Finite Group Databasebeta

← #20273289

Subgroups — #20273289

Order 6784.

218 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 61 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#16811154C211yesmaximal, characteristic
2#16810673C211yesmaximal, characteristic, complemented
3#16810490C211yesmaximal, characteristic, complemented
4#16810361C211yesmaximal, characteristic, complemented
5#16810359C211yesmaximal, characteristic, complemented
6#16810357C211yesmaximal, characteristic
7C53×#327C211yesmaximal, characteristic
8#13133210—21no—
9#13133208C411yescharacteristic, complemented
10#13133204C411yescharacteristic, complemented
11#13133198—21no—
12#13133196C2211yescharacteristic
13#13133182C411yescharacteristic, complemented
14D4×Dic53—21no—
15D4×Dic53C411yescharacteristic, complemented
16#13133173—21no—
17C2×C4×Dic53C2211yescharacteristic
18#13133146—21no—
19C22⋊#88322C2211yescharacteristic
20C8×Dic53—21no—
21#13133124—21no—
22#13133124—21no—
23#13133124C2211yescharacteristic
24#121×C53C2211yescharacteristic
25C22⋊C8×C53C2211yescharacteristic
26#95×C53C2211yescharacteristic
27C2×D4×C53—21no—
28C2×D4×C53C2×C411yescharacteristic
29C22×C4×C53—21no—
30C22×C4×C53C2311yescharacteristic
31C22×Dic53—21no—
32C22×Dic53—21no—
33C22×Dic53D411yescharacteristic
34C22×Dic53D411yescharacteristic
35C2×Dic106C2×C411yescharacteristic
36C2×C8×C53—21no—
37C4⋊C4×C53C2×C411yescharacteristic
38C22⋊C4×C53—21no—
39C22⋊Dic53—21no—
40C22⋊Dic53—21no—
41C22⋊Dic53—21no—
42C22⋊Dic53—21no—
43C4⋊Dic53—21no—
44C4⋊Dic53C2×C411yescharacteristic
45#11719919—21no—
46#11719919—21no—
47#11719919—21no—
48C4×Dic53—21no—
49C4×Dic53C2×C411yescharacteristic
50C4×Dic53C2×C411yescharacteristic
51C4×Dic53D411yescharacteristic
52C4×Dic53D411yescharacteristic
53C2×#88322—21no—
54C23×C53—21no—
55C23×C53C2×D411yescharacteristic
56D4×C53—41no—
57D4×C53—41no—
58D4×C53—41no—
59C2×C4×C53—21no—
60C2×C4×C53—41no—
61C2×C4×C53—41no—
62C2×C4×C53—41no—
63C2×C4×C53—41no—
64C2×C4×C53C22×C411yescharacteristic
65C2×C4×C53C2×D411yescharacteristic
66C2×Dic53—21no—
67C2×Dic53—21no—
68C2×Dic53—21no—
69C2×Dic53—21no—
70C2×Dic53—41no—
71C2×Dic53—42no—
72C2×Dic53—41no—
73C2×Dic53—41no—
74C2×Dic53—41no—
75C2×Dic53C22⋊C411yescharacteristic
76C2×Dic53C22⋊C411yescharacteristic
77C2×Dic53C22⋊C411yescharacteristic
78C2×Dic53C22⋊C411yescharacteristic
79Dic106—41no—
80C8×C53—41no—
81#88322—41no—
82C22×C53—21no—
83C22×C53—21no—
84C22×C53—41no—
85C22×C53—41no—
86C22×C53—41no—
87C22×C53C2×C22⋊C411yescharacteristic
88C22×C53#13511yescharacteristic
89C22×C53#13511yescharacteristic
90C4×C53—21no—
91C4×C53—21no—
92C4×C53—41no—
93C4×C53—81no—
94Dic53—21no—
95Dic53—21no—
96Dic53—21no—
97Dic53—21no—
98Dic53—41no—
99Dic53—41no—
100Dic53—81no—
101#1521—531nomaximal, Sylow
102C2×C53—21no—
103C2×C53—21no—
104C2×C53—81no—
105C2×C53#41011yescharacteristic
106C2×C53#41611yescharacteristic
107C2×C53#42111yescharacteristic
108#529—531no—
109#431—531no—
110#383—531no—
111#328—531no—
112#328—531no—
113#327—531no—
114#327D5311yescharacteristic
115C53#152111yescharacteristic, complemented, Sylow
116#126—1061no—
117#125—531no—
118#124—531no—
119#123—1061no—
120C22⋊Q8—531no—
121#121C2×D5311yescharacteristic
122C4×Q8—531no—
123C4×D4—531no—
124C4×D4—1061no—
125#117—1061no—
126C2×C42—531no—
127C4⋊C8—1061no—
128C22⋊C8—531no—
129C22⋊C8C2×D5311yescharacteristic
130C4×C8—1061no—
131#95—531no—
132#95—1061no—
133#95—1061no—
134#95C2×D5311yescharacteristic
135C2×Q8—531no—
136C2×D4—21no—
137C2×D4C4×D5311yescharacteristic
138C22×C4—21no—
139C22×C4—531no—
140C22×C4—531no—
141C22×C4—1061no—
142C22×C4—1061no—
143C22×C4C22×D5311yescharacteristic
144C2×C8—21no—
145C2×C8—1061no—
146C4⋊C4—531no—
147C4⋊C4—1061no—
148C4⋊C4—1061no—
149C4⋊C4—1061no—
150C4⋊C4—1061no—
151C4⋊C4C4×D5311yescharacteristic
152C22⋊C4—21no—
153C22⋊C4—1061no—
154C22⋊C4—1061no—
155C22⋊C4—1061no—
156C22⋊C4—1061no—
157C42—531no—
158C42—531no—
159C42—531no—
160C42—531no—
161C42—1061no—
162C23—21no—
163C23D4×D5311yescharacteristic
164Q8—2121no—
165D4—41no—
166D4—41no—
167D4—41no—
168C2×C4—21no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—531no—
174C2×C4—531no—
175C2×C4—531no—
176C2×C4—531no—
177C2×C4—1061no—
178C2×C4—1061no—
179C2×C4—1061no—
180C2×C4—1061no—
181C2×C4—2121no—
182C2×C4—2122no—
183C2×C4—2121no—
184C2×C4—2121no—
185C2×C4—2121no—
186C2×C4C2×C4×D5311yescharacteristic
187C2×C4D4×D5311yescharacteristic
188C8—41no—
189C8—2121no—
190C22—21no—
191C22—21no—
192C22—41no—
193C22—41no—
194C22—41no—
195C22C22⋊C4×D5311yescharacteristic
196C22#1313323611yescharacteristic
197C22#1313323811yescharacteristic
198C4—21no—
199C4—21no—
200C4—41no—
201C4—81no—
202C4—1061no—
203C4—1061no—
204C4—1061no—
205C4—1061no—
206C4—4241no—
207C4—2121no—
208C4—2121no—
209C2—21no—
210C2—21no—
211C2—81no—
212C2#1681059511yescharacteristic
213C2#1681061611yescharacteristic
214C2#1681063811yescharacteristic
subgroupG/H
centerC22C22⋊C4×D53
commutatorC2×C4×C53C22×C4
FrattiniC22×C4C22×D53
FittingC53×#327C2
socleC22×C53C2×C22⋊C4
hypercenter#327D53
hypocenterC53#1521
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #327

Complements

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

NH ≅ G/Ntype
2. #16810673C2semidirect
3. #16810490C2semidirect
4. #16810361C2semidirect
5. #16810359C2semidirect
9. #13133208C4semidirect
10. #13133204C4semidirect
13. #13133182C4semidirect
15. D4×Dic53C4semidirect
115. C53#1521semidirect