Finite Group Databasebeta

← #20290992

Subgroups — #20290992

Order 6784.

286 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 54 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×#13133276C212yesmaximal, complemented
2#16811530C212yesmaximal, complemented
3#16811528C212yesmaximal, complemented
4#16811192C211yesmaximal, characteristic
5C2×C22⋊#6157—22no—
6#13133276—22no—
7#13133276C2214yes—
8C2×#11719945C411yescharacteristic, complemented
9C22×D212C2211yescharacteristic
10#13133214C411yescharacteristic, complemented
11#13133214C2211yescharacteristic
12#13133205C411yescharacteristic, complemented
13#13133205C2211yescharacteristic
14#13133192—22no—
15#13133187—22no—
16#117×C53C411yescharacteristic, complemented
17C23×D53D412yes—
18C22×#6157—42no—
19C22×C4×C53C2×C411yescharacteristic
20C2×C22⋊D53—21no—
21#11719945—42no—
22C2×D212—22no—
23C2×D212—22no—
24C2×D212—42no—
25C2×D212C2×C411yescharacteristic
26C2×D212C2×C411yescharacteristic, complemented
27C2×D212D412yes—
28C2×D212C2311yescharacteristic
29C2×C4×D53—21no—
30C2×Dic106C2×C411yescharacteristic, complemented
31C22⋊#6157—24no—
32C22⋊#6157—42no—
33C4⋊C4×C53—21no—
34C22⋊C4×C53—21no—
35C42×C53C2×C411yescharacteristic, complemented
36C42×C53C2×C411yescharacteristic, complemented
37#11719921—22no—
38#11719921—22no—
39C23×C53C22⋊C411yescharacteristic
40C22×D53—22no—
41C22×D53—22no—
42C22×D53—21no—
43C22×D53—82no—
44C22×D53C22⋊C412yes—
45C22×D53C2×D412yes—
46C2×#6157—44no—
47C2×#6157—84no—
48C2×C4×C53—22no—
49C2×C4×C53—21no—
50C2×C4×C53—21no—
51C2×C4×C53C22⋊C411yescharacteristic
52C2×C4×C53C22×C411yescharacteristic
53C22⋊D53—42no—
54C2×Dic53—21no—
55D212—41no—
56D212—42no—
57D212—41no—
58D212—41no—
59D212—82no—
60D212—82no—
61C4×D53—42no—
62Dic106—41no—
63C22×C53—41no—
64C22×C53#9912yes—
65C22×C53C2×C22⋊C411yescharacteristic
66C2×D53—22no—
67C2×D53—22no—
68C2×D53—44no—
69C2×D53—42no—
70C2×D53—42no—
71C2×D53—42no—
72C2×D53—41no—
73C2×D53—82no—
74#6157—84no—
75C4×C53—21no—
76C4×C53—21no—
77C4×C53—41no—
78C4×C53—41no—
79Dic53—41no—
80#2127—531nomaximal, Sylow
81C2×C53—21no—
82C2×C53—22no—
83C2×C53C2×#9911yescharacteristic
84D53—42no—
85D53—42no—
86D53—81no—
87#535—531no—
88C2×#99—532no—
89#354—532no—
90#353—532no—
91C53#212711yescharacteristic, complemented, Sylow
92C2×#41—531no—
93C22×D4—531no—
94#127—531no—
95#127—531no—
96#124—531no—
97#124—531no—
98#121—1062no—
99#120—1062no—
100#117#615711yescharacteristic, complemented
101C2×C22⋊C4—1062no—
102#99—534no—
103#99—1062no—
104C24—532no—
105#41—2122no—
106C2×Q8—531no—
107C2×D4—532no—
108C2×D4—531no—
109C2×D4—531no—
110C2×D4—531no—
111C2×D4—1062no—
112C2×D4—1062no—
113C2×D4—1061no—
114C2×D4—2122no—
115C22×C4—1061no—
116C22×C4—2122no—
117C22×C4C2×#615711yescharacteristic
118C4⋊C4—21no—
119C22⋊C4—21no—
120C22⋊C4—1062no—
121C22⋊C4—1062no—
122C22⋊C4—1064no—
123C22⋊C4—2122no—
124C42C2×#615711yescharacteristic, complemented
125C42C2×#615711yescharacteristic, complemented
126C23—532no—
127C23—532no—
128C23—1062no—
129C23—1062no—
130C23—1061no—
131C23—4242no—
132C23C22⋊#615711yescharacteristic
133Q8—2121no—
134D4—4242no—
135D4—4242no—
136D4—2122no—
137D4—2121no—
138D4—2121no—
139D4—2121no—
140D4—2122no—
141C2×C4—22no—
142C2×C4—21no—
143C2×C4—21no—
144C2×C4—1061no—
145C2×C4—4244no—
146C2×C4—2122no—
147C2×C4—2124no—
148C2×C4C22⋊#615711yescharacteristic
149C2×C4C22×#615711yescharacteristic
150C22—41no—
151C22—1062no—
152C22—1062no—
153C22—4242no—
154C22—2122no—
155C22—2122no—
156C22—2122no—
157C22—2124no—
158C22—2121no—
159C22#1313327612yes—
160C22C2×C22⋊#615711yescharacteristic
161C4—21no—
162C4—21no—
163C4—41no—
164C4—41no—
165C4—4244no—
166C4—2121no—
167C2—21no—
168C2—22no—
169C2—4241no—
170C2—2122no—
171C2—2122no—
172C2C2×#1313327611yescharacteristic
subgroupG/H
centerC2C2×#13133276
commutatorC2×C4×C53C22×C4
FrattiniC2×C4C22×#6157
Fitting#117×C53C4
socleC2×C53C2×#99
hypercenter#117#6157
hypocenterC53#2127
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C23 ◁ C22×C4 ◁ #117

Complements

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

NH ≅ G/Ntype
1. C2×#13133276C2semidirect
2. #16811530C2semidirect
3. #16811528C2semidirect
8. C2×#11719945C4semidirect
10. #13133214C4semidirect
12. #13133205C4semidirect
16. #117×C53C4semidirect
26. C2×D212C2×C4semidirect
30. C2×Dic106C2×C4semidirect
35. C42×C53C2×C4semidirect
36. C42×C53C2×C4semidirect
91. C53#2127semidirect
100. #117#6157semidirect
124. C42C2×#6157semidirect
125. C42C2×#6157semidirect