Finite Group Databasebeta

← #20273693

Subgroups — #20273693

Order 6784.

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

normal subgroups only

This group has 260 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#16810779C211yesmaximal, characteristic, complemented
3#16810762C211yesmaximal, characteristic, complemented
4#16810359C211yesmaximal, characteristic, complemented
5#16810356C211yesmaximal, characteristic, complemented
6#16810347C211yesmaximal, characteristic, complemented
7C53×#393C211yesmaximal, characteristic, complemented
8C2×#11719925—21no—
9C2×#11719924—21no—
10#13133210—21no—
11#13133203—21no—
12#13133199—21no—
13#13133196C2211yescharacteristic
14#13133191C2211yescharacteristic
15#13133190—21no—
16#13133186—21no—
17C4×C22⋊D53—21no—
18#13133141—21no—
19#13133141—21no—
20C22⋊#88322C2211yescharacteristic
21C22⋊#88322C2211yescharacteristic
22C22×Q8×C53C2211yescharacteristic
23C2×C22⋊C4×C53—21no—
24C2×C22⋊C4×C53C2211yescharacteristic
25#95×C53—21no—
26#95×C53C2211yescharacteristic
27C24×C53D411yescharacteristic
28C2×Q8×C53—21no—
29C2×Q8×C53—21no—
30C2×Q8×C53—42no—
31C22×C4×C53—21no—
32C22×C4×C53—21no—
33C22×C4×C53D411yescharacteristic
34C22×C4×C53C2311yescharacteristic
35C2×C22⋊D53—21no—
36C2×C22⋊D53—21no—
37C2×C22⋊D53—21no—
38C2×D212D411yescharacteristic
39C2×C4×D53—21no—
40C2×Dic106D411yescharacteristic
41C22⋊C4×C53—21no—
42C22⋊C4×C53—21no—
43C22⋊C4×C53—42no—
44C22⋊Dic53—21no—
45C22⋊Dic53—21no—
46C22⋊Dic53—21no—
47#11719925—42no—
48#11719924—42no—
49#11719921—21no—
50#11719921—21no—
51C4⋊Dic53D411yescharacteristic
52C4⋊Dic53D411yescharacteristic
53#11719919—21no—
54#11719919—21no—
55C4×Dic53—21no—
56C2×#88322—21no—
57C2×#88322—21no—
58C23×C53—21no—
59C23×C53—21no—
60C23×C53—21no—
61C23×C53—41no—
62C23×C53—41no—
63C23×C53C2×D411yescharacteristic
64C22×D53—21no—
65Q8×C53—22no—
66Q8×C53—22no—
67Q8×C53—41no—
68Q8×C53—41no—
69C2×C4×C53—22no—
70C2×C4×C53—21no—
71C2×C4×C53—41no—
72C2×C4×C53—41no—
73C2×C4×C53—41no—
74C2×C4×C53—41no—
75C2×C4×C53—41no—
76C2×C4×C53—41no—
77C2×C4×C53—41no—
78C2×C4×C53C2×D411yescharacteristic
79C2×C4×C53C2×D411yescharacteristic
80C22⋊D53—41no—
81C22⋊D53—41no—
82C22⋊D53—81no—
83C22⋊D53—81no—
84C2×Dic53—21no—
85C2×Dic53—21no—
86C2×Dic53—21no—
87D212—41no—
88C4×D53—81no—
89Dic106—41no—
90#88322—41no—
91#88322—41no—
92C22×C53—21no—
93C22×C53—21no—
94C22×C53—42no—
95C22×C53—41no—
96C22×C53—41no—
97C22×C53—41no—
98C22×C53—81no—
99C22×C53#12011yescharacteristic
100C22×C53#13511yescharacteristic
101C22×C53#13711yescharacteristic
102C2×D53—41no—
103C2×D53—41no—
104C2×D53—41no—
105C4×C53—21no—
106C4×C53—21no—
107C4×C53—41no—
108C4×C53—42no—
109C4×C53—81no—
110Dic53—41no—
111Dic53—41no—
112Dic53—41no—
113#1606—531nomaximal, Sylow
114C2×C53—21no—
115C2×C53—21no—
116C2×C53—81no—
117C2×C53#45211yescharacteristic
118C2×C53#45511yescharacteristic
119C2×C53B(4,𝔽2)11yescharacteristic
120D53—81no—
121#538—531no—
122#451—531no—
123#448—531no—
124#393D5311yescharacteristic, complemented
125#328—531no—
126#327—531no—
127#323—531no—
128C53#160611yescharacteristic, complemented, Sylow
129C22×Q8C2×D5311yescharacteristic
130C2×Q16—1061no—
131C2×SD16—1061no—
132#126—1061no—
133#124—1061no—
134#123—1061no—
135C22⋊Q8—531no—
136#121—531no—
137#121—1061no—
138#120—1061no—
139C4×D4—1061no—
140C2×C22⋊C4—21no—
141C2×C22⋊C4C2×D5311yescharacteristic
142#103—1061no—
143#103—1061no—
144C22⋊C8—531no—
145C22⋊C8—531no—
146#95—21no—
147#95C2×D5311yescharacteristic
148C24C22⋊D5311yescharacteristic
149C2×Q8—21no—
150C2×Q8—21no—
151C2×Q8—42no—
152C2×Q8—531no—
153C2×D4—531no—
154C2×D4—1061no—
155C2×D4—1061no—
156C2×D4—1061no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—1061no—
160C22×C4C22⋊D5311yescharacteristic
161C22×C4C22×D5311yescharacteristic
162Q16—2122no—
163SD16—2122no—
164C2×C8—1061no—
165C2×C8—1061no—
166C4⋊C4—531no—
167C4⋊C4—531no—
168C4⋊C4—1061no—
169C4⋊C4—1061no—
170C22⋊C4—21no—
171C22⋊C4—21no—
172C22⋊C4—42no—
173C22⋊C4—1061no—
174C22⋊C4—1061no—
175C22⋊C4—1061no—
176C22⋊C4—1061no—
177C22⋊C4—1061no—
178C42—1061no—
179C23—21no—
180C23—21no—
181C23—21no—
182C23—41no—
183C23—41no—
184C23—1061no—
185C23C2×C22⋊D5311yescharacteristic
186Q8—22no—
187Q8—22no—
188Q8—41no—
189Q8—41no—
190Q8—2121no—
191D4—4241no—
192D4—4241no—
193D4—2121no—
194D4—2121no—
195D4—2121no—
196C2×C4—22no—
197C2×C4—21no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—1061no—
206C2×C4—1061no—
207C2×C4—1061no—
208C2×C4—4241no—
209C2×C4D4×D5311yescharacteristic
210C2×C4D4×D5311yescharacteristic
211C8—2121no—
212C8—2121no—
213C22—21no—
214C22—21no—
215C22—42no—
216C22—41no—
217C22—41no—
218C22—41no—
219C22—81no—
220C22—2121no—
221C22—2121no—
222C22—2121no—
223C22#1313318511yescharacteristic
224C22#1313323711yescharacteristic
225C22#1313324811yescharacteristic
226C4—21no—
227C4—21no—
228C4—41no—
229C4—42no—
230C4—81no—
231C4—2121no—
232C4—2121no—
233C4—2121no—
234C2—21no—
235C2—21no—
236C2—81no—
237C2—4241no—
238C2#1681078311yescharacteristic
239C2#1681080011yescharacteristic
240C2#1681080911yescharacteristic
subgroupG/H
centerC22#13133185
commutatorC22×C4×C53C23
FrattiniC22×C4C22×D53
FittingC53×#393C2
socleC22×C53#120
hypercenter#393D53
hypocenterC53#1606
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #393

Complements

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

NH ≅ G/Ntype
2. #16810779C2semidirect
3. #16810762C2semidirect
4. #16810359C2semidirect
5. #16810356C2semidirect
6. #16810347C2semidirect
7. C53×#393C2semidirect
124. #393D53semidirect
128. C53#1606semidirect