Finite Group Databasebeta

← #20286642

Subgroups — #20286642

Order 6784.

398 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 115 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#16811339C211yesmaximal, characteristic, complemented
2#16811276C211yesmaximal, characteristic, complemented
3#16810963C212yesmaximal, complemented
4#16810910C211yesmaximal, characteristic, complemented
5#16810855C212yesmaximal, complemented
6#16810825C211yesmaximal, characteristic, complemented
7#16810771C212yesmaximal, complemented
8C4×D424C211yesmaximal, characteristic, complemented
9C8×D212C211yesmaximal, characteristic, complemented
10#16810618C212yesmaximal, complemented
11C53×#487C211yesmaximal, characteristic, complemented
12C2×#11719949C2212yes—
13C2×#11719924C2212yes—
14C2×D424C2211yescharacteristic
15C2×C8×D53C2212yes—
16#13133206—22no—
17#13133202—22no—
18#13133193C2212yes—
19#13133192—21no—
20#13133192—21no—
21#13133192C2212yes—
22C4×D212C2211yescharacteristic
23C4×D212C2211yescharacteristic
24C4×D212C2211yescharacteristic
25#13133175—21no—
26#13133175C2212yes—
27C4⋊C4×D53—21no—
28#13133151C2211yescharacteristic
29#13133148C2212yes—
30C4⋊#88322C2211yescharacteristic
31#13133141C2212yes—
32#13133139C2211yescharacteristic
33#13133139C2211yescharacteristic
34#13133137C2212yes—
35#13133129C2212yes—
36C4⋊Q8×C53C2211yescharacteristic
37C4⋊Q8×C53C2211yescharacteristic
38#103×C53C2212yes—
39#103×C53C2212yes—
40C4×C8×C53C2211yescharacteristic
41C2×Q8×C53—21no—
42C2×Q8×C53C2312yes—
43#11719949—24no—
44#11719949—42no—
45C2×D212—22no—
46C2×D212—22no—
47C2×D212D411yescharacteristic
48C2×D212C2311yescharacteristic
49C2×D212C2311yescharacteristic
50C2×C4×D53—21no—
51C2×C4×D53—21no—
52C2×C4×D53—22no—
53C2×C4×D53C2312yes—
54C2×C8×C53C2311yescharacteristic
55C2×C8×C53C2311yescharacteristic
56C4⋊C4×C53—21no—
57C4⋊C4×C53—21no—
58C4⋊C4×C53—21no—
59C4⋊C4×C53C2312yes—
60C42×C53C2311yescharacteristic
61#11719924—42no—
62#11719921—22no—
63#11719921—22no—
64#11719921—21no—
65#11719921—21no—
66#11719921—22no—
67#11719921D412yes—
68C4⋊Dic53D411yescharacteristic
69C4⋊Dic53C2311yescharacteristic
70C4⋊Dic53C2311yescharacteristic
71#11719919—22no—
72C4×Dic53—22no—
73C2×#88322C2312yes—
74D424—22no—
75C8×D53—42no—
76C22×D53—21no—
77C22×D53—21no—
78C22×D53C2×D412yes—
79Q8×C53—22no—
80Q8×C53—22no—
81Q8×C53—81no—
82C2×C4×C53—22no—
83C2×C4×C53—22no—
84C2×C4×C53C2×D411yescharacteristic
85C2×C4×C53C2×D411yescharacteristic
86C2×C4×C53C2411yescharacteristic
87C2×Dic53—21no—
88C2×Dic53—21no—
89C2×Dic53C2×D412yes—
90D212—22no—
91D212—21no—
92D212—21no—
93D212—21no—
94D212—44no—
95D212—82no—
96D212#4112yes—
97C4×D53—22no—
98C4×D53—22no—
99C4×D53—44no—
100C4×D53—42no—
101C4×D53—42no—
102C8×C53—21no—
103C8×C53—21no—
104#88322—22no—
105C22×C53C22×D411yescharacteristic
106C2×D53—22no—
107C2×D53—22no—
108C2×D53—22no—
109C2×D53—22no—
110C2×D53—41no—
111C2×D53—41no—
112C2×D53—41no—
113C2×D53—41no—
114C2×D53—41no—
115C4×C53—21no—
116C4×C53—42no—
117C4×C53—42no—
118C4×C53#13512yes—
119C4×C53C2×#4111yescharacteristic
120C4×C53H(5,𝔽2)11yescharacteristic
121Dic53—22no—
122Dic53—41no—
123Dic53—41no—
124#3303—531nomaximal, Sylow
125C2×C53#54611yescharacteristic
126C2×C53C2×#13511yescharacteristic
127C2×C53#57711yescharacteristic
128D53—42no—
129D53—42no—
130D53—81no—
131#555—531no—
132#547—531no—
133#487D5311yescharacteristic, complemented
134#485—532no—
135#476—531no—
136#465—532no—
137#461—531no—
138#449—532no—
139C4×D8—531no—
140C8×D4—531no—
141#416—532no—
142C53#330311yescharacteristic, complemented, Sylow
143C2×#41—532no—
144C2×SD16—532no—
145C2×D8—531no—
146C22×C8—532no—
147C4⋊Q8C2×D5311yescharacteristic
148C4⋊Q8C2×D5311yescharacteristic
149#124—1062no—
150#123—1062no—
151C22⋊Q8—532no—
152#121—532no—
153#121—1061no—
154#121—1061no—
155C4×D4—531no—
156C4×D4—531no—
157C4×D4—531no—
158#117—532no—
159#117—1061no—
160C2×C4⋊C4—1061no—
161#107—531no—
162#106—532no—
163C4⋊C8—531no—
164#103—532no—
165#103C2×D5312yes—
166#103C2×D5312yes—
167#102—531no—
168#102—531no—
169#102—532no—
170C22⋊C8—532no—
171C4×C8C2×D5311yescharacteristic
172#41—1064no—
173#41—2122no—
174C2×Q8—21no—
175C2×Q8C22×D5312yes—
176C2×D4—531no—
177C2×D4—531no—
178C2×D4—531no—
179C2×D4—1062no—
180C2×D4—1062no—
181C22×C4—532no—
182C22×C4—1062no—
183C22×C4—1061no—
184C22×C4—1061no—
185SD16—2122no—
186D8—1062no—
187C2×C8—532no—
188C2×C8—2122no—
189C2×C8C22×D5311yescharacteristic
190C2×C8C22×D5311yescharacteristic
191C4⋊C4—21no—
192C4⋊C4—21no—
193C4⋊C4—21no—
194C4⋊C4—531no—
195C4⋊C4—531no—
196C4⋊C4—531no—
197C4⋊C4—1062no—
198C4⋊C4C22×D5312yes—
199C22⋊C4—532no—
200C22⋊C4—1062no—
201C22⋊C4—1062no—
202C22⋊C4—1062no—
203C22⋊C4—1061no—
204C22⋊C4—1061no—
205C42—1062no—
206C42C22×D5311yescharacteristic
207C23—532no—
208C23—1061no—
209C23—1061no—
210Q8—22no—
211Q8—22no—
212Q8—81no—
213D4—532no—
214D4—1062no—
215D4—1061no—
216D4—1061no—
217D4—1061no—
218D4—4242no—
219D4—2124no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—532no—
223C2×C4—1062no—
224C2×C4—1062no—
225C2×C4—1061no—
226C2×C4—1061no—
227C2×C4—2124no—
228C2×C4—2122no—
229C2×C4—2122no—
230C2×C4D4×D5311yescharacteristic
231C2×C4D4×D5311yescharacteristic
232C2×C4C23×D5311yescharacteristic
233C8—21no—
234C8—21no—
235C8—1062no—
236C22—1062no—
237C22—1062no—
238C22—1062no—
239C22—1062no—
240C22—2121no—
241C22—2121no—
242C22—2121no—
243C22—2121no—
244C22—2121no—
245C22C2×D4×D5311yescharacteristic
246C4—21no—
247C4—42no—
248C4—42no—
249C4—1062no—
250C4—2121no—
251C4—2121no—
252C4#1313324012yes—
253C4#41×D5311yescharacteristic
254C4#1313326311yescharacteristic
255C2—4241no—
256C2—2122no—
257C2—2122no—
258C2#1681126611yescharacteristic
259C2C2×#1313324011yescharacteristic
260C2#1681142911yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#487C2
socleC22×C53C22×D4
hypercenter#487D53
hypocenterC53#3303
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #487

Complements

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

NH ≅ G/Ntype
1. #16811339C2semidirect
2. #16811276C2semidirect
3. #16810963C2semidirect
4. #16810910C2semidirect
5. #16810855C2semidirect
6. #16810825C2semidirect
7. #16810771C2semidirect
8. C4×D424C2semidirect
9. C8×D212C2semidirect
10. #16810618C2semidirect
11. C53×#487C2semidirect
133. #487D53semidirect
142. C53#3303semidirect