Finite Group Databasebeta

← #20274822

Subgroups — #20274822

Order 6784.

296 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 73 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#16811096C211yesmaximal, characteristic
2C2×#13133131C211yesmaximal, characteristic
3C2×#13133131C211yesmaximal, characteristic
4#16810390C211yesmaximal, characteristic, complemented
5#16810389C211yesmaximal, characteristic, complemented
6#16810389C211yesmaximal, characteristic, complemented
7C53×#342C211yesmaximal, characteristic, complemented
8C22×C22⋊D53C2211yescharacteristic
9C4×C22⋊D53—22no—
10#13133177—22no—
11#13133173C412yescomplemented
12C2×C22⋊Dic53C2211yescharacteristic
13C2×C22⋊Dic53C2211yescharacteristic
14C2×C22⋊Dic53C2211yescharacteristic
15C22⋊C4×D53C412yescomplemented
16#13133131—22no—
17#13133131—22no—
18C2×C22⋊C4×C53C2211yescharacteristic
19C2×C22⋊C4×C53C2211yescharacteristic
20C2×C22⋊C4×C53C2211yescharacteristic
21C24×C53C2311yescharacteristic
22C23×D53C2×C411yescharacteristic
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⋊D53D412yes—
31C2×C22⋊D53D412yes—
32C22×Dic53—21no—
33C22×Dic53—21no—
34C22×Dic53—21no—
35C22×Dic53C2×C411yescharacteristic
36C2×C4×D53—22no—
37C22⋊C4×C53—21no—
38C22⋊C4×C53—21no—
39C22⋊C4×C53—21no—
40C22⋊C4×C53—21no—
41C22⋊C4×C53—21no—
42C22⋊C4×C53C2×C412yes—
43C22⋊Dic53—21no—
44C22⋊Dic53—21no—
45C22⋊Dic53—21no—
46C22⋊Dic53—21no—
47C22⋊Dic53—21no—
48C22⋊Dic53C2×C412yes—
49#11719921—22no—
50#11719919—22no—
51C4×Dic53—22no—
52C23×C53—81no—
53C23×C53C22×C411yescharacteristic
54C23×C53C2×D411yescharacteristic
55C23×C53C2×D411yescharacteristic
56C23×C53#4111yescharacteristic
57C23×C53#4111yescharacteristic
58C23×C53#4111yescharacteristic
59C23×C53#4111yescharacteristic
60C22×D53—21no—
61C22×D53—21no—
62C22×D53—81no—
63C22×D53C22⋊C412yes—
64C2×C4×C53—22no—
65C2×C4×C53—41no—
66C2×C4×C53—41no—
67C2×C4×C53—81no—
68C2×C4×C53—81no—
69C2×C4×C53—81no—
70C22⋊D53—22no—
71C22⋊D53—22no—
72C22⋊D53—22no—
73C22⋊D53—22no—
74C2×Dic53—21no—
75C2×Dic53—21no—
76C2×Dic53—22no—
77C2×Dic53—41no—
78C2×Dic53—41no—
79C2×Dic53—81no—
80C2×Dic53—81no—
81C2×Dic53—81no—
82C2×Dic53C22⋊C412yes—
83C4×D53—82no—
84C22×C53—41no—
85C22×C53—41no—
86C22×C53—41no—
87C22×C53—41no—
88C22×C53—41no—
89C22×C53—41no—
90C22×C53—41no—
91C22×C53C2×C22⋊C411yescharacteristic
92C22×C53#11711yescharacteristic
93C22×C53#11711yescharacteristic
94C22×C53#12311yescharacteristic
95C22×C53#12311yescharacteristic
96C22×C53#12311yescharacteristic
97C22×C53#12311yescharacteristic
98C2×D53—22no—
99C2×D53—42no—
100C2×D53—81no—
101C2×D53—81no—
102C4×C53—42no—
103C4×C53—81no—
104C4×C53—81no—
105Dic53—22no—
106Dic53—42no—
107Dic53—81no—
108Dic53—81no—
109#1828—531nomaximal, Sylow
110C2×C53—21no—
111C2×C53—21no—
112C2×C53—21no—
113C2×C53—21no—
114C2×C53—21no—
115C2×C53—21no—
116C2×C53—21no—
117C2×C53#38111yescharacteristic
118D53—42no—
119#518—531no—
120C2×#99—531no—
121C2×#99—531no—
122#342—531no—
123#342—531no—
124#342—531no—
125#342D5311yescharacteristic, complemented
126C53#182811yescharacteristic, complemented, Sylow
127C22×D4—531no—
128C4×D4—1062no—
129C4×D4—1062no—
130#117—532no—
131C2×C22⋊C4—531no—
132C2×C22⋊C4—532no—
133C2×C22⋊C4—531no—
134C2×C22⋊C4—531no—
135C2×C22⋊C4C2×D5311yescharacteristic
136C2×C22⋊C4C2×D5311yescharacteristic
137C2×C22⋊C4C2×D5311yescharacteristic
138#99—1062no—
139#99—1062no—
140C24—531no—
141C24C22×D5311yescharacteristic
142C2×D4—532no—
143C2×D4—532no—
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—1062no—
153C22×C4—1061no—
154C22×C4—1061no—
155C22×C4—1061no—
156C4⋊C4—1062no—
157C22⋊C4—21no—
158C22⋊C4—21no—
159C22⋊C4—21no—
160C22⋊C4—21no—
161C22⋊C4—21no—
162C22⋊C4—532no—
163C22⋊C4—1062no—
164C22⋊C4—1061no—
165C22⋊C4—1061no—
166C22⋊C4—1061no—
167C22⋊C4—1061no—
168C22⋊C4—1061no—
169C22⋊C4C4×D5312yes—
170C42—1062no—
171C23—81no—
172C23—532no—
173C23—1061no—
174C23—1061no—
175C23—4241no—
176C23C2×C4×D5311yescharacteristic
177C23#1171994511yescharacteristic
178C23#1171994511yescharacteristic
179C23D4×D5311yescharacteristic
180C23D4×D5311yescharacteristic
181C23#1171994711yescharacteristic
182C23#1171994911yescharacteristic
183D4—1062no—
184D4—1062no—
185D4—1062no—
186D4—1062no—
187C2×C4—22no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—81no—
191C2×C4—81no—
192C2×C4—81no—
193C2×C4—532no—
194C2×C4—1061no—
195C2×C4—1061no—
196C2×C4—1062no—
197C2×C4—4242no—
198C2×C4—4241no—
199C2×C4—4241no—
200C2×C4—4241no—
201C2×C4—2121no—
202C2×C4—2121no—
203C22—41no—
204C22—41no—
205C22—41no—
206C22—41no—
207C22—41no—
208C22—41no—
209C22—41no—
210C22—1062no—
211C22—4241no—
212C22—4241no—
213C22—2122no—
214C22C22⋊C4×D5311yescharacteristic
215C22#1313317411yescharacteristic
216C22#1313317511yescharacteristic
217C22#1313319911yescharacteristic
218C22#1313319911yescharacteristic
219C22#1313320211yescharacteristic
220C22#1313320211yescharacteristic
221C4—42no—
222C4—81no—
223C4—81no—
224C4—1062no—
225C4—4241no—
226C4—4241no—
227C4—2122no—
228C2—21no—
229C2—21no—
230C2—21no—
231C2—21no—
232C2—21no—
233C2—21no—
234C2—21no—
235C2—2122no—
236C2#1681048511yescharacteristic
subgroupG/H
centerC2#16810485
commutatorC23×C53C22×C4
FrattiniC24C22×D53
FittingC53×#342C2
socleC2×C53#381
hypercenter#342D53
hypocenterC53#1828
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C24 ◁ #342

Complements

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

NH ≅ G/Ntype
4. #16810390C2semidirect
5. #16810389C2semidirect
6. #16810389C2semidirect
7. C53×#342C2semidirect
11. #13133173C4semidirect
15. C22⋊C4×D53C4semidirect
125. #342D53semidirect
126. C53#1828semidirect