Finite Group Databasebeta

← #20273707

Subgroups — #20273707

Order 6784.

228 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 51 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#16811248C211yesmaximal, characteristic
2#16810954C211yesmaximal, characteristic, complemented
3#16810951C211yesmaximal, characteristic, complemented
4#16810550C211yesmaximal, characteristic
5#16810359C211yesmaximal, characteristic, complemented
6#16810346C211yesmaximal, characteristic
7C53×#328C211yesmaximal, characteristic, complemented
8#13133197—21no—
9#13133196C2211yescharacteristic
10#13133195—21no—
11#13133195—21no—
12#13133194—21no—
13#13133193—21no—
14C2×C4⋊Dic53C2211yescharacteristic
15C2×#11719921—21no—
16C2×#11719921C2211yescharacteristic
17C22⋊C4×D53—21no—
18#13133149—21no—
19#13133148—21no—
20#13133142—21no—
21#13133141—21no—
22C22⋊#88322C2211yescharacteristic
23#13133124—21no—
24C22⋊Q8×C53C2211yescharacteristic
25C22⋊C8×C53C2211yescharacteristic
26#95×C53C2211yescharacteristic
27C23×D53D411yescharacteristic
28C2×Q8×C53D411yescharacteristic
29C22×C4×C53—21no—
30C22×C4×C53C2311yescharacteristic
31C22×Dic53—21no—
32C22×Dic53D411yescharacteristic
33C2×C4×D53—21no—
34C2×C4×D53—21no—
35C2×Dic106—21no—
36C2×Dic106D411yescharacteristic
37C2×C8×C53—21no—
38C4⋊C4×C53—21no—
39C4⋊C4×C53D411yescharacteristic
40C22⋊C4×C53—21no—
41C22⋊Dic53—21no—
42#11719921—21no—
43#11719921—21no—
44#11719921—21no—
45#11719921—21no—
46#11719921—42no—
47C4⋊Dic53—21no—
48C4⋊Dic53—21no—
49C4⋊Dic53—21no—
50C4⋊Dic53D411yescharacteristic
51#11719919—21no—
52#11719919—21no—
53#11719919—21no—
54C2×#88322—21no—
55C23×C53C2×D411yescharacteristic
56C22×D53—21no—
57C22×D53—21no—
58C22×D53—41no—
59C22×D53—41no—
60C22×D53SD1612yes—
61Q8×C53—41no—
62C2×C4×C53—21no—
63C2×C4×C53—21no—
64C2×C4×C53—41no—
65C2×C4×C53—41no—
66C2×C4×C53—41no—
67C2×C4×C53—41no—
68C2×C4×C53C2×D411yescharacteristic
69C2×C4×C53C2×D411yescharacteristic
70C2×Dic53—21no—
71C2×Dic53—21no—
72C2×Dic53—21no—
73C2×Dic53—41no—
74C2×Dic53—41no—
75C2×Dic53—41no—
76C2×Dic53—41no—
77C4×D53—81no—
78C4×D53—81no—
79Dic106—41no—
80Dic106—81no—
81C8×C53—41no—
82#88322—41no—
83C22×C53—21no—
84C22×C53—21no—
85C22×C53#12011yescharacteristic
86C22×C53C2×SD1611yescharacteristic
87C22×C53#13711yescharacteristic
88C2×D53—22no—
89C2×D53—41no—
90C2×D53—41no—
91C2×D53—42no—
92C2×D53—81no—
93C4×C53—21no—
94C4×C53—21no—
95C4×C53—41no—
96C4×C53—41no—
97C4×C53—81no—
98Dic53—41no—
99Dic53—41no—
100Dic53—41no—
101Dic53—81no—
102#1608—531nomaximal, Sylow
103C2×C53—21no—
104C2×C53—21no—
105C2×C53#44811yescharacteristic
106C2×C53#45511yescharacteristic
107C2×C53#45811yescharacteristic
108D53—42no—
109#543—531no—
110#483—531no—
111#483—531no—
112#397—531no—
113#328—531no—
114#328D5311yescharacteristic, complemented
115#323—531no—
116C53#160811yescharacteristic, complemented, Sylow
117C22⋊Q8—531no—
118C22⋊Q8—1061no—
119C22⋊Q8—1061no—
120C22⋊Q8—1061no—
121C22⋊Q8—1061no—
122C22⋊Q8—1061no—
123C22⋊Q8C2×D5311yescharacteristic
124C2×C4⋊C4—531no—
125C2×C22⋊C4—531no—
126C2×C22⋊C4—1061no—
127C2×C22⋊C4—1061no—
128#106—1061no—
129#106—1061no—
130#103—1061no—
131#103—1061no—
132C22⋊C8—531no—
133C22⋊C8C2×D5311yescharacteristic
134#95—1061no—
135#95C2×D5311yescharacteristic
136C24—531no—
137C2×Q8—531no—
138C2×Q8—1061no—
139C2×Q8D21211yescharacteristic
140C22×C4—21no—
141C22×C4—531no—
142C22×C4—1061no—
143C22×C4—1061no—
144C22×C4—1061no—
145C22×C4C22×D5311yescharacteristic
146C2×C8—21no—
147C2×C8—1061no—
148C4⋊C4—21no—
149C4⋊C4—531no—
150C4⋊C4—1061no—
151C4⋊C4—1061no—
152C4⋊C4—1061no—
153C4⋊C4—1061no—
154C4⋊C4—1061no—
155C4⋊C4—1061no—
156C4⋊C4D21211yescharacteristic
157C22⋊C4—21no—
158C22⋊C4—1061no—
159C22⋊C4—1061no—
160C22⋊C4—1061no—
161C22⋊C4—1061no—
162C22⋊C4—1061no—
163C22⋊C4—2122no—
164C23—532no—
165C23—1061no—
166C23—1061no—
167C23—2121no—
168C23—2121no—
169C23D4×D5311yescharacteristic
170Q8—41no—
171Q8—4241no—
172Q8—2121no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—1061no—
180C2×C4—1061no—
181C2×C4—1061no—
182C2×C4—4241no—
183C2×C4—4241no—
184C2×C4—2121no—
185C2×C4—2121no—
186C2×C4—2121no—
187C2×C4—2121no—
188C2×C4C2×D21211yescharacteristic
189C2×C4D4×D5311yescharacteristic
190C8—41no—
191C8—2121no—
192C22—21no—
193C22—21no—
194C22—1062no—
195C22—4241no—
196C22—2122no—
197C22—2121no—
198C22—2121no—
199C22#1313318711yescharacteristic
200C22SD16×D5311yescharacteristic
201C22#1313324911yescharacteristic
202C4—21no—
203C4—21no—
204C4—41no—
205C4—41no—
206C4—81no—
207C4—4241no—
208C4—2121no—
209C4—2121no—
210C4—2121no—
211C2—21no—
212C2—21no—
213C2—2122no—
214C2#1681076311yescharacteristic
215C2#1681080211yescharacteristic
216C2#1681081411yescharacteristic
subgroupG/H
centerC22#13133187
commutatorC22×C4×C53C23
FrattiniC22×C4C22×D53
FittingC53×#328C2
socleC22×C53#120
hypercenter#328D53
hypocenterC53#1608
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #328

Complements

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

NH ≅ G/Ntype
2. #16810954C2semidirect
3. #16810951C2semidirect
5. #16810359C2semidirect
7. C53×#328C2semidirect
114. #328D53semidirect
116. C53#1608semidirect