Finite Group Databasebeta

← #20273650

Subgroups — #20273650

Order 6784.

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

normal subgroups only

This group has 310 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#16810540C211yesmaximal, characteristic, complemented
3#16810357C212yesmaximal, complemented
4#16810349C211yesmaximal, characteristic, complemented
5C53×#537C211yesmaximal, characteristic, complemented
6C22×D212C2211yescharacteristic
7C2×#11719922—22no—
8C2×#11719921—21no—
9C2×#11719921—21no—
10#13133137—22no—
11C22⋊#88322C2212yes—
12#13133124C2211yescharacteristic
13C2×#41×C53—21no—
14C22⋊Q8×C53—22no—
15#121×C53—22no—
16#121×C53—21no—
17#121×C53C2212yes—
18C2×C4⋊C4×C53C2211yescharacteristic
19C23×D53—21no—
20#41×C53—42no—
21#41×C53—42no—
22C2×Q8×C53—21no—
23C2×D4×C53—21no—
24C2×D4×C53—22no—
25C2×D4×C53—21no—
26C2×D4×C53D412yes—
27C22×C4×C53—22no—
28C22×C4×C53D411yescharacteristic
29C22×C4×C53D411yescharacteristic
30C22×C4×C53C2311yescharacteristic
31C22×Dic53—21no—
32C2×D212—22no—
33C2×D212—42no—
34C4⋊C4×C53—21no—
35C4⋊C4×C53—22no—
36C4⋊C4×C53D412yes—
37C22⋊C4×C53—22no—
38C22⋊C4×C53—22no—
39#11719922—44no—
40#11719921—42no—
41#11719921—42no—
42C2×#88322—22no—
43C23×C53—22no—
44C23×C53C2×D411yescharacteristic
45C22×D53—22no—
46C22×D53—41no—
47C22×D53—41no—
48C22×D53—41no—
49C22×D53—41no—
50C22×D53—41no—
51C22×D53—41no—
52Q8×C53—42no—
53D4×C53—42no—
54D4×C53—42no—
55D4×C53—42no—
56D4×C53—42no—
57D4×C53—81no—
58C2×C4×C53—21no—
59C2×C4×C53—21no—
60C2×C4×C53—22no—
61C2×C4×C53—21no—
62C2×C4×C53—41no—
63C2×C4×C53—44no—
64C2×C4×C53—41no—
65C2×C4×C53D812yescomplemented
66C2×C4×C53C2×D412yes—
67C2×Dic53—41no—
68C2×Dic53—41no—
69C2×Dic53—41no—
70D212—24no—
71D212—42no—
72#88322—42no—
73C22×C53—21no—
74C22×C53—21no—
75C22×C53—42no—
76C22×C53—42no—
77C22×C53—42no—
78C22×C53#12011yescharacteristic
79C22×C53C2×D811yescharacteristic
80C22×C53AGL(1,ℤ/8)11yescharacteristic
81C2×D53—42no—
82C2×D53—42no—
83C2×D53—42no—
84C2×D53—81no—
85C2×D53—81no—
86C2×D53—81no—
87C2×D53—81no—
88C4×C53—22no—
89C4×C53—22no—
90C4×C53—42no—
91C4×C53—41no—
92Dic53—81no—
93#1598—531nomaximal, Sylow
94C2×C53—21no—
95C2×C53—21no—
96C2×C53—82no—
97C2×C53#44711yescharacteristic
98C2×C53#45411yescharacteristic
99C2×C53B(4,𝔽2)11yescharacteristic
100D53—82no—
101#537D5311yescharacteristic, complemented
102#447—532no—
103#394—531no—
104#327—532no—
105#324—531no—
106C53#159811yescharacteristic, complemented, Sylow
107C2×#41—21no—
108C22×D4—531no—
109C2×D8—1062no—
110C22⋊Q8—22no—
111#121—22no—
112#121—21no—
113#121C2×D5312yes—
114C2×C4⋊C4C2×D5311yescharacteristic
115C2×C22⋊C4—1061no—
116C2×C22⋊C4—1061no—
117#102—1062no—
118C22⋊C8—532no—
119#95—531no—
120C24—1061no—
121#41—42no—
122#41—42no—
123C2×Q8—21no—
124C2×D4—21no—
125C2×D4—22no—
126C2×D4—21no—
127C2×D4—1062no—
128C2×D4—2122no—
129C2×D4C22⋊D5312yes—
130C22×C4—22no—
131C22×C4—1061no—
132C22×C4C22⋊D5311yescharacteristic
133C22×C4C22⋊D5311yescharacteristic
134C22×C4C22×D5311yescharacteristic
135D8—2124no—
136C2×C8—1062no—
137C4⋊C4—21no—
138C4⋊C4—22no—
139C4⋊C4C22⋊D5312yes—
140C22⋊C4—22no—
141C22⋊C4—22no—
142C22⋊C4—2122no—
143C22⋊C4—2122no—
144C23—22no—
145C23—1062no—
146C23—2121no—
147C23—2121no—
148C23—2121no—
149C23—2121no—
150C23—2121no—
151C23—2121no—
152C23C2×C22⋊D5311yescharacteristic
153Q8—42no—
154D4—42no—
155D4—42no—
156D4—42no—
157D4—42no—
158D4—81no—
159D4—1064no—
160D4—2122no—
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—22no—
165C2×C4—41no—
166C2×C4—44no—
167C2×C4—41no—
168C2×C4—2121no—
169C2×C4—2121no—
170C2×C4—2121no—
171C2×C4#1171992212yescomplemented
172C2×C4C2×C22⋊D5312yes—
173C8—2122no—
174C22—21no—
175C22—21no—
176C22—42no—
177C22—42no—
178C22—42no—
179C22—4241no—
180C22—4241no—
181C22—4241no—
182C22—4241no—
183C22—2122no—
184C22—2122no—
185C22—2122no—
186C22#1313318611yescharacteristic
187C22C2×#1171992211yescharacteristic
188C22#1313324311yescharacteristic
189C4—22no—
190C4—22no—
191C4—42no—
192C4—41no—
193C4—4241no—
194C2—21no—
195C2—21no—
196C2—82no—
197C2—4242no—
198C2#1681075711yescharacteristic
199C2#1681079611yescharacteristic
200C2#1681081011yescharacteristic
subgroupG/H
centerC22#13133186
commutatorC22×C4×C53C23
FrattiniC22×C4C22×D53
FittingC53×#537C2
socleC22×C53#120
hypercenter#537D53
hypocenterC53#1598
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #537

Complements

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

NH ≅ G/Ntype
1. #16810756C2semidirect
2. #16810540C2semidirect
3. #16810357C2semidirect
4. #16810349C2semidirect
5. C53×#537C2semidirect
65. C2×C4×C53D8semidirect
101. #537D53semidirect
106. C53#1598semidirect
171. C2×C4#11719922semidirect