Finite Group Databasebeta

← #27303181

Subgroups — #27303181

Order 13568.

370 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 99 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#20285239C211yesmaximal, characteristic
2#20276435C212yesmaximal
3#20272822C212yesmaximal
4#20272799C211yesmaximal, characteristic
5C53×#1971C211yesmaximal, characteristic
6#16811180—21no—
7C2×#13133157C2212yes—
8#16811032—24no—
9#16810661—21no—
10#16810661C2212yes—
11#16810427—24no—
12C22×Q16×C53C2211yescharacteristic
13C2×#106×C53C2211yescharacteristic
14C2×#103×C53—21no—
15C2×#97×C53C2211yescharacteristic
16C53×#391—21no—
17#13133217—22no—
18#13133209—22no—
19#13133196—42no—
20C4×Dic106—42no—
21#13133176—22no—
22#13133157—22no—
23#13133157—22no—
24C2×#11719908—22no—
25#13133151—22no—
26#13133151D414yes—
27#13133149—22no—
28#13133149—22no—
29C22×Q8×C53—21no—
30C2×Q16×C53—21no—
31C2×Q16×C53—22no—
32C2×Q16×C53—22no—
33C2×Q16×C53Q812yescomplemented
34C22×C8×C53D411yescharacteristic
35C22×C8×C53C2311yescharacteristic
36C2×C4⋊C4×C53—21no—
37C2×C42×C53D411yescharacteristic
38#106×C53—22no—
39#103×C53—42no—
40#97×C53—22no—
41#95×C53—42no—
42C2×Q8×C53—22no—
43C2×Q8×C53—41no—
44C2×Q8×C53—41no—
45C2×Q8×C53—41no—
46C2×Q8×C53—41no—
47C2×Q8×C53—41no—
48C22×C4×C53—41no—
49C22×C4×C53—41no—
50C22×C4×C53D812yes—
51C22×C4×C53C2×D411yescharacteristic
52C2×Dic106—42no—
53Q16×C53—24no—
54Q16×C53—42no—
55C2×C8×C53—21no—
56C2×C8×C53—21no—
57C2×C8×C53SD1612yes—
58C2×C8×C53C2×D412yes—
59C2×C8×C53C2×Q811yescharacteristic
60C2×C8×C53#4112yes—
61C2×C8×C53#4111yescharacteristic
62C4⋊C4×C53—41no—
63C4⋊C4×C53—41no—
64C42×C53—21no—
65C42×C53—21no—
66C22⋊Dic53—42no—
67C4⋊Dic53—24no—
68C4⋊Dic53—42no—
69#11719919—44no—
70C4×Dic53—42no—
71#11719908—42no—
72C23×C53C2×D811yescharacteristic
73Q8×C53—42no—
74Q8×C53—42no—
75Q8×C53—81no—
76Q8×C53—81no—
77C2×C4×C53—22no—
78C2×C4×C53—42no—
79C2×C4×C53—42no—
80C2×C4×C53—81no—
81C2×C4×C53—81no—
82C2×C4×C53—82no—
83C2×C4×C53—81no—
84C2×C4×C53#12011yescharacteristic
85C2×C4×C53C22⋊Q811yescharacteristic
86C2×C4×C53#12411yescharacteristic
87C2×C4×C53C2×SD1611yescharacteristic
88C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
89C2×C4×C53#13711yescharacteristic
90C2×Dic53—44no—
91Dic106—84no—
92C8×C53—22no—
93C8×C53C22⋊Q812yes—
94C8×C53#12412yes—
95#13368—531nomaximal, Sylow
96C22×C53—22no—
97C22×C53#44711yescharacteristic
98C22×C53#47411yescharacteristic
99C22×C53#48611yescharacteristic
100C4×C53—42no—
101C4×C53—82no—
102C4×C53—161no—
103C4×C53#39311yescharacteristic
104C4×C53#44811yescharacteristic
105C4×C53#47611yescharacteristic
106C4×C53#48911yescharacteristic
107Dic53—84no—
108#3162—531no—
109#2141—532no—
110#1971D5311yescharacteristic
111#1386—532no—
112#1377—531no—
113C2×C53—21no—
114C2×C53—21no—
115C2×C53#202311yescharacteristic
116C2×C53#219112yes—
117C22×Q16C2×D5311yescharacteristic
118#533—1061no—
119C2×M32—532no—
120#501—1064no—
121#427—532no—
122#427—1061no—
123C2×#106C2×D5311yescharacteristic
124C2×#103—21no—
125C2×#97C2×D5311yescharacteristic
126#391—21no—
127#358—1064no—
128C53#1336811yescharacteristic, complemented, Sylow
129C22×Q8—21no—
130C2×Q16—21no—
131C2×Q16—22no—
132C2×Q16—22no—
133C2×Q16Dic10612yescomplemented
134C22×C8C22⋊D5311yescharacteristic
135C22×C8C22×D5311yescharacteristic
136C4⋊Q8—1062no—
137#125—1062no—
138C22⋊Q8—2122no—
139C4×Q8—2122no—
140#117—1062no—
141C2×C4⋊C4—21no—
142C2×C42C22⋊D5311yescharacteristic
143M32—1062no—
144M32—1062no—
145C2×C16—1062no—
146#107—534no—
147#107—1062no—
148#106—22no—
149#106—1062no—
150#106—1062no—
151#103—42no—
152#97—22no—
153#95—42no—
154C2×Q8—22no—
155C2×Q8—41no—
156C2×Q8—41no—
157C2×Q8—41no—
158C2×Q8—41no—
159C2×Q8—41no—
160C2×Q8—2122no—
161C22×C4—41no—
162C22×C4—41no—
163C22×C4#1171992212yes—
164C22×C4C2×C22⋊D5311yescharacteristic
165Q16—24no—
166Q16—42no—
167C2×C8—21no—
168C2×C8—21no—
169C2×C8#1171992312yes—
170C2×C8C2×Dic10611yescharacteristic
171C2×C8#1171994511yescharacteristic
172C2×C8D4×D5312yes—
173C2×C8#1171994712yes—
174C4⋊C4—41no—
175C4⋊C4—41no—
176C4⋊C4—1064no—
177C4⋊C4—2124no—
178C4⋊C4—2122no—
179C22⋊C4—2122no—
180C42—21no—
181C42—21no—
182C42—2122no—
183C16—2122no—
184C23C2×#1171992211yescharacteristic
185Q8—42no—
186Q8—42no—
187Q8—81no—
188Q8—81no—
189Q8—4244no—
190C2×C4—22no—
191C2×C4—42no—
192C2×C4—42no—
193C2×C4—81no—
194C2×C4—81no—
195C2×C4—82no—
196C2×C4—81no—
197C2×C4—2124no—
198C2×C4#1313318511yescharacteristic
199C2×C4#1313319611yescharacteristic
200C2×C4#1313320411yescharacteristic
201C2×C4C2×#1171992311yescharacteristic
202C2×C4#1313324211yescharacteristic
203C2×C4#1313324711yescharacteristic
204C8—22no—
205C8#1313319412yes—
206C8#1313320312yes—
207C22—22no—
208C22#1681075611yescharacteristic
209C22#1681090311yescharacteristic
210C22#1681096911yescharacteristic
211C4—42no—
212C4—82no—
213C4—161no—
214C4—4244no—
215C4#1681053511yescharacteristic
216C4#1681076111yescharacteristic
217C4#1681091311yescharacteristic
218C4#1681098211yescharacteristic
219C2—21no—
220C2—21no—
221C2#2027585111yescharacteristic
222C2#2027666512yes—
subgroupG/H
centerC22#16810756
commutator, cosocleC22×C8×C53C23
FrattiniC22×C8C22×D53
FittingC53×#1971C2
socleC22×C53#447
prime radicalC23×C53C2×D8
hypercenter#1971D53
hypocenterC53#13368
LCSG ▷ C22×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#97 ◁ #1971

Complements

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

NH ≅ G/Ntype
33. C2×Q16×C53Q8semidirect
128. C53#13368semidirect
133. C2×Q16Dic106semidirect

Maximal subgroups

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

subgroupindex
#202852392
#202764352
#202728222
#202727992
C53×#19712
#1336853