Finite Group Databasebeta

← #20276069

Subgroups — #20276069

Order 6784.

314 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 75 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#16811249C211yesmaximal, characteristic
2#16810498C212yesmaximal
3#16810495C211yesmaximal, characteristic
4#16810389C212yesmaximal
5C53×#342C211yesmaximal, characteristic
6C23×Dic53C2211yescharacteristic
7#13133196—21no—
8#13133196—21no—
9#13133194—22no—
10#13133194—22no—
11C2×C4⋊Dic53—22no—
12C2×#11719919—21no—
13C2×C22⋊Dic53C2211yescharacteristic
14C2×C22⋊Dic53C2212yes—
15#13133124—21no—
16#13133124—22no—
17C2×C22⋊C4×C53C2212yes—
18C2×C22⋊C4×C53C2211yescharacteristic
19C24×C53C2311yescharacteristic
20C22×C4×C53—22no—
21C22×C4×C53D411yescharacteristic
22C22×C4×C53D411yescharacteristic
23C22×Dic53—21no—
24C22×Dic53—21no—
25C22×Dic53—22no—
26C22×Dic53—42no—
27C22×Dic53D412yes—
28C22×Dic53D412yes—
29C2×Dic106—21no—
30C2×Dic106—21no—
31C22⋊C4×C53—22no—
32C22⋊C4×C53—22no—
33C22⋊C4×C53Q814yescomplemented
34C22⋊Dic53—22no—
35C22⋊Dic53—22no—
36C22⋊Dic53—21no—
37C22⋊Dic53Q812yescomplemented
38C4⋊Dic53—21no—
39C4⋊Dic53—21no—
40C4⋊Dic53—44no—
41#11719919—22no—
42#11719919—22no—
43#11719919—42no—
44C23×C53—41no—
45C23×C53—41no—
46C23×C53C2×D411yescharacteristic
47C23×C53C2×D412yes—
48C23×C53C2×Q812yes—
49C23×C53C2×Q811yescharacteristic
50C23×C53#4111yescharacteristic
51C2×C4×C53—21no—
52C2×C4×C53—21no—
53C2×C4×C53—24no—
54C2×C4×C53—42no—
55C2×C4×C53—42no—
56C2×C4×C53—41no—
57C2×C4×C53—41no—
58C2×Dic53—22no—
59C2×Dic53—24no—
60C2×Dic53—22no—
61C2×Dic53—22no—
62C2×Dic53—22no—
63C2×Dic53—41no—
64C2×Dic53—41no—
65C2×Dic53—42no—
66C2×Dic53—41no—
67C2×Dic53—41no—
68C2×Dic53—42no—
69C2×Dic53—42no—
70Dic106—81no—
71Dic106—81no—
72C22×C53—21no—
73C22×C53—22no—
74C22×C53—21no—
75C22×C53—22no—
76C22×C53—42no—
77C22×C53—41no—
78C22×C53—41no—
79C22×C53#12011yescharacteristic
80C22×C53C22⋊Q812yes—
81C22×C53C22⋊Q811yescharacteristic
82C22×C53C4⋊Q812yes—
83C22×C53C4⋊Q811yescharacteristic
84C4×C53—44no—
85C4×C53—41no—
86C4×C53—41no—
87Dic53—42no—
88Dic53—42no—
89Dic53—42no—
90Dic53—42no—
91#2062—531nomaximal, Sylow
92C2×C53—21no—
93C2×C53—22no—
94C2×C53—22no—
95C2×C53—21no—
96C2×C53#39511yescharacteristic
97C2×C53B(4,𝔽2)11yescharacteristic
98C2×C53#45811yescharacteristic
99#543—531no—
100#385—532no—
101#385—531no—
102#342—532no—
103#342D5311yescharacteristic
104C53#206211yescharacteristic, complemented, Sylow
105C23×C4—531no—
106C22⋊Q8—1062no—
107C22⋊Q8—1062no—
108C22⋊Q8—1061no—
109C22⋊Q8—1061no—
110C2×C4⋊C4—1062no—
111C2×C4⋊C4—1061no—
112C2×C22⋊C4—531no—
113C2×C22⋊C4—532no—
114C2×C22⋊C4C2×D5311yescharacteristic
115C2×C22⋊C4C2×D5312yes—
116#95—1062no—
117#95—1061no—
118C24C22×D5311yescharacteristic
119C2×Q8—1061no—
120C2×Q8—1061no—
121C22×C4—22no—
122C22×C4—532no—
123C22×C4—532no—
124C22×C4—1061no—
125C22×C4—1061no—
126C22×C4—1062no—
127C22×C4—2122no—
128C22×C4D21211yescharacteristic
129C22×C4D21211yescharacteristic
130C4⋊C4—1062no—
131C4⋊C4—1062no—
132C4⋊C4—1061no—
133C4⋊C4—1061no—
134C4⋊C4—2122no—
135C4⋊C4—2124no—
136C22⋊C4—22no—
137C22⋊C4—22no—
138C22⋊C4—532no—
139C22⋊C4—1062no—
140C22⋊C4—1062no—
141C22⋊C4—1061no—
142C22⋊C4Dic10614yescomplemented
143C23—41no—
144C23—41no—
145C23C2×Dic10612yes—
146C23C2×D21211yescharacteristic
147C23D4×D5312yes—
148C23#1171994711yescharacteristic
149C23Q8×D5311yescharacteristic
150Q8—4241no—
151Q8—4241no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—24no—
155C2×C4—42no—
156C2×C4—42no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—1062no—
160C2×C4—1064no—
161C2×C4—1062no—
162C2×C4—1062no—
163C2×C4—1062no—
164C2×C4—2121no—
165C2×C4—2121no—
166C2×C4—2121no—
167C2×C4—2122no—
168C2×C4—2121no—
169C2×C4—2122no—
170C2×C4—2122no—
171C22—21no—
172C22—22no—
173C22—21no—
174C22—22no—
175C22—42no—
176C22—41no—
177C22—41no—
178C22#1313318711yescharacteristic
179C22#1313319412yes—
180C22#1313319711yescharacteristic
181C22#1313321612yes—
182C22#1313321711yescharacteristic
183C4—44no—
184C4—41no—
185C4—41no—
186C4—2122no—
187C4—2122no—
188C4—2122no—
189C4—2122no—
190C2—22no—
191C2—21no—
192C2—22no—
193C2—21no—
194C2#1681054411yescharacteristic
195C2#1681081111yescharacteristic
196C2#1681081411yescharacteristic
subgroupG/H
centerC22#13133187
commutatorC24×C53C23
FrattiniC24C22×D53
FittingC53×#342C2
socleC22×C53#120
hypercenter#342D53
hypocenterC53#2062
LCSG ▷ C24×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C24 ◁ #342

Complements

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

NH ≅ G/Ntype
33. C22⋊C4×C53Q8semidirect
37. C22⋊Dic53Q8semidirect
104. C53#2062semidirect
142. C22⋊C4Dic106semidirect