Finite Group Databasebeta

← #20286166

Subgroups — #20286166

Order 6784.

322 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 109 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#16811233C212yesmaximal
2#16810933C212yesmaximal, complemented
3#16810928C212yesmaximal, complemented
4#16810881C211yesmaximal, characteristic
5#16810862C212yesmaximal
6#16810832C212yesmaximal, complemented
7#16810831C212yesmaximal, complemented
8#16810651C211yesmaximal, characteristic
9C53×#470C211yesmaximal, characteristic
10C2×Dic212C2211yescharacteristic
11C2×#11719925C2212yes—
12C22×#88322C2211yescharacteristic
13#13133215C2212yes—
14#13133210—22no—
15#13133210—22no—
16#13133208—22no—
17#13133208C2212yes—
18#13133207—22no—
19#13133196C2212yes—
20Q8×Dic53C2212yes—
21#13133182C2212yes—
22#13133176C2211yescharacteristic
23#13133150C2212yes—
24#13133149C2211yescharacteristic
25#13133146C2212yes—
26#13133146C2212yes—
27#13133142C2212yes—
28#13133141C2212yes—
29#13133140C2212yes—
30C2×Q16×C53C2211yescharacteristic
31C2×M16×C53C2211yescharacteristic
32C22⋊Q8×C53C2212yes—
33#106×C53C2211yescharacteristic
34#103×C53C2212yes—
35C2×Q8×C53C2312yes—
36C22×C4×C53C2311yescharacteristic
37C2×Dic106C2312yes—
38Q16×C53—41no—
39M16×C53—41no—
40C2×C8×C53C2311yescharacteristic
41C2×C8×C53C2311yescharacteristic
42C4⋊C4×C53—22no—
43C4⋊C4×C53C2312yes—
44C22⋊C4×C53—22no—
45C22⋊Dic53—22no—
46C22⋊Dic53D411yescharacteristic
47C22⋊Dic53D411yescharacteristic
48#11719925—42no—
49C4⋊Dic53—22no—
50C4⋊Dic53—22no—
51C4⋊Dic53D412yes—
52C4⋊Dic53C2312yes—
53#11719919—22no—
54#11719919—22no—
55#11719919—22no—
56#11719919—22no—
57#11719919—22no—
58C4×Dic53—22no—
59C4×Dic53—22no—
60C4×Dic53C2312yes—
61C2×#88322—41no—
62C2×#88322C2312yes—
63Dic212—41no—
64C23×C53C2×D411yescharacteristic
65Q8×C53—22no—
66Q8×C53—22no—
67C2×C4×C53—21no—
68C2×C4×C53—21no—
69C2×C4×C53—22no—
70C2×C4×C53—22no—
71C2×C4×C53C2×D411yescharacteristic
72C2×C4×C53C2411yescharacteristic
73C2×Dic53—22no—
74C2×Dic53—22no—
75C2×Dic53C2×D412yes—
76C2×Dic53C2×D412yes—
77Dic106—22no—
78Dic106—22no—
79C8×C53—21no—
80C8×C53—21no—
81#88322—22no—
82C22×C53—21no—
83C22×C53—22no—
84C22×C53C22×D411yescharacteristic
85C4×C53—21no—
86C4×C53—42no—
87C4×C53—42no—
88C4×C53H(5,𝔽2)11yescharacteristic
89C4×C532−1+411yescharacteristic
90Dic53—22no—
91Dic53—22no—
92Dic53—42no—
93Dic53—42no—
94#3261—531nomaximal, Sylow
95C2×C53—41no—
96C2×C53#53711yescharacteristic
97C2×C53#57812yes—
98#541—532no—
99#479—532no—
100#479—532no—
101#470—531no—
102#470D5311yescharacteristic
103#467—532no—
104#462—532no—
105#462—532no—
106#424—531no—
107C53#326111yescharacteristic, complemented, Sylow
108C2×Q16—532no—
109C2×Q16—531no—
110C2×Q16C2×D5311yescharacteristic
111C2×M16C2×D5311yescharacteristic
112C22×C8—531no—
113C4⋊Q8—532no—
114#126—1062no—
115#126—1062no—
116#125—532no—
117#125—1062no—
118#125—1062no—
119C22⋊Q8—532no—
120C22⋊Q8C2×D5312yes—
121C4×Q8—532no—
122C4×Q8—532no—
123#117—531no—
124#107—532no—
125#106—531no—
126#106C2×D5311yescharacteristic
127C4⋊C8—532no—
128C4⋊C8—532no—
129#103—532no—
130#103—532no—
131#103—532no—
132#103C2×D5312yes—
133C2×Q8—532no—
134C2×Q8C22×D5312yes—
135C22×C4C22×D5311yescharacteristic
136Q16—41no—
137Q16—2121no—
138Q16—2122no—
139M16—41no—
140C2×C8—532no—
141C2×C8—2121no—
142C2×C8C22×D5311yescharacteristic
143C2×C8C22×D5311yescharacteristic
144C4⋊C4—22no—
145C4⋊C4—532no—
146C4⋊C4—532no—
147C4⋊C4—1062no—
148C4⋊C4—1062no—
149C4⋊C4—1062no—
150C4⋊C4—1062no—
151C4⋊C4—1062no—
152C4⋊C4—1062no—
153C4⋊C4—1062no—
154C4⋊C4C22×D5312yes—
155C22⋊C4—22no—
156C22⋊C4—531no—
157C22⋊C4—531no—
158C22⋊C4—1062no—
159C42—532no—
160C42—1062no—
161C42—1062no—
162C23D4×D5311yescharacteristic
163Q8—22no—
164Q8—22no—
165Q8—1062no—
166Q8—1062no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—532no—
172C2×C4—532no—
173C2×C4—1062no—
174C2×C4—1062no—
175C2×C4D4×D5311yescharacteristic
176C2×C4C23×D5311yescharacteristic
177C8—21no—
178C8—21no—
179C8—1062no—
180C22—22no—
181C22—21no—
182C22C2×D4×D5311yescharacteristic
183C4—21no—
184C4—42no—
185C4—42no—
186C4—1062no—
187C4—1062no—
188C4—2122no—
189C4—2122no—
190C4#1313326211yescharacteristic
191C4#1313326411yescharacteristic
192C2—41no—
193C2#1681120011yescharacteristic
194C2#1681143912yes—
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#470C2
socleC22×C53C22×D4
hypercenter#470D53
hypocenterC53#3261
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #470

Complements

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

NH ≅ G/Ntype
2. #16810933C2semidirect
3. #16810928C2semidirect
6. #16810832C2semidirect
7. #16810831C2semidirect
107. C53#3261semidirect