Finite Group Databasebeta

← #20285853

Subgroups — #20285853

Order 6784.

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

normal subgroups only

This group has 370 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#16811028C211yesmaximal, characteristic, complemented
4#16811021C211yesmaximal, characteristic, complemented
5#16810970C211yesmaximal, characteristic, complemented
6#16810970C211yesmaximal, characteristic, complemented
7#16810948C214yesmaximal, complemented
8#16810924C214yesmaximal, complemented
9C53×#431C211yesmaximal, characteristic, complemented
10C2×#11719945—21no—
11#13133217C2211yescharacteristic
12#13133217C2211yescharacteristic
13#13133217C2211yescharacteristic
14#13133214C2211yescharacteristic
15#13133209C2212yes—
16#13133205—21no—
17#13133196—21no—
18#13133196—21no—
19#13133191—21no—
20#13133191C2212yes—
21C4×Dic106—21no—
22C4×Dic106—21no—
23C4×D212—21no—
24C4×D212C2212yes—
25#13133176C2212yes—
26#13133151C2214yes—
27#13133149C2214yes—
28#13133139C2214yes—
29#13133139C2214yes—
30C2×C42×C53C2211yescharacteristic
31C4⋊C8×C53C2212yes—
32C22⋊C8×C53C2212yes—
33C4×C8×C53C2211yescharacteristic
34C4×C8×C53C2211yescharacteristic
35C22×C4×C53D411yescharacteristic
36C22×C4×C53D411yescharacteristic
37C22×C4×C53C2311yescharacteristic
38C2×C22⋊D53—22no—
39#11719945—44no—
40C2×D212—21no—
41C2×D212—21no—
42C2×D212C2312yes—
43C2×C4×D53—22no—
44C2×Dic106—21no—
45C2×Dic106—21no—
46C2×Dic106—21no—
47C2×C8×C53C2312yes—
48C2×C8×C53C2312yes—
49C42×C53D411yescharacteristic
50C42×C53D411yescharacteristic
51C42×C53C2311yescharacteristic
52C42×C53C2311yescharacteristic
53C22⋊Dic53—22no—
54#11719921—22no—
55#11719921—22no—
56C4⋊Dic53—21no—
57C4⋊Dic53—21no—
58C4⋊Dic53—21no—
59C4⋊Dic53C2312yes—
60C4⋊Dic53C2312yes—
61C4⋊Dic53C2312yes—
62#11719919—22no—
63#11719919—22no—
64C4×Dic53—22no—
65C23×C53C2×D411yescharacteristic
66C22×D53—22no—
67C2×C4×C53—22no—
68C2×C4×C53—22no—
69C2×C4×C53—22no—
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—24no—
81D212—42no—
82D212—42no—
83C4×D53—44no—
84Dic106—42no—
85Dic106—42no—
86Dic106—42no—
87C8×C53—22no—
88C8×C53—22no—
89C22×C53—21no—
90C22×C53—21no—
91C22×C53—21no—
92C22×C53C22×D411yescharacteristic
93C2×D53—42no—
94C2×D53—42no—
95C2×D53—42no—
96C4×C53—21no—
97C4×C53—21no—
98C4×C53—21no—
99C4×C53#13512yes—
100C4×C53AGL(1,ℤ/8)12yes—
101C4×C532−1+412yes—
102Dic53—42no—
103Dic53—42no—
104Dic53—42no—
105#3228—531nomaximal, Sylow
106C2×C53—41no—
107C2×C53#53611yescharacteristic
108C2×C53C2×#13511yescharacteristic
109C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
110D53—82no—
111#533—531no—
112#532—531no—
113#500—531no—
114#498—531no—
115#486—531no—
116#486—531no—
117#482—534no—
118#478—534no—
119#431D5311yescharacteristic, complemented
120C53#322811yescharacteristic, complemented, Sylow
121C2×#41—1061no—
122C4⋊Q8—531no—
123C4⋊Q8—531no—
124C4⋊Q8—531no—
125#127—531no—
126#125—532no—
127#124—1061no—
128C22⋊Q8—1061no—
129C22⋊Q8—1061no—
130#121—532no—
131#121—1061no—
132C4×Q8—1061no—
133C4×Q8—1061no—
134C4×D4—532no—
135C4×D4—1061no—
136#117—532no—
137C2×C42C2×D5311yescharacteristic
138#107—534no—
139#106—534no—
140C4⋊C8C2×D5312yes—
141#102—534no—
142#102—534no—
143C22⋊C8C2×D5312yes—
144C4×C8C2×D5311yescharacteristic
145C4×C8C2×D5311yescharacteristic
146#41—2124no—
147C2×Q8—1061no—
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
158C2×C8C22×D5312yes—
159C2×C8C22×D5312yes—
160C4⋊C4—532no—
161C4⋊C4—532no—
162C4⋊C4—532no—
163C4⋊C4—1061no—
164C4⋊C4—1062no—
165C4⋊C4—1062no—
166C4⋊C4—1061no—
167C4⋊C4—1061no—
168C22⋊C4—1062no—
169C22⋊C4—1062no—
170C22⋊C4—1062no—
171C42—1062no—
172C42D21211yescharacteristic
173C42D21211yescharacteristic
174C42C22×D5311yescharacteristic
175C42C22×D5311yescharacteristic
176C23—1062no—
177C23C2×D21211yescharacteristic
178Q8—2122no—
179Q8—2122no—
180Q8—2122no—
181D4—1064no—
182D4—2124no—
183D4—2122no—
184D4—2122no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—1062no—
189C2×C4—1062no—
190C2×C4—1062no—
191C2×C4—2124no—
192C2×C4C2×D21211yescharacteristic
193C2×C4C2×D21211yescharacteristic
194C2×C4C2×D21211yescharacteristic
195C2×C4C2×D21211yescharacteristic
196C2×C4C2×D21211yescharacteristic
197C2×C4C23×D5311yescharacteristic
198C8—22no—
199C8—22no—
200C22—21no—
201C22—21no—
202C22—21no—
203C22—2122no—
204C22—2122no—
205C22—2122no—
206C22C22×D21211yescharacteristic
207C4—21no—
208C4—21no—
209C4—21no—
210C4—2122no—
211C4—2122no—
212C4—2122no—
213C4#1313323912yes—
214C4#1313324512yes—
215C4#1313326412yes—
216C2—41no—
217C2—4242no—
218C2#1681119911yescharacteristic
219C2C2×#1313323911yescharacteristic
220C2C2×#1313324511yescharacteristic
subgroupG/H
centerC22C22×D212
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#431C2
socleC22×C53C22×D4
hypercenter#431D53
hypocenterC53#3228
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #431

Complements

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

NH ≅ G/Ntype
1. #16811180C2semidirect
3. #16811028C2semidirect
4. #16811021C2semidirect
5. #16810970C2semidirect
6. #16810970C2semidirect
7. #16810948C2semidirect
8. #16810924C2semidirect
9. C53×#431C2semidirect
119. #431D53semidirect
120. C53#3228semidirect