Finite Group Databasebeta

← C67×#34967 #30714218

Subgroups — C67×#34967 #30714218

Order 17152.

2164 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C67×#2522C214yesmaximal, complemented
2C67×#2516C214yesmaximal, complemented
3C67×#2464C214yesmaximal, complemented
4C67×#2457C214yesmaximal, complemented
5C67×#2402C211yesmaximal, characteristic, complemented
6C67×#2397C211yesmaximal, characteristic, complemented
7C67×#2392C212yesmaximal, complemented
8C4×#123×C67C214yesmaximal, complemented
9C4×#121×C67C214yesmaximal, complemented
10C67×#2298C211yesmaximal, characteristic, complemented
11C42×D4×C67C212yesmaximal, complemented
12C2×#123×C67C2214yes—
13C2×#121×C67C2214yes—
14C2×C4×D4×C67C2212yes—
15C2×C4×D4×C67C2214yes—
16C2×C4×D4×C67C2214yes—
17C2×C4×D4×C67C2214yes—
18C2×#117×C67C2211yescharacteristic
19C22×C42×C67C2214yes—
20#400×C67C2214yes—
21#399×C67C2214yes—
22#398×C67C2212yes—
23#398×C67C2214yes—
24#397×C67C2212yes—
25#396×C67C2214yes—
26#394×C67C2214yes—
27#394×C67C2212yes—
28#392×C67C2212yes—
29#392×C67C2214yes—
30#390×C67C2214yes—
31#389×C67C2214yes—
32#388×C67C2214yes—
33#388×C67C2214yes—
34#388×C67C2214yes—
35#388×C67C2214yes—
36#388×C67C2214yes—
37#387×C67C2214yes—
38#387×C67C2214yes—
39#387×C67C2214yes—
40#386×C67C2214yes—
41#386×C67C2214yes—
42#386×C67C2214yes—
43#385×C67C2214yes—
44#381×C67C2214yes—
45#380×C67C2214yes—
46C4×C4⋊C4×C67C2214yes—
47C4×C4⋊C4×C67C2212yes—
48C4×C4⋊C4×C67C2212yes—
49C4×C4⋊C4×C67C2214yes—
50C4×C22⋊C4×C67C2214yes—
51C4×C22⋊C4×C67C2214yes—
52C4×C22⋊C4×C67C2214yes—
53C4×C22⋊C4×C67C2212yes—
54C4×C22⋊C4×C67C2214yes—
55#376×C67C2211yescharacteristic
56C43×C67C2211yescharacteristic
57C22×D4×C67C2312yes—
58C22×D4×C67C2314yes—
59C22×D4×C67C2314yes—
60C23×C4×C67C2314yes—
61C23×C4×C67C2311yescharacteristic
62C23×C4×C67C2314yes—
63C23×C4×C67C2314yes—
64#123×C67—216no—
65#121×C67—216no—
66C4×D4×C67—216no—
67C4×D4×C67—28no—
68C4×D4×C67—48no—
69C4×D4×C67—48no—
70#117×C67—24no—
71C2×C4⋊C4×C67C2312yes—
72C2×C4⋊C4×C67C2314yes—
73C2×C4⋊C4×C67C2312yes—
74C2×C4⋊C4×C67C2314yes—
75C2×C4⋊C4×C67C2314yes—
76C2×C4⋊C4×C67C2314yes—
77C2×C4⋊C4×C67C2314yes—
78C2×C4⋊C4×C67C2314yes—
79C2×C22⋊C4×C67C2312yes—
80C2×C22⋊C4×C67C2314yes—
81C2×C22⋊C4×C67C2314yes—
82C2×C22⋊C4×C67C2314yes—
83C2×C22⋊C4×C67C2314yes—
84C2×C22⋊C4×C67C2314yes—
85C2×C22⋊C4×C67C2314yes—
86C2×C22⋊C4×C67C2314yes—
87C2×C22⋊C4×C67C2314yes—
88C2×C22⋊C4×C67C2314yes—
89C2×C22⋊C4×C67C2314yes—
90C2×C42×C67—48no—
91C2×C42×C67C2314yes—
92C2×C42×C67C2312yes—
93C2×C42×C67C2314yes—
94C2×C42×C67C2311yescharacteristic
95C2×C42×C67C2312yes—
96C2×C42×C67C2314yes—
97C2×C42×C67C2314yes—
98C2×C42×C67C2311yescharacteristic
99#95×C67C2314yes—
100#95×C67C2314yes—
101#95×C67C2314yes—
102#95×C67C2312yes—
103#95×C67C2312yes—
104#95×C67C2314yes—
105#95×C67C2314yes—
106#95×C67C2314yes—
107#95×C67C2314yes—
108#95×C67C2314yes—
109#95×C67C2314yes—
110C24×C67C2411yescharacteristic
111C24×C67C2414yes—
112C2×D4×C67—28no—
113C2×D4×C67—28no—
114C2×D4×C67—28no—
115C2×D4×C67—28no—
116C2×D4×C67—24no—
117C2×D4×C67—28no—
118C2×D4×C67—44no—
119C2×D4×C67—44no—
120C22×C4×C67—28no—
121C22×C4×C67—22no—
122C22×C4×C67—28no—
123C22×C4×C67—28no—
124C22×C4×C67—28no—
125C22×C4×C67—28no—
126C22×C4×C67—28no—
127C22×C4×C67—28no—
128C22×C4×C67—24no—
129C22×C4×C67—44no—
130C22×C4×C67—44no—
131C22×C4×C67C2414yes—
132C22×C4×C67C2414yes—
133C22×C4×C67C2412yes—
134C22×C4×C67C2414yes—
135C22×C4×C67C2411yescharacteristic
136C22×C4×C67C2414yes—
137C22×C4×C67C2412yes—
138C22×C4×C67C2414yes—
139C22×C4×C67C2411yescharacteristic
140C4⋊C4×C67—24no—
141C4⋊C4×C67—28no—
142C4⋊C4×C67—24no—
143C4⋊C4×C67—28no—
144C4⋊C4×C67—28no—
145C4⋊C4×C67—28no—
146C4⋊C4×C67—44no—
147C4⋊C4×C67—44no—
148C22⋊C4×C67—28no—
149C22⋊C4×C67—24no—
150C22⋊C4×C67—28no—
151C22⋊C4×C67—28no—
152C22⋊C4×C67—28no—
153C22⋊C4×C67—28no—
154C22⋊C4×C67—44no—
155C22⋊C4×C67—44no—
156C22⋊C4×C67—44no—
157C22⋊C4×C67—44no—
158C22⋊C4×C67#41116yes—
159C42×C67—22no—
160C42×C67—28no—
161C42×C67—24no—
162C42×C67—28no—
163C42×C67—24no—
164C42×C67—44no—
165C42×C67—44no—
166C42×C67#4114yes—
167C23×C67—24no—
168C23×C67—24no—
169C23×C67—24no—
170C23×C67—24no—
171C23×C67—24no—
172C23×C67—24no—
173C23×C67—21no—
174C23×C67—24no—
175C23×C67—22no—
176C23×C67C2×#4118yes—
177C23×C67C2511yescharacteristic
178D4×C67—216no—
179D4×C67—416no—
180D4×C67—416no—
181C2×C4×C67—24no—
182C2×C4×C67—24no—
183C2×C4×C67—24no—
184C2×C4×C67—24no—
185C2×C4×C67—24no—
186C2×C4×C67—22no—
187C2×C4×C67—24no—
188C2×C4×C67—24no—
189C2×C4×C67—24no—
190C2×C4×C67—24no—
191C2×C4×C67—24no—
192C2×C4×C67—21no—
193C2×C4×C67—24no—
194C2×C4×C67—24no—
195C2×C4×C67—24no—
196C2×C4×C67—24no—
197C2×C4×C67—24no—
198C2×C4×C67—22no—
199C2×C4×C67—416no—
200C2×C4×C67—416no—
201C2×C4×C67—416no—
202C2×C4×C67—44no—
203C2×C4×C67C2×#4118yes—
204C2×C4×C67C2×#4114yes—
205C2×C4×C67C2×#4118yes—
206C2×C4×C67C2×#4112yes—
207C2×C4×C67H(5,𝔽2)14yes—
208C22×C67—28no—
209C22×C67—28no—
210C22×C67—28no—
211C22×C67—44no—
212C22×C67—44no—
213C22×C67—44no—
214C22×C67—44no—
215C22×C67—42no—
216C22×C67—41no—
217C22×C67—44no—
218C22×C67C22×#4111yescharacteristic
219C22×C67C22×#4114yes—
220C22×C67C2×H(5,𝔽2)12yes—
221C4×C67—24no—
222C4×C67—28no—
223C4×C67—28no—
224C4×C67—42no—
225C4×C67—44no—
226C4×C67—41no—
227C4×C67—44no—
228C4×C67—44no—
229C4×C67#54014yes—
230#34967C6711yesmaximal, characteristic, direct product factor, Sylow
231C2×C67—48no—
232C2×C67—81no—
233C2×C67#344412yes—
234C2×C67C2×#54011yescharacteristic
235C2×C67#348114yes—
236#2522C2×C6714yescomplemented
237#2516C2×C6714yescomplemented
238#2464C2×C6714yescomplemented
239#2457C2×C6714yescomplemented
240#2402C2×C6711yescharacteristic, complemented
241#2397C2×C6711yescharacteristic, complemented
242#2392C2×C6712yescomplemented
243C4×#123C2×C6714yescomplemented
244C4×#121C2×C6714yescomplemented
245#2298C2×C6711yescharacteristic, complemented
246C42×D4C2×C6712yescomplemented
247C67#3496711yescharacteristic, direct product factor, Sylow
248C2×#123C22×C6714yes—
249C2×#121C22×C6714yes—
250C2×C4×D4C22×C6712yes—
251C2×C4×D4C22×C6714yes—
252C2×C4×D4C22×C6714yes—
253C2×C4×D4C22×C6714yes—
254C2×#117C22×C6711yescharacteristic
255C22×C42C22×C6714yes—
256#400C22×C6714yes—
257#399C22×C6714yes—
258#398C22×C6712yes—
259#398C22×C6714yes—
260#397C22×C6712yes—
261#396C22×C6714yes—
262#394C22×C6712yes—
263#394C22×C6714yes—
264#392C22×C6712yes—
265#392C22×C6714yes—
266#390C22×C6714yes—
267#389C22×C6714yes—
268#388C22×C6714yes—
269#388C22×C6714yes—
270#388C22×C6714yes—
271#388C22×C6714yes—
272#388C22×C6714yes—
273#387C22×C6714yes—
274#387C22×C6714yes—
275#387C22×C6714yes—
276#386C22×C6714yes—
277#386C22×C6714yes—
278#386C22×C6714yes—
279#385C22×C6714yes—
280#381C22×C6714yes—
281#380C22×C6714yes—
282C4×C4⋊C4C22×C6712yes—
283C4×C4⋊C4C22×C6714yes—
284C4×C4⋊C4C22×C6714yes—
285C4×C4⋊C4C22×C6712yes—
286C4×C22⋊C4C22×C6714yes—
287C4×C22⋊C4C22×C6714yes—
288C4×C22⋊C4C22×C6714yes—
289C4×C22⋊C4C22×C6714yes—
290C4×C22⋊C4C22×C6712yes—
291#376C22×C6711yescharacteristic
292C43C22×C6711yescharacteristic
293C22×D4C23×C6712yes—
294C22×D4C23×C6714yes—
295C22×D4C23×C6714yes—
296C23×C4C23×C6711yescharacteristic
297C23×C4C23×C6714yes—
298C23×C4C23×C6714yes—
299C23×C4C23×C6714yes—
300#123—216no—
301#121—216no—
302C4×D4—216no—
303C4×D4—28no—
304C4×D4—48no—
305C4×D4—48no—
306#117—24no—
307C2×C4⋊C4C23×C6712yes—
308C2×C4⋊C4C23×C6714yes—
309C2×C4⋊C4C23×C6712yes—
310C2×C4⋊C4C23×C6714yes—
311C2×C4⋊C4C23×C6714yes—
312C2×C4⋊C4C23×C6714yes—
313C2×C4⋊C4C23×C6714yes—
314C2×C4⋊C4C23×C6714yes—
315C2×C22⋊C4C23×C6712yes—
316C2×C22⋊C4C23×C6714yes—
317C2×C22⋊C4C23×C6714yes—
318C2×C22⋊C4C23×C6714yes—
319C2×C22⋊C4C23×C6714yes—
320C2×C22⋊C4C23×C6714yes—
321C2×C22⋊C4C23×C6714yes—
322C2×C22⋊C4C23×C6714yes—
323C2×C22⋊C4C23×C6714yes—
324C2×C22⋊C4C23×C6714yes—
325C2×C22⋊C4C23×C6714yes—
326C2×C42—48no—
327C2×C42C23×C6711yescharacteristic
328C2×C42C23×C6712yes—
329C2×C42C23×C6714yes—
330C2×C42C23×C6714yes—
331C2×C42C23×C6712yes—
332C2×C42C23×C6714yes—
333C2×C42C23×C6714yes—
334C2×C42C23×C6711yescharacteristic
335#95C23×C6714yes—
336#95C23×C6714yes—
337#95C23×C6714yes—
338#95C23×C6712yes—
339#95C23×C6712yes—
340#95C23×C6714yes—
341#95C23×C6714yes—
342#95C23×C6714yes—
343#95C23×C6714yes—
344#95C23×C6714yes—
345#95C23×C6714yes—
346C24C24×C6711yescharacteristic
347C24C24×C6714yes—
348C2×D4—28no—
349C2×D4—28no—
350C2×D4—28no—
351C2×D4—28no—
352C2×D4—24no—
353C2×D4—28no—
354C2×D4—44no—
355C2×D4—44no—
356C22×C4—28no—
357C22×C4—22no—
358C22×C4—28no—
359C22×C4—28no—
360C22×C4—28no—
361C22×C4—28no—
362C22×C4—28no—
363C22×C4—28no—
364C22×C4—24no—
365C22×C4—44no—
366C22×C4—44no—
367C22×C4C24×C6714yes—
368C22×C4C24×C6712yes—
369C22×C4C24×C6714yes—
370C22×C4C24×C6714yes—
371C22×C4C24×C6711yescharacteristic
372C22×C4C24×C6714yes—
373C22×C4C24×C6712yes—
374C22×C4C24×C6714yes—
375C22×C4C24×C6711yescharacteristic
376C4⋊C4—24no—
377C4⋊C4—28no—
378C4⋊C4—24no—
379C4⋊C4—28no—
380C4⋊C4—28no—
381C4⋊C4—28no—
382C4⋊C4—44no—
383C4⋊C4—44no—
384C22⋊C4—28no—
385C22⋊C4—24no—
386C22⋊C4—28no—
387C22⋊C4—28no—
388C22⋊C4—28no—
389C22⋊C4—28no—
390C22⋊C4—44no—
391C22⋊C4—44no—
392C22⋊C4—44no—
393C22⋊C4—44no—
394C22⋊C4#41×C67116yes—
395C42—22no—
396C42—28no—
397C42—24no—
398C42—28no—
399C42—24no—
400C42—44no—
401C42—44no—
402C42#41×C6714yes—
403C23—24no—
404C23—24no—
405C23—24no—
406C23—24no—
407C23—24no—
408C23—24no—
409C23—21no—
410C23—24no—
411C23—22no—
412C23C2×#41×C6718yes—
413C23C25×C6711yescharacteristic
414D4—216no—
415D4—416no—
416D4—416no—
417C2×C4—24no—
418C2×C4—24no—
419C2×C4—24no—
420C2×C4—24no—
421C2×C4—24no—
422C2×C4—22no—
423C2×C4—24no—
424C2×C4—24no—
425C2×C4—24no—
426C2×C4—24no—
427C2×C4—24no—
428C2×C4—21no—
429C2×C4—24no—
430C2×C4—24no—
431C2×C4—24no—
432C2×C4—24no—
433C2×C4—24no—
434C2×C4—22no—
435C2×C4—416no—
436C2×C4—416no—
437C2×C4—416no—
438C2×C4—44no—
439C2×C4C2×#41×C6718yes—
440C2×C4C2×#41×C6714yes—
441C2×C4C2×#41×C6718yes—
442C2×C4C2×#41×C6712yes—
443C2×C4H(5,𝔽2)×C6714yes—
444C22—28no—
445C22—28no—
446C22—28no—
447C22—44no—
448C22—44no—
449C22—44no—
450C22—44no—
451C22—42no—
452C22—41no—
453C22—44no—
454C22C22×#41×C6711yescharacteristic
455C22C22×#41×C6714yes—
456C22C2×H(5,𝔽2)×C6712yes—
457C4—24no—
458C4—28no—
459C4—28no—
460C4—42no—
461C4—44no—
462C4—41no—
463C4—44no—
464C4—44no—
465C4#540×C6714yes—
466C2—48no—
467C2—81no—
468C2C67×#344412yes—
469C2C2×#540×C6711yescharacteristic
470C2C67×#348114yes—
subgroupG/H
centerC22×C4×C67C24
commutator, Frattini, cosocleC23C25×C67
socleC23×C67C25
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C22×C4×C67 ◁ G

Complements

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

NH ≅ G/Ntype
1. C67×#2522C2semidirect
2. C67×#2516C2semidirect
3. C67×#2464C2semidirect
4. C67×#2457C2semidirect
5. C67×#2402C2semidirect
6. C67×#2397C2semidirect
7. C67×#2392C2semidirect
8. C4×#123×C67C2semidirect
9. C4×#121×C67C2semidirect
10. C67×#2298C2semidirect
11. C42×D4×C67C2semidirect
230. #34967C67direct (N × H)
236. #2522C2×C67semidirect
237. #2516C2×C67semidirect
238. #2464C2×C67semidirect
239. #2457C2×C67semidirect
240. #2402C2×C67semidirect
241. #2397C2×C67semidirect
242. #2392C2×C67semidirect
243. C4×#123C2×C67semidirect
244. C4×#121C2×C67semidirect
245. #2298C2×C67semidirect
246. C42×D4C2×C67semidirect
247. C67#34967direct (N × H)

Maximal subgroups

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

subgroupindex
C67×#25222
C67×#25162
C67×#24642
C67×#24572
C67×#24022
C67×#23972
C67×#23922
C4×#123×C672
C4×#121×C672
C67×#22982
C42×D4×C672
#3496767