Finite Group Databasebeta

← #20290236

Subgroups — #20290236

Order 6784.

208 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 62 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#16811643C211yesmaximal, characteristic
2C2×#13133284C211yesmaximal, characteristic
3#16811482C212yesmaximal, complemented
4#16811482C212yesmaximal, complemented
5#16811176C211yesmaximal, characteristic
6C22×#88324—21no—
7#13133284—21no—
8#13133284C2212yes—
9#13133284C2212yes—
10C22⋊#88324—21no—
11#13133272—21no—
12#13133216C2211yescharacteristic
13#13133216C2211yescharacteristic
14#13133208C411yescharacteristic
15#13133208C411yescharacteristic
16#13133194—21no—
17#13133194—21no—
18C4×Dic106—21no—
19C4×Dic106—21no—
20#13133176C411yescharacteristic
21C2×C4×Dic53C2211yescharacteristic
22#117×C53C411yescharacteristic
23C22×C4×C53C2×C411yescharacteristic
24C22×Dic53D411yescharacteristic
25C22×Dic53D411yescharacteristic
26C2×Dic106—21no—
27C2×Dic106—21no—
28C2×#88324—22no—
29C2×#88324—22no—
30C2×#88324—42no—
31C4⋊C4×C53C2×C411yescharacteristic
32C4⋊C4×C53C2×C411yescharacteristic
33C22⋊C4×C53—21no—
34C42×C53—21no—
35C22⋊Dic53—21no—
36C4⋊Dic53—21no—
37C4⋊Dic53—21no—
38C4⋊Dic53C2×C411yescharacteristic
39C4⋊Dic53C2×C411yescharacteristic
40#11719919—21no—
41#11719919—21no—
42#11719919—21no—
43#11719919—21no—
44C4×Dic53—21no—
45C4×Dic53C2×C411yescharacteristic
46C4×Dic53D411yescharacteristic
47C4×Dic53D411yescharacteristic
48C4×Dic53C2311yescharacteristic
49C23×C53C22⋊C411yescharacteristic
50C2×C4×C53—22no—
51C2×C4×C53—21no—
52C2×C4×C53—21no—
53C2×C4×C53C22⋊C411yescharacteristic
54C2×C4×C53C22×C411yescharacteristic
55C2×Dic53—22no—
56C2×Dic53—21no—
57C2×Dic53—21no—
58C2×Dic53—21no—
59C2×Dic53—21no—
60C2×Dic53C22⋊C411yescharacteristic
61C2×Dic53C22⋊C411yescharacteristic
62C2×Dic53C2×D411yescharacteristic
63C2×Dic53C2×D411yescharacteristic
64Dic106—42no—
65Dic106—41no—
66Dic106—41no—
67#88324—42no—
68#88324—42no—
69C22×C53—21no—
70C22×C53—21no—
71C22×C53#10112yes—
72C22×C53C2×C22⋊C411yescharacteristic
73C4×C53—21no—
74C4×C53—21no—
75C4×C53—41no—
76C4×C53—41no—
77Dic53—21no—
78Dic53—21no—
79Dic53—41no—
80Dic53—41no—
81Dic53#13512yes—
82Dic53AGL(1,ℤ/8)11yescharacteristic
83Dic53#13711yescharacteristic
84#1552—531nomaximal, Sylow
85C2×C53—22no—
86C2×C53C2×#10111yescharacteristic
87C2×C53#41611yescharacteristic
88C2×C53#41711yescharacteristic
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—531no—
100C22⋊Q8—1061no—
101C22⋊Q8—1061no—
102C4×Q8—1061no—
103C4×Q8—1061no—
104#117—531no—
105#117#615711yescharacteristic
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×C4C2×#615711yescharacteristic
117C2×C8—1062no—
118C2×C8—1062no—
119C2×C8—2122no—
120C4⋊C4—531no—
121C4⋊C4—531no—
122C4⋊C4—1061no—
123C4⋊C4—1061no—
124C4⋊C4—1061no—
125C4⋊C4—1061no—
126C4⋊C4—1061no—
127C4⋊C4—1061no—
128C4⋊C4C2×#615711yescharacteristic
129C4⋊C4C2×#615711yescharacteristic
130C22⋊C4—21no—
131C22⋊C4—1061no—
132C42—21no—
133C42—531no—
134C42—531no—
135C42—531no—
136C42—531no—
137C42—1061no—
138C23C22⋊#615711yescharacteristic
139Q8—2122no—
140Q8—2121no—
141Q8—2121no—
142C2×C4—22no—
143C2×C4—21no—
144C2×C4—21no—
145C2×C4—531no—
146C2×C4—531no—
147C2×C4—531no—
148C2×C4—531no—
149C2×C4—1061no—
150C2×C4—1062no—
151C2×C4—1061no—
152C2×C4—1061no—
153C2×C4—1061no—
154C2×C4C22⋊#615711yescharacteristic
155C2×C4C22×#615711yescharacteristic
156C8—2122no—
157C8—2122no—
158C22—21no—
159C22—21no—
160C22#1313327812yes—
161C22C2×C22⋊#615711yescharacteristic
162C4—21no—
163C4—21no—
164C4—41no—
165C4—41no—
166C4—532no—
167C4—531no—
168C4—531no—
169C4—1061no—
170C4—1061no—
171C4—2121no—
172C4—2121no—
173C2—22no—
174C2C2×#1313327811yescharacteristic
175C2#1681161311yescharacteristic
176C2#1681161411yescharacteristic
subgroupG/H
centerC22C2×C22⋊#6157
commutatorC2×C4×C53C22×C4
FrattiniC2×C4C22×#6157
Fitting#117×C53C4
socleC22×C53C2×C22⋊C4
hypercenter#117#6157
hypocenterC53#1552
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #117

Complements

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

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