Finite Group Databasebeta

← C67×#61695 #30740946

Subgroups — C67×#61695 #30740946

Order 17152.

2018 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C67×#3501C211yesmaximal, characteristic, complemented
2C2×#547×C67C211yesmaximal, characteristic
3C67×#3335C218yesmaximal, complemented
4C67×#3298C218yesmaximal, complemented
5C67×#3229C212yesmaximal, complemented
6C67×#3187C214yesmaximal, complemented
7C67×#3101C212yesmaximal, complemented
8C2×#483×C67C214yesmaximal, complemented
9C67×#2972C211yesmaximal, characteristic, complemented
10C22×#41×C67—21no—
11#554×C67—24no—
12#554×C67C2214yescomplemented
13D4×Q8×C67C2214yescomplemented
14#547×C67—24no—
15#547×C67C2218yes—
16#544×C67—21no—
17#543×C67—22no—
18#533×C67C2212yes—
19C2×C4⋊Q8×C67C2211yescharacteristic
20C2×#123×C67—22no—
21C2×C22⋊Q8×C67—21no—
22C2×C22⋊Q8×C67C2214yes—
23C2×#121×C67C2212yes—
24#518×C67—21no—
25C2×C4×D4×C67C2211yescharacteristic
26C2×C4×D4×C67C2211yescharacteristic
27C22×C4⋊C4×C67C2212yes—
28#498×C67C2214yescomplemented
29#487×C67C2214yescomplemented
30#483×C67C2214yes—
31#483×C67C2218yes—
32#483×C67C22116yescomplemented
33#481×C67C2218yescomplemented
34#476×C67C2214yescomplemented
35#475×C67C2218yescomplemented
36#465×C67C22116yescomplemented
37#461×C67C2214yescomplemented
38#450×C67C2214yes—
39#448×C67C2218yescomplemented
40C4×SD16×C67C2218yescomplemented
41C8×D4×C67C2218yescomplemented
42#431×C67C2212yes—
43C2×#106×C67C2218yescomplemented
44C2×#103×C67C2218yescomplemented
45C2×C22⋊C8×C67C2214yes—
46C2×#41×C67—28no—
47C2×#41×C67—48no—
48C22×Q8×C67—21no—
49C22×Q8×C67C2314yes—
50C22×D4×C67—22no—
51C22×D4×C67D411yescharacteristic
52C22×D4×C67C2311yescharacteristic
53C23×C4×C67—22no—
54C23×C4×C67—21no—
55C23×C4×C67C2312yes—
56C2×SD16×C67C2318yes—
57C22×C8×C67C2318yes—
58C4⋊Q8×C67—24no—
59C4⋊Q8×C67—22no—
60C4⋊Q8×C67C2312yes—
61C4⋊Q8×C67C2314yes—
62C4⋊Q8×C67C2312yes—
63#125×C67—24no—
64#125×C67—22no—
65#123×C67—216no—
66C22⋊Q8×C67—24no—
67C22⋊Q8×C67—24no—
68C22⋊Q8×C67—22no—
69C22⋊Q8×C67—28no—
70C22⋊Q8×C67—28no—
71C22⋊Q8×C67—28no—
72C22⋊Q8×C67C2314yes—
73C22⋊Q8×C67C2318yes—
74#121×C67—24no—
75#121×C67C2318yes—
76C4×Q8×C67—22no—
77C4×Q8×C67C2314yes—
78C4×D4×C67—24no—
79C4×D4×C67—24no—
80C4×D4×C67—22no—
81C4×D4×C67D414yes—
82C4×D4×C67C2314yes—
83C4×D4×C67C2314yes—
84#117×C67—22no—
85C2×C4⋊C4×C67—22no—
86C2×C4⋊C4×C67—24no—
87C2×C4⋊C4×C67—24no—
88C2×C4⋊C4×C67—21no—
89C2×C4⋊C4×C67D411yescharacteristic
90C2×C4⋊C4×C67C2314yes—
91C2×C4⋊C4×C67C2311yescharacteristic
92C2×C4⋊C4×C67C2312yes—
93C2×C4⋊C4×C67C2318yes—
94C2×C22⋊C4×C67—24no—
95C2×C22⋊C4×C67—21no—
96C2×C22⋊C4×C67—22no—
97C2×C22⋊C4×C67D412yes—
98C2×C42×C67C2311yescharacteristic
99#106×C67C23116yes—
100#106×C67C2318yes—
101C4⋊C8×C67C2314yes—
102#103×C67C23116yes—
103#103×C67C2318yes—
104#102×C67C2318yes—
105C22⋊C8×C67C2318yes—
106C22⋊C8×C67C2314yes—
107C4×C8×C67C2314yes—
108C24×C67—21no—
109C24×C67C2×D412yes—
110#41×C67—432no—
111C2×Q8×C67—24no—
112C2×Q8×C67—22no—
113C2×Q8×C67—24no—
114C2×Q8×C67—48no—
115C2×Q8×C67—44no—
116C2×Q8×C67C2414yes—
117C2×D4×C67—28no—
118C2×D4×C67—21no—
119C2×D4×C67—42no—
120C2×D4×C67—48no—
121C2×D4×C67SD1618yescomplemented
122C2×D4×C67C2×D414yes—
123C2×D4×C67C2412yes—
124C22×C4×C67—21no—
125C22×C4×C67—22no—
126C22×C4×C67—28no—
127C22×C4×C67—24no—
128C22×C4×C67—24no—
129C22×C4×C67—24no—
130C22×C4×C67—44no—
131C22×C4×C67—48no—
132C22×C4×C67—44no—
133C22×C4×C67C2×D411yescharacteristic
134C22×C4×C67C2×D412yes—
135C22×C4×C67C2×D414yes—
136C22×C4×C67C2×D411yescharacteristic
137C22×C4×C67C2411yescharacteristic
138C22×C4×C67C2414yes—
139SD16×C67—48no—
140C2×C8×C67—48no—
141C2×C8×C67C2418yes—
142C4⋊C4×C67—21no—
143C4⋊C4×C67—22no—
144C4⋊C4×C67—28no—
145C4⋊C4×C67—24no—
146C4⋊C4×C67—28no—
147C4⋊C4×C67—24no—
148C4⋊C4×C67—24no—
149C4⋊C4×C67—22no—
150C4⋊C4×C67—22no—
151C4⋊C4×C67—22no—
152C4⋊C4×C67—24no—
153C4⋊C4×C67C2×D414yes—
154C4⋊C4×C67C2414yes—
155C4⋊C4×C67C2412yes—
156C4⋊C4×C67C2414yes—
157C22⋊C4×C67—24no—
158C22⋊C4×C67—28no—
159C22⋊C4×C67—24no—
160C22⋊C4×C67—28no—
161C22⋊C4×C67C2×D418yes—
162C42×C67—24no—
163C42×C67C2×D412yes—
164C42×C67C2412yes—
165C23×C67—22no—
166C23×C67—42no—
167C23×C67—42no—
168C23×C67—42no—
169C23×C67—42no—
170C23×C67—42no—
171C23×C67C2×SD1618yes—
172C23×C67C22×D411yescharacteristic
173C23×C67C22×D414yes—
174Q8×C67—28no—
175Q8×C67—48no—
176Q8×C67—82no—
177Q8×C67—84no—
178D4×C67—24no—
179D4×C67—42no—
180D4×C67—416no—
181D4×C67C2×SD16116yes—
182C2×C4×C67—28no—
183C2×C4×C67—22no—
184C2×C4×C67—24no—
185C2×C4×C67—24no—
186C2×C4×C67—22no—
187C2×C4×C67—24no—
188C2×C4×C67—44no—
189C2×C4×C67—42no—
190C2×C4×C67—41no—
191C2×C4×C67—42no—
192C2×C4×C67—44no—
193C2×C4×C67—416no—
194C2×C4×C67—48no—
195C2×C4×C67—84no—
196C2×C4×C67—84no—
197C2×C4×C67C2×SD1614yes—
198C2×C4×C67C22×D411yescharacteristic
199C2×C4×C67C22×D412yes—
200C2×C4×C67C22×D414yes—
201C2×C4×C67C22×D412yes—
202C2×C4×C67C2511yescharacteristic
203C8×C67—28no—
204C22×C67—21no—
205C22×C67—24no—
206C22×C67—21no—
207C22×C67—24no—
208C22×C67—48no—
209C22×C67—42no—
210C22×C67—42no—
211C22×C67—42no—
212C22×C67—84no—
213C22×C67C22×SD1612yes—
214C22×C67C22×SD1618yes—
215C22×C67C23×D411yescharacteristic
216C4×C67—22no—
217C4×C67—21no—
218C4×C67—24no—
219C4×C67—44no—
220C4×C67—44no—
221C4×C67—42no—
222C4×C67—44no—
223C4×C67C22×SD1614yes—
224C4×C67#58512yes—
225#61695C6711yesmaximal, characteristic, direct product factor, Sylow
226C2×C67—22no—
227C2×C67—28no—
228C2×C67—82no—
229C2×C67#348511yescharacteristic
230C2×C67C23×SD1611yescharacteristic
231C2×C67#358611yescharacteristic
232#3501C2×C6711yescharacteristic, complemented
233C2×#547C2×C6711yescharacteristic
234#3335C2×C6718yescomplemented
235#3298C2×C6718yescomplemented
236#3229C2×C6712yescomplemented
237#3187C2×C6714yescomplemented
238#3101C2×C6712yescomplemented
239C2×#483C2×C6714yescomplemented
240#2972C2×C6711yescharacteristic, complemented
241C67#6169511yescharacteristic, direct product factor, Sylow
242C22×#41—21no—
243#554—24no—
244#554C22×C6714yescomplemented
245D4×Q8C22×C6714yescomplemented
246#547—24no—
247#547C22×C6718yes—
248#544—21no—
249#543—22no—
250#533C22×C6712yes—
251C2×C4⋊Q8C22×C6711yescharacteristic
252C2×#123—22no—
253C2×C22⋊Q8—21no—
254C2×C22⋊Q8C22×C6714yes—
255C2×#121C22×C6712yes—
256#518—21no—
257C2×C4×D4C22×C6711yescharacteristic
258C2×C4×D4C22×C6711yescharacteristic
259C22×C4⋊C4C22×C6712yes—
260#498C22×C6714yescomplemented
261#487C22×C6714yescomplemented
262#483C22×C6714yes—
263#483C22×C6718yes—
264#483C22×C67116yescomplemented
265#481C22×C6718yescomplemented
266#476C22×C6714yescomplemented
267#475C22×C6718yescomplemented
268#465C22×C67116yescomplemented
269#461C22×C6714yescomplemented
270#450C22×C6714yes—
271#448C22×C6718yescomplemented
272C4×SD16C22×C6718yescomplemented
273C8×D4C22×C6718yescomplemented
274#431C22×C6712yes—
275C2×#106C22×C6718yescomplemented
276C2×#103C22×C6718yescomplemented
277C2×C22⋊C8C22×C6714yes—
278C2×#41—28no—
279C2×#41—48no—
280C22×Q8—21no—
281C22×Q8C23×C6714yes—
282C22×D4—22no—
283C22×D4D4×C6711yescharacteristic
284C22×D4C23×C6711yescharacteristic
285C23×C4—22no—
286C23×C4—21no—
287C23×C4C23×C6712yes—
288C2×SD16C23×C6718yes—
289C22×C8C23×C6718yes—
290C4⋊Q8—24no—
291C4⋊Q8—22no—
292C4⋊Q8C23×C6712yes—
293C4⋊Q8C23×C6714yes—
294C4⋊Q8C23×C6712yes—
295#125—24no—
296#125—22no—
297#123—216no—
298C22⋊Q8—24no—
299C22⋊Q8—24no—
300C22⋊Q8—22no—
301C22⋊Q8—28no—
302C22⋊Q8—28no—
303C22⋊Q8—28no—
304C22⋊Q8C23×C6714yes—
305C22⋊Q8C23×C6718yes—
306#121—24no—
307#121C23×C6718yes—
308C4×Q8—22no—
309C4×Q8C23×C6714yes—
310C4×D4—24no—
311C4×D4—24no—
312C4×D4—22no—
313C4×D4D4×C6714yes—
314C4×D4C23×C6714yes—
315C4×D4C23×C6714yes—
316#117—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—24no—
319C2×C4⋊C4—24no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4D4×C6711yescharacteristic
322C2×C4⋊C4C23×C6711yescharacteristic
323C2×C4⋊C4C23×C6712yes—
324C2×C4⋊C4C23×C6718yes—
325C2×C4⋊C4C23×C6714yes—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—21no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4D4×C6712yes—
330C2×C42C23×C6711yescharacteristic
331#106C23×C6718yes—
332#106C23×C67116yes—
333C4⋊C8C23×C6714yes—
334#103C23×C6718yes—
335#103C23×C67116yes—
336#102C23×C6718yes—
337C22⋊C8C23×C6714yes—
338C22⋊C8C23×C6718yes—
339C4×C8C23×C6714yes—
340C24—21no—
341C24C2×D4×C6712yes—
342#41—432no—
343C2×Q8—24no—
344C2×Q8—22no—
345C2×Q8—24no—
346C2×Q8—48no—
347C2×Q8—44no—
348C2×Q8C24×C6714yes—
349C2×D4—28no—
350C2×D4—21no—
351C2×D4—42no—
352C2×D4—48no—
353C2×D4SD16×C6718yescomplemented
354C2×D4C2×D4×C6714yes—
355C2×D4C24×C6712yes—
356C22×C4—21no—
357C22×C4—22no—
358C22×C4—28no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—44no—
363C22×C4—48no—
364C22×C4—44no—
365C22×C4C2×D4×C6711yescharacteristic
366C22×C4C2×D4×C6712yes—
367C22×C4C2×D4×C6714yes—
368C22×C4C2×D4×C6711yescharacteristic
369C22×C4C24×C6711yescharacteristic
370C22×C4C24×C6714yes—
371SD16—48no—
372C2×C8—48no—
373C2×C8C24×C6718yes—
374C4⋊C4—21no—
375C4⋊C4—22no—
376C4⋊C4—28no—
377C4⋊C4—24no—
378C4⋊C4—28no—
379C4⋊C4—24no—
380C4⋊C4—24no—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—22no—
384C4⋊C4—24no—
385C4⋊C4C2×D4×C6714yes—
386C4⋊C4C24×C6714yes—
387C4⋊C4C24×C6712yes—
388C4⋊C4C24×C6714yes—
389C22⋊C4—24no—
390C22⋊C4—28no—
391C22⋊C4—24no—
392C22⋊C4—28no—
393C22⋊C4C2×D4×C6718yes—
394C42—24no—
395C42C2×D4×C6712yes—
396C42C24×C6712yes—
397C23—22no—
398C23—42no—
399C23—42no—
400C23—42no—
401C23—42no—
402C23—42no—
403C23C2×SD16×C6718yes—
404C23C22×D4×C6711yescharacteristic
405C23C22×D4×C6714yes—
406Q8—28no—
407Q8—48no—
408Q8—84no—
409Q8—82no—
410D4—24no—
411D4—42no—
412D4—416no—
413D4C2×SD16×C67116yes—
414C2×C4—28no—
415C2×C4—22no—
416C2×C4—24no—
417C2×C4—24no—
418C2×C4—22no—
419C2×C4—24no—
420C2×C4—44no—
421C2×C4—42no—
422C2×C4—41no—
423C2×C4—42no—
424C2×C4—44no—
425C2×C4—416no—
426C2×C4—48no—
427C2×C4—84no—
428C2×C4—84no—
429C2×C4C2×SD16×C6714yes—
430C2×C4C22×D4×C6711yescharacteristic
431C2×C4C22×D4×C6712yes—
432C2×C4C22×D4×C6714yes—
433C2×C4C22×D4×C6712yes—
434C2×C4C25×C6711yescharacteristic
435C8—28no—
436C22—21no—
437C22—24no—
438C22—21no—
439C22—24no—
440C22—48no—
441C22—42no—
442C22—42no—
443C22—42no—
444C22—84no—
445C22C22×SD16×C6712yes—
446C22C22×SD16×C6718yes—
447C22C23×D4×C6711yescharacteristic
448C4—22no—
449C4—21no—
450C4—24no—
451C4—42no—
452C4—44no—
453C4—44no—
454C4—44no—
455C4C22×SD16×C6714yes—
456C4#585×C6712yes—
457C2—22no—
458C2—28no—
459C2—82no—
460C2C67×#348511yescharacteristic
461C2C23×SD16×C6711yescharacteristic
462C2C67×#358611yescharacteristic
subgroupG/H
center, socleC22×C67C23×D4
commutator, Frattini, cosocleC2×C4C25×C67
prime radicalC2×#547×C67C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C67 ◁ C2×C4×D4×C67 ◁ G

Complements

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

NH ≅ G/Ntype
1. C67×#3501C2semidirect
3. C67×#3335C2semidirect
4. C67×#3298C2semidirect
5. C67×#3229C2semidirect
6. C67×#3187C2semidirect
7. C67×#3101C2semidirect
8. C2×#483×C67C2semidirect
9. C67×#2972C2semidirect
12. #554×C67C22semidirect
13. D4×Q8×C67C22semidirect
28. #498×C67C22semidirect
29. #487×C67C22semidirect
32. #483×C67C22semidirect
33. #481×C67C22semidirect
34. #476×C67C22semidirect
35. #475×C67C22semidirect
36. #465×C67C22semidirect
37. #461×C67C22semidirect
39. #448×C67C22semidirect
40. C4×SD16×C67C22semidirect
41. C8×D4×C67C22semidirect
43. C2×#106×C67C22semidirect
44. C2×#103×C67C22semidirect
121. C2×D4×C67SD16semidirect
225. #61695C67direct (N × H)
232. #3501C2×C67semidirect
234. #3335C2×C67semidirect
235. #3298C2×C67semidirect
236. #3229C2×C67semidirect
237. #3187C2×C67semidirect
238. #3101C2×C67semidirect
239. C2×#483C2×C67semidirect
240. #2972C2×C67semidirect
241. C67#61695direct (N × H)
244. #554C22×C67semidirect
245. D4×Q8C22×C67semidirect
260. #498C22×C67semidirect
261. #487C22×C67semidirect
264. #483C22×C67semidirect
265. #481C22×C67semidirect
266. #476C22×C67semidirect
267. #475C22×C67semidirect
268. #465C22×C67semidirect
269. #461C22×C67semidirect
271. #448C22×C67semidirect
272. C4×SD16C22×C67semidirect
273. C8×D4C22×C67semidirect
275. C2×#106C22×C67semidirect
276. C2×#103C22×C67semidirect
353. C2×D4SD16×C67semidirect

Maximal subgroups

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

subgroupindex
C67×#35012
C2×#547×C672
C67×#33352
C67×#32982
C67×#32292
C67×#31872
C67×#31012
C2×#483×C672
C67×#29722
#6169567