Finite Group Databasebeta

← #20291015

Subgroups — #20291015

Order 6784.

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

normal subgroups only

This group has 262 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
1#16811610C212yesmaximal, complemented
2#16811532C212yesmaximal, complemented
3#16811531C212yesmaximal, complemented
4#16811186C211yesmaximal, characteristic
5C2×#11719940—22no—
6#13133278C2214yescomplemented
7#13133277—22no—
8C2×#11719947C2211yescharacteristic
9C2×D4×D53C411yescharacteristic
10#13133213C411yescharacteristic
11#13133205C2211yescharacteristic
12#13133204C2211yescharacteristic
13#13133189—22no—
14#13133185—22no—
15#13133174C411yescharacteristic
16#127×C53C411yescharacteristic
17C23×D53—21no—
18C2×D4×C53—22no—
19C2×D4×C53C2×C411yescharacteristic
20C2×D4×C53C2×C411yescharacteristic
21C2×C22⋊D53—22no—
22C2×C22⋊D53—21no—
23C2×C22⋊D53D412yes—
24C22×Dic53D412yes—
25#11719947—42no—
26D4×D53—42no—
27C2×D212C2×C411yescharacteristic
28C2×C4×D53C2×C411yescharacteristic
29C2×Dic106C2311yescharacteristic
30#11719940—24no—
31#11719940—42no—
32C2×#88324—42no—
33C42×C53C2×C411yescharacteristic
34C22⋊Dic53—22no—
35#11719921—21no—
36#11719921—22no—
37#11719919—21no—
38C4×Dic53C2×C411yescharacteristic
39C23×C53—21no—
40C23×C53C22⋊C412yes—
41C22×D53—21no—
42C22×D53—42no—
43C22×D53—82no—
44C22×D53C22⋊C411yescharacteristic
45D4×C53—41no—
46D4×C53—41no—
47D4×C53—82no—
48C2×C4×C53—21no—
49C2×C4×C53C22×C411yescharacteristic
50C22⋊D53—42no—
51C22⋊D53—42no—
52C22⋊D53—82no—
53C2×Dic53—22no—
54C2×Dic53—22no—
55C2×Dic53—21no—
56C2×Dic53C22⋊C411yescharacteristic
57C2×Dic53C2×D412yes—
58D212—41no—
59C4×D53—22no—
60Dic106—41no—
61#88324—44no—
62C22×C53—22no—
63C22×C53—42no—
64C22×C53—42no—
65C22×C53—41no—
66C22×C53C2×C22⋊C411yescharacteristic
67C2×D53—21no—
68C2×D53—41no—
69C2×D53—42no—
70C2×D53—41no—
71C2×D53—84no—
72C4×C53—21no—
73C4×C53—41no—
74Dic53—22no—
75Dic53—41no—
76Dic53#9912yes—
77#2134—531nomaximal, Sylow
78C2×C53—21no—
79C2×C53—42no—
80C2×C53—81no—
81C2×C53C2×#9911yescharacteristic
82D53—41no—
83D53—81no—
84#535—531no—
85#413—532no—
86#355—532no—
87#355—532no—
88C53#213411yescharacteristic, complemented, Sylow
89C2×#41—531no—
90C22×D4—531no—
91C2×M16—1062no—
92#127—531no—
93#127#615711yescharacteristic
94#124—531no—
95#124—531no—
96#121—1062no—
97#120—1062no—
98#117—531no—
99#101—534no—
100#100—1062no—
101C24—1061no—
102#41—2122no—
103C2×Q8—531no—
104C2×D4—22no—
105C2×D4—532no—
106C2×D4—531no—
107C2×D4—1061no—
108C2×D4—1062no—
109C2×D4—2122no—
110C2×D4C2×#615711yescharacteristic
111C2×D4C2×#615711yescharacteristic
112C22×C4—532no—
113C22×C4—531no—
114M16—1064no—
115M16—2122no—
116C2×C8—2122no—
117C4⋊C4—1061no—
118C22⋊C4—1062no—
119C22⋊C4—1061no—
120C22⋊C4—1062no—
121C42—531no—
122C42C2×#615711yescharacteristic
123C23—21no—
124C23—531no—
125C23—1061no—
126C23—4242no—
127C23—2122no—
128C23C22⋊#615712yes—
129Q8—2121no—
130D4—41no—
131D4—41no—
132D4—82no—
133D4—4242no—
134D4—2121no—
135D4—2122no—
136D4—2122no—
137C2×C4—21no—
138C2×C4—531no—
139C2×C4—532no—
140C2×C4—1062no—
141C2×C4—1062no—
142C2×C4—1062no—
143C2×C4—1061no—
144C2×C4C22×#615711yescharacteristic
145C8—2124no—
146C22—22no—
147C22—42no—
148C22—42no—
149C22—41no—
150C22—1061no—
151C22—4244no—
152C22—2122no—
153C22—2121no—
154C22—2121no—
155C22C2×C22⋊#615711yescharacteristic
156C4—21no—
157C4—41no—
158C4—532no—
159C4—1062no—
160C4—2121no—
161C2—21no—
162C2—42no—
163C2—81no—
164C2—4241no—
165C2—2121no—
166C2D53⋊#9911yescharacteristic
subgroupG/H
centerC2D53⋊#99
commutatorC2×C4×C53C22×C4
FrattiniC2×C4C22×#6157
Fitting#127×C53C4
socleC2×C53C2×#99
hypercenter#127#6157
hypocenterC53#2134
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. #16811610C2semidirect
2. #16811532C2semidirect
3. #16811531C2semidirect
6. #13133278C22semidirect
88. C53#2134semidirect