Finite Group Databasebeta

← #20282602

Subgroups — #20282602

Order 6784.

376 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 376 conjugacy classes of subgroups — too many to draw. Only its 133 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#16810564C212yesmaximal
2#16810562C211yesmaximal, characteristic
3#16810553C212yesmaximal
4#16810530C212yesmaximal
5#16810528C211yesmaximal, characteristic
6#16810527C212yesmaximal
7#16810509C212yesmaximal
8#16810506C212yesmaximal
9C53×#395C211yesmaximal, characteristic
10C22×Dic106C2211yescharacteristic
11C2×C4⋊Dic53C2212yes—
12C2×C4⋊Dic53C2211yescharacteristic
13C2×#11719919C2212yes—
14C2×#11719919C2212yes—
15C2×C4×Dic53C2211yescharacteristic
16C2×C4×Dic53C2212yes—
17#13133124C2212yes—
18#13133124C2212yes—
19#13133124C2212yes—
20#13133124C2212yes—
21#13133124C2212yes—
22#13133124C2212yes—
23#13133124C2212yes—
24#13133124C2212yes—
25#13133124C2211yescharacteristic
26C22×Q8×C53C2211yescharacteristic
27C2×C4⋊C4×C53C2212yes—
28C2×C4⋊C4×C53C2211yescharacteristic
29#95×C53C2212yes—
30#95×C53C2211yescharacteristic
31C2×Q8×C53—24no—
32C2×Q8×C53—41no—
33C22×C4×C53C2312yes—
34C22×C4×C53C2311yescharacteristic
35C22×C4×C53C2311yescharacteristic
36C22×C4×C53C2311yescharacteristic
37C22×C4×C53C2312yes—
38C22×Dic53C2312yes—
39C22×Dic53C2312yes—
40C22×Dic53C2312yes—
41C22×Dic53C2312yes—
42C2×Dic106—41no—
43C2×Dic106—41no—
44C2×Dic106Q814yescomplemented
45C4⋊C4×C53—24no—
46C4⋊C4×C53—41no—
47C4⋊Dic53—24no—
48C4⋊Dic53—41no—
49#11719919—42no—
50#11719919—42no—
51C4×Dic53—42no—
52C4×Dic53—41no—
53C23×C53C2411yescharacteristic
54Q8×C53—44no—
55C2×C4×C53—22no—
56C2×C4×C53—21no—
57C2×C4×C53—21no—
58C2×C4×C53—22no—
59C2×C4×C53—22no—
60C2×C4×C53—22no—
61C2×C4×C53—22no—
62C2×C4×C53—21no—
63C2×C4×C53—22no—
64C2×C4×C53—21no—
65C2×C4×C53—22no—
66C2×C4×C53C2×Q812yes—
67C2×C4×C53#4114yes—
68C2×Dic53—22no—
69C2×Dic53—22no—
70C2×Dic53—22no—
71C2×Dic53—22no—
72C2×Dic53—22no—
73C2×Dic53—22no—
74C2×Dic53—22no—
75C2×Dic53—22no—
76C2×Dic53—22no—
77C2×Dic53—24no—
78C2×Dic53C2×Q814yes—
79Dic106—44no—
80C22×C53C22×Q811yescharacteristic
81C22×C53C2×#4112yes—
82C22×C53H(5,𝔽2)11yescharacteristic
83C22×C53H(5,𝔽2)11yescharacteristic
84C22×C532−1+412yes—
85C4×C53—22no—
86C4×C53—24no—
87C4×C53—42no—
88C4×C53—41no—
89C4×C53—41no—
90Dic53—24no—
91Dic53—42no—
92Dic53—42no—
93Dic53—42no—
94#2799—531nomaximal, Sylow
95C2×C53#54212yes—
96C2×C53#54311yescharacteristic
97C2×C53#55111yescharacteristic
98C2×C53#55712yes—
99C2×C53#56311yescharacteristic
100#400—531no—
101#400—532no—
102#398—532no—
103#395D5311yescharacteristic
104#391—532no—
105#391—531no—
106#391—532no—
107#387—532no—
108#387—532no—
109C53#279911yescharacteristic, complemented, Sylow
110C22×Q8—531no—
111C22×Q8C2×D5311yescharacteristic
112C2×C4⋊C4—532no—
113C2×C4⋊C4—532no—
114C2×C4⋊C4—532no—
115C2×C4⋊C4—531no—
116C2×C4⋊C4C2×D5312yes—
117C2×C4⋊C4C2×D5311yescharacteristic
118C2×C42—531no—
119C2×C42—532no—
120#95—532no—
121#95—532no—
122#95—532no—
123#95—532no—
124#95—532no—
125#95—532no—
126#95—532no—
127#95—532no—
128#95—531no—
129#95C2×D5312yes—
130#95C2×D5311yescharacteristic
131C2×Q8—24no—
132C2×Q8—41no—
133C2×Q8—534no—
134C2×Q8—2121no—
135C2×Q8—2121no—
136C22×C4—532no—
137C22×C4—532no—
138C22×C4—532no—
139C22×C4—532no—
140C22×C4C22×D5311yescharacteristic
141C22×C4C22×D5311yescharacteristic
142C22×C4C22×D5312yes—
143C22×C4C22×D5312yes—
144C22×C4C22×D5311yescharacteristic
145C4⋊C4—24no—
146C4⋊C4—41no—
147C4⋊C4—1064no—
148C4⋊C4—2122no—
149C4⋊C4—2122no—
150C4⋊C4—2121no—
151C42—2122no—
152C42—2121no—
153C23C23×D5311yescharacteristic
154Q8—44no—
155Q8—2124no—
156C2×C4—22no—
157C2×C4—22no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—22no—
161C2×C4—22no—
162C2×C4—22no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—534no—
168C2×C4—1062no—
169C2×C4—1062no—
170C2×C4—1062no—
171C2×C4—1062no—
172C2×C4—1062no—
173C2×C4—1062no—
174C2×C4—1062no—
175C2×C4—1064no—
176C2×C4—1062no—
177C2×C4—1062no—
178C2×C4#1171994714yes—
179C2×C4Q8×D5312yes—
180C22C2×Q8×D5311yescharacteristic
181C22C2×#1171994712yes—
182C22#1313326211yescharacteristic
183C22#1313326311yescharacteristic
184C22#1313326512yes—
185C4—24no—
186C4—22no—
187C4—41no—
188C4—41no—
189C4—42no—
190C4—1064no—
191C4—2122no—
192C4—2122no—
193C4—2122no—
194C2#1681124612yes—
195C2#1681124811yescharacteristic
196C2#1681130411yescharacteristic
197C2#1681134812yes—
198C2#1681137211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#395C2
hypercenter#395D53
hypocenterC53#2799
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #395

Complements

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

NH ≅ G/Ntype
44. C2×Dic106Q8semidirect
109. C53#2799semidirect