Finite Group Databasebeta

← #27360391

Subgroups — #27360391

Order 13568.

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

normal subgroups only

This group has 918 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
1C2×#16811353C211yesmaximal, characteristic
2#20284841C212yesmaximal
3#20273933C218yesmaximal, complemented
4#20273441C211yesmaximal, characteristic
5#20273163C212yesmaximal
6C53×#3139C211yesmaximal, characteristic
7#16811353—24no—
8#16811353—22no—
9#16811353C2214yes—
10C2×#13133216—22no—
11C2×#13133215—22no—
12C2×C4×Dic106C2212yes—
13C2×#13133182—22no—
14#16810951—22no—
15#16810905C2214yes—
16#16810830C2214yes—
17#16810778—22no—
18#16810728—24no—
19#16810680C2212yes—
20#16810372C2214yes—
21#16810355C2218yes—
22C53×#533C2211yescharacteristic
23C2×C4⋊Q8×C53C2211yescharacteristic
24C53×#491—22no—
25C53×#487C2214yes—
26C53×#419—22no—
27C4×M16×C53C2211yescharacteristic
28C22×Dic106—22no—
29C22×Dic106D412yes—
30C2×#11719925—24no—
31#13133216—28no—
32#13133216—28no—
33#13133215—28no—
34#13133215—28no—
35C4×Dic106—22no—
36C4×Dic106—22no—
37C4×Dic106D414yes—
38C4×Dic106C2314yes—
39#13133182—28no—
40#13133182—28no—
41C2×C4⋊Dic53D412yes—
42C2×#11719919—24no—
43C2×#11719919—22no—
44C2×C4×Dic53—22no—
45C2×C4×Dic53—22no—
46#13133148—24no—
47C4⋊#88322C2314yes—
48#13133141—24no—
49#13133140—24no—
50C22⋊#88322—22no—
51#13133128—22no—
52C22×Q8×C53D412yes—
53C2×M16×C53—21no—
54C4⋊Q8×C53—22no—
55C4⋊Q8×C53D412yes—
56C4⋊Q8×C53C2312yes—
57C4⋊Q8×C53C2312yes—
58#125×C53—21no—
59C22⋊Q8×C53—22no—
60C4×Q8×C53—22no—
61#117×C53—21no—
62C2×C4⋊C4×C53—22no—
63C2×C42×C53C2311yescharacteristic
64#103×C53—28no—
65#97×C53—21no—
66C4×C8×C53C2312yes—
67C2×Q8×C53—22no—
68C2×Q8×C53—22no—
69C2×Q8×C53—44no—
70C2×Q8×C53C2×D414yes—
71C22×C4×C53—22no—
72C22×C4×C53C2×D412yes—
73C22×C4×C53C2×D411yescharacteristic
74C22×Dic53—22no—
75C22×Dic53—22no—
76C2×Dic106—22no—
77C2×Dic106—28no—
78C2×Dic106—28no—
79C2×Dic106—84no—
80C2×Dic106C2×D414yes—
81M16×C53—42no—
82C2×C8×C53—22no—
83C4⋊C4×C53—24no—
84C4⋊C4×C53—24no—
85C4⋊C4×C53—22no—
86C4⋊C4×C53—22no—
87C4⋊C4×C53—42no—
88C22⋊C4×C53—41no—
89C42×C53—41no—
90C42×C53C2×D412yes—
91C42×C53C2×D411yescharacteristic
92C42×C53C2411yescharacteristic
93#11719925—48no—
94C4⋊Dic53—22no—
95C4⋊Dic53C2×D414yes—
96#11719919—216no—
97#11719919—24no—
98#11719919—24no—
99C4×Dic53—28no—
100C4×Dic53—24no—
101C4×Dic53—24no—
102C2×#88322—24no—
103C23×C53#12011yescharacteristic
104Q8×C53—44no—
105Q8×C53—44no—
106Q8×C53—82no—
107C2×C4×C53—22no—
108C2×C4×C53—24no—
109C2×C4×C53—24no—
110C2×C4×C53—44no—
111C2×C4×C53—42no—
112C2×C4×C53#12012yes—
113C2×C4×C53#12011yescharacteristic
114C2×C4×C53C22×D412yes—
115C2×C4×C53C22×D411yescharacteristic
116C2×Dic53—28no—
117C2×Dic53—24no—
118C2×Dic53—24no—
119C2×Dic53—44no—
120Dic106—28no—
121Dic106—24no—
122Dic106—88no—
123Dic106#13718yes—
124C8×C53—42no—
125#88322—44no—
126#18551—531nomaximal, Sylow
127C22×C53—22no—
128C22×C53#45512yes—
129C22×C53C2×#12011yescharacteristic
130C4×C53—22no—
131C4×C53—21no—
132C4×C53—21no—
133C4×C53—44no—
134C4×C53—82no—
135C4×C53C2×#13714yes—
136Dic53—28no—
137Dic53—44no—
138C2×Q82—531no—
139#3139D5311yescharacteristic
140#3113—532no—
141#1645—538no—
142#1558—531no—
143#1491—532no—
144C2×C53—22no—
145C2×C53#300412yes—
146C2×C53C2×#45511yescharacteristic
147Q82—534no—
148Q82—1064no—
149Q82—1062no—
150#533C2×D5311yescharacteristic
151C2×C4⋊Q8—1062no—
152C2×C4⋊Q8—1062no—
153C2×C4⋊Q8C2×D5311yescharacteristic
154C2×C4×Q8—532no—
155C2×C4×Q8—1062no—
156#491—22no—
157#487C2×D5314yes—
158#483—1062no—
159#475—534no—
160#462—534no—
161#451—1062no—
162#441—1064no—
163#432—532no—
164#419—22no—
165C4×M16C2×D5311yescharacteristic
166#333—534no—
167Q8⋊C8—538no—
168C53#1855111yescharacteristic, complemented, Sylow
169C22×Q8—532no—
170C22×Q8—1062no—
171C22×Q8C22⋊D5312yes—
172C2×Q16—1064no—
173C2×M16—21no—
174C4⋊Q8—22no—
175C4⋊Q8—1068no—
176C4⋊Q8—1068no—
177C4⋊Q8—1068no—
178C4⋊Q8—1068no—
179C4⋊Q8C22⋊D5312yes—
180C4⋊Q8C22×D5312yes—
181C4⋊Q8C22×D5312yes—
182#125—21no—
183C22⋊Q8—22no—
184C4×Q8—22no—
185C4×Q8—534no—
186C4×Q8—534no—
187C4×Q8—1068no—
188C4×Q8—1068no—
189C4×Q8—1062no—
190C4×Q8—1062no—
191#117—21no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—532no—
194C2×C4⋊C4—1064no—
195C2×C4⋊C4—1062no—
196C2×C42—1062no—
197C2×C42—1062no—
198C2×C42C22×D5311yescharacteristic
199#106—1064no—
200C4⋊C8—534no—
201#103—28no—
202#103—1064no—
203#103—1064no—
204C22⋊C8—1062no—
205#97—21no—
206#97—1062no—
207C4×C8C22×D5312yes—
208C2×Q8—22no—
209C2×Q8—22no—
210C2×Q8—44no—
211C2×Q8—534no—
212C2×Q8—1068no—
213C2×Q8—1068no—
214C2×Q8—1062no—
215C2×Q8—4244no—
216C2×Q8C2×C22⋊D5314yes—
217C22×C4—22no—
218C22×C4—1062no—
219C22×C4—1062no—
220C22×C4D4×D5312yes—
221C22×C4C2×C22⋊D5311yescharacteristic
222Q16—2128no—
223M16—42no—
224C2×C8—22no—
225C2×C8—1064no—
226C4⋊C4—24no—
227C4⋊C4—24no—
228C4⋊C4—22no—
229C4⋊C4—22no—
230C4⋊C4—42no—
231C4⋊C4—534no—
232C4⋊C4—10616no—
233C4⋊C4—1062no—
234C4⋊C4—1064no—
235C4⋊C4—1064no—
236C22⋊C4—41no—
237C42—41no—
238C42—1068no—
239C42—1064no—
240C42—1064no—
241C42D4×D5312yes—
242C42C2×C22⋊D5311yescharacteristic
243C42C23×D5311yescharacteristic
244C23#1313318511yescharacteristic
245Q8—44no—
246Q8—44no—
247Q8—82no—
248Q8—538no—
249Q8—1068no—
250Q8—1064no—
251Q8—4248no—
252C2×C4—22no—
253C2×C4—24no—
254C2×C4—24no—
255C2×C4—44no—
256C2×C4—42no—
257C2×C4—1068no—
258C2×C4—1064no—
259C2×C4—1064no—
260C2×C4—2124no—
261C2×C4#1313318511yescharacteristic
262C2×C4#1313318512yes—
263C2×C4C2×D4×D5312yes—
264C2×C4C22×C22⋊D5311yescharacteristic
265C8—42no—
266C8—2124no—
267C22—22no—
268C22#1681080012yes—
269C22C2×#1313318511yescharacteristic
270C4—22no—
271C4—21no—
272C4—21no—
273C4—44no—
274C4—82no—
275C4—1068no—
276C4—2124no—
277C4#137×D5314yes—
278C2—22no—
279C2#2028394712yes—
280C2C2×#1681080011yescharacteristic
subgroupG/H
centerC22C2×#13133185
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#3139C2
socleC22×C53C2×#120
prime radicalC23×C53#120
hypercenter#3139D53
hypocenterC53#18551
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3139

Complements

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

NH ≅ G/Ntype
3. #20273933C2semidirect
168. C53#18551semidirect

Maximal subgroups

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

subgroupindex
C2×#168113532
#202848412
#202739332
#202734412
#202731632
C53×#31392
#1855153