Finite Group Databasebeta

← C22×#16810391 #27381533

Subgroups — C22×#16810391 #27381533

Order 13568.

2494 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#11719917C211yesmaximal, characteristic
2C23×C4⋊Dic53C211yesmaximal, characteristic
3C2×#16810391C2112yesmaximal, direct product factor
4C23×M16×C53C211yesmaximal, characteristic
5C24×Dic53C412yes—
6C22×#11719917—23no—
7C22×#11719917—24no—
8C22×#11719917C418yescomplemented
9C22×#11719917C2216yes—
10C23×#88322—21no—
11C22×C4⋊Dic53—21no—
12C22×C4⋊Dic53—23no—
13C22×C4⋊Dic53—21no—
14C22×C4⋊Dic53—23no—
15C22×C4⋊Dic53C416yes—
16C22×C4⋊Dic53C2216yes—
17#16810391C22116yesdirect product factor
18C24×C4×C53C2211yescharacteristic
19C22×M16×C53—24no—
20C22×M16×C53—23no—
21C22×M16×C53C418yescomplemented
22C22×M16×C53C2216yes—
23C23×C8×C53—21no—
24C23×Dic53—26no—
25C23×Dic53—22no—
26C23×Dic53—48no—
27C23×Dic53C2×C4112yes—
28C2×#11719917—224no—
29C2×#11719917—212no—
30C2×#11719917—26no—
31C2×#11719917C2×C4124yescomplemented
32C2×#11719917C2314yes—
33C22×#88322—28no—
34C22×#88322—26no—
35C2×C4⋊Dic53—212no—
36C2×C4⋊Dic53—26no—
37C2×C4⋊Dic53—26no—
38C2×C4⋊Dic53—26no—
39C2×C4⋊Dic53—26no—
40C2×C4⋊Dic53—26no—
41C2×C4⋊Dic53—26no—
42C2×C4⋊Dic53C2×C4112yes—
43C2×C4⋊Dic53C2314yes—
44C25×C53C2×C411yescharacteristic
45C23×C4×C53C2×C413yes—
46C23×C4×C53C2×C414yes—
47C23×C4×C53C2×C414yes—
48C23×C4×C53D414yes—
49C23×C4×C53D411yescharacteristic
50C23×C4×C53D413yes—
51C23×C4×C53D411yescharacteristic
52C23×C4×C53D413yes—
53C23×C4×C53Q814yes—
54C23×C4×C53C2313yes—
55C2×M16×C53—224no—
56C2×M16×C53—212no—
57C2×M16×C53—26no—
58C2×M16×C53C2×C4124yescomplemented
59C2×M16×C53C2314yes—
60C22×C8×C53—28no—
61C22×C8×C53—26no—
62C24×C53—24no—
63C24×C53—24no—
64C24×C53C4214yes—
65C24×C53C22⋊C413yes—
66C24×C53C22⋊C411yescharacteristic
67C24×C53C4⋊C414yes—
68C24×C53C22×C413yes—
69C22×C4×C53C42112yes—
70C22×C4×C53C22⋊C416yes—
71C22×C4×C53C22⋊C413yes—
72C22×C4×C53C22⋊C413yes—
73C22×C4×C53C22⋊C414yes—
74C22×C4×C53C22⋊C4112yes—
75C22×C4×C53C22⋊C414yes—
76C22×C4×C53C22⋊C4112yes—
77C22×C4×C53C4⋊C4112yes—
78C22×C4×C53C4⋊C4112yes—
79C22×C4×C53C4⋊C414yes—
80C22×C4×C53C4⋊C4112yes—
81C22×C4×C53C4⋊C414yes—
82C22×C4×C53C22×C413yes—
83C22×C4×C53C22×C416yes—
84C22×C4×C53C22×C416yes—
85C22×C4×C53C2×D416yes—
86C22×C4×C53C2×D413yes—
87C22×C4×C53C2×D413yes—
88C22×C4×C53C2×D413yes—
89C22×C4×C53C2×D413yes—
90C22×C4×C53C2×Q816yes—
91C22×C4×C53C2411yescharacteristic
92C22×Dic53—212no—
93C22×Dic53—212no—
94C22×Dic53—424no—
95C22×Dic53—48no—
96C22×Dic53—424no—
97C22×Dic53C22×C418yes—
98M16×C53—224no—
99M16×C53—24no—
100M16×C53C22×C418yescomplemented
101C2×C8×C53—224no—
102C2×C8×C53—24no—
103C4⋊Dic53—212no—
104C4⋊Dic53—212no—
105C4⋊Dic53—24no—
106C4⋊Dic53—24no—
107#11719917—224no—
108#11719917—24no—
109#11719917C22×C418yescomplemented
110C2×#88322—224no—
111C2×#88322—24no—
112C23×C53—216no—
113C23×C53—216no—
114C23×C53—26no—
115C23×C53—26no—
116C23×C53—26no—
117C23×C53—26no—
118C23×C53#95112yes—
119C23×C53#9514yes—
120C23×C53#10014yes—
121C23×C53#10114yes—
122C23×C53C2×C4216yes—
123C23×C53C2×C22⋊C413yes—
124C23×C53C2×C22⋊C413yes—
125C23×C53C2×C4⋊C416yes—
126C23×C53C23×C411yescharacteristic
127C2×C4×C53#95124yes—
128C2×C4×C53#95112yes—
129C2×C4×C53#95112yes—
130C2×C4×C53C2×C4216yes—
131C2×C4×C53C2×C22⋊C416yes—
132C2×C4×C53C2×C22⋊C416yes—
133C2×C4×C53C2×C22⋊C413yes—
134C2×C4×C53C2×C22⋊C413yes—
135C2×C4×C53C2×C22⋊C416yes—
136C2×C4×C53C2×C22⋊C416yes—
137C2×C4×C53C2×C4⋊C416yes—
138C2×C4×C53C2×C4⋊C416yes—
139C2×C4×C53C2×C4⋊C416yes—
140C2×C4×C53C2×C4⋊C416yes—
141C2×C4×C53C2×C4⋊C416yes—
142C2×C4×C53C23×C411yescharacteristic
143C2×C4×C53C23×C411yescharacteristic
144C2×C4×C53C22×D411yescharacteristic
145C2×C4×C53C22×D411yescharacteristic
146C2×C4×C53C22×D411yescharacteristic
147C2×C4×C53C22×Q811yescharacteristic
148C2×Dic53—28no—
149C2×Dic53—424no—
150C2×Dic53—424no—
151C2×Dic53—48no—
152C8×C53—28no—
153#88322—28no—
154C22×#343—531nomaximal, Sylow
155C22×C53—23no—
156C22×C53—21no—
157C22×C53—23no—
158C22×C53—21no—
159C22×C53—224no—
160C22×C53—212no—
161C22×C53—212no—
162C22×C53#343116yescomplemented
163C22×C53C2×#9516yes—
164C22×C53C2×#9516yes—
165C22×C53C2×#10016yes—
166C22×C53C2×#10116yes—
167C22×C53C22×C4211yescharacteristic
168C22×C53C22×C22⋊C411yescharacteristic
169C22×C53C22×C4⋊C411yescharacteristic
170C4×C53C2×#9516yes—
171C4×C53C2×#9516yes—
172C4×C53C22×C22⋊C411yescharacteristic
173C4×C53C22×C22⋊C411yescharacteristic
174C4×C53C22×C4⋊C411yescharacteristic
175C4×C53C22×C4⋊C411yescharacteristic
176Dic53—48no—
177C23×M16—531no—
178C23×M16D5311yescharacteristic
179C23×C4⋊C4—531no—
180C2×#343—5312no—
181C2×C53—21no—
182C2×C53—26no—
183C2×C53—21no—
184C2×C53C2×#343112yescomplemented
185C2×C53C22×#9511yescharacteristic
186C2×C53C22×#10011yescharacteristic
187C2×C53C22×#10111yescharacteristic
188C24×C4—532no—
189C24×C4C2×D5311yescharacteristic
190C22×M16—23no—
191C22×M16—24no—
192C22×M16—538no—
193C22×M16—536no—
194C22×M16—1063no—
195C22×M16—1064no—
196C22×M16Dic5318yescomplemented
197C22×M16C2×D5316yes—
198C23×C8—21no—
199C23×C8—1061no—
200C22×C4⋊C4—536no—
201C22×C4⋊C4—536no—
202C22×C4⋊C4—1063no—
203C22×C4⋊C4—1061no—
204C22×C4⋊C4—1063no—
205C22×C4⋊C4—1061no—
206#343—5316no—
207C53C22×#34311yescharacteristic, complemented, Sylow
208C25C4×D5311yescharacteristic
209C23×C4—5312no—
210C23×C4—1066no—
211C23×C4—1062no—
212C23×C4—2128no—
213C23×C4Dic10614yes—
214C23×C4C4×D5314yes—
215C23×C4C4×D5313yes—
216C23×C4D21213yes—
217C23×C4D21211yescharacteristic
218C23×C4C2×Dic5314yes—
219C23×C4C22⋊D5313yes—
220C23×C4C22⋊D5311yescharacteristic
221C23×C4C22⋊D5314yes—
222C23×C4C22×D5313yes—
223C2×M16—212no—
224C2×M16—26no—
225C2×M16—224no—
226C2×M16—5324no—
227C2×M16—534no—
228C2×M16—1066no—
229C2×M16—10612no—
230C2×M16—10624no—
231C2×M16C2×Dic53124yescomplemented
232C2×M16C22×D5314yes—
233C22×C8—26no—
234C22×C8—28no—
235C22×C8—1066no—
236C22×C8—1068no—
237C2×C4⋊C4—5312no—
238C2×C4⋊C4—534no—
239C2×C4⋊C4—1066no—
240C2×C4⋊C4—1066no—
241C2×C4⋊C4—1066no—
242C2×C4⋊C4—1066no—
243C2×C4⋊C4—10612no—
244C2×C4⋊C4—1066no—
245C2×C4⋊C4—1066no—
246C24—24no—
247C24—24no—
248C24C4×Dic5314yes—
249C24#1171991914yes—
250C24#1171992113yes—
251C24#1171992111yescharacteristic
252C24C2×C4×D5313yes—
253C22×C4—538no—
254C22×C4—10612no—
255C22×C4—10612no—
256C22×C4—21224no—
257C22×C4—2128no—
258C22×C4—21224no—
259C22×C4C4×Dic53112yes—
260C22×C4#11719919112yes—
261C22×C4#11719919112yes—
262C22×C4#1171991914yes—
263C22×C4C4⋊Dic53112yes—
264C22×C4C4⋊Dic5314yes—
265C22×C4#1171992116yes—
266C22×C4#1171992113yes—
267C22×C4#1171992113yes—
268C22×C4#1171992114yes—
269C22×C4#11719921112yes—
270C22×C4C22⋊Dic5314yes—
271C22×C4C22⋊Dic53112yes—
272C22×C4C2×Dic10616yes—
273C22×C4C2×C4×D5313yes—
274C22×C4C2×C4×D5316yes—
275C22×C4C2×D21213yes—
276C22×C4C2×D21213yes—
277C22×C4C22×Dic5316yes—
278C22×C4C2×C22⋊D5316yes—
279C22×C4C2×C22⋊D5313yes—
280C22×C4C2×C22⋊D5313yes—
281C22×C4C23×D5311yescharacteristic
282M16—224no—
283M16—24no—
284M16—538no—
285M16—10624no—
286M16—1064no—
287M16C22×Dic5318yescomplemented
288C2×C8—224no—
289C2×C8—24no—
290C2×C8—10624no—
291C2×C8—1064no—
292C4⋊C4—10612no—
293C4⋊C4—10612no—
294C4⋊C4—1064no—
295C4⋊C4—1064no—
296C23—26no—
297C23—26no—
298C23—26no—
299C23—26no—
300C23—216no—
301C23—216no—
302C23#13133124112yes—
303C23#1313312414yes—
304C23#1313313314yes—
305C23#1313313514yes—
306C23C2×C4×Dic5316yes—
307C23C2×#1171992113yes—
308C23C2×#1171992113yes—
309C23C2×#1171991916yes—
310C23C22×C4×D5311yescharacteristic
311C2×C4—1068no—
312C2×C4—21224no—
313C2×C4—21224no—
314C2×C4—2128no—
315C2×C4#13133124124yes—
316C2×C4#13133124112yes—
317C2×C4#13133124112yes—
318C2×C4C2×C4×Dic5316yes—
319C2×C4C2×#1171992116yes—
320C2×C4C2×#1171992116yes—
321C2×C4C2×#1171992113yes—
322C2×C4C2×#1171992113yes—
323C2×C4C2×C22⋊Dic5316yes—
324C2×C4C2×C22⋊Dic5316yes—
325C2×C4C2×#1171991916yes—
326C2×C4C2×#1171991916yes—
327C2×C4C2×#1171991916yes—
328C2×C4C2×C4⋊Dic5316yes—
329C2×C4C2×C4⋊Dic5316yes—
330C2×C4C22×C4×D5311yescharacteristic
331C2×C4C23×Dic5311yescharacteristic
332C2×C4C22×C22⋊D5311yescharacteristic
333C2×C4C22×C22⋊D5311yescharacteristic
334C2×C4C22×D21211yescharacteristic
335C2×C4C22×Dic10611yescharacteristic
336C8—28no—
337C8—1068no—
338C22—224no—
339C22—212no—
340C22—212no—
341C22—23no—
342C22—21no—
343C22—23no—
344C22—21no—
345C22#16810391116yesdirect product factor
346C22C2×#1313312416yes—
347C22C2×#1313312416yes—
348C22C2×#1313313316yes—
349C22C2×#1313313516yes—
350C22C22×C4×Dic5311yescharacteristic
351C22C22×#1171992111yescharacteristic
352C22C22×#1171991911yescharacteristic
353C4—2128no—
354C4C2×#1313312416yes—
355C4C2×#1313312416yes—
356C4C22×#1171992111yescharacteristic
357C4C22×C22⋊Dic5311yescharacteristic
358C4C22×#1171991911yescharacteristic
359C4C22×C4⋊Dic5311yescharacteristic
360C2—21no—
361C2—21no—
362C2—26no—
363C2C2×#16810391112yesdirect product factor
364C2C22×#1313312411yescharacteristic
365C2C22×#1313313311yescharacteristic
366C2C22×#1313313511yescharacteristic
subgroupG/H
centerC24#11719921
commutatorC22×C53C22×C42
FrattiniC22×C4C23×D53
FittingC23×M16×C53C2
socleC24×C53C22⋊C4
prime radicalC25×C53C2×C4
cosocleC22×C4×C53C24
hypercenterC23×M16D53
hypocenterC53C22×#343
LCSG ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C24 ◁ C24×C4 ◁ C23×M16

Complements

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

NH ≅ G/Ntype
3. C2×#16810391C2direct (N × H)
8. C22×#11719917C4semidirect
17. #16810391C22direct (N × H)
21. C22×M16×C53C4semidirect
31. C2×#11719917C2×C4semidirect
58. C2×M16×C53C2×C4semidirect
100. M16×C53C22×C4semidirect
109. #11719917C22×C4semidirect
162. C22×C53#343semidirect
184. C2×C53C2×#343semidirect
196. C22×M16Dic53semidirect
207. C53C22×#343semidirect
231. C2×M16C2×Dic53semidirect
287. M16C22×Dic53semidirect
345. C22#16810391direct (N × H)
363. C2C2×#16810391direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#117199172
C23×C4⋊Dic532
C2×#168103912
C23×M16×C532
C22×#34353