Finite Group Databasebeta

← #20284644

Subgroups — #20284644

Order 6784.

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

normal subgroups only

This group has 414 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×#13133176C211yesmaximal, characteristic, complemented
2#16810961C214yesmaximal, complemented
3#16810944C214yesmaximal, complemented
4#16810666C212yesmaximal, complemented
5#16810624C212yesmaximal, complemented
6#16810587C211yesmaximal, characteristic, complemented
7C53×#525C211yesmaximal, characteristic
8C2×#11719917C2212yes—
9#13133176—22no—
10#13133176C2214yes—
11C2×C4⋊Dic53C2212yes—
12C2×C22⋊Dic53—21no—
13C2×C4×Dic53—21no—
14#13133150C2214yescomplemented
15#13133148C2214yescomplemented
16#13133141C2214yescomplemented
17#13133138C2214yescomplemented
18C22⋊#88322C2214yes—
19C2×#41×C53C2211yescharacteristic
20C23×C4×C53C2211yescharacteristic
21#123×C53—21no—
22C22⋊Q8×C53C2212yes—
23#121×C53C2212yes—
24#120×C53—21no—
25C4×D4×C53—22no—
26#117×C53C2211yescharacteristic
27C24×C53D411yescharacteristic
28#41×C53—24no—
29C2×Q8×C53C2311yescharacteristic
30C2×D4×C53—22no—
31C2×D4×C53—21no—
32C2×D4×C53C2311yescharacteristic
33C22×C4×C53—24no—
34C22×C4×C53—21no—
35C22×C4×C53D412yes—
36C22×C4×C53D411yescharacteristic
37C22×C4×C53C2312yes—
38C22×C4×C53C2311yescharacteristic
39C22×Dic53—22no—
40C4⋊C4×C53—22no—
41C4⋊C4×C53C2312yes—
42C22⋊C4×C53—21no—
43C22⋊C4×C53—22no—
44C22⋊C4×C53—22no—
45C42×C53—21no—
46C22⋊Dic53—24no—
47C4⋊Dic53—22no—
48C4⋊Dic53C2314yes—
49C4×Dic53—24no—
50#11719917—24no—
51C2×#88322C2314yes—
52C23×C53—22no—
53C23×C53—21no—
54C23×C53—42no—
55C23×C53C2×D412yes—
56C23×C53C2×D411yescharacteristic
57Q8×C53—22no—
58D4×C53—22no—
59D4×C53—44no—
60D4×C53—42no—
61C2×C4×C53—22no—
62C2×C4×C53—24no—
63C2×C4×C53—24no—
64C2×C4×C53—21no—
65C2×C4×C53—22no—
66C2×C4×C53—42no—
67C2×C4×C53C2×D412yes—
68C2×C4×C53C2×D411yescharacteristic
69C2×C4×C53#4114yes—
70C2×C4×C53C2411yescharacteristic
71C2×Dic53—24no—
72C2×Dic53—44no—
73#88322—24no—
74C22×C53—22no—
75C22×C53—22no—
76C22×C53—22no—
77C22×C53—22no—
78C22×C53—44no—
79C22×C53—41no—
80C22×C53—42no—
81C22×C53#12312yes—
82C22×C53C22×D411yescharacteristic
83C4×C53—22no—
84C4×C53—41no—
85C4×C53—42no—
86C4×C53#12312yes—
87C4×C53C2×#4112yes—
88Dic53—44no—
89#3093—531nomaximal, Sylow
90C2×C53—22no—
91C2×C53—42no—
92C2×C53—81no—
93C2×C53C2×#12311yescharacteristic
94C2×C53#57512yes—
95#525D5311yescharacteristic
96C2×#117—531no—
97#485—534no—
98#482—534no—
99#428—532no—
100#417—532no—
101#407—531no—
102C53#309311yescharacteristic, complemented, Sylow
103C2×#41C2×D5311yescharacteristic
104C23×C4C2×D5311yescharacteristic
105C2×M16—532no—
106#123—21no—
107C22⋊Q8C2×D5312yes—
108#121C2×D5312yes—
109#120—21no—
110C4×D4—22no—
111#117—534no—
112#117—1062no—
113#117C2×D5311yescharacteristic
114C2×C4⋊C4—532no—
115C2×C22⋊C4—1061no—
116C2×C42—1061no—
117#107—534no—
118#106—534no—
119#103—534no—
120#102—534no—
121C22⋊C8—534no—
122C24C22⋊D5311yescharacteristic
123#41—24no—
124C2×Q8C22×D5311yescharacteristic
125C2×D4—22no—
126C2×D4—21no—
127C2×D4C22×D5311yescharacteristic
128C22×C4—24no—
129C22×C4—21no—
130C22×C4—1062no—
131C22×C4C22⋊D5311yescharacteristic
132C22×C4C22⋊D5312yes—
133C22×C4C22×D5312yes—
134C22×C4C22×D5311yescharacteristic
135M16—1064no—
136C2×C8—534no—
137C4⋊C4—22no—
138C4⋊C4—534no—
139C4⋊C4—1062no—
140C4⋊C4C22×D5312yes—
141C22⋊C4—21no—
142C22⋊C4—22no—
143C22⋊C4—22no—
144C22⋊C4—1064no—
145C42—21no—
146C42—1064no—
147C23—22no—
148C23—21no—
149C23—42no—
150C23C2×C22⋊D5312yes—
151C23C2×C22⋊D5311yescharacteristic
152Q8—22no—
153D4—22no—
154D4—44no—
155D4—42no—
156C2×C4—22no—
157C2×C4—24no—
158C2×C4—24no—
159C2×C4—22no—
160C2×C4—21no—
161C2×C4—42no—
162C2×C4—1064no—
163C2×C4—2124no—
164C2×C4#1171994714yes—
165C2×C4C2×C22⋊D5312yes—
166C2×C4C2×C22⋊D5311yescharacteristic
167C2×C4C23×D5311yescharacteristic
168C8—1064no—
169C22—22no—
170C22—22no—
171C22—22no—
172C22—22no—
173C22—44no—
174C22—41no—
175C22—42no—
176C22#1313319812yes—
177C22C22×C22⋊D5311yescharacteristic
178C4—22no—
179C4—41no—
180C4—42no—
181C4—2124no—
182C4#1313319812yes—
183C4C2×#1171994712yes—
184C2—22no—
185C2—42no—
186C2—81no—
187C2C2×#1313319811yescharacteristic
188C2#1681142112yes—
subgroupG/H
centerC22C22×C22⋊D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#525C2
socleC22×C53C22×D4
hypercenter#525D53
hypocenterC53#3093
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #525

Complements

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

NH ≅ G/Ntype
1. C2×#13133176C2semidirect
2. #16810961C2semidirect
3. #16810944C2semidirect
4. #16810666C2semidirect
5. #16810624C2semidirect
6. #16810587C2semidirect
14. #13133150C22semidirect
15. #13133148C22semidirect
16. #13133141C22semidirect
17. #13133138C22semidirect
102. C53#3093semidirect