Finite Group Databasebeta

← #20273201

Subgroups — #20273201

Order 6784.

238 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 81 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#16810676C211yesmaximal, characteristic, complemented
2#16810469C211yesmaximal, characteristic, complemented
3#16810360C212yesmaximal, complemented
4#16810357C212yesmaximal
5C53×#529C211yesmaximal, characteristic
6C2×C4×Dic53—21no—
7C4⋊#88322—21no—
8C22⋊#88322C2212yes—
9C4×#88322—21no—
10#13133124—21no—
11#13133124C2212yes—
12#126×C53—21no—
13#125×C53C411yescharacteristic, complemented
14#124×C53C411yescharacteristic, complemented
15#123×C53—21no—
16C22⋊Q8×C53C2211yescharacteristic
17#121×C53C2211yescharacteristic
18C4×Q8×C53C411yescharacteristic, complemented
19C4×D4×C53—21no—
20C4×D4×C53C411yescharacteristic, complemented
21#117×C53—21no—
22C2×C42×C53C2211yescharacteristic
23C2×Q8×C53C2×C411yescharacteristic
24C2×D4×C53—21no—
25C2×D4×C53C2×C411yescharacteristic
26C22×C4×C53—21no—
27C22×C4×C53D411yescharacteristic
28C22×C4×C53D411yescharacteristic
29C22×C4×C53C2311yescharacteristic
30C22×Dic53—22no—
31C4⋊C4×C53—21no—
32C4⋊C4×C53—21no—
33C4⋊C4×C53—21no—
34C4⋊C4×C53—21no—
35C4⋊C4×C53C2×C411yescharacteristic
36C4⋊C4×C53C2×C411yescharacteristic
37C22⋊C4×C53—21no—
38C22⋊C4×C53—21no—
39C22⋊C4×C53—21no—
40C22⋊C4×C53—21no—
41C22⋊C4×C53—21no—
42C42×C53—21no—
43C42×C53C2×C411yescharacteristic
44C42×C53C2×C411yescharacteristic
45C42×C53D411yescharacteristic
46C42×C53D411yescharacteristic
47C4×Dic53—24no—
48C2×#88322—22no—
49C23×C53—21no—
50C23×C53C2×D411yescharacteristic
51Q8×C53—41no—
52D4×C53—41no—
53D4×C53—42no—
54C2×C4×C53—22no—
55C2×C4×C53—22no—
56C2×C4×C53—21no—
57C2×C4×C53—21no—
58C2×C4×C53—21no—
59C2×C4×C53—41no—
60C2×C4×C53—42no—
61C2×C4×C53C22⋊C411yescharacteristic
62C2×C4×C53C22⋊C411yescharacteristic
63C2×C4×C53C22⋊C411yescharacteristic
64C2×C4×C53C22⋊C411yescharacteristic
65C2×C4×C53C22×C411yescharacteristic
66C2×C4×C53C2×D411yescharacteristic
67C2×Dic53—24no—
68C2×Dic53—42no—
69C2×Dic53—42no—
70#88322—42no—
71C22×C53—21no—
72C22×C53—21no—
73C22×C53—41no—
74C22×C53—41no—
75C22×C53—41no—
76C22×C53C2×C22⋊C411yescharacteristic
77C22×C53#13512yes—
78C4×C53—21no—
79C4×C53—21no—
80C4×C53—21no—
81C4×C53—21no—
82C4×C53—21no—
83C4×C53—41no—
84C4×C53—41no—
85C4×C53—41no—
86C4×C53#10412yes—
87Dic53—44no—
88#1500—531nomaximal, Sylow
89C2×C53—22no—
90C2×C53—81no—
91C2×C53#41011yescharacteristic
92C2×C53#41611yescharacteristic
93C2×C53C2×#10411yescharacteristic
94#529D5311yescharacteristic
95#431—531no—
96#376—531no—
97#328—532no—
98#327—532no—
99C53#150011yescharacteristic, complemented, Sylow
100#126—21no—
101#125Dic5311yescharacteristic, complemented
102#124Dic5311yescharacteristic, complemented
103#123—21no—
104C22⋊Q8C2×D5311yescharacteristic
105#121C2×D5311yescharacteristic
106C4×Q8Dic5311yescharacteristic, complemented
107C4×D4—21no—
108C4×D4Dic5311yescharacteristic, complemented
109#117—21no—
110C2×C42—1061no—
111C2×C42C2×D5311yescharacteristic
112C4⋊C8—1061no—
113C22⋊C8—532no—
114C4×C8—1061no—
115#95—532no—
116#95—1061no—
117C2×Q8C2×Dic5311yescharacteristic
118C2×D4—21no—
119C2×D4C2×Dic5311yescharacteristic
120C22×C4—21no—
121C22×C4—1062no—
122C22×C4C22⋊D5311yescharacteristic
123C22×C4C22⋊D5311yescharacteristic
124C22×C4C22×D5311yescharacteristic
125C2×C8—1062no—
126C4⋊C4—21no—
127C4⋊C4—21no—
128C4⋊C4—21no—
129C4⋊C4—21no—
130C4⋊C4C2×Dic5311yescharacteristic
131C4⋊C4C2×Dic5311yescharacteristic
132C22⋊C4—21no—
133C22⋊C4—21no—
134C22⋊C4—21no—
135C22⋊C4—21no—
136C22⋊C4—21no—
137C42—21no—
138C42—1064no—
139C42C2×Dic5311yescharacteristic
140C42C2×Dic5311yescharacteristic
141C42C22⋊D5311yescharacteristic
142C42C22⋊D5311yescharacteristic
143C23—21no—
144C23C2×C22⋊D5311yescharacteristic
145Q8—41no—
146D4—41no—
147D4—42no—
148C2×C4—22no—
149C2×C4—22no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—41no—
154C2×C4—42no—
155C2×C4—1064no—
156C2×C4—2122no—
157C2×C4—2122no—
158C2×C4C22⋊Dic5311yescharacteristic
159C2×C4C22⋊Dic5311yescharacteristic
160C2×C4C22⋊Dic5311yescharacteristic
161C2×C4C22⋊Dic5311yescharacteristic
162C2×C4C22×Dic5311yescharacteristic
163C2×C4C2×C22⋊D5311yescharacteristic
164C8—2122no—
165C22—21no—
166C22—21no—
167C22—41no—
168C22—41no—
169C22—41no—
170C22C2×C22⋊Dic5311yescharacteristic
171C22#1313323712yes—
172C4—21no—
173C4—21no—
174C4—21no—
175C4—21no—
176C4—21no—
177C4—41no—
178C4—41no—
179C4—41no—
180C4—2124no—
181C4#1313314412yes—
182C2—22no—
183C2—81no—
184C2#1681059711yescharacteristic
185C2#1681061911yescharacteristic
186C2C2×#1313314411yescharacteristic
subgroupG/H
centerC2×C4C22⋊Dic53
commutatorC2×C4×C53C22×C4
FrattiniC22×C4C22×D53
FittingC53×#529C2
socleC22×C53C2×C22⋊C4
hypercenter#529D53
hypocenterC53#1500
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #529

Complements

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

NH ≅ G/Ntype
1. #16810676C2semidirect
2. #16810469C2semidirect
3. #16810360C2semidirect
13. #125×C53C4semidirect
14. #124×C53C4semidirect
18. C4×Q8×C53C4semidirect
20. C4×D4×C53C4semidirect
99. C53#1500semidirect
101. #125Dic53semidirect
102. #124Dic53semidirect
106. C4×Q8Dic53semidirect
108. C4×D4Dic53semidirect