Finite Group Databasebeta

← #20273741

Subgroups — #20273741

Order 6784.

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

normal subgroups only

This group has 236 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#16810959C211yesmaximal, characteristic, complemented
3#16810956C211yesmaximal, characteristic, complemented
4#16810560C211yesmaximal, 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⋊Dic53—21no—
15C2×C4⋊Dic53C2211yescharacteristic
16C2×#11719921—21no—
17C2×#11719921C2211yescharacteristic
18C22⋊C4×D53—21no—
19#13133151—21no—
20#13133150—21no—
21#13133142—21no—
22#13133141—21no—
23C22⋊#88322C2211yescharacteristic
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⋊Dic53—42no—
51C4⋊Dic53D411yescharacteristic
52#11719919—21no—
53#11719919—21no—
54#11719919—21no—
55C2×#88322—21no—
56C23×C53C2×D411yescharacteristic
57C22×D53—21no—
58C22×D53—21no—
59C22×D53—41no—
60C22×D53—41no—
61C22×D53Q1612yes—
62Q8×C53—41no—
63C2×C4×C53—22no—
64C2×C4×C53—21no—
65C2×C4×C53—21no—
66C2×C4×C53—41no—
67C2×C4×C53—41no—
68C2×C4×C53—41no—
69C2×C4×C53C2×D411yescharacteristic
70C2×C4×C53C2×D411yescharacteristic
71C2×Dic53—21no—
72C2×Dic53—21no—
73C2×Dic53—21no—
74C2×Dic53—41no—
75C2×Dic53—41no—
76C2×Dic53—41no—
77C2×Dic53—41no—
78C4×D53—81no—
79C4×D53—81no—
80Dic106—41no—
81Dic106—81no—
82C8×C53—41no—
83#88322—41no—
84C22×C53—21no—
85C22×C53—21no—
86C22×C53#12011yescharacteristic
87C22×C53C2×Q1611yescharacteristic
88C22×C53#13711yescharacteristic
89C2×D53—22no—
90C2×D53—41no—
91C2×D53—41no—
92C2×D53—42no—
93C2×D53—81no—
94C4×C53—21no—
95C4×C53—21no—
96C4×C53—42no—
97C4×C53—41no—
98C4×C53—41no—
99Dic53—41no—
100Dic53—41no—
101Dic53—41no—
102Dic53—81no—
103#1614—531nomaximal, Sylow
104C2×C53—21no—
105C2×C53—21no—
106C2×C53#45111yescharacteristic
107C2×C53#45311yescharacteristic
108C2×C53#45811yescharacteristic
109D53—42no—
110#543—531no—
111#484—531no—
112#484—531no—
113#399—531no—
114#328—531no—
115#328D5311yescharacteristic, complemented
116#323—531no—
117C53#161411yescharacteristic, complemented, Sylow
118C22⋊Q8—531no—
119C22⋊Q8—1061no—
120C22⋊Q8—1061no—
121C22⋊Q8—1061no—
122C22⋊Q8—1061no—
123C22⋊Q8—1061no—
124C22⋊Q8C2×D5311yescharacteristic
125C2×C4⋊C4—531no—
126C2×C4⋊C4—1061no—
127C2×C22⋊C4—531no—
128C2×C22⋊C4—1061no—
129C2×C22⋊C4—1061no—
130#107—1061no—
131#107—1061no—
132#103—1061no—
133#103—1061no—
134C22⋊C8—531no—
135C22⋊C8C2×D5311yescharacteristic
136#95C2×D5311yescharacteristic
137C24—531no—
138C2×Q8—531no—
139C2×Q8—1061no—
140C2×Q8D21211yescharacteristic
141C22×C4—21no—
142C22×C4—531no—
143C22×C4—1061no—
144C22×C4—1061no—
145C22×C4—1061no—
146C22×C4C22×D5311yescharacteristic
147C2×C8—21no—
148C2×C8—1061no—
149C4⋊C4—21no—
150C4⋊C4—531no—
151C4⋊C4—1061no—
152C4⋊C4—1061no—
153C4⋊C4—1061no—
154C4⋊C4—1061no—
155C4⋊C4—1061no—
156C4⋊C4—1061no—
157C4⋊C4—2122no—
158C4⋊C4D21211yescharacteristic
159C22⋊C4—21no—
160C22⋊C4—1061no—
161C22⋊C4—1061no—
162C22⋊C4—1061no—
163C22⋊C4—1061no—
164C22⋊C4—1061no—
165C22⋊C4—2122no—
166C23—532no—
167C23—1061no—
168C23—1061no—
169C23—2121no—
170C23—2121no—
171C23D4×D5311yescharacteristic
172Q8—41no—
173Q8—4241no—
174Q8—2121no—
175C2×C4—22no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—1061no—
182C2×C4—1061no—
183C2×C4—1061no—
184C2×C4—4241no—
185C2×C4—4241no—
186C2×C4—2121no—
187C2×C4—2121no—
188C2×C4—2121no—
189C2×C4—2121no—
190C2×C4C2×D21211yescharacteristic
191C2×C4D4×D5311yescharacteristic
192C8—41no—
193C8—2121no—
194C22—21no—
195C22—21no—
196C22—1062no—
197C22—4241no—
198C22—2122no—
199C22—2121no—
200C22—2121no—
201C22#1313318711yescharacteristic
202C22Q16×D5311yescharacteristic
203C22#1313324611yescharacteristic
204C4—21no—
205C4—21no—
206C4—42no—
207C4—41no—
208C4—41no—
209C4—4241no—
210C4—2121no—
211C4—2121no—
212C4—2121no—
213C2—21no—
214C2—21no—
215C2—2122no—
216C2#1681078011yescharacteristic
217C2#1681079211yescharacteristic
218C2#1681081411yescharacteristic
subgroupG/H
centerC22#13133187
commutatorC22×C4×C53C23
FrattiniC22×C4C22×D53
FittingC53×#328C2
socleC22×C53#120
hypercenter#328D53
hypocenterC53#1614
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. #16810959C2semidirect
3. #16810956C2semidirect
5. #16810359C2semidirect
7. C53×#328C2semidirect
115. #328D53semidirect
117. C53#1614semidirect