Finite Group Databasebeta

← C67×#33681 #30712932

Subgroups — C67×#33681 #30712932

Order 17152.

532 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C67×#3175C211yesmaximal, characteristic
2C67×#2950C211yesmaximal, characteristic, complemented
3C67×#2237C211yesmaximal, characteristic, complemented
4C67×#2234C211yesmaximal, characteristic
5C67×#2227C211yesmaximal, characteristic, complemented
6C67×#2225C211yesmaximal, characteristic, complemented
7C67×#2211C211yesmaximal, characteristic, complemented
8C67×#2211C211yesmaximal, characteristic, complemented
9C67×#2188C211yesmaximal, characteristic, complemented
10C67×#2185C211yesmaximal, characteristic
11C67×#2176C211yesmaximal, characteristic, complemented
12C67×#2175C211yesmaximal, characteristic, complemented
13C4×Q32×C67C211yesmaximal, characteristic, complemented
14C4×SD32×C67C211yesmaximal, characteristic, complemented
15C67×#2164C211yesmaximal, characteristic, complemented
16C22×Q16×C67C2211yescharacteristic
17#542×C67—21no—
18#519×C67—21no—
19C2×Q32×C67C2211yescharacteristic
20C2×Q32×C67C2211yescharacteristic
21C2×SD32×C67C2211yescharacteristic
22C2×SD32×C67C2211yescharacteristic
23#500×C67C2211yescharacteristic
24#495×C67C2211yescharacteristic
25#489×C67—21no—
26#468×C67—21no—
27#467×C67C2211yescharacteristic
28#466×C67C2211yescharacteristic
29#441×C67—21no—
30#441×C67—21no—
31#440×C67—21no—
32C4×Q16×C67C2211yescharacteristic
33C4×Q16×C67C2211yescharacteristic
34C4×Q16×C67C2211yescharacteristic
35C4×Q16×C67C2211yescharacteristic
36C4×Q16×C67C2211yescharacteristic
37C4×D8×C67C2211yescharacteristic
38#427×C67C2211yescharacteristic
39#419×C67—21no—
40#405×C67C2211yescharacteristic
41#367×C67C2211yescharacteristic
42#367×C67C2211yescharacteristic
43#366×C67C2211yescharacteristic
44#366×C67C2211yescharacteristic
45C4⋊C16×C67C2211yescharacteristic
46C4⋊C16×C67C2211yescharacteristic
47#358×C67C2211yescharacteristic
48#358×C67C2211yescharacteristic
49#358×C67C2211yescharacteristic
50#358×C67C2211yescharacteristic
51#358×C67C2211yescharacteristic
52#358×C67C2211yescharacteristic
53#357×C67C2211yescharacteristic
54#357×C67C2211yescharacteristic
55C22⋊C16×C67C2211yescharacteristic
56C22⋊C16×C67C2211yescharacteristic
57#346×C67C2211yescharacteristic
58C4×C16×C67C2211yescharacteristic
59C22×Q8×C67—21no—
60C2×Q16×C67—22no—
61C2×Q16×C67—22no—
62C2×Q16×C67—21no—
63C2×Q16×C67C2311yescharacteristic
64C2×Q16×C67C2311yescharacteristic
65C2×Q16×C67C2311yescharacteristic
66C2×SD16×C67—21no—
67C2×D8×C67C2311yescharacteristic
68C2×M16×C67D411yescharacteristic
69C22×C8×C67C2311yescharacteristic
70C4⋊Q8×C67—21no—
71#126×C67—41no—
72#124×C67—21no—
73#124×C67—41no—
74#123×C67—41no—
75C22⋊Q8×C67—21no—
76C22⋊Q8×C67—41no—
77#121×C67—21no—
78C4×Q8×C67—21no—
79C4×Q8×C67—21no—
80C4×Q8×C67—21no—
81C4×Q8×C67—21no—
82C4×Q8×C67—21no—
83C4×Q8×C67—41no—
84C4×Q8×C67—41no—
85C4×D4×C67—21no—
86#117×C67—21no—
87#117×C67—41no—
88#117×C67—41no—
89#117×C67D411yescharacteristic
90Q32×C67—22no—
91Q32×C67—22no—
92SD32×C67—22no—
93SD32×C67—22no—
94C2×C16×C67C2311yescharacteristic
95C2×C16×C67C2311yescharacteristic
96C2×C16×C67C2311yescharacteristic
97C2×C16×C67C2311yescharacteristic
98#107×C67—21no—
99#107×C67C2311yescharacteristic
100#107×C67C2311yescharacteristic
101#107×C67C2311yescharacteristic
102#107×C67C2311yescharacteristic
103#106×C67—21no—
104#103×C67—21no—
105#103×C67—21no—
106#103×C67—21no—
107#103×C67—21no—
108#103×C67—21no—
109#103×C67—21no—
110#102×C67—21no—
111#102×C67—21no—
112#97×C67D411yescharacteristic
113#97×C67D411yescharacteristic
114C4×C8×C67C2311yescharacteristic
115C4×C8×C67C2311yescharacteristic
116C2×Q8×C67—24no—
117C2×Q8×C67—21no—
118C2×Q8×C67—21no—
119C2×Q8×C67—21no—
120C2×Q8×C67—42no—
121C2×Q8×C67—41no—
122C2×Q8×C67—41no—
123C2×D4×C67—21no—
124C2×D4×C67—41no—
125C22×C4×C67—41no—
126C22×C4×C67—41no—
127C22×C4×C67C2×D411yescharacteristic
128Q16×C67—24no—
129Q16×C67—21no—
130Q16×C67—21no—
131Q16×C67—21no—
132Q16×C67—21no—
133Q16×C67#4112yes—
134Q16×C67#4112yes—
135SD16×C67—42no—
136D8×C67—21no—
137D8×C67—21no—
138M16×C67—41no—
139C2×C8×C67—21no—
140C2×C8×C67—21no—
141C2×C8×C67C2×D411yescharacteristic
142C2×C8×C67C2×D411yescharacteristic
143C2×C8×C67C2×D411yescharacteristic
144C2×C8×C67C2411yescharacteristic
145C4⋊C4×C67—21no—
146C4⋊C4×C67—21no—
147C4⋊C4×C67—21no—
148C4⋊C4×C67—21no—
149C4⋊C4×C67—41no—
150C4⋊C4×C67—41no—
151C4⋊C4×C67—41no—
152C4⋊C4×C67—41no—
153C4⋊C4×C67—41no—
154C4⋊C4×C67—41no—
155C4⋊C4×C67—41no—
156C4⋊C4×C67D811yescharacteristic
157C4⋊C4×C67D811yescharacteristic
158C22⋊C4×C67—41no—
159C22⋊C4×C67—41no—
160C22⋊C4×C67—41no—
161C22⋊C4×C67—41no—
162C22⋊C4×C67—41no—
163C22⋊C4×C67—41no—
164C22⋊C4×C67D811yescharacteristic
165C22⋊C4×C67D811yescharacteristic
166C42×C67—41no—
167C42×C67—41no—
168C42×C67—41no—
169C42×C67—41no—
170C42×C67—41no—
171C42×C67—41no—
172C42×C67C2×D411yescharacteristic
173C42×C67C2×D411yescharacteristic
174C16×C67—21no—
175C16×C67—21no—
176C16×C67—21no—
177C16×C67—21no—
178C23×C67—41no—
179C23×C67C2×D811yescharacteristic
180Q8×C67—22no—
181Q8×C67—22no—
182Q8×C67—44no—
183Q8×C67—41no—
184Q8×C67—41no—
185Q8×C67—41no—
186Q8×C67—41no—
187Q8×C67—161no—
188D4×C67—41no—
189D4×C67—41no—
190D4×C67—161no—
191C2×C4×C67—21no—
192C2×C4×C67—21no—
193C2×C4×C67—44no—
194C2×C4×C67—41no—
195C2×C4×C67—41no—
196C2×C4×C67—41no—
197C2×C4×C67—41no—
198C2×C4×C67—41no—
199C2×C4×C67—41no—
200C2×C4×C67—41no—
201C2×C4×C67—161no—
202C2×C4×C67C2×D811yescharacteristic
203C2×C4×C67C2×D811yescharacteristic
204C2×C4×C67C2×D811yescharacteristic
205C2×C4×C67C2×D811yescharacteristic
206C2×C4×C67C2×D811yescharacteristic
207C2×C4×C67C22×D411yescharacteristic
208C8×C67—21no—
209C8×C67—21no—
210C8×C67—21no—
211C8×C67C2×#4111yescharacteristic
212C8×C67C2×#4111yescharacteristic
213C22×C67—21no—
214C22×C67—21no—
215C22×C67—21no—
216C22×C67—81no—
217C22×C67—81no—
218C22×C67—81no—
219C22×C67C22×D811yescharacteristic
220C4×C67—21no—
221C4×C67—21no—
222C4×C67—21no—
223C4×C67—21no—
224C4×C67—21no—
225C4×C67—42no—
226C4×C67—42no—
227C4×C67—81no—
228C4×C67—81no—
229C4×C67—81no—
230C4×C67—81no—
231C4×C67—81no—
232C4×C67#52511yescharacteristic
233C4×C67#57511yescharacteristic
234#33681C6711yesmaximal, characteristic, direct product factor, Sylow
235C2×C67—41no—
236C2×C67—161no—
237C2×C67#310011yescharacteristic
238C2×C67#341611yescharacteristic
239C2×C67#341711yescharacteristic
240#3175C2×C6711yescharacteristic
241#2950C2×C6711yescharacteristic, complemented
242#2237C2×C6711yescharacteristic, complemented
243#2234C2×C6711yescharacteristic
244#2227C2×C6711yescharacteristic, complemented
245#2225C2×C6711yescharacteristic, complemented
246#2211C2×C6711yescharacteristic, complemented
247#2211C2×C6711yescharacteristic, complemented
248#2188C2×C6711yescharacteristic, complemented
249#2185C2×C6711yescharacteristic
250#2176C2×C6711yescharacteristic, complemented
251#2175C2×C6711yescharacteristic, complemented
252C4×Q32C2×C6711yescharacteristic, complemented
253C4×SD32C2×C6711yescharacteristic, complemented
254#2164C2×C6711yescharacteristic, complemented
255C67#3368111yescharacteristic, direct product factor, Sylow
256C22×Q16C22×C6711yescharacteristic
257#542—21no—
258#519—21no—
259C2×Q32C22×C6711yescharacteristic
260C2×Q32C22×C6711yescharacteristic
261C2×SD32C22×C6711yescharacteristic
262C2×SD32C22×C6711yescharacteristic
263#500C22×C6711yescharacteristic
264#495C22×C6711yescharacteristic
265#489—21no—
266#468—21no—
267#467C22×C6711yescharacteristic
268#466C22×C6711yescharacteristic
269#441—21no—
270#441—21no—
271#440—21no—
272C4×Q16C22×C6711yescharacteristic
273C4×Q16C22×C6711yescharacteristic
274C4×Q16C22×C6711yescharacteristic
275C4×Q16C22×C6711yescharacteristic
276C4×Q16C22×C6711yescharacteristic
277C4×D8C22×C6711yescharacteristic
278#427C22×C6711yescharacteristic
279#419—21no—
280#405C22×C6711yescharacteristic
281#367C22×C6711yescharacteristic
282#367C22×C6711yescharacteristic
283#366C22×C6711yescharacteristic
284#366C22×C6711yescharacteristic
285C4⋊C16C22×C6711yescharacteristic
286C4⋊C16C22×C6711yescharacteristic
287#358C22×C6711yescharacteristic
288#358C22×C6711yescharacteristic
289#358C22×C6711yescharacteristic
290#358C22×C6711yescharacteristic
291#358C22×C6711yescharacteristic
292#358C22×C6711yescharacteristic
293#357C22×C6711yescharacteristic
294#357C22×C6711yescharacteristic
295C22⋊C16C22×C6711yescharacteristic
296C22⋊C16C22×C6711yescharacteristic
297#346C22×C6711yescharacteristic
298C4×C16C22×C6711yescharacteristic
299C22×Q8—21no—
300C2×Q16—22no—
301C2×Q16—22no—
302C2×Q16—21no—
303C2×Q16C23×C6711yescharacteristic
304C2×Q16C23×C6711yescharacteristic
305C2×Q16C23×C6711yescharacteristic
306C2×SD16—21no—
307C2×D8C23×C6711yescharacteristic
308C2×M16D4×C6711yescharacteristic
309C22×C8C23×C6711yescharacteristic
310C4⋊Q8—21no—
311#126—41no—
312#124—21no—
313#124—41no—
314#123—41no—
315C22⋊Q8—21no—
316C22⋊Q8—41no—
317#121—21no—
318C4×Q8—21no—
319C4×Q8—21no—
320C4×Q8—21no—
321C4×Q8—21no—
322C4×Q8—21no—
323C4×Q8—41no—
324C4×Q8—41no—
325C4×D4—21no—
326#117—21no—
327#117—41no—
328#117—41no—
329#117D4×C6711yescharacteristic
330Q32—22no—
331Q32—22no—
332SD32—22no—
333SD32—22no—
334C2×C16C23×C6711yescharacteristic
335C2×C16C23×C6711yescharacteristic
336C2×C16C23×C6711yescharacteristic
337C2×C16C23×C6711yescharacteristic
338#107—21no—
339#107C23×C6711yescharacteristic
340#107C23×C6711yescharacteristic
341#107C23×C6711yescharacteristic
342#107C23×C6711yescharacteristic
343#106—21no—
344#103—21no—
345#103—21no—
346#103—21no—
347#103—21no—
348#103—21no—
349#103—21no—
350#102—21no—
351#102—21no—
352#97D4×C6711yescharacteristic
353#97D4×C6711yescharacteristic
354C4×C8C23×C6711yescharacteristic
355C4×C8C23×C6711yescharacteristic
356C2×Q8—24no—
357C2×Q8—21no—
358C2×Q8—21no—
359C2×Q8—21no—
360C2×Q8—42no—
361C2×Q8—41no—
362C2×Q8—41no—
363C2×D4—21no—
364C2×D4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4C2×D4×C6711yescharacteristic
368Q16—24no—
369Q16—21no—
370Q16—21no—
371Q16—21no—
372Q16—21no—
373Q16#41×C6712yes—
374Q16#41×C6712yes—
375SD16—42no—
376D8—21no—
377D8—21no—
378M16—41no—
379C2×C8—21no—
380C2×C8—21no—
381C2×C8C2×D4×C6711yescharacteristic
382C2×C8C2×D4×C6711yescharacteristic
383C2×C8C2×D4×C6711yescharacteristic
384C2×C8C24×C6711yescharacteristic
385C4⋊C4—21no—
386C4⋊C4—21no—
387C4⋊C4—21no—
388C4⋊C4—21no—
389C4⋊C4—41no—
390C4⋊C4—41no—
391C4⋊C4—41no—
392C4⋊C4—41no—
393C4⋊C4—41no—
394C4⋊C4—41no—
395C4⋊C4—41no—
396C4⋊C4D8×C6711yescharacteristic
397C4⋊C4D8×C6711yescharacteristic
398C22⋊C4—41no—
399C22⋊C4—41no—
400C22⋊C4—41no—
401C22⋊C4—41no—
402C22⋊C4—41no—
403C22⋊C4—41no—
404C22⋊C4D8×C6711yescharacteristic
405C22⋊C4D8×C6711yescharacteristic
406C42—41no—
407C42—41no—
408C42—41no—
409C42—41no—
410C42—41no—
411C42—41no—
412C42C2×D4×C6711yescharacteristic
413C42C2×D4×C6711yescharacteristic
414C16—21no—
415C16—21no—
416C16—21no—
417C16—21no—
418C23—41no—
419C23C2×D8×C6711yescharacteristic
420Q8—22no—
421Q8—22no—
422Q8—44no—
423Q8—41no—
424Q8—41no—
425Q8—41no—
426Q8—41no—
427Q8—161no—
428D4—41no—
429D4—41no—
430D4—161no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—44no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—161no—
442C2×C4C2×D8×C6711yescharacteristic
443C2×C4C2×D8×C6711yescharacteristic
444C2×C4C2×D8×C6711yescharacteristic
445C2×C4C2×D8×C6711yescharacteristic
446C2×C4C2×D8×C6711yescharacteristic
447C2×C4C22×D4×C6711yescharacteristic
448C8—21no—
449C8—21no—
450C8—21no—
451C8C2×#41×C6711yescharacteristic
452C8C2×#41×C6711yescharacteristic
453C22—21no—
454C22—21no—
455C22—21no—
456C22—81no—
457C22—81no—
458C22—81no—
459C22C22×D8×C6711yescharacteristic
460C4—21no—
461C4—21no—
462C4—21no—
463C4—21no—
464C4—21no—
465C4—42no—
466C4—42no—
467C4—81no—
468C4—81no—
469C4—81no—
470C4—81no—
471C4—81no—
472C4#525×C6711yescharacteristic
473C4#575×C6711yescharacteristic
474C2—41no—
475C2—161no—
476C2C67×#310011yescharacteristic
477C2C67×#341611yescharacteristic
478C2C67×#341711yescharacteristic
subgroupG/H
center, socleC22×C67C22×D8
commutator, Frattini, cosocleC2×C8C24×C67
prime radical#466×C67C22
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C67 ◁ #117×C67 ◁ #405×C67 ◁ G

Complements

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

NH ≅ G/Ntype
2. C67×#2950C2semidirect
3. C67×#2237C2semidirect
5. C67×#2227C2semidirect
6. C67×#2225C2semidirect
7. C67×#2211C2semidirect
8. C67×#2211C2semidirect
9. C67×#2188C2semidirect
11. C67×#2176C2semidirect
12. C67×#2175C2semidirect
13. C4×Q32×C67C2semidirect
14. C4×SD32×C67C2semidirect
15. C67×#2164C2semidirect
234. #33681C67direct (N × H)
241. #2950C2×C67semidirect
242. #2237C2×C67semidirect
244. #2227C2×C67semidirect
245. #2225C2×C67semidirect
246. #2211C2×C67semidirect
247. #2211C2×C67semidirect
248. #2188C2×C67semidirect
250. #2176C2×C67semidirect
251. #2175C2×C67semidirect
252. C4×Q32C2×C67semidirect
253. C4×SD32C2×C67semidirect
254. #2164C2×C67semidirect
255. C67#33681direct (N × H)

Maximal subgroups

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

subgroupindex
C67×#31752
C67×#29502
C67×#22372
C67×#22342
C67×#22272
C67×#22252
C67×#22112
C67×#21882
C67×#21852
C67×#21762
C67×#21752
C4×Q32×C672
C4×SD32×C672
C67×#21642
#3368167