Finite Group Databasebeta

← #27298494

Subgroups — #27298494

Order 13568.

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

normal subgroups only

This group has 196 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#20276956C211yesmaximal, characteristic, complemented
2#20276892C211yesmaximal, characteristic, complemented
3#20276788C211yesmaximal, characteristic
4#20276556C211yesmaximal, characteristic, complemented
5#20272782C211yesmaximal, characteristic, complemented
6#20272777C211yesmaximal, characteristic, complemented
7C53×#1357C211yesmaximal, characteristic, complemented
8#16811059—21no—
9#16811041—21no—
10#16811025C2211yescharacteristic
11#16810741C2211yescharacteristic
12#16810446—21no—
13#16810444C2211yescharacteristic
14#16810436—21no—
15#16810434—21no—
16#16810424—21no—
17#16810424—21no—
18#16810408—21no—
19#16810399C2211yescharacteristic
20C53×#493C2211yescharacteristic
21C53×#364C2211yescharacteristic
22C53×#364C2211yescharacteristic
23#13133239D411yescharacteristic
24#13133223—21no—
25#13133209—21no—
26#13133161—41no—
27#13133158—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×D8×C53—21no—
38C2×D8×C53—21no—
39#127×C53—21no—
40M32×C53—21no—
41M32×C53—21no—
42C4×C8×C53C2311yescharacteristic
43C2×D4×C53—41no—
44C2×D4×C53—81no—
45#11719945—21no—
46D8×C53—41no—
47D8×C53—41no—
48D8×C53—41no—
49D8×C53—41no—
50C2×C8×C53C2×D411yescharacteristic
51C2×C8×C53C2×D411yescharacteristic
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—
66C23×C53—81no—
67D4×C53—81no—
68D4×C53—81no—
69D4×C53—162no—
70C2×C4×C53—21no—
71C2×C4×C53#12111yescharacteristic
72C22⋊D53—41no—
73C2×Dic53—81no—
74D212—21no—
75C4×D53—41no—
76Dic106—21no—
77C8×C53C2×SD1611yescharacteristic
78C8×C53#13511yescharacteristic
79C8×C53#13511yescharacteristic
80C8×C53AGL(1,ℤ/8)11yescharacteristic
81#88322—41no—
82#12465—531nomaximal, Sylow
83C22×C53—162no—
84C22×C53—161no—
85C22×C53#47211yescharacteristic
86C2×D53—41no—
87C4×C53—41no—
88C4×C53#46311yescharacteristic
89C4×C53#46511yescharacteristic
90Dic53—41no—
91Dic53—161no—
92#2244—531no—
93#2231—531no—
94#2213—531no—
95#2171—531no—
96#1369—531no—
97#1366—531no—
98#1357D5311yescharacteristic, complemented
99C2×C53—21no—
100C2×C53—321no—
101C2×C53#167711yescharacteristic
102D53—81no—
103#509—1061no—
104#504—1061no—
105#499—531no—
106#493C2×D5311yescharacteristic
107#443—531no—
108#365—1061no—
109#364—531no—
110#364C2×D5311yescharacteristic
111#364C2×D5311yescharacteristic
112#361—1061no—
113#360—1061no—
114#357—1061no—
115#357—1061no—
116#350—1061no—
117#346—531no—
118C53#1246511yescharacteristic, complemented, Sylow
119#135—531no—
120C2×D8—21no—
121C2×D8—21no—
122#131—1061no—
123#127—21no—
124#125—1061no—
125SD32—2121no—
126D16—2121no—
127M32—21no—
128M32—21no—
129M32—1061no—
130M32—2121no—
131C2×C16—531no—
132C2×C16—531no—
133C2×C16—2121no—
134#108—531no—
135#107—1061no—
136#106—1061no—
137#104—1061no—
138C4×C8C22×D5311yescharacteristic
139#41—1061no—
140C2×D4—41no—
141C2×D4—81no—
142Q16—531no—
143SD16—1061no—
144D8—41no—
145D8—41no—
146D8—41no—
147D8—41no—
148D8—531no—
149M16—1061no—
150M16—2121no—
151C2×C8—2121no—
152C2×C8D4×D5311yescharacteristic
153C2×C8D4×D5311yescharacteristic
154C4⋊C4—4241no—
155C4⋊C4—2121no—
156C42#1171994511yescharacteristic
157C16—41no—
158C16—41no—
159C16—1061no—
160C16—1061no—
161C16—2121no—
162C23—81no—
163Q8—1061no—
164D4—81no—
165D4—81no—
166D4—162no—
167D4—1061no—
168D4—2121no—
169C2×C4—21no—
170C2×C4—4241no—
171C2×C4—2121no—
172C2×C4#1313319011yescharacteristic
173C8—2121no—
174C8SD16×D5311yescharacteristic
175C8#1313323611yescharacteristic
176C8#1313323811yescharacteristic
177C8#1313324111yescharacteristic
178C22—162no—
179C22—161no—
180C22—2121no—
181C22#1681089211yescharacteristic
182C4—41no—
183C4—8481no—
184C4—2121no—
185C4#1681083711yescharacteristic
186C4#1681085111yescharacteristic
187C2—21no—
188C2—321no—
189C2—4241no—
190C2#2027412311yescharacteristic
subgroupG/H
centerC2#20274123
commutator, cosocleC4×C8×C53C23
FrattiniC4×C8C22×D53
FittingC53×#1357C2
socleC2×C53#1677
prime radical#20276788C2
hypercenter#1357D53
hypocenterC53#12465
LCSG ▷ C4×C8×C53 ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ #1357

Complements

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

NH ≅ G/Ntype
1. #20276956C2semidirect
2. #20276892C2semidirect
4. #20276556C2semidirect
5. #20272782C2semidirect
6. #20272777C2semidirect
7. C53×#1357C2semidirect
98. #1357D53semidirect
118. C53#12465semidirect

Maximal subgroups

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

subgroupindex
#202769562
#202768922
#202767882
#202765562
#202727822
#202727772
C53×#13572
#1246553