Finite Group Databasebeta

← #27381466

Subgroups — #27381466

Order 13568.

2036 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811390C211yesmaximal, characteristic
2#20287918C211yesmaximal, characteristic
3C2×#16810379C2112yesmaximal, complemented
4C53×#3428C211yesmaximal, characteristic
5#16811390—24no—
6#16811390—24no—
7C2×#13133224C418yescomplemented
8C22×#11719917C2216yes—
9#16811106—24no—
10#16811101—21no—
11#16811098—23no—
12#41×Dic53C418yescomplemented
13C2×#13133176C2216yes—
14#16810379C22116yes—
15C22×#41×C53C2211yescharacteristic
16C53×#520—24no—
17C53×#519—21no—
18C53×#518—23no—
19C4×#41×C53C418yescomplemented
20C2×#117×C53C2216yes—
21#13133224—232no—
22C2×#11719917—212no—
23C2×#11719917—212no—
24C2×#11719917C2×C4112yescomplemented
25C2×#11719917C2314yes—
26C22×#88322C2×C4112yescomplemented
27Q8×Dic53—28no—
28D4×Dic53—224no—
29#13133176—26no—
30#13133176—26no—
31#13133176C2×C4112yescomplemented
32#13133176C2314yes—
33C2×C4⋊Dic53—26no—
34C2×C22⋊Dic53—26no—
35C2×C4×Dic53C2×C4112yescomplemented
36C2×#41×C53C2×C414yes—
37C2×#41×C53C2×C414yes—
38C2×#41×C53C2×C414yes—
39C2×#41×C53D414yes—
40C2×#41×C53D414yes—
41C2×#41×C53Q814yes—
42C22×Q8×C53D411yescharacteristic
43C22×D4×C53D413yes—
44C23×C4×C53C2313yes—
45C4×Q8×C53—28no—
46C4×D4×C53—224no—
47#117×C53—26no—
48#117×C53—26no—
49#117×C53C2×C4112yescomplemented
50#117×C53C2314yes—
51C2×C4⋊C4×C53—26no—
52C2×C22⋊C4×C53—26no—
53C2×C42×C53C2×C4112yescomplemented
54C24×C53C2×D413yes—
55#41×C53C42116yes—
56#41×C53C22⋊C4116yes—
57#41×C53C4⋊C4116yes—
58#41×C53C4⋊C4116yes—
59C2×Q8×C53C22⋊C414yes—
60C2×Q8×C53C22⋊C414yes—
61C2×Q8×C53C4⋊C414yes—
62C2×D4×C53C22⋊C4112yes—
63C2×D4×C53C22⋊C4112yes—
64C2×D4×C53C4⋊C4112yes—
65C22×C4×C53—212no—
66C22×C4×C53C22×C416yes—
67C22×C4×C53C22×C416yes—
68C22×C4×C53C22×C416yes—
69C22×C4×C53C2×D413yes—
70C22×C4×C53C2×D416yes—
71C22×C4×C53C2×D416yes—
72C22×C4×C53C2×Q816yes—
73C22×C4×C53C2411yescharacteristic
74C22×Dic53—212no—
75C4⋊C4×C53—212no—
76C4⋊C4×C53—24no—
77C22⋊C4×C53—212no—
78C22⋊C4×C53—24no—
79C42×C53—212no—
80C42×C53C22×C418yes—
81C22⋊Dic53—212no—
82C22⋊Dic53—24no—
83C4⋊Dic53—212no—
84C4⋊Dic53—24no—
85C4×Dic53—212no—
86C4×Dic53C22×C418yes—
87#11719917—224no—
88#11719917—24no—
89#11719917—24no—
90C2×#88322—224no—
91C2×#88322C22×C418yes—
92C23×C53—83no—
93C23×C53C2×C22⋊C416yes—
94C23×C53C2×C22⋊C416yes—
95C23×C53C2×C4⋊C416yes—
96C23×C53C22×D411yescharacteristic
97Q8×C53#95116yes—
98D4×C53#95148yes—
99C2×C4×C53—28no—
100C2×C4×C53—812no—
101C2×C4×C53C2×C42112yes—
102C2×C4×C53C2×C22⋊C4112yes—
103C2×C4×C53C2×C22⋊C416yes—
104C2×C4×C53C2×C22⋊C416yes—
105C2×C4×C53C2×C4⋊C4112yes—
106C2×C4×C53C2×C4⋊C416yes—
107C2×C4×C53C2×C4⋊C4112yes—
108C2×C4×C53C23×C411yescharacteristic
109C2×C4×C53C23×C411yescharacteristic
110C2×C4×C53C23×C411yescharacteristic
111C2×C4×C53C22×D411yescharacteristic
112C2×C4×C53C22×D411yescharacteristic
113C2×C4×C53C22×Q811yescharacteristic
114C2×Dic53—28no—
115C2×Dic53—812no—
116#88322—28no—
117#20338—531nomaximal, Sylow
118C22×C53—46no—
119C22×C53—46no—
120C22×C53—41no—
121C22×C53—46no—
122C22×C53C2×#95112yes—
123C22×C53C22×C22⋊C411yescharacteristic
124C22×C53C22×C22⋊C411yescharacteristic
125C22×C53C22×C4⋊C411yescharacteristic
126C4×C53—48no—
127C4×C53C2×#95112yes—
128C4×C53C22×C4211yescharacteristic
129C4×C53C22×C22⋊C411yescharacteristic
130C4×C53C22×C4⋊C411yescharacteristic
131C4×C53C22×C4⋊C411yescharacteristic
132Dic53—48no—
133C2×#568—531no—
134#3428—531no—
135#3428D5311yescharacteristic
136C2×H(3,ℤ/4)—5312no—
137C2×C53—21no—
138C2×C53—21no—
139C2×C53—21no—
140C2×C53—212no—
141C2×C53C22×#9511yescharacteristic
142C22×#41C2×D5311yescharacteristic
143#568—1064no—
144#568—1064no—
145C2×#131—538no—
146C22×M16—536no—
147#520—24no—
148#520—1064no—
149#519—21no—
150#519—1061no—
151#518—23no—
152#518—1063no—
153C4×#41—538no—
154C4×#41Dic5318yescomplemented
155C2×#117—536no—
156C2×#117C2×D5316yes—
157H(3,ℤ/4)—5316no—
158C53#2033811yescharacteristic, complemented, Sylow
159C2×#41Dic10614yes—
160C2×#41C4×D5314yes—
161C2×#41C4×D5314yes—
162C2×#41D21214yes—
163C2×#41C2×Dic5314yes—
164C2×#41C22⋊D5314yes—
165C22×Q8C22⋊D5311yescharacteristic
166C22×D4C22⋊D5313yes—
167C23×C4C22×D5313yes—
168#131—10632no—
169C2×M16—5312no—
170C2×M16—534no—
171C2×M16—10612no—
172C2×M16—10612no—
173C22×C8—5312no—
174C4×Q8—28no—
175C4×Q8—1068no—
176C4×D4—224no—
177C4×D4—10624no—
178#117—26no—
179#117—26no—
180#117—5312no—
181#117—534no—
182#117—1066no—
183#117—1066no—
184#117C2×Dic53112yescomplemented
185#117C22×D5314yes—
186C2×C4⋊C4—26no—
187C2×C4⋊C4—1066no—
188C2×C22⋊C4—26no—
189C2×C22⋊C4—1066no—
190C2×C42—5312no—
191C2×C42C2×Dic53112yescomplemented
192C24C2×C22⋊D5313yes—
193#41C4×Dic53116yes—
194#41#11719919116yes—
195#41C4⋊Dic53116yes—
196#41#11719921116yes—
197C2×Q8#1171991914yes—
198C2×Q8#1171992114yes—
199C2×Q8C22⋊Dic5314yes—
200C2×D4#11719919112yes—
201C2×D4#11719921112yes—
202C2×D4C22⋊Dic53112yes—
203C22×C4—212no—
204C22×C4—10612no—
205C22×C4C2×Dic10616yes—
206C22×C4C2×C4×D5316yes—
207C22×C4C2×C4×D5316yes—
208C22×C4C2×D21216yes—
209C22×C4C22×Dic5316yes—
210C22×C4C2×C22⋊D5313yes—
211C22×C4C2×C22⋊D5316yes—
212C22×C4C23×D5311yescharacteristic
213M16—10624no—
214M16—1064no—
215M16—1064no—
216C2×C8—538no—
217C2×C8—10624no—
218C4⋊C4—212no—
219C4⋊C4—24no—
220C4⋊C4—10612no—
221C4⋊C4—1064no—
222C22⋊C4—212no—
223C22⋊C4—24no—
224C22⋊C4—10612no—
225C22⋊C4—1064no—
226C42—212no—
227C42—538no—
228C42—10612no—
229C42C22×Dic5318yes—
230C23—83no—
231C23C2×#1171992116yes—
232C23C2×C22⋊Dic5316yes—
233C23C2×#1171991916yes—
234C23C22×C22⋊D5311yescharacteristic
235Q8#13133124116yes—
236D4#13133124148yes—
237C2×C4—28no—
238C2×C4—812no—
239C2×C4—1068no—
240C2×C4—42412no—
241C2×C4C2×C4×Dic53112yes—
242C2×C4C2×#11719921112yes—
243C2×C4C2×#1171992116yes—
244C2×C4C2×C22⋊Dic5316yes—
245C2×C4C2×#11719919112yes—
246C2×C4C2×#1171991916yes—
247C2×C4C2×C4⋊Dic53112yes—
248C2×C4C22×C4×D5311yescharacteristic
249C2×C4C22×C4×D5311yescharacteristic
250C2×C4C23×Dic5311yescharacteristic
251C2×C4C22×C22⋊D5311yescharacteristic
252C2×C4C22×D21211yescharacteristic
253C2×C4C22×Dic10611yescharacteristic
254C8—1068no—
255C22—46no—
256C22—46no—
257C22—41no—
258C22—46no—
259C22C2×#13133124112yes—
260C22C22×#1171992111yescharacteristic
261C22C22×C22⋊Dic5311yescharacteristic
262C22C22×#1171991911yescharacteristic
263C4—48no—
264C4—2128no—
265C4C2×#13133124112yes—
266C4C22×C4×Dic5311yescharacteristic
267C4C22×#1171992111yescharacteristic
268C4C22×#1171991911yescharacteristic
269C4C22×C4⋊Dic5311yescharacteristic
270C2—21no—
271C2—21no—
272C2—21no—
273C2—212no—
274C2C22×#1313312411yescharacteristic
subgroupG/H
centerC4C22×C4×Dic53
commutatorC4×C53C22×C42
FrattiniC22×C4C23×D53
FittingC53×#3428C2
socleC2×C53C22×#95
prime radicalC22×#41×C53C22
cosocleC22×C4×C53C24
hypercenter#3428D53
hypocenterC53#20338
LCSG ▷ C4×C53 ▷ C53
UCSC1 ◁ C4 ◁ #3428

Complements

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

NH ≅ G/Ntype
3. C2×#16810379C2semidirect
7. C2×#13133224C4semidirect
12. #41×Dic53C4semidirect
19. C4×#41×C53C4semidirect
24. C2×#11719917C2×C4semidirect
26. C22×#88322C2×C4semidirect
31. #13133176C2×C4semidirect
35. C2×C4×Dic53C2×C4semidirect
49. #117×C53C2×C4semidirect
53. C2×C42×C53C2×C4semidirect
154. C4×#41Dic53semidirect
158. C53#20338semidirect
184. #117C2×Dic53semidirect
191. C2×C42C2×Dic53semidirect

Maximal subgroups

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

subgroupindex
C2×#168113902
#202879182
C2×#168103792
C53×#34282
#2033853