Finite Group Databasebeta

← #20275669

Subgroups — #20275669

Order 6784.

260 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 99 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×#13133208C211yesmaximal, characteristic
2C2×#13133146C211yesmaximal, characteristic
3C4⋊C4×Dic53C211yesmaximal, characteristic
4#16810386C211yesmaximal, characteristic
5#16810385C211yesmaximal, characteristic
6#16810385C211yesmaximal, characteristic
7C53×#340C211yesmaximal, characteristic
8C22×#88322C2211yescharacteristic
9#13133208—22no—
10#13133208C414yescomplemented
11C2×C4⋊Dic53—21no—
12C2×C4⋊Dic53C2211yescharacteristic
13C2×C4⋊Dic53C2211yescharacteristic
14C2×#11719919—21no—
15C2×C4×Dic53—21no—
16C2×C4×Dic53C2211yescharacteristic
17#13133146—22no—
18#13133124—21no—
19C22×C8×C53C2211yescharacteristic
20C2×C4⋊C4×C53C2211yescharacteristic
21C2×C4⋊C4×C53C2211yescharacteristic
22C22×C4×C53—21no—
23C22×C4×C53—21no—
24C22×C4×C53C2311yescharacteristic
25C22×Dic53—21no—
26C22×Dic53—21no—
27C22×Dic53D411yescharacteristic
28C22×Dic53D411yescharacteristic
29C2×C8×C53—21no—
30C2×C8×C53—21no—
31C2×C8×C53—21no—
32C4⋊C4×C53—21no—
33C4⋊C4×C53—21no—
34C4⋊C4×C53C2×C412yes—
35C4⋊C4×C53Q812yes—
36C4⋊Dic53—21no—
37C4⋊Dic53—21no—
38C4⋊Dic53—24no—
39C4⋊Dic53C2×C412yes—
40C4⋊Dic53Q812yes—
41#11719919—24no—
42C4×Dic53—42no—
43C4×Dic53C2×C412yes—
44C4×Dic53D412yes—
45C2×#88322—21no—
46C2×#88322—21no—
47C2×#88322—21no—
48C23×C53C2×D411yescharacteristic
49C2×C4×C53—22no—
50C2×C4×C53—22no—
51C2×C4×C53—41no—
52C2×C4×C53—41no—
53C2×C4×C53—41no—
54C2×C4×C53—41no—
55C2×C4×C53C22×C411yescharacteristic
56C2×C4×C53C2×D411yescharacteristic
57C2×C4×C53C2×Q811yescharacteristic
58C2×C4×C53C2×Q811yescharacteristic
59C2×C4×C53#4111yescharacteristic
60C2×C4×C53#4111yescharacteristic
61C2×Dic53—21no—
62C2×Dic53—21no—
63C2×Dic53—21no—
64C2×Dic53—21no—
65C2×Dic53—22no—
66C2×Dic53—22no—
67C2×Dic53—41no—
68C2×Dic53—41no—
69C2×Dic53—41no—
70C2×Dic53—41no—
71C2×Dic53C22⋊C412yes—
72C2×Dic53C22⋊C412yes—
73C8×C53—41no—
74#88322—41no—
75C22×C53C2×C22⋊C411yescharacteristic
76C22×C53C22⋊Q811yescharacteristic
77C22×C53C22⋊Q811yescharacteristic
78C22×C53#13511yescharacteristic
79C22×C53#13511yescharacteristic
80C22×C53AGL(1,ℤ/8)11yescharacteristic
81C22×C53#13711yescharacteristic
82C4×C53—42no—
83C4×C53—42no—
84C4×C53C4×Q811yescharacteristic
85C4×C53C4×Q811yescharacteristic
86C4×C53#12411yescharacteristic
87C4×C53C4⋊Q811yescharacteristic
88Dic53—22no—
89Dic53—22no—
90Dic53—42no—
91Dic53—42no—
92#1989—531nomaximal, Sylow
93C2×C53#39111yescharacteristic
94C2×C53#41611yescharacteristic
95C2×C53#41711yescharacteristic
96C2×C53#47811yescharacteristic
97C2×C53#47811yescharacteristic
98C2×C53#47911yescharacteristic
99C2×C53#47911yescharacteristic
100C2×#125—531no—
101C2×C4⋊C8—531no—
102C4×C4⋊C4—531no—
103#340—531no—
104#340—531no—
105#340—531no—
106#340D5311yescharacteristic
107C53#198911yescharacteristic, complemented, Sylow
108C22×C8—531no—
109C22×C8C2×D5311yescharacteristic
110#125—534no—
111#125—1062no—
112C2×C4⋊C4—531no—
113C2×C4⋊C4—531no—
114C2×C4⋊C4—1061no—
115C2×C4⋊C4—1061no—
116C2×C4⋊C4C2×D5311yescharacteristic
117C2×C4⋊C4C2×D5311yescharacteristic
118C2×C42—531no—
119C2×C42—1061no—
120C4⋊C8—1062no—
121#95—1061no—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—531no—
125C22×C4—531no—
126C22×C4—1061no—
127C22×C4—1061no—
128C22×C4C22×D5311yescharacteristic
129C2×C8—21no—
130C2×C8—21no—
131C2×C8—21no—
132C2×C8—1061no—
133C2×C8—1061no—
134C2×C8—1061no—
135C4⋊C4—21no—
136C4⋊C4—21no—
137C4⋊C4—532no—
138C4⋊C4—532no—
139C4⋊C4—1064no—
140C4⋊C4—1064no—
141C4⋊C4—1061no—
142C4⋊C4—1061no—
143C4⋊C4Dic10612yes—
144C4⋊C4C4×D5312yes—
145C42—532no—
146C42—532no—
147C42—2122no—
148C23D4×D5311yescharacteristic
149C2×C4—22no—
150C2×C4—22no—
151C2×C4—41no—
152C2×C4—41no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—532no—
156C2×C4—532no—
157C2×C4—1061no—
158C2×C4—1061no—
159C2×C4—1061no—
160C2×C4—1061no—
161C2×C4—1062no—
162C2×C4—1062no—
163C2×C4—2121no—
164C2×C4—2121no—
165C2×C4—2121no—
166C2×C4—2121no—
167C2×C4C2×Dic10611yescharacteristic
168C2×C4C2×C4×D5311yescharacteristic
169C2×C4#1171994511yescharacteristic
170C2×C4D4×D5311yescharacteristic
171C2×C4#1171994711yescharacteristic
172C2×C4Q8×D5311yescharacteristic
173C8—41no—
174C8—2121no—
175C22C22⋊C4×D5311yescharacteristic
176C22#1313319411yescharacteristic
177C22#1313319511yescharacteristic
178C22#1313323811yescharacteristic
179C22#1313324011yescharacteristic
180C22#1313324111yescharacteristic
181C22#1313324611yescharacteristic
182C4—42no—
183C4—42no—
184C4—1062no—
185C4—1062no—
186C4—2122no—
187C4—2122no—
188C4#1313318211yescharacteristic
189C4C4×Dic10611yescharacteristic
190C4#1313320311yescharacteristic
191C4#1313321611yescharacteristic
192C2#1681052711yescharacteristic
193C2#1681061811yescharacteristic
194C2#1681062111yescharacteristic
195C2#1681092511yescharacteristic
196C2#1681092611yescharacteristic
197C2#1681093211yescharacteristic
198C2#1681093311yescharacteristic
subgroupG/H
centerC23D4×D53
commutatorC2×C4×C53C22×C4
FrattiniC22×C4C22×D53
FittingC53×#340C2
socleC23×C53C2×D4
hypercenter#340D53
hypocenterC53#1989
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C4 ◁ #340

Complements

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

NH ≅ G/Ntype
10. #13133208C4semidirect
107. C53#1989semidirect