Finite Group Databasebeta

← #27297450

Subgroups — #27297450

Order 13568.

286 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 55 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#20286154C211yesmaximal, characteristic
2#20276929C212yesmaximal, complemented
3#20272705C212yesmaximal, complemented
4#20272668C211yesmaximal, characteristic, complemented
5C53×#2103C211yesmaximal, characteristic, complemented
6#16811228—21no—
7#16810924—22no—
8#16810876C2212yes—
9#16810859—21no—
10#16810819—22no—
11#16810447—22no—
12#16810435—22no—
13#16810405C2212yes—
14C53×#430C2211yescharacteristic
15C53×#424C2211yescharacteristic
16C53×#344—21no—
17C53×#341—21no—
18C53×#338C2211yescharacteristic
19C2×D424—21no—
20C2×D424D412yes—
21#13133214—21no—
22#13133209—21no—
23#13133202—42no—
24#13133191—22no—
25#13133190—42no—
26C4×D212—22no—
27#13133157—22no—
28#13133151—21no—
29#13133149D412yes—
30#13133139—22no—
31#13133139—22no—
32C2×M16×C53—21no—
33C2×M16×C53—21no—
34C2×M16×C53C2311yescharacteristic
35C22×C8×C53D411yescharacteristic
36#117×C53D411yescharacteristic
37#108×C53—22no—
38C4⋊C8×C53—22no—
39C22×C4×C53C2×D411yescharacteristic
40C2×C22⋊D53—42no—
41C2×D212—22no—
42C2×D212—42no—
43C2×C4×D53—42no—
44M16×C53—41no—
45M16×C53—41no—
46M16×C53—41no—
47M16×C53—41no—
48M16×C53—41no—
49C2×C8×C53—21no—
50C2×C8×C53—41no—
51C2×C8×C53—41no—
52C2×C8×C53—41no—
53C2×C8×C53C2×D412yes—
54C4⋊C4×C53—21no—
55C22⋊C4×C53SD1612yes—
56C42×C53—21no—
57#11719921—42no—
58#11719921—42no—
59C4⋊Dic53—22no—
60#11719919—42no—
61D424—42no—
62D424—81no—
63#11719908—42no—
64C23×C53C2×SD1611yescharacteristic
65C22×D53—42no—
66C2×C4×C53—21no—
67C2×C4×C53—21no—
68C2×C4×C53—22no—
69C2×C4×C53#12011yescharacteristic
70C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
71C22⋊D53—162no—
72C2×Dic53—42no—
73D212—42no—
74D212—42no—
75D212—162no—
76C4×D53—162no—
77C8×C53—22no—
78C8×C53—41no—
79C8×C53—81no—
80C8×C53—81no—
81#12304—531nomaximal, Sylow
82C22×C53—21no—
83C22×C53—41no—
84C22×C53#45011yescharacteristic
85C2×D53—82no—
86C2×D53—82no—
87C2×D53—82no—
88C4×C53—21no—
89C4×C53—42no—
90C4×C53#45411yescharacteristic
91C4×C53#45811yescharacteristic
92Dic53—82no—
93#3259—531no—
94#2238—532no—
95#2103D5311yescharacteristic, complemented
96#1341—532no—
97#1328—531no—
98C2×C53—21no—
99C2×C53—41no—
100C2×C53#161611yescharacteristic
101D53—162no—
102#540—1061no—
103#478—1062no—
104#469—532no—
105#466—1061no—
106#459—1062no—
107#430C2×D5311yescharacteristic
108#424C2×D5311yescharacteristic
109#365—1062no—
110#361—1062no—
111#349—532no—
112#344—21no—
113#341—21no—
114#338C2×D5311yescharacteristic
115C53#1230411yescharacteristic, complemented, Sylow
116C2×D8—532no—
117C2×D8—1061no—
118C2×M16—21no—
119C2×M16—21no—
120C2×M16C22×D5311yescharacteristic
121C22×C8C22⋊D5311yescharacteristic
122#127—1061no—
123#125—1061no—
124#123—2122no—
125#121—1062no—
126#121—2122no—
127C4×D4—1062no—
128#117C22⋊D5311yescharacteristic
129M32—1062no—
130#108—22no—
131#107—1061no—
132#106—532no—
133C4⋊C8—22no—
134#102—1062no—
135#102—1062no—
136C2×D4—1062no—
137C2×D4—2122no—
138C2×D4—2122no—
139C22×C4—2122no—
140C22×C4C2×C22⋊D5311yescharacteristic
141D8—4241no—
142D8—2122no—
143M16—41no—
144M16—41no—
145M16—41no—
146M16—41no—
147M16—41no—
148C2×C8—21no—
149C2×C8—41no—
150C2×C8—41no—
151C2×C8—41no—
152C2×C8D4×D5312yes—
153C4⋊C4—21no—
154C4⋊C4—1062no—
155C4⋊C4—2122no—
156C22⋊C4—2122no—
157C22⋊C4—2122no—
158C22⋊C4#1171992412yes—
159C42—21no—
160C16—2122no—
161C23—2122no—
162C23C2×#1171992411yescharacteristic
163D4—8482no—
164D4—8482no—
165D4—2122no—
166D4—2122no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—22no—
170C2×C4—8482no—
171C2×C4—2122no—
172C2×C4#1313318511yescharacteristic
173C2×C4#1313324211yescharacteristic
174C8—22no—
175C8—41no—
176C8—81no—
177C8—81no—
178C22—21no—
179C22—41no—
180C22—4242no—
181C22—4242no—
182C22—4242no—
183C22#1681077311yescharacteristic
184C4—21no—
185C4—42no—
186C4—4242no—
187C4#1681079511yescharacteristic
188C4#1681081211yescharacteristic
189C2—21no—
190C2—41no—
191C2—8482no—
192C2#2027375111yescharacteristic
subgroupG/H
centerC2#20273751
commutator, cosocleC2×M16×C53C23
FrattiniC2×M16C22×D53
FittingC53×#2103C2
socleC2×C53#1616
prime radical#20286154C2
hypercenter#2103D53
hypocenterC53#12304
LCSG ▷ C2×M16×C53 ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ #2103

Complements

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

NH ≅ G/Ntype
2. #20276929C2semidirect
3. #20272705C2semidirect
4. #20272668C2semidirect
5. C53×#2103C2semidirect
95. #2103D53semidirect
115. C53#12304semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#202861542
#202769292
#202727052
#202726682
C53×#21032
#1230453