Finite Group Databasebeta

← #27297439

Subgroups — #27297439

Order 13568.

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

normal subgroups only

This group has 310 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#20276932C212yesmaximal, complemented
2#20276268C211yesmaximal, characteristic
3#20272703C212yesmaximal, complemented
4#20272653C211yesmaximal, characteristic
5C53×#3202C211yesmaximal, characteristic
6#16810672C2211yescharacteristic
7#16810649—21no—
8#16810640—21no—
9#16810445—22no—
10#16810439—22no—
11#16810405C2212yes—
12#16810380C2211yescharacteristic
13C53×#536—21no—
14C53×#485—22no—
15C53×#470C2212yes—
16C53×#467—21no—
17C53×#451—22no—
18C53×#408C2211yescharacteristic
19C2×#11719917—21no—
20#13133176—21no—
21#13133157—22no—
22#13133153—22no—
23#13133146—42no—
24#13133144—42no—
25C2×#41×C53D411yescharacteristic
26C22×Q8×C53—21no—
27C2×Q16×C53—21no—
28C2×Q16×C53D412yes—
29C2×M16×C53C2311yescharacteristic
30C22×C8×C53D411yescharacteristic
31C4⋊Q8×C53—41no—
32#124×C53—41no—
33#123×C53—41no—
34#123×C53—41no—
35C22⋊Q8×C53—22no—
36C22⋊Q8×C53—22no—
37#117×C53—21no—
38#107×C53—21no—
39#106×C53D412yes—
40#103×C53—22no—
41#103×C53—22no—
42C22⋊C8×C53—22no—
43#41×C53—41no—
44#41×C53—41no—
45C2×Q8×C53—22no—
46C2×Q8×C53—41no—
47C2×Q8×C53—81no—
48C2×Q8×C53—81no—
49C2×Q8×C53SD1611yescharacteristic
50C2×D4×C53—21no—
51C2×D4×C53SD1611yescharacteristic
52C22×C4×C53—21no—
53C22×C4×C53—41no—
54C22×C4×C53C2×D411yescharacteristic
55Q16×C53—42no—
56Q16×C53—81no—
57M16×C53—41no—
58C2×C8×C53—21no—
59C2×C8×C53—41no—
60C2×C8×C53C2×D412yes—
61C4⋊C4×C53—22no—
62C4⋊C4×C53—42no—
63C4⋊C4×C53—42no—
64C22⋊C4×C53—41no—
65C22⋊C4×C53—42no—
66C22⋊C4×C53—42no—
67C42×C53—41no—
68C22⋊Dic53—41no—
69C4⋊Dic53—41no—
70C4×Dic53—22no—
71#11719917—41no—
72#11719917—41no—
73C2×#88322—41no—
74#11719908—42no—
75C23×C53—21no—
76C23×C53AGL(1,ℤ/8)11yescharacteristic
77Q8×C53—41no—
78Q8×C53—42no—
79Q8×C53—42no—
80Q8×C53—82no—
81D4×C53—41no—
82D4×C53—81no—
83C2×C4×C53—21no—
84C2×C4×C53—21no—
85C2×C4×C53—21no—
86C2×C4×C53—41no—
87C2×C4×C53—41no—
88C2×C4×C53—42no—
89C2×C4×C53—42no—
90C2×C4×C53—82no—
91C2×C4×C53#12011yescharacteristic
92C2×C4×C53C2×SD1611yescharacteristic
93C2×Dic53—42no—
94C8×C53—22no—
95C8×C53—41no—
96#88322—81no—
97#12302—531nomaximal, Sylow
98C22×C53—21no—
99C22×C53—41no—
100C22×C53—42no—
101C22×C53—41no—
102C22×C53#45011yescharacteristic
103C4×C53—21no—
104C4×C53—41no—
105C4×C53—82no—
106C4×C53—82no—
107C4×C53#45311yescharacteristic
108C4×C53#45811yescharacteristic
109Dic53—82no—
110#3202D5311yescharacteristic
111#2239—532no—
112#2098—531no—
113#1340—532no—
114#1321—531no—
115C2×C53—21no—
116C2×C53—41no—
117C2×C53—81no—
118C2×C53#161311yescharacteristic
119#536—21no—
120#485—22no—
121#470C2×D5312yes—
122#467—21no—
123#451—22no—
124#430—531no—
125#424—1061no—
126#421—1061no—
127#408C2×D5311yescharacteristic
128#365—1062no—
129#362—1062no—
130#349—532no—
131H(3,ℤ/4)—531no—
132C53#1230211yescharacteristic, complemented, Sylow
133C2×#41C22⋊D5311yescharacteristic
134C22×Q8—21no—
135C2×Q16—21no—
136C2×Q16C22⋊D5312yes—
137C2×M16—1061no—
138C2×M16C22×D5311yescharacteristic
139C22×C8C22⋊D5311yescharacteristic
140C4⋊Q8—41no—
141#124—41no—
142#123—41no—
143#123—41no—
144C22⋊Q8—22no—
145C22⋊Q8—22no—
146#117—21no—
147#117—1061no—
148M32—1062no—
149#108—1062no—
150#107—21no—
151#106C22⋊D5312yes—
152C4⋊C8—2122no—
153#104—2122no—
154#103—22no—
155#103—22no—
156C22⋊C8—22no—
157#41—41no—
158#41—41no—
159C2×Q8—22no—
160C2×Q8—41no—
161C2×Q8—81no—
162C2×Q8—81no—
163C2×Q8#1171992311yescharacteristic
164C2×D4—21no—
165C2×D4#1171992311yescharacteristic
166C22×C4—21no—
167C22×C4—41no—
168C22×C4C2×C22⋊D5311yescharacteristic
169Q16—42no—
170Q16—81no—
171M16—41no—
172M16—2121no—
173M16—2121no—
174C2×C8—21no—
175C2×C8—41no—
176C2×C8—2121no—
177C2×C8C2×C22⋊D5312yes—
178C4⋊C4—22no—
179C4⋊C4—42no—
180C4⋊C4—42no—
181C4⋊C4—2121no—
182C22⋊C4—41no—
183C22⋊C4—42no—
184C22⋊C4—42no—
185C22⋊C4—2121no—
186C42—41no—
187C42—1062no—
188C16—2122no—
189C23—21no—
190C23#1313324311yescharacteristic
191Q8—41no—
192Q8—42no—
193Q8—42no—
194Q8—82no—
195D4—41no—
196D4—81no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—82no—
205C2×C4—2122no—
206C2×C4#1313318611yescharacteristic
207C2×C4C2×#1171992311yescharacteristic
208C8—22no—
209C8—41no—
210C8—4241no—
211C22—21no—
212C22—41no—
213C22—42no—
214C22—41no—
215C22#1681077411yescharacteristic
216C4—21no—
217C4—41no—
218C4—82no—
219C4—82no—
220C4—4242no—
221C4#1681079111yescharacteristic
222C4#1681081311yescharacteristic
223C2—21no—
224C2—41no—
225C2—81no—
226C2#2027373511yescharacteristic
subgroupG/H
centerC2#20273735
commutator, cosocleC2×M16×C53C23
FrattiniC2×M16C22×D53
FittingC53×#3202C2
socleC2×C53#1613
prime radicalC2×#41×C53D4
hypercenter#3202D53
hypocenterC53#12302
LCSG ▷ C2×M16×C53 ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ #3202

Complements

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

NH ≅ G/Ntype
1. #20276932C2semidirect
3. #20272703C2semidirect
132. C53#12302semidirect

Maximal subgroups

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

subgroupindex
#202769322
#202762682
#202727032
#202726532
C53×#32022
#1230253