Finite Group Databasebeta

← C22×#16863685 #31723293

Subgroups — C22×#16863685 #31723293

Order 17152.

2790 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14524042C211yesmaximal, characteristic, complemented
2C22×#14524005C211yesmaximal, characteristic
3C2×#16863685C2124yesmaximal, direct product factor
4C22×#14523974C211yesmaximal, characteristic, complemented
5C22×Dic67⋊C8C211yesmaximal, characteristic, complemented
6C22×#14523965C211yesmaximal, characteristic
7C22×#14523963C211yesmaximal, characteristic
8C22×#106×C67C211yesmaximal, characteristic, complemented
9C23×Dic134—21no—
10C23×D268C2211yescharacteristic
11C23×C4×D67—21no—
12C23×C67⋊C8C2211yescharacteristic
13C2×#14524042—26no—
14C2×#14524042C22112yescomplemented
15C2×#14524005—26no—
16C2×#14524005C22112yes—
17C22×C4⋊Dic67C2211yescharacteristic
18C22×Dic67⋊C4—21no—
19C22×(C2×D67)⋊C4—21no—
20C22×C4×Dic67C2211yescharacteristic
21#16863685C22164yesdirect product factor
22C2×#14523974C22112yescomplemented
23C2×Dic67⋊C8C22112yescomplemented
24C2×#14523965C22112yes—
25C2×#14523963C22112yes—
26C23×C8×C67C2211yescharacteristic
27C22×C4⋊C4×C67C2211yescharacteristic
28C22×C4⋊C4×C67C2211yescharacteristic
29C2×#106×C67C22112yescomplemented
30C24×D67—21no—
31C22×Dic134—212no—
32C22×Dic134—48no—
33C22×D268—23no—
34C22×D268—24no—
35C22×D268Q818yescomplemented
36C22×D268C2316yes—
37C23×Dic67—21no—
38C23×Dic67D411yescharacteristic
39C23×Dic67D411yescharacteristic
40C22×C4×D67—212no—
41C22×C4×D67—48no—
42C22×C67⋊C8—24no—
43C22×C67⋊C8C2316yes—
44#14524042—224no—
45#14524042C23116yescomplemented
46#14524005—224no—
47#14524005C23116yes—
48C2×C4⋊Dic67—23no—
49C2×C4⋊Dic67C2316yes—
50C2×Dic67⋊C4—212no—
51C2×(C2×D67)⋊C4—212no—
52C2×C4×Dic67D416yes—
53C2×C4×Dic67C2316yes—
54#14523974C23116yescomplemented
55Dic67⋊C8C23116yescomplemented
56#14523965C23116yes—
57#14523963C23116yes—
58C23×C4×C67—21no—
59C23×C4×C67—21no—
60C23×C4×C67C2311yescharacteristic
61C22×C8×C67—24no—
62C22×C8×C67C2316yes—
63C2×C4⋊C4×C67—23no—
64C2×C4⋊C4×C67—23no—
65C2×C4⋊C4×C67C2316yes—
66C2×C4⋊C4×C67C2316yes—
67#106×C67C23116yescomplemented
68C23×D67—28no—
69C23×D67—26no—
70C23×D67—44no—
71C23×D67—44no—
72C2×Dic134—216no—
73C2×Dic134—448no—
74C2×D268—224no—
75C2×D268—26no—
76C2×D268—212no—
77C2×D268C2×Q8124yescomplemented
78C2×D268C2414yes—
79C22×Dic67—24no—
80C22×Dic67—26no—
81C22×Dic67—44no—
82C22×Dic67SD1618yes—
83C22×Dic67C2×D416yes—
84C22×Dic67C2×D416yes—
85C2×C4×D67—216no—
86C2×C4×D67—448no—
87C2×C67⋊C8—212no—
88C2×C67⋊C8C2414yes—
89C4⋊Dic67—26no—
90C4⋊Dic67C2414yes—
91Dic67⋊C4—216no—
92(C2×D67)⋊C4—216no—
93C4×Dic67C2×D4112yes—
94C4×Dic67C2414yes—
95C24×C67C2×D411yescharacteristic
96C22×C4×C67—26no—
97C22×C4×C67—26no—
98C22×C4×C67—44no—
99C22×C4×C67—44no—
100C22×C4×C67C2×D413yes—
101C22×C4×C67C2×Q814yes—
102C22×C4×C67#4114yes—
103C22×C4×C67C2413yes—
104C2×C8×C67—212no—
105C2×C8×C67C2414yes—
106C4⋊C4×C67—26no—
107C4⋊C4×C67—26no—
108C4⋊C4×C67C2414yes—
109C4⋊C4×C67C2414yes—
110C23×C67C22⋊Q814yes—
111C23×C67C2×SD1614yes—
112C23×C67AGL(1,ℤ/8)14yes—
113C23×C67C22×D413yes—
114C22×D67—224no—
115C22×D67—24no—
116C22×D67—412no—
117C22×D67—412no—
118C22×D67—432no—
119C2×C4×C67—24no—
120C2×C4×C67—24no—
121C2×C4×C67—412no—
122C2×C4×C67—412no—
123C2×C4×C67C22⋊Q8112yes—
124C2×C4×C67C22×D413yes—
125C2×C4×C67C22×Q816yes—
126C2×C4×C67C2×#4116yes—
127C2×C4×C67C2511yescharacteristic
128C2×Dic67—212no—
129C2×Dic67—24no—
130C2×Dic67—412no—
131C2×Dic67C2×SD16124yes—
132C2×Dic67C22×D414yes—
133C2×Dic67C22×D414yes—
134D268—224no—
135D268—24no—
136D268C22×Q818yescomplemented
137C4×D67—432no—
138Dic134—432no—
139C8×C67—24no—
140C67⋊C8—24no—
141C22×C67#476116yescomplemented
142C22×C67C2×C22⋊Q816yes—
143C22×C67C22×SD1616yes—
144C22×C67C2×AGL(1,ℤ/8)16yes—
145C22×C67C23×D411yescharacteristic
146C2×D67—28no—
147C2×D67—44no—
148C2×D67—44no—
149C2×D67—448no—
150C4×C67—44no—
151C4×C67—44no—
152C4×C67C2×C22⋊Q816yes—
153C4×C67C23×Q811yescharacteristic
154C4×C67C22×#4111yescharacteristic
155Dic67—24no—
156Dic67—44no—
157Dic67C22×SD1618yes—
158C22×#476—671nomaximal, Sylow
159C2×C67C2×#476112yescomplemented
160C2×C67C22×C22⋊Q811yescharacteristic
161C2×C67C23×SD1611yescharacteristic
162C2×C67C22×AGL(1,ℤ/8)11yescharacteristic
163D67—48no—
164C22×C4⋊Q8—671no—
165C22×C4×D4—671no—
166C2×#476—6724no—
167C22×#106—671no—
168C22×#106D6711yescharacteristic, complemented
169C22×C4⋊C8—671no—
170C22×#102—671no—
171C22×#102—671no—
172C67C22×#47611yescharacteristic, complemented, Sylow
173C23×Q8—1341no—
174C23×D4—671no—
175C24×C4—1341no—
176C23×C8—671no—
177C23×C8C2×D6711yescharacteristic
178C2×C4⋊Q8—6712no—
179C2×C4⋊Q8—1346no—
180C2×C4×D4—6712no—
181C2×C4×D4—1346no—
182C22×C4⋊C4—671no—
183C22×C4⋊C4—1341no—
184C22×C4⋊C4C2×D6711yescharacteristic
185C22×C4⋊C4C2×D6711yescharacteristic
186C22×C22⋊C4—1341no—
187C22×C42—671no—
188#476—6764no—
189C2×#106—6712no—
190C2×#106C2×D67112yescomplemented
191C2×C4⋊C8—6712no—
192C2×#102—6712no—
193C2×#102—6712no—
194C25—1341no—
195C22×Q8—2688no—
196C22×Q8—13412no—
197C22×D4—678no—
198C22×D4—676no—
199C22×D4—1343no—
200C22×D4—1344no—
201C23×C4—21no—
202C23×C4—21no—
203C23×C4—2688no—
204C23×C4—671no—
205C23×C4—671no—
206C23×C4—13412no—
207C23×C4—1341no—
208C23×C4C22×D6711yescharacteristic
209C22×C8—24no—
210C22×C8—676no—
211C22×C8—1344no—
212C22×C8C22×D6716yes—
213C4⋊Q8—6716no—
214C4⋊Q8—13424no—
215C4×D4—6716no—
216C4×D4—13424no—
217C2×C4⋊C4—23no—
218C2×C4⋊C4—23no—
219C2×C4⋊C4—676no—
220C2×C4⋊C4—13412no—
221C2×C4⋊C4—1343no—
222C2×C4⋊C4C22×D6716yes—
223C2×C4⋊C4C22×D6716yes—
224C2×C22⋊C4—13412no—
225C2×C42—676no—
226C2×C42—676no—
227#106—6716no—
228#106C22×D67116yescomplemented
229C4⋊C8—6716no—
230#102—6716no—
231#102—6716no—
232C24—2684no—
233C24—2684no—
234C24—1348no—
235C24—1346no—
236C24D4×D6711yescharacteristic
237C2×Q8—26848no—
238C2×Q8—13416no—
239C2×D4—6724no—
240C2×D4—674no—
241C2×D4—13424no—
242C2×D4—1346no—
243C2×D4—13412no—
244C22×C4—26no—
245C22×C4—26no—
246C22×C4—44no—
247C22×C4—44no—
248C22×C4—26848no—
249C22×C4—2684no—
250C22×C4—678no—
251C22×C4—676no—
252C22×C4—676no—
253C22×C4—13416no—
254C22×C4—1346no—
255C22×C4—1344no—
256C22×C4D4×D6713yes—
257C22×C4Q8×D6714yes—
258C22×C4#1176162814yes—
259C22×C4C23×D6713yes—
260C2×C8—212no—
261C2×C8—674no—
262C2×C8—13412no—
263C2×C8C23×D6714yes—
264C4⋊C4—26no—
265C4⋊C4—26no—
266C4⋊C4—674no—
267C4⋊C4—13416no—
268C4⋊C4—1346no—
269C4⋊C4C23×D6714yes—
270C4⋊C4C23×D6714yes—
271C22⋊C4—13416no—
272C42—6712no—
273C42—674no—
274C23—26812no—
275C23—26812no—
276C23—26832no—
277C23—13424no—
278C23—1344no—
279C23#1452402114yes—
280C23SD16×D6714yes—
281C23#1452407014yes—
282C23C2×D4×D6713yes—
283Q8—26832no—
284D4—678no—
285D4—13424no—
286D4—1344no—
287C2×C4—24no—
288C2×C4—24no—
289C2×C4—412no—
290C2×C4—412no—
291C2×C4—26832no—
292C2×C4—26812no—
293C2×C4—6724no—
294C2×C4—674no—
295C2×C4—674no—
296C2×C4—1344no—
297C2×C4—13412no—
298C2×C4#14524021112yes—
299C2×C4C2×D4×D6713yes—
300C2×C4C2×Q8×D6716yes—
301C2×C4C2×#1176162816yes—
302C2×C4C24×D6711yescharacteristic
303C8—24no—
304C8—1344no—
305C22—2684no—
306C22—2684no—
307C22—26848no—
308C22—1348no—
309C22#16863685116yesdirect product factor
310C22C2×#1452402116yes—
311C22C2×SD16×D6716yes—
312C22C2×#1452407016yes—
313C22C22×D4×D6711yescharacteristic
314C4—44no—
315C4—44no—
316C4—2684no—
317C4—678no—
318C4—1344no—
319C4C2×#1452402116yes—
320C4C22×Q8×D6711yescharacteristic
321C4C22×#1176162811yescharacteristic
322C2—2688no—
323C2C2×#16863685112yesdirect product factor
324C2C22×#1452402111yescharacteristic
325C2C22×SD16×D6711yescharacteristic
326C2C22×#1452407011yescharacteristic
subgroupG/H
centerC24D4×D67
commutator, cosocleC2×C4×C67C25
FrattiniC2×C4C24×D67
FittingC22×#106×C67C2
socleC24×C67C2×D4
prime radicalC23×D268C22
hypercenterC22×#106D67
hypocenterC67C22×#476
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C24 ◁ C23×C4 ◁ C22×#106

Complements

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

NH ≅ G/Ntype
1. C22×#14524042C2semidirect
3. C2×#16863685C2direct (N × H)
4. C22×#14523974C2semidirect
5. C22×Dic67⋊C8C2semidirect
8. C22×#106×C67C2semidirect
14. C2×#14524042C22semidirect
21. #16863685C22direct (N × H)
22. C2×#14523974C22semidirect
23. C2×Dic67⋊C8C22semidirect
29. C2×#106×C67C22semidirect
35. C22×D268Q8semidirect
45. #14524042C23semidirect
54. #14523974C23semidirect
55. Dic67⋊C8C23semidirect
67. #106×C67C23semidirect
77. C2×D268C2×Q8semidirect
136. D268C22×Q8semidirect
141. C22×C67#476semidirect
159. C2×C67C2×#476semidirect
168. C22×#106D67semidirect
172. C67C22×#476semidirect
190. C2×#106C2×D67semidirect
228. #106C22×D67semidirect
309. C22#16863685direct (N × H)
323. C2C2×#16863685direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145240422
C22×#145240052
C2×#168636852
C22×#145239742
C22×Dic67⋊C82
C22×#145239652
C22×#145239632
C22×#106×C672
C22×#47667