Finite Group Databasebeta

← #20275819

Subgroups — #20275819

Order 6784.

350 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 69 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#16811423C211yesmaximal, characteristic, complemented
2#16811421C212yesmaximal, complemented
3#16810589C212yesmaximal, complemented
4#16810381C211yesmaximal, characteristic, complemented
5C53×#408C211yesmaximal, characteristic, complemented
6C2×#11719945C2212yes—
7#13133250—22no—
8#13133248—22no—
9#13133247—22no—
10#13133245—22no—
11#13133243—22no—
12#13133242—22no—
13#13133239—22no—
14#13133237—24no—
15C2×#11719917C2212yes—
16#13133129—24no—
17C2×#41×C53C2211yescharacteristic
18C2×M16×C53C2211yescharacteristic
19C22×C8×C53C2211yescharacteristic
20C22⋊C8×C53—22no—
21#41×C53—22no—
22#41×C53—22no—
23C2×Q8×C53D411yescharacteristic
24C2×D4×C53—21no—
25C2×D4×C53D411yescharacteristic
26C22×C4×C53—21no—
27C22×C4×C53C2311yescharacteristic
28C2×C22⋊D53—22no—
29#11719945—22no—
30#11719945—22no—
31#11719945—22no—
32#11719945—22no—
33C2×D212D412yes—
34C2×C4×D53—22no—
35C2×Dic106D412yes—
36M16×C53—22no—
37C2×C8×C53—21no—
38C2×C8×C53—22no—
39C2×C8×C53D412yes—
40#11719925—24no—
41#11719924—24no—
42#11719923—24no—
43#11719922—24no—
44#11719917—22no—
45#11719917—22no—
46C2×#88322D414yes—
47Dic212—22no—
48D424—22no—
49#11719913—24no—
50C23×C53—21no—
51C23×C53#4111yescharacteristic
52C22×D53—22no—
53Q8×C53—22no—
54D4×C53—22no—
55D4×C53—42no—
56C2×C4×C53—21no—
57C2×C4×C53—41no—
58C2×C4×C53—41no—
59C2×C4×C53C2×D411yescharacteristic
60C2×C4×C53C2×D412yes—
61C2×C4×C53C2×D412yes—
62C2×C4×C53C2×D411yescharacteristic
63C22⋊D53—42no—
64C22⋊D53—42no—
65C2×Dic53—22no—
66D212—22no—
67D212—22no—
68C4×D53—42no—
69C4×D53—42no—
70Dic106—22no—
71Dic106—22no—
72C8×C53—22no—
73C8×C53—41no—
74#88322—24no—
75C22×C53—41no—
76C22×C53—42no—
77C22×C53—41no—
78C22×C53#12112yes—
79C22×C53#12111yescharacteristic
80C2×D53—44no—
81C2×D53—42no—
82C4×C53—41no—
83C4×C53#12011yescharacteristic
84C4×C53#12012yes—
85C4×C53#12711yescharacteristic
86Dic53—42no—
87#2017—531nomaximal, Sylow
88C2×C53—22no—
89C2×C53—21no—
90C2×C53—81no—
91C2×C53#39211yescharacteristic
92D53—82no—
93#575—532no—
94#575—531no—
95#408—532no—
96#408D5311yescharacteristic, complemented
97#338—531no—
98C53#201711yescharacteristic, complemented, Sylow
99C2×#41—532no—
100C2×#41C2×D5311yescharacteristic
101#137—1062no—
102#137—1062no—
103#137—1062no—
104AGL(1,ℤ/8)—1062no—
105AGL(1,ℤ/8)—1062no—
106AGL(1,ℤ/8)—1062no—
107#135—1064no—
108#135—1062no—
109C2×M16—532no—
110C2×M16C2×D5311yescharacteristic
111C22×C8C2×D5311yescharacteristic
112C22⋊C8—22no—
113C22⋊C8—1064no—
114#41—22no—
115#41—22no—
116#41—1062no—
117#41—1062no—
118#41—1062no—
119#41—1062no—
120C2×Q8—532no—
121C2×Q8D21211yescharacteristic
122C2×D4—21no—
123C2×D4—532no—
124C2×D4—1062no—
125C2×D4D21211yescharacteristic
126C22×C4—21no—
127C22×C4—1062no—
128C22×C4C22×D5311yescharacteristic
129Q16—1062no—
130Q16—1064no—
131SD16—1064no—
132SD16—1064no—
133SD16—1064no—
134D8—1062no—
135D8—1064no—
136M16—22no—
137M16—1062no—
138M16—1062no—
139C2×C8—22no—
140C2×C8—21no—
141C2×C8—534no—
142C2×C8C22⋊D5312yes—
143C23—21no—
144C23—1062no—
145C23#1171994511yescharacteristic
146Q8—22no—
147Q8—1062no—
148Q8—1062no—
149D4—22no—
150D4—42no—
151D4—1062no—
152D4—1062no—
153D4—2122no—
154D4—2122no—
155C2×C4—21no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—1062no—
159C2×C4—2122no—
160C2×C4—2122no—
161C2×C4C2×D21211yescharacteristic
162C2×C4D4×D5312yes—
163C2×C4D4×D5312yes—
164C2×C4C2×C22⋊D5311yescharacteristic
165C8—22no—
166C8—41no—
167C8—1064no—
168C22—41no—
169C22—42no—
170C22—41no—
171C22—2124no—
172C22—2122no—
173C22#1313319012yes—
174C22#1313319111yescharacteristic
175C4—41no—
176C4—2122no—
177C4#1313318511yescharacteristic
178C4#1313318712yes—
179C4#1313321311yescharacteristic
180C2—21no—
181C2—22no—
182C2—81no—
183C2—4242no—
184C2#1681053211yescharacteristic
subgroupG/H
centerC4#13133213
commutatorC22×C4×C53C23
FrattiniC22×C4C22×D53
FittingC53×#408C2
socleC2×C53#392
hypercenter#408D53
hypocenterC53#2017
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C4 ◁ C22×C4 ◁ #408

Complements

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

NH ≅ G/Ntype
1. #16811423C2semidirect
2. #16811421C2semidirect
3. #16810589C2semidirect
4. #16810381C2semidirect
5. C53×#408C2semidirect
96. #408D53semidirect
98. C53#2017semidirect