Finite Group Databasebeta

← #20275890

Subgroups — #20275890

Order 6784.

244 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 244 conjugacy classes of subgroups — too many to draw. Only its 63 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#16811176C211yesmaximal, characteristic, complemented
2#16810639C211yesmaximal, characteristic
3C2×#13133143C211yesmaximal, characteristic
4#16810596C211yesmaximal, characteristic
5#16810384C211yesmaximal, characteristic, complemented
6#16810380C211yesmaximal, characteristic, complemented
7C53×#344C211yesmaximal, characteristic, complemented
8C2×#11719945C2211yescharacteristic
9#13133216—21no—
10#13133208—21no—
11#13133194—22no—
12C4×Dic106—22no—
13#13133176C2211yescharacteristic
14#13133176C2211yescharacteristic
15C2×C4×Dic53C2211yescharacteristic
16#13133143—21no—
17#13133143—21no—
18#13133143—22no—
19#13133131—22no—
20C2×M16×C53C2211yescharacteristic
21C2×M16×C53C2211yescharacteristic
22#117×C53C2211yescharacteristic
23C22×C4×C53C2311yescharacteristic
24C2×C22⋊D53—21no—
25C22×Dic53—21no—
26#11719945—21no—
27#11719945—21no—
28#11719945—21no—
29#11719945—21no—
30C2×D212Q811yescharacteristic, complemented
31C2×C4×D53—21no—
32C2×Dic106—22no—
33C2×Dic106Q811yescharacteristic, complemented
34M16×C53—21no—
35M16×C53—21no—
36M16×C53—21no—
37M16×C53—21no—
38C2×C8×C53—21no—
39C2×C8×C53—21no—
40C4⋊C4×C53—21no—
41C22⋊C4×C53D412yes—
42C42×C53—21no—
43C22⋊Dic53—21no—
44C22⋊Dic53—21no—
45C4⋊Dic53—22no—
46C4⋊Dic53—21no—
47C4⋊Dic53—21no—
48#11719919—22no—
49#11719919—22no—
50C4×Dic53—21no—
51C4×Dic53—21no—
52C4×Dic53D411yescharacteristic
53C4×Dic53D411yescharacteristic
54C4×Dic53D412yes—
55C23×C53C2×D411yescharacteristic
56C22×D53—21no—
57C2×C4×C53—22no—
58C2×C4×C53C2×D411yescharacteristic
59C2×C4×C53C2×D411yescharacteristic
60C2×C4×C53C2×Q811yescharacteristic
61C2×C4×C53#4111yescharacteristic
62C2×C4×C53#4111yescharacteristic
63C2×C4×C53#4111yescharacteristic
64C22⋊D53—41no—
65C22⋊D53—41no—
66C2×Dic53—21no—
67C2×Dic53—22no—
68C2×Dic53—21no—
69C2×Dic53—21no—
70C2×Dic53—42no—
71C2×Dic53—41no—
72D212—21no—
73D212—21no—
74C4×D53—41no—
75C4×D53—41no—
76Dic106—21no—
77Dic106—21no—
78Dic106—44no—
79C8×C53—41no—
80C8×C53—41no—
81C22×C53—41no—
82C22×C53#12011yescharacteristic
83C22×C53C22⋊Q811yescharacteristic
84C22×C53#12411yescharacteristic
85C2×D53—41no—
86C2×D53—81no—
87C4×C53—42no—
88C4×C53C22⋊Q811yescharacteristic
89C4×C53C22⋊Q811yescharacteristic
90C4×C53#12411yescharacteristic
91C4×C53#12411yescharacteristic
92Dic53—22no—
93Dic53—42no—
94Dic53—41no—
95Dic53—41no—
96Dic53—81no—
97#2031—531nomaximal, Sylow
98C2×C53—21no—
99C2×C53—21no—
100C2×C53—21no—
101C2×C53#39311yescharacteristic
102D53—81no—
103#533—531no—
104#421—531no—
105C2×#104—531no—
106#410—531no—
107#344D5311yescharacteristic, complemented
108#339—531no—
109H(3,ℤ/4)—531no—
110C53#203111yescharacteristic, complemented, Sylow
111C2×#41—531no—
112C2×M16C2×D5311yescharacteristic
113C2×M16C2×D5311yescharacteristic
114C4⋊Q8—1061no—
115#125—1061no—
116C22⋊Q8—1062no—
117C4×Q8—1062no—
118#117—531no—
119#117—531no—
120#117C2×D5311yescharacteristic
121C2×C42—531no—
122#104—1061no—
123#104—1061no—
124#104—1062no—
125#99—1062no—
126#41—1061no—
127#41—1061no—
128#41—1061no—
129#41—1061no—
130C2×Q8—531no—
131C2×Q8—1062no—
132C2×D4—531no—
133C2×D4—1061no—
134C22×C4—1061no—
135C22×C4—1061no—
136C22×C4C22×D5311yescharacteristic
137M16—21no—
138M16—21no—
139M16—21no—
140M16—21no—
141C2×C8—21no—
142C2×C8—21no—
143C4⋊C4—21no—
144C4⋊C4—1062no—
145C4⋊C4—1062no—
146C4⋊C4—1062no—
147C4⋊C4—1061no—
148C4⋊C4—1061no—
149C22⋊C4—1061no—
150C22⋊C4—1061no—
151C22⋊C4D21212yes—
152C42—21no—
153C42—531no—
154C42—531no—
155C42—532no—
156C42—1061no—
157C42—1061no—
158C23—1061no—
159C23C2×D21211yescharacteristic
160Q8—1061no—
161Q8—1061no—
162Q8—2124no—
163D4—1061no—
164D4—1061no—
165D4—2121no—
166D4—2121no—
167C2×C4—22no—
168C2×C4—1061no—
169C2×C4—1061no—
170C2×C4—1062no—
171C2×C4—1061no—
172C2×C4—2122no—
173C2×C4—2121no—
174C2×C4—2121no—
175C2×C4—2121no—
176C2×C4#1171994511yescharacteristic
177C2×C4#1171994511yescharacteristic
178C2×C4D4×D5311yescharacteristic
179C2×C4D4×D5311yescharacteristic
180C2×C4#1171994711yescharacteristic
181C2×C4Q8×D5311yescharacteristic
182C8—41no—
183C8—41no—
184C22—41no—
185C22—4241no—
186C22—2121no—
187C22#1313318711yescharacteristic
188C22#1313319711yescharacteristic
189C22#1313320511yescharacteristic
190C4—42no—
191C4—1062no—
192C4—4241no—
193C4—2121no—
194C4—2122no—
195C4—2121no—
196C4#1313319511yescharacteristic
197C4#1313319511yescharacteristic
198C4#1313320311yescharacteristic
199C4#1313320311yescharacteristic
200C2—21no—
201C2—21no—
202C2—21no—
203C2—4241no—
204C2#1681053611yescharacteristic
subgroupG/H
centerC4#13133203
commutatorC22×C4×C53C23
FrattiniC22×C4C22×D53
FittingC53×#344C2
socleC2×C53#393
hypercenter#344D53
hypocenterC53#2031
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C4 ◁ C22×C4 ◁ #344

Complements

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

NH ≅ G/Ntype
1. #16811176C2semidirect
5. #16810384C2semidirect
6. #16810380C2semidirect
7. C53×#344C2semidirect
30. C2×D212Q8semidirect
33. C2×Dic106Q8semidirect
107. #344D53semidirect
110. C53#2031semidirect