Finite Group Databasebeta

← #20273673

Subgroups — #20273673

Order 6784.

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

normal subgroups only

This group has 256 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#16810781C211yesmaximal, characteristic, complemented
3#16810778C211yesmaximal, characteristic, complemented
4#16810536C211yesmaximal, characteristic
5#16810359C211yesmaximal, characteristic, complemented
6#16810346C211yesmaximal, characteristic
7C53×#328C211yesmaximal, characteristic, complemented
8C22×Dic106C2211yescharacteristic
9C2×Dic212—21no—
10C2×#11719925—21no—
11#13133197—21no—
12#13133196C2211yescharacteristic
13#13133195—21no—
14#13133195—21no—
15#13133194—21no—
16#13133193—21no—
17C2×#11719921—21no—
18C2×#11719921C2211yescharacteristic
19C22⋊C4×D53—21no—
20#13133142—21no—
21#13133140—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×Dic106—21no—
37C2×Dic106—21no—
38C2×Dic106—42no—
39C2×Dic106D411yescharacteristic
40C2×C8×C53—21no—
41C4⋊C4×C53—21no—
42C4⋊C4×C53D411yescharacteristic
43C22⋊C4×C53—21no—
44C22⋊Dic53—21no—
45#11719925—42no—
46#11719921—21no—
47#11719921—21no—
48#11719921—21no—
49#11719921—21no—
50#11719921—42no—
51C4⋊Dic53—21no—
52C4⋊Dic53D411yescharacteristic
53#11719919—21no—
54#11719919—21no—
55#11719919—21no—
56C2×#88322—21no—
57Dic212—42no—
58C23×C53C2×D411yescharacteristic
59C22×D53—21no—
60C22×D53—21no—
61C22×D53—41no—
62C22×D53—41no—
63C22×D53Q1612yescomplemented
64Q8×C53—41no—
65C2×C4×C53—21no—
66C2×C4×C53—21no—
67C2×C4×C53—41no—
68C2×C4×C53—41no—
69C2×C4×C53—41no—
70C2×C4×C53—41no—
71C2×C4×C53C2×D411yescharacteristic
72C2×C4×C53C2×D411yescharacteristic
73C2×Dic53—22no—
74C2×Dic53—21no—
75C2×Dic53—21no—
76C2×Dic53—21no—
77C2×Dic53—41no—
78C2×Dic53—41no—
79C2×Dic53—41no—
80C4×D53—81no—
81C4×D53—81no—
82Dic106—22no—
83Dic106—22no—
84Dic106—41no—
85Dic106—41no—
86Dic106—41no—
87Dic106—81no—
88C8×C53—41no—
89#88322—41no—
90C22×C53—21no—
91C22×C53—21no—
92C22×C53#12011yescharacteristic
93C22×C53C2×Q1611yescharacteristic
94C22×C53#13711yescharacteristic
95C2×D53—22no—
96C2×D53—41no—
97C2×D53—41no—
98C2×D53—42no—
99C2×D53—81no—
100C4×C53—21no—
101C4×C53—21no—
102C4×C53—41no—
103C4×C53—41no—
104C4×C53—81no—
105Dic53—41no—
106Dic53—41no—
107Dic53—42no—
108Dic53—41no—
109#1602—531nomaximal, Sylow
110C2×C53—21no—
111C2×C53—21no—
112C2×C53#45111yescharacteristic
113C2×C53#45511yescharacteristic
114C2×C53B(4,𝔽2)11yescharacteristic
115D53—42no—
116#543—531no—
117#451—531no—
118#451—531no—
119#393—531no—
120#328—531no—
121#328D5311yescharacteristic, complemented
122#323—531no—
123C53#160211yescharacteristic, complemented, Sylow
124C22×Q8—531no—
125C2×Q16—1061no—
126C2×Q16—1061no—
127C22⋊Q8—531no—
128C22⋊Q8—1061no—
129C22⋊Q8—1061no—
130C22⋊Q8—1061no—
131C22⋊Q8—1061no—
132C22⋊Q8—1061no—
133C22⋊Q8C2×D5311yescharacteristic
134C2×C22⋊C4—531no—
135C2×C22⋊C4—1061no—
136C2×C22⋊C4—1061no—
137#103—1061no—
138#103—1061no—
139C22⋊C8—531no—
140C22⋊C8C2×D5311yescharacteristic
141#95—1061no—
142#95C2×D5311yescharacteristic
143C24—531no—
144C2×Q8—531no—
145C2×Q8—1061no—
146C2×Q8—1061no—
147C2×Q8—1061no—
148C2×Q8—2122no—
149C2×Q8D21211yescharacteristic
150C22×C4—21no—
151C22×C4—531no—
152C22×C4—1061no—
153C22×C4—1061no—
154C22×C4—1061no—
155C22×C4C22×D5311yescharacteristic
156Q16—2122no—
157Q16—2122no—
158C2×C8—21no—
159C2×C8—1061no—
160C4⋊C4—21no—
161C4⋊C4—531no—
162C4⋊C4—1061no—
163C4⋊C4—1061no—
164C4⋊C4—1061no—
165C4⋊C4—1061no—
166C4⋊C4D21211yescharacteristic
167C22⋊C4—21no—
168C22⋊C4—1061no—
169C22⋊C4—1061no—
170C22⋊C4—1061no—
171C22⋊C4—1061no—
172C22⋊C4—1061no—
173C22⋊C4—2122no—
174C23—532no—
175C23—1061no—
176C23—1061no—
177C23—2121no—
178C23—2121no—
179C23D4×D5311yescharacteristic
180Q8—41no—
181Q8—1062no—
182Q8—1062no—
183Q8—4241no—
184Q8—2121no—
185Q8—2121no—
186Q8—2121no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—1062no—
194C2×C4—1061no—
195C2×C4—1061no—
196C2×C4—1061no—
197C2×C4—4241no—
198C2×C4—4241no—
199C2×C4—2121no—
200C2×C4—2121no—
201C2×C4—2121no—
202C2×C4C2×D21211yescharacteristic
203C2×C4D4×D5311yescharacteristic
204C8—41no—
205C8—2121no—
206C22—21no—
207C22—21no—
208C22—1062no—
209C22—4241no—
210C22—2122no—
211C22—2121no—
212C22—2121no—
213C22#1313318711yescharacteristic
214C22Q16×D5311yescharacteristic
215C22#1313324611yescharacteristic
216C4—21no—
217C4—21no—
218C4—41no—
219C4—41no—
220C4—81no—
221C4—2121no—
222C4—2121no—
223C4—2121no—
224C4—2122no—
225C2—21no—
226C2—21no—
227C2—2122no—
228C2#1681078011yescharacteristic
229C2#1681080211yescharacteristic
230C2#1681081111yescharacteristic
subgroupG/H
centerC22#13133187
commutatorC22×C4×C53C23
FrattiniC22×C4C22×D53
FittingC53×#328C2
socleC22×C53#120
hypercenter#328D53
hypocenterC53#1602
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. #16810781C2semidirect
3. #16810778C2semidirect
5. #16810359C2semidirect
7. C53×#328C2semidirect
63. C22×D53Q16semidirect
121. #328D53semidirect
123. C53#1602semidirect