Finite Group Databasebeta

← #20276126

Subgroups — #20276126

Order 6784.

238 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 83 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
1C2×#13133141C211yesmaximal, characteristic
2#16810530C211yesmaximal, characteristic
3#16810492C211yesmaximal, characteristic
4#16810386C211yesmaximal, characteristic
5#16810385C211yesmaximal, characteristic
6#16810377C211yesmaximal, characteristic
7C2×#103×C53C211yesmaximal, characteristic
8C22×#88322C2211yescharacteristic
9C2×C4⋊Dic53C2211yescharacteristic
10C2×C4⋊Dic53C2211yescharacteristic
11C2×#11719919—21no—
12C2×#11719919—21no—
13C2×C4×Dic53C2211yescharacteristic
14#13133141—22no—
15#13133124—21no—
16#13133124—21no—
17C22×Q8×C53C2211yescharacteristic
18C22×C8×C53C2211yescharacteristic
19C2×C4⋊C4×C53C2211yescharacteristic
20#103×C53—22no—
21C2×Q8×C53—21no—
22C2×Q8×C53—21no—
23C2×Q8×C53—21no—
24C2×Q8×C53—21no—
25C2×Q8×C53—21no—
26C2×Q8×C53D412yes—
27C22×C4×C53—21no—
28C22×C4×C53—21no—
29C22×C4×C53C2311yescharacteristic
30C22×Dic53—21no—
31C22×Dic53—21no—
32C22×Dic53D411yescharacteristic
33C22×Dic53D411yescharacteristic
34C2×C8×C53—21no—
35C2×C8×C53—21no—
36C2×C8×C53—21no—
37C4⋊C4×C53—21no—
38C4⋊C4×C53—21no—
39C4⋊Dic53—21no—
40C4⋊Dic53—21no—
41C4⋊Dic53—21no—
42C4⋊Dic53—21no—
43#11719919—42no—
44#11719919—42no—
45C4×Dic53—21no—
46C4×Dic53—21no—
47C2×#88322—21no—
48C2×#88322—21no—
49C2×#88322—21no—
50C23×C53C2×D411yescharacteristic
51Q8×C53—22no—
52Q8×C53—22no—
53Q8×C53—41no—
54Q8×C53—41no—
55C2×C4×C53—22no—
56C2×C4×C53—41no—
57C2×C4×C53—41no—
58C2×C4×C53—41no—
59C2×C4×C53—41no—
60C2×C4×C53—41no—
61C2×C4×C53C2×D411yescharacteristic
62C2×C4×C53#4111yescharacteristic
63C2×C4×C53#4111yescharacteristic
64C2×C4×C53#4111yescharacteristic
65C2×C4×C53#4111yescharacteristic
66C2×C4×C53#4111yescharacteristic
67C2×Dic53—21no—
68C2×Dic53—21no—
69C2×Dic53—21no—
70C2×Dic53—21no—
71C2×Dic53—41no—
72C2×Dic53—41no—
73C2×Dic53—41no—
74C2×Dic53—41no—
75C2×Dic53—41no—
76C2×Dic53—41no—
77C2×Dic53SD1612yes—
78C2×Dic53Q1612yes—
79C8×C53—41no—
80#88322—41no—
81C22×C53#12111yescharacteristic
82C22×C53#12311yescharacteristic
83C22×C53#12411yescharacteristic
84C22×C53C2×SD1611yescharacteristic
85C22×C53C2×Q1611yescharacteristic
86C22×C53AGL(1,ℤ/8)11yescharacteristic
87C22×C53#13711yescharacteristic
88C4×C53—42no—
89C4×C53—81no—
90C4×C53#12311yescharacteristic
91C4×C53#12311yescharacteristic
92C4×C53#12611yescharacteristic
93C4×C53#12611yescharacteristic
94Dic53—22no—
95Dic53—22no—
96Dic53—81no—
97Dic53—81no—
98#2072—531nomaximal, Sylow
99C2×C53#39711yescharacteristic
100C2×C53#46011yescharacteristic
101C2×C53#46211yescharacteristic
102C2×C53#48311yescharacteristic
103C2×C53#48411yescharacteristic
104C2×C53#48711yescharacteristic
105C2×C53#48911yescharacteristic
106C2×#103—531no—
107C2×#103D5311yescharacteristic
108#391—531no—
109#384—531no—
110#340—531no—
111#340—531no—
112#336—531no—
113C53#207211yescharacteristic, complemented, Sylow
114C22×Q8C2×D5311yescharacteristic
115C22×C8—531no—
116C22×C8C2×D5311yescharacteristic
117C2×C4⋊C4—531no—
118C2×C4⋊C4—531no—
119C2×C4⋊C4—1061no—
120C2×C4⋊C4—1061no—
121C2×C4⋊C4C2×D5311yescharacteristic
122C2×C42—531no—
123#103—22no—
124#103—1062no—
125#95—1061no—
126#95—1061no—
127C2×Q8—21no—
128C2×Q8—21no—
129C2×Q8—21no—
130C2×Q8—21no—
131C2×Q8—21no—
132C2×Q8C22⋊D5312yes—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—531no—
136C22×C4—531no—
137C22×C4—1061no—
138C22×C4—1061no—
139C22×C4C22×D5311yescharacteristic
140C2×C8—21no—
141C2×C8—21no—
142C2×C8—21no—
143C2×C8—1061no—
144C2×C8—1061no—
145C2×C8—1061no—
146C4⋊C4—21no—
147C4⋊C4—21no—
148C4⋊C4—1061no—
149C4⋊C4—1061no—
150C4⋊C4—1061no—
151C4⋊C4—1061no—
152C4⋊C4—2122no—
153C4⋊C4—2122no—
154C42—1061no—
155C42—1061no—
156C23D4×D5311yescharacteristic
157Q8—22no—
158Q8—22no—
159Q8—41no—
160Q8—41no—
161C2×C4—22no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—532no—
168C2×C4—532no—
169C2×C4—1061no—
170C2×C4—1061no—
171C2×C4—1061no—
172C2×C4—1061no—
173C2×C4—2121no—
174C2×C4—2121no—
175C2×C4—2121no—
176C2×C4—2121no—
177C2×C4—2121no—
178C2×C4—2121no—
179C2×C4#1171994511yescharacteristic
180C2×C4#1171994511yescharacteristic
181C2×C4#1171994711yescharacteristic
182C2×C4#1171994711yescharacteristic
183C2×C4#1171994711yescharacteristic
184C2×C4C2×C22⋊D5311yescharacteristic
185C8—41no—
186C8—2121no—
187C22#1313318911yescharacteristic
188C22#1313319911yescharacteristic
189C22#1313320311yescharacteristic
190C22SD16×D5311yescharacteristic
191C22Q16×D5311yescharacteristic
192C22#1313324411yescharacteristic
193C22#1313324911yescharacteristic
194C4—42no—
195C4—81no—
196C4—1062no—
197C4—1062no—
198C4—4241no—
199C4—4241no—
200C4#1313319811yescharacteristic
201C4#1313320011yescharacteristic
202C4#1313321011yescharacteristic
203C4#1313321011yescharacteristic
204C2#1681055111yescharacteristic
205C2#1681082111yescharacteristic
206C2#1681083111yescharacteristic
207C2#1681095211yescharacteristic
208C2#1681095711yescharacteristic
209C2#1681097211yescharacteristic
210C2#1681098411yescharacteristic
subgroupG/H
centerC23D4×D53
commutatorC22×C4×C53C23
FrattiniC22×C4C22×D53
FittingC2×#103×C53C2
socleC23×C53C2×D4
hypercenterC2×#103D53
hypocenterC53#2072
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#103

Complements

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

NH ≅ G/Ntype
113. C53#2072semidirect