Finite Group Databasebeta

← #20275682

Subgroups — #20275682

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×#13133127C211yesmaximal, characteristic
3#16810492C211yesmaximal, 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×#11719919—21no—
16C2×#11719919—21no—
17C2×C4×Dic53C2211yescharacteristic
18#13133127—22no—
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×C53Q812yes—
32C4⋊C4×C53—21no—
33C4⋊C4×C53—21no—
34C4⋊C4×C53—21no—
35C4⋊C4×C53C2×C412yes—
36C4⋊Dic53—21no—
37C4⋊Dic53—24no—
38C4⋊Dic53—21no—
39C4⋊Dic53—21no—
40C4⋊Dic53C2×C412yes—
41#11719919—24no—
42#11719919—42no—
43#11719919—42no—
44C4×Dic53C2×C412yes—
45C4×Dic53D412yes—
46C2×#88322—21no—
47C2×#88322—21no—
48C2×#88322Q812yes—
49C23×C53C2×D411yescharacteristic
50C2×C4×C53—22no—
51C2×C4×C53—41no—
52C2×C4×C53—41no—
53C2×C4×C53—41no—
54C2×C4×C53—41no—
55C2×C4×C53—41no—
56C2×C4×C53C22×C411yescharacteristic
57C2×C4×C53C2×D411yescharacteristic
58C2×C4×C53C2×Q811yescharacteristic
59C2×C4×C53C2×Q811yescharacteristic
60C2×C4×C53#4111yescharacteristic
61C2×C4×C53#4111yescharacteristic
62C2×Dic53—21no—
63C2×Dic53—21no—
64C2×Dic53—21no—
65C2×Dic53—21no—
66C2×Dic53—22no—
67C2×Dic53—41no—
68C2×Dic53—41no—
69C2×Dic53—41no—
70C2×Dic53—41no—
71C2×Dic53—41no—
72C2×Dic53C22⋊C412yes—
73C2×Dic53C22⋊C412yes—
74C8×C53—22no—
75#88322—22no—
76C22×C53C2×C22⋊C411yescharacteristic
77C22×C53#12411yescharacteristic
78C22×C53C4⋊Q811yescharacteristic
79C22×C53AGL(1,ℤ/8)11yescharacteristic
80C22×C53AGL(1,ℤ/8)11yescharacteristic
81C22×C53#13711yescharacteristic
82C22×C53#13711yescharacteristic
83C4×C53—42no—
84C4×C53—81no—
85C4×C53C4×Q811yescharacteristic
86C4×C53C4×Q811yescharacteristic
87C4×C53C22⋊Q811yescharacteristic
88C4×C53C22⋊Q811yescharacteristic
89Dic53—22no—
90Dic53—22no—
91Dic53—42no—
92Dic53—81no—
93#1992—531nomaximal, Sylow
94C2×C53#39111yescharacteristic
95C2×C53#41711yescharacteristic
96C2×C53#41711yescharacteristic
97C2×C53#49011yescharacteristic
98C2×C53#49111yescharacteristic
99C2×C53#50111yescharacteristic
100C2×C53#50111yescharacteristic
101C2×#125—531no—
102C2×#97—531no—
103#384—531no—
104#340—531no—
105#340—531no—
106#340—531no—
107#340D5311yescharacteristic
108C53#199211yescharacteristic, complemented, Sylow
109C22×C8—531no—
110C22×C8C2×D5311yescharacteristic
111#125—534no—
112#125—1062no—
113C2×C4⋊C4—531no—
114C2×C4⋊C4—531no—
115C2×C4⋊C4—1061no—
116C2×C4⋊C4—1061no—
117C2×C4⋊C4—1061no—
118C2×C4⋊C4—1061no—
119C2×C4⋊C4C2×D5311yescharacteristic
120C2×C4⋊C4C2×D5311yescharacteristic
121C2×C42—531no—
122#97—1062no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—531no—
126C22×C4—531no—
127C22×C4—1061no—
128C22×C4—1061no—
129C22×C4C22×D5311yescharacteristic
130C2×C8—21no—
131C2×C8—21no—
132C2×C8—532no—
133C2×C8—1061no—
134C2×C8—1061no—
135C2×C8Dic10612yes—
136C4⋊C4—21no—
137C4⋊C4—21no—
138C4⋊C4—21no—
139C4⋊C4—532no—
140C4⋊C4—1064no—
141C4⋊C4—1064no—
142C4⋊C4—1061no—
143C4⋊C4—1061no—
144C4⋊C4—1061no—
145C4⋊C4—2122no—
146C4⋊C4—2122no—
147C4⋊C4C4×D5312yes—
148C42—532no—
149C42—532no—
150C23D4×D5311yescharacteristic
151C2×C4—22no—
152C2×C4—41no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—532no—
158C2×C4—532no—
159C2×C4—1061no—
160C2×C4—1061no—
161C2×C4—1061no—
162C2×C4—1061no—
163C2×C4—1062no—
164C2×C4—2121no—
165C2×C4—2121no—
166C2×C4—2121no—
167C2×C4—2121no—
168C2×C4—2121no—
169C2×C4C2×Dic10611yescharacteristic
170C2×C4C2×C4×D5311yescharacteristic
171C2×C4#1171994511yescharacteristic
172C2×C4D4×D5311yescharacteristic
173C2×C4#1171994711yescharacteristic
174C2×C4Q8×D5311yescharacteristic
175C8—22no—
176C8—1062no—
177C22C22⋊C4×D5311yescharacteristic
178C22#1313320311yescharacteristic
179C22#1313321611yescharacteristic
180C22#1313324111yescharacteristic
181C22#1313324411yescharacteristic
182C22#1313324611yescharacteristic
183C22#1313324911yescharacteristic
184C4—42no—
185C4—81no—
186C4—1062no—
187C4—1062no—
188C4—4241no—
189C4—2122no—
190C4#1313318211yescharacteristic
191C4C4×Dic10611yescharacteristic
192C4#1313319411yescharacteristic
193C4#1313319511yescharacteristic
194C2#1681052711yescharacteristic
195C2#1681062111yescharacteristic
196C2#1681062311yescharacteristic
197C2#1681098611yescharacteristic
198C2#1681099011yescharacteristic
199C2#1681103011yescharacteristic
200C2#1681103111yescharacteristic
subgroupG/H
centerC23D4×D53
commutatorC2×C4×C53C22×C4
FrattiniC22×C4C22×D53
FittingC53×#340C2
socleC23×C53C2×D4
hypercenter#340D53
hypocenterC53#1992
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
108. C53#1992semidirect