Finite Group Databasebeta

← #20276110

Subgroups — #20276110

Order 6784.

248 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 89 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×#13133150C211yesmaximal, characteristic
2C2×C4⋊#88322C211yesmaximal, characteristic
3C2×#13133141C211yesmaximal, characteristic
4C2×#13133141C211yesmaximal, characteristic
5#16810521C211yesmaximal, characteristic
6#16810385C211yesmaximal, characteristic
7C53×#391C211yesmaximal, characteristic
8C22×#88322C2211yescharacteristic
9C22×#88322C2211yescharacteristic
10C2×C4⋊Dic53C2211yescharacteristic
11C2×C4⋊Dic53C2211yescharacteristic
12C2×#11719919—21no—
13#13133150—22no—
14C4⋊#88322—22no—
15#13133141—22no—
16#13133141—22no—
17#13133124—21no—
18C22×Q8×C53C2211yescharacteristic
19C2×C4⋊C4×C53C2211yescharacteristic
20C2×C42×C53C2211yescharacteristic
21#95×C53—21no—
22#95×C53—21no—
23C2×Q8×C53—21no—
24C2×Q8×C53—21no—
25C2×Q8×C53—21no—
26C2×Q8×C53—21no—
27C2×Q8×C53—21no—
28C2×Q8×C53Q812yes—
29C22×C4×C53—21no—
30C22×C4×C53—21no—
31C22×C4×C53D411yescharacteristic
32C22×C4×C53D411yescharacteristic
33C22×C4×C53C2311yescharacteristic
34C22×Dic53—21no—
35C22×Dic53—21no—
36C4⋊C4×C53—21no—
37C4⋊C4×C53—21no—
38C42×C53—21no—
39C42×C53—21no—
40C4⋊Dic53—21no—
41C4⋊Dic53—21no—
42C4⋊Dic53—21no—
43C4⋊Dic53D412yes—
44#11719919—42no—
45C2×#88322—21no—
46C2×#88322—21no—
47C2×#88322—21no—
48C2×#88322—21no—
49C2×#88322—21no—
50C2×#88322D412yes—
51C23×C53C2×D411yescharacteristic
52Q8×C53—22no—
53Q8×C53—22no—
54Q8×C53—41no—
55Q8×C53—41no—
56C2×C4×C53—21no—
57C2×C4×C53—21no—
58C2×C4×C53—21no—
59C2×C4×C53—22no—
60C2×C4×C53—21no—
61C2×C4×C53—21no—
62C2×C4×C53—41no—
63C2×C4×C53—41no—
64C2×C4×C53—41no—
65C2×C4×C53—41no—
66C2×C4×C53—41no—
67C2×C4×C53Q1612yes—
68C2×C4×C53C2×D411yescharacteristic
69C2×C4×C53C2×D411yescharacteristic
70C2×C4×C53C2×Q811yescharacteristic
71C2×C4×C53#4111yescharacteristic
72C2×C4×C53#4111yescharacteristic
73C2×C4×C53#4111yescharacteristic
74C2×Dic53—22no—
75C2×Dic53—41no—
76C2×Dic53—41no—
77C2×Dic53—41no—
78C2×Dic53—41no—
79C2×Dic53—41no—
80#88322—22no—
81#88322—41no—
82C22×C53#12111yescharacteristic
83C22×C53#12111yescharacteristic
84C22×C53C22⋊Q811yescharacteristic
85C22×C53C2×Q1611yescharacteristic
86C22×C53#13511yescharacteristic
87C22×C53AGL(1,ℤ/8)11yescharacteristic
88C22×C53#13711yescharacteristic
89C4×C53—22no—
90C4×C53—42no—
91C4×C53—41no—
92C4×C53—81no—
93C4×C53#12111yescharacteristic
94C4×C53C22⋊Q811yescharacteristic
95C4×C53C22⋊Q811yescharacteristic
96C4×C53#12611yescharacteristic
97Dic53—42no—
98Dic53—81no—
99#2068—531nomaximal, Sylow
100C2×C53#39611yescharacteristic
101C2×C53#46211yescharacteristic
102C2×C53#46311yescharacteristic
103C2×C53#46711yescharacteristic
104C2×C53#46811yescharacteristic
105C2×C53#47711yescharacteristic
106C2×C53#47911yescharacteristic
107C2×#107—531no—
108C2×C4⋊C8—531no—
109C2×#103—531no—
110C2×#103—531no—
111#391D5311yescharacteristic
112#389—531no—
113#340—531no—
114C53#206811yescharacteristic, complemented, Sylow
115C22×Q8C2×D5311yescharacteristic
116C22×C8—531no—
117C22×C8—531no—
118C2×C4⋊C4—531no—
119C2×C4⋊C4—531no—
120C2×C4⋊C4—1061no—
121C2×C4⋊C4C2×D5311yescharacteristic
122C2×C42C2×D5311yescharacteristic
123#107—1062no—
124C4⋊C8—1062no—
125#103—1062no—
126#103—1062no—
127#95—21no—
128#95—21no—
129#95—1061no—
130C2×Q8—21no—
131C2×Q8—21no—
132C2×Q8—21no—
133C2×Q8—21no—
134C2×Q8—21no—
135C2×Q8Dic10612yes—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—1061no—
139C22×C4—1061no—
140C22×C4C22⋊D5311yescharacteristic
141C22×C4C22⋊D5311yescharacteristic
142C22×C4C22×D5311yescharacteristic
143C2×C8—532no—
144C2×C8—1061no—
145C2×C8—1061no—
146C2×C8—1061no—
147C2×C8—1061no—
148C2×C8—1061no—
149C4⋊C4—21no—
150C4⋊C4—21no—
151C4⋊C4—532no—
152C4⋊C4—1061no—
153C4⋊C4—1061no—
154C4⋊C4—1061no—
155C4⋊C4—2122no—
156C42—21no—
157C42—21no—
158C23C2×C22⋊D5311yescharacteristic
159Q8—22no—
160Q8—22no—
161Q8—41no—
162Q8—41no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—22no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—1062no—
175C2×C4—2121no—
176C2×C4—2121no—
177C2×C4—2121no—
178C2×C4—2121no—
179C2×C4—2121no—
180C2×C4#1171992512yes—
181C2×C4C2×Dic10611yescharacteristic
182C2×C4#1171994511yescharacteristic
183C2×C4D4×D5311yescharacteristic
184C2×C4D4×D5311yescharacteristic
185C2×C4#1171994711yescharacteristic
186C2×C4#1171994711yescharacteristic
187C8—1062no—
188C8—2121no—
189C22#1313318811yescharacteristic
190C22#1313318911yescharacteristic
191C22#1313319611yescharacteristic
192C22C2×#1171992511yescharacteristic
193C22#1313323711yescharacteristic
194C22#1313324211yescharacteristic
195C22#1313324811yescharacteristic
196C4—22no—
197C4—42no—
198C4—41no—
199C4—81no—
200C4—4241no—
201C4—2122no—
202C4#1313319011yescharacteristic
203C4#1313319411yescharacteristic
204C4#1313319411yescharacteristic
205C4#1313321011yescharacteristic
206C2#1681054611yescharacteristic
207C2#1681083011yescharacteristic
208C2#1681083511yescharacteristic
209C2#1681086211yescharacteristic
210C2#1681087111yescharacteristic
211C2#1681091811yescharacteristic
212C2#1681093011yescharacteristic
subgroupG/H
centerC23C2×C22⋊D53
commutatorC22×C4×C53C23
FrattiniC22×C4C22×D53
FittingC53×#391C2
socleC23×C53C2×D4
hypercenter#391D53
hypocenterC53#2068
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #391

Complements

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

NH ≅ G/Ntype
114. C53#2068semidirect