Finite Group Databasebeta

← #27304912

Subgroups — #27304912

Order 13568.

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

normal subgroups only

This group has 216 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#20277030C211yesmaximal, characteristic, complemented
2#20277028C211yesmaximal, characteristic, complemented
3#20276963C211yesmaximal, characteristic, complemented
4#20276595C211yesmaximal, characteristic, complemented
5#20272912C211yesmaximal, characteristic, complemented
6#20272904C211yesmaximal, characteristic, complemented
7C53×#1419C211yesmaximal, characteristic, complemented
8#16811055C2211yescharacteristic
9C2×D848C2211yescharacteristic
10C2×#13133158—21no—
11#16810998—21no—
12#16810739—21no—
13#16810462—21no—
14#16810461—21no—
15D1696—21no—
16#16810458—21no—
17#16810457C2211yescharacteristic
18#16810429—21no—
19C16×Dic53C2211yescharacteristic
20C2×D16×C53C2211yescharacteristic
21C53×M64C2211yescharacteristic
22C53×#368C2211yescharacteristic
23#13133239—21no—
24C2×D424—21no—
25C2×#11719922—41no—
26#13133225—41no—
27#13133213—41no—
28Dic424D411yescharacteristic, complemented
29#13133162—21no—
30D848—21no—
31D848—21no—
32D848D411yescharacteristic, complemented
33#13133158—42no—
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—
45C2×C22⋊D53—82no—
46#11719945—41no—
47C2×D212—41no—
48D8×C53—41no—
49D8×C53—41no—
50M16×C53—41no—
51C2×C8×C53C2×D411yescharacteristic
52#11719922—81no—
53#11719922—81no—
54C4×Dic53D811yescharacteristic
55C2×#88322D811yescharacteristic
56Dic212—21no—
57D424—21no—
58D424—41no—
59D424—41no—
60#11719913—41no—
61#11719912—81no—
62C8×D53—81no—
63C16×C53C2×D411yescharacteristic
64C16×C53#4111yescharacteristic
65#11719908—21no—
66C23×C53—81no—
67C22×D53—81no—
68D4×C53—81no—
69D4×C53—81no—
70C2×C4×C53C2×D811yescharacteristic
71C22⋊D53—81no—
72C22⋊D53—164no—
73C2×Dic53—21no—
74D212—41no—
75D212—81no—
76D212—81no—
77C4×D53—81no—
78Dic106—41no—
79C8×C53—81no—
80C8×C53#12111yescharacteristic
81C8×C53AGL(1,ℤ/8)11yescharacteristic
82#88322D1611yescharacteristic
83#88322D1611yescharacteristic
84#13692—531nomaximal, Sylow
85C22×C53—162no—
86C22×C53—161no—
87C22×C53#50911yescharacteristic
88C2×D53—81no—
89C2×D53—162no—
90C2×D53—161no—
91C4×C53#45911yescharacteristic
92C4×C53C2×D1611yescharacteristic
93Dic53—22no—
94Dic53—81no—
95#2263—531no—
96#2263—531no—
97#2246—531no—
98#2178—531no—
99#1422—531no—
100#1419—531no—
101#1419D5311yescharacteristic, complemented
102C2×C53—21no—
103C2×C53—321no—
104C2×C53#220611yescharacteristic
105D53—161no—
106D53—321no—
107#508—531no—
108C2×D16—531no—
109C2×D16—1061no—
110C2×D16C2×D5311yescharacteristic
111#493—1061no—
112#443—1061no—
113SD64—1061no—
114SD64—1061no—
115D32—1061no—
116D32—1061no—
117M64—531no—
118M64C2×D5311yescharacteristic
119#368C2×D5311yescharacteristic
120#359—1061no—
121C4×C16—531no—
122C53#1369211yescharacteristic, complemented, Sylow
123#135—1061no—
124C2×D8—21no—
125C2×D8—1061no—
126C2×D8—2121no—
127#131—2121no—
128#127—2121no—
129Q32—531no—
130SD32—1061no—
131D16—21no—
132D16—21no—
133D16—531no—
134D16—1061no—
135D16—1061no—
136D16—2122no—
137C2×C16—531no—
138C2×C16C22×D5311yescharacteristic
139#108—21no—
140#104—2121no—
141C4×C8—531no—
142C32—21no—
143C32—1061no—
144#41—2121no—
145C2×D4—41no—
146C2×D4—4242no—
147C2×D4—2121no—
148Q16—1061no—
149SD16—2121no—
150D8—41no—
151D8—41no—
152D8—1061no—
153D8—4241no—
154D8—4241no—
155D8—2121no—
156D8—2121no—
157M16—41no—
158M16—4241no—
159C2×C8—531no—
160C2×C8—4241no—
161C2×C8D4×D5311yescharacteristic
162C42—531no—
163C16—1061no—
164C16#1171994511yescharacteristic
165C16D4×D5311yescharacteristic
166C23—81no—
167C23—4241no—
168Q8—2121no—
169D4—81no—
170D4—81no—
171D4—8484no—
172D4—4241no—
173D4—4241no—
174D4—4241no—
175D4—2121no—
176C2×C4—1061no—
177C2×C4—4241no—
178C2×C4D8×D5311yescharacteristic
179C8—81no—
180C8—531no—
181C8—531no—
182C8#1313319011yescharacteristic
183C8#1313324411yescharacteristic
184C22—162no—
185C22—161no—
186C22—8482no—
187C22—8481no—
188C22—4241no—
189C22#1681106011yescharacteristic
190C4—1062no—
191C4—4241no—
192C4#1681081711yescharacteristic
193C4D16×D5311yescharacteristic
194C2—21no—
195C2—321no—
196C2—8481no—
197C2—16961no—
198C2#2027674911yescharacteristic
subgroupG/H
centerC2#20276749
commutator, cosocleC2×C16×C53C23
FrattiniC2×C16C22×D53
FittingC53×#1419C2
socleC2×C53#2206
hypercenter#1419D53
hypocenterC53#13692
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. #20277030C2semidirect
2. #20277028C2semidirect
3. #20276963C2semidirect
4. #20276595C2semidirect
5. #20272912C2semidirect
6. #20272904C2semidirect
7. C53×#1419C2semidirect
28. Dic424D4semidirect
32. D848D4semidirect
101. #1419D53semidirect
122. C53#13692semidirect

Maximal subgroups

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

subgroupindex
#202770302
#202770282
#202769632
#202765952
#202729122
#202729042
C53×#14192
#1369253