Finite Group Databasebeta

← #30855574

Subgroups — #30855574

Order 17152.

1602 conjugacy classes of subgroups; only the 275 normal subgroups are shown.

normal subgroups only

This group has 1602 conjugacy classes of subgroups — too many to draw. Only its 275 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16863870C211yesmaximal, characteristic
2#22562786C212yesmaximal
3#22560286C211yesmaximal, characteristic, complemented
4#22560284C212yesmaximal, complemented
5#22560282C214yesmaximal, complemented
6#342×D67C214yesmaximal, complemented
7C67×#1820C211yesmaximal, characteristic, complemented
8C23×D268C2211yescharacteristic
9#16863870—28no—
10C2×C4×D268C414yescomplemented
11C2×#14524005C414yescomplemented
12C2×#14524002C2211yescharacteristic
13C2×C22⋊C4×D67C2214yes—
14C2×C22⋊C4×D67C2212yes—
15C2×C22⋊C4×D67C2212yes—
16#16863274—28no—
17#16863259C2212yes—
18#16863251C2214yes—
19#16863163C2214yes—
20#16863162C2218yes—
21C2×#117×C67C2211yescharacteristic
22#381×C67C2212yes—
23#342×C67C2214yes—
24C24×D67C2311yescharacteristic
25C24×D67C2311yescharacteristic
26C22×D268—28no—
27C22×D268C2×C412yes—
28C22×D268C2×C412yes—
29C22×D268D418yes—
30C22×C4×D67—24no—
31C22×C4×D67C2×C414yes—
32C22×C4×D67C2×C414yescomplemented
33C4×D268—216no—
34#14524005—216no—
35#14524002—24no—
36C2×C4⋊Dic67C2×C412yescomplemented
37C2×C22⋊Dic67C2314yes—
38C2×C22⋊Dic67C2312yes—
39C2×(C2×D67)⋊C4—24no—
40C2×(C2×D67)⋊C4—28no—
41C2×(C2×D67)⋊C4C2×C414yes—
42C2×(C2×D67)⋊C4C2×C414yescomplemented
43C22⋊C4×D67—28no—
44C22⋊C4×D67—28no—
45C22⋊C4×D67—216no—
46C2×C4×Dic67C2×C412yescomplemented
47C67⋊#95—24no—
48C23×C4×C67C2311yescharacteristic
49#117×C67—24no—
50C2×C4⋊C4×C67C2×C412yescomplemented
51C2×C22⋊C4×C67C2314yes—
52C2×C22⋊C4×C67C2312yes—
53C2×C42×C67C2×C412yescomplemented
54#95×C67—24no—
55C23×D67—24no—
56C23×D67—82no—
57C23×D67—82no—
58C23×D67C22×C412yes—
59C23×D67C22×C412yes—
60C23×D67C2×D414yes—
61C23×D67C2×D414yes—
62C23×D67#4118yes—
63C2×D268—28no—
64C2×D268—28no—
65C2×D268—216no—
66C2×D268—216no—
67C2×D268C22⋊C418yes—
68C2×D268C22⋊C418yes—
69C22×Dic67—24no—
70C22×Dic67C22×C414yes—
71C2×C4×D67—28no—
72C2×C4×D67—28no—
73C2×C4×D67—48no—
74C2×C4×D67—88no—
75C2×C4×D67—84no—
76C2×C4×D67—84no—
77C4⋊Dic67—24no—
78C22⋊Dic67—24no—
79C22⋊Dic67—28no—
80(C2×D67)⋊C4—28no—
81(C2×D67)⋊C4—28no—
82(C2×D67)⋊C4—48no—
83(C2×D67)⋊C4—88no—
84C4×Dic67—24no—
85C24×C67C2411yescharacteristic
86C22×C4×C67—24no—
87C22×C4×C67—24no—
88C22×C4×C67C22×C411yescharacteristic
89C22×C4×C67C22×C411yescharacteristic
90C22×C4×C67C22×C414yes—
91C22×C4×C67C2×D414yes—
92C4⋊C4×C67—24no—
93C22⋊C4×C67—24no—
94C22⋊C4×C67—28no—
95C42×C67—24no—
96C23×C67—42no—
97C23×C67C23×C411yescharacteristic
98C23×C67C22×D412yes—
99C23×C67C2×#4114yes—
100C22×D67—24no—
101C22×D67—28no—
102C22×D67—44no—
103C22×D67—44no—
104C22×D67—48no—
105C22×D67—48no—
106C22×D67—88no—
107C22×D67—88no—
108C22×D67C2×C22⋊C414yes—
109C22×D67C2×C22⋊C414yes—
110C22×D67#11718yes—
111C22×D67#12318yes—
112C22×D67#12318yes—
113C2×C4×C67—22no—
114C2×C4×C67—22no—
115C2×C4×C67—28no—
116C2×C4×C67—24no—
117C2×C4×C67—44no—
118C2×C4×C67—48no—
119C2×C4×C67—44no—
120C2×C4×C67C2×C22⋊C412yes—
121C2×C4×C67C2×C22⋊C412yes—
122C2×Dic67—24no—
123C2×Dic67—44no—
124C2×Dic67—48no—
125C2×Dic67—44no—
126D268—216no—
127D268—216no—
128C4×D67—88no—
129C4×D67—88no—
130C4×D67—816no—
131C22×C67—22no—
132C22×C67—24no—
133C22×C67—44no—
134C22×C67C22×C22⋊C411yescharacteristic
135C22×C67C2×#11712yes—
136C22×C67C2×#12314yes—
137C2×D67—28no—
138C2×D67—28no—
139C2×D67—24no—
140C2×D67—48no—
141C2×D67—48no—
142C2×D67—48no—
143C2×D67—48no—
144C2×D67—88no—
145C2×D67#38118yes—
146C4×C67—22no—
147C4×C67—22no—
148C4×C67—44no—
149C4×C67—84no—
150Dic67—44no—
151Dic67—84no—
152#20689—671nomaximal, Sylow
153C2×C67—24no—
154C2×C67—22no—
155C2×C67C2×#38111yescharacteristic
156C2×C67#288312yes—
157D67—28no—
158D67—44no—
159C2×#518—671no—
160#2308—672no—
161#1820—672no—
162#1820—671no—
163#1820D6711yescharacteristic, complemented
164#1819—674no—
165C2×#342—674no—
166C67#2068911yescharacteristic, complemented, Sylow
167C23×D4—671no—
168#518—1348no—
169C2×C4×D4—674no—
170C2×C4×D4—674no—
171C2×#117—671no—
172C2×#117C2×D6711yescharacteristic
173C22×C22⋊C4—672no—
174C22×C22⋊C4—672no—
175C22×C22⋊C4—674no—
176#386—1348no—
177#381—672no—
178#381C2×D6712yes—
179#379—674no—
180#342—678no—
181#342—674no—
182#342C2×D6714yes—
183C25—671no—
184C25—671no—
185C22×D4—672no—
186C22×D4—672no—
187C22×D4—678no—
188C22×D4—1348no—
189C23×C4—674no—
190C23×C4—674no—
191C23×C4—1344no—
192C23×C4C22×D6711yescharacteristic
193C4×D4—13416no—
194C4×D4—13416no—
195#117—24no—
196#117—1344no—
197C2×C4⋊C4—672no—
198C2×C4⋊C4C4×D6712yescomplemented
199C2×C22⋊C4—672no—
200C2×C22⋊C4—674no—
201C2×C22⋊C4—674no—
202C2×C22⋊C4—674no—
203C2×C22⋊C4—13416no—
204C2×C22⋊C4—1348no—
205C2×C22⋊C4—1348no—
206C2×C22⋊C4—1344no—
207C2×C22⋊C4—1348no—
208C2×C22⋊C4C22×D6714yes—
209C2×C22⋊C4C22×D6712yes—
210C2×C42—672no—
211C2×C42C4×D6712yescomplemented
212#95—24no—
213#95—1344no—
214C24—5362no—
215C24—5362no—
216C24—672no—
217C24—674no—
218C24—672no—
219C24—674no—
220C24—678no—
221C24—1344no—
222C24C23×D6711yescharacteristic
223C2×D4—678no—
224C2×D4—678no—
225C2×D4—1348no—
226C2×D4—1348no—
227C2×D4—13416no—
228C2×D4—13416no—
229C22×C4—24no—
230C22×C4—24no—
231C22×C4—2688no—
232C22×C4—5368no—
233C22×C4—5364no—
234C22×C4—5364no—
235C22×C4—674no—
236C22×C4—1348no—
237C22×C4—1348no—
238C22×C4—1344no—
239C22×C4C2×C4×D6711yescharacteristic
240C22×C4C2×C4×D6711yescharacteristic
241C22×C4C2×C4×D6714yes—
242C22×C4D4×D6714yes—
243C4⋊C4—24no—
244C4⋊C4—1344no—
245C22⋊C4—24no—
246C22⋊C4—28no—
247C22⋊C4—2688no—
248C22⋊C4—5368no—
249C22⋊C4—1348no—
250C22⋊C4—1348no—
251C22⋊C4—1348no—
252C22⋊C4—1344no—
253C42—24no—
254C42—1344no—
255C23—42no—
256C23—2684no—
257C23—2684no—
258C23—2688no—
259C23—2688no—
260C23—5368no—
261C23—5368no—
262C23—678no—
263C23—678no—
264C23—678no—
265C23—674no—
266C23—674no—
267C23—1344no—
268C23—1348no—
269C23C22×C4×D6711yescharacteristic
270C23C2×D4×D6712yes—
271C23#41×D6714yes—
272D4—13416no—
273D4—13416no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—28no—
277C2×C4—24no—
278C2×C4—44no—
279C2×C4—48no—
280C2×C4—44no—
281C2×C4—2684no—
282C2×C4—2688no—
283C2×C4—2684no—
284C2×C4—5368no—
285C2×C4—5368no—
286C2×C4—53616no—
287C2×C4—1344no—
288C2×C4C22⋊C4×D6712yes—
289C2×C4C22⋊C4×D6712yes—
290C22—22no—
291C22—24no—
292C22—44no—
293C22—2688no—
294C22—2688no—
295C22—2688no—
296C22—2688no—
297C22—5368no—
298C22—678no—
299C22—1344no—
300C22—1348no—
301C22—1348no—
302C22C2×C22⋊C4×D6711yescharacteristic
303C22#117×D6712yes—
304C22#123×D6714yes—
305C4—22no—
306C4—22no—
307C4—44no—
308C4—84no—
309C4—2684no—
310C4—5364no—
311C2—24no—
312C2—22no—
313C2—2684no—
314C2—1348no—
315C2#381×D6711yescharacteristic
316C2#2256866812yes—
subgroupG/H
centerC22C2×C22⋊C4×D67
commutatorC23×C67C23×C4
FrattiniC24C23×D67
FittingC67×#1820C2
socleC22×C67C22×C22⋊C4
prime radicalC23×D268C22
cosocleC24×C67C24
hypercenter#1820D67
hypocenterC67#20689
LCSG ▷ C23×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1820

Complements

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

NH ≅ G/Ntype
3. #22560286C2semidirect
4. #22560284C2semidirect
5. #22560282C2semidirect
6. #342×D67C2semidirect
7. C67×#1820C2semidirect
10. C2×C4×D268C4semidirect
11. C2×#14524005C4semidirect
32. C22×C4×D67C2×C4semidirect
36. C2×C4⋊Dic67C2×C4semidirect
42. C2×(C2×D67)⋊C4C2×C4semidirect
46. C2×C4×Dic67C2×C4semidirect
50. C2×C4⋊C4×C67C2×C4semidirect
53. C2×C42×C67C2×C4semidirect
163. #1820D67semidirect
166. C67#20689semidirect
198. C2×C4⋊C4C4×D67semidirect
211. C2×C42C4×D67semidirect

Maximal subgroups

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

subgroupindex
C2×#168638702
#225627862
#225602862
#225602842
#225602822
#342×D672
C67×#18202
#2068967