Finite Group Databasebeta

← C2×#22566145 #31022931

Subgroups — C2×#22566145 #31022931

Order 17152.

2806 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14524043C211yesmaximal, characteristic, complemented
2C22×#14524031C211yesmaximal, characteristic
3#22566145C2116yesmaximal, direct product factor
4C2×#16863323C214yesmaximal, complemented
5C2×#16863309C214yesmaximal, complemented
6C2×#16863302C212yesmaximal, complemented
7C2×#16863298C212yesmaximal
8C2×#376×C67C211yesmaximal, characteristic, complemented
9C23×Dic134C2211yescharacteristic
10C23×Dic134C2212yes—
11C23×D268C2211yescharacteristic
12C2×#14524043—28no—
13C2×#14524043C2218yescomplemented
14C2×#14524031—28no—
15C2×#14524031C2218yes—
16C22×C4⋊Dic67C2212yes—
17C22×C4⋊Dic67C2212yes—
18C22×(C2×D67)⋊C4C2212yes—
19C22×(C2×D67)⋊C4C2212yes—
20C22×(C2×D67)⋊C4C2218yes—
21#16863323C22132yescomplemented
22#16863309C22132yescomplemented
23#16863302C22116yescomplemented
24#16863298C22116yes—
25C2×C67⋊#95C2218yes—
26C22×C42×C67C2212yes—
27C22×C42×C67C2211yescharacteristic
28#376×C67C2218yescomplemented
29C2×#95×C67C2212yes—
30C2×#95×C67C2212yes—
31C24×D67C2312yes—
32C22×Dic134—24no—
33C22×Dic134—28no—
34C22×Dic134—28no—
35C22×Dic134—48no—
36C22×Dic134C2314yes—
37C22×Dic134C2318yes—
38C22×D268—24no—
39C22×D268—28no—
40C22×D268C2314yes—
41C23×Dic67C2312yes—
42C23×Dic67C2314yes—
43#14524043—232no—
44#14524031—232no—
45C2×C4⋊Dic67—28no—
46C2×C4⋊Dic67—28no—
47C2×C4⋊Dic67C2318yes—
48C2×C4⋊Dic67C2318yes—
49C2×(C2×D67)⋊C4—28no—
50C2×(C2×D67)⋊C4—28no—
51C2×(C2×D67)⋊C4—416no—
52C2×(C2×D67)⋊C4C2318yes—
53C2×(C2×D67)⋊C4C2318yes—
54C2×(C2×D67)⋊C4C23132yes—
55C67⋊#95C23132yes—
56C23×C4×C67C2314yes—
57C23×C4×C67C2311yescharacteristic
58C23×C4×C67C2311yescharacteristic
59C23×C4×C67C2311yescharacteristic
60C2×C42×C67—28no—
61C2×C42×C67D418yes—
62C2×C42×C67C2318yes—
63C2×C42×C67C2314yes—
64#95×C67C2318yes—
65#95×C67C2318yes—
66C23×D67—24no—
67C23×D67—28no—
68C23×D67—28no—
69C23×D67—28no—
70C23×D67C2414yes—
71C2×Dic134—28no—
72C2×Dic134—216no—
73C2×Dic134—216no—
74C2×Dic134—416no—
75C2×Dic134—432no—
76C2×Dic134—416no—
77C2×D268—28no—
78C2×D268—216no—
79C2×D268—416no—
80C22×Dic67—24no—
81C22×Dic67—28no—
82C22×Dic67—28no—
83C22×Dic67—216no—
84C22×Dic67C2414yes—
85C22×Dic67C2418yes—
86C4⋊Dic67—216no—
87C4⋊Dic67—216no—
88(C2×D67)⋊C4—216no—
89(C2×D67)⋊C4—216no—
90(C2×D67)⋊C4—432no—
91C24×C67C2411yescharacteristic
92C22×C4×C67—24no—
93C22×C4×C67—24no—
94C22×C4×C67—28no—
95C22×C4×C67—28no—
96C22×C4×C67—28no—
97C22×C4×C67C2×D414yes—
98C22×C4×C67C2×D414yes—
99C22×C4×C67C2×D414yes—
100C22×C4×C67#4118yes—
101C22×C4×C67C2418yes—
102C22×C4×C67C2412yes—
103C22×C4×C67C2412yes—
104C22×C4×C67C2412yes—
105C42×C67—216no—
106C42×C67C2×D4116yes—
107C23×C67C22×D412yes—
108C23×C67C2×#4114yes—
109C23×C67H(5,𝔽2)14yes—
110C23×C672−1+414yes—
111C23×C67C2511yescharacteristic
112C22×D67—24no—
113C22×D67—28no—
114C22×D67—28no—
115C22×D67—28no—
116C22×D67—48no—
117C22×D67—48no—
118C22×D67—48no—
119C22×D67—432no—
120C2×C4×C67—28no—
121C2×C4×C67—28no—
122C2×C4×C67—24no—
123C2×C4×C67—24no—
124C2×C4×C67—28no—
125C2×C4×C67—28no—
126C2×C4×C67—28no—
127C2×C4×C67—416no—
128C2×C4×C67#124116yes—
129C2×C4×C67C22×D414yes—
130C2×C4×C67C22×D414yes—
131C2×C4×C67C22×D414yes—
132C2×C4×C67C2×#4118yes—
133C2×Dic67—24no—
134C2×Dic67—28no—
135C2×Dic67—28no—
136C2×Dic67—216no—
137C2×Dic67—48no—
138C2×Dic67—416no—
139D268—416no—
140Dic134—432no—
141Dic134—416no—
142C22×C67C2×#12414yes—
143C22×C67#53514yes—
144C22×C67#53614yes—
145C22×C67#542116yes—
146C22×C67C23×D411yescharacteristic
147C22×C67C22×#4112yes—
148C22×C67C2×H(5,𝔽2)12yes—
149C22×C67C2×2−1+412yes—
150C2×D67—44no—
151C2×D67—44no—
152C2×D67—44no—
153C2×D67—416no—
154C2×D67—816no—
155C4×C67—24no—
156C4×C67—24no—
157C4×C67—48no—
158C4×C67C2×#12418yes—
159Dic67—44no—
160Dic67—48no—
161C2×#2640—671nomaximal, Sylow
162C2×C67#264018yescomplemented
163C2×C67C22×#12411yescharacteristic
164C2×C67C2×#53511yescharacteristic
165C2×C67C2×#53611yescharacteristic
166C2×C67C2×#54214yes—
167D67—84no—
168C22×C4⋊Q8—671no—
169C22×#124—671no—
170#2640—6716no—
171C2×#397—674no—
172C2×#393—674no—
173C2×#391—672no—
174C2×#390—672no—
175C2×#376D6711yescharacteristic, complemented
176C67C2×#264011yescharacteristic, complemented, Sylow
177C23×Q8—672no—
178C23×Q8—671no—
179C23×D4—671no—
180C2×C4⋊Q8—678no—
181C2×C4⋊Q8—1348no—
182C2×#124—678no—
183C2×#124—1348no—
184C22×C4⋊C4—672no—
185C22×C4⋊C4—672no—
186C22×C22⋊C4—678no—
187C22×C22⋊C4—672no—
188C22×C22⋊C4—672no—
189C22×C42C2×D6712yes—
190C22×C42C2×D6711yescharacteristic
191#397—6732no—
192#393—6732no—
193#391—6716no—
194#390—6716no—
195#376C2×D6718yescomplemented
196C2×#95—678no—
197C2×#95C2×D6712yes—
198C2×#95C2×D6712yes—
199C25—672no—
200C22×Q8—2688no—
201C22×Q8—678no—
202C22×Q8—674no—
203C22×Q8—1344no—
204C22×Q8—1348no—
205C22×Q8—1348no—
206C22×D4—674no—
207C22×D4—1344no—
208C22×D4—1348no—
209C23×C4—674no—
210C23×C4—672no—
211C23×C4C22×D6714yes—
212C23×C4C22×D6711yescharacteristic
213C23×C4C22×D6711yescharacteristic
214C23×C4C22×D6711yescharacteristic
215C4⋊Q8—13432no—
216#124—13432no—
217C2×C4⋊C4—678no—
218C2×C4⋊C4—678no—
219C2×C4⋊C4—1348no—
220C2×C4⋊C4—1348no—
221C2×C22⋊C4—26816no—
222C2×C22⋊C4—6732no—
223C2×C22⋊C4—678no—
224C2×C22⋊C4—678no—
225C2×C22⋊C4—1348no—
226C2×C22⋊C4—1348no—
227C2×C42—28no—
228C2×C42D26818yes—
229C2×C42C22×D6718yes—
230C2×C42C22×D6714yes—
231#95—6732no—
232#95C22×D6718yes—
233#95C22×D6718yes—
234C24—674no—
235C24—1344no—
236C24—1348no—
237C24—1348no—
238C24—1348no—
239C24C23×D6711yescharacteristic
240C2×Q8—26832no—
241C2×Q8—26816no—
242C2×Q8—26816no—
243C2×Q8—1348no—
244C2×Q8—13416no—
245C2×Q8—13416no—
246C2×D4—26816no—
247C2×D4—1348no—
248C2×D4—13416no—
249C22×C4—24no—
250C22×C4—24no—
251C22×C4—28no—
252C22×C4—28no—
253C22×C4—28no—
254C22×C4—674no—
255C22×C4—678no—
256C22×C4—1344no—
257C22×C4—1348no—
258C22×C4—1348no—
259C22×C4—13416no—
260C22×C4C2×D26814yes—
261C22×C4C2×D26814yes—
262C22×C4C2×D26814yes—
263C22×C4#1176162818yes—
264C22×C4C23×D6718yes—
265C22×C4C23×D6712yes—
266C22×C4C23×D6712yes—
267C22×C4C23×D6712yes—
268C4⋊C4—13416no—
269C4⋊C4—13416no—
270C22⋊C4—26832no—
271C22⋊C4—13416no—
272C22⋊C4—13416no—
273C42—216no—
274C42C2×D268116yes—
275C23—26832no—
276C23—2688no—
277C23—2688no—
278C23—2688no—
279C23—1344no—
280C23—1348no—
281C23—1348no—
282C23—1348no—
283C23C22×D26812yes—
284C23C2×#1176162814yes—
285C23#1452408914yes—
286C23#1452409014yes—
287C23C24×D6711yescharacteristic
288Q8—26832no—
289Q8—26816no—
290D4—26816no—
291C2×C4—28no—
292C2×C4—24no—
293C2×C4—28no—
294C2×C4—24no—
295C2×C4—28no—
296C2×C4—28no—
297C2×C4—28no—
298C2×C4—416no—
299C2×C4—2688no—
300C2×C4—26816no—
301C2×C4—1344no—
302C2×C4—1348no—
303C2×C4—1348no—
304C2×C4—13416no—
305C2×C4#14524031116yes—
306C2×C4C22×D26814yes—
307C2×C4C22×D26814yes—
308C2×C4C22×D26814yes—
309C2×C4C2×#1176162818yes—
310C22—26816no—
311C22—2684no—
312C22—2684no—
313C22—2684no—
314C22—53616no—
315C22C2×#1452403114yes—
316C22#1686396514yes—
317C22#1686397214yes—
318C22#16864017116yes—
319C22C23×D26811yescharacteristic
320C22C22×#1176162812yes—
321C22C2×#1452408912yes—
322C22C2×#1452409012yes—
323C4—24no—
324C4—24no—
325C4—48no—
326C4—2684no—
327C4—2688no—
328C4C2×#1452403118yes—
329C2—5364no—
330C2#2256614518yesdirect product factor
331C2C22×#1452403111yescharacteristic
332C2C2×#1686396511yescharacteristic
333C2C2×#1686397211yescharacteristic
334C2C2×#1686401714yes—
subgroupG/H
centerC24C23×D67
commutator, cosocleC23×C67C25
FrattiniC23C24×D67
FittingC2×#376×C67C2
socleC24×C67C24
prime radicalC23×D268C22
hypercenterC2×#376D67
hypocenterC67C2×#2640
LCSG ▷ C23×C67 ▷ C67
UCSC1 ◁ C24 ◁ C2×#376

Complements

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

NH ≅ G/Ntype
1. C22×#14524043C2semidirect
3. #22566145C2direct (N × H)
4. C2×#16863323C2semidirect
5. C2×#16863309C2semidirect
6. C2×#16863302C2semidirect
8. C2×#376×C67C2semidirect
13. C2×#14524043C22semidirect
21. #16863323C22semidirect
22. #16863309C22semidirect
23. #16863302C22semidirect
28. #376×C67C22semidirect
162. C2×C67#2640semidirect
175. C2×#376D67semidirect
176. C67C2×#2640semidirect
195. #376C2×D67semidirect
330. C2#22566145direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145240432
C22×#145240312
#225661452
C2×#168633232
C2×#168633092
C2×#168633022
C2×#168632982
C2×#376×C672
C2×#264067