Finite Group Databasebeta

← #27296686

Subgroups — #27296686

Order 13568.

372 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 59 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#20282987C211yesmaximal, characteristic, complemented
2#20273684C212yesmaximal
3#20272744C212yesmaximal, complemented
4#20272659C211yesmaximal, characteristic, complemented
5C53×#2040C211yesmaximal, characteristic
6#16810761—22no—
7#16810565—22no—
8#16810562—21no—
9#16810544—22no—
10#16810543C2212yes—
11#16810489—21no—
12#16810357—22no—
13#16810349C2212yes—
14C53×#537C2211yescharacteristic
15C2×#104×C53—21no—
16C53×#417—21no—
17C53×#382C2211yescharacteristic
18C53×#343C2211yescharacteristic
19C22×Dic106D412yes—
20C2×#11719923—42no—
21C2×C4⋊Dic53—21no—
22C2×C4⋊Dic53—21no—
23C2×C4⋊Dic53—21no—
24C2×#11719919—22no—
25#13133140—42no—
26C22⋊#88322—22no—
27#13133124—22no—
28#13133124—22no—
29#13133124—21no—
30#13133124—22no—
31#13133124D412yes—
32C2×#41×C53—21no—
33C2×M16×C53—21no—
34C22⋊Q8×C53—41no—
35#121×C53—22no—
36#121×C53—41no—
37C2×C4⋊C4×C53D411yescharacteristic
38C2×C4⋊C4×C53C2311yescharacteristic
39C2×C42×C53D411yescharacteristic
40#104×C53—42no—
41#103×C53—41no—
42#102×C53—41no—
43#95×C53—22no—
44#41×C53—41no—
45#41×C53—41no—
46#41×C53—41no—
47#41×C53—41no—
48C2×Q8×C53—21no—
49C2×D4×C53—21no—
50C2×D4×C53—22no—
51C2×D4×C53—41no—
52C22×C4×C53—21no—
53C22×C4×C53—21no—
54C22×C4×C53—41no—
55C22×C4×C53C2×D411yescharacteristic
56C22×C4×C53C2×D412yes—
57C22×Dic53—22no—
58C22×Dic53—22no—
59C2×Dic106—42no—
60C2×Dic106—82no—
61M16×C53—41no—
62M16×C53—41no—
63C2×C8×C53—41no—
64C4⋊C4×C53—22no—
65C4⋊C4×C53—41no—
66C4⋊C4×C53—41no—
67C22⋊C4×C53—42no—
68C42×C53—21no—
69C42×C53SD1612yescomplemented
70#11719923—84no—
71C4⋊Dic53—81no—
72C4⋊Dic53—81no—
73C4⋊Dic53—81no—
74#11719919—82no—
75C2×#88322—42no—
76C23×C53—41no—
77C23×C53#12011yescharacteristic
78Q8×C53—41no—
79Q8×C53—41no—
80D4×C53—41no—
81D4×C53—41no—
82D4×C53—82no—
83D4×C53—81no—
84D4×C53—81no—
85C2×C4×C53—21no—
86C2×C4×C53—21no—
87C2×C4×C53—22no—
88C2×C4×C53—22no—
89C2×C4×C53—41no—
90C2×C4×C53—41no—
91C2×C4×C53—42no—
92C2×C4×C53—42no—
93C2×C4×C53—41no—
94C2×C4×C53—41no—
95C2×C4×C53—81no—
96C2×C4×C53—81no—
97C2×C4×C53C2×SD1611yescharacteristic
98C2×C4×C53#13711yescharacteristic
99C2×Dic53—42no—
100C2×Dic53—42no—
101C2×Dic53—42no—
102C2×Dic53—42no—
103C2×Dic53—42no—
104C2×Dic53—42no—
105Dic106—44no—
106Dic106—82no—
107C8×C53—81no—
108#88322—82no—
109#12184—531nomaximal, Sylow
110C22×C53—21no—
111C22×C53—21no—
112C22×C53—81no—
113C22×C53—81no—
114C22×C53—81no—
115C22×C53#44811yescharacteristic
116C22×C53#45411yescharacteristic
117C22×C53B(4,𝔽2)11yescharacteristic
118C4×C53—21no—
119C4×C53—21no—
120C4×C53—42no—
121C4×C53—42no—
122C4×C53—81no—
123C4×C53—81no—
124Dic53—82no—
125Dic53—82no—
126#2844—531no—
127#2040D5311yescharacteristic
128#1604—532no—
129#1354—532no—
130#1324—531no—
131C2×C53—21no—
132C2×C53—21no—
133C2×C53—161no—
134C2×C53#159911yescharacteristic
135C2×C53#219911yescharacteristic
136C2×C53#220311yescharacteristic
137#537C2×D5311yescharacteristic
138#448—1062no—
139C2×#104—21no—
140#417—21no—
141#400—1062no—
142#400—1061no—
143#395—532no—
144#395—1062no—
145#382—1061no—
146#382C2×D5311yescharacteristic
147#343C2×D5311yescharacteristic
148#327—1062no—
149#324—532no—
150C53#1218411yescharacteristic, complemented, Sylow
151C2×#41—21no—
152C22×Q8—532no—
153C2×SD16—2122no—
154C2×M16—21no—
155C22⋊Q8—41no—
156#121—22no—
157#121—41no—
158C2×C4⋊C4—1061no—
159C2×C4⋊C4—1061no—
160C2×C4⋊C4—1061no—
161C2×C4⋊C4—1062no—
162C2×C4⋊C4C22⋊D5311yescharacteristic
163C2×C4⋊C4C22×D5311yescharacteristic
164C2×C42C22⋊D5311yescharacteristic
165#104—42no—
166#103—41no—
167#103—2122no—
168#102—41no—
169C22⋊C8—1062no—
170#95—22no—
171#95—532no—
172#95—1062no—
173#95—1062no—
174#95—1061no—
175#95—1062no—
176#41—41no—
177#41—41no—
178#41—41no—
179#41—41no—
180C2×Q8—21no—
181C2×Q8—4242no—
182C2×Q8—2122no—
183C2×D4—21no—
184C2×D4—22no—
185C2×D4—41no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—41no—
189C22×C4—1062no—
190C22×C4—1062no—
191C22×C4D4×D5312yes—
192C22×C4C2×C22⋊D5311yescharacteristic
193SD16—4244no—
194M16—41no—
195M16—41no—
196C2×C8—41no—
197C2×C8—2122no—
198C4⋊C4—22no—
199C4⋊C4—41no—
200C4⋊C4—41no—
201C4⋊C4—4242no—
202C4⋊C4—4241no—
203C4⋊C4—4241no—
204C4⋊C4—4241no—
205C22⋊C4—42no—
206C42—21no—
207C42#1171992312yescomplemented
208C23—41no—
209C23#1313318511yescharacteristic
210Q8—41no—
211Q8—41no—
212Q8—4242no—
213Q8—2124no—
214D4—41no—
215D4—41no—
216D4—82no—
217D4—81no—
218D4—81no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—81no—
230C2×C4—81no—
231C2×C4—2122no—
232C2×C4—2122no—
233C2×C4—2122no—
234C2×C4—2122no—
235C2×C4—2122no—
236C2×C4—2122no—
237C2×C4C2×#1171992311yescharacteristic
238C2×C4#1313324711yescharacteristic
239C8—81no—
240C8—4242no—
241C22—21no—
242C22—21no—
243C22—81no—
244C22—81no—
245C22—81no—
246C22#1681076111yescharacteristic
247C22#1681079511yescharacteristic
248C22#1681080911yescharacteristic
249C4—21no—
250C4—21no—
251C4—42no—
252C4—42no—
253C4—81no—
254C4—81no—
255C4—4242no—
256C4—4242no—
257C2—21no—
258C2—21no—
259C2—161no—
260C2#2027365411yescharacteristic
261C2#2027671511yescharacteristic
262C2#2027673511yescharacteristic
subgroupG/H
centerC22#16810809
commutator, cosocleC2×C4⋊C4×C53C23
FrattiniC2×C4⋊C4C22×D53
FittingC53×#2040C2
socleC22×C53B(4,𝔽2)
prime radicalC53×#537C22
hypercenter#2040D53
hypocenterC53#12184
LCSG ▷ C2×C4⋊C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2040

Complements

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

NH ≅ G/Ntype
1. #20282987C2semidirect
3. #20272744C2semidirect
4. #20272659C2semidirect
69. C42×C53SD16semidirect
150. C53#12184semidirect
207. C42#11719923semidirect

Maximal subgroups

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

subgroupindex
#202829872
#202736842
#202727442
#202726592
C53×#20402
#1218453