Finite Group Databasebeta

← #20280485

Subgroups — #20280485

Order 6784.

528 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 157 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×#13133215C211yesmaximal, characteristic, complemented
2C2×#13133196C211yesmaximal, characteristic
3C2×#13133176C211yesmaximal, characteristic
4#16810556C214yesmaximal, complemented
5#16810551C214yesmaximal
6#16810530C211yesmaximal, characteristic, complemented
7#16810528C211yesmaximal, characteristic, complemented
8#16810488C211yesmaximal, characteristic, complemented
9C2×C22⋊Q8×C53C211yesmaximal, characteristic
10C22×Dic106C2211yescharacteristic
11#13133215—24no—
12#13133196—24no—
13#13133176—24no—
14C2×C4⋊Dic53C2211yescharacteristic
15C2×C4⋊Dic53C2212yes—
16C2×#11719919C2214yes—
17C2×#11719919C2212yes—
18C2×C22⋊Dic53C2212yes—
19C2×C22⋊Dic53C2212yes—
20C2×C4×Dic53C2211yescharacteristic
21C2×C4×Dic53C2211yescharacteristic
22#13133124C2214yes—
23#13133124C2214yes—
24#13133124C2214yes—
25C22×Q8×C53C2211yescharacteristic
26C23×C4×C53C2211yescharacteristic
27C22⋊Q8×C53—24no—
28C2×C4⋊C4×C53C2211yescharacteristic
29C2×C4⋊C4×C53C2212yes—
30C2×C22⋊C4×C53C2212yes—
31C24×C53C2311yescharacteristic
32C2×Q8×C53—22no—
33C2×Q8×C53—22no—
34C2×Q8×C53—22no—
35C22×C4×C53—22no—
36C22×C4×C53—22no—
37C22×C4×C53D414yes—
38C22×C4×C53C2312yes—
39C22×C4×C53C2312yes—
40C22×C4×C53C2311yescharacteristic
41C22×C4×C53C2311yescharacteristic
42C22×Dic53C2312yes—
43C22×Dic53C2312yes—
44C22×Dic53C2312yes—
45C22×Dic53C2312yes—
46C2×Dic106—22no—
47C2×Dic106—22no—
48C2×Dic106—22no—
49C4⋊C4×C53—24no—
50C4⋊C4×C53—22no—
51C22⋊C4×C53—24no—
52C22⋊Dic53—24no—
53C22⋊Dic53—24no—
54C4⋊Dic53—22no—
55C4⋊Dic53—24no—
56#11719919—24no—
57#11719919—28no—
58C4×Dic53—22no—
59C4×Dic53D414yes—
60C23×C53—21no—
61C23×C53—21no—
62C23×C53—22no—
63C23×C53—22no—
64C23×C53C2×D412yes—
65C23×C53C2411yescharacteristic
66Q8×C53—44no—
67C2×C4×C53—21no—
68C2×C4×C53—21no—
69C2×C4×C53—28no—
70C2×C4×C53—22no—
71C2×C4×C53—22no—
72C2×C4×C53—22no—
73C2×C4×C53—22no—
74C2×C4×C53—24no—
75C2×C4×C53C2×D412yes—
76C2×C4×C53C2×D412yes—
77C2×C4×C53C2×D412yes—
78C2×C4×C53#4112yes—
79C2×Dic53—22no—
80C2×Dic53—22no—
81C2×Dic53—22no—
82C2×Dic53—22no—
83C2×Dic53—22no—
84C2×Dic53—22no—
85C2×Dic53—22no—
86C2×Dic53—22no—
87C2×Dic53—22no—
88C2×Dic53—24no—
89C2×Dic53C2×D414yes—
90Dic106—44no—
91C22×C53—22no—
92C22×C53—22no—
93C22×C53—22no—
94C22×C53—42no—
95C22×C53—42no—
96C22×C53C22×D411yescharacteristic
97C22×C53C22×D411yescharacteristic
98C22×C53C2×#4111yescharacteristic
99C22×C532−1+412yes—
100C22×C532−1+412yes—
101C4×C53—22no—
102C4×C53—42no—
103C4×C53—42no—
104C4×C53#12114yes—
105Dic53—24no—
106Dic53—42no—
107Dic53—42no—
108Dic53—42no—
109#2621—531nomaximal, Sylow
110C2×C53—42no—
111C2×C53C2×#12111yescharacteristic
112C2×C53#53611yescharacteristic
113C2×C53#53612yes—
114C2×C53#53611yescharacteristic
115C2×C53#54112yes—
116C2×C4⋊Q8—531no—
117C2×C22⋊Q8—531no—
118C2×C22⋊Q8D5311yescharacteristic
119C2×#117—531no—
120#398—534no—
121#397—534no—
122#391—531no—
123#391—531no—
124#382—531no—
125C53#262111yescharacteristic, complemented, Sylow
126C22×Q8—531no—
127C22×Q8C2×D5311yescharacteristic
128C23×C4C2×D5311yescharacteristic
129C4⋊Q8—1064no—
130C22⋊Q8—24no—
131C22⋊Q8—1064no—
132#117—1064no—
133C2×C4⋊C4—532no—
134C2×C4⋊C4—532no—
135C2×C4⋊C4—531no—
136C2×C4⋊C4—534no—
137C2×C4⋊C4C2×D5311yescharacteristic
138C2×C4⋊C4C2×D5312yes—
139C2×C22⋊C4—532no—
140C2×C22⋊C4—532no—
141C2×C22⋊C4C2×D5312yes—
142C2×C42—531no—
143C2×C42—531no—
144#95—534no—
145#95—534no—
146#95—534no—
147C24C22×D5311yescharacteristic
148C2×Q8—22no—
149C2×Q8—22no—
150C2×Q8—22no—
151C2×Q8—1062no—
152C2×Q8—1062no—
153C2×Q8—1062no—
154C22×C4—22no—
155C22×C4—22no—
156C22×C4—532no—
157C22×C4—532no—
158C22×C4—532no—
159C22×C4—532no—
160C22×C4C22⋊D5314yes—
161C22×C4C22×D5312yes—
162C22×C4C22×D5312yes—
163C22×C4C22×D5311yescharacteristic
164C22×C4C22×D5311yescharacteristic
165C4⋊C4—22no—
166C4⋊C4—24no—
167C4⋊C4—1064no—
168C4⋊C4—1068no—
169C4⋊C4—1064no—
170C4⋊C4—1062no—
171C22⋊C4—24no—
172C22⋊C4—1064no—
173C22⋊C4—1064no—
174C42—534no—
175C42—1062no—
176C23—21no—
177C23—21no—
178C23—22no—
179C23—22no—
180C23C2×C22⋊D5312yes—
181C23C23×D5311yescharacteristic
182Q8—44no—
183Q8—2124no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—28no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—24no—
192C2×C4—534no—
193C2×C4—1062no—
194C2×C4—1062no—
195C2×C4—1062no—
196C2×C4—1062no—
197C2×C4—1062no—
198C2×C4—1062no—
199C2×C4—1062no—
200C2×C4—1062no—
201C2×C4—1062no—
202C2×C4—1064no—
203C2×C4D4×D5312yes—
204C2×C4#1171994712yes—
205C2×C4C2×C22⋊D5312yes—
206C2×C4C2×C22⋊D5312yes—
207C22—22no—
208C22—22no—
209C22—22no—
210C22—42no—
211C22—42no—
212C22C2×D4×D5311yescharacteristic
213C22C22×C22⋊D5311yescharacteristic
214C22C2×#1171994711yescharacteristic
215C22#1313326412yes—
216C22#1313326512yes—
217C4—22no—
218C4—42no—
219C4—42no—
220C4—1064no—
221C4—2122no—
222C4—2122no—
223C4—2122no—
224C4#1313318914yes—
225C2—42no—
226C2C2×#1313318911yescharacteristic
227C2#1681119411yescharacteristic
228C2#1681119612yes—
229C2#1681119711yescharacteristic
230C2#1681123312yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC2×C22⋊Q8×C53C2
hypercenterC2×C22⋊Q8D53
hypocenterC53#2621
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×#13133215C2semidirect
4. #16810556C2semidirect
6. #16810530C2semidirect
7. #16810528C2semidirect
8. #16810488C2semidirect
125. C53#2621semidirect