Finite Group Databasebeta

← #20274942

Subgroups — #20274942

Order 6784.

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

normal subgroups only

This group has 302 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
1#16811103C211yesmaximal, characteristic
2#16810666C211yesmaximal, characteristic, complemented
3C2×#13133145C211yesmaximal, characteristic
4#16810625C211yesmaximal, characteristic
5#16810493C211yesmaximal, characteristic, complemented
6#16810391C211yesmaximal, characteristic, complemented
7C53×#339C211yesmaximal, characteristic, complemented
8C2×#11719945C2211yescharacteristic
9C2×#11719917C2211yescharacteristic
10C4×Dic106—21no—
11C4×D212—21no—
12#13133177—22no—
13#13133175—21no—
14C2×C4⋊Dic53—22no—
15C2×C4⋊Dic53C2211yescharacteristic
16C2×C4⋊Dic53C2211yescharacteristic
17C4⋊C4×D53—21no—
18#13133150—21no—
19#13133148—21no—
20#13133145C414yescomplemented
21#13133142—21no—
22#13133139—21no—
23C2×M16×C53C2211yescharacteristic
24#117×C53C2211yescharacteristic
25C2×C42×C53C2211yescharacteristic
26C22×C4×C53—21no—
27C22×C4×C53C2311yescharacteristic
28C2×C22⋊D53—21no—
29C22×Dic53—21no—
30C22×Dic53D412yes—
31#11719945—21no—
32#11719945—21no—
33#11719945C2×C412yes—
34#11719945Q812yes—
35C2×D212D411yescharacteristic
36C2×C4×D53—21no—
37C2×C4×D53—22no—
38C2×Dic106D411yescharacteristic
39M16×C53—21no—
40M16×C53—21no—
41C2×C8×C53—21no—
42C4⋊C4×C53D411yescharacteristic
43C4⋊C4×C53D411yescharacteristic
44C22⋊C4×C53—21no—
45C42×C53—21no—
46C42×C53—21no—
47C42×C53C2×C412yescomplemented
48#11719921—22no—
49C4⋊Dic53—21no—
50C4⋊Dic53—21no—
51C4⋊Dic53—21no—
52C4⋊Dic53—21no—
53C4⋊Dic53—44no—
54#11719919—22no—
55C4×Dic53—22no—
56#11719917—21no—
57#11719917C2×C412yescomplemented
58C2×#88322—21no—
59C23×C53C2×D411yescharacteristic
60C22×D53—21no—
61C2×C4×C53—22no—
62C2×C4×C53—22no—
63C2×C4×C53—22no—
64C2×C4×C53—21no—
65C2×C4×C53—21no—
66C2×C4×C53C22×C411yescharacteristic
67C2×C4×C53C2×D411yescharacteristic
68C2×C4×C53C2×D411yescharacteristic
69C2×C4×C53C2×Q811yescharacteristic
70C2×C4×C53#4111yescharacteristic
71C2×C4×C53#4111yescharacteristic
72C22⋊D53—22no—
73C22⋊D53—22no—
74C2×Dic53—21no—
75C2×Dic53—22no—
76C2×Dic53—42no—
77C2×Dic53—42no—
78C2×Dic53—41no—
79D212—21no—
80D212C4⋊C412yescomplemented
81C4×D53—22no—
82C4×D53—22no—
83C4×D53—82no—
84Dic106—21no—
85Dic106C4⋊C412yescomplemented
86C8×C53—41no—
87#88322—22no—
88C22×C53—21no—
89C22×C53—21no—
90C22×C53C4×D411yescharacteristic
91C22×C53#12011yescharacteristic
92C22×C53C22⋊Q811yescharacteristic
93C2×D53—22no—
94C2×D53—41no—
95C2×D53—41no—
96C4×C53—24no—
97C4×C53—41no—
98C4×C53—41no—
99C4×C53C2×C4⋊C411yescharacteristic
100C4×C53C4×D411yescharacteristic
101C4×C53C22⋊Q811yescharacteristic
102C4×C53#12311yescharacteristic
103Dic53—22no—
104Dic53—42no—
105Dic53—81no—
106#1859—531nomaximal, Sylow
107C2×C53—21no—
108C2×C53—21no—
109C2×C53#38511yescharacteristic
110C2×C53#45311yescharacteristic
111C2×C53#45611yescharacteristic
112D53—42no—
113#520—531no—
114#428—531no—
115C2×#104—531no—
116#417—531no—
117#384—531no—
118#343—531no—
119#339D5311yescharacteristic, complemented
120C53#185911yescharacteristic, complemented, Sylow
121C2×#41—531no—
122C2×M16—531no—
123C2×M16C2×D5311yescharacteristic
124C4×Q8—1061no—
125C4×D4—1061no—
126C4×D4—1062no—
127#117—1061no—
128#117C2×D5311yescharacteristic
129C2×C4⋊C4—531no—
130C2×C4⋊C4—531no—
131C2×C4⋊C4—1061no—
132C2×C4⋊C4—1062no—
133C2×C42C2×D5311yescharacteristic
134#107—1061no—
135#106—1061no—
136#104—534no—
137#103—1061no—
138#102—1061no—
139#41—532no—
140#41—532no—
141#41—1061no—
142#41—1061no—
143C2×Q8—531no—
144C2×D4—531no—
145C2×D4—1061no—
146C22×C4—21no—
147C22×C4—532no—
148C22×C4—1062no—
149C22×C4—1061no—
150C22×C4—1061no—
151C22×C4C22×D5311yescharacteristic
152M16—21no—
153M16—21no—
154M16—532no—
155M16—1061no—
156C2×C8—21no—
157C2×C8—1061no—
158C4⋊C4—1062no—
159C4⋊C4—1061no—
160C4⋊C4—1061no—
161C4⋊C4—1061no—
162C4⋊C4—1061no—
163C4⋊C4—2124no—
164C4⋊C4D21211yescharacteristic
165C4⋊C4D21211yescharacteristic
166C22⋊C4—21no—
167C22⋊C4—1062no—
168C42—21no—
169C42—21no—
170C42—1062no—
171C42C4×D5312yescomplemented
172C23—1061no—
173C23D4×D5311yescharacteristic
174Q8—532no—
175Q8—1061no—
176D4—532no—
177D4—1062no—
178D4—1062no—
179D4—1061no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—1061no—
186C2×C4—1062no—
187C2×C4—1062no—
188C2×C4—1062no—
189C2×C4—4242no—
190C2×C4—2122no—
191C2×C4—2121no—
192C2×C4—2122no—
193C2×C4C2×C4×D5311yescharacteristic
194C2×C4C2×D21211yescharacteristic
195C2×C4#1171994511yescharacteristic
196C2×C4D4×D5311yescharacteristic
197C2×C4#1171994711yescharacteristic
198C2×C4Q8×D5311yescharacteristic
199C8—41no—
200C8—1062no—
201C22—21no—
202C22—21no—
203C22—1062no—
204C22—2121no—
205C22—2121no—
206C22#1313317711yescharacteristic
207C22#1313318711yescharacteristic
208C22#1313319511yescharacteristic
209C4—24no—
210C4—41no—
211C4—41no—
212C4—1062no—
213C4—4241no—
214C4—2122no—
215C4C4⋊C4×D5311yescharacteristic
216C4C4×D21211yescharacteristic
217C4#1313319711yescharacteristic
218C4#1313319911yescharacteristic
219C2—21no—
220C2—21no—
221C2—2122no—
222C2#1681049711yescharacteristic
223C2#1681079211yescharacteristic
224C2#1681080711yescharacteristic
subgroupG/H
centerC22#13133187
commutatorC2×C4×C53C22×C4
FrattiniC22×C4C22×D53
FittingC53×#339C2
socleC22×C53#120
hypercenter#339D53
hypocenterC53#1859
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #339

Complements

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

NH ≅ G/Ntype
2. #16810666C2semidirect
5. #16810493C2semidirect
6. #16810391C2semidirect
7. C53×#339C2semidirect
20. #13133145C4semidirect
47. C42×C53C2×C4semidirect
57. #11719917C2×C4semidirect
80. D212C4⋊C4semidirect
85. Dic106C4⋊C4semidirect
119. #339D53semidirect
120. C53#1859semidirect
171. C42C4×D53semidirect