Finite Group Databasebeta

← C2×#22563321 #31015510

Subgroups — C2×#22563321 #31015510

Order 17152.

3158 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22563321C2116yesmaximal, direct product factor
2C2×#16863303C212yesmaximal, complemented
3C2×#16863289C214yesmaximal, complemented
4C2×#16863276C212yesmaximal
5C2×C22⋊C4×Dic67C214yesmaximal, complemented
6C2×#16863242C212yesmaximal, complemented
7C22×#124×C67C211yesmaximal, characteristic
8C24×Dic67C2212yes—
9C22×C4⋊Dic67C2212yes—
10C22×C22⋊Dic67C2214yes—
11C22×C22⋊Dic67C2212yes—
12C22×C4×Dic67C2214yes—
13C22×C4×Dic67C2212yes—
14#16863303C22116yescomplemented
15#16863289C22132yescomplemented
16#16863276C22116yes—
17C22⋊C4×Dic67C22132yescomplemented
18#16863242C22116yescomplemented
19C2×C67⋊#95C2214yes—
20C2×C67⋊#95C2212yes—
21C2×C67⋊#95C2214yes—
22C2×C67⋊#95C2212yes—
23C23×Q8×C67C2211yescharacteristic
24C23×D4×C67C2211yescharacteristic
25C2×#124×C67C4116yescomplemented
26C2×#124×C67C2218yes—
27C22×C22⋊C4×C67C2214yes—
28C22×C42×C67C2211yescharacteristic
29C23×Dic67—216no—
30C23×Dic67—44no—
31C23×Dic67C2314yes—
32C23×Dic67C2318yes—
33C23×Dic67C2314yes—
34C2×C4⋊Dic67—44no—
35C2×C4⋊Dic67C2318yes—
36C2×C22⋊Dic67—216no—
37C2×C22⋊Dic67—44no—
38C2×C22⋊Dic67C23116yes—
39C2×C22⋊Dic67C2318yes—
40C2×C4×Dic67—216no—
41C2×C4×Dic67—44no—
42C2×C4×Dic67C23116yes—
43C2×C4×Dic67C2318yes—
44C67⋊#95C2318yes—
45C67⋊#95C23116yes—
46C67⋊#95C2318yes—
47C67⋊#95C23116yes—
48C25×C67C2312yes—
49C22×Q8×C67—42no—
50C22×Q8×C67—42no—
51C22×Q8×C67C2×C418yescomplemented
52C22×Q8×C67C2314yes—
53C22×D4×C67—42no—
54C22×D4×C67—42no—
55C22×D4×C67C2×C418yes—
56C22×D4×C67C2314yes—
57C23×C4×C67C2311yescharacteristic
58C23×C4×C67C2312yes—
59C23×C4×C67C2312yes—
60#124×C67C2×C4164yescomplemented
61C2×C22⋊C4×C67C2×C4132yescomplemented
62C2×C22⋊C4×C67C23116yes—
63C2×C42×C67C2×C418yescomplemented
64C2×C42×C67C2314yes—
65C22×Dic67—28no—
66C22×Dic67—28no—
67C22×Dic67—28no—
68C22×Dic67—28no—
69C22×Dic67—28no—
70C22×Dic67—232no—
71C22×Dic67—48no—
72C22×Dic67—432no—
73C22×Dic67#41116yes—
74C22×Dic67C2418yes—
75C22×Dic67C2418yes—
76C4⋊Dic67—48no—
77C22⋊Dic67—232no—
78C22⋊Dic67—48no—
79C4×Dic67—232no—
80C4×Dic67—48no—
81C24×C67—28no—
82C24×C67—28no—
83C24×C67—24no—
84C24×C67—24no—
85C24×C67C22×C418yes—
86C24×C67C2414yes—
87C24×C67C2411yescharacteristic
88C2×Q8×C67—416no—
89C2×Q8×C67—44no—
90C2×Q8×C67—44no—
91C2×Q8×C67C22×C4116yescomplemented
92C2×D4×C67—416no—
93C2×D4×C67—44no—
94C2×D4×C67—44no—
95C2×D4×C67C22×C4116yes—
96C22×C4×C67—28no—
97C22×C4×C67—22no—
98C22×C4×C67—22no—
99C22×C4×C67—24no—
100C22×C4×C67—24no—
101C22×C4×C67C22×C418yes—
102C22×C4×C67C22×C414yes—
103C22×C4×C67C22×C418yes—
104C22×C4×C67C2414yes—
105C22×C4×C67C2412yes—
106C22×C4×C67C2414yes—
107C22⋊C4×C67C22×C4164yescomplemented
108C42×C67C22×C4116yescomplemented
109C23×C67—232no—
110C23×C67—216no—
111C23×C67—28no—
112C23×C67—28no—
113C23×C67—24no—
114C23×C67—24no—
115C23×C67—48no—
116C23×C67—48no—
117C23×C67—416no—
118C23×C67C23×C418yes—
119C23×C67C23×C412yes—
120C23×C67C2×#4118yes—
121C23×C67H(5,𝔽2)12yes—
122C23×C67H(5,𝔽2)12yes—
123C23×C67C2511yescharacteristic
124Q8×C67—416no—
125D4×C67—416no—
126C2×C4×C67—216no—
127C2×C4×C67—28no—
128C2×C4×C67—216no—
129C2×C4×C67—28no—
130C2×C4×C67—22no—
131C2×C4×C67—22no—
132C2×C4×C67—24no—
133C2×C4×C67—24no—
134C2×C4×C67C23×C418yes—
135C2×C4×C67C23×C414yes—
136C2×C4×C67C23×C418yes—
137C2×Dic67—232no—
138C2×Dic67—28no—
139C2×Dic67—28no—
140C2×Dic67—28no—
141C2×Dic67—28no—
142C2×Dic67—28no—
143C2×Dic67—432no—
144C2×Dic67—416no—
145C2×Dic67C2×#41116yes—
146C22×C67—216no—
147C22×C67—28no—
148C22×C67—432no—
149C22×C67—44no—
150C22×C67—44no—
151C22×C67—48no—
152C22×C67C4×#4118yes—
153C22×C67#51814yes—
154C22×C67#55118yes—
155C22×C67#55518yes—
156C22×C67C24×C411yescharacteristic
157C22×C67C22×#4114yes—
158C22×C67C2×H(5,𝔽2)11yescharacteristic
159C22×C67C2×H(5,𝔽2)11yescharacteristic
160C4×C67—28no—
161C4×C67—24no—
162C4×C67—28no—
163Dic67—216no—
164Dic67—48no—
165C2×#2372—671nomaximal, Sylow
166C2×C67—48no—
167C2×C67#237218yescomplemented
168C2×C67C2×C4×#4112yes—
169C2×C67C2×#51811yescharacteristic
170C2×C67C2×#55112yes—
171C2×C67C2×#55512yes—
172C22×#124D6711yescharacteristic
173#2372—6716no—
174C2×#391—672no—
175C2×#388—674no—
176C2×#386—672no—
177C2×C4×C22⋊C4—674no—
178C2×#376—672no—
179C67C2×#237211yescharacteristic, complemented, Sylow
180C23×Q8C2×D6711yescharacteristic
181C23×D4C2×D6711yescharacteristic
182C24×C4—672no—
183C2×#124Dic67116yescomplemented
184C2×#124C2×D6718yes—
185C22×C4⋊C4—672no—
186C22×C22⋊C4—672no—
187C22×C22⋊C4—674no—
188C22×C22⋊C4C2×D6714yes—
189C22×C42—672no—
190C22×C42—674no—
191C22×C42C2×D6711yescharacteristic
192#391—6716no—
193#388—6732no—
194#386—6716no—
195C4×C22⋊C4—6732no—
196#376—6716no—
197C2×#95—672no—
198C2×#95—674no—
199C2×#95—672no—
200C2×#95—674no—
201C25C22×D6712yes—
202C22×Q8—42no—
203C22×Q8—42no—
204C22×Q8C2×Dic6718yescomplemented
205C22×Q8C22×D6714yes—
206C22×D4—42no—
207C22×D4—42no—
208C22×D4C2×Dic6718yes—
209C22×D4C22×D6714yes—
210C23×C4—2684no—
211C23×C4—678no—
212C23×C4—674no—
213C23×C4—674no—
214C23×C4—13416no—
215C23×C4C22×D6712yes—
216C23×C4C22×D6712yes—
217C23×C4C22×D6711yescharacteristic
218#124C2×Dic67164yescomplemented
219C2×C4⋊C4—2684no—
220C2×C4⋊C4—678no—
221C2×C22⋊C4—2684no—
222C2×C22⋊C4—678no—
223C2×C22⋊C4—6716no—
224C2×C22⋊C4—13416no—
225C2×C22⋊C4C2×Dic67132yescomplemented
226C2×C22⋊C4C22×D67116yes—
227C2×C42—2684no—
228C2×C42—678no—
229C2×C42—6716no—
230C2×C42—13416no—
231C2×C42C2×Dic6718yescomplemented
232C2×C42C22×D6714yes—
233#95—678no—
234#95—6716no—
235#95—678no—
236#95—6716no—
237C24—28no—
238C24—28no—
239C24—24no—
240C24—24no—
241C24C22×Dic6718yes—
242C24C23×D6714yes—
243C24C23×D6711yescharacteristic
244C2×Q8—416no—
245C2×Q8—44no—
246C2×Q8—44no—
247C2×Q8C22×Dic67116yescomplemented
248C2×D4—416no—
249C2×D4—44no—
250C2×D4—44no—
251C2×D4C22×Dic67116yes—
252C22×C4—28no—
253C22×C4—24no—
254C22×C4—24no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—2688no—
258C22×C4—26832no—
259C22×C4—6716no—
260C22×C4—678no—
261C22×C4—678no—
262C22×C4—13432no—
263C22×C4—1348no—
264C22×C4—1348no—
265C22×C4—1348no—
266C22×C4—1348no—
267C22×C4—1348no—
268C22×C4C22×Dic6718yes—
269C22×C4C22×Dic6714yes—
270C22×C4C22×Dic6718yes—
271C22×C4C23×D6712yes—
272C22×C4C23×D6714yes—
273C22×C4C23×D6714yes—
274C4⋊C4—2688no—
275C22⋊C4—2688no—
276C22⋊C4—13432no—
277C22⋊C4C22×Dic67164yescomplemented
278C42—2688no—
279C42—13432no—
280C42C22×Dic67116yescomplemented
281C23—232no—
282C23—216no—
283C23—28no—
284C23—28no—
285C23—24no—
286C23—24no—
287C23—48no—
288C23—48no—
289C23—416no—
290C23C23×Dic6718yes—
291C23C23×Dic6712yes—
292C23#41×D6718yes—
293C23#1452408812yes—
294C23#1452408812yes—
295C23C24×D6711yescharacteristic
296Q8—416no—
297D4—416no—
298C2×C4—216no—
299C2×C4—28no—
300C2×C4—216no—
301C2×C4—28no—
302C2×C4—24no—
303C2×C4—24no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—26816no—
307C2×C4—26832no—
308C2×C4—6716no—
309C2×C4—1348no—
310C2×C4—1348no—
311C2×C4—1348no—
312C2×C4—13432no—
313C2×C4—1348no—
314C2×C4—1348no—
315C2×C4C23×Dic6718yes—
316C2×C4C23×Dic6714yes—
317C2×C4C23×Dic6718yes—
318C22—216no—
319C22—28no—
320C22—44no—
321C22—44no—
322C22—48no—
323C22—432no—
324C22#41×Dic6718yes—
325C22#1686387114yes—
326C22#1686407618yes—
327C22#1686410818yes—
328C22C24×Dic6711yescharacteristic
329C22C2×#41×D6714yes—
330C22C2×#1452408811yescharacteristic
331C22C2×#1452408811yescharacteristic
332C4—28no—
333C4—24no—
334C4—28no—
335C4—2688no—
336C4—13416no—
337C2—48no—
338C2#2256332118yesdirect product factor
339C2C2×#41×Dic6712yes—
340C2C2×#1686387111yescharacteristic
341C2C2×#1686407612yes—
342C2C2×#1686410812yes—
subgroupG/H
centerC24C23×D67
commutatorC22×C67C24×C4
FrattiniC23C24×D67
FittingC22×#124×C67C2
socleC24×C67C24
prime radicalC23×D4×C67C22
cosocleC23×C67C25
hypercenterC22×#124D67
hypocenterC67C2×#2372
LCSG ▷ C22×C67 ▷ C67
UCSC1 ◁ C24 ◁ C22×#124

Complements

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

NH ≅ G/Ntype
1. #22563321C2direct (N × H)
2. C2×#16863303C2semidirect
3. C2×#16863289C2semidirect
5. C2×C22⋊C4×Dic67C2semidirect
6. C2×#16863242C2semidirect
14. #16863303C22semidirect
15. #16863289C22semidirect
17. C22⋊C4×Dic67C22semidirect
18. #16863242C22semidirect
25. C2×#124×C67C4semidirect
51. C22×Q8×C67C2×C4semidirect
60. #124×C67C2×C4semidirect
61. C2×C22⋊C4×C67C2×C4semidirect
63. C2×C42×C67C2×C4semidirect
91. C2×Q8×C67C22×C4semidirect
107. C22⋊C4×C67C22×C4semidirect
108. C42×C67C22×C4semidirect
167. C2×C67#2372semidirect
179. C67C2×#2372semidirect
183. C2×#124Dic67semidirect
204. C22×Q8C2×Dic67semidirect
218. #124C2×Dic67semidirect
225. C2×C22⋊C4C2×Dic67semidirect
231. C2×C42C2×Dic67semidirect
247. C2×Q8C22×Dic67semidirect
277. C22⋊C4C22×Dic67semidirect
280. C42C22×Dic67semidirect
338. C2#22563321direct (N × H)

Maximal subgroups

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

subgroupindex
#225633212
C2×#168633032
C2×#168632892
C2×#168632762
C2×C22⋊C4×Dic672
C2×#168632422
C22×#124×C672
C2×#237267