Finite Group Databasebeta

← #20273657

Subgroups — #20273657

Order 6784.

266 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 53 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#16811254C211yesmaximal, characteristic, complemented
2#16810776C212yesmaximal
3#16810540C211yesmaximal, characteristic
4#16810361C212yesmaximal, complemented
5C53×#324C211yesmaximal, characteristic, complemented
6C22×D212C2211yescharacteristic
7C2×#11719913—22no—
8#13133216—22no—
9#13133208—22no—
10#13133196—21no—
11#13133196C2212yes—
12#13133176—21no—
13C2×#11719921—21no—
14C2×#11719921—21no—
15#13133139—22no—
16#13133124C2211yescharacteristic
17C2×C4⋊C4×C53C2211yescharacteristic
18C22⋊C8×C53C2212yes—
19C23×D53—21no—
20C22×C4×C53D411yescharacteristic
21C22×C4×C53D411yescharacteristic
22C22×C4×C53C2311yescharacteristic
23C22×Dic53—21no—
24C2×D212—22no—
25C2×D212—42no—
26C2×Dic106—21no—
27C2×Dic106D412yes—
28C2×C8×C53—22no—
29C4⋊C4×C53—22no—
30C22⋊Dic53—22no—
31#11719921—42no—
32#11719921—42no—
33C4⋊Dic53—21no—
34C4⋊Dic53—21no—
35C4⋊Dic53—21no—
36C4⋊Dic53D412yes—
37#11719919—22no—
38#11719919—22no—
39C4×Dic53—22no—
40#11719913—44no—
41C23×C53C2×D411yescharacteristic
42C22×D53—22no—
43C22×D53—41no—
44C22×D53—41no—
45C22×D53—41no—
46C22×D53—41no—
47C22×D53—41no—
48C22×D53—41no—
49C2×C4×C53—21no—
50C2×C4×C53—21no—
51C2×C4×C53—21no—
52C2×C4×C53—41no—
53C2×C4×C53—41no—
54C2×C4×C53SD1612yescomplemented
55C2×C4×C53C2×D412yes—
56C2×Dic53—22no—
57C2×Dic53—22no—
58C2×Dic53—41no—
59C2×Dic53—41no—
60C2×Dic53—41no—
61D212—24no—
62D212—42no—
63Dic106—42no—
64Dic106—81no—
65C8×C53—42no—
66C22×C53—21no—
67C22×C53—21no—
68C22×C53#12011yescharacteristic
69C22×C53C2×SD1611yescharacteristic
70C22×C53#13711yescharacteristic
71C2×D53—42no—
72C2×D53—42no—
73C2×D53—42no—
74C2×D53—81no—
75C2×D53—81no—
76C2×D53—81no—
77C2×D53—81no—
78C4×C53—22no—
79C4×C53—22no—
80C4×C53—41no—
81Dic53—42no—
82Dic53—42no—
83Dic53—81no—
84#1599—531nomaximal, Sylow
85C2×C53—21no—
86C2×C53—21no—
87C2×C53#44811yescharacteristic
88C2×C53#45411yescharacteristic
89C2×C53B(4,𝔽2)11yescharacteristic
90D53—82no—
91#544—531no—
92#450—532no—
93#394—531no—
94#328—532no—
95#324D5311yescharacteristic, complemented
96C53#159911yescharacteristic, complemented, Sylow
97C22×D4—531no—
98C2×SD16—1062no—
99C4⋊Q8—1062no—
100#125—1062no—
101C22⋊Q8—532no—
102C22⋊Q8—1061no—
103#117—1061no—
104C2×C4⋊C4C2×D5311yescharacteristic
105C2×C22⋊C4—1061no—
106C2×C22⋊C4—1061no—
107#102—1062no—
108C22⋊C8C2×D5312yes—
109#95—531no—
110C24—1061no—
111C2×Q8—532no—
112C2×Q8—1061no—
113C2×D4—1062no—
114C2×D4—2122no—
115C22×C4—1061no—
116C22×C4D21211yescharacteristic
117C22×C4D21211yescharacteristic
118C22×C4C22×D5311yescharacteristic
119SD16—2124no—
120C2×C8—22no—
121C4⋊C4—22no—
122C4⋊C4—532no—
123C4⋊C4—1061no—
124C4⋊C4—1061no—
125C4⋊C4—1061no—
126C4⋊C4—1062no—
127C4⋊C4—1062no—
128C22⋊C4—1062no—
129C22⋊C4—2122no—
130C22⋊C4—2122no—
131C42—1062no—
132C23—1062no—
133C23—2121no—
134C23—2121no—
135C23—2121no—
136C23—2121no—
137C23—2121no—
138C23—2121no—
139C23C2×D21211yescharacteristic
140Q8—4241no—
141Q8—2122no—
142D4—1064no—
143D4—2122no—
144C2×C4—21no—
145C2×C4—21no—
146C2×C4—21no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4—1062no—
150C2×C4—1062no—
151C2×C4—2121no—
152C2×C4—2121no—
153C2×C4—2121no—
154C2×C4#1171991312yescomplemented
155C2×C4D4×D5312yes—
156C8—42no—
157C22—21no—
158C22—21no—
159C22—4241no—
160C22—4241no—
161C22—4241no—
162C22—4241no—
163C22—2122no—
164C22—2122no—
165C22—2122no—
166C22#1313318711yescharacteristic
167C22C2×#1171991311yescharacteristic
168C22#1313325011yescharacteristic
169C4—22no—
170C4—22no—
171C4—41no—
172C4—4241no—
173C4—2122no—
174C4—2122no—
175C2—21no—
176C2—21no—
177C2—4242no—
178C2#1681076411yescharacteristic
179C2#1681079811yescharacteristic
180C2#1681081111yescharacteristic
subgroupG/H
centerC22#13133187
commutatorC22×C4×C53C23
FrattiniC22×C4C22×D53
FittingC53×#324C2
socleC22×C53#120
hypercenter#324D53
hypocenterC53#1599
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #324

Complements

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

NH ≅ G/Ntype
1. #16811254C2semidirect
4. #16810361C2semidirect
5. C53×#324C2semidirect
54. C2×C4×C53SD16semidirect
95. #324D53semidirect
96. C53#1599semidirect
154. C2×C4#11719913semidirect