Finite Group Databasebeta

← #20276388

Subgroups — #20276388

Order 6784.

238 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 55 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#16811193C211yesmaximal, characteristic
2#16810595C212yesmaximal
3#16810418C211yesmaximal, characteristic, complemented
4#16810417C212yesmaximal, complemented
5C53×#354C211yesmaximal, characteristic, complemented
6C2×#11719947C2211yescharacteristic
7C2×Q8×D53C411yescharacteristic, complemented
8#13133217C411yescharacteristic, complemented
9#13133215C411yescharacteristic, complemented
10#13133204C2211yescharacteristic
11#13133199—22no—
12#13133194—22no—
13#13133174C411yescharacteristic, complemented
14#13133173—22no—
15#13133132C2212yes—
16#13133131—22no—
17#124×C53C2211yescharacteristic
18#99×C53C2212yes—
19C2×Q8×C53C2×C411yescharacteristic
20C2×D4×C53C2311yescharacteristic
21C2×C22⋊D53D412yes—
22C22×Dic53D412yes—
23Q8×D53—42no—
24#11719947—42no—
25C2×C4×D53—21no—
26C2×C4×D53C2×C411yescharacteristic
27C2×Dic106—21no—
28C2×Dic106C2×C411yescharacteristic
29C2×Dic106C2×C411yescharacteristic
30C22⋊C4×C53—22no—
31C22⋊C4×C53—22no—
32C42×C53C2×C411yescharacteristic
33C22⋊Dic53—22no—
34C22⋊Dic53—22no—
35#11719921—21no—
36C4⋊Dic53—22no—
37#11719919—21no—
38#11719919—22no—
39#11719919—42no—
40C4×Dic53—42no—
41C4×Dic53C2×C411yescharacteristic
42C23×C53C2×D412yes—
43C22×D53C22⋊C411yescharacteristic
44Q8×C53—41no—
45D4×C53—41no—
46C2×C4×C53—21no—
47C2×C4×C53—21no—
48C2×C4×C53—42no—
49C2×C4×C53C22×C411yescharacteristic
50C22⋊D53—42no—
51C2×Dic53—22no—
52C2×Dic53—22no—
53C2×Dic53—21no—
54C2×Dic53—21no—
55C2×Dic53—42no—
56C2×Dic53C22⋊C412yes—
57C2×Dic53C22⋊C411yescharacteristic
58C4×D53—22no—
59C4×D53—42no—
60Dic106—41no—
61Dic106—42no—
62Dic106—41no—
63C22×C53—22no—
64C22×C53—42no—
65C22×C53C2×C22⋊C411yescharacteristic
66C2×D53—21no—
67C2×D53—41no—
68C4×C53—21no—
69C4×C53—41no—
70C4×C53—41no—
71C4×C53—82no—
72Dic53—22no—
73Dic53—41no—
74Dic53—41no—
75Dic53—82no—
76Dic53#9912yes—
77#2129—531nomaximal, Sylow
78C2×C53—21no—
79C2×C53—42no—
80C2×C53C2×#9911yescharacteristic
81D53—41no—
82#536—531no—
83#410—532no—
84#354—531no—
85#354—532no—
86#354D5311yescharacteristic, complemented
87C53#212911yescharacteristic, complemented, Sylow
88C2×#41—531no—
89C22×Q8—531no—
90C4⋊Q8—531no—
91C4⋊Q8—531no—
92#124—531no—
93#124C2×D5311yescharacteristic
94#123—1062no—
95C22⋊Q8—1062no—
96#117—531no—
97#117—1062no—
98#99—532no—
99#99—1062no—
100#99C2×D5312yes—
101#41—2122no—
102C2×Q8—531no—
103C2×Q8—531no—
104C2×Q8—1061no—
105C2×Q8—2122no—
106C2×Q8C4×D5311yescharacteristic
107C2×D4—532no—
108C2×D4C22×D5311yescharacteristic
109C22×C4—532no—
110C22×C4—531no—
111C22×C4—1061no—
112C4⋊C4—1061no—
113C4⋊C4—1062no—
114C4⋊C4—1062no—
115C4⋊C4—2122no—
116C22⋊C4—22no—
117C22⋊C4—22no—
118C22⋊C4—1061no—
119C22⋊C4—1062no—
120C22⋊C4—1062no—
121C42—531no—
122C42—2122no—
123C42C4×D5311yescharacteristic
124C23—531no—
125C23D4×D5312yes—
126Q8—41no—
127Q8—2121no—
128Q8—2121no—
129Q8—2122no—
130D4—41no—
131D4—2122no—
132C2×C4—21no—
133C2×C4—21no—
134C2×C4—42no—
135C2×C4—531no—
136C2×C4—532no—
137C2×C4—1062no—
138C2×C4—1062no—
139C2×C4—1062no—
140C2×C4—1061no—
141C2×C4—1061no—
142C2×C4—2122no—
143C2×C4—2122no—
144C2×C4C2×C4×D5311yescharacteristic
145C22—22no—
146C22—42no—
147C22—1061no—
148C22—2121no—
149C22C22⋊C4×D5311yescharacteristic
150C4—21no—
151C4—41no—
152C4—41no—
153C4—82no—
154C4—532no—
155C4—1062no—
156C4—4242no—
157C4—2121no—
158C4—2121no—
159C2—21no—
160C2—42no—
161C2—2121no—
162C2#99×D5311yescharacteristic
subgroupG/H
centerC2#99×D53
commutatorC2×C4×C53C22×C4
FrattiniC2×D4C22×D53
FittingC53×#354C2
socleC2×C53C2×#99
hypercenter#354D53
hypocenterC53#2129
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #354

Complements

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

NH ≅ G/Ntype
3. #16810418C2semidirect
4. #16810417C2semidirect
5. C53×#354C2semidirect
7. C2×Q8×D53C4semidirect
8. #13133217C4semidirect
9. #13133215C4semidirect
13. #13133174C4semidirect
86. #354D53semidirect
87. C53#2129semidirect