Finite Group Databasebeta

← #20278712

Subgroups — #20278712

Order 6784.

668 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 668 conjugacy classes of subgroups — too many to draw. Only its 145 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×#13133193C212yesmaximal, complemented
2C2×#13133177C212yesmaximal, complemented
3#16810539C212yesmaximal, complemented
4#16810532C211yesmaximal, characteristic
5#16810512C212yesmaximal, complemented
6#16810510C212yesmaximal, complemented
7#16810501C212yesmaximal, complemented
8#16810495C211yesmaximal, characteristic, complemented
9C53×#393C211yesmaximal, characteristic, complemented
10C22×D212C2211yescharacteristic
11C22×C22⋊D53C2212yes—
12C23×Dic53C2211yescharacteristic
13C22×C4×D53C2212yes—
14#13133193—28no—
15#13133177—28no—
16C2×C4⋊Dic53C2212yes—
17C2×#11719919C2212yes—
18C2×#11719919C2212yes—
19C2×C22⋊Dic53C2211yescharacteristic
20C2×#11719921C2212yes—
21C2×#11719921C2212yes—
22C2×#11719921C2212yes—
23C2×#11719921C2212yes—
24C2×C4×Dic53C2212yes—
25#13133124C2212yes—
26#13133124C2211yescharacteristic
27#13133124C2212yes—
28C22×Q8×C53C2211yescharacteristic
29C2×C22⋊C4×C53C2212yes—
30C2×C22⋊C4×C53C2211yescharacteristic
31#95×C53C2212yes—
32#95×C53C2211yescharacteristic
33C24×C53C2311yescharacteristic
34C23×D53C2312yes—
35C2×Q8×C53—24no—
36C2×Q8×C53—41no—
37C22×C4×C53C2311yescharacteristic
38C22×C4×C53C2312yes—
39C22×C4×C53C2311yescharacteristic
40C22×C4×C53C2312yes—
41C2×C22⋊D53—24no—
42C2×C22⋊D53—24no—
43C2×C22⋊D53—24no—
44C22×Dic53—24no—
45C22×Dic53—41no—
46C22×Dic53C2312yes—
47C22×Dic53C2312yes—
48C22×Dic53C2312yes—
49C2×D212—24no—
50C2×D212—41no—
51C2×C4×D53—24no—
52C2×C4×D53—24no—
53C2×C4×D53—42no—
54C22⋊C4×C53—24no—
55C22⋊C4×C53—41no—
56C22⋊Dic53—22no—
57#11719921—24no—
58#11719921—24no—
59#11719921—24no—
60#11719921—24no—
61C4⋊Dic53—24no—
62#11719919—24no—
63#11719919D418yescomplemented
64C4×Dic53—24no—
65C23×C53—22no—
66C23×C53—22no—
67C23×C53—21no—
68C23×C53—21no—
69C23×C53#4112yes—
70C23×C53C2411yescharacteristic
71C22×D53—22no—
72C22×D53—22no—
73C22×D53—22no—
74C22×D53—22no—
75C22×D53—22no—
76C22×D53—22no—
77C22×D53#4114yes—
78Q8×C53—44no—
79C2×C4×C53—22no—
80C2×C4×C53—22no—
81C2×C4×C53—22no—
82C2×C4×C53—22no—
83C2×C4×C53—22no—
84C2×C4×C53—22no—
85C2×C4×C53—21no—
86C2×C4×C53—21no—
87C2×C4×C53—22no—
88C2×C4×C53C2×D414yes—
89C22⋊D53—216no—
90C2×Dic53—22no—
91C2×Dic53—22no—
92C2×Dic53—22no—
93C2×Dic53—22no—
94C2×Dic53—22no—
95C2×Dic53—22no—
96C2×Dic53—22no—
97C2×Dic53—44no—
98C2×Dic53C2×D414yes—
99C2×Dic53C2×D414yes—
100D212—44no—
101C4×D53—48no—
102C22×C53—24no—
103C22×C53—22no—
104C22×C53—41no—
105C22×C53—42no—
106C22×C53—41no—
107C22×C53C22×D412yes—
108C22×C53C2×#4112yes—
109C22×C53C2×#4111yescharacteristic
110C22×C53H(5,𝔽2)11yescharacteristic
111C22×C532−1+411yescharacteristic
112C2×D53—24no—
113C2×D53—24no—
114C2×D53—24no—
115C2×D53—42no—
116C2×D53—42no—
117C2×D53—42no—
118C2×D53—42no—
119C4×C53—24no—
120C4×C53—41no—
121C4×C53—42no—
122C4×C53—41no—
123Dic53—24no—
124Dic53—24no—
125Dic53—42no—
126#2463—531nomaximal, Sylow
127C2×C53—42no—
128C2×C53#52512yes—
129C2×C53#53911yescharacteristic
130C2×C53D4211yescharacteristic
131C2×C53#54812yes—
132C2×C53#55111yescharacteristic
133D53—44no—
134C2×C22⋊Q8—532no—
135C2×C4×D4—532no—
136#394—532no—
137#393D5311yescharacteristic, complemented
138#392—531no—
139#388—532no—
140#387—532no—
141#386—532no—
142#385—531no—
143C53#246311yescharacteristic, complemented, Sylow
144C22×Q8C2×D5311yescharacteristic
145C22×D4—532no—
146C22×D4—531no—
147C23×C4—531no—
148C23×C4—532no—
149C22⋊Q8—1068no—
150C4×D4—1068no—
151C2×C4⋊C4—532no—
152C2×C4⋊C4—532no—
153C2×C4⋊C4—532no—
154C2×C22⋊C4—531no—
155C2×C22⋊C4—532no—
156C2×C22⋊C4—532no—
157C2×C22⋊C4—532no—
158C2×C22⋊C4—532no—
159C2×C22⋊C4C2×D5312yes—
160C2×C22⋊C4C2×D5311yescharacteristic
161C2×C42—532no—
162#95—532no—
163#95—531no—
164#95—532no—
165#95C2×D5312yes—
166#95C2×D5311yescharacteristic
167C24—532no—
168C24C22×D5311yescharacteristic
169C2×Q8—24no—
170C2×Q8—41no—
171C2×D4—1064no—
172C2×D4—1064no—
173C2×D4—1064no—
174C2×D4—1064no—
175C2×D4—2121no—
176C22×C4—532no—
177C22×C4—532no—
178C22×C4—532no—
179C22×C4—1064no—
180C22×C4—1064no—
181C22×C4—1064no—
182C22×C4—2122no—
183C22×C4—2121no—
184C22×C4C22×D5311yescharacteristic
185C22×C4C22×D5312yes—
186C22×C4C22×D5311yescharacteristic
187C22×C4C22×D5312yes—
188C4⋊C4—538no—
189C4⋊C4—1064no—
190C4⋊C4—1064no—
191C22⋊C4—24no—
192C22⋊C4—41no—
193C22⋊C4—1064no—
194C22⋊C4—1064no—
195C22⋊C4—1064no—
196C22⋊C4—1062no—
197C22⋊C4—1064no—
198C42—1064no—
199C23—22no—
200C23—22no—
201C23—21no—
202C23—21no—
203C23—534no—
204C23—1062no—
205C23—1062no—
206C23—1062no—
207C23—1062no—
208C23—1062no—
209C23—1062no—
210C23#1171994912yes—
211C23C23×D5311yescharacteristic
212Q8—44no—
213D4—10616no—
214D4—2124no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—22no—
224C2×C4—534no—
225C2×C4—534no—
226C2×C4—1062no—
227C2×C4—1062no—
228C2×C4—1062no—
229C2×C4—1062no—
230C2×C4—1062no—
231C2×C4—1062no—
232C2×C4—1062no—
233C2×C4—2128no—
234C2×C4—2124no—
235C2×C4D4×D5314yes—
236C22—24no—
237C22—22no—
238C22—42no—
239C22—41no—
240C22—41no—
241C22—1064no—
242C22—1064no—
243C22—1064no—
244C22—2122no—
245C22—2122no—
246C22—2122no—
247C22—2122no—
248C22C2×D4×D5312yes—
249C22#41×D5312yes—
250C22C2×#1171994911yescharacteristic
251C22#1313326211yescharacteristic
252C22#1313326411yescharacteristic
253C4—24no—
254C4—42no—
255C4—41no—
256C4—41no—
257C4—1064no—
258C4—1064no—
259C4—2122no—
260C2—42no—
261C2—2124no—
262C2#1681113612yes—
263C2#1681122211yescharacteristic
264C2#1681125511yescharacteristic
265C2#1681128112yes—
266C2#1681130311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#393C2
hypercenter#393D53
hypocenterC53#2463
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #393

Complements

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

NH ≅ G/Ntype
1. C2×#13133193C2semidirect
2. C2×#13133177C2semidirect
3. #16810539C2semidirect
5. #16810512C2semidirect
6. #16810510C2semidirect
7. #16810501C2semidirect
8. #16810495C2semidirect
9. C53×#393C2semidirect
63. #11719919D4semidirect
137. #393D53semidirect
143. C53#2463semidirect