Finite Group Databasebeta

← #20277986

Subgroups — #20277986

Order 6784.

528 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 159 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×Q8×Dic53C211yesmaximal, characteristic, complemented
2C2×#13133175C211yesmaximal, characteristic
3#16810562C211yesmaximal, characteristic, complemented
4#16810560C211yesmaximal, characteristic
5#16810555C212yesmaximal, complemented
6#16810545C212yesmaximal
7#16810534C211yesmaximal, characteristic
8C4⋊C4×Dic53C212yesmaximal, complemented
9#16810474C211yesmaximal, characteristic, complemented
10#16810470C212yesmaximal
11C53×#395C211yesmaximal, characteristic, complemented
12C22×C4×D53C2211yescharacteristic
13Q8×Dic53—24no—
14#13133175—24no—
15C2×C4⋊Dic53C2212yes—
16C2×C4⋊Dic53C2211yescharacteristic
17C2×C4⋊Dic53C2212yes—
18C2×C4⋊Dic53C2211yescharacteristic
19C2×#11719919C2212yes—
20C2×#11719921C2212yes—
21C2×#11719921C2211yescharacteristic
22C2×#11719921C2211yescharacteristic
23C2×#11719921C2212yes—
24C2×C4×Dic53C2211yescharacteristic
25C2×C4×Dic53C2212yes—
26C2×C4×Dic53C2212yes—
27C2×C4×Dic53C2211yescharacteristic
28#13133124C2212yes—
29#13133124C2211yescharacteristic
30#13133124C2212yes—
31#13133124C2212yes—
32C22×Q8×C53C2211yescharacteristic
33C2×C4⋊C4×C53C2212yes—
34C2×C4⋊C4×C53C2211yescharacteristic
35#95×C53C2212yes—
36#95×C53C2211yescharacteristic
37C23×D53C2311yescharacteristic
38C2×Q8×C53—22no—
39C2×Q8×C53—24no—
40C22×C4×C53C2311yescharacteristic
41C22×C4×C53C2311yescharacteristic
42C22×C4×C53C2311yescharacteristic
43C22×C4×C53C2312yes—
44C22×C4×C53C2312yes—
45C22×Dic53C2312yes—
46C22×Dic53C2312yes—
47C22×Dic53C2311yescharacteristic
48C22×Dic53C2311yescharacteristic
49C22×Dic53C2311yescharacteristic
50C2×C4×D53—22no—
51C2×C4×D53—22no—
52C2×C4×D53Q814yescomplemented
53C4⋊C4×C53—22no—
54C4⋊C4×C53—24no—
55#11719921—22no—
56#11719921—24no—
57#11719921—24no—
58#11719921—22no—
59C4⋊Dic53—22no—
60C4⋊Dic53—24no—
61C4⋊Dic53—22no—
62C4⋊Dic53—24no—
63#11719919—24no—
64C4×Dic53—22no—
65C4×Dic53—24no—
66C4×Dic53—24no—
67C4×Dic53D414yes—
68C23×C53C2411yescharacteristic
69C22×D53—21no—
70C22×D53—21no—
71C22×D53—22no—
72C22×D53—22no—
73C22×D53C2×Q812yes—
74Q8×C53—28no—
75C2×C4×C53—21no—
76C2×C4×C53—22no—
77C2×C4×C53—21no—
78C2×C4×C53—22no—
79C2×C4×C53—22no—
80C2×C4×C53—22no—
81C2×C4×C53—22no—
82C2×C4×C53—21no—
83C2×C4×C53—22no—
84C2×C4×C53C2×D412yes—
85C2×C4×C53C2×Q812yes—
86C2×C4×C53#4114yes—
87C2×C4×C53#4114yes—
88C2×Dic53—22no—
89C2×Dic53—22no—
90C2×Dic53—21no—
91C2×Dic53—22no—
92C2×Dic53—22no—
93C2×Dic53—22no—
94C2×Dic53—22no—
95C2×Dic53—22no—
96C2×Dic53—22no—
97C2×Dic53C2×D412yes—
98C2×Dic53C2×D412yes—
99C2×Dic53C2×Q812yes—
100C2×Dic53#4112yes—
101C4×D53—28no—
102C22×C53C22×D411yescharacteristic
103C22×C53C22×Q811yescharacteristic
104C22×C53C2×#4111yescharacteristic
105C22×C53C2×#4112yes—
106C22×C53C2×#4112yes—
107C2×D53—22no—
108C2×D53—24no—
109C2×D53—41no—
110C2×D53—42no—
111C2×D53—41no—
112C4×C53—22no—
113C4×C53—22no—
114C4×C53—24no—
115C4×C53—24no—
116C4×C53—41no—
117Dic53—22no—
118Dic53—42no—
119Dic53—42no—
120Dic53—41no—
121Dic53C22⋊Q814yes—
122#2396—531nomaximal, Sylow
123C2×C53C2×C22⋊Q811yescharacteristic
124C2×C53#52511yescharacteristic
125C2×C53#52912yes—
126C2×C53#53211yescharacteristic
127C2×C53#53312yes—
128D53—42no—
129C2×C4×Q8—531no—
130C2×#117—531no—
131#400—531no—
132#399—531no—
133#398—532no—
134#396—532no—
135#395D5311yescharacteristic, complemented
136#393—531no—
137C4×C4⋊C4—532no—
138C4×C4⋊C4—531no—
139C4×C22⋊C4—532no—
140C53#239611yescharacteristic, complemented, Sylow
141C22×Q8C2×D5311yescharacteristic
142C23×C4—531no—
143C4×Q8—1064no—
144#117—1064no—
145C2×C4⋊C4—531no—
146C2×C4⋊C4—532no—
147C2×C4⋊C4—531no—
148C2×C4⋊C4—532no—
149C2×C4⋊C4—532no—
150C2×C4⋊C4C2×D5311yescharacteristic
151C2×C4⋊C4C2×D5312yes—
152C2×C22⋊C4—531no—
153C2×C22⋊C4—531no—
154C2×C22⋊C4—532no—
155C2×C22⋊C4—532no—
156C2×C42—531no—
157C2×C42—531no—
158C2×C42—532no—
159C2×C42—532no—
160#95—532no—
161#95—532no—
162#95—531no—
163#95—532no—
164#95C2×D5312yes—
165#95C2×D5311yescharacteristic
166C24—531no—
167C2×Q8—22no—
168C2×Q8—24no—
169C22×C4—531no—
170C22×C4—531no—
171C22×C4—534no—
172C22×C4—532no—
173C22×C4—531no—
174C22×C4—532no—
175C22×C4—1062no—
176C22×C4—1062no—
177C22×C4C22×D5311yescharacteristic
178C22×C4C22×D5311yescharacteristic
179C22×C4C22×D5312yes—
180C22×C4C22×D5311yescharacteristic
181C22×C4C22×D5312yes—
182C4⋊C4—22no—
183C4⋊C4—24no—
184C4⋊C4—1062no—
185C4⋊C4—1064no—
186C4⋊C4—1062no—
187C4⋊C4—1064no—
188C4⋊C4—1064no—
189C22⋊C4—1062no—
190C22⋊C4—1062no—
191C22⋊C4—1064no—
192C22⋊C4—1064no—
193C42—534no—
194C42—1062no—
195C42—1064no—
196C42—1064no—
197C23—532no—
198C23—1061no—
199C23—1061no—
200C23—1062no—
201C23—1062no—
202C23C23×D5311yescharacteristic
203Q8—28no—
204C2×C4—22no—
205C2×C4—21no—
206C2×C4—22no—
207C2×C4—21no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—21no—
212C2×C4—22no—
213C2×C4—532no—
214C2×C4—532no—
215C2×C4—532no—
216C2×C4—532no—
217C2×C4—1068no—
218C2×C4—1062no—
219C2×C4—1061no—
220C2×C4—1062no—
221C2×C4—1062no—
222C2×C4—1062no—
223C2×C4—1062no—
224C2×C4—1062no—
225C2×C4—1062no—
226C2×C4—1062no—
227C2×C4D4×D5312yes—
228C2×C4#1171994714yes—
229C2×C4Q8×D5312yes—
230C2×C4#1171994914yes—
231C22—1062no—
232C22—1064no—
233C22—2121no—
234C22—2122no—
235C22—2121no—
236C22C2×D4×D5311yescharacteristic
237C22C2×Q8×D5311yescharacteristic
238C22#41×D5311yescharacteristic
239C22C2×#1171994712yes—
240C22C2×#1171994912yes—
241C4—22no—
242C4—22no—
243C4—24no—
244C4—24no—
245C4—41no—
246C4—534no—
247C4—1062no—
248C4—2122no—
249C4—2121no—
250C4—2122no—
251C2—2122no—
252C2C22⋊Q8×D5311yescharacteristic
253C2#1681112911yescharacteristic
254C2#1681115912yes—
255C2#1681117411yescharacteristic
256C2#1681117812yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#395C2
hypercenter#395D53
hypocenterC53#2396
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #395

Complements

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

NH ≅ G/Ntype
1. C2×Q8×Dic53C2semidirect
3. #16810562C2semidirect
5. #16810555C2semidirect
8. C4⋊C4×Dic53C2semidirect
9. #16810474C2semidirect
11. C53×#395C2semidirect
52. C2×C4×D53Q8semidirect
135. #395D53semidirect
140. C53#2396semidirect