Finite Group Databasebeta

← #27298343

Subgroups — #27298343

Order 13568.

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

normal subgroups only

This group has 224 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#20276971C211yesmaximal, characteristic, complemented
2#20276797C211yesmaximal, characteristic, complemented
3#20276792C211yesmaximal, characteristic
4#20276553C211yesmaximal, characteristic, complemented
5#20272753C211yesmaximal, characteristic, complemented
6#20272751C211yesmaximal, characteristic, complemented
7C53×#1373C211yesmaximal, characteristic, complemented
8#16811066—21no—
9#16811061—21no—
10C2×#13133162—21no—
11C2×#13133162—21no—
12#16811039—21no—
13#16811003C2211yescharacteristic
14#16811000C2211yescharacteristic
15#16810741C2211yescharacteristic
16#16810444C2211yescharacteristic
17#16810440—21no—
18#16810437—21no—
19#16810406—21no—
20C53×#364C2211yescharacteristic
21C53×#364C2211yescharacteristic
22C4×C16×C53C2211yescharacteristic
23#13133239D411yescharacteristic
24C2×Dic212—21no—
25C2×Dic212—21no—
26C2×D424—21no—
27C2×D424—21no—
28#13133223—21no—
29#13133217—21no—
30#13133214—21no—
31Dic424—41no—
32#13133162—42no—
33#13133162—41no—
34#13133162—41no—
35#13133162—41no—
36#13133162—41no—
37D848—41no—
38#13133157—21no—
39#13133156—41no—
40C16×D53—41no—
41#13133153D411yescharacteristic
42#13133145—21no—
43M32×C53—21no—
44M32×C53—21no—
45C2×C16×C53D411yescharacteristic
46C2×C16×C53D411yescharacteristic
47C4×C8×C53C2311yescharacteristic
48#11719945—21no—
49C2×D212—41no—
50C2×D212—81no—
51C2×Dic106—41no—
52C2×C8×C53C2×D411yescharacteristic
53C2×C8×C53C2×D411yescharacteristic
54C42×C53#4111yescharacteristic
55C4⋊Dic53—81no—
56#11719917—21no—
57Dic212—41no—
58Dic212—41no—
59Dic212—41no—
60Dic212—41no—
61Dic212D811yescharacteristic
62D424—41no—
63D424—41no—
64D424—41no—
65D424—41no—
66D424D811yescharacteristic
67#11719913—21no—
68#11719912—41no—
69C8×D53—41no—
70C16×C53—21no—
71C16×C53—41no—
72C16×C53—41no—
73C16×C53D811yescharacteristic
74C16×C53D811yescharacteristic
75#11719908—41no—
76C22×D53—81no—
77C2×C4×C53—21no—
78C2×C4×C53#12111yescharacteristic
79C22⋊D53—41no—
80C2×Dic53—81no—
81D212—21no—
82D212—81no—
83D212—81no—
84D212—162no—
85C4×D53—41no—
86Dic106—21no—
87Dic106—81no—
88Dic106—81no—
89C8×C53C2×D811yescharacteristic
90C8×C53C2×D811yescharacteristic
91C8×C53#13511yescharacteristic
92C8×C53AGL(1,ℤ/8)11yescharacteristic
93#88322—41no—
94#12438—531nomaximal, Sylow
95C22×C53#47211yescharacteristic
96C2×D53—41no—
97C2×D53—162no—
98C2×D53—161no—
99C4×C53—22no—
100C4×C53#45911yescharacteristic
101C4×C53#46611yescharacteristic
102Dic53—41no—
103Dic53—161no—
104#2248—531no—
105#2214—531no—
106#2213—531no—
107#2170—531no—
108#1373D5311yescharacteristic, complemented
109#1358—531no—
110#1357—531no—
111C2×C53—21no—
112C2×C53#166711yescharacteristic
113D53—81no—
114D53—321no—
115#510—1061no—
116#509—1061no—
117C2×SD32—1061no—
118C2×SD32—1061no—
119#504—1061no—
120#494—531no—
121#493—531no—
122#443—531no—
123#364—531no—
124#364C2×D5311yescharacteristic
125#364C2×D5311yescharacteristic
126#362—1061no—
127#361—1061no—
128#350—1061no—
129C4×C16C2×D5311yescharacteristic
130C53#1243811yescharacteristic, complemented, Sylow
131#135—531no—
132C2×Q16—1061no—
133C2×Q16—1061no—
134C2×D8—1061no—
135C2×D8—1061no—
136#131—1061no—
137C4⋊Q8—1061no—
138#127—1061no—
139Q32—2121no—
140SD32—2122no—
141SD32—2121no—
142SD32—2121no—
143SD32—2121no—
144SD32—2121no—
145D16—2121no—
146M32—21no—
147M32—21no—
148M32—1061no—
149M32—2121no—
150C2×C16—2121no—
151C2×C16D21211yescharacteristic
152C2×C16D21211yescharacteristic
153#108—531no—
154#104—1061no—
155C4×C8C22×D5311yescharacteristic
156#41—1061no—
157C2×Q8—2121no—
158C2×D4—4241no—
159C2×D4—2121no—
160Q16—531no—
161Q16—2121no—
162Q16—2121no—
163Q16—2121no—
164Q16—2121no—
165SD16—1061no—
166D8—531no—
167D8—2121no—
168D8—2121no—
169D8—2121no—
170D8—2121no—
171M16—1061no—
172M16—2121no—
173C2×C8—2121no—
174C2×C8C2×D21211yescharacteristic
175C2×C8D4×D5311yescharacteristic
176C4⋊C4—4241no—
177C42#1171994911yescharacteristic
178C16—21no—
179C16—41no—
180C16—41no—
181C16—2121no—
182C16D42411yescharacteristic
183C16D42411yescharacteristic
184C23—4241no—
185Q8—1061no—
186Q8—4241no—
187Q8—4241no—
188D4—8482no—
189D4—1061no—
190D4—4241no—
191D4—4241no—
192D4—2121no—
193C2×C4—21no—
194C2×C4—4241no—
195C2×C4—2121no—
196C2×C4#1313319211yescharacteristic
197C8—2121no—
198C8D8×D5311yescharacteristic
199C8C2×D42411yescharacteristic
200C8#1313324011yescharacteristic
201C8#1313324511yescharacteristic
202C22—8482no—
203C22—8481no—
204C22—2121no—
205C22#1681089411yescharacteristic
206C4—22no—
207C4—8481no—
208C4—2121no—
209C4#1681081911yescharacteristic
210C4#1681086011yescharacteristic
211C2—21no—
212C2—16961no—
213C2—4241no—
214C2#2027407111yescharacteristic
subgroupG/H
centerC2#20274071
commutator, cosocleC4×C8×C53C23
FrattiniC4×C8C22×D53
FittingC53×#1373C2
socleC2×C53#1667
prime radical#20276792C2
hypercenter#1373D53
hypocenterC53#12438
LCSG ▷ C4×C8×C53 ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ #1373

Complements

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

NH ≅ G/Ntype
1. #20276971C2semidirect
2. #20276797C2semidirect
4. #20276553C2semidirect
5. #20272753C2semidirect
6. #20272751C2semidirect
7. C53×#1373C2semidirect
108. #1373D53semidirect
130. C53#12438semidirect

Maximal subgroups

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

subgroupindex
#202769712
#202767972
#202767922
#202765532
#202727532
#202727512
C53×#13732
#1243853