Finite Group Databasebeta

← #20273638

Subgroups — #20273638

Order 6784.

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

normal subgroups only

This group has 316 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#16810773C212yesmaximal, complemented
2#16810532C211yesmaximal, characteristic
3#16810360C212yesmaximal, complemented
4#16810347C211yesmaximal, characteristic, complemented
5C53×#543C211yesmaximal, characteristic, complemented
6C22×D212C2211yescharacteristic
7C22×C22⋊D53—21no—
8C2×#11719924—22no—
9C2×#11719921—21no—
10#13133137—22no—
11C22⋊#88322C2212yes—
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
20C23×D53—21no—
21C2×Q8×C53—21no—
22C2×Q8×C53D412yescomplemented
23C22×C4×C53—21no—
24C22×C4×C53—21no—
25C22×C4×C53D411yescharacteristic
26C22×C4×C53C2311yescharacteristic
27C2×C22⋊D53—42no—
28C2×C22⋊D53—42no—
29C2×C22⋊D53—42no—
30C22×Dic53—21no—
31C2×D212—22no—
32C2×D212—42no—
33C4⋊C4×C53—22no—
34C4⋊C4×C53—21no—
35C4⋊C4×C53—22no—
36C4⋊C4×C53D412yescomplemented
37C22⋊C4×C53—22no—
38C22⋊C4×C53—22no—
39C22⋊C4×C53—22no—
40#11719924—44no—
41#11719921—42no—
42C2×#88322—22no—
43C23×C53—21no—
44C23×C53—21no—
45C23×C53—41no—
46C23×C53—41no—
47C23×C53SD1612yescomplemented
48C23×C53C2×D411yescharacteristic
49C22×D53—22no—
50C22×D53—41no—
51C22×D53—41no—
52C22×D53—41no—
53C22×D53—41no—
54C22×D53—41no—
55C22×D53—41no—
56Q8×C53—42no—
57Q8×C53—81no—
58C2×C4×C53—22no—
59C2×C4×C53—22no—
60C2×C4×C53—21no—
61C2×C4×C53—41no—
62C2×C4×C53—41no—
63C2×C4×C53—81no—
64C2×C4×C53—81no—
65C2×C4×C53C2×D412yes—
66C22⋊D53—48no—
67C2×Dic53—22no—
68C2×Dic53—41no—
69C2×Dic53—41no—
70D212—24no—
71D212—42no—
72#88322—42no—
73C22×C53—21no—
74C22×C53—21no—
75C22×C53—22no—
76C22×C53—42no—
77C22×C53—41no—
78C22×C53—41no—
79C22×C53—81no—
80C22×C53#12011yescharacteristic
81C22×C53C2×SD1611yescharacteristic
82C22×C53#13711yescharacteristic
83C2×D53—42no—
84C2×D53—42no—
85C2×D53—42no—
86C2×D53—81no—
87C2×D53—81no—
88C2×D53—81no—
89C2×D53—81no—
90C4×C53—22no—
91C4×C53—41no—
92C4×C53—42no—
93C4×C53—42no—
94Dic53—42no—
95#1596—531nomaximal, Sylow
96C2×C53—21no—
97C2×C53—21no—
98C2×C53—42no—
99C2×C53#44811yescharacteristic
100C2×C53#45311yescharacteristic
101C2×C53B(4,𝔽2)11yescharacteristic
102D53—82no—
103#543D5311yescharacteristic, complemented
104#450—532no—
105#392—531no—
106#328—532no—
107#323—531no—
108C53#159611yescharacteristic, complemented, Sylow
109C22×D4—531no—
110C22×D4—1061no—
111C2×SD16—1062no—
112C22⋊Q8—22no—
113C22⋊Q8—21no—
114C22⋊Q8—22no—
115C22⋊Q8C2×D5312yes—
116C2×C22⋊C4—21no—
117C2×C22⋊C4—1061no—
118C2×C22⋊C4C2×D5311yescharacteristic
119#102—1062no—
120C22⋊C8—532no—
121#95—531no—
122C24—1061no—
123C24C22⋊D5311yescharacteristic
124C2×Q8—21no—
125C2×Q8C22⋊D5312yescomplemented
126C2×D4—1062no—
127C2×D4—2122no—
128C2×D4—2122no—
129C2×D4—2122no—
130C2×D4—2122no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—1061no—
134C22×C4C22⋊D5311yescharacteristic
135C22×C4C22×D5311yescharacteristic
136SD16—2124no—
137C2×C8—1062no—
138C4⋊C4—21no—
139C4⋊C4—22no—
140C4⋊C4—22no—
141C4⋊C4C22⋊D5312yescomplemented
142C22⋊C4—22no—
143C22⋊C4—22no—
144C22⋊C4—22no—
145C22⋊C4—2122no—
146C23—21no—
147C23—21no—
148C23—41no—
149C23—41no—
150C23—1062no—
151C23—2121no—
152C23—2121no—
153C23—2121no—
154C23—2121no—
155C23—2121no—
156C23—2121no—
157C23#1171992412yescomplemented
158C23C2×C22⋊D5311yescharacteristic
159Q8—42no—
160Q8—81no—
161D4—1064no—
162D4—2122no—
163D4—2128no—
164C2×C4—22no—
165C2×C4—21no—
166C2×C4—22no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—81no—
170C2×C4—81no—
171C2×C4—1062no—
172C2×C4—2121no—
173C2×C4—2121no—
174C2×C4C2×C22⋊D5312yes—
175C8—2122no—
176C22—21no—
177C22—21no—
178C22—22no—
179C22—42no—
180C22—41no—
181C22—41no—
182C22—81no—
183C22—4241no—
184C22—4241no—
185C22—4241no—
186C22—4241no—
187C22—2122no—
188C22—2122no—
189C22—2122no—
190C22#1313318611yescharacteristic
191C22C2×#1171992411yescharacteristic
192C22#1313324811yescharacteristic
193C4—22no—
194C4—41no—
195C4—42no—
196C4—42no—
197C4—2122no—
198C2—21no—
199C2—21no—
200C2—42no—
201C2—4242no—
202C2#1681076211yescharacteristic
203C2#1681079111yescharacteristic
204C2#1681081011yescharacteristic
subgroupG/H
centerC22#13133186
commutatorC22×C4×C53C23
FrattiniC22×C4C22×D53
FittingC53×#543C2
socleC22×C53#120
hypercenter#543D53
hypocenterC53#1596
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. #16810773C2semidirect
3. #16810360C2semidirect
4. #16810347C2semidirect
5. C53×#543C2semidirect
22. C2×Q8×C53D4semidirect
36. C4⋊C4×C53D4semidirect
47. C23×C53SD16semidirect
103. #543D53semidirect
108. C53#1596semidirect
125. C2×Q8C22⋊D53semidirect
141. C4⋊C4C22⋊D53semidirect
157. C23#11719924semidirect