Finite Group Databasebeta

← #20277258

Subgroups — #20277258

Order 6784.

652 conjugacy classes of subgroups; only the 213 normal subgroups are shown.

normal subgroups only

This group has 652 conjugacy classes of subgroups — too many to draw. Only its 213 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
1C2×#13133206C211yesmaximal, characteristic
2C2×Q8×Dic53C211yesmaximal, characteristic, complemented
3C2×#13133179C211yesmaximal, characteristic
4#16810525C211yesmaximal, characteristic
5#16810515C214yesmaximal, complemented
6#16810488C211yesmaximal, characteristic, complemented
7#16810470C214yesmaximal, complemented
8C42×Dic53C211yesmaximal, characteristic, complemented
9C53×#391C211yesmaximal, characteristic, complemented
10C22×D212C2211yescharacteristic
11C22×C4×D53C2212yes—
12#13133206C418yescomplemented
13Q8×Dic53—24no—
14#13133179—24no—
15C2×C4⋊Dic53C2212yes—
16C2×C4⋊Dic53C2211yescharacteristic
17C2×#11719921C2212yes—
18C2×#11719921C2214yes—
19C2×#11719921C2214yes—
20C2×C4×Dic53C2212yes—
21C2×C4×Dic53C2212yes—
22C2×C4×Dic53C2212yes—
23C2×C4×Dic53C2211yescharacteristic
24C2×C4×Dic53C2211yescharacteristic
25#13133124C2214yes—
26C22×Q8×C53C2211yescharacteristic
27C2×C4⋊C4×C53C2211yescharacteristic
28C2×C42×C53C2211yescharacteristic
29#95×C53C2214yes—
30C23×D53C2312yes—
31C2×Q8×C53—22no—
32C2×Q8×C53—22no—
33C2×Q8×C53C2×C414yescomplemented
34C22×C4×C53C2312yes—
35C22×C4×C53C2312yes—
36C22×C4×C53C2312yes—
37C22×C4×C53C2311yescharacteristic
38C22×Dic53C2312yes—
39C22×Dic53C2312yes—
40C22×Dic53C2311yescharacteristic
41C22×Dic53C2311yescharacteristic
42C2×D212—22no—
43C2×D212—22no—
44C2×D212C2×C414yes—
45C2×C4×D53—24no—
46C2×C4×D53—28no—
47C4⋊C4×C53—22no—
48C42×C53—22no—
49#11719921—28no—
50#11719921—24no—
51#11719921C2×C4116yescomplemented
52C4⋊Dic53—22no—
53C4⋊Dic53—24no—
54C4×Dic53—24no—
55C4×Dic53—24no—
56C4×Dic53—24no—
57C4×Dic53C2×C414yescomplemented
58C4×Dic53D414yes—
59C23×C53C2411yescharacteristic
60C22×D53—22no—
61C22×D53—24no—
62C22×D53—22no—
63C22×D53—24no—
64C22×D53C22×C414yes—
65Q8×C53—28no—
66C2×C4×C53—22no—
67C2×C4×C53—22no—
68C2×C4×C53—22no—
69C2×C4×C53—22no—
70C2×C4×C53—22no—
71C2×C4×C53—24no—
72C2×C4×C53C22×C414yes—
73C2×C4×C53C22×C412yes—
74C2×C4×C53C2×D412yes—
75C2×C4×C53#4112yes—
76C2×C4×C53#4114yes—
77C2×Dic53—22no—
78C2×Dic53—22no—
79C2×Dic53—22no—
80C2×Dic53—22no—
81C2×Dic53—24no—
82C2×Dic53C22×C414yes—
83C2×Dic53C2×D412yes—
84C2×Dic53C2×D412yes—
85C2×Dic53#4114yes—
86D212—28no—
87C4×D53—48no—
88C22×C53C23×C411yescharacteristic
89C22×C53C22×D411yescharacteristic
90C22×C53C2×#4112yes—
91C22×C53C2×#4111yescharacteristic
92C22×C53C2×#4112yes—
93C2×D53—24no—
94C2×D53—28no—
95C2×D53—42no—
96C2×D53—44no—
97C2×D53—42no—
98C4×C53—24no—
99C4×C53—24no—
100C4×C53—42no—
101C4×C53C4×D414yes—
102Dic53—24no—
103Dic53—22no—
104Dic53—42no—
105Dic53#12414yescomplemented
106C4×#124—531nomaximal, Sylow
107C2×C53C2×C4×D411yescharacteristic
108C2×C53C4×#4112yes—
109C2×C53C2×#12411yescharacteristic
110C2×C53#52912yes—
111C2×C53#53211yescharacteristic
112D53—44no—
113C2×#124—531no—
114C2×C4×Q8—531no—
115C2×C4×D4—531no—
116#391D5311yescharacteristic, complemented
117#390—531no—
118#388—534no—
119#382—531no—
120C4×C22⋊C4—534no—
121C43—531no—
122C53C4×#12411yescharacteristic, complemented, Sylow
123C22×Q8C2×D5311yescharacteristic
124C22×D4—531no—
125C23×C4—532no—
126#124—538no—
127C4×Q8—1064no—
128C4×D4—1064no—
129C2×C4⋊C4—531no—
130C2×C4⋊C4—532no—
131C2×C4⋊C4C2×D5311yescharacteristic
132C2×C22⋊C4—534no—
133C2×C22⋊C4—532no—
134C2×C22⋊C4—534no—
135C2×C42—531no—
136C2×C42—531no—
137C2×C42—532no—
138C2×C42—532no—
139C2×C42—532no—
140C2×C42C2×D5311yescharacteristic
141#95—534no—
142#95C2×D5314yes—
143C24—532no—
144C2×Q8—22no—
145C2×Q8—22no—
146C2×Q8C4×D5314yescomplemented
147C2×D4—534no—
148C2×D4—1062no—
149C2×D4—1062no—
150C22×C4—531no—
151C22×C4—532no—
152C22×C4—531no—
153C22×C4—532no—
154C22×C4—1064no—
155C22×C4—1068no—
156C22×C4C22×D5312yes—
157C22×C4C22×D5312yes—
158C22×C4C22×D5312yes—
159C22×C4C22×D5311yescharacteristic
160C4⋊C4—22no—
161C4⋊C4—1064no—
162C4⋊C4—1062no—
163C22⋊C4—5316no—
164C22⋊C4—1064no—
165C22⋊C4—1068no—
166C42—22no—
167C42—534no—
168C42—534no—
169C42—1064no—
170C42—1064no—
171C42—1064no—
172C23—534no—
173C23—1062no—
174C23—1064no—
175C23—1062no—
176C23—1064no—
177C23C23×D5311yescharacteristic
178Q8—28no—
179D4—1068no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—24no—
186C2×C4—532no—
187C2×C4—534no—
188C2×C4—534no—
189C2×C4—532no—
190C2×C4—1062no—
191C2×C4—1062no—
192C2×C4—1062no—
193C2×C4—1062no—
194C2×C4—1064no—
195C2×C4—2128no—
196C2×C4C2×C4×D5314yes—
197C2×C4C2×C4×D5312yes—
198C2×C4D4×D5312yes—
199C2×C4#1171994712yes—
200C2×C4#1171994914yes—
201C22—1064no—
202C22—1068no—
203C22—2122no—
204C22—2124no—
205C22—2122no—
206C22C22×C4×D5311yescharacteristic
207C22C2×D4×D5311yescharacteristic
208C22#41×D5312yes—
209C22C2×#1171994711yescharacteristic
210C22C2×#1171994912yes—
211C4—24no—
212C4—24no—
213C4—42no—
214C4—534no—
215C4—1064no—
216C4—1062no—
217C4—2122no—
218C4#1313317714yes—
219C2—2124no—
220C2C2×#1313317711yescharacteristic
221C2C4×#1171994912yes—
222C2#124×D5311yescharacteristic
223C2#1681115912yes—
224C2#1681117411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutatorC22×C53C23×C4
FittingC53×#391C2
socleC23×C53C24
hypercenter#391D53
hypocenterC53C4×#124
LCSG ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
2. C2×Q8×Dic53C2semidirect
5. #16810515C2semidirect
6. #16810488C2semidirect
7. #16810470C2semidirect
8. C42×Dic53C2semidirect
9. C53×#391C2semidirect
12. #13133206C4semidirect
33. C2×Q8×C53C2×C4semidirect
51. #11719921C2×C4semidirect
57. C4×Dic53C2×C4semidirect
105. Dic53#124semidirect
116. #391D53semidirect
122. C53C4×#124semidirect
146. C2×Q8C4×D53semidirect