Finite Group Databasebeta

← C67×#10241 #30689492

Subgroups — C67×#10241 #30689492

Order 17152.

590 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C67×#3227C211yesmaximal, characteristic, complemented
2C67×#3036C211yesmaximal, characteristic
3C67×#2563C211yesmaximal, characteristic, complemented
4C67×#1549C211yesmaximal, characteristic, complemented
5C67×#1502C211yesmaximal, characteristic, complemented
6C67×#1478C211yesmaximal, characteristic, complemented
7C67×#1303C211yesmaximal, characteristic, complemented
8C2×AGL(1,ℤ/8)×C67—21no—
9#532×C67C2211yescharacteristic
10C2×C4⋊Q8×C67C2211yescharacteristic
11C2×C4×D4×C67C2211yescharacteristic
12#498×C67—21no—
13#486×C67—21no—
14#481×C67—22no—
15#476×C67—22no—
16#463×C67—22no—
17#461×C67—21no—
18#459×C67—21no—
19#431×C67C2211yescharacteristic
20#431×C67C2211yescharacteristic
21#423×C67C2211yescharacteristic
22#417×C67—21no—
23#393×C67—22no—
24#390×C67—21no—
25#382×C67—21no—
26#380×C67—22no—
27C4×C4⋊C4×C67C2211yescharacteristic
28#332×C67—22no—
29#327×C67—22no—
30D4⋊C8×C67—22no—
31C2×#41×C67—21no—
32C22×Q8×C67—21no—
33C22×D4×C67D411yescharacteristic
34C23×C4×C67—21no—
35AGL(1,ℤ/8)×C67—42no—
36AGL(1,ℤ/8)×C67—42no—
37C2×SD16×C67—22no—
38C2×D8×C67—22no—
39C2×M16×C67—21no—
40C4⋊Q8×C67—21no—
41C4⋊Q8×C67—21no—
42C4⋊Q8×C67—21no—
43C4⋊Q8×C67—21no—
44C4⋊Q8×C67D411yescharacteristic
45#127×C67D411yescharacteristic
46#124×C67—21no—
47#121×C67—21no—
48#121×C67—21no—
49C4×D4×C67—21no—
50C4×D4×C67—21no—
51C4×D4×C67—21no—
52C4×D4×C67—21no—
53C4×D4×C67—21no—
54C4×D4×C67—21no—
55C2×C4⋊C4×C67—21no—
56C2×C4⋊C4×C67—21no—
57C2×C4⋊C4×C67D412yes—
58C2×C4⋊C4×C67D411yescharacteristic
59C2×C22⋊C4×C67—21no—
60C2×C22⋊C4×C67—24no—
61C2×C42×C67—21no—
62C2×C42×C67C2311yescharacteristic
63#106×C67—24no—
64C4⋊C8×C67—21no—
65C4⋊C8×C67—21no—
66C4⋊C8×C67D412yes—
67C4⋊C8×C67D412yes—
68#103×C67—22no—
69#102×C67—22no—
70#102×C67—24no—
71C22⋊C8×C67—21no—
72C22⋊C8×C67—21no—
73C4×C8×C67—21no—
74C4×C8×C67—21no—
75#95×C67—24no—
76#95×C67D412yes—
77C24×C67—21no—
78#41×C67—41no—
79#41×C67—41no—
80#41×C67—41no—
81#41×C67—41no—
82C2×Q8×C67—21no—
83C2×Q8×C67—41no—
84C2×Q8×C67—41no—
85C2×Q8×C67—82no—
86C2×D4×C67—21no—
87C2×D4×C67—21no—
88C2×D4×C67—21no—
89C2×D4×C67—21no—
90C2×D4×C67—21no—
91C2×D4×C67—21no—
92C2×D4×C67—21no—
93C2×D4×C67—21no—
94C2×D4×C67—41no—
95C2×D4×C67SD1612yes—
96C22×C4×C67—22no—
97C22×C4×C67—22no—
98C22×C4×C67—21no—
99C22×C4×C67—44no—
100C22×C4×C67—41no—
101C22×C4×C67—41no—
102C22×C4×C67—41no—
103C22×C4×C67C2×D411yescharacteristic
104C22×C4×C67C2×D411yescharacteristic
105C22×C4×C67C2×D411yescharacteristic
106SD16×C67—44no—
107D8×C67—44no—
108M16×C67—41no—
109M16×C67—41no—
110C2×C8×C67—22no—
111C2×C8×C67—22no—
112C2×C8×C67—41no—
113C4⋊C4×C67—21no—
114C4⋊C4×C67—22no—
115C4⋊C4×C67—21no—
116C4⋊C4×C67—21no—
117C4⋊C4×C67—22no—
118C4⋊C4×C67—21no—
119C4⋊C4×C67—42no—
120C4⋊C4×C67—41no—
121C4⋊C4×C67—41no—
122C22⋊C4×C67—41no—
123C22⋊C4×C67—41no—
124C22⋊C4×C67—41no—
125C22⋊C4×C67—41no—
126C22⋊C4×C67—84no—
127C42×C67—41no—
128C42×C67—41no—
129C42×C67C2×D411yescharacteristic
130C42×C67C2×D411yescharacteristic
131C42×C67C2×D411yescharacteristic
132C42×C67#4111yescharacteristic
133C23×C67—22no—
134C23×C67—42no—
135C23×C67—42no—
136C23×C67—41no—
137C23×C67—41no—
138C23×C67—41no—
139C23×C67#12011yescharacteristic
140Q8×C67—41no—
141Q8×C67—41no—
142Q8×C67—82no—
143Q8×C67—82no—
144D4×C67—22no—
145D4×C67—22no—
146D4×C67—41no—
147D4×C67—41no—
148D4×C67—41no—
149D4×C67—42no—
150D4×C67—41no—
151D4×C67—81no—
152D4×C67—81no—
153D4×C67—81no—
154C2×C4×C67—22no—
155C2×C4×C67—41no—
156C2×C4×C67—42no—
157C2×C4×C67—44no—
158C2×C4×C67—42no—
159C2×C4×C67—42no—
160C2×C4×C67—41no—
161C2×C4×C67—41no—
162C2×C4×C67—42no—
163C2×C4×C67—42no—
164C2×C4×C67—82no—
165C2×C4×C67—82no—
166C2×C4×C67—81no—
167C2×C4×C67—81no—
168C2×C4×C67#12011yescharacteristic
169C2×C4×C67#12011yescharacteristic
170C2×C4×C67#12111yescharacteristic
171C2×C4×C67#12111yescharacteristic
172C2×C4×C67#12111yescharacteristic
173C2×C4×C67#12711yescharacteristic
174C2×C4×C67C2×SD1611yescharacteristic
175C2×C4×C67#13511yescharacteristic
176C2×C4×C67AGL(1,ℤ/8)11yescharacteristic
177C2×C4×C67#13711yescharacteristic
178C8×C67—42no—
179C8×C67—42no—
180C8×C67—81no—
181C22×C67—21no—
182C22×C67—21no—
183C22×C67—44no—
184C22×C67—42no—
185C22×C67—82no—
186C22×C67—81no—
187C22×C67—81no—
188C22×C67—81no—
189C22×C67—81no—
190C22×C67—81no—
191C22×C67#39211yescharacteristic
192C22×C67#44811yescharacteristic
193C22×C67#44911yescharacteristic
194C4×C67—21no—
195C4×C67—21no—
196C4×C67—21no—
197C4×C67—82no—
198C4×C67—81no—
199C4×C67—81no—
200C4×C67—82no—
201C4×C67#45011yescharacteristic
202C4×C67#45211yescharacteristic
203C4×C67#45412yes—
204C4×C67#46511yescharacteristic
205C4×C67#46811yescharacteristic
206#10241C6711yesmaximal, characteristic, direct product factor, Sylow
207C2×C67—21no—
208C2×C67—21no—
209C2×C67—82no—
210C2×C67—161no—
211C2×C67#200011yescharacteristic
212C2×C67#200311yescharacteristic
213C2×C67#200811yescharacteristic
214#3227C2×C6711yescharacteristic, complemented
215#3036C2×C6711yescharacteristic
216#2563C2×C6711yescharacteristic, complemented
217#1549C2×C6711yescharacteristic, complemented
218#1502C2×C6711yescharacteristic, complemented
219#1478C2×C6711yescharacteristic, complemented
220#1303C2×C6711yescharacteristic, complemented
221C67#1024111yescharacteristic, direct product factor, Sylow
222C2×AGL(1,ℤ/8)—21no—
223#532C22×C6711yescharacteristic
224C2×C4⋊Q8C22×C6711yescharacteristic
225C2×C4×D4C22×C6711yescharacteristic
226#498—21no—
227#486—21no—
228#481—22no—
229#476—22no—
230#463—22no—
231#461—21no—
232#459—21no—
233#431C22×C6711yescharacteristic
234#431C22×C6711yescharacteristic
235#423C22×C6711yescharacteristic
236#417—21no—
237#393—22no—
238#390—21no—
239#382—21no—
240#380—22no—
241C4×C4⋊C4C22×C6711yescharacteristic
242#332—22no—
243#327—22no—
244D4⋊C8—22no—
245C2×#41—21no—
246C22×Q8—21no—
247C22×D4D4×C6711yescharacteristic
248C23×C4—21no—
249AGL(1,ℤ/8)—42no—
250AGL(1,ℤ/8)—42no—
251C2×SD16—22no—
252C2×D8—22no—
253C2×M16—21no—
254C4⋊Q8—21no—
255C4⋊Q8—21no—
256C4⋊Q8—21no—
257C4⋊Q8—21no—
258C4⋊Q8D4×C6711yescharacteristic
259#127D4×C6711yescharacteristic
260#124—21no—
261#121—21no—
262#121—21no—
263C4×D4—21no—
264C4×D4—21no—
265C4×D4—21no—
266C4×D4—21no—
267C4×D4—21no—
268C4×D4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4D4×C6711yescharacteristic
272C2×C4⋊C4D4×C6712yes—
273C2×C22⋊C4—21no—
274C2×C22⋊C4—24no—
275C2×C42—21no—
276C2×C42C23×C6711yescharacteristic
277#106—24no—
278C4⋊C8—21no—
279C4⋊C8—21no—
280C4⋊C8D4×C6712yes—
281C4⋊C8D4×C6712yes—
282#103—22no—
283#102—22no—
284#102—24no—
285C22⋊C8—21no—
286C22⋊C8—21no—
287C4×C8—21no—
288C4×C8—21no—
289#95—24no—
290#95D4×C6712yes—
291C24—21no—
292#41—41no—
293#41—41no—
294#41—41no—
295#41—41no—
296C2×Q8—21no—
297C2×Q8—41no—
298C2×Q8—41no—
299C2×Q8—82no—
300C2×D4—21no—
301C2×D4—21no—
302C2×D4—21no—
303C2×D4—21no—
304C2×D4—21no—
305C2×D4—21no—
306C2×D4—21no—
307C2×D4—21no—
308C2×D4—41no—
309C2×D4SD16×C6712yes—
310C22×C4—21no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—44no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4C2×D4×C6711yescharacteristic
318C22×C4C2×D4×C6711yescharacteristic
319C22×C4C2×D4×C6711yescharacteristic
320SD16—44no—
321D8—44no—
322M16—41no—
323M16—41no—
324C2×C8—22no—
325C2×C8—22no—
326C2×C8—41no—
327C4⋊C4—21no—
328C4⋊C4—22no—
329C4⋊C4—21no—
330C4⋊C4—21no—
331C4⋊C4—22no—
332C4⋊C4—21no—
333C4⋊C4—42no—
334C4⋊C4—41no—
335C4⋊C4—41no—
336C22⋊C4—41no—
337C22⋊C4—41no—
338C22⋊C4—41no—
339C22⋊C4—41no—
340C22⋊C4—84no—
341C42—41no—
342C42—41no—
343C42C2×D4×C6711yescharacteristic
344C42C2×D4×C6711yescharacteristic
345C42C2×D4×C6711yescharacteristic
346C42#41×C6711yescharacteristic
347C23—22no—
348C23—42no—
349C23—42no—
350C23—41no—
351C23—41no—
352C23—41no—
353C23#120×C6711yescharacteristic
354Q8—41no—
355Q8—41no—
356Q8—82no—
357Q8—82no—
358D4—22no—
359D4—22no—
360D4—41no—
361D4—41no—
362D4—41no—
363D4—42no—
364D4—41no—
365D4—81no—
366D4—81no—
367D4—81no—
368C2×C4—22no—
369C2×C4—41no—
370C2×C4—42no—
371C2×C4—44no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—42no—
378C2×C4—82no—
379C2×C4—82no—
380C2×C4—81no—
381C2×C4—81no—
382C2×C4#120×C6711yescharacteristic
383C2×C4#120×C6711yescharacteristic
384C2×C4#121×C6711yescharacteristic
385C2×C4#121×C6711yescharacteristic
386C2×C4#121×C6711yescharacteristic
387C2×C4#127×C6711yescharacteristic
388C2×C4C2×SD16×C6711yescharacteristic
389C2×C4#135×C6711yescharacteristic
390C2×C4AGL(1,ℤ/8)×C6711yescharacteristic
391C2×C4#137×C6711yescharacteristic
392C8—42no—
393C8—42no—
394C8—81no—
395C22—21no—
396C22—21no—
397C22—44no—
398C22—42no—
399C22—82no—
400C22—81no—
401C22—81no—
402C22—81no—
403C22—81no—
404C22—81no—
405C22#392×C6711yescharacteristic
406C22#448×C6711yescharacteristic
407C22#449×C6711yescharacteristic
408C4—21no—
409C4—21no—
410C4—21no—
411C4—82no—
412C4—81no—
413C4—81no—
414C4—82no—
415C4#450×C6711yescharacteristic
416C4#452×C6711yescharacteristic
417C4#454×C6712yes—
418C4#465×C6711yescharacteristic
419C4#468×C6711yescharacteristic
420C2—21no—
421C2—21no—
422C2—82no—
423C2—161no—
424C2C67×#200011yescharacteristic
425C2C67×#200311yescharacteristic
426C2C67×#200811yescharacteristic
subgroupG/H
center, socleC22×C67#392
commutator, Frattini, cosocleC2×C42C23×C67
prime radicalC67×#3036C2
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C67 ◁ C2×C42×C67 ◁ G

Complements

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

NH ≅ G/Ntype
1. C67×#3227C2semidirect
3. C67×#2563C2semidirect
4. C67×#1549C2semidirect
5. C67×#1502C2semidirect
6. C67×#1478C2semidirect
7. C67×#1303C2semidirect
206. #10241C67direct (N × H)
214. #3227C2×C67semidirect
216. #2563C2×C67semidirect
217. #1549C2×C67semidirect
218. #1502C2×C67semidirect
219. #1478C2×C67semidirect
220. #1303C2×C67semidirect
221. C67#10241direct (N × H)

Maximal subgroups

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

subgroupindex
C67×#32272
C67×#30362
C67×#25632
C67×#15492
C67×#15022
C67×#14782
C67×#13032
#1024167