Finite Group Databasebeta

← #26120421

Subgroups — #26120421

Order 12032.

370 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 75 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#20254769C211yesmaximal, characteristic
2#20242108C212yesmaximal, complemented
3#20241998C212yesmaximal, complemented
4#20241299C211yesmaximal, characteristic, complemented
5C47×#1838C211yesmaximal, characteristic, complemented
6#15689673C2212yes—
7#15689414—22no—
8#15689376—22no—
9#15689319—22no—
10#15689180C2212yes—
11#15688868—22no—
12#15688866—22no—
13#15688862—24no—
14C47×#423C2211yescharacteristic
15C47×#423C2211yescharacteristic
16C2×#97×C47C2211yescharacteristic
17C47×#341—22no—
18C2×#10625350—22no—
19C2×D376—22no—
20#13094024D412yes—
21#13094021D412yes—
22#13094012—22no—
23#13093998—22no—
24#13093998—22no—
25C4×D188—22no—
26C4×D188—22no—
27#13093956—22no—
28C4⋊(C47⋊C8)—22no—
29#13093946—24no—
30C22⋊(C47⋊C8)—22no—
31#13093935—22no—
32C2×M16×C47—21no—
33C2×M16×C47D412yes—
34C22×C8×C47—21no—
35C2×C42×C47C2311yescharacteristic
36C4⋊C8×C47—22no—
37C4⋊C8×C47—22no—
38#97×C47—21no—
39#97×C47D412yes—
40C22×C4×C47C2×D411yescharacteristic
41C22×C4×C47#4112yes—
42C2×C22⋊D47—42no—
43#10625350—84no—
44C2×D188—22no—
45C2×D188—22no—
46C2×D188—22no—
47C2×C4×D47—42no—
48C2×Dic94—22no—
49M16×C47—42no—
50M16×C47—42no—
51C2×C8×C47—22no—
52C2×C8×C47—22no—
53C2×C8×C47—41no—
54C2×C8×C47—41no—
55C2×C8×C47—41no—
56C42×C47C2×D412yes—
57C42×C47C2×D411yescharacteristic
58C42×C47#4111yescharacteristic
59(C2×D47)⋊C4—42no—
60(C2×D47)⋊C4—42no—
61C4⋊Dic47—22no—
62C4⋊Dic47—22no—
63C2×C47⋊C8—42no—
64D376—82no—
65C23×C47#12411yescharacteristic
66C22×D47—42no—
67C2×C4×C47—21no—
68C2×C4×C47—21no—
69C2×C4×C47—42no—
70C2×C4×C47#12011yescharacteristic
71C2×C4×C47#12111yescharacteristic
72C2×C4×C47#12112yes—
73C2×C4×C47#12312yes—
74C22⋊D47—84no—
75C2×Dic47—42no—
76D188—42no—
77D188—42no—
78D188—82no—
79D188—82no—
80C4×D47—84no—
81Dic94—82no—
82C8×C47—42no—
83C8×C47—42no—
84C8×C47—81no—
85C47⋊C8—82no—
86#10525—471nomaximal, Sylow
87C22×C47—22no—
88C22×C47—21no—
89C22×C47#39411yescharacteristic
90C2×D47—82no—
91C2×D47—82no—
92C2×D47—82no—
93C4×C47—22no—
94C4×C47—22no—
95C4×C47—21no—
96C4×C47#45411yescharacteristic
97C4×C47#45511yescharacteristic
98Dic47—82no—
99#3253—471no—
100#1838D4711yescharacteristic, complemented
101#1557—472no—
102#1530—472no—
103#1307—471no—
104C2×C47—41no—
105C2×C47#204011yescharacteristic
106C2×C47#204812yes—
107D47—162no—
108#532—472no—
109#476—942no—
110#469—942no—
111#459—942no—
112#432—472no—
113#423C2×D4711yescharacteristic
114#423C2×D4711yescharacteristic
115C2×#97C2×D4711yescharacteristic
116#341—22no—
117#332—942no—
118#331—942no—
119#329—944no—
120C47#1052511yescharacteristic, complemented, Sylow
121C2×#41—942no—
122C2×D8—942no—
123C2×M16—21no—
124C2×M16C22⋊D4712yes—
125C22×C8—21no—
126C4⋊Q8—472no—
127#127—472no—
128#124—942no—
129#121—942no—
130#121—942no—
131C4×D4—942no—
132C4×D4—942no—
133C2×C42C22×D4711yescharacteristic
134#106—942no—
135C4⋊C8—22no—
136C4⋊C8—22no—
137C4⋊C8—942no—
138#102—944no—
139C22⋊C8—942no—
140#97—21no—
141#97—942no—
142#97D18812yes—
143#41—3764no—
144C2×Q8—942no—
145C2×D4—942no—
146C2×D4—942no—
147C2×D4—942no—
148C2×D4—1882no—
149C22×C4—1882no—
150C22×C4#1062535412yes—
151C22×C4C2×C22⋊D4711yescharacteristic
152D8—3762no—
153M16—42no—
154M16—42no—
155C2×C8—22no—
156C2×C8—22no—
157C2×C8—41no—
158C2×C8—41no—
159C2×C8—41no—
160C2×C8—1882no—
161C4⋊C4—942no—
162C4⋊C4—942no—
163C22⋊C4—1882no—
164C22⋊C4—1882no—
165C42C2×D18811yescharacteristic
166C42#1062535011yescharacteristic
167C42D4×D4712yes—
168C23—1882no—
169C23#1309401311yescharacteristic
170Q8—3762no—
171D4—3762no—
172D4—3762no—
173D4—3764no—
174D4—1882no—
175D4—1882no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—42no—
179C2×C4—3764no—
180C2×C4—1882no—
181C2×C4#1309399211yescharacteristic
182C2×C4#1309399811yescharacteristic
183C2×C4#1309399912yes—
184C2×C4#1309400912yes—
185C8—42no—
186C8—42no—
187C8—81no—
188C8—3762no—
189C22—21no—
190C22—22no—
191C22—3762no—
192C22—3762no—
193C22—3762no—
194C22#1568904011yescharacteristic
195C4—21no—
196C4—22no—
197C4—22no—
198C4—3762no—
199C4#1568929511yescharacteristic
200C4#1568930011yescharacteristic
201C2—41no—
202C2—7522no—
203C2#2024461711yescharacteristic
204C2#2024466312yes—
subgroupG/H
centerC22#15689040
commutator, cosocleC2×C42×C47C23
FrattiniC2×C42C22×D47
FittingC47×#1838C2
socleC22×C47#394
prime radical#20254769C2
hypercenter#1838D47
hypocenterC47#10525
LCSG ▷ C2×C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1838

Complements

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

NH ≅ G/Ntype
2. #20242108C2semidirect
3. #20241998C2semidirect
4. #20241299C2semidirect
5. C47×#1838C2semidirect
100. #1838D47semidirect
120. C47#10525semidirect

Maximal subgroups

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

subgroupindex
#202547692
#202421082
#202419982
#202412992
C47×#18382
#1052547