Finite Group Databasebeta

← #20282750

Subgroups — #20282750

Order 6784.

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

normal subgroups only

This group has 444 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
1C2×#13133196C211yesmaximal, characteristic
2#16810562C212yesmaximal, complemented
3#16810561C212yesmaximal, complemented
4#16810551C214yesmaximal
5#16810543C212yesmaximal, complemented
6#16810542C212yesmaximal, complemented
7#16810486C211yesmaximal, characteristic
8C2×C22⋊Q8×C53C211yesmaximal, characteristic
9C22×Dic106C2211yescharacteristic
10#13133196—24no—
11C2×C4⋊Dic53C2211yescharacteristic
12C2×#11719919C2212yes—
13C2×#11719919C2214yes—
14C2×C22⋊Dic53C2212yes—
15C2×C22⋊Dic53C2212yes—
16#13133124C2214yes—
17#13133124C2212yes—
18#13133124C2214yes—
19#13133124C2212yes—
20#13133124C2214yes—
21C22×Q8×C53C2211yescharacteristic
22C23×C4×C53C2211yescharacteristic
23C22⋊Q8×C53—24no—
24C2×C4⋊C4×C53C2212yes—
25C2×C4⋊C4×C53C2211yescharacteristic
26C2×C22⋊C4×C53C2212yes—
27C24×C53C2311yescharacteristic
28C2×Q8×C53—22no—
29C2×Q8×C53—42no—
30C22×C4×C53—42no—
31C22×C4×C53D414yes—
32C22×C4×C53C2312yes—
33C22×C4×C53C2312yes—
34C22×C4×C53C2311yescharacteristic
35C22×C4×C53C2311yescharacteristic
36C22×Dic53C2314yes—
37C22×Dic53C2312yes—
38C22×Dic53C2312yes—
39C2×Dic106—22no—
40C2×Dic106—42no—
41C4⋊C4×C53—24no—
42C4⋊C4×C53—22no—
43C22⋊C4×C53—24no—
44C22⋊Dic53—24no—
45C22⋊Dic53—42no—
46C4⋊Dic53—22no—
47#11719919—24no—
48#11719919—44no—
49C23×C53—22no—
50C23×C53—22no—
51C23×C53—22no—
52C23×C53C2×D412yes—
53C23×C53C2411yescharacteristic
54Q8×C53—44no—
55C2×C4×C53—22no—
56C2×C4×C53—22no—
57C2×C4×C53—22no—
58C2×C4×C53—22no—
59C2×C4×C53—24no—
60C2×C4×C53—24no—
61C2×C4×C53—44no—
62C2×C4×C53C2×D412yes—
63C2×C4×C53C2×D412yes—
64C2×Dic53—22no—
65C2×Dic53—22no—
66C2×Dic53—24no—
67C2×Dic53—24no—
68C2×Dic53—24no—
69C2×Dic53—24no—
70C2×Dic53—24no—
71Dic106—44no—
72C22×C53—22no—
73C22×C53—22no—
74C22×C53—22no—
75C22×C53—44no—
76C22×C53C22×D411yescharacteristic
77C22×C53H(5,𝔽2)12yes—
78C22×C532−1+412yes—
79C22×C532−1+412yes—
80C4×C53—22no—
81C4×C53—22no—
82C4×C53—42no—
83C4×C53—42no—
84Dic53—42no—
85Dic53—42no—
86Dic53—44no—
87#2817—531nomaximal, Sylow
88C2×C53—42no—
89C2×C53#53411yescharacteristic
90C2×C53#53611yescharacteristic
91C2×C53#53611yescharacteristic
92C2×C53#56314yes—
93C2×C22⋊Q8—531no—
94C2×C22⋊Q8D5311yescharacteristic
95#400—532no—
96#400—532no—
97#397—534no—
98#395—532no—
99#395—532no—
100#381—531no—
101C53#281711yescharacteristic, complemented, Sylow
102C22×Q8—531no—
103C22×Q8C2×D5311yescharacteristic
104C23×C4C2×D5311yescharacteristic
105C22⋊Q8—24no—
106C22⋊Q8—1064no—
107C2×C4⋊C4—532no—
108C2×C4⋊C4—531no—
109C2×C4⋊C4—534no—
110C2×C4⋊C4C2×D5312yes—
111C2×C4⋊C4C2×D5311yescharacteristic
112C2×C22⋊C4—532no—
113C2×C22⋊C4—532no—
114C2×C22⋊C4C2×D5312yes—
115#95—532no—
116#95—532no—
117#95—534no—
118#95—534no—
119#95—534no—
120C24C22×D5311yescharacteristic
121C2×Q8—22no—
122C2×Q8—42no—
123C2×Q8—1062no—
124C2×Q8—2122no—
125C22×C4—42no—
126C22×C4—534no—
127C22×C4—532no—
128C22×C4—532no—
129C22×C4C22⋊D5314yes—
130C22×C4C22×D5312yes—
131C22×C4C22×D5312yes—
132C22×C4C22×D5311yescharacteristic
133C22×C4C22×D5311yescharacteristic
134C4⋊C4—22no—
135C4⋊C4—24no—
136C4⋊C4—1064no—
137C4⋊C4—1062no—
138C4⋊C4—2124no—
139C22⋊C4—24no—
140C22⋊C4—1064no—
141C22⋊C4—2122no—
142C23—22no—
143C23—22no—
144C23—22no—
145C23C2×C22⋊D5312yes—
146C23C23×D5311yescharacteristic
147Q8—44no—
148Q8—2124no—
149C2×C4—22no—
150C2×C4—22no—
151C2×C4—22no—
152C2×C4—22no—
153C2×C4—24no—
154C2×C4—24no—
155C2×C4—44no—
156C2×C4—1062no—
157C2×C4—1062no—
158C2×C4—1064no—
159C2×C4—1064no—
160C2×C4—1064no—
161C2×C4—1064no—
162C2×C4—1064no—
163C2×C4C2×C22⋊D5312yes—
164C2×C4C2×C22⋊D5312yes—
165C22—22no—
166C22—22no—
167C22—22no—
168C22—44no—
169C22C22×C22⋊D5311yescharacteristic
170C22#1313326212yes—
171C22#1313326412yes—
172C22#1313326512yes—
173C4—22no—
174C4—22no—
175C4—42no—
176C4—42no—
177C4—2124no—
178C4—2122no—
179C4—2122no—
180C2—42no—
181C2#1681118211yescharacteristic
182C2#1681119411yescharacteristic
183C2#1681119711yescharacteristic
184C2#1681137414yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC2×C22⋊Q8×C53C2
hypercenterC2×C22⋊Q8D53
hypocenterC53#2817
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. #16810562C2semidirect
3. #16810561C2semidirect
5. #16810543C2semidirect
6. #16810542C2semidirect
101. C53#2817semidirect