Finite Group Databasebeta

← #30864797

Subgroups — #30864797

Order 17152.

1294 conjugacy classes of subgroups; only the 235 normal subgroups are shown.

normal subgroups only

This group has 1294 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×#16863972C211yesmaximal, characteristic
2#22562794C212yesmaximal, complemented
3#22560666C218yesmaximal, complemented
4#22560286C211yesmaximal, characteristic, complemented
5#22560126C212yesmaximal
6C67×#1820C211yesmaximal, characteristic, complemented
7C22×#11761628C2211yescharacteristic
8C23×Dic134C2211yescharacteristic
9#16863972—28no—
10C2×#14524043C412yescomplemented
11C2×#14524031C412yescomplemented
12C2×#14524027C2214yes—
13C2×#14524023C414yescomplemented
14C2×#14523999C2214yes—
15C2×#14523957C2218yes—
16#16863300—28no—
17#16863259C2212yes—
18#16863257C2214yes—
19#16863163C2214yes—
20C2×#117×C67C2211yescharacteristic
21#381×C67C2212yes—
22#342×C67C2214yes—
23C2×#11761628—24no—
24C2×#11761628—44no—
25C22×Dic134—44no—
26C22×Dic134C2×C411yescharacteristic
27C22×Dic134C2×C412yes—
28C22×Dic134C2×C412yes—
29C22×Dic134D418yes—
30C22×D268C2×C411yescharacteristic
31C22×C22⋊D67C2312yes—
32C23×Dic67C2312yes—
33C22×C4×D67C2×C412yes—
34#14524043—28no—
35#14524031—28no—
36#14524027—28no—
37#14524027—28no—
38#14524023—216no—
39#14523999—48no—
40C2×C4⋊Dic67C2×C418yes—
41C2×Dic67⋊C4—24no—
42C2×C22⋊Dic67C2314yes—
43C2×(C2×D67)⋊C4C2×C418yes—
44C2×C4×Dic67—24no—
45#14523957—48no—
46C67⋊#95—28no—
47C67⋊#95—24no—
48C23×C4×C67C2311yescharacteristic
49#117×C67—24no—
50C2×C4⋊C4×C67C2×C412yes—
51C2×C22⋊C4×C67C2314yes—
52C2×C22⋊C4×C67C2312yes—
53C2×C42×C67C2×C412yes—
54#95×C67—24no—
55C23×D67C22×C412yes—
56#11761628—416no—
57C2×Dic134—22no—
58C2×Dic134—44no—
59C2×Dic134—44no—
60C2×Dic134—48no—
61C2×Dic134—48no—
62C2×Dic134—42no—
63C2×Dic134C22⋊C418yes—
64C2×Dic134C22⋊C418yes—
65C2×D268—22no—
66C2×D268—42no—
67C2×C22⋊D67—24no—
68C2×C22⋊D67—28no—
69C22×Dic67—28no—
70C22×Dic67—24no—
71C22×Dic67C22×C412yes—
72C22×Dic67C22×C414yes—
73C22×Dic67C2×D414yes—
74C22×Dic67C2×D414yes—
75C2×C4×D67—24no—
76C2×C4×D67—44no—
77C4⋊Dic67—216no—
78Dic67⋊C4—48no—
79C22⋊Dic67—28no—
80(C2×D67)⋊C4—216no—
81C4×Dic67—48no—
82C24×C67C2411yescharacteristic
83C22×C4×C67—24no—
84C22×C4×C67—24no—
85C22×C4×C67C22×C411yescharacteristic
86C22×C4×C67C22×C411yescharacteristic
87C22×C4×C67C22×C414yes—
88C22×C4×C67C2×D414yes—
89C4⋊C4×C67—24no—
90C22⋊C4×C67—24no—
91C22⋊C4×C67—28no—
92C42×C67—24no—
93C23×C67—42no—
94C23×C67C23×C411yescharacteristic
95C23×C67C22×D412yes—
96C23×C67H(5,𝔽2)14yes—
97C22×D67—22no—
98C22×D67—24no—
99C22×D67—44no—
100C2×C4×C67—22no—
101C2×C4×C67—22no—
102C2×C4×C67—28no—
103C2×C4×C67—24no—
104C2×C4×C67—44no—
105C2×C4×C67—48no—
106C2×C4×C67—44no—
107C2×C4×C67C2×C22⋊C412yes—
108C2×C4×C67C2×C22⋊C412yes—
109C22⋊D67—48no—
110C2×Dic67—28no—
111C2×Dic67—24no—
112C2×Dic67—24no—
113C2×Dic67—28no—
114C2×Dic67—22no—
115C2×Dic67—24no—
116C2×Dic67—48no—
117C2×Dic67—44no—
118C2×Dic67C2×C22⋊C418yes—
119D268—44no—
120C4×D67—48no—
121Dic134—48no—
122Dic134—48no—
123Dic134—44no—
124C22×C67—24no—
125C22×C67—22no—
126C22×C67—44no—
127C22×C67C22×C22⋊C411yescharacteristic
128C22×C67#51812yes—
129C22×C67#53414yes—
130C2×D67—42no—
131C2×D67—44no—
132C2×D67—84no—
133C4×C67—22no—
134C4×C67—22no—
135C4×C67—44no—
136C4×C67—84no—
137Dic67—28no—
138Dic67—42no—
139Dic67—84no—
140#21751—671nomaximal, Sylow
141C2×C67—24no—
142C2×C67—22no—
143C2×C67#230811yescharacteristic
144C2×C67#288412yes—
145D67—82no—
146C2×#536—671no—
147#2310—672no—
148#1899—678no—
149#1820—671no—
150#1820D6711yescharacteristic, complemented
151#1782—672no—
152C67#2175111yescharacteristic, complemented, Sylow
153C22×#41—671no—
154C23×Q8—671no—
155#536—1348no—
156C2×C4⋊Q8—672no—
157C2×#124—672no—
158C2×#123—674no—
159C2×C22⋊Q8—674no—
160C2×#117—674no—
161C2×#117C2×D6711yescharacteristic
162C2×#99—678no—
163#391—1348no—
164#381—672no—
165#381—674no—
166#381C2×D6712yes—
167#342—674no—
168#342C2×D6714yes—
169C2×#41—2684no—
170C2×#41—1344no—
171C22×Q8—2684no—
172C22×Q8—672no—
173C22×Q8—672no—
174C22×Q8—671no—
175C22×Q8—678no—
176C22×D4—672no—
177C22×D4—671no—
178C23×C4—672no—
179C23×C4—672no—
180C23×C4C22×D6711yescharacteristic
181C4⋊Q8—1348no—
182#124—1348no—
183#123—1348no—
184#123—1348no—
185C22⋊Q8—13416no—
186#117—24no—
187#117—2688no—
188C2×C4⋊C4—678no—
189C2×C4⋊C4—1344no—
190C2×C4⋊C4C4×D6712yes—
191C2×C22⋊C4—678no—
192C2×C22⋊C4—674no—
193C2×C22⋊C4C22×D6714yes—
194C2×C22⋊C4C22×D6712yes—
195C2×C42—1344no—
196C2×C42C4×D6712yes—
197#99—2688no—
198#95—24no—
199#95—1348no—
200#95—1344no—
201C24—672no—
202C24C23×D6711yescharacteristic
203#41—26816no—
204C2×Q8—2684no—
205C2×Q8—2684no—
206C2×Q8—2688no—
207C2×Q8—2688no—
208C2×Q8—2682no—
209C2×Q8—678no—
210C2×Q8—678no—
211C2×Q8—1342no—
212C2×D4—2682no—
213C2×D4—1344no—
214C2×D4—1348no—
215C2×D4—1342no—
216C22×C4—24no—
217C22×C4—24no—
218C22×C4—2684no—
219C22×C4—674no—
220C22×C4—674no—
221C22×C4—674no—
222C22×C4—672no—
223C22×C4—1344no—
224C22×C4—1344no—
225C22×C4—1348no—
226C22×C4C2×C4×D6711yescharacteristic
227C22×C4C2×C4×D6711yescharacteristic
228C22×C4C2×C4×D6714yes—
229C22×C4D4×D6714yes—
230C4⋊C4—24no—
231C4⋊C4—2688no—
232C4⋊C4—13416no—
233C22⋊C4—24no—
234C22⋊C4—28no—
235C22⋊C4—13416no—
236C22⋊C4—1348no—
237C42—24no—
238C42—2688no—
239C23—42no—
240C23—2684no—
241C23—1342no—
242C23—1344no—
243C23C22×C4×D6711yescharacteristic
244C23C2×D4×D6712yes—
245C23#1452408914yes—
246Q8—2684no—
247Q8—2688no—
248Q8—2688no—
249D4—2684no—
250D4—2688no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—28no—
254C2×C4—24no—
255C2×C4—44no—
256C2×C4—48no—
257C2×C4—44no—
258C2×C4—2688no—
259C2×C4—2688no—
260C2×C4—2684no—
261C2×C4—678no—
262C2×C4—1342no—
263C2×C4—1344no—
264C2×C4—1348no—
265C2×C4—1344no—
266C2×C4—1344no—
267C2×C4—1348no—
268C2×C4C22⋊C4×D6712yes—
269C2×C4C22⋊C4×D6712yes—
270C22—24no—
271C22—22no—
272C22—44no—
273C22—2684no—
274C22—2682no—
275C22—5364no—
276C22C2×C22⋊C4×D6711yescharacteristic
277C22#1686387012yes—
278C22#1686395714yes—
279C4—22no—
280C4—22no—
281C4—44no—
282C4—84no—
283C4—2682no—
284C4—5364no—
285C4—1348no—
286C2—24no—
287C2—22no—
288C2—5362no—
289C2#2256278611yescharacteristic
290C2#2256867112yes—
subgroupG/H
centerC22C2×C22⋊C4×D67
commutatorC23×C67C23×C4
FrattiniC24C23×D67
FittingC67×#1820C2
socleC22×C67C22×C22⋊C4
prime radicalC22×#11761628C22
cosocleC24×C67C24
hypercenter#1820D67
hypocenterC67#21751
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. #22562794C2semidirect
3. #22560666C2semidirect
4. #22560286C2semidirect
6. C67×#1820C2semidirect
10. C2×#14524043C4semidirect
11. C2×#14524031C4semidirect
13. C2×#14524023C4semidirect
150. #1820D67semidirect
152. C67#21751semidirect

Maximal subgroups

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

subgroupindex
C2×#168639722
#225627942
#225606662
#225602862
#225601262
C67×#18202
#2175167