Finite Group Databasebeta

← #27285416

Subgroups — #27285416

Order 13568.

600 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 600 conjugacy classes of subgroups — too many to draw. Only its 97 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
1C2×#16810982C212yesmaximal, complemented
2#20275534C211yesmaximal, characteristic, complemented
3C2×#16810363C212yesmaximal, complemented
4#20272596C211yesmaximal, characteristic, complemented
5C53×#2819C211yesmaximal, characteristic
6C2×#13133217C2211yescharacteristic
7#16810982—24no—
8#16810982—24no—
9C2×#13133141—22no—
10C2×#13133140—22no—
11C2×#13133138—22no—
12C2×#13133128C2211yescharacteristic
13C2×#13133128C2212yes—
14#16810363—24no—
15C2×#124×C53C2212yes—
16C53×#394—22no—
17C53×#393—22no—
18C53×#392—22no—
19C53×#382C2211yescharacteristic
20C22×Dic106—21no—
21C22×#88322—22no—
22C22×#88322—21no—
23#13133217—21no—
24#13133217—22no—
25#13133217D412yescomplemented
26C2×C4⋊Dic53—22no—
27#13133141—28no—
28#13133140—44no—
29#13133138—28no—
30#13133128—22no—
31#13133128—22no—
32#13133128D414yescomplemented
33C22×Q8×C53D412yes—
34C22×D4×C53—22no—
35C22×D4×C53D412yes—
36#124×C53—22no—
37#124×C53—22no—
38#124×C53—22no—
39#124×C53—22no—
40C2×C4⋊C4×C53D412yes—
41C2×C22⋊C4×C53—24no—
42C2×C22⋊C4×C53—22no—
43C2×C22⋊C4×C53—22no—
44C2×C42×C53C2311yescharacteristic
45#95×C53—22no—
46#95×C53—22no—
47C24×C53—22no—
48C2×Q8×C53—22no—
49C2×Q8×C53—22no—
50C2×Q8×C53—44no—
51C2×D4×C53—22no—
52C2×D4×C53—22no—
53C2×D4×C53—44no—
54C2×D4×C53—82no—
55C2×D4×C53—84no—
56C22×C4×C53—22no—
57C22×C4×C53—22no—
58C22×C4×C53C2×D412yes—
59C22×C4×C53C2×D411yescharacteristic
60C22×Dic53—41no—
61C2×Dic106—22no—
62C2×Dic106—42no—
63C2×Dic106—42no—
64C2×Dic106—41no—
65C4⋊C4×C53—42no—
66C22⋊C4×C53—42no—
67C22⋊C4×C53—42no—
68C22⋊C4×C53—42no—
69C22⋊C4×C53—42no—
70C22⋊C4×C53—84no—
71C42×C53C2×D411yescharacteristic
72C42×C53C2×D412yes—
73C42×C53#4111yescharacteristic
74C4⋊Dic53—24no—
75C4⋊Dic53—42no—
76C2×#88322—22no—
77C2×#88322—24no—
78C2×#88322—41no—
79C2×#88322—44no—
80C2×#88322—41no—
81C23×C53—42no—
82C23×C53—42no—
83C23×C53—42no—
84C23×C53—44no—
85C23×C53—44no—
86C23×C53#12011yescharacteristic
87Q8×C53—44no—
88Q8×C53—82no—
89D4×C53—44no—
90D4×C53—82no—
91D4×C53—88no—
92C2×C4×C53—21no—
93C2×C4×C53—21no—
94C2×C4×C53—24no—
95C2×C4×C53—42no—
96C2×C4×C53—42no—
97C2×C4×C53—44no—
98C2×C4×C53—44no—
99C2×C4×C53#12012yes—
100C2×C4×C53#12111yescharacteristic
101C2×C4×C53#12112yes—
102C2×C4×C53#12711yescharacteristic
103C2×Dic53—42no—
104C2×Dic53—81no—
105C2×Dic53—81no—
106Dic106—44no—
107Dic106—82no—
108#88322—44no—
109#88322—42no—
110#10085—531nomaximal, Sylow
111C22×C53—84no—
112C22×C53—84no—
113C22×C53—82no—
114C22×C53—82no—
115C22×C53—82no—
116C22×C53#39211yescharacteristic
117C22×C53#45311yescharacteristic
118C22×C53#45411yescharacteristic
119C22×C53#45514yes—
120C4×C53—22no—
121C4×C53—22no—
122C4×C53—21no—
123C4×C53—21no—
124C4×C53—82no—
125C4×C53—82no—
126Dic53—82no—
127C2×#489—532no—
128#2819D5311yescharacteristic
129#1960—531no—
130C2×#329—532no—
131#1295—531no—
132C2×C53—162no—
133C2×C53#165912yes—
134C2×C53#166212yes—
135C2×C53#200412yes—
136C2×C53#200611yescharacteristic
137C2×C4⋊Q8—531no—
138C2×#124C2×D5312yes—
139#489—1064no—
140#489—1064no—
141C2×#103—1062no—
142C2×#103—1062no—
143C2×#102—1062no—
144C2×#97—532no—
145C2×#97—531no—
146#394—22no—
147#393—22no—
148#392—22no—
149#382C2×D5311yescharacteristic
150#329—1064no—
151C53#1008511yescharacteristic, complemented, Sylow
152C22×Q8—1061no—
153C22×Q8C22⋊D5312yes—
154C22×D4—22no—
155C22×D4C22⋊D5312yes—
156C22×C8—1062no—
157C22×C8—1061no—
158C4⋊Q8—532no—
159C4⋊Q8—1062no—
160C4⋊Q8—1061no—
161#124—22no—
162#124—22no—
163#124—22no—
164#124—22no—
165C2×C4⋊C4—1062no—
166C2×C4⋊C4C22⋊D5312yes—
167C2×C22⋊C4—22no—
168C2×C22⋊C4—22no—
169C2×C22⋊C4—24no—
170C2×C42C22×D5311yescharacteristic
171#103—1068no—
172#103—2124no—
173#102—1068no—
174#97—534no—
175#97—1062no—
176#97—1062no—
177#95—22no—
178#95—22no—
179C24—22no—
180C2×Q8—22no—
181C2×Q8—22no—
182C2×Q8—44no—
183C2×Q8—1062no—
184C2×Q8—2122no—
185C2×Q8—2122no—
186C2×Q8—2121no—
187C2×D4—22no—
188C2×D4—22no—
189C2×D4—44no—
190C2×D4—82no—
191C2×D4—84no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—2121no—
195C22×C4C2×C22⋊D5312yes—
196C22×C4C2×C22⋊D5311yescharacteristic
197C2×C8—1062no—
198C2×C8—1064no—
199C2×C8—2121no—
200C2×C8—2124no—
201C2×C8—2121no—
202C4⋊C4—42no—
203C4⋊C4—1064no—
204C4⋊C4—2122no—
205C22⋊C4—42no—
206C22⋊C4—42no—
207C22⋊C4—42no—
208C22⋊C4—42no—
209C22⋊C4—84no—
210C42D4×D5311yescharacteristic
211C42D4×D5312yes—
212C42#1171994711yescharacteristic
213C23—42no—
214C23—42no—
215C23—42no—
216C23—44no—
217C23—44no—
218C23#1313318611yescharacteristic
219Q8—44no—
220Q8—82no—
221Q8—4242no—
222Q8—2124no—
223D4—44no—
224D4—82no—
225D4—88no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—24no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—44no—
232C2×C4—44no—
233C2×C4—4241no—
234C2×C4—4241no—
235C2×C4—2122no—
236C2×C4#1313318512yes—
237C2×C4#1313318811yescharacteristic
238C2×C4#1313318912yes—
239C2×C4#1313321311yescharacteristic
240C8—2124no—
241C8—2122no—
242C22—82no—
243C22—82no—
244C22—82no—
245C22—84no—
246C22—84no—
247C22#1681053111yescharacteristic
248C22#1681079111yescharacteristic
249C22#1681079611yescharacteristic
250C22#1681080114yes—
251C4—22no—
252C4—22no—
253C4—21no—
254C4—21no—
255C4—82no—
256C4—82no—
257C4—4242no—
258C2—162no—
259C2#2027402112yes—
260C2#2027403812yes—
261C2#2027574212yes—
262C2#2027575711yescharacteristic
subgroupG/H
centerC23#13133186
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
Fitting, prime radicalC53×#2819C2
socleC23×C53#120
hypercenter#2819D53
hypocenterC53#10085
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2819

Complements

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

NH ≅ G/Ntype
1. C2×#16810982C2semidirect
2. #20275534C2semidirect
3. C2×#16810363C2semidirect
4. #20272596C2semidirect
25. #13133217D4semidirect
32. #13133128D4semidirect
151. C53#10085semidirect

Maximal subgroups

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

subgroupindex
C2×#168109822
#202755342
C2×#168103632
#202725962
C53×#28192
#1008553