Finite Group Databasebeta

← #20276659

Subgroups — #20276659

Order 6784.

226 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 47 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#16811420C211yesmaximal, characteristic, complemented
2#16811059C211yesmaximal, characteristic, complemented
3#16810873C211yesmaximal, characteristic, complemented
4#16810434C211yesmaximal, characteristic, complemented
5#16810433C211yesmaximal, characteristic, complemented
6#16810404C211yesmaximal, characteristic, complemented
7C53×#509C211yesmaximal, characteristic, complemented
8#41×D53—21no—
9#13133249—21no—
10#13133246—21no—
11#13133244—21no—
12#13133241—21no—
13#13133239C2211yescharacteristic
14#13133237—21no—
15C2×#11719912C2211yescharacteristic
16#13133188—21no—
17#13133161—21no—
18#13133158—21no—
19#13133157C2211yescharacteristic, complemented
20#13133149C2211yescharacteristic, complemented
21#13133138—21no—
22#135×C53C2211yescharacteristic
23C2×D8×C53C2211yescharacteristic, complemented
24SD32×C53—21no—
25D16×C53—21no—
26M32×C53C2211yescharacteristic, complemented
27#41×C53—21no—
28C2×D4×C53—21no—
29C2×C22⋊D53—41no—
30#11719949—21no—
31Q8×D53—21no—
32#11719947—21no—
33#11719947—41no—
34D4×D53—21no—
35D4×D53—41no—
36#11719945—21no—
37#11719945—41no—
38C2×C4×D53—41no—
39C2×C4×D53D411yescharacteristic
40Q16×C53D411yescharacteristic, complemented
41SD16×C53—21no—
42D8×C53—21no—
43D8×C53—21no—
44D8×C53D411yescharacteristic, complemented
45C2×C8×C53C2311yescharacteristic
46C22⋊Dic53—41no—
47#11719925—21no—
48#11719924—21no—
49#11719923—21no—
50#11719922—21no—
51C4⋊Dic53—21no—
52C2×#88322D411yescharacteristic
53Dic212D411yescharacteristic, complemented
54D424D411yescharacteristic, complemented
55#11719913—21no—
56#11719912—21no—
57#11719912—21no—
58C16×C53—21no—
59#11719908—21no—
60C23×C53—41no—
61C22×D53—41no—
62C22×D53D811yescharacteristic
63Q8×C53—21no—
64D4×C53—21no—
65D4×C53—41no—
66D4×C53—41no—
67D4×C53—41no—
68C2×C4×C53—41no—
69C2×C4×C53C2×D411yescharacteristic
70C22⋊D53—41no—
71C22⋊D53—41no—
72C22⋊D53—41no—
73C22⋊D53—82no—
74C2×Dic53—41no—
75C2×Dic53—41no—
76C2×Dic53D811yescharacteristic
77D212—21no—
78D212—41no—
79C4×D53—21no—
80C4×D53—21no—
81C4×D53—41no—
82C4×D53—41no—
83C4×D53—41no—
84Dic106—21no—
85Dic106—41no—
86C8×C53C2×D411yescharacteristic
87C8×C53C2×D411yescharacteristic
88#88322—21no—
89C22×C53—41no—
90C22×C53—81no—
91C22×C53—81no—
92C22×C53—81no—
93C22×C53C2×D811yescharacteristic
94C2×D53—21no—
95C2×D53—41no—
96C2×D53—41no—
97C2×D53—82no—
98C4×C53—41no—
99C4×C53#12011yescharacteristic
100C4×C53AGL(1,ℤ/8)11yescharacteristic
101Dic53—21no—
102Dic53—41no—
103Dic53—81no—
104#2190—531nomaximal, Sylow
105C2×C53—21no—
106C2×C53—81no—
107C2×C53—161no—
108C2×C53#44711yescharacteristic
109D53—41no—
110D53—81no—
111#575—531no—
112#509—531no—
113#509D5311yescharacteristic, complemented
114#469—531no—
115#360—531no—
116#360—531no—
117#349—531no—
118C53#219011yescharacteristic, complemented, Sylow
119C2×#41—1061no—
120#137—1061no—
121#137—1061no—
122AGL(1,ℤ/8)—1061no—
123AGL(1,ℤ/8)—1061no—
124#135—531no—
125#135—1061no—
126#135C2×D5311yescharacteristic
127C2×D8C2×D5311yescharacteristic, complemented
128C2×M16—531no—
129#121—1061no—
130SD32—21no—
131SD32—1061no—
132D16—21no—
133D16—1061no—
134M32—531no—
135M32C2×D5311yescharacteristic, complemented
136#106—531no—
137#102—1061no—
138#41—21no—
139#41—1061no—
140#41—1061no—
141#41—1061no—
142#41—2121no—
143#41—2121no—
144C2×Q8—1061no—
145C2×D4—21no—
146C2×D4—1061no—
147C2×D4—2121no—
148C2×D4—2121no—
149C22×C4—531no—
150C22×C4—2121no—
151Q16—531no—
152Q16—1061no—
153Q16C22⋊D5311yescharacteristic, complemented
154SD16—21no—
155SD16—1061no—
156SD16—1061no—
157SD16—1061no—
158D8—21no—
159D8—21no—
160D8—531no—
161D8—1061no—
162D8C22⋊D5311yescharacteristic, complemented
163M16—1061no—
164M16—1061no—
165C2×C8—531no—
166C2×C8C22×D5311yescharacteristic
167C4⋊C4—1061no—
168C22⋊C4—2121no—
169C16—21no—
170C16—1061no—
171C23—41no—
172C23—531no—
173C23—2121no—
174Q8—21no—
175Q8—1061no—
176Q8—2121no—
177D4—21no—
178D4—41no—
179D4—41no—
180D4—41no—
181D4—1061no—
182D4—4242no—
183D4—2121no—
184D4—2121no—
185D4—2121no—
186D4—2121no—
187C2×C4—41no—
188C2×C4—531no—
189C2×C4—1061no—
190C2×C4—1061no—
191C2×C4—2121no—
192C2×C4—2121no—
193C2×C4—2121no—
194C2×C4—2121no—
195C2×C4—2121no—
196C2×C4D4×D5311yescharacteristic
197C8—1061no—
198C8D4×D5311yescharacteristic
199C8C2×C22⋊D5311yescharacteristic
200C22—41no—
201C22—81no—
202C22—81no—
203C22—81no—
204C22—1061no—
205C22—4242no—
206C22—2121no—
207C22—2121no—
208C22D8×D5311yescharacteristic
209C4—41no—
210C4—1061no—
211C4—4241no—
212C4—2121no—
213C4#1313318511yescharacteristic
214C4#1313324111yescharacteristic
215C2—21no—
216C2—81no—
217C2—161no—
218C2—4241no—
219C2—2121no—
220C2#1681075511yescharacteristic
subgroupG/H
centerC2#16810755
commutatorC2×C8×C53C23
FrattiniC2×C8C22×D53
FittingC53×#509C2
socleC2×C53#447
hypercenter#509D53
hypocenterC53#2190
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #509

Complements

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

NH ≅ G/Ntype
1. #16811420C2semidirect
2. #16811059C2semidirect
3. #16810873C2semidirect
4. #16810434C2semidirect
5. #16810433C2semidirect
6. #16810404C2semidirect
7. C53×#509C2semidirect
19. #13133157C22semidirect
20. #13133149C22semidirect
23. C2×D8×C53C22semidirect
26. M32×C53C22semidirect
40. Q16×C53D4semidirect
44. D8×C53D4semidirect
53. Dic212D4semidirect
54. D424D4semidirect
113. #509D53semidirect
118. C53#2190semidirect
127. C2×D8C2×D53semidirect
135. M32C2×D53semidirect
153. Q16C22⋊D53semidirect
162. D8C22⋊D53semidirect