Finite Group Databasebeta

← #26120418

Subgroups — #26120418

Order 12032.

370 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 77 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#20243528C211yesmaximal, characteristic, complemented
2#20242110C212yesmaximal, complemented
3#20241999C212yesmaximal, complemented
4#20241299C211yesmaximal, characteristic, complemented
5C47×#3253C211yesmaximal, characteristic
6#15689180C2212yes—
7#15689147C2211yescharacteristic
8C2×#13093935C2211yescharacteristic
9#15688887—22no—
10#15688869—22no—
11#15688867—22no—
12#15688863—24no—
13C47×#532C2212yes—
14C47×#476—22no—
15C47×#469—22no—
16C47×#459—22no—
17C47×#423C2211yescharacteristic
18C2×#10625329—21no—
19C22×C47⋊C8—21no—
20C4⋊(C47⋊C8)—22no—
21C4⋊(C47⋊C8)—22no—
22C22⋊(C47⋊C8)—22no—
23#13093935—21no—
24#13093935—22no—
25#13093935D412yes—
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—84no—
41C2×Q8×C47—22no—
42C2×D4×C47—22no—
43C2×D4×C47—22no—
44C2×D4×C47—22no—
45C2×D4×C47—42no—
46C22×C4×C47—42no—
47C22×C4×C47C2×D411yescharacteristic
48C22×C4×C47#4112yes—
49D8×C47—82no—
50M16×C47—42no—
51C2×C8×C47—22no—
52C4⋊C4×C47—22no—
53C4⋊C4×C47—22no—
54C22⋊C4×C47—42no—
55C22⋊C4×C47—42no—
56C42×C47C2×D412yes—
57C42×C47C2×D411yescharacteristic
58C42×C47#4111yescharacteristic
59#10625329—42no—
60C2×C47⋊C8—22no—
61C2×C47⋊C8—41no—
62C2×C47⋊C8—41no—
63C2×C47⋊C8—41no—
64C2×C47⋊C8—42no—
65C23×C47—42no—
66C23×C47#12411yescharacteristic
67Q8×C47—82no—
68D4×C47—42no—
69D4×C47—42no—
70D4×C47—82no—
71D4×C47—82no—
72D4×C47—84no—
73C2×C4×C47—21no—
74C2×C4×C47—21no—
75C2×C4×C47—42no—
76C2×C4×C47—42no—
77C2×C4×C47—84no—
78C2×C4×C47#12011yescharacteristic
79C2×C4×C47#12111yescharacteristic
80C2×C4×C47#12112yes—
81C2×C4×C47#12312yes—
82C8×C47—42no—
83C47⋊C8—42no—
84C47⋊C8—82no—
85C47⋊C8—81no—
86#10525—471nomaximal, Sylow
87C22×C47—22no—
88C22×C47—21no—
89C22×C47—82no—
90C22×C47—82no—
91C22×C47—82no—
92C22×C47#39411yescharacteristic
93C4×C47—22no—
94C4×C47—22no—
95C4×C47—21no—
96C4×C47—82no—
97C4×C47#45411yescharacteristic
98C4×C47#45511yescharacteristic
99#3253D4711yescharacteristic
100#1838—471no—
101#1557—472no—
102#1530—472no—
103#1307—471no—
104C2×C47—41no—
105C2×C47—162no—
106C2×C47#204011yescharacteristic
107C2×C47#204812yes—
108#532C2×D4712yes—
109#476—22no—
110#469—22no—
111#459—22no—
112#432—472no—
113#423—471no—
114#423C2×D4711yescharacteristic
115C2×#97—471no—
116#341—942no—
117#332—942no—
118#331—942no—
119#329—944no—
120C47#1052511yescharacteristic, complemented, Sylow
121C2×#41—22no—
122C2×D8—22no—
123C2×M16—941no—
124C2×M16C22⋊D4712yes—
125C22×C8—941no—
126C4⋊Q8C22⋊D4712yes—
127#127C22⋊D4712yes—
128#124—22no—
129#121—22no—
130#121—22no—
131C4×D4—22no—
132C4×D4—22no—
133C2×C42C22×D4711yescharacteristic
134#106—22no—
135C4⋊C8—22no—
136C4⋊C8—942no—
137C4⋊C8—942no—
138#102—24no—
139C22⋊C8—942no—
140#97—472no—
141#97—942no—
142#97—941no—
143#41—84no—
144C2×Q8—22no—
145C2×D4—22no—
146C2×D4—22no—
147C2×D4—22no—
148C2×D4—42no—
149C22×C4—42no—
150C22×C4#1062535212yes—
151C22×C4C2×C22⋊D4711yescharacteristic
152D8—82no—
153M16—42no—
154M16—1882no—
155C2×C8—22no—
156C2×C8—942no—
157C2×C8—1881no—
158C2×C8—1882no—
159C2×C8—1881no—
160C2×C8—1881no—
161C4⋊C4—22no—
162C4⋊C4—22no—
163C22⋊C4—42no—
164C22⋊C4—42no—
165C42D4×D4711yescharacteristic
166C42#1062535211yescharacteristic
167C42C2×C22⋊D4712yes—
168C23—42no—
169C23#1309401111yescharacteristic
170Q8—82no—
171D4—42no—
172D4—42no—
173D4—82no—
174D4—82no—
175D4—84no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—42no—
179C2×C4—42no—
180C2×C4—84no—
181C2×C4#1309399311yescharacteristic
182C2×C4#1309399511yescharacteristic
183C2×C4#1309399612yes—
184C2×C4#1309400512yes—
185C8—42no—
186C8—3762no—
187C8—3761no—
188C8—1882no—
189C22—21no—
190C22—22no—
191C22—82no—
192C22—82no—
193C22—82no—
194C22#1568903711yescharacteristic
195C4—21no—
196C4—22no—
197C4—22no—
198C4—82no—
199C4#1568929611yescharacteristic
200C4#1568930111yescharacteristic
201C2—41no—
202C2—162no—
203C2#2024461411yescharacteristic
204C2#2024465812yes—
subgroupG/H
centerC22#15689037
commutator, cosocleC2×C42×C47C23
FrattiniC2×C42C22×D47
Fitting, prime radicalC47×#3253C2
socleC22×C47#394
hypercenter#3253D47
hypocenterC47#10525
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. #20243528C2semidirect
2. #20242110C2semidirect
3. #20241999C2semidirect
4. #20241299C2semidirect
120. C47#10525semidirect

Maximal subgroups

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

subgroupindex
#202435282
#202421102
#202419992
#202412992
C47×#32532
#1052547