Finite Group Databasebeta

← #26104053

Subgroups — #26104053

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#20245053C211yesmaximal, characteristic
3C47×#1962C211yesmaximal, characteristic
4#15689107—21no—
5#15689097—21no—
6#15688922C412yescomplemented
7#15688920—21no—
8#15688918—21no—
9#15688916C412yescomplemented
10C47×#532C2211yescharacteristic
11C47×#423C411yescharacteristic, complemented
12C2×#104×C47—21no—
13C2×#104×C47—21no—
14C43×C47C411yescharacteristic, complemented
15C2×#10625329—41no—
16#13093983—41no—
17#13093943—22no—
18#13093941—41no—
19#13093939—22no—
20#13093939—41no—
21C2×#41×C47D411yescharacteristic
22C2×#41×C47D411yescharacteristic
23C2×M16×C47—21no—
24C4⋊Q8×C47C2×C411yescharacteristic
25#127×C47C2×C411yescharacteristic
26#124×C47D411yescharacteristic
27#124×C47Q811yescharacteristic
28#121×C47—41no—
29C4×D4×C47—41no—
30C2×C42×C47—21no—
31C2×C42×C47—41no—
32C2×C42×C47C2×C411yescharacteristic
33C4⋊C8×C47—41no—
34#104×C47—42no—
35#104×C47—42no—
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×C47—41no—
51C22×C4×C47C22⋊C411yescharacteristic
52M16×C47—41no—
53M16×C47—41no—
54C2×C8×C47—41no—
55C4⋊C4×C47—41no—
56C22⋊C4×C47—41no—
57C22⋊C4×C47—41no—
58C42×C47—21no—
59C42×C47—22no—
60C42×C47—41no—
61C42×C47—44no—
62C42×C47C4211yescharacteristic
63C42×C47C4⋊C411yescharacteristic
64C22⋊Dic47—42no—
65C4⋊Dic47—81no—
66C4×Dic47—81no—
67#10625329—42no—
68#10625329—81no—
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—42no—
88C2×C4×C47—44no—
89C2×C4×C47—41no—
90C2×C4×C47—41no—
91C2×C4×C47—41no—
92C2×C4×C47—41no—
93C2×C4×C47#9511yescharacteristic
94C2×C4×C47#9911yescharacteristic
95C2×Dic47—82no—
96C8×C47—81no—
97C47⋊C8—82no—
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—44no—
110C4×C47—41no—
111C4×C47—41no—
112C4×C47#35311yescharacteristic
113C4×C47#35411yescharacteristic
114Dic47—162no—
115#2132—471no—
116#2126—471no—
117#1962D4711yescharacteristic
118C2×C47—21no—
119C2×C47—41no—
120C2×C47—81no—
121C2×C47—81no—
122C2×C47#139311yescharacteristic
123#532C2×D4711yescharacteristic
124#423Dic4711yescharacteristic, complemented
125C2×#104—21no—
126C2×#104—21no—
127#413—941no—
128#410—941no—
129C43Dic4711yescharacteristic, complemented
130#356—472no—
131#355—941no—
132#354—941no—
133#353—472no—
134C47#749711yescharacteristic, complemented, Sylow
135C2×#41C22⋊D4711yescharacteristic
136C2×#41C22⋊D4711yescharacteristic
137C2×M16—21no—
138C2×M16—1881no—
139C4⋊Q8C4×D4711yescharacteristic
140#127C4×D4711yescharacteristic
141#124Dic9411yescharacteristic
142#124D18811yescharacteristic
143#121—41no—
144C4×D4—41no—
145#117—1881no—
146C2×C42—21no—
147C2×C42—41no—
148C2×C42C2×Dic4711yescharacteristic
149C4⋊C8—41no—
150#104—42no—
151#104—42no—
152#101—942no—
153#100—1881no—
154#99—942no—
155#99—1881no—
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×D4Dic47⋊C411yescharacteristic
166C2×D4(C2×D47)⋊C411yescharacteristic
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—41no—
171C22×C4C22⋊Dic4711yescharacteristic
172M16—41no—
173M16—41no—
174M16—3761no—
175M16—1882no—
176C2×C8—41no—
177C2×C8—3761no—
178C4⋊C4—41no—
179C4⋊C4—3761no—
180C22⋊C4—41no—
181C22⋊C4—41no—
182C22⋊C4—1882no—
183C42—21no—
184C42—22no—
185C42—41no—
186C42—44no—
187C42—3761no—
188C42C4×Dic4711yescharacteristic
189C42C4⋊Dic4711yescharacteristic
190C23—21no—
191C23—21no—
192C23#1309393911yescharacteristic
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—42no—
208C2×C4—44no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—3762no—
214C2×C4C47⋊#9511yescharacteristic
215C2×C4#1309393911yescharacteristic
216C8—81no—
217C8—3762no—
218C22—21no—
219C22—41no—
220C22—41no—
221C22—41no—
222C22—81no—
223C22—81no—
224C22#1568889011yescharacteristic
225C4—22no—
226C4—21no—
227C4—41no—
228C4—44no—
229C4—41no—
230C4—41no—
231C4—7522no—
232C4#1568891611yescharacteristic
233C4#1568891811yescharacteristic
234C2—21no—
235C2—41no—
236C2—81no—
237C2—81no—
238C2#2024151311yescharacteristic
subgroupG/H
centerC4#15688916
commutatorC42×C47C42
Frattini#532C2×D47
FittingC47×#1962C2
socleC2×C47#1393
prime radical, cosocleC47×#532C22
hypercenter#1962D47
hypocenterC47#7497
LCSG ▷ C42×C47 ▷ C2×C4×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C4 ◁ C2×C4 ◁ C22×C4 ◁ #532 ◁ #1962

Complements

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

NH ≅ G/Ntype
6. #15688922C4semidirect
9. #15688916C4semidirect
11. C47×#423C4semidirect
14. C43×C47C4semidirect
124. #423Dic47semidirect
129. C43Dic47semidirect
134. C47#7497semidirect

Maximal subgroups

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

subgroupindex
#202450732
#202450532
C47×#19622
#749747