Finite Group Databasebeta

← C67×#33691 #30712942

Subgroups — C67×#33691 #30712942

Order 17152.

524 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C67×#3173C211yesmaximal, characteristic, complemented
2C67×#2955C211yesmaximal, characteristic, complemented
3C67×#2239C211yesmaximal, characteristic, complemented
4C67×#2238C211yesmaximal, characteristic, complemented
5C67×#2230C212yesmaximal, complemented
6C67×#2212C212yesmaximal, complemented
7C67×#2191C211yesmaximal, characteristic, complemented
8C67×#2190C211yesmaximal, characteristic
9C67×AGL(1,ℤ/16)C212yesmaximal, complemented
10C67×#2180C212yesmaximal, complemented
11C67×#2166C211yesmaximal, characteristic, complemented
12#575×C67C2211yescharacteristic
13C2×#131×C67—21no—
14#542×C67—21no—
15#510×C67C2212yes—
16#509×C67C2212yes—
17#498×C67C2211yescharacteristic, complemented
18#495×C67C2211yescharacteristic, complemented
19#489×C67—21no—
20#470×C67C2211yescharacteristic
21#469×C67C2211yescharacteristic
22#465×C67—21no—
23#444×C67—21no—
24#444×C67C2212yes—
25#443×C67—21no—
26#443×C67C2212yes—
27#442×C67C2211yescharacteristic, complemented
28#441×C67C2211yescharacteristic, complemented
29C4×SD16×C67—21no—
30#430×C67C2211yescharacteristic
31#421×C67—21no—
32#405×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—41no—
54#131×C67—41no—
55C2×M16×C67—21no—
56C2×M16×C67—41no—
57C2×M16×C67C2311yescharacteristic
58C22×C8×C67—41no—
59C22×C8×C67D411yescharacteristic
60C4⋊Q8×C67—21no—
61#126×C67—41no—
62#124×C67—21no—
63#124×C67—41no—
64#123×C67—41no—
65C22⋊Q8×C67—21no—
66C22⋊Q8×C67—41no—
67#121×C67—21no—
68C4×Q8×C67—21no—
69C4×D4×C67—21no—
70#117×C67D411yescharacteristic
71Q32×C67—22no—
72SD32×C67—22no—
73SD32×C67—22no—
74D16×C67—22no—
75M32×C67C2312yes—
76M32×C67C2312yes—
77#108×C67—21no—
78#108×C67C2312yes—
79#106×C67—21no—
80#106×C67C2311yescharacteristic
81#106×C67C2311yescharacteristic
82#104×C67—22no—
83#104×C67—22no—
84#103×C67—21no—
85#103×C67—21no—
86#102×C67—21no—
87#102×C67—21no—
88#97×C67D411yescharacteristic
89#97×C67C2311yescharacteristic
90C4×C8×C67D411yescharacteristic
91C4×C8×C67C2311yescharacteristic
92#41×C67—22no—
93#41×C67—22no—
94#41×C67—41no—
95#41×C67—41no—
96C2×Q8×C67—21no—
97C2×Q8×C67—21no—
98C2×Q8×C67—41no—
99C2×D4×C67—21no—
100C2×D4×C67—21no—
101C2×D4×C67—41no—
102C2×D4×C67—41no—
103C22×C4×C67—41no—
104C22×C4×C67—41no—
105C22×C4×C67C2×D411yescharacteristic
106Q16×C67—21no—
107Q16×C67—22no—
108Q16×C67#4112yes—
109SD16×C67—22no—
110SD16×C67—22no—
111SD16×C67—41no—
112SD16×C67—41no—
113D8×C67—21no—
114D8×C67—22no—
115D8×C67#4112yes—
116M16×C67—22no—
117M16×C67—22no—
118M16×C67—42no—
119M16×C67—41no—
120M16×C67—42no—
121C2×C8×C67—21no—
122C2×C8×C67—21no—
123C2×C8×C67—42no—
124C2×C8×C67—41no—
125C2×C8×C67—42no—
126C2×C8×C67C2×D411yescharacteristic
127C2×C8×C67C2×D411yescharacteristic
128C2×C8×C67C2×D411yescharacteristic
129C2×C8×C67C2411yescharacteristic
130C4⋊C4×C67—21no—
131C4⋊C4×C67—21no—
132C4⋊C4×C67—21no—
133C4⋊C4×C67—41no—
134C4⋊C4×C67—41no—
135C4⋊C4×C67—41no—
136C22⋊C4×C67—21no—
137C22⋊C4×C67—41no—
138C22⋊C4×C67—41no—
139C22⋊C4×C67—41no—
140C22⋊C4×C67—41no—
141C22⋊C4×C67—41no—
142C42×C67—41no—
143C42×C67C2×D411yescharacteristic
144C42×C67C2×D411yescharacteristic
145C16×C67—22no—
146C16×C67—22no—
147C23×C67—41no—
148C23×C67—41no—
149C23×C67AGL(1,ℤ/8)11yescharacteristic
150Q8×C67—22no—
151Q8×C67—41no—
152Q8×C67—41no—
153Q8×C67—41no—
154Q8×C67—161no—
155D4×C67—22no—
156D4×C67—41no—
157D4×C67—42no—
158D4×C67—42no—
159D4×C67—41no—
160D4×C67—41no—
161D4×C67—161no—
162C2×C4×C67—21no—
163C2×C4×C67—21no—
164C2×C4×C67—21no—
165C2×C4×C67—21no—
166C2×C4×C67—42no—
167C2×C4×C67—41no—
168C2×C4×C67—42no—
169C2×C4×C67—41no—
170C2×C4×C67—41no—
171C2×C4×C67—41no—
172C2×C4×C67—161no—
173C2×C4×C67AGL(1,ℤ/8)11yescharacteristic
174C2×C4×C67C22×D411yescharacteristic
175C8×C67—21no—
176C8×C67—21no—
177C8×C67—42no—
178C8×C67AGL(1,ℤ/8)11yescharacteristic
179C8×C67AGL(1,ℤ/8)11yescharacteristic
180C8×C67C2×#4112yes—
181C22×C67—21no—
182C22×C67—41no—
183C22×C67—42no—
184C22×C67—81no—
185C22×C67—81no—
186C22×C67—82no—
187C22×C67—81no—
188C22×C67C2×AGL(1,ℤ/8)11yescharacteristic
189C4×C67—21no—
190C4×C67—41no—
191C4×C67—41no—
192C4×C67—42no—
193C4×C67—81no—
194C4×C67—81no—
195C4×C67—81no—
196C4×C67#52511yescharacteristic
197C4×C67C2×AGL(1,ℤ/8)11yescharacteristic
198#33691C6711yesmaximal, characteristic, direct product factor, Sylow
199C2×C67—21no—
200C2×C67—41no—
201C2×C67—82no—
202C2×C67—161no—
203C2×C67#310911yescharacteristic
204#3173C2×C6711yescharacteristic, complemented
205#2955C2×C6711yescharacteristic, complemented
206#2239C2×C6711yescharacteristic, complemented
207#2238C2×C6711yescharacteristic, complemented
208#2230C2×C6712yescomplemented
209#2212C2×C6712yescomplemented
210#2191C2×C6711yescharacteristic, complemented
211#2190C2×C6711yescharacteristic
212AGL(1,ℤ/16)C2×C6712yescomplemented
213#2180C2×C6712yescomplemented
214#2166C2×C6711yescharacteristic, complemented
215C67#3369111yescharacteristic, direct product factor, Sylow
216#575C22×C6711yescharacteristic
217C2×#131—21no—
218#542—21no—
219#510C22×C6712yes—
220#509C22×C6712yes—
221#498C22×C6711yescharacteristic, complemented
222#495C22×C6711yescharacteristic, complemented
223#489—21no—
224#470C22×C6711yescharacteristic
225#469C22×C6711yescharacteristic
226#465—21no—
227#444—21no—
228#444C22×C6712yes—
229#443—21no—
230#443C22×C6712yes—
231#442C22×C6711yescharacteristic, complemented
232#441C22×C6711yescharacteristic, complemented
233C4×SD16—21no—
234#430C22×C6711yescharacteristic
235#421—21no—
236#405C22×C6711yescharacteristic
237#365C22×C6712yescomplemented
238#365C22×C6712yescomplemented
239#364C22×C6712yescomplemented
240#362C22×C6712yescomplemented
241#361C22×C6712yescomplemented
242#360C22×C6712yes—
243#360C22×C6712yes—
244#349C22×C6711yescharacteristic
245#349C22×C6711yescharacteristic
246#347C22×C6712yescomplemented
247C2×#41—21no—
248#137—22no—
249AGL(1,ℤ/8)—22no—
250#135—21no—
251#135C23×C6712yes—
252C2×Q16C23×C6711yescharacteristic
253C2×SD16—21no—
254C2×D8C23×C6711yescharacteristic
255#131—22no—
256#131—22no—
257#131—41no—
258#131—41no—
259C2×M16—21no—
260C2×M16—41no—
261C2×M16C23×C6711yescharacteristic
262C22×C8—41no—
263C22×C8D4×C6711yescharacteristic
264C4⋊Q8—21no—
265#126—41no—
266#124—21no—
267#124—41no—
268#123—41no—
269C22⋊Q8—21no—
270C22⋊Q8—41no—
271#121—21no—
272C4×Q8—21no—
273C4×D4—21no—
274#117D4×C6711yescharacteristic
275Q32—22no—
276SD32—22no—
277SD32—22no—
278D16—22no—
279M32C23×C6712yes—
280M32C23×C6712yes—
281#108—21no—
282#108C23×C6712yes—
283#106—21no—
284#106C23×C6711yescharacteristic
285#106C23×C6711yescharacteristic
286#104—22no—
287#104—22no—
288#103—21no—
289#103—21no—
290#102—21no—
291#102—21no—
292#97D4×C6711yescharacteristic
293#97C23×C6711yescharacteristic
294C4×C8D4×C6711yescharacteristic
295C4×C8C23×C6711yescharacteristic
296#41—22no—
297#41—22no—
298#41—41no—
299#41—41no—
300C2×Q8—21no—
301C2×Q8—21no—
302C2×Q8—41no—
303C2×D4—21no—
304C2×D4—21no—
305C2×D4—41no—
306C2×D4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4C2×D4×C6711yescharacteristic
310Q16—21no—
311Q16—22no—
312Q16#41×C6712yes—
313SD16—22no—
314SD16—22no—
315SD16—41no—
316SD16—41no—
317D8—21no—
318D8—22no—
319D8#41×C6712yes—
320M16—22no—
321M16—22no—
322M16—42no—
323M16—41no—
324M16—42no—
325C2×C8—21no—
326C2×C8—21no—
327C2×C8—42no—
328C2×C8—41no—
329C2×C8—42no—
330C2×C8C2×D4×C6711yescharacteristic
331C2×C8C2×D4×C6711yescharacteristic
332C2×C8C2×D4×C6711yescharacteristic
333C2×C8C24×C6711yescharacteristic
334C4⋊C4—21no—
335C4⋊C4—21no—
336C4⋊C4—21no—
337C4⋊C4—41no—
338C4⋊C4—41no—
339C4⋊C4—41no—
340C22⋊C4—21no—
341C22⋊C4—41no—
342C22⋊C4—41no—
343C22⋊C4—41no—
344C22⋊C4—41no—
345C22⋊C4—41no—
346C42—41no—
347C42C2×D4×C6711yescharacteristic
348C42C2×D4×C6711yescharacteristic
349C16—22no—
350C16—22no—
351C23—41no—
352C23—41no—
353C23AGL(1,ℤ/8)×C6711yescharacteristic
354Q8—22no—
355Q8—41no—
356Q8—41no—
357Q8—41no—
358Q8—161no—
359D4—22no—
360D4—41no—
361D4—42no—
362D4—42no—
363D4—41no—
364D4—41no—
365D4—161no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—42no—
371C2×C4—41no—
372C2×C4—42no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—161no—
377C2×C4AGL(1,ℤ/8)×C6711yescharacteristic
378C2×C4C22×D4×C6711yescharacteristic
379C8—21no—
380C8—21no—
381C8—42no—
382C8AGL(1,ℤ/8)×C6711yescharacteristic
383C8AGL(1,ℤ/8)×C6711yescharacteristic
384C8C2×#41×C6712yes—
385C22—21no—
386C22—41no—
387C22—42no—
388C22—81no—
389C22—81no—
390C22—82no—
391C22—81no—
392C22C2×AGL(1,ℤ/8)×C6711yescharacteristic
393C4—21no—
394C4—41no—
395C4—41no—
396C4—42no—
397C4—81no—
398C4—81no—
399C4—81no—
400C4#525×C6711yescharacteristic
401C4C2×AGL(1,ℤ/8)×C6711yescharacteristic
402C2—21no—
403C2—41no—
404C2—82no—
405C2—161no—
406C2C67×#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 ◁ #405×C67 ◁ G

Complements

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

NH ≅ G/Ntype
1. C67×#3173C2semidirect
2. C67×#2955C2semidirect
3. C67×#2239C2semidirect
4. C67×#2238C2semidirect
5. C67×#2230C2semidirect
6. C67×#2212C2semidirect
7. C67×#2191C2semidirect
9. C67×AGL(1,ℤ/16)C2semidirect
10. C67×#2180C2semidirect
11. C67×#2166C2semidirect
17. #498×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
198. #33691C67direct (N × H)
204. #3173C2×C67semidirect
205. #2955C2×C67semidirect
206. #2239C2×C67semidirect
207. #2238C2×C67semidirect
208. #2230C2×C67semidirect
209. #2212C2×C67semidirect
210. #2191C2×C67semidirect
212. AGL(1,ℤ/16)C2×C67semidirect
213. #2180C2×C67semidirect
214. #2166C2×C67semidirect
215. C67#33691direct (N × H)
221. #498C22×C67semidirect
222. #495C22×C67semidirect
231. #442C22×C67semidirect
232. #441C22×C67semidirect
237. #365C22×C67semidirect
238. #365C22×C67semidirect
239. #364C22×C67semidirect
240. #362C22×C67semidirect
241. #361C22×C67semidirect
246. #347C22×C67semidirect

Maximal subgroups

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

subgroupindex
C67×#31732
C67×#29552
C67×#22392
C67×#22382
C67×#22302
C67×#22122
C67×#21912
C67×#21902
C67×AGL(1,ℤ/16)2
C67×#21802
C67×#21662
#3369167