Finite Group Databasebeta

← #20285852

Subgroups — #20285852

Order 6784.

370 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 117 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#16811178C211yesmaximal, characteristic, complemented
2#16811174C211yesmaximal, characteristic
3#16811026C211yesmaximal, characteristic, complemented
4#16810971C211yesmaximal, characteristic, complemented
5#16810970C211yesmaximal, characteristic, complemented
6#16810949C214yesmaximal, complemented
7#16810922C214yesmaximal, complemented
8#16810674C211yesmaximal, characteristic, complemented
9C53×#498C211yesmaximal, characteristic, complemented
10C2×#11719949—21no—
11#13133217C2211yescharacteristic
12#13133214C2211yescharacteristic
13#13133208C2212yes—
14#13133206—21no—
15#13133197—21no—
16#13133193—21no—
17#13133192—21no—
18#13133192C2212yes—
19Q8×Dic53—21no—
20#13133182—21no—
21#13133179—21no—
22#13133179C2212yes—
23#13133175C2212yes—
24C42×D53C2211yescharacteristic
25#13133150C2214yes—
26#13133146C2212yes—
27#13133139C2214yes—
28#13133137C2214yes—
29#13133129C2212yes—
30C4×#88322C2211yescharacteristic
31C4⋊Q8×C53C2211yescharacteristic
32C4⋊Q8×C53C2211yescharacteristic
33#106×C53C2214yes—
34C4×C8×C53C2211yescharacteristic
35C2×Q8×C53—21no—
36C2×Q8×C53—21no—
37#11719949—44no—
38C2×D212—22no—
39C2×D212—21no—
40C2×D212—21no—
41C2×D212C2312yes—
42C2×C4×D53—22no—
43C2×C4×D53D411yescharacteristic
44C2×C4×D53D411yescharacteristic
45C2×C4×D53C2311yescharacteristic
46C2×Dic106—21no—
47C2×C8×C53C2312yes—
48C4⋊C4×C53—21no—
49C4⋊C4×C53—21no—
50C4⋊C4×C53C2312yes—
51C4⋊C4×C53C2312yes—
52C42×C53C2311yescharacteristic
53#11719921—22no—
54#11719921—22no—
55#11719921—22no—
56C4⋊Dic53—21no—
57C4⋊Dic53—22no—
58C4⋊Dic53C2312yes—
59#11719919—22no—
60C4×Dic53—22no—
61C4×Dic53D411yescharacteristic
62C4×Dic53D411yescharacteristic
63C4×Dic53C2311yescharacteristic
64C2×#88322C2312yes—
65C22×D53—22no—
66C22×D53C2×D411yescharacteristic
67Q8×C53—42no—
68Q8×C53—42no—
69C2×C4×C53—22no—
70C2×C4×C53—22no—
71C2×C4×C53C2×D411yescharacteristic
72C2×C4×C53C2×D411yescharacteristic
73C2×C4×C53C2411yescharacteristic
74C2×Dic53—22no—
75C2×Dic53C2×D411yescharacteristic
76C2×Dic53C2×D411yescharacteristic
77C2×Dic53C2×D411yescharacteristic
78D212—24no—
79D212—44no—
80D212—42no—
81D212—42no—
82C4×D53—22no—
83C4×D53—22no—
84C4×D53—22no—
85C4×D53—44no—
86Dic106—42no—
87C8×C53—22no—
88#88322—22no—
89C22×C53C22×D411yescharacteristic
90C2×D53—21no—
91C2×D53—21no—
92C2×D53—21no—
93C2×D53—42no—
94C2×D53—42no—
95C2×D53—42no—
96C4×C53—42no—
97C4×C53—42no—
98C4×C53#13512yes—
99C4×C53AGL(1,ℤ/8)12yes—
100C4×C532−1+412yes—
101Dic53—21no—
102Dic53—21no—
103Dic53—21no—
104Dic53—42no—
105#3228—531nomaximal, Sylow
106C2×C53#53611yescharacteristic
107C2×C53C2×#13511yescharacteristic
108C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
109D53—41no—
110D53—82no—
111#533—531no—
112#532—531no—
113#500—531no—
114#498D5311yescharacteristic, complemented
115#486—531no—
116#486—531no—
117#482—534no—
118#478—534no—
119#431—531no—
120C53#322811yescharacteristic, complemented, Sylow
121C2×#41—1061no—
122C4⋊Q8—531no—
123C4⋊Q8C2×D5311yescharacteristic
124C4⋊Q8C2×D5311yescharacteristic
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×C42—531no—
138#107—534no—
139#106C2×D5314yes—
140C4⋊C8—532no—
141#102—534no—
142#102—534no—
143C22⋊C8—532no—
144C4×C8—531no—
145C4×C8C2×D5311yescharacteristic
146#41—2124no—
147C2×Q8—21no—
148C2×Q8—21no—
149C2×Q8—1061no—
150C2×D4—532no—
151C2×D4—1062no—
152C2×D4—1061no—
153C2×D4—1061no—
154C22×C4—531no—
155C22×C4—531no—
156C22×C4—531no—
157C22×C4—1062no—
158C2×C8—532no—
159C2×C8C22×D5312yes—
160C4⋊C4—21no—
161C4⋊C4—21no—
162C4⋊C4—532no—
163C4⋊C4—1062no—
164C4⋊C4—1062no—
165C4⋊C4—1061no—
166C4⋊C4C22×D5312yes—
167C4⋊C4C22×D5312yes—
168C22⋊C4—1062no—
169C22⋊C4—1062no—
170C22⋊C4—1062no—
171C42—531no—
172C42—531no—
173C42—531no—
174C42—1062no—
175C42C22×D5311yescharacteristic
176C23—531no—
177C23—1062no—
178Q8—42no—
179Q8—42no—
180Q8—2122no—
181D4—1064no—
182D4—2124no—
183D4—2122no—
184D4—2122no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—531no—
188C2×C4—531no—
189C2×C4—531no—
190C2×C4—1062no—
191C2×C4—1062no—
192C2×C4—1062no—
193C2×C4—1062no—
194C2×C4—2124no—
195C2×C4D4×D5311yescharacteristic
196C2×C4D4×D5311yescharacteristic
197C2×C4C23×D5311yescharacteristic
198C8—22no—
199C8—1062no—
200C22—1061no—
201C22—1061no—
202C22—1061no—
203C22—2122no—
204C22—2122no—
205C22—2122no—
206C22C2×D4×D5311yescharacteristic
207C4—42no—
208C4—42no—
209C4—1061no—
210C4—1061no—
211C4—1061no—
212C4—2122no—
213C4#1313323812yes—
214C4#1313324412yes—
215C4#1313326512yes—
216C2—4242no—
217C2—2121no—
218C2#1681119811yescharacteristic
219C2C2×#1313323811yescharacteristic
220C2C2×#1313324411yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#498C2
socleC22×C53C22×D4
hypercenter#498D53
hypocenterC53#3228
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #498

Complements

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

NH ≅ G/Ntype
1. #16811178C2semidirect
3. #16811026C2semidirect
4. #16810971C2semidirect
5. #16810970C2semidirect
6. #16810949C2semidirect
7. #16810922C2semidirect
8. #16810674C2semidirect
9. C53×#498C2semidirect
114. #498D53semidirect
120. C53#3228semidirect