Finite Group Databasebeta

← C22×#16863164 #30853272

Subgroups — C22×#16863164 #30853272

Order 17152.

2494 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#11761612C211yesmaximal, characteristic
2C23×C4⋊Dic67C211yesmaximal, characteristic
3C2×#16863164C2112yesmaximal, direct product factor
4C23×M16×C67C211yesmaximal, characteristic
5C24×Dic67C412yes—
6C22×#11761612—23no—
7C22×#11761612—24no—
8C22×#11761612C418yescomplemented
9C22×#11761612C2216yes—
10C23×C67⋊C8—21no—
11C22×C4⋊Dic67—21no—
12C22×C4⋊Dic67—23no—
13C22×C4⋊Dic67—21no—
14C22×C4⋊Dic67—23no—
15C22×C4⋊Dic67C416yes—
16C22×C4⋊Dic67C2216yes—
17#16863164C22116yesdirect product factor
18C24×C4×C67C2211yescharacteristic
19C22×M16×C67—24no—
20C22×M16×C67—23no—
21C22×M16×C67C418yescomplemented
22C22×M16×C67C2216yes—
23C23×C8×C67—21no—
24C23×Dic67—26no—
25C23×Dic67—22no—
26C23×Dic67—48no—
27C23×Dic67C2×C4112yes—
28C2×#11761612—224no—
29C2×#11761612—212no—
30C2×#11761612—26no—
31C2×#11761612C2×C4124yescomplemented
32C2×#11761612C2314yes—
33C22×C67⋊C8—28no—
34C22×C67⋊C8—26no—
35C2×C4⋊Dic67—212no—
36C2×C4⋊Dic67—26no—
37C2×C4⋊Dic67—26no—
38C2×C4⋊Dic67—26no—
39C2×C4⋊Dic67—26no—
40C2×C4⋊Dic67—26no—
41C2×C4⋊Dic67—26no—
42C2×C4⋊Dic67C2×C4112yes—
43C2×C4⋊Dic67C2314yes—
44C25×C67C2×C411yescharacteristic
45C23×C4×C67C2×C413yes—
46C23×C4×C67C2×C414yes—
47C23×C4×C67C2×C414yes—
48C23×C4×C67D411yescharacteristic
49C23×C4×C67D413yes—
50C23×C4×C67D414yes—
51C23×C4×C67D411yescharacteristic
52C23×C4×C67D413yes—
53C23×C4×C67Q814yes—
54C23×C4×C67C2313yes—
55C2×M16×C67—224no—
56C2×M16×C67—212no—
57C2×M16×C67—26no—
58C2×M16×C67C2×C4124yescomplemented
59C2×M16×C67C2314yes—
60C22×C8×C67—28no—
61C22×C8×C67—26no—
62C22×Dic67—212no—
63C22×Dic67—212no—
64C22×Dic67—424no—
65C22×Dic67—48no—
66C22×Dic67—424no—
67C22×Dic67C22×C418yes—
68#11761612—224no—
69#11761612—24no—
70#11761612C22×C418yescomplemented
71C2×C67⋊C8—224no—
72C2×C67⋊C8—24no—
73C4⋊Dic67—212no—
74C4⋊Dic67—212no—
75C4⋊Dic67—24no—
76C4⋊Dic67—24no—
77C24×C67—24no—
78C24×C67—24no—
79C24×C67C4214yes—
80C24×C67C22⋊C413yes—
81C24×C67C22⋊C411yescharacteristic
82C24×C67C4⋊C414yes—
83C24×C67C22×C413yes—
84C22×C4×C67C42112yes—
85C22×C4×C67C22⋊C416yes—
86C22×C4×C67C22⋊C413yes—
87C22×C4×C67C22⋊C413yes—
88C22×C4×C67C22⋊C414yes—
89C22×C4×C67C22⋊C4112yes—
90C22×C4×C67C22⋊C414yes—
91C22×C4×C67C22⋊C4112yes—
92C22×C4×C67C4⋊C4112yes—
93C22×C4×C67C4⋊C4112yes—
94C22×C4×C67C4⋊C414yes—
95C22×C4×C67C4⋊C4112yes—
96C22×C4×C67C4⋊C414yes—
97C22×C4×C67C22×C413yes—
98C22×C4×C67C22×C416yes—
99C22×C4×C67C22×C416yes—
100C22×C4×C67C2×D416yes—
101C22×C4×C67C2×D413yes—
102C22×C4×C67C2×D413yes—
103C22×C4×C67C2×D413yes—
104C22×C4×C67C2×D413yes—
105C22×C4×C67C2×Q816yes—
106C22×C4×C67C2411yescharacteristic
107M16×C67—224no—
108M16×C67—24no—
109M16×C67C22×C418yescomplemented
110C2×C8×C67—224no—
111C2×C8×C67—24no—
112C23×C67—216no—
113C23×C67—216no—
114C23×C67—26no—
115C23×C67—26no—
116C23×C67—26no—
117C23×C67—26no—
118C23×C67#95112yes—
119C23×C67#9514yes—
120C23×C67#10014yes—
121C23×C67#10114yes—
122C23×C67C2×C4216yes—
123C23×C67C2×C22⋊C413yes—
124C23×C67C2×C22⋊C413yes—
125C23×C67C2×C4⋊C416yes—
126C23×C67C23×C411yescharacteristic
127C2×C4×C67#95124yes—
128C2×C4×C67#95112yes—
129C2×C4×C67#95112yes—
130C2×C4×C67C2×C4216yes—
131C2×C4×C67C2×C22⋊C416yes—
132C2×C4×C67C2×C22⋊C416yes—
133C2×C4×C67C2×C22⋊C413yes—
134C2×C4×C67C2×C22⋊C413yes—
135C2×C4×C67C2×C22⋊C416yes—
136C2×C4×C67C2×C22⋊C416yes—
137C2×C4×C67C2×C4⋊C416yes—
138C2×C4×C67C2×C4⋊C416yes—
139C2×C4×C67C2×C4⋊C416yes—
140C2×C4×C67C2×C4⋊C416yes—
141C2×C4×C67C2×C4⋊C416yes—
142C2×C4×C67C23×C411yescharacteristic
143C2×C4×C67C23×C411yescharacteristic
144C2×C4×C67C22×D411yescharacteristic
145C2×C4×C67C22×D411yescharacteristic
146C2×C4×C67C22×D411yescharacteristic
147C2×C4×C67C22×Q811yescharacteristic
148C2×Dic67—28no—
149C2×Dic67—424no—
150C2×Dic67—424no—
151C2×Dic67—48no—
152C8×C67—28no—
153C67⋊C8—28no—
154C22×C67—23no—
155C22×C67—21no—
156C22×C67—23no—
157C22×C67—21no—
158C22×C67—224no—
159C22×C67—212no—
160C22×C67—212no—
161C22×C67#343116yescomplemented
162C22×C67C2×#9516yes—
163C22×C67C2×#9516yes—
164C22×C67C2×#10016yes—
165C22×C67C2×#10116yes—
166C22×C67C22×C4211yescharacteristic
167C22×C67C22×C22⋊C411yescharacteristic
168C22×C67C22×C4⋊C411yescharacteristic
169C4×C67C2×#9516yes—
170C4×C67C2×#9516yes—
171C4×C67C22×C22⋊C411yescharacteristic
172C4×C67C22×C22⋊C411yescharacteristic
173C4×C67C22×C4⋊C411yescharacteristic
174C4×C67C22×C4⋊C411yescharacteristic
175Dic67—48no—
176C22×#343—671nomaximal, Sylow
177C2×C67—21no—
178C2×C67—26no—
179C2×C67—21no—
180C2×C67C2×#343112yescomplemented
181C2×C67C22×#9511yescharacteristic
182C2×C67C22×#10011yescharacteristic
183C2×C67C22×#10111yescharacteristic
184C23×M16—671no—
185C23×M16D6711yescharacteristic
186C23×C4⋊C4—671no—
187C2×#343—6712no—
188C67C22×#34311yescharacteristic, complemented, Sylow
189C24×C4—672no—
190C24×C4C2×D6711yescharacteristic
191C22×M16—24no—
192C22×M16—23no—
193C22×M16—678no—
194C22×M16—676no—
195C22×M16—1343no—
196C22×M16—1344no—
197C22×M16Dic6718yescomplemented
198C22×M16C2×D6716yes—
199C23×C8—21no—
200C23×C8—1341no—
201C22×C4⋊C4—676no—
202C22×C4⋊C4—676no—
203C22×C4⋊C4—1343no—
204C22×C4⋊C4—1341no—
205C22×C4⋊C4—1343no—
206C22×C4⋊C4—1341no—
207#343—6716no—
208C25C4×D6711yescharacteristic
209C23×C4—2688no—
210C23×C4—6712no—
211C23×C4—1346no—
212C23×C4—1342no—
213C23×C4Dic13414yes—
214C23×C4C4×D6713yes—
215C23×C4C4×D6714yes—
216C23×C4D26813yes—
217C23×C4D26811yescharacteristic
218C23×C4C2×Dic6714yes—
219C23×C4C22⋊D6714yes—
220C23×C4C22⋊D6713yes—
221C23×C4C22⋊D6711yescharacteristic
222C23×C4C22×D6713yes—
223C2×M16—224no—
224C2×M16—212no—
225C2×M16—26no—
226C2×M16—6724no—
227C2×M16—674no—
228C2×M16—13424no—
229C2×M16—13412no—
230C2×M16—1346no—
231C2×M16C2×Dic67124yescomplemented
232C2×M16C22×D6714yes—
233C22×C8—28no—
234C22×C8—26no—
235C22×C8—1348no—
236C22×C8—1346no—
237C2×C4⋊C4—6712no—
238C2×C4⋊C4—674no—
239C2×C4⋊C4—13412no—
240C2×C4⋊C4—1346no—
241C2×C4⋊C4—1346no—
242C2×C4⋊C4—1346no—
243C2×C4⋊C4—1346no—
244C2×C4⋊C4—1346no—
245C2×C4⋊C4—1346no—
246C24—24no—
247C24—24no—
248C24C4×Dic6714yes—
249C24(C2×D67)⋊C413yes—
250C24(C2×D67)⋊C411yescharacteristic
251C24Dic67⋊C414yes—
252C24C2×C4×D6713yes—
253C22×C4—26824no—
254C22×C4—2688no—
255C22×C4—26824no—
256C22×C4—678no—
257C22×C4—13412no—
258C22×C4—13412no—
259C22×C4C4×Dic67112yes—
260C22×C4(C2×D67)⋊C416yes—
261C22×C4(C2×D67)⋊C413yes—
262C22×C4(C2×D67)⋊C413yes—
263C22×C4(C2×D67)⋊C414yes—
264C22×C4(C2×D67)⋊C4112yes—
265C22×C4C22⋊Dic6714yes—
266C22×C4C22⋊Dic67112yes—
267C22×C4Dic67⋊C4112yes—
268C22×C4Dic67⋊C4112yes—
269C22×C4Dic67⋊C414yes—
270C22×C4C4⋊Dic67112yes—
271C22×C4C4⋊Dic6714yes—
272C22×C4C2×C4×D6713yes—
273C22×C4C2×C4×D6716yes—
274C22×C4C22×Dic6716yes—
275C22×C4C2×C22⋊D6716yes—
276C22×C4C2×C22⋊D6713yes—
277C22×C4C2×C22⋊D6713yes—
278C22×C4C2×D26813yes—
279C22×C4C2×D26813yes—
280C22×C4C2×Dic13416yes—
281C22×C4C23×D6711yescharacteristic
282M16—224no—
283M16—24no—
284M16—678no—
285M16—13424no—
286M16—1344no—
287M16C22×Dic6718yescomplemented
288C2×C8—224no—
289C2×C8—24no—
290C2×C8—13424no—
291C2×C8—1344no—
292C4⋊C4—13412no—
293C4⋊C4—13412no—
294C4⋊C4—1344no—
295C4⋊C4—1344no—
296C23—26no—
297C23—26no—
298C23—26no—
299C23—26no—
300C23—216no—
301C23—216no—
302C23C67⋊#95112yes—
303C23C67⋊#9514yes—
304C23#1452395914yes—
305C23#1452396114yes—
306C23C2×C4×Dic6716yes—
307C23C2×(C2×D67)⋊C413yes—
308C23C2×(C2×D67)⋊C413yes—
309C23C2×Dic67⋊C416yes—
310C23C22×C4×D6711yescharacteristic
311C2×C4—26824no—
312C2×C4—26824no—
313C2×C4—2688no—
314C2×C4—1348no—
315C2×C4C67⋊#95124yes—
316C2×C4C67⋊#95112yes—
317C2×C4C67⋊#95112yes—
318C2×C4C2×C4×Dic6716yes—
319C2×C4C2×(C2×D67)⋊C413yes—
320C2×C4C2×(C2×D67)⋊C413yes—
321C2×C4C2×(C2×D67)⋊C416yes—
322C2×C4C2×(C2×D67)⋊C416yes—
323C2×C4C2×C22⋊Dic6716yes—
324C2×C4C2×C22⋊Dic6716yes—
325C2×C4C2×Dic67⋊C416yes—
326C2×C4C2×Dic67⋊C416yes—
327C2×C4C2×Dic67⋊C416yes—
328C2×C4C2×C4⋊Dic6716yes—
329C2×C4C2×C4⋊Dic6716yes—
330C2×C4C22×C4×D6711yescharacteristic
331C2×C4C23×Dic6711yescharacteristic
332C2×C4C22×C22⋊D6711yescharacteristic
333C2×C4C22×C22⋊D6711yescharacteristic
334C2×C4C22×D26811yescharacteristic
335C2×C4C22×Dic13411yescharacteristic
336C8—28no—
337C8—1348no—
338C22—224no—
339C22—212no—
340C22—212no—
341C22—23no—
342C22—21no—
343C22—23no—
344C22—21no—
345C22#16863164116yesdirect product factor
346C22C2×C67⋊#9516yes—
347C22C2×C67⋊#9516yes—
348C22C2×#1452395916yes—
349C22C2×#1452396116yes—
350C22C22×C4×Dic6711yescharacteristic
351C22C22×(C2×D67)⋊C411yescharacteristic
352C22C22×Dic67⋊C411yescharacteristic
353C4—2688no—
354C4C2×C67⋊#9516yes—
355C4C2×C67⋊#9516yes—
356C4C22×(C2×D67)⋊C411yescharacteristic
357C4C22×C22⋊Dic6711yescharacteristic
358C4C22×Dic67⋊C411yescharacteristic
359C4C22×C4⋊Dic6711yescharacteristic
360C2—26no—
361C2—21no—
362C2—21no—
363C2C2×#16863164112yesdirect product factor
364C2C22×C67⋊#9511yescharacteristic
365C2C22×#1452395911yescharacteristic
366C2C22×#1452396111yescharacteristic
subgroupG/H
centerC24(C2×D67)⋊C4
commutatorC22×C67C22×C42
FrattiniC22×C4C23×D67
FittingC23×M16×C67C2
socleC24×C67C22⋊C4
prime radicalC25×C67C2×C4
cosocleC22×C4×C67C24
hypercenterC23×M16D67
hypocenterC67C22×#343
LCSG ▷ C22×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C24 ◁ C24×C4 ◁ C23×M16

Complements

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

NH ≅ G/Ntype
3. C2×#16863164C2direct (N × H)
8. C22×#11761612C4semidirect
17. #16863164C22direct (N × H)
21. C22×M16×C67C4semidirect
31. C2×#11761612C2×C4semidirect
58. C2×M16×C67C2×C4semidirect
70. #11761612C22×C4semidirect
109. M16×C67C22×C4semidirect
161. C22×C67#343semidirect
180. C2×C67C2×#343semidirect
188. C67C22×#343semidirect
197. C22×M16Dic67semidirect
231. C2×M16C2×Dic67semidirect
287. M16C22×Dic67semidirect
345. C22#16863164direct (N × H)
363. C2C2×#16863164direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#117616122
C23×C4⋊Dic672
C2×#168631642
C23×M16×C672
C22×#34367