Finite Group Databasebeta

← #20282539

Subgroups — #20282539

Order 6784.

344 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 121 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#16810564C212yesmaximal
3#16810562C211yesmaximal, characteristic
4#16810510C212yesmaximal
5#16810509C212yesmaximal
6#16810504C212yesmaximal
7#16810504C212yesmaximal
8#16810492C211yesmaximal, characteristic
9C53×#398C211yesmaximal, characteristic
10C2×C4⋊Dic53C2211yescharacteristic
11C2×C4⋊Dic53C2211yescharacteristic
12C2×#11719919C2212yes—
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
26C2×C4⋊C4×C53C2212yes—
27C2×C4⋊C4×C53C2211yescharacteristic
28C2×C4⋊C4×C53C2211yescharacteristic
29#95×C53C2212yes—
30#95×C53C2211yescharacteristic
31C22×C4×C53C2312yes—
32C22×C4×C53C2311yescharacteristic
33C22×C4×C53C2312yes—
34C22×C4×C53C2311yescharacteristic
35C22×C4×C53C2311yescharacteristic
36C22×Dic53C2312yes—
37C22×Dic53C2312yes—
38C22×Dic53C2312yes—
39C22×Dic53C2312yes—
40C4⋊C4×C53—24no—
41C4⋊C4×C53—41no—
42C4⋊C4×C53—41no—
43C4⋊Dic53—41no—
44C4⋊Dic53—41no—
45#11719919—24no—
46#11719919—24no—
47#11719919—24no—
48C4×Dic53—22no—
49C4×Dic53—42no—
50C23×C53C2411yescharacteristic
51C2×C4×C53—21no—
52C2×C4×C53—21no—
53C2×C4×C53—22no—
54C2×C4×C53—22no—
55C2×C4×C53—22no—
56C2×C4×C53—22no—
57C2×C4×C53—22no—
58C2×C4×C53—22no—
59C2×C4×C53—21no—
60C2×C4×C53—21no—
61C2×C4×C53—22no—
62C2×C4×C53—22no—
63C2×C4×C53#4112yes—
64C2×Dic53—22no—
65C2×Dic53—22no—
66C2×Dic53—22no—
67C2×Dic53—22no—
68C2×Dic53—22no—
69C2×Dic53—22no—
70C2×Dic53—22no—
71C2×Dic53—22no—
72C2×Dic53—22no—
73C2×Dic53—24no—
74C2×Dic53#4114yes—
75C22×C53C2×#4112yes—
76C22×C53C2×#4111yescharacteristic
77C22×C53H(5,𝔽2)11yescharacteristic
78C22×C532−1+411yescharacteristic
79C22×C532−1+412yes—
80C4×C53—22no—
81C4×C53—42no—
82C4×C53—42no—
83C4×C53—41no—
84C4×C53—41no—
85Dic53—24no—
86Dic53—42no—
87Dic53—42no—
88Dic53—42no—
89#2794—531nomaximal, Sylow
90C2×C53#53911yescharacteristic
91C2×C53#54112yes—
92C2×C53#55212yes—
93C2×C53#55911yescharacteristic
94C2×C53#56311yescharacteristic
95#400—531no—
96#400—532no—
97#400—532no—
98#398D5311yescharacteristic
99#387—532no—
100#387—532no—
101#387—532no—
102#387—532no—
103#384—531no—
104C53#279411yescharacteristic, complemented, Sylow
105C2×C4⋊C4—532no—
106C2×C4⋊C4—532no—
107C2×C4⋊C4—532no—
108C2×C4⋊C4—531no—
109C2×C4⋊C4—531no—
110C2×C4⋊C4C2×D5312yes—
111C2×C4⋊C4C2×D5311yescharacteristic
112C2×C4⋊C4C2×D5311yescharacteristic
113C2×C42—531no—
114C2×C42—532no—
115#95—532no—
116#95—532no—
117#95—532no—
118#95—532no—
119#95—532no—
120#95—532no—
121#95—532no—
122#95—532no—
123#95—531no—
124#95C2×D5312yes—
125#95C2×D5311yescharacteristic
126C22×C4—532no—
127C22×C4—532no—
128C22×C4—532no—
129C22×C4—532no—
130C22×C4C22×D5312yes—
131C22×C4C22×D5312yes—
132C22×C4C22×D5311yescharacteristic
133C22×C4C22×D5311yescharacteristic
134C22×C4C22×D5311yescharacteristic
135C4⋊C4—24no—
136C4⋊C4—41no—
137C4⋊C4—41no—
138C4⋊C4—1064no—
139C4⋊C4—1064no—
140C4⋊C4—1064no—
141C4⋊C4—2121no—
142C4⋊C4—2121no—
143C42—1062no—
144C42—2122no—
145C23C23×D5311yescharacteristic
146C2×C4—21no—
147C2×C4—21no—
148C2×C4—22no—
149C2×C4—22no—
150C2×C4—22no—
151C2×C4—22no—
152C2×C4—22no—
153C2×C4—22no—
154C2×C4—21no—
155C2×C4—21no—
156C2×C4—22no—
157C2×C4—22no—
158C2×C4—534no—
159C2×C4—1062no—
160C2×C4—1062no—
161C2×C4—1062no—
162C2×C4—1062no—
163C2×C4—1062no—
164C2×C4—1062no—
165C2×C4—1062no—
166C2×C4—1062no—
167C2×C4—1064no—
168C2×C4—1062no—
169C2×C4#1171994712yes—
170C22#41×D5312yes—
171C22C2×#1171994711yescharacteristic
172C22#1313326211yescharacteristic
173C22#1313326411yescharacteristic
174C22#1313326512yes—
175C4—22no—
176C4—42no—
177C4—41no—
178C4—41no—
179C4—42no—
180C4—1064no—
181C4—2122no—
182C4—2122no—
183C4—2122no—
184C2#1681121811yescharacteristic
185C2#1681123212yes—
186C2#1681130712yes—
187C2#1681135711yescharacteristic
188C2#1681137211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#398C2
hypercenter#398D53
hypocenterC53#2794
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
104. C53#2794semidirect