Finite Group Databasebeta

← #27288871

Subgroups — #27288871

Order 13568.

310 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 85 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#20275547C211yesmaximal, characteristic
2#20273365C212yesmaximal
3#20272612C212yesmaximal, complemented
4#20272544C211yesmaximal, characteristic, complemented
5C53×#1832C211yesmaximal, characteristic, complemented
6#16811173C2211yescharacteristic
7#16810676C2212yes—
8#16810647C2211yescharacteristic
9C2×#13133145—22no—
10#16810370—22no—
11#16810366—22no—
12C2×C4⋊C8×C53C2212yes—
13C53×#376C2211yescharacteristic
14C53×#336—22no—
15C2×#11719945—21no—
16C2×#11719917—21no—
17#13133217D411yescharacteristic
18#13133214D411yescharacteristic
19#13133205—21no—
20#13133191—22no—
21C4×D212—22no—
22C4⋊#88322—22no—
23C4⋊#88322—22no—
24#13133145—44no—
25C22⋊#88322—22no—
26C4×#88322—22no—
27C22×C8×C53—22no—
28C2×C42×C53D412yes—
29C2×C42×C53C2311yescharacteristic
30C4⋊C8×C53—22no—
31C4⋊C8×C53—22no—
32#95×C53—22no—
33C22×C4×C53—22no—
34C22×C4×C53C2×D411yescharacteristic
35C22×C4×C53#4112yes—
36C2×C22⋊D53—41no—
37#11719945—42no—
38#11719945—42no—
39C2×D212—21no—
40C2×D212—22no—
41C2×C4×D53—41no—
42C2×Dic106—21no—
43C2×C8×C53—42no—
44C2×C8×C53—42no—
45C2×C8×C53—42no—
46C42×C53—22no—
47C42×C53—22no—
48C42×C53C2×D411yescharacteristic
49C42×C53#4111yescharacteristic
50C42×C53#4112yes—
51#11719921—42no—
52C4⋊Dic53—22no—
53#11719917—42no—
54C2×#88322—42no—
55C2×#88322—41no—
56C23×C53#12311yescharacteristic
57C22×D53—41no—
58C2×C4×C53—42no—
59C2×C4×C53—44no—
60C2×C4×C53—42no—
61C2×C4×C53#12111yescharacteristic
62C2×C4×C53#12312yes—
63C2×C4×C53#12411yescharacteristic
64C2×C4×C53#12612yes—
65C2×C4×C53#13512yes—
66C2×C4×C53AGL(1,ℤ/8)12yes—
67C22⋊D53—82no—
68C2×Dic53—41no—
69D212—42no—
70D212—82no—
71C4×D53—81no—
72C4×D53—81no—
73Dic106—42no—
74C8×C53—82no—
75#88322—82no—
76#88322—81no—
77#10629—531nomaximal, Sylow
78C22×C53—22no—
79C22×C53#39711yescharacteristic
80C22×C53#48212yes—
81C2×D53—81no—
82C2×D53—82no—
83C4×C53—22no—
84C4×C53—22no—
85C4×C53—82no—
86C4×C53#46312yes—
87C4×C53#48811yescharacteristic
88C4×C53#49011yescharacteristic
89Dic53—81no—
90#1963—531no—
91#1832D5311yescharacteristic, complemented
92#1540—532no—
93#1301—532no—
94#1274—531no—
95C2×C53—22no—
96C2×C53#207311yescharacteristic
97C2×C53#208012yes—
98D53—161no—
99#532—531no—
100#431—532no—
101#423—531no—
102C2×C4⋊C8C2×D5312yes—
103C2×#104—1062no—
104#376C2×D5311yescharacteristic
105#336—22no—
106#332—1062no—
107#331—1062no—
108C53#1062911yescharacteristic, complemented, Sylow
109C2×#41—1061no—
110C2×M16—1061no—
111C22×C8—22no—
112C4⋊Q8—531no—
113#127—531no—
114#124—1061no—
115#121—1062no—
116C4×D4—1062no—
117C2×C42C22⋊D5312yes—
118C2×C42C22×D5311yescharacteristic
119C4⋊C8—22no—
120C4⋊C8—22no—
121C4⋊C8—1062no—
122C4⋊C8—1062no—
123#104—2124no—
124C22⋊C8—1062no—
125C4×C8—1062no—
126#95—22no—
127#41—2122no—
128#41—2122no—
129C2×Q8—1061no—
130C2×D4—1061no—
131C2×D4—1062no—
132C2×D4—2121no—
133C22×C4—22no—
134C22×C4—2121no—
135C22×C4#1171994512yes—
136C22×C4C2×C22⋊D5311yescharacteristic
137M16—2122no—
138C2×C8—42no—
139C2×C8—42no—
140C2×C8—42no—
141C2×C8—2122no—
142C2×C8—2121no—
143C4⋊C4—1062no—
144C22⋊C4—2122no—
145C42—22no—
146C42—22no—
147C42D4×D5311yescharacteristic
148C42#1171994711yescharacteristic
149C42#1171994912yes—
150C23—2121no—
151C23#1313320011yescharacteristic
152Q8—2122no—
153D4—4242no—
154D4—4242no—
155D4—2122no—
156C2×C4—42no—
157C2×C4—44no—
158C2×C4—42no—
159C2×C4—4241no—
160C2×C4—4241no—
161C2×C4—2121no—
162C2×C4#1313318811yescharacteristic
163C2×C4#1313320212yes—
164C2×C4#1313320611yescharacteristic
165C2×C4#1313321212yes—
166C2×C4#1313323712yes—
167C2×C4#1313324212yes—
168C8—82no—
169C8—4242no—
170C8—4241no—
171C22—22no—
172C22—4241no—
173C22—4242no—
174C22#1681054911yescharacteristic
175C22#1681094712yes—
176C4—22no—
177C4—22no—
178C4—82no—
179C4—4241no—
180C4#1681083512yes—
181C4#1681098011yescharacteristic
182C4#1681098911yescharacteristic
183C2—22no—
184C2—8481no—
185C2#2027612911yescharacteristic
186C2#2027616612yes—
subgroupG/H
centerC2×C4#13133206
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#1832C2
socleC22×C53#397
prime radical#16811173C22
hypercenter#1832D53
hypocenterC53#10629
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1832

Complements

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

NH ≅ G/Ntype
3. #20272612C2semidirect
4. #20272544C2semidirect
5. C53×#1832C2semidirect
91. #1832D53semidirect
108. C53#10629semidirect

Maximal subgroups

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

subgroupindex
#202755472
#202733652
#202726122
#202725442
C53×#18322
#1062953