Finite Group Databasebeta

← #20290237

Subgroups — #20290237

Order 6784.

208 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 60 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#16811646C211yesmaximal, characteristic
2D53⋊(C4⋊C8)C211yesmaximal, characteristic
3#16811482C212yesmaximal, complemented
4#16811482C212yesmaximal, complemented
5#16811179C211yesmaximal, characteristic
6C2×D53⋊C8—21no—
7#13133284C2212yes—
8#13133284C2212yes—
9C4⋊#88324—21no—
10#13133274—21no—
11#13133272—21no—
12#13133216C2211yescharacteristic
13#13133216C2211yescharacteristic
14#13133209C411yescharacteristic
15#13133195—21no—
16#13133195—21no—
17#13133182—21no—
18#13133182—21no—
19#13133175C411yescharacteristic
20#13133175C411yescharacteristic
21C42×D53C2211yescharacteristic
22#125×C53C411yescharacteristic
23C2×C4×D53C2×C411yescharacteristic
24C2×C4×D53D411yescharacteristic
25C2×C4×D53D411yescharacteristic
26C2×Dic106—21no—
27C2×Dic106—21no—
28C2×#88324—22no—
29C2×#88324—22no—
30D53⋊C8—42no—
31C4⋊C4×C53—21no—
32C4⋊C4×C53—21no—
33C4⋊C4×C53C2×C411yescharacteristic
34C4⋊C4×C53C2×C411yescharacteristic
35C42×C53C2×C411yescharacteristic
36#11719921—21no—
37#11719921—21no—
38C4⋊Dic53C2×C411yescharacteristic
39C4⋊Dic53C2×C411yescharacteristic
40#11719919—21no—
41#11719919—21no—
42#11719919—21no—
43#11719919—21no—
44C4×Dic53—21no—
45C4×Dic53—21no—
46C4×Dic53D411yescharacteristic
47C4×Dic53D411yescharacteristic
48C4×Dic53C2311yescharacteristic
49C22×D53C22⋊C411yescharacteristic
50C2×C4×C53—21no—
51C2×C4×C53—21no—
52C2×C4×C53C22⋊C411yescharacteristic
53C2×C4×C53C22⋊C411yescharacteristic
54C2×C4×C53C22×C411yescharacteristic
55C2×Dic53—21no—
56C2×Dic53—21no—
57C2×Dic53C22⋊C411yescharacteristic
58C2×Dic53C2×D411yescharacteristic
59C2×Dic53C2×D411yescharacteristic
60C4×D53—22no—
61C4×D53—22no—
62C4×D53—21no—
63C4×D53—21no—
64Dic106—42no—
65Dic106—41no—
66Dic106—41no—
67#88324—42no—
68#88324—42no—
69C22×C53C2×C22⋊C411yescharacteristic
70C2×D53—21no—
71C2×D53—21no—
72C2×D53#10112yes—
73C4×C53—21no—
74C4×C53—21no—
75C4×C53—21no—
76C4×C53—41no—
77C4×C53—41no—
78Dic53—21no—
79Dic53—41no—
80Dic53—41no—
81Dic53#13512yes—
82Dic53AGL(1,ℤ/8)11yescharacteristic
83Dic53#13711yescharacteristic
84#1552—531nomaximal, Sylow
85C2×C53C2×#10111yescharacteristic
86C2×C53#41611yescharacteristic
87C2×C53#41711yescharacteristic
88D53—22no—
89#533—531no—
90#432—531no—
91C2×C4⋊C8—531no—
92#332—532no—
93#332—532no—
94C53#155211yescharacteristic, complemented, Sylow
95C22×C8—1061no—
96C4⋊Q8—531no—
97C4⋊Q8—531no—
98#125—531no—
99#125#615711yescharacteristic
100C22⋊Q8—1061no—
101C22⋊Q8—1061no—
102C4×Q8—1061no—
103C4×Q8—1061no—
104#117—531no—
105#117—531no—
106C2×C42—531no—
107C4⋊C8—532no—
108C4⋊C8—532no—
109C4⋊C8—1061no—
110C22⋊C8—1061no—
111#97—1061no—
112C2×Q8—1061no—
113C2×Q8—1061no—
114C22×C4—531no—
115C22×C4—531no—
116C22×C4—531no—
117C2×C8—1062no—
118C2×C8—1062no—
119C2×C8—2122no—
120C4⋊C4—21no—
121C4⋊C4—21no—
122C4⋊C4—531no—
123C4⋊C4—531no—
124C4⋊C4—1061no—
125C4⋊C4—1061no—
126C4⋊C4—1061no—
127C4⋊C4—1061no—
128C4⋊C4C2×#615711yescharacteristic
129C4⋊C4C2×#615711yescharacteristic
130C22⋊C4—1061no—
131C22⋊C4—1061no—
132C42—531no—
133C42—531no—
134C42—531no—
135C42—1061no—
136C42—1061no—
137C42C2×#615711yescharacteristic
138C23—531no—
139Q8—2122no—
140Q8—2121no—
141Q8—2121no—
142C2×C4—21no—
143C2×C4—21no—
144C2×C4—531no—
145C2×C4—531no—
146C2×C4—531no—
147C2×C4—1061no—
148C2×C4—1062no—
149C2×C4—1061no—
150C2×C4—1062no—
151C2×C4—1061no—
152C2×C4—1061no—
153C2×C4C22⋊#615711yescharacteristic
154C2×C4C22⋊#615711yescharacteristic
155C2×C4C22×#615711yescharacteristic
156C8—2122no—
157C8—2122no—
158C22—532no—
159C22—1061no—
160C22—1061no—
161C22C2×C22⋊#615711yescharacteristic
162C4—21no—
163C4—21no—
164C4—21no—
165C4—41no—
166C4—41no—
167C4—532no—
168C4—531no—
169C4—531no—
170C4—1061no—
171C4—2121no—
172C4—2121no—
173C2—1062no—
174C2D53⋊#10111yescharacteristic
175C2#1681161311yescharacteristic
176C2#1681161411yescharacteristic
subgroupG/H
centerC22C2×C22⋊#6157
commutatorC2×C4×C53C22×C4
FrattiniC2×C4C22×#6157
Fitting#125×C53C4
socleC22×C53C2×C22⋊C4
hypercenter#125#6157
hypocenterC53#1552
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #125

Complements

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

NH ≅ G/Ntype
3. #16811482C2semidirect
4. #16811482C2semidirect
94. C53#1552semidirect