Finite Group Databasebeta

← #27304905

Subgroups — #27304905

Order 13568.

200 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 51 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#20277035C211yesmaximal, characteristic, complemented
2#20277028C211yesmaximal, characteristic
3#20276973C211yesmaximal, characteristic, complemented
4#20276595C211yesmaximal, characteristic, complemented
5#20272912C211yesmaximal, characteristic, complemented
6#20272907C211yesmaximal, characteristic, complemented
7C53×#1419C211yesmaximal, characteristic, complemented
8#16811055C2211yescharacteristic
9C2×Dic424C2211yescharacteristic
10C2×#13133161—21no—
11#16811006—21no—
12#16810739—21no—
13Dic848—21no—
14#16810462—21no—
15#16810461—21no—
16#16810458—21no—
17#16810457C2211yescharacteristic
18#16810429—21no—
19C16×Dic53C2211yescharacteristic
20C2×D16×C53C2211yescharacteristic
21C53×M64C2211yescharacteristic
22C53×#368C2211yescharacteristic
23#13133239—21no—
24C2×Dic212—21no—
25C2×#11719923—41no—
26#13133225—41no—
27#13133204—41no—
28Dic424—21no—
29Dic424—21no—
30Dic424D411yescharacteristic
31#13133162—21no—
32#13133161—42no—
33D848D411yescharacteristic
34C2×#11719908D411yescharacteristic
35#13133143—41no—
36C8×Dic53D411yescharacteristic
37#13133123—21no—
38C2×D8×C53—21no—
39D16×C53—21no—
40D16×C53—21no—
41C2×C16×C53C2311yescharacteristic
42#108×C53—21no—
43C32×C53—21no—
44C2×D4×C53—41no—
45#11719945—41no—
46C2×Dic106—41no—
47D8×C53—41no—
48D8×C53—41no—
49M16×C53—41no—
50C2×C8×C53C2×D411yescharacteristic
51C22⋊Dic53—82no—
52#11719923—81no—
53#11719923—81no—
54C4×Dic53D811yescharacteristic
55C2×#88322D811yescharacteristic
56Dic212—21no—
57Dic212—41no—
58Dic212—41no—
59D424—21no—
60#11719913—41no—
61#11719912—81no—
62C8×D53—81no—
63C16×C53C2×D411yescharacteristic
64C16×C53#4111yescharacteristic
65#11719908—21no—
66C23×C53—81no—
67D4×C53—81no—
68D4×C53—81no—
69C2×C4×C53C2×D811yescharacteristic
70C22⋊D53—81no—
71C2×Dic53—21no—
72C2×Dic53—81no—
73D212—41no—
74C4×D53—81no—
75Dic106—41no—
76Dic106—81no—
77Dic106—81no—
78C8×C53—81no—
79C8×C53#12111yescharacteristic
80C8×C53AGL(1,ℤ/8)11yescharacteristic
81#88322D1611yescharacteristic
82#88322D1611yescharacteristic
83#13691—531nomaximal, Sylow
84C22×C53—162no—
85C22×C53—161no—
86C22×C53#50911yescharacteristic
87C2×D53—81no—
88C4×C53#45911yescharacteristic
89C4×C53C2×D1611yescharacteristic
90Dic53—41no—
91Dic53—81no—
92Dic53—161no—
93#2264—531no—
94#2263—531no—
95#2249—531no—
96#2178—531no—
97#1422—531no—
98#1420—531no—
99#1419D5311yescharacteristic, complemented
100C2×C53—21no—
101C2×C53—321no—
102C2×C53#220611yescharacteristic
103D53—161no—
104#508—531no—
105C2×Q32—531no—
106C2×SD32—1061no—
107C2×D16C2×D5311yescharacteristic
108#495—1061no—
109#443—1061no—
110Q64—1061no—
111SD64—1061no—
112SD64—1061no—
113D32—1061no—
114M64—531no—
115M64C2×D5311yescharacteristic
116#368C2×D5311yescharacteristic
117#359—1061no—
118C4×C16—531no—
119C53#1369111yescharacteristic, complemented, Sylow
120#135—1061no—
121C2×Q16—1061no—
122C2×SD16—2121no—
123C2×D8—21no—
124#131—2121no—
125#124—2121no—
126Q32—531no—
127Q32—1061no—
128Q32—1061no—
129SD32—1061no—
130SD32—2122no—
131D16—21no—
132D16—21no—
133D16—531no—
134C2×C16—531no—
135C2×C16C22×D5311yescharacteristic
136#108—21no—
137#104—2121no—
138C4×C8—531no—
139C32—21no—
140C32—1061no—
141#41—2121no—
142C2×Q8—2121no—
143C2×D4—41no—
144Q16—1061no—
145Q16—2121no—
146Q16—2121no—
147SD16—4241no—
148SD16—4241no—
149SD16—2121no—
150D8—41no—
151D8—41no—
152D8—1061no—
153M16—41no—
154M16—4241no—
155C2×C8—531no—
156C2×C8—4241no—
157C2×C8D4×D5311yescharacteristic
158C22⋊C4—4242no—
159C42—531no—
160C16—1061no—
161C16#1171994511yescharacteristic
162C16D4×D5311yescharacteristic
163C23—81no—
164Q8—4241no—
165Q8—4241no—
166Q8—2121no—
167D4—81no—
168D4—81no—
169D4—4241no—
170D4—2121no—
171C2×C4—1061no—
172C2×C4—4241no—
173C2×C4—4241no—
174C2×C4D8×D5311yescharacteristic
175C8—81no—
176C8—531no—
177C8—531no—
178C8#1313319011yescharacteristic
179C8#1313324411yescharacteristic
180C22—162no—
181C22—161no—
182C22—4241no—
183C22#1681106011yescharacteristic
184C4—8481no—
185C4—4241no—
186C4—2121no—
187C4#1681081711yescharacteristic
188C4D16×D5311yescharacteristic
189C2—21no—
190C2—321no—
191C2—8481no—
192C2#2027674911yescharacteristic
subgroupG/H
centerC2#20276749
commutator, cosocleC2×C16×C53C23
FrattiniC2×C16C22×D53
FittingC53×#1419C2
socleC2×C53#2206
prime radical#20277028C2
hypercenter#1419D53
hypocenterC53#13691
LCSG ▷ C2×C16×C53 ▷ C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ C2×C16 ◁ #1419

Complements

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

NH ≅ G/Ntype
1. #20277035C2semidirect
3. #20276973C2semidirect
4. #20276595C2semidirect
5. #20272912C2semidirect
6. #20272907C2semidirect
7. C53×#1419C2semidirect
99. #1419D53semidirect
119. C53#13691semidirect

Maximal subgroups

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

subgroupindex
#202770352
#202770282
#202769732
#202765952
#202729122
#202729072
C53×#14192
#1369153