Finite Group Databasebeta

← #20286103

Subgroups — #20286103

Order 6784.

386 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 125 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, complemented
2#16811173C211yesmaximal, characteristic
3#16810919C212yesmaximal, complemented
4#16810904C212yesmaximal, complemented
5#16810869C214yesmaximal, complemented
6#16810829C212yesmaximal, complemented
7#16810824C212yesmaximal, complemented
8C53×#423C211yesmaximal, characteristic, complemented
9C2×#11719945—21no—
10C2×#11719913C2214yes—
11#13133217C2211yescharacteristic
12#13133217C2211yescharacteristic
13#13133217C2211yescharacteristic
14#13133214C2211yescharacteristic
15#13133209—21no—
16#13133205—21no—
17#13133196—21no—
18#13133196C2212yes—
19#13133191—21no—
20#13133191C2212yes—
21C4×Dic106—21no—
22C4×Dic106C2212yes—
23C4×D212—21no—
24C4×D212C2212yes—
25#13133176—21no—
26#13133151C2214yes—
27#13133142C2214yes—
28#13133139C2214yes—
29C2×M16×C53C2212yes—
30C2×C42×C53C2211yescharacteristic
31C4⋊C8×C53C2212yes—
32C4⋊C8×C53C2212yes—
33C22×C4×C53D411yescharacteristic
34C22×C4×C53D411yescharacteristic
35C22×C4×C53C2311yescharacteristic
36C2×C22⋊D53—22no—
37#11719945—44no—
38C2×D212—21no—
39C2×D212—21no—
40C2×D212C2312yes—
41C2×C4×D53—22no—
42C2×Dic106—21no—
43C2×Dic106—21no—
44C2×Dic106C2312yes—
45M16×C53—42no—
46C2×C8×C53C2314yes—
47C42×C53D411yescharacteristic
48C42×C53D411yescharacteristic
49C42×C53C2311yescharacteristic
50C42×C53C2311yescharacteristic
51C22⋊Dic53—22no—
52#11719921—22no—
53#11719921—22no—
54C4⋊Dic53—21no—
55C4⋊Dic53—21no—
56C4⋊Dic53—21no—
57C4⋊Dic53—21no—
58C4⋊Dic53C2312yes—
59C4⋊Dic53C2312yes—
60#11719919—22no—
61#11719919—22no—
62C4×Dic53—22no—
63#11719913—44no—
64C23×C53C2×D411yescharacteristic
65C22×D53—22no—
66C2×C4×C53—22no—
67C2×C4×C53—22no—
68C2×C4×C53—21no—
69C2×C4×C53—21no—
70C2×C4×C53C2×D411yescharacteristic
71C2×C4×C53C2×D411yescharacteristic
72C2×C4×C53C2×D411yescharacteristic
73C2×C4×C53C2×D411yescharacteristic
74C2×C4×C53C2×D411yescharacteristic
75C2×C4×C53C2411yescharacteristic
76C22⋊D53—44no—
77C2×Dic53—22no—
78C2×Dic53—22no—
79C2×Dic53—22no—
80D212—22no—
81D212—22no—
82D212—42no—
83D212—42no—
84C4×D53—44no—
85Dic106—22no—
86Dic106—22no—
87Dic106—42no—
88Dic106—42no—
89C8×C53—24no—
90C22×C53—21no—
91C22×C53—21no—
92C22×C53—21no—
93C22×C53C22×D411yescharacteristic
94C2×D53—42no—
95C2×D53—42no—
96C2×D53—42no—
97C4×C53—21no—
98C4×C53—21no—
99C4×C53—21no—
100C4×C53AGL(1,ℤ/8)12yes—
101C4×C53#13712yes—
102C4×C53H(5,𝔽2)11yescharacteristic
103C4×C532−1+411yescharacteristic
104Dic53—42no—
105Dic53—42no—
106Dic53—42no—
107#3254—531nomaximal, Sylow
108C2×C53—41no—
109C2×C53#53711yescharacteristic
110C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
111C2×C53C2×#13711yescharacteristic
112D53—82no—
113#533—531no—
114#532—531no—
115#477—532no—
116#474—532no—
117#468—534no—
118#461—532no—
119#460—532no—
120#423D5311yescharacteristic, complemented
121C53#325411yescharacteristic, complemented, Sylow
122C2×#41—1061no—
123C2×SD16—534no—
124C2×M16C2×D5312yes—
125C4⋊Q8—531no—
126C4⋊Q8—531no—
127C4⋊Q8—531no—
128#127—531no—
129#125—1061no—
130#124—1061no—
131C22⋊Q8—532no—
132C22⋊Q8—1061no—
133#121—532no—
134#121—1061no—
135C4×Q8—532no—
136C4×Q8—1061no—
137C4×D4—532no—
138C4×D4—1061no—
139#117—1061no—
140C2×C42C2×D5311yescharacteristic
141#107—534no—
142C4⋊C8C2×D5312yes—
143C4⋊C8C2×D5312yes—
144#103—534no—
145#102—534no—
146#41—2124no—
147C2×Q8—532no—
148C2×Q8—1061no—
149C2×Q8—1061no—
150C2×D4—532no—
151C2×D4—1062no—
152C2×D4—1061no—
153C2×D4—1061no—
154C22×C4—1062no—
155C22×C4D21211yescharacteristic
156C22×C4D21211yescharacteristic
157C22×C4C22×D5311yescharacteristic
158SD16—2124no—
159M16—42no—
160C2×C8C22×D5314yes—
161C4⋊C4—532no—
162C4⋊C4—532no—
163C4⋊C4—1061no—
164C4⋊C4—1061no—
165C4⋊C4—1061no—
166C4⋊C4—1062no—
167C4⋊C4—1062no—
168C4⋊C4—1061no—
169C22⋊C4—1062no—
170C22⋊C4—1062no—
171C22⋊C4—1062no—
172C42—1062no—
173C42D21211yescharacteristic
174C42D21211yescharacteristic
175C42C22×D5311yescharacteristic
176C42C22×D5311yescharacteristic
177C23—1062no—
178C23C2×D21211yescharacteristic
179Q8—1062no—
180Q8—1062no—
181Q8—2122no—
182Q8—2122no—
183D4—1062no—
184D4—1062no—
185D4—2122no—
186D4—2124no—
187D4—2122no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—1062no—
193C2×C4—1062no—
194C2×C4—1062no—
195C2×C4—2124no—
196C2×C4C2×D21211yescharacteristic
197C2×C4C2×D21211yescharacteristic
198C2×C4C2×D21211yescharacteristic
199C2×C4C2×D21211yescharacteristic
200C2×C4C2×D21211yescharacteristic
201C2×C4C23×D5311yescharacteristic
202C8—24no—
203C22—21no—
204C22—21no—
205C22—21no—
206C22—2122no—
207C22—2122no—
208C22—2122no—
209C22C22×D21211yescharacteristic
210C4—21no—
211C4—21no—
212C4—21no—
213C4—2122no—
214C4—2122no—
215C4—2122no—
216C4#1313324512yes—
217C4#1313325012yes—
218C4#1313326211yescharacteristic
219C4#1313326511yescharacteristic
220C2—41no—
221C2—4242no—
222C2#1681120411yescharacteristic
223C2C2×#1313324511yescharacteristic
224C2C2×#1313325011yescharacteristic
subgroupG/H
centerC22C22×D212
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#423C2
socleC22×C53C22×D4
hypercenter#423D53
hypocenterC53#3254
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #423

Complements

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

NH ≅ G/Ntype
1. #16811180C2semidirect
3. #16810919C2semidirect
4. #16810904C2semidirect
5. #16810869C2semidirect
6. #16810829C2semidirect
7. #16810824C2semidirect
8. C53×#423C2semidirect
120. #423D53semidirect
121. C53#3254semidirect