Finite Group Databasebeta

← #20275893

Subgroups — #20275893

Order 6784.

278 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 63 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×#13133131C211yesmaximal, characteristic
3C2×#13133131C211yesmaximal, characteristic
4C2×#13133131C211yesmaximal, characteristic
5#16810390C211yesmaximal, characteristic, complemented
6#16810389C211yesmaximal, characteristic, complemented
7C53×#342C211yesmaximal, characteristic, complemented
8C22×C22⋊D53C2211yescharacteristic
9#13133196—22no—
10#13133194—22no—
11#13133194—22no—
12C2×C22⋊Dic53C2211yescharacteristic
13C2×C22⋊Dic53C2211yescharacteristic
14C2×C22⋊Dic53C2211yescharacteristic
15#13133131—22no—
16#13133131—22no—
17#13133131—22no—
18C2×C22⋊C4×C53C2211yescharacteristic
19C2×C22⋊C4×C53C2211yescharacteristic
20C2×C22⋊C4×C53C2211yescharacteristic
21C24×C53C2311yescharacteristic
22C23×D53Q811yescharacteristic, complemented
23C22×C4×C53—21no—
24C22×C4×C53—21no—
25C22×C4×C53—21no—
26C2×C22⋊D53—21no—
27C2×C22⋊D53—21no—
28C2×C22⋊D53—21no—
29C2×C22⋊D53—21no—
30C2×C22⋊D53—21no—
31C2×C22⋊D53—21no—
32C22×Dic53—21no—
33C22×Dic53—21no—
34C22×Dic53—21no—
35C22×Dic53Q811yescharacteristic, complemented
36C2×Dic106—22no—
37C22⋊C4×C53—21no—
38C22⋊C4×C53—21no—
39C22⋊C4×C53—21no—
40C22⋊C4×C53—21no—
41C22⋊C4×C53—21no—
42C22⋊C4×C53D412yes—
43C22⋊Dic53—21no—
44C22⋊Dic53—21no—
45C22⋊Dic53—21no—
46C22⋊Dic53—21no—
47C22⋊Dic53D412yes—
48C22⋊Dic53D412yes—
49C4⋊Dic53—22no—
50#11719919—24no—
51C23×C53—81no—
52C23×C53C2×D411yescharacteristic
53C23×C53C2×D411yescharacteristic
54C23×C53C2×D411yescharacteristic
55C23×C53C2×Q811yescharacteristic
56C23×C53#4111yescharacteristic
57C23×C53#4111yescharacteristic
58C23×C53#4111yescharacteristic
59C22×D53—21no—
60C22×D53—21no—
61C22×D53—21no—
62C22×D53—81no—
63C2×C4×C53—22no—
64C2×C4×C53—41no—
65C2×C4×C53—41no—
66C2×C4×C53—81no—
67C2×C4×C53—81no—
68C2×C4×C53—81no—
69C22⋊D53—41no—
70C22⋊D53—41no—
71C22⋊D53—41no—
72C22⋊D53—41no—
73C2×Dic53—22no—
74C2×Dic53—22no—
75C2×Dic53—21no—
76C2×Dic53—21no—
77C2×Dic53—21no—
78C2×Dic53—41no—
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#12011yescharacteristic
91C22×C53C22⋊Q811yescharacteristic
92C22×C53C22⋊Q811yescharacteristic
93C22×C53C22⋊Q811yescharacteristic
94C22×C53#12411yescharacteristic
95C22×C53#12411yescharacteristic
96C22×C53#12411yescharacteristic
97C2×D53—41no—
98C2×D53—81no—
99C2×D53—81no—
100C2×D53—81no—
101C4×C53—42no—
102C4×C53—81no—
103C4×C53—81no—
104Dic53—42no—
105Dic53—42no—
106Dic53—41no—
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#39311yescharacteristic
117D53—81no—
118#543—531no—
119C2×#99—531no—
120C2×#99—531no—
121C2×#99—531no—
122#342—531no—
123#342—531no—
124#342D5311yescharacteristic, complemented
125C53#203211yescharacteristic, complemented, Sylow
126C22×D4—531no—
127C22⋊Q8—1062no—
128C22⋊Q8—1062no—
129C22⋊Q8—1062no—
130C2×C22⋊C4—531no—
131C2×C22⋊C4—531no—
132C2×C22⋊C4—531no—
133C2×C22⋊C4C2×D5311yescharacteristic
134C2×C22⋊C4C2×D5311yescharacteristic
135C2×C22⋊C4C2×D5311yescharacteristic
136#99—1062no—
137#99—1062no—
138#99—1062no—
139C24—531no—
140C24C22×D5311yescharacteristic
141C2×Q8—1062no—
142C2×D4—1061no—
143C2×D4—1061no—
144C2×D4—1061no—
145C2×D4—1061no—
146C2×D4—1061no—
147C2×D4—1061no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—531no—
152C22×C4—1061no—
153C22×C4—1061no—
154C22×C4—1061no—
155C4⋊C4—1064no—
156C4⋊C4—1062no—
157C22⋊C4—21no—
158C22⋊C4—21no—
159C22⋊C4—21no—
160C22⋊C4—21no—
161C22⋊C4—21no—
162C22⋊C4—532no—
163C22⋊C4—532no—
164C22⋊C4—1061no—
165C22⋊C4—1061no—
166C22⋊C4—1061no—
167C22⋊C4—1061no—
168C22⋊C4D21212yes—
169C23—81no—
170C23—1061no—
171C23—1061no—
172C23—1061no—
173C23—4241no—
174C23C2×D21211yescharacteristic
175C23#1171994511yescharacteristic
176C23#1171994511yescharacteristic
177C23D4×D5311yescharacteristic
178C23D4×D5311yescharacteristic
179C23#1171994711yescharacteristic
180C23Q8×D5311yescharacteristic
181Q8—4242no—
182D4—2121no—
183D4—2121no—
184D4—2121no—
185D4—2121no—
186C2×C4—22no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—81no—
190C2×C4—81no—
191C2×C4—81no—
192C2×C4—1062no—
193C2×C4—1062no—
194C2×C4—1061no—
195C2×C4—1061no—
196C2×C4—1061no—
197C2×C4—4241no—
198C2×C4—4241no—
199C2×C4—4241no—
200C2×C4—2121no—
201C22—41no—
202C22—41no—
203C22—41no—
204C22—41no—
205C22—41no—
206C22—41no—
207C22—41no—
208C22—4241no—
209C22—4241no—
210C22—4241no—
211C22—2121no—
212C22#1313318711yescharacteristic
213C22#1313319511yescharacteristic
214C22#1313319511yescharacteristic
215C22#1313319711yescharacteristic
216C22#1313320311yescharacteristic
217C22#1313320311yescharacteristic
218C22#1313320511yescharacteristic
219C4—42no—
220C4—81no—
221C4—81no—
222C4—4241no—
223C4—2122no—
224C4—2122no—
225C4—2121no—
226C2—21no—
227C2—21no—
228C2—21no—
229C2—21no—
230C2—21no—
231C2—21no—
232C2—21no—
233C2—4241no—
234C2#1681053611yescharacteristic
subgroupG/H
centerC2#16810536
commutatorC24×C53C23
FrattiniC24C22×D53
FittingC53×#342C2
socleC2×C53#393
hypercenter#342D53
hypocenterC53#2032
LCSG ▷ C24×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C24 ◁ #342

Complements

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

NH ≅ G/Ntype
1. #16811249C2semidirect
5. #16810390C2semidirect
6. #16810389C2semidirect
7. C53×#342C2semidirect
22. C23×D53Q8semidirect
35. C22×Dic53Q8semidirect
124. #342D53semidirect
125. C53#2032semidirect