Finite Group Databasebeta

← #20273649

Subgroups — #20273649

Order 6784.

310 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 53 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#16811204C211yesmaximal, characteristic, complemented
2#16810757C212yesmaximal, complemented
3#16810356C212yesmaximal, complemented
4#16810349C211yesmaximal, characteristic, complemented
5C53×#394C211yesmaximal, characteristic, complemented
6C2×#11719945—21no—
7C2×#11719922—22no—
8#13133197—22no—
9#13133192—22no—
10#13133191—21no—
11#13133191C2212yes—
12#13133138—22no—
13C22⋊#88322C2212yes—
14C22×D4×C53C2211yescharacteristic
15C2×C4⋊C4×C53C2211yescharacteristic
16C2×C22⋊C4×C53—21no—
17C2×C22⋊C4×C53—21no—
18#95×C53C2211yescharacteristic
19C24×C53—21no—
20C2×D4×C53—22no—
21C2×D4×C53—42no—
22C22×C4×C53—21no—
23C22×C4×C53D411yescharacteristic
24C22×C4×C53D411yescharacteristic
25C22×C4×C53C2311yescharacteristic
26C2×C22⋊D53—22no—
27#11719945—42no—
28#11719945—42no—
29C2×D212—21no—
30C2×D212—21no—
31C2×D212D412yes—
32C2×C4×D53—22no—
33C2×Dic106—21no—
34C4⋊C4×C53—22no—
35C22⋊C4×C53—42no—
36C22⋊C4×C53—42no—
37#11719922—44no—
38#11719921—22no—
39#11719921—22no—
40C4⋊Dic53—21no—
41C4⋊Dic53D412yes—
42C2×#88322—22no—
43C23×C53—22no—
44C23×C53—41no—
45C23×C53—41no—
46C23×C53—41no—
47C23×C53—41no—
48C23×C53—41no—
49C23×C53—41no—
50C23×C53C2×D411yescharacteristic
51C22×D53—22no—
52D4×C53—24no—
53D4×C53—42no—
54C2×C4×C53—21no—
55C2×C4×C53—21no—
56C2×C4×C53—21no—
57C2×C4×C53—41no—
58C2×C4×C53—41no—
59C2×C4×C53—41no—
60C2×C4×C53—41no—
61C2×C4×C53—41no—
62C2×C4×C53D812yescomplemented
63C2×C4×C53C2×D412yes—
64C22⋊D53—42no—
65C22⋊D53—42no—
66C2×Dic53—22no—
67D212—42no—
68D212—42no—
69D212—81no—
70C4×D53—44no—
71Dic106—42no—
72#88322—42no—
73C22×C53—21no—
74C22×C53—21no—
75C22×C53—42no—
76C22×C53—42no—
77C22×C53—42no—
78C22×C53—81no—
79C22×C53—81no—
80C22×C53—81no—
81C22×C53—81no—
82C22×C53#12011yescharacteristic
83C22×C53C2×D811yescharacteristic
84C22×C53AGL(1,ℤ/8)11yescharacteristic
85C2×D53—42no—
86C2×D53—42no—
87C2×D53—42no—
88C4×C53—22no—
89C4×C53—22no—
90C4×C53—41no—
91C4×C53—81no—
92Dic53—42no—
93#1598—531nomaximal, Sylow
94C2×C53—21no—
95C2×C53—21no—
96C2×C53—82no—
97C2×C53#44711yescharacteristic
98C2×C53#45411yescharacteristic
99C2×C53B(4,𝔽2)11yescharacteristic
100D53—82no—
101#537—531no—
102#447—532no—
103#394D5311yescharacteristic, complemented
104#327—532no—
105#324—531no—
106C53#159811yescharacteristic, complemented, Sylow
107C2×#41—1061no—
108C22×D4C2×D5311yescharacteristic
109C2×D8—1062no—
110C22⋊Q8—1062no—
111#121—532no—
112#121—1062no—
113#121—1061no—
114C2×C4⋊C4C2×D5311yescharacteristic
115C2×C22⋊C4—21no—
116C2×C22⋊C4—21no—
117#102—1062no—
118C22⋊C8—532no—
119#95C2×D5311yescharacteristic
120C24—21no—
121#41—2122no—
122#41—2122no—
123C2×Q8—1061no—
124C2×D4—22no—
125C2×D4—42no—
126C2×D4—532no—
127C2×D4—1061no—
128C2×D4—1061no—
129C2×D4—1062no—
130C22×C4—21no—
131C22×C4—1062no—
132C22×C4C22⋊D5311yescharacteristic
133C22×C4C22⋊D5311yescharacteristic
134C22×C4C22×D5311yescharacteristic
135D8—2124no—
136C2×C8—1062no—
137C4⋊C4—22no—
138C4⋊C4—532no—
139C4⋊C4—1061no—
140C22⋊C4—42no—
141C22⋊C4—42no—
142C22⋊C4—1062no—
143C22⋊C4—1062no—
144C23—22no—
145C23—41no—
146C23—41no—
147C23—41no—
148C23—41no—
149C23—41no—
150C23—41no—
151C23—1062no—
152C23C2×C22⋊D5311yescharacteristic
153Q8—2122no—
154D4—24no—
155D4—42no—
156D4—4241no—
157D4—2122no—
158D4—2122no—
159D4—2122no—
160D4—2122no—
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—1062no—
170C2×C4—2124no—
171C2×C4#1171992212yescomplemented
172C2×C4D4×D5312yes—
173C8—2122no—
174C22—21no—
175C22—21no—
176C22—42no—
177C22—42no—
178C22—42no—
179C22—81no—
180C22—81no—
181C22—81no—
182C22—81no—
183C22—2122no—
184C22—2122no—
185C22—2122no—
186C22#1313318511yescharacteristic
187C22C2×#1171992211yescharacteristic
188C22#1313324211yescharacteristic
189C4—22no—
190C4—22no—
191C4—41no—
192C4—81no—
193C4—2122no—
194C2—21no—
195C2—21no—
196C2—82no—
197C2—4242no—
198C2#1681075611yescharacteristic
199C2#1681079511yescharacteristic
200C2#1681080911yescharacteristic
subgroupG/H
centerC22#13133185
commutatorC22×C4×C53C23
FrattiniC22×C4C22×D53
FittingC53×#394C2
socleC22×C53#120
hypercenter#394D53
hypocenterC53#1598
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #394

Complements

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

NH ≅ G/Ntype
1. #16811204C2semidirect
2. #16810757C2semidirect
3. #16810356C2semidirect
4. #16810349C2semidirect
5. C53×#394C2semidirect
62. C2×C4×C53D8semidirect
103. #394D53semidirect
106. C53#1598semidirect
171. C2×C4#11719922semidirect