Finite Group Databasebeta

← C67×#24986 #30704237

Subgroups — C67×#24986 #30704237

Order 17152.

642 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C67×#3067C211yesmaximal, characteristic
2C67×#2943C211yesmaximal, characteristic, complemented
3C67×#2080C212yesmaximal, complemented
4C67×#2053C212yesmaximal, complemented
5C67×#1976C211yesmaximal, characteristic, complemented
6C67×#1946C211yesmaximal, characteristic, complemented
7C67×#1906C212yesmaximal, complemented
8C67×#1833C211yesmaximal, characteristic, complemented
9C67×#1827C212yesmaximal, complemented
10C67×#1812C211yesmaximal, characteristic, complemented
11C67×#1765C211yesmaximal, characteristic, complemented
12#575×C67C2212yes—
13C2×#135×C67C2211yescharacteristic
14C2×#131×C67C2211yescharacteristic
15C4×#41×C67C2212yes—
16#473×C67—21no—
17#472×C67—21no—
18#471×C67—22no—
19#442×C67—21no—
20#441×C67—21no—
21#440×C67—22no—
22C2×#108×C67C2211yescharacteristic
23#427×C67C2211yescharacteristic
24#427×C67C2211yescharacteristic
25#424×C67C2212yes—
26#421×C67C2212yes—
27C2×#104×C67C2211yescharacteristic
28#416×C67C2211yescharacteristic
29#416×C67C2212yes—
30#413×C67C2212yes—
31#410×C67C2212yes—
32#410×C67C2212yes—
33#405×C67C2211yescharacteristic
34C2×#97×C67C2211yescharacteristic
35#344×C67C2212yes—
36#341×C67C2211yescharacteristic
37#339×C67C2211yescharacteristic
38#339×C67C2212yes—
39H(3,ℤ/4)×C67C2212yes—
40H(3,ℤ/4)×C67C2212yes—
41C2×#41×C67C2312yes—
42C2×#41×C67C2311yescharacteristic
43#137×C67—22no—
44#137×C67—22no—
45AGL(1,ℤ/8)×C67—22no—
46AGL(1,ℤ/8)×C67—22no—
47#135×C67—21no—
48#135×C67—21no—
49#135×C67—21no—
50#135×C67—21no—
51#135×C67—42no—
52C2×Q16×C67D411yescharacteristic
53C2×SD16×C67D412yes—
54C2×D8×C67D411yescharacteristic
55#131×C67—21no—
56#131×C67—21no—
57#131×C67—21no—
58#131×C67—21no—
59C2×M16×C67—21no—
60C2×M16×C67C2311yescharacteristic
61C2×M16×C67C2312yes—
62C2×M16×C67C2311yescharacteristic
63C22×C8×C67—21no—
64C22×C8×C67C2311yescharacteristic
65C22×C8×C67C2311yescharacteristic
66C4×Q8×C67—22no—
67C4×D4×C67—22no—
68C4×D4×C67—42no—
69#117×C67—22no—
70#117×C67C2312yes—
71#117×C67C2311yescharacteristic
72#117×C67C2312yes—
73C2×C42×C67—22no—
74C2×C42×C67C2311yescharacteristic
75#108×C67—21no—
76#108×C67—21no—
77#107×C67—21no—
78#107×C67—21no—
79#106×C67—21no—
80#106×C67—21no—
81C4⋊C8×C67—42no—
82#104×C67—22no—
83#104×C67—42no—
84#103×C67—22no—
85#103×C67—21no—
86#102×C67—22no—
87#102×C67—21no—
88#101×C67—22no—
89#100×C67—22no—
90#99×C67—42no—
91#99×C67—42no—
92#97×C67—21no—
93#97×C67—21no—
94#97×C67—21no—
95C4×C8×C67—21no—
96#41×C67—21no—
97#41×C67—21no—
98#41×C67—21no—
99#41×C67—22no—
100#41×C67—22no—
101#41×C67—22no—
102#41×C67—22no—
103#41×C67—21no—
104C2×Q8×C67C2×D412yes—
105C2×Q8×C67#4111yescharacteristic
106C2×D4×C67—22no—
107C2×D4×C67—21no—
108C2×D4×C67C2×D412yes—
109C2×D4×C67#4111yescharacteristic
110C22×C4×C67—21no—
111C22×C4×C67—22no—
112C22×C4×C67—21no—
113C22×C4×C67—42no—
114C22×C4×C67C2411yescharacteristic
115Q16×C67—21no—
116Q16×C67—21no—
117Q16×C67—42no—
118SD16×C67—22no—
119SD16×C67—22no—
120SD16×C67—42no—
121SD16×C67—42no—
122D8×C67—21no—
123D8×C67—21no—
124D8×C67—42no—
125M16×C67—21no—
126M16×C67—22no—
127M16×C67—21no—
128M16×C67—21no—
129M16×C67—22no—
130M16×C67—21no—
131M16×C67—41no—
132M16×C67—41no—
133C2×C8×C67—21no—
134C2×C8×C67—21no—
135C2×C8×C67—21no—
136C2×C8×C67—21no—
137C2×C8×C67—21no—
138C2×C8×C67—22no—
139C2×C8×C67—41no—
140C2×C8×C67—41no—
141C2×C8×C67C2×D411yescharacteristic
142C2×C8×C67C2×D411yescharacteristic
143C2×C8×C67#4111yescharacteristic
144C2×C8×C67#4111yescharacteristic
145C2×C8×C67#4111yescharacteristic
146C2×C8×C67#4111yescharacteristic
147C4⋊C4×C67—22no—
148C4⋊C4×C67—21no—
149C4⋊C4×C67—22no—
150C4⋊C4×C67—42no—
151C22⋊C4×C67—22no—
152C22⋊C4×C67—22no—
153C22⋊C4×C67—21no—
154C22⋊C4×C67—42no—
155C42×C67—21no—
156C42×C67—22no—
157C42×C67—21no—
158C42×C67—22no—
159C42×C67—42no—
160C42×C67#4112yes—
161C23×C67—22no—
162C23×C67—21no—
163C23×C67H(5,𝔽2)11yescharacteristic
164Q8×C67—22no—
165Q8×C67—21no—
166Q8×C67—21no—
167Q8×C67—22no—
168D4×C67—22no—
169D4×C67—21no—
170D4×C67—21no—
171D4×C67—22no—
172D4×C67—42no—
173D4×C67—41no—
174D4×C67—41no—
175D4×C67—42no—
176C2×C4×C67—22no—
177C2×C4×C67—22no—
178C2×C4×C67—21no—
179C2×C4×C67—41no—
180C2×C4×C67—42no—
181C2×C4×C67—42no—
182C2×C4×C67—41no—
183C2×C4×C67—42no—
184C2×C4×C67—42no—
185C2×C4×C67—41no—
186C2×C4×C67—81no—
187C2×C4×C67—84no—
188C2×C4×C67C22×D411yescharacteristic
189C2×C4×C67C2×#4111yescharacteristic
190C2×C4×C67C2×#4111yescharacteristic
191C2×C4×C67C2×#4111yescharacteristic
192C2×C4×C67C2×#4111yescharacteristic
193C2×C4×C67H(5,𝔽2)11yescharacteristic
194C8×C67—41no—
195C8×C67—41no—
196C8×C67—42no—
197C8×C67—41no—
198C8×C67#12311yescharacteristic
199C8×C67#12311yescharacteristic
200C8×C67#12311yescharacteristic
201C8×C67#12311yescharacteristic
202C22×C67—41no—
203C22×C67—41no—
204C22×C67—42no—
205C22×C67—82no—
206C22×C67—81no—
207C22×C67#54011yescharacteristic
208C22×C67#54611yescharacteristic
209C22×C67#55311yescharacteristic
210C4×C67—42no—
211C4×C67—42no—
212C4×C67—41no—
213C4×C67—82no—
214C4×C67—81no—
215C4×C67—82no—
216C4×C67C2×#12311yescharacteristic
217C4×C67#52911yescharacteristic
218C4×C67#54611yescharacteristic
219C4×C67#55511yescharacteristic
220#24986C6711yesmaximal, characteristic, direct product factor, Sylow
221C2×C67—21no—
222C2×C67—21no—
223C2×C67—21no—
224C2×C67—82no—
225C2×C67—81no—
226C2×C67#242011yescharacteristic
227#3067C2×C6711yescharacteristic
228#2943C2×C6711yescharacteristic, complemented
229#2080C2×C6712yescomplemented
230#2053C2×C6712yescomplemented
231#1976C2×C6711yescharacteristic, complemented
232#1946C2×C6711yescharacteristic, complemented
233#1906C2×C6712yescomplemented
234#1833C2×C6711yescharacteristic, complemented
235#1827C2×C6712yescomplemented
236#1812C2×C6711yescharacteristic, complemented
237#1765C2×C6711yescharacteristic, complemented
238C67#2498611yescharacteristic, direct product factor, Sylow
239#575C22×C6712yes—
240C2×#135C22×C6711yescharacteristic
241C2×#131C22×C6711yescharacteristic
242C4×#41C22×C6712yes—
243#473—21no—
244#472—21no—
245#471—22no—
246#442—21no—
247#441—21no—
248#440—22no—
249C2×#108C22×C6711yescharacteristic
250#427C22×C6711yescharacteristic
251#427C22×C6711yescharacteristic
252#424C22×C6712yes—
253#421C22×C6712yes—
254C2×#104C22×C6711yescharacteristic
255#416C22×C6711yescharacteristic
256#416C22×C6712yes—
257#413C22×C6712yes—
258#410C22×C6712yes—
259#410C22×C6712yes—
260#405C22×C6711yescharacteristic
261C2×#97C22×C6711yescharacteristic
262#344C22×C6712yes—
263#341C22×C6711yescharacteristic
264#339C22×C6711yescharacteristic
265#339C22×C6712yes—
266H(3,ℤ/4)C22×C6712yes—
267H(3,ℤ/4)C22×C6712yes—
268C2×#41C23×C6712yes—
269C2×#41C23×C6711yescharacteristic
270#137—22no—
271#137—22no—
272AGL(1,ℤ/8)—22no—
273AGL(1,ℤ/8)—22no—
274#135—21no—
275#135—21no—
276#135—21no—
277#135—21no—
278#135—42no—
279C2×Q16D4×C6711yescharacteristic
280C2×SD16D4×C6712yes—
281C2×D8D4×C6711yescharacteristic
282#131—21no—
283#131—21no—
284#131—21no—
285#131—21no—
286C2×M16—21no—
287C2×M16C23×C6711yescharacteristic
288C2×M16C23×C6712yes—
289C2×M16C23×C6711yescharacteristic
290C22×C8—21no—
291C22×C8C23×C6711yescharacteristic
292C22×C8C23×C6711yescharacteristic
293C4×Q8—22no—
294C4×D4—22no—
295C4×D4—42no—
296#117—22no—
297#117C23×C6712yes—
298#117C23×C6711yescharacteristic
299#117C23×C6712yes—
300C2×C42—22no—
301C2×C42C23×C6711yescharacteristic
302#108—21no—
303#108—21no—
304#107—21no—
305#107—21no—
306#106—21no—
307#106—21no—
308C4⋊C8—42no—
309#104—22no—
310#104—42no—
311#103—22no—
312#103—21no—
313#102—22no—
314#102—21no—
315#101—22no—
316#100—22no—
317#99—42no—
318#99—42no—
319#97—21no—
320#97—21no—
321#97—21no—
322C4×C8—21no—
323#41—21no—
324#41—21no—
325#41—21no—
326#41—22no—
327#41—22no—
328#41—22no—
329#41—22no—
330#41—21no—
331C2×Q8C2×D4×C6712yes—
332C2×Q8#41×C6711yescharacteristic
333C2×D4—21no—
334C2×D4—22no—
335C2×D4C2×D4×C6712yes—
336C2×D4#41×C6711yescharacteristic
337C22×C4—21no—
338C22×C4—22no—
339C22×C4—21no—
340C22×C4—42no—
341C22×C4C24×C6711yescharacteristic
342Q16—21no—
343Q16—21no—
344Q16—42no—
345SD16—22no—
346SD16—22no—
347SD16—42no—
348SD16—42no—
349D8—21no—
350D8—21no—
351D8—42no—
352M16—21no—
353M16—22no—
354M16—21no—
355M16—21no—
356M16—22no—
357M16—21no—
358M16—41no—
359M16—41no—
360C2×C8—21no—
361C2×C8—21no—
362C2×C8—21no—
363C2×C8—21no—
364C2×C8—21no—
365C2×C8—22no—
366C2×C8—41no—
367C2×C8—41no—
368C2×C8C2×D4×C6711yescharacteristic
369C2×C8C2×D4×C6711yescharacteristic
370C2×C8#41×C6711yescharacteristic
371C2×C8#41×C6711yescharacteristic
372C2×C8#41×C6711yescharacteristic
373C2×C8#41×C6711yescharacteristic
374C4⋊C4—22no—
375C4⋊C4—21no—
376C4⋊C4—22no—
377C4⋊C4—42no—
378C22⋊C4—22no—
379C22⋊C4—21no—
380C22⋊C4—22no—
381C22⋊C4—42no—
382C42—21no—
383C42—21no—
384C42—22no—
385C42—22no—
386C42—42no—
387C42#41×C6712yes—
388C23—22no—
389C23—21no—
390C23H(5,𝔽2)×C6711yescharacteristic
391Q8—22no—
392Q8—21no—
393Q8—21no—
394Q8—22no—
395D4—22no—
396D4—21no—
397D4—21no—
398D4—22no—
399D4—41no—
400D4—42no—
401D4—41no—
402D4—42no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—21no—
406C2×C4—41no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—41no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—41no—
413C2×C4—81no—
414C2×C4—84no—
415C2×C4C22×D4×C6711yescharacteristic
416C2×C4C2×#41×C6711yescharacteristic
417C2×C4C2×#41×C6711yescharacteristic
418C2×C4C2×#41×C6711yescharacteristic
419C2×C4C2×#41×C6711yescharacteristic
420C2×C4H(5,𝔽2)×C6711yescharacteristic
421C8—41no—
422C8—41no—
423C8—42no—
424C8—41no—
425C8#123×C6711yescharacteristic
426C8#123×C6711yescharacteristic
427C8#123×C6711yescharacteristic
428C8#123×C6711yescharacteristic
429C22—41no—
430C22—41no—
431C22—42no—
432C22—82no—
433C22—81no—
434C22#540×C6711yescharacteristic
435C22#546×C6711yescharacteristic
436C22#553×C6711yescharacteristic
437C4—42no—
438C4—42no—
439C4—41no—
440C4—82no—
441C4—82no—
442C4—81no—
443C4C2×#123×C6711yescharacteristic
444C4#529×C6711yescharacteristic
445C4#546×C6711yescharacteristic
446C4#555×C6711yescharacteristic
447C2—21no—
448C2—21no—
449C2—21no—
450C2—82no—
451C2—81no—
452C2C67×#242011yescharacteristic
subgroupG/H
centerC4×C67#555
commutator, Frattini, cosocleC22×C4C24×C67
socleC2×C67#2420
prime radicalC67×#3067C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C4×C67 ◁ C22×C4×C67 ◁ G

Complements

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

NH ≅ G/Ntype
2. C67×#2943C2semidirect
3. C67×#2080C2semidirect
4. C67×#2053C2semidirect
5. C67×#1976C2semidirect
6. C67×#1946C2semidirect
7. C67×#1906C2semidirect
8. C67×#1833C2semidirect
9. C67×#1827C2semidirect
10. C67×#1812C2semidirect
11. C67×#1765C2semidirect
220. #24986C67direct (N × H)
228. #2943C2×C67semidirect
229. #2080C2×C67semidirect
230. #2053C2×C67semidirect
231. #1976C2×C67semidirect
232. #1946C2×C67semidirect
233. #1906C2×C67semidirect
234. #1833C2×C67semidirect
235. #1827C2×C67semidirect
236. #1812C2×C67semidirect
237. #1765C2×C67semidirect
238. C67#24986direct (N × H)

Maximal subgroups

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

subgroupindex
C67×#30672
C67×#29432
C67×#20802
C67×#20532
C67×#19762
C67×#19462
C67×#19062
C67×#18332
C67×#18272
C67×#18122
C67×#17652
#2498667