Finite Group Databasebeta

← #27287182

Subgroups — #27287182

Order 13568.

416 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 101 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#20275692C211yesmaximal, characteristic
2#20273278C212yesmaximal
3#20272573C212yesmaximal
4#20272548C211yesmaximal, characteristic
5C53×#2862C211yesmaximal, characteristic
6#16811180C2211yescharacteristic
7#16810676C2212yes—
8#16810647C2211yescharacteristic
9#16810383—21no—
10#16810383—21no—
11#16810359—24no—
12C2×C4⋊Q8×C53—21no—
13C53×#391—24no—
14C53×#382—21no—
15C53×#382C2212yes—
16C43×C53C2211yescharacteristic
17C2×#11719917—21no—
18#13133217Q812yescomplemented
19#13133209—21no—
20#13133196—22no—
21C4×Dic106—22no—
22#13133176—21no—
23C4⋊#88322—22no—
24C4⋊#88322—22no—
25C22⋊#88322—22no—
26C4×#88322—22no—
27C22×Q8×C53—22no—
28C4⋊Q8×C53—42no—
29C4⋊Q8×C53—42no—
30C2×C4⋊C4×C53—22no—
31C2×C4⋊C4×C53—22no—
32C2×C42×C53—22no—
33C2×C42×C53D411yescharacteristic
34C2×C42×C53D411yescharacteristic
35C2×C42×C53C2311yescharacteristic
36#95×C53—24no—
37#95×C53—22no—
38#95×C53D414yes—
39C2×Q8×C53—44no—
40C2×Q8×C53—48no—
41C22×C4×C53—21no—
42C22×C4×C53—21no—
43C22×C4×C53—24no—
44C22×C4×C53C2×D412yes—
45C22×C4×C53C2×D411yescharacteristic
46C2×Dic106—22no—
47C4⋊C4×C53—44no—
48C4⋊C4×C53—42no—
49C4⋊C4×C53—42no—
50C42×C53—21no—
51C42×C53—21no—
52C42×C53—44no—
53C42×C53SD1612yes—
54C42×C53Q1612yes—
55C42×C53C2×Q811yescharacteristic
56C42×C53#4111yescharacteristic
57C42×C53#4112yes—
58C22⋊Dic53—41no—
59C4⋊Dic53—22no—
60C4⋊Dic53—22no—
61#11719919—42no—
62C4×Dic53—41no—
63#11719917—41no—
64#11719917—41no—
65C2×#88322—42no—
66C2×#88322—41no—
67C23×C53#12011yescharacteristic
68Q8×C53—88no—
69C2×C4×C53—24no—
70C2×C4×C53—22no—
71C2×C4×C53—22no—
72C2×C4×C53—42no—
73C2×C4×C53—42no—
74C2×C4×C53—48no—
75C2×C4×C53—44no—
76C2×C4×C53C22⋊Q812yes—
77C2×C4×C53C22⋊Q811yescharacteristic
78C2×C4×C53#12412yes—
79C2×C4×C53#12411yescharacteristic
80C2×C4×C53C2×SD1611yescharacteristic
81C2×C4×C53C2×Q1611yescharacteristic
82C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
83C2×C4×C53#13711yescharacteristic
84C2×Dic53—42no—
85Dic106—82no—
86#88322—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—84no—
98C4×C53#47511yescharacteristic
99C4×C53#47711yescharacteristic
100C4×C53#48711yescharacteristic
101C4×C53#48911yescharacteristic
102Dic53—82no—
103#2862D5311yescharacteristic
104#1994—531no—
105#1518—532no—
106#1286—532no—
107#1276—531no—
108C2×C53—21no—
109C2×C53—21no—
110C2×C53#202411yescharacteristic
111C2×C53#203112yes—
112#533—531no—
113C2×C4⋊Q8—21no—
114#431—532no—
115#423—531no—
116#391—24no—
117#382—21no—
118#382C2×D5312yes—
119C43C2×D5311yescharacteristic
120#339—1061no—
121#339—1061no—
122#328—1064no—
123C53#1036311yescharacteristic, complemented, Sylow
124C22×Q8—22no—
125C2×M16—1061no—
126C4⋊Q8—42no—
127C4⋊Q8—42no—
128C4⋊Q8—532no—
129#125—1061no—
130C22⋊Q8—1062no—
131C4×Q8—1062no—
132#117—1061no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C42—22no—
136C2×C42C22⋊D5311yescharacteristic
137C2×C42C22⋊D5311yescharacteristic
138C2×C42C22×D5311yescharacteristic
139C4⋊C8—1062no—
140C4⋊C8—1062no—
141C22⋊C8—1062no—
142C4×C8—1062no—
143#95—24no—
144#95—22no—
145#95C22⋊D5314yes—
146C2×Q8—44no—
147C2×Q8—48no—
148C2×Q8—1062no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—24no—
152C22×C4C2×C22⋊D5312yes—
153C22×C4C2×C22⋊D5311yescharacteristic
154M16—2121no—
155M16—2121no—
156C2×C8—2122no—
157C2×C8—2121no—
158C4⋊C4—44no—
159C4⋊C4—42no—
160C4⋊C4—42no—
161C4⋊C4—1062no—
162C4⋊C4—1062no—
163C4⋊C4—2122no—
164C22⋊C4—2121no—
165C42—21no—
166C42—21no—
167C42—44no—
168C42—2121no—
169C42#1171992312yes—
170C42#1171992512yes—
171C42Q8×D5311yescharacteristic
172C42#1171994911yescharacteristic
173C42#1171994912yes—
174C23#1313318611yescharacteristic
175Q8—88no—
176Q8—4242no—
177C2×C4—24no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—42no—
181C2×C4—42no—
182C2×C4—48no—
183C2×C4—44no—
184C2×C4—2122no—
185C2×C4#1313319312yes—
186C2×C4#1313319311yescharacteristic
187C2×C4#1313320612yes—
188C2×C4#1313320611yescharacteristic
189C2×C4C2×#1171992311yescharacteristic
190C2×C4C2×#1171992511yescharacteristic
191C2×C4#1313324311yescharacteristic
192C2×C4#1313324811yescharacteristic
193C8—4242no—
194C8—4241no—
195C22—22no—
196C22#1681053411yescharacteristic
197C22#1681077411yescharacteristic
198C22#1681077911yescharacteristic
199C4—24no—
200C4—22no—
201C4—22no—
202C4—42no—
203C4—84no—
204C4—4242no—
205C4#1681090511yescharacteristic
206C4#1681091511yescharacteristic
207C4#1681097511yescharacteristic
208C4#1681098511yescharacteristic
209C2—21no—
210C2—21no—
211C2#2027585311yescharacteristic
212C2#2027588412yes—
subgroupG/H
centerC22#16810534
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#2862C2
socleC22×C53#393
prime radicalC23×C53#120
hypercenter#2862D53
hypocenterC53#10363
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2862

Complements

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

NH ≅ G/Ntype
18. #13133217Q8semidirect
123. C53#10363semidirect

Maximal subgroups

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

subgroupindex
#202756922
#202732782
#202725732
#202725482
C53×#28622
#1036353