Finite Group Databasebeta

← C67×#34594 #30713845

Subgroups — C67×#34594 #30713845

Order 17152.

2362 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#529×C67C212yesmaximal, complemented
2C67×#2664C214yesmaximal, complemented
3C67×#2662C214yesmaximal, complemented
4C67×#2659C211yesmaximal, characteristic, complemented
5C67×#2656C211yesmaximal, characteristic
6C67×#2639C212yesmaximal, complemented
7C67×#2618C214yesmaximal, complemented
8C67×#2396C214yesmaximal, complemented
9C67×#2392C214yesmaximal, complemented
10C4×#121×C67C214yesmaximal, complemented
11C2×C4×C4⋊C4×C67C211yesmaximal, characteristic, complemented
12#529×C67—216no—
13C2×#126×C67C2214yes—
14C2×#125×C67C2212yes—
15C2×#124×C67C2212yes—
16C2×#123×C67C2214yes—
17C2×C22⋊Q8×C67C2212yes—
18C2×#121×C67C2214yes—
19C2×#121×C67C2212yes—
20C2×C4×Q8×C67C2212yes—
21C2×C4×D4×C67C2214yes—
22C2×C4×D4×C67C2212yes—
23C2×C4×D4×C67C2214yes—
24C2×#117×C67C2214yes—
25C22×C4⋊C4×C67C2212yes—
26C22×C42×C67C2212yes—
27C22×C42×C67C2211yescharacteristic
28#400×C67C2214yes—
29#399×C67C2214yes—
30#398×C67C2218yes—
31#398×C67C2214yes—
32#397×C67C2214yes—
33#396×C67C2218yes—
34#396×C67C2218yes—
35#395×C67C2214yes—
36#394×C67C2218yes—
37#394×C67C2214yes—
38#393×C67C2214yes—
39#392×C67C2214yes—
40#390×C67C2214yes—
41#389×C67C2214yes—
42#388×C67C2218yes—
43#386×C67C2218yes—
44#380×C67C2214yes—
45C4×C4⋊C4×C67C2214yes—
46C4×C4⋊C4×C67C2214yes—
47C4×C4⋊C4×C67C2214yes—
48C4×C22⋊C4×C67C2218yes—
49C2×#95×C67C2211yescharacteristic
50C2×#95×C67C2211yescharacteristic
51C22×Q8×C67C2312yes—
52C22×D4×C67C2314yes—
53C22×D4×C67C2312yes—
54C22×D4×C67C2314yes—
55C23×C4×C67C2314yes—
56C23×C4×C67C2311yescharacteristic
57C23×C4×C67C2311yescharacteristic
58C23×C4×C67C2312yes—
59C23×C4×C67C2311yescharacteristic
60C23×C4×C67C2312yes—
61#126×C67—216no—
62#125×C67—28no—
63#124×C67—28no—
64#123×C67—216no—
65C22⋊Q8×C67—28no—
66#121×C67—28no—
67#121×C67—216no—
68C4×Q8×C67—28no—
69C4×D4×C67—216no—
70C4×D4×C67—28no—
71C4×D4×C67—48no—
72#117×C67—216no—
73C2×C4⋊C4×C67—28no—
74C2×C4⋊C4×C67C2318yes—
75C2×C4⋊C4×C67C2314yes—
76C2×C4⋊C4×C67C2314yes—
77C2×C4⋊C4×C67C2312yes—
78C2×C4⋊C4×C67C2314yes—
79C2×C4⋊C4×C67C2312yes—
80C2×C4⋊C4×C67C2314yes—
81C2×C4⋊C4×C67C2314yes—
82C2×C4⋊C4×C67C2314yes—
83C2×C4⋊C4×C67C2314yes—
84C2×C22⋊C4×C67C2314yes—
85C2×C22⋊C4×C67C2314yes—
86C2×C22⋊C4×C67C2314yes—
87C2×C22⋊C4×C67C2314yes—
88C2×C22⋊C4×C67C2314yes—
89C2×C22⋊C4×C67C2318yes—
90C2×C22⋊C4×C67C2318yes—
91C2×C22⋊C4×C67C2318yes—
92C2×C42×C67—44no—
93C2×C42×C67D418yescomplemented
94C2×C42×C67C2314yes—
95C2×C42×C67C2314yes—
96C2×C42×C67C2314yes—
97C2×C42×C67C2312yes—
98C2×C42×C67C2312yes—
99#95×C67C2314yes—
100#95×C67C2314yes—
101#95×C67C2318yes—
102#95×C67C2318yes—
103#95×C67C2318yes—
104C24×C67C2414yes—
105C24×C67C2411yescharacteristic
106C2×Q8×C67—24no—
107C2×Q8×C67—44no—
108C2×D4×C67—28no—
109C2×D4×C67—216no—
110C2×D4×C67—24no—
111C2×D4×C67—28no—
112C2×D4×C67—44no—
113C2×D4×C67—44no—
114C2×D4×C67—44no—
115C22×C4×C67—24no—
116C22×C4×C67—22no—
117C22×C4×C67—22no—
118C22×C4×C67—24no—
119C22×C4×C67—28no—
120C22×C4×C67—28no—
121C22×C4×C67—28no—
122C22×C4×C67—42no—
123C22×C4×C67—42no—
124C22×C4×C67—42no—
125C22×C4×C67—42no—
126C22×C4×C67—42no—
127C22×C4×C67C2×D414yes—
128C22×C4×C67C2×D414yes—
129C22×C4×C67C2×D414yes—
130C22×C4×C67#4118yes—
131C22×C4×C67C2414yes—
132C22×C4×C67C2414yes—
133C22×C4×C67C2414yes—
134C22×C4×C67C2414yes—
135C22×C4×C67C2411yescharacteristic
136C22×C4×C67C2412yes—
137C22×C4×C67C2412yes—
138C22×C4×C67C2414yes—
139C22×C4×C67C2411yescharacteristic
140C4⋊C4×C67—28no—
141C4⋊C4×C67—28no—
142C4⋊C4×C67—24no—
143C4⋊C4×C67—28no—
144C4⋊C4×C67—24no—
145C4⋊C4×C67—28no—
146C4⋊C4×C67—216no—
147C4⋊C4×C67—28no—
148C4⋊C4×C67—44no—
149C4⋊C4×C67—44no—
150C22⋊C4×C67—28no—
151C22⋊C4×C67—28no—
152C22⋊C4×C67—28no—
153C22⋊C4×C67—28no—
154C22⋊C4×C67—28no—
155C22⋊C4×C67—216no—
156C22⋊C4×C67—48no—
157C22⋊C4×C67—48no—
158C42×C67—28no—
159C42×C67—28no—
160C42×C67—44no—
161C42×C67C2×D418yes—
162C42×C67C2×D418yes—
163C23×C67—21no—
164C23×C67—21no—
165C23×C67—22no—
166C23×C67—24no—
167C23×C67—28no—
168C23×C67—24no—
169C23×C67—24no—
170C23×C67—28no—
171C23×C67C22×D412yes—
172C23×C67C2×#4114yes—
173C23×C67C2511yescharacteristic
174Q8×C67—48no—
175D4×C67—48no—
176D4×C67—416no—
177D4×C67—416no—
178C2×C4×C67—28no—
179C2×C4×C67—24no—
180C2×C4×C67—24no—
181C2×C4×C67—21no—
182C2×C4×C67—21no—
183C2×C4×C67—24no—
184C2×C4×C67—24no—
185C2×C4×C67—28no—
186C2×C4×C67—24no—
187C2×C4×C67—24no—
188C2×C4×C67—24no—
189C2×C4×C67—28no—
190C2×C4×C67—24no—
191C2×C4×C67—24no—
192C2×C4×C67—24no—
193C2×C4×C67—24no—
194C2×C4×C67—24no—
195C2×C4×C67—28no—
196C2×C4×C67—44no—
197C2×C4×C67—48no—
198C2×C4×C67—48no—
199C2×C4×C67—416no—
200C2×C4×C67C22×D414yes—
201C2×C4×C67C22×D412yes—
202C2×C4×C67C22×D414yes—
203C2×C4×C67C22×D412yes—
204C2×C4×C67C2×#4118yes—
205C2×C4×C67H(5,𝔽2)12yes—
206C2×C4×C672−1+412yes—
207C22×C67—22no—
208C22×C67—22no—
209C22×C67—24no—
210C22×C67—24no—
211C22×C67—44no—
212C22×C67—44no—
213C22×C67—44no—
214C22×C67—48no—
215C22×C67—48no—
216C22×C67#53214yes—
217C22×C67C23×D411yescharacteristic
218C22×C67C22×#4112yes—
219C22×C67C2×H(5,𝔽2)11yescharacteristic
220C22×C67C2×2−1+411yescharacteristic
221C22×C67#58512yes—
222C4×C67—24no—
223C4×C67—22no—
224C4×C67—24no—
225C4×C67—28no—
226C4×C67—44no—
227C4×C67—44no—
228C4×C67—44no—
229C4×C67—44no—
230C4×C67#53714yes—
231#34594C6711yesmaximal, characteristic, direct product factor, Sylow
232C2×C67—24no—
233C2×C67—84no—
234C2×C67C2×#53211yescharacteristic
235C2×C67C2×#53711yescharacteristic
236C2×C67#344911yescharacteristic
237C2×C67#345912yes—
238C2×C67#346112yes—
239C2×#529C2×C6712yescomplemented
240#2664C2×C6714yescomplemented
241#2662C2×C6714yescomplemented
242#2659C2×C6711yescharacteristic, complemented
243#2656C2×C6711yescharacteristic
244#2639C2×C6712yescomplemented
245#2618C2×C6714yescomplemented
246#2396C2×C6714yescomplemented
247#2392C2×C6714yescomplemented
248C4×#121C2×C6714yescomplemented
249C2×C4×C4⋊C4C2×C6711yescharacteristic, complemented
250C67#3459411yescharacteristic, direct product factor, Sylow
251#529—216no—
252C2×#126C22×C6714yes—
253C2×#125C22×C6712yes—
254C2×#124C22×C6712yes—
255C2×#123C22×C6714yes—
256C2×C22⋊Q8C22×C6712yes—
257C2×#121C22×C6714yes—
258C2×#121C22×C6712yes—
259C2×C4×Q8C22×C6712yes—
260C2×C4×D4C22×C6712yes—
261C2×C4×D4C22×C6714yes—
262C2×C4×D4C22×C6714yes—
263C2×#117C22×C6714yes—
264C22×C4⋊C4C22×C6712yes—
265C22×C42C22×C6712yes—
266C22×C42C22×C6711yescharacteristic
267#400C22×C6714yes—
268#399C22×C6714yes—
269#398C22×C6718yes—
270#398C22×C6714yes—
271#397C22×C6714yes—
272#396C22×C6718yes—
273#396C22×C6718yes—
274#395C22×C6714yes—
275#394C22×C6718yes—
276#394C22×C6714yes—
277#393C22×C6714yes—
278#392C22×C6714yes—
279#390C22×C6714yes—
280#389C22×C6714yes—
281#388C22×C6718yes—
282#386C22×C6718yes—
283#380C22×C6714yes—
284C4×C4⋊C4C22×C6714yes—
285C4×C4⋊C4C22×C6714yes—
286C4×C4⋊C4C22×C6714yes—
287C4×C22⋊C4C22×C6718yes—
288C2×#95C22×C6711yescharacteristic
289C2×#95C22×C6711yescharacteristic
290C22×Q8C23×C6712yes—
291C22×D4C23×C6714yes—
292C22×D4C23×C6712yes—
293C22×D4C23×C6714yes—
294C23×C4C23×C6714yes—
295C23×C4C23×C6711yescharacteristic
296C23×C4C23×C6711yescharacteristic
297C23×C4C23×C6712yes—
298C23×C4C23×C6711yescharacteristic
299C23×C4C23×C6712yes—
300#126—216no—
301#125—28no—
302#124—28no—
303#123—216no—
304C22⋊Q8—28no—
305#121—28no—
306#121—216no—
307C4×Q8—28no—
308C4×D4—216no—
309C4×D4—28no—
310C4×D4—48no—
311#117—216no—
312C2×C4⋊C4—28no—
313C2×C4⋊C4C23×C6718yes—
314C2×C4⋊C4C23×C6714yes—
315C2×C4⋊C4C23×C6714yes—
316C2×C4⋊C4C23×C6712yes—
317C2×C4⋊C4C23×C6714yes—
318C2×C4⋊C4C23×C6712yes—
319C2×C4⋊C4C23×C6714yes—
320C2×C4⋊C4C23×C6714yes—
321C2×C4⋊C4C23×C6714yes—
322C2×C4⋊C4C23×C6714yes—
323C2×C22⋊C4C23×C6714yes—
324C2×C22⋊C4C23×C6714yes—
325C2×C22⋊C4C23×C6714yes—
326C2×C22⋊C4C23×C6714yes—
327C2×C22⋊C4C23×C6714yes—
328C2×C22⋊C4C23×C6718yes—
329C2×C22⋊C4C23×C6718yes—
330C2×C22⋊C4C23×C6718yes—
331C2×C42—44no—
332C2×C42D4×C6718yescomplemented
333C2×C42C23×C6714yes—
334C2×C42C23×C6714yes—
335C2×C42C23×C6714yes—
336C2×C42C23×C6712yes—
337C2×C42C23×C6712yes—
338#95C23×C6714yes—
339#95C23×C6714yes—
340#95C23×C6718yes—
341#95C23×C6718yes—
342#95C23×C6718yes—
343C24C24×C6714yes—
344C24C24×C6711yescharacteristic
345C2×Q8—24no—
346C2×Q8—44no—
347C2×D4—28no—
348C2×D4—216no—
349C2×D4—24no—
350C2×D4—28no—
351C2×D4—44no—
352C2×D4—44no—
353C2×D4—44no—
354C22×C4—24no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—28no—
359C22×C4—28no—
360C22×C4—28no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4C2×D4×C6714yes—
367C22×C4C2×D4×C6714yes—
368C22×C4C2×D4×C6714yes—
369C22×C4#41×C6718yes—
370C22×C4C24×C6714yes—
371C22×C4C24×C6714yes—
372C22×C4C24×C6714yes—
373C22×C4C24×C6714yes—
374C22×C4C24×C6712yes—
375C22×C4C24×C6711yescharacteristic
376C22×C4C24×C6712yes—
377C22×C4C24×C6714yes—
378C22×C4C24×C6711yescharacteristic
379C4⋊C4—28no—
380C4⋊C4—28no—
381C4⋊C4—24no—
382C4⋊C4—28no—
383C4⋊C4—24no—
384C4⋊C4—28no—
385C4⋊C4—216no—
386C4⋊C4—28no—
387C4⋊C4—44no—
388C4⋊C4—44no—
389C22⋊C4—28no—
390C22⋊C4—28no—
391C22⋊C4—28no—
392C22⋊C4—28no—
393C22⋊C4—28no—
394C22⋊C4—216no—
395C22⋊C4—48no—
396C22⋊C4—48no—
397C42—28no—
398C42—28no—
399C42—44no—
400C42C2×D4×C6718yes—
401C42C2×D4×C6718yes—
402C23—21no—
403C23—21no—
404C23—22no—
405C23—24no—
406C23—28no—
407C23—24no—
408C23—24no—
409C23—28no—
410C23C22×D4×C6712yes—
411C23C2×#41×C6714yes—
412C23C25×C6711yescharacteristic
413Q8—48no—
414D4—48no—
415D4—416no—
416D4—416no—
417C2×C4—28no—
418C2×C4—24no—
419C2×C4—24no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—24no—
423C2×C4—24no—
424C2×C4—28no—
425C2×C4—24no—
426C2×C4—24no—
427C2×C4—24no—
428C2×C4—28no—
429C2×C4—24no—
430C2×C4—24no—
431C2×C4—24no—
432C2×C4—24no—
433C2×C4—24no—
434C2×C4—28no—
435C2×C4—44no—
436C2×C4—48no—
437C2×C4—48no—
438C2×C4—416no—
439C2×C4C22×D4×C6714yes—
440C2×C4C22×D4×C6712yes—
441C2×C4C22×D4×C6714yes—
442C2×C4C22×D4×C6712yes—
443C2×C4C2×#41×C6718yes—
444C2×C4H(5,𝔽2)×C6712yes—
445C2×C42−1+4×C6712yes—
446C22—22no—
447C22—22no—
448C22—24no—
449C22—24no—
450C22—44no—
451C22—44no—
452C22—44no—
453C22—48no—
454C22—48no—
455C22#532×C6714yes—
456C22C23×D4×C6711yescharacteristic
457C22C22×#41×C6712yes—
458C22C2×H(5,𝔽2)×C6711yescharacteristic
459C22C2×2−1+4×C6711yescharacteristic
460C22#585×C6712yes—
461C4—24no—
462C4—22no—
463C4—24no—
464C4—28no—
465C4—44no—
466C4—44no—
467C4—44no—
468C4—44no—
469C4#537×C6714yes—
470C2—24no—
471C2—84no—
472C2C2×#532×C6711yescharacteristic
473C2C2×#537×C6711yescharacteristic
474C2C67×#344911yescharacteristic
475C2C67×#345912yes—
476C2C67×#346112yes—
subgroupG/H
centerC22×C4×C67C24
commutator, Frattini, cosocleC23C25×C67
socleC23×C67C25
prime radicalC67×#2656C2
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C22×C4×C67 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×#529×C67C2semidirect
2. C67×#2664C2semidirect
3. C67×#2662C2semidirect
4. C67×#2659C2semidirect
6. C67×#2639C2semidirect
7. C67×#2618C2semidirect
8. C67×#2396C2semidirect
9. C67×#2392C2semidirect
10. C4×#121×C67C2semidirect
11. C2×C4×C4⋊C4×C67C2semidirect
93. C2×C42×C67D4semidirect
231. #34594C67direct (N × H)
239. C2×#529C2×C67semidirect
240. #2664C2×C67semidirect
241. #2662C2×C67semidirect
242. #2659C2×C67semidirect
244. #2639C2×C67semidirect
245. #2618C2×C67semidirect
246. #2396C2×C67semidirect
247. #2392C2×C67semidirect
248. C4×#121C2×C67semidirect
249. C2×C4×C4⋊C4C2×C67semidirect
250. C67#34594direct (N × H)
332. C2×C42D4×C67semidirect

Maximal subgroups

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

subgroupindex
C2×#529×C672
C67×#26642
C67×#26622
C67×#26592
C67×#26562
C67×#26392
C67×#26182
C67×#23962
C67×#23922
C4×#121×C672
C2×C4×C4⋊C4×C672
#3459467