Finite Group Databasebeta

← C67×#33904 #30713155

Subgroups — C67×#33904 #30713155

Order 17152.

482 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C67×#3339C211yesmaximal, characteristic
2C67×#3336C211yesmaximal, characteristic, complemented
3C67×#2253C211yesmaximal, characteristic, complemented
4C67×#2237C211yesmaximal, characteristic, complemented
5C67×#2236C211yesmaximal, characteristic, complemented
6C67×#2234C211yesmaximal, characteristic, complemented
7C67×#2232C211yesmaximal, characteristic, complemented
8C67×#2229C211yesmaximal, characteristic, complemented
9C67×#2228C211yesmaximal, characteristic, complemented
10C67×#2217C211yesmaximal, characteristic, complemented
11C67×#2217C211yesmaximal, characteristic, complemented
12C4×SD32×C67C211yesmaximal, characteristic, complemented
13D4×C16×C67C211yesmaximal, characteristic, complemented
14C67×#2158C211yesmaximal, characteristic, complemented
15C2×#367×C67C211yesmaximal, characteristic, complemented
16#553×C67—21no—
17#552×C67—21no—
18C2×SD32×C67C2211yescharacteristic
19C22×C16×C67C2211yescharacteristic
20C22×C16×C67C2211yescharacteristic
21#499×C67C2211yescharacteristic
22#499×C67C2211yescharacteristic
23#485×C67—21no—
24#483×C67—21no—
25#482×C67—21no—
26#481×C67—21no—
27#479×C67—21no—
28#478×C67—21no—
29#467×C67C2211yescharacteristic
30#467×C67C2211yescharacteristic
31#466×C67C2211yescharacteristic
32#466×C67C2211yescharacteristic
33C4×Q16×C67C2211yescharacteristic
34C4×D8×C67C2211yescharacteristic
35C8×D4×C67C2211yescharacteristic
36#427×C67C2211yescharacteristic
37#427×C67C2211yescharacteristic
38C2×#107×C67C2211yescharacteristic
39C2×#107×C67C2211yescharacteristic
40#367×C67C2211yescharacteristic
41#367×C67C2211yescharacteristic
42#367×C67C2211yescharacteristic
43#367×C67C2211yescharacteristic
44#367×C67C2211yescharacteristic
45#366×C67C2211yescharacteristic
46#366×C67C2211yescharacteristic
47#366×C67C2211yescharacteristic
48#366×C67C2211yescharacteristic
49C4⋊C16×C67C2211yescharacteristic
50#358×C67C2211yescharacteristic
51#358×C67C2211yescharacteristic
52#358×C67C2211yescharacteristic
53#357×C67C2211yescharacteristic
54#357×C67C2211yescharacteristic
55#357×C67C2211yescharacteristic
56C22⋊C16×C67C2211yescharacteristic
57C22⋊C16×C67C2211yescharacteristic
58C4×C16×C67C2211yescharacteristic
59C2×Q16×C67C2311yescharacteristic
60C2×D8×C67C2311yescharacteristic
61C22×C8×C67C2311yescharacteristic
62C22×C8×C67C2311yescharacteristic
63#126×C67—41no—
64#126×C67—41no—
65#125×C67—21no—
66#125×C67—21no—
67#125×C67—41no—
68#123×C67—41no—
69#123×C67—41no—
70C22⋊Q8×C67—21no—
71C22⋊Q8×C67—21no—
72#121×C67—21no—
73#121×C67—21no—
74#121×C67—41no—
75C4×Q8×C67—21no—
76C4×D4×C67—21no—
77C4×D4×C67—41no—
78C4×D4×C67D411yescharacteristic
79#117×C67—21no—
80#117×C67—21no—
81#117×C67—41no—
82C2×C4⋊C4×C67—21no—
83C2×C4⋊C4×C67—21no—
84SD32×C67—22no—
85C2×C16×C67—22no—
86C2×C16×C67—22no—
87C2×C16×C67C2311yescharacteristic
88C2×C16×C67C2311yescharacteristic
89C2×C16×C67C2311yescharacteristic
90C2×C16×C67C2311yescharacteristic
91#107×C67—21no—
92#107×C67—21no—
93#107×C67C2311yescharacteristic
94#107×C67C2311yescharacteristic
95#107×C67C2311yescharacteristic
96#107×C67C2311yescharacteristic
97#107×C67C2311yescharacteristic
98#107×C67C2311yescharacteristic
99#106×C67—21no—
100#106×C67—21no—
101#106×C67—21no—
102#106×C67—21no—
103C4⋊C8×C67D411yescharacteristic
104#103×C67—21no—
105#103×C67—21no—
106#103×C67—21no—
107#102×C67—21no—
108#102×C67—21no—
109#102×C67—21no—
110C22⋊C8×C67D411yescharacteristic
111C22⋊C8×C67D411yescharacteristic
112C4×C8×C67C2311yescharacteristic
113C2×Q8×C67—21no—
114C2×D4×C67—21no—
115C2×D4×C67—41no—
116C2×D4×C67—41no—
117C2×D4×C67D811yescharacteristic
118C22×C4×C67—41no—
119C22×C4×C67—41no—
120C22×C4×C67—41no—
121C22×C4×C67C2×D411yescharacteristic
122C22×C4×C67C2×D411yescharacteristic
123Q16×C67—21no—
124Q16×C67—21no—
125D8×C67—21no—
126D8×C67—21no—
127C2×C8×C67—21no—
128C2×C8×C67—21no—
129C2×C8×C67—21no—
130C2×C8×C67—21no—
131C2×C8×C67C2×D411yescharacteristic
132C2×C8×C67C2×D411yescharacteristic
133C2×C8×C67C2×D411yescharacteristic
134C2×C8×C67C2411yescharacteristic
135C4⋊C4×C67—21no—
136C4⋊C4×C67—21no—
137C4⋊C4×C67—21no—
138C4⋊C4×C67—21no—
139C4⋊C4×C67—21no—
140C4⋊C4×C67—21no—
141C4⋊C4×C67—41no—
142C4⋊C4×C67—41no—
143C4⋊C4×C67—41no—
144C4⋊C4×C67—41no—
145C4⋊C4×C67—41no—
146C4⋊C4×C67—41no—
147C4⋊C4×C67—41no—
148C4⋊C4×C67—41no—
149C4⋊C4×C67—41no—
150C4⋊C4×C67D811yescharacteristic
151C22⋊C4×C67—41no—
152C22⋊C4×C67—41no—
153C22⋊C4×C67—41no—
154C22⋊C4×C67—41no—
155C22⋊C4×C67—41no—
156C22⋊C4×C67—41no—
157C22⋊C4×C67—41no—
158C22⋊C4×C67D811yescharacteristic
159C22⋊C4×C67D811yescharacteristic
160C42×C67—41no—
161C42×C67—41no—
162C42×C67—41no—
163C42×C67C2×D411yescharacteristic
164C16×C67—21no—
165C16×C67—21no—
166C16×C67—21no—
167C16×C67#4112yes—
168C23×C67—41no—
169C23×C67C2×D811yescharacteristic
170C23×C67C2×D811yescharacteristic
171Q8×C67—41no—
172Q8×C67—41no—
173D4×C67—22no—
174D4×C67—41no—
175D4×C67—41no—
176D4×C67—82no—
177D4×C67—161no—
178C2×C4×C67—21no—
179C2×C4×C67—21no—
180C2×C4×C67—21no—
181C2×C4×C67—21no—
182C2×C4×C67—41no—
183C2×C4×C67—41no—
184C2×C4×C67—41no—
185C2×C4×C67—41no—
186C2×C4×C67—41no—
187C2×C4×C67—41no—
188C2×C4×C67—41no—
189C2×C4×C67—82no—
190C2×C4×C67—161no—
191C2×C4×C67—161no—
192C2×C4×C67C2×D811yescharacteristic
193C2×C4×C67C2×D811yescharacteristic
194C2×C4×C67C2×D811yescharacteristic
195C2×C4×C67C2×D811yescharacteristic
196C2×C4×C67C22×D411yescharacteristic
197C8×C67—21no—
198C8×C67—21no—
199C8×C67—21no—
200C8×C67C2×#4111yescharacteristic
201C8×C672−1+411yescharacteristic
202C22×C67—21no—
203C22×C67—21no—
204C22×C67—21no—
205C22×C67—21no—
206C22×C67—21no—
207C22×C67—81no—
208C22×C67—81no—
209C22×C67—81no—
210C22×C67#50812yes—
211C22×C67C22×D811yescharacteristic
212C4×C67—21no—
213C4×C67—21no—
214C4×C67—21no—
215C4×C67—21no—
216C4×C67—81no—
217C4×C67—81no—
218C4×C67—81no—
219C4×C67—81no—
220C4×C67—81no—
221C4×C67—81no—
222C4×C67—81no—
223C4×C67#54711yescharacteristic
224C4×C67#57811yescharacteristic
225#33904C6711yesmaximal, characteristic, direct product factor, Sylow
226C2×C67—22no—
227C2×C67—41no—
228C2×C67—161no—
229C2×C67#333411yescharacteristic
230C2×C67C2×#50811yescharacteristic
231C2×C67#341611yescharacteristic
232#3339C2×C6711yescharacteristic
233#3336C2×C6711yescharacteristic, complemented
234#2253C2×C6711yescharacteristic, complemented
235#2237C2×C6711yescharacteristic, complemented
236#2236C2×C6711yescharacteristic, complemented
237#2234C2×C6711yescharacteristic, complemented
238#2232C2×C6711yescharacteristic, complemented
239#2229C2×C6711yescharacteristic, complemented
240#2228C2×C6711yescharacteristic, complemented
241#2217C2×C6711yescharacteristic, complemented
242#2217C2×C6711yescharacteristic, complemented
243C4×SD32C2×C6711yescharacteristic, complemented
244D4×C16C2×C6711yescharacteristic, complemented
245#2158C2×C6711yescharacteristic, complemented
246C2×#367C2×C6711yescharacteristic, complemented
247C67#3390411yescharacteristic, direct product factor, Sylow
248#553—21no—
249#552—21no—
250C2×SD32C22×C6711yescharacteristic
251C22×C16C22×C6711yescharacteristic
252C22×C16C22×C6711yescharacteristic
253#499C22×C6711yescharacteristic
254#499C22×C6711yescharacteristic
255#485—21no—
256#483—21no—
257#482—21no—
258#481—21no—
259#479—21no—
260#478—21no—
261#467C22×C6711yescharacteristic
262#467C22×C6711yescharacteristic
263#466C22×C6711yescharacteristic
264#466C22×C6711yescharacteristic
265C4×Q16C22×C6711yescharacteristic
266C4×D8C22×C6711yescharacteristic
267C8×D4C22×C6711yescharacteristic
268#427C22×C6711yescharacteristic
269#427C22×C6711yescharacteristic
270C2×#107C22×C6711yescharacteristic
271C2×#107C22×C6711yescharacteristic
272#367C22×C6711yescharacteristic
273#367C22×C6711yescharacteristic
274#367C22×C6711yescharacteristic
275#367C22×C6711yescharacteristic
276#367C22×C6711yescharacteristic
277#366C22×C6711yescharacteristic
278#366C22×C6711yescharacteristic
279#366C22×C6711yescharacteristic
280#366C22×C6711yescharacteristic
281C4⋊C16C22×C6711yescharacteristic
282#358C22×C6711yescharacteristic
283#358C22×C6711yescharacteristic
284#358C22×C6711yescharacteristic
285#357C22×C6711yescharacteristic
286#357C22×C6711yescharacteristic
287#357C22×C6711yescharacteristic
288C22⋊C16C22×C6711yescharacteristic
289C22⋊C16C22×C6711yescharacteristic
290C4×C16C22×C6711yescharacteristic
291C2×Q16C23×C6711yescharacteristic
292C2×D8C23×C6711yescharacteristic
293C22×C8C23×C6711yescharacteristic
294C22×C8C23×C6711yescharacteristic
295#126—41no—
296#126—41no—
297#125—21no—
298#125—21no—
299#125—41no—
300#123—41no—
301#123—41no—
302C22⋊Q8—21no—
303C22⋊Q8—21no—
304#121—21no—
305#121—21no—
306#121—41no—
307C4×Q8—21no—
308C4×D4—21no—
309C4×D4—41no—
310C4×D4D4×C6711yescharacteristic
311#117—21no—
312#117—21no—
313#117—41no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316SD32—22no—
317C2×C16—22no—
318C2×C16—22no—
319C2×C16C23×C6711yescharacteristic
320C2×C16C23×C6711yescharacteristic
321C2×C16C23×C6711yescharacteristic
322C2×C16C23×C6711yescharacteristic
323#107—21no—
324#107—21no—
325#107C23×C6711yescharacteristic
326#107C23×C6711yescharacteristic
327#107C23×C6711yescharacteristic
328#107C23×C6711yescharacteristic
329#107C23×C6711yescharacteristic
330#107C23×C6711yescharacteristic
331#106—21no—
332#106—21no—
333#106—21no—
334#106—21no—
335C4⋊C8D4×C6711yescharacteristic
336#103—21no—
337#103—21no—
338#103—21no—
339#102—21no—
340#102—21no—
341#102—21no—
342C22⋊C8D4×C6711yescharacteristic
343C22⋊C8D4×C6711yescharacteristic
344C4×C8C23×C6711yescharacteristic
345C2×Q8—21no—
346C2×D4—21no—
347C2×D4—41no—
348C2×D4—41no—
349C2×D4D8×C6711yescharacteristic
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4C2×D4×C6711yescharacteristic
354C22×C4C2×D4×C6711yescharacteristic
355Q16—21no—
356Q16—21no—
357D8—21no—
358D8—21no—
359C2×C8—21no—
360C2×C8—21no—
361C2×C8—21no—
362C2×C8—21no—
363C2×C8C2×D4×C6711yescharacteristic
364C2×C8C2×D4×C6711yescharacteristic
365C2×C8C2×D4×C6711yescharacteristic
366C2×C8C24×C6711yescharacteristic
367C4⋊C4—21no—
368C4⋊C4—21no—
369C4⋊C4—21no—
370C4⋊C4—21no—
371C4⋊C4—21no—
372C4⋊C4—21no—
373C4⋊C4—41no—
374C4⋊C4—41no—
375C4⋊C4—41no—
376C4⋊C4—41no—
377C4⋊C4—41no—
378C4⋊C4—41no—
379C4⋊C4—41no—
380C4⋊C4—41no—
381C4⋊C4—41no—
382C4⋊C4D8×C6711yescharacteristic
383C22⋊C4—41no—
384C22⋊C4—41no—
385C22⋊C4—41no—
386C22⋊C4—41no—
387C22⋊C4—41no—
388C22⋊C4—41no—
389C22⋊C4—41no—
390C22⋊C4D8×C6711yescharacteristic
391C22⋊C4D8×C6711yescharacteristic
392C42—41no—
393C42—41no—
394C42—41no—
395C42C2×D4×C6711yescharacteristic
396C16—21no—
397C16—21no—
398C16—21no—
399C16#41×C6712yes—
400C23—41no—
401C23C2×D8×C6711yescharacteristic
402C23C2×D8×C6711yescharacteristic
403Q8—41no—
404Q8—41no—
405D4—22no—
406D4—41no—
407D4—41no—
408D4—82no—
409D4—161no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—82no—
422C2×C4—161no—
423C2×C4—161no—
424C2×C4C2×D8×C6711yescharacteristic
425C2×C4C2×D8×C6711yescharacteristic
426C2×C4C2×D8×C6711yescharacteristic
427C2×C4C2×D8×C6711yescharacteristic
428C2×C4C22×D4×C6711yescharacteristic
429C8—21no—
430C8—21no—
431C8—21no—
432C8C2×#41×C6711yescharacteristic
433C82−1+4×C6711yescharacteristic
434C22—21no—
435C22—21no—
436C22—21no—
437C22—21no—
438C22—21no—
439C22—81no—
440C22—81no—
441C22—81no—
442C22#508×C6712yes—
443C22C22×D8×C6711yescharacteristic
444C4—21no—
445C4—21no—
446C4—21no—
447C4—21no—
448C4—81no—
449C4—81no—
450C4—81no—
451C4—81no—
452C4—81no—
453C4—81no—
454C4—81no—
455C4#547×C6711yescharacteristic
456C4#578×C6711yescharacteristic
457C2—22no—
458C2—41no—
459C2—161no—
460C2C67×#333411yescharacteristic
461C2C2×#508×C6711yescharacteristic
462C2C67×#341611yescharacteristic
subgroupG/H
center, socleC22×C67C22×D8
commutator, Frattini, cosocleC2×C8C24×C67
prime radicalC67×#3339C2
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C67 ◁ C4×D4×C67 ◁ C8×D4×C67 ◁ G

Complements

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

NH ≅ G/Ntype
2. C67×#3336C2semidirect
3. C67×#2253C2semidirect
4. C67×#2237C2semidirect
5. C67×#2236C2semidirect
6. C67×#2234C2semidirect
7. C67×#2232C2semidirect
8. C67×#2229C2semidirect
9. C67×#2228C2semidirect
10. C67×#2217C2semidirect
11. C67×#2217C2semidirect
12. C4×SD32×C67C2semidirect
13. D4×C16×C67C2semidirect
14. C67×#2158C2semidirect
15. C2×#367×C67C2semidirect
225. #33904C67direct (N × H)
233. #3336C2×C67semidirect
234. #2253C2×C67semidirect
235. #2237C2×C67semidirect
236. #2236C2×C67semidirect
237. #2234C2×C67semidirect
238. #2232C2×C67semidirect
239. #2229C2×C67semidirect
240. #2228C2×C67semidirect
241. #2217C2×C67semidirect
242. #2217C2×C67semidirect
243. C4×SD32C2×C67semidirect
244. D4×C16C2×C67semidirect
245. #2158C2×C67semidirect
246. C2×#367C2×C67semidirect
247. C67#33904direct (N × H)

Maximal subgroups

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

subgroupindex
C67×#33392
C67×#33362
C67×#22532
C67×#22372
C67×#22362
C67×#22342
C67×#22322
C67×#22292
C67×#22282
C67×#22172
C4×SD32×C672
D4×C16×C672
C67×#21582
C2×#367×C672
#3390467