Finite Group Databasebeta

← #20287069

Subgroups — #20287069

Order 6784.

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

normal subgroups only

This group has 388 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#16811334C211yesmaximal, characteristic, complemented
2#16811278C211yesmaximal, characteristic
3#16811019C211yesmaximal, characteristic, complemented
4#16810955C212yesmaximal, complemented
5#16810940C212yesmaximal, complemented
6#16810909C211yesmaximal, characteristic, complemented
7#16810851C212yesmaximal, complemented
8C4×#11719923C211yesmaximal, characteristic, complemented
9#16810688C211yesmaximal, characteristic, complemented
10#106×D53C212yesmaximal, complemented
11C53×#476C211yesmaximal, characteristic, complemented
12C2×#11719945—21no—
13C2×#11719923C2211yescharacteristic
14C2×C8×D53C2212yes—
15#13133217C2211yescharacteristic
16#13133208—21no—
17#13133199—22no—
18#13133197C2212yes—
19#13133196—21no—
20#13133193—22no—
21#13133188C2212yes—
22C4×Dic106C2211yescharacteristic
23C4×D212C2211yescharacteristic
24#13133179—21no—
25C4⋊C4×D53C2212yes—
26C4⋊C4×D53C2212yes—
27#13133149C2212yes—
28#13133148C2212yes—
29#13133148C2212yes—
30#13133148C2211yescharacteristic
31#13133142C2212yes—
32#13133140C2211yescharacteristic
33#13133138C2211yescharacteristic
34#13133129C2212yes—
35C4×#88322C2211yescharacteristic
36C4⋊Q8×C53C2211yescharacteristic
37C4×D4×C53C2211yescharacteristic
38#106×C53C2212yes—
39C4⋊C8×C53C2211yescharacteristic
40#102×C53C2212yes—
41C2×Q8×C53—21no—
42C2×D4×C53C2311yescharacteristic
43C22×C4×C53—21no—
44C2×C22⋊D53—22no—
45#11719945—44no—
46C2×D212D411yescharacteristic
47C2×C4×D53—22no—
48C2×C4×D53—22no—
49C2×C4×D53C2312yes—
50C2×Dic106—21no—
51C2×Dic106C2311yescharacteristic
52C2×C8×C53C2312yes—
53C4⋊C4×C53—21no—
54C4⋊C4×C53C2312yes—
55C4⋊C4×C53C2311yescharacteristic
56C22⋊C4×C53—21no—
57C42×C53C2311yescharacteristic
58C22⋊Dic53—22no—
59#11719923—22no—
60#11719921—22no—
61#11719921D412yes—
62C4⋊Dic53—21no—
63C4⋊Dic53—22no—
64C4⋊Dic53D411yescharacteristic
65C4⋊Dic53C2311yescharacteristic
66C4⋊Dic53C2312yes—
67#11719919—21no—
68#11719919—22no—
69#11719919—22no—
70C4×Dic53—21no—
71C2×#88322C2311yescharacteristic
72C2×#88322C2311yescharacteristic
73C8×D53—24no—
74C23×C53—21no—
75C22×D53C2×D412yes—
76Q8×C53—81no—
77D4×C53—21no—
78D4×C53—21no—
79C2×C4×C53—21no—
80C2×C4×C53—22no—
81C2×C4×C53—42no—
82C2×C4×C53C2×D411yescharacteristic
83C2×C4×C53C2×D411yescharacteristic
84C2×C4×C53C2411yescharacteristic
85C22⋊D53—44no—
86C2×Dic53—22no—
87C2×Dic53—21no—
88C2×Dic53—21no—
89C2×Dic53C2×D412yes—
90D212SD1614yescomplemented
91C4×D53—22no—
92C4×D53—22no—
93C4×D53—44no—
94C4×D53—82no—
95Dic106—21no—
96Dic106—21no—
97Dic106—42no—
98C8×C53—22no—
99#88322—21no—
100#88322#4112yescomplemented
101C22×C53—41no—
102C22×C53—41no—
103C22×C53—41no—
104C22×C53C22×D411yescharacteristic
105C2×D53—22no—
106C2×D53—22no—
107C2×D53C2×SD1614yes—
108C4×C53—21no—
109C4×C53—41no—
110C4×C53—42no—
111C4×C53C2×SD1612yes—
112C4×C53C2×#4111yescharacteristic
113C4×C532−1+411yescharacteristic
114Dic53—22no—
115Dic53—41no—
116Dic53—42no—
117Dic53—41no—
118#3335—531nomaximal, Sylow
119C2×C53—81no—
120C2×C53#54711yescharacteristic
121C2×C53C22×SD1611yescharacteristic
122C2×C53#57711yescharacteristic
123D53—24no—
124#554—531no—
125#547—531no—
126#498—531no—
127#483—532no—
128#481—532no—
129#476D5311yescharacteristic, complemented
130#475—531no—
131#465—532no—
132C4×SD16—531no—
133C8×D4—531no—
134C2×#106—532no—
135C53#333511yescharacteristic, complemented, Sylow
136C2×#41—1061no—
137C2×SD16—531no—
138C22×C8—532no—
139C4⋊Q8—531no—
140C4⋊Q8C2×D5311yescharacteristic
141#125—1061no—
142#123—1062no—
143C22⋊Q8—532no—
144C22⋊Q8—1062no—
145C22⋊Q8—1061no—
146#121—532no—
147C4×Q8—531no—
148C4×D4—531no—
149C4×D4—1061no—
150C4×D4C2×D5311yescharacteristic
151C2×C4⋊C4—532no—
152C2×C4⋊C4—532no—
153#106—532no—
154#106—531no—
155#106—532no—
156#106—532no—
157#106C2×D5312yes—
158C4⋊C8C2×D5311yescharacteristic
159#103—532no—
160#103—531no—
161#102—531no—
162#102C2×D5312yes—
163C22⋊C8—532no—
164C4×C8—531no—
165#41—2124no—
166C2×Q8—21no—
167C2×Q8—531no—
168C2×Q8—1061no—
169C2×D4—531no—
170C2×D4—1062no—
171C2×D4C22×D5311yescharacteristic
172C22×C4—21no—
173C22×C4—532no—
174C22×C4—1062no—
175C22×C4—1062no—
176SD16—1062no—
177C2×C8—531no—
178C2×C8—531no—
179C2×C8—1064no—
180C2×C8C22×D5312yes—
181C4⋊C4—21no—
182C4⋊C4—531no—
183C4⋊C4—531no—
184C4⋊C4—532no—
185C4⋊C4—1062no—
186C4⋊C4—1062no—
187C4⋊C4—1062no—
188C4⋊C4—1061no—
189C4⋊C4—1061no—
190C4⋊C4C22×D5312yes—
191C4⋊C4C22×D5311yescharacteristic
192C22⋊C4—21no—
193C22⋊C4—532no—
194C22⋊C4—1062no—
195C22⋊C4—1062no—
196C42—1061no—
197C42C22×D5311yescharacteristic
198C23—21no—
199C23—532no—
200Q8—81no—
201Q8—1061no—
202Q8—1061no—
203Q8—2122no—
204D4—21no—
205D4—21no—
206D4—534no—
207D4—2124no—
208C2×C4—21no—
209C2×C4—22no—
210C2×C4—42no—
211C2×C4—532no—
212C2×C4—1062no—
213C2×C4—1062no—
214C2×C4—1061no—
215C2×C4—1061no—
216C2×C4—1062no—
217C2×C4—4242no—
218C2×C4—2124no—
219C2×C4D4×D5311yescharacteristic
220C2×C4D4×D5311yescharacteristic
221C2×C4C23×D5311yescharacteristic
222C8—22no—
223C8—532no—
224C8—1061no—
225C22—41no—
226C22—41no—
227C22—41no—
228C22—534no—
229C22—1062no—
230C22—1062no—
231C22C2×D4×D5311yescharacteristic
232C4—21no—
233C4—41no—
234C4—42no—
235C4—1062no—
236C4—2121no—
237C4—2122no—
238C4—2121no—
239C4SD16×D5312yes—
240C4#41×D5311yescharacteristic
241C4#1313326511yescharacteristic
242C2—81no—
243C2—1064no—
244C2#1681127511yescharacteristic
245C2C2×SD16×D5311yescharacteristic
246C2#1681143311yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#476C2
socleC22×C53C22×D4
hypercenter#476D53
hypocenterC53#3335
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #476

Complements

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

NH ≅ G/Ntype
1. #16811334C2semidirect
3. #16811019C2semidirect
4. #16810955C2semidirect
5. #16810940C2semidirect
6. #16810909C2semidirect
7. #16810851C2semidirect
8. C4×#11719923C2semidirect
9. #16810688C2semidirect
10. #106×D53C2semidirect
11. C53×#476C2semidirect
90. D212SD16semidirect
100. #88322#41semidirect
129. #476D53semidirect
135. C53#3335semidirect