Finite Group Databasebeta

← C67×#33914 #30713165

Subgroups — C67×#33914 #30713165

Order 17152.

488 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C67×#3341C211yesmaximal, characteristic, complemented
2C67×#3339C211yesmaximal, characteristic, complemented
3C67×#2255C211yesmaximal, characteristic, complemented
4C67×#2234C211yesmaximal, characteristic, complemented
5C67×#2234C211yesmaximal, characteristic, complemented
6C67×#2234C211yesmaximal, characteristic, complemented
7C67×#2232C211yesmaximal, characteristic, complemented
8C67×#2228C211yesmaximal, characteristic, complemented
9C67×#2224C211yesmaximal, characteristic, complemented
10C67×#2218C211yesmaximal, characteristic, complemented
11C67×#2215C211yesmaximal, characteristic, complemented
12C67×#2177C211yesmaximal, characteristic, complemented
13C67×#2168C211yesmaximal, characteristic, complemented
14C67×#2159C211yesmaximal, characteristic, complemented
15C67×#2158C211yesmaximal, characteristic, complemented
16#555×C67—21no—
17#553×C67—21no—
18C2×D16×C67C2211yescharacteristic
19C2×M32×C67C2211yescharacteristic
20C22×C16×C67C2211yescharacteristic
21#500×C67C2211yescharacteristic
22#499×C67C2211yescharacteristic
23#482×C67—21no—
24#482×C67—21no—
25#482×C67—21no—
26#481×C67—21no—
27#478×C67—21no—
28#476×C67—21no—
29#466×C67C2211yescharacteristic
30#466×C67C2211yescharacteristic
31#466×C67C2211yescharacteristic
32#466×C67C2211yescharacteristic
33C4×D8×C67C2211yescharacteristic
34C4×D8×C67C2211yescharacteristic
35C8×D4×C67C2211yescharacteristic
36#427×C67C2211yescharacteristic
37#427×C67C2211yescharacteristic
38#427×C67C2211yescharacteristic
39C2×#107×C67C2211yescharacteristic
40#367×C67C2211yescharacteristic
41#367×C67C2211yescharacteristic
42#367×C67C2211yescharacteristic
43#367×C67C2211yescharacteristic
44#366×C67C2211yescharacteristic
45#366×C67C2211yescharacteristic
46#366×C67C2211yescharacteristic
47#366×C67C2211yescharacteristic
48#366×C67C2211yescharacteristic
49C4⋊C16×C67C2211yescharacteristic
50#357×C67C2211yescharacteristic
51#357×C67C2211yescharacteristic
52#357×C67C2211yescharacteristic
53#357×C67C2211yescharacteristic
54#357×C67C2211yescharacteristic
55#357×C67C2211yescharacteristic
56C22⋊C16×C67C2211yescharacteristic
57C22⋊C16×C67C2211yescharacteristic
58#346×C67C2211yescharacteristic
59C2×D8×C67C2311yescharacteristic
60C2×D8×C67C2311yescharacteristic
61C22×C8×C67C2311yescharacteristic
62C22×C8×C67C2311yescharacteristic
63C4⋊Q8×C67—21no—
64#126×C67—41no—
65#125×C67—21no—
66#124×C67—41no—
67#124×C67—41no—
68#123×C67—41no—
69#121×C67—21no—
70#121×C67—21no—
71#121×C67—21no—
72#121×C67—21no—
73#121×C67—41no—
74#121×C67—41no—
75C4×D4×C67—21no—
76C4×D4×C67—21no—
77C4×D4×C67—41no—
78C4×D4×C67D411yescharacteristic
79#117×C67—21no—
80#117×C67—21no—
81#117×C67—21no—
82#117×C67—41no—
83C2×C4⋊C4×C67—21no—
84D16×C67—22no—
85M32×C67—22no—
86C2×C16×C67—22no—
87C2×C16×C67C2311yescharacteristic
88C2×C16×C67C2311yescharacteristic
89C2×C16×C67C2311yescharacteristic
90C2×C16×C67C2311yescharacteristic
91#107×C67—21no—
92#107×C67—21no—
93#107×C67C2311yescharacteristic
94#107×C67C2311yescharacteristic
95#107×C67C2311yescharacteristic
96#107×C67C2311yescharacteristic
97#107×C67C2311yescharacteristic
98#107×C67C2311yescharacteristic
99#106×C67—21no—
100#106×C67—21no—
101#106×C67—21no—
102#106×C67—21no—
103C4⋊C8×C67D411yescharacteristic
104#102×C67—21no—
105#102×C67—21no—
106#102×C67—21no—
107#102×C67—21no—
108#102×C67—21no—
109#102×C67—21no—
110C22⋊C8×C67D411yescharacteristic
111C22⋊C8×C67D411yescharacteristic
112C4×C8×C67C2311yescharacteristic
113C2×Q8×C67—41no—
114C2×D4×C67—21no—
115C2×D4×C67—21no—
116C2×D4×C67—41no—
117C2×D4×C67—41no—
118C2×D4×C67—41no—
119C2×D4×C67—41no—
120C2×D4×C67D811yescharacteristic
121C22×C4×C67—41no—
122C22×C4×C67—41no—
123C22×C4×C67—41no—
124C22×C4×C67C2×D411yescharacteristic
125C22×C4×C67C2×D411yescharacteristic
126D8×C67—21no—
127D8×C67—21no—
128D8×C67—21no—
129D8×C67—21no—
130C2×C8×C67—21no—
131C2×C8×C67—21no—
132C2×C8×C67—21no—
133C2×C8×C67—21no—
134C2×C8×C67C2×D411yescharacteristic
135C2×C8×C67C2×D411yescharacteristic
136C2×C8×C67C2×D411yescharacteristic
137C2×C8×C67C2411yescharacteristic
138C4⋊C4×C67—21no—
139C4⋊C4×C67—21no—
140C4⋊C4×C67—21no—
141C4⋊C4×C67—21no—
142C4⋊C4×C67—21no—
143C4⋊C4×C67—21no—
144C4⋊C4×C67—41no—
145C4⋊C4×C67—41no—
146C4⋊C4×C67—41no—
147C4⋊C4×C67—41no—
148C4⋊C4×C67D811yescharacteristic
149C22⋊C4×C67—41no—
150C22⋊C4×C67—41no—
151C22⋊C4×C67—41no—
152C22⋊C4×C67—41no—
153C22⋊C4×C67—41no—
154C22⋊C4×C67—41no—
155C22⋊C4×C67—41no—
156C22⋊C4×C67—41no—
157C22⋊C4×C67—41no—
158C22⋊C4×C67D811yescharacteristic
159C22⋊C4×C67D811yescharacteristic
160C42×C67—41no—
161C42×C67—41no—
162C42×C67—41no—
163C42×C67C2×D411yescharacteristic
164C16×C67—21no—
165C16×C67—21no—
166C16×C67—21no—
167C16×C67#4112yes—
168C23×C67—41no—
169C23×C67—41no—
170C23×C67C2×D811yescharacteristic
171C23×C67C2×D811yescharacteristic
172Q8×C67—161no—
173D4×C67—41no—
174D4×C67—41no—
175D4×C67—41no—
176D4×C67—41no—
177D4×C67—41no—
178D4×C67—82no—
179D4×C67—161no—
180D4×C67—161no—
181D4×C67—161no—
182C2×C4×C67—21no—
183C2×C4×C67—21no—
184C2×C4×C67—21no—
185C2×C4×C67—21no—
186C2×C4×C67—41no—
187C2×C4×C67—41no—
188C2×C4×C67—41no—
189C2×C4×C67—41no—
190C2×C4×C67—41no—
191C2×C4×C67—41no—
192C2×C4×C67—82no—
193C2×C4×C67—161no—
194C2×C4×C67—161no—
195C2×C4×C67C2×D811yescharacteristic
196C2×C4×C67C2×D811yescharacteristic
197C2×C4×C67C2×D811yescharacteristic
198C2×C4×C67C2×D811yescharacteristic
199C2×C4×C67C22×D411yescharacteristic
200C8×C67—21no—
201C8×C67—21no—
202C8×C67—21no—
203C8×C67C2×#4111yescharacteristic
204C8×C672−1+411yescharacteristic
205C22×C67—21no—
206C22×C67—21no—
207C22×C67—21no—
208C22×C67—21no—
209C22×C67—21no—
210C22×C67—21no—
211C22×C67—81no—
212C22×C67—81no—
213C22×C67—81no—
214C22×C67—81no—
215C22×C67—81no—
216C22×C67—81no—
217C22×C67C22×D811yescharacteristic
218C4×C67—21no—
219C4×C67—21no—
220C4×C67—21no—
221C4×C67—21no—
222C4×C67—81no—
223C4×C67—81no—
224C4×C67—81no—
225C4×C67—81no—
226C4×C67—81no—
227C4×C67—81no—
228C4×C67#54711yescharacteristic
229C4×C67#57811yescharacteristic
230#33914C6711yesmaximal, characteristic, direct product factor, Sylow
231C2×C67—41no—
232C2×C67—41no—
233C2×C67—161no—
234C2×C67—161no—
235C2×C67#333411yescharacteristic
236C2×C67#341411yescharacteristic
237C2×C67#341511yescharacteristic
238#3341C2×C6711yescharacteristic, complemented
239#3339C2×C6711yescharacteristic, complemented
240#2255C2×C6711yescharacteristic, complemented
241#2234C2×C6711yescharacteristic, complemented
242#2234C2×C6711yescharacteristic, complemented
243#2234C2×C6711yescharacteristic, complemented
244#2232C2×C6711yescharacteristic, complemented
245#2228C2×C6711yescharacteristic, complemented
246#2224C2×C6711yescharacteristic, complemented
247#2218C2×C6711yescharacteristic, complemented
248#2215C2×C6711yescharacteristic, complemented
249#2177C2×C6711yescharacteristic, complemented
250#2168C2×C6711yescharacteristic, complemented
251#2159C2×C6711yescharacteristic, complemented
252#2158C2×C6711yescharacteristic, complemented
253C67#3391411yescharacteristic, direct product factor, Sylow
254#555—21no—
255#553—21no—
256C2×D16C22×C6711yescharacteristic
257C2×M32C22×C6711yescharacteristic
258C22×C16C22×C6711yescharacteristic
259#500C22×C6711yescharacteristic
260#499C22×C6711yescharacteristic
261#482—21no—
262#482—21no—
263#482—21no—
264#481—21no—
265#478—21no—
266#476—21no—
267#466C22×C6711yescharacteristic
268#466C22×C6711yescharacteristic
269#466C22×C6711yescharacteristic
270#466C22×C6711yescharacteristic
271C4×D8C22×C6711yescharacteristic
272C4×D8C22×C6711yescharacteristic
273C8×D4C22×C6711yescharacteristic
274#427C22×C6711yescharacteristic
275#427C22×C6711yescharacteristic
276#427C22×C6711yescharacteristic
277C2×#107C22×C6711yescharacteristic
278#367C22×C6711yescharacteristic
279#367C22×C6711yescharacteristic
280#367C22×C6711yescharacteristic
281#367C22×C6711yescharacteristic
282#366C22×C6711yescharacteristic
283#366C22×C6711yescharacteristic
284#366C22×C6711yescharacteristic
285#366C22×C6711yescharacteristic
286#366C22×C6711yescharacteristic
287C4⋊C16C22×C6711yescharacteristic
288#357C22×C6711yescharacteristic
289#357C22×C6711yescharacteristic
290#357C22×C6711yescharacteristic
291#357C22×C6711yescharacteristic
292#357C22×C6711yescharacteristic
293#357C22×C6711yescharacteristic
294C22⋊C16C22×C6711yescharacteristic
295C22⋊C16C22×C6711yescharacteristic
296#346C22×C6711yescharacteristic
297C2×D8C23×C6711yescharacteristic
298C2×D8C23×C6711yescharacteristic
299C22×C8C23×C6711yescharacteristic
300C22×C8C23×C6711yescharacteristic
301C4⋊Q8—21no—
302#126—41no—
303#125—21no—
304#124—41no—
305#124—41no—
306#123—41no—
307#121—21no—
308#121—21no—
309#121—21no—
310#121—21no—
311#121—41no—
312#121—41no—
313C4×D4—21no—
314C4×D4—21no—
315C4×D4—41no—
316C4×D4D4×C6711yescharacteristic
317#117—21no—
318#117—21no—
319#117—21no—
320#117—41no—
321C2×C4⋊C4—21no—
322D16—22no—
323M32—22no—
324C2×C16—22no—
325C2×C16C23×C6711yescharacteristic
326C2×C16C23×C6711yescharacteristic
327C2×C16C23×C6711yescharacteristic
328C2×C16C23×C6711yescharacteristic
329#107—21no—
330#107—21no—
331#107C23×C6711yescharacteristic
332#107C23×C6711yescharacteristic
333#107C23×C6711yescharacteristic
334#107C23×C6711yescharacteristic
335#107C23×C6711yescharacteristic
336#107C23×C6711yescharacteristic
337#106—21no—
338#106—21no—
339#106—21no—
340#106—21no—
341C4⋊C8D4×C6711yescharacteristic
342#102—21no—
343#102—21no—
344#102—21no—
345#102—21no—
346#102—21no—
347#102—21no—
348C22⋊C8D4×C6711yescharacteristic
349C22⋊C8D4×C6711yescharacteristic
350C4×C8C23×C6711yescharacteristic
351C2×Q8—41no—
352C2×D4—21no—
353C2×D4—21no—
354C2×D4—41no—
355C2×D4—41no—
356C2×D4—41no—
357C2×D4—41no—
358C2×D4D8×C6711yescharacteristic
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4C2×D4×C6711yescharacteristic
363C22×C4C2×D4×C6711yescharacteristic
364D8—21no—
365D8—21no—
366D8—21no—
367D8—21no—
368C2×C8—21no—
369C2×C8—21no—
370C2×C8—21no—
371C2×C8—21no—
372C2×C8C2×D4×C6711yescharacteristic
373C2×C8C2×D4×C6711yescharacteristic
374C2×C8C2×D4×C6711yescharacteristic
375C2×C8C24×C6711yescharacteristic
376C4⋊C4—21no—
377C4⋊C4—21no—
378C4⋊C4—21no—
379C4⋊C4—21no—
380C4⋊C4—21no—
381C4⋊C4—21no—
382C4⋊C4—41no—
383C4⋊C4—41no—
384C4⋊C4—41no—
385C4⋊C4—41no—
386C4⋊C4D8×C6711yescharacteristic
387C22⋊C4—41no—
388C22⋊C4—41no—
389C22⋊C4—41no—
390C22⋊C4—41no—
391C22⋊C4—41no—
392C22⋊C4—41no—
393C22⋊C4—41no—
394C22⋊C4—41no—
395C22⋊C4—41no—
396C22⋊C4D8×C6711yescharacteristic
397C22⋊C4D8×C6711yescharacteristic
398C42—41no—
399C42—41no—
400C42—41no—
401C42C2×D4×C6711yescharacteristic
402C16—21no—
403C16—21no—
404C16—21no—
405C16#41×C6712yes—
406C23—41no—
407C23—41no—
408C23C2×D8×C6711yescharacteristic
409C23C2×D8×C6711yescharacteristic
410Q8—161no—
411D4—41no—
412D4—41no—
413D4—41no—
414D4—41no—
415D4—41no—
416D4—82no—
417D4—161no—
418D4—161no—
419D4—161no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—82no—
431C2×C4—161no—
432C2×C4—161no—
433C2×C4C2×D8×C6711yescharacteristic
434C2×C4C2×D8×C6711yescharacteristic
435C2×C4C2×D8×C6711yescharacteristic
436C2×C4C2×D8×C6711yescharacteristic
437C2×C4C22×D4×C6711yescharacteristic
438C8—21no—
439C8—21no—
440C8—21no—
441C8C2×#41×C6711yescharacteristic
442C82−1+4×C6711yescharacteristic
443C22—21no—
444C22—21no—
445C22—21no—
446C22—21no—
447C22—21no—
448C22—21no—
449C22—81no—
450C22—81no—
451C22—81no—
452C22—81no—
453C22—81no—
454C22—81no—
455C22C22×D8×C6711yescharacteristic
456C4—21no—
457C4—21no—
458C4—21no—
459C4—21no—
460C4—81no—
461C4—81no—
462C4—81no—
463C4—81no—
464C4—81no—
465C4—81no—
466C4#547×C6711yescharacteristic
467C4#578×C6711yescharacteristic
468C2—41no—
469C2—41no—
470C2—161no—
471C2—161no—
472C2C67×#333411yescharacteristic
473C2C67×#341411yescharacteristic
474C2C67×#341511yescharacteristic
subgroupG/H
center, socleC22×C67C22×D8
commutator, Frattini, cosocleC2×C8C24×C67
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C67 ◁ C4×D4×C67 ◁ C8×D4×C67 ◁ G

Complements

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

NH ≅ G/Ntype
1. C67×#3341C2semidirect
2. C67×#3339C2semidirect
3. C67×#2255C2semidirect
4. C67×#2234C2semidirect
5. C67×#2234C2semidirect
6. C67×#2234C2semidirect
7. C67×#2232C2semidirect
8. C67×#2228C2semidirect
9. C67×#2224C2semidirect
10. C67×#2218C2semidirect
11. C67×#2215C2semidirect
12. C67×#2177C2semidirect
13. C67×#2168C2semidirect
14. C67×#2159C2semidirect
15. C67×#2158C2semidirect
230. #33914C67direct (N × H)
238. #3341C2×C67semidirect
239. #3339C2×C67semidirect
240. #2255C2×C67semidirect
241. #2234C2×C67semidirect
242. #2234C2×C67semidirect
243. #2234C2×C67semidirect
244. #2232C2×C67semidirect
245. #2228C2×C67semidirect
246. #2224C2×C67semidirect
247. #2218C2×C67semidirect
248. #2215C2×C67semidirect
249. #2177C2×C67semidirect
250. #2168C2×C67semidirect
251. #2159C2×C67semidirect
252. #2158C2×C67semidirect
253. C67#33914direct (N × H)

Maximal subgroups

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

subgroupindex
C67×#33412
C67×#33392
C67×#22552
C67×#22342
C67×#22322
C67×#22282
C67×#22242
C67×#22182
C67×#22152
C67×#21772
C67×#21682
C67×#21592
C67×#21582
#3391467