Finite Group Databasebeta

← #26103829

Subgroups — #26103829

Order 12032.

228 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 41 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#20244612C211yesmaximal, characteristic
2#20241334C211yesmaximal, characteristic, complemented
3C47×#1296C211yesmaximal, characteristic, complemented
4#15689704C411yescharacteristic
5C2×#13093952—21no—
6#15689122—21no—
7#15688892C411yescharacteristic
8#15688849—21no—
9C2×C4⋊C8×C47—21no—
10C47×#384C2211yescharacteristic
11C2×#10625350—21no—
12C2×#10625329—21no—
13#13094004—41no—
14#13093999—41no—
15#13093998—41no—
16#13093952—41no—
17#13093952—41no—
18#13093947—41no—
19#13093944—41no—
20C22⋊(C47⋊C8)—41no—
21C22×C8×C47—41no—
22C2×C4⋊C4×C47—41no—
23C2×C4⋊C4×C47C2×C411yescharacteristic
24C2×C4⋊C4×C47D411yescharacteristic
25C2×C42×C47D411yescharacteristic
26C4⋊C8×C47—22no—
27C4⋊C8×C47—41no—
28C22×C4×C47—21no—
29C22×C4×C47—21no—
30C22×C4×C47—21no—
31C22×C4×C47C22⋊C411yescharacteristic
32C2×C22⋊D47—41no—
33#10625350—41no—
34#10625350—41no—
35#10625350—41no—
36#10625350—41no—
37C2×D188—21no—
38C2×D188—41no—
39C2×C4×D47—41no—
40C2×Dic94—21no—
41C2×C8×C47—42no—
42C2×C8×C47—81no—
43C2×C8×C47—81no—
44C4⋊C4×C47—41no—
45C4⋊C4×C47—41no—
46C4⋊C4×C47—81no—
47C4⋊C4×C47—81no—
48C42×C47—21no—
49C42×C47D811yescharacteristic
50C42×C47SD1611yescharacteristic
51(C2×D47)⋊C4—42no—
52C4⋊Dic47—41no—
53#10625329—41no—
54#10625329—41no—
55C2×C47⋊C8—41no—
56C2×C47⋊C8—81no—
57C23×C47#9911yescharacteristic
58C22×D47—41no—
59C2×C4×C47—21no—
60C2×C4×C47—21no—
61C2×C4×C47—21no—
62C2×C4×C47—21no—
63C2×C4×C47—21no—
64C2×C4×C47—21no—
65C2×C4×C47—21no—
66C2×C4×C47—21no—
67C2×C4×C47—41no—
68C2×C4×C47—41no—
69C2×C4×C47—81no—
70C2×C4×C47—81no—
71C2×C4×C47#10211yescharacteristic
72C2×C4×C47#10411yescharacteristic
73C22⋊D47—81no—
74C22⋊D47—81no—
75C2×Dic47—41no—
76D188—41no—
77D188—41no—
78D188—161no—
79C4×D47—81no—
80C4×D47—81no—
81Dic94—41no—
82Dic94—41no—
83C8×C47—82no—
84C47⋊C8—81no—
85C47⋊C8—161no—
86#7417—471nomaximal, Sylow
87C22×C47—21no—
88C22×C47—21no—
89C22×C47#32711yescharacteristic
90C22×C47AΣL(1,𝔽16)11yescharacteristic
91C22×C47#35511yescharacteristic
92C2×D47—81no—
93C2×D47—81no—
94C2×D47—81no—
95C4×C47—21no—
96C4×C47—21no—
97C4×C47—21no—
98C4×C47—21no—
99C4×C47—21no—
100C4×C47—21no—
101C4×C47—81no—
102C4×C47—81no—
103Dic47—81no—
104#2039—471no—
105#1325—471no—
106#1296D4711yescharacteristic, complemented
107C2×C47—21no—
108C2×C47—21no—
109C2×C47#134511yescharacteristic
110C2×C47#140211yescharacteristic
111C2×C47#140511yescharacteristic
112D47—161no—
113#537—471no—
114C2×C4⋊C8—21no—
115C2×#104—941no—
116#417—941no—
117#384C2×D4711yescharacteristic
118#343—471no—
119#324—941no—
120C47#741711yescharacteristic, complemented, Sylow
121C2×#41—941no—
122C2×M16—941no—
123C22×C8—41no—
124C22⋊Q8—1881no—
125#121—1881no—
126#121—1881no—
127C2×C4⋊C4—41no—
128C2×C4⋊C4C4×D4711yescharacteristic
129C2×C4⋊C4D18811yescharacteristic
130C2×C42C22⋊D4711yescharacteristic
131C4⋊C8—22no—
132C4⋊C8—41no—
133#104—1881no—
134#104—1881no—
135#103—1881no—
136#102—1881no—
137C22⋊C8—1881no—
138#41—1881no—
139#41—1881no—
140#41—1881no—
141#41—1881no—
142C2×Q8—941no—
143C2×D4—941no—
144C2×D4—1881no—
145C2×D4—1881no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—1881no—
150C22×C4(C2×D47)⋊C411yescharacteristic
151M16—1881no—
152M16—1881no—
153C2×C8—42no—
154C2×C8—81no—
155C2×C8—81no—
156C2×C8—3761no—
157C2×C8—1881no—
158C4⋊C4—41no—
159C4⋊C4—41no—
160C4⋊C4—81no—
161C4⋊C4—81no—
162C4⋊C4—1881no—
163C22⋊C4—1882no—
164C42—21no—
165C42#1062533411yescharacteristic
166C42#1062533611yescharacteristic
167C23—1881no—
168C23#1309393811yescharacteristic
169Q8—1881no—
170Q8—1881no—
171D4—3761no—
172D4—3761no—
173D4—1881no—
174D4—1881no—
175D4—7521no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—81no—
187C2×C4—81no—
188C2×C4—3761no—
189C2×C4—3761no—
190C2×C4—1881no—
191C2×C4#1309394411yescharacteristic
192C2×C4#1309395211yescharacteristic
193C8—82no—
194C8—3761no—
195C8—7521no—
196C22—21no—
197C22—21no—
198C22—3761no—
199C22—3761no—
200C22—3761no—
201C22#1568885611yescharacteristic
202C22#1568891111yescharacteristic
203C22#1568891911yescharacteristic
204C4—21no—
205C4—21no—
206C4—21no—
207C4—21no—
208C4—21no—
209C4—21no—
210C4—81no—
211C4—81no—
212C4—3761no—
213C2—21no—
214C2—21no—
215C2—7521no—
216C2#2024139011yescharacteristic
217C2#2024153411yescharacteristic
218C2#2024154311yescharacteristic
subgroupG/H
centerC22#15688911
commutatorC2×C4⋊C4×C47C2×C4
Frattini#384C2×D47
FittingC47×#1296C2
socleC22×C47AΣL(1,𝔽16)
prime radical#15689704C4
cosocleC47×#384C22
hypercenter#1296D47
hypocenterC47#7417
LCSG ▷ C2×C4⋊C4×C47 ▷ C22×C4×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ #384 ◁ #1296

Complements

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

NH ≅ G/Ntype
2. #20241334C2semidirect
3. C47×#1296C2semidirect
106. #1296D47semidirect
120. C47#7417semidirect

Maximal subgroups

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

subgroupindex
#202446122
#202413342
C47×#12962
#741747