Finite Group Databasebeta

← #20285092

Subgroups — #20285092

Order 6784.

380 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 125 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#16811178C211yesmaximal, characteristic
2C4⋊Q8×D53C211yesmaximal, characteristic
3#16810990C214yesmaximal, complemented
4#16810975C211yesmaximal, characteristic, complemented
5#16810974C211yesmaximal, characteristic, complemented
6#16810973C211yesmaximal, characteristic, complemented
7#16810630C214yesmaximal
8C4×#11719912C211yesmaximal, characteristic
9C53×#487C211yesmaximal, characteristic, complemented
10C2×Q8×D53C2212yes—
11C2×#11719912C2212yes—
12#13133217C2211yescharacteristic
13#13133217C2211yescharacteristic
14#13133216—21no—
15#13133216—21no—
16#13133215C2212yes—
17#13133208C2212yes—
18#13133197—21no—
19#13133193—21no—
20Q8×Dic53—21no—
21#13133182—21no—
22#13133175C2212yes—
23C4⋊C4×D53—21no—
24C4⋊C4×D53—21no—
25C42×D53C2211yescharacteristic
26#13133142C2214yes—
27#13133141C2214yes—
28#13133140C2214yes—
29#13133127C2212yes—
30C4×#88322C2211yescharacteristic
31C4⋊Q8×C53C2211yescharacteristic
32C4⋊Q8×C53C2211yescharacteristic
33#103×C53C2214yes—
34C4×C8×C53C2211yescharacteristic
35C2×Q8×C53—21no—
36C2×Q8×C53C2312yes—
37Q8×D53—28no—
38C2×C4×D53—22no—
39C2×C4×D53D411yescharacteristic
40C2×C4×D53D411yescharacteristic
41C2×C4×D53C2311yescharacteristic
42C2×Dic106—21no—
43C2×Dic106—22no—
44C2×Dic106C2312yes—
45C2×C8×C53C2312yes—
46C4⋊C4×C53—21no—
47C4⋊C4×C53—21no—
48C4⋊C4×C53—21no—
49C4⋊C4×C53C2312yes—
50C42×C53C2311yescharacteristic
51#11719921—22no—
52C4⋊Dic53—21no—
53C4⋊Dic53—21no—
54C4⋊Dic53—21no—
55C4⋊Dic53—22no—
56C4⋊Dic53C2312yes—
57#11719919—22no—
58#11719919—22no—
59#11719919—22no—
60C4×Dic53—22no—
61C4×Dic53D411yescharacteristic
62C4×Dic53D411yescharacteristic
63C4×Dic53C2311yescharacteristic
64C2×#88322C2312yes—
65#11719912—24no—
66C22×D53C2×D411yescharacteristic
67Q8×C53—24no—
68Q8×C53—42no—
69C2×C4×C53—22no—
70C2×C4×C53—22no—
71C2×C4×C53C2×D411yescharacteristic
72C2×C4×C53C2×D411yescharacteristic
73C2×C4×C53C2411yescharacteristic
74C2×Dic53—22no—
75C2×Dic53—22no—
76C2×Dic53C2×D411yescharacteristic
77C2×Dic53C2×D411yescharacteristic
78C2×Dic53C2×D411yescharacteristic
79C4×D53—22no—
80C4×D53—22no—
81C4×D53—44no—
82C4×D53#4114yes—
83Dic106—24no—
84Dic106—42no—
85Dic106—44no—
86C8×C53—22no—
87#88322—22no—
88C22×C53C22×D411yescharacteristic
89C2×D53—21no—
90C2×D53—21no—
91C2×D53#12412yes—
92C4×C53—42no—
93C4×C53—42no—
94C4×C53#13712yes—
95C4×C53#13712yes—
96C4×C53C2×#4112yes—
97Dic53—21no—
98Dic53—21no—
99Dic53—42no—
100Dic53—42no—
101Dic53#12412yes—
102#3139—531nomaximal, Sylow
103C2×C53C2×#12411yescharacteristic
104C2×C53C2×#13711yescharacteristic
105C2×C53C2×#13711yescharacteristic
106D53—22no—
107#533—531no—
108C2×C4⋊Q8—531no—
109#491—534no—
110#487—531no—
111#487—531no—
112#487—531no—
113#487D5311yescharacteristic, complemented
114#419—534no—
115C4×M16—531no—
116C53#313911yescharacteristic, complemented, Sylow
117C22×Q8—532no—
118C2×M16—532no—
119C4⋊Q8—532no—
120C4⋊Q8—531no—
121C4⋊Q8—531no—
122C4⋊Q8—1061no—
123C4⋊Q8—1061no—
124C4⋊Q8C2×D5311yescharacteristic
125C4⋊Q8C2×D5311yescharacteristic
126#125—532no—
127C22⋊Q8—1061no—
128C22⋊Q8—1061no—
129C4×Q8—1061no—
130C4×Q8—1061no—
131#117—532no—
132C2×C4⋊C4—1061no—
133C2×C4⋊C4—1061no—
134C2×C42—531no—
135#103—534no—
136#103—534no—
137#103—534no—
138#103C2×D5314yes—
139#97—532no—
140C4×C8—531no—
141C4×C8C2×D5311yescharacteristic
142C2×Q8—21no—
143C2×Q8—532no—
144C2×Q8—1062no—
145C2×Q8—1068no—
146C2×Q8—1061no—
147C2×Q8C22×D5312yes—
148C22×C4—531no—
149C22×C4—531no—
150C22×C4—531no—
151C22×C4—1062no—
152M16—1064no—
153C2×C8—532no—
154C2×C8C22×D5312yes—
155C4⋊C4—21no—
156C4⋊C4—21no—
157C4⋊C4—21no—
158C4⋊C4—532no—
159C4⋊C4—1062no—
160C4⋊C4—1062no—
161C4⋊C4—1062no—
162C4⋊C4—1062no—
163C4⋊C4—1061no—
164C4⋊C4—1061no—
165C4⋊C4—1061no—
166C4⋊C4C22×D5312yes—
167C22⋊C4—1062no—
168C42—531no—
169C42—531no—
170C42—531no—
171C42—1062no—
172C42C22×D5311yescharacteristic
173C23—531no—
174Q8—24no—
175Q8—42no—
176Q8—1064no—
177Q8—2124no—
178Q8—2122no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—531no—
182C2×C4—534no—
183C2×C4—531no—
184C2×C4—531no—
185C2×C4—1062no—
186C2×C4—1062no—
187C2×C4—1062no—
188C2×C4—1062no—
189C2×C4—2124no—
190C2×C4D4×D5311yescharacteristic
191C2×C4D4×D5311yescharacteristic
192C2×C4C23×D5311yescharacteristic
193C8—22no—
194C8—1062no—
195C22—532no—
196C22—1061no—
197C22—1061no—
198C22C2×D4×D5311yescharacteristic
199C4—42no—
200C4—42no—
201C4—532no—
202C4—1061no—
203C4—1061no—
204C4—2122no—
205C4—2122no—
206C4#1313324612yes—
207C4#1313324912yes—
208C4#41×D5312yes—
209C2—1062no—
210C2#124×D5311yescharacteristic
211C2C2×#1313324611yescharacteristic
212C2C2×#1313324911yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#487C2
socleC22×C53C22×D4
hypercenter#487D53
hypocenterC53#3139
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
3. #16810990C2semidirect
4. #16810975C2semidirect
5. #16810974C2semidirect
6. #16810973C2semidirect
9. C53×#487C2semidirect
113. #487D53semidirect
116. C53#3139semidirect