Finite Group Databasebeta

← #20275161

Subgroups — #20275161

Order 6784.

358 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 89 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#16811134C211yesmaximal, characteristic
2C2×#13133176C211yesmaximal, characteristic, complemented
3C2×#13133143C212yesmaximal, complemented
4#16810384C212yesmaximal, complemented
5C53×#407C211yesmaximal, characteristic, complemented
6C2×#11719945C2211yescharacteristic
7#13133200—21no—
8#13133196C412yescomplemented
9#13133191C412yescomplemented
10#13133186—21no—
11C4×C22⋊D53—22no—
12#13133176—24no—
13#13133176C2211yescharacteristic
14C2×C4⋊Dic53—21no—
15C2×C22⋊Dic53—21no—
16C2×C4×Dic53C2212yes—
17#13133143—24no—
18C23×C4×C53C2211yescharacteristic
19C2×M16×C53C2212yes—
20C22⋊C8×C53—22no—
21C24×C53D411yescharacteristic
22C22×C4×C53—22no—
23C22×C4×C53—22no—
24C22×C4×C53C2×C412yes—
25C22×C4×C53D411yescharacteristic
26C22×C4×C53D412yes—
27C22×C4×C53C2311yescharacteristic
28C2×C22⋊D53—22no—
29C2×C22⋊D53—21no—
30C22×Dic53—22no—
31#11719945—22no—
32#11719945—22no—
33C2×D212C2×C411yescharacteristic
34C2×C4×D53—21no—
35C2×Dic106C2×C411yescharacteristic
36M16×C53—22no—
37M16×C53—22no—
38C2×C8×C53—22no—
39C22⋊Dic53—24no—
40C22⋊Dic53—22no—
41C22⋊Dic53—21no—
42#11719921—22no—
43C4⋊Dic53—24no—
44C4⋊Dic53C2×C412yes—
45#11719919—22no—
46C4×Dic53—22no—
47C4×Dic53—21no—
48C4×Dic53D414yescomplemented
49C23×C53—21no—
50C23×C53—21no—
51C23×C53—42no—
52C23×C53C22⋊C412yes—
53C23×C53C2×D411yescharacteristic
54C22×D53—21no—
55C2×C4×C53—21no—
56C2×C4×C53—21no—
57C2×C4×C53—24no—
58C2×C4×C53—24no—
59C2×C4×C53C22⋊C412yes—
60C2×C4×C53C22×C411yescharacteristic
61C2×C4×C53C2×D412yes—
62C2×C4×C53C2×D411yescharacteristic
63C2×C4×C53#4112yes—
64C22⋊D53—44no—
65C22⋊D53—41no—
66C22⋊D53—41no—
67C2×Dic53—21no—
68C2×Dic53—22no—
69C2×Dic53—24no—
70C2×Dic53—44no—
71D212—22no—
72C4×D53—42no—
73Dic106—22no—
74C8×C53—42no—
75C22×C53—22no—
76C22×C53—22no—
77C22×C53—22no—
78C22×C53—22no—
79C22×C53—42no—
80C22×C53—42no—
81C22×C53C2×C22⋊C411yescharacteristic
82C22×C53#12011yescharacteristic
83C22×C53#12311yescharacteristic
84C2×D53—41no—
85C2×D53—42no—
86C4×C53—22no—
87C4×C53C4×D412yes—
88C4×C53#12112yes—
89Dic53—44no—
90Dic53—41no—
91Dic53—42no—
92#1897—531nomaximal, Sylow
93C2×C53—21no—
94C2×C53—21no—
95C2×C53—42no—
96C2×C53#38611yescharacteristic
97C2×C53#42112yes—
98D53—81no—
99#525—531no—
100C2×#117—531no—
101C2×#104—532no—
102#407D5311yescharacteristic, complemented
103#339—532no—
104C53#189711yescharacteristic, complemented, Sylow
105C2×#41—531no—
106C23×C4C2×D5311yescharacteristic
107C2×M16C2×D5312yes—
108#123—1061no—
109C22⋊Q8—532no—
110#121—532no—
111#120—1061no—
112C4×D4—1062no—
113#117—531no—
114#117—1064no—
115C2×C4⋊C4—1061no—
116C2×C22⋊C4—1061no—
117C2×C42—532no—
118#104—1064no—
119C22⋊C8—22no—
120C24D21211yescharacteristic
121#41—1062no—
122#41—1062no—
123C2×Q8—531no—
124C2×D4—531no—
125C2×D4—1061no—
126C2×D4—1062no—
127C22×C4—22no—
128C22×C4—22no—
129C22×C4—1062no—
130C22×C4—1061no—
131C22×C4C4×D5312yes—
132C22×C4D21211yescharacteristic
133C22×C4C22⋊D5312yes—
134C22×C4C22×D5311yescharacteristic
135M16—22no—
136M16—22no—
137C2×C8—22no—
138C4⋊C4—532no—
139C4⋊C4—1064no—
140C4⋊C4—1062no—
141C22⋊C4—1064no—
142C22⋊C4—1061no—
143C22⋊C4—1062no—
144C22⋊C4—1062no—
145C42—534no—
146C42—1062no—
147C42—1061no—
148C23—21no—
149C23—21no—
150C23—42no—
151C23—1061no—
152C23#1171992112yes—
153C23C2×D21211yescharacteristic
154Q8—1062no—
155D4—1062no—
156D4—2124no—
157D4—2121no—
158D4—2121no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—24no—
162C2×C4—24no—
163C2×C4—1062no—
164C2×C4—1061no—
165C2×C4—1064no—
166C2×C4—2124no—
167C2×C4—2122no—
168C2×C4#1171992112yes—
169C2×C4C2×C4×D5311yescharacteristic
170C2×C4D4×D5312yes—
171C2×C4#1171994712yes—
172C2×C4C2×C22⋊D5311yescharacteristic
173C8—42no—
174C22—22no—
175C22—22no—
176C22—22no—
177C22—22no—
178C22—42no—
179C22—42no—
180C22—2121no—
181C22—2122no—
182C22C2×#1171992111yescharacteristic
183C22#1313318711yescharacteristic
184C22#1313320111yescharacteristic
185C4—22no—
186C4—2124no—
187C4—2121no—
188C4—2122no—
189C4#1313317712yes—
190C4#1313318912yes—
191C2—21no—
192C2—21no—
193C2—42no—
194C2—4241no—
195C2#1681050011yescharacteristic
196C2#1681063912yes—
subgroupG/H
centerC2×C4C2×C22⋊D53
commutatorC2×C4×C53C22×C4
FrattiniC22×C4C22×D53
FittingC53×#407C2
socleC22×C53C2×C22⋊C4
hypercenter#407D53
hypocenterC53#1897
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
2. C2×#13133176C2semidirect
3. C2×#13133143C2semidirect
4. #16810384C2semidirect
5. C53×#407C2semidirect
8. #13133196C4semidirect
9. #13133191C4semidirect
48. C4×Dic53D4semidirect
102. #407D53semidirect
104. C53#1897semidirect