Finite Group Databasebeta

← #20273670

Subgroups — #20273670

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#16810779C211yesmaximal, characteristic, complemented
3#16810534C211yesmaximal, characteristic
4#16810361C211yesmaximal, characteristic, complemented
5#16810360C211yesmaximal, characteristic, complemented
6#16810346C211yesmaximal, characteristic
7C53×#451C211yesmaximal, characteristic, complemented
8C2×#11719925—21no—
9#13133197—21no—
10#13133196C2211yescharacteristic
11#13133195—21no—
12#13133195—21no—
13#13133194—21no—
14#13133193—21no—
15C2×#11719921—21no—
16C2×#11719921C2211yescharacteristic
17C22⋊C4×D53—21no—
18#13133141—21no—
19C22⋊#88322C2211yescharacteristic
20#13133124—21no—
21#13133124C2211yescharacteristic
22C22×Q8×C53C2211yescharacteristic
23C2×Q16×C53—21no—
24C22⋊Q8×C53C2211yescharacteristic
25#103×C53—21no—
26C22⋊C8×C53C2211yescharacteristic
27C23×D53D411yescharacteristic
28C2×Q8×C53—21no—
29C2×Q8×C53—21no—
30C2×Q8×C53—42no—
31C2×Q8×C53D411yescharacteristic
32C22×C4×C53—21no—
33C22×C4×C53C2311yescharacteristic
34C22×Dic53—21no—
35C22×Dic53D411yescharacteristic
36C2×C4×D53—21no—
37C2×C4×D53—21no—
38C2×Dic106—21no—
39C2×Dic106D411yescharacteristic
40Q16×C53—42no—
41C2×C8×C53—21no—
42C4⋊C4×C53—21no—
43C4⋊C4×C53D411yescharacteristic
44C22⋊C4×C53—21no—
45C22⋊Dic53—21no—
46#11719925—42no—
47#11719921—21no—
48#11719921—21no—
49#11719921—21no—
50#11719921—21no—
51#11719921—42no—
52C4⋊Dic53—21no—
53C4⋊Dic53D411yescharacteristic
54#11719919—21no—
55#11719919—21no—
56#11719919—21no—
57C2×#88322—21no—
58C23×C53C2×D411yescharacteristic
59C22×D53—21no—
60C22×D53—21no—
61C22×D53—41no—
62C22×D53—41no—
63C22×D53Q1612yescomplemented
64Q8×C53—22no—
65Q8×C53—22no—
66Q8×C53—41no—
67Q8×C53—41no—
68Q8×C53—41no—
69C2×C4×C53—22no—
70C2×C4×C53—21no—
71C2×C4×C53—21no—
72C2×C4×C53—41no—
73C2×C4×C53—41no—
74C2×C4×C53—41no—
75C2×C4×C53C2×D411yescharacteristic
76C2×C4×C53C2×D411yescharacteristic
77C2×Dic53—21no—
78C2×Dic53—21no—
79C2×Dic53—21no—
80C2×Dic53—41no—
81C2×Dic53—41no—
82C2×Dic53—41no—
83C2×Dic53—41no—
84C4×D53—81no—
85C4×D53—81no—
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—42no—
105Dic53—41no—
106Dic53—41no—
107Dic53—41no—
108Dic53—81no—
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#451D5311yescharacteristic, complemented
119#393—531no—
120#328—531no—
121#328—531no—
122#323—531no—
123C53#160211yescharacteristic, complemented, Sylow
124C22×Q8C2×D5311yescharacteristic
125C2×Q16—21no—
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—21no—
138#103—1061no—
139C22⋊C8—531no—
140C22⋊C8C2×D5311yescharacteristic
141#95—531no—
142#95—1061no—
143C24—531no—
144C2×Q8—21no—
145C2×Q8—21no—
146C2×Q8—42no—
147C2×Q8—531no—
148C2×Q8—1061no—
149C2×Q8C22⋊D5311yescharacteristic
150C22×C4—21no—
151C22×C4—531no—
152C22×C4—1061no—
153C22×C4—1061no—
154C22×C4—1061no—
155C22×C4C22×D5311yescharacteristic
156Q16—42no—
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⋊C4C22⋊D5311yescharacteristic
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—22no—
181Q8—22no—
182Q8—41no—
183Q8—41no—
184Q8—41no—
185Q8—4241no—
186Q8—2121no—
187C2×C4—22no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—1061no—
194C2×C4—1061no—
195C2×C4—1061no—
196C2×C4—4241no—
197C2×C4—4241no—
198C2×C4—2121no—
199C2×C4—2121no—
200C2×C4—2121no—
201C2×C4—2121no—
202C2×C4D4×D5311yescharacteristic
203C2×C4C2×C22⋊D5311yescharacteristic
204C8—41no—
205C8—2121no—
206C22—21no—
207C22—21no—
208C22—1062no—
209C22—4241no—
210C22—2122no—
211C22—2121no—
212C22—2121no—
213C22#1313318511yescharacteristic
214C22Q16×D5311yescharacteristic
215C22#1313324911yescharacteristic
216C4—21no—
217C4—21no—
218C4—41no—
219C4—42no—
220C4—41no—
221C4—4241no—
222C4—2121no—
223C4—2121no—
224C4—2121no—
225C2—21no—
226C2—21no—
227C2—2122no—
228C2#1681077711yescharacteristic
229C2#1681079911yescharacteristic
230C2#1681080911yescharacteristic
subgroupG/H
centerC22#13133185
commutatorC22×C4×C53C23
FrattiniC22×C4C22×D53
FittingC53×#451C2
socleC22×C53#120
hypercenter#451D53
hypocenterC53#1602
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
2. #16810779C2semidirect
4. #16810361C2semidirect
5. #16810360C2semidirect
7. C53×#451C2semidirect
63. C22×D53Q16semidirect
118. #451D53semidirect
123. C53#1602semidirect