Finite Group Databasebeta

← #26102981

Subgroups — #26102981

Order 12032.

264 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 47 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#20241312C211yesmaximal, characteristic
2#20241265C211yesmaximal, characteristic
3C47×#2001C211yesmaximal, characteristic
4#15689087—21no—
5C22⋊(C47⋊C16)—21no—
6C22×Q16×C47C411yescharacteristic
7C2×C22⋊Q8×C47—21no—
8C2×#103×C47—21no—
9C2×C22⋊C8×C47C2211yescharacteristic
10C47×#340C411yescharacteristic
11C2×#10625329—41no—
12C2×C47⋊C16—41no—
13C22⋊(C47⋊C8)—22no—
14C22⋊(C47⋊C8)—41no—
15C22×Q8×C47—21no—
16C23×C4×C47D411yescharacteristic
17C2×Q16×C47—41no—
18C2×Q16×C47—42no—
19C22×C8×C47C2×C411yescharacteristic
20C22×C8×C47D411yescharacteristic
21C22⋊Q8×C47—42no—
22C22⋊Q8×C47—42no—
23C2×C4⋊C4×C47—21no—
24C2×C4⋊C4×C47—41no—
25C2×C22⋊C4×C47—41no—
26#103×C47—42no—
27C22⋊C8×C47—41no—
28C24×C47SD1611yescharacteristic
29C2×Q8×C47—22no—
30C2×Q8×C47—41no—
31C2×Q8×C47—41no—
32C2×Q8×C47—41no—
33C2×Q8×C47—41no—
34C2×Q8×C47—41no—
35C22×C4×C47—21no—
36C22×C4×C47—21no—
37C22×C4×C47—41no—
38C22×C4×C47—41no—
39C22×C4×C47—41no—
40C22×C4×C47—41no—
41C22×C4×C47C22⋊C411yescharacteristic
42C22×C4×C47D811yescharacteristic
43Q16×C47—42no—
44Q16×C47—42no—
45C2×C8×C47—21no—
46C2×C8×C47—21no—
47C2×C8×C47—41no—
48C2×C8×C47—41no—
49C4⋊C4×C47—22no—
50C4⋊C4×C47—41no—
51C4⋊C4×C47—44no—
52C22⋊C4×C47—44no—
53#10625329—81no—
54#10625329—81no—
55C2×C47⋊C8—42no—
56C47⋊C16—81no—
57C23×C47—21no—
58C23×C47—21no—
59C23×C47—21no—
60C23×C47—41no—
61C23×C47—41no—
62C23×C47#10211yescharacteristic
63Q8×C47—42no—
64Q8×C47—42no—
65Q8×C47—81no—
66Q8×C47—81no—
67C2×C4×C47—21no—
68C2×C4×C47—21no—
69C2×C4×C47—21no—
70C2×C4×C47—41no—
71C2×C4×C47—41no—
72C2×C4×C47—41no—
73C2×C4×C47—42no—
74C2×C4×C47—42no—
75C2×C4×C47—81no—
76C2×C4×C47—81no—
77C2×C4×C47—81no—
78C2×C4×C47—81no—
79C2×C4×C47—81no—
80C2×C4×C47#9911yescharacteristic
81C2×C4×C47#10411yescharacteristic
82C8×C47—41no—
83C8×C47—41no—
84C47⋊C8—82no—
85#7086—471nomaximal, Sylow
86C22×C47—21no—
87C22×C47—21no—
88C22×C47—42no—
89C22×C47—41no—
90C22×C47—41no—
91C22×C47—41no—
92C22×C47—81no—
93C22×C47#32711yescharacteristic
94C22×C47#35911yescharacteristic
95C22×C47#36211yescharacteristic
96C4×C47—21no—
97C4×C47—41no—
98C4×C47—82no—
99C4×C47—82no—
100C4×C47AΣL(1,𝔽16)11yescharacteristic
101C4×C47#35311yescharacteristic
102#2001D4711yescharacteristic
103#1314—471no—
104#1293—471no—
105C2×C47—21no—
106C2×C47—21no—
107C2×C47—81no—
108C2×C47#134211yescharacteristic
109C2×C47#134311yescharacteristic
110C2×C47#135011yescharacteristic
111C22×Q16Dic4711yescharacteristic
112C2×C22⋊Q8—21no—
113C2×#103—21no—
114#407—941no—
115C2×C22⋊C8C2×D4711yescharacteristic
116C22⋊C16—941no—
117#340Dic4711yescharacteristic
118C47#708611yescharacteristic, complemented, Sylow
119C22×Q8—21no—
120C23×C4C22⋊D4711yescharacteristic
121C2×Q16—41no—
122C2×Q16—42no—
123C2×M16—1881no—
124C22×C8C2×Dic4711yescharacteristic
125C22×C8C22⋊D4711yescharacteristic
126C22⋊Q8—42no—
127C22⋊Q8—42no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4—41no—
130C2×C22⋊C4—41no—
131C2×C16—1881no—
132#103—42no—
133C22⋊C8—41no—
134C22⋊C8—942no—
135C22⋊C8—1881no—
136C24#1062533511yescharacteristic
137C2×Q8—22no—
138C2×Q8—41no—
139C2×Q8—41no—
140C2×Q8—41no—
141C2×Q8—41no—
142C2×Q8—41no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—41no—
146C22×C4—41no—
147C22×C4—41no—
148C22×C4—41no—
149C22×C4#1062533411yescharacteristic
150C22×C4C22⋊Dic4711yescharacteristic
151Q16—42no—
152Q16—42no—
153M16—3761no—
154M16—3761no—
155C2×C8—21no—
156C2×C8—21no—
157C2×C8—41no—
158C2×C8—41no—
159C2×C8—1882no—
160C4⋊C4—22no—
161C4⋊C4—41no—
162C4⋊C4—44no—
163C22⋊C4—44no—
164C16—3761no—
165C23—21no—
166C23—21no—
167C23—21no—
168C23—41no—
169C23—41no—
170C23#1309394511yescharacteristic
171Q8—42no—
172Q8—42no—
173Q8—81no—
174Q8—81no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—42no—
182C2×C4—42no—
183C2×C4—81no—
184C2×C4—81no—
185C2×C4—81no—
186C2×C4—81no—
187C2×C4—81no—
188C2×C4#1309393911yescharacteristic
189C2×C4#1309395111yescharacteristic
190C8—41no—
191C8—41no—
192C8—3762no—
193C22—21no—
194C22—21no—
195C22—42no—
196C22—41no—
197C22—41no—
198C22—41no—
199C22—81no—
200C22#1568885711yescharacteristic
201C22#1568893011yescharacteristic
202C22#1568893911yescharacteristic
203C4—21no—
204C4—41no—
205C4—82no—
206C4—82no—
207C4#1568891011yescharacteristic
208C4#1568891611yescharacteristic
209C2—21no—
210C2—21no—
211C2—81no—
212C2#2024138211yescharacteristic
213C2#2024138511yescharacteristic
214C2#2024140611yescharacteristic
subgroupG/H
centerC22#15688857
commutatorC22×C8×C47C2×C4
FrattiniC2×C22⋊C8C2×D47
FittingC47×#2001C2
socleC22×C47#327
prime radicalC24×C47SD16
cosocleC2×C22⋊C8×C47C22
hypercenter#2001D47
hypocenterC47#7086
LCSG ▷ C22×C8×C47 ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C22⋊C8 ◁ #2001

Complements

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

NH ≅ G/Ntype
118. C47#7086semidirect

Maximal subgroups

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

subgroupindex
#202413122
#202412652
C47×#20012
#708647