Finite Group Databasebeta

← #27271634

Subgroups — #27271634

Order 13568.

198 conjugacy classes of subgroups; only the 37 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 37 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#20276002C211yesmaximal, characteristic
2#20272666C211yesmaximal, characteristic, complemented
3C53×#1309C211yesmaximal, characteristic, complemented
4#16811423C411yescharacteristic
5C2×#13133145—21no—
6#16810605—21no—
7#16810405—21no—
8#16810381C411yescharacteristic
9C53×#423—21no—
10C53×#423C2211yescharacteristic
11C2×#11719945—21no—
12#13133250—41no—
13#13133245—41no—
14#13133239—41no—
15C2×#11719917—21no—
16#13133157—41no—
17#13133145—41no—
18#13133145—41no—
19#13133135—41no—
20#13133133—41no—
21C2×M16×C53—22no—
22C2×M16×C53C2×C411yescharacteristic
23C2×M16×C53D411yescharacteristic
24C2×C42×C53D411yescharacteristic
25C4⋊C8×C53—41no—
26C4⋊C8×C53—41no—
27C4⋊C8×C53—41no—
28C22×C4×C53—21no—
29C22×C4×C53C22⋊C411yescharacteristic
30C2×C22⋊D53—41no—
31#11719945—41no—
32#11719945—41no—
33#11719945—41no—
34#11719945—41no—
35C2×D212—21no—
36C2×C4×D53—41no—
37C2×Dic106—21no—
38M16×C53—41no—
39M16×C53—41no—
40M16×C53—82no—
41C2×C8×C53—21no—
42C2×C8×C53—21no—
43C2×C8×C53—42no—
44C42×C53—21no—
45C42×C53D811yescharacteristic
46C42×C53SD1611yescharacteristic
47#11719917—41no—
48#11719917—41no—
49C2×#88322—41no—
50Dic212—41no—
51D424—41no—
52#11719913—42no—
53#11719908—81no—
54C23×C53#10411yescharacteristic
55C22×D53—41no—
56C2×C4×C53—21no—
57C2×C4×C53—21no—
58C2×C4×C53—21no—
59C2×C4×C53—21no—
60C2×C4×C53—41no—
61C2×C4×C53—41no—
62C2×C4×C53#9911yescharacteristic
63C2×C4×C53#10211yescharacteristic
64C22⋊D53—81no—
65C22⋊D53—81no—
66C2×Dic53—41no—
67D212—41no—
68D212—41no—
69C4×D53—81no—
70C4×D53—81no—
71Dic106—41no—
72Dic106—41no—
73C8×C53—41no—
74C8×C53—41no—
75C8×C53—82no—
76#88322—81no—
77#7414—531nomaximal, Sylow
78C22×C53—21no—
79C22×C53—41no—
80C22×C53#32711yescharacteristic
81C2×D53—81no—
82C2×D53—161no—
83C4×C53—21no—
84C4×C53—21no—
85C4×C53—21no—
86C4×C53—41no—
87C4×C53AΣL(1,𝔽16)11yescharacteristic
88C4×C53#35511yescharacteristic
89Dic53—81no—
90#2050—531no—
91#1327—531no—
92#1309D5311yescharacteristic, complemented
93C2×C53—21no—
94C2×C53—41no—
95C2×C53#134511yescharacteristic
96D53—161no—
97#575—531no—
98#423—21no—
99#423C2×D5311yescharacteristic
100C2×#104—1061no—
101#413—1061no—
102#349—1061no—
103#338—531no—
104C53#741411yescharacteristic, complemented, Sylow
105C2×#41—1061no—
106#137—2121no—
107AGL(1,ℤ/8)—2121no—
108#135—2121no—
109C2×M16—22no—
110C2×M16—1061no—
111C2×M16C4×D5311yescharacteristic
112C2×M16D21211yescharacteristic
113C2×C42C22⋊D5311yescharacteristic
114M32—2121no—
115C4⋊C8—41no—
116C4⋊C8—41no—
117C4⋊C8—41no—
118#104—2121no—
119#104—2121no—
120#101—2121no—
121#100—2121no—
122#41—2121no—
123#41—2121no—
124#41—2121no—
125#41—2121no—
126C2×Q8—1061no—
127C2×D4—1061no—
128C2×D4—2121no—
129C22×C4—21no—
130C22×C4—2121no—
131C22×C4#1171992111yescharacteristic
132Q16—2121no—
133SD16—2122no—
134D8—2121no—
135M16—41no—
136M16—41no—
137M16—82no—
138M16—2121no—
139M16—2121no—
140C2×C8—21no—
141C2×C8—21no—
142C2×C8—42no—
143C2×C8—2121no—
144C42—21no—
145C42#1171992211yescharacteristic
146C42#1171992411yescharacteristic
147C16—4241no—
148C23—2121no—
149C23#1313314511yescharacteristic
150Q8—2121no—
151Q8—2121no—
152D4—4241no—
153D4—4241no—
154D4—2121no—
155D4—2121no—
156C2×C4—21no—
157C2×C4—21no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—4241no—
163C2×C4—4241no—
164C2×C4—2121no—
165C2×C4#1313313111yescharacteristic
166C2×C4#1313313711yescharacteristic
167C8—41no—
168C8—41no—
169C8—82no—
170C8—4241no—
171C22—21no—
172C22—41no—
173C22—8481no—
174C22—4241no—
175C22#1681035611yescharacteristic
176C4—21no—
177C4—21no—
178C4—21no—
179C4—41no—
180C4—4241no—
181C4#1681041111yescharacteristic
182C4#1681041911yescharacteristic
183C2—21no—
184C2—41no—
185C2—8481no—
186C2#2027271711yescharacteristic
subgroupG/H
centerC2#20272717
commutatorC2×M16×C53C2×C4
Frattini#423C2×D53
FittingC53×#1309C2
socleC2×C53#1345
prime radical#16811423C4
cosocleC53×#423C22
hypercenter#1309D53
hypocenterC53#7414
LCSG ▷ C2×M16×C53 ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ #423 ◁ #1309

Complements

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

NH ≅ G/Ntype
2. #20272666C2semidirect
3. C53×#1309C2semidirect
92. #1309D53semidirect
104. C53#7414semidirect

Maximal subgroups

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

subgroupindex
#202760022
#202726662
C53×#13092
#741453