Finite Group Databasebeta

← #27276651

Subgroups — #27276651

Order 13568.

728 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 728 conjugacy classes of subgroups — too many to draw. Only its 139 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#20275693C212yesmaximal, complemented
2#20275544C211yesmaximal, characteristic
3#20273453C212yesmaximal, complemented
4#20273450C211yesmaximal, characteristic
5C53×#3550C211yesmaximal, characteristic
6#16810647C2212yes—
7#16810647C2211yescharacteristic
8C2×#13133144—22no—
9C42×Dic53C2211yescharacteristic
10#16810382—24no—
11#16810372—21no—
12#16810369—22no—
13#16810369—22no—
14#16810367—21no—
15C53×#563C414yescomplemented
16C53×#559—21no—
17Q82×C53—21no—
18C53×#556—22no—
19D4×Q8×C53—22no—
20C53×#542—24no—
21C53×#536—22no—
22C53×#533C2212yes—
23C53×#532C2211yescharacteristic
24C2×#11719917—21no—
25C2×#11719917—22no—
26C2×C4×Dic53—21no—
27C2×C4×Dic53—22no—
28C4⋊#88322—22no—
29C4⋊#88322—22no—
30C4⋊#88322—22no—
31#13133144—44no—
32C2×#41×C53D412yes—
33C22×Q8×C53—22no—
34C4⋊Q8×C53—24no—
35C4⋊Q8×C53—24no—
36C4⋊Q8×C53D412yes—
37C4⋊Q8×C53D412yes—
38C4⋊Q8×C53D411yescharacteristic
39#127×C53D411yescharacteristic
40#126×C53—28no—
41#125×C53C2×C414yes—
42#124×C53—24no—
43#124×C53—24no—
44#124×C53C2×C412yes—
45#123×C53—24no—
46#123×C53—24no—
47C22⋊Q8×C53—24no—
48C22⋊Q8×C53—28no—
49C22⋊Q8×C53—22no—
50C22⋊Q8×C53—22no—
51#121×C53—22no—
52C4×Q8×C53—24no—
53C4×Q8×C53—22no—
54C4×Q8×C53—22no—
55C4×D4×C53—24no—
56C4×D4×C53—22no—
57#117×C53—22no—
58#117×C53D414yes—
59C2×C42×C53C2311yescharacteristic
60#41×C53—42no—
61#41×C53—42no—
62C2×Q8×C53—24no—
63C2×Q8×C53—22no—
64C2×Q8×C53—22no—
65C2×Q8×C53—82no—
66C2×Q8×C53—82no—
67C2×Q8×C53C22⋊C412yes—
68C2×D4×C53—22no—
69C2×D4×C53—22no—
70C2×D4×C53C22⋊C412yes—
71C22×C4×C53—24no—
72C22×C4×C53—22no—
73C22×C4×C53C2×D412yes—
74C22×C4×C53C2×D411yescharacteristic
75C22×Dic53—41no—
76C4⋊C4×C53—28no—
77C4⋊C4×C53—24no—
78C4⋊C4×C53—24no—
79C4⋊C4×C53—22no—
80C4⋊C4×C53—22no—
81C4⋊C4×C53—22no—
82C4⋊C4×C53C22⋊C414yes—
83C4⋊C4×C53C22⋊C414yes—
84C22⋊C4×C53—24no—
85C22⋊C4×C53—28no—
86C22⋊C4×C53—24no—
87C22⋊C4×C53—24no—
88C42×C53—24no—
89C42×C53—24no—
90C42×C53C22×C411yescharacteristic
91C42×C53C2×D412yes—
92C42×C53C2×D411yescharacteristic
93C4×Dic53—22no—
94C4×Dic53—24no—
95C4×Dic53—42no—
96C4×Dic53—41no—
97#11719917—42no—
98#11719917—44no—
99C2×#88322—41no—
100C2×#88322—42no—
101C23×C53—22no—
102C23×C53#12011yescharacteristic
103Q8×C53—42no—
104Q8×C53—42no—
105Q8×C53—42no—
106Q8×C53—42no—
107Q8×C53—42no—
108Q8×C53—84no—
109D4×C53—42no—
110D4×C53—42no—
111D4×C53—42no—
112D4×C53—82no—
113C2×C4×C53—21no—
114C2×C4×C53—22no—
115C2×C4×C53—21no—
116C2×C4×C53—22no—
117C2×C4×C53—24no—
118C2×C4×C53—22no—
119C2×C4×C53—24no—
120C2×C4×C53—42no—
121C2×C4×C53—42no—
122C2×C4×C53—84no—
123C2×C4×C53C2×C22⋊C411yescharacteristic
124C2×C4×C53C2×C22⋊C412yes—
125C2×C4×C53#12011yescharacteristic
126C2×C4×C53#12012yes—
127C2×Dic53—42no—
128C2×Dic53—44no—
129#88322—81no—
130#88322—82no—
131#8531—531nomaximal, Sylow
132C22×C53—21no—
133C22×C53—22no—
134C22×C53—42no—
135C22×C53—44no—
136C22×C53#37911yescharacteristic
137C4×C53—21no—
138C4×C53—22no—
139C4×C53—42no—
140C4×C53—44no—
141C4×C53—44no—
142C4×C53#45311yescharacteristic
143C4×C53#45412yes—
144C4×C53#45511yescharacteristic
145C4×C53#45612yes—
146Dic53—44no—
147#3550D5311yescharacteristic
148#1994—532no—
149#1962—531no—
150#1561—532no—
151#1560—531no—
152C2×C53—41no—
153C2×C53—82no—
154C2×C53#179211yescharacteristic
155C2×C53#179312yes—
156#563Dic5314yescomplemented
157#559—21no—
158Q82—21no—
159#556—22no—
160D4×Q8—22no—
161#542—24no—
162#536—22no—
163#533C2×D5312yes—
164#532C2×D5311yescharacteristic
165#423—532no—
166#423—531no—
167C2×#104—1062no—
168C43—531no—
169#339—1064no—
170#333—1061no—
171#332—1062no—
172#332—1062no—
173#331—1061no—
174C53#853111yescharacteristic, complemented, Sylow
175C2×#41C22⋊D5312yes—
176C22×Q8—22no—
177C2×M16—1062no—
178C2×M16—1061no—
179C4⋊Q8—24no—
180C4⋊Q8—24no—
181C4⋊Q8C22⋊D5311yescharacteristic
182C4⋊Q8C22⋊D5312yes—
183C4⋊Q8C22⋊D5312yes—
184#127C22⋊D5311yescharacteristic
185#126—28no—
186#125C2×Dic5314yes—
187#124—24no—
188#124—24no—
189#124C2×Dic5312yes—
190#123—24no—
191#123—24no—
192C22⋊Q8—24no—
193C22⋊Q8—22no—
194C22⋊Q8—22no—
195C22⋊Q8—28no—
196#121—22no—
197C4×Q8—24no—
198C4×Q8—22no—
199C4×Q8—22no—
200C4×D4—24no—
201C4×D4—22no—
202#117—22no—
203#117C22⋊D5314yes—
204C2×C42—1062no—
205C2×C42—1061no—
206C2×C42C22×D5311yescharacteristic
207C4⋊C8—1062no—
208C4⋊C8—1062no—
209C4⋊C8—1062no—
210#104—2124no—
211#41—42no—
212#41—42no—
213C2×Q8—24no—
214C2×Q8—22no—
215C2×Q8—22no—
216C2×Q8—82no—
217C2×Q8—82no—
218C2×Q8C22⋊Dic5312yes—
219C2×D4—22no—
220C2×D4—22no—
221C2×D4C22⋊Dic5312yes—
222C22×C4—24no—
223C22×C4—22no—
224C22×C4—2121no—
225C22×C4C2×C22⋊D5312yes—
226C22×C4C2×C22⋊D5311yescharacteristic
227M16—2122no—
228M16—2124no—
229C2×C8—2121no—
230C2×C8—2122no—
231C4⋊C4—28no—
232C4⋊C4—24no—
233C4⋊C4—24no—
234C4⋊C4—22no—
235C4⋊C4—22no—
236C4⋊C4—22no—
237C4⋊C4C22⋊Dic5314yes—
238C4⋊C4C22⋊Dic5314yes—
239C22⋊C4—24no—
240C22⋊C4—28no—
241C22⋊C4—24no—
242C22⋊C4—24no—
243C42—24no—
244C42—24no—
245C42—1062no—
246C42—1064no—
247C42—2122no—
248C42—2121no—
249C42C22×Dic5311yescharacteristic
250C42C2×C22⋊D5312yes—
251C42C2×C22⋊D5311yescharacteristic
252C23—22no—
253C23#1313318611yescharacteristic
254Q8—42no—
255Q8—42no—
256Q8—42no—
257Q8—42no—
258Q8—42no—
259Q8—84no—
260D4—42no—
261D4—42no—
262D4—42no—
263D4—82no—
264C2×C4—21no—
265C2×C4—22no—
266C2×C4—21no—
267C2×C4—22no—
268C2×C4—24no—
269C2×C4—22no—
270C2×C4—24no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—84no—
274C2×C4—2122no—
275C2×C4—2124no—
276C2×C4C2×C22⋊Dic5312yes—
277C2×C4C2×C22⋊Dic5311yescharacteristic
278C2×C4#1313318611yescharacteristic
279C2×C4#1313318612yes—
280C8—4241no—
281C8—4242no—
282C22—21no—
283C22—22no—
284C22—42no—
285C22—44no—
286C22#1681047911yescharacteristic
287C4—21no—
288C4—22no—
289C4—42no—
290C4—44no—
291C4—44no—
292C4—2124no—
293C4#1681079111yescharacteristic
294C4#1681079612yes—
295C4#1681080111yescharacteristic
296C4#1681080612yes—
297C2—41no—
298C2—82no—
299C2#2027466911yescharacteristic
300C2#2027467412yes—
subgroupG/H
centerC22#16810479
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#3550C2
socleC22×C53#379
prime radicalC53×#532C22
cosocleC2×C42×C53C23
hypercenter#3550D53
hypocenterC53#8531
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3550

Complements

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

NH ≅ G/Ntype
1. #20275693C2semidirect
3. #20273453C2semidirect
15. C53×#563C4semidirect
156. #563Dic53semidirect
174. C53#8531semidirect

Maximal subgroups

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

subgroupindex
#202756932
#202755442
#202734532
#202734502
C53×#35502
#853153