Finite Group Databasebeta

← #20277426

Subgroups — #20277426

Order 6784.

612 conjugacy classes of subgroups; only the 205 normal subgroups are shown.

normal subgroups only

This group has 612 conjugacy classes of subgroups — too many to draw. Only its 205 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×#13133205C211yesmaximal, characteristic
2C2×C4×Dic106C211yesmaximal, characteristic, complemented
3C2×C4×D212C211yesmaximal, characteristic
4#16810529C211yesmaximal, characteristic, complemented
5#16810525C211yesmaximal, characteristic
6#16810515C214yesmaximal, complemented
7#16810497C214yesmaximal, complemented
8#16810493C211yesmaximal, characteristic, complemented
9C53×#376C211yesmaximal, characteristic, complemented
10C22×Dic106C2211yescharacteristic
11C22×D212C2211yescharacteristic
12C22×C4×D53C2212yes—
13#13133205C418yescomplemented
14C4×Dic106—24no—
15C4×D212—24no—
16C2×C4⋊Dic53C2211yescharacteristic
17C2×C4⋊Dic53C2211yescharacteristic
18C2×C4⋊Dic53C2214yes—
19C2×#11719919C2212yes—
20C2×#11719921C2212yes—
21C2×#11719921C2214yes—
22C2×#11719921C2214yes—
23C2×C4×Dic53C2212yes—
24#13133124C2214yes—
25C2×C42×C53C2211yescharacteristic
26C2×C42×C53C2211yescharacteristic
27C2×C42×C53C2211yescharacteristic
28#95×C53C2212yes—
29#95×C53C2212yes—
30C23×D53C2312yes—
31C22×C4×C53C2312yes—
32C22×C4×C53C2312yes—
33C22×C4×C53C2311yescharacteristic
34C22×C4×C53C2311yescharacteristic
35C22×C4×C53C2311yescharacteristic
36C22×Dic53C2312yes—
37C22×Dic53C2312yes—
38C22×Dic53C2312yes—
39C2×D212—22no—
40C2×D212—22no—
41C2×D212C2×C414yes—
42C2×C4×D53—24no—
43C2×C4×D53—44no—
44C2×Dic106—22no—
45C2×Dic106—22no—
46C2×Dic106C2×C414yescomplemented
47C42×C53—22no—
48C42×C53C2×C414yescomplemented
49C42×C53D414yes—
50#11719921—24no—
51#11719921—44no—
52#11719921C2×C4116yescomplemented
53C4⋊Dic53—22no—
54C4⋊Dic53—22no—
55C4⋊Dic53—44no—
56#11719919—24no—
57C4×Dic53—24no—
58C23×C53C2411yescharacteristic
59C22×D53—22no—
60C22×D53—22no—
61C22×D53—24no—
62C22×D53—24no—
63C22×D53C22×C414yes—
64C2×C4×C53—22no—
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×C53C22×C414yes—
72C2×C4×C53C22×C412yes—
73C2×C4×C53C2×D412yes—
74C2×C4×C53C2×D412yes—
75C2×C4×C53C2×D412yes—
76C2×C4×C53#4112yes—
77C2×Dic53—22no—
78C2×Dic53—22no—
79C2×Dic53—22no—
80C2×Dic53—22no—
81C2×Dic53—24no—
82C2×Dic53—24no—
83C2×Dic53C22×C414yes—
84D212—28no—
85C4×D53—48no—
86Dic106—28no—
87C22×C53C23×C411yescharacteristic
88C22×C53C22×D411yescharacteristic
89C22×C53C2×#4111yescharacteristic
90C22×C53H(5,𝔽2)12yes—
91C22×C532−1+412yes—
92C2×D53—24no—
93C2×D53—28no—
94C2×D53—42no—
95C2×D53—42no—
96C2×D53—44no—
97C4×C53—24no—
98C4×C53—22no—
99C4×C53—22no—
100C4×C53—42no—
101C4×C53C4×D414yes—
102Dic53—24no—
103Dic53—42no—
104Dic53—42no—
105#2326—531nomaximal, Sylow
106C2×C53C2×C4×D411yescharacteristic
107C2×C53#52012yes—
108C2×C53#53511yescharacteristic
109C2×C53#53611yescharacteristic
110C2×C53#54212yes—
111D53—44no—
112C2×#124—531no—
113C2×C4×Q8—531no—
114C2×C4×D4—531no—
115#391—531no—
116#390—531no—
117#388—534no—
118#385—534no—
119#384—531no—
120#376D5311yescharacteristic, complemented
121C53#232611yescharacteristic, complemented, Sylow
122C22×Q8—531no—
123C22×D4—531no—
124C23×C4—532no—
125#124—538no—
126C4×Q8—1064no—
127C4×D4—1064no—
128C2×C4⋊C4—531no—
129C2×C4⋊C4—532no—
130C2×C4⋊C4—531no—
131C2×C4⋊C4—534no—
132C2×C22⋊C4—534no—
133C2×C22⋊C4—532no—
134C2×C22⋊C4—534no—
135C2×C42—532no—
136C2×C42C2×D5311yescharacteristic
137C2×C42C2×D5311yescharacteristic
138C2×C42C2×D5311yescharacteristic
139#95—534no—
140#95C2×D5312yes—
141#95C2×D5312yes—
142C24—532no—
143C2×Q8—534no—
144C2×Q8—1062no—
145C2×Q8—1062no—
146C2×D4—534no—
147C2×D4—1062no—
148C2×D4—1062no—
149C22×C4—532no—
150C22×C4—532no—
151C22×C4—532no—
152C22×C4—1064no—
153C22×C4—2124no—
154C22×C4C22×D5312yes—
155C22×C4C22×D5312yes—
156C22×C4C22×D5311yescharacteristic
157C22×C4C22×D5311yescharacteristic
158C22×C4C22×D5311yescharacteristic
159C4⋊C4—1062no—
160C4⋊C4—1064no—
161C4⋊C4—1062no—
162C4⋊C4—2124no—
163C22⋊C4—5316no—
164C22⋊C4—1064no—
165C22⋊C4—2124no—
166C42—22no—
167C42—1064no—
168C42C4×D5314yescomplemented
169C42D21214yes—
170C23—534no—
171C23—1062no—
172C23—1062no—
173C23—1064no—
174C23—1064no—
175C23C23×D5311yescharacteristic
176Q8—1068no—
177D4—1068no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—534no—
186C2×C4—1062no—
187C2×C4—1062no—
188C2×C4—1062no—
189C2×C4—1062no—
190C2×C4—1064no—
191C2×C4—1064no—
192C2×C4—2128no—
193C2×C4C2×C4×D5314yes—
194C2×C4C2×C4×D5312yes—
195C2×C4C2×D21212yes—
196C2×C4C2×D21212yes—
197C2×C4C2×D21212yes—
198C2×C4#1171994512yes—
199C22—1064no—
200C22—1068no—
201C22—2122no—
202C22—2122no—
203C22—2124no—
204C22C22×C4×D5311yescharacteristic
205C22C22×D21211yescharacteristic
206C22C2×#1171994511yescharacteristic
207C22#1313326312yes—
208C22#1313326412yes—
209C4—24no—
210C4—22no—
211C4—22no—
212C4—42no—
213C4—1064no—
214C4—2122no—
215C4—2122no—
216C4C4×D21214yes—
217C2—2124no—
218C2C2×C4×D21211yescharacteristic
219C2#1681110312yes—
220C2#1681119211yescharacteristic
221C2#1681119911yescharacteristic
222C2#1681124412yes—
subgroupG/H
center, FrattiniC23C23×D53
commutatorC22×C53C23×C4
FittingC53×#376C2
socleC23×C53C24
hypercenter#376D53
hypocenterC53#2326
LCSG ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ #376

Complements

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

NH ≅ G/Ntype
2. C2×C4×Dic106C2semidirect
4. #16810529C2semidirect
6. #16810515C2semidirect
7. #16810497C2semidirect
8. #16810493C2semidirect
9. C53×#376C2semidirect
13. #13133205C4semidirect
46. C2×Dic106C2×C4semidirect
48. C42×C53C2×C4semidirect
52. #11719921C2×C4semidirect
120. #376D53semidirect
121. C53#2326semidirect
168. C42C4×D53semidirect