Finite Group Databasebeta

← #30857669

Subgroups — #30857669

Order 17152.

1866 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16863960C211yesmaximal, characteristic
2C22×#14523959C212yesmaximal, complemented
3#22568654C211yesmaximal, characteristic, complemented
4#22560672C218yesmaximal, complemented
5#22559947C212yesmaximal, complemented
6C2×#568×C67C211yesmaximal, characteristic, complemented
7C23×D268C2211yescharacteristic
8C22×#11761612C2212yes—
9#16863960—24no—
10#16863960—24no—
11C2×#14524039C412yes—
12C2×#14524032C412yes—
13C2×#14524017C2214yes—
14C2×#14524011C414yes—
15C2×#14524002C2211yescharacteristic
16C2×#14523959—28no—
17C2×#14523959C2218yes—
18#16863363—24no—
19#16863360C2214yescomplemented
20#16863164C2218yescomplemented
21C22×#41×C67C2211yescharacteristic
22#568×C67—28no—
23C2×#131×C67—24no—
24C22×M16×C67C2212yes—
25C22×M16×C67C2214yescomplemented
26C24×D67C2×C412yes—
27C22×D268—24no—
28C22×D268—22no—
29C22×D268—22no—
30C22×D268—24no—
31C22×D268C2×C412yes—
32C22×D268C2312yes—
33C22×C22⋊D67C2×C418yes—
34C2×#11761612—22no—
35C2×#11761612—24no—
36C2×#11761612—24no—
37C2×#11761612C2314yes—
38C22×C67⋊C8—22no—
39#14524039—28no—
40#14524032—28no—
41#14524017—28no—
42#14524017—28no—
43#14524011—216no—
44#14524002—22no—
45#14524002—22no—
46C2×C4⋊Dic67C2312yes—
47C2×C22⋊Dic67C2×C412yes—
48C2×(C2×D67)⋊C4C2×C418yes—
49C2×C4×Dic67C2×C412yes—
50#14523959—216no—
51C22⋊(C67⋊C8)—28no—
52C2×#41×C67—24no—
53C2×#41×C67—24no—
54C2×#41×C67D414yes—
55C2×#41×C67D414yes—
56C22×Q8×C67C2×C411yescharacteristic
57C22×D4×C67C2×C412yes—
58C22×D4×C67C2×C411yescharacteristic
59C23×C4×C67C2312yes—
60C23×C4×C67C2311yescharacteristic
61#131×C67—232no—
62C2×M16×C67—22no—
63C2×M16×C67—24no—
64C2×M16×C67—24no—
65C2×M16×C67—216no—
66C2×M16×C67D418yescomplemented
67C2×M16×C67C2314yes—
68C22×C8×C67—22no—
69C22×C8×C67—24no—
70C23×D67—24no—
71C23×D67—28no—
72C23×D67—84no—
73C23×D67C22×C414yes—
74C2×D268—24no—
75C2×D268—28no—
76C2×D268—24no—
77C2×D268—22no—
78C2×D268—22no—
79C2×D268—24no—
80C2×D268—44no—
81C2×D268—44no—
82C2×D268—44no—
83C2×D268—44no—
84C2×C22⋊D67—216no—
85C2×C22⋊D67—88no—
86C22×Dic67C22×C414yes—
87#11761612—24no—
88#11761612—24no—
89#11761612—44no—
90C2×C67⋊C8—24no—
91C2×C67⋊C8—44no—
92C4⋊Dic67—22no—
93C4⋊Dic67—22no—
94C22⋊Dic67—24no—
95(C2×D67)⋊C4—216no—
96C4×Dic67—24no—
97C24×C67C22×C412yes—
98C24×C67C22×C411yescharacteristic
99#41×C67—28no—
100#41×C67—28no—
101#41×C67—28no—
102#41×C67—28no—
103C2×Q8×C67—24no—
104C2×Q8×C67C22⋊C414yes—
105C2×D4×C67—28no—
106C2×D4×C67—24no—
107C2×D4×C67C22⋊C418yes—
108C2×D4×C67C22⋊C414yes—
109C22×C4×C67—24no—
110C22×C4×C67—24no—
111C22×C4×C67C22×C412yes—
112C22×C4×C67C22×C411yescharacteristic
113C22×C4×C67C2×D414yes—
114C22×C4×C67C2×D412yes—
115C22×C4×C67C2×D414yes—
116C22×C4×C67C2×D414yes—
117C22×C4×C67C2×D412yes—
118C22×C4×C67#4114yes—
119C22×C4×C67C2411yescharacteristic
120M16×C67—216no—
121M16×C67—28no—
122M16×C67—24no—
123M16×C67C2×D418yes—
124C2×C8×C67—216no—
125C2×C8×C67—28no—
126C2×C8×C67—24no—
127C23×C67—24no—
128C23×C67—42no—
129C23×C67—44no—
130C23×C67C2×C22⋊C414yes—
131C23×C67C2×C22⋊C412yes—
132C23×C67C4×D414yes—
133C23×C67C23×C411yescharacteristic
134C22×D67—24no—
135C22×D67—28no—
136C22×D67—48no—
137C22×D67—48no—
138C22×D67—88no—
139C22×D67—816no—
140Q8×C67—28no—
141D4×C67—216no—
142D4×C67—28no—
143C2×C4×C67—24no—
144C2×C4×C67—24no—
145C2×C4×C67—28no—
146C2×C4×C67—24no—
147C2×C4×C67C2×C22⋊C414yes—
148C2×C4×C67C2×C22⋊C412yes—
149C2×C4×C67C4×D414yes—
150C2×C4×C67#12014yes—
151C2×C4×C67#12114yes—
152C2×C4×C67#12114yes—
153C2×C4×C67#12314yes—
154C2×C4×C67C22×D412yes—
155C2×C4×C67C22×D411yescharacteristic
156C2×C4×C67C22×D411yescharacteristic
157C2×C4×C67C2×#4112yes—
158C22⋊D67—816no—
159C2×Dic67—24no—
160C2×Dic67—44no—
161D268—42no—
162D268—44no—
163D268—42no—
164D268—48no—
165C8×C67—28no—
166C67⋊C8—44no—
167C22×C67—22no—
168C22×C67—24no—
169C22×C67—28no—
170C22×C67—24no—
171C22×C67—44no—
172C22×C67—84no—
173C22×C67#38614yes—
174C22×C67C22×C22⋊C411yescharacteristic
175C22×C67C2×C4×D412yes—
176C2×D67—48no—
177C2×D67—44no—
178C2×D67—816no—
179C2×D67—88no—
180C4×C67—24no—
181C4×C67#38614yes—
182C4×C67C2×#12011yescharacteristic
183C4×C67C2×#12112yes—
184C4×C67C2×#12311yescharacteristic
185Dic67—44no—
186#20990—671nomaximal, Sylow
187C2×C67—22no—
188C2×C67—24no—
189C2×C67—44no—
190C2×C67C2×#38611yescharacteristic
191C2×C67#288812yes—
192D67—84no—
193C2×#568D6711yescharacteristic, complemented
194C2×#535—671no—
195C22×#100—672no—
196#2879—671no—
197#1900—678no—
198#1742—672no—
199C67#2099011yescharacteristic, complemented, Sylow
200C22×#41C2×D6711yescharacteristic
201C23×D4—671no—
202#568—28no—
203C2×#131—24no—
204C22×M16—672no—
205C22×M16C2×D6712yes—
206C22×M16C2×D6714yescomplemented
207#535—1344no—
208#535—1344no—
209C2×#127—672no—
210C2×#124—672no—
211C2×#121—674no—
212C2×#120—674no—
213C2×#117—671no—
214C2×#100—678no—
215C2×#100—1348no—
216#408—1344no—
217#407—674no—
218#343—678no—
219C25—672no—
220C2×#41—24no—
221C2×#41—24no—
222C2×#41D26814yes—
223C2×#41C22⋊D6714yes—
224C22×Q8C4×D6711yescharacteristic
225C22×D4—678no—
226C22×D4—672no—
227C22×D4—672no—
228C22×D4—1342no—
229C22×D4—1344no—
230C22×D4—1342no—
231C22×D4—1344no—
232C22×D4C4×D6712yes—
233C22×D4C4×D6711yescharacteristic
234C23×C4C22×D6712yes—
235C23×C4C22×D6711yescharacteristic
236#131—232no—
237C2×M16—22no—
238C2×M16—24no—
239C2×M16—24no—
240C2×M16—216no—
241C2×M16—674no—
242C2×M16—1344no—
243C2×M16—1344no—
244C2×M16—1342no—
245C2×M16C22⋊D6718yescomplemented
246C2×M16C22×D6714yes—
247C22×C8—22no—
248C22×C8—24no—
249C22×C8—1342no—
250#127—1348no—
251#124—1348no—
252#121—1348no—
253#121—1348no—
254#120—13416no—
255#117—1342no—
256#117—1342no—
257C2×C4⋊C4—672no—
258C2×C22⋊C4—672no—
259C2×C22⋊C4—678no—
260C2×C42—672no—
261#100—13416no—
262C22⋊C8—1348no—
263C24—5364no—
264C24—674no—
265C24—1344no—
266C24—1348no—
267C24C2×C4×D6712yes—
268C24C2×C4×D6711yescharacteristic
269#41—28no—
270#41—28no—
271#41—28no—
272#41—28no—
273C2×Q8—24no—
274C2×Q8(C2×D67)⋊C414yes—
275C2×D4—28no—
276C2×D4—24no—
277C2×D4—2684no—
278C2×D4—2684no—
279C2×D4—2684no—
280C2×D4—2684no—
281C2×D4—5368no—
282C2×D4—13416no—
283C2×D4—1344no—
284C2×D4—1348no—
285C2×D4—1344no—
286C2×D4—1342no—
287C2×D4—1342no—
288C2×D4—1344no—
289C2×D4(C2×D67)⋊C418yes—
290C2×D4(C2×D67)⋊C414yes—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—674no—
294C22×C4C2×C4×D6712yes—
295C22×C4C2×C4×D6711yescharacteristic
296C22×C4C2×C22⋊D6714yes—
297C22×C4C2×C22⋊D6712yes—
298C22×C4C2×C22⋊D6714yes—
299C22×C4C2×D26814yes—
300C22×C4C2×D26812yes—
301C22×C4#1176162814yes—
302C22×C4C23×D6711yescharacteristic
303M16—216no—
304M16—28no—
305M16—24no—
306M16—2684no—
307M16—1344no—
308M16—1344no—
309M16C2×C22⋊D6718yes—
310C2×C8—216no—
311C2×C8—28no—
312C2×C8—24no—
313C2×C8—2684no—
314C2×C8—1344no—
315C4⋊C4—1342no—
316C4⋊C4—1342no—
317C22⋊C4—1344no—
318C22⋊C4—13416no—
319C42—1344no—
320C23—24no—
321C23—42no—
322C23—44no—
323C23—2688no—
324C23—2688no—
325C23—5368no—
326C23—53616no—
327C23—1344no—
328C23—1348no—
329C23C2×(C2×D67)⋊C414yes—
330C23C2×(C2×D67)⋊C412yes—
331C23C4×C22⋊D6714yes—
332C23C22×C4×D6711yescharacteristic
333Q8—28no—
334D4—216no—
335D4—28no—
336D4—2688no—
337D4—2682no—
338D4—2684no—
339D4—2682no—
340D4—53616no—
341C2×C4—24no—
342C2×C4—24no—
343C2×C4—28no—
344C2×C4—24no—
345C2×C4—2684no—
346C2×C4—1344no—
347C2×C4C2×(C2×D67)⋊C414yes—
348C2×C4C2×(C2×D67)⋊C412yes—
349C2×C4C4×C22⋊D6714yes—
350C2×C4#1452401214yes—
351C2×C4#1452401714yes—
352C2×C4#1452401714yes—
353C2×C4#1452402614yes—
354C2×C4C22×C22⋊D6712yes—
355C2×C4C22×C22⋊D6711yescharacteristic
356C2×C4C22×D26811yescharacteristic
357C2×C4C2×#1176162812yes—
358C8—28no—
359C8—2684no—
360C22—22no—
361C22—24no—
362C22—24no—
363C22—28no—
364C22—44no—
365C22—84no—
366C22—2684no—
367C22—2688no—
368C22—5368no—
369C22—53616no—
370C22#1686327514yes—
371C22C22×(C2×D67)⋊C411yescharacteristic
372C22C2×C4×C22⋊D6712yes—
373C4—24no—
374C4—2684no—
375C4#1686327514yes—
376C4C2×#1452401211yescharacteristic
377C4C2×#1452401712yes—
378C4C2×#1452402611yescharacteristic
379C2—22no—
380C2—24no—
381C2—44no—
382C2—5364no—
383C2C2×#1686327511yescharacteristic
384C2#2256868712yes—
subgroupG/H
centerC22C22×(C2×D67)⋊C4
commutatorC23×C67C23×C4
FrattiniC22×C4C23×D67
FittingC2×#568×C67C2
socleC22×C67C22×C22⋊C4
prime radicalC2×#16863960C2
cosocleC22×C4×C67C24
hypercenterC2×#568D67
hypocenterC67#20990
LCSG ▷ C23×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C22×#41 ◁ C2×#568

Complements

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

NH ≅ G/Ntype
2. C22×#14523959C2semidirect
3. #22568654C2semidirect
4. #22560672C2semidirect
5. #22559947C2semidirect
6. C2×#568×C67C2semidirect
19. #16863360C22semidirect
20. #16863164C22semidirect
25. C22×M16×C67C22semidirect
66. C2×M16×C67D4semidirect
193. C2×#568D67semidirect
199. C67#20990semidirect
206. C22×M16C2×D67semidirect
245. C2×M16C22⋊D67semidirect

Maximal subgroups

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

subgroupindex
C2×#168639602
C22×#145239592
#225686542
#225606722
#225599472
C2×#568×C672
#2099067