Finite Group Databasebeta

← C67×#33911 #30713162

Subgroups — C67×#33911 #30713162

Order 17152.

472 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C67×#3338C211yesmaximal, characteristic
2C67×#3336C211yesmaximal, characteristic
3C67×#2255C211yesmaximal, characteristic, complemented
4C67×#2237C211yesmaximal, characteristic, complemented
5C67×#2236C211yesmaximal, characteristic, complemented
6C67×#2235C211yesmaximal, characteristic, complemented
7C67×#2235C211yesmaximal, characteristic, complemented
8C67×#2229C211yesmaximal, characteristic, complemented
9C67×#2225C211yesmaximal, characteristic, complemented
10C67×#2219C211yesmaximal, characteristic, complemented
11C67×#2216C211yesmaximal, characteristic, complemented
12C67×#2176C211yesmaximal, characteristic, complemented
13C67×#2168C211yesmaximal, characteristic
14C67×#2159C211yesmaximal, characteristic, complemented
15C2×#367×C67C211yesmaximal, characteristic, complemented
16#554×C67—21no—
17#552×C67—21no—
18C2×Q32×C67C2211yescharacteristic
19C2×M32×C67C2211yescharacteristic
20C22×C16×C67C2211yescharacteristic
21#500×C67C2211yescharacteristic
22#499×C67C2211yescharacteristic
23#485×C67—21no—
24#484×C67—21no—
25#484×C67—21no—
26#483×C67—21no—
27#479×C67—21no—
28#477×C67—21no—
29#467×C67C2211yescharacteristic
30#467×C67C2211yescharacteristic
31#467×C67C2211yescharacteristic
32#467×C67C2211yescharacteristic
33C4×Q16×C67C2211yescharacteristic
34C4×Q16×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#358×C67C2211yescharacteristic
51#358×C67C2211yescharacteristic
52#358×C67C2211yescharacteristic
53#358×C67C2211yescharacteristic
54#358×C67C2211yescharacteristic
55#358×C67C2211yescharacteristic
56C22⋊C16×C67C2211yescharacteristic
57C22⋊C16×C67C2211yescharacteristic
58#346×C67C2211yescharacteristic
59C2×Q16×C67C2311yescharacteristic
60C2×Q16×C67C2311yescharacteristic
61C22×C8×C67C2311yescharacteristic
62C22×C8×C67C2311yescharacteristic
63C4⋊Q8×C67—21no—
64#126×C67—41no—
65#125×C67—21no—
66#125×C67—41no—
67#125×C67—41no—
68#123×C67—41no—
69C22⋊Q8×C67—21no—
70C22⋊Q8×C67—21no—
71C22⋊Q8×C67—21no—
72C22⋊Q8×C67—21no—
73C22⋊Q8×C67—41no—
74C22⋊Q8×C67—41no—
75C4×Q8×C67—21no—
76C4×Q8×C67—21no—
77C4×D4×C67—41no—
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—
84Q32×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#103×C67—21no—
108#103×C67—21no—
109#103×C67—21no—
110C22⋊C8×C67D411yescharacteristic
111C22⋊C8×C67D411yescharacteristic
112C4×C8×C67C2311yescharacteristic
113C2×Q8×C67—21no—
114C2×Q8×C67—21no—
115C2×Q8×C67—41no—
116C2×D4×C67D811yescharacteristic
117C22×C4×C67—41no—
118C22×C4×C67—41no—
119C22×C4×C67—41no—
120C22×C4×C67C2×D411yescharacteristic
121C22×C4×C67C2×D411yescharacteristic
122Q16×C67—21no—
123Q16×C67—21no—
124Q16×C67—21no—
125Q16×C67—21no—
126C2×C8×C67—21no—
127C2×C8×C67—21no—
128C2×C8×C67—21no—
129C2×C8×C67—21no—
130C2×C8×C67C2×D411yescharacteristic
131C2×C8×C67C2×D411yescharacteristic
132C2×C8×C67C2×D411yescharacteristic
133C2×C8×C67C2411yescharacteristic
134C4⋊C4×C67—21no—
135C4⋊C4×C67—21no—
136C4⋊C4×C67—21no—
137C4⋊C4×C67—21no—
138C4⋊C4×C67—21no—
139C4⋊C4×C67—21no—
140C4⋊C4×C67—41no—
141C4⋊C4×C67—41no—
142C4⋊C4×C67—41no—
143C4⋊C4×C67—41no—
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×C67D811yescharacteristic
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×C67C2×D811yescharacteristic
169C23×C67C2×D811yescharacteristic
170Q8×C67—41no—
171Q8×C67—41no—
172Q8×C67—41no—
173Q8×C67—41no—
174Q8×C67—161no—
175D4×C67—22no—
176C2×C4×C67—21no—
177C2×C4×C67—21no—
178C2×C4×C67—21no—
179C2×C4×C67—21no—
180C2×C4×C67—41no—
181C2×C4×C67—41no—
182C2×C4×C67—41no—
183C2×C4×C67—41no—
184C2×C4×C67—41no—
185C2×C4×C67—41no—
186C2×C4×C67—41no—
187C2×C4×C67—41no—
188C2×C4×C67—82no—
189C2×C4×C67—161no—
190C2×C4×C67—161no—
191C2×C4×C67C2×D811yescharacteristic
192C2×C4×C67C2×D811yescharacteristic
193C2×C4×C67C2×D811yescharacteristic
194C2×C4×C67C2×D811yescharacteristic
195C2×C4×C67C22×D411yescharacteristic
196C8×C67—21no—
197C8×C67—21no—
198C8×C67—21no—
199C8×C67C2×#4111yescharacteristic
200C8×C672−1+411yescharacteristic
201C22×C67—21no—
202C22×C67—21no—
203C22×C67—21no—
204C22×C67—21no—
205C22×C67—21no—
206C22×C67#51012yes—
207C22×C67C22×D811yescharacteristic
208C4×C67—21no—
209C4×C67—21no—
210C4×C67—21no—
211C4×C67—21no—
212C4×C67—81no—
213C4×C67—81no—
214C4×C67—81no—
215C4×C67—81no—
216C4×C67—81no—
217C4×C67—81no—
218C4×C67—81no—
219C4×C67—81no—
220C4×C67#54711yescharacteristic
221C4×C67#57811yescharacteristic
222#33911C6711yesmaximal, characteristic, direct product factor, Sylow
223C2×C67—22no—
224C2×C67—41no—
225C2×C67#333411yescharacteristic
226C2×C67C2×#51011yescharacteristic
227C2×C67#341611yescharacteristic
228#3338C2×C6711yescharacteristic
229#3336C2×C6711yescharacteristic
230#2255C2×C6711yescharacteristic, complemented
231#2237C2×C6711yescharacteristic, complemented
232#2236C2×C6711yescharacteristic, complemented
233#2235C2×C6711yescharacteristic, complemented
234#2235C2×C6711yescharacteristic, complemented
235#2229C2×C6711yescharacteristic, complemented
236#2225C2×C6711yescharacteristic, complemented
237#2219C2×C6711yescharacteristic, complemented
238#2216C2×C6711yescharacteristic, complemented
239#2176C2×C6711yescharacteristic, complemented
240#2168C2×C6711yescharacteristic
241#2159C2×C6711yescharacteristic, complemented
242C2×#367C2×C6711yescharacteristic, complemented
243C67#3391111yescharacteristic, direct product factor, Sylow
244#554—21no—
245#552—21no—
246C2×Q32C22×C6711yescharacteristic
247C2×M32C22×C6711yescharacteristic
248C22×C16C22×C6711yescharacteristic
249#500C22×C6711yescharacteristic
250#499C22×C6711yescharacteristic
251#485—21no—
252#484—21no—
253#484—21no—
254#483—21no—
255#479—21no—
256#477—21no—
257#467C22×C6711yescharacteristic
258#467C22×C6711yescharacteristic
259#467C22×C6711yescharacteristic
260#467C22×C6711yescharacteristic
261C4×Q16C22×C6711yescharacteristic
262C4×Q16C22×C6711yescharacteristic
263C8×D4C22×C6711yescharacteristic
264#427C22×C6711yescharacteristic
265C2×#107C22×C6711yescharacteristic
266C2×#107C22×C6711yescharacteristic
267C2×#107C22×C6711yescharacteristic
268#367C22×C6711yescharacteristic
269#367C22×C6711yescharacteristic
270#367C22×C6711yescharacteristic
271#367C22×C6711yescharacteristic
272#367C22×C6711yescharacteristic
273#367C22×C6711yescharacteristic
274#366C22×C6711yescharacteristic
275#366C22×C6711yescharacteristic
276#366C22×C6711yescharacteristic
277C4⋊C16C22×C6711yescharacteristic
278#358C22×C6711yescharacteristic
279#358C22×C6711yescharacteristic
280#358C22×C6711yescharacteristic
281#358C22×C6711yescharacteristic
282#358C22×C6711yescharacteristic
283#358C22×C6711yescharacteristic
284C22⋊C16C22×C6711yescharacteristic
285C22⋊C16C22×C6711yescharacteristic
286#346C22×C6711yescharacteristic
287C2×Q16C23×C6711yescharacteristic
288C2×Q16C23×C6711yescharacteristic
289C22×C8C23×C6711yescharacteristic
290C22×C8C23×C6711yescharacteristic
291C4⋊Q8—21no—
292#126—41no—
293#125—21no—
294#125—41no—
295#125—41no—
296#123—41no—
297C22⋊Q8—21no—
298C22⋊Q8—21no—
299C22⋊Q8—21no—
300C22⋊Q8—21no—
301C22⋊Q8—41no—
302C22⋊Q8—41no—
303C4×Q8—21no—
304C4×Q8—21no—
305C4×D4—41no—
306C4×D4D4×C6711yescharacteristic
307#117—21no—
308#117—41no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312Q32—22no—
313M32—22no—
314C2×C16—22no—
315C2×C16C23×C6711yescharacteristic
316C2×C16C23×C6711yescharacteristic
317C2×C16C23×C6711yescharacteristic
318C2×C16C23×C6711yescharacteristic
319#107—21no—
320#107—21no—
321#107—21no—
322#107—21no—
323#107C23×C6711yescharacteristic
324#107C23×C6711yescharacteristic
325#107C23×C6711yescharacteristic
326#107C23×C6711yescharacteristic
327#107C23×C6711yescharacteristic
328#107C23×C6711yescharacteristic
329#106—21no—
330#106—21no—
331C4⋊C8D4×C6711yescharacteristic
332#103—21no—
333#103—21no—
334#103—21no—
335#103—21no—
336#103—21no—
337#103—21no—
338C22⋊C8D4×C6711yescharacteristic
339C22⋊C8D4×C6711yescharacteristic
340C4×C8C23×C6711yescharacteristic
341C2×Q8—21no—
342C2×Q8—21no—
343C2×Q8—41no—
344C2×D4D8×C6711yescharacteristic
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4C2×D4×C6711yescharacteristic
349C22×C4C2×D4×C6711yescharacteristic
350Q16—21no—
351Q16—21no—
352Q16—21no—
353Q16—21no—
354C2×C8—21no—
355C2×C8—21no—
356C2×C8—21no—
357C2×C8—21no—
358C2×C8C2×D4×C6711yescharacteristic
359C2×C8C2×D4×C6711yescharacteristic
360C2×C8C2×D4×C6711yescharacteristic
361C2×C8C24×C6711yescharacteristic
362C4⋊C4—21no—
363C4⋊C4—21no—
364C4⋊C4—21no—
365C4⋊C4—21no—
366C4⋊C4—21no—
367C4⋊C4—21no—
368C4⋊C4—41no—
369C4⋊C4—41no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C4⋊C4—41no—
374C4⋊C4—41no—
375C4⋊C4—41no—
376C4⋊C4—41no—
377C4⋊C4—41no—
378C4⋊C4—41no—
379C4⋊C4—41no—
380C4⋊C4D8×C6711yescharacteristic
381C22⋊C4—41no—
382C22⋊C4—41no—
383C22⋊C4—41no—
384C22⋊C4—41no—
385C22⋊C4—41no—
386C22⋊C4D8×C6711yescharacteristic
387C22⋊C4D8×C6711yescharacteristic
388C42—41no—
389C42—41no—
390C42—41no—
391C42C2×D4×C6711yescharacteristic
392C16—21no—
393C16—21no—
394C16—21no—
395C16#41×C6712yes—
396C23C2×D8×C6711yescharacteristic
397C23C2×D8×C6711yescharacteristic
398Q8—41no—
399Q8—41no—
400Q8—41no—
401Q8—41no—
402Q8—161no—
403D4—22no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—82no—
417C2×C4—161no—
418C2×C4—161no—
419C2×C4C2×D8×C6711yescharacteristic
420C2×C4C2×D8×C6711yescharacteristic
421C2×C4C2×D8×C6711yescharacteristic
422C2×C4C2×D8×C6711yescharacteristic
423C2×C4C22×D4×C6711yescharacteristic
424C8—21no—
425C8—21no—
426C8—21no—
427C8C2×#41×C6711yescharacteristic
428C82−1+4×C6711yescharacteristic
429C22—21no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22—21no—
434C22#510×C6712yes—
435C22C22×D8×C6711yescharacteristic
436C4—21no—
437C4—21no—
438C4—21no—
439C4—21no—
440C4—81no—
441C4—81no—
442C4—81no—
443C4—81no—
444C4—81no—
445C4—81no—
446C4—81no—
447C4—81no—
448C4#547×C6711yescharacteristic
449C4#578×C6711yescharacteristic
450C2—22no—
451C2—41no—
452C2C67×#333411yescharacteristic
453C2C2×#510×C6711yescharacteristic
454C2C67×#341611yescharacteristic
subgroupG/H
center, socleC22×C67C22×D8
commutator, Frattini, cosocleC2×C8C24×C67
prime radicalC2×D4×C67D8
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
3. C67×#2255C2semidirect
4. C67×#2237C2semidirect
5. C67×#2236C2semidirect
6. C67×#2235C2semidirect
7. C67×#2235C2semidirect
8. C67×#2229C2semidirect
9. C67×#2225C2semidirect
10. C67×#2219C2semidirect
11. C67×#2216C2semidirect
12. C67×#2176C2semidirect
14. C67×#2159C2semidirect
15. C2×#367×C67C2semidirect
222. #33911C67direct (N × H)
230. #2255C2×C67semidirect
231. #2237C2×C67semidirect
232. #2236C2×C67semidirect
233. #2235C2×C67semidirect
234. #2235C2×C67semidirect
235. #2229C2×C67semidirect
236. #2225C2×C67semidirect
237. #2219C2×C67semidirect
238. #2216C2×C67semidirect
239. #2176C2×C67semidirect
241. #2159C2×C67semidirect
242. C2×#367C2×C67semidirect
243. C67#33911direct (N × H)

Maximal subgroups

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

subgroupindex
C67×#33382
C67×#33362
C67×#22552
C67×#22372
C67×#22362
C67×#22352
C67×#22292
C67×#22252
C67×#22192
C67×#22162
C67×#21762
C67×#21682
C67×#21592
C2×#367×C672
#3391167