Finite Group Databasebeta

← #20273311

Subgroups — #20273311

Order 6784.

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

normal subgroups only

This group has 264 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#124×D53C211yesmaximal, characteristic
2#16810678C211yesmaximal, characteristic, complemented
3#16810575C211yesmaximal, characteristic, complemented
4#16810363C211yesmaximal, characteristic, complemented
5#16810362C211yesmaximal, characteristic, complemented
6#16810362C211yesmaximal, characteristic, complemented
7C53×#329C211yesmaximal, characteristic, complemented
8C2×Q8×D53C411yescharacteristic
9C2×D4×D53C411yescharacteristic
10C2×#11719912—21no—
11#13133206C411yescharacteristic
12#13133205C2211yescharacteristic
13#13133204C411yescharacteristic
14#13133203—21no—
15#13133203—21no—
16C22⋊C4×D53—21no—
17C22⋊C4×D53—21no—
18C42×D53C2211yescharacteristic
19#13133146—21no—
20#13133129—21no—
21#13133128C2211yescharacteristic, complemented
22#13133128C2211yescharacteristic, complemented
23C8×Dic53—21no—
24#124×C53C2211yescharacteristic
25#97×C53C2211yescharacteristic, complemented
26#97×C53C2211yescharacteristic, complemented
27C23×D53—21no—
28C2×Q8×C53C2×C411yescharacteristic
29C2×D4×C53C2×C411yescharacteristic
30C2×C22⋊D53—21no—
31Q8×D53—42no—
32D4×D53—42no—
33C2×D212C2×C411yescharacteristic
34C2×C4×D53—21no—
35C2×C4×D53C2×C411yescharacteristic
36C2×C4×D53D411yescharacteristic
37C2×C4×D53D411yescharacteristic
38C2×Dic106—21no—
39C2×Dic106C2×C411yescharacteristic
40C2×C8×C53—21no—
41C2×C8×C53—21no—
42C22⋊C4×C53—21no—
43C22⋊C4×C53—21no—
44C42×C53C2311yescharacteristic
45C22⋊Dic53—21no—
46C22⋊Dic53—21no—
47#11719921—21no—
48#11719921—21no—
49#11719921—21no—
50#11719921—21no—
51C4×Dic53C2×C411yescharacteristic
52C4×Dic53D411yescharacteristic
53C4×Dic53D411yescharacteristic
54C2×#88322—21no—
55C2×#88322—21no—
56#11719912—24no—
57C23×C53—21no—
58C22×D53—21no—
59C22×D53—42no—
60C22×D53—42no—
61C22×D53—42no—
62C22×D53C22⋊C411yescharacteristic
63Q8×C53—41no—
64D4×C53—41no—
65C2×C4×C53—21no—
66C2×C4×C53C22×C411yescharacteristic
67C2×C4×C53C2×D411yescharacteristic
68C2×C4×C53C2×D411yescharacteristic
69C22⋊D53—42no—
70C2×Dic53—21no—
71C2×Dic53C22⋊C411yescharacteristic
72C2×Dic53C22⋊C411yescharacteristic
73C2×Dic53C22⋊C411yescharacteristic
74D212—41no—
75C4×D53—22no—
76C4×D53—22no—
77C4×D53—41no—
78C4×D53—42no—
79Dic106—42no—
80Dic106—41no—
81C8×C53—22no—
82C8×C53—41no—
83#88322—22no—
84#88322—41no—
85C22×C53—41no—
86C22×C53—41no—
87C22×C53—41no—
88C22×C53C2×C22⋊C411yescharacteristic
89C2×D53—21no—
90C2×D53—21no—
91C2×D53—41no—
92C2×D53—41no—
93C2×D53—41no—
94C2×D53—84no—
95C2×D53#10412yes—
96C4×C53—21no—
97C4×C53—21no—
98C4×C53—21no—
99C4×C53—41no—
100Dic53—21no—
101Dic53—21no—
102Dic53—41no—
103Dic53#10012yes—
104#1527—531nomaximal, Sylow
105C2×C53—81no—
106C2×C53C2×#10011yescharacteristic
107C2×C53C2×#10411yescharacteristic
108C2×C53#42111yescharacteristic
109D53—22no—
110D53—81no—
111C2×#124—531no—
112#432—531no—
113C4×M16—531no—
114#329—531no—
115#329—531no—
116#329—531no—
117#329D5311yescharacteristic, complemented
118C53#152711yescharacteristic, complemented, Sylow
119C22×Q8—531no—
120C22×D4—531no—
121C2×M16—1061no—
122#124—531no—
123#124—531no—
124#124—531no—
125#124—1061no—
126#124—1061no—
127#124C2×D5311yescharacteristic
128C2×C22⋊C4—1061no—
129C2×C22⋊C4—1061no—
130C2×C42—531no—
131C4⋊C8—1061no—
132C22⋊C8—1061no—
133#97—531no—
134#97—531no—
135#97C2×D5311yescharacteristic, complemented
136#97C2×D5311yescharacteristic, complemented
137C4×C8—1061no—
138C24—1061no—
139C2×Q8—531no—
140C2×Q8—1061no—
141C2×Q8—2122no—
142C2×Q8C4×D5311yescharacteristic
143C2×D4—531no—
144C2×D4—1061no—
145C2×D4—2122no—
146C2×D4C4×D5311yescharacteristic
147C22×C4—531no—
148C22×C4—531no—
149C22×C4—531no—
150C22×C4—1061no—
151M16—1064no—
152C2×C8—21no—
153C2×C8—21no—
154C2×C8—1061no—
155C2×C8—1061no—
156C22⋊C4—21no—
157C22⋊C4—21no—
158C22⋊C4—1061no—
159C22⋊C4—1061no—
160C22⋊C4—1061no—
161C22⋊C4—1061no—
162C22⋊C4—1061no—
163C22⋊C4—1061no—
164C42—531no—
165C42—531no—
166C42—531no—
167C42C22×D5311yescharacteristic
168C23—21no—
169C23—531no—
170C23—1061no—
171C23—2122no—
172C23—2122no—
173C23—2122no—
174Q8—41no—
175Q8—2121no—
176Q8—2122no—
177D4—41no—
178D4—2121no—
179D4—2122no—
180C2×C4—21no—
181C2×C4—531no—
182C2×C4—531no—
183C2×C4—531no—
184C2×C4—1062no—
185C2×C4—1062no—
186C2×C4—1061no—
187C2×C4—2121no—
188C2×C4—2122no—
189C2×C4C2×C4×D5311yescharacteristic
190C2×C4D4×D5311yescharacteristic
191C2×C4D4×D5311yescharacteristic
192C8—22no—
193C8—41no—
194C8—1062no—
195C8—2121no—
196C22—41no—
197C22—41no—
198C22—41no—
199C22—532no—
200C22—1061no—
201C22—1061no—
202C22—4244no—
203C22—2121no—
204C22—2121no—
205C22—2121no—
206C22C22⋊C4×D5311yescharacteristic
207C4—21no—
208C4—21no—
209C4—21no—
210C4—41no—
211C4—532no—
212C4—1061no—
213C4—1061no—
214C4—2121no—
215C2—81no—
216C2—1062no—
217C2—4241no—
218C2#100×D5311yescharacteristic
219C2#104×D5311yescharacteristic
220C2#1681063811yescharacteristic
subgroupG/H
centerC22C22⋊C4×D53
commutatorC2×C4×C53C22×C4
FrattiniC42C22×D53
FittingC53×#329C2
socleC22×C53C2×C22⋊C4
hypercenter#329D53
hypocenterC53#1527
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #329

Complements

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

NH ≅ G/Ntype
2. #16810678C2semidirect
3. #16810575C2semidirect
4. #16810363C2semidirect
5. #16810362C2semidirect
6. #16810362C2semidirect
7. C53×#329C2semidirect
21. #13133128C22semidirect
22. #13133128C22semidirect
25. #97×C53C22semidirect
26. #97×C53C22semidirect
117. #329D53semidirect
118. C53#1527semidirect
135. #97C2×D53semidirect
136. #97C2×D53semidirect