Finite Group Databasebeta

← #20285923

Subgroups — #20285923

Order 6784.

324 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 121 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#16811180C211yesmaximal, characteristic
2C2×#13133209C211yesmaximal, characteristic
3#16811032C212yesmaximal, complemented
4#16810992C212yesmaximal, complemented
5#16810959C212yesmaximal
6#16810954C212yesmaximal
7#16810931C214yesmaximal, complemented
8C53×#432C211yesmaximal, characteristic
9#13133217C2211yescharacteristic
10#13133217C2211yescharacteristic
11#13133209—21no—
12#13133209—21no—
13#13133209—21no—
14#13133209C2212yes—
15#13133209C2212yes—
16#13133196—21no—
17#13133196C2212yes—
18C4×Dic106—21no—
19C4×Dic106C2212yes—
20#13133176—21no—
21C2×C4⋊Dic53C2211yescharacteristic
22C2×C4⋊Dic53C2211yescharacteristic
23C2×#11719919—21no—
24C2×#11719919—21no—
25#13133151C2214yes—
26#13133149C2214yes—
27#13133142C2214yes—
28#13133142C2214yes—
29C2×C42×C53C2211yescharacteristic
30C4⋊C8×C53C2212yes—
31C22⋊C8×C53C2212yes—
32#97×C53C2212yes—
33C22×C4×C53D411yescharacteristic
34C22×C4×C53D411yescharacteristic
35C22×C4×C53C2311yescharacteristic
36C22×Dic53—21no—
37C22×Dic53—21no—
38C2×Dic106—21no—
39C2×Dic106C2312yes—
40C2×C8×C53C2314yes—
41C42×C53D411yescharacteristic
42C42×C53D411yescharacteristic
43C42×C53C2311yescharacteristic
44C42×C53C2311yescharacteristic
45C22⋊Dic53—22no—
46C4⋊Dic53—21no—
47C4⋊Dic53—21no—
48C4⋊Dic53—21no—
49C4⋊Dic53—21no—
50C4⋊Dic53—21no—
51C4⋊Dic53C2312yes—
52C4⋊Dic53C2312yes—
53C4⋊Dic53C2312yes—
54#11719919—22no—
55#11719919—22no—
56#11719919—22no—
57#11719919—22no—
58#11719919—22no—
59#11719919—22no—
60C4×Dic53—22no—
61C23×C53C2×D411yescharacteristic
62C2×C4×C53—22no—
63C2×C4×C53—22no—
64C2×C4×C53—41no—
65C2×C4×C53C2×D411yescharacteristic
66C2×C4×C53C2×D411yescharacteristic
67C2×C4×C53C2×D411yescharacteristic
68C2×C4×C53C2×D411yescharacteristic
69C2×C4×C53C2×D411yescharacteristic
70C2×C4×C53C2411yescharacteristic
71C2×Dic53—22no—
72C2×Dic53—22no—
73C2×Dic53—22no—
74C2×Dic53—22no—
75C2×Dic53—42no—
76C2×Dic53—42no—
77Dic106—24no—
78Dic106—42no—
79C8×C53—24no—
80C22×C53—21no—
81C22×C53—21no—
82C22×C53#13712yes—
83C22×C53C22×D411yescharacteristic
84C4×C53—21no—
85C4×C53—21no—
86C4×C53—21no—
87C4×C53—21no—
88C4×C53—21no—
89C4×C532−1+412yes—
90Dic53—42no—
91Dic53—42no—
92Dic53—42no—
93Dic53—42no—
94#3236—531nomaximal, Sylow
95C2×C53—22no—
96C2×C53#53611yescharacteristic
97C2×C53C2×#13711yescharacteristic
98C2×C53#57511yescharacteristic
99#533—531no—
100C2×#125—531no—
101#501—532no—
102#491—532no—
103#484—532no—
104#483—532no—
105#479—534no—
106#432D5311yescharacteristic
107C53#323611yescharacteristic, complemented, Sylow
108C4⋊Q8—531no—
109C4⋊Q8—531no—
110#125—532no—
111#125—532no—
112#125—1061no—
113#125—1061no—
114#125—1061no—
115C22⋊Q8—532no—
116C22⋊Q8—1061no—
117C4×Q8—532no—
118C4×Q8—1061no—
119#117—1061no—
120C2×C4⋊C4—531no—
121C2×C4⋊C4—531no—
122C2×C4⋊C4—1061no—
123C2×C4⋊C4—1061no—
124C2×C42C2×D5311yescharacteristic
125#107—534no—
126#106—534no—
127C4⋊C8C2×D5312yes—
128#103—534no—
129#103—534no—
130C22⋊C8C2×D5312yes—
131#97C2×D5312yes—
132C2×Q8—532no—
133C2×Q8—1061no—
134C22×C4—1061no—
135C22×C4—1061no—
136C22×C4D21211yescharacteristic
137C22×C4D21211yescharacteristic
138C22×C4C22×D5311yescharacteristic
139C2×C8C22×D5314yes—
140C4⋊C4—532no—
141C4⋊C4—532no—
142C4⋊C4—532no—
143C4⋊C4—1061no—
144C4⋊C4—1061no—
145C4⋊C4—1062no—
146C4⋊C4—1062no—
147C4⋊C4—1062no—
148C4⋊C4—1062no—
149C4⋊C4—1061no—
150C4⋊C4—1061no—
151C4⋊C4—1062no—
152C4⋊C4—1062no—
153C4⋊C4—1061no—
154C22⋊C4—1062no—
155C42—1062no—
156C42D21211yescharacteristic
157C42D21211yescharacteristic
158C42C22×D5311yescharacteristic
159C42C22×D5311yescharacteristic
160C23C2×D21211yescharacteristic
161Q8—1064no—
162Q8—2122no—
163C2×C4—22no—
164C2×C4—22no—
165C2×C4—41no—
166C2×C4—1062no—
167C2×C4—1062no—
168C2×C4—1062no—
169C2×C4—1062no—
170C2×C4—2122no—
171C2×C4—2122no—
172C2×C4C2×D21211yescharacteristic
173C2×C4C2×D21211yescharacteristic
174C2×C4C2×D21211yescharacteristic
175C2×C4C2×D21211yescharacteristic
176C2×C4C2×D21211yescharacteristic
177C2×C4C23×D5311yescharacteristic
178C8—24no—
179C22—21no—
180C22—21no—
181C22#1313325012yes—
182C22C22×D21211yescharacteristic
183C4—21no—
184C4—21no—
185C4—21no—
186C4—21no—
187C4—21no—
188C4—2122no—
189C4—2122no—
190C4—2122no—
191C4—2122no—
192C4#1313326412yes—
193C2—22no—
194C2#1681119911yescharacteristic
195C2C2×#1313325011yescharacteristic
196C2#1681142311yescharacteristic
subgroupG/H
centerC22C22×D212
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#432C2
socleC22×C53C22×D4
hypercenter#432D53
hypocenterC53#3236
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #432

Complements

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

NH ≅ G/Ntype
3. #16811032C2semidirect
4. #16810992C2semidirect
7. #16810931C2semidirect
107. C53#3236semidirect