Finite Group Databasebeta

← #20273706

Subgroups — #20273706

Order 6784.

228 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 57 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#16810951C212yesmaximal, complemented
2#16810548C211yesmaximal, characteristic
3#16810360C212yesmaximal, complemented
4#16810347C211yesmaximal, characteristic
5C53×#543C211yesmaximal, characteristic
6C2×C4⋊Dic53C2211yescharacteristic
7C2×C22⋊Dic53—21no—
8#13133148—22no—
9#13133141—22no—
10C22⋊#88322C2212yes—
11#13133124—21no—
12#13133124C2211yescharacteristic
13C22⋊Q8×C53—22no—
14C22⋊Q8×C53—22no—
15C22⋊Q8×C53—21no—
16C22⋊Q8×C53C2212yes—
17C2×C22⋊C4×C53—21no—
18C2×C22⋊C4×C53C2211yescharacteristic
19C24×C53D411yescharacteristic
20C2×Q8×C53—21no—
21C2×Q8×C53D412yes—
22C22×C4×C53—21no—
23C22×C4×C53—21no—
24C22×C4×C53D411yescharacteristic
25C22×C4×C53C2311yescharacteristic
26C22×Dic53—21no—
27C22×Dic53—21no—
28C4⋊C4×C53—22no—
29C4⋊C4×C53—21no—
30C4⋊C4×C53—22no—
31C4⋊C4×C53D412yes—
32C22⋊C4×C53—22no—
33C22⋊C4×C53—22no—
34C22⋊C4×C53—22no—
35C22⋊Dic53—42no—
36C4⋊Dic53—22no—
37C2×#88322—22no—
38C23×C53—21no—
39C23×C53—21no—
40C23×C53—41no—
41C23×C53—41no—
42C23×C53SD1612yes—
43C23×C53C2×D411yescharacteristic
44Q8×C53—42no—
45Q8×C53—81no—
46C2×C4×C53—22no—
47C2×C4×C53—22no—
48C2×C4×C53—21no—
49C2×C4×C53—41no—
50C2×C4×C53—41no—
51C2×C4×C53—81no—
52C2×C4×C53—81no—
53C2×C4×C53C2×D412yes—
54C2×Dic53—41no—
55C2×Dic53—41no—
56C2×Dic53—41no—
57C2×Dic53—41no—
58C2×Dic53—41no—
59C2×Dic53—41no—
60#88322—42no—
61C22×C53—21no—
62C22×C53—21no—
63C22×C53—22no—
64C22×C53—42no—
65C22×C53—41no—
66C22×C53—41no—
67C22×C53—81no—
68C22×C53#12011yescharacteristic
69C22×C53C2×SD1611yescharacteristic
70C22×C53#13711yescharacteristic
71C4×C53—22no—
72C4×C53—41no—
73C4×C53—42no—
74C4×C53—42no—
75Dic53—81no—
76Dic53—81no—
77#1608—531nomaximal, Sylow
78C2×C53—21no—
79C2×C53—21no—
80C2×C53—42no—
81C2×C53#44811yescharacteristic
82C2×C53#45511yescharacteristic
83C2×C53#45811yescharacteristic
84#543D5311yescharacteristic
85#483—532no—
86#397—531no—
87#328—532no—
88#323—531no—
89C53#160811yescharacteristic, complemented, Sylow
90C22⋊Q8—22no—
91C22⋊Q8—21no—
92C22⋊Q8—22no—
93C22⋊Q8C2×D5312yes—
94C2×C4⋊C4—531no—
95C2×C22⋊C4—21no—
96C2×C22⋊C4—1061no—
97C2×C22⋊C4C2×D5311yescharacteristic
98#106—1062no—
99#103—1062no—
100C22⋊C8—532no—
101#95—531no—
102#95—1061no—
103C24C22⋊D5311yescharacteristic
104C2×Q8—21no—
105C2×Q8C22⋊D5312yes—
106C22×C4—21no—
107C22×C4—21no—
108C22×C4—1061no—
109C22×C4—1061no—
110C22×C4C22⋊D5311yescharacteristic
111C22×C4C22×D5311yescharacteristic
112C2×C8—1062no—
113C4⋊C4—21no—
114C4⋊C4—22no—
115C4⋊C4—22no—
116C4⋊C4—1062no—
117C4⋊C4C22⋊D5312yes—
118C22⋊C4—22no—
119C22⋊C4—22no—
120C22⋊C4—22no—
121C22⋊C4—2122no—
122C23—21no—
123C23—21no—
124C23—41no—
125C23—41no—
126C23#1171992412yes—
127C23C2×C22⋊D5311yescharacteristic
128Q8—42no—
129Q8—81no—
130C2×C4—22no—
131C2×C4—21no—
132C2×C4—22no—
133C2×C4—41no—
134C2×C4—41no—
135C2×C4—81no—
136C2×C4—81no—
137C2×C4—2121no—
138C2×C4—2121no—
139C2×C4—2121no—
140C2×C4—2121no—
141C2×C4—2121no—
142C2×C4—2121no—
143C2×C4C2×C22⋊D5312yes—
144C8—2122no—
145C22—21no—
146C22—21no—
147C22—22no—
148C22—42no—
149C22—41no—
150C22—41no—
151C22—81no—
152C22#1313318611yescharacteristic
153C22C2×#1171992411yescharacteristic
154C22#1313324811yescharacteristic
155C4—22no—
156C4—41no—
157C4—42no—
158C4—42no—
159C4—4241no—
160C4—4241no—
161C2—21no—
162C2—21no—
163C2—42no—
164C2#1681076211yescharacteristic
165C2#1681080111yescharacteristic
166C2#1681081311yescharacteristic
subgroupG/H
centerC22#13133186
commutatorC22×C4×C53C23
FrattiniC22×C4C22×D53
FittingC53×#543C2
socleC22×C53#120
hypercenter#543D53
hypocenterC53#1608
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #543

Complements

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

NH ≅ G/Ntype
1. #16810951C2semidirect
3. #16810360C2semidirect
89. C53#1608semidirect