Finite Group Databasebeta

← #30864284

Subgroups — #30864284

Order 17152.

1398 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16863772C211yesmaximal, characteristic
2C2×#16863506C214yesmaximal, complemented
3C4×#14524040C211yesmaximal, characteristic, complemented
4#22561084C214yesmaximal, complemented
5#22561016C211yesmaximal, characteristic, complemented
6#22561014C211yesmaximal, characteristic, complemented
7#22560365C211yesmaximal, characteristic, complemented
8C4×#14523954C211yesmaximal, characteristic, complemented
9C4×#127×C67C211yesmaximal, characteristic, complemented
10C22×#11761613C2214yes—
11C2×#14524040C2211yescharacteristic
12C2×C4×D268—22no—
13C2×C4×D268C2212yes—
14#16863772C418yescomplemented
15#16863506—216no—
16C2×#14523974C2214yes—
17C2×#14523964C2214yes—
18C2×#14523963C2214yes—
19C2×#14523954C2212yes—
20C2×#14523954C2212yes—
21C2×C4×C67⋊C8C2211yescharacteristic
22C2×C4×C67⋊C8C2211yescharacteristic
23#16863298—22no—
24#16863298C2212yes—
25#16863266C2211yescharacteristic
26C2×#127×C67C2211yescharacteristic
27C2×C4×D4×C67—22no—
28C2×C4×D4×C67C2212yes—
29#390×C67—22no—
30#390×C67C2212yes—
31#384×C67C2211yescharacteristic
32C43×C67C2211yescharacteristic
33C22×D268—22no—
34C22×D268C2312yes—
35C22×C4×D67—22no—
36C2×#11761613—28no—
37C2×#11761613—28no—
38C2×#11761613C2×C4116yescomplemented
39C22×C67⋊C8C2312yes—
40C22×C67⋊C8C2312yes—
41#14524040—22no—
42#14524040C2×C414yescomplemented
43C4×D268—28no—
44C4×D268—48no—
45C2×C4⋊Dic67—22no—
46C2×C4⋊Dic67C2312yes—
47C2×(C2×D67)⋊C4—22no—
48C2×(C2×D67)⋊C4—24no—
49#14523974—28no—
50#14523964—28no—
51#14523963—28no—
52#14523954—24no—
53#14523954D418yescomplemented
54C4×C67⋊C8—22no—
55C4×C67⋊C8C2×C414yescomplemented
56C22×D4×C67—22no—
57C22×D4×C67C2312yes—
58C23×C4×C67—22no—
59#127×C67—22no—
60#127×C67C2×C414yescomplemented
61C4×D4×C67—28no—
62C4×D4×C67—48no—
63C2×C4⋊C4×C67—22no—
64C2×C4⋊C4×C67C2312yes—
65C2×C22⋊C4×C67—24no—
66C2×C22⋊C4×C67—22no—
67C2×C42×C67D412yes—
68C2×C42×C67D412yes—
69C2×C42×C67C2311yescharacteristic
70C2×C42×C67C2311yescharacteristic
71C2×C42×C67C2311yescharacteristic
72C23×D67—22no—
73C2×D268—24no—
74C2×D268—24no—
75C2×D268—28no—
76C2×D268—22no—
77C2×D268—44no—
78C2×D268—44no—
79C2×D268C22×C414yes—
80C22×Dic67—22no—
81C2×C4×D67—48no—
82C2×C4×D67—44no—
83#11761613—232no—
84C2×C67⋊C8—22no—
85C2×C67⋊C8—22no—
86C2×C67⋊C8C22×C414yes—
87C2×C67⋊C8C2×D414yes—
88C2×C67⋊C8C2×D414yes—
89C2×C67⋊C8#4114yes—
90C4⋊Dic67—24no—
91C4⋊Dic67—44no—
92(C2×D67)⋊C4—48no—
93(C2×D67)⋊C4—44no—
94C24×C67—22no—
95C2×D4×C67—28no—
96C2×D4×C67—24no—
97C2×D4×C67—24no—
98C2×D4×C67—22no—
99C2×D4×C67—44no—
100C2×D4×C67—44no—
101C2×D4×C67C22×C414yes—
102C22×C4×C67—22no—
103C22×C4×C67—48no—
104C22×C4×C67—44no—
105C22×C4×C67C2×D412yes—
106C22×C4×C67C2×D411yescharacteristic
107C22×C4×C67C2×D411yescharacteristic
108C22×C4×C67C2×D412yes—
109C22×C4×C67C2411yescharacteristic
110C4⋊C4×C67—24no—
111C4⋊C4×C67—44no—
112C22⋊C4×C67—48no—
113C22⋊C4×C67—44no—
114C42×C67—24no—
115C42×C67—24no—
116C42×C67—22no—
117C42×C67C22×C412yes—
118C42×C67C2×D414yes—
119C42×C67#4112yes—
120C23×C67—24no—
121C23×C67—42no—
122C23×C67—42no—
123C23×C67—44no—
124C23×C67—44no—
125C23×C67C22×D411yescharacteristic
126C22×D67—24no—
127C22×D67—42no—
128C22×D67—42no—
129C22×D67—44no—
130C22×D67—44no—
131D4×C67—28no—
132D4×C67—44no—
133D4×C67—416no—
134C2×C4×C67—22no—
135C2×C4×C67—22no—
136C2×C4×C67—22no—
137C2×C4×C67—22no—
138C2×C4×C67—22no—
139C2×C4×C67—42no—
140C2×C4×C67—44no—
141C2×C4×C67—88no—
142C2×C4×C67C4×D414yes—
143C2×C4×C67#12712yes—
144C2×C4×C67#12712yes—
145C2×C4×C67AGL(1,ℤ/8)14yes—
146C2×C4×C67AGL(1,ℤ/8)14yes—
147C2×C4×C67C23×C411yescharacteristic
148C2×C4×C67C22×D412yes—
149C2×C4×C67C2×#4112yes—
150C2×C4×C67C2×#4111yescharacteristic
151C2×Dic67—42no—
152C2×Dic67—44no—
153D268—28no—
154D268—44no—
155D268—416no—
156C4×D67—88no—
157C67⋊C8—24no—
158C67⋊C8C4×D418yescomplemented
159C22×C67—44no—
160C22×C67—48no—
161C22×C67—82no—
162C22×C67—82no—
163C22×C67—84no—
164C22×C67C2×C4×D411yescharacteristic
165C22×C67C2×#12711yescharacteristic
166C22×C67#53211yescharacteristic
167C22×C67C2×AGL(1,ℤ/8)12yes—
168C22×C67C2×AGL(1,ℤ/8)12yes—
169C2×D67—44no—
170C2×D67—48no—
171C2×D67—82no—
172C2×D67—82no—
173C2×D67—84no—
174C4×C67—22no—
175C4×C67—22no—
176C4×C67—42no—
177C4×C67—82no—
178C4×C67#44218yescomplemented
179C4×C67C2×C4×D412yes—
180C4×C67#53212yes—
181Dic67—82no—
182#21690—671nomaximal, Sylow
183C2×C67—84no—
184C2×C67C4×#12711yescharacteristic
185C2×C67C2×#44212yes—
186C2×C67#315111yescharacteristic
187C2×C67#315111yescharacteristic
188C2×C67#318412yes—
189D67—84no—
190C2×#493—671no—
191C2×#442—674no—
192C4×#127—671no—
193C4×#127D6711yescharacteristic, complemented
194#1972—674no—
195#1957—671no—
196#1957—671no—
197#1844—671no—
198C4×#97—671no—
199C67#2169011yescharacteristic, complemented, Sylow
200C22×D8—674no—
201C2×#127—671no—
202C2×#127C2×D6711yescharacteristic
203C2×C4×D4—22no—
204C2×C4×D4—672no—
205C2×C4×D4—1342no—
206C2×C4×D4C2×D6712yes—
207#493—678no—
208#442—13416no—
209C2×#106—674no—
210C2×#102—674no—
211C2×#102—674no—
212C2×#97—672no—
213C2×#97—672no—
214C2×C4×C8—671no—
215C2×C4×C8—671no—
216#390—22no—
217#390—672no—
218#390—1342no—
219#390C2×D6712yes—
220#384—671no—
221#384C2×D6711yescharacteristic
222C43C2×D6711yescharacteristic
223C22×D4—22no—
224C22×D4—672no—
225C22×D4—1342no—
226C22×D4C22×D6712yes—
227C23×C4—22no—
228C23×C4—1342no—
229C2×D8—6716no—
230C2×D8—1348no—
231C2×D8—1348no—
232C22×C8—672no—
233C22×C8—672no—
234#127—22no—
235#127—674no—
236#127—1342no—
237#127C4×D6714yescomplemented
238C4×D4—28no—
239C4×D4—48no—
240C4×D4—2688no—
241C4×D4—1348no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—672no—
244C2×C4⋊C4—1342no—
245C2×C4⋊C4C22×D6712yes—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—1342no—
249C2×C22⋊C4—1344no—
250C2×C42C22⋊D6712yes—
251C2×C42C22⋊D6712yes—
252C2×C42C22×D6711yescharacteristic
253C2×C42C22×D6711yescharacteristic
254C2×C42C22×D6711yescharacteristic
255#106—1348no—
256#102—1348no—
257#102—1348no—
258#97—678no—
259#97—1344no—
260C4×C8—674no—
261C4×C8—1342no—
262C24—22no—
263C24—1342no—
264C2×D4—28no—
265C2×D4—24no—
266C2×D4—24no—
267C2×D4—22no—
268C2×D4—44no—
269C2×D4—44no—
270C2×D4—2684no—
271C2×D4—2684no—
272C2×D4—674no—
273C2×D4—1348no—
274C2×D4—1344no—
275C2×D4—1344no—
276C2×D4—1342no—
277C2×D4C2×C4×D6714yes—
278C22×C4—22no—
279C22×C4—48no—
280C22×C4—44no—
281C22×C4—2688no—
282C22×C4—2684no—
283C22×C4—1342no—
284C22×C4C2×C22⋊D6712yes—
285C22×C4C2×C22⋊D6711yescharacteristic
286C22×C4C2×C22⋊D6711yescharacteristic
287C22×C4C2×C22⋊D6712yes—
288C22×C4C23×D6711yescharacteristic
289D8—13432no—
290C2×C8—674no—
291C2×C8—674no—
292C2×C8—674no—
293C2×C8—674no—
294C2×C8—1342no—
295C2×C8—1342no—
296C4⋊C4—24no—
297C4⋊C4—44no—
298C4⋊C4—2684no—
299C4⋊C4—1344no—
300C22⋊C4—48no—
301C22⋊C4—44no—
302C22⋊C4—2688no—
303C22⋊C4—2684no—
304C42—24no—
305C42—24no—
306C42—22no—
307C42C2×C4×D6712yes—
308C42D4×D6714yes—
309C42#1176162812yes—
310C23—24no—
311C23—44no—
312C23—44no—
313C23—42no—
314C23—42no—
315C23—2684no—
316C23—2684no—
317C23—2682no—
318C23—2682no—
319C23—1344no—
320C23C22×C22⋊D6711yescharacteristic
321D4—28no—
322D4—416no—
323D4—44no—
324D4—26816no—
325D4—2684no—
326D4—1348no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—44no—
333C2×C4—42no—
334C2×C4—88no—
335C2×C4—2684no—
336C2×C4—2682no—
337C2×C4—5368no—
338C2×C4C4×C22⋊D6714yes—
339C2×C4#1452403912yes—
340C2×C4#1452403912yes—
341C2×C4#1452406814yes—
342C2×C4#1452406914yes—
343C2×C4C22×C4×D6711yescharacteristic
344C2×C4C2×D4×D6712yes—
345C2×C4#41×D6712yes—
346C2×C4C2×#1176162811yescharacteristic
347C8—678no—
348C8—1344no—
349C22—48no—
350C22—44no—
351C22—82no—
352C22—82no—
353C22—84no—
354C22—2688no—
355C22—2684no—
356C22—5362no—
357C22—5362no—
358C22—5364no—
359C22C2×C4×C22⋊D6711yescharacteristic
360C22C2×#1452403911yescharacteristic
361C22#1686394211yescharacteristic
362C22C2×#1452406812yes—
363C22C2×#1452406912yes—
364C4—22no—
365C4—22no—
366C4—42no—
367C4—82no—
368C4—5362no—
369C4#1686350618yescomplemented
370C4C4×D4×D6712yes—
371C4#1686394312yes—
372C2—84no—
373C2—5364no—
374C2C4×#1452403911yescharacteristic
375C2C2×#1686350612yes—
376C2#2257065911yescharacteristic
377C2#2257066211yescharacteristic
378C2#2257094312yes—
subgroupG/H
centerC23C22×C22⋊D67
commutatorC2×C4×C67C23×C4
FrattiniC22×C4C23×D67
FittingC4×#127×C67C2
socleC23×C67C22×D4
prime radicalC2×#16863772C2
cosocleC22×C4×C67C24
hypercenterC4×#127D67
hypocenterC67#21690
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C23 ◁ C43 ◁ C4×#127

Complements

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

NH ≅ G/Ntype
2. C2×#16863506C2semidirect
3. C4×#14524040C2semidirect
4. #22561084C2semidirect
5. #22561016C2semidirect
6. #22561014C2semidirect
7. #22560365C2semidirect
8. C4×#14523954C2semidirect
9. C4×#127×C67C2semidirect
14. #16863772C4semidirect
38. C2×#11761613C2×C4semidirect
42. #14524040C2×C4semidirect
53. #14523954D4semidirect
55. C4×C67⋊C8C2×C4semidirect
60. #127×C67C2×C4semidirect
158. C67⋊C8C4×D4semidirect
178. C4×C67#442semidirect
193. C4×#127D67semidirect
199. C67#21690semidirect
237. #127C4×D67semidirect
369. C4#16863506semidirect

Maximal subgroups

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

subgroupindex
C2×#168637722
C2×#168635062
C4×#145240402
#225610842
#225610162
#225610142
#225603652
C4×#145239542
C4×#127×C672
#2169067