Finite Group Databasebeta

← C67×#12749 #30692000

Subgroups — C67×#12749 #30692000

Order 17152.

882 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#578×C67C211yesmaximal, characteristic
2C2×#362×C67C211yesmaximal, characteristic, complemented
3C2×#359×C67C211yesmaximal, characteristic, complemented
4C67×#2139C211yesmaximal, characteristic, complemented
5C2×#358×C67C211yesmaximal, characteristic, complemented
6C2×#350×C67C211yesmaximal, characteristic, complemented
7C67×#1816C211yesmaximal, characteristic, complemented
8C2×2−1+4×C67—21no—
9#578×C67—22no—
10#578×C67—22no—
11#578×C67—22no—
12#578×C67C414yescomplemented
13C2×#137×C67—22no—
14C2×#137×C67—21no—
15C2×#135×C67—21no—
16C2×#135×C67C2211yescharacteristic
17C22×Q16×C67—21no—
18C22×Q16×C67C2211yescharacteristic
19C2×#131×C67C2211yescharacteristic
20C2×M32×C67C2211yescharacteristic
21C22×C16×C67C2211yescharacteristic
22C2×#108×C67C2211yescharacteristic
23C2×#107×C67C2211yescharacteristic
24#417×C67—21no—
25#362×C67—22no—
26#359×C67—22no—
27#358×C67—22no—
28#358×C67—21no—
29#357×C67—21no—
30#350×C67—22no—
31#343×C67—21no—
322−1+4×C67—24no—
332−1+4×C67—24no—
342−1+4×C67—42no—
352−1+4×C67—42no—
36C2×#41×C67—22no—
37C2×#41×C67—21no—
38C2×#41×C67—42no—
39C2×#41×C67—41no—
40C2×#41×C67D411yescharacteristic
41C22×Q8×C67—22no—
42C22×Q8×C67—21no—
43C22×Q8×C67—41no—
44#137×C67—24no—
45#137×C67—28no—
46#137×C67—28no—
47#137×C67—24no—
48#135×C67—24no—
49#135×C67—24no—
50#135×C67—21no—
51#135×C67—21no—
52#135×C67C2×C412yes—
53#135×C67D412yes—
54C2×Q16×C67—22no—
55C2×Q16×C67—24no—
56C2×Q16×C67—24no—
57C2×Q16×C67—21no—
58C2×Q16×C67—21no—
59C2×Q16×C67—21no—
60C2×Q16×C67C2×C412yes—
61C2×Q16×C67D411yescharacteristic
62C2×Q16×C67D412yes—
63C2×Q16×C67D412yes—
64C2×SD16×C67—22no—
65C2×SD16×C67—21no—
66C2×D8×C67D411yescharacteristic
67#131×C67—21no—
68#131×C67—21no—
69#131×C67C2×C412yes—
70#131×C67D412yes—
71C2×M16×C67—21no—
72C2×M16×C67—21no—
73C2×M16×C67D411yescharacteristic
74C22×C8×C67—21no—
75C22×C8×C67C2311yescharacteristic
76C2×C4⋊C4×C67—21no—
77M32×C67—21no—
78M32×C67—21no—
79C2×C16×C67—21no—
80C2×C16×C67—21no—
81C2×C16×C67—21no—
82C2×C16×C67—21no—
83#108×C67—21no—
84#108×C67—21no—
85#107×C67—21no—
86#107×C67—21no—
87#103×C67—41no—
88#102×C67—41no—
89#41×C67—21no—
90#41×C67—21no—
91#41×C67—28no—
92#41×C67—22no—
93#41×C67—22no—
94#41×C67—44no—
95#41×C67—48no—
96#41×C67—48no—
97#41×C67—44no—
98#41×C67—44no—
99#41×C67—41no—
100#41×C67—41no—
101#41×C67C22⋊C412yes—
102#41×C67SD1612yes—
103C2×Q8×C67—28no—
104C2×Q8×C67—22no—
105C2×Q8×C67—22no—
106C2×Q8×C67—21no—
107C2×Q8×C67—22no—
108C2×Q8×C67—42no—
109C2×Q8×C67—44no—
110C2×Q8×C67—44no—
111C2×Q8×C67—44no—
112C2×Q8×C67—41no—
113C2×Q8×C67—41no—
114C2×Q8×C67—41no—
115C2×Q8×C67D811yescharacteristic
116C2×D4×C67—21no—
117C2×D4×C67—21no—
118C2×D4×C67—42no—
119C2×D4×C67—41no—
120C2×D4×C67D811yescharacteristic
121C22×C4×C67—21no—
122C22×C4×C67—42no—
123C22×C4×C67—42no—
124C22×C4×C67—41no—
125C22×C4×C67—41no—
126C22×C4×C67—41no—
127C22×C4×C67C2×D411yescharacteristic
128Q16×C67—28no—
129Q16×C67—22no—
130Q16×C67—21no—
131Q16×C67—22no—
132Q16×C67—22no—
133Q16×C67C22⋊C412yes—
134Q16×C67C22⋊C414yes—
135SD16×C67—28no—
136SD16×C67—22no—
137SD16×C67—22no—
138D8×C67—21no—
139D8×C67C22⋊C412yes—
140M16×C67—21no—
141M16×C67—22no—
142M16×C67—22no—
143M16×C67—41no—
144M16×C67—41no—
145M16×C67C22⋊C412yes—
146C2×C8×C67—21no—
147C2×C8×C67—22no—
148C2×C8×C67—22no—
149C2×C8×C67—41no—
150C2×C8×C67C22×C411yescharacteristic
151C2×C8×C67C2×D411yescharacteristic
152C2×C8×C67C2×D411yescharacteristic
153C2×C8×C67C2×D411yescharacteristic
154C2×C8×C67C2×D411yescharacteristic
155C2×C8×C67C2×D411yescharacteristic
156C4⋊C4×C67—41no—
157C4⋊C4×C67—41no—
158C16×C67—41no—
159C16×C67—41no—
160C23×C67—21no—
161C23×C67—41no—
162C23×C67AGL(1,ℤ/8)11yescharacteristic
163Q8×C67—21no—
164Q8×C67—24no—
165Q8×C67—22no—
166Q8×C67—48no—
167Q8×C67—42no—
168Q8×C67—42no—
169Q8×C67—41no—
170Q8×C67—42no—
171Q8×C67#10212yes—
172D4×C67—22no—
173D4×C67—22no—
174D4×C67—22no—
175D4×C67—21no—
176D4×C67—48no—
177D4×C67—42no—
178D4×C67—42no—
179D4×C67—41no—
180D4×C67#10212yes—
181C2×C4×C67—21no—
182C2×C4×C67—22no—
183C2×C4×C67—22no—
184C2×C4×C67—48no—
185C2×C4×C67—48no—
186C2×C4×C67—42no—
187C2×C4×C67—42no—
188C2×C4×C67—41no—
189C2×C4×C67—42no—
190C2×C4×C67—42no—
191C2×C4×C67—42no—
192C2×C4×C67—81no—
193C2×C4×C67—161no—
194C2×C4×C67C2×C22⋊C411yescharacteristic
195C2×C4×C67#12011yescharacteristic
196C2×C4×C67#12011yescharacteristic
197C2×C4×C67C2×D811yescharacteristic
198C2×C4×C67C2×SD1611yescharacteristic
199C2×C4×C67AGL(1,ℤ/8)11yescharacteristic
200C8×C67—22no—
201C8×C67—81no—
202C8×C67C2×C22⋊C411yescharacteristic
203C8×C67C2×C22⋊C411yescharacteristic
204C8×C67#12011yescharacteristic
205C8×C67#12011yescharacteristic
206C22×C67—21no—
207C22×C67—21no—
208C22×C67—22no—
209C22×C67—41no—
210C22×C67—41no—
211C22×C67—42no—
212C22×C67—81no—
213C22×C67—81no—
214C22×C67#41811yescharacteristic
215C22×C67#44711yescharacteristic
216C22×C67#45011yescharacteristic
217C4×C67—22no—
218C4×C67—44no—
219C4×C67—42no—
220C4×C67—161no—
221C4×C67#37911yescharacteristic
222C4×C67C2×#10211yescharacteristic
223C4×C67#44711yescharacteristic
224C4×C67#45011yescharacteristic
225#12749C6711yesmaximal, characteristic, direct product factor, Sylow
226C2×C67—21no—
227C2×C67—21no—
228C2×C67—42no—
229C2×C67—82no—
230C2×C67#178611yescharacteristic
231C2×C67#219311yescharacteristic
232C2×C67#219511yescharacteristic
233C2×#578C2×C6711yescharacteristic
234C2×#362C2×C6711yescharacteristic, complemented
235C2×#359C2×C6711yescharacteristic, complemented
236#2139C2×C6711yescharacteristic, complemented
237C2×#358C2×C6711yescharacteristic, complemented
238C2×#350C2×C6711yescharacteristic, complemented
239#1816C2×C6711yescharacteristic, complemented
240C67#1274911yescharacteristic, direct product factor, Sylow
241C2×2−1+4—21no—
242#578—22no—
243#578—22no—
244#578—22no—
245#578C4×C6714yescomplemented
246C2×#137—22no—
247C2×#137—21no—
248C2×#135—21no—
249C2×#135C22×C6711yescharacteristic
250C22×Q16—21no—
251C22×Q16C22×C6711yescharacteristic
252C2×#131C22×C6711yescharacteristic
253C2×M32C22×C6711yescharacteristic
254C22×C16C22×C6711yescharacteristic
255C2×#108C22×C6711yescharacteristic
256C2×#107C22×C6711yescharacteristic
257#417—21no—
258#362—22no—
259#359—22no—
260#358—22no—
261#358—21no—
262#357—21no—
263#350—22no—
264#343—21no—
2652−1+4—24no—
2662−1+4—24no—
2672−1+4—42no—
2682−1+4—42no—
269C2×#41—22no—
270C2×#41—21no—
271C2×#41—42no—
272C2×#41—41no—
273C2×#41D4×C6711yescharacteristic
274C22×Q8—22no—
275C22×Q8—21no—
276C22×Q8—41no—
277#137—24no—
278#137—28no—
279#137—28no—
280#137—24no—
281#135—24no—
282#135—24no—
283#135—21no—
284#135—21no—
285#135C2×C4×C6712yes—
286#135D4×C6712yes—
287C2×Q16—22no—
288C2×Q16—24no—
289C2×Q16—24no—
290C2×Q16—21no—
291C2×Q16—21no—
292C2×Q16—21no—
293C2×Q16C2×C4×C6712yes—
294C2×Q16D4×C6711yescharacteristic
295C2×Q16D4×C6712yes—
296C2×Q16D4×C6712yes—
297C2×SD16—22no—
298C2×SD16—21no—
299C2×D8D4×C6711yescharacteristic
300#131—21no—
301#131—21no—
302#131C2×C4×C6712yes—
303#131D4×C6712yes—
304C2×M16—21no—
305C2×M16—21no—
306C2×M16D4×C6711yescharacteristic
307C22×C8—21no—
308C22×C8C23×C6711yescharacteristic
309C2×C4⋊C4—21no—
310M32—21no—
311M32—21no—
312C2×C16—21no—
313C2×C16—21no—
314C2×C16—21no—
315C2×C16—21no—
316#108—21no—
317#108—21no—
318#107—21no—
319#107—21no—
320#103—41no—
321#102—41no—
322#41—21no—
323#41—21no—
324#41—28no—
325#41—22no—
326#41—22no—
327#41—44no—
328#41—48no—
329#41—48no—
330#41—44no—
331#41—44no—
332#41—41no—
333#41—41no—
334#41C22⋊C4×C6712yes—
335#41SD16×C6712yes—
336C2×Q8—28no—
337C2×Q8—22no—
338C2×Q8—22no—
339C2×Q8—21no—
340C2×Q8—22no—
341C2×Q8—42no—
342C2×Q8—44no—
343C2×Q8—44no—
344C2×Q8—44no—
345C2×Q8—41no—
346C2×Q8—41no—
347C2×Q8—41no—
348C2×Q8D8×C6711yescharacteristic
349C2×D4—21no—
350C2×D4—21no—
351C2×D4—42no—
352C2×D4—41no—
353C2×D4D8×C6711yescharacteristic
354C22×C4—21no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4C2×D4×C6711yescharacteristic
361Q16—28no—
362Q16—22no—
363Q16—21no—
364Q16—22no—
365Q16—22no—
366Q16C22⋊C4×C6712yes—
367Q16C22⋊C4×C6714yes—
368SD16—28no—
369SD16—22no—
370SD16—22no—
371D8—21no—
372D8C22⋊C4×C6712yes—
373M16—21no—
374M16—22no—
375M16—22no—
376M16—41no—
377M16—41no—
378M16C22⋊C4×C6712yes—
379C2×C8—21no—
380C2×C8—22no—
381C2×C8—22no—
382C2×C8—41no—
383C2×C8C22×C4×C6711yescharacteristic
384C2×C8C2×D4×C6711yescharacteristic
385C2×C8C2×D4×C6711yescharacteristic
386C2×C8C2×D4×C6711yescharacteristic
387C2×C8C2×D4×C6711yescharacteristic
388C2×C8C2×D4×C6711yescharacteristic
389C4⋊C4—41no—
390C4⋊C4—41no—
391C16—41no—
392C16—41no—
393C23—21no—
394C23—41no—
395C23AGL(1,ℤ/8)×C6711yescharacteristic
396Q8—21no—
397Q8—24no—
398Q8—22no—
399Q8—48no—
400Q8—41no—
401Q8—42no—
402Q8—42no—
403Q8—42no—
404Q8#102×C6712yes—
405D4—22no—
406D4—22no—
407D4—22no—
408D4—21no—
409D4—48no—
410D4—42no—
411D4—42no—
412D4—41no—
413D4#102×C6712yes—
414C2×C4—21no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—48no—
418C2×C4—48no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—41no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—81no—
426C2×C4—161no—
427C2×C4C2×C22⋊C4×C6711yescharacteristic
428C2×C4#120×C6711yescharacteristic
429C2×C4#120×C6711yescharacteristic
430C2×C4C2×D8×C6711yescharacteristic
431C2×C4C2×SD16×C6711yescharacteristic
432C2×C4AGL(1,ℤ/8)×C6711yescharacteristic
433C8—22no—
434C8—81no—
435C8C2×C22⋊C4×C6711yescharacteristic
436C8C2×C22⋊C4×C6711yescharacteristic
437C8#120×C6711yescharacteristic
438C8#120×C6711yescharacteristic
439C22—21no—
440C22—21no—
441C22—22no—
442C22—41no—
443C22—41no—
444C22—42no—
445C22—81no—
446C22—81no—
447C22#418×C6711yescharacteristic
448C22#447×C6711yescharacteristic
449C22#450×C6711yescharacteristic
450C4—22no—
451C4—44no—
452C4—42no—
453C4—161no—
454C4#379×C6711yescharacteristic
455C4C2×#102×C6711yescharacteristic
456C4#447×C6711yescharacteristic
457C4#450×C6711yescharacteristic
458C2—21no—
459C2—21no—
460C2—42no—
461C2—82no—
462C2C67×#178611yescharacteristic
463C2C67×#219311yescharacteristic
464C2C67×#219511yescharacteristic
subgroupG/H
center, socleC22×C67#447
commutatorC2×C8C22×C4×C67
Frattini, cosocleC22×C8C23×C67
prime radicalC2×#578×C67C2
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C67 ◁ C22×C4×C67 ◁ C2×#131×C67 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×#362×C67C2semidirect
3. C2×#359×C67C2semidirect
4. C67×#2139C2semidirect
5. C2×#358×C67C2semidirect
6. C2×#350×C67C2semidirect
7. C67×#1816C2semidirect
12. #578×C67C4semidirect
225. #12749C67direct (N × H)
234. C2×#362C2×C67semidirect
235. C2×#359C2×C67semidirect
236. #2139C2×C67semidirect
237. C2×#358C2×C67semidirect
238. C2×#350C2×C67semidirect
239. #1816C2×C67semidirect
240. C67#12749direct (N × H)
245. #578C4×C67semidirect

Maximal subgroups

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

subgroupindex
C2×#578×C672
C2×#362×C672
C2×#359×C672
C67×#21392
C2×#358×C672
C2×#350×C672
C67×#18162
#1274967