Finite Group Databasebeta

← #20279666

Subgroups — #20279666

Order 6784.

676 conjugacy classes of subgroups; only the 185 normal subgroups are shown.

normal subgroups only

This group has 676 conjugacy classes of subgroups — too many to draw. Only its 185 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×#13133194C214yesmaximal, complemented
3C2×D4×Dic53C211yesmaximal, characteristic
4#16810559C214yesmaximal, complemented
5#16810526C211yesmaximal, characteristic
6#16810493C211yesmaximal, characteristic, complemented
7#16810492C211yesmaximal, characteristic, complemented
8C4×#11719919C211yesmaximal, characteristic, complemented
9C53×#390C211yesmaximal, characteristic
10C22×Dic106C2212yes—
11C23×Dic53C2212yes—
12#13133216—24no—
13#13133194—216no—
14D4×Dic53—24no—
15C2×C4⋊Dic53C2211yescharacteristic
16C2×C4⋊Dic53C2214yes—
17C2×C4⋊Dic53C2211yescharacteristic
18C2×#11719919C2212yes—
19C2×#11719919C2214yes—
20C2×#11719919C2212yes—
21C2×C22⋊Dic53C2214yes—
22C2×C22⋊Dic53C2212yes—
23C2×C4×Dic53C2211yescharacteristic
24C2×C4×Dic53C2211yescharacteristic
25#13133124C2212yes—
26C22×D4×C53C2211yescharacteristic
27C2×C4⋊C4×C53C2211yescharacteristic
28C2×C22⋊C4×C53C2214yes—
29C2×C42×C53C2211yescharacteristic
30C24×C53C2312yes—
31C2×D4×C53—22no—
32C2×D4×C53—22no—
33C2×D4×C53Q814yescomplemented
34C22×C4×C53C2312yes—
35C22×C4×C53C2312yes—
36C22×C4×C53C2311yescharacteristic
37C22×Dic53—24no—
38C22×Dic53—28no—
39C22×Dic53C2312yes—
40C22×Dic53C2312yes—
41C22×Dic53C2311yescharacteristic
42C22×Dic53C2312yes—
43C22×Dic53C2311yescharacteristic
44C2×Dic106—28no—
45C2×Dic106—24no—
46C4⋊C4×C53—22no—
47C22⋊C4×C53—28no—
48C42×C53—22no—
49C22⋊Dic53—24no—
50C22⋊Dic53—28no—
51C4⋊Dic53—22no—
52C4⋊Dic53—28no—
53C4⋊Dic53—22no—
54#11719919—28no—
55#11719919—24no—
56#11719919D418yescomplemented
57C4×Dic53—22no—
58C4×Dic53D414yes—
59C23×C53—22no—
60C23×C53—24no—
61C23×C53—22no—
62C23×C53—24no—
63C23×C53C2×Q814yes—
64C23×C53C2411yescharacteristic
65D4×C53—28no—
66C2×C4×C53—22no—
67C2×C4×C53—22no—
68C2×C4×C53—24no—
69C2×C4×C53—22no—
70C2×C4×C53C2×D414yes—
71C2×C4×C53C2×D412yes—
72C2×C4×C53C2×Q812yes—
73C2×C4×C53#4112yes—
74C2×Dic53—22no—
75C2×Dic53—22no—
76C2×Dic53—22no—
77C2×Dic53—22no—
78C2×Dic53—24no—
79C2×Dic53—24no—
80C2×Dic53—22no—
81C2×Dic53—48no—
82C2×Dic53C2×D412yes—
83C2×Dic53C2×D414yes—
84C2×Dic53C2×D412yes—
85C2×Dic53C2×D414yes—
86Dic106—48no—
87C22×C53—24no—
88C22×C53—28no—
89C22×C53—42no—
90C22×C53—42no—
91C22×C53—44no—
92C22×C53C22×D411yescharacteristic
93C22×C53C22×D412yes—
94C22×C53C22×Q811yescharacteristic
95C22×C53C2×#4111yescharacteristic
96C22×C532−1+412yes—
97C4×C53—24no—
98C4×C53—42no—
99C4×C53C22⋊Q814yescomplemented
100Dic53—22no—
101Dic53—24no—
102Dic53—42no—
103Dic53—42no—
104Dic53#12714yes—
105#2542—531nomaximal, Sylow
106C2×C53—44no—
107C2×C53C2×C22⋊Q811yescharacteristic
108C2×C53C2×#12711yescharacteristic
109C2×C53#53611yescharacteristic
110C2×C53#54712yes—
111C2×C53D4×Q812yes—
112C2×C4⋊Q8—531no—
113C2×C22⋊Q8—534no—
114C2×C4×D4—531no—
115#399—534no—
116#390—531no—
117#390D5311yescharacteristic
118#384—531no—
119#384—531no—
120C4×C4⋊C4—531no—
121C53#254211yescharacteristic, complemented, Sylow
122C22×Q8—532no—
123C22×D4C2×D5311yescharacteristic
124C23×C4—532no—
125C4⋊Q8—1064no—
126C22⋊Q8—10616no—
127C4×D4—1064no—
128C2×C4⋊C4—532no—
129C2×C4⋊C4—531no—
130C2×C4⋊C4—534no—
131C2×C4⋊C4—534no—
132C2×C4⋊C4—532no—
133C2×C4⋊C4—531no—
134C2×C4⋊C4C2×D5311yescharacteristic
135C2×C22⋊C4—532no—
136C2×C22⋊C4—534no—
137C2×C22⋊C4C2×D5314yes—
138C2×C42—531no—
139C2×C42—531no—
140C2×C42C2×D5311yescharacteristic
141#95—532no—
142C24C22×D5312yes—
143C2×Q8—1068no—
144C2×Q8—1064no—
145C2×D4—22no—
146C2×D4—22no—
147C2×D4Dic10614yescomplemented
148C22×C4—531no—
149C22×C4—531no—
150C22×C4—532no—
151C22×C4—532no—
152C22×C4—532no—
153C22×C4—1064no—
154C22×C4—1068no—
155C22×C4C22×D5312yes—
156C22×C4C22×D5312yes—
157C22×C4C22×D5311yescharacteristic
158C4⋊C4—22no—
159C4⋊C4—538no—
160C4⋊C4—1068no—
161C4⋊C4—1064no—
162C4⋊C4—1062no—
163C4⋊C4—1068no—
164C4⋊C4—1062no—
165C22⋊C4—28no—
166C22⋊C4—1064no—
167C22⋊C4—1068no—
168C42—22no—
169C42—534no—
170C42—1062no—
171C23—24no—
172C23—22no—
173C23—22no—
174C23—24no—
175C23C2×Dic10614yes—
176C23C23×D5311yescharacteristic
177Q8—2128no—
178D4—28no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—24no—
182C2×C4—22no—
183C2×C4—532no—
184C2×C4—534no—
185C2×C4—532no—
186C2×C4—534no—
187C2×C4—1062no—
188C2×C4—1062no—
189C2×C4—1062no—
190C2×C4—1062no—
191C2×C4—1064no—
192C2×C4—1064no—
193C2×C4—1062no—
194C2×C4—2128no—
195C2×C4C2×Dic10612yes—
196C2×C4D4×D5314yes—
197C2×C4D4×D5312yes—
198C2×C4#1171994712yes—
199C22—28no—
200C22—24no—
201C22—44no—
202C22—42no—
203C22—42no—
204C22C2×D4×D5312yes—
205C22C2×D4×D5311yescharacteristic
206C22C22×Dic10611yescharacteristic
207C22C2×#1171994711yescharacteristic
208C22#1313326412yes—
209C4—24no—
210C4—42no—
211C4—534no—
212C4—1064no—
213C4—1062no—
214C4—2122no—
215C4—2122no—
216C4#1313319414yescomplemented
217C2—44no—
218C2C2×#1313319411yescharacteristic
219C2#127×D5311yescharacteristic
220C2#1681119311yescharacteristic
221C2#1681127312yes—
222C2D4×Dic10612yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#390C2
hypercenter#390D53
hypocenterC53#2542
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
1. C2×#13133216C2semidirect
2. C2×#13133194C2semidirect
4. #16810559C2semidirect
6. #16810493C2semidirect
7. #16810492C2semidirect
8. C4×#11719919C2semidirect
33. C2×D4×C53Q8semidirect
56. #11719919D4semidirect
99. C4×C53C22⋊Q8semidirect
121. C53#2542semidirect
147. C2×D4Dic106semidirect
216. C4#13133194semidirect