Finite Group Databasebeta

← C67×#33060 #30712311

Subgroups — C67×#33060 #30712311

Order 17152.

822 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
1C2×#486×C67C211yesmaximal, characteristic, complemented
2C67×#3079C211yesmaximal, characteristic, complemented
3C67×#3038C211yesmaximal, characteristic
4C4×#131×C67C211yesmaximal, characteristic, complemented
5C67×#1997C211yesmaximal, characteristic, complemented
6C67×#1965C211yesmaximal, characteristic, complemented
7C67×#1931C214yesmaximal, complemented
8C67×#1907C214yesmaximal, complemented
9C67×#1837C211yesmaximal, characteristic, complemented
10C2×AGL(1,ℤ/8)×C67C2212yes—
11C2×#131×C67C2212yes—
12C2×C4⋊Q8×C67C2211yescharacteristic
13C2×#127×C67C2211yescharacteristic
14C4×#41×C67C2211yescharacteristic
15#486×C67—21no—
16#486×C67—21no—
17#486×C67—22no—
18#477×C67—21no—
19#476×C67—21no—
20#475×C67—21no—
21#474×C67—21no—
22#460×C67—21no—
23#460×C67—21no—
24#459×C67—21no—
25#459×C67—21no—
26#428×C67C2212yes—
27#423×C67C2211yescharacteristic
28#423×C67C2211yescharacteristic
29#418×C67C2212yes—
30#417×C67C2212yes—
31C2×#102×C67C2214yes—
32C2×#100×C67C2214yes—
33#405×C67—22no—
34#405×C67C2212yes—
35C4×M16×C67—21no—
36C4×M16×C67C2211yescharacteristic
37C2×C4×C8×C67—21no—
38C2×C4×C8×C67C2211yescharacteristic
39#343×C67C2214yes—
40#341×C67C2214yes—
41C2×#41×C67C2311yescharacteristic
42C22×Q8×C67—21no—
43C22×D4×C67—21no—
44C22×D4×C67—21no—
45C22×D4×C67C2312yes—
46AGL(1,ℤ/8)×C67—28no—
47C2×SD16×C67—22no—
48C2×D8×C67—22no—
49#131×C67—24no—
50#131×C67—24no—
51C2×M16×C67—22no—
52C2×M16×C67C2312yes—
53C2×M16×C67C2312yes—
54C2×M16×C67C2312yes—
55C22×C8×C67—22no—
56C22×C8×C67C2312yes—
57C4⋊Q8×C67—21no—
58C4⋊Q8×C67—21no—
59C4⋊Q8×C67—21no—
60C4⋊Q8×C67—21no—
61#127×C67—21no—
62#127×C67—21no—
63#127×C67—21no—
64#127×C67—21no—
65C4×Q8×C67D411yescharacteristic
66C4×Q8×C67D411yescharacteristic
67C4×D4×C67—21no—
68C4×D4×C67—21no—
69C4×D4×C67D411yescharacteristic
70C4×D4×C67D411yescharacteristic
71#117×C67—21no—
72#117×C67C2311yescharacteristic
73C2×C4⋊C4×C67—21no—
74C2×C4⋊C4×C67C2312yes—
75C2×C42×C67—21no—
76C2×C42×C67C2311yescharacteristic
77#107×C67—22no—
78#106×C67—22no—
79C4⋊C8×C67—21no—
80C4⋊C8×C67—21no—
81C4⋊C8×C67—21no—
82C4⋊C8×C67—21no—
83#103×C67—22no—
84#102×C67—22no—
85#102×C67—24no—
86#102×C67—24no—
87#102×C67—22no—
88#102×C67—22no—
89#100×C67—28no—
90#97×C67—22no—
91#97×C67—21no—
92C4×C8×C67—22no—
93C4×C8×C67—21no—
94C4×C8×C67D411yescharacteristic
95C4×C8×C67D411yescharacteristic
96C4×C8×C67D412yes—
97C24×C67—22no—
98#41×C67—22no—
99#41×C67#4114yes—
100C2×Q8×C67—41no—
101C2×Q8×C67—41no—
102C2×Q8×C67—82no—
103C2×Q8×C67C2×D411yescharacteristic
104C2×D4×C67—21no—
105C2×D4×C67—22no—
106C2×D4×C67—22no—
107C2×D4×C67—24no—
108C2×D4×C67—24no—
109C2×D4×C67—41no—
110C2×D4×C67—41no—
111C2×D4×C67—41no—
112C2×D4×C67—41no—
113C2×D4×C67—82no—
114C2×D4×C67—82no—
115C2×D4×C67C2×D411yescharacteristic
116C22×C4×C67—21no—
117C22×C4×C67—21no—
118C22×C4×C67—21no—
119C22×C4×C67—22no—
120C22×C4×C67#4111yescharacteristic
121C22×C4×C67#4111yescharacteristic
122C22×C4×C67C2411yescharacteristic
123SD16×C67—44no—
124D8×C67—44no—
125M16×C67—28no—
126M16×C67—22no—
127M16×C67—24no—
128M16×C67—24no—
129M16×C67#4114yes—
130C2×C8×C67—28no—
131C2×C8×C67—22no—
132C2×C8×C67—22no—
133C2×C8×C67—22no—
134C2×C8×C67—22no—
135C2×C8×C67—22no—
136C2×C8×C67C2×D412yes—
137C2×C8×C67C2×D412yes—
138C4⋊C4×C67—21no—
139C4⋊C4×C67—21no—
140C4⋊C4×C67—22no—
141C4⋊C4×C67—22no—
142C4⋊C4×C67—41no—
143C4⋊C4×C67—41no—
144C4⋊C4×C67C2×D411yescharacteristic
145C4⋊C4×C67C2×D411yescharacteristic
146C22⋊C4×C67—21no—
147C22⋊C4×C67—21no—
148C22⋊C4×C67—21no—
149C42×C67—21no—
150C42×C67—21no—
151C42×C67—21no—
152C42×C67C2×D411yescharacteristic
153C42×C67C2×D411yescharacteristic
154C42×C67C2×D411yescharacteristic
155C42×C67C2×D411yescharacteristic
156C23×C67—21no—
157C23×C67—42no—
158C23×C67—44no—
159C23×C67—82no—
160C23×C67—82no—
161C23×C67C2×#4111yescharacteristic
162Q8×C67—21no—
163Q8×C67—82no—
164Q8×C67—82no—
165Q8×C67#12412yes—
166D4×C67—24no—
167D4×C67—21no—
168D4×C67—44no—
169D4×C67—42no—
170D4×C67—42no—
171D4×C67—82no—
172D4×C67—82no—
173D4×C67—82no—
174D4×C67—82no—
175D4×C67#12412yes—
176C2×C4×C67—21no—
177C2×C4×C67—24no—
178C2×C4×C67—22no—
179C2×C4×C67—22no—
180C2×C4×C67—21no—
181C2×C4×C67—21no—
182C2×C4×C67—42no—
183C2×C4×C67—42no—
184C2×C4×C67—42no—
185C2×C4×C67—82no—
186C2×C4×C67#12111yescharacteristic
187C2×C4×C67#12111yescharacteristic
188C2×C4×C67#12111yescharacteristic
189C2×C4×C67#12111yescharacteristic
190C2×C4×C67C22×D411yescharacteristic
191C2×C4×C67C22×D411yescharacteristic
192C2×C4×C67C2×#4112yes—
193C2×C4×C67C2×#4112yes—
194C8×C67—22no—
195C8×C67—24no—
196C8×C67—22no—
197C8×C67—42no—
198C8×C67—42no—
199C22×C67—21no—
200C22×C67—21no—
201C22×C67—22no—
202C22×C67—41no—
203C22×C67—41no—
204C22×C67—82no—
205C22×C67—82no—
206C22×C67—82no—
207C22×C67—164no—
208C22×C67C2×#12111yescharacteristic
209C22×C67#52912yes—
210C4×C67—22no—
211C4×C67—21no—
212C4×C67—21no—
213C4×C67—41no—
214C4×C67—41no—
215C4×C67—82no—
216C4×C67#47112yes—
217C4×C67#47212yes—
218C4×C67#52512yes—
219C4×C67C2×#12411yescharacteristic
220C4×C67#53211yescharacteristic
221#33060C6711yesmaximal, characteristic, direct product factor, Sylow
222C2×C67—22no—
223C2×C67—42no—
224C2×C67—162no—
225C2×C67#285011yescharacteristic
226C2×C67C2×#47111yescharacteristic
227C2×C67C2×#47211yescharacteristic
228C2×#486C2×C6711yescharacteristic, complemented
229#3079C2×C6711yescharacteristic, complemented
230#3038C2×C6711yescharacteristic
231C4×#131C2×C6711yescharacteristic, complemented
232#1997C2×C6711yescharacteristic, complemented
233#1965C2×C6711yescharacteristic, complemented
234#1931C2×C6714yescomplemented
235#1907C2×C6714yescomplemented
236#1837C2×C6711yescharacteristic, complemented
237C67#3306011yescharacteristic, direct product factor, Sylow
238C2×AGL(1,ℤ/8)C22×C6712yes—
239C2×#131C22×C6712yes—
240C2×C4⋊Q8C22×C6711yescharacteristic
241C2×#127C22×C6711yescharacteristic
242C4×#41C22×C6711yescharacteristic
243#486—21no—
244#486—21no—
245#486—22no—
246#477—21no—
247#476—21no—
248#475—21no—
249#474—21no—
250#460—21no—
251#460—21no—
252#459—21no—
253#459—21no—
254#428C22×C6712yes—
255#423C22×C6711yescharacteristic
256#423C22×C6711yescharacteristic
257#418C22×C6712yes—
258#417C22×C6712yes—
259C2×#102C22×C6714yes—
260C2×#100C22×C6714yes—
261#405—22no—
262#405C22×C6712yes—
263C4×M16—21no—
264C4×M16C22×C6711yescharacteristic
265C2×C4×C8—21no—
266C2×C4×C8C22×C6711yescharacteristic
267#343C22×C6714yes—
268#341C22×C6714yes—
269C2×#41C23×C6711yescharacteristic
270C22×Q8—21no—
271C22×D4—21no—
272C22×D4—21no—
273C22×D4C23×C6712yes—
274AGL(1,ℤ/8)—28no—
275C2×SD16—22no—
276C2×D8—22no—
277#131—24no—
278#131—24no—
279C2×M16—22no—
280C2×M16C23×C6712yes—
281C2×M16C23×C6712yes—
282C2×M16C23×C6712yes—
283C22×C8—22no—
284C22×C8C23×C6712yes—
285C4⋊Q8—21no—
286C4⋊Q8—21no—
287C4⋊Q8—21no—
288C4⋊Q8—21no—
289#127—21no—
290#127—21no—
291#127—21no—
292#127—21no—
293C4×Q8D4×C6711yescharacteristic
294C4×Q8D4×C6711yescharacteristic
295C4×D4—21no—
296C4×D4—21no—
297C4×D4D4×C6711yescharacteristic
298C4×D4D4×C6711yescharacteristic
299#117—21no—
300#117C23×C6711yescharacteristic
301C2×C4⋊C4—21no—
302C2×C4⋊C4C23×C6712yes—
303C2×C42—21no—
304C2×C42C23×C6711yescharacteristic
305#107—22no—
306#106—22no—
307C4⋊C8—21no—
308C4⋊C8—21no—
309C4⋊C8—21no—
310C4⋊C8—21no—
311#103—22no—
312#102—22no—
313#102—24no—
314#102—24no—
315#102—22no—
316#102—22no—
317#100—28no—
318#97—22no—
319#97—21no—
320C4×C8—22no—
321C4×C8—21no—
322C4×C8D4×C6711yescharacteristic
323C4×C8D4×C6711yescharacteristic
324C4×C8D4×C6712yes—
325C24—22no—
326#41—22no—
327#41#41×C6714yes—
328C2×Q8—41no—
329C2×Q8—41no—
330C2×Q8—82no—
331C2×Q8C2×D4×C6711yescharacteristic
332C2×D4—21no—
333C2×D4—22no—
334C2×D4—22no—
335C2×D4—24no—
336C2×D4—24no—
337C2×D4—41no—
338C2×D4—41no—
339C2×D4—41no—
340C2×D4—41no—
341C2×D4—82no—
342C2×D4—82no—
343C2×D4C2×D4×C6711yescharacteristic
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—22no—
348C22×C4#41×C6711yescharacteristic
349C22×C4#41×C6711yescharacteristic
350C22×C4C24×C6711yescharacteristic
351SD16—44no—
352D8—44no—
353M16—28no—
354M16—22no—
355M16—24no—
356M16—24no—
357M16#41×C6714yes—
358C2×C8—28no—
359C2×C8—22no—
360C2×C8—22no—
361C2×C8—22no—
362C2×C8—22no—
363C2×C8—22no—
364C2×C8C2×D4×C6712yes—
365C2×C8C2×D4×C6712yes—
366C4⋊C4—21no—
367C4⋊C4—21no—
368C4⋊C4—22no—
369C4⋊C4—22no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C4⋊C4C2×D4×C6711yescharacteristic
373C4⋊C4C2×D4×C6711yescharacteristic
374C22⋊C4—21no—
375C22⋊C4—21no—
376C22⋊C4—21no—
377C42—21no—
378C42—21no—
379C42—21no—
380C42C2×D4×C6711yescharacteristic
381C42C2×D4×C6711yescharacteristic
382C42C2×D4×C6711yescharacteristic
383C42C2×D4×C6711yescharacteristic
384C23—21no—
385C23—42no—
386C23—44no—
387C23—82no—
388C23—82no—
389C23C2×#41×C6711yescharacteristic
390Q8—21no—
391Q8—82no—
392Q8—82no—
393Q8#124×C6712yes—
394D4—24no—
395D4—21no—
396D4—44no—
397D4—42no—
398D4—42no—
399D4—82no—
400D4—82no—
401D4—82no—
402D4—82no—
403D4#124×C6712yes—
404C2×C4—21no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—24no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—82no—
414C2×C4#121×C6711yescharacteristic
415C2×C4#121×C6711yescharacteristic
416C2×C4#121×C6711yescharacteristic
417C2×C4#121×C6711yescharacteristic
418C2×C4C22×D4×C6711yescharacteristic
419C2×C4C22×D4×C6711yescharacteristic
420C2×C4C2×#41×C6712yes—
421C2×C4C2×#41×C6712yes—
422C8—22no—
423C8—24no—
424C8—22no—
425C8—42no—
426C8—42no—
427C22—21no—
428C22—21no—
429C22—22no—
430C22—41no—
431C22—41no—
432C22—82no—
433C22—82no—
434C22—82no—
435C22—164no—
436C22C2×#121×C6711yescharacteristic
437C22#529×C6712yes—
438C4—21no—
439C4—21no—
440C4—22no—
441C4—41no—
442C4—41no—
443C4—82no—
444C4#471×C6712yes—
445C4#472×C6712yes—
446C4#525×C6712yes—
447C4C2×#124×C6711yescharacteristic
448C4#532×C6711yescharacteristic
449C2—22no—
450C2—42no—
451C2—162no—
452C2C67×#285011yescharacteristic
453C2C2×#471×C6711yescharacteristic
454C2C2×#472×C6711yescharacteristic
subgroupG/H
center, socleC22×C67C2×#121
commutator, Frattini, cosocleC22×C4C24×C67
prime radicalC67×#3038C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C67 ◁ C2×C42×C67 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×#486×C67C2semidirect
2. C67×#3079C2semidirect
4. C4×#131×C67C2semidirect
5. C67×#1997C2semidirect
6. C67×#1965C2semidirect
7. C67×#1931C2semidirect
8. C67×#1907C2semidirect
9. C67×#1837C2semidirect
221. #33060C67direct (N × H)
228. C2×#486C2×C67semidirect
229. #3079C2×C67semidirect
231. C4×#131C2×C67semidirect
232. #1997C2×C67semidirect
233. #1965C2×C67semidirect
234. #1931C2×C67semidirect
235. #1907C2×C67semidirect
236. #1837C2×C67semidirect
237. C67#33060direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#486×C672
C67×#30792
C67×#30382
C4×#131×C672
C67×#19972
C67×#19652
C67×#19312
C67×#19072
C67×#18372
#3306067