Finite Group Databasebeta

← #20290763

Subgroups — #20290763

Order 6784.

314 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 66 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#16811691C211yesmaximal, characteristic, complemented
2#16811621C212yesmaximal
3D53⋊#101C211yesmaximal, characteristic, complemented
4#16811490C212yesmaximal, complemented
5#16811193C211yesmaximal, characteristic
6D53⋊M16C2212yes—
7Q8×#6157—24no—
8#13133296—22no—
9#13133296C2212yes—
10#13133281—24no—
11#13133278—22no—
12C2×#11719947C2211yescharacteristic
13C2×Q8×D53C2211yescharacteristic
14#13133217C411yescharacteristic, complemented
15#13133215C411yescharacteristic, complemented
16#13133204C411yescharacteristic, complemented
17#13133199—22no—
18#13133194—22no—
19#13133174C2211yescharacteristic
20#124×C53C411yescharacteristic, complemented
21C2×Q8×C53C2×C411yescharacteristic
22C2×D4×C53C2×C411yescharacteristic
23C2×C22⋊D53—21no—
24C22×Dic53—21no—
25Q8×D53—22no—
26Q8×D53—22no—
27#11719947—22no—
28#11719947—22no—
29C2×C4×D53D412yes—
30C2×C4×D53C2311yescharacteristic
31C2×Dic106C2×C411yescharacteristic
32C2×Dic106C2×C411yescharacteristic
33C2×Dic106D412yes—
34C22⋊#6157—22no—
35#11719940—22no—
36C4⋊#6157—24no—
37C4⋊#6157—22no—
38C4×#6157—24no—
39C4×#6157D414yes—
40#11719936—22no—
41D53⋊C8—22no—
42C22⋊C4×C53—22no—
43C42×C53C2×C411yescharacteristic
44C22⋊Dic53—22no—
45#11719921—21no—
46C4⋊Dic53—22no—
47#11719919—22no—
48#11719919—21no—
49C4×Dic53C2×C411yescharacteristic
50C23×C53—21no—
51C22×D53C2×D411yescharacteristic
52C2×#6157—24no—
53Q8×C53—22no—
54D4×C53—22no—
55C2×C4×C53—21no—
56C2×C4×C53C22⋊C412yes—
57C2×C4×C53C22×C411yescharacteristic
58C22⋊D53—41no—
59C22⋊D53—41no—
60C2×Dic53—21no—
61C2×Dic53—21no—
62C2×Dic53—42no—
63C2×Dic53C22⋊C412yes—
64C2×Dic53C2×D411yescharacteristic
65C4×D53—22no—
66C4×D53—22no—
67C4×D53C2×D412yes—
68C4×D53#4112yes—
69Dic106—22no—
70Dic106—24no—
71Dic106—22no—
72#88324—42no—
73C22×C53—42no—
74C22×C53—41no—
75C22×C53C2×C22⋊C411yescharacteristic
76C2×D53—41no—
77C2×D53#12011yescharacteristic
78C2×D53#12311yescharacteristic
79#6157—44no—
80C4×C53—22no—
81C4×C53—41no—
82C4×C53C4×D412yes—
83Dic53—22no—
84Dic53—41no—
85Dic53—41no—
86Dic53#12112yes—
87#1912—531nomaximal, Sylow
88C2×C53—21no—
89C2×C53—81no—
90C2×C53#38611yescharacteristic
91D53—21no—
92D53—21no—
93#536—531no—
94#519—531no—
95#421—532no—
96C2×#101—531no—
97H(3,ℤ/4)—532no—
98C53#191211yescharacteristic, complemented, Sylow
99C2×#41—531no—
100C22×Q8—531no—
101C2×M16—532no—
102C4⋊Q8—531no—
103C4⋊Q8—531no—
104#124—531no—
105#124#615711yescharacteristic, complemented
106#123—1062no—
107C22⋊Q8—1062no—
108C4×Q8—1064no—
109#117—532no—
110#117—531no—
111#117—1062no—
112#104—1064no—
113#101—1062no—
114#41—1062no—
115#41—1062no—
116C2×Q8—532no—
117C2×Q8—531no—
118C2×Q8—531no—
119C2×Q8—1062no—
120C2×Q8—1062no—
121C2×Q8C2×#615711yescharacteristic
122C2×D4—1061no—
123C2×D4C2×#615711yescharacteristic
124C22×C4—531no—
125C22×C4—532no—
126C22×C4—1061no—
127M16—1062no—
128M16—1062no—
129C2×C8—1062no—
130C4⋊C4—1061no—
131C4⋊C4—1062no—
132C4⋊C4—1062no—
133C4⋊C4—1064no—
134C4⋊C4—1062no—
135C22⋊C4—22no—
136C22⋊C4—1061no—
137C22⋊C4—1062no—
138C22⋊C4—1062no—
139C42—534no—
140C42—531no—
141C42—1064no—
142C42C2×#615711yescharacteristic
143C23—21no—
144C23—531no—
145Q8—22no—
146Q8—1064no—
147Q8—1062no—
148Q8—1062no—
149D4—22no—
150D4—2121no—
151D4—2121no—
152C2×C4—21no—
153C2×C4—531no—
154C2×C4—532no—
155C2×C4—532no—
156C2×C4—532no—
157C2×C4—1062no—
158C2×C4—1062no—
159C2×C4—1064no—
160C2×C4—1061no—
161C2×C4—1061no—
162C2×C4—2122no—
163C2×C4C22⋊#615712yes—
164C2×C4C22×#615711yescharacteristic
165C8—2122no—
166C22—42no—
167C22—41no—
168C22—531no—
169C22—531no—
170C22—2121no—
171C22C2×C22⋊#615711yescharacteristic
172C4—22no—
173C4—41no—
174C4—532no—
175C4—1062no—
176C4—2121no—
177C4—2124no—
178C4—2121no—
179C4D4×#615712yes—
180C2—21no—
181C2—81no—
182C2—1061no—
183C2—1061no—
184C2#1681157111yescharacteristic
subgroupG/H
centerC2#16811571
commutatorC2×C4×C53C22×C4
FrattiniC2×C4C22×#6157
Fitting#124×C53C4
socleC2×C53#386
hypercenter#124#6157
hypocenterC53#1912
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ #124

Complements

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

NH ≅ G/Ntype
1. #16811691C2semidirect
3. D53⋊#101C2semidirect
4. #16811490C2semidirect
14. #13133217C4semidirect
15. #13133215C4semidirect
16. #13133204C4semidirect
20. #124×C53C4semidirect
98. C53#1912semidirect
105. #124#6157semidirect