Finite Group Databasebeta

← #20275192

Subgroups — #20275192

Order 6784.

324 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 79 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×#13133243C211yesmaximal, characteristic
2#16811389C211yesmaximal, characteristic, complemented
3#16810641C211yesmaximal, characteristic, complemented
4C2×#13133134C211yesmaximal, characteristic, complemented
5#16810393C211yesmaximal, characteristic, complemented
6#16810381C211yesmaximal, characteristic, complemented
7C53×#418C211yesmaximal, characteristic, complemented
8C2×#11719945C2211yescharacteristic
9#13133243—22no—
10#13133243—22no—
11C2×#11719923C412yescomplemented
12C2×#11719922C412yescomplemented
13#13133225—22no—
14#13133223—22no—
15C2×#11719917C2211yescharacteristic
16C2×#11719917C2211yescharacteristic
17C2×#11719917C2211yescharacteristic
18M16×D53—22no—
19#13133145—22no—
20#13133134—22no—
21C22×D4×C53C2211yescharacteristic
22C2×M16×C53C2211yescharacteristic
23#117×C53C2211yescharacteristic
24#102×C53—22no—
25C24×C53—21no—
26C2×D4×C53—21no—
27C2×D4×C53—21no—
28C2×D4×C53—21no—
29C2×D4×C53—21no—
30C2×D4×C53—21no—
31C2×D4×C53C2×C412yes—
32C22×C4×C53C2311yescharacteristic
33C2×C22⋊D53—21no—
34#11719945—21no—
35#11719945—21no—
36#11719945D412yes—
37#11719945D412yes—
38C2×D212C2×C411yescharacteristic
39C2×C4×D53—21no—
40C2×Dic106C2×C411yescharacteristic
41M16×C53—21no—
42M16×C53—21no—
43C2×C8×C53D412yes—
44C4⋊C4×C53—21no—
45C22⋊C4×C53—21no—
46C42×C53—21no—
47#11719923—24no—
48#11719922—24no—
49#11719917—21no—
50#11719917—21no—
51#11719917—21no—
52#11719917—21no—
53#11719917—21no—
54#11719917D412yes—
55C2×#88322—21no—
56C2×#88322—21no—
57C2×#88322C2×C412yes—
58#11719912—24no—
59C8×D53—24no—
60C23×C53—22no—
61C23×C53—41no—
62C23×C53—41no—
63C23×C53—82no—
64C23×C53#4111yescharacteristic
65C22×D53—21no—
66D4×C53—22no—
67D4×C53—22no—
68D4×C53—41no—
69D4×C53—41no—
70C2×C4×C53—41no—
71C2×C4×C53C22×C411yescharacteristic
72C2×C4×C53C2×D411yescharacteristic
73C2×C4×C53C2×D411yescharacteristic
74C2×C4×C53C2×D411yescharacteristic
75C2×C4×C53C2×D411yescharacteristic
76C2×C4×C53#4111yescharacteristic
77C22⋊D53—22no—
78C22⋊D53—22no—
79C2×Dic53—21no—
80D212—21no—
81D212C22⋊C412yes—
82C4×D53—22no—
83C4×D53—22no—
84Dic106—21no—
85Dic106C22⋊C412yes—
86C8×C53—22no—
87#88322—22no—
88#88322—22no—
89#88322—41no—
90C22×C53—41no—
91C22×C53—42no—
92C22×C53—82no—
93C22×C53—82no—
94C22×C53—82no—
95C22×C53C4×D411yescharacteristic
96C22×C53#12111yescharacteristic
97C22×C53#12311yescharacteristic
98C2×D53—22no—
99C2×D53—81no—
100C4×C53—81no—
101C4×C53C2×C22⋊C411yescharacteristic
102C4×C53C4×D411yescharacteristic
103C4×C53#12011yescharacteristic
104C4×C53#12111yescharacteristic
105Dic53—22no—
106#1903—531nomaximal, Sylow
107C2×C53—21no—
108C2×C53—21no—
109C2×C53—21no—
110C2×C53—82no—
111C2×C53#38611yescharacteristic
112D53—42no—
113C2×AGL(1,ℤ/8)—531no—
114#568—531no—
115#421—531no—
116#418D5311yescharacteristic, complemented
117C2×#100—531no—
118#344—531no—
119#338—531no—
120C53#190311yescharacteristic, complemented, Sylow
121C2×#41—531no—
122C22×D4C2×D5311yescharacteristic
123AGL(1,ℤ/8)—1062no—
124AGL(1,ℤ/8)—1062no—
125C2×SD16—532no—
126C2×D8—532no—
127#131—1062no—
128#131—1062no—
129C2×M16—531no—
130C2×M16—531no—
131C2×M16—531no—
132C2×M16—1062no—
133C2×M16C2×D5311yescharacteristic
134#117C2×D5311yescharacteristic
135#104—1062no—
136#102—22no—
137#100—1062no—
138C24—21no—
139#41—532no—
140#41—532no—
141#41—1061no—
142#41—1061no—
143C2×Q8—531no—
144C2×D4—21no—
145C2×D4—21no—
146C2×D4—21no—
147C2×D4—21no—
148C2×D4—21no—
149C2×D4—531no—
150C2×D4—1061no—
151C2×D4C4×D5312yes—
152C22×C4—1061no—
153C22×C4C22×D5311yescharacteristic
154SD16—1064no—
155D8—1064no—
156M16—21no—
157M16—21no—
158M16—532no—
159M16—1064no—
160M16—1061no—
161M16—1061no—
162M16—1061no—
163M16—1061no—
164M16—1061no—
165C2×C8—532no—
166C2×C8—1064no—
167C2×C8—1061no—
168C2×C8—1061no—
169C2×C8C22⋊D5312yes—
170C4⋊C4—21no—
171C22⋊C4—21no—
172C42—21no—
173C23—22no—
174C23—41no—
175C23—41no—
176C23—82no—
177C23—1061no—
178C23#1171994711yescharacteristic
179Q8—532no—
180Q8—1061no—
181D4—22no—
182D4—22no—
183D4—41no—
184D4—41no—
185D4—532no—
186D4—1062no—
187D4—1062no—
188D4—1061no—
189C2×C4—41no—
190C2×C4—1061no—
191C2×C4—1062no—
192C2×C4—1062no—
193C2×C4C2×C4×D5311yescharacteristic
194C2×C4#1171994511yescharacteristic
195C2×C4D4×D5311yescharacteristic
196C2×C4D4×D5311yescharacteristic
197C2×C4D4×D5311yescharacteristic
198C2×C4C2×C22⋊D5311yescharacteristic
199C8—22no—
200C8—1062no—
201C8—1062no—
202C8—2121no—
203C22—41no—
204C22—42no—
205C22—82no—
206C22—82no—
207C22—82no—
208C22—1062no—
209C22—4241no—
210C22#1313317711yescharacteristic
211C22#1313318811yescharacteristic
212C22#1313319911yescharacteristic
213C4—81no—
214C4—1062no—
215C4C22⋊C4×D5311yescharacteristic
216C4C4×C22⋊D5311yescharacteristic
217C4#1313318511yescharacteristic
218C4#1313319011yescharacteristic
219C2—21no—
220C2—21no—
221C2—21no—
222C2—82no—
223C2—2122no—
224C2#1681049911yescharacteristic
subgroupG/H
centerC2#16810499
commutatorC2×C4×C53C22×C4
FrattiniC22×C4C22×D53
FittingC53×#418C2
socleC2×C53#386
hypercenter#418D53
hypocenterC53#1903
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
2. #16811389C2semidirect
3. #16810641C2semidirect
4. C2×#13133134C2semidirect
5. #16810393C2semidirect
6. #16810381C2semidirect
7. C53×#418C2semidirect
11. C2×#11719923C4semidirect
12. C2×#11719922C4semidirect
116. #418D53semidirect
120. C53#1903semidirect