Finite Group Databasebeta

← C67×#33690 #30712941

Subgroups — C67×#33690 #30712941

Order 17152.

526 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C67×#3183C211yesmaximal, characteristic, complemented
2C67×#2956C211yesmaximal, characteristic, complemented
3C67×#2239C211yesmaximal, characteristic, complemented
4C67×#2238C211yesmaximal, characteristic, complemented
5C67×#2231C212yesmaximal, complemented
6C67×#2212C212yesmaximal, complemented
7C67×#2191C211yesmaximal, characteristic, complemented
8C67×#2190C211yesmaximal, characteristic
9C67×AGL(1,ℤ/16)C212yesmaximal, complemented
10C67×#2180C212yesmaximal, complemented
11C67×#2165C211yesmaximal, characteristic, complemented
12#575×C67C2211yescharacteristic
13#568×C67—21no—
14#529×C67—21no—
15#510×C67C2212yes—
16#509×C67C2212yes—
17#499×C67C2211yescharacteristic, complemented
18#495×C67C2211yescharacteristic, complemented
19#488×C67—21no—
20#470×C67C2211yescharacteristic
21#469×C67C2211yescharacteristic
22#468×C67—21no—
23#444×C67—21no—
24#444×C67C2212yes—
25#443×C67—21no—
26#443×C67C2212yes—
27#442×C67C2211yescharacteristic, complemented
28#441×C67C2211yescharacteristic, complemented
29#440×C67—21no—
30#430×C67C2211yescharacteristic
31C2×#104×C67—21no—
32C4×M16×C67C2211yescharacteristic
33#365×C67C2212yescomplemented
34#365×C67C2212yescomplemented
35#364×C67C2212yescomplemented
36#362×C67C2212yescomplemented
37#361×C67C2212yescomplemented
38#360×C67C2212yes—
39#360×C67C2212yes—
40#349×C67C2211yescharacteristic
41#349×C67C2211yescharacteristic
42#347×C67C2212yescomplemented
43C2×#41×C67—21no—
44#137×C67—22no—
45AGL(1,ℤ/8)×C67—22no—
46#135×C67—21no—
47#135×C67C2312yes—
48C2×Q16×C67C2311yescharacteristic
49C2×SD16×C67—21no—
50C2×D8×C67C2311yescharacteristic
51#131×C67—22no—
52#131×C67—22no—
53#131×C67—42no—
54C2×M16×C67—21no—
55C2×M16×C67—42no—
56C2×M16×C67D411yescharacteristic
57C2×M16×C67C2311yescharacteristic
58#126×C67—41no—
59#125×C67—21no—
60#124×C67—21no—
61#123×C67—41no—
62C22⋊Q8×C67—21no—
63#121×C67—21no—
64C4×Q8×C67—21no—
65C4×D4×C67—21no—
66C4×D4×C67—41no—
67#117×C67—41no—
68C2×C42×C67D411yescharacteristic
69Q32×C67—22no—
70SD32×C67—22no—
71SD32×C67—22no—
72D16×C67—22no—
73M32×C67C2312yes—
74M32×C67C2312yes—
75#108×C67—21no—
76#108×C67C2312yes—
77#107×C67—21no—
78#106×C67C2311yescharacteristic
79#106×C67C2311yescharacteristic
80#104×C67—22no—
81#104×C67—22no—
82#103×C67—21no—
83#103×C67—21no—
84#102×C67—21no—
85#102×C67—21no—
86#97×C67D411yescharacteristic
87#97×C67C2311yescharacteristic
88C4×C8×C67D411yescharacteristic
89C4×C8×C67C2311yescharacteristic
90#41×C67—22no—
91#41×C67—22no—
92#41×C67—41no—
93#41×C67—41no—
94C2×Q8×C67—21no—
95C2×Q8×C67—21no—
96C2×D4×C67—21no—
97C2×D4×C67—21no—
98C2×D4×C67—41no—
99C2×D4×C67—41no—
100C22×C4×C67—21no—
101C22×C4×C67—41no—
102C22×C4×C67—41no—
103C22×C4×C67C2×D411yescharacteristic
104Q16×C67—21no—
105Q16×C67—22no—
106Q16×C67#4112yes—
107SD16×C67—22no—
108SD16×C67—22no—
109SD16×C67—42no—
110D8×C67—21no—
111D8×C67—22no—
112D8×C67#4112yes—
113M16×C67—22no—
114M16×C67—22no—
115M16×C67—22no—
116M16×C67—42no—
117M16×C67—41no—
118M16×C67—42no—
119C2×C8×C67—42no—
120C2×C8×C67—41no—
121C2×C8×C67—42no—
122C2×C8×C67C2×D411yescharacteristic
123C2×C8×C67C2×D411yescharacteristic
124C2×C8×C67C2×D411yescharacteristic
125C2×C8×C67C2411yescharacteristic
126C4⋊C4×C67—21no—
127C4⋊C4×C67—21no—
128C4⋊C4×C67—41no—
129C4⋊C4×C67—41no—
130C4⋊C4×C67—41no—
131C4⋊C4×C67—41no—
132C22⋊C4×C67—41no—
133C22⋊C4×C67—41no—
134C22⋊C4×C67—41no—
135C22⋊C4×C67—41no—
136C22⋊C4×C67—41no—
137C42×C67—21no—
138C42×C67—41no—
139C42×C67C2×D411yescharacteristic
140C42×C67C2×D411yescharacteristic
141C16×C67—22no—
142C16×C67—22no—
143C23×C67—41no—
144C23×C67—41no—
145C23×C67AGL(1,ℤ/8)11yescharacteristic
146Q8×C67—22no—
147Q8×C67—41no—
148Q8×C67—41no—
149Q8×C67—41no—
150D4×C67—22no—
151D4×C67—41no—
152D4×C67—42no—
153D4×C67—42no—
154D4×C67—41no—
155D4×C67—41no—
156D4×C67—82no—
157C2×C4×C67—21no—
158C2×C4×C67—21no—
159C2×C4×C67—21no—
160C2×C4×C67—21no—
161C2×C4×C67—42no—
162C2×C4×C67—41no—
163C2×C4×C67—42no—
164C2×C4×C67—41no—
165C2×C4×C67—41no—
166C2×C4×C67—41no—
167C2×C4×C67—81no—
168C2×C4×C67—82no—
169C2×C4×C67AGL(1,ℤ/8)11yescharacteristic
170C2×C4×C67C22×D411yescharacteristic
171C8×C67—21no—
172C8×C67—21no—
173C8×C67—42no—
174C8×C67AGL(1,ℤ/8)12yes—
175C8×C67C2×#4112yes—
176C22×C67—21no—
177C22×C67—41no—
178C22×C67—42no—
179C22×C67—81no—
180C22×C67—81no—
181C22×C67—81no—
182C22×C67—161no—
183C22×C67C2×AGL(1,ℤ/8)11yescharacteristic
184C4×C67—21no—
185C4×C67—41no—
186C4×C67—41no—
187C4×C67—42no—
188C4×C67—81no—
189C4×C67—81no—
190C4×C67—81no—
191C4×C67#52511yescharacteristic
192C4×C67C2×AGL(1,ℤ/8)11yescharacteristic
193#33690C6711yesmaximal, characteristic, direct product factor, Sylow
194C2×C67—21no—
195C2×C67—41no—
196C2×C67—82no—
197C2×C67—161no—
198C2×C67#310911yescharacteristic
199#3183C2×C6711yescharacteristic, complemented
200#2956C2×C6711yescharacteristic, complemented
201#2239C2×C6711yescharacteristic, complemented
202#2238C2×C6711yescharacteristic, complemented
203#2231C2×C6712yescomplemented
204#2212C2×C6712yescomplemented
205#2191C2×C6711yescharacteristic, complemented
206#2190C2×C6711yescharacteristic
207AGL(1,ℤ/16)C2×C6712yescomplemented
208#2180C2×C6712yescomplemented
209#2165C2×C6711yescharacteristic, complemented
210C67#3369011yescharacteristic, direct product factor, Sylow
211#575C22×C6711yescharacteristic
212#568—21no—
213#529—21no—
214#510C22×C6712yes—
215#509C22×C6712yes—
216#499C22×C6711yescharacteristic, complemented
217#495C22×C6711yescharacteristic, complemented
218#488—21no—
219#470C22×C6711yescharacteristic
220#469C22×C6711yescharacteristic
221#468—21no—
222#444—21no—
223#444C22×C6712yes—
224#443—21no—
225#443C22×C6712yes—
226#442C22×C6711yescharacteristic, complemented
227#441C22×C6711yescharacteristic, complemented
228#440—21no—
229#430C22×C6711yescharacteristic
230C2×#104—21no—
231C4×M16C22×C6711yescharacteristic
232#365C22×C6712yescomplemented
233#365C22×C6712yescomplemented
234#364C22×C6712yescomplemented
235#362C22×C6712yescomplemented
236#361C22×C6712yescomplemented
237#360C22×C6712yes—
238#360C22×C6712yes—
239#349C22×C6711yescharacteristic
240#349C22×C6711yescharacteristic
241#347C22×C6712yescomplemented
242C2×#41—21no—
243#137—22no—
244AGL(1,ℤ/8)—22no—
245#135—21no—
246#135C23×C6712yes—
247C2×Q16C23×C6711yescharacteristic
248C2×SD16—21no—
249C2×D8C23×C6711yescharacteristic
250#131—22no—
251#131—22no—
252#131—42no—
253C2×M16—21no—
254C2×M16—42no—
255C2×M16D4×C6711yescharacteristic
256C2×M16C23×C6711yescharacteristic
257#126—41no—
258#125—21no—
259#124—21no—
260#123—41no—
261C22⋊Q8—21no—
262#121—21no—
263C4×Q8—21no—
264C4×D4—21no—
265C4×D4—41no—
266#117—41no—
267C2×C42D4×C6711yescharacteristic
268Q32—22no—
269SD32—22no—
270SD32—22no—
271D16—22no—
272M32C23×C6712yes—
273M32C23×C6712yes—
274#108—21no—
275#108C23×C6712yes—
276#107—21no—
277#106C23×C6711yescharacteristic
278#106C23×C6711yescharacteristic
279#104—22no—
280#104—22no—
281#103—21no—
282#103—21no—
283#102—21no—
284#102—21no—
285#97D4×C6711yescharacteristic
286#97C23×C6711yescharacteristic
287C4×C8D4×C6711yescharacteristic
288C4×C8C23×C6711yescharacteristic
289#41—22no—
290#41—22no—
291#41—41no—
292#41—41no—
293C2×Q8—21no—
294C2×Q8—21no—
295C2×D4—21no—
296C2×D4—21no—
297C2×D4—41no—
298C2×D4—41no—
299C22×C4—21no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4C2×D4×C6711yescharacteristic
303Q16—21no—
304Q16—22no—
305Q16#41×C6712yes—
306SD16—22no—
307SD16—22no—
308SD16—42no—
309D8—21no—
310D8—22no—
311D8#41×C6712yes—
312M16—22no—
313M16—22no—
314M16—22no—
315M16—42no—
316M16—41no—
317M16—42no—
318C2×C8—42no—
319C2×C8—41no—
320C2×C8—42no—
321C2×C8C2×D4×C6711yescharacteristic
322C2×C8C2×D4×C6711yescharacteristic
323C2×C8C2×D4×C6711yescharacteristic
324C2×C8C24×C6711yescharacteristic
325C4⋊C4—21no—
326C4⋊C4—21no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4—41no—
331C22⋊C4—41no—
332C22⋊C4—41no—
333C22⋊C4—41no—
334C22⋊C4—41no—
335C22⋊C4—41no—
336C42—21no—
337C42—41no—
338C42C2×D4×C6711yescharacteristic
339C42C2×D4×C6711yescharacteristic
340C16—22no—
341C16—22no—
342C23—41no—
343C23—41no—
344C23AGL(1,ℤ/8)×C6711yescharacteristic
345Q8—22no—
346Q8—41no—
347Q8—41no—
348Q8—41no—
349D4—22no—
350D4—41no—
351D4—42no—
352D4—42no—
353D4—41no—
354D4—41no—
355D4—82no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—42no—
361C2×C4—41no—
362C2×C4—42no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—81no—
367C2×C4—82no—
368C2×C4AGL(1,ℤ/8)×C6711yescharacteristic
369C2×C4C22×D4×C6711yescharacteristic
370C8—21no—
371C8—21no—
372C8—42no—
373C8AGL(1,ℤ/8)×C6712yes—
374C8C2×#41×C6712yes—
375C22—21no—
376C22—41no—
377C22—42no—
378C22—81no—
379C22—81no—
380C22—81no—
381C22—161no—
382C22C2×AGL(1,ℤ/8)×C6711yescharacteristic
383C4—21no—
384C4—41no—
385C4—41no—
386C4—42no—
387C4—81no—
388C4—81no—
389C4—81no—
390C4#525×C6711yescharacteristic
391C4C2×AGL(1,ℤ/8)×C6711yescharacteristic
392C2—21no—
393C2—41no—
394C2—82no—
395C2—161no—
396C2C67×#310911yescharacteristic
subgroupG/H
center, socleC2×C67#3109
commutator, Frattini, cosocleC2×C8C24×C67
prime radicalC67×#2190C2
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C67 ◁ C2×C4×C67 ◁ C4×M16×C67 ◁ G

Complements

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

NH ≅ G/Ntype
1. C67×#3183C2semidirect
2. C67×#2956C2semidirect
3. C67×#2239C2semidirect
4. C67×#2238C2semidirect
5. C67×#2231C2semidirect
6. C67×#2212C2semidirect
7. C67×#2191C2semidirect
9. C67×AGL(1,ℤ/16)C2semidirect
10. C67×#2180C2semidirect
11. C67×#2165C2semidirect
17. #499×C67C22semidirect
18. #495×C67C22semidirect
27. #442×C67C22semidirect
28. #441×C67C22semidirect
33. #365×C67C22semidirect
34. #365×C67C22semidirect
35. #364×C67C22semidirect
36. #362×C67C22semidirect
37. #361×C67C22semidirect
42. #347×C67C22semidirect
193. #33690C67direct (N × H)
199. #3183C2×C67semidirect
200. #2956C2×C67semidirect
201. #2239C2×C67semidirect
202. #2238C2×C67semidirect
203. #2231C2×C67semidirect
204. #2212C2×C67semidirect
205. #2191C2×C67semidirect
207. AGL(1,ℤ/16)C2×C67semidirect
208. #2180C2×C67semidirect
209. #2165C2×C67semidirect
210. C67#33690direct (N × H)
216. #499C22×C67semidirect
217. #495C22×C67semidirect
226. #442C22×C67semidirect
227. #441C22×C67semidirect
232. #365C22×C67semidirect
233. #365C22×C67semidirect
234. #364C22×C67semidirect
235. #362C22×C67semidirect
236. #361C22×C67semidirect
241. #347C22×C67semidirect

Maximal subgroups

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

subgroupindex
C67×#31832
C67×#29562
C67×#22392
C67×#22382
C67×#22312
C67×#22122
C67×#21912
C67×#21902
C67×AGL(1,ℤ/16)2
C67×#21802
C67×#21652
#3369067