Finite Group Databasebeta

← #20275117

Subgroups — #20275117

Order 6784.

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

normal subgroups only

This group has 344 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×#13133205C211yesmaximal, characteristic
2#16811100C211yesmaximal, characteristic, complemented
3#16810629C211yesmaximal, characteristic
4#16810627C211yesmaximal, characteristic
5C2×#13133135C211yesmaximal, characteristic, complemented
6#16810383C211yesmaximal, characteristic, complemented
7C53×#339C211yesmaximal, characteristic, complemented
8C22×Dic106C2211yescharacteristic
9C22×D212C2211yescharacteristic
10C2×#11719917C2211yescharacteristic
11#13133205—22no—
12#13133205C414yescomplemented
13C4×Dic106—22no—
14#13133182—22no—
15#13133176C2211yescharacteristic
16#13133173—22no—
17C2×#11719921—22no—
18#13133139—22no—
19#13133137—22no—
20#13133135—22no—
21C2×M16×C53C2211yescharacteristic
22#117×C53C2211yescharacteristic
23C2×C42×C53C2211yescharacteristic
24C23×D53—21no—
25C22×C4×C53—21no—
26C22×C4×C53C2311yescharacteristic
27C22×Dic53D412yes—
28C2×D212—21no—
29C2×D212—21no—
30C2×D212—21no—
31C2×D212—21no—
32C2×D212—21no—
33C2×D212C2×C412yes—
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—
46C42×C53—21no—
47C42×C53C2×C412yes—
48C22⋊Dic53—21no—
49#11719921—28no—
50C4⋊Dic53D412yes—
51#11719919—24no—
52C4×Dic53—24no—
53C4×Dic53—21no—
54#11719917—21no—
55#11719917—21no—
56C2×#88322—21no—
57C23×C53C2×D411yescharacteristic
58C22×D53—22no—
59C22×D53—44no—
60C22×D53—41no—
61C22×D53—41no—
62C2×C4×C53—22no—
63C2×C4×C53—22no—
64C2×C4×C53—42no—
65C2×C4×C53C22×C411yescharacteristic
66C2×C4×C53C2×D411yescharacteristic
67C2×C4×C53C2×D411yescharacteristic
68C2×C4×C53C2×D411yescharacteristic
69C2×C4×C53#4111yescharacteristic
70C2×C4×C53#4111yescharacteristic
71C2×Dic53—22no—
72C2×Dic53—22no—
73C2×Dic53—22no—
74C2×Dic53C22⋊C414yes—
75D212—22no—
76D212—22no—
77D212—41no—
78D212—41no—
79Dic106—22no—
80Dic106—22no—
81Dic106—22no—
82Dic106—22no—
83C8×C53—41no—
84#88322—41no—
85C22×C53—22no—
86C22×C53C2×C22⋊C411yescharacteristic
87C22×C53#12011yescharacteristic
88C22×C53#12311yescharacteristic
89C2×D53—44no—
90C2×D53—42no—
91C2×D53—82no—
92C2×D53—82no—
93C4×C53—42no—
94C4×C53—42no—
95C4×C53C4×D411yescharacteristic
96C4×C53C4×D411yescharacteristic
97C4×C53#12111yescharacteristic
98C4×C53#12111yescharacteristic
99Dic53—24no—
100Dic53—42no—
101#1888—531nomaximal, Sylow
102C2×C53—21no—
103C2×C53—21no—
104C2×C53#38611yescharacteristic
105C2×C53#45412yes—
106D53—82no—
107C2×#124—531no—
108#519—531no—
109#418—531no—
110#418—531no—
111C2×#101—531no—
112#339—531no—
113#339D5311yescharacteristic, complemented
114C53#188811yescharacteristic, complemented, Sylow
115C22×Q8—531no—
116C22×D4—531no—
117C2×M16—531no—
118C2×M16C2×D5311yescharacteristic
119#124—534no—
120#124—1062no—
121C4×Q8—1062no—
122C4×Q8—1062no—
123#117—531no—
124#117—1062no—
125#117C2×D5311yescharacteristic
126C2×C22⋊C4—1062no—
127C2×C42C2×D5311yescharacteristic
128#102—1062no—
129#102—1062no—
130#101—1062no—
131C24—1061no—
132C2×Q8—532no—
133C2×Q8—532no—
134C2×Q8—1061no—
135C2×Q8—1061no—
136C2×Q8—1061no—
137C2×Q8—1061no—
138C2×D4—532no—
139C2×D4—1061no—
140C2×D4—1061no—
141C2×D4—1061no—
142C2×D4—1061no—
143C2×D4—1061no—
144C22×C4—21no—
145C22×C4—532no—
146C22×C4C22×D5311yescharacteristic
147M16—21no—
148M16—21no—
149M16—1061no—
150M16—1061no—
151C2×C8—21no—
152C2×C8—1061no—
153C4⋊C4—532no—
154C4⋊C4—1064no—
155C4⋊C4D21212yes—
156C22⋊C4—21no—
157C22⋊C4—1068no—
158C22⋊C4—1061no—
159C42—21no—
160C42—21no—
161C42—1064no—
162C42—1061no—
163C42C4×D5312yes—
164C23—1062no—
165C23—2124no—
166C23—2121no—
167C23—2121no—
168C23D4×D5311yescharacteristic
169Q8—1062no—
170Q8—1062no—
171Q8—1062no—
172Q8—1062no—
173D4—1062no—
174D4—1062no—
175D4—2121no—
176D4—2121no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—42no—
180C2×C4—534no—
181C2×C4—1062no—
182C2×C4—1062no—
183C2×C4—1062no—
184C2×C4C2×C4×D5311yescharacteristic
185C2×C4C2×D21211yescharacteristic
186C2×C4#1171994511yescharacteristic
187C2×C4D4×D5311yescharacteristic
188C2×C4D4×D5311yescharacteristic
189C2×C4#1171994911yescharacteristic
190C8—41no—
191C8—2121no—
192C22—22no—
193C22—4242no—
194C22—4242no—
195C22—2124no—
196C22—2122no—
197C22C22⋊C4×D5311yescharacteristic
198C22#1313318711yescharacteristic
199C22#1313320211yescharacteristic
200C4—42no—
201C4—42no—
202C4—1064no—
203C4—2122no—
204C4#1313317911yescharacteristic
205C4C4×D21211yescharacteristic
206C4#1313319011yescharacteristic
207C4#1313319211yescharacteristic
208C2—21no—
209C2—21no—
210C2—4242no—
211C2#1681050111yescharacteristic
212C2#1681079712yes—
subgroupG/H
centerC22#13133187
commutatorC2×C4×C53C22×C4
FrattiniC22×C4C22×D53
FittingC53×#339C2
socleC22×C53#120
hypercenter#339D53
hypocenterC53#1888
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. #16811100C2semidirect
5. C2×#13133135C2semidirect
6. #16810383C2semidirect
7. C53×#339C2semidirect
12. #13133205C4semidirect
113. #339D53semidirect
114. C53#1888semidirect