Finite Group Databasebeta

← #27297456

Subgroups — #27297456

Order 13568.

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

normal subgroups only

This group has 262 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
2#20276295C211yesmaximal, characteristic
3#20272709C212yesmaximal
4#20272668C211yesmaximal, characteristic
5C53×#3261C211yesmaximal, characteristic
6#16810672C2211yescharacteristic
7#16810445—22no—
8#16810439—22no—
9#16810405C2212yes—
10#16810393—21no—
11#16810388—21no—
12#16810381C2211yescharacteristic
13C53×#541—21no—
14C53×#479—22no—
15C53×#470C2212yes—
16C53×#467—21no—
17C53×#462—22no—
18C53×#424C2211yescharacteristic
19C2×#11719917—21no—
20C2×#11719917—21no—
21#13133157—22no—
22#13133153—22no—
23C2×Q16×C53—21no—
24C2×Q16×C53D412yes—
25C2×M16×C53C2311yescharacteristic
26C22×C8×C53D411yescharacteristic
27C4⋊Q8×C53—21no—
28#126×C53—42no—
29#125×C53—21no—
30#125×C53—42no—
31C22⋊Q8×C53—22no—
32C4×Q8×C53—22no—
33#117×C53D411yescharacteristic
34#107×C53—21no—
35#106×C53D412yes—
36C4⋊C8×C53—22no—
37#103×C53—22no—
38#103×C53—22no—
39C2×Q8×C53—22no—
40C22×C4×C53C2×D411yescharacteristic
41Q16×C53—42no—
42Q16×C53—81no—
43M16×C53—41no—
44C2×C8×C53—21no—
45C2×C8×C53—41no—
46C2×C8×C53C2×D412yes—
47C4⋊C4×C53—21no—
48C4⋊C4×C53—22no—
49C4⋊C4×C53—42no—
50C4⋊C4×C53—42no—
51C4⋊C4×C53—42no—
52C4⋊C4×C53—42no—
53C22⋊C4×C53—42no—
54C22⋊C4×C53SD1612yes—
55C42×C53—21no—
56C42×C53—42no—
57#11719917—41no—
58#11719917—41no—
59#11719917—41no—
60#11719917—41no—
61C2×#88322—41no—
62C2×#88322—41no—
63#11719908—42no—
64C23×C53C2×SD1611yescharacteristic
65Q8×C53—42no—
66Q8×C53—42no—
67C2×C4×C53—21no—
68C2×C4×C53—21no—
69C2×C4×C53—22no—
70C2×C4×C53—42no—
71C2×C4×C53—42no—
72C2×C4×C53#12011yescharacteristic
73C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
74C8×C53—22no—
75C8×C53—41no—
76#88322—81no—
77#88322—81no—
78#12305—531nomaximal, Sylow
79C22×C53—21no—
80C22×C53—41no—
81C22×C53#45011yescharacteristic
82C4×C53—21no—
83C4×C53—42no—
84C4×C53—82no—
85C4×C53—82no—
86C4×C53#45411yescharacteristic
87C4×C53#45811yescharacteristic
88#3261D5311yescharacteristic
89#2239—532no—
90#2103—531no—
91#1342—532no—
92#1328—531no—
93C2×C53—21no—
94C2×C53—41no—
95C2×C53#161611yescharacteristic
96#541—21no—
97#479—22no—
98#470C2×D5312yes—
99#467—21no—
100#462—22no—
101#430—531no—
102#424C2×D5311yescharacteristic
103#365—1062no—
104#362—1062no—
105#349—532no—
106#344—1061no—
107#341—1061no—
108#338—531no—
109C53#1230511yescharacteristic, complemented, Sylow
110C2×Q16—21no—
111C2×Q16C22⋊D5312yes—
112C2×M16—1061no—
113C2×M16—1061no—
114C2×M16C22×D5311yescharacteristic
115C22×C8C22⋊D5311yescharacteristic
116C4⋊Q8—21no—
117#126—42no—
118#125—21no—
119#125—42no—
120C22⋊Q8—22no—
121C4×Q8—22no—
122#117C22⋊D5311yescharacteristic
123M32—1062no—
124#108—1062no—
125#107—21no—
126#106C22⋊D5312yes—
127C4⋊C8—22no—
128#103—22no—
129#103—22no—
130C2×Q8—22no—
131C22×C4C2×C22⋊D5311yescharacteristic
132Q16—42no—
133Q16—81no—
134M16—41no—
135M16—2121no—
136M16—2121no—
137M16—2121no—
138M16—2121no—
139C2×C8—21no—
140C2×C8—41no—
141C2×C8—2121no—
142C2×C8—2121no—
143C2×C8C2×C22⋊D5312yes—
144C4⋊C4—21no—
145C4⋊C4—22no—
146C4⋊C4—42no—
147C4⋊C4—42no—
148C4⋊C4—42no—
149C4⋊C4—42no—
150C22⋊C4—42no—
151C22⋊C4#1171992312yes—
152C42—21no—
153C42—42no—
154C16—2122no—
155C23C2×#1171992311yescharacteristic
156Q8—42no—
157Q8—42no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—22no—
161C2×C4—42no—
162C2×C4—42no—
163C2×C4#1313318611yescharacteristic
164C2×C4#1313324311yescharacteristic
165C8—22no—
166C8—41no—
167C8—4241no—
168C8—4241no—
169C22—21no—
170C22—41no—
171C22#1681077411yescharacteristic
172C4—21no—
173C4—42no—
174C4—82no—
175C4—82no—
176C4#1681079611yescharacteristic
177C4#1681081311yescharacteristic
178C2—21no—
179C2—41no—
180C2#2027375211yescharacteristic
subgroupG/H
centerC2#20273752
commutator, cosocleC2×M16×C53C23
FrattiniC2×M16C22×D53
FittingC53×#3261C2
socleC2×C53#1616
prime radicalC23×C53C2×SD16
hypercenter#3261D53
hypocenterC53#12305
LCSG ▷ C2×M16×C53 ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ #3261

Complements

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

NH ≅ G/Ntype
109. C53#12305semidirect

Maximal subgroups

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

subgroupindex
#202769322
#202762952
#202727092
#202726682
C53×#32612
#1230553