Finite Group Databasebeta

← #20275892

Subgroups — #20275892

Order 6784.

278 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 65 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#16811249C211yesmaximal, characteristic, complemented
2C2×#13133132C211yesmaximal, characteristic
3C2×#13133132C211yesmaximal, characteristic
4#16810390C211yesmaximal, characteristic, complemented
5#16810389C211yesmaximal, characteristic, complemented
6#16810389C211yesmaximal, characteristic, complemented
7C2×#99×C53C211yesmaximal, characteristic
8#13133196—22no—
9#13133194—22no—
10#13133194—22no—
11C2×C22⋊Dic53C2211yescharacteristic
12C2×C22⋊Dic53C2211yescharacteristic
13C2×C22⋊Dic53C2211yescharacteristic
14C2×C22⋊Dic53C2211yescharacteristic
15#13133132—22no—
16#13133132—22no—
17C22×D4×C53C2211yescharacteristic
18C2×C22⋊C4×C53C2211yescharacteristic
19C2×C22⋊C4×C53C2211yescharacteristic
20#99×C53—22no—
21C24×C53Q811yescharacteristic, complemented
22C24×C53C2311yescharacteristic
23C2×D4×C53—21no—
24C2×D4×C53—21no—
25C2×D4×C53—21no—
26C2×D4×C53—21no—
27C2×D4×C53—21no—
28C2×D4×C53—21no—
29C22×C4×C53—21no—
30C22×C4×C53—21no—
31C22×C4×C53Q811yescharacteristic, complemented
32C22×Dic53—21no—
33C22×Dic53—21no—
34C22×Dic53—21no—
35C22×Dic53—21no—
36C2×Dic106—22no—
37C22⋊C4×C53—21no—
38C22⋊C4×C53—21no—
39C22⋊C4×C53—21no—
40C22⋊C4×C53D412yes—
41C22⋊Dic53—21no—
42C22⋊Dic53—21no—
43C22⋊Dic53—21no—
44C22⋊Dic53—21no—
45C22⋊Dic53—21no—
46C22⋊Dic53—21no—
47C22⋊Dic53D412yes—
48C22⋊Dic53D412yes—
49C4⋊Dic53—22no—
50#11719919—24no—
51C23×C53—21no—
52C23×C53—21no—
53C23×C53—21no—
54C23×C53—81no—
55C23×C53—81no—
56C23×C53C2×D411yescharacteristic
57C23×C53C2×D411yescharacteristic
58C23×C53C2×D411yescharacteristic
59C23×C53C2×Q811yescharacteristic
60C23×C53#4111yescharacteristic
61C23×C53#4111yescharacteristic
62C23×C53#4111yescharacteristic
63D4×C53—41no—
64D4×C53—41no—
65D4×C53—41no—
66D4×C53—41no—
67C2×C4×C53—22no—
68C2×C4×C53—21no—
69C2×C4×C53—21no—
70C2×C4×C53—21no—
71C2×C4×C53—41no—
72C2×C4×C53—81no—
73C2×C4×C53—81no—
74C2×Dic53—22no—
75C2×Dic53—22no—
76C2×Dic53—41no—
77C2×Dic53—41no—
78C2×Dic53—81no—
79C2×Dic53—81no—
80C2×Dic53—81no—
81C2×Dic53—81no—
82Dic106—82no—
83C22×C53—41no—
84C22×C53—41no—
85C22×C53—41no—
86C22×C53—41no—
87C22×C53—41no—
88C22×C53—41no—
89C22×C53—41no—
90C22×C53—41no—
91C22×C53—81no—
92C22×C53—81no—
93C22×C53—81no—
94C22×C53#12011yescharacteristic
95C22×C53C22⋊Q811yescharacteristic
96C22×C53C22⋊Q811yescharacteristic
97C22×C53C22⋊Q811yescharacteristic
98C22×C53#12411yescharacteristic
99C22×C53#12411yescharacteristic
100C22×C53#12411yescharacteristic
101C4×C53—42no—
102C4×C53—41no—
103C4×C53—81no—
104Dic53—42no—
105Dic53—42no—
106Dic53—81no—
107Dic53—81no—
108#2032—531nomaximal, Sylow
109C2×C53—21no—
110C2×C53—21no—
111C2×C53—21no—
112C2×C53—21no—
113C2×C53—21no—
114C2×C53—21no—
115C2×C53—21no—
116C2×C53—81no—
117C2×C53#39311yescharacteristic
118#543—531no—
119C2×#99—531no—
120C2×#99—531no—
121C2×#99D5311yescharacteristic
122#342—531no—
123#342—531no—
124#342—531no—
125C53#203211yescharacteristic, complemented, Sylow
126C22×D4C2×D5311yescharacteristic
127C22⋊Q8—1062no—
128C22⋊Q8—1062no—
129C22⋊Q8—1062no—
130C2×C22⋊C4—531no—
131C2×C22⋊C4—531no—
132C2×C22⋊C4—531no—
133C2×C22⋊C4—531no—
134C2×C22⋊C4C2×D5311yescharacteristic
135C2×C22⋊C4C2×D5311yescharacteristic
136#99—22no—
137#99—1062no—
138#99—1062no—
139C24Dic10611yescharacteristic, complemented
140C24C22×D5311yescharacteristic
141C2×Q8—1062no—
142C2×D4—21no—
143C2×D4—21no—
144C2×D4—21no—
145C2×D4—21no—
146C2×D4—21no—
147C2×D4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—1061no—
151C22×C4—1061no—
152C22×C4—1061no—
153C22×C4—1061no—
154C22×C4Dic10611yescharacteristic, complemented
155C4⋊C4—1064no—
156C4⋊C4—1062no—
157C22⋊C4—21no—
158C22⋊C4—21no—
159C22⋊C4—21no—
160C22⋊C4—532no—
161C22⋊C4—532no—
162C22⋊C4—1061no—
163C22⋊C4—1061no—
164C22⋊C4—1061no—
165C22⋊C4—1061no—
166C22⋊C4—1061no—
167C22⋊C4—1061no—
168C22⋊C4C22⋊D5312yes—
169C23—21no—
170C23—21no—
171C23—21no—
172C23—81no—
173C23—81no—
174C23C2×Dic10611yescharacteristic
175C23#1171994511yescharacteristic
176C23D4×D5311yescharacteristic
177C23D4×D5311yescharacteristic
178C23#1171994711yescharacteristic
179C23#1171994711yescharacteristic
180C23C2×C22⋊D5311yescharacteristic
181Q8—4242no—
182D4—41no—
183D4—41no—
184D4—41no—
185D4—41no—
186C2×C4—22no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—41no—
191C2×C4—81no—
192C2×C4—81no—
193C2×C4—1062no—
194C2×C4—1062no—
195C2×C4—4241no—
196C2×C4—4241no—
197C2×C4—4241no—
198C2×C4—4241no—
199C2×C4—2121no—
200C2×C4—2121no—
201C22—41no—
202C22—41no—
203C22—41no—
204C22—41no—
205C22—41no—
206C22—41no—
207C22—41no—
208C22—41no—
209C22—81no—
210C22—81no—
211C22—81no—
212C22#1313318511yescharacteristic
213C22#1313319411yescharacteristic
214C22#1313319411yescharacteristic
215C22#1313319611yescharacteristic
216C22#1313320311yescharacteristic
217C22#1313320311yescharacteristic
218C22#1313320411yescharacteristic
219C4—42no—
220C4—41no—
221C4—81no—
222C4—4241no—
223C4—4241no—
224C4—2122no—
225C4—2122no—
226C2—21no—
227C2—21no—
228C2—21no—
229C2—21no—
230C2—21no—
231C2—21no—
232C2—21no—
233C2—81no—
234C2#1681053511yescharacteristic
subgroupG/H
centerC2#16810535
commutatorC24×C53C23
FrattiniC24C22×D53
FittingC2×#99×C53C2
socleC2×C53#393
hypercenterC2×#99D53
hypocenterC53#2032
LCSG ▷ C24×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C24 ◁ C2×#99

Complements

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

NH ≅ G/Ntype
1. #16811249C2semidirect
4. #16810390C2semidirect
5. #16810389C2semidirect
6. #16810389C2semidirect
21. C24×C53Q8semidirect
31. C22×C4×C53Q8semidirect
125. C53#2032semidirect
139. C24Dic106semidirect
154. C22×C4Dic106semidirect