Finite Group Databasebeta

← #30855241

Subgroups — #30855241

Order 17152.

1462 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16863370C212yesmaximal
2#22562827C212yesmaximal, complemented
3#22560197C218yesmaximal, complemented
4#22559942C212yesmaximal, complemented
5C2×#520×C67C211yesmaximal, characteristic
6C2×#14524002C2214yes—
7#16863370—28no—
8C2×#14523958C2218yes—
9#16863295—28no—
10#16863256C2218yes—
11#16863163C2218yes—
12C22×#41×C67C2211yescharacteristic
13#520×C67—28no—
14C2×C4×Q8×C67C412yescomplemented
15C2×C4×D4×C67C412yescomplemented
16C2×C4×D4×C67C2214yes—
17C2×#117×C67C412yescomplemented
18C2×#117×C67C2211yescharacteristic
19C22×C4⋊C4×C67C412yescomplemented
20C22×C4⋊C4×C67C2211yescharacteristic
21#14524002—28no—
22#14524002—28no—
23C2×C4⋊Dic67—24no—
24C2×C4⋊Dic67—28no—
25C2×C22⋊Dic67C2318yes—
26C2×C4×Dic67—24no—
27#14523958—216no—
28C67⋊#95—28no—
29C2×#41×C67—24no—
30C2×#41×C67—24no—
31C2×#41×C67D414yes—
32C2×#41×C67Q814yes—
33C22×Q8×C67C2×C411yescharacteristic
34C22×D4×C67C2×C411yescharacteristic
35C22×D4×C67C2312yes—
36C23×C4×C67C2×C412yes—
37C23×C4×C67C2×C414yes—
38C23×C4×C67C2312yes—
39C23×C4×C67C2311yescharacteristic
40C4×Q8×C67—28no—
41C4×D4×C67—216no—
42C4×D4×C67—28no—
43#117×C67—24no—
44#117×C67—28no—
45C2×C4⋊C4×C67—28no—
46C2×C4⋊C4×C67C2×C411yescharacteristic
47C2×C4⋊C4×C67C2×C414yes—
48C2×C4⋊C4×C67C2×C411yescharacteristic
49C2×C4⋊C4×C67C2×C412yes—
50C2×C4⋊C4×C67C2×C412yes—
51C2×C4⋊C4×C67D418yes—
52C2×C22⋊C4×C67C2×C414yes—
53C2×C22⋊C4×C67C2312yes—
54C2×C42×C67C2×C414yes—
55C2×C42×C67C2×C412yes—
56C22×Dic67—28no—
57C4⋊Dic67—48no—
58C4⋊Dic67—88no—
59C22⋊Dic67—28no—
60C22⋊Dic67—28no—
61C4×Dic67—48no—
62C24×C67C22×C412yes—
63C24×C67C2411yescharacteristic
64#41×C67—216no—
65#41×C67—216no—
66C2×Q8×C67—22no—
67C2×Q8×C67—22no—
68C2×Q8×C67C4⋊C414yescomplemented
69C2×D4×C67—24no—
70C2×D4×C67—24no—
71C2×D4×C67—22no—
72C2×D4×C67—22no—
73C2×D4×C67C4⋊C414yescomplemented
74C2×D4×C67C2×D414yes—
75C2×D4×C67C2×Q814yes—
76C22×C4×C67—24no—
77C22×C4×C67—24no—
78C22×C4×C67—28no—
79C22×C4×C67—82no—
80C22×C4×C67—84no—
81C22×C4×C67C4⋊C418yescomplemented
82C22×C4×C67C22×C412yes—
83C22×C4×C67C22×C412yes—
84C22×C4×C67C22×C412yes—
85C22×C4×C67C22×C414yes—
86C22×C4×C67C22×C411yescharacteristic
87C22×C4×C67C22×C411yescharacteristic
88C22×C4×C67C2×D418yes—
89C22×C4×C67C2×D414yes—
90C22×C4×C67C2×D412yes—
91C22×C4×C67C2×Q812yes—
92C22×C4×C67#4114yes—
93C4⋊C4×C67—22no—
94C4⋊C4×C67—28no—
95C4⋊C4×C67—22no—
96C4⋊C4×C67C22⋊C418yes—
97C4⋊C4×C67C22⋊C418yes—
98C22⋊C4×C67—24no—
99C22⋊C4×C67—28no—
100C42×C67—28no—
101C42×C67—24no—
102C23×C67—22no—
103C23×C67—22no—
104C23×C67—41no—
105C23×C67—41no—
106C23×C67—42no—
107C23×C67—42no—
108C23×C67C2×C4⋊C414yes—
109C23×C67C23×C411yescharacteristic
110C23×C67C22×D412yes—
111C23×C67C22×D411yescharacteristic
112C23×C67C22×Q811yescharacteristic
113C23×C67C2×#4112yes—
114Q8×C67—28no—
115D4×C67—28no—
116D4×C67—28no—
117D4×C67—28no—
118C2×C4×C67—22no—
119C2×C4×C67—22no—
120C2×C4×C67—216no—
121C2×C4×C67—22no—
122C2×C4×C67—22no—
123C2×C4×C67—44no—
124C2×C4×C67—48no—
125C2×C4×C67—42no—
126C2×C4×C67—42no—
127C2×C4×C67—88no—
128C2×C4×C67C2×C22⋊C418yes—
129C2×C4×C67C2×C22⋊C412yes—
130C2×C4×C67C2×C22⋊C412yes—
131C2×C4×C67C2×C4⋊C414yes—
132C2×C4×C67C2×C4⋊C412yes—
133C2×C4×C67C2×C4⋊C412yes—
134C2×C4×C67#11712yes—
135C2×C4×C67#11712yes—
136C2×C4×C67#12118yes—
137C2×C4×C67C22⋊Q818yes—
138C2×Dic67—48no—
139C2×Dic67—48no—
140C2×Dic67—48no—
141C22×C67—22no—
142C22×C67—21no—
143C22×C67—22no—
144C22×C67—21no—
145C22×C67—24no—
146C22×C67—41no—
147C22×C67—41no—
148C22×C67—42no—
149C22×C67—44no—
150C22×C67—44no—
151C22×C67—42no—
152C22×C67—42no—
153C22×C67C22×C22⋊C411yescharacteristic
154C22×C67C22×C4⋊C411yescharacteristic
155C22×C67C2×#11711yescharacteristic
156C22×C67C2×#12112yes—
157C22×C67C2×C22⋊Q812yes—
158C4×C67—28no—
159C4×C67—24no—
160C4×C67—42no—
161C4×C67—42no—
162C4×C67#38014yes—
163C4×C67#38014yes—
164Dic67—88no—
165#20634—671nomaximal, Sylow
166C2×C67—22no—
167C2×C67—21no—
168C2×C67—21no—
169C2×C67—22no—
170C2×C67—44no—
171C2×C67C2×#38011yescharacteristic
172C2×C67#288311yescharacteristic
173C2×C67#288411yescharacteristic
174C2×#520D6711yescharacteristic
175C2×#410—672no—
176#2316—672no—
177#1799—678no—
178#1740—672no—
179C67#2063411yescharacteristic, complemented, Sylow
180C22×#41C2×D6711yescharacteristic
181#520—28no—
182C2×C4×Q8Dic6712yescomplemented
183C2×C4×D4Dic6712yescomplemented
184C2×C4×D4C2×D6714yes—
185C2×#117—674no—
186C2×#117Dic6712yescomplemented
187C2×#117C2×D6711yescharacteristic
188C22×C4⋊C4Dic6712yescomplemented
189C22×C4⋊C4C2×D6711yescharacteristic
190#410—1348no—
191C2×#99—678no—
192#389—1348no—
193#380—678no—
194#342—678no—
195C2×#41—24no—
196C2×#41—24no—
197C2×#41Dic13414yes—
198C2×#41D26814yes—
199C22×Q8C2×Dic6711yescharacteristic
200C22×D4C2×Dic6711yescharacteristic
201C22×D4C22×D6712yes—
202C23×C4C2×Dic6712yes—
203C23×C4C2×Dic6714yes—
204C23×C4C22×D6712yes—
205C23×C4C22×D6711yescharacteristic
206C4×Q8—28no—
207C4×D4—216no—
208C4×D4—28no—
209#117—24no—
210#117—28no—
211#117—1348no—
212#117—1348no—
213C2×C4⋊C4—28no—
214C2×C4⋊C4—1344no—
215C2×C4⋊C4—1348no—
216C2×C4⋊C4C2×Dic6711yescharacteristic
217C2×C4⋊C4C2×Dic6714yes—
218C2×C4⋊C4C2×Dic6711yescharacteristic
219C2×C4⋊C4C2×Dic6712yes—
220C2×C4⋊C4C2×Dic6712yes—
221C2×C4⋊C4C22⋊D6718yes—
222C2×C22⋊C4—678no—
223C2×C22⋊C4C2×Dic6714yes—
224C2×C22⋊C4C22×D6712yes—
225C2×C42—1344no—
226C2×C42C2×Dic6714yes—
227C2×C42C2×Dic6712yes—
228#99—13416no—
229#95—1348no—
230C24C22×Dic6712yes—
231C24C23×D6711yescharacteristic
232#41—216no—
233#41—216no—
234C2×Q8—22no—
235C2×Q8—22no—
236C2×Q8C4⋊Dic6714yescomplemented
237C2×D4—24no—
238C2×D4—24no—
239C2×D4—22no—
240C2×D4—22no—
241C2×D4C4⋊Dic6714yescomplemented
242C2×D4C2×D26814yes—
243C2×D4C2×Dic13414yes—
244C22×C4—24no—
245C22×C4—24no—
246C22×C4—28no—
247C22×C4—82no—
248C22×C4—84no—
249C22×C4—1348no—
250C22×C4C4⋊Dic6718yescomplemented
251C22×C4C22×Dic6712yes—
252C22×C4C22×Dic6712yes—
253C22×C4C22×Dic6712yes—
254C22×C4C22×Dic6714yes—
255C22×C4C22×Dic6711yescharacteristic
256C22×C4C22×Dic6711yescharacteristic
257C22×C4C2×C22⋊D6718yes—
258C22×C4C2×C22⋊D6714yes—
259C22×C4C2×D26812yes—
260C22×C4C2×Dic13412yes—
261C22×C4#1176162814yes—
262C4⋊C4—22no—
263C4⋊C4—28no—
264C4⋊C4—22no—
265C4⋊C4—2688no—
266C4⋊C4—5368no—
267C4⋊C4C22⋊Dic6718yes—
268C4⋊C4C22⋊Dic6718yes—
269C22⋊C4—24no—
270C22⋊C4—28no—
271C22⋊C4—1348no—
272C22⋊C4—1348no—
273C42—28no—
274C42—24no—
275C42—2688no—
276C23—22no—
277C23—22no—
278C23—41no—
279C23—41no—
280C23—42no—
281C23—42no—
282C23C2×C4⋊Dic6714yes—
283C23C23×Dic6711yescharacteristic
284C23C22×C22⋊D6712yes—
285C23C22×D26811yescharacteristic
286C23C22×Dic13411yescharacteristic
287C23C2×#1176162812yes—
288Q8—28no—
289D4—28no—
290D4—28no—
291D4—28no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—216no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—48no—
298C2×C4—44no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—88no—
302C2×C4—2688no—
303C2×C4—2688no—
304C2×C4—2688no—
305C2×C4C2×C22⋊Dic6718yes—
306C2×C4C2×C22⋊Dic6712yes—
307C2×C4C2×C22⋊Dic6712yes—
308C2×C4C2×C4⋊Dic6714yes—
309C2×C4C2×C4⋊Dic6712yes—
310C2×C4C2×C4⋊Dic6712yes—
311C2×C4#1452400212yes—
312C2×C4#1452400212yes—
313C2×C4#1452401718yes—
314C2×C4#1452402218yes—
315C22—21no—
316C22—22no—
317C22—21no—
318C22—22no—
319C22—24no—
320C22—41no—
321C22—41no—
322C22—42no—
323C22—44no—
324C22—44no—
325C22—42no—
326C22—42no—
327C22C22×C22⋊Dic6711yescharacteristic
328C22C22×C4⋊Dic6711yescharacteristic
329C22C2×#1452400211yescharacteristic
330C22C2×#1452401712yes—
331C22C2×#1452402212yes—
332C4—24no—
333C4—28no—
334C4—42no—
335C4—42no—
336C4—5368no—
337C4#1686325614yes—
338C4#1686325614yes—
339C2—22no—
340C2—21no—
341C2—21no—
342C2—22no—
343C2—44no—
344C2C2×#1686325611yescharacteristic
345C2#2256867011yescharacteristic
346C2#2256867311yescharacteristic
subgroupG/H
centerC22C22×C22⋊Dic67
commutatorC23×C67C23×C4
FrattiniC24C23×D67
FittingC2×#520×C67C2
socleC22×C67C22×C22⋊C4
prime radicalC22×#41×C67C22
cosocleC24×C67C24
hypercenterC2×#520D67
hypocenterC67#20634
LCSG ▷ C23×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C22×C4⋊C4 ◁ C2×#520

Complements

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

NH ≅ G/Ntype
2. #22562827C2semidirect
3. #22560197C2semidirect
4. #22559942C2semidirect
14. C2×C4×Q8×C67C4semidirect
15. C2×C4×D4×C67C4semidirect
17. C2×#117×C67C4semidirect
19. C22×C4⋊C4×C67C4semidirect
68. C2×Q8×C67C4⋊C4semidirect
73. C2×D4×C67C4⋊C4semidirect
81. C22×C4×C67C4⋊C4semidirect
179. C67#20634semidirect
182. C2×C4×Q8Dic67semidirect
183. C2×C4×D4Dic67semidirect
186. C2×#117Dic67semidirect
188. C22×C4⋊C4Dic67semidirect
236. C2×Q8C4⋊Dic67semidirect
241. C2×D4C4⋊Dic67semidirect
250. C22×C4C4⋊Dic67semidirect

Maximal subgroups

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

subgroupindex
C2×#168633702
#225628272
#225601972
#225599422
C2×#520×C672
#2063467