Finite Group Databasebeta

← #28309814

Subgroups — #28309814

Order 13568.

360 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 64 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#20291084C212yesmaximal, complemented
2#20289725C212yesmaximal, complemented
3#20289724C212yesmaximal, complemented
4#20285175C211yesmaximal, characteristic
5#16811546C2214yes—
6#16811519—22no—
7#16811515—22no—
8C2×#13133247—22no—
9C2×#13133217C411yescharacteristic
10#16811018—22no—
11#16811002C411yescharacteristic
12#16811002C2211yescharacteristic
13#16810995C411yescharacteristic
14#16810995C2211yescharacteristic
15C4×#11719917C2211yescharacteristic
16C53×#532C411yescharacteristic
17#13133291—24no—
18C22×Dic106—21no—
19#13133247—48no—
20C2×#11719925—22no—
21C2×#11719925—22no—
22C2×#11719923—22no—
23C2×#11719923—22no—
24C2×#11719917D412yes—
25#13133217—21no—
26#13133217—41no—
27#13133217C2×C411yescharacteristic
28#13133217C2×C411yescharacteristic
29C2×C4⋊Dic53—41no—
30#13133128D412yes—
31C4×#88322C2×C411yescharacteristic
32C4×#88322C2311yescharacteristic
33C2×#41×C53—21no—
34C4⋊Q8×C53C2×C411yescharacteristic
35#127×C53C2×C411yescharacteristic
36#124×C53—21no—
37#121×C53—41no—
38C4×D4×C53—41no—
39C2×C42×C53C2×C411yescharacteristic
40#41×C53—44no—
41C2×Q8×C53—21no—
42C2×D4×C53—21no—
43C2×D4×C53—41no—
44C2×D4×C53—41no—
45C22×C4×C53—21no—
46C22×C4×C53—41no—
47C22×C4×C53C22⋊C411yescharacteristic
48C22×Dic53—41no—
49C2×Dic106—21no—
50C2×Dic106—21no—
51C2×Dic106—44no—
52C2×Dic106—41no—
53C4⋊C4×C53—41no—
54C22⋊C4×C53—42no—
55C42×C53—21no—
56C42×C53C22⋊C411yescharacteristic
57C42×C53C22×C411yescharacteristic
58#11719925—44no—
59#11719925—44no—
60#11719923—44no—
61#11719923—44no—
62C4⋊Dic53—42no—
63C4⋊Dic53—41no—
64C4⋊Dic53—41no—
65#11719917—24no—
66C2×#88322C22⋊C412yes—
67C2×#88322C2×D412yes—
68#11719910—44no—
69C23×C53—41no—
70C23×C53#10011yescharacteristic
71Q8×C53—42no—
72D4×C53—42no—
73D4×C53—82no—
74D4×C53—82no—
75C2×C4×C53—21no—
76C2×C4×C53—22no—
77C2×C4×C53—21no—
78C2×C4×C53—42no—
79C2×C4×C53—41no—
80C2×C4×C53—82no—
81C2×C4×C53#10011yescharacteristic
82C2×C4×C53C2×C22⋊C411yescharacteristic
83C2×Dic53—41no—
84C2×Dic53—41no—
85C2×Dic53—82no—
86Dic106—42no—
87Dic106—42no—
88Dic106—82no—
89#88322—22no—
90#88322AGL(1,ℤ/8)14yes—
91#11993—531nomaximal, Sylow
92C22×C53—21no—
93C22×C53—41no—
94C22×C53—82no—
95C22×C53—81no—
96C22×C53C2×#10011yescharacteristic
97C4×C53—22no—
98C4×C53—21no—
99C4×C53—41no—
100C4×C53—81no—
101C4×C53#41812yes—
102Dic53—81no—
103Dic53—81no—
104#3152—531no—
105#2165—532no—
106#1359—532no—
107#1358—532no—
108C2×C53—21no—
109C2×C53—41no—
110C2×C53—161no—
111C2×C53#155511yescharacteristic
112C2×#137—1062no—
113#532#615711yescharacteristic
114C2×C4⋊Q8—531no—
115#497—1062no—
116#494—531no—
117#494—531no—
118#492—531no—
119#492—531no—
120C4×M16—531no—
121#364—534no—
122#349—1062no—
123#347—1062no—
124C53#1199311yescharacteristic, complemented, Sylow
125C2×#41—21no—
126C22×Q8—1061no—
127#137—2128no—
128C2×Q16—1062no—
129C2×Q16—1062no—
130C2×SD16—1062no—
131C2×SD16—1062no—
132C2×M16—532no—
133C4⋊Q8—531no—
134C4⋊Q8—531no—
135C4⋊Q8—1061no—
136C4⋊Q8—2121no—
137C4⋊Q8C2×#615711yescharacteristic
138#127C2×#615711yescharacteristic
139#124—21no—
140#121—41no—
141C4×D4—41no—
142C2×C4⋊C4—2121no—
143C2×C42C2×#615711yescharacteristic
144M32—1064no—
145#97—532no—
146C4×C8—531no—
147C4×C8—531no—
148#41—44no—
149C2×Q8—21no—
150C2×Q8—1061no—
151C2×Q8—1061no—
152C2×Q8—2124no—
153C2×Q8—2121no—
154C2×D4—21no—
155C2×D4—41no—
156C2×D4—41no—
157C22×C4—21no—
158C22×C4—41no—
159C22×C4—2121no—
160C22×C4C22⋊#615711yescharacteristic
161Q16—2124no—
162Q16—2124no—
163SD16—2124no—
164SD16—2124no—
165M16—1064no—
166C2×C8—532no—
167C2×C8—532no—
168C4⋊C4—41no—
169C4⋊C4—2121no—
170C4⋊C4—2121no—
171C4⋊C4—2122no—
172C22⋊C4—42no—
173C42—21no—
174C42C22⋊#615711yescharacteristic
175C42C22×#615711yescharacteristic
176C16—2124no—
177C23—41no—
178C23#1313327711yescharacteristic
179Q8—42no—
180Q8—4242no—
181Q8—2122no—
182Q8—2122no—
183D4—42no—
184D4—82no—
185D4—82no—
186C2×C4—21no—
187C2×C4—22no—
188C2×C4—21no—
189C2×C4—42no—
190C2×C4—41no—
191C2×C4—82no—
192C2×C4—4242no—
193C2×C4—2121no—
194C2×C4—2121no—
195C2×C4#1313327711yescharacteristic
196C2×C4C2×C22⋊#615711yescharacteristic
197C8—534no—
198C8—1062no—
199C22—21no—
200C22—41no—
201C22—82no—
202C22—81no—
203C22C2×#1313327711yescharacteristic
204C4—22no—
205C4—21no—
206C4—41no—
207C4—81no—
208C4—4241no—
209C4—4241no—
210C4#1681161512yes—
211C2—21no—
212C2—41no—
213C2—161no—
214C2#2029024211yescharacteristic
subgroupG/H
centerC2#20290242
commutatorC42×C53C22×C4
FrattiniC42C22×#6157
Fitting, prime radicalC53×#532C4
socleC2×C53#1555
cosocleC4×#88322C23
hypercenter#532#6157
hypocenterC53#11993
LCSG ▷ C42×C53 ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C42 ◁ #532

Complements

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

NH ≅ G/Ntype
1. #20291084C2semidirect
2. #20289725C2semidirect
3. #20289724C2semidirect
124. C53#11993semidirect

Maximal subgroups

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

subgroupindex
#202910842
#202897252
#202897242
#202851752
#1199353