Finite Group Databasebeta

← #20282679

Subgroups — #20282679

Order 6784.

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

normal subgroups only

This group has 392 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
1#16810556C212yesmaximal
2#16810555C212yesmaximal
3#16810544C212yesmaximal
4#16810530C212yesmaximal
5#16810529C211yesmaximal, characteristic
6#16810521C211yesmaximal, characteristic
7#16810520C212yesmaximal
8#16810506C212yesmaximal
9C53×#391C211yesmaximal, characteristic
10C22×Dic106C2211yescharacteristic
11C2×C4⋊Dic53C2211yescharacteristic
12C2×C4⋊Dic53C2211yescharacteristic
13C2×C4⋊Dic53C2212yes—
14C2×C4⋊Dic53C2212yes—
15C2×C4⋊Dic53C2211yescharacteristic
16C2×#11719919C2212yes—
17C2×#11719919C2212yes—
18C2×#11719919C2212yes—
19C2×C4×Dic53C2212yes—
20#13133124C2212yes—
21#13133124C2212yes—
22#13133124C2212yes—
23#13133124C2212yes—
24#13133124C2212yes—
25#13133124C2212yes—
26C22×Q8×C53C2211yescharacteristic
27C2×C4⋊C4×C53C2211yescharacteristic
28C2×C42×C53C2211yescharacteristic
29#95×C53C2212yes—
30#95×C53C2212yes—
31C2×Q8×C53—41no—
32C2×Q8×C53—41no—
33C2×Q8×C53Q814yescomplemented
34C22×C4×C53C2312yes—
35C22×C4×C53C2311yescharacteristic
36C22×C4×C53C2312yes—
37C22×C4×C53C2311yescharacteristic
38C22×C4×C53C2311yescharacteristic
39C22×Dic53C2312yes—
40C22×Dic53C2312yes—
41C22×Dic53C2312yes—
42C22×Dic53C2312yes—
43C2×Dic106—22no—
44C2×Dic106—41no—
45C2×Dic106—41no—
46C4⋊C4×C53—41no—
47C42×C53—41no—
48C4⋊Dic53—42no—
49C4⋊Dic53—42no—
50C4⋊Dic53—41no—
51C4⋊Dic53—41no—
52C4⋊Dic53Q814yescomplemented
53#11719919—24no—
54#11719919—42no—
55#11719919—42no—
56C4×Dic53—42no—
57C23×C53C2411yescharacteristic
58Q8×C53—44no—
59C2×C4×C53—21no—
60C2×C4×C53—21no—
61C2×C4×C53—22no—
62C2×C4×C53—22no—
63C2×C4×C53—21no—
64C2×C4×C53—21no—
65C2×C4×C53—21no—
66C2×C4×C53—21no—
67C2×C4×C53—22no—
68C2×C4×C53—22no—
69C2×C4×C53—22no—
70C2×C4×C53C2×Q814yes—
71C2×C4×C53C2×Q812yes—
72C2×C4×C53C2×Q812yes—
73C2×C4×C53#4112yes—
74C2×Dic53—22no—
75C2×Dic53—22no—
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×Dic53C2×Q814yes—
86Dic106—44no—
87C22×C53C22×Q811yescharacteristic
88C22×C53C22×Q811yescharacteristic
89C22×C53C2×#4111yescharacteristic
90C22×C53H(5,𝔽2)11yescharacteristic
91C22×C53H(5,𝔽2)11yescharacteristic
92C22×C53H(5,𝔽2)11yescharacteristic
93C22×C532−1+411yescharacteristic
94C4×C53—22no—
95C4×C53—24no—
96C4×C53—22no—
97C4×C53—22no—
98C4×C53—42no—
99Dic53—24no—
100Dic53—42no—
101Dic53—42no—
102Dic53—42no—
103#2806—531nomaximal, Sylow
104C2×C53#53811yescharacteristic
105C2×C53#54311yescharacteristic
106C2×C53#54411yescharacteristic
107C2×C53#55411yescharacteristic
108C2×C53#55711yescharacteristic
109C2×C53Q8211yescharacteristic
110C2×C53#56211yescharacteristic
111#398—532no—
112#398—532no—
113#395—532no—
114#391—532no—
115#391—531no—
116#391D5311yescharacteristic
117#389—531no—
118#389—532no—
119#387—532no—
120C53#280611yescharacteristic, complemented, Sylow
121C22×Q8—531no—
122C22×Q8C2×D5311yescharacteristic
123C2×C4⋊C4—531no—
124C2×C4⋊C4—532no—
125C2×C4⋊C4—532no—
126C2×C4⋊C4—532no—
127C2×C4⋊C4—531no—
128C2×C4⋊C4—531no—
129C2×C4⋊C4—532no—
130C2×C4⋊C4—532no—
131C2×C4⋊C4C2×D5311yescharacteristic
132C2×C42—532no—
133C2×C42C2×D5311yescharacteristic
134#95—532no—
135#95—532no—
136#95—532no—
137#95—532no—
138#95—532no—
139#95—532no—
140#95C2×D5312yes—
141#95C2×D5312yes—
142C2×Q8—41no—
143C2×Q8—41no—
144C2×Q8—1062no—
145C2×Q8—2121no—
146C2×Q8—2121no—
147C2×Q8Dic10614yescomplemented
148C22×C4—532no—
149C22×C4—532no—
150C22×C4—532no—
151C22×C4—532no—
152C22×C4C22×D5312yes—
153C22×C4C22×D5311yescharacteristic
154C22×C4C22×D5312yes—
155C22×C4C22×D5311yescharacteristic
156C22×C4C22×D5311yescharacteristic
157C4⋊C4—41no—
158C4⋊C4—534no—
159C4⋊C4—1064no—
160C4⋊C4—2122no—
161C4⋊C4—2122no—
162C4⋊C4—2121no—
163C4⋊C4—2122no—
164C4⋊C4—2121no—
165C4⋊C4—2122no—
166C42—41no—
167C42—2122no—
168C23C23×D5311yescharacteristic
169Q8—44no—
170Q8—2124no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—22no—
175C2×C4—21no—
176C2×C4—22no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—534no—
183C2×C4—1062no—
184C2×C4—1062no—
185C2×C4—1062no—
186C2×C4—1062no—
187C2×C4—1062no—
188C2×C4—1062no—
189C2×C4—1062no—
190C2×C4—1062no—
191C2×C4—1062no—
192C2×C4—1062no—
193C2×C4—1062no—
194C2×C4C2×Dic10612yes—
195C2×C4C2×Dic10614yes—
196C2×C4Q8×D5312yes—
197C2×C4#1171994912yes—
198C22C2×Q8×D5311yescharacteristic
199C22C22×Dic10611yescharacteristic
200C22C2×#1171994911yescharacteristic
201C22#1313326211yescharacteristic
202C22#1313326211yescharacteristic
203C22#1313326311yescharacteristic
204C22#1313326411yescharacteristic
205C4—22no—
206C4—22no—
207C4—24no—
208C4—22no—
209C4—42no—
210C4—1064no—
211C4—2122no—
212C4—2122no—
213C4—2122no—
214C2#1681121311yescharacteristic
215C2#1681124911yescharacteristic
216C2#1681125211yescharacteristic
217C2#1681133411yescharacteristic
218C2#1681135211yescharacteristic
219C2Q8×Dic10611yescharacteristic
220C2#1681136711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#391C2
hypercenter#391D53
hypocenterC53#2806
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
33. C2×Q8×C53Q8semidirect
52. C4⋊Dic53Q8semidirect
120. C53#2806semidirect
147. C2×Q8Dic106semidirect