Finite Group Databasebeta

← #20279727

Subgroups — #20279727

Order 6784.

544 conjugacy classes of subgroups; only the 169 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 169 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×#13133194C212yesmaximal
2#16810556C212yesmaximal, complemented
3#16810543C212yesmaximal, complemented
4#16810527C212yesmaximal, complemented
5#16810520C212yesmaximal, complemented
6#16810495C212yesmaximal
7C4×C22⋊Dic53C211yesmaximal, characteristic
8C2×#13133124C211yesmaximal, characteristic
9C53×#380C211yesmaximal, characteristic
10C22×Dic106C2212yes—
11C23×Dic53C2212yes—
12#13133194—28no—
13C2×C4⋊Dic53C2212yes—
14C2×#11719919C2212yes—
15C2×#11719919C2212yes—
16C2×#11719919C2212yes—
17C2×#11719919C2212yes—
18C2×C22⋊Dic53C2211yescharacteristic
19C2×C22⋊Dic53C2211yescharacteristic
20C2×C4×Dic53C2212yes—
21#13133124C2212yes—
22#13133124C2212yes—
23#13133124C2212yes—
24#13133124C2211yescharacteristic
25#13133124C2212yes—
26#13133124C2211yescharacteristic
27C23×C4×C53C2211yescharacteristic
28C2×C4⋊C4×C53C2212yes—
29C2×C22⋊C4×C53C2212yes—
30#95×C53C2211yescharacteristic
31#95×C53C2211yescharacteristic
32C24×C53C2311yescharacteristic
33C22×C4×C53—42no—
34C22×C4×C53Q814yescomplemented
35C22×C4×C53C2312yes—
36C22×C4×C53C2312yes—
37C22×C4×C53C2311yescharacteristic
38C22×C4×C53C2311yescharacteristic
39C22×Dic53—24no—
40C22×Dic53—24no—
41C22×Dic53—42no—
42C22×Dic53C2312yes—
43C22×Dic53C2312yes—
44C22×Dic53C2312yes—
45C22×Dic53C2312yes—
46C2×Dic106—24no—
47C2×Dic106—24no—
48C2×Dic106—42no—
49C4⋊C4×C53—42no—
50C22⋊C4×C53—24no—
51C22⋊Dic53D414yes—
52C22⋊Dic53Q814yescomplemented
53C4⋊Dic53—24no—
54#11719919—24no—
55#11719919—24no—
56#11719919—24no—
57#11719919—42no—
58C4×Dic53—42no—
59C23×C53—22no—
60C23×C53—22no—
61C23×C53C2×D412yes—
62C23×C53C2×Q812yes—
63C23×C53C2×Q812yes—
64C23×C53C2411yescharacteristic
65C2×C4×C53—22no—
66C2×C4×C53—22no—
67C2×C4×C53—22no—
68C2×C4×C53—22no—
69C2×C4×C53—22no—
70C2×C4×C53—22no—
71C2×C4×C53—22no—
72C2×C4×C53—22no—
73C2×C4×C53—44no—
74C2×C4×C53C2×Q812yes—
75C2×C4×C53C2×Q812yes—
76C2×Dic53—22no—
77C2×Dic53—22no—
78C2×Dic53—22no—
79C2×Dic53—22no—
80C2×Dic53—22no—
81C2×Dic53—22no—
82C2×Dic53—22no—
83C2×Dic53—22no—
84C2×Dic53—22no—
85C2×Dic53—48no—
86C2×Dic53C2×D414yes—
87C2×Dic53C2×Q814yes—
88C2×Dic53#4114yes—
89Dic106—48no—
90C22×C53—22no—
91C22×C53—22no—
92C22×C53—24no—
93C22×C53—24no—
94C22×C53C4⋊Q814yes—
95C22×C53C22×D411yescharacteristic
96C22×C53C22×Q811yescharacteristic
97C22×C53C22×Q811yescharacteristic
98C22×C53C2×#4112yes—
99C22×C53H(5,𝔽2)12yes—
100C4×C53—22no—
101C4×C53—22no—
102C4×C53—42no—
103C4×C53—42no—
104Dic53—24no—
105Dic53—24no—
106Dic53—24no—
107Dic53—42no—
108#2549—531nomaximal, Sylow
109C2×C53—24no—
110C2×C53C2×C4⋊Q811yescharacteristic
111C2×C53#53311yescharacteristic
112C2×C53#54311yescharacteristic
113C2×C53#54612yes—
114C2×C53#55412yes—
115C2×C22⋊Q8—532no—
116#398—532no—
117#395—532no—
118#391—532no—
119#389—532no—
120#385—532no—
121#380D5311yescharacteristic
122C4×C22⋊C4—531no—
123C2×#95—531no—
124C53#254911yescharacteristic, complemented, Sylow
125C22×Q8—532no—
126C23×C4—532no—
127C23×C4C2×D5311yescharacteristic
128C22⋊Q8—1068no—
129C2×C4⋊C4—532no—
130C2×C4⋊C4—532no—
131C2×C4⋊C4—532no—
132C2×C4⋊C4—532no—
133C2×C4⋊C4—532no—
134C2×C4⋊C4C2×D5312yes—
135C2×C22⋊C4—531no—
136C2×C22⋊C4—531no—
137C2×C22⋊C4C2×D5312yes—
138C2×C42—532no—
139#95—532no—
140#95—531no—
141#95—531no—
142#95—532no—
143#95—532no—
144#95—532no—
145#95C2×D5311yescharacteristic
146#95C2×D5311yescharacteristic
147C24C22×D5311yescharacteristic
148C2×Q8—1064no—
149C2×Q8—1064no—
150C2×Q8—2122no—
151C22×C4—42no—
152C22×C4—532no—
153C22×C4—532no—
154C22×C4—532no—
155C22×C4—532no—
156C22×C4—1064no—
157C22×C4—1064no—
158C22×C4—2122no—
159C22×C4Dic10614yescomplemented
160C22×C4C22×D5312yes—
161C22×C4C22×D5312yes—
162C22×C4C22×D5311yescharacteristic
163C22×C4C22×D5311yescharacteristic
164C4⋊C4—42no—
165C4⋊C4—1064no—
166C4⋊C4—1064no—
167C4⋊C4—1064no—
168C4⋊C4—1064no—
169C4⋊C4—2122no—
170C22⋊C4—24no—
171C22⋊C4—534no—
172C22⋊C4—534no—
173C42—2122no—
174C23—22no—
175C23—22no—
176C23C2×Dic10612yes—
177C23D4×D5312yes—
178C23Q8×D5312yes—
179C23C23×D5311yescharacteristic
180Q8—2128no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—44no—
190C2×C4—534no—
191C2×C4—534no—
192C2×C4—534no—
193C2×C4—1062no—
194C2×C4—1062no—
195C2×C4—1062no—
196C2×C4—1062no—
197C2×C4—1062no—
198C2×C4—1062no—
199C2×C4—1062no—
200C2×C4—1062no—
201C2×C4—1062no—
202C2×C4—2128no—
203C2×C4C2×Dic10612yes—
204C2×C4C2×Dic10612yes—
205C22—24no—
206C22—22no—
207C22—22no—
208C22—24no—
209C22#1313321614yes—
210C22C2×D4×D5311yescharacteristic
211C22C2×Q8×D5311yescharacteristic
212C22C22×Dic10611yescharacteristic
213C22#41×D5312yes—
214C22#1313326212yes—
215C4—22no—
216C4—22no—
217C4—42no—
218C4—42no—
219C4—1064no—
220C4—1064no—
221C4—1064no—
222C4—2122no—
223C2—24no—
224C2C2×#1313321611yescharacteristic
225C2#1681117611yescharacteristic
226C2#1681124911yescharacteristic
227C2#1681126212yes—
228C2#1681132712yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#380C2
hypercenter#380D53
hypocenterC53#2549
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #380

Complements

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

NH ≅ G/Ntype
2. #16810556C2semidirect
3. #16810543C2semidirect
4. #16810527C2semidirect
5. #16810520C2semidirect
34. C22×C4×C53Q8semidirect
52. C22⋊Dic53Q8semidirect
124. C53#2549semidirect
159. C22×C4Dic106semidirect