Finite Group Databasebeta

← #27281457

Subgroups — #27281457

Order 13568.

558 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 558 conjugacy classes of subgroups — too many to draw. Only its 135 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#20279707C211yesmaximal, characteristic
2#20274707C211yesmaximal, characteristic
3#20273386C212yesmaximal, complemented
4#20272617C212yesmaximal, complemented
5C53×#2972C211yesmaximal, characteristic
6C2×#13133217C2211yescharacteristic
7C2×#13133204—21no—
8#16810647C2212yes—
9C2×#13133134—21no—
10#16810548C414yes—
11#16810528—21no—
12#16810483—22no—
13C4×C4⋊Dic53C2211yescharacteristic
14#16810392—21no—
15#16810368—24no—
16C2×C4×D4×C53C2211yescharacteristic
17C8×D4×C53—24no—
18C53×#431C2212yes—
19C2×C22⋊C8×C53—22no—
20C22×Dic106—21no—
21C2×#11719917—22no—
22#13133217—22no—
23#13133217—22no—
24#13133204—42no—
25#13133204—42no—
26C2×C4⋊Dic53—21no—
27C2×C4⋊Dic53—21no—
28C2×C4⋊Dic53C2×C412yes—
29C2×C22⋊Dic53—24no—
30C2×C4×Dic53—21no—
31C4⋊#88322—22no—
32C4⋊#88322—22no—
33#13133134—42no—
34#13133124—24no—
35#13133124C2×C412yes—
36C22×D4×C53D411yescharacteristic
37C23×C4×C53D412yes—
38C22×C8×C53—24no—
39C4×D4×C53—22no—
40C4×D4×C53—22no—
41C2×C4⋊C4×C53D411yescharacteristic
42C2×C22⋊C4×C53C2×C412yes—
43C2×C42×C53C2311yescharacteristic
44C4⋊C8×C53D414yes—
45C22⋊C8×C53—24no—
46C22⋊C8×C53—22no—
47C4×C8×C53—22no—
48C24×C53C22⋊C412yes—
49C2×D4×C53—22no—
50C2×D4×C53—22no—
51C2×D4×C53D812yes—
52C2×D4×C53SD1612yes—
53C22×C4×C53—22no—
54C22×C4×C53—22no—
55C22×C4×C53—44no—
56C22×C4×C53C22⋊C412yes—
57C22×C4×C53C22×C411yescharacteristic
58C22×C4×C53C2×D411yescharacteristic
59C22×C4×C53C2×D411yescharacteristic
60C22×Dic53—22no—
61C22×Dic53—22no—
62C2×Dic106—42no—
63C2×Dic106—41no—
64C2×Dic106—41no—
65C2×Dic106—42no—
66C2×C8×C53—24no—
67C2×C8×C53—84no—
68C4⋊C4×C53—22no—
69C22⋊C4×C53—24no—
70C42×C53C2×D412yes—
71C42×C53#4112yes—
72C22⋊Dic53—48no—
73C4⋊Dic53—24no—
74C4⋊Dic53—42no—
75C4⋊Dic53—41no—
76C4⋊Dic53—41no—
77C4×Dic53—41no—
78C4×Dic53—41no—
79#11719917—42no—
80#11719917—42no—
81C2×#88322—42no—
82C23×C53—22no—
83C23×C53—22no—
84C23×C53—42no—
85C23×C53—42no—
86C23×C53#10214yes—
87C23×C53C2×C22⋊C411yescharacteristic
88D4×C53—24no—
89D4×C53—24no—
90C2×C4×C53—22no—
91C2×C4×C53—22no—
92C2×C4×C53—41no—
93C2×C4×C53—44no—
94C2×C4×C53—42no—
95C2×C4×C53—82no—
96C2×C4×C53C4×D412yes—
97C2×C4×C53#12011yescharacteristic
98C2×C4×C53#12112yes—
99C2×C4×C53#12311yescharacteristic
100C2×C4×C53C2×D811yescharacteristic
101C2×C4×C53C2×SD1611yescharacteristic
102C2×C4×C53#13512yes—
103C2×Dic53—42no—
104C2×Dic53—42no—
105C2×Dic53—42no—
106C2×Dic53—42no—
107C2×Dic53—42no—
108C2×Dic53—42no—
109Dic106—82no—
110Dic106—82no—
111C8×C53—44no—
112#88322—82no—
113#9393—531nomaximal, Sylow
114C22×C53—21no—
115C22×C53—21no—
116C22×C53—24no—
117C22×C53—44no—
118C22×C53—42no—
119C22×C53—42no—
120C22×C53—82no—
121C22×C53#38611yescharacteristic
122C22×C53C2×#10211yescharacteristic
123C22×C53#41611yescharacteristic
124C4×C53—21no—
125C4×C53—22no—
126C4×C53—42no—
127C4×C53#45512yes—
128C4×C53#46512yes—
129C4×C53#46612yes—
130Dic53—82no—
131Dic53—82no—
132#2972D5311yescharacteristic
133#2547—531no—
134#1800—531no—
135#1546—532no—
136#1303—532no—
137C2×C53—21no—
138C2×C53—21no—
139C2×C53—44no—
140C2×C53#188511yescharacteristic
141C2×C53#189311yescharacteristic
142C2×C53#191511yescharacteristic
143C2×C4⋊Q8—531no—
144C2×#124—1061no—
145C2×C4×D4C2×D5311yescharacteristic
146C8×D4—24no—
147#431C2×D5312yes—
148#423—532no—
149C2×#100—1061no—
150C2×C22⋊C8—22no—
151#397—534no—
152#391—1061no—
153#380—1062no—
154C4×C4⋊C4—531no—
155#343—1061no—
156#332—1064no—
157C53#939311yescharacteristic, complemented, Sylow
158C22×Q8—1061no—
159C22×D4D21211yescharacteristic
160C23×C4C22⋊D5312yes—
161C2×M16—1062no—
162C22×C8—24no—
163C4⋊Q8—1062no—
164C4⋊Q8—1062no—
165#124—2122no—
166#124—2122no—
167C4×D4—22no—
168C4×D4—22no—
169C2×C4⋊C4—532no—
170C2×C4⋊C4—1061no—
171C2×C4⋊C4—1061no—
172C2×C4⋊C4D21211yescharacteristic
173C2×C22⋊C4—1064no—
174C2×C22⋊C4C4×D5312yes—
175C2×C42—1061no—
176C2×C42C22×D5311yescharacteristic
177C4⋊C8—1062no—
178C4⋊C8—1062no—
179C4⋊C8C22⋊D5314yes—
180#100—2122no—
181C22⋊C8—24no—
182C22⋊C8—22no—
183C4×C8—22no—
184#95—532no—
185#95—1064no—
186C24#1171992112yes—
187C2×Q8—2122no—
188C2×Q8—2121no—
189C2×Q8—2121no—
190C2×Q8—2122no—
191C2×D4—22no—
192C2×D4—22no—
193C2×D4#1171991312yes—
194C2×D4D42412yes—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—44no—
198C22×C4—1062no—
199C22×C4—1062no—
200C22×C4#1171992112yes—
201C22×C4C2×C4×D5311yescharacteristic
202C22×C4C2×D21211yescharacteristic
203C22×C4C2×C22⋊D5311yescharacteristic
204M16—2122no—
205M16—2122no—
206C2×C8—24no—
207C2×C8—84no—
208C2×C8—2122no—
209C4⋊C4—22no—
210C4⋊C4—1064no—
211C4⋊C4—2122no—
212C4⋊C4—2121no—
213C4⋊C4—2121no—
214C22⋊C4—24no—
215C22⋊C4—2128no—
216C42—2121no—
217C42—2121no—
218C42#1171994512yes—
219C42C2×C22⋊D5312yes—
220C23—22no—
221C23—22no—
222C23—42no—
223C23—42no—
224C23#1313313914yes—
225C23C2×#1171992111yescharacteristic
226Q8—4242no—
227Q8—4242no—
228D4—24no—
229D4—24no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—41no—
233C2×C4—44no—
234C2×C4—42no—
235C2×C4—82no—
236C2×C4—2122no—
237C2×C4—2122no—
238C2×C4—2122no—
239C2×C4—2122no—
240C2×C4—2122no—
241C2×C4—2122no—
242C2×C4C4×C22⋊D5312yes—
243C2×C4#1313318611yescharacteristic
244C2×C4#1313319112yes—
245C2×C4#1313320011yescharacteristic
246C2×C4C2×D42411yescharacteristic
247C2×C4C2×#1171991311yescharacteristic
248C2×C4#1313323912yes—
249C8—44no—
250C8—4242no—
251C22—21no—
252C22—21no—
253C22—24no—
254C22—44no—
255C22—42no—
256C22—42no—
257C22—82no—
258C22#1681050211yescharacteristic
259C22C2×#1313313911yescharacteristic
260C22#1681062011yescharacteristic
261C4—22no—
262C4—21no—
263C4—42no—
264C4—4242no—
265C4—4242no—
266C4#1681080112yes—
267C4#1681085212yes—
268C4#1681085912yes—
269C2—21no—
270C2—21no—
271C2—44no—
272C2#2027510311yescharacteristic
273C2#2027514311yescharacteristic
274C2#2027526711yescharacteristic
subgroupG/H
centerC22#16810502
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#2972C2
socleC22×C53#386
prime radicalC22×D4×C53D4
cosocleC2×C42×C53C23
hypercenter#2972D53
hypocenterC53#9393
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2972

Complements

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

NH ≅ G/Ntype
3. #20273386C2semidirect
4. #20272617C2semidirect
157. C53#9393semidirect

Maximal subgroups

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

subgroupindex
#202797072
#202747072
#202733862
#202726172
C53×#29722
#939353