Finite Group Databasebeta

← #27366011

Subgroups — #27366011

Order 13568.

920 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 920 conjugacy classes of subgroups — too many to draw. Only its 171 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#20285726C212yesmaximal, complemented
2#20285237C211yesmaximal, characteristic, complemented
3#20274032C218yesmaximal, complemented
4#20273353C211yesmaximal, characteristic, complemented
5#20273326C212yesmaximal, complemented
6C53×#3548C211yesmaximal, characteristic
7#16811180C2211yescharacteristic
8#16811033C2214yes—
9#16811029—22no—
10#16811018C2214yes—
11#16810982C2214yes—
12#16810830—22no—
13#16810820—22no—
14#16810785—24no—
15#16810680C2212yes—
16#16810659—22no—
17C2×#13133128C2211yescharacteristic
18#16810365C2214yes—
19#16810363C2218yes—
20C53×#562C2214yes—
21C53×#554—22no—
22C53×#550—22no—
23C53×#547—22no—
24C53×#540—22no—
25C53×#537—24no—
26C53×#533C2211yescharacteristic
27C53×#532C2212yes—
28C2×#11719925—24no—
29C2×#11719923—24no—
30C22×#88322—21no—
31#13133217C2312yes—
32#13133209—21no—
33#13133196—22no—
34C4×Dic106—22no—
35#13133176—21no—
36#13133150—24no—
37#13133148—24no—
38C4⋊#88322—22no—
39#13133141—24no—
40#13133138—24no—
41C22⋊#88322—22no—
42#13133128—21no—
43#13133128C2314yes—
44#13133128C2312yes—
45C2×#41×C53—24no—
46C2×#41×C53D414yes—
47C4⋊Q8×C53D412yes—
48C4⋊Q8×C53D412yes—
49#127×C53—22no—
50#127×C53D412yes—
51#125×C53—22no—
52#125×C53C2312yes—
53#124×C53C2314yes—
54#123×C53—28no—
55C22⋊Q8×C53—24no—
56C22⋊Q8×C53—28no—
57C22⋊Q8×C53—22no—
58#121×C53—28no—
59#121×C53—24no—
60#121×C53—28no—
61#121×C53—22no—
62#121×C53—22no—
63C4×Q8×C53—22no—
64C4×D4×C53—24no—
65C4×D4×C53—22no—
66C4×D4×C53—22no—
67#117×C53D412yes—
68C2×C4⋊C4×C53—24no—
69C2×C42×C53C2311yescharacteristic
70#41×C53—84no—
71#41×C53—88no—
72C2×Q8×C53—22no—
73C2×Q8×C53C2×D414yes—
74C2×D4×C53—28no—
75C2×D4×C53—24no—
76C2×D4×C53—22no—
77C2×D4×C53—22no—
78C2×D4×C53C2×D414yes—
79C22×C4×C53—28no—
80C22×C4×C53—24no—
81C22×C4×C53C2×D411yescharacteristic
82C22×C4×C53C2×D412yes—
83C2×Dic106—22no—
84C4⋊C4×C53—28no—
85C4⋊C4×C53—22no—
86C4⋊C4×C53—22no—
87C4⋊C4×C53—22no—
88C4⋊C4×C53—24no—
89C4⋊C4×C53C2×D414yes—
90C22⋊C4×C53—28no—
91C22⋊C4×C53—22no—
92C22⋊C4×C53—24no—
93C22⋊C4×C53—24no—
94C42×C53—22no—
95C42×C53C2×D411yescharacteristic
96C42×C53C2×D412yes—
97C42×C53C2411yescharacteristic
98C22⋊Dic53—41no—
99#11719925—84no—
100#11719923—84no—
101C4⋊Dic53—22no—
102C4⋊Dic53—22no—
103#11719919—42no—
104C4×Dic53—41no—
105C2×#88322—24no—
106C2×#88322—22no—
107C2×#88322—24no—
108C23×C53—24no—
109C23×C53#12011yescharacteristic
110Q8×C53—44no—
111Q8×C53—44no—
112D4×C53—44no—
113D4×C53—44no—
114D4×C53—44no—
115D4×C53—84no—
116D4×C53—88no—
117C2×C4×C53—22no—
118C2×C4×C53—24no—
119C2×C4×C53—24no—
120C2×C4×C53—42no—
121C2×C4×C53—48no—
122C2×C4×C53—88no—
123C2×C4×C53#12012yes—
124C2×C4×C53#12011yescharacteristic
125C2×C4×C53C22×D412yes—
126C2×C4×C53C22×D411yescharacteristic
127C2×Dic53—42no—
128Dic106—42no—
129Dic106—42no—
130#88322—24no—
131#88322—44no—
132#19065—531nomaximal, Sylow
133C22×C53—21no—
134C22×C53—22no—
135C22×C53—44no—
136C22×C53—44no—
137C22×C53—44no—
138C22×C53C2×#12011yescharacteristic
139C4×C53—21no—
140C4×C53—22no—
141C4×C53—22no—
142C4×C53—44no—
143C4×C53—44no—
144C4×C53#45612yes—
145Dic53—82no—
146#3548D5311yescharacteristic
147#3216—532no—
148#3162—531no—
149#1661—538no—
150#1537—531no—
151#1530—532no—
152C2×C53—41no—
153C2×C53—84no—
154C2×C53C2×#45611yescharacteristic
155C2×C53#301912yes—
156#562C2×D5314yes—
157#554—22no—
158#550—22no—
159#547—22no—
160#540—22no—
161#537—24no—
162#533—531no—
163#533C2×D5311yescharacteristic
164#532C2×D5312yes—
165#501—534no—
166#501—1062no—
167#497—534no—
168#489—534no—
169#462—1062no—
170#460—1062no—
171#452—1064no—
172#432—532no—
173#427—1062no—
174C2×#97—531no—
175#330—534no—
176#329—538no—
177C53#1906511yescharacteristic, complemented, Sylow
178C2×#41—24no—
179C2×#41C22⋊D5314yes—
180C2×Q16—1064no—
181C2×SD16—1064no—
182C22×C8—1061no—
183C4⋊Q8—532no—
184C4⋊Q8C22⋊D5312yes—
185C4⋊Q8C22⋊D5312yes—
186#127—22no—
187#127C22⋊D5312yes—
188#125—22no—
189#125—1061no—
190#125C22×D5312yes—
191#124C22×D5314yes—
192#123—28no—
193C22⋊Q8—24no—
194C22⋊Q8—28no—
195C22⋊Q8—22no—
196C22⋊Q8—1062no—
197#121—28no—
198#121—24no—
199#121—28no—
200#121—22no—
201#121—22no—
202C4×Q8—22no—
203C4×Q8—1062no—
204C4×D4—24no—
205C4×D4—22no—
206C4×D4—22no—
207#117—1061no—
208#117C22⋊D5312yes—
209C2×C4⋊C4—24no—
210C2×C42C22×D5311yescharacteristic
211#107—1064no—
212#106—1064no—
213C4⋊C8—1062no—
214#103—1064no—
215#102—1064no—
216C22⋊C8—1062no—
217#97—532no—
218#97—534no—
219#97—1061no—
220#41—84no—
221#41—88no—
222C2×Q8—22no—
223C2×Q8—1062no—
224C2×Q8C2×C22⋊D5314yes—
225C2×D4—28no—
226C2×D4—24no—
227C2×D4—22no—
228C2×D4—22no—
229C2×D4C2×C22⋊D5314yes—
230C22×C4—28no—
231C22×C4—24no—
232C22×C4C2×C22⋊D5311yescharacteristic
233C22×C4C2×C22⋊D5312yes—
234Q16—4244no—
235SD16—4244no—
236C2×C8—1064no—
237C2×C8—1062no—
238C2×C8—1064no—
239C4⋊C4—28no—
240C4⋊C4—22no—
241C4⋊C4—22no—
242C4⋊C4—22no—
243C4⋊C4—24no—
244C4⋊C4—1062no—
245C4⋊C4—1062no—
246C4⋊C4—2122no—
247C4⋊C4C2×C22⋊D5314yes—
248C22⋊C4—28no—
249C22⋊C4—22no—
250C22⋊C4—24no—
251C22⋊C4—24no—
252C22⋊C4—2121no—
253C42—22no—
254C42—2121no—
255C42C2×C22⋊D5311yescharacteristic
256C42C2×C22⋊D5312yes—
257C42C23×D5311yescharacteristic
258C23—24no—
259C23#1313318611yescharacteristic
260Q8—44no—
261Q8—44no—
262Q8—2122no—
263Q8—2122no—
264D4—44no—
265D4—44no—
266D4—44no—
267D4—84no—
268D4—88no—
269C2×C4—22no—
270C2×C4—24no—
271C2×C4—24no—
272C2×C4—42no—
273C2×C4—48no—
274C2×C4—88no—
275C2×C4—2122no—
276C2×C4#1313318612yes—
277C2×C4#1313318611yescharacteristic
278C2×C4C22×C22⋊D5311yescharacteristic
279C2×C4C22×C22⋊D5312yes—
280C8—1064no—
281C8—2124no—
282C22—21no—
283C22—22no—
284C22—44no—
285C22—44no—
286C22—44no—
287C22C2×#1313318611yescharacteristic
288C4—21no—
289C4—22no—
290C4—22no—
291C4—44no—
292C4—44no—
293C4—4242no—
294C4#1681080612yes—
295C2—41no—
296C2—84no—
297C2C2×#1681080611yescharacteristic
298C2#2028407212yes—
subgroupG/H
centerC22C2×#13133186
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
Fitting, prime radicalC53×#3548C2
socleC22×C53C2×#120
hypercenter#3548D53
hypocenterC53#19065
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3548

Complements

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

NH ≅ G/Ntype
1. #20285726C2semidirect
2. #20285237C2semidirect
3. #20274032C2semidirect
4. #20273353C2semidirect
5. #20273326C2semidirect
177. C53#19065semidirect

Maximal subgroups

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

subgroupindex
#202857262
#202852372
#202740322
#202733532
#202733262
C53×#35482
#1906553