Finite Group Databasebeta

← #20279667

Subgroups — #20279667

Order 6784.

676 conjugacy classes of subgroups; only the 177 normal subgroups are shown.

normal subgroups only

This group has 676 conjugacy classes of subgroups — too many to draw. Only its 177 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×#13133216C211yesmaximal, characteristic, complemented
2C2×#13133195C214yesmaximal, complemented
3C2×#13133179C211yesmaximal, characteristic
4#16810558C214yesmaximal, complemented
5#16810525C211yesmaximal, characteristic
6#16810524C211yesmaximal, characteristic
7#16810492C211yesmaximal, characteristic, complemented
8C4×#11719919C211yesmaximal, characteristic, complemented
9C53×#384C211yesmaximal, characteristic, complemented
10C22×Dic106C2212yes—
11C22×D212C2211yescharacteristic
12C22×C4×D53C2212yes—
13#13133216—24no—
14#13133195—216no—
15#13133179—24no—
16C2×C4⋊Dic53C2211yescharacteristic
17C2×#11719919C2212yes—
18C2×#11719919C2214yes—
19C2×#11719919C2212yes—
20C2×#11719921C2214yes—
21C2×#11719921C2214yes—
22C2×#11719921C2212yes—
23C2×C4×Dic53C2211yescharacteristic
24C2×C4×Dic53C2211yescharacteristic
25#13133124C2212yes—
26C2×C4⋊C4×C53C2211yescharacteristic
27C2×C4⋊C4×C53C2214yes—
28C2×C4⋊C4×C53C2211yescharacteristic
29C2×C42×C53C2211yescharacteristic
30C23×D53C2312yes—
31C22×C4×C53C2312yes—
32C22×C4×C53C2312yes—
33C22×C4×C53C2312yes—
34C22×C4×C53C2311yescharacteristic
35C22×Dic53C2312yes—
36C22×Dic53C2311yescharacteristic
37C22×Dic53C2312yes—
38C22×Dic53C2311yescharacteristic
39C2×D212—22no—
40C2×D212—22no—
41C2×D212Q814yescomplemented
42C2×C4×D53—24no—
43C2×C4×D53—28no—
44C2×Dic106—28no—
45C2×Dic106—24no—
46C4⋊C4×C53—22no—
47C4⋊C4×C53—28no—
48C4⋊C4×C53—22no—
49C42×C53—22no—
50#11719921—28no—
51#11719921—24no—
52#11719921—28no—
53C4⋊Dic53—22no—
54#11719919—28no—
55#11719919—24no—
56#11719919D418yescomplemented
57C4×Dic53—22no—
58C4×Dic53D414yes—
59C23×C53C2411yescharacteristic
60C22×D53—22no—
61C22×D53—24no—
62C22×D53—22no—
63C22×D53—24no—
64C22×D53C2×Q814yes—
65C2×C4×C53—22no—
66C2×C4×C53—22no—
67C2×C4×C53—22no—
68C2×C4×C53—24no—
69C2×C4×C53—24no—
70C2×C4×C53—22no—
71C2×C4×C53C2×D414yes—
72C2×C4×C53C2×D412yes—
73C2×C4×C53C2×Q812yes—
74C2×C4×C53#4112yes—
75C2×Dic53—22no—
76C2×Dic53—22no—
77C2×Dic53—22no—
78C2×Dic53—22no—
79C2×Dic53—24no—
80C2×Dic53C2×D412yes—
81C2×Dic53C2×D414yes—
82C2×Dic53C2×D412yes—
83C2×Dic53C2×D414yes—
84D212—28no—
85C4×D53—48no—
86Dic106—48no—
87C22×C53C22×D411yescharacteristic
88C22×C53C22×D412yes—
89C22×C53C22×Q811yescharacteristic
90C22×C53C2×#4111yescharacteristic
91C22×C532−1+412yes—
92C2×D53—24no—
93C2×D53—28no—
94C2×D53—42no—
95C2×D53—42no—
96C2×D53—44no—
97C4×C53—24no—
98C4×C53—42no—
99C4×C53—42no—
100C4×C53C22⋊Q814yescomplemented
101Dic53—22no—
102Dic53—24no—
103Dic53—42no—
104Dic53#12714yes—
105#2542—531nomaximal, Sylow
106C2×C53C2×C22⋊Q811yescharacteristic
107C2×C53C2×#12711yescharacteristic
108C2×C53#53611yescharacteristic
109C2×C53#54712yes—
110C2×C53D4×Q812yes—
111D53—44no—
112C2×C4⋊Q8—531no—
113C2×C22⋊Q8—534no—
114C2×C4×D4—531no—
115#399—534no—
116#390—531no—
117#390—531no—
118#384—531no—
119#384D5311yescharacteristic, complemented
120C4×C4⋊C4—531no—
121C53#254211yescharacteristic, complemented, Sylow
122C22×Q8—532no—
123C22×D4—531no—
124C23×C4—532no—
125C4⋊Q8—1064no—
126C22⋊Q8—10616no—
127C4×D4—1064no—
128C2×C4⋊C4—532no—
129C2×C4⋊C4—534no—
130C2×C4⋊C4—532no—
131C2×C4⋊C4—531no—
132C2×C4⋊C4C2×D5311yescharacteristic
133C2×C4⋊C4C2×D5314yes—
134C2×C4⋊C4C2×D5311yescharacteristic
135C2×C22⋊C4—532no—
136C2×C22⋊C4—534no—
137C2×C22⋊C4—534no—
138C2×C42—531no—
139C2×C42—531no—
140C2×C42C2×D5311yescharacteristic
141#95—532no—
142C24—532no—
143C2×Q8—1068no—
144C2×Q8—1064no—
145C2×D4—534no—
146C2×D4—1062no—
147C2×D4—1062no—
148C22×C4—531no—
149C22×C4—531no—
150C22×C4—532no—
151C22×C4—532no—
152C22×C4—1064no—
153C22×C4—1068no—
154C22×C4C22×D5312yes—
155C22×C4C22×D5312yes—
156C22×C4C22×D5312yes—
157C22×C4C22×D5311yescharacteristic
158C4⋊C4—22no—
159C4⋊C4—28no—
160C4⋊C4—22no—
161C4⋊C4—538no—
162C4⋊C4—1068no—
163C4⋊C4—1064no—
164C4⋊C4—1062no—
165C22⋊C4—1064no—
166C22⋊C4—1068no—
167C22⋊C4—1068no—
168C42—22no—
169C42—534no—
170C42—1062no—
171C23—534no—
172C23—1064no—
173C23—1062no—
174C23—1062no—
175C23—1064no—
176C23C23×D5311yescharacteristic
177Q8—2128no—
178D4—1068no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—24no—
183C2×C4—24no—
184C2×C4—22no—
185C2×C4—532no—
186C2×C4—534no—
187C2×C4—532no—
188C2×C4—534no—
189C2×C4—1062no—
190C2×C4—1062no—
191C2×C4—1062no—
192C2×C4—1064no—
193C2×C4—1062no—
194C2×C4—2128no—
195C2×C4#1171994512yes—
196C2×C4D4×D5314yes—
197C2×C4D4×D5312yes—
198C2×C4Q8×D5312yes—
199C22—1068no—
200C22—1064no—
201C22—2124no—
202C22—2122no—
203C22—2122no—
204C22C2×D4×D5312yes—
205C22C2×D4×D5311yescharacteristic
206C22C2×Q8×D5311yescharacteristic
207C22C2×#1171994511yescharacteristic
208C22#1313326512yes—
209C4—24no—
210C4—42no—
211C4—42no—
212C4—534no—
213C4—1064no—
214C4—1062no—
215C4—2122no—
216C4#1313319514yescomplemented
217C2—2124no—
218C2C2×#1313319511yescharacteristic
219C2#127×D5311yescharacteristic
220C2#1681119811yescharacteristic
221C2#1681127412yes—
222C2#1681128812yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#384C2
hypercenter#384D53
hypocenterC53#2542
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #384

Complements

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

NH ≅ G/Ntype
1. C2×#13133216C2semidirect
2. C2×#13133195C2semidirect
4. #16810558C2semidirect
7. #16810492C2semidirect
8. C4×#11719919C2semidirect
9. C53×#384C2semidirect
41. C2×D212Q8semidirect
56. #11719919D4semidirect
100. C4×C53C22⋊Q8semidirect
119. #384D53semidirect
121. C53#2542semidirect
216. C4#13133195semidirect