Finite Group Databasebeta

← #20275888

Subgroups — #20275888

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#16811180C211yesmaximal, characteristic, complemented
2#16810639C211yesmaximal, characteristic
3C2×#13133145C211yesmaximal, characteristic
4#16810596C211yesmaximal, characteristic
5#16810394C211yesmaximal, characteristic, complemented
6#16810379C211yesmaximal, characteristic, complemented
7C53×#339C211yesmaximal, characteristic, complemented
8C2×#11719945C2211yescharacteristic
9C2×#11719917C2211yescharacteristic
10#13133217—21no—
11#13133209—21no—
12#13133196—22no—
13C4×Dic106—22no—
14#13133176C2211yescharacteristic
15#13133176C2211yescharacteristic
16#13133145—21no—
17#13133145—21no—
18#13133143—22no—
19#13133131—22no—
20C2×M16×C53C2211yescharacteristic
21#117×C53C2211yescharacteristic
22C2×C42×C53C2211yescharacteristic
23C22×C4×C53—21no—
24C22×C4×C53C2311yescharacteristic
25C2×C22⋊D53—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—
36C2×C8×C53—21no—
37C4⋊C4×C53—21no—
38C22⋊C4×C53—21no—
39C42×C53—21no—
40C42×C53—21no—
41C42×C53D411yescharacteristic
42C42×C53D411yescharacteristic
43C22⋊Dic53—21no—
44C22⋊Dic53D412yes—
45C4⋊Dic53—22no—
46C4⋊Dic53—21no—
47C4⋊Dic53—21no—
48#11719919—24no—
49C4×Dic53—21no—
50C4×Dic53D412yes—
51#11719917—21no—
52#11719917—21no—
53C2×#88322—21no—
54C23×C53C2×D411yescharacteristic
55C22×D53—21no—
56C2×C4×C53—21no—
57C2×C4×C53—21no—
58C2×C4×C53—42no—
59C2×C4×C53—41no—
60C2×C4×C53C2×D411yescharacteristic
61C2×C4×C53C2×D411yescharacteristic
62C2×C4×C53C2×Q811yescharacteristic
63C2×C4×C53#4111yescharacteristic
64C2×C4×C53#4111yescharacteristic
65C2×C4×C53#4111yescharacteristic
66C22⋊D53—41no—
67C22⋊D53—41no—
68C2×Dic53—22no—
69C2×Dic53—22no—
70C2×Dic53—21no—
71D212—21no—
72D212—21no—
73C4×D53—41no—
74C4×D53—41no—
75Dic106—21no—
76Dic106—21no—
77Dic106—44no—
78C8×C53—41no—
79#88322—41no—
80C22×C53—41no—
81C22×C53#12011yescharacteristic
82C22×C53C22⋊Q811yescharacteristic
83C22×C53#12411yescharacteristic
84C2×D53—41no—
85C2×D53—81no—
86C4×C53—22no—
87C4×C53—41no—
88C4×C53—81no—
89C4×C53C22⋊Q811yescharacteristic
90C4×C53C22⋊Q811yescharacteristic
91C4×C53#12411yescharacteristic
92C4×C53#12411yescharacteristic
93Dic53—42no—
94Dic53—42no—
95Dic53—41no—
96#2031—531nomaximal, Sylow
97C2×C53—21no—
98C2×C53—21no—
99C2×C53—21no—
100C2×C53#39311yescharacteristic
101D53—81no—
102#533—531no—
103#421—531no—
104C2×#104—531no—
105#410—531no—
106#344—531no—
107#339D5311yescharacteristic, complemented
108H(3,ℤ/4)—531no—
109C53#203111yescharacteristic, complemented, Sylow
110C2×#41—531no—
111C2×M16—531no—
112C2×M16C2×D5311yescharacteristic
113C4⋊Q8—1061no—
114#125—1061no—
115C22⋊Q8—1062no—
116C4×Q8—1062no—
117#117—531no—
118#117—531no—
119#117C2×D5311yescharacteristic
120C2×C42C2×D5311yescharacteristic
121#104—1062no—
122#104—1061no—
123#104—1061no—
124#99—1062no—
125#41—1061no—
126#41—1061no—
127#41—1061no—
128#41—1061no—
129C2×Q8—531no—
130C2×Q8—1062no—
131C2×D4—531no—
132C2×D4—1061no—
133C22×C4—21no—
134C22×C4—1061no—
135C22×C4C22×D5311yescharacteristic
136M16—21no—
137M16—21no—
138M16—1061no—
139M16—1061no—
140C2×C8—21no—
141C2×C8—1061no—
142C4⋊C4—21no—
143C4⋊C4—1064no—
144C4⋊C4—1062no—
145C4⋊C4—1061no—
146C4⋊C4—1061no—
147C22⋊C4—21no—
148C22⋊C4—532no—
149C22⋊C4—1061no—
150C42—21no—
151C42—21no—
152C42—532no—
153C42—1061no—
154C42D21211yescharacteristic
155C42D21211yescharacteristic
156C23—1061no—
157C23D4×D5311yescharacteristic
158Q8—1061no—
159Q8—1061no—
160Q8—2124no—
161D4—1061no—
162D4—1061no—
163D4—2121no—
164D4—2121no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—42no—
168C2×C4—41no—
169C2×C4—1062no—
170C2×C4—1062no—
171C2×C4—1061no—
172C2×C4—2121no—
173C2×C4—2121no—
174C2×C4C2×D21211yescharacteristic
175C2×C4#1171994511yescharacteristic
176C2×C4#1171994511yescharacteristic
177C2×C4D4×D5311yescharacteristic
178C2×C4#1171994711yescharacteristic
179C2×C4Q8×D5311yescharacteristic
180C8—41no—
181C8—2121no—
182C22—41no—
183C22—4241no—
184C22—2121no—
185C22#1313318711yescharacteristic
186C22#1313319511yescharacteristic
187C22#1313320311yescharacteristic
188C4—22no—
189C4—41no—
190C4—81no—
191C4—2122no—
192C4—2122no—
193C4—2121no—
194C4#1313319511yescharacteristic
195C4#1313319711yescharacteristic
196C4#1313320311yescharacteristic
197C4#1313320511yescharacteristic
198C2—21no—
199C2—21no—
200C2—21no—
201C2—4241no—
202C2#1681053611yescharacteristic
subgroupG/H
centerC4#13133203
commutatorC22×C4×C53C23
FrattiniC22×C4C22×D53
FittingC53×#339C2
socleC2×C53#393
hypercenter#339D53
hypocenterC53#2031
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C4 ◁ C22×C4 ◁ #339

Complements

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

NH ≅ G/Ntype
1. #16811180C2semidirect
5. #16810394C2semidirect
6. #16810379C2semidirect
7. C53×#339C2semidirect
30. C2×D212Q8semidirect
33. C2×Dic106Q8semidirect
107. #339D53semidirect
109. C53#2031semidirect