Finite Group Databasebeta

← #27273190

Subgroups — #27273190

Order 13568.

478 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 95 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#20276243C211yesmaximal, characteristic
2#20276186C212yesmaximal, complemented
3#20272538C212yesmaximal, complemented
4#20272536C211yesmaximal, characteristic
5C53×#2683C211yesmaximal, characteristic
6C2×#13133150—21no—
7C2×#13133150—22no—
8C2×#13133138—21no—
9#16810493C2211yescharacteristic
10#16810385—22no—
11#16810378C2211yescharacteristic
12#16810378C2212yes—
13C2×C22⋊Q8×C53—22no—
14C53×#396—22no—
15C53×#395—21no—
16C53×#390C2211yescharacteristic
17C53×#389C2212yes—
18C53×#386—21no—
19C22×#88322—21no—
20C22×#88322—22no—
21C2×C4⋊Dic53—21no—
22C2×C4⋊Dic53—22no—
23#13133150—42no—
24#13133150—44no—
25#13133138—42no—
26C22×Q8×C53—21no—
27C22×D4×C53D411yescharacteristic
28C23×C4×C53—21no—
29C22⋊Q8×C53—44no—
30C22⋊Q8×C53—44no—
31C2×C4⋊C4×C53—22no—
32C2×C4⋊C4×C53—22no—
33C2×C4⋊C4×C53D412yes—
34C2×C4⋊C4×C53D412yes—
35C2×C4⋊C4×C53D411yescharacteristic
36C2×C22⋊C4×C53—22no—
37C2×C22⋊C4×C53—21no—
38C2×C22⋊C4×C53—22no—
39C2×C42×C53C2311yescharacteristic
40#95×C53—21no—
41#95×C53—22no—
42C24×C53—21no—
43C2×Q8×C53—44no—
44C2×Q8×C53—81no—
45C2×D4×C53—21no—
46C2×D4×C53—41no—
47C2×D4×C53—41no—
48C2×D4×C53Q1612yes—
49C22×C4×C53—22no—
50C22×C4×C53—22no—
51C22×C4×C53—21no—
52C22×C4×C53—44no—
53C22×C4×C53—81no—
54C22×C4×C53C2×D412yes—
55C22×C4×C53C2×D411yescharacteristic
56C22×Dic53—41no—
57C4⋊C4×C53—22no—
58C4⋊C4×C53—44no—
59C4⋊C4×C53—42no—
60C4⋊C4×C53—41no—
61C4⋊C4×C53—42no—
62C4⋊C4×C53—42no—
63C4⋊C4×C53D814yes—
64C22⋊C4×C53—44no—
65C22⋊C4×C53—42no—
66C22⋊C4×C53—42no—
67C22⋊C4×C53—81no—
68C42×C53—41no—
69C4⋊Dic53—41no—
70C4⋊Dic53—41no—
71C4⋊Dic53—42no—
72C4⋊Dic53—42no—
73C2×#88322—22no—
74C2×#88322—24no—
75C2×#88322—41no—
76C2×#88322—42no—
77C2×#88322—42no—
78C2×#88322—41no—
79C23×C53—22no—
80C23×C53—42no—
81C23×C53—41no—
82C23×C53—42no—
83C23×C53—41no—
84C23×C53#12011yescharacteristic
85Q8×C53—84no—
86D4×C53—42no—
87D4×C53—81no—
88C2×C4×C53—21no—
89C2×C4×C53—22no—
90C2×C4×C53—21no—
91C2×C4×C53—22no—
92C2×C4×C53—24no—
93C2×C4×C53—42no—
94C2×C4×C53—42no—
95C2×C4×C53—42no—
96C2×C4×C53—42no—
97C2×C4×C53—41no—
98C2×C4×C53—42no—
99C2×C4×C53—42no—
100C2×C4×C53—84no—
101C2×C4×C53C2×D812yes—
102C2×C4×C53C2×Q1611yescharacteristic
103C2×C4×C53AGL(1,ℤ/8)12yes—
104C2×C4×C53#13711yescharacteristic
105C2×Dic53—82no—
106C2×Dic53—81no—
107#88322—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—44no—
125C4×C53—82no—
126C4×C53—81no—
127Dic53—161no—
128#2683D5311yescharacteristic
129#2093—531no—
130#2083—532no—
131#1271—532no—
132#1270—531no—
133C2×C53—82no—
134C2×C53#161012yes—
135C2×C53#161411yescharacteristic
136C2×C53#161911yescharacteristic
137C2×C53#164912yes—
138C2×C53#166011yescharacteristic
139C2×C22⋊Q8—22no—
140C2×#107—1062no—
141C2×#107—1061no—
142C2×#102—1061no—
143#396—22no—
144#395—21no—
145#390C2×D5311yescharacteristic
146#389C2×D5312yes—
147#386—21no—
148#384—531no—
149#340—1062no—
150#336—532no—
151#336—531no—
152C53#779711yescharacteristic, complemented, Sylow
153C22×Q8—21no—
154C22×D4C22⋊D5311yescharacteristic
155C23×C4—21no—
156C22×C8—1062no—
157C22×C8—1061no—
158C22⋊Q8—44no—
159C22⋊Q8—44no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—1062no—
163C2×C4⋊C4—1061no—
164C2×C4⋊C4C22⋊D5312yes—
165C2×C4⋊C4C22⋊D5312yes—
166C2×C4⋊C4C22⋊D5311yescharacteristic
167C2×C22⋊C4—22no—
168C2×C22⋊C4—22no—
169C2×C22⋊C4—21no—
170C2×C42C22×D5311yescharacteristic
171#107—2122no—
172#107—2124no—
173#102—2122no—
174#95—21no—
175#95—22no—
176C24—21no—
177C2×Q8—44no—
178C2×Q8—81no—
179C2×D4—21no—
180C2×D4—41no—
181C2×D4—41no—
182C2×D4#1171992512yes—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—21no—
186C22×C4—44no—
187C22×C4—81no—
188C22×C4—2121no—
189C22×C4C2×C22⋊D5312yes—
190C22×C4C2×C22⋊D5311yescharacteristic
191C2×C8—1062no—
192C2×C8—1064no—
193C2×C8—2122no—
194C2×C8—2121no—
195C2×C8—2122no—
196C2×C8—2121no—
197C4⋊C4—22no—
198C4⋊C4—41no—
199C4⋊C4—42no—
200C4⋊C4—44no—
201C4⋊C4—42no—
202C4⋊C4—42no—
203C4⋊C4—2121no—
204C4⋊C4—2121no—
205C4⋊C4—2122no—
206C4⋊C4—2122no—
207C4⋊C4#1171992214yes—
208C22⋊C4—44no—
209C22⋊C4—42no—
210C22⋊C4—42no—
211C22⋊C4—81no—
212C42—41no—
213C23—22no—
214C23—42no—
215C23—41no—
216C23—42no—
217C23—41no—
218C23#1313318611yescharacteristic
219Q8—84no—
220D4—42no—
221D4—81no—
222C2×C4—21no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—21no—
226C2×C4—24no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—41no—
232C2×C4—42no—
233C2×C4—42no—
234C2×C4—84no—
235C2×C4—4242no—
236C2×C4—4241no—
237C2×C4C2×#1171992212yes—
238C2×C4C2×#1171992511yescharacteristic
239C2×C4#1313324312yes—
240C2×C4#1313324811yescharacteristic
241C8—2124no—
242C8—2122no—
243C22—44no—
244C22—42no—
245C22—82no—
246C22—81no—
247C22—81no—
248C22#1681075712yes—
249C22#1681077911yescharacteristic
250C22#1681079111yescharacteristic
251C22#1681080612yes—
252C22#1681081311yescharacteristic
253C4—22no—
254C4—22no—
255C4—21no—
256C4—21no—
257C4—44no—
258C4—82no—
259C4—81no—
260C4—8481no—
261C2—82no—
262C2#2027371812yes—
263C2#2027374011yescharacteristic
264C2#2027376911yescharacteristic
265C2#2027396412yes—
266C2#2027402611yescharacteristic
subgroupG/H
centerC23#13133186
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#2683C2
socleC23×C53#120
prime radicalC22×D4×C53D4
hypercenter#2683D53
hypocenterC53#7797
LCSG ▷ C2×C42×C53 ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2683

Complements

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

NH ≅ G/Ntype
2. #20276186C2semidirect
3. #20272538C2semidirect
152. C53#7797semidirect

Maximal subgroups

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

subgroupindex
#202762432
#202761862
#202725382
#202725362
C53×#26832
#779753