Finite Group Databasebeta

← #20275133

Subgroups — #20275133

Order 6784.

342 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 89 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#16811130C211yesmaximal, characteristic
2#16810607C211yesmaximal, characteristic, complemented
3#16810586C212yesmaximal, complemented
4#16810394C212yesmaximal, complemented
5C53×#568C211yesmaximal, characteristic, complemented
6C22×C4×D53C2211yescharacteristic
7C2×#11719917C2212yes—
8#13133198C412yes—
9#13133193—21no—
10#13133188—21no—
11#13133185C412yes—
12C4×C22⋊D53—22no—
13#13133176C2211yescharacteristic
14#13133136—21no—
15#13133134—21no—
16#13133129—24no—
17C2×#41×C53C2211yescharacteristic
18#131×C53—24no—
19C2×M16×C53—22no—
20C2×M16×C53C2212yes—
21C23×D53C2×C411yescharacteristic
22#41×C53—22no—
23#41×C53—22no—
24C2×Q8×C53D411yescharacteristic
25C2×D4×C53C2×C412yes—
26C2×D4×C53D411yescharacteristic
27C22×C4×C53D412yes—
28C22×C4×C53C2311yescharacteristic
29C2×C22⋊D53—22no—
30C22×Dic53C2×C411yescharacteristic
31C2×C4×D53—22no—
32C2×C4×D53—21no—
33C2×C4×D53—21no—
34C2×C4×D53—22no—
35M16×C53—24no—
36M16×C53—22no—
37M16×C53—22no—
38C2×C8×C53—24no—
39C2×C8×C53D414yescomplemented
40C22⋊Dic53—22no—
41C22⋊Dic53C2×C412yes—
42#11719921—22no—
43C4⋊Dic53—21no—
44#11719919—22no—
45C4×Dic53—21no—
46#11719917—22no—
47#11719917—22no—
48C2×#88322—22no—
49C23×C53C22⋊C412yes—
50C23×C53C22×C411yescharacteristic
51C22×D53—21no—
52C22×D53—22no—
53C22×D53—42no—
54Q8×C53—22no—
55D4×C53—24no—
56D4×C53—22no—
57C2×C4×C53—22no—
58C2×C4×C53—22no—
59C2×C4×C53C22⋊C412yes—
60C2×C4×C53C2×D411yescharacteristic
61C2×C4×C53C2×D412yes—
62C2×C4×C53C2×D411yescharacteristic
63C2×C4×C53#4112yes—
64C22⋊D53—44no—
65C2×Dic53—21no—
66C2×Dic53—22no—
67C2×Dic53—22no—
68C4×D53—41no—
69C4×D53—42no—
70C4×D53—41no—
71C8×C53—24no—
72#88322—42no—
73C22×C53—22no—
74C22×C53—41no—
75C22×C53—42no—
76C22×C53C2×C22⋊C411yescharacteristic
77C22×C53C4×D412yes—
78C2×D53—22no—
79C2×D53—24no—
80C2×D53—41no—
81C2×D53—42no—
82C2×D53—41no—
83C4×C53—22no—
84C4×C53#12011yescharacteristic
85C4×C53#12112yes—
86C4×C53#12311yescharacteristic
87Dic53—42no—
88Dic53—41no—
89#1891—531nomaximal, Sylow
90C2×C53—21no—
91C2×C53—22no—
92C2×C53—42no—
93C2×C53#38611yescharacteristic
94D53—42no—
95#568D5311yescharacteristic, complemented
96#525—531no—
97#413—531no—
98#407—532no—
99#344—532no—
100C53#189111yescharacteristic, complemented, Sylow
101C2×#41C2×D5311yescharacteristic
102C23×C4—531no—
103#131—24no—
104C2×M16—22no—
105C2×M16—532no—
106C2×M16C2×D5312yes—
107#123—532no—
108C22⋊Q8—1061no—
109#121—1061no—
110#120—532no—
111C4×D4—1062no—
112#117—531no—
113#101—1061no—
114#100—1061no—
115C22⋊C8—1064no—
116C24—531no—
117#41—22no—
118#41—22no—
119C2×Q8D21211yescharacteristic
120C2×D4—1062no—
121C2×D4C4×D5312yes—
122C2×D4D21211yescharacteristic
123C22×C4—531no—
124C22×C4—1061no—
125C22×C4—1062no—
126C22×C4—1062no—
127C22×C4—1061no—
128C22×C4C22⋊D5312yes—
129C22×C4C22×D5311yescharacteristic
130M16—24no—
131M16—22no—
132M16—22no—
133M16—1062no—
134M16—1062no—
135C2×C8—24no—
136C2×C8—1062no—
137C2×C8C22⋊D5314yescomplemented
138C4⋊C4—1061no—
139C4⋊C4—1062no—
140C22⋊C4—532no—
141C22⋊C4—1062no—
142C22⋊C4—1062no—
143C42—1061no—
144C23—1061no—
145C23—1062no—
146C23—2122no—
147C23#1171992112yes—
148C23C2×C4×D5311yescharacteristic
149Q8—22no—
150D4—24no—
151D4—22no—
152D4—2124no—
153C2×C4—22no—
154C2×C4—22no—
155C2×C4—1062no—
156C2×C4—1061no—
157C2×C4—1062no—
158C2×C4—2121no—
159C2×C4—2122no—
160C2×C4—2121no—
161C2×C4#1171992112yes—
162C2×C4C2×D21211yescharacteristic
163C2×C4#1171994512yes—
164C2×C4C2×C22⋊D5312yes—
165C2×C4C2×C22⋊D5311yescharacteristic
166C8—24no—
167C8—2122no—
168C22—22no—
169C22—41no—
170C22—42no—
171C22—1062no—
172C22—1064no—
173C22—2121no—
174C22—2122no—
175C22—2121no—
176C22C2×#1171992111yescharacteristic
177C22C4×C22⋊D5312yes—
178C4—22no—
179C4—2122no—
180C4—2121no—
181C4#1313318611yescharacteristic
182C4#1313319112yes—
183C4#1313320011yescharacteristic
184C2—21no—
185C2—22no—
186C2—42no—
187C2—2122no—
188C2#1681050211yescharacteristic
subgroupG/H
centerC4#13133186
commutatorC23×C53C22×C4
FrattiniC22×C4C22×D53
FittingC53×#568C2
socleC2×C53#386
hypercenter#568D53
hypocenterC53#1891
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C4 ◁ C22×C4 ◁ #568

Complements

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

NH ≅ G/Ntype
2. #16810607C2semidirect
3. #16810586C2semidirect
4. #16810394C2semidirect
5. C53×#568C2semidirect
39. C2×C8×C53D4semidirect
95. #568D53semidirect
100. C53#1891semidirect
137. C2×C8C22⋊D53semidirect