Finite Group Databasebeta

← #20290905

Subgroups — #20290905

Order 6784.

324 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 74 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#16811565C211yesmaximal, characteristic
2C4×C22⋊#6157C211yesmaximal, characteristic
3#16811501C214yesmaximal
4C2×#13133197C211yesmaximal, characteristic
5C2×C4⋊#6157—21no—
6C2×C22⋊#6157C2212yes—
7C2×C22⋊#6157C2212yes—
8C2×C4×#6157—21no—
9(C2×#6157)⋊C4—21no—
10(C2×#6157)⋊C4—21no—
11C22×Dic106C411yescharacteristic, complemented
12C22×C4×D53C2211yescharacteristic
13#13133197—21no—
14#13133197—21no—
15#13133197—21no—
16#13133197—21no—
17C2×C4⋊Dic53C412yescomplemented
18C2×#11719921C2212yes—
19C2×C4⋊C4×C53C411yescharacteristic, complemented
20C23×D53C2311yescharacteristic
21C22×#6157—22no—
22C22×#6157—22no—
23C22×C4×C53C2×C411yescharacteristic
24C22×C4×C53C2×C412yes—
25C22×Dic53C2×C412yes—
26C22×Dic53C2×C411yescharacteristic
27C2×C4×D53—42no—
28C2×C4×D53D411yescharacteristic
29C2×C4×D53D411yescharacteristic
30C2×C4×D53D411yescharacteristic
31C2×C4×D53D411yescharacteristic
32C2×Dic106—21no—
33C2×Dic106—21no—
34C2×Dic106—42no—
35C22⋊#6157—22no—
36C22⋊#6157—24no—
37C22⋊#6157Q814yescomplemented
38C4⋊#6157—42no—
39C4×#6157—42no—
40C4⋊C4×C53—21no—
41C4⋊C4×C53—21no—
42#11719921—22no—
43#11719921—22no—
44C4⋊Dic53—22no—
45C4⋊Dic53—22no—
46C23×C53C22×C411yescharacteristic
47C22×D53—41no—
48C22×D53—41no—
49C22×D53C2×D411yescharacteristic
50C22×D53C2×D411yescharacteristic
51C22×D53C2×Q812yes—
52C22×D53#4112yes—
53C2×#6157—24no—
54C2×#6157—42no—
55C2×#6157—42no—
56C2×#6157—42no—
57C2×#6157—42no—
58C2×#6157—42no—
59C2×C4×C53—22no—
60C2×C4×C53—22no—
61C2×C4×C53—42no—
62C2×C4×C53C22⋊C411yescharacteristic
63C2×C4×C53C22⋊C411yescharacteristic
64C2×Dic53—22no—
65C2×Dic53—41no—
66C2×Dic53—42no—
67C2×Dic53C22⋊C411yescharacteristic
68C2×Dic53C22⋊C411yescharacteristic
69C4×D53—22no—
70C4×D53—22no—
71C4×D53—41no—
72C4×D53—41no—
73Dic106—42no—
74Dic106—42no—
75C22×C53—21no—
76C22×C53—21no—
77C22×C53C2×C22⋊C411yescharacteristic
78C22×C53C4×Q812yes—
79C2×D53—21no—
80C2×D53—21no—
81C2×D53—21no—
82C2×D53—21no—
83C2×D53—42no—
84C2×D53—42no—
85C2×D53C22⋊Q812yes—
86C2×D53#12411yescharacteristic
87C2×D53C4⋊Q811yescharacteristic
88#6157—44no—
89#6157—82no—
90C4×C53—21no—
91C4×C53—42no—
92C4×C53#9912yes—
93Dic53—21no—
94Dic53—21no—
95Dic53—42no—
96#1996—531nomaximal, Sylow
97C2×C53—22no—
98C2×C53#39111yescharacteristic
99C2×C53C2×#9911yescharacteristic
100C2×C53#41011yescharacteristic
101D53—22no—
102D53—21no—
103D53—21no—
104C2×C22⋊Q8—531no—
105#380—531no—
106C4×C22⋊C4—531no—
107#342—534no—
108C53#199611yescharacteristic, complemented, Sylow
109C22×Q8—531no—
110C23×C4—531no—
111C22⋊Q8—1061no—
112C22⋊Q8—1061no—
113C22⋊Q8—1061no—
114C22⋊Q8—1061no—
115C2×C4⋊C4—532no—
116C2×C4⋊C4—1061no—
117C2×C4⋊C4#615711yescharacteristic, complemented
118C2×C22⋊C4—532no—
119C2×C22⋊C4—532no—
120C2×C22⋊C4—532no—
121C2×C42—1061no—
122#95—1061no—
123#95—1061no—
124C24—531no—
125C2×Q8—1061no—
126C2×Q8—1061no—
127C2×Q8—2122no—
128C22×C4—531no—
129C22×C4—531no—
130C22×C4—531no—
131C22×C4—531no—
132C22×C4—531no—
133C22×C4—532no—
134C22×C4—1062no—
135C22×C4—1062no—
136C22×C4—2122no—
137C22×C4C2×#615712yes—
138C22×C4C2×#615711yescharacteristic
139C4⋊C4—21no—
140C4⋊C4—21no—
141C4⋊C4—1062no—
142C4⋊C4—1062no—
143C4⋊C4—2122no—
144C22⋊C4—534no—
145C22⋊C4—1062no—
146C22⋊C4—1062no—
147C22⋊C4—1062no—
148C22⋊C4—1064no—
149C42—2122no—
150C23—531no—
151C23—531no—
152C23—532no—
153C23—532no—
154C23—2121no—
155C23—2121no—
156C23C22×#615711yescharacteristic
157Q8—2122no—
158Q8—2122no—
159C2×C4—22no—
160C2×C4—22no—
161C2×C4—42no—
162C2×C4—531no—
163C2×C4—531no—
164C2×C4—1062no—
165C2×C4—1062no—
166C2×C4—1064no—
167C2×C4—1062no—
168C2×C4—2121no—
169C2×C4—2121no—
170C2×C4—2121no—
171C2×C4—2122no—
172C2×C4—2122no—
173C2×C4—2122no—
174C2×C4—2122no—
175C2×C4—2122no—
176C2×C4—2122no—
177C2×C4C22⋊#615711yescharacteristic
178C2×C4C22⋊#615711yescharacteristic
179C22—21no—
180C22—21no—
181C22—532no—
182C22—531no—
183C22—531no—
184C22—1061no—
185C22—1061no—
186C22—1061no—
187C22—1061no—
188C22—2122no—
189C22—2122no—
190C22C2×C22⋊#615711yescharacteristic
191C22Q8×#615712yes—
192C4—21no—
193C4—42no—
194C4—1061no—
195C4—1061no—
196C4—4242no—
197C4—2124no—
198C4—2122no—
199C4#1313327612yes—
200C2—22no—
201C2—1061no—
202C2—1062no—
203C2—1061no—
204C2(C2×#6157)⋊Q811yescharacteristic
205C2C2×#1313327611yescharacteristic
206C2#1681160311yescharacteristic
subgroupG/H
centerC22C2×C22⋊#6157
commutatorC23×C53C22×C4
FrattiniC23C22×#6157
FittingC2×C4⋊C4×C53C4
socleC22×C53C2×C22⋊C4
hypercenterC2×C4⋊C4#6157
hypocenterC53#1996
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
11. C22×Dic106C4semidirect
17. C2×C4⋊Dic53C4semidirect
19. C2×C4⋊C4×C53C4semidirect
37. C22⋊#6157Q8semidirect
108. C53#1996semidirect
117. C2×C4⋊C4#6157semidirect