Finite Group Databasebeta

← #27297468

Subgroups — #27297468

Order 13568.

318 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 318 conjugacy classes of subgroups — too many to draw. Only its 59 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#20276927C212yesmaximal, complemented
2#20276290C211yesmaximal, characteristic, complemented
3#20272700C212yesmaximal, complemented
4#20272666C211yesmaximal, characteristic, complemented
5C53×#3253C211yesmaximal, characteristic
6#16810672C2211yescharacteristic
7#16810445—22no—
8#16810436—22no—
9#16810405C2212yes—
10#16810395—21no—
11#16810383—21no—
12#16810380C2211yescharacteristic
13C53×#532—21no—
14C53×#476—22no—
15C53×#469—21no—
16C53×#469C2212yes—
17C53×#459—22no—
18C53×#423C2211yescharacteristic
19C2×#11719917—21no—
20#13133176—21no—
21#13133157—22no—
22#13133153—22no—
23C2×#41×C53—41no—
24C2×D8×C53—21no—
25C2×D8×C53D412yes—
26C2×M16×C53D411yescharacteristic
27C2×M16×C53C2311yescharacteristic
28C4⋊Q8×C53—21no—
29#127×C53—21no—
30#124×C53—41no—
31#121×C53—22no—
32#121×C53—41no—
33C4×D4×C53—22no—
34C4×D4×C53—41no—
35C2×C42×C53D411yescharacteristic
36#106×C53—21no—
37#106×C53D412yes—
38C4⋊C8×C53—22no—
39#102×C53—22no—
40#102×C53—22no—
41#41×C53—84no—
42C2×Q8×C53—41no—
43C2×D4×C53—22no—
44C2×D4×C53—41no—
45C2×D4×C53—41no—
46C2×D4×C53—42no—
47C22×C4×C53—21no—
48C22×C4×C53—42no—
49C22×C4×C53C2×D411yescharacteristic
50D8×C53—42no—
51D8×C53—81no—
52M16×C53—41no—
53M16×C53—41no—
54C2×C8×C53—21no—
55C2×C8×C53C2×D412yes—
56C4⋊C4×C53—22no—
57C4⋊C4×C53—41no—
58C22⋊C4×C53—42no—
59C22⋊C4×C53—42no—
60C42×C53—21no—
61C42×C53SD1611yescharacteristic
62C42×C53SD1611yescharacteristic
63C22⋊Dic53—41no—
64C4⋊Dic53—41no—
65C4×Dic53—41no—
66#11719917—41no—
67#11719917—41no—
68C2×#88322—41no—
69#11719908—42no—
70C23×C53—42no—
71C23×C53AGL(1,ℤ/8)11yescharacteristic
72Q8×C53—82no—
73D4×C53—42no—
74D4×C53—42no—
75D4×C53—82no—
76D4×C53—82no—
77D4×C53—84no—
78C2×C4×C53—21no—
79C2×C4×C53—21no—
80C2×C4×C53—21no—
81C2×C4×C53—21no—
82C2×C4×C53—42no—
83C2×C4×C53—42no—
84C2×C4×C53—84no—
85C2×C4×C53#12011yescharacteristic
86C2×C4×C53C2×SD1611yescharacteristic
87C2×Dic53—81no—
88C8×C53—22no—
89C8×C53—41no—
90#88322—81no—
91#12307—531nomaximal, Sylow
92C22×C53—21no—
93C22×C53—41no—
94C22×C53—82no—
95C22×C53—82no—
96C22×C53—82no—
97C22×C53#45011yescharacteristic
98C4×C53—22no—
99C4×C53—21no—
100C4×C53—41no—
101C4×C53—82no—
102C4×C53#45411yescharacteristic
103C4×C53#45811yescharacteristic
104Dic53—161no—
105#3253D5311yescharacteristic
106#2238—532no—
107#2102—531no—
108#1339—532no—
109#1327—531no—
110C2×C53—21no—
111C2×C53—41no—
112C2×C53—162no—
113C2×C53#161611yescharacteristic
114#532—21no—
115#476—22no—
116#469—21no—
117#469C2×D5312yes—
118#459—22no—
119#430—531no—
120#423C2×D5311yescharacteristic
121#365—1062no—
122#361—1062no—
123#349—532no—
124#344—1061no—
125#339—1061no—
126H(3,ℤ/4)—531no—
127C53#1230711yescharacteristic, complemented, Sylow
128C2×#41—41no—
129C2×D8—21no—
130C2×D8C22⋊D5312yes—
131C2×M16—1061no—
132C2×M16C22⋊D5311yescharacteristic
133C2×M16C22×D5311yescharacteristic
134C4⋊Q8—21no—
135#127—21no—
136#124—41no—
137#121—22no—
138#121—41no—
139C4×D4—22no—
140C4×D4—41no—
141#117—1061no—
142C2×C42C22⋊D5311yescharacteristic
143M32—1062no—
144#108—1062no—
145#106—21no—
146#106C22⋊D5312yes—
147C4⋊C8—22no—
148#102—22no—
149#102—22no—
150#41—84no—
151C2×Q8—41no—
152C2×D4—22no—
153C2×D4—41no—
154C2×D4—41no—
155C2×D4—42no—
156C22×C4—21no—
157C22×C4—42no—
158C22×C4C2×C22⋊D5311yescharacteristic
159D8—42no—
160D8—81no—
161M16—41no—
162M16—41no—
163M16—2121no—
164M16—2121no—
165C2×C8—21no—
166C2×C8—2121no—
167C2×C8C2×C22⋊D5312yes—
168C4⋊C4—22no—
169C4⋊C4—41no—
170C4⋊C4—2121no—
171C22⋊C4—42no—
172C22⋊C4—42no—
173C22⋊C4—2121no—
174C42—21no—
175C42—2121no—
176C42#1171992311yescharacteristic
177C42#1171992311yescharacteristic
178C16—2122no—
179C23—42no—
180C23#1313324311yescharacteristic
181Q8—82no—
182D4—42no—
183D4—42no—
184D4—82no—
185D4—82no—
186D4—84no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—42no—
192C2×C4—42no—
193C2×C4—84no—
194C2×C4—4241no—
195C2×C4#1313318611yescharacteristic
196C2×C4C2×#1171992311yescharacteristic
197C8—22no—
198C8—41no—
199C8—4241no—
200C22—21no—
201C22—41no—
202C22—82no—
203C22—82no—
204C22—82no—
205C22#1681077411yescharacteristic
206C4—22no—
207C4—21no—
208C4—41no—
209C4—82no—
210C4—8481no—
211C4#1681079611yescharacteristic
212C4#1681081311yescharacteristic
213C2—21no—
214C2—41no—
215C2—162no—
216C2#2027375211yescharacteristic
subgroupG/H
centerC2#20273752
commutator, cosocleC2×M16×C53C23
FrattiniC2×M16C22×D53
Fitting, prime radicalC53×#3253C2
socleC2×C53#1616
hypercenter#3253D53
hypocenterC53#12307
LCSG ▷ C2×M16×C53 ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ #3253

Complements

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

NH ≅ G/Ntype
1. #20276927C2semidirect
2. #20276290C2semidirect
3. #20272700C2semidirect
4. #20272666C2semidirect
127. C53#12307semidirect

Maximal subgroups

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

subgroupindex
#202769272
#202762902
#202727002
#202726662
C53×#32532
#1230753