Finite Group Databasebeta

← #20290991

Subgroups — #20290991

Order 6784.

286 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 52 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
1D53⋊#99C212yesmaximal, complemented
2#16811530C212yesmaximal, complemented
3#16811527C212yesmaximal, complemented
4#16811186C211yesmaximal, characteristic
5C2×C22⋊#6157—22no—
6#13133276C2214yes—
7#13133275—22no—
8C2×#11719947C411yescharacteristic, complemented
9C2×D4×D53C2211yescharacteristic
10#13133213C2211yescharacteristic
11#13133205C2211yescharacteristic
12#13133204C411yescharacteristic, complemented
13#13133189—22no—
14#13133185—22no—
15#13133174C411yescharacteristic, complemented
16#127×C53C411yescharacteristic, complemented
17C23×D53D412yes—
18C22×#6157—42no—
19C2×D4×C53—22no—
20C2×D4×C53C2×C411yescharacteristic
21C2×D4×C53C2×C411yescharacteristic, complemented
22C2×C22⋊D53—21no—
23C2×C22⋊D53—22no—
24C2×C22⋊D53D412yes—
25C22×Dic53—21no—
26#11719947—42no—
27D4×D53—42no—
28C2×D212C2311yescharacteristic
29C2×C4×D53C2×C411yescharacteristic
30C2×Dic106C2×C411yescharacteristic, complemented
31C22⋊#6157—24no—
32C22⋊#6157—42no—
33C42×C53C2×C411yescharacteristic, complemented
34C22⋊Dic53—22no—
35#11719921—21no—
36#11719921—22no—
37#11719919—21no—
38C4×Dic53C2×C411yescharacteristic, complemented
39C23×C53—21no—
40C23×C53C22⋊C412yes—
41C22×D53—22no—
42C22×D53—22no—
43C22×D53—82no—
44C22×D53C22⋊C411yescharacteristic
45C22×D53C2×D412yes—
46C2×#6157—44no—
47C2×#6157—84no—
48D4×C53—41no—
49D4×C53—41no—
50D4×C53—82no—
51C2×C4×C53—21no—
52C2×C4×C53C22×C411yescharacteristic
53C22⋊D53—42no—
54C22⋊D53—42no—
55C22⋊D53—82no—
56C2×Dic53—21no—
57C2×Dic53—21no—
58C2×Dic53—42no—
59C2×Dic53C22⋊C411yescharacteristic
60D212—41no—
61C4×D53—22no—
62Dic106—41no—
63C22×C53—22no—
64C22×C53—42no—
65C22×C53—42no—
66C22×C53—41no—
67C22×C53C2×C22⋊C411yescharacteristic
68C2×D53—22no—
69C2×D53—41no—
70C2×D53—44no—
71C2×D53—42no—
72C2×D53—82no—
73C2×D53#9912yes—
74#6157—84no—
75C4×C53—21no—
76C4×C53—41no—
77Dic53—21no—
78Dic53—41no—
79Dic53—41no—
80#2127—531nomaximal, Sylow
81C2×C53—21no—
82C2×C53—42no—
83C2×C53—81no—
84C2×C53C2×#9911yescharacteristic
85D53—22no—
86D53—42no—
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#615711yescharacteristic, complemented
96#124—531no—
97#124—531no—
98#121—1062no—
99#120—1062no—
100#117—531no—
101C2×C22⋊C4—1062no—
102#99—534no—
103#99—1062no—
104C24—532no—
105#41—2122no—
106C2×Q8—531no—
107C2×D4—22no—
108C2×D4—532no—
109C2×D4—531no—
110C2×D4—1062no—
111C2×D4—1061no—
112C2×D4—2122no—
113C2×D4C2×#615711yescharacteristic
114C2×D4C2×#615711yescharacteristic, complemented
115C22×C4—531no—
116C22×C4—1061no—
117C22×C4—2122no—
118C4⋊C4—1061no—
119C22⋊C4—1061no—
120C22⋊C4—1062no—
121C22⋊C4—1062no—
122C22⋊C4—1064no—
123C22⋊C4—2122no—
124C42—531no—
125C42C2×#615711yescharacteristic, complemented
126C23—21no—
127C23—531no—
128C23—532no—
129C23—1062no—
130C23—1062no—
131C23—4242no—
132C23C22⋊#615712yes—
133Q8—2121no—
134D4—41no—
135D4—41no—
136D4—82no—
137D4—4242no—
138D4—2121no—
139D4—2122no—
140D4—2122no—
141C2×C4—21no—
142C2×C4—531no—
143C2×C4—1062no—
144C2×C4—1061no—
145C2×C4—1061no—
146C2×C4—4244no—
147C2×C4—2122no—
148C2×C4—2124no—
149C2×C4C22×#615711yescharacteristic
150C22—22no—
151C22—42no—
152C22—42no—
153C22—41no—
154C22—532no—
155C22—1062no—
156C22—4242no—
157C22—2121no—
158C22—2122no—
159C22—2124no—
160C22C2×C22⋊#615711yescharacteristic
161C4—21no—
162C4—41no—
163C4—1061no—
164C4—4244no—
165C4—2121no—
166C4—2121no—
167C2—21no—
168C2—42no—
169C2—81no—
170C2—1062no—
171C2—2122no—
172C2D53⋊#9911yescharacteristic
subgroupG/H
centerC2D53⋊#99
commutatorC2×C4×C53C22×C4
FrattiniC2×C4C22×#6157
Fitting#127×C53C4
socleC2×C53C2×#99
hypercenter#127#6157
hypocenterC53#2127
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #127

Complements

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

NH ≅ G/Ntype
1. D53⋊#99C2semidirect
2. #16811530C2semidirect
3. #16811527C2semidirect
8. C2×#11719947C4semidirect
12. #13133204C4semidirect
15. #13133174C4semidirect
16. #127×C53C4semidirect
21. C2×D4×C53C2×C4semidirect
30. C2×Dic106C2×C4semidirect
33. C42×C53C2×C4semidirect
38. C4×Dic53C2×C4semidirect
91. C53#2127semidirect
95. #127#6157semidirect
114. C2×D4C2×#6157semidirect
125. C42C2×#6157semidirect