Finite Group Databasebeta

← C67×#9210 #30688461

Subgroups — C67×#9210 #30688461

Order 17152.

510 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C67×#3035C211yesmaximal, characteristic
2C67×#2965C211yesmaximal, characteristic, complemented
3C67×#1806C211yesmaximal, characteristic, complemented
4C67×#1502C211yesmaximal, characteristic, complemented
5C67×#1492C211yesmaximal, characteristic, complemented
6C67×#1301C211yesmaximal, characteristic, complemented
7C67×#1279C211yesmaximal, characteristic, complemented
8C2×#135×C67—21no—
9#532×C67C2211yescharacteristic
10C4×#41×C67C2211yescharacteristic
11#464×C67—21no—
12#463×C67—21no—
13#462×C67C411yescharacteristic, complemented
14#461×C67C411yescharacteristic, complemented
15#460×C67C411yescharacteristic, complemented
16#459×C67C411yescharacteristic, complemented
17#446×C67—21no—
18#436×C67—21no—
19#435×C67—22no—
20#431×C67—21no—
21#431×C67C2211yescharacteristic
22#423×C67C2211yescharacteristic
23C2×C4⋊C8×C67C2211yescharacteristic
24C2×#104×C67—21no—
25#416×C67—21no—
26C4×M16×C67—21no—
27C2×C4×C8×C67C2211yescharacteristic
28C4×C4⋊C4×C67C2211yescharacteristic
29#339×C67—21no—
30#327×C67—22no—
31D4⋊C8×C67—21no—
32D4⋊C8×C67—21no—
33C2×#41×C67—21no—
34C2×#41×C67D411yescharacteristic
35#135×C67—42no—
36#135×C67—42no—
37C2×Q16×C67—21no—
38C2×SD16×C67—21no—
39C2×SD16×C67—21no—
40C2×D8×C67—21no—
41C2×M16×C67—22no—
42C2×M16×C67—21no—
43C22×C8×C67—21no—
44C22×C8×C67—21no—
45C4⋊Q8×C67C2×C411yescharacteristic
46#127×C67C2×C411yescharacteristic
47#124×C67—21no—
48#121×C67—21no—
49#121×C67—21no—
50C4×Q8×C67C2×C411yescharacteristic
51C4×Q8×C67D411yescharacteristic
52C4×D4×C67—21no—
53C4×D4×C67—21no—
54C4×D4×C67—21no—
55C4×D4×C67—21no—
56C4×D4×C67C2×C411yescharacteristic
57C4×D4×C67D411yescharacteristic
58#117×C67—21no—
59#117×C67D411yescharacteristic
60C2×C4⋊C4×C67—21no—
61C2×C42×C67—21no—
62C2×C42×C67—21no—
63C2×C42×C67C2311yescharacteristic
64C4⋊C8×C67—22no—
65C4⋊C8×C67—21no—
66C4⋊C8×C67—21no—
67C4⋊C8×C67—21no—
68C4⋊C8×C67—21no—
69C4⋊C8×C67C2×C411yescharacteristic
70C4⋊C8×C67C2×C411yescharacteristic
71#104×C67—42no—
72#103×C67—21no—
73#103×C67—21no—
74#102×C67—21no—
75#102×C67—21no—
76C22⋊C8×C67—22no—
77C22⋊C8×C67D412yes—
78#97×C67—22no—
79C4×C8×C67—21no—
80C4×C8×C67—21no—
81C4×C8×C67D411yescharacteristic
82C4×C8×C67D411yescharacteristic
83#95×C67—21no—
84#41×C67—21no—
85#41×C67—21no—
86#41×C67—21no—
87#41×C67—21no—
88#41×C67—41no—
89#41×C67—41no—
90#41×C67—41no—
91#41×C67—41no—
92C2×Q8×C67—21no—
93C2×Q8×C67C22⋊C411yescharacteristic
94C2×D4×C67—21no—
95C2×D4×C67—21no—
96C2×D4×C67—21no—
97C2×D4×C67—21no—
98C2×D4×C67—41no—
99C2×D4×C67C22⋊C411yescharacteristic
100C22×C4×C67—21no—
101C22×C4×C67—21no—
102C22×C4×C67—21no—
103C22×C4×C67—41no—
104C22×C4×C67—41no—
105C22×C4×C67C2×D411yescharacteristic
106C22×C4×C67C2×D411yescharacteristic
107C22×C4×C67#4111yescharacteristic
108Q16×C67—42no—
109SD16×C67—42no—
110SD16×C67—42no—
111D8×C67—42no—
112M16×C67—44no—
113M16×C67—41no—
114M16×C67—41no—
115C2×C8×C67—21no—
116C2×C8×C67—21no—
117C2×C8×C67—21no—
118C2×C8×C67—21no—
119C2×C8×C67—22no—
120C2×C8×C67—42no—
121C2×C8×C67—41no—
122C2×C8×C67—41no—
123C2×C8×C67—41no—
124C4⋊C4×C67—21no—
125C4⋊C4×C67—21no—
126C4⋊C4×C67—22no—
127C4⋊C4×C67—21no—
128C4⋊C4×C67—21no—
129C4⋊C4×C67—41no—
130C4⋊C4×C67C22⋊C411yescharacteristic
131C4⋊C4×C67C22⋊C411yescharacteristic
132C22⋊C4×C67—21no—
133C22⋊C4×C67—21no—
134C22⋊C4×C67—21no—
135C22⋊C4×C67—41no—
136C22⋊C4×C67—41no—
137C42×C67—21no—
138C42×C67—21no—
139C42×C67—21no—
140C42×C67—41no—
141C42×C67—41no—
142C42×C67C22×C411yescharacteristic
143C42×C67C2×D411yescharacteristic
144C42×C67C2×D411yescharacteristic
145C42×C67#4111yescharacteristic
146C23×C67—21no—
147C23×C67—41no—
148C23×C67#12111yescharacteristic
149Q8×C67—21no—
150Q8×C67—21no—
151Q8×C67—41no—
152Q8×C67—41no—
153D4×C67—21no—
154D4×C67—21no—
155D4×C67—41no—
156D4×C67—41no—
157D4×C67—41no—
158D4×C67—41no—
159D4×C67—41no—
160D4×C67—41no—
161D4×C67—81no—
162D4×C67—81no—
163D4×C67—81no—
164C2×C4×C67—21no—
165C2×C4×C67—21no—
166C2×C4×C67—21no—
167C2×C4×C67—21no—
168C2×C4×C67—21no—
169C2×C4×C67—41no—
170C2×C4×C67—42no—
171C2×C4×C67—41no—
172C2×C4×C67—41no—
173C2×C4×C67—41no—
174C2×C4×C67—41no—
175C2×C4×C67—42no—
176C2×C4×C67—41no—
177C2×C4×C67—81no—
178C2×C4×C67—81no—
179C2×C4×C67—81no—
180C2×C4×C67—81no—
181C2×C4×C67C2×C22⋊C411yescharacteristic
182C2×C4×C67C4×D411yescharacteristic
183C2×C4×C67C4×D411yescharacteristic
184C2×C4×C67#12011yescharacteristic
185C2×C4×C67#12111yescharacteristic
186C2×C4×C67#12311yescharacteristic
187C2×C4×C67AGL(1,ℤ/8)11yescharacteristic
188C2×C4×C67AGL(1,ℤ/8)11yescharacteristic
189C2×C4×C67AGL(1,ℤ/8)11yescharacteristic
190C2×C4×C67#13711yescharacteristic
191C8×C67—22no—
192C8×C67—41no—
193C8×C67—42no—
194C8×C67—41no—
195C8×C67—41no—
196C8×C67—81no—
197C22×C67—21no—
198C22×C67—21no—
199C22×C67—41no—
200C22×C67—41no—
201C22×C67—41no—
202C22×C67—81no—
203C22×C67—81no—
204C22×C67—81no—
205C22×C67#38611yescharacteristic
206C22×C67#46811yescharacteristic
207C22×C67#46911yescharacteristic
208C4×C67—21no—
209C4×C67—21no—
210C4×C67—21no—
211C4×C67—41no—
212C4×C67—41no—
213C4×C67—41no—
214C4×C67—82no—
215C4×C67—81no—
216C4×C67#41711yescharacteristic
217C4×C67#41811yescharacteristic
218C4×C67#44311yescharacteristic
219C4×C67#44411yescharacteristic
220C4×C67#46811yescharacteristic
221C4×C67#46911yescharacteristic
222#9210C6711yesmaximal, characteristic, direct product factor, Sylow
223C2×C67—21no—
224C2×C67—21no—
225C2×C67—81no—
226C2×C67—161no—
227C2×C67#187611yescharacteristic
228C2×C67#189511yescharacteristic
229C2×C67#190611yescharacteristic
230#3035C2×C6711yescharacteristic
231#2965C2×C6711yescharacteristic, complemented
232#1806C2×C6711yescharacteristic, complemented
233#1502C2×C6711yescharacteristic, complemented
234#1492C2×C6711yescharacteristic, complemented
235#1301C2×C6711yescharacteristic, complemented
236#1279C2×C6711yescharacteristic, complemented
237C67#921011yescharacteristic, direct product factor, Sylow
238C2×#135—21no—
239#532C22×C6711yescharacteristic
240C4×#41C22×C6711yescharacteristic
241#464—21no—
242#463—21no—
243#462C4×C6711yescharacteristic, complemented
244#461C4×C6711yescharacteristic, complemented
245#460C4×C6711yescharacteristic, complemented
246#459C4×C6711yescharacteristic, complemented
247#446—21no—
248#436—21no—
249#435—22no—
250#431—21no—
251#431C22×C6711yescharacteristic
252#423C22×C6711yescharacteristic
253C2×C4⋊C8C22×C6711yescharacteristic
254C2×#104—21no—
255#416—21no—
256C4×M16—21no—
257C2×C4×C8C22×C6711yescharacteristic
258C4×C4⋊C4C22×C6711yescharacteristic
259#339—21no—
260#327—22no—
261D4⋊C8—21no—
262D4⋊C8—21no—
263C2×#41—21no—
264C2×#41D4×C6711yescharacteristic
265#135—42no—
266#135—42no—
267C2×Q16—21no—
268C2×SD16—21no—
269C2×SD16—21no—
270C2×D8—21no—
271C2×M16—22no—
272C2×M16—21no—
273C22×C8—21no—
274C22×C8—21no—
275C4⋊Q8C2×C4×C6711yescharacteristic
276#127C2×C4×C6711yescharacteristic
277#124—21no—
278#121—21no—
279#121—21no—
280C4×Q8C2×C4×C6711yescharacteristic
281C4×Q8D4×C6711yescharacteristic
282C4×D4—21no—
283C4×D4—21no—
284C4×D4—21no—
285C4×D4—21no—
286C4×D4C2×C4×C6711yescharacteristic
287C4×D4D4×C6711yescharacteristic
288#117—21no—
289#117D4×C6711yescharacteristic
290C2×C4⋊C4—21no—
291C2×C42—21no—
292C2×C42—21no—
293C2×C42C23×C6711yescharacteristic
294C4⋊C8—22no—
295C4⋊C8—21no—
296C4⋊C8—21no—
297C4⋊C8—21no—
298C4⋊C8—21no—
299C4⋊C8C2×C4×C6711yescharacteristic
300C4⋊C8C2×C4×C6711yescharacteristic
301#104—42no—
302#103—21no—
303#103—21no—
304#102—21no—
305#102—21no—
306C22⋊C8—22no—
307C22⋊C8D4×C6712yes—
308#97—22no—
309C4×C8—21no—
310C4×C8—21no—
311C4×C8D4×C6711yescharacteristic
312C4×C8D4×C6711yescharacteristic
313#95—21no—
314#41—21no—
315#41—21no—
316#41—21no—
317#41—21no—
318#41—41no—
319#41—41no—
320#41—41no—
321#41—41no—
322C2×Q8—21no—
323C2×Q8C22⋊C4×C6711yescharacteristic
324C2×D4—21no—
325C2×D4—21no—
326C2×D4—21no—
327C2×D4—21no—
328C2×D4—41no—
329C2×D4C22⋊C4×C6711yescharacteristic
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4C2×D4×C6711yescharacteristic
336C22×C4C2×D4×C6711yescharacteristic
337C22×C4#41×C6711yescharacteristic
338Q16—42no—
339SD16—42no—
340SD16—42no—
341D8—42no—
342M16—44no—
343M16—41no—
344M16—41no—
345C2×C8—21no—
346C2×C8—21no—
347C2×C8—21no—
348C2×C8—21no—
349C2×C8—22no—
350C2×C8—42no—
351C2×C8—41no—
352C2×C8—41no—
353C2×C8—41no—
354C4⋊C4—21no—
355C4⋊C4—21no—
356C4⋊C4—22no—
357C4⋊C4—21no—
358C4⋊C4—21no—
359C4⋊C4—41no—
360C4⋊C4C22⋊C4×C6711yescharacteristic
361C4⋊C4C22⋊C4×C6711yescharacteristic
362C22⋊C4—21no—
363C22⋊C4—21no—
364C22⋊C4—21no—
365C22⋊C4—41no—
366C22⋊C4—41no—
367C42—21no—
368C42—21no—
369C42—21no—
370C42—41no—
371C42—41no—
372C42C22×C4×C6711yescharacteristic
373C42C2×D4×C6711yescharacteristic
374C42C2×D4×C6711yescharacteristic
375C42#41×C6711yescharacteristic
376C23—21no—
377C23—41no—
378C23#121×C6711yescharacteristic
379Q8—21no—
380Q8—21no—
381Q8—41no—
382Q8—41no—
383D4—21no—
384D4—21no—
385D4—41no—
386D4—41no—
387D4—41no—
388D4—41no—
389D4—41no—
390D4—41no—
391D4—81no—
392D4—81no—
393D4—81no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—41no—
400C2×C4—42no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—42no—
406C2×C4—41no—
407C2×C4—81no—
408C2×C4—81no—
409C2×C4—81no—
410C2×C4—81no—
411C2×C4C2×C22⋊C4×C6711yescharacteristic
412C2×C4C4×D4×C6711yescharacteristic
413C2×C4C4×D4×C6711yescharacteristic
414C2×C4#120×C6711yescharacteristic
415C2×C4#121×C6711yescharacteristic
416C2×C4#123×C6711yescharacteristic
417C2×C4AGL(1,ℤ/8)×C6711yescharacteristic
418C2×C4AGL(1,ℤ/8)×C6711yescharacteristic
419C2×C4AGL(1,ℤ/8)×C6711yescharacteristic
420C2×C4#137×C6711yescharacteristic
421C8—22no—
422C8—41no—
423C8—42no—
424C8—41no—
425C8—41no—
426C8—81no—
427C22—21no—
428C22—21no—
429C22—41no—
430C22—41no—
431C22—41no—
432C22—81no—
433C22—81no—
434C22—81no—
435C22#386×C6711yescharacteristic
436C22#468×C6711yescharacteristic
437C22#469×C6711yescharacteristic
438C4—21no—
439C4—21no—
440C4—21no—
441C4—41no—
442C4—41no—
443C4—41no—
444C4—82no—
445C4—81no—
446C4#417×C6711yescharacteristic
447C4#418×C6711yescharacteristic
448C4#443×C6711yescharacteristic
449C4#444×C6711yescharacteristic
450C4#468×C6711yescharacteristic
451C4#469×C6711yescharacteristic
452C2—21no—
453C2—21no—
454C2—81no—
455C2—161no—
456C2C67×#187611yescharacteristic
457C2C67×#189511yescharacteristic
458C2C67×#190611yescharacteristic
subgroupG/H
centerC2×C4×C67#121
commutatorC42C22×C4×C67
Frattini, cosocleC2×C42C23×C67
socleC22×C67#386
prime radicalC67×#3035C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4×C67 ◁ C2×C42×C67 ◁ G

Complements

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

NH ≅ G/Ntype
2. C67×#2965C2semidirect
3. C67×#1806C2semidirect
4. C67×#1502C2semidirect
5. C67×#1492C2semidirect
6. C67×#1301C2semidirect
7. C67×#1279C2semidirect
13. #462×C67C4semidirect
14. #461×C67C4semidirect
15. #460×C67C4semidirect
16. #459×C67C4semidirect
222. #9210C67direct (N × H)
231. #2965C2×C67semidirect
232. #1806C2×C67semidirect
233. #1502C2×C67semidirect
234. #1492C2×C67semidirect
235. #1301C2×C67semidirect
236. #1279C2×C67semidirect
237. C67#9210direct (N × H)
243. #462C4×C67semidirect
244. #461C4×C67semidirect
245. #460C4×C67semidirect
246. #459C4×C67semidirect

Maximal subgroups

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

subgroupindex
C67×#30352
C67×#29652
C67×#18062
C67×#15022
C67×#14922
C67×#13012
C67×#12792
#921067