Finite Group Databasebeta

← #27287222

Subgroups — #27287222

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#20275696C211yesmaximal, characteristic
2#20273213C212yesmaximal
3#20272618C212yesmaximal
4#20272544C211yesmaximal, characteristic
5C53×#2399C211yesmaximal, characteristic
6#16811180C2211yescharacteristic
7#16810676C2212yes—
8#16810647C2211yescharacteristic
9#16810383—21no—
10#16810383—21no—
11#16810359—22no—
12#16810359—22no—
13C2×C4×Q8×C53—21no—
14C53×#398—22no—
15C53×#395—22no—
16C4×C4⋊C4×C53—21no—
17C4×C4⋊C4×C53C2212yes—
18C53×#376C2211yescharacteristic
19C2×#11719917—21no—
20#13133217Q812yescomplemented
21#13133209—21no—
22#13133196—22no—
23C4×Dic106—22no—
24#13133176—21no—
25C4⋊#88322—22no—
26C4⋊#88322—22no—
27C22⋊#88322—22no—
28C4×#88322—22no—
29C22×Q8×C53—21no—
30C4×Q8×C53—42no—
31C4×Q8×C53—42no—
32C2×C4⋊C4×C53—22no—
33C2×C4⋊C4×C53—21no—
34C2×C4⋊C4×C53—22no—
35C2×C4⋊C4×C53—22no—
36C2×C42×C53—22no—
37C2×C42×C53D411yescharacteristic
38C2×C42×C53D411yescharacteristic
39C2×C42×C53C2311yescharacteristic
40#95×C53—21no—
41#95×C53—21no—
42#95×C53—22no—
43#95×C53D412yes—
44#95×C53D412yes—
45C2×Q8×C53—42no—
46C2×Q8×C53—44no—
47C22×C4×C53—21no—
48C22×C4×C53—21no—
49C22×C4×C53—22no—
50C22×C4×C53—22no—
51C22×C4×C53C2×D412yes—
52C22×C4×C53C2×D411yescharacteristic
53C2×Dic106—22no—
54C4⋊C4×C53—44no—
55C4⋊C4×C53—44no—
56C4⋊C4×C53—44no—
57C4⋊C4×C53—42no—
58C42×C53—21no—
59C42×C53—21no—
60C42×C53—42no—
61C42×C53—42no—
62C42×C53SD1612yes—
63C42×C53Q1612yes—
64C42×C53C2×Q811yescharacteristic
65C42×C53#4112yes—
66C42×C53#4111yescharacteristic
67C22⋊Dic53—41no—
68C4⋊Dic53—22no—
69C4⋊Dic53—22no—
70#11719919—42no—
71C4×Dic53—41no—
72#11719917—41no—
73#11719917—41no—
74C2×#88322—42no—
75C2×#88322—41no—
76C23×C53#12011yescharacteristic
77Q8×C53—48no—
78C2×C4×C53—24no—
79C2×C4×C53—22no—
80C2×C4×C53—22no—
81C2×C4×C53—42no—
82C2×C4×C53—42no—
83C2×C4×C53—42no—
84C2×C4×C53—42no—
85C2×C4×C53—42no—
86C2×C4×C53—42no—
87C2×C4×C53—42no—
88C2×C4×C53—42no—
89C2×C4×C53C22⋊Q812yes—
90C2×C4×C53C22⋊Q811yescharacteristic
91C2×C4×C53#12412yes—
92C2×C4×C53#12411yescharacteristic
93C2×C4×C53C2×SD1611yescharacteristic
94C2×C4×C53C2×Q1611yescharacteristic
95C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
96C2×C4×C53#13711yescharacteristic
97C2×Dic53—42no—
98Dic106—82no—
99#88322—82no—
100#88322—81no—
101#10371—531nomaximal, Sylow
102C22×C53—22no—
103C22×C53#39311yescharacteristic
104C22×C53#45011yescharacteristic
105C22×C53#45111yescharacteristic
106C4×C53—22no—
107C4×C53—22no—
108C4×C53—42no—
109C4×C53—44no—
110C4×C53—42no—
111C4×C53—82no—
112C4×C53#47511yescharacteristic
113C4×C53#47711yescharacteristic
114C4×C53#48711yescharacteristic
115C4×C53#48911yescharacteristic
116Dic53—82no—
117#2399D5311yescharacteristic
118#1995—531no—
119#1503—532no—
120#1303—532no—
121#1274—531no—
122C2×C53—21no—
123C2×C53—21no—
124C2×C53#202411yescharacteristic
125C2×C53#203112yes—
126#533—531no—
127C2×C4×Q8—21no—
128#431—532no—
129#423—531no—
130#398—22no—
131#395—22no—
132C4×C4⋊C4—21no—
133C4×C4⋊C4C2×D5312yes—
134#376C2×D5311yescharacteristic
135#339—1061no—
136#339—1061no—
137#328—1062no—
138#328—1062no—
139C53#1037111yescharacteristic, complemented, Sylow
140C22×Q8—21no—
141C2×M16—1061no—
142C4⋊Q8—532no—
143#125—1061no—
144C22⋊Q8—1062no—
145C4×Q8—42no—
146C4×Q8—42no—
147C4×Q8—1062no—
148#117—1061no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C42—22no—
154C2×C42C22⋊D5311yescharacteristic
155C2×C42C22⋊D5311yescharacteristic
156C2×C42C22×D5311yescharacteristic
157C4⋊C8—1062no—
158C4⋊C8—1062no—
159C22⋊C8—1062no—
160C4×C8—1062no—
161#95—21no—
162#95—21no—
163#95—22no—
164#95C22⋊D5312yes—
165#95C22⋊D5312yes—
166C2×Q8—42no—
167C2×Q8—44no—
168C2×Q8—1062no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4C2×C22⋊D5312yes—
174C22×C4C2×C22⋊D5311yescharacteristic
175M16—2121no—
176M16—2121no—
177C2×C8—2122no—
178C2×C8—2121no—
179C4⋊C4—44no—
180C4⋊C4—42no—
181C4⋊C4—44no—
182C4⋊C4—44no—
183C4⋊C4—1062no—
184C4⋊C4—1062no—
185C4⋊C4—2122no—
186C22⋊C4—2121no—
187C42—21no—
188C42—21no—
189C42—42no—
190C42—42no—
191C42—2121no—
192C42#1171992312yes—
193C42#1171992512yes—
194C42Q8×D5311yescharacteristic
195C42#1171994911yescharacteristic
196C42#1171994912yes—
197C23#1313318611yescharacteristic
198Q8—48no—
199Q8—4242no—
200C2×C4—24no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—42no—
211C2×C4—2122no—
212C2×C4#1313319312yes—
213C2×C4#1313319311yescharacteristic
214C2×C4#1313320612yes—
215C2×C4#1313320611yescharacteristic
216C2×C4C2×#1171992311yescharacteristic
217C2×C4C2×#1171992511yescharacteristic
218C2×C4#1313324311yescharacteristic
219C2×C4#1313324811yescharacteristic
220C8—4242no—
221C8—4241no—
222C22—22no—
223C22#1681053411yescharacteristic
224C22#1681077411yescharacteristic
225C22#1681077911yescharacteristic
226C4—22no—
227C4—22no—
228C4—42no—
229C4—44no—
230C4—42no—
231C4—82no—
232C4—4242no—
233C4#1681090511yescharacteristic
234C4#1681091511yescharacteristic
235C4#1681097511yescharacteristic
236C4#1681098511yescharacteristic
237C2—21no—
238C2—21no—
239C2#2027585311yescharacteristic
240C2#2027588412yes—
subgroupG/H
centerC2×C4#13133206
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#2399C2
socleC22×C53#393
prime radicalC23×C53#120
hypercenter#2399D53
hypocenterC53#10371
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2399

Complements

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

NH ≅ G/Ntype
20. #13133217Q8semidirect
139. C53#10371semidirect

Maximal subgroups

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

subgroupindex
#202756962
#202732132
#202726182
#202725442
C53×#23992
#1037153