Finite Group Databasebeta

← #26120452

Subgroups — #26120452

Order 12032.

386 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 85 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#20243525C211yesmaximal, characteristic, complemented
2#20242123C212yesmaximal, complemented
3#20241839C212yesmaximal, complemented
4C47⋊#1309C211yesmaximal, characteristic, complemented
5C47×#3253C211yesmaximal, characteristic
6#15689147C2211yescharacteristic
7#15689147C2212yes—
8C2×C4×C47⋊C8C2211yescharacteristic
9#15688887—22no—
10#15688872—22no—
11#15688867—22no—
12D4⋊(C47⋊C8)—24no—
13C47×#532C2212yes—
14C47×#476—22no—
15C47×#469—22no—
16C47×#459—22no—
17C47×#423C2211yescharacteristic
18C2×#10625329—21no—
19C2×#10625329—22no—
20C22×C47⋊C8—21no—
21C4⋊(C47⋊C8)—22no—
22C4⋊(C47⋊C8)—22no—
23C4⋊(C47⋊C8)—22no—
24C4×C47⋊C8—21no—
25C4×C47⋊C8D412yes—
26C2×#41×C47—22no—
27C2×D8×C47—22no—
28C2×M16×C47D412yes—
29C4⋊Q8×C47D412yes—
30#127×C47D412yes—
31#124×C47—22no—
32#121×C47—22no—
33#121×C47—22no—
34C4×D4×C47—22no—
35C4×D4×C47—22no—
36C2×C42×C47C2311yescharacteristic
37#106×C47—22no—
38C4⋊C8×C47—22no—
39#102×C47—24no—
40#41×C47—82no—
41#41×C47—82no—
42C2×Q8×C47—22no—
43C2×D4×C47—22no—
44C2×D4×C47—22no—
45C2×D4×C47—22no—
46C2×D4×C47—42no—
47C22×C4×C47—42no—
48C22×C4×C47C2×D411yescharacteristic
49C22×C4×C47#4112yes—
50D8×C47—82no—
51M16×C47—42no—
52C2×C8×C47—22no—
53C4⋊C4×C47—22no—
54C4⋊C4×C47—22no—
55C22⋊C4×C47—42no—
56C22⋊C4×C47—42no—
57C42×C47C2×D412yes—
58C42×C47C2×D411yescharacteristic
59C42×C47#4111yescharacteristic
60#10625329—42no—
61#10625329—82no—
62C2×C47⋊C8—22no—
63C2×C47⋊C8—41no—
64C2×C47⋊C8—41no—
65C2×C47⋊C8—41no—
66C2×C47⋊C8—42no—
67C23×C47—42no—
68C23×C47#12411yescharacteristic
69Q8×C47—82no—
70D4×C47—42no—
71D4×C47—42no—
72D4×C47—42no—
73D4×C47—42no—
74D4×C47—82no—
75D4×C47—84no—
76C2×C4×C47—21no—
77C2×C4×C47—21no—
78C2×C4×C47—22no—
79C2×C4×C47—22no—
80C2×C4×C47—42no—
81C2×C4×C47—82no—
82C2×C4×C47—82no—
83C2×C4×C47#12011yescharacteristic
84C2×C4×C47#12111yescharacteristic
85C2×C4×C47#12112yes—
86C2×C4×C47#12312yes—
87C8×C47—42no—
88C47⋊C8—42no—
89C47⋊C8—81no—
90C47⋊C8—82no—
91#10531—471nomaximal, Sylow
92C22×C47—22no—
93C22×C47—21no—
94C22×C47—82no—
95C22×C47—82no—
96C22×C47—82no—
97C22×C47#39411yescharacteristic
98C4×C47—22no—
99C4×C47—21no—
100C4×C47—82no—
101C4×C47#45411yescharacteristic
102C4×C47#45511yescharacteristic
103C4×C47#47112yes—
104C4×C47#47212yes—
105#3253D4711yescharacteristic
106#1837—471no—
107#1560—472no—
108#1492—472no—
109#1309—471no—
110C2×C47—41no—
111C2×C47—162no—
112C2×C47#204011yescharacteristic
113C2×C47#205112yes—
114#532C2×D4712yes—
115#476—22no—
116#469—22no—
117#459—22no—
118#423—471no—
119#423—472no—
120#423C2×D4711yescharacteristic
121C2×C4×C8—471no—
122#341—942no—
123#333—942no—
124#331—942no—
125D4⋊C8—944no—
126C47#1053111yescharacteristic, complemented, Sylow
127C2×#41—22no—
128C2×D8—22no—
129C2×M16—942no—
130C2×M16—941no—
131C2×M16C22⋊D4712yes—
132C22×C8—941no—
133C4⋊Q8C22⋊D4712yes—
134#127C22⋊D4712yes—
135#124—22no—
136#121—22no—
137#121—22no—
138C4×D4—22no—
139C4×D4—22no—
140C2×C42C22×D4711yescharacteristic
141#106—22no—
142C4⋊C8—22no—
143C4⋊C8—942no—
144C4⋊C8—942no—
145C4⋊C8—942no—
146#102—24no—
147C4×C8—472no—
148C4×C8—941no—
149#41—82no—
150#41—82no—
151C2×Q8—22no—
152C2×D4—22no—
153C2×D4—22no—
154C2×D4—22no—
155C2×D4—42no—
156C22×C4—42no—
157C22×C4#1062535212yes—
158C22×C4C2×C22⋊D4711yescharacteristic
159D8—82no—
160M16—42no—
161M16—3762no—
162M16—1882no—
163C2×C8—22no—
164C2×C8—942no—
165C2×C8—1881no—
166C2×C8—1882no—
167C2×C8—1881no—
168C2×C8—1881no—
169C4⋊C4—22no—
170C4⋊C4—22no—
171C22⋊C4—42no—
172C22⋊C4—42no—
173C42D4×D4711yescharacteristic
174C42#1062535211yescharacteristic
175C42C2×C22⋊D4712yes—
176C23—42no—
177C23#1309401111yescharacteristic
178Q8—82no—
179D4—42no—
180D4—42no—
181D4—42no—
182D4—42no—
183D4—82no—
184D4—84no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—42no—
190C2×C4—82no—
191C2×C4—82no—
192C2×C4#1309399311yescharacteristic
193C2×C4#1309399511yescharacteristic
194C2×C4#1309399612yes—
195C2×C4#1309400512yes—
196C8—42no—
197C8—3761no—
198C8—3762no—
199C8—1882no—
200C22—21no—
201C22—22no—
202C22—82no—
203C22—82no—
204C22—82no—
205C22#1568903711yescharacteristic
206C4—21no—
207C4—22no—
208C4—82no—
209C4#1568929611yescharacteristic
210C4#1568930111yescharacteristic
211C4#1568938412yes—
212C4#1568939112yes—
213C2—41no—
214C2—162no—
215C2#2024461411yescharacteristic
216C2#2024467712yes—
subgroupG/H
centerC22#15689037
commutator, cosocleC2×C42×C47C23
FrattiniC2×C42C22×D47
Fitting, prime radicalC47×#3253C2
socleC22×C47#394
hypercenter#3253D47
hypocenterC47#10531
LCSG ▷ C2×C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3253

Complements

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

NH ≅ G/Ntype
1. #20243525C2semidirect
2. #20242123C2semidirect
3. #20241839C2semidirect
4. C47⋊#1309C2semidirect
126. C47#10531semidirect

Maximal subgroups

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

subgroupindex
#202435252
#202421232
#202418392
C47⋊#13092
C47×#32532
#1053147