Finite Group Databasebeta

← #20281089

Subgroups — #20281089

Order 6784.

592 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 592 conjugacy classes of subgroups — too many to draw. Only its 149 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×#13133195C212yesmaximal, complemented
2C2×#13133193C212yesmaximal, complemented
3#16810547C212yesmaximal, complemented
4#16810545C212yesmaximal, complemented
5#16810543C211yesmaximal, characteristic, complemented
6#16810524C211yesmaximal, characteristic
7#16810520C212yesmaximal, complemented
8#16810501C212yesmaximal
9C53×#395C211yesmaximal, characteristic, complemented
10C22×Dic106C2211yescharacteristic
11C22×D212C2211yescharacteristic
12C22×C4×D53C2212yes—
13#13133195—28no—
14#13133193—28no—
15C2×C4⋊Dic53C2212yes—
16C2×#11719919C2212yes—
17C2×#11719919C2212yes—
18C2×#11719919C2212yes—
19C2×#11719919C2212yes—
20C2×#11719921C2212yes—
21C2×#11719921C2212yes—
22C2×#11719921C2212yes—
23C2×#11719921C2212yes—
24C2×#11719921C2212yes—
25C2×C4×Dic53C2211yescharacteristic
26#13133124C2212yes—
27#13133124C2211yescharacteristic
28C22×Q8×C53C2211yescharacteristic
29C2×C4⋊C4×C53C2212yes—
30C2×C4⋊C4×C53C2211yescharacteristic
31#95×C53C2212yes—
32#95×C53C2211yescharacteristic
33C23×D53C2312yes—
34C2×Q8×C53—24no—
35C2×Q8×C53—41no—
36C22×C4×C53C2312yes—
37C22×C4×C53C2311yescharacteristic
38C22×C4×C53C2311yescharacteristic
39C22×C4×C53C2312yes—
40C22×C4×C53C2311yescharacteristic
41C22×Dic53C2312yes—
42C22×Dic53C2312yes—
43C22×Dic53C2312yes—
44C2×D212—41no—
45C2×D212—41no—
46C2×D212Q814yescomplemented
47C2×C4×D53—24no—
48C2×C4×D53—24no—
49C2×C4×D53—42no—
50C2×Dic106—24no—
51C2×Dic106—41no—
52C4⋊C4×C53—24no—
53C4⋊C4×C53—41no—
54#11719921—24no—
55#11719921—24no—
56#11719921—24no—
57#11719921—24no—
58#11719921—42no—
59C4⋊Dic53—24no—
60#11719919—24no—
61#11719919—24no—
62#11719919—42no—
63#11719919D418yescomplemented
64C4×Dic53—41no—
65C23×C53C2411yescharacteristic
66C22×D53—22no—
67C22×D53—22no—
68C22×D53—22no—
69C22×D53—22no—
70C22×D53—22no—
71C22×D53—22no—
72C22×D53C2×Q814yes—
73Q8×C53—44no—
74C2×C4×C53—22no—
75C2×C4×C53—22no—
76C2×C4×C53—21no—
77C2×C4×C53—21no—
78C2×C4×C53—22no—
79C2×C4×C53—22no—
80C2×C4×C53—22no—
81C2×C4×C53—21no—
82C2×C4×C53—21no—
83C2×C4×C53—22no—
84C2×C4×C53—22no—
85C2×C4×C53C2×D414yes—
86C2×C4×C53C2×Q812yes—
87C2×Dic53—22no—
88C2×Dic53—22no—
89C2×Dic53—22no—
90C2×Dic53—22no—
91C2×Dic53—22no—
92C2×Dic53—22no—
93C2×Dic53—22no—
94C2×Dic53C2×D414yes—
95C2×Dic53C2×D414yes—
96D212—44no—
97C4×D53—48no—
98Dic106—44no—
99C22×C53C22×D412yes—
100C22×C53C22×Q811yescharacteristic
101C22×C53H(5,𝔽2)11yescharacteristic
102C22×C53H(5,𝔽2)11yescharacteristic
103C22×C532−1+412yes—
104C2×D53—24no—
105C2×D53—24no—
106C2×D53—24no—
107C2×D53—42no—
108C2×D53—42no—
109C2×D53—42no—
110C2×D53—42no—
111C4×C53—22no—
112C4×C53—24no—
113C4×C53—42no—
114C4×C53—41no—
115C4×C53—41no—
116Dic53—24no—
117Dic53—24no—
118Dic53—42no—
119#2683—531nomaximal, Sylow
120C2×C53#53712yes—
121C2×C53#54311yescharacteristic
122C2×C53D4211yescharacteristic
123C2×C53D4×Q812yes—
124C2×C53#56311yescharacteristic
125D53—44no—
126C2×C22⋊Q8—532no—
127C2×C22⋊Q8—532no—
128#396—532no—
129#396—532no—
130#395—531no—
131#395D5311yescharacteristic, complemented
132#390—531no—
133#389—532no—
134#386—532no—
135C53#268311yescharacteristic, complemented, Sylow
136C22×Q8—531no—
137C22×Q8C2×D5311yescharacteristic
138C22×D4—531no—
139C23×C4—532no—
140C22⋊Q8—1068no—
141C22⋊Q8—1068no—
142C2×C4⋊C4—532no—
143C2×C4⋊C4—532no—
144C2×C4⋊C4—532no—
145C2×C4⋊C4—532no—
146C2×C4⋊C4—532no—
147C2×C4⋊C4C2×D5312yes—
148C2×C4⋊C4C2×D5311yescharacteristic
149C2×C22⋊C4—532no—
150C2×C22⋊C4—532no—
151C2×C22⋊C4—532no—
152C2×C22⋊C4—532no—
153C2×C22⋊C4—532no—
154C2×C42—531no—
155#95—532no—
156#95—531no—
157#95C2×D5312yes—
158#95C2×D5311yescharacteristic
159C24—532no—
160C2×Q8—24no—
161C2×Q8—41no—
162C2×Q8—1064no—
163C2×Q8—2121no—
164C2×D4—534no—
165C2×D4—2121no—
166C2×D4—2121no—
167C22×C4—532no—
168C22×C4—532no—
169C22×C4—532no—
170C22×C4—1064no—
171C22×C4—1064no—
172C22×C4—2122no—
173C22×C4C22×D5311yescharacteristic
174C22×C4C22×D5312yes—
175C22×C4C22×D5311yescharacteristic
176C22×C4C22×D5312yes—
177C22×C4C22×D5311yescharacteristic
178C4⋊C4—24no—
179C4⋊C4—41no—
180C4⋊C4—538no—
181C4⋊C4—1064no—
182C4⋊C4—1064no—
183C4⋊C4—1064no—
184C4⋊C4—2122no—
185C22⋊C4—1064no—
186C22⋊C4—1064no—
187C22⋊C4—1064no—
188C22⋊C4—1064no—
189C22⋊C4—2122no—
190C42—2121no—
191C23—534no—
192C23—1062no—
193C23—1062no—
194C23—1062no—
195C23—1062no—
196C23—1062no—
197C23—1062no—
198C23C23×D5311yescharacteristic
199Q8—44no—
200Q8—2124no—
201D4—2124no—
202C2×C4—22no—
203C2×C4—21no—
204C2×C4—22no—
205C2×C4—21no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—534no—
214C2×C4—534no—
215C2×C4—1062no—
216C2×C4—1062no—
217C2×C4—1062no—
218C2×C4—1062no—
219C2×C4—1062no—
220C2×C4—1062no—
221C2×C4—1062no—
222C2×C4—2128no—
223C2×C4D4×D5314yes—
224C2×C4Q8×D5312yes—
225C22—1064no—
226C22—1064no—
227C22—1064no—
228C22—2122no—
229C22—2122no—
230C22—2122no—
231C22—2122no—
232C22C2×D4×D5312yes—
233C22C2×Q8×D5311yescharacteristic
234C22#1313326211yescharacteristic
235C22#1313326311yescharacteristic
236C22#1313326512yes—
237C4—22no—
238C4—24no—
239C4—42no—
240C4—41no—
241C4—41no—
242C4—1064no—
243C4—1064no—
244C4—2122no—
245C2—2124no—
246C2#1681120312yes—
247C2#1681124811yescharacteristic
248C2#1681125511yescharacteristic
249C2#1681128812yes—
250C2#1681137211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#395C2
hypercenter#395D53
hypocenterC53#2683
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #395

Complements

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

NH ≅ G/Ntype
1. C2×#13133195C2semidirect
2. C2×#13133193C2semidirect
3. #16810547C2semidirect
4. #16810545C2semidirect
5. #16810543C2semidirect
7. #16810520C2semidirect
9. C53×#395C2semidirect
46. C2×D212Q8semidirect
63. #11719919D4semidirect
131. #395D53semidirect
135. C53#2683semidirect