yescharacteristic
225C4—22no—
226C4—21no—
227C4—41no—
228C4—41no—
229C4—41no—
230C4—162no—
231C4—1884no—
232C4#1568891511yescharacteristic
233C4#1568891711yescharacteristic
234C2—21no—
235C2—41no—
236C2—81no—
237C2—81no—
238C2#2024151211yescharacteristic
subgroupG/H
centerC4#15688915
commutatorC42×C47C42
Frattini#532C2×D47
FittingC47×#2126C2
socleC2×C47#1393
prime radical, cosocleC47×#532C22
hypercenter#2126D47
hypocenterC47#7497
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. #15688922C4semidirect
14. C47×#353C4semidirect
133. #353Dic47semidirect
134. C47#7497semidirect

Maximal subgroups

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

subgroupindex
#202450732
#202442172
C47×#21262
#749747