Finite Group Databasebeta

← #20275105

Subgroups — #20275105

Order 6784.

324 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 83 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
1C2×#13133215C211yesmaximal, characteristic
2#16811100C211yesmaximal, characteristic
3#16810632C211yesmaximal, characteristic
4C2×#13133135C211yesmaximal, characteristic
5#16810384C211yesmaximal, characteristic
6#16810382C211yesmaximal, characteristic
7C53×#419C211yesmaximal, characteristic
8C22×Dic106C2211yescharacteristic
9C2×#11719917C2211yescharacteristic
10#13133215—22no—
11#13133215C414yescomplemented
12C4×Dic106—22no—
13#13133182—22no—
14#13133176C2211yescharacteristic
15#13133173—22no—
16C2×#11719919—22no—
17C2×C4×Dic53C2211yescharacteristic
18#13133141—22no—
19#13133135—22no—
20C22×Q8×C53C2211yescharacteristic
21C2×M16×C53C2211yescharacteristic
22#117×C53C2211yescharacteristic
23#103×C53—22no—
24C2×Q8×C53—21no—
25C2×Q8×C53—21no—
26C2×Q8×C53—21no—
27C2×Q8×C53—21no—
28C2×Q8×C53—21no—
29C2×Q8×C53C2×C412yes—
30C22×C4×C53—21no—
31C22×C4×C53C2311yescharacteristic
32C22×Dic53—21no—
33C22×Dic53D412yes—
34C2×Dic106—21no—
35C2×Dic106—21no—
36C2×Dic106—21no—
37C2×Dic106—21no—
38C2×Dic106C2×C412yes—
39C2×Dic106D412yes—
40M16×C53—21no—
41M16×C53—21no—
42C2×C8×C53—21no—
43C4⋊C4×C53D412yes—
44C22⋊C4×C53—21no—
45C42×C53—21no—
46C22⋊Dic53—21no—
47C4⋊Dic53D412yes—
48#11719919—24no—
49#11719919—28no—
50C4×Dic53—24no—
51C4×Dic53—21no—
52C4×Dic53—21no—
53C4×Dic53C2×C412yes—
54#11719917—21no—
55#11719917—21no—
56C2×#88322—21no—
57C23×C53C2×D411yescharacteristic
58Q8×C53—22no—
59Q8×C53—22no—
60Q8×C53—41no—
61Q8×C53—41no—
62C2×C4×C53—22no—
63C2×C4×C53—22no—
64C2×C4×C53—41no—
65C2×C4×C53—41no—
66C2×C4×C53C22×C411yescharacteristic
67C2×C4×C53C2×D411yescharacteristic
68C2×C4×C53C2×D411yescharacteristic
69C2×C4×C53C2×D411yescharacteristic
70C2×C4×C53#4111yescharacteristic
71C2×C4×C53#4111yescharacteristic
72C2×Dic53—24no—
73C2×Dic53—22no—
74C2×Dic53—22no—
75C2×Dic53—22no—
76C2×Dic53—22no—
77C2×Dic53C22⋊C414yes—
78Dic106—22no—
79Dic106—22no—
80Dic106—22no—
81Dic106—22no—
82C8×C53—41no—
83#88322—41no—
84C22×C53—22no—
85C22×C53C2×C22⋊C411yescharacteristic
86C22×C53#12011yescharacteristic
87C22×C53#12311yescharacteristic
88C4×C53—42no—
89C4×C53—42no—
90C4×C53C4×D411yescharacteristic
91C4×C53C4×D411yescharacteristic
92C4×C53#12111yescharacteristic
93C4×C53#12111yescharacteristic
94Dic53—24no—
95Dic53—24no—
96Dic53—42no—
97#1886—531nomaximal, Sylow
98C2×C53—21no—
99C2×C53—21no—
100C2×C53#38611yescharacteristic
101C2×C53#45612yes—
102C2×C4⋊Q8—531no—
103#519—531no—
104#419—531no—
105#419D5311yescharacteristic
106C2×#101—531no—
107#339—531no—
108#339—531no—
109C53#188611yescharacteristic, complemented, Sylow
110C22×Q8—531no—
111C22×Q8C2×D5311yescharacteristic
112C2×M16—531no—
113C2×M16C2×D5311yescharacteristic
114C4⋊Q8—534no—
115C4⋊Q8—1062no—
116C4×Q8—1062no—
117C4×Q8—1062no—
118#117—531no—
119#117—1062no—
120#117C2×D5311yescharacteristic
121C2×C4⋊C4—1062no—
122C2×C42—531no—
123#103—22no—
124#103—1062no—
125#101—1062no—
126C2×Q8—21no—
127C2×Q8—21no—
128C2×Q8—21no—
129C2×Q8—21no—
130C2×Q8—21no—
131C2×Q8—532no—
132C2×Q8—532no—
133C2×Q8—1061no—
134C2×Q8—1061no—
135C2×Q8—1061no—
136C2×Q8—1061no—
137C2×Q8C4×D5312yes—
138C22×C4—21no—
139C22×C4—532no—
140C22×C4—1061no—
141C22×C4C22×D5311yescharacteristic
142M16—21no—
143M16—21no—
144M16—1061no—
145M16—1061no—
146C2×C8—21no—
147C2×C8—1061no—
148C4⋊C4—532no—
149C4⋊C4—1068no—
150C4⋊C4—1064no—
151C4⋊C4C22⋊D5312yes—
152C22⋊C4—21no—
153C22⋊C4—1061no—
154C42—21no—
155C42—532no—
156C42—1064no—
157C42—1061no—
158C42—1061no—
159C23D4×D5311yescharacteristic
160Q8—22no—
161Q8—22no—
162Q8—41no—
163Q8—41no—
164Q8—1062no—
165Q8—1062no—
166Q8—1062no—
167Q8—1062no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—534no—
173C2×C4—1064no—
174C2×C4—1062no—
175C2×C4—1062no—
176C2×C4—1062no—
177C2×C4—1062no—
178C2×C4C2×C4×D5311yescharacteristic
179C2×C4#1171994511yescharacteristic
180C2×C4D4×D5311yescharacteristic
181C2×C4D4×D5311yescharacteristic
182C2×C4#1171994711yescharacteristic
183C2×C4C2×C22⋊D5311yescharacteristic
184C8—41no—
185C8—2121no—
186C22—22no—
187C22C22⋊C4×D5311yescharacteristic
188C22#1313318511yescharacteristic
189C22#1313319911yescharacteristic
190C4—42no—
191C4—42no—
192C4—1064no—
193C4—1064no—
194C4—2122no—
195C4#1313317711yescharacteristic
196C4C4×C22⋊D5311yescharacteristic
197C4#1313318811yescharacteristic
198C4#1313319011yescharacteristic
199C2—21no—
200C2—21no—
201C2#1681049911yescharacteristic
202C2#1681080412yes—
subgroupG/H
centerC22#13133185
commutatorC2×C4×C53C22×C4
FrattiniC22×C4C22×D53
FittingC53×#419C2
socleC22×C53#120
hypercenter#419D53
hypocenterC53#1886
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
11. #13133215C4semidirect
109. C53#1886semidirect