Finite Group Databasebeta

← #27298501

Subgroups — #27298501

Order 13568.

186 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 186 conjugacy classes of subgroups — too many to draw. Only its 53 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#20276960C211yesmaximal, characteristic, complemented
2#20276892C211yesmaximal, characteristic
3#20276793C211yesmaximal, characteristic
4#20276556C211yesmaximal, characteristic
5#20272782C211yesmaximal, characteristic, complemented
6#20272777C211yesmaximal, characteristic, complemented
7C53×#1358C211yesmaximal, characteristic, complemented
8#16811064—21no—
9#16811041—21no—
10#16811025C2211yescharacteristic
11#16810741C2211yescharacteristic
12#16810446—21no—
13#16810444C2211yescharacteristic
14#16810439—21no—
15#16810434—21no—
16#16810427—21no—
17#16810427—21no—
18#16810408—21no—
19#16810399C2211yescharacteristic
20C53×#494C2211yescharacteristic
21C53×#364C2211yescharacteristic
22C53×#364C2211yescharacteristic
23#13133239D411yescharacteristic
24#13133223—21no—
25#13133209—21no—
26#13133163—41no—
27#13133160—41no—
28#13133157—21no—
29#13133156—41no—
30C2×#11719908D411yescharacteristic
31C2×#11719908D411yescharacteristic
32C16×D53—41no—
33#13133153D411yescharacteristic
34#13133151—21no—
35#13133149—21no—
36#13133145—21no—
37C2×Q16×C53—21no—
38C2×Q16×C53—21no—
39C4⋊Q8×C53—21no—
40M32×C53—21no—
41M32×C53—21no—
42C4×C8×C53C2311yescharacteristic
43C2×Q8×C53—41no—
44#11719945—21no—
45Q16×C53—41no—
46Q16×C53—41no—
47Q16×C53—41no—
48Q16×C53—41no—
49C2×C8×C53C2×D411yescharacteristic
50C2×C8×C53C2×D411yescharacteristic
51C4⋊C4×C53—81no—
52C42×C53#4111yescharacteristic
53C4⋊Dic53—41no—
54#11719919—81no—
55#11719917—21no—
56Dic212SD1611yescharacteristic
57D424SD1611yescharacteristic
58#11719913—21no—
59#11719912—41no—
60C8×D53—41no—
61C16×C53—41no—
62C16×C53—41no—
63#11719908—21no—
64#11719908—21no—
65#11719908—41no—
66Q8×C53—81no—
67Q8×C53—81no—
68C2×C4×C53—21no—
69C2×C4×C53—81no—
70C2×C4×C53#12111yescharacteristic
71C22⋊D53—41no—
72C2×Dic53—81no—
73D212—21no—
74C4×D53—41no—
75Dic106—21no—
76C8×C53C2×SD1611yescharacteristic
77C8×C53#13511yescharacteristic
78C8×C53#13511yescharacteristic
79C8×C53AGL(1,ℤ/8)11yescharacteristic
80#88322—41no—
81#12466—531nomaximal, Sylow
82C22×C53#47211yescharacteristic
83C2×D53—41no—
84C4×C53—41no—
85C4×C53—161no—
86C4×C53#46311yescharacteristic
87C4×C53#46511yescharacteristic
88Dic53—41no—
89Dic53—161no—
90#2245—531no—
91#2231—531no—
92#2214—531no—
93#2171—531no—
94#1369—531no—
95#1366—531no—
96#1358D5311yescharacteristic, complemented
97C2×C53—21no—
98C2×C53#167711yescharacteristic
99D53—81no—
100#510—1061no—
101#504—1061no—
102#499—531no—
103#494C2×D5311yescharacteristic
104#443—531no—
105#365—1061no—
106#364—531no—
107#364C2×D5311yescharacteristic
108#364C2×D5311yescharacteristic
109#362—1061no—
110#360—1061no—
111#358—1061no—
112#358—1061no—
113#350—1061no—
114#346—531no—
115C53#1246611yescharacteristic, complemented, Sylow
116#135—531no—
117C2×Q16—21no—
118C2×Q16—21no—
119#131—1061no—
120C4⋊Q8—21no—
121#125—1061no—
122Q32—2121no—
123SD32—2121no—
124M32—21no—
125M32—21no—
126M32—1061no—
127M32—2121no—
128C2×C16—531no—
129C2×C16—531no—
130C2×C16—2121no—
131#108—531no—
132#107—1061no—
133#106—1061no—
134#104—1061no—
135C4×C8C22×D5311yescharacteristic
136#41—1061no—
137C2×Q8—41no—
138Q16—41no—
139Q16—41no—
140Q16—41no—
141Q16—41no—
142Q16—531no—
143SD16—1061no—
144D8—531no—
145M16—1061no—
146M16—2121no—
147C2×C8—2121no—
148C2×C8D4×D5311yescharacteristic
149C2×C8D4×D5311yescharacteristic
150C4⋊C4—81no—
151C4⋊C4—4241no—
152C4⋊C4—2121no—
153C42#1171994511yescharacteristic
154C16—41no—
155C16—41no—
156C16—1061no—
157C16—1061no—
158C16—2121no—
159Q8—81no—
160Q8—81no—
161Q8—1061no—
162D4—1061no—
163D4—2121no—
164C2×C4—21no—
165C2×C4—81no—
166C2×C4—4241no—
167C2×C4—2121no—
168C2×C4#1313319011yescharacteristic
169C8—2121no—
170C8SD16×D5311yescharacteristic
171C8#1313323611yescharacteristic
172C8#1313323811yescharacteristic
173C8#1313324111yescharacteristic
174C22—2121no—
175C22#1681089211yescharacteristic
176C4—41no—
177C4—161no—
178C4—8481no—
179C4—2121no—
180C4#1681083711yescharacteristic
181C4#1681085111yescharacteristic
182C2—21no—
183C2—4241no—
184C2#2027412311yescharacteristic
subgroupG/H
centerC2#20274123
commutator, cosocleC4×C8×C53C23
FrattiniC4×C8C22×D53
FittingC53×#1358C2
socleC2×C53#1677
prime radical#13133239D4
hypercenter#1358D53
hypocenterC53#12466
LCSG ▷ C4×C8×C53 ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ #1358

Complements

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

NH ≅ G/Ntype
1. #20276960C2semidirect
5. #20272782C2semidirect
6. #20272777C2semidirect
7. C53×#1358C2semidirect
96. #1358D53semidirect
115. C53#12466semidirect

Maximal subgroups

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

subgroupindex
#202769602
#202768922
#202767932
#202765562
#202727822
#202727772
C53×#13582
#1246653