Finite Group Databasebeta

← #27273192

Subgroups — #27273192

Order 13568.

478 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 87 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#20281087C211yesmaximal, characteristic
2#20276243C211yesmaximal, characteristic
3#20276188C212yesmaximal, complemented
4#20272539C212yesmaximal, complemented
5C53×#1270C211yesmaximal, characteristic
6C2×#13133194—22no—
7C2×#13133151—22no—
8C2×#13133150—21no—
9C2×#13133138—21no—
10#16810546—22no—
11#16810543—21no—
12#16810521C2212yes—
13#16810503—21no—
14#16810493C2211yescharacteristic
15#16810386—22no—
16#16810378C2211yescharacteristic
17C53×#390C2211yescharacteristic
18C53×#336C2212yes—
19C22×Dic106—21no—
20C23×Dic53—21no—
21C22×#88322—21no—
22#13133194—44no—
23#13133194—44no—
24C2×C4⋊Dic53—21no—
25C2×C4⋊Dic53—22no—
26C2×C4⋊Dic53D412yes—
27C2×C4⋊Dic53D412yes—
28C2×#11719919—22no—
29C2×#11719919—22no—
30C2×C22⋊Dic53—22no—
31C2×C22⋊Dic53—21no—
32#13133151—44no—
33#13133150—42no—
34#13133138—42no—
35#13133124—21no—
36#13133124—22no—
37C22×D4×C53D411yescharacteristic
38C22×C8×C53—22no—
39C2×C4⋊C4×C53D411yescharacteristic
40C2×C22⋊C4×C53—22no—
41C2×C42×C53C2311yescharacteristic
42C24×C53—21no—
43C2×D4×C53—21no—
44C2×D4×C53—41no—
45C2×D4×C53—41no—
46C2×D4×C53Q1612yes—
47C22×C4×C53—21no—
48C22×C4×C53C2×D412yes—
49C22×C4×C53C2×D411yescharacteristic
50C22×Dic53—22no—
51C22×Dic53—22no—
52C22×Dic53—44no—
53C22×Dic53—41no—
54C22×Dic53—81no—
55C2×Dic106—44no—
56C2×Dic106—81no—
57C2×C8×C53—24no—
58C2×C8×C53—42no—
59C2×C8×C53—42no—
60C4⋊C4×C53—41no—
61C22⋊C4×C53—42no—
62C22⋊C4×C53—42no—
63C42×C53—41no—
64C22⋊Dic53—44no—
65C22⋊Dic53—81no—
66C4⋊Dic53—22no—
67C4⋊Dic53—42no—
68C4⋊Dic53—41no—
69C4⋊Dic53—41no—
70C4⋊Dic53—42no—
71C4⋊Dic53—42no—
72C4⋊Dic53D814yes—
73#11719919—44no—
74#11719919—42no—
75#11719919—42no—
76C2×#88322—22no—
77C2×#88322—41no—
78C2×#88322—41no—
79C23×C53—22no—
80C23×C53—42no—
81C23×C53—41no—
82C23×C53—42no—
83C23×C53—41no—
84C23×C53#12011yescharacteristic
85D4×C53—42no—
86D4×C53—81no—
87C2×C4×C53—21no—
88C2×C4×C53—22no—
89C2×C4×C53—21no—
90C2×C4×C53—22no—
91C2×C4×C53—42no—
92C2×C4×C53—41no—
93C2×C4×C53C2×D812yes—
94C2×C4×C53C2×Q1611yescharacteristic
95C2×C4×C53AGL(1,ℤ/8)12yes—
96C2×C4×C53#13711yescharacteristic
97C2×Dic53—24no—
98C2×Dic53—42no—
99C2×Dic53—42no—
100C2×Dic53—42no—
101C2×Dic53—42no—
102C2×Dic53—42no—
103C2×Dic53—84no—
104C2×Dic53—82no—
105C2×Dic53—81no—
106Dic106—84no—
107C8×C53—44no—
108#88322—42no—
109#7797—531nomaximal, Sylow
110C22×C53—42no—
111C22×C53—44no—
112C22×C53—82no—
113C22×C53—81no—
114C22×C53—81no—
115C22×C53#44712yes—
116C22×C53#45111yescharacteristic
117C22×C53#45311yescharacteristic
118C22×C53#45612yes—
119C22×C53#45811yescharacteristic
120C4×C53—22no—
121C4×C53—22no—
122C4×C53—21no—
123C4×C53—21no—
124C4×C53—81no—
125Dic53—44no—
126Dic53—82no—
127Dic53—161no—
128#2683—531no—
129#2093—531no—
130#2083—532no—
131#1271—532no—
132#1270D5311yescharacteristic
133C2×C53—82no—
134C2×C53#161012yes—
135C2×C53#161411yescharacteristic
136C2×C53#161911yescharacteristic
137C2×C53#164912yes—
138C2×C53#166011yescharacteristic
139C2×C22⋊Q8—1062no—
140C2×#107—1062no—
141C2×#107—1061no—
142C2×#102—1061no—
143#396—1062no—
144#395—1061no—
145#390C2×D5311yescharacteristic
146#389—532no—
147#386—1061no—
148#384—531no—
149#340—1062no—
150#336—531no—
151#336C2×D5312yes—
152C53#779711yescharacteristic, complemented, Sylow
153C22×Q8—1061no—
154C22×D4D21211yescharacteristic
155C23×C4—1061no—
156C22×C8—22no—
157C22×C8—1061no—
158C22⋊Q8—2124no—
159C22⋊Q8—2124no—
160C2×C4⋊C4—532no—
161C2×C4⋊C4—532no—
162C2×C4⋊C4—1062no—
163C2×C4⋊C4—1062no—
164C2×C4⋊C4—1061no—
165C2×C4⋊C4—1062no—
166C2×C4⋊C4D21211yescharacteristic
167C2×C22⋊C4—22no—
168C2×C22⋊C4—1061no—
169C2×C22⋊C4—1062no—
170C2×C42C22×D5311yescharacteristic
171#107—2122no—
172#107—2124no—
173#102—2122no—
174#95—1061no—
175#95—1062no—
176C24—21no—
177C2×Q8—4241no—
178C2×Q8—2124no—
179C2×D4—21no—
180C2×D4—41no—
181C2×D4—41no—
182C2×D4Dic21212yes—
183C22×C4—21no—
184C22×C4—1062no—
185C22×C4—1062no—
186C22×C4—4241no—
187C22×C4—2124no—
188C22×C4—2121no—
189C22×C4C2×D21211yescharacteristic
190C22×C4D4×D5312yes—
191C2×C8—24no—
192C2×C8—42no—
193C2×C8—42no—
194C2×C8—1062no—
195C2×C8—2121no—
196C2×C8—2121no—
197C4⋊C4—41no—
198C4⋊C4—534no—
199C4⋊C4—1062no—
200C4⋊C4—2121no—
201C4⋊C4—2121no—
202C4⋊C4—2122no—
203C4⋊C4—2124no—
204C4⋊C4—2122no—
205C4⋊C4—2122no—
206C4⋊C4—2122no—
207C4⋊C4—2122no—
208C22⋊C4—42no—
209C22⋊C4—42no—
210C22⋊C4—4241no—
211C22⋊C4—2124no—
212C42—41no—
213C23—22no—
214C23—42no—
215C23—41no—
216C23—42no—
217C23—41no—
218C23#1313318711yescharacteristic
219Q8—4244no—
220D4—42no—
221D4—81no—
222C2×C4—21no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—21no—
226C2×C4—42no—
227C2×C4—41no—
228C2×C4—1064no—
229C2×C4—4242no—
230C2×C4—4241no—
231C2×C4—4244no—
232C2×C4—2122no—
233C2×C4—2122no—
234C2×C4—2122no—
235C2×C4—2122no—
236C2×C4—2122no—
237C2×C4D8×D5312yes—
238C2×C4C2×Dic21211yescharacteristic
239C2×C4#1313324412yes—
240C2×C4#1313325011yescharacteristic
241C8—44no—
242C8—2122no—
243C22—44no—
244C22—42no—
245C22—82no—
246C22—81no—
247C22—81no—
248C22#1681075812yes—
249C22#1681078111yescharacteristic
250C22#1681079311yescharacteristic
251C22#1681080712yes—
252C22#1681081411yescharacteristic
253C4—22no—
254C4—22no—
255C4—21no—
256C4—21no—
257C4—81no—
258C4—8481no—
259C4—4242no—
260C4—2124no—
261C2—82no—
262C2#2027371912yes—
263C2#2027374211yescharacteristic
264C2#2027377011yescharacteristic
265C2#2027396712yes—
266C2#2027402711yescharacteristic
subgroupG/H
centerC23#13133187
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#1270C2
socleC23×C53#120
prime radicalC22×D4×C53D4
hypercenter#1270D53
hypocenterC53#7797
LCSG ▷ C2×C42×C53 ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1270

Complements

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

NH ≅ G/Ntype
3. #20276188C2semidirect
4. #20272539C2semidirect
152. C53#7797semidirect

Maximal subgroups

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

subgroupindex
#202810872
#202762432
#202761882
#202725392
C53×#12702
#779753