Finite Group Databasebeta

← #20273765

Subgroups — #20273765

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#16811211C211yesmaximal, characteristic
2#16810959C211yesmaximal, characteristic, complemented
3#16810942C211yesmaximal, characteristic, complemented
4#16810559C211yesmaximal, characteristic, complemented
5#16810361C211yesmaximal, characteristic, complemented
6#16810358C211yesmaximal, characteristic, complemented
7C53×#323C211yesmaximal, characteristic, complemented
8#13133210—21no—
9#13133203—21no—
10#13133199—21no—
11#13133196C2211yescharacteristic
12#13133191C2211yescharacteristic
13#13133190—21no—
14#13133186—21no—
15C4×C22⋊D53—21no—
16C2×C4⋊Dic53C2211yescharacteristic
17C2×#11719919—21no—
18C2×C22⋊Dic53—21no—
19#13133151—21no—
20#13133149—21no—
21#13133142—21no—
22#13133139—21no—
23#13133124C2211yescharacteristic
24C2×C22⋊C4×C53C2211yescharacteristic
25C22⋊C8×C53C2211yescharacteristic
26C22⋊C8×C53C2211yescharacteristic
27C24×C53D411yescharacteristic
28C22×C4×C53D411yescharacteristic
29C22×C4×C53C2311yescharacteristic
30C2×C22⋊D53—21no—
31C2×C22⋊D53—21no—
32C2×C22⋊D53—21no—
33C22×Dic53—21no—
34C22×Dic53—21no—
35C2×D212D411yescharacteristic
36C2×C4×D53—21no—
37C2×Dic106D411yescharacteristic
38C2×C8×C53—21no—
39C2×C8×C53—21no—
40C22⋊C4×C53—21no—
41C22⋊C4×C53—21no—
42C22⋊Dic53—21no—
43C22⋊Dic53—21no—
44C22⋊Dic53—21no—
45C22⋊Dic53—42no—
46#11719921—21no—
47#11719921—21no—
48C4⋊Dic53—21no—
49C4⋊Dic53—21no—
50C4⋊Dic53D411yescharacteristic
51C4⋊Dic53D411yescharacteristic
52#11719919—21no—
53#11719919—21no—
54#11719919—42no—
55C4×Dic53—21no—
56C23×C53—21no—
57C23×C53—21no—
58C23×C53—21no—
59C23×C53—41no—
60C23×C53—41no—
61C23×C53C2×D411yescharacteristic
62C22×D53—21no—
63C2×C4×C53—21no—
64C2×C4×C53—41no—
65C2×C4×C53—41no—
66C2×C4×C53C2×D411yescharacteristic
67C2×C4×C53C2×D411yescharacteristic
68C22⋊D53—41no—
69C22⋊D53—41no—
70C22⋊D53—81no—
71C22⋊D53—81no—
72C2×Dic53—22no—
73C2×Dic53—21no—
74C2×Dic53—21no—
75C2×Dic53—21no—
76C2×Dic53—41no—
77C2×Dic53—41no—
78C2×Dic53—41no—
79C2×Dic53—41no—
80C2×Dic53—41no—
81D212—41no—
82C4×D53—81no—
83Dic106—41no—
84C8×C53—41no—
85C8×C53—41no—
86C22×C53—21no—
87C22×C53—21no—
88C22×C53—42no—
89C22×C53—41no—
90C22×C53—41no—
91C22×C53—41no—
92C22×C53—81no—
93C22×C53#12011yescharacteristic
94C22×C53#13511yescharacteristic
95C22×C53#13711yescharacteristic
96C2×D53—41no—
97C2×D53—41no—
98C2×D53—41no—
99C4×C53—21no—
100C4×C53—21no—
101C4×C53—41no—
102Dic53—42no—
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#327—531no—
121#323D5311yescharacteristic, complemented
122C53#161811yescharacteristic, complemented, Sylow
123#126—1061no—
124#124—1061no—
125#123—1061no—
126C22⋊Q8—531no—
127#121—531no—
128#121—1061no—
129#120—1061no—
130C4×D4—1061no—
131C2×C4⋊C4—531no—
132C2×C4⋊C4—1061no—
133C2×C22⋊C4—1061no—
134C2×C22⋊C4C2×D5311yescharacteristic
135#107—1061no—
136#106—1061no—
137#103—1061no—
138#102—1061no—
139C22⋊C8C2×D5311yescharacteristic
140C22⋊C8C2×D5311yescharacteristic
141#95—531no—
142C24D21211yescharacteristic
143C2×Q8—531no—
144C2×D4—531no—
145C2×D4—1061no—
146C2×D4—1061no—
147C2×D4—1061no—
148C22×C4—1061no—
149C22×C4—1061no—
150C22×C4—1061no—
151C22×C4D21211yescharacteristic
152C22×C4C22×D5311yescharacteristic
153C2×C8—21no—
154C2×C8—21no—
155C4⋊C4—531no—
156C4⋊C4—531no—
157C4⋊C4—1061no—
158C4⋊C4—1061no—
159C4⋊C4—1061no—
160C4⋊C4—1061no—
161C4⋊C4—2122no—
162C22⋊C4—21no—
163C22⋊C4—21no—
164C22⋊C4—1061no—
165C22⋊C4—1061no—
166C22⋊C4—1061no—
167C22⋊C4—1061no—
168C22⋊C4—1061no—
169C22⋊C4—2122no—
170C42—1061no—
171C23—21no—
172C23—21no—
173C23—21no—
174C23—41no—
175C23—41no—
176C23—1061no—
177C23C2×D21211yescharacteristic
178Q8—2121no—
179D4—4241no—
180D4—4241no—
181D4—2121no—
182D4—2121no—
183D4—2121no—
184C2×C4—21no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—1062no—
188C2×C4—1061no—
189C2×C4—1061no—
190C2×C4—1061no—
191C2×C4—4241no—
192C2×C4—2121no—
193C2×C4—2121no—
194C2×C4—2121no—
195C2×C4—2121no—
196C2×C4—2121no—
197C2×C4D4×D5311yescharacteristic
198C2×C4D4×D5311yescharacteristic
199C8—41no—
200C8—41no—
201C22—21no—
202C22—21no—
203C22—42no—
204C22—41no—
205C22—41no—
206C22—41no—
207C22—81no—
208C22—2121no—
209C22—2121no—
210C22—2121no—
211C22#1313318711yescharacteristic
212C22#1313323911yescharacteristic
213C22#1313325011yescharacteristic
214C4—21no—
215C4—21no—
216C4—41no—
217C4—4241no—
218C4—2122no—
219C4—2121no—
220C4—2121no—
221C4—2121no—
222C2—21no—
223C2—21no—
224C2—81no—
225C2—4241no—
226C2#1681078711yescharacteristic
227C2#1681079311yescharacteristic
228C2#1681081411yescharacteristic
subgroupG/H
centerC22#13133187
commutatorC22×C4×C53C23
FrattiniC22×C4C22×D53
FittingC53×#323C2
socleC22×C53#120
hypercenter#323D53
hypocenterC53#1618
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #323

Complements

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

NH ≅ G/Ntype
2. #16810959C2semidirect
3. #16810942C2semidirect
4. #16810559C2semidirect
5. #16810361C2semidirect
6. #16810358C2semidirect
7. C53×#323C2semidirect
121. #323D53semidirect
122. C53#1618semidirect