Finite Group Databasebeta

← #20287354

Subgroups — #20287354

Order 6784.

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

normal subgroups only

This group has 370 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#16811348C211yesmaximal, characteristic, complemented
2#16811288C211yesmaximal, characteristic
3#16810972C212yesmaximal, complemented
4#16810906C212yesmaximal, complemented
5#16810905C211yesmaximal, characteristic, complemented
6#16810827C212yesmaximal
7#16810825C211yesmaximal, characteristic
8C8×Dic106C211yesmaximal, characteristic, complemented
9C4×#11719913C211yesmaximal, characteristic
10#16810708C212yesmaximal
11C53×#487C211yesmaximal, characteristic, complemented
12C2×Q8×D53—21no—
13C2×#11719913C2211yescharacteristic
14C2×#11719924C2212yes—
15#13133216C2212yes—
16#13133215C2212yes—
17#13133208—21no—
18#13133207—22no—
19#13133195—22no—
20#13133193—21no—
21Q8×Dic53C2212yes—
22C4×Dic106C2211yescharacteristic
23C4×Dic106C2211yescharacteristic
24#13133182—21no—
25C4×D212C2211yescharacteristic
26#13133179C2212yes—
27#13133149C2211yescharacteristic
28#13133148C2212yes—
29C4⋊#88322C2211yescharacteristic
30#13133146C2212yes—
31#13133142C2211yescharacteristic
32#13133141C2212yes—
33#13133140C2212yes—
34#13133139C2211yescharacteristic
35C8×Dic53C2212yes—
36C4⋊Q8×C53C2211yescharacteristic
37C4⋊Q8×C53C2211yescharacteristic
38#103×C53C2212yes—
39#103×C53C2212yes—
40C4×C8×C53C2211yescharacteristic
41C2×Q8×C53—21no—
42C2×Q8×C53C2312yes—
43Q8×D53—44no—
44C2×D212C2311yescharacteristic
45C2×C4×D53—21no—
46C2×C4×D53—22no—
47C2×Dic106—22no—
48C2×Dic106D411yescharacteristic
49C2×Dic106C2311yescharacteristic
50C2×C8×C53C2311yescharacteristic
51C2×C8×C53C2311yescharacteristic
52C4⋊C4×C53—21no—
53C4⋊C4×C53—21no—
54C4⋊C4×C53—21no—
55C4⋊C4×C53C2312yes—
56C42×C53C2311yescharacteristic
57#11719924—24no—
58#11719921—21no—
59#11719921—22no—
60C4⋊Dic53—22no—
61C4⋊Dic53D411yescharacteristic
62C4⋊Dic53C2311yescharacteristic
63C4⋊Dic53C2311yescharacteristic
64#11719919—21no—
65#11719919—22no—
66#11719919—22no—
67#11719919—22no—
68#11719919D412yes—
69C4×Dic53—21no—
70C4×Dic53—22no—
71C4×Dic53C2312yes—
72C2×#88322C2312yes—
73#11719913—22no—
74C22×D53—21no—
75Q8×C53—22no—
76Q8×C53—22no—
77Q8×C53—42no—
78C2×C4×C53—22no—
79C2×C4×C53—22no—
80C2×C4×C53C2×D411yescharacteristic
81C2×C4×C53C2×D411yescharacteristic
82C2×C4×C53C2411yescharacteristic
83C2×Dic53—21no—
84C2×Dic53—21no—
85C2×Dic53—21no—
86C2×Dic53C2×D412yes—
87C2×Dic53C2×D412yes—
88D212—21no—
89D212#4112yes—
90C4×D53—42no—
91C4×D53—44no—
92Dic106—21no—
93Dic106—21no—
94Dic106—44no—
95Dic106SD1614yescomplemented
96C8×C53—21no—
97C8×C53—21no—
98#88322—22no—
99C22×C53C22×D411yescharacteristic
100C2×D53—22no—
101C2×D53—41no—
102C2×D53—41no—
103C4×C53—21no—
104C4×C53—42no—
105C4×C53—42no—
106C4×C53C2×SD1612yes—
107C4×C53C2×#4111yescharacteristic
108C4×C532−1+411yescharacteristic
109Dic53—22no—
110Dic53—41no—
111Dic53—41no—
112Dic53—41no—
113Dic53C2×SD1614yes—
114#3359—531nomaximal, Sylow
115C2×C53#54811yescharacteristic
116C2×C53C22×SD1611yescharacteristic
117C2×C53#57811yescharacteristic
118D53—42no—
119#557—531no—
120D4×Q8—531no—
121#487—532no—
122#487D5311yescharacteristic, complemented
123#475—531no—
124#475—532no—
125#461—531no—
126#461—532no—
127C8×Q8—531no—
128C4×SD16—532no—
129C4×SD16—531no—
130C53#335911yescharacteristic, complemented, Sylow
131C22×Q8—1061no—
132C2×SD16—532no—
133C2×SD16—531no—
134C4⋊Q8—532no—
135C4⋊Q8—532no—
136C4⋊Q8C2×D5311yescharacteristic
137C4⋊Q8C2×D5311yescharacteristic
138#125—1062no—
139#125—1061no—
140C22⋊Q8—1062no—
141C22⋊Q8—1061no—
142C4×Q8—531no—
143C4×Q8—532no—
144C4×Q8—531no—
145C4×Q8—1061no—
146C4×D4—532no—
147C4×D4—531no—
148#106—531no—
149#106—532no—
150C4⋊C8—532no—
151C4⋊C8—531no—
152#103—532no—
153#103—531no—
154#103—532no—
155#103C2×D5312yes—
156#103C2×D5312yes—
157#102—531no—
158C4×C8—532no—
159C4×C8C2×D5311yescharacteristic
160C2×Q8—21no—
161C2×Q8—531no—
162C2×Q8—531no—
163C2×Q8—1062no—
164C2×Q8—2124no—
165C2×Q8C22×D5312yes—
166C2×D4—531no—
167C22×C4—1062no—
168C22×C4—1061no—
169SD16—1062no—
170SD16—1064no—
171C2×C8—532no—
172C2×C8C22×D5311yescharacteristic
173C2×C8C22×D5311yescharacteristic
174C4⋊C4—21no—
175C4⋊C4—21no—
176C4⋊C4—21no—
177C4⋊C4—532no—
178C4⋊C4—531no—
179C4⋊C4—531no—
180C4⋊C4—531no—
181C4⋊C4—1062no—
182C4⋊C4—1062no—
183C4⋊C4—1062no—
184C4⋊C4—1062no—
185C4⋊C4—1061no—
186C4⋊C4C22×D5312yes—
187C22⋊C4—1062no—
188C22⋊C4—1061no—
189C42—532no—
190C42—1062no—
191C42—1061no—
192C42C22×D5311yescharacteristic
193C23—1061no—
194Q8—22no—
195Q8—22no—
196Q8—42no—
197Q8—534no—
198Q8—1061no—
199Q8—1061no—
200Q8—2124no—
201D4—532no—
202D4—1061no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—532no—
206C2×C4—532no—
207C2×C4—1061no—
208C2×C4—1061no—
209C2×C4—1061no—
210C2×C4—2124no—
211C2×C4—2122no—
212C2×C4D4×D5311yescharacteristic
213C2×C4D4×D5311yescharacteristic
214C2×C4C23×D5311yescharacteristic
215C8—21no—
216C8—21no—
217C8—1062no—
218C22—1062no—
219C22—2121no—
220C22—2121no—
221C22C2×D4×D5311yescharacteristic
222C4—21no—
223C4—42no—
224C4—42no—
225C4—534no—
226C4—1062no—
227C4—2121no—
228C4—2121no—
229C4—2121no—
230C4SD16×D5312yes—
231C4#41×D5311yescharacteristic
232C4#1313326411yescharacteristic
233C2—2122no—
234C2#1681128111yescharacteristic
235C2C2×SD16×D5311yescharacteristic
236C2#1681143611yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#487C2
socleC22×C53C22×D4
hypercenter#487D53
hypocenterC53#3359
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. #16811348C2semidirect
3. #16810972C2semidirect
4. #16810906C2semidirect
5. #16810905C2semidirect
8. C8×Dic106C2semidirect
11. C53×#487C2semidirect
95. Dic106SD16semidirect
122. #487D53semidirect
130. C53#3359semidirect