Finite Group Databasebeta

← #20283177

Subgroups — #20283177

Order 6784.

716 conjugacy classes of subgroups; only the 209 normal subgroups are shown.

normal subgroups only

This group has 716 conjugacy classes of subgroups — too many to draw. Only its 209 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
1C2×#13133264C211yesmaximal, characteristic
2#16811100C212yesmaximal, complemented
3#16810597C211yesmaximal, characteristic, complemented
4#16810596C212yesmaximal, complemented
5#16810595C218yesmaximal, complemented
6C53×#410C211yesmaximal, characteristic, complemented
7#13133264—24no—
8#13133264—24no—
9C2×#11719945C412yescomplemented
10C2×#11719945C2211yescharacteristic
11C2×#11719947C412yescomplemented
12C2×#11719947C2214yes—
13C22×Dic106C412yescomplemented
14C22×Dic106C2211yescharacteristic
15C2×Q8×D53C412yescomplemented
16C4×Dic106—24no—
17#13133182—24no—
18#13133176C2212yes—
19#13133173C2218yes—
20#13133132C2214yes—
21#13133131C2218yes—
22C2×#41×C53C2211yescharacteristic
23#117×C53C2212yes—
24#99×C53C2214yes—
25#41×C53—22no—
26#41×C53—22no—
27C2×Q8×C53C2×C411yescharacteristic
28C2×D4×C53C2×C411yescharacteristic
29C2×D4×C53C2312yes—
30C22×C4×C53C2×C412yes—
31C22×C4×C53C2311yescharacteristic
32C2×C22⋊D53C2×C414yes—
33C2×C22⋊D53C2312yes—
34C22×Dic53C2×C414yes—
35C22×Dic53C2312yes—
36Q8×D53—28no—
37#11719947—216no—
38#11719947—28no—
39#11719945—22no—
40#11719945—22no—
41#11719945—24no—
42#11719945—24no—
43C2×D212C2×C411yescharacteristic
44C2×C4×D53C2×C412yes—
45C2×C4×D53C2×C414yes—
46C2×Dic106—24no—
47C2×Dic106—24no—
48C2×Dic106C2×C414yes—
49C2×Dic106C2×C411yescharacteristic
50C2×Dic106C2×C412yes—
51C2×Dic106C2×C412yes—
52C2×Dic106D414yes—
53C2×Dic106D414yes—
54C4⋊C4×C53—22no—
55C22⋊C4×C53C2314yes—
56C42×C53—22no—
57C22⋊Dic53C2314yes—
58C4⋊Dic53—22no—
59#11719919—28no—
60C4×Dic53—22no—
61C4×Dic53—28no—
62C23×C53C22×C412yes—
63C23×C53C2411yescharacteristic
64C22×D53C22×C412yes—
65Q8×C53—22no—
66D4×C53—24no—
67D4×C53—22no—
68C2×C4×C53—22no—
69C2×C4×C53—22no—
70C2×C4×C53—24no—
71C2×C4×C53C22×C412yes—
72C2×C4×C53C22×C411yescharacteristic
73C2×C4×C53C22×C411yescharacteristic
74C2×C4×C53C2×D412yes—
75C2×C4×C53C2×D412yes—
76C22⋊D53—24no—
77C22⋊D53—28no—
78C2×Dic53—28no—
79C2×Dic53—24no—
80C2×Dic53C22×C412yes—
81C2×Dic53C22×C414yes—
82C2×Dic53C2×D418yes—
83D212—21no—
84D212—21no—
85C4×D53—24no—
86C4×D53—28no—
87Dic106—21no—
88Dic106—21no—
89Dic106—28no—
90Dic106C22⋊C414yes—
91Dic106C22⋊C414yes—
92Dic106C22⋊C418yes—
93C22×C53—22no—
94C22×C53—41no—
95C22×C53—42no—
96C22×C53C23×C411yescharacteristic
97C22×C53C22×D412yes—
98C2×D53—22no—
99C2×D53—42no—
100C4×C53—22no—
101C4×C53—44no—
102C4×C53C2×C22⋊C411yescharacteristic
103C4×C53C2×C22⋊C411yescharacteristic
104C4×C53C2×C22⋊C412yes—
105Dic53—22no—
106Dic53—44no—
107Dic53C2×C22⋊C418yes—
108#2884—531nomaximal, Sylow
109C2×C53—22no—
110C2×C53—21no—
111C2×C53—42no—
112C2×C53C22×C22⋊C411yescharacteristic
113D53—42no—
114C2×2−1+4—531no—
115#519—532no—
116#410—531no—
117#410—532no—
118#410—538no—
119#410D5311yescharacteristic, complemented
120C53#288411yescharacteristic, complemented, Sylow
1212−1+4—1064no—
1222−1+4—1064no—
123C2×#41—532no—
124C2×#41—534no—
125C2×#41—532no—
126C2×#41—531no—
127C2×#41C2×D5311yescharacteristic
128C22×Q8—532no—
129C22×Q8—532no—
130C22×Q8—531no—
131C4×Q8—1064no—
132C4×Q8—1064no—
133#117—532no—
134#117—538no—
135#117C2×D5312yes—
136#99—534no—
137#99—538no—
138#99C2×D5314yes—
139#41—22no—
140#41—22no—
141#41—1064no—
142#41—1064no—
143#41—10616no—
144#41—1068no—
145#41—1062no—
146#41—1062no—
147C2×Q8—532no—
148C2×Q8—532no—
149C2×Q8—534no—
150C2×Q8—531no—
151C2×Q8—534no—
152C2×Q8—534no—
153C2×Q8—1064no—
154C2×Q8—1064no—
155C2×Q8—1068no—
156C2×Q8C4×D5311yescharacteristic
157C2×D4—534no—
158C2×D4—532no—
159C2×D4—531no—
160C2×D4C4×D5311yescharacteristic
161C2×D4C22×D5312yes—
162C22×C4—532no—
163C22×C4—534no—
164C22×C4—534no—
165C22×C4—532no—
166C22×C4C4×D5312yes—
167C22×C4C22×D5311yescharacteristic
168C4⋊C4—22no—
169C4⋊C4—1068no—
170C4⋊C4—1062no—
171C22⋊C4—534no—
172C22⋊C4C22×D5314yes—
173C42—22no—
174C42—1068no—
175C42—1062no—
176C23—532no—
177C23C2×C4×D5312yes—
178C23C23×D5311yescharacteristic
179Q8—22no—
180Q8—534no—
181Q8—534no—
182Q8—538no—
183Q8—1068no—
184Q8—1061no—
185Q8—1061no—
186D4—24no—
187D4—22no—
188D4—1068no—
189D4—1064no—
190D4—1061no—
191D4—1061no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—24no—
195C2×C4—534no—
196C2×C4—538no—
197C2×C4—532no—
198C2×C4—1068no—
199C2×C4—1068no—
200C2×C4—1064no—
201C2×C4—1064no—
202C2×C4C2×C4×D5312yes—
203C2×C4C2×C4×D5311yescharacteristic
204C2×C4C2×C4×D5311yescharacteristic
205C2×C4D4×D5312yes—
206C2×C4D4×D5312yes—
207C22—22no—
208C22—41no—
209C22—42no—
210C22—1062no—
211C22—2122no—
212C22C22×C4×D5311yescharacteristic
213C22C2×D4×D5312yes—
214C4—22no—
215C4—44no—
216C4—538no—
217C4—1062no—
218C4—2124no—
219C4C22⋊C4×D5311yescharacteristic
220C4C22⋊C4×D5311yescharacteristic
221C4C22⋊C4×D5312yes—
222C2—22no—
223C2—21no—
224C2—42no—
225C2—2122no—
226C2C2×C22⋊C4×D5311yescharacteristic
subgroupG/H
centerC2C2×C22⋊C4×D53
commutatorC22×C53C23×C4
FrattiniC23C23×D53
FittingC53×#410C2
socleC2×C53C22×C22⋊C4
hypercenter#410D53
hypocenterC53#2884
LCSG ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C22×C4 ◁ #410

Complements

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

NH ≅ G/Ntype
2. #16811100C2semidirect
3. #16810597C2semidirect
4. #16810596C2semidirect
5. #16810595C2semidirect
6. C53×#410C2semidirect
9. C2×#11719945C4semidirect
11. C2×#11719947C4semidirect
13. C22×Dic106C4semidirect
15. C2×Q8×D53C4semidirect
119. #410D53semidirect
120. C53#2884semidirect