Finite Group Databasebeta

← C22×#16863616 #31722637

Subgroups — C22×#16863616 #31722637

Order 17152.

2918 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#11761618C211yesmaximal, characteristic, complemented
2C22×#14524030C211yesmaximal, characteristic
3C22×Q8×Dic67C211yesmaximal, characteristic, complemented
4C2×#16863616C2124yesmaximal, direct product factor
5C22×Dic67⋊C8C211yesmaximal, characteristic, complemented
6C22×#14523968C211yesmaximal, characteristic, complemented
7C22×#14523964C211yesmaximal, characteristic
8C23×SD16×C67C211yesmaximal, characteristic
9C23×Dic134C2211yescharacteristic
10C22×#11761618—28no—
11C22×#11761618C22112yescomplemented
12C23×C67⋊C8C2211yescharacteristic
13C2×#14524030—26no—
14C2×#14524030C22112yes—
15C2×Q8×Dic67—26no—
16C2×Q8×Dic67C22112yescomplemented
17C22×C4⋊Dic67—21no—
18C22×C4⋊Dic67C2211yescharacteristic
19C22×C22⋊Dic67—21no—
20C22×C22⋊Dic67—21no—
21C22×C4×Dic67—21no—
22C22×C4×Dic67C2211yescharacteristic
23#16863616C22164yesdirect product factor
24C2×Dic67⋊C8C22112yescomplemented
25C2×#14523968C22112yescomplemented
26C2×#14523964C22112yes—
27C23×Q8×C67C2211yescharacteristic
28C23×D4×C67C2211yescharacteristic
29C22×SD16×C67—28no—
30C22×SD16×C67C22112yes—
31C23×C8×C67C2211yescharacteristic
32C22×Dic134—23no—
33C22×Dic134—24no—
34C22×Dic134—24no—
35C22×Dic134C2316yes—
36C23×Dic67—21no—
37C23×Dic67—21no—
38C23×Dic67D411yescharacteristic
39C23×Dic67D411yescharacteristic
40C2×#11761618—248no—
41C2×#11761618C23116yescomplemented
42C22×C67⋊C8—24no—
43C22×C67⋊C8C2316yes—
44#14524030—224no—
45#14524030C23116yes—
46Q8×Dic67—224no—
47Q8×Dic67C23116yescomplemented
48C2×C4⋊Dic67—212no—
49C2×C4⋊Dic67—23no—
50C2×C4⋊Dic67C2316yes—
51C2×C22⋊Dic67—212no—
52C2×C22⋊Dic67—212no—
53C2×C4×Dic67—212no—
54C2×C4×Dic67D416yes—
55C2×C4×Dic67C2316yes—
56Dic67⋊C8C23116yescomplemented
57#14523968C23116yescomplemented
58#14523964C23116yes—
59C25×C67—21no—
60C22×Q8×C67—23no—
61C22×Q8×C67—24no—
62C22×Q8×C67D418yes—
63C22×Q8×C67C2316yes—
64C22×D4×C67—24no—
65C22×D4×C67—23no—
66C22×D4×C67—24no—
67C22×D4×C67C2316yes—
68C23×C4×C67—21no—
69C23×C4×C67C2311yescharacteristic
70C2×SD16×C67—248no—
71C2×SD16×C67C23116yes—
72C22×C8×C67—24no—
73C22×C8×C67C2316yes—
74C2×Dic134—212no—
75C2×Dic134—212no—
76C2×Dic134—26no—
77C2×Dic134—412no—
78C2×Dic134C2414yes—
79C22×Dic67—24no—
80C22×Dic67—24no—
81C22×Dic67—26no—
82C22×Dic67—26no—
83C22×Dic67—44no—
84C22×Dic67—44no—
85C22×Dic67C2×D416yes—
86C22×Dic67C2×D416yes—
87#11761618—232no—
88C2×C67⋊C8—212no—
89C2×C67⋊C8C2414yes—
90C4⋊Dic67—26no—
91C4⋊Dic67—216no—
92C4⋊Dic67C2414yes—
93C22⋊Dic67—216no—
94C22⋊Dic67—216no—
95C4×Dic67—216no—
96C4×Dic67C2×D4112yes—
97C4×Dic67C2414yes—
98C24×C67—26no—
99C24×C67—44no—
100C24×C67—44no—
101C24×C67—44no—
102C24×C67C2×D411yescharacteristic
103C2×Q8×C67—224no—
104C2×Q8×C67—26no—
105C2×Q8×C67—212no—
106C2×Q8×C67C2×D4124yes—
107C2×Q8×C67C2414yes—
108C2×D4×C67—212no—
109C2×D4×C67—26no—
110C2×D4×C67—212no—
111C2×D4×C67—412no—
112C2×D4×C67C2414yes—
113C22×C4×C67—28no—
114C22×C4×C67—26no—
115C22×C4×C67C2×D413yes—
116C22×C4×C67C2×D414yes—
117C22×C4×C67#4114yes—
118C22×C4×C67C2413yes—
119SD16×C67—232no—
120C2×C8×C67—212no—
121C2×C8×C67C2414yes—
122C23×C67—24no—
123C23×C67—412no—
124C23×C67—412no—
125C23×C67—412no—
126C23×C67—816no—
127C23×C67#12114yes—
128C23×C67#13514yes—
129C23×C67#13714yes—
130C23×C67C22×D413yes—
131Q8×C67—224no—
132Q8×C67—24no—
133Q8×C67C22×D418yes—
134D4×C67—24no—
135D4×C67—24no—
136D4×C67—412no—
137C2×C4×C67—224no—
138C2×C4×C67—24no—
139C2×C4×C67#121112yes—
140C2×C4×C67C22×D413yes—
141C2×C4×C67C22×D416yes—
142C2×C4×C67C2×#4116yes—
143C2×C4×C67C2511yescharacteristic
144C2×Dic67—212no—
145C2×Dic67—212no—
146C2×Dic67—24no—
147C2×Dic67—24no—
148C2×Dic67—412no—
149C2×Dic67—412no—
150C2×Dic67C22×D414yes—
151C2×Dic67C22×D414yes—
152Dic134—24no—
153Dic134—24no—
154Dic134—412no—
155C8×C67—24no—
156C67⋊C8—24no—
157C22×C67—44no—
158C22×C67—44no—
159C22×C67—44no—
160C22×C67—824no—
161C22×C67#464116yescomplemented
162C22×C67C2×#12116yes—
163C22×C67C2×#13516yes—
164C22×C67C2×#13716yes—
165C22×C67C23×D411yescharacteristic
166C4×C67—28no—
167C4×C67C2×#12116yes—
168C4×C67C23×D411yescharacteristic
169C4×C67C22×#4111yescharacteristic
170Dic67—24no—
171Dic67—24no—
172Dic67—44no—
173Dic67—44no—
174C22×#464—671nomaximal, Sylow
175C2×C67—84no—
176C2×C67C2×#464112yescomplemented
177C2×C67C22×#12111yescharacteristic
178C2×C67C22×#13511yescharacteristic
179C2×C67C22×#13711yescharacteristic
180C23×Q16—671no—
181C23×SD16D6711yescharacteristic
182C22×#124—671no—
183C22×C4×Q8—671no—
184C2×#464—6724no—
185C22×C4⋊C8—671no—
186C22×#103—671no—
187C22×#102—671no—
188C67C22×#46411yescharacteristic, complemented, Sylow
189C23×Q8—671no—
190C23×Q8C2×D6711yescharacteristic
191C23×D4C2×D6711yescharacteristic
192C22×Q16—6712no—
193C22×Q16—1348no—
194C22×SD16—28no—
195C22×SD16C2×D67112yes—
196C23×C8—671no—
197C23×C8C2×D6711yescharacteristic
198C2×#124—6712no—
199C2×#124—1346no—
200C2×C4×Q8—6712no—
201C2×C4×Q8—1346no—
202C22×C4⋊C4—671no—
203C22×C4⋊C4—1341no—
204C22×C22⋊C4—1341no—
205C22×C22⋊C4—1341no—
206C22×C42—671no—
207C22×C42—1341no—
208#464—6764no—
209C2×C4⋊C8—6712no—
210C2×#103—6712no—
211C2×#102—6712no—
212C25—21no—
213C22×Q8—23no—
214C22×Q8—24no—
215C22×Q8—676no—
216C22×Q8—1344no—
217C22×Q8—1344no—
218C22×Q8—1343no—
219C22×Q8C22⋊D6718yes—
220C22×Q8C22×D6716yes—
221C22×D4—23no—
222C22×D4—24no—
223C22×D4—24no—
224C22×D4C22×D6716yes—
225C23×C4—21no—
226C23×C4—671no—
227C23×C4—671no—
228C23×C4—1341no—
229C23×C4—1341no—
230C23×C4C22×D6711yescharacteristic
231C2×Q16—6716no—
232C2×Q16—13448no—
233C2×SD16—248no—
234C2×SD16C22×D67116yes—
235C22×C8—24no—
236C22×C8—676no—
237C22×C8—1344no—
238C22×C8C22×D6716yes—
239#124—6716no—
240#124—13424no—
241C4×Q8—6716no—
242C4×Q8—13424no—
243C2×C4⋊C4—676no—
244C2×C4⋊C4—13412no—
245C2×C4⋊C4—1343no—
246C2×C22⋊C4—13412no—
247C2×C22⋊C4—13412no—
248C2×C42—676no—
249C2×C42—676no—
250C2×C42—13412no—
251C4⋊C8—6716no—
252#103—6716no—
253#102—6716no—
254C24—26no—
255C24—44no—
256C24—44no—
257C24—44no—
258C24D4×D6711yescharacteristic
259C2×Q8—224no—
260C2×Q8—26no—
261C2×Q8—212no—
262C2×Q8—26812no—
263C2×Q8—674no—
264C2×Q8—13412no—
265C2×Q8—13412no—
266C2×Q8—1346no—
267C2×Q8C2×C22⋊D67124yes—
268C2×Q8C23×D6714yes—
269C2×D4—26no—
270C2×D4—212no—
271C2×D4—212no—
272C2×D4—412no—
273C2×D4C23×D6714yes—
274C22×C4—28no—
275C22×C4—26no—
276C22×C4—2684no—
277C22×C4—2684no—
278C22×C4—676no—
279C22×C4—676no—
280C22×C4—1346no—
281C22×C4—1346no—
282C22×C4—1344no—
283C22×C4—1344no—
284C22×C4D4×D6713yes—
285C22×C4C2×C22⋊D6714yes—
286C22×C4#1176162714yes—
287C22×C4C23×D6713yes—
288Q16—13432no—
289SD16—232no—
290C2×C8—212no—
291C2×C8—674no—
292C2×C8—13412no—
293C2×C8C23×D6714yes—
294C4⋊C4—674no—
295C4⋊C4—13416no—
296C4⋊C4—1346no—
297C22⋊C4—13416no—
298C22⋊C4—13416no—
299C42—6712no—
300C42—674no—
301C42—13416no—
302C23—24no—
303C23—412no—
304C23—412no—
305C23—412no—
306C23—816no—
307C23#1452401514yes—
308C23#1452406414yes—
309C23#1452407214yes—
310C23C2×D4×D6713yes—
311Q8—224no—
312Q8—24no—
313Q8—26812no—
314Q8—1344no—
315Q8—1344no—
316Q8C22×C22⋊D6718yes—
317D4—24no—
318D4—24no—
319D4—412no—
320C2×C4—224no—
321C2×C4—24no—
322C2×C4—26812no—
323C2×C4—26812no—
324C2×C4—674no—
325C2×C4—674no—
326C2×C4—1344no—
327C2×C4—1344no—
328C2×C4—13412no—
329C2×C4—13412no—
330C2×C4#14524015112yes—
331C2×C4C2×D4×D6713yes—
332C2×C4C22×C22⋊D6716yes—
333C2×C4C2×#1176162716yes—
334C2×C4C24×D6711yescharacteristic
335C8—24no—
336C8—1344no—
337C22—44no—
338C22—44no—
339C22—44no—
340C22—824no—
341C22#16863616116yesdirect product factor
342C22C2×#1452401516yes—
343C22C2×#1452406416yes—
344C22C2×#1452407216yes—
345C22C22×D4×D6711yescharacteristic
346C4—28no—
347C4—2684no—
348C4—2684no—
349C4—1344no—
350C4—1344no—
351C4C2×#1452401516yes—
352C4C23×C22⋊D6711yescharacteristic
353C4C22×#1176162711yescharacteristic
354C2—84no—
355C2C2×#16863616112yesdirect product factor
356C2C22×#1452401511yescharacteristic
357C2C22×#1452406411yescharacteristic
358C2C22×#1452407211yescharacteristic
subgroupG/H
centerC24D4×D67
commutator, cosocleC2×C4×C67C25
FrattiniC2×C4C24×D67
FittingC23×SD16×C67C2
socleC24×C67C2×D4
prime radicalC23×D4×C67C22
hypercenterC23×SD16D67
hypocenterC67C22×#464
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C24 ◁ C23×C4 ◁ C23×SD16

Complements

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

NH ≅ G/Ntype
1. C23×#11761618C2semidirect
3. C22×Q8×Dic67C2semidirect
4. C2×#16863616C2direct (N × H)
5. C22×Dic67⋊C8C2semidirect
6. C22×#14523968C2semidirect
11. C22×#11761618C22semidirect
16. C2×Q8×Dic67C22semidirect
23. #16863616C22direct (N × H)
24. C2×Dic67⋊C8C22semidirect
25. C2×#14523968C22semidirect
41. C2×#11761618C23semidirect
47. Q8×Dic67C23semidirect
56. Dic67⋊C8C23semidirect
57. #14523968C23semidirect
161. C22×C67#464semidirect
176. C2×C67C2×#464semidirect
188. C67C22×#464semidirect
341. C22#16863616direct (N × H)
355. C2C2×#16863616direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#117616182
C22×#145240302
C22×Q8×Dic672
C2×#168636162
C22×Dic67⋊C82
C22×#145239682
C22×#145239642
C23×SD16×C672
C22×#46467