Finite Group Databasebeta

← #27281485

Subgroups — #27281485

Order 13568.

514 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 514 conjugacy classes of subgroups — too many to draw. Only its 115 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#20283756C211yesmaximal, characteristic
2#20279888C211yesmaximal, characteristic
3#20273397C212yesmaximal, complemented
4#20272616C212yesmaximal, complemented
5C53×#1800C211yesmaximal, characteristic
6C2×#13133217C2211yescharacteristic
7#16810701—24no—
8#16810676C2212yes—
9#16810587—22no—
10#16810535C414yes—
11#16810526—21no—
12#16810488—21no—
13#16810483—22no—
14C4×C4⋊Dic53C2211yescharacteristic
15#16810371—24no—
16C2×C4×D4×C53C2211yescharacteristic
17C53×#423C2212yes—
18C2×#100×C53—21no—
19C53×#343—21no—
20C22×Dic106C2×C412yes—
21C2×#11719917—24no—
22#13133217—22no—
23#13133217—22no—
24C2×C4⋊Dic53—21no—
25C2×C4⋊Dic53—21no—
26C2×C4⋊Dic53—21no—
27C2×C22⋊Dic53—24no—
28C2×C4×Dic53—21no—
29C4⋊#88322D414yes—
30C22⋊#88322—24no—
31C22⋊#88322—22no—
32C4×#88322—22no—
33#13133124—24no—
34#13133124C2×C412yes—
35C22×D4×C53D411yescharacteristic
36C23×C4×C53D412yes—
37C2×M16×C53—22no—
38C4×D4×C53—22no—
39C4×D4×C53—22no—
40C2×C4⋊C4×C53D411yescharacteristic
41C2×C22⋊C4×C53C2×C412yes—
42C2×C42×C53C2311yescharacteristic
43C4⋊C8×C53—22no—
44C4⋊C8×C53—22no—
45#100×C53—42no—
46C24×C53C22⋊C412yes—
47C2×D4×C53—22no—
48C2×D4×C53—21no—
49C2×D4×C53—22no—
50C2×D4×C53—21no—
51C22×C4×C53—22no—
52C22×C4×C53—22no—
53C22×C4×C53—44no—
54C22×C4×C53C22⋊C412yes—
55C22×C4×C53C22×C411yescharacteristic
56C22×C4×C53C2×D411yescharacteristic
57C22×C4×C53C2×D411yescharacteristic
58C22×Dic53—22no—
59C22×Dic53—22no—
60C2×Dic106—24no—
61C2×Dic106—82no—
62M16×C53—42no—
63M16×C53—42no—
64C2×C8×C53—42no—
65C4⋊C4×C53—22no—
66C22⋊C4×C53—24no—
67C42×C53C2×D412yes—
68C42×C53#4112yes—
69C22⋊Dic53—84no—
70C4⋊Dic53—42no—
71C4⋊Dic53—41no—
72C4⋊Dic53—42no—
73C4⋊Dic53—41no—
74C4×Dic53—41no—
75C4×Dic53—41no—
76#11719917—84no—
77C2×#88322—24no—
78C23×C53—22no—
79C23×C53—22no—
80C23×C53—22no—
81C23×C53—42no—
82C23×C53—42no—
83C23×C53C2×C22⋊C411yescharacteristic
84D4×C53—42no—
85D4×C53—42no—
86C2×C4×C53—22no—
87C2×C4×C53—22no—
88C2×C4×C53—41no—
89C2×C4×C53—44no—
90C2×C4×C53—42no—
91C2×C4×C53—82no—
92C2×C4×C53C4×D412yes—
93C2×C4×C53#12011yescharacteristic
94C2×C4×C53#12112yes—
95C2×C4×C53#12311yescharacteristic
96C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
97C2×C4×C53#13712yes—
98C2×C4×C53#13711yescharacteristic
99C2×Dic53—44no—
100C2×Dic53—42no—
101C2×Dic53—42no—
102C2×Dic53—42no—
103C2×Dic53—42no—
104Dic106—84no—
105C8×C53—82no—
106#88322—44no—
107#9399—531nomaximal, Sylow
108C22×C53—21no—
109C22×C53—21no—
110C22×C53—44no—
111C22×C53—42no—
112C22×C53—42no—
113C22×C53—42no—
114C22×C53—82no—
115C22×C53#38611yescharacteristic
116C22×C53#41711yescharacteristic
117C22×C53#41911yescharacteristic
118C4×C53—21no—
119C4×C53—22no—
120C4×C53—42no—
121C4×C53#45512yes—
122C4×C53#46812yes—
123C4×C53#47012yes—
124Dic53—82no—
125Dic53—82no—
126#2975—531no—
127#2563—531no—
128#1800D5311yescharacteristic
129#1548—532no—
130#1303—532no—
131C2×C53—21no—
132C2×C53—21no—
133C2×C53—82no—
134C2×C53#188511yescharacteristic
135C2×C53#189411yescharacteristic
136C2×C53#191411yescharacteristic
137C2×C4⋊Q8—531no—
138C2×C4×D4C2×D5311yescharacteristic
139#436—1064no—
140#431—532no—
141#423C2×D5312yes—
142C2×#100—21no—
143#407—1062no—
144#393—534no—
145#390—1061no—
146#382—1061no—
147#380—1062no—
148C4×C4⋊C4—531no—
149#343—21no—
150#333—1064no—
151C53#939911yescharacteristic, complemented, Sylow
152C22×Q8—532no—
153C22×D4C22⋊D5311yescharacteristic
154C23×C4D21212yes—
155C2×M16—22no—
156C2×M16—1064no—
157C4⋊Q8—1062no—
158C4⋊Q8—1062no—
159C4×D4—22no—
160C4×D4—22no—
161C2×C4⋊C4—1061no—
162C2×C4⋊C4—1061no—
163C2×C4⋊C4—1061no—
164C2×C4⋊C4C22⋊D5311yescharacteristic
165C2×C22⋊C4—1064no—
166C2×C22⋊C4C4×D5312yes—
167C2×C42—1061no—
168C2×C42C22×D5311yescharacteristic
169C4⋊C8—22no—
170C4⋊C8—22no—
171C4⋊C8—534no—
172#100—42no—
173C22⋊C8—1064no—
174C22⋊C8—1062no—
175C4×C8—1062no—
176#95—532no—
177#95—1064no—
178C24#1171992112yes—
179C2×Q8—1064no—
180C2×Q8—4242no—
181C2×D4—22no—
182C2×D4—21no—
183C2×D4—22no—
184C2×D4—21no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—44no—
188C22×C4—1062no—
189C22×C4—1062no—
190C22×C4#1171992112yes—
191C22×C4C2×C4×D5311yescharacteristic
192C22×C4C2×D21211yescharacteristic
193C22×C4C2×C22⋊D5311yescharacteristic
194M16—42no—
195M16—42no—
196M16—4244no—
197C2×C8—42no—
198C2×C8—1064no—
199C4⋊C4—22no—
200C4⋊C4—2122no—
201C4⋊C4—2121no—
202C4⋊C4—2122no—
203C4⋊C4—2121no—
204C22⋊C4—24no—
205C22⋊C4—4244no—
206C42—2121no—
207C42—2121no—
208C42D4×D5312yes—
209C42#1171994712yes—
210C23—22no—
211C23—22no—
212C23—22no—
213C23—42no—
214C23—42no—
215C23C2×#1171992111yescharacteristic
216Q8—4244no—
217D4—42no—
218D4—42no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—41no—
222C2×C4—44no—
223C2×C4—42no—
224C2×C4—82no—
225C2×C4—2122no—
226C2×C4—2124no—
227C2×C4—2122no—
228C2×C4—2122no—
229C2×C4—2122no—
230C2×C4#1313317712yes—
231C2×C4#1313318711yescharacteristic
232C2×C4#1313318912yes—
233C2×C4#1313320111yescharacteristic
234C2×C4#1313324311yescharacteristic
235C2×C4#1313324712yes—
236C2×C4#1313324811yescharacteristic
237C8—82no—
238C8—2124no—
239C22—21no—
240C22—21no—
241C22—44no—
242C22—42no—
243C22—42no—
244C22—42no—
245C22—82no—
246C22#1681050011yescharacteristic
247C22#1681062211yescharacteristic
248C22#1681063111yescharacteristic
249C4—22no—
250C4—21no—
251C4—42no—
252C4—4242no—
253C4—4242no—
254C4#1681080312yes—
255C4#1681086712yes—
256C4#1681087912yes—
257C2—21no—
258C2—21no—
259C2—82no—
260C2#2027510111yescharacteristic
261C2#2027514611yescharacteristic
262C2#2027526011yescharacteristic
subgroupG/H
centerC22#16810500
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1800C2
socleC22×C53#386
prime radicalC22×D4×C53D4
cosocleC2×C42×C53C23
hypercenter#1800D53
hypocenterC53#9399
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1800

Complements

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

NH ≅ G/Ntype
3. #20273397C2semidirect
4. #20272616C2semidirect
151. C53#9399semidirect

Maximal subgroups

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

subgroupindex
#202837562
#202798882
#202733972
#202726162
C53×#18002
#939953