Finite Group Databasebeta

← #26120416

Subgroups — #26120416

Order 12032.

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

normal subgroups only

This group has 386 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×#1849C211yesmaximal, characteristic, complemented
6#15689673C2212yes—
7#15689414—22no—
8#15689376—22no—
9#15689319—22no—
10#15689180C2212yes—
11#15688868—22no—
12#15688866—22no—
13#15688862—24no—
14C2×#108×C47—22no—
15C47×#423C2211yescharacteristic
16C47×#423C2211yescharacteristic
17C2×#97×C47C2211yescharacteristic
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
36#108×C47—44no—
37C4⋊C8×C47—22no—
38C4⋊C8×C47—22no—
39#97×C47—21no—
40#97×C47D412yes—
41C22×C4×C47C2×D411yescharacteristic
42C22×C4×C47#4112yes—
43C2×C22⋊D47—42no—
44#10625350—84no—
45C2×D188—22no—
46C2×D188—22no—
47C2×D188—22no—
48C2×C4×D47—42no—
49C2×Dic94—22no—
50M16×C47—42no—
51M16×C47—42no—
52C2×C8×C47—22no—
53C2×C8×C47—22no—
54C2×C8×C47—22no—
55C2×C8×C47—22no—
56C2×C8×C47—41no—
57C42×C47C2×D412yes—
58C42×C47C2×D411yescharacteristic
59C42×C47#4111yescharacteristic
60(C2×D47)⋊C4—42no—
61(C2×D47)⋊C4—42no—
62C4⋊Dic47—22no—
63C4⋊Dic47—22no—
64C2×C47⋊C8—42no—
65D376—82no—
66C23×C47#12411yescharacteristic
67C22×D47—42no—
68C2×C4×C47—21no—
69C2×C4×C47—21no—
70C2×C4×C47—42no—
71C2×C4×C47#12011yescharacteristic
72C2×C4×C47#12111yescharacteristic
73C2×C4×C47#12112yes—
74C2×C4×C47#12312yes—
75C22⋊D47—84no—
76C2×Dic47—42no—
77D188—42no—
78D188—42no—
79D188—82no—
80D188—82no—
81C4×D47—84no—
82Dic94—82no—
83C8×C47—24no—
84C8×C47—42no—
85C8×C47—81no—
86C47⋊C8—82no—
87#10524—471nomaximal, Sylow
88C22×C47—22no—
89C22×C47—21no—
90C22×C47#39411yescharacteristic
91C2×D47—82no—
92C2×D47—82no—
93C2×D47—82no—
94C4×C47—22no—
95C4×C47—22no—
96C4×C47—21no—
97C4×C47#45311yescharacteristic
98C4×C47#45611yescharacteristic
99Dic47—82no—
100#3253—471no—
101#1849D4711yescharacteristic, complemented
102#1557—472no—
103#1530—472no—
104#1307—471no—
105C2×C47—41no—
106C2×C47#203911yescharacteristic
107C2×C47#204812yes—
108D47—162no—
109#532—472no—
110#476—942no—
111#469—942no—
112#459—942no—
113#432—472no—
114C2×#108—22no—
115#423C2×D4711yescharacteristic
116#423C2×D4711yescharacteristic
117C2×#97C2×D4711yescharacteristic
118#332—942no—
119#331—942no—
120#329—944no—
121C47#1052411yescharacteristic, complemented, Sylow
122C2×#41—942no—
123C2×D8—942no—
124C2×M16—21no—
125C2×M16C22⋊D4712yes—
126C22×C8—21no—
127C4⋊Q8—472no—
128#127—472no—
129#124—942no—
130#121—942no—
131#121—942no—
132C4×D4—942no—
133C4×D4—942no—
134C2×C42C22×D4711yescharacteristic
135#108—44no—
136#106—942no—
137C4⋊C8—22no—
138C4⋊C8—22no—
139C4⋊C8—942no—
140#102—944no—
141C22⋊C8—942no—
142#97—21no—
143#97—942no—
144#97D18812yes—
145#41—3764no—
146C2×Q8—942no—
147C2×D4—942no—
148C2×D4—942no—
149C2×D4—942no—
150C2×D4—1882no—
151C22×C4—1882no—
152C22×C4#1062535412yes—
153C22×C4C2×C22⋊D4711yescharacteristic
154D8—3762no—
155M16—42no—
156M16—42no—
157C2×C8—22no—
158C2×C8—22no—
159C2×C8—22no—
160C2×C8—22no—
161C2×C8—41no—
162C2×C8—1882no—
163C4⋊C4—942no—
164C4⋊C4—942no—
165C22⋊C4—1882no—
166C22⋊C4—1882no—
167C42C2×D18811yescharacteristic
168C42#1062535011yescharacteristic
169C42D4×D4712yes—
170C23—1882no—
171C23#1309401311yescharacteristic
172Q8—3762no—
173D4—3762no—
174D4—3762no—
175D4—3764no—
176D4—1882no—
177D4—1882no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—42no—
181C2×C4—3764no—
182C2×C4—1882no—
183C2×C4#1309399211yescharacteristic
184C2×C4#1309399811yescharacteristic
185C2×C4#1309399912yes—
186C2×C4#1309400912yes—
187C8—24no—
188C8—42no—
189C8—81no—
190C8—3762no—
191C22—21no—
192C22—22no—
193C22—3762no—
194C22—3762no—
195C22—3762no—
196C22#1568904011yescharacteristic
197C4—21no—
198C4—22no—
199C4—22no—
200C4—3762no—
201C4#1568929011yescharacteristic
202C4#1568930511yescharacteristic
203C2—41no—
204C2—7522no—
205C2#2024461211yescharacteristic
206C2#2024466312yes—
subgroupG/H
centerC22#15689040
commutator, cosocleC2×C42×C47C23
FrattiniC2×C42C22×D47
FittingC47×#1849C2
socleC22×C47#394
prime radical#20254769C2
hypercenter#1849D47
hypocenterC47#10524
LCSG ▷ C2×C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1849

Complements

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

NH ≅ G/Ntype
2. #20242108C2semidirect
3. #20241998C2semidirect
4. #20241299C2semidirect
5. C47×#1849C2semidirect
101. #1849D47semidirect
121. C47#10524semidirect

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×#18492
#1052447