Finite Group Databasebeta

← C2×#20274431 #27381491

Subgroups — C2×#20274431 #27381491

Order 13568.

2398 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C24×#88322C211yesmaximal, characteristic
2C22×#13133176C211yesmaximal, characteristic
3#20274431C218yesmaximal, direct product factor
4C2×#16810385C214yesmaximal, complemented
5C22×#117×C53C211yesmaximal, characteristic
6C23×#88322—24no—
7C23×#88322—24no—
8C23×#88322C418yescomplemented
9C23×#88322C2212yes—
10C23×#88322C2214yes—
11C2×#13133176—24no—
12C2×#13133176—22no—
13C2×#13133176C418yescomplemented
14C2×#13133176C2214yes—
15C22×C4⋊Dic53C2212yes—
16C22×C22⋊Dic53—21no—
17C22×C4×Dic53—21no—
18#16810385C22116yescomplemented
19C24×C4×C53C2211yescharacteristic
20C2×#117×C53—24no—
21C2×#117×C53—22no—
22C2×#117×C53C418yescomplemented
23C2×#117×C53C2214yes—
24C22×C4⋊C4×C53C2212yes—
25C22×C22⋊C4×C53—21no—
26C22×C42×C53—21no—
27C23×Dic53—22no—
28C22×#88322—232no—
29C22×#88322—24no—
30C22×#88322—28no—
31C22×#88322—24no—
32C22×#88322—28no—
33C22×#88322C2×C418yes—
34C22×#88322C2×C4116yescomplemented
35C22×#88322C2314yes—
36#13133176—216no—
37#13133176—28no—
38#13133176C2×C4116yescomplemented
39C2×C4⋊Dic53—24no—
40C2×C4⋊Dic53—22no—
41C2×C4⋊Dic53C2×C418yes—
42C2×C4⋊Dic53C2314yes—
43C2×C22⋊Dic53—28no—
44C2×C22⋊Dic53—24no—
45C2×C4×Dic53—28no—
46C2×C4×Dic53—24no—
47C25×C53D411yescharacteristic
48C23×C4×C53—22no—
49C23×C4×C53C2×C414yes—
50C23×C4×C53C2×C414yes—
51C23×C4×C53C2×C414yes—
52C23×C4×C53D412yes—
53C23×C4×C53D411yescharacteristic
54C23×C4×C53D414yes—
55C23×C4×C53D414yes—
56C23×C4×C53Q814yes—
57C23×C4×C53C2311yescharacteristic
58C23×C4×C53C2312yes—
59#117×C53—216no—
60#117×C53—28no—
61#117×C53C2×C4116yescomplemented
62C2×C4⋊C4×C53—24no—
63C2×C4⋊C4×C53—22no—
64C2×C4⋊C4×C53C2×C418yes—
65C2×C4⋊C4×C53C2314yes—
66C2×C22⋊C4×C53—28no—
67C2×C22⋊C4×C53—24no—
68C2×C42×C53—28no—
69C2×C42×C53—24no—
70C24×C53—28no—
71C24×C53C22⋊C414yes—
72C24×C53C22⋊C414yes—
73C24×C53C4⋊C414yes—
74C24×C53C2×D412yes—
75C24×C53C2×D411yescharacteristic
76C22×C4×C53—28no—
77C22×C4×C53—24no—
78C22×C4×C53—48no—
79C22×C4×C53C42116yes—
80C22×C4×C53C22⋊C414yes—
81C22×C4×C53C22⋊C418yes—
82C22×C4×C53C22⋊C414yes—
83C22×C4×C53C22⋊C418yes—
84C22×C4×C53C22⋊C4116yes—
85C22×C4×C53C4⋊C418yes—
86C22×C4×C53C4⋊C414yes—
87C22×C4×C53C4⋊C4116yes—
88C22×C4×C53C4⋊C4116yes—
89C22×C4×C53C22×C412yes—
90C22×C4×C53C22×C414yes—
91C22×C4×C53C22×C412yes—
92C22×C4×C53C22×C414yes—
93C22×C4×C53C22×C412yes—
94C22×C4×C53C22×C414yes—
95C22×C4×C53C2×D412yes—
96C22×C4×C53C2×D411yescharacteristic
97C22×C4×C53C2×D412yes—
98C22×C4×C53C2×D414yes—
99C22×C4×C53C2×D412yes—
100C22×C4×C53C2×D414yes—
101C22×C4×C53C2×Q814yes—
102C22×C4×C53C2×Q812yes—
103C22×C4×C53C2411yescharacteristic
104C22×Dic53—28no—
105C22×Dic53—24no—
106C22×Dic53—48no—
107C4⋊C4×C53—28no—
108C4⋊C4×C53—24no—
109C4⋊C4×C53C22×C418yes—
110C22⋊C4×C53—216no—
111C42×C53—216no—
112C22⋊Dic53—216no—
113C4⋊Dic53—28no—
114C4⋊Dic53—24no—
115C4⋊Dic53C22×C418yes—
116C4×Dic53—216no—
117C2×#88322—216no—
118C2×#88322—232no—
119C2×#88322—24no—
120C2×#88322—24no—
121C2×#88322C22×C418yes—
122C23×C53—28no—
123C23×C53—216no—
124C23×C53—216no—
125C23×C53—216no—
126C23×C53#95116yes—
127C23×C53C2×C22⋊C414yes—
128C23×C53C2×C22⋊C412yes—
129C23×C53C2×C22⋊C414yes—
130C23×C53C2×C22⋊C412yes—
131C23×C53C2×C4⋊C414yes—
132C23×C53C2×C4⋊C412yes—
133C23×C53#13518yes—
134C23×C53C22×D411yescharacteristic
135C2×C4×C53—28no—
136C2×C4×C53—48no—
137C2×C4×C53—416no—
138C2×C4×C53#95132yes—
139C2×C4×C53#95116yes—
140C2×C4×C53C2×C4218yes—
141C2×C4×C53C2×C4214yes—
142C2×C4×C53C2×C22⋊C418yes—
143C2×C4×C53C2×C22⋊C414yes—
144C2×C4×C53C2×C22⋊C414yes—
145C2×C4×C53C2×C22⋊C412yes—
146C2×C4×C53C2×C22⋊C414yes—
147C2×C4×C53C2×C22⋊C412yes—
148C2×C4×C53C2×C4⋊C418yes—
149C2×C4×C53C2×C4⋊C414yes—
150C2×C4×C53C2×C4⋊C412yes—
151C2×C4×C53C2×C4⋊C414yes—
152C2×C4×C53C2×C4⋊C418yes—
153C2×C4×C53C2×C4⋊C414yes—
154C2×C4×C53C23×C411yescharacteristic
155C2×C4×C53C23×C411yescharacteristic
156C2×C4×C53C23×C411yescharacteristic
157C2×C4×C53C22×D411yescharacteristic
158C2×C4×C53C22×D411yescharacteristic
159C2×C4×C53C22×Q811yescharacteristic
160C2×Dic53—28no—
161C2×Dic53—48no—
162C2×Dic53—416no—
163#88322—28no—
164C2×#1735—531nomaximal, Sylow
165C22×C53—28no—
166C22×C53—28no—
167C22×C53—28no—
168C22×C53—232no—
169C22×C53C2×#9518yes—
170C22×C53C2×#9514yes—
171C22×C53#41618yes—
172C22×C53#41618yes—
173C22×C53#42718yes—
174C22×C53C22×C22⋊C411yescharacteristic
175C22×C53C22×C22⋊C411yescharacteristic
176C22×C53C22×C4⋊C411yescharacteristic
177C22×C53C2×#13514yes—
178C4×C53—48no—
179C4×C53C2×#9518yes—
180C4×C53C2×#9514yes—
181C4×C53C22×C4211yescharacteristic
182C4×C53C22×C22⋊C411yescharacteristic
183C4×C53C22×C4⋊C411yescharacteristic
184C4×C53C22×C4⋊C411yescharacteristic
185Dic53—48no—
186C24×C8—531no—
187C22×#117—531no—
188C22×#117D5311yescharacteristic
189#1735—538no—
190C2×#340—534no—
191C2×C53—28no—
192C2×C53#173518yescomplemented
193C2×C53C22×#9511yescharacteristic
194C2×C53C2×#41612yes—
195C2×C53C2×#41612yes—
196C2×C53C2×#42712yes—
197C24×C4C2×D5311yescharacteristic
198C23×C8—538no—
199C23×C8—532no—
200C23×C8—534no—
201C23×C8—1064no—
202C23×C8—1064no—
203C2×#117—22no—
204C2×#117—24no—
205C2×#117—538no—
206C2×#117—534no—
207C2×#117—1064no—
208C2×#117—1062no—
209C2×#117Dic5318yescomplemented
210C2×#117C2×D5314yes—
211C22×C4⋊C4—532no—
212C22×C4⋊C4C2×D5312yes—
213C22×C22⋊C4—21no—
214C22×C22⋊C4—1061no—
215C22×C42—21no—
216C22×C42—1061no—
217#340—5316no—
218C53C2×#173511yescharacteristic, complemented, Sylow
219C25C22⋊D5311yescharacteristic
220C23×C4—22no—
221C23×C4—1062no—
222C23×C4Dic10614yes—
223C23×C4C4×D5314yes—
224C23×C4C4×D5314yes—
225C23×C4D21214yes—
226C23×C4C2×Dic5314yes—
227C23×C4C22⋊D5314yes—
228C23×C4C22⋊D5312yes—
229C23×C4C22⋊D5311yescharacteristic
230C23×C4C22×D5311yescharacteristic
231C23×C4C22×D5312yes—
232C22×C8—538no—
233C22×C8—5316no—
234C22×C8—534no—
235C22×C8—1064no—
236C22×C8—1068no—
237C22×C8—1064no—
238C22×C8—1068no—
239C22×C8—10632no—
240#117—28no—
241#117—216no—
242#117—5316no—
243#117—1068no—
244#117—10616no—
245#117C2×Dic53116yescomplemented
246C2×C4⋊C4—22no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—538no—
249C2×C4⋊C4—534no—
250C2×C4⋊C4—1064no—
251C2×C4⋊C4—1062no—
252C2×C4⋊C4C2×Dic5318yes—
253C2×C4⋊C4C22×D5314yes—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—1064no—
257C2×C22⋊C4—1068no—
258C2×C42—24no—
259C2×C42—28no—
260C2×C42—1064no—
261C2×C42—1068no—
262C24—28no—
263C24#1171991914yes—
264C24#1171992114yes—
265C24C22⋊Dic5314yes—
266C24C2×C22⋊D5312yes—
267C24C2×C22⋊D5311yescharacteristic
268C22×C4—28no—
269C22×C4—24no—
270C22×C4—48no—
271C22×C4—1068no—
272C22×C4—1064no—
273C22×C4—2128no—
274C22×C4C4×Dic53116yes—
275C22×C4#1171991918yes—
276C22×C4#1171991914yes—
277C22×C4#11719919116yes—
278C22×C4C4⋊Dic53116yes—
279C22×C4#1171992114yes—
280C22×C4#1171992118yes—
281C22×C4#11719921116yes—
282C22×C4C22⋊Dic5314yes—
283C22×C4C22⋊Dic5318yes—
284C22×C4C2×Dic10614yes—
285C22×C4C2×Dic10612yes—
286C22×C4C2×C4×D5314yes—
287C22×C4C2×C4×D5312yes—
288C22×C4C2×C4×D5312yes—
289C22×C4C2×C4×D5314yes—
290C22×C4C2×D21212yes—
291C22×C4C2×D21214yes—
292C22×C4C22×Dic5312yes—
293C22×C4C22×Dic5314yes—
294C22×C4C2×C22⋊D5312yes—
295C22×C4C2×C22⋊D5311yescharacteristic
296C22×C4C2×C22⋊D5312yes—
297C22×C4C2×C22⋊D5314yes—
298C22×C4C23×D5311yescharacteristic
299C2×C8—538no—
300C2×C8—10616no—
301C2×C8—10632no—
302C2×C8—1064no—
303C2×C8—1064no—
304C4⋊C4—28no—
305C4⋊C4—24no—
306C4⋊C4—538no—
307C4⋊C4—1068no—
308C4⋊C4—1064no—
309C4⋊C4C22×Dic5318yes—
310C22⋊C4—216no—
311C22⋊C4—10616no—
312C42—216no—
313C42—10616no—
314C23—216no—
315C23—216no—
316C23—28no—
317C23—216no—
318C23#13133124116yes—
319C23C2×#1171992114yes—
320C23C2×#1171992112yes—
321C23C2×C22⋊Dic5314yes—
322C23C2×C22⋊Dic5312yes—
323C23C2×#1171991914yes—
324C23C2×#1171991912yes—
325C23#1313323718yes—
326C23C22×C22⋊D5311yescharacteristic
327C2×C4—28no—
328C2×C4—416no—
329C2×C4—48no—
330C2×C4—1068no—
331C2×C4—21216no—
332C2×C4—2128no—
333C2×C4#13133124132yes—
334C2×C4#13133124116yes—
335C2×C4C2×C4×Dic5318yes—
336C2×C4C2×C4×Dic5314yes—
337C2×C4C2×#1171992118yes—
338C2×C4C2×#1171992114yes—
339C2×C4C2×#1171992114yes—
340C2×C4C2×#1171992112yes—
341C2×C4C2×C22⋊Dic5314yes—
342C2×C4C2×C22⋊Dic5312yes—
343C2×C4C2×#1171991918yes—
344C2×C4C2×#1171991914yes—
345C2×C4C2×#1171991914yes—
346C2×C4C2×#1171991912yes—
347C2×C4C2×C4⋊Dic5318yes—
348C2×C4C2×C4⋊Dic5314yes—
349C2×C4C22×C4×D5311yescharacteristic
350C2×C4C22×C4×D5311yescharacteristic
351C2×C4C23×Dic5311yescharacteristic
352C2×C4C22×C22⋊D5311yescharacteristic
353C2×C4C22×D21211yescharacteristic
354C2×C4C22×Dic10611yescharacteristic
355C8—1068no—
356C22—28no—
357C22—28no—
358C22—28no—
359C22—232no—
360C22C2×#1313312418yes—
361C22C2×#1313312414yes—
362C22#1681061718yes—
363C22#1681061918yes—
364C22#1681065918yes—
365C22C22×#1171992111yescharacteristic
366C22C22×C22⋊Dic5311yescharacteristic
367C22C22×#1171991911yescharacteristic
368C22C2×#1313323714yes—
369C4—48no—
370C4—2128no—
371C4C2×#1313312418yes—
372C4C2×#1313312414yes—
373C4C22×C4×Dic5311yescharacteristic
374C4C22×#1171992111yescharacteristic
375C4C22×#1171991911yescharacteristic
376C4C22×C4⋊Dic5311yescharacteristic
377C2—28no—
378C2#2027443118yesdirect product factor
379C2C22×#1313312411yescharacteristic
380C2C2×#1681061712yes—
381C2C2×#1681061912yes—
382C2C2×#1681065912yes—
subgroupG/H
centerC23×C4C22⋊D53
commutatorC4×C53C22×C42
FrattiniC22×C4C23×D53
FittingC22×#117×C53C2
socleC24×C53C2×D4
prime radicalC25×C53D4
cosocleC22×C4×C53C24
hypercenterC22×#117D53
hypocenterC53C2×#1735
LCSG ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23×C4 ◁ C24×C4 ◁ C22×#117

Complements

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

NH ≅ G/Ntype
3. #20274431C2direct (N × H)
4. C2×#16810385C2semidirect
8. C23×#88322C4semidirect
13. C2×#13133176C4semidirect
18. #16810385C22semidirect
22. C2×#117×C53C4semidirect
34. C22×#88322C2×C4semidirect
38. #13133176C2×C4semidirect
61. #117×C53C2×C4semidirect
192. C2×C53#1735semidirect
209. C2×#117Dic53semidirect
218. C53C2×#1735semidirect
245. #117C2×Dic53semidirect
378. C2#20274431direct (N × H)

Maximal subgroups

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

subgroupindex
C24×#883222
C22×#131331762
#202744312
C2×#168103852
C22×#117×C532
C2×#173553