Finite Group Databasebeta

← #20286579

Subgroups — #20286579

Order 6784.

420 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 123 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#16811288C211yesmaximal, characteristic, complemented
2#16811276C211yesmaximal, characteristic
3#16810952C212yesmaximal, complemented
4#16810905C211yesmaximal, characteristic, complemented
5#16810855C212yesmaximal, complemented
6#16810825C211yesmaximal, characteristic, complemented
7#16810763C212yesmaximal, complemented
8C4×#11719913C211yesmaximal, characteristic, complemented
9C8×D212C211yesmaximal, characteristic, complemented
10#103×D53C212yesmaximal, complemented
11C53×#487C211yesmaximal, characteristic, complemented
12C2×#11719949—21no—
13C2×Q8×D53C2212yes—
14C2×#11719913C2211yescharacteristic
15C2×#11719924C2212yes—
16C2×C8×D53C2212yes—
17#13133216—21no—
18#13133202—22no—
19#13133195—22no—
20#13133193—21no—
21#13133193C2212yes—
22#13133192C2212yes—
23C4×Dic106C2211yescharacteristic
24C4×D212C2211yescharacteristic
25C4×D212C2211yescharacteristic
26#13133179—21no—
27C4⋊C4×D53C2212yes—
28#13133149C2211yescharacteristic
29#13133148C2212yes—
30C4⋊#88322C2211yescharacteristic
31#13133142C2211yescharacteristic
32#13133141C2212yes—
33#13133140C2212yes—
34#13133139C2211yescharacteristic
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—44no—
44Q8×D53—24no—
45Q8×D53—42no—
46C2×D212—22no—
47C2×D212D411yescharacteristic
48C2×D212C2311yescharacteristic
49C2×C4×D53—22no—
50C2×C4×D53—22no—
51C2×C4×D53—21no—
52C2×C4×D53C2312yes—
53C2×Dic106—22no—
54C2×Dic106C2311yescharacteristic
55C2×C8×C53C2311yescharacteristic
56C2×C8×C53C2311yescharacteristic
57C4⋊C4×C53—21no—
58C4⋊C4×C53—21no—
59C4⋊C4×C53—21no—
60C4⋊C4×C53C2312yes—
61C42×C53C2311yescharacteristic
62#11719924—42no—
63#11719921—22no—
64#11719921—21no—
65#11719921—22no—
66#11719921D412yes—
67C4⋊Dic53D411yescharacteristic
68C4⋊Dic53C2311yescharacteristic
69C4⋊Dic53C2311yescharacteristic
70#11719919—21no—
71#11719919—22no—
72#11719919—22no—
73C4×Dic53—21no—
74C2×#88322C2312yes—
75#11719913—22no—
76C8×D53—42no—
77C22×D53—21no—
78C22×D53C2×D412yes—
79Q8×C53—22no—
80Q8×C53—22no—
81Q8×C53—42no—
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×Dic53—21no—
90C2×Dic53C2×D412yes—
91D212—21no—
92D212—21no—
93D212—44no—
94D212SD1614yescomplemented
95C4×D53—22no—
96C4×D53—22no—
97C4×D53—44no—
98C4×D53—44no—
99C4×D53—42no—
100Dic106—21no—
101Dic106—44no—
102Dic106#4112yescomplemented
103C8×C53—21no—
104C8×C53—21no—
105#88322—22no—
106C22×C53C22×D411yescharacteristic
107C2×D53—22no—
108C2×D53—22no—
109C2×D53—41no—
110C2×D53—41no—
111C2×D53—41no—
112C2×D53C2×SD1614yes—
113C4×C53—21no—
114C4×C53—42no—
115C4×C53—42no—
116C4×C53C2×SD1612yes—
117C4×C53C2×#4111yescharacteristic
118C4×C53H(5,𝔽2)11yescharacteristic
119Dic53—22no—
120Dic53—22no—
121Dic53—41no—
122Dic53—41no—
123#3298—531nomaximal, Sylow
124C2×C53#54611yescharacteristic
125C2×C53C22×SD1611yescharacteristic
126C2×C53#57811yescharacteristic
127D53—24no—
128D53—81no—
129D4×Q8—531no—
130#547—531no—
131#487D5311yescharacteristic, complemented
132#483—532no—
133#475—531no—
134#465—532no—
135#461—531no—
136#448—532no—
137C4×SD16—531no—
138C8×D4—531no—
139C2×#103—532no—
140C53#329811yescharacteristic, complemented, Sylow
141C2×#41—1061no—
142C22×Q8—532no—
143C2×SD16—532no—
144C2×SD16—531no—
145C22×C8—532no—
146C4⋊Q8—1061no—
147C4⋊Q8C2×D5311yescharacteristic
148C4⋊Q8C2×D5311yescharacteristic
149#123—1062no—
150C22⋊Q8—532no—
151C22⋊Q8—1062no—
152C22⋊Q8—1061no—
153#121—532no—
154C4×Q8—531no—
155C4×D4—531no—
156C4×D4—531no—
157C4×D4—1061no—
158C2×C4⋊C4—532no—
159#106—531no—
160#106—532no—
161C4⋊C8—531no—
162#103—532no—
163#103—531no—
164#103—532no—
165#103C2×D5312yes—
166#103C2×D5312yes—
167#102—531no—
168C22⋊C8—532no—
169C4×C8C2×D5311yescharacteristic
170#41—2124no—
171C2×Q8—21no—
172C2×Q8—531no—
173C2×Q8—1062no—
174C2×Q8—1064no—
175C2×Q8—2122no—
176C2×Q8C22×D5312yes—
177C2×D4—531no—
178C2×D4—531no—
179C2×D4—1062no—
180C22×C4—532no—
181C22×C4—1062no—
182C22×C4—1062no—
183C22×C4—1061no—
184SD16—1062no—
185SD16—2122no—
186C2×C8—532no—
187C2×C8—2122no—
188C2×C8C22×D5311yescharacteristic
189C2×C8C22×D5311yescharacteristic
190C4⋊C4—21no—
191C4⋊C4—21no—
192C4⋊C4—21no—
193C4⋊C4—531no—
194C4⋊C4—531no—
195C4⋊C4—531no—
196C4⋊C4—1062no—
197C4⋊C4—1062no—
198C4⋊C4—1061no—
199C4⋊C4C22×D5312yes—
200C22⋊C4—532no—
201C22⋊C4—1062no—
202C22⋊C4—1062no—
203C22⋊C4—1061no—
204C42—1061no—
205C42C22×D5311yescharacteristic
206C23—532no—
207C23—1061no—
208Q8—22no—
209Q8—22no—
210Q8—42no—
211Q8—532no—
212Q8—1061no—
213Q8—2124no—
214D4—534no—
215D4—1061no—
216D4—1061no—
217D4—2124no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—532no—
221C2×C4—1061no—
222C2×C4—1062no—
223C2×C4—1062no—
224C2×C4—1061no—
225C2×C4—1061no—
226C2×C4—2124no—
227C2×C4—2124no—
228C2×C4—2122no—
229C2×C4D4×D5311yescharacteristic
230C2×C4D4×D5311yescharacteristic
231C2×C4C23×D5311yescharacteristic
232C8—21no—
233C8—21no—
234C8—1062no—
235C22—534no—
236C22—1062no—
237C22—1062no—
238C22—2121no—
239C22—2121no—
240C22—2121no—
241C22C2×D4×D5311yescharacteristic
242C4—21no—
243C4—42no—
244C4—42no—
245C4—1062no—
246C4—1062no—
247C4—2121no—
248C4—2121no—
249C4SD16×D5312yes—
250C4#41×D5311yescharacteristic
251C4#1313326311yescharacteristic
252C2—1064no—
253C2—4241no—
254C2#1681126611yescharacteristic
255C2C2×SD16×D5311yescharacteristic
256C2#1681143611yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#487C2
socleC22×C53C22×D4
hypercenter#487D53
hypocenterC53#3298
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. #16811288C2semidirect
3. #16810952C2semidirect
4. #16810905C2semidirect
5. #16810855C2semidirect
6. #16810825C2semidirect
7. #16810763C2semidirect
8. C4×#11719913C2semidirect
9. C8×D212C2semidirect
10. #103×D53C2semidirect
11. C53×#487C2semidirect
94. D212SD16semidirect
102. Dic106#41semidirect
131. #487D53semidirect
140. C53#3298semidirect