Finite Group Databasebeta

← #20290756

Subgroups — #20290756

Order 6784.

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

normal subgroups only

This group has 312 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#16811701C211yesmaximal, characteristic
2#16811616C212yesmaximal, complemented
3#16811603C211yesmaximal, characteristic
4#16811490C212yesmaximal, complemented
5#16811193C211yesmaximal, characteristic
6#13133304—24no—
7D53⋊M16C2212yes—
8C2×#11719940—22no—
9#13133296C2212yes—
10#13133280—24no—
11#13133275—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⋊D53D412yes—
24C22×Dic53D412yes—
25Q8×D53—22no—
26Q8×D53—22no—
27#11719947—22no—
28#11719947—22no—
29C2×C4×D53—21no—
30C2×C4×D53C2311yescharacteristic
31C2×Dic106—21no—
32C2×Dic106C2×C411yescharacteristic
33C2×Dic106C2×C411yescharacteristic
34C22⋊#6157—22no—
35#11719940—24no—
36#11719940—22no—
37C2×#88324—24no—
38C4⋊#6157—22no—
39C4×#6157—22no—
40#11719936—22no—
41D53⋊C8D414yes—
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×C53C22⋊C412yes—
51C22×D53C2×D411yescharacteristic
52C2×#6157—42no—
53Q8×C53—22no—
54D4×C53—22no—
55C2×C4×C53—21no—
56C2×C4×C53—21no—
57C2×C4×C53C22×C411yescharacteristic
58C22⋊D53—24no—
59C2×Dic53—22no—
60C2×Dic53—22no—
61C2×Dic53—21no—
62C2×Dic53—21no—
63C2×Dic53C22⋊C412yes—
64C2×Dic53C2×D411yescharacteristic
65C4×D53—42no—
66C4×D53C2×D412yes—
67C4×D53#4112yes—
68Dic106—22no—
69Dic106—22no—
70Dic106—41no—
71Dic106—41no—
72#88324—24no—
73C22×C53—22no—
74C22×C53—42no—
75C22×C53C2×C22⋊C411yescharacteristic
76C2×D53—41no—
77C2×D53#12112yes—
78#6157—82no—
79C4×C53—41no—
80C4×C53—41no—
81C4×C53C4×D412yes—
82Dic53—22no—
83Dic53—41no—
84Dic53—41no—
85Dic53#12011yescharacteristic
86Dic53#12311yescharacteristic
87#1909—531nomaximal, Sylow
88C2×C53—21no—
89C2×C53—42no—
90C2×C53#38611yescharacteristic
91D53—22no—
92#568—531no—
93#536—531no—
94#419—532no—
95#410—531no—
96H(3,ℤ/4)—532no—
97C53#190911yescharacteristic, complemented, Sylow
98C2×#41—531no—
99C22×Q8—531no—
100#131—1064no—
101C2×M16—532no—
102C2×M16—1062no—
103C4⋊Q8—531no—
104C4⋊Q8—531no—
105#124—531no—
106#124#615711yescharacteristic, complemented
107#123—1062no—
108C22⋊Q8—1062no—
109#117—531no—
110#117—532no—
111#103—1064no—
112#99—1062no—
113#41—1062no—
114#41—1062no—
115C2×Q8—531no—
116C2×Q8—531no—
117C2×Q8—1061no—
118C2×Q8—1062no—
119C2×Q8—1062no—
120C2×Q8C2×#615711yescharacteristic
121C2×D4—532no—
122C2×D4C2×#615711yescharacteristic
123C22×C4—531no—
124C22×C4—532no—
125C22×C4—1061no—
126M16—1062no—
127M16—1062no—
128M16—1064no—
129C2×C8—534no—
130C2×C8—1064no—
131C4⋊C4—1061no—
132C4⋊C4—1062no—
133C4⋊C4—1062no—
134C4⋊C4—1062no—
135C22⋊C4—22no—
136C22⋊C4—1061no—
137C22⋊C4—1062no—
138C22⋊C4—1062no—
139C42—531no—
140C42—1062no—
141C42C2×#615711yescharacteristic
142C23—531no—
143C23C22⋊#615712yes—
144Q8—22no—
145Q8—1062no—
146Q8—1062no—
147Q8—2121no—
148Q8—2121no—
149D4—22no—
150D4—1064no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—531no—
154C2×C4—532no—
155C2×C4—532no—
156C2×C4—532no—
157C2×C4—1062no—
158C2×C4—1062no—
159C2×C4—1061no—
160C2×C4—1061no—
161C2×C4—2122no—
162C2×C4—2122no—
163C2×C4C22×#615711yescharacteristic
164C8—1064no—
165C22—22no—
166C22—42no—
167C22—532no—
168C22—2121no—
169C22C2×C22⋊#615711yescharacteristic
170C4—41no—
171C4—41no—
172C4—531no—
173C4—531no—
174C4—1062no—
175C4—4242no—
176C4—2121no—
177C4—2121no—
178C4D4×#615712yes—
179C2—21no—
180C2—42no—
181C2—1062no—
182C2#1681157111yescharacteristic
subgroupG/H
centerC2#16811571
commutatorC2×C4×C53C22×C4
FrattiniC2×C4C22×#6157
Fitting#124×C53C4
socleC2×C53#386
hypercenter#124#6157
hypocenterC53#1909
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
2. #16811616C2semidirect
4. #16811490C2semidirect
14. #13133217C4semidirect
15. #13133215C4semidirect
16. #13133204C4semidirect
20. #124×C53C4semidirect
97. C53#1909semidirect
106. #124#6157semidirect