Finite Group Databasebeta

← #20273764

Subgroups — #20273764

Order 6784.

240 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 49 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#16811210C211yesmaximal, characteristic
2#16810959C211yesmaximal, characteristic, complemented
3#16810939C211yesmaximal, characteristic, complemented
4#16810560C211yesmaximal, characteristic, complemented
5#16810359C211yesmaximal, characteristic, complemented
6#16810346C211yesmaximal, characteristic, complemented
7C53×#327C211yesmaximal, characteristic, complemented
8#13133212—21no—
9#13133203—21no—
10#13133201—21no—
11#13133196C2211yescharacteristic
12#13133189—21no—
13#13133185—21no—
14D4×Dic53—21no—
15C2×C4⋊Dic53—21no—
16C2×C4⋊Dic53C2211yescharacteristic
17C2×#11719921—21no—
18C2×#11719921C2211yescharacteristic
19#13133151—21no—
20#13133148—21no—
21#13133142—21no—
22#13133138—21no—
23C22⋊#88322C2211yescharacteristic
24#121×C53C2211yescharacteristic
25C22⋊C8×C53C2211yescharacteristic
26#95×C53C2211yescharacteristic
27C23×D53D411yescharacteristic
28C2×D4×C53—21no—
29C2×D4×C53D411yescharacteristic
30C22×C4×C53—21no—
31C22×C4×C53C2311yescharacteristic
32C2×C22⋊D53—21no—
33C2×C22⋊D53—21no—
34C22×Dic53—21no—
35C22×Dic53—21no—
36C22×Dic53D411yescharacteristic
37C2×Dic106D411yescharacteristic
38C2×C8×C53—21no—
39C4⋊C4×C53D411yescharacteristic
40C22⋊C4×C53—21no—
41C22⋊Dic53—21no—
42C22⋊Dic53—21no—
43#11719921—21no—
44#11719921—21no—
45#11719921—21no—
46#11719921—21no—
47#11719921—42no—
48C4⋊Dic53—21no—
49C4⋊Dic53—21no—
50C4⋊Dic53—21no—
51C4⋊Dic53—42no—
52C4⋊Dic53D411yescharacteristic
53#11719919—21no—
54C4×Dic53—21no—
55C2×#88322—21no—
56C23×C53—21no—
57C23×C53C2×D411yescharacteristic
58C22×D53—21no—
59C22×D53—21no—
60C22×D53—21no—
61C22×D53—41no—
62C22×D53—41no—
63D4×C53—41no—
64D4×C53—41no—
65D4×C53—41no—
66C2×C4×C53—22no—
67C2×C4×C53—21no—
68C2×C4×C53—41no—
69C2×C4×C53—41no—
70C2×C4×C53—41no—
71C2×C4×C53C2×D411yescharacteristic
72C2×C4×C53C2×D411yescharacteristic
73C22⋊D53—81no—
74C22⋊D53—81no—
75C2×Dic53—21no—
76C2×Dic53—21no—
77C2×Dic53—21no—
78C2×Dic53—41no—
79C2×Dic53—41no—
80C2×Dic53—41no—
81C2×Dic53—41no—
82C2×Dic53—81no—
83Dic106—41no—
84C8×C53—41no—
85#88322—41no—
86C22×C53—21no—
87C22×C53—21no—
88C22×C53—41no—
89C22×C53—41no—
90C22×C53—41no—
91C22×C53#12011yescharacteristic
92C22×C53#13511yescharacteristic
93C22×C53#13711yescharacteristic
94C2×D53—42no—
95C2×D53—41no—
96C2×D53—41no—
97C2×D53—41no—
98C2×D53—81no—
99C4×C53—21no—
100C4×C53—21no—
101C4×C53—42no—
102C4×C53—41no—
103Dic53—41no—
104Dic53—41no—
105Dic53—41no—
106Dic53—81no—
107#1618—531nomaximal, Sylow
108C2×C53—21no—
109C2×C53—21no—
110C2×C53—81no—
111C2×C53#45211yescharacteristic
112C2×C53#45311yescharacteristic
113C2×C53#45811yescharacteristic
114D53—81no—
115#538—531no—
116#484—531no—
117#481—531no—
118#399—531no—
119#328—531no—
120#327D5311yescharacteristic, complemented
121#323—531no—
122C53#161811yescharacteristic, complemented, Sylow
123#126—1061no—
124#124—1061no—
125#123—1061no—
126C22⋊Q8—531no—
127#121—1061no—
128#121C2×D5311yescharacteristic
129#120—1061no—
130C4×D4—1061no—
131C2×C4⋊C4—531no—
132C2×C4⋊C4—1061no—
133C2×C22⋊C4—531no—
134C2×C22⋊C4—1061no—
135#107—1061no—
136#106—1061no—
137#103—1061no—
138#102—1061no—
139C22⋊C8—531no—
140C22⋊C8C2×D5311yescharacteristic
141#95C2×D5311yescharacteristic
142C24—531no—
143C2×Q8—531no—
144C2×D4—21no—
145C2×D4—1061no—
146C2×D4—1061no—
147C2×D4D21211yescharacteristic
148C22×C4—21no—
149C22×C4—531no—
150C22×C4—1061no—
151C22×C4—1061no—
152C22×C4C22×D5311yescharacteristic
153C2×C8—21no—
154C2×C8—1061no—
155C4⋊C4—531no—
156C4⋊C4—1061no—
157C4⋊C4—1061no—
158C4⋊C4—1061no—
159C4⋊C4—1061no—
160C4⋊C4—2122no—
161C4⋊C4D21211yescharacteristic
162C22⋊C4—21no—
163C22⋊C4—1061no—
164C22⋊C4—1061no—
165C22⋊C4—1061no—
166C22⋊C4—1061no—
167C22⋊C4—1061no—
168C22⋊C4—1061no—
169C22⋊C4—2122no—
170C42—1061no—
171C23—21no—
172C23—1061no—
173C23—1061no—
174C23—1061no—
175C23—2121no—
176C23—2121no—
177C23D4×D5311yescharacteristic
178Q8—2121no—
179D4—41no—
180D4—41no—
181D4—41no—
182D4—4241no—
183D4—4241no—
184C2×C4—22no—
185C2×C4—21no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—1061no—
190C2×C4—1061no—
191C2×C4—1061no—
192C2×C4—4241no—
193C2×C4—2121no—
194C2×C4—2121no—
195C2×C4—2121no—
196C2×C4—2121no—
197C2×C4C2×D21211yescharacteristic
198C2×C4D4×D5311yescharacteristic
199C8—41no—
200C8—2121no—
201C22—21no—
202C22—21no—
203C22—41no—
204C22—41no—
205C22—41no—
206C22—4241no—
207C22—2122no—
208C22—2121no—
209C22—2121no—
210C22—2121no—
211C22#1313318711yescharacteristic
212C22#1313323611yescharacteristic
213C22#1313324611yescharacteristic
214C4—21no—
215C4—21no—
216C4—42no—
217C4—41no—
218C4—4241no—
219C4—2121no—
220C4—2121no—
221C4—2121no—
222C2—21no—
223C2—21no—
224C2—81no—
225C2—4241no—
226C2#1681078611yescharacteristic
227C2#1681079211yescharacteristic
228C2#1681081411yescharacteristic
subgroupG/H
centerC22#13133187
commutatorC22×C4×C53C23
FrattiniC22×C4C22×D53
FittingC53×#327C2
socleC22×C53#120
hypercenter#327D53
hypocenterC53#1618
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #327

Complements

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

NH ≅ G/Ntype
2. #16810959C2semidirect
3. #16810939C2semidirect
4. #16810560C2semidirect
5. #16810359C2semidirect
6. #16810346C2semidirect
7. C53×#327C2semidirect
120. #327D53semidirect
122. C53#1618semidirect