Finite Group Databasebeta

← C67×#33680 #30712931

Subgroups — C67×#33680 #30712931

Order 17152.

534 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×#3177C211yesmaximal, characteristic, complemented
2C67×#2951C211yesmaximal, characteristic
3C67×#2236C211yesmaximal, characteristic, complemented
4C67×#2235C211yesmaximal, characteristic, complemented
5C67×#2229C211yesmaximal, characteristic, complemented
6C67×#2228C211yesmaximal, characteristic, complemented
7C67×#2209C211yesmaximal, characteristic, complemented
8C67×#2208C211yesmaximal, characteristic, complemented
9C67×#2189C211yesmaximal, characteristic
10C67×#2189C211yesmaximal, characteristic
11C67×#2176C211yesmaximal, characteristic, complemented
12C67×#2175C211yesmaximal, characteristic, complemented
13C4×Q32×C67C211yesmaximal, characteristic, complemented
14C4×SD32×C67C211yesmaximal, characteristic, complemented
15C67×#2164C211yesmaximal, characteristic, complemented
16C2×#135×C67C2211yescharacteristic
17#541×C67—21no—
18#520×C67—21no—
19C2×Q32×C67C2211yescharacteristic
20C2×Q32×C67C2211yescharacteristic
21C2×SD32×C67C2211yescharacteristic
22C2×SD32×C67C2211yescharacteristic
23#499×C67C2211yescharacteristic
24#494×C67C2211yescharacteristic
25#491×C67—21no—
26#470×C67—21no—
27#467×C67C2211yescharacteristic
28#467×C67C2211yescharacteristic
29#441×C67—21no—
30#440×C67—21no—
31#440×C67—21no—
32C4×Q16×C67C2211yescharacteristic
33C4×Q16×C67C2211yescharacteristic
34C4×Q16×C67C2211yescharacteristic
35C4×Q16×C67C2211yescharacteristic
36C4×D8×C67C2211yescharacteristic
37C4×D8×C67C2211yescharacteristic
38C2×#107×C67C2211yescharacteristic
39#417×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
59C2×#41×C67—21no—
60#135×C67—22no—
61#135×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#126×C67—41no—
73#125×C67—21no—
74#125×C67—41no—
75#125×C67—41no—
76C22⋊Q8×C67—21no—
77C22⋊Q8×C67—21no—
78C4×Q8×C67—21no—
79C4×Q8×C67—21no—
80C4×Q8×C67—21no—
81C4×Q8×C67—21no—
82C4×D4×C67—21no—
83C4×D4×C67—21no—
84C4×D4×C67—41no—
85C4×D4×C67—41no—
86#117×C67—41no—
87#117×C67D411yescharacteristic
88C2×C4⋊C4×C67—21no—
89C2×C4⋊C4×C67—41no—
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
116#41×C67—24no—
117#41×C67—42no—
118C2×Q8×C67—21no—
119C2×Q8×C67—21no—
120C2×Q8×C67—21no—
121C2×D4×C67—21no—
122C2×D4×C67—41no—
123C22×C4×C67—41no—
124C22×C4×C67—41no—
125C22×C4×C67—41no—
126C22×C4×C67—41no—
127C22×C4×C67C2×D411yescharacteristic
128Q16×C67—21no—
129Q16×C67—21no—
130Q16×C67—21no—
131Q16×C67—21no—
132Q16×C67—21no—
133Q16×C67—42no—
134Q16×C67#4112yes—
135SD16×C67—24no—
136D8×C67—21no—
137D8×C67#4112yes—
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×C67—41no—
157C4⋊C4×C67—41no—
158C4⋊C4×C67—41no—
159C4⋊C4×C67—41no—
160C4⋊C4×C67D811yescharacteristic
161C4⋊C4×C67D811yescharacteristic
162C22⋊C4×C67—41no—
163C22⋊C4×C67—41no—
164C22⋊C4×C67—41no—
165C22⋊C4×C67—41no—
166C22⋊C4×C67D811yescharacteristic
167C22⋊C4×C67D811yescharacteristic
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—41no—
182Q8×C67—41no—
183Q8×C67—41no—
184Q8×C67—41no—
185Q8×C67—41no—
186D4×C67—22no—
187D4×C67—44no—
188D4×C67—41no—
189C2×C4×C67—21no—
190C2×C4×C67—21no—
191C2×C4×C67—44no—
192C2×C4×C67—41no—
193C2×C4×C67—41no—
194C2×C4×C67—41no—
195C2×C4×C67—41no—
196C2×C4×C67—41no—
197C2×C4×C67—41no—
198C2×C4×C67—41no—
199C2×C4×C67—161no—
200C2×C4×C67—161no—
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—42no—
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—81no—
227C4×C67—81no—
228C4×C67—81no—
229C4×C67—81no—
230C4×C67—81no—
231C4×C67—81no—
232C4×C67#52511yescharacteristic
233C4×C67#57511yescharacteristic
234#33680C6711yesmaximal, characteristic, direct product factor, Sylow
235C2×C67—41no—
236C2×C67—82no—
237C2×C67#310011yescharacteristic
238C2×C67#341611yescharacteristic
239C2×C67#341711yescharacteristic
240#3177C2×C6711yescharacteristic, complemented
241#2951C2×C6711yescharacteristic
242#2236C2×C6711yescharacteristic, complemented
243#2235C2×C6711yescharacteristic, complemented
244#2229C2×C6711yescharacteristic, complemented
245#2228C2×C6711yescharacteristic, complemented
246#2209C2×C6711yescharacteristic, complemented
247#2208C2×C6711yescharacteristic, complemented
248#2189C2×C6711yescharacteristic
249#2189C2×C6711yescharacteristic
250#2176C2×C6711yescharacteristic, complemented
251#2175C2×C6711yescharacteristic, complemented
252C4×Q32C2×C6711yescharacteristic, complemented
253C4×SD32C2×C6711yescharacteristic, complemented
254#2164C2×C6711yescharacteristic, complemented
255C67#3368011yescharacteristic, direct product factor, Sylow
256C2×#135C22×C6711yescharacteristic
257#541—21no—
258#520—21no—
259C2×Q32C22×C6711yescharacteristic
260C2×Q32C22×C6711yescharacteristic
261C2×SD32C22×C6711yescharacteristic
262C2×SD32C22×C6711yescharacteristic
263#499C22×C6711yescharacteristic
264#494C22×C6711yescharacteristic
265#491—21no—
266#470—21no—
267#467C22×C6711yescharacteristic
268#467C22×C6711yescharacteristic
269#441—21no—
270#440—21no—
271#440—21no—
272C4×Q16C22×C6711yescharacteristic
273C4×Q16C22×C6711yescharacteristic
274C4×Q16C22×C6711yescharacteristic
275C4×Q16C22×C6711yescharacteristic
276C4×D8C22×C6711yescharacteristic
277C4×D8C22×C6711yescharacteristic
278C2×#107C22×C6711yescharacteristic
279#417—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
299C2×#41—21no—
300#135—22no—
301#135—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#126—41no—
313#125—21no—
314#125—41no—
315#125—41no—
316C22⋊Q8—21no—
317C22⋊Q8—21no—
318C4×Q8—21no—
319C4×Q8—21no—
320C4×Q8—21no—
321C4×Q8—21no—
322C4×D4—21no—
323C4×D4—21no—
324C4×D4—41no—
325C4×D4—41no—
326#117—41no—
327#117D4×C6711yescharacteristic
328C2×C4⋊C4—21no—
329C2×C4⋊C4—41no—
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
356#41—24no—
357#41—42no—
358C2×Q8—21no—
359C2×Q8—21no—
360C2×Q8—21no—
361C2×D4—21no—
362C2×D4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4C2×D4×C6711yescharacteristic
368Q16—21no—
369Q16—21no—
370Q16—21no—
371Q16—21no—
372Q16—21no—
373Q16—42no—
374Q16#41×C6712yes—
375SD16—24no—
376D8—21no—
377D8#41×C6712yes—
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⋊C4—41no—
397C4⋊C4—41no—
398C4⋊C4—41no—
399C4⋊C4—41no—
400C4⋊C4D8×C6711yescharacteristic
401C4⋊C4D8×C6711yescharacteristic
402C22⋊C4—41no—
403C22⋊C4—41no—
404C22⋊C4—41no—
405C22⋊C4—41no—
406C22⋊C4D8×C6711yescharacteristic
407C22⋊C4D8×C6711yescharacteristic
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—41no—
422Q8—41no—
423Q8—41no—
424Q8—41no—
425Q8—41no—
426D4—22no—
427D4—44no—
428D4—41no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—44no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—161no—
440C2×C4—161no—
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—42no—
457C22—81no—
458C22—81no—
459C22C22×D8×C6711yescharacteristic
460C4—21no—
461C4—21no—
462C4—21no—
463C4—21no—
464C4—21no—
465C4—42no—
466C4—81no—
467C4—81no—
468C4—81no—
469C4—81no—
470C4—81no—
471C4—81no—
472C4#525×C6711yescharacteristic
473C4#575×C6711yescharacteristic
474C2—41no—
475C2—82no—
476C2C67×#310011yescharacteristic
477C2C67×#341611yescharacteristic
478C2C67×#341711yescharacteristic
subgroupG/H
center, socleC22×C67C22×D8
commutator, Frattini, cosocleC2×C8C24×C67
prime radicalC2×#135×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
1. C67×#3177C2semidirect
3. C67×#2236C2semidirect
4. C67×#2235C2semidirect
5. C67×#2229C2semidirect
6. C67×#2228C2semidirect
7. C67×#2209C2semidirect
8. C67×#2208C2semidirect
11. C67×#2176C2semidirect
12. C67×#2175C2semidirect
13. C4×Q32×C67C2semidirect
14. C4×SD32×C67C2semidirect
15. C67×#2164C2semidirect
234. #33680C67direct (N × H)
240. #3177C2×C67semidirect
242. #2236C2×C67semidirect
243. #2235C2×C67semidirect
244. #2229C2×C67semidirect
245. #2228C2×C67semidirect
246. #2209C2×C67semidirect
247. #2208C2×C67semidirect
250. #2176C2×C67semidirect
251. #2175C2×C67semidirect
252. C4×Q32C2×C67semidirect
253. C4×SD32C2×C67semidirect
254. #2164C2×C67semidirect
255. C67#33680direct (N × H)

Maximal subgroups

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

subgroupindex
C67×#31772
C67×#29512
C67×#22362
C67×#22352
C67×#22292
C67×#22282
C67×#22092
C67×#22082
C67×#21892
C67×#21762
C67×#21752
C4×Q32×C672
C4×SD32×C672
C67×#21642
#3368067