Finite Group Databasebeta

← #20273633

Subgroups — #20273633

Order 6784.

368 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 57 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#16810756C212yesmaximal, complemented
2#16810532C211yesmaximal, characteristic, complemented
3#16810357C212yesmaximal, complemented
4#16810347C211yesmaximal, characteristic, complemented
5C53×#534C211yesmaximal, characteristic, complemented
6C22×D212C2211yescharacteristic, complemented
7C22×C22⋊D53—21no—
8C2×#11719922—22no—
9C2×#11719921—21no—
10#13133137—22no—
11C22⋊#88322C2212yescomplemented
12#13133124C2211yescharacteristic, complemented
13C22×D4×C53—21no—
14#123×C53—22no—
15#121×C53—21no—
16#121×C53C2212yes—
17#120×C53—22no—
18C2×C22⋊C4×C53C2211yescharacteristic
19C24×C53—21no—
20C24×C53D411yescharacteristic
21C23×D53—21no—
22C2×D4×C53—22no—
23C2×D4×C53—22no—
24C2×D4×C53—21no—
25C2×D4×C53—42no—
26C2×D4×C53—42no—
27C2×D4×C53D412yescomplemented
28C22×C4×C53—21no—
29C22×C4×C53D411yescharacteristic
30C22×C4×C53C2311yescharacteristic
31C2×C22⋊D53—42no—
32C2×C22⋊D53—42no—
33C2×C22⋊D53—42no—
34C22×Dic53—21no—
35C2×D212—22no—
36C2×D212—42no—
37C4⋊C4×C53—21no—
38C4⋊C4×C53D412yescomplemented
39C22⋊C4×C53—22no—
40C22⋊C4×C53—22no—
41C22⋊C4×C53—22no—
42#11719922—44no—
43#11719921—42no—
44C2×#88322—22no—
45C23×C53—21no—
46C23×C53—21no—
47C23×C53—22no—
48C23×C53—41no—
49C23×C53—41no—
50C23×C53—42no—
51C23×C53—81no—
52C23×C53—81no—
53C23×C53D812yescomplemented
54C23×C53C2×D411yescharacteristic
55C22×D53—22no—
56C22×D53—41no—
57C22×D53—41no—
58C22×D53—41no—
59C22×D53—41no—
60C22×D53—41no—
61C22×D53—41no—
62D4×C53—42no—
63D4×C53—44no—
64D4×C53—42no—
65D4×C53—42no—
66D4×C53—81no—
67C2×C4×C53—22no—
68C2×C4×C53—21no—
69C2×C4×C53—41no—
70C2×C4×C53—42no—
71C2×C4×C53—41no—
72C2×C4×C53C2×D412yes—
73C22⋊D53—48no—
74C2×Dic53—22no—
75C2×Dic53—41no—
76C2×Dic53—41no—
77D212—24no—
78D212—42no—
79#88322—42no—
80C22×C53—21no—
81C22×C53—21no—
82C22×C53—22no—
83C22×C53—42no—
84C22×C53—41no—
85C22×C53—41no—
86C22×C53—42no—
87C22×C53—42no—
88C22×C53—42no—
89C22×C53—81no—
90C22×C53—84no—
91C22×C53#12011yescharacteristic
92C22×C53C2×D811yescharacteristic
93C22×C53AGL(1,ℤ/8)11yescharacteristic
94C2×D53—42no—
95C2×D53—42no—
96C2×D53—42no—
97C2×D53—81no—
98C2×D53—81no—
99C2×D53—81no—
100C2×D53—81no—
101C4×C53—22no—
102C4×C53—42no—
103C4×C53—41no—
104Dic53—42no—
105#1595—531nomaximal, Sylow
106C2×C53—21no—
107C2×C53—21no—
108C2×C53—42no—
109C2×C53—82no—
110C2×C53#44711yescharacteristic
111C2×C53#45311yescharacteristic
112C2×C53B(4,𝔽2)11yescharacteristic
113D53—82no—
114#534D5311yescharacteristic, complemented
115#447—532no—
116#392—531no—
117#327—532no—
118#323—531no—
119C53#159511yescharacteristic, complemented, Sylow
120C22×D4—21no—
121C22×D4—531no—
122C22×D4—1061no—
123C2×D8—1062no—
124#123—22no—
125#121—21no—
126#121C2×D5312yes—
127#120—22no—
128C2×C22⋊C4—1061no—
129C2×C22⋊C4C2×D5311yescharacteristic
130#102—1062no—
131C22⋊C8—532no—
132#95—531no—
133C24—21no—
134C24—1061no—
135C24C22⋊D5311yescharacteristic
136C2×D4—22no—
137C2×D4—22no—
138C2×D4—21no—
139C2×D4—42no—
140C2×D4—42no—
141C2×D4—1062no—
142C2×D4—2122no—
143C2×D4—2122no—
144C2×D4—2122no—
145C2×D4—2122no—
146C2×D4C22⋊D5312yescomplemented
147C22×C4—21no—
148C22×C4—1061no—
149C22×C4C22⋊D5311yescharacteristic
150C22×C4C22×D5311yescharacteristic
151D8—2124no—
152C2×C8—1062no—
153C4⋊C4—21no—
154C4⋊C4C22⋊D5312yescomplemented
155C22⋊C4—22no—
156C22⋊C4—22no—
157C22⋊C4—22no—
158C22⋊C4—2122no—
159C23—21no—
160C23—21no—
161C23—22no—
162C23—41no—
163C23—41no—
164C23—42no—
165C23—81no—
166C23—81no—
167C23—1062no—
168C23—2121no—
169C23—2121no—
170C23—2121no—
171C23—2121no—
172C23—2121no—
173C23—2121no—
174C23#1171992212yescomplemented
175C23C2×C22⋊D5311yescharacteristic
176D4—42no—
177D4—44no—
178D4—42no—
179D4—42no—
180D4—81no—
181D4—1064no—
182D4—2122no—
183D4—2128no—
184C2×C4—21no—
185C2×C4—22no—
186C2×C4—41no—
187C2×C4—42no—
188C2×C4—41no—
189C2×C4—1062no—
190C2×C4—2121no—
191C2×C4—2121no—
192C2×C4C2×C22⋊D5312yes—
193C8—2122no—
194C22—21no—
195C22—21no—
196C22—22no—
197C22—42no—
198C22—41no—
199C22—41no—
200C22—42no—
201C22—42no—
202C22—42no—
203C22—81no—
204C22—84no—
205C22—4241no—
206C22—4241no—
207C22—4241no—
208C22—4241no—
209C22—2122no—
210C22—2122no—
211C22—2122no—
212C22#1313318611yescharacteristic
213C22C2×#1171992211yescharacteristic
214C22#1313324311yescharacteristic
215C4—22no—
216C4—42no—
217C4—41no—
218C4—2122no—
219C2—21no—
220C2—21no—
221C2—42no—
222C2—82no—
223C2—4242no—
224C2#1681075711yescharacteristic
225C2#1681079111yescharacteristic
226C2#1681081011yescharacteristic
subgroupG/H
centerC22#13133186
commutatorC22×C4×C53C23
FrattiniC22×C4C22×D53
FittingC53×#534C2
socleC22×C53#120
hypercenter#534D53
hypocenterC53#1595
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #534

Complements

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

NH ≅ G/Ntype
1. #16810756C2semidirect
2. #16810532C2semidirect
3. #16810357C2semidirect
4. #16810347C2semidirect
5. C53×#534C2semidirect
6. C22×D212C22semidirect
11. C22⋊#88322C22semidirect
12. #13133124C22semidirect
27. C2×D4×C53D4semidirect
38. C4⋊C4×C53D4semidirect
53. C23×C53D8semidirect
114. #534D53semidirect
119. C53#1595semidirect
146. C2×D4C22⋊D53semidirect
154. C4⋊C4C22⋊D53semidirect
174. C23#11719922semidirect