Finite Group Databasebeta

← C67×#33912 #30713163

Subgroups — C67×#33912 #30713163

Order 17152.

516 conjugacy classes of subgroups.

The subgroup lattice has 482 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C67×#3339C211yesmaximal, characteristic, complemented
2C67×#3334C211yesmaximal, characteristic, complemented
3C67×#2255C211yesmaximal, characteristic, complemented
4C67×#2234C211yesmaximal, characteristic, complemented
5C67×#2233C211yesmaximal, characteristic, complemented
6C67×#2233C211yesmaximal, characteristic, complemented
7C67×#2232C211yesmaximal, characteristic, complemented
8C67×#2228C211yesmaximal, characteristic, complemented
9C67×#2226C211yesmaximal, characteristic, complemented
10C67×#2218C211yesmaximal, characteristic, complemented
11C67×#2215C211yesmaximal, characteristic, complemented
12C67×#2177C211yesmaximal, characteristic, complemented
13C67×#2168C211yesmaximal, characteristic, complemented
14C67×#2159C211yesmaximal, characteristic, complemented
15C2×#367×C67C211yesmaximal, characteristic, complemented
16#553×C67—21no—
17#547×C67—21no—
18C2×D16×C67C2211yescharacteristic
19C2×M32×C67C2211yescharacteristic
20C22×C16×C67C2211yescharacteristic
21#500×C67C2211yescharacteristic
22#499×C67C2211yescharacteristic
23#482×C67—21no—
24#481×C67—21no—
25#480×C67—21no—
26#480×C67—21no—
27#478×C67—21no—
28#474×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
37C2×#107×C67C2211yescharacteristic
38C2×#107×C67C2211yescharacteristic
39C2×#107×C67C2211yescharacteristic
40#367×C67C2211yescharacteristic
41#367×C67C2211yescharacteristic
42#367×C67C2211yescharacteristic
43#367×C67C2211yescharacteristic
44#367×C67C2211yescharacteristic
45#367×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×#41×C67—41no—
60C2×D8×C67C2311yescharacteristic
61C2×D8×C67C2311yescharacteristic
62C22×C8×C67C2311yescharacteristic
63C22×C8×C67C2311yescharacteristic
64C4⋊Q8×C67—21no—
65#126×C67—41no—
66#125×C67—21no—
67#123×C67—41no—
68#123×C67—41no—
69#123×C67—41no—
70C22⋊Q8×C67—41no—
71#121×C67—21no—
72#121×C67—21no—
73#121×C67—21no—
74#121×C67—21no—
75#121×C67—41no—
76C4×D4×C67—21no—
77C4×D4×C67—21no—
78C4×D4×C67—41no—
79C4×D4×C67D411yescharacteristic
80#117×C67—21no—
81C2×C4⋊C4×C67—21no—
82C2×C4⋊C4×C67—21no—
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×C67—21no—
94#107×C67—21no—
95#107×C67C2311yescharacteristic
96#107×C67C2311yescharacteristic
97#107×C67C2311yescharacteristic
98#107×C67C2311yescharacteristic
99#107×C67C2311yescharacteristic
100#107×C67C2311yescharacteristic
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
113#41×C67—84no—
114C2×Q8×C67—41no—
115C2×D4×C67—21no—
116C2×D4×C67—21no—
117C2×D4×C67—41no—
118C2×D4×C67—41no—
119C2×D4×C67—41no—
120C2×D4×C67—41no—
121C2×D4×C67D811yescharacteristic
122C22×C4×C67—41no—
123C22×C4×C67—41no—
124C22×C4×C67—41no—
125C22×C4×C67—41no—
126C22×C4×C67—41no—
127C22×C4×C67C2×D411yescharacteristic
128C22×C4×C67C2×D411yescharacteristic
129D8×C67—21no—
130D8×C67—21no—
131D8×C67—21no—
132D8×C67—21no—
133C2×C8×C67—21no—
134C2×C8×C67—21no—
135C2×C8×C67—21no—
136C2×C8×C67—21no—
137C2×C8×C67C2×D411yescharacteristic
138C2×C8×C67C2×D411yescharacteristic
139C2×C8×C67C2×D411yescharacteristic
140C2×C8×C67C2411yescharacteristic
141C4⋊C4×C67—21no—
142C4⋊C4×C67—21no—
143C4⋊C4×C67—21no—
144C4⋊C4×C67—21no—
145C4⋊C4×C67—21no—
146C4⋊C4×C67—21no—
147C4⋊C4×C67—41no—
148C4⋊C4×C67—41no—
149C4⋊C4×C67—41no—
150C4⋊C4×C67—41no—
151C4⋊C4×C67—41no—
152C4⋊C4×C67—41no—
153C4⋊C4×C67D811yescharacteristic
154C22⋊C4×C67—41no—
155C22⋊C4×C67—41no—
156C22⋊C4×C67—41no—
157C22⋊C4×C67—41no—
158C22⋊C4×C67—41no—
159C22⋊C4×C67—41no—
160C22⋊C4×C67—41no—
161C22⋊C4×C67D811yescharacteristic
162C22⋊C4×C67D811yescharacteristic
163C42×C67—41no—
164C42×C67C2×D411yescharacteristic
165C16×C67—21no—
166C16×C67—21no—
167C16×C67—21no—
168C16×C67#4112yescomplemented
169C23×C67—41no—
170C23×C67—41no—
171C23×C67C2×D811yescharacteristic
172C23×C67C2×D811yescharacteristic
173Q8×C67—82no—
174D4×C67—22no—
175D4×C67—41no—
176D4×C67—41no—
177D4×C67—41no—
178D4×C67—41no—
179D4×C67—82no—
180D4×C67—82no—
181D4×C67—82no—
182D4×C67—161no—
183C2×C4×C67—21no—
184C2×C4×C67—21no—
185C2×C4×C67—21no—
186C2×C4×C67—21no—
187C2×C4×C67—41no—
188C2×C4×C67—41no—
189C2×C4×C67—41no—
190C2×C4×C67—41no—
191C2×C4×C67—41no—
192C2×C4×C67—41no—
193C2×C4×C67—82no—
194C2×C4×C67—82no—
195C2×C4×C67—82no—
196C2×C4×C67—161no—
197C2×C4×C67—161no—
198C2×C4×C67C2×D811yescharacteristic
199C2×C4×C67C2×D811yescharacteristic
200C2×C4×C67C2×D811yescharacteristic
201C2×C4×C67C2×D811yescharacteristic
202C2×C4×C67C22×D411yescharacteristic
203C8×C67—21no—
204C8×C67—21no—
205C8×C67—21no—
206C8×C67C2×#4111yescharacteristic
207C8×C672−1+411yescharacteristic
208C22×C67—21no—
209C22×C67—21no—
210C22×C67—21no—
211C22×C67—21no—
212C22×C67—21no—
213C22×C67—81no—
214C22×C67—81no—
215C22×C67—81no—
216C22×C67—81no—
217C22×C67—81no—
218C22×C67—81no—
219C22×C67#50912yes—
220C22×C67C22×D811yescharacteristic
221C4×C67—21no—
222C4×C67—21no—
223C4×C67—21no—
224C4×C67—21no—
225C4×C67—81no—
226C4×C67—81no—
227C4×C67—81no—
228C4×C67—81no—
229C4×C67—81no—
230C4×C67—81no—
231C4×C67#54711yescharacteristic
232C4×C67#57811yescharacteristic
233#33912C6711yesmaximal, characteristic, direct product factor, Sylow
234C2×C67—22no—
235C2×C67—41no—
236C2×C67—161no—
237C2×C67—161no—
238C2×C67#333411yescharacteristic
239C2×C67C2×#50911yescharacteristic
240C2×C67#341611yescharacteristic
241#3339C2×C6711yescharacteristic, complemented
242#3334C2×C6711yescharacteristic, complemented
243#2255C2×C6711yescharacteristic, complemented
244#2234C2×C6711yescharacteristic, complemented
245#2233C2×C6711yescharacteristic, complemented
246#2233C2×C6711yescharacteristic, complemented
247#2232C2×C6711yescharacteristic, complemented
248#2228C2×C6711yescharacteristic, complemented
249#2226C2×C6711yescharacteristic, complemented
250#2218C2×C6711yescharacteristic, complemented
251#2215C2×C6711yescharacteristic, complemented
252#2177C2×C6711yescharacteristic, complemented
253#2168C2×C6711yescharacteristic, complemented
254#2159C2×C6711yescharacteristic, complemented
255C2×#367C2×C6711yescharacteristic, complemented
256C67#3391211yescharacteristic, direct product factor, Sylow
257#553—21no—
258#547—21no—
259C2×D16C22×C6711yescharacteristic
260C2×M32C22×C6711yescharacteristic
261C22×C16C22×C6711yescharacteristic
262#500C22×C6711yescharacteristic
263#499C22×C6711yescharacteristic
264#482—21no—
265#481—21no—
266#480—21no—
267#480—21no—
268#478—21no—
269#474—21no—
270#466C22×C6711yescharacteristic
271#466C22×C6711yescharacteristic
272#466C22×C6711yescharacteristic
273#466C22×C6711yescharacteristic
274C4×D8C22×C6711yescharacteristic
275C4×D8C22×C6711yescharacteristic
276C8×D4C22×C6711yescharacteristic
277#427C22×C6711yescharacteristic
278C2×#107C22×C6711yescharacteristic
279C2×#107C22×C6711yescharacteristic
280C2×#107C22×C6711yescharacteristic
281#367C22×C6711yescharacteristic
282#367C22×C6711yescharacteristic
283#367C22×C6711yescharacteristic
284#367C22×C6711yescharacteristic
285#367C22×C6711yescharacteristic
286#367C22×C6711yescharacteristic
287#366C22×C6711yescharacteristic
288#366C22×C6711yescharacteristic
289#366C22×C6711yescharacteristic
290C4⋊C16C22×C6711yescharacteristic
291#357C22×C6711yescharacteristic
292#357C22×C6711yescharacteristic
293#357C22×C6711yescharacteristic
294#357C22×C6711yescharacteristic
295#357C22×C6711yescharacteristic
296#357C22×C6711yescharacteristic
297C22⋊C16C22×C6711yescharacteristic
298C22⋊C16C22×C6711yescharacteristic
299#346C22×C6711yescharacteristic
300C2×#41—41no—
301C2×D8C23×C6711yescharacteristic
302C2×D8C23×C6711yescharacteristic
303C22×C8C23×C6711yescharacteristic
304C22×C8C23×C6711yescharacteristic
305C4⋊Q8—21no—
306#126—41no—
307#125—21no—
308#123—41no—
309#123—41no—
310#123—41no—
311C22⋊Q8—41no—
312#121—21no—
313#121—21no—
314#121—21no—
315#121—21no—
316#121—41no—
317C4×D4—21no—
318C4×D4—21no—
319C4×D4—41no—
320C4×D4D4×C6711yescharacteristic
321#117—21no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—21no—
325D16—22no—
326M32—22no—
327C2×C16—22no—
328C2×C16C23×C6711yescharacteristic
329C2×C16C23×C6711yescharacteristic
330C2×C16C23×C6711yescharacteristic
331C2×C16C23×C6711yescharacteristic
332#107—21no—
333#107—21no—
334#107—21no—
335#107—21no—
336#107C23×C6711yescharacteristic
337#107C23×C6711yescharacteristic
338#107C23×C6711yescharacteristic
339#107C23×C6711yescharacteristic
340#107C23×C6711yescharacteristic
341#107C23×C6711yescharacteristic
342#106—21no—
343#106—21no—
344C4⋊C8D4×C6711yescharacteristic
345#102—21no—
346#102—21no—
347#102—21no—
348#102—21no—
349#102—21no—
350#102—21no—
351C22⋊C8D4×C6711yescharacteristic
352C22⋊C8D4×C6711yescharacteristic
353C4×C8C23×C6711yescharacteristic
354#41—84no—
355C2×Q8—41no—
356C2×D4—21no—
357C2×D4—21no—
358C2×D4—41no—
359C2×D4—41no—
360C2×D4—41no—
361C2×D4—41no—
362C2×D4D8×C6711yescharacteristic
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4C2×D4×C6711yescharacteristic
369C22×C4C2×D4×C6711yescharacteristic
370D8—21no—
371D8—21no—
372D8—21no—
373D8—21no—
374C2×C8—21no—
375C2×C8—21no—
376C2×C8—21no—
377C2×C8—21no—
378C2×C8C2×D4×C6711yescharacteristic
379C2×C8C2×D4×C6711yescharacteristic
380C2×C8C2×D4×C6711yescharacteristic
381C2×C8C24×C6711yescharacteristic
382C4⋊C4—21no—
383C4⋊C4—21no—
384C4⋊C4—21no—
385C4⋊C4—21no—
386C4⋊C4—21no—
387C4⋊C4—21no—
388C4⋊C4—41no—
389C4⋊C4—41no—
390C4⋊C4—41no—
391C4⋊C4—41no—
392C4⋊C4—41no—
393C4⋊C4—41no—
394C4⋊C4D8×C6711yescharacteristic
395C22⋊C4—41no—
396C22⋊C4—41no—
397C22⋊C4—41no—
398C22⋊C4—41no—
399C22⋊C4—41no—
400C22⋊C4—41no—
401C22⋊C4—41no—
402C22⋊C4D8×C6711yescharacteristic
403C22⋊C4D8×C6711yescharacteristic
404C42—41no—
405C42C2×D4×C6711yescharacteristic
406C16—21no—
407C16—21no—
408C16—21no—
409C16#41×C6712yescomplemented
410C23—41no—
411C23—41no—
412C23C2×D8×C6711yescharacteristic
413C23C2×D8×C6711yescharacteristic
414Q8—82no—
415D4—22no—
416D4—41no—
417D4—41no—
418D4—41no—
419D4—41no—
420D4—82no—
421D4—82no—
422D4—82no—
423D4—161no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—82no—
435C2×C4—82no—
436C2×C4—82no—
437C2×C4—161no—
438C2×C4—161no—
439C2×C4C2×D8×C6711yescharacteristic
440C2×C4C2×D8×C6711yescharacteristic
441C2×C4C2×D8×C6711yescharacteristic
442C2×C4C2×D8×C6711yescharacteristic
443C2×C4C22×D4×C6711yescharacteristic
444C8—21no—
445C8—21no—
446C8—21no—
447C8C2×#41×C6711yescharacteristic
448C82−1+4×C6711yescharacteristic
449C22—21no—
450C22—21no—
451C22—21no—
452C22—21no—
453C22—21no—
454C22—81no—
455C22—81no—
456C22—81no—
457C22—81no—
458C22—81no—
459C22—81no—
460C22#509×C6712yes—
461C22C22×D8×C6711yescharacteristic
462C4—21no—
463C4—21no—
464C4—21no—
465C4—21no—
466C4—81no—
467C4—81no—
468C4—81no—
469C4—81no—
470C4—81no—
471C4—81no—
472C4#547×C6711yescharacteristic
473C4#578×C6711yescharacteristic
474C2—22no—
475C2—41no—
476C2—161no—
477C2—161no—
478C2C67×#333411yescharacteristic
479C2C2×#509×C6711yescharacteristic
480C2C67×#341611yescharacteristic
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×#3339C2semidirect
2. C67×#3334C2semidirect
3. C67×#2255C2semidirect
4. C67×#2234C2semidirect
5. C67×#2233C2semidirect
6. C67×#2233C2semidirect
7. C67×#2232C2semidirect
8. C67×#2228C2semidirect
9. C67×#2226C2semidirect
10. C67×#2218C2semidirect
11. C67×#2215C2semidirect
12. C67×#2177C2semidirect
13. C67×#2168C2semidirect
14. C67×#2159C2semidirect
15. C2×#367×C67C2semidirect
168. C16×C67#41semidirect
233. #33912C67direct (N × H)
241. #3339C2×C67semidirect
242. #3334C2×C67semidirect
243. #2255C2×C67semidirect
244. #2234C2×C67semidirect
245. #2233C2×C67semidirect
246. #2233C2×C67semidirect
247. #2232C2×C67semidirect
248. #2228C2×C67semidirect
249. #2226C2×C67semidirect
250. #2218C2×C67semidirect
251. #2215C2×C67semidirect
252. #2177C2×C67semidirect
253. #2168C2×C67semidirect
254. #2159C2×C67semidirect
255. C2×#367C2×C67semidirect
256. C67#33912direct (N × H)
409. C16#41×C67semidirect

Maximal subgroups

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

subgroupindex
C67×#33392
C67×#33342
C67×#22552
C67×#22342
C67×#22332
C67×#22322
C67×#22282
C67×#22262
C67×#22182
C67×#22152
C67×#21772
C67×#21682
C67×#21592
C2×#367×C672
#3391267