Finite Group Databasebeta

← #30864792

Subgroups — #30864792

Order 17152.

1606 conjugacy classes of subgroups; only the 235 normal subgroups are shown.

normal subgroups only

This group has 1606 conjugacy classes of subgroups — too many to draw. Only its 235 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×#16863965C211yesmaximal, characteristic
2#22562786C212yesmaximal, complemented
3#22560661C218yesmaximal, complemented
4#22560286C211yesmaximal, characteristic, complemented
5#22560126C212yesmaximal, complemented
6C67×#1820C211yesmaximal, characteristic, complemented
7C22×#11761628C2211yescharacteristic
8C23×D268C2211yescharacteristic
9#16863965—28no—
10C2×#14524040C412yescomplemented
11C2×#14524031C412yescomplemented
12C2×#14524018C414yescomplemented
13C2×#14524013C2214yes—
14C2×C22⋊C4×D67C2214yes—
15C2×#14523957C2218yes—
16#16863274—28no—
17#16863259C2212yes—
18#16863251C2214yes—
19#16863163C2214yes—
20C2×#117×C67C2211yescharacteristic
21#381×C67C2212yes—
22#342×C67C2214yes—
23C24×D67C2312yes—
24C2×#11761628—24no—
25C2×#11761628—44no—
26C22×Dic134C2×C411yescharacteristic, complemented
27C22×D268—44no—
28C22×D268C2×C412yes—
29C22×D268C2×C412yes—
30C22×D268C2×C411yescharacteristic, complemented
31C22×D268C2×C418yescomplemented
32C22×D268D418yes—
33C22×C22⋊D67C2312yes—
34C22×C4×D67—24no—
35C22×C4×D67C2×C412yescomplemented
36#14524040—28no—
37#14524031—28no—
38#14524018—216no—
39#14524013—28no—
40#14524013—28no—
41C2×C22⋊Dic67C2314yes—
42C2×(C2×D67)⋊C4—24no—
43C2×(C2×D67)⋊C4—28no—
44C2×(C2×D67)⋊C4C2×C418yescomplemented
45C22⋊C4×D67—48no—
46#14523957—48no—
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—28no—
56C23×D67—84no—
57C23×D67C22×C412yes—
58C23×D67C22×C414yes—
59C23×D67C2×D414yes—
60C23×D67C2×D414yes—
61#11761628—416no—
62C2×Dic134—22no—
63C2×Dic134—42no—
64C2×D268—22no—
65C2×D268—216no—
66C2×D268—44no—
67C2×D268—44no—
68C2×D268—48no—
69C2×D268—48no—
70C2×D268—42no—
71C2×D268—88no—
72C2×D268C22⋊C418yes—
73C2×D268C22⋊C418yes—
74C2×C22⋊D67—24no—
75C2×C22⋊D67—28no—
76C22×Dic67—24no—
77C22×Dic67C22×C412yes—
78C2×C4×D67—24no—
79C2×C4×D67—44no—
80C2×C4×D67—48no—
81C2×C4×D67—88no—
82C22⋊Dic67—28no—
83(C2×D67)⋊C4—216no—
84(C2×D67)⋊C4—48no—
85(C2×D67)⋊C4—88no—
86C24×C67C2411yescharacteristic
87C22×C4×C67—24no—
88C22×C4×C67—24no—
89C22×C4×C67C22×C411yescharacteristic
90C22×C4×C67C22×C411yescharacteristic
91C22×C4×C67C22×C414yes—
92C22×C4×C67C2×D414yes—
93C4⋊C4×C67—24no—
94C22⋊C4×C67—24no—
95C22⋊C4×C67—28no—
96C42×C67—24no—
97C23×C67—42no—
98C23×C67C23×C411yescharacteristic
99C23×C67C22×D412yes—
100C23×C67H(5,𝔽2)14yes—
101C22×D67—24no—
102C22×D67—24no—
103C22×D67—28no—
104C22×D67—28no—
105C22×D67—22no—
106C22×D67—24no—
107C22×D67—48no—
108C22×D67—48no—
109C22×D67—48no—
110C22×D67—44no—
111C22×D67—816no—
112C22×D67C2×C22⋊C418yes—
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—
122C22⋊D67—48no—
123C2×Dic67—22no—
124C2×Dic67—24no—
125C2×Dic67—48no—
126C2×Dic67—44no—
127D268—48no—
128D268—48no—
129D268—44no—
130D268—816no—
131C4×D67—48no—
132C4×D67—816no—
133Dic134—44no—
134C22×C67—24no—
135C22×C67—22no—
136C22×C67—44no—
137C22×C67C22×C22⋊C411yescharacteristic
138C22×C67#51812yes—
139C22×C67#53414yes—
140C2×D67—28no—
141C2×D67—216no—
142C2×D67—48no—
143C2×D67—48no—
144C2×D67—42no—
145C2×D67—44no—
146C2×D67—88no—
147C2×D67—88no—
148C2×D67—84no—
149C4×C67—22no—
150C4×C67—22no—
151C4×C67—44no—
152C4×C67—84no—
153Dic67—42no—
154Dic67—84no—
155#21750—671nomaximal, Sylow
156C2×C67—24no—
157C2×C67—22no—
158C2×C67#230811yescharacteristic
159C2×C67#288312yes—
160D67—48no—
161D67—82no—
162C2×#535—671no—
163#2308—672no—
164#1898—678no—
165#1820—671no—
166#1820D6711yescharacteristic, complemented
167#1782—672no—
168C67#2175011yescharacteristic, complemented, Sylow
169C22×#41—671no—
170C23×D4—671no—
171#535—1348no—
172C2×#127—672no—
173C2×#124—672no—
174C2×#121—674no—
175C2×#120—674no—
176C2×#117C2×D6711yescharacteristic
177C22×C22⋊C4—674no—
178C2×#99—678no—
179#386—1348no—
180#381—672no—
181#381C2×D6712yes—
182#379—674no—
183#342—674no—
184#342C2×D6714yes—
185C25—672no—
186C2×#41—2684no—
187C2×#41—1344no—
188C22×Q8—671no—
189C22×D4—2684no—
190C22×D4—672no—
191C22×D4—672no—
192C22×D4—672no—
193C22×D4—671no—
194C22×D4—678no—
195C22×D4—678no—
196C23×C4—672no—
197C23×C4—1344no—
198C23×C4C22×D6711yescharacteristic
199#127—1348no—
200#124—1348no—
201#121—13416no—
202#120—1348no—
203#120—1348no—
204#117—24no—
205C2×C4⋊C4C4×D6712yescomplemented
206C2×C22⋊C4—2688no—
207C2×C22⋊C4—678no—
208C2×C22⋊C4—674no—
209C2×C22⋊C4—1344no—
210C2×C22⋊C4—1348no—
211C2×C22⋊C4C22×D6714yes—
212C2×C22⋊C4C22×D6712yes—
213C2×C42C4×D6712yescomplemented
214#99—2688no—
215#95—24no—
216#95—1344no—
217C24—5364no—
218C24—674no—
219C24—674no—
220C24—674no—
221C24—672no—
222C24—1348no—
223C24C23×D6711yescharacteristic
224#41—26816no—
225C2×Q8—2682no—
226C2×Q8—1342no—
227C2×D4—2684no—
228C2×D4—2684no—
229C2×D4—2688no—
230C2×D4—2688no—
231C2×D4—2682no—
232C2×D4—5368no—
233C2×D4—678no—
234C2×D4—678no—
235C2×D4—13416no—
236C2×D4—1344no—
237C2×D4—1348no—
238C2×D4—1342no—
239C22×C4—24no—
240C22×C4—24no—
241C22×C4—2684no—
242C22×C4—2688no—
243C22×C4—5368no—
244C22×C4—672no—
245C22×C4—1344no—
246C22×C4—1344no—
247C22×C4C2×C4×D6711yescharacteristic
248C22×C4C2×C4×D6711yescharacteristic
249C22×C4C2×C4×D6714yes—
250C22×C4D4×D6714yes—
251C4⋊C4—24no—
252C22⋊C4—24no—
253C22⋊C4—28no—
254C22⋊C4—2688no—
255C22⋊C4—5368no—
256C22⋊C4—13416no—
257C22⋊C4—1348no—
258C42—24no—
259C23—42no—
260C23—2688no—
261C23—2684no—
262C23—2688no—
263C23—2688no—
264C23—53616no—
265C23—678no—
266C23—1342no—
267C23—1344no—
268C23—1344no—
269C23—1344no—
270C23—1348no—
271C23—1348no—
272C23C22×C4×D6711yescharacteristic
273C23C2×D4×D6712yes—
274C23#1452408914yes—
275Q8—2684no—
276D4—2684no—
277D4—2688no—
278D4—2688no—
279D4—2688no—
280D4—53616no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—28no—
284C2×C4—24no—
285C2×C4—44no—
286C2×C4—48no—
287C2×C4—44no—
288C2×C4—2688no—
289C2×C4—2688no—
290C2×C4—2684no—
291C2×C4—53616no—
292C2×C4—1342no—
293C2×C4—1344no—
294C2×C4C22⋊C4×D6712yes—
295C2×C4C22⋊C4×D6712yes—
296C22—24no—
297C22—22no—
298C22—44no—
299C22—2688no—
300C22—2688no—
301C22—2684no—
302C22—2682no—
303C22—5368no—
304C22—5368no—
305C22—5364no—
306C22—13416no—
307C22—1348no—
308C22C2×C22⋊C4×D6711yescharacteristic
309C22#1686387012yes—
310C22#1686395714yes—
311C4—22no—
312C4—22no—
313C4—44no—
314C4—84no—
315C4—2682no—
316C4—5364no—
317C2—24no—
318C2—22no—
319C2—2688no—
320C2—5362no—
321C2#2256278611yescharacteristic
322C2#2256866812yes—
subgroupG/H
centerC22C2×C22⋊C4×D67
commutatorC23×C67C23×C4
FrattiniC24C23×D67
FittingC67×#1820C2
socleC22×C67C22×C22⋊C4
prime radicalC2×#16863965C2
cosocleC24×C67C24
hypercenter#1820D67
hypocenterC67#21750
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
2. #22562786C2semidirect
3. #22560661C2semidirect
4. #22560286C2semidirect
5. #22560126C2semidirect
6. C67×#1820C2semidirect
10. C2×#14524040C4semidirect
11. C2×#14524031C4semidirect
12. C2×#14524018C4semidirect
26. C22×Dic134C2×C4semidirect
30. C22×D268C2×C4semidirect
31. C22×D268C2×C4semidirect
35. C22×C4×D67C2×C4semidirect
44. C2×(C2×D67)⋊C4C2×C4semidirect
50. C2×C4⋊C4×C67C2×C4semidirect
53. C2×C42×C67C2×C4semidirect
166. #1820D67semidirect
168. C67#21750semidirect
205. C2×C4⋊C4C4×D67semidirect
213. C2×C42C4×D67semidirect

Maximal subgroups

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

subgroupindex
C2×#168639652
#225627862
#225606612
#225602862
#225601262
C67×#18202
#2175067