Finite Group Databasebeta

← #20276341

Subgroups — #20276341

Order 6784.

220 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 75 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#131×D53C211yesmaximal, characteristic
2C2×#13133156C211yesmaximal, characteristic, complemented
3M32×D53C211yesmaximal, characteristic, complemented
4#16810408C211yesmaximal, characteristic, complemented
5#16810407C211yesmaximal, characteristic, complemented
6#16810406C211yesmaximal, characteristic, complemented
7C53×#350C211yesmaximal, characteristic, complemented
8#41×D53C411yescharacteristic
9#13133225—21no—
10#13133225C411yescharacteristic
11#13133224C411yescharacteristic
12#13133223—21no—
13#13133223C2211yescharacteristic
14M16×D53—21no—
15M16×D53C411yescharacteristic
16C2×C8×D53—21no—
17C2×C8×D53C2211yescharacteristic
18#13133157C2211yescharacteristic, complemented
19#13133156—21no—
20#13133156—21no—
21#13133156—21no—
22C2×#11719908C2211yescharacteristic, complemented
23C16×D53—21no—
24#131×C53C2211yescharacteristic
25M32×C53C2211yescharacteristic, complemented
26C2×C16×C53C2211yescharacteristic, complemented
27#41×C53C2×C411yescharacteristic
28#11719949C811yescharacteristic
29Q8×D53C811yescharacteristic
30#11719947—21no—
31#11719947C811yescharacteristic
32D4×D53—21no—
33D4×D53C811yescharacteristic
34#11719945—21no—
35#11719945C2×C411yescharacteristic
36C2×C4×D53—21no—
37C2×C4×D53C2×C411yescharacteristic
38M16×C53—21no—
39M16×C53C2×C411yescharacteristic
40C2×C8×C53—21no—
41C2×C8×C53C2311yescharacteristic
42#11719917—21no—
43#11719917C2×C411yescharacteristic
44C2×#88322—21no—
45C2×#88322C2×C411yescharacteristic
46#11719912—21no—
47#11719912—21no—
48#11719912—21no—
49C8×D53—21no—
50C8×D53—21no—
51C8×D53—21no—
52C8×D53D411yescharacteristic
53C8×D53D411yescharacteristic
54C8×D53D411yescharacteristic
55C8×D53D411yescharacteristic
56C16×C53—21no—
57C16×C53—21no—
58#11719908—21no—
59#11719908—21no—
60C22×D53—21no—
61C22×D53M1611yescharacteristic
62Q8×C53C2×C811yescharacteristic
63D4×C53—21no—
64D4×C53C2×C811yescharacteristic
65C2×C4×C53—21no—
66C2×C4×C53C22×C411yescharacteristic
67C22⋊D53—21no—
68C22⋊D53—21no—
69C22⋊D53—21no—
70C2×Dic53—21no—
71C2×Dic53M1611yescharacteristic
72D212—21no—
73D212C2×C811yescharacteristic
74C4×D53—21no—
75C4×D53—21no—
76C4×D53—21no—
77C4×D53C22⋊C411yescharacteristic
78C4×D53C22⋊C411yescharacteristic
79C4×D53C2×C811yescharacteristic
80C4×D53C2×C811yescharacteristic
81Dic106—21no—
82Dic106C2×C811yescharacteristic
83C8×C53—21no—
84C8×C53C2×D411yescharacteristic
85C8×C53C2×D411yescharacteristic
86#88322—21no—
87#88322C22⋊C411yescharacteristic
88#88322C22⋊C411yescharacteristic
89C22×C53—21no—
90C22×C53C2×M1611yescharacteristic
91C2×D53—21no—
92C2×D53—41no—
93C2×D53—42no—
94C2×D53C22⋊C811yescharacteristic
95C2×D53C22⋊C811yescharacteristic
96C4×C53—21no—
97C4×C53C2×C22⋊C411yescharacteristic
98C4×C53C22×C811yescharacteristic
99Dic53—21no—
100Dic53C22⋊C811yescharacteristic
101Dic53C22⋊C811yescharacteristic
102#2117—531nomaximal, Sylow
103C2×C53—21no—
104C2×C53—41no—
105C2×C53C2×C22⋊C811yescharacteristic
106D53—21no—
107D53—21no—
108D53—41no—
109C2×#131—531no—
110C2×M32—531no—
111C2×M32—531no—
112#350—531no—
113#350—531no—
114#350—531no—
115#350D5311yescharacteristic, complemented
116C53#211711yescharacteristic, complemented, Sylow
117C2×#41—531no—
118#131—531no—
119#131—531no—
120#131—531no—
121#131—1061no—
122#131—1061no—
123#131C2×D5311yescharacteristic
124C2×M16—531no—
125C2×M16—1061no—
126C22×C8—531no—
127C22×C8—1061no—
128M32—531no—
129M32—1061no—
130M32—1061no—
131M32—1061no—
132M32C2×D5311yescharacteristic, complemented
133C2×C16—531no—
134C2×C16—1061no—
135C2×C16C2×D5311yescharacteristic, complemented
136#41—531no—
137#41—531no—
138#41—531no—
139#41—1061no—
140#41—1061no—
141#41C4×D5311yescharacteristic
142C2×Q8—531no—
143C2×D4—531no—
144C2×D4—1061no—
145C22×C4—531no—
146C22×C4—1061no—
147M16—21no—
148M16—531no—
149M16—1061no—
150M16—1061no—
151M16—1061no—
152M16—1061no—
153M16C4×D5311yescharacteristic
154C2×C8—21no—
155C2×C8—531no—
156C2×C8—531no—
157C2×C8—531no—
158C2×C8—531no—
159C2×C8—531no—
160C2×C8—1061no—
161C2×C8—1061no—
162C2×C8—1061no—
163C2×C8—1061no—
164C2×C8C22×D5311yescharacteristic
165C16—21no—
166C16—21no—
167C16—1061no—
168C16—1061no—
169C23—531no—
170C23—1061no—
171Q8—531no—
172Q8—1061no—
173Q8C8×D5311yescharacteristic
174D4—21no—
175D4—531no—
176D4—1061no—
177D4—1061no—
178D4—1061no—
179D4—1061no—
180D4C8×D5311yescharacteristic
181C2×C4—21no—
182C2×C4—531no—
183C2×C4—531no—
184C2×C4—531no—
185C2×C4—531no—
186C2×C4—531no—
187C2×C4—1061no—
188C2×C4—1061no—
189C2×C4—1061no—
190C2×C4—1061no—
191C2×C4C2×C4×D5311yescharacteristic
192C8—21no—
193C8—531no—
194C8—531no—
195C8—1061no—
196C8D4×D5311yescharacteristic
197C8D4×D5311yescharacteristic
198C22—21no—
199C22—531no—
200C22—531no—
201C22—1061no—
202C22—2121no—
203C22—2122no—
204C22M16×D5311yescharacteristic
205C4—21no—
206C4—531no—
207C4—531no—
208C4—1061no—
209C4C22⋊C4×D5311yescharacteristic
210C4C2×C8×D5311yescharacteristic
211C2—21no—
212C2—41no—
213C2—1061no—
214C2—1061no—
215C2—2121no—
216C2C22⋊C8×D5311yescharacteristic
subgroupG/H
centerC8D4×D53
commutatorC4×C53C22×C8
FrattiniC2×C8C22×D53
FittingC53×#350C2
socleC2×C53C2×C22⋊C8
hypercenter#350D53
hypocenterC53#2117
LCSG ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C8 ◁ C2×C8 ◁ #350

Complements

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

NH ≅ G/Ntype
2. C2×#13133156C2semidirect
3. M32×D53C2semidirect
4. #16810408C2semidirect
5. #16810407C2semidirect
6. #16810406C2semidirect
7. C53×#350C2semidirect
18. #13133157C22semidirect
22. C2×#11719908C22semidirect
25. M32×C53C22semidirect
26. C2×C16×C53C22semidirect
115. #350D53semidirect
116. C53#2117semidirect
132. M32C2×D53semidirect
135. C2×C16C2×D53semidirect