Finite Group Databasebeta

← C67×#33910 #30713161

Subgroups — C67×#33910 #30713161

Order 17152.

504 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C67×#3336C211yesmaximal, characteristic, complemented
2C67×#3334C211yesmaximal, characteristic
3C67×#2255C211yesmaximal, characteristic, complemented
4C67×#2236C211yesmaximal, characteristic, complemented
5C67×#2235C211yesmaximal, characteristic, complemented
6C67×#2234C211yesmaximal, characteristic, complemented
7C67×#2232C211yesmaximal, characteristic, complemented
8C67×#2228C211yesmaximal, characteristic, complemented
9C67×#2227C211yesmaximal, characteristic, complemented
10C67×#2220C211yesmaximal, characteristic, complemented
11C67×#2217C211yesmaximal, characteristic, complemented
12C67×#2175C211yesmaximal, characteristic, complemented
13C67×#2168C211yesmaximal, characteristic, complemented
14C67×#2159C211yesmaximal, characteristic, complemented
15C2×#366×C67C211yesmaximal, characteristic, complemented
16#552×C67—21no—
17#547×C67—21no—
18C2×SD32×C67C2211yescharacteristic
19C2×M32×C67C2211yescharacteristic
20C22×C16×C67C2211yescharacteristic
21#500×C67C2211yescharacteristic
22#499×C67C2211yescharacteristic
23#485×C67—21no—
24#483×C67—21no—
25#480×C67—21no—
26#480×C67—21no—
27#479×C67—21no—
28#474×C67—21no—
29#467×C67C2211yescharacteristic
30#467×C67C2211yescharacteristic
31#466×C67C2211yescharacteristic
32#466×C67C2211yescharacteristic
33C4×Q16×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#366×C67C2211yescharacteristic
44#366×C67C2211yescharacteristic
45#366×C67C2211yescharacteristic
46#366×C67C2211yescharacteristic
47#366×C67C2211yescharacteristic
48#366×C67C2211yescharacteristic
49C4⋊C16×C67C2211yescharacteristic
50#358×C67C2211yescharacteristic
51#358×C67C2211yescharacteristic
52#358×C67C2211yescharacteristic
53#357×C67C2211yescharacteristic
54#357×C67C2211yescharacteristic
55#357×C67C2211yescharacteristic
56C22⋊C16×C67C2211yescharacteristic
57C22⋊C16×C67C2211yescharacteristic
58#346×C67C2211yescharacteristic
59C2×#41×C67—41no—
60C2×Q16×C67C2311yescharacteristic
61C2×D8×C67C2311yescharacteristic
62C22×C8×C67C2311yescharacteristic
63C22×C8×C67C2311yescharacteristic
64C4⋊Q8×C67—21no—
65#126×C67—41no—
66#125×C67—21no—
67#125×C67—41no—
68#123×C67—41no—
69#123×C67—41no—
70#123×C67—41no—
71C22⋊Q8×C67—21no—
72C22⋊Q8×C67—21no—
73C22⋊Q8×C67—41no—
74#121×C67—21no—
75#121×C67—21no—
76C4×Q8×C67—21no—
77C4×D4×C67—21no—
78C4×D4×C67D411yescharacteristic
79#117×C67—21no—
80#117×C67—41no—
81C2×C4⋊C4×C67—21no—
82C2×C4⋊C4×C67—21no—
83C2×C4⋊C4×C67—21no—
84SD32×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#103×C67—21no—
105#103×C67—21no—
106#103×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—21no—
115C2×Q8×C67—41no—
116C2×D4×C67—21no—
117C2×D4×C67—41no—
118C2×D4×C67—41no—
119C2×D4×C67D811yescharacteristic
120C22×C4×C67—41no—
121C22×C4×C67—41no—
122C22×C4×C67—41no—
123C22×C4×C67—41no—
124C22×C4×C67C2×D411yescharacteristic
125C22×C4×C67C2×D411yescharacteristic
126Q16×C67—21no—
127Q16×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×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×C67D811yescharacteristic
161C22⋊C4×C67D811yescharacteristic
162C42×C67—41no—
163C42×C67—41no—
164C42×C67C2×D411yescharacteristic
165C16×C67—21no—
166C16×C67—21no—
167C16×C67—21no—
168C16×C67#4112yescomplemented
169C23×C67—41no—
170C23×C67C2×D811yescharacteristic
171C23×C67C2×D811yescharacteristic
172Q8×C67—41no—
173Q8×C67—41no—
174Q8×C67—82no—
175D4×C67—22no—
176D4×C67—41no—
177D4×C67—41no—
178D4×C67—82no—
179D4×C67—82no—
180C2×C4×C67—21no—
181C2×C4×C67—21no—
182C2×C4×C67—21no—
183C2×C4×C67—21no—
184C2×C4×C67—41no—
185C2×C4×C67—41no—
186C2×C4×C67—41no—
187C2×C4×C67—41no—
188C2×C4×C67—41no—
189C2×C4×C67—41no—
190C2×C4×C67—41no—
191C2×C4×C67—82no—
192C2×C4×C67—82no—
193C2×C4×C67—82no—
194C2×C4×C67—82no—
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—81no—
211C22×C67—81no—
212C22×C67—81no—
213C22×C67#50912yes—
214C22×C67C22×D811yescharacteristic
215C4×C67—21no—
216C4×C67—21no—
217C4×C67—21no—
218C4×C67—21no—
219C4×C67—81no—
220C4×C67—81no—
221C4×C67—81no—
222C4×C67—81no—
223C4×C67—81no—
224C4×C67—81no—
225C4×C67—81no—
226C4×C67#54711yescharacteristic
227C4×C67#57811yescharacteristic
228#33910C6711yesmaximal, characteristic, direct product factor, Sylow
229C2×C67—22no—
230C2×C67—41no—
231C2×C67—161no—
232C2×C67#333411yescharacteristic
233C2×C67C2×#50911yescharacteristic
234C2×C67#341711yescharacteristic
235#3336C2×C6711yescharacteristic, complemented
236#3334C2×C6711yescharacteristic
237#2255C2×C6711yescharacteristic, complemented
238#2236C2×C6711yescharacteristic, complemented
239#2235C2×C6711yescharacteristic, complemented
240#2234C2×C6711yescharacteristic, complemented
241#2232C2×C6711yescharacteristic, complemented
242#2228C2×C6711yescharacteristic, complemented
243#2227C2×C6711yescharacteristic, complemented
244#2220C2×C6711yescharacteristic, complemented
245#2217C2×C6711yescharacteristic, complemented
246#2175C2×C6711yescharacteristic, complemented
247#2168C2×C6711yescharacteristic, complemented
248#2159C2×C6711yescharacteristic, complemented
249C2×#366C2×C6711yescharacteristic, complemented
250C67#3391011yescharacteristic, direct product factor, Sylow
251#552—21no—
252#547—21no—
253C2×SD32C22×C6711yescharacteristic
254C2×M32C22×C6711yescharacteristic
255C22×C16C22×C6711yescharacteristic
256#500C22×C6711yescharacteristic
257#499C22×C6711yescharacteristic
258#485—21no—
259#483—21no—
260#480—21no—
261#480—21no—
262#479—21no—
263#474—21no—
264#467C22×C6711yescharacteristic
265#467C22×C6711yescharacteristic
266#466C22×C6711yescharacteristic
267#466C22×C6711yescharacteristic
268C4×Q16C22×C6711yescharacteristic
269C4×D8C22×C6711yescharacteristic
270C8×D4C22×C6711yescharacteristic
271#427C22×C6711yescharacteristic
272C2×#107C22×C6711yescharacteristic
273C2×#107C22×C6711yescharacteristic
274C2×#107C22×C6711yescharacteristic
275#367C22×C6711yescharacteristic
276#367C22×C6711yescharacteristic
277#367C22×C6711yescharacteristic
278#366C22×C6711yescharacteristic
279#366C22×C6711yescharacteristic
280#366C22×C6711yescharacteristic
281#366C22×C6711yescharacteristic
282#366C22×C6711yescharacteristic
283#366C22×C6711yescharacteristic
284C4⋊C16C22×C6711yescharacteristic
285#358C22×C6711yescharacteristic
286#358C22×C6711yescharacteristic
287#358C22×C6711yescharacteristic
288#357C22×C6711yescharacteristic
289#357C22×C6711yescharacteristic
290#357C22×C6711yescharacteristic
291C22⋊C16C22×C6711yescharacteristic
292C22⋊C16C22×C6711yescharacteristic
293#346C22×C6711yescharacteristic
294C2×#41—41no—
295C2×Q16C23×C6711yescharacteristic
296C2×D8C23×C6711yescharacteristic
297C22×C8C23×C6711yescharacteristic
298C22×C8C23×C6711yescharacteristic
299C4⋊Q8—21no—
300#126—41no—
301#125—21no—
302#125—41no—
303#123—41no—
304#123—41no—
305#123—41no—
306C22⋊Q8—21no—
307C22⋊Q8—21no—
308C22⋊Q8—41no—
309#121—21no—
310#121—21no—
311C4×Q8—21no—
312C4×D4—21no—
313C4×D4D4×C6711yescharacteristic
314#117—21no—
315#117—41no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319SD32—22no—
320M32—22no—
321C2×C16—22no—
322C2×C16C23×C6711yescharacteristic
323C2×C16C23×C6711yescharacteristic
324C2×C16C23×C6711yescharacteristic
325C2×C16C23×C6711yescharacteristic
326#107—21no—
327#107—21no—
328#107—21no—
329#107—21no—
330#107C23×C6711yescharacteristic
331#107C23×C6711yescharacteristic
332#107C23×C6711yescharacteristic
333#107C23×C6711yescharacteristic
334#107C23×C6711yescharacteristic
335#107C23×C6711yescharacteristic
336#106—21no—
337#106—21no—
338C4⋊C8D4×C6711yescharacteristic
339#103—21no—
340#103—21no—
341#103—21no—
342#102—21no—
343#102—21no—
344#102—21no—
345C22⋊C8D4×C6711yescharacteristic
346C22⋊C8D4×C6711yescharacteristic
347C4×C8C23×C6711yescharacteristic
348#41—84no—
349C2×Q8—21no—
350C2×Q8—41no—
351C2×D4—21no—
352C2×D4—41no—
353C2×D4—41no—
354C2×D4D8×C6711yescharacteristic
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4C2×D4×C6711yescharacteristic
360C22×C4C2×D4×C6711yescharacteristic
361Q16—21no—
362Q16—21no—
363D8—21no—
364D8—21no—
365C2×C8—21no—
366C2×C8—21no—
367C2×C8—21no—
368C2×C8—21no—
369C2×C8C2×D4×C6711yescharacteristic
370C2×C8C2×D4×C6711yescharacteristic
371C2×C8C2×D4×C6711yescharacteristic
372C2×C8C24×C6711yescharacteristic
373C4⋊C4—21no—
374C4⋊C4—21no—
375C4⋊C4—21no—
376C4⋊C4—21no—
377C4⋊C4—21no—
378C4⋊C4—21no—
379C4⋊C4—41no—
380C4⋊C4—41no—
381C4⋊C4—41no—
382C4⋊C4—41no—
383C4⋊C4—41no—
384C4⋊C4—41no—
385C4⋊C4—41no—
386C4⋊C4—41no—
387C4⋊C4—41no—
388C4⋊C4D8×C6711yescharacteristic
389C22⋊C4—41no—
390C22⋊C4—41no—
391C22⋊C4—41no—
392C22⋊C4—41no—
393C22⋊C4—41no—
394C22⋊C4—41no—
395C22⋊C4D8×C6711yescharacteristic
396C22⋊C4D8×C6711yescharacteristic
397C42—41no—
398C42—41no—
399C42C2×D4×C6711yescharacteristic
400C16—21no—
401C16—21no—
402C16—21no—
403C16#41×C6712yescomplemented
404C23—41no—
405C23C2×D8×C6711yescharacteristic
406C23C2×D8×C6711yescharacteristic
407Q8—41no—
408Q8—41no—
409Q8—82no—
410D4—22no—
411D4—41no—
412D4—41no—
413D4—82no—
414D4—82no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—82no—
427C2×C4—82no—
428C2×C4—82no—
429C2×C4—82no—
430C2×C4C2×D8×C6711yescharacteristic
431C2×C4C2×D8×C6711yescharacteristic
432C2×C4C2×D8×C6711yescharacteristic
433C2×C4C2×D8×C6711yescharacteristic
434C2×C4C22×D4×C6711yescharacteristic
435C8—21no—
436C8—21no—
437C8—21no—
438C8C2×#41×C6711yescharacteristic
439C82−1+4×C6711yescharacteristic
440C22—21no—
441C22—21no—
442C22—21no—
443C22—21no—
444C22—21no—
445C22—81no—
446C22—81no—
447C22—81no—
448C22#509×C6712yes—
449C22C22×D8×C6711yescharacteristic
450C4—21no—
451C4—21no—
452C4—21no—
453C4—21no—
454C4—81no—
455C4—81no—
456C4—81no—
457C4—81no—
458C4—81no—
459C4—81no—
460C4—81no—
461C4#547×C6711yescharacteristic
462C4#578×C6711yescharacteristic
463C2—22no—
464C2—41no—
465C2—161no—
466C2C67×#333411yescharacteristic
467C2C2×#509×C6711yescharacteristic
468C2C67×#341711yescharacteristic
subgroupG/H
center, socleC22×C67C22×D8
commutator, Frattini, cosocleC2×C8C24×C67
prime radicalC67×#3334C2
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×#3336C2semidirect
3. C67×#2255C2semidirect
4. C67×#2236C2semidirect
5. C67×#2235C2semidirect
6. C67×#2234C2semidirect
7. C67×#2232C2semidirect
8. C67×#2228C2semidirect
9. C67×#2227C2semidirect
10. C67×#2220C2semidirect
11. C67×#2217C2semidirect
12. C67×#2175C2semidirect
13. C67×#2168C2semidirect
14. C67×#2159C2semidirect
15. C2×#366×C67C2semidirect
168. C16×C67#41semidirect
228. #33910C67direct (N × H)
235. #3336C2×C67semidirect
237. #2255C2×C67semidirect
238. #2236C2×C67semidirect
239. #2235C2×C67semidirect
240. #2234C2×C67semidirect
241. #2232C2×C67semidirect
242. #2228C2×C67semidirect
243. #2227C2×C67semidirect
244. #2220C2×C67semidirect
245. #2217C2×C67semidirect
246. #2175C2×C67semidirect
247. #2168C2×C67semidirect
248. #2159C2×C67semidirect
249. C2×#366C2×C67semidirect
250. C67#33910direct (N × H)
403. C16#41×C67semidirect

Maximal subgroups

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

subgroupindex
C67×#33362
C67×#33342
C67×#22552
C67×#22362
C67×#22352
C67×#22342
C67×#22322
C67×#22282
C67×#22272
C67×#22202
C67×#22172
C67×#21752
C67×#21682
C67×#21592
C2×#366×C672
#3391067