Finite Group Databasebeta

← #27287186

Subgroups — #27287186

Order 13568.

416 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 97 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#20283093C211yesmaximal, characteristic
2#20275692C211yesmaximal, characteristic
3#20273280C212yesmaximal
4#20272574C212yesmaximal
5C53×#1276C211yesmaximal, characteristic
6#16811180C2211yescharacteristic
7C2×#13133217—21no—
8#16810647C2211yescharacteristic
9#16810529—24no—
10#16810489—21no—
11#16810489C2212yes—
12#16810383—21no—
13#16810383—21no—
14#16810361—24no—
15C53×#431C2212yes—
16C43×C53C2211yescharacteristic
17C22×Dic106—22no—
18C2×#11719917—21no—
19#13133217—42no—
20#13133217—42no—
21#13133217Q812yescomplemented
22#13133209—21no—
23#13133196—22no—
24C4×Dic106—22no—
25#13133176—21no—
26C2×C4⋊Dic53—22no—
27C2×C4⋊Dic53—22no—
28C4⋊#88322—22no—
29#13133124—24no—
30#13133124—22no—
31#13133124D414yes—
32C2×C42×C53—22no—
33C2×C42×C53D411yescharacteristic
34C2×C42×C53D411yescharacteristic
35C2×C42×C53C2311yescharacteristic
36C4⋊C8×C53—22no—
37C22⋊C8×C53—22no—
38C4×C8×C53—22no—
39C22×C4×C53—21no—
40C22×C4×C53—21no—
41C22×C4×C53C2×D412yes—
42C22×C4×C53C2×D411yescharacteristic
43C22×Dic53—24no—
44C2×Dic106—22no—
45C2×Dic106—44no—
46C2×Dic106—48no—
47C2×C8×C53—42no—
48C42×C53—21no—
49C42×C53—21no—
50C42×C53—44no—
51C42×C53SD1612yes—
52C42×C53Q1612yes—
53C42×C53C2×Q811yescharacteristic
54C42×C53#4112yes—
55C42×C53#4111yescharacteristic
56C22⋊Dic53—41no—
57C4⋊Dic53—22no—
58C4⋊Dic53—22no—
59C4⋊Dic53—44no—
60C4⋊Dic53—42no—
61C4⋊Dic53—42no—
62#11719919—42no—
63C4×Dic53—41no—
64#11719917—41no—
65#11719917—41no—
66C2×#88322—41no—
67C23×C53#12011yescharacteristic
68C2×C4×C53—24no—
69C2×C4×C53—22no—
70C2×C4×C53—22no—
71C2×C4×C53—42no—
72C2×C4×C53—42no—
73C2×C4×C53C22⋊Q812yes—
74C2×C4×C53C22⋊Q811yescharacteristic
75C2×C4×C53#12412yes—
76C2×C4×C53#12411yescharacteristic
77C2×C4×C53C2×SD1611yescharacteristic
78C2×C4×C53C2×Q1611yescharacteristic
79C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
80C2×C4×C53#13711yescharacteristic
81C2×Dic53—48no—
82C2×Dic53—42no—
83C2×Dic53—44no—
84Dic106—88no—
85Dic106—82no—
86C8×C53—82no—
87#88322—81no—
88#10363—531nomaximal, Sylow
89C22×C53—22no—
90C22×C53#39311yescharacteristic
91C22×C53#45011yescharacteristic
92C22×C53#45111yescharacteristic
93C4×C53—24no—
94C4×C53—22no—
95C4×C53—22no—
96C4×C53—42no—
97C4×C53#47511yescharacteristic
98C4×C53#47711yescharacteristic
99C4×C53#48711yescharacteristic
100C4×C53#48911yescharacteristic
101Dic53—84no—
102Dic53—82no—
103#2862—531no—
104#1994—531no—
105#1518—532no—
106#1286—532no—
107#1276D5311yescharacteristic
108C2×C53—21no—
109C2×C53—21no—
110C2×C53#202411yescharacteristic
111C2×C53#203112yes—
112#533—531no—
113C2×C4⋊Q8—1061no—
114#431C2×D5312yes—
115#423—531no—
116#391—1064no—
117#382—532no—
118#382—1061no—
119C43C2×D5311yescharacteristic
120#339—1061no—
121#339—1061no—
122#328—1064no—
123C53#1036311yescharacteristic, complemented, Sylow
124C22×Q8—1062no—
125C2×M16—1061no—
126C4⋊Q8—532no—
127C4⋊Q8—2122no—
128C4⋊Q8—2122no—
129#125—1061no—
130C22⋊Q8—1062no—
131C4×Q8—1062no—
132#117—1061no—
133C2×C4⋊C4—1062no—
134C2×C4⋊C4—1062no—
135C2×C42—22no—
136C2×C42D21211yescharacteristic
137C2×C42D21211yescharacteristic
138C2×C42C22×D5311yescharacteristic
139C4⋊C8—22no—
140C4⋊C8—1062no—
141C22⋊C8—22no—
142C4×C8—22no—
143#95—534no—
144#95—1064no—
145#95—1062no—
146C2×Q8—1062no—
147C2×Q8—2124no—
148C2×Q8—2128no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—1064no—
152C22×C4C2×D21211yescharacteristic
153C22×C4D4×D5312yes—
154M16—2121no—
155M16—2121no—
156C2×C8—42no—
157C2×C8—2121no—
158C4⋊C4—1062no—
159C4⋊C4—1062no—
160C4⋊C4—2124no—
161C4⋊C4—2122no—
162C4⋊C4—2122no—
163C4⋊C4—2122no—
164C22⋊C4—2121no—
165C42—21no—
166C42—21no—
167C42—44no—
168C42—2121no—
169C42#1171991312yes—
170C42Dic21212yes—
171C42#1171994512yes—
172C42#1171994711yescharacteristic
173C42Q8×D5311yescharacteristic
174C23#1313318711yescharacteristic
175Q8—4242no—
176Q8—4248no—
177C2×C4—24no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—42no—
181C2×C4—42no—
182C2×C4—2128no—
183C2×C4—2124no—
184C2×C4—2122no—
185C2×C4#1313319512yes—
186C2×C4#1313319711yescharacteristic
187C2×C4#1313320312yes—
188C2×C4#1313320511yescharacteristic
189C2×C4C2×#1171991311yescharacteristic
190C2×C4C2×Dic21211yescharacteristic
191C2×C4#1313324511yescharacteristic
192C2×C4#1313325011yescharacteristic
193C8—82no—
194C8—4241no—
195C22—22no—
196C22#1681053611yescharacteristic
197C22#1681077611yescharacteristic
198C22#1681078111yescharacteristic
199C4—24no—
200C4—22no—
201C4—22no—
202C4—42no—
203C4—4244no—
204C4—4242no—
205C4#1681090911yescharacteristic
206C4#1681091911yescharacteristic
207C4#1681097411yescharacteristic
208C4#1681098311yescharacteristic
209C2—21no—
210C2—21no—
211C2#2027585711yescharacteristic
212C2#2027588812yes—
subgroupG/H
centerC22#16810536
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#1276C2
socleC22×C53#393
prime radicalC23×C53#120
hypercenter#1276D53
hypocenterC53#10363
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1276

Complements

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

NH ≅ G/Ntype
21. #13133217Q8semidirect
123. C53#10363semidirect

Maximal subgroups

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

subgroupindex
#202830932
#202756922
#202732802
#202725742
C53×#12762
#1036353