Finite Group Databasebeta

← C22×#16810385 #27381487

Subgroups — C22×#16810385 #27381487

Order 13568.

3006 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C24×#88322C211yesmaximal, characteristic
2C23×C4⋊Dic53C211yesmaximal, characteristic
3C2×#16810385C2112yesmaximal, direct product factor
4C23×C4⋊C4×C53C211yesmaximal, characteristic
5C24×Dic53—21no—
6C23×#88322—24no—
7C23×#88322—24no—
8C23×#88322C418yescomplemented
9C23×#88322C2216yes—
10C22×C4⋊Dic53—24no—
11C22×C4⋊Dic53—23no—
12C22×C4⋊Dic53C418yescomplemented
13C22×C4⋊Dic53C2216yes—
14#16810385C22116yesdirect product factor
15C24×C4×C53—21no—
16C24×C4×C53C2211yescharacteristic
17C22×C4⋊C4×C53—23no—
18C22×C4⋊C4×C53—24no—
19C22×C4⋊C4×C53C418yescomplemented
20C22×C4⋊C4×C53C2216yes—
21C23×Dic53—28no—
22C23×Dic53—26no—
23C23×Dic53—44no—
24C23×Dic53—44no—
25C22×#88322—232no—
26C22×#88322—212no—
27C22×#88322—212no—
28C22×#88322C2×C4124yescomplemented
29C22×#88322C2314yes—
30C2×C4⋊Dic53—224no—
31C2×C4⋊Dic53—212no—
32C2×C4⋊Dic53—26no—
33C2×C4⋊Dic53C2×C4124yescomplemented
34C2×C4⋊Dic53C2314yes—
35C25×C53D411yescharacteristic
36C23×C4×C53—28no—
37C23×C4×C53—26no—
38C23×C4×C53—44no—
39C23×C4×C53—44no—
40C23×C4×C53C2×C414yes—
41C23×C4×C53C2×C414yes—
42C23×C4×C53C2×C414yes—
43C23×C4×C53D413yes—
44C23×C4×C53D414yes—
45C23×C4×C53D414yes—
46C23×C4×C53Q814yes—
47C23×C4×C53C2313yes—
48C2×C4⋊C4×C53—224no—
49C2×C4⋊C4×C53—212no—
50C2×C4⋊C4×C53—26no—
51C2×C4⋊C4×C53C2×C4124yescomplemented
52C2×C4⋊C4×C53C2314yes—
53C24×C53C22⋊C414yes—
54C24×C53C22⋊C414yes—
55C24×C53C4⋊C414yes—
56C24×C53D814yes—
57C24×C53SD1614yes—
58C24×C53SD1614yes—
59C24×C53Q1614yes—
60C24×C53C2×D413yes—
61C22×C4×C53—224no—
62C22×C4×C53—24no—
63C22×C4×C53—432no—
64C22×C4×C53—412no—
65C22×C4×C53—412no—
66C22×C4×C53C42116yes—
67C22×C4×C53C22⋊C4112yes—
68C22×C4×C53C22⋊C4112yes—
69C22×C4×C53C22⋊C4116yes—
70C22×C4×C53C4⋊C4112yes—
71C22×C4×C53C4⋊C4116yes—
72C22×C4×C53C4⋊C4116yes—
73C22×C4×C53C22×C416yes—
74C22×C4×C53C22×C416yes—
75C22×C4×C53C22×C416yes—
76C22×C4×C53C2×D413yes—
77C22×C4×C53C2×D416yes—
78C22×C4×C53C2×D416yes—
79C22×C4×C53C2×Q816yes—
80C22×C4×C53C2411yescharacteristic
81C22×Dic53—224no—
82C22×Dic53—24no—
83C22×Dic53—432no—
84C22×Dic53—412no—
85C22×Dic53—412no—
86C4⋊C4×C53—224no—
87C4⋊C4×C53—24no—
88C4⋊C4×C53C22×C418yescomplemented
89C4⋊Dic53—224no—
90C4⋊Dic53—24no—
91C4⋊Dic53C22×C418yescomplemented
92C2×#88322—248no—
93C2×#88322—24no—
94C2×#88322—24no—
95C2×#88322C22×C418yescomplemented
96C23×C53#95116yes—
97C23×C53#102116yes—
98C23×C53#102116yes—
99C23×C53#103116yes—
100C23×C53#103116yes—
101C23×C53#106116yes—
102C23×C53#107116yes—
103C23×C53C2×C22⋊C416yes—
104C23×C53C2×C22⋊C416yes—
105C23×C53C2×C4⋊C416yes—
106C23×C53C2×D816yes—
107C23×C53C2×SD1616yes—
108C23×C53C2×SD1616yes—
109C23×C53C2×Q1616yes—
110C23×C53C22×D411yescharacteristic
111C2×C4×C53—28no—
112C2×C4×C53—44no—
113C2×C4×C53—44no—
114C2×C4×C53—448no—
115C2×C4×C53#95148yes—
116C2×C4×C53C2×C42112yes—
117C2×C4×C53C2×C22⋊C4112yes—
118C2×C4×C53C2×C22⋊C416yes—
119C2×C4×C53C2×C22⋊C416yes—
120C2×C4×C53C2×C4⋊C4112yes—
121C2×C4×C53C2×C4⋊C416yes—
122C2×C4×C53C2×C4⋊C4112yes—
123C2×C4×C53C23×C411yescharacteristic
124C2×C4×C53C23×C411yescharacteristic
125C2×C4×C53C23×C411yescharacteristic
126C2×C4×C53C22×D411yescharacteristic
127C2×C4×C53C22×D411yescharacteristic
128C2×C4×C53C22×Q811yescharacteristic
129C2×Dic53—28no—
130C2×Dic53—44no—
131C2×Dic53—44no—
132C2×Dic53—448no—
133#88322—28no—
134C22×#340—531nomaximal, Sylow
135C22×C53#340164yescomplemented
136C22×C53C2×#95112yes—
137C22×C53C2×#102112yes—
138C22×C53C2×#102112yes—
139C22×C53C2×#103112yes—
140C22×C53C2×#103112yes—
141C22×C53C2×#106112yes—
142C22×C53C2×#107112yes—
143C22×C53C22×C22⋊C411yescharacteristic
144C22×C53C22×C22⋊C411yescharacteristic
145C22×C53C22×C4⋊C411yescharacteristic
146C22×C53C22×D811yescharacteristic
147C22×C53C22×SD1611yescharacteristic
148C22×C53C22×SD1611yescharacteristic
149C22×C53C22×Q1611yescharacteristic
150C4×C53—48no—
151C4×C53C2×#95112yes—
152C4×C53C22×C4211yescharacteristic
153C4×C53C22×C22⋊C411yescharacteristic
154C4×C53C22×C4⋊C411yescharacteristic
155C4×C53C22×C4⋊C411yescharacteristic
156Dic53—48no—
157C24×C8—531no—
158C23×C4⋊C4—531no—
159C23×C4⋊C4D5311yescharacteristic
160C2×#340—5312no—
161C2×C53C2×#340124yescomplemented
162C2×C53C22×#9511yescharacteristic
163C2×C53C22×#10211yescharacteristic
164C2×C53C22×#10211yescharacteristic
165C2×C53C22×#10311yescharacteristic
166C2×C53C22×#10311yescharacteristic
167C2×C53C22×#10611yescharacteristic
168C2×C53C22×#10711yescharacteristic
169C24×C4—21no—
170C24×C4—1061no—
171C24×C4C2×D5311yescharacteristic
172C23×C8—538no—
173C23×C8—536no—
174C23×C8—1064no—
175C23×C8—1064no—
176C22×C4⋊C4—24no—
177C22×C4⋊C4—23no—
178C22×C4⋊C4—538no—
179C22×C4⋊C4—536no—
180C22×C4⋊C4—1064no—
181C22×C4⋊C4—1063no—
182C22×C4⋊C4Dic5318yescomplemented
183C22×C4⋊C4C2×D5316yes—
184#340—5316no—
185C53C22×#34011yescharacteristic, complemented, Sylow
186C25C22⋊D5311yescharacteristic
187C23×C4—26no—
188C23×C4—28no—
189C23×C4—44no—
190C23×C4—44no—
191C23×C4—1066no—
192C23×C4—1068no—
193C23×C4—2124no—
194C23×C4—2124no—
195C23×C4Dic10614yes—
196C23×C4C4×D5314yes—
197C23×C4C4×D5314yes—
198C23×C4D21214yes—
199C23×C4C2×Dic5314yes—
200C23×C4C22⋊D5314yes—
201C23×C4C22⋊D5313yes—
202C23×C4C22×D5313yes—
203C22×C8—5324no—
204C22×C8—534no—
205C22×C8—10612no—
206C22×C8—10612no—
207C22×C8—10632no—
208C2×C4⋊C4—26no—
209C2×C4⋊C4—212no—
210C2×C4⋊C4—224no—
211C2×C4⋊C4—5324no—
212C2×C4⋊C4—534no—
213C2×C4⋊C4—10612no—
214C2×C4⋊C4—1066no—
215C2×C4⋊C4—10624no—
216C2×C4⋊C4C2×Dic53124yescomplemented
217C2×C4⋊C4C22×D5314yes—
218C24#1171991914yes—
219C24#1171992114yes—
220C24#1171992214yes—
221C24#1171992314yes—
222C24#1171992414yes—
223C24#1171992514yes—
224C24C22⋊Dic5314yes—
225C24C2×C22⋊D5313yes—
226C22×C4—224no—
227C22×C4—24no—
228C22×C4—432no—
229C22×C4—412no—
230C22×C4—412no—
231C22×C4—10624no—
232C22×C4—1064no—
233C22×C4—21232no—
234C22×C4—21212no—
235C22×C4—21212no—
236C22×C4C4×Dic53116yes—
237C22×C4#11719919112yes—
238C22×C4#11719919116yes—
239C22×C4C4⋊Dic53116yes—
240C22×C4#11719921112yes—
241C22×C4#11719921116yes—
242C22×C4C22⋊Dic53112yes—
243C22×C4C2×Dic10616yes—
244C22×C4C2×C4×D5316yes—
245C22×C4C2×C4×D5316yes—
246C22×C4C2×D21216yes—
247C22×C4C22×Dic5316yes—
248C22×C4C2×C22⋊D5313yes—
249C22×C4C2×C22⋊D5316yes—
250C22×C4C23×D5311yescharacteristic
251C2×C8—538no—
252C2×C8—10648no—
253C2×C8—1064no—
254C2×C8—1064no—
255C4⋊C4—224no—
256C4⋊C4—24no—
257C4⋊C4—538no—
258C4⋊C4—10624no—
259C4⋊C4—1064no—
260C4⋊C4C22×Dic5318yescomplemented
261C23#13133124116yes—
262C23#13133137116yes—
263C23#13133138116yes—
264C23#13133140116yes—
265C23#13133141116yes—
266C23#13133148116yes—
267C23#13133150116yes—
268C23C2×#1171992116yes—
269C23C2×C22⋊Dic5316yes—
270C23C2×#1171991916yes—
271C23C2×#1171992216yes—
272C23C2×#1171992416yes—
273C23C2×#1171992316yes—
274C23C2×#1171992516yes—
275C23C22×C22⋊D5311yescharacteristic
276C2×C4—28no—
277C2×C4—448no—
278C2×C4—44no—
279C2×C4—44no—
280C2×C4—1068no—
281C2×C4—21248no—
282C2×C4—2124no—
283C2×C4—2124no—
284C2×C4#13133124148yes—
285C2×C4C2×C4×Dic53112yes—
286C2×C4C2×#11719921112yes—
287C2×C4C2×#1171992116yes—
288C2×C4C2×C22⋊Dic5316yes—
289C2×C4C2×#11719919112yes—
290C2×C4C2×#1171991916yes—
291C2×C4C2×C4⋊Dic53112yes—
292C2×C4C22×C4×D5311yescharacteristic
293C2×C4C22×C4×D5311yescharacteristic
294C2×C4C23×Dic5311yescharacteristic
295C2×C4C22×C22⋊D5311yescharacteristic
296C2×C4C22×D21211yescharacteristic
297C2×C4C22×Dic10611yescharacteristic
298C8—1068no—
299C22#16810385164yesdirect product factor
300C22C2×#13133124112yes—
301C22C2×#13133137112yes—
302C22C2×#13133138112yes—
303C22C2×#13133140112yes—
304C22C2×#13133141112yes—
305C22C2×#13133148112yes—
306C22C2×#13133150112yes—
307C22C22×#1171992111yescharacteristic
308C22C22×C22⋊Dic5311yescharacteristic
309C22C22×#1171991911yescharacteristic
310C22C22×#1171992211yescharacteristic
311C22C22×#1171992411yescharacteristic
312C22C22×#1171992311yescharacteristic
313C22C22×#1171992511yescharacteristic
314C4—48no—
315C4—2128no—
316C4C2×#13133124112yes—
317C4C22×C4×Dic5311yescharacteristic
318C4C22×#1171992111yescharacteristic
319C4C22×#1171991911yescharacteristic
320C4C22×C4⋊Dic5311yescharacteristic
321C2C2×#16810385124yesdirect product factor
322C2C22×#1313312411yescharacteristic
323C2C22×#1313313711yescharacteristic
324C2C22×#1313313811yescharacteristic
325C2C22×#1313314011yescharacteristic
326C2C22×#1313314111yescharacteristic
327C2C22×#1313314811yescharacteristic
328C2C22×#1313315011yescharacteristic
subgroupG/H
centerC25C22⋊D53
commutatorC4×C53C22×C42
FrattiniC22×C4C23×D53
FittingC23×C4⋊C4×C53C2
socle, prime radicalC25×C53D4
cosocleC22×C4×C53C24
hypercenterC23×C4⋊C4D53
hypocenterC53C22×#340
LCSG ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C25 ◁ C24×C4 ◁ C23×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. C2×#16810385C2direct (N × H)
8. C23×#88322C4semidirect
12. C22×C4⋊Dic53C4semidirect
14. #16810385C22direct (N × H)
19. C22×C4⋊C4×C53C4semidirect
28. C22×#88322C2×C4semidirect
33. C2×C4⋊Dic53C2×C4semidirect
51. C2×C4⋊C4×C53C2×C4semidirect
88. C4⋊C4×C53C22×C4semidirect
91. C4⋊Dic53C22×C4semidirect
95. C2×#88322C22×C4semidirect
135. C22×C53#340semidirect
161. C2×C53C2×#340semidirect
182. C22×C4⋊C4Dic53semidirect
185. C53C22×#340semidirect
216. C2×C4⋊C4C2×Dic53semidirect
260. C4⋊C4C22×Dic53semidirect
299. C22#16810385direct (N × H)
321. C2C2×#16810385direct (N × H)

Maximal subgroups

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

subgroupindex
C24×#883222
C23×C4⋊Dic532
C2×#168103852
C23×C4⋊C4×C532
C22×#34053