Finite Group Databasebeta

← #20282660

Subgroups — #20282660

Order 6784.

480 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 129 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#16810562C211yesmaximal, characteristic, complemented
2#16810536C212yesmaximal, complemented
3#16810536C212yesmaximal, complemented
4#16810527C212yesmaximal, complemented
5#16810524C211yesmaximal, characteristic
6#16810501C212yesmaximal
7#16810497C212yesmaximal, complemented
8#16810494C212yesmaximal, complemented
9C53×#400C211yesmaximal, characteristic, complemented
10C22×Dic106C2212yes—
11C22×D212C2211yescharacteristic
12C22×C4×D53C2212yes—
13C2×C4⋊Dic53C2211yescharacteristic
14C2×#11719919C2212yes—
15C2×#11719921C2212yes—
16C2×#11719921C2212yes—
17C2×#11719921C2212yes—
18C2×#11719921C2212yes—
19C2×#11719921C2212yes—
20C2×C4×Dic53C2211yescharacteristic
21#13133124C2212yes—
22#13133124C2212yes—
23#13133124C2212yes—
24#13133124C2211yescharacteristic
25#13133124C2212yes—
26C2×C4⋊C4×C53C2211yescharacteristic
27C2×C4⋊C4×C53C2211yescharacteristic
28#95×C53C2212yes—
29#95×C53C2212yes—
30#95×C53C2211yescharacteristic
31C23×D53C2312yes—
32C22×C4×C53C2312yes—
33C22×C4×C53C2311yescharacteristic
34C22×C4×C53C2312yes—
35C22×C4×C53C2311yescharacteristic
36C22×C4×C53C2311yescharacteristic
37C22×Dic53C2312yes—
38C22×Dic53C2312yes—
39C22×Dic53C2312yes—
40C2×D212—41no—
41C2×D212—41no—
42C2×D212Q814yescomplemented
43C2×C4×D53—42no—
44C2×C4×D53—42no—
45C2×C4×D53—42no—
46C2×Dic106—24no—
47C2×Dic106—42no—
48C2×Dic106—42no—
49C4⋊C4×C53—41no—
50C4⋊C4×C53—41no—
51#11719921—24no—
52#11719921—24no—
53#11719921—42no—
54#11719921—42no—
55#11719921—42no—
56C4⋊Dic53—41no—
57#11719919—42no—
58C4×Dic53—41no—
59C23×C53C2411yescharacteristic
60C22×D53—22no—
61C22×D53—22no—
62C22×D53—22no—
63C22×D53—22no—
64C22×D53—22no—
65C22×D53—22no—
66C22×D53C2×Q814yes—
67C2×C4×C53—21no—
68C2×C4×C53—21no—
69C2×C4×C53—22no—
70C2×C4×C53—22no—
71C2×C4×C53—22no—
72C2×C4×C53—22no—
73C2×C4×C53—22no—
74C2×C4×C53—22no—
75C2×C4×C53—22no—
76C2×C4×C53—22no—
77C2×C4×C53—21no—
78C2×C4×C53—21no—
79C2×C4×C53C2×Q812yes—
80C2×Dic53—22no—
81C2×Dic53—22no—
82C2×Dic53—22no—
83C2×Dic53—22no—
84C2×Dic53—22no—
85C2×Dic53—22no—
86C2×Dic53—22no—
87C2×Dic53—22no—
88C2×Dic53#4114yes—
89D212—44no—
90C4×D53—48no—
91Dic106—48no—
92C22×C53C22×Q811yescharacteristic
93C22×C53C2×#4112yes—
94C22×C53H(5,𝔽2)11yescharacteristic
95C22×C53H(5,𝔽2)11yescharacteristic
96C22×C53H(5,𝔽2)12yes—
97C2×D53—24no—
98C2×D53—24no—
99C2×D53—24no—
100C2×D53—42no—
101C2×D53—42no—
102C2×D53—42no—
103C2×D53—42no—
104C4×C53—22no—
105C4×C53—42no—
106C4×C53—41no—
107C4×C53—41no—
108C4×C53—42no—
109Dic53—24no—
110Dic53—42no—
111Dic53—42no—
112#2804—531nomaximal, Sylow
113C2×C53#53812yes—
114C2×C53#54311yescharacteristic
115C2×C53#55412yes—
116C2×C53#55911yescharacteristic
117C2×C53H(3,𝔽4)11yescharacteristic
118D53—44no—
119#400—531no—
120#400D5311yescharacteristic, complemented
121#393—532no—
122#393—532no—
123#391—532no—
124#390—531no—
125#386—532no—
126#385—532no—
127#385—532no—
128C53#280411yescharacteristic, complemented, Sylow
129C22×Q8—532no—
130C22×D4—531no—
131C23×C4—532no—
132C2×C4⋊C4—532no—
133C2×C4⋊C4—531no—
134C2×C4⋊C4C2×D5311yescharacteristic
135C2×C4⋊C4C2×D5311yescharacteristic
136C2×C22⋊C4—532no—
137C2×C22⋊C4—532no—
138C2×C22⋊C4—532no—
139C2×C22⋊C4—532no—
140C2×C22⋊C4—532no—
141C2×C42—531no—
142#95—532no—
143#95—532no—
144#95—532no—
145#95—532no—
146#95—531no—
147#95C2×D5312yes—
148#95C2×D5312yes—
149#95C2×D5311yescharacteristic
150C24—532no—
151C2×Q8—1064no—
152C2×Q8—2122no—
153C2×Q8—2122no—
154C2×D4—534no—
155C2×D4—2121no—
156C2×D4—2121no—
157C22×C4—532no—
158C22×C4—532no—
159C22×C4—532no—
160C22×C4—2122no—
161C22×C4—2122no—
162C22×C4—2122no—
163C22×C4C22×D5311yescharacteristic
164C22×C4C22×D5312yes—
165C22×C4C22×D5311yescharacteristic
166C22×C4C22×D5312yes—
167C22×C4C22×D5311yescharacteristic
168C4⋊C4—41no—
169C4⋊C4—41no—
170C4⋊C4—2122no—
171C4⋊C4—2121no—
172C22⋊C4—1064no—
173C22⋊C4—1064no—
174C22⋊C4—2122no—
175C22⋊C4—2122no—
176C22⋊C4—2122no—
177C42—2121no—
178C23—534no—
179C23—1062no—
180C23—1062no—
181C23—1062no—
182C23—1062no—
183C23—1062no—
184C23—1062no—
185C23C23×D5311yescharacteristic
186Q8—2128no—
187D4—2124no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—534no—
201C2×C4—1062no—
202C2×C4—1062no—
203C2×C4—1062no—
204C2×C4—1062no—
205C2×C4—1062no—
206C2×C4—1062no—
207C2×C4—1062no—
208C2×C4—1062no—
209C2×C4—2128no—
210C2×C4Q8×D5312yes—
211C22—1064no—
212C22—1064no—
213C22—1064no—
214C22—2122no—
215C22—2122no—
216C22—2122no—
217C22—2122no—
218C22C2×Q8×D5311yescharacteristic
219C22#41×D5312yes—
220C22#1313326211yescharacteristic
221C22#1313326312yes—
222C22#1313326311yescharacteristic
223C4—22no—
224C4—41no—
225C4—41no—
226C4—42no—
227C4—42no—
228C4—1064no—
229C4—2122no—
230C4—2122no—
231C2—2124no—
232C2#1681121212yes—
233C2#1681124811yescharacteristic
234C2#1681132912yes—
235C2#1681135711yescharacteristic
236C2#1681136511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#400C2
hypercenter#400D53
hypocenterC53#2804
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
1. #16810562C2semidirect
2. #16810536C2semidirect
3. #16810536C2semidirect
4. #16810527C2semidirect
7. #16810497C2semidirect
8. #16810494C2semidirect
9. C53×#400C2semidirect
42. C2×D212Q8semidirect
120. #400D53semidirect
128. C53#2804semidirect